motion planning and tracking control for distributed

Motion planning and tracking control fordistributed parameter systems

Lecture Notes for the Spring School „An Introduction to Modelling andControl of Systems governed by PDEs“, Oaxaca (MX)

Prof. Dr.–Ing. habil. Thomas Meurer

May 2019

Chair of Automatic ControlKiel University

Motion planning and tracking control for distributed parameter systems

Lecture Notes for the Spring School „An Introduction to Modelling and Control of Systemsgoverned by PDEs“, Oaxaca (MX), May 2019

Prof. Dr.–Ing. habil. Thomas Meurer

Chair of Automatic ControlKiel UniversityInstitute of Electrical Engineering and Information ScienceFaculty of EngineeringKaiserstraße 2D–24143 Kiel

k [email protected]–kiel.dem http://www.control.tf.uni–kiel.de

© Chair of Automatic Control, Kiel University

Preface

These lecture notes are prepared for the Spring School during the 2019 CPDE to be held in Oaxaca,Mexico from May 16th to 18th, 2019 and serve as a brief introduction to the analysis and control designfor systems governed by partial differential equations (PDEs). The topics address some extensions offlatness, backstepping and Lyapunov–based control to PDEs. It is in particular our desire to show boththeoretically and in certain selected experimental setups, that a combination of these approachesleads to a systematic procedure for the design of stabilizing and robustifying tracking controllers forcertain classes of systems governed by (nonlinear) PDEs.

The content of the lecture notes exceeds what is covered during the school to serve as a referenceand source of further activities. The interested reader is also referred to my book Control of Higher–Dimensional PDEs: Flatness and Backstepping Designs, Series: Communications and Control Engi-neering, Springer–Verlag, 2012.

Thomas Meurer

v

Contents

1 Introduction 11.1 Control of PDE systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Selected examples of distributed parameter systems . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Diffusion–convection–reaction systems . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Flexible structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Abstract formulation of linear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Flatness–based trajectory planning and feedforward control design 212.1 Finite–dimensional nonlinear control systems . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Distributed parameter control systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Trajectory planning for PDE systems . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Operational calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Flatness–based trajectory planning for the linear heat equation . . . . . . . . . 242.3.2 Flatness–based trajectory planning for the linear wave equation . . . . . . . . . 29

2.4 Riesz spectral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.1 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.2 Flatness–based state and input parametrization . . . . . . . . . . . . . . . . . . 362.4.3 Application to the linear heat and wave equation with in–domain control . . . 41

2.5 Extension to nonlinear problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Feedback stabilization and observer design using backstepping 533.1 Introduction to Lyapunov’s stability theory . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Feedback control for a linear diffusion–reaction system using backstepping . . . . . . . 56

3.2.1 Stabilization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.2 Kernel computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.3 Solution of the kernel PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.4 Inverse transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.5 State–feedback controller and closed–loop stability . . . . . . . . . . . . . . . . 59

3.3 Tracking control for a semilinear diffusion–reaction system using flatness and back-stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3.1 Flatness–based trajectory planning and feedforward control . . . . . . . . . . . 613.3.2 Stabilization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3.3 Kernel computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3.4 Solution of the kernel PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Extensions of PDE backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Lyapunov–based feedback stabilization and observer design 774.1 Extensions to Lyapunov’s stability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Vibration suppression for a flexible beam . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

vii

4.2.1 Well–posedness of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2.2 Lyapunov–based feedback control design . . . . . . . . . . . . . . . . . . . . . . 814.2.3 Lyapunov–based state–observer design . . . . . . . . . . . . . . . . . . . . . . . . 834.2.4 Stability of the composite system . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

viii

1Introduction

Partial differential equations (PDEs) can be considered as the fundamental mathematical descriptionof many technical processes. In general, this distributed parameter description becomes an essentialingredient of the modeling and analysis process if the spatial or property–related distribution of theprocess variables can no longer be neglected. Following the exposition in [63] some characteristicexamples are summarized below:

• chemical or biochemical reactors [42] including three-way catalysts for exhaust gas after–treatment in automotive applications, (reactive) distillation and adsorption processes [114], oractivated sludge processes for wastewater treatment [52];

• thermal systems [6] or the reheating and cooling of metal slabs during the steel processing toachieve desired metallurgical changes [119];

• electrochemical systems including fuel cells [115] and Li–ion or Li–polymer battery devices forenergy production and storage [14, 28];

• smart materials, adaptive structures and resonant systems [61, 8, 81];

• flexible structures in aerospace and mechanical applications such as adaptive or flapping wingstructures [113], micro–mechanic bending cantilevers in atomic force microscopes [10], ordeformable mirrors in adaptive optics [87];

• fluid dynamical systems [1, 9], mixing processes and coupled fluid–structure interactions;

• wave propagation in optical fibers [105] and traffic congestion [124, 32];

• energy production in fusion reactors [116, 3];

• multi–agent systems [68, 25, 63, 83, 80, 24].

The dynamic operation of these distributed parameter systems (DPSs) essentially relies on the incor-poration of suitable control strategies to influence the system dynamics and to enlarge the operatingrange.

1.1 Control of PDE systems

The so–called two–degrees–of–freedom (2DOF) control concept, that forms the basis for these lecturenotes, is schematically shown in Figure 1.1. Given the distributed parameter system Σ∞ with outputy(t ), the control structure comprises trajectory planning Σ∗ and feedforward control Σ f f to imposethe desired output trajectory y∗(t) by means of u∗(t) and feedback control Σe

f bto provide the state–feedback ue (t) in terms of the estimated error state xe (t) obtained from an error system observerΣe

ob processing the tracking error y e (t) = y(t)− y∗(t). In the nominal case of an exact plant modelΣ∞, no exogenous disturbances and perfectly known initial conditions the feedforward control incombination with trajectory planning ensures that the output y(t ) exactly tracks a prescribed desired

1

Σ∗ Σff

Σefb

Σeob

Σ∞y∗

u∗ u y

ye −xe

ue

Figure 1.1: Block diagram of the 2DOF control concept for the DPS Σ∞ with trajectory planning Σ∗, feed-forward control Σ f f , error feedback control Σe

f b , error system observer Σeob for tracking control

y → y∗.

trajectory y∗(t ). To account for model uncertainties, disturbances and instability a feedback controlleris designed to stabilize the tracking error dynamics, i.e. the evolution of the deviation between theactual state and its desired value. The latter is herein known from the flatness–based state andinput parametrization. Moreover, it is in general necessary to integrate a distributed parameterstate–observer to estimate the unmeasured state evolution from the available measurements.

The model–based control and observer design for distributed parameter systems1 can be in generalclassified into early and late lumping techniques. While in the early lumping approach, the governingPDEs are reduced to a finite–dimensional description by making use of suitable approximation andmodel reduction techniques prior to the feedback control design the late lumping approach directlyexploits the system formulation in terms of PDEs. In the following, only late lumping design techniquesare addressed and if needed combined with suitable approximation methods for implementation.Following the schematics in Figure 1.1 the lecture notes cover

• flatness–based trajectory planning and feedforward control for PDEs,

• backstepping techniques for state–feedback control and observer design and

• Lyapunov– or passivity–based output feedback control.

Introductions and a brief literature survey are provided in the individual chapters. Since mathematicalmodels are the common starting point for any development, in the following a summary of selectedPDE control problems is provided, that serve as basis for the application of flatness–, backstepping–and passivity–based control as well as their combination.

1.2 Selected examples of distributed parameter systems

Subsequently, distributed parameter models are summarized covering selected control examples.Where directly possible MATLAB code is provided for the numerical simulation and the later evaluationof the control concepts.

1.2.1 Heat equation

The heat equation describing non–convective heat transfer can be rather easily derived by makinguse of the first law of thermodynamics taking into account Fourier’s law of heat conduction. This

1This section does not aim to provide a comprehensive review of existing analysis and control design techniques for PDEsystems. For this, the reader is referred, e.g., to [17, 57, 118, 63].

2 Chapter 1 Introduction

yields the spatial–temporal evolution of the temperature field x(z , t ) in an incompressible volumeΩaccording to

ρc(x(z , t ))∂t x(z , t ) =∇· (λ(x(z , t ))∇x(z , t ))+W (x(z , t ),u(z , t ), z , t ). (1.1)

Herein, ρ is the density of the material, c(x(z , t )) is the specific heat capacity and λ(x(z , t )) is thethermal conductivity. The heat source W (x(z , t),u(z , t), z , t) is used to summarize dissipative andhence irreversible energy conversions with u(z , t) denoting an external quantity. This arises, e.g.,when modeling the ohmic resistance of a heat and electricity conduction material or the energy richradiation absorbed by the body. For an isotropic material with constant coefficients (1.1) reducesto

ρc∂t x(z , t ) =λ∆x(z , t )+W (x(z , t ),u(z , t ), z , t ). (1.2)

To completely describe the temperature field the PDE has to be complemented by boundary condi-tions and an initial condition. For this, three types of boundary conditions on (z , t) ∈ ∂Ω×R+ aredistinguished in general, see, e.g., [6]:

1. The temperature is given as a function of time and surface position, i.e.

x(z , t ) = v (z , t ). (1.3)

This is called a boundary condition of first type or a Dirichlet condition.

2. The heat flux normal to the surface is prescribed as a function of time and surface positionwith

q(z , t ) ·n=−λ(x(z , t ))∇x(z , t ) ·n=−λ(x(z , t ))∂nx(z , t ) = v (z , t ). (1.4)

This boundary condition of second type is also called a Neumann condition.

3. When the body of temperature x(z , t ) is in contact with another medium of temperature v (z , t ),energy is exchanged along the boundary. In this case, different scenarios have to be distin-guished depending on either firm or loose contact between conducting solid bodies or thecontact with a fluid. If the conducting body of temperature x(z , t ) is in contact with a fluid oftemperature v (z , t) a thermal boundary layer develops along the interface. Let α denote theconvective heat transfer coefficient, then a balance of energy along the boundary yields

−λ(x(z , t ))∂nx(z , t ) =α(x(z , t )− v (z , t )

). (1.5)

Boundary conditions of the form (1.5) are called mixed or Robin conditions.

If the bodyΩ reduces to the lineΩ = [0,1], then (1.1) or (1.2) reduce to well–know one–dimensionalheat equation in a single spatial coordinate z . Here, ∇· (λ(x(z , t ))∇x(z , t )) in (1.1) has to be replacedby ∂z (λ(x(z , t ))∂z x(z , t )) while λ∆x(z , t ) simplifies to λ∂2

z x(z , t ) in (1.2).

For further considerations including both analytical and numerical solution methods or extensions toconvective heat transfer, the reader is referred to, e.g., [13, 11, 6] and the many references therein.

1.2 Selected examples of distributed parameter systems 3

Remark 1.1: Numerical solution using MATLAB

The one–dimensional heat equation can be directly solved numerically using the function pdepeof MATLAB. This is exemplarily shown in the listing below for the equations

c∂t x(z , t ) = ∂z (a∂z x(z , t ))+ f (x(z , t ))+u(z , t ), z ∈ (0,1), t > 0 (1.6a)

∂z x(0, t ) = 0, x(1, t ) = v (t ), t ≥ 0 (1.6b)

x(z ,0) = 0, z ∈ [0,1] (1.6c)

with

f (x(z , t )) = 5(x(z , t ))2 −x(z , t )

u(z , t ) = 0

v (t ) = (t −0.5)σ(t −0.5)−2(t −1)σ(t −1)+ (t −1.5)σ(t −1.5)

(1.6d)

Herein, σ(t ) denotes the Heaviside function.

function out=heateqn()%%Numerical solution of the linear heat equation using MATLAB.%(c) Thomas Meurer, CAU Kiel

%------------------------% MAIN

%Preparationsm=0;zmax=1.0;tmax=2.0;z=linspace(0,zmax,101);t=linspace(0,tmax,201);

%System parameters and functionsa=1.0;f=inline(’-x+5*x^2’,’x’);u=inline(’0’,’z’,’t’);v=inline(’(t-0.5).*stepfun(t,0.5)-2*(t-1).*stepfun(t,1)...+(t-1.5).*stepfun(t,1.5)’,’t’);

%Solver callsol=pdepe(m,@heat_pde,@heat_ic,@heat_bc,z,t,[],a,f,u,v);x=sol(:,:,1);

%Outputout.z = z; out.t = t; out.x = x;

%Surface plotfigure; mesh(z,t,x); xlabel(’z’); ylabel(’t’);


%------------------------% SUBFUNCTIONS

function [c,g,s]=heat_pde(z,t,x,DxDz,a,f,u,v)%Partial differential equationc = 1.0;g = a*DxDz;s = f(x) + u(z,t);

function u0=heat_ic(z,a,f,u,v);%Initial conditionx0 = 0.0;

function [pl,ql,pr,qr]=heat_bc(zl,xl,zr,xr,t,a,f,u,v)%Boundary conditionspl=0.0;ql=1.0;pr=xr-v(t);qr=0.0;

The solution is thereby determined by a so–called method–of–lines approach, where the PDE issemi–discretized in the spatial coordinate.

1.2.2 Diffusion–convection–reaction systems

Fixed–bed or tubular reactors are among the most common types of chemical conversion devicesin chemical engineering. Herein, a rather complex interplay of diffusive, convective and reactiveeffects arises, that has to be captured in mathematical process models. Figure 1.2 shows a schematic(differential) control volume of a fixed–bed reactor, that is packed by some catalyst. Typically reactantsare led through the reactor in gaseous state with chemical reactions being initiated at the surface ofthe catalyst.

z z +dzz

Catalyst

Voidage

Cooling jacket

T, w j

p,ρ, v

Figure 1.2: Differential reactor element with inflow temperature T , mass fraction w j , pressure p, density ρ,fluid velocity v .


Taking into account global mass balance, componentwise mass balances, energy or enthalpy balance,impulse balances, and thermodynamics equations of state the reactor modeling results in a descriptionin terms of coupled hyperbolic (and/or parabolic) PDEs. Depending on the desired level of detail andresolution highly complex models are obtained (see, e.g., [42]), that are hardly accessible for controldesign. Hence, different simplifications are typically introduced related to fluid flow in the reactor (e.g.,plug flow assumption), dimension (e.g., one–dimensional compared to three–dimensional spatialdomain) or the parametrization of the reaction rates (e.g., linear or nonlinear dependency on massfractions or molar concentrations, respectively).

As an example, a distributed parameter description of the form

∂t x(z , t ) = ∂z [D(z , x(z , t ))∂z x(z , t ))]+ f (z , x(z , t ),∂z x(z , t ),u(t )), z ∈ (0,L), t > 0 (1.7a)

g 0(x(0, t ),∂z x(0, t ), v 0(t )) = 0,

g L(x(L, t ),∂z x(L, t ), v L(t )) = 0,t ≥ 0 (1.7b)

x(z ,0) = x(0, z ), z ∈ [0,L] (1.7c)

does seem appropriate to reflect the dominating dynamics. Herein, D(z , x(z , t )) is the diffusion ma-trix, f (z , x(z , t ),∂z x(z , t ),u(t )) represents reactive and convective effects, arising nonlinearities, andinhomogeneous terms u(t ), g 0(·) and g L(·) refer to in general non–autonomous boundary conditions,and x(0, z ) is the initial reactor state. The state vector x(z , t ) may comprise mass fractions (or molarconcentrations) and temperatures. Considering the reactor in a neighborhood of an operating profile,i.e., a steady state solution of (1.7), then linearization of (1.7) yields a (local) system formulation in theform of a linear diffusion–convection–reaction system

∂t x(z , t ) = D(z )∂2z x(z , t )+C (z )∂z x(z , t )+R(z )x(z , t )+B(z )u(t ), z ∈ (0,L), t > 0 (1.8a)

G0x(0, t )+F0∂z x(0, t )) = H0v 0(t ),

GL x(L, t )+FL∂z x(L, t )) = HL v L(t ),t ≥ 0 (1.8b)

x(z ,0) = x0(z ), z ∈ [0,L]. (1.8c)

Herein, D(z ), C (z ) and R(z ) denote diffusion, convection and reaction matrices, respectively. Forfurther details on the mathematical modeling of reaction engineering applications using distributedparameter systems the reader is referred, e.g., to [4, 11, 42].

Remark 1.2: Numerical solution using MATLAB

For the one–dimensional setting with z ∈ [0,L], L ∈R+ the function pdepe of MATLAB can beused for the numerical solution of both the nonlinear PDE system (1.7) and the linear PDEsystem (1.8). For this, the assumption has to be made, that

g 0(x(0, t ),∂z x(0, t ), v 0(t )) = D(0, x(0, t ))∂z x(0, t )+p0(x(0, t ), v 0(t ))

g L(x(L, t ),∂z x(L, t ), v L(t )) = D(L, x(L, t ))∂z x(L, t )+pL(x(L, t ), v L(t ))(1.9)

A similar relationship has to hold for (1.8b). The following listing provides an implementation of(1.7), (1.9). Here, the arising matrix– and vector–valued functions are assumed to be available onthe MATLAB path with the corresponding number of input arguments.

function out=dcrs()%


%Numerical solution of a vector-valued diffusion-convection-reaction%system using MATLAB.%(c) Thomas Meurer, CAU Kiel

%------------------------% MAIN

%Preparationsm=0;zmax=1.0;tmax=2.0;z=linspace(0,zmax,101);t=linspace(0,tmax,201);

%System parameters and functionsh.D = @fun_D;h.f = @fun_f;h.p0 = @fun_p0;h.pL = @fun_pL;h.x0 = @fun_x0;%% alternatively define inline or as subfunctions

%Solver callsol=pdepe(m,@dcrs_pde,@dcrs_ic,@dcrs_bc,z,t,[],h);

%Outputout.z = z; out.t = t; out.x = sol;

%------------------------% SUBFUNCTIONS

function [c,g,s]=dcrs_pde(z,t,x,DxDz,h)%Partial differential equationc = ones(size(x,1),1);g = h.D(z,x)*DxDz;s = h.f(z,x,DxDz,t);

function x0=dcrs_ic(z,h);%Initial conditionx0 = h.x0(z);

function [pl,ql,pr,qr]=dcrs_bc(zl,xl,zr,xr,t,h)%Boundary conditionsE = ones(size(xl,1),1);pl = h.p0(xl,t);ql = E;pr = h.pL(xr,t);qr = E;


As alternative numerical solution techniques consider, e.g., the weighted residual or Galerkinapproach [20] or finite difference techniques [117].

1.2.3 Flexible structures

Flexible structures with embedded actuators and sensors arise in a broad variety in different appli-cations including large scale manipulators [33], lightweight robotics and adaptive structures [61, 8,26, 84, 81, 102, 107, 88, 44], or fluid–structure interaction [46, 47]. For a detailed introduction to themathematical modeling of flexible structures using, e.g., the extended Hamilton’s principle, the readeris referred to [85, 62, 31, 86].

In the following, only the wave equation and the Euler–Bernoulli beam equation are presented asexamples for PDEs modeling flexible structures. Considering, e.g., the transversal motion of a string,the torsional motion of a rod or the longitudinal deflection of a beam mathematical modeling leads tothe wave equation which, in the linear case, is governed by

∂2t x(z , t ) = c2∂2

z x(z , t )−α∂t x(z , t )−β∂z x(z , t )+γx(z , t )+b(z )u(t ), z ∈ (0,L), t > 0 (1.10a)

∂z x(0, t )+p0x(0, t ) = v0(t ),

∂z x(L, t )+pL x(L, t ) = vL0(t ),t ≥ 0 (1.10b)

x(z ,0) = x0(z ), ∂t x(z ,0) = x1(z ), z ∈ [0,L]. (1.10c)

Here, c denotes the speed of wave propagation (for a discussion on dispersion, phase and groupvelocity the reader is referred to, e.g., [124]), α denotes a temporal damping coefficient, β is a spatialdamping coefficient, and γ, p0, pL are some parameters. The wave equation is a prototypical exampleof a hyperbolic PDE which, contrary to the heat equation (1.1), exhibits a finite speed of propagation.Due to this property hyperbolic PDEs cannot be directly solved using standard numerical tools such asthe pdepe function of MATLAB but require special emphasis when discretizing the PDE and boundaryconditions.

To motivate the Euler–Bernoulli beam equation consider the cantilevered beam structure with tipmass depicted in Figure 1.3. The beam is actuated by pairs of piezoelectric patch actuators, wherethe patches on the front side ( f s) and the patches on the back side (bs) are bonded symmetricallyonto the beam structure. The mounted piezoelectric actuators allow to locally induce bending strains

z1

z3

z2

Lc

bc

hc

z1p,1 z1

p,2

mtm , Itm

bp

Lp

x(Lc , t )

Figure 1.3: Cantilever beam with pairs of patches.

within the patch covered intervals [z1p,k , z1

p,k +Lp ] of the beam domain defined by

Λεk (z1) =(%ε(z1 − z1

p,k )−%ε(z1 − z1p,k −Lp )

), (1.11)


where %ε(z1) represents a (possibly smooth) transition function from %ε(z1) = 0 for z1 < −ε/2 to%ε(z1) = 1 for z1 > ε/2. Here, Lc and Lp denote the length of the beam and the patches, respectively.Considering the individual patch contributions to stiffness, damping, and inertia, this configurationresults in a beam model with spatially varying parameters such that the governing equations of motionfor the beam deflection x(z , t ) are given by

µ(z1)∂2t x(z1, t )+γe (z1)∂t x(z1, t )+∂2

z1

(E I (z1)∂2

z1 x(z1, t ))=−

m∑k=1

Γk (z1)uk (t ) (1.12a)

with the boundary conditions

x(z1, t ) = 0,∂z1 x(z1, t ) = 0, z1 = 0 (1.12b)

E I (z1)∂2z1 x(z1, t )+ Itm∂

2t ∂z1 x(z1, t ) = 0,

∂z1

(E I (z1)∂2

z1 x(z1, t ))−mtm∂

2t x(z1, t ) = 0.

z1 = Lc (1.12c)

and the initial conditions

x(z1,0) = x0(z1), ∂t x(z1,0) = v0(z1). (1.12d)

The coefficients µ(z1), γe (z1), E I (z1) denote mass per unit length, viscous damping and stiffness,respectively, and are given by

µ(z1) =µc +2m∑

k=1Λεk (z1)µp (1.13)

γe (z1) = γec +2

m∑k=1

Λεk (z1)γep (1.14)

E I (z1) = E Ic +2m∑

k=1Λεk (z1)E Ip . (1.15)

The in–domain actuation is given in terms of

Γk (z1) = Γp,k∂2z1Λ

εk (z1), (1.16)

where Γp,k summarizes patch actuator specific parameters. Additionally, mtm denotes the massand Itm the inertia of the tip mass. Here, the indices c, p and tm indicate the contributions of thecarrier layer, the patches and the tip mass, respectively. The equations of motion (1.12) can be directlydetermined by means of the extended Hamilton’s principle using calculus of variations [100]. Thedistributed parameter system (1.12) is a so–called biharmonic PDE due to the arising forth orderdifferentiation in z1. For the numerical solution of (1.12) either weighted residual methods, Galerkinmethods, finite element methods, or the Rayleigh–Ritz ansatz can be applied, see, e.g., [62].

1.3 Abstract formulation of linear PDEs

For the mathematical analysis and control design it is often advantageous to rewrite the govern-ing PDEs as a so–called abstract Cauchy problem [17, 118]. Given a linear distributed parametersystems the abstract formulation resembles the well–known state space formulation for linear finite–dimensional systems and takes the form

x(t ) =Ax(t )+Bu(t ), t > 0 (1.17a)

1.3 Abstract formulation of linear PDEs 9

x(0) = x0 ∈D(A). (1.17b)

Here, a state vector x(t) ∈ X from some Hilbert space X is introduced with A referring to a lineardifferential operator, that maps elements of its domain D(A) to X . The operator B denotes the inputoperator which maps the input u(t ) ∈U to X .

In the following it is assumed, that the reader is familiar with basics on linear function spaces. For thesake of completeness some examples of Hilbert spaces, i.e., normed linear spaces equipped with aninner product ⟨·, ·⟩X , are provided, that are useful for the subsequent analysis:

• Space of Lebesgue measurable functions Lp (a,b): Let p ≥ 1 be a fixed integer and let a,b ∈R. Wedenote by Lp (a,b) the set of measurable functions x(t ) with

∫ ba |x(t )|p dt <∞ equipped with the

norm2

‖x‖Lp =(∫ b

a|x(t )|p dt

) 1p

. (1.18)

For p = 2 an inner product on L2(a,b) can be introduced by

⟨x(t ), y(t )⟩L2 =∫ b

ax(t )y(t )dt , ⟨x(t ), x(t )⟩L2 = ‖x‖2

L2 (1.19)

given x(t ), y(t ) ∈ L2. If p =∞, then the norm is defined as

‖x‖L∞ = ess supt∈[a,b]

|x(t )| (1.20)

provided, that esssupt∈[a,b] |x(t )| <∞.

• Sobolev spaces H p (a,b): Let p ≥ 1 be a fixed integer and let a,b ∈R. The subspace of L2(a,b)defined by

H p (a,b) =

x(t ) ∈ L2(a,b) : ∂ jt x(t ) ∈ L2(a,b), j = 0,1, . . . , p

(1.21)

equipped with the inner product

⟨x(t ), y(t )⟩H p =p∑

j=0⟨∂ j

t x(t ),∂ jt y(t )⟩L2 (1.22)

is a Hilbert space. It can be shown, that H p (a,b) is the completion of C p (a,b) or C∞(a,b)function with respect to the norm (1.22). Note also the embedding H p+1(a,b) ⊂ H p (a,b). Fordetails on Sobolev spaces and their properties the reader is referred to [2].

With these preparations the examples introduced before, i.e., the heat equation, the wave equationand the Euler–Bernoulli beam, can be transferred into the abstract formulation (1.17).

Example 1.1 (Heat equation). Consider the linear heat equation, i.e.,

c∂t x(z , t ) = a∂2z x(z , t ))+u(z , t ), z ∈ (0,1), t > 0 (1.23a)

∂z x(0, t = 0, x(1, t ) = 0, t ≥ 0 (1.23b)

x(z ,0) = 0, z ∈ [0,1]. (1.23c)

2One actually needs to consider equivalence classes since ‖x‖Lp = 0 implies x(t ) = 0 only almost everywhere. For detailsconsult, e.g., [17].


The consideration of inhomogeneous boundary conditions requires special emphasis in the contextof the abstract setting so that only homogeneous boundary conditions are assumed for the sake ofsimplicity. Taking x(t) = x(·, t), u(t) = u(·, t), x0 = x0(·), and introducing X = L2(0,1) as solutionspace we obtain

x(t ) =Ax(t )+bu(t ), t > 0 (1.24a)

x(0) = x0 (1.24b)

with

Ax = a

c∂2

z x, b = 1

c(1.24c)

and domain

D(A) = x ∈ X : ∂2

z x ∈ L2(0,1) with ∂z x(0) = 0, x(1) = 0

. (1.24d)

Example 1.2 (Wave equation). Consider (1.10) for homogeneous boundary conditions, i.e.,

∂2t x(z , t ) = c2∂2

z x(z , t )−α∂t x(z , t )−β∂z x(z , t )+γx(z , t )+b(z )u(t ), (z , t ) ∈ (0,L), t > 0

(1.25a)

x(0, t ) = 0,

∂z x(L, t ) = 0,t ≥ 0 (1.25b)

x(z ,0) = x0(z ), ∂t x(z ,0) = x1(z ), z ∈ [0,L]. (1.25c)

As is common for finite–dimensional second order systems a state vector has to be introduced toobtain a formulation as coupled system of first order ODEs or PDEs, respectively. Hence, let

x(t ) =[

x1(t )x2(t )

]=

[x(·, t )∂t x(·, t )

](1.26)

and consider X = H 10 (0,L)×L2(0,L) with H 1

0 (0,L) = x ∈ H 1(0,L) : x(0) = 0 as solution space withinner product and induced norm

⟨x , y⟩X = c2⟨x1, y1⟩H 1 +⟨x2, y2⟩L2 , ‖x‖2X = ⟨x , x⟩X (1.27)

for x , y ∈ X . The motivation to introduce this particular Hilbert space is given by consideration of theenergy stored in the (undamped) system which directly corresponds to 1

2 ⟨x , x⟩X as sum of potentialand kinetic energy. With this, we obtain

x(t ) =Ax(t )+bu(t ), t > 0 (1.28a)

x(0) = x0 =[x0 x1

]T ∈D(A) (1.28b)

with

Ax =[

x2

c2∂2z x1 −αx2 −β∂z x1 +γx1

], b =

[0

b(z )

](1.28c)

and domain

D(A) = x ∈ X : x1 ∈ (H 2(0,L)∩H 10 (0,L)), x2 ∈ H 1(0,L) with ∂z x1(L) = 0. (1.28d)

1.3 Abstract formulation of linear PDEs 11

Example 1.3 (Euler–Bernoulli beam equation). The transfer of (1.12) into the abstract form (1.17)requires special emphasis due to the dynamic boundary conditions induced by the attached endmass. To take this into account introduce the state vector

x(t ) =

x1(t )x2(t )x3(t )x4(t )

=

x(·, t )∂t x(·, t )∂t x(Lc , t

∂t∂z x(Lc , t )

(1.29)

defined on the Hilbert space X = H 20 (0,Lc )×L2(0,Lc )×R2 with H 2

0 (0,Lc ) = H 2(0,Lc ) : x(0) = ∂z x(0) =0. We make use of the inner product

⟨x , y⟩X =∫ Lc

0

(µx2 y2 +E I∂2

z x1∂2z y1

)dz +mtm x3 y3 + Itm x4 y4 (1.30)

with the norm induced by ‖x‖X =√⟨x , x⟩X . With this, X as defined above becomes a Hilbert space.As in the previous example the norm can be associated with the sum of kinetic and potential energystored in the undamped system. The abstract formulation follows as

x(t ) =Ax(t )+Bu(t ), t > 0 (1.31a)

x(0) = x0 =[x0(z ) v0(z ) v0(Lc ) ∂z v0(Lc )

]T ∈D(A) (1.31b)

with

Ax =

x2

− 1µ(z )

(γe (z )x2 +∂2

z

(E I (z )∂2

z x1))

1mtm

∂z

(E I (z )∂2

z x1) |z=Lc

− 1Itm

(E I (z )∂2

z x1) |z=Lc

, B=− 1

µ(z )

0 . . . 0

Γ1(z ) . . . Γm(z )0 . . . 00 . . . 0

(1.31c)

and domain

D(A) = x ∈ X : x1 ∈ (H 4(0,Lc )∩H 20 (0,Lc )), x2 ∈ H 2

0 (0,Lc ), x3 ∈R, x4 ∈Rwith

x3 = x2(Lc ), x4 = ∂z x2(Lc ). (1.31d)


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[99] B. Schörkhuber, T. Meurer, and A. Jüngel. „Flatness–based trajectory planning for semilinearparabolic PDEs“. In: Proc. IEEE Conference on Decision and Control (CDC). Maui (HI), USA,Dec. 2012, pp. 3538–3543 (cit. on pp. 44, 61).

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References 19

2Flatness–based trajectory planning andfeedforward control design

The notion of (differential) flatness, as introduced by M. Fliess and co–workers [22, 21], has turnedout to provide a useful approach for solving trajectory tracking problems [90, 59, 76, 60, 93, 108].Originally, the concept was formulated for finite–dimensional nonlinear systems but many aspectshave been successfully generalized to certain classes of infinite-dimensional systems. In this chapter,we provide an introduction to flatness–based techniques by first summarizing some results obtainedfor the finite–dimensional case. This is followed by illustrating different extensions of this approach toinfinite–dimensional systems governed by distributed parameter systems.

2.1 Finite–dimensional nonlinear control systems

Roughly speaking, flatness means, that there exists a so-called flat or basic output such that all systemvariables (states, inputs and outputs) can be parametrized in terms of this flat output and its timederivatives1. For the sake of simplicity let us consider a nonlinear system of the form

x = f (x ,u) (2.1)

with the state x(t ) ∈X ⊂Rn and the input u(t ) ∈U ⊂Rm . The system (2.1) is called flat if there exists aso-called flat or basic output ξ(t ) such that [22, 89, 93]

(i) the components of ξ(t ) are functions of x(t ) and u(t ) and their time derivatives2

ξ=φ(x ,u, u, . . . ,u(γ)) , (2.2)

(ii) the components of ξ(t ) are not related by any differential or algebraic equation of the form3

ϕ(ξ, ξ, . . . ,ξ(δ))= 0 , (2.3)

(iii) and all system variables can be parametrized by the flat output and its time derivatives

x =ψ1

(ξ, ξ, . . . ,ξ(β))

u =ψ2

(ξ, ξ, . . . ,ξ(β+1)).

(2.4)

1As will be shown in subsequent sections for infinite–dimensional systems governed by PDEs an infinite number of timederivatives of the flat output may be required.

2Here and subsequently we will denote the γth time derivative of a function f (t ) in the form f (γ) = ∂γt f (t ) .3This condition is equivalent to dimu = dimξ.

21

Example 2.1 ((Linear time invariant system). A linear time-invariant system is flat if and only ifit is controllable. A controllable linear system can be always transformed into the controllabilitynormal form which for SISO systems is given by

x1

x2...

xn−1

xn

︸︷︷︸

x

=

0 1 0 . . . 00 0 1 . . . 0...

.... . .

. . ....

0 0 . . . 0 1−a0 −a1 . . . −an−2 −an−1

︸︷︷︸

A

x1

x2...

xn−1

xn

︸︷︷︸

x

+

00...01

︸︷︷︸

b

u (2.5)

with the coefficients a j , j = 1, . . . ,n, of the characteristic polynomial of the matrix A ∈Rn×n . Obvi-ously, ξ(t ) = x1(t ) serves as a flat output and the state and input parametrization according to (2.4)reads

x j = ξ( j−1), j = 1,2, . . . ,n (2.6)

u =n∑

j=0a jξ

( j ) (2.7)

with an = 1 and ξ(0)(t ) = ξ(t ).

In view of the analysis of distributed parameter systems some essential properties of flat finite–dimensional nonlinear control systems are summarized below:

• Steady state analysis: The state and input parametrizations (2.4) can be similarly used to analyzethe steady state behavior, i.e. let ξ(t ) = ξs , ξ( j ) = 0, j ≥ 1 and consider 0 = f (x s ,us), then

x s =ψ1(ξs ,0, . . . ,0) = ψ1(ξs)

us =ψ2(ξs ,0, . . . ,0) = ψ2(ξs).

This property will play a substantial role for distributed parameter systems.

• Uniqueness of flat outputs: Flat systems may have more than a single flat output fulfilling theconditions (2.2)–(2.4). In this case (differential) relations exist, that allow to transfer one flatoutput into the other.

• Uncontrolled systems: Free (uncontrolled) systems x(t) = f (x(t)) are not flat. Assume forthe moment, that there exists a flat output ξ(t) with x(t) = ψ1

(ξ(t), ξ(t), . . . ,ξ(β)(t)

). Then

substitution of this expression into the differential equation yields a differential equation of theform (2.3) which contradicts the assumption.

• Linear time invariant systems: For linear time invariant systems flatness and controllability areequivalent (see Example 2.1).

• Existence of flat outputs: For the flatness of nonlinear SISO systems necessary and sufficientconditions exist which are well-known from the theory of exact input–state linearization. Infact for SISO systems exact input–state linearizability and flatness are equivalent. For MIMOsystems also necessary and sufficient conditions are available for a nonlinear system to beexactly input–state linearizable, see, e.g., [38, 77]. In the last years much effort has been madein finding conditions for a MIMO system to be flat, see, e.g., [54, 96, 97]. However, if a MIMOsystem is exact input–state linearizable, then it is also flat.

22 Chapter 2 Flatness–based trajectory planning and feedforward control design

• Flatness–based feedforward control: Let ξ∗(t) ∈C n(R) denote a desired trajectory for the flatoutput. Then substitution of ξ∗(t ) into (2.4) directly yields the feedforward control u∗(t ), that isrequired to realize x(t ) → x∗(t ) in open–loop without integration of any differential equation,i.e.

x∗ =ψ1

(ξ∗, ξ∗, . . . ,ξ∗ (β))

u∗ =ψ2

(ξ∗, ξ∗, . . . ,ξ∗ (β+1)).

(2.8)

The determined trajectories x(t ) and u(t ) can be in advance utilized to analyze the fulfillment ofstate and input constraints. In addition, x∗(t ) can be considered as reference path to determinetracking controllers for (2.1).

• Flatness–based tracking control: Flat nonlinear systems can be exactly feedback linearized sothat linear methods can be considered for stabilization and control. Quasi–static or dynamicstate–feedback control can be utilized for the asymptotic stabilization of the tracking errore(t) = ξ(t)−ξ∗(t) or ex (t) = x(t)− x∗(t), respectively. By transferring (2.1) into the Brunovskýnormal form desired eigenvalues can be assigned for the tracking error dynamics.

It is obvious from the discussion, that flat systems have very pleasing properties to solve the trajectorytracking problem for finite–dimensional nonlinear systems. Hence, the question arises if similarproperties can be deduced when extending flatness–based methods to distributed parameter systems.This is the main topic of this chapter.

2.2 Distributed parameter control systems

The underlying idea of equivalence and flatness, i.e. the existence of a one–to–one correspondencebetween trajectories of systems, can be also adapted to systems governed by PDEs [51, 92, 94, 65,125]. Hence, the recent work on the flatness concept has mainly dealt with its extension to trajectoryplanning for boundary controlled linear and certain nonlinear distributed parameter systems in asingle and multiple spatial coordinate(s).

Remark 2.1

It is nowadays rather common to refer to a basic output instead of a flat output when consider-ing flatness–based methods for distributed parameter systems. Hence, in the following, bothnotations are used interchangeably.

2.2.1 Trajectory planning for PDE systems

For parabolic and biharmonic PDEs the application of operational calculus using basically the Laplacetransform or formal power series yields the state and input parametrization in terms of fractionaldifferentiation operators or infinite power series representations4. The arising series coefficientsdepend on successive time derivatives of the basic output. This requires to restrict the basic outputto a certain Gevrey class to ensure uniform convergence of the series. Examples concern trajectoryplanning for the linear heat equation [51] and for the linear diffusion equation with spatially dependentcoefficients [51, 58] in several state variables [23, 65]. In addition, certain semi– and quasi–lineardiffusion–convection–reaction systems modeling tubular reactors are considered, e.g., in [58, 74, 65, 75,120], while a moving boundary problem (Stefan problem) is studied in [19, 95]. A further generalizationfor semi–linear PDEs is considered in [98], where formal integration is used to determine the state

4This introductory literature review is a condensed version of the respective section in [63].

2.2 Distributed parameter control systems 23

and input parametrization. Besides parabolic PDEs, results on the trajectory planning for hyperbolicsystems exhibiting wave dynamics are available [79, 94, 125, 123].

Solutions to the trajectory planning problem for PDEs defined on higher–dimensional domains areprovided, e.g., in [94] for the control of the temperature evolution inside a cylinder. By exploiting therotational symmetry of the domain the problem is thereby reduced to two decoupled 1–dimensionalsystems. The motion of a fluid represented by linearized wave equations under the shallow waterapproximation inside a moving tank being subject to controlled translations and rotations is analyzedin [78]. The solution to the trajectory planning problem is obtained by in principle superimposingthe solution of two decoupled 1–dimensional problems. First computations with entire functions aresuggested in [91] for the 2– and 3–dimensional wave equation with a finite–dimensional control actingsimultaneously on all of the domain’s boundary. It is, however, shown that the resulting flatness–basedparametrizations diverge in general.

By considering so–called Riesz spectral systems a rather generic approach for flatness–based trajectoryplanning for distributed parameter systems is developed in [66, 63], that covers both boundary and in–domain control. Here, a particular re–formulation of the resolvent operator is used to systematicallyconstruct a basic output. Convergence of the differential state and input parametrizations is analyzedby making use of entire function theory and essentially relies on the distribution of the eigenvalues ofthe system operator. For boundary controlled diffusion–convection–reaction systems with spatiallyand time varying parameters defined on a parallelepiped domain a solution to the trajectory planningproblem is provided in [72] by considering a formal integration of the PDE.

Based on this short survey of the available techniques in the following selected techniques are sum-marized and evaluated for the benchmark examples introduced in Section 1.2.

2.3 Operational calculus

Application of operational calculus, i.e., integral transformations such as the Laplace transform orthe Mikusinski calculus, enables to transfer linear initial boundary value problems into ordinarydifferential equations. As is shown subsequently for different examples, this allows to determine abasic output for the original distributed parameter formulation.

Remark 2.2

It should be mentioned, that the main focus of this section is the development of the underlyingideas. For the generalization of the design approach based on operational calculus to rathergeneric formulations of linear distributed parameter control problems the reader is referred tothe literature cited at respective positions in the text.

2.3.1 Flatness–based trajectory planning for the linear heat equation

The temperature distribution x(z , t ) for the heated rod shown below with ideal insulation at z = 0 andboundary input u(t ) an z = 1 is considered.


x(z , t )

z0 1

u(t )

Taking into account the presentation in Section 1.2.1 the spatial–temporal evolution of x(z , t) isgoverned by the linear heat equation

∂t x(z , t ) = k∂2z x(z , t ), z ∈ (0,1), t > 0 (2.9a)

∂z x(0, t ) = 0, x(1, t ) = u(t ), t > 0 (2.9b)

x(z ,0) = x0(z ), z ∈ [0,1] (2.9c)

with k =λ/(ρc). For the sake of simplicity it is assumed, that all variables are dimensionless which canbe easily achieved by proper normalization. Let x0(z ) = 0, then application of the Laplace transformto (2.9) results in the boundary value problem

∂2z x(z , s) = s

kx(z , s), ∂z x(0, s) = 0, x(1, s) = u(s) (2.10)

which admits the solution

x(z , s) = g (z , s)u(s), g (z , s) = cosh(µ(s)z )

cosh(µ(s))(2.11)

with µ(s) =ps/k. Based on the transfer function g (z , s) we verify the existence of a basic output. Let

u(s) = cosh(µ(s))ξ(s), then (2.11) implies

x(z , s) = cosh(µ(s)z )ξ(s), u(s) = cosh(µ(s))ξ(s). (2.12)

Due to the formal relation with (2.3) the quantity ξ(s) can be considered a basic output in the opera-tional domain. For the determination of the time–domain parametrizations the series expansion ofthe cosh–function is considered, i.e.,

cosh(µ(s)z ) =∞∑

n=0

(µ(s)z )2n

(2n)!=

∞∑n=0

( s

k

)n z2n

(2n)!.

This yields with (2.12) the state and input parametrizations

x(z , t ) =∞∑

n=0

z2n

kn(2n)!∂n

t ξ(t ), u(t ) =∞∑

n=0

1

kn(2n)!∂n

t ξ(t ). (2.13)

In addition, the introduced formal quantity ξ(s) or ξ(t ), respectively, admits the interpretation ξ(t ) =x(0, t ) which can be verified by direct substitution in (2.12).

2.3.1.1 Convergence analysis

The state and input parametrizations (2.13) impose certain restrictions on any admissible trajectoryfor the basic output. In particular it is required, that

2.3 Operational calculus 25

• ξ(t ) is infinitely often continuously differentiable, i.e., it is a smooth function and

• the growth of the derivatives of ξ(t) is sufficiently bounded so that convergence of the series(2.13) can be ensured.

To address both issues the notion of a Gevrey class function is required [36].

Definition 2.1: Gevrey class

The function ξ(t) :Ω→R is in GD,α(Ω), the class of Gevrey functions of order α, if ξ(t) ∈C∞(Ω)and there exists a positive constant D such that

supt∈Ω

|∂nt ξ(t )| ≤ Dn+1(n!)α (2.14)

for all n ∈N∪ 0.

Recall the Cauchy–Hadamard theorem for the radius of convergence % of the power series∑

n an zn ,i.e.,

% =

0 if limn→∞ |an | 1

n →∞∞ if limn→∞ |an | 1

n = 01

limsupn→∞ |an | 1n

else.

Herein, limsupn→∞ |an | 1n denotes the largest accumulation point of the sequence (|an | 1

n )n [34]. Withthese preliminaries it is a rather easy task to proof the following result.

Proposition 2.1. Let ξ(t ) ∈GD,α(R), then the series

x(z , t ) =∞∑

n=0

z2n

kn(2n)!∂n

t ξ(t ) (2.15)

converges uniformly with infinite radius of convergence % in z if α< 2 and with radius of convergence% = 2

pk/D if α= 2.

Proof. Since ξ(t ) is by definition a Gevrey class function of order α and hence fulfills (2.14), the series(2.15) for k > 0 and z ∈ [0,1] can be bounded according to

|x(z , t )| =∣∣∣∣ ∞∑n=0

z2n

kn(2n)!∂n

t ξ(t )

∣∣∣∣≤ ∞∑n=0

Dn+1

kn

(n!)α

(2n)!z2n = D

∞∑n=0

(n!)α

(2n)!

(Dz2

k

)n.

Let η= Dz2/k, then the last expression can be interpreted as a power series in η. Taking into accountthe Cauchy–Hadamard theorem its radius of convergence %η in the variable η can be determined as

%η =

∞, α< 2

4, α= 2

0, α> 2.

To verify this result the Stirling formula can be used which provides the asymptotic relation n! ∼p2πnn+ 1

2 /en for n À 1. In view of η= Dz2/k the radius of convergence in z follows as %z =√

k%η/D

and proves the claim.


2.3.1.2 Admissible trajectories for the basic output

As has been shown before the flatness–based approach essentially relies on planning admissibletrajectories for the flat output taking into account the conditions formulated in Proposition 2.1. Inaddition, two goals have to be distinguished:

• Finite time transitions between steady states: On rather common control task is the realization oftransitions between steady states within a desired time interval t ∈ [0,T ]. For the heat equation(2.9) this requires the determination of the feedforward control

u∗(t ), t ∈ [0,T ] : x0(z ) = x∗(z ,0) → x(z ,T ) = x∗T (z ). (2.16)

Here, the initial and final profile x0(z ) and xT (z ), respectively, are assumed to be solutionsxs(z ;us) of the boundary value problem associated with (2.9), i.e.,

k∂2z xs(z ) = 0, z ∈ (0,1)

∂z xs(0) = 0, xs(1) = us

with x∗0 (z ) = xs(z ;u∗

0 ) and x∗T (z ) = xs(z ;u∗

T ).

Due to the flatness property the transition can be alternatively formulated in terms of flat outputξ(t) = x(0, t). Let ξ∗0 and ξ∗T denote desired initial and final values, then steady state solutionsxs(z ;ξs) fulfill the boundary value problem

k∂2z xs(z ) = 0, z ∈ (0,1)

∂z xs(0) = 0, xs(0) = ξs .

Hence, the transition can be formulated as finding the trajectory ξ∗(t ) ∈GD,α(R) connecting ξ∗0and ξ∗T , i.e.,

ξ∗(t ), t ∈ [0,T ] : ξ∗0 = ξ∗(0) → ξ∗(T ) = ξ∗T with ∂nt ξ

∗(t )|t∈0,T = 0. (2.17)

For the explicit realization of ξ∗(t ) satisfying the specification (2.17) and hence x∗(z , t ) as wellas u∗(t) by evaluating (2.13) it is required, that ξ∗(t) is infinitely often differentiable in t butlocally non–analytic at t = 0 and t = T . As is shown below this restricts the Gevrey order α ofξ∗(t ) to the interval α ∈ (1,2).


To fulfill these restrictions in the following we make use of the Gevrey class function

ξ∗(t ) = ξ∗0 +(ξ∗T −ξ∗0

)Θω,T (t ) (2.18)

with

Θω,T (t ) =

0, t ≤ 0

1, t ≥ T∫ t0 θω,T (τ)dτ∫ T0 θω,T (τ)dτ

, t ∈ (0,T )

(2.19)

and

θω,T (t ) =

0, t 6∈ (0,T )

exp(− ([

1− tT

] tT

)−ω), t ∈ (0,T ).

(2.20)

In particular,Θω,T (t ) is of Gevrey order α= 1+1/ω [37, 58]. The parameter ω and the transitiontime T can be used to adjust the slope of ξ∗(t ). Figure 2.1 shows ξ∗(t ) defined in (2.18) for twodifferent values of ω to illustrate this effect.

• Finite time transitions between arbitrary states: If the initial and final profiles x∗(z ,0) andx∗(z ,T ) do not correspond to steady state profiles of the considered distributed parametersystem, then trajectory planning is significantly complicated. Explicit computations are notprovided here but the interested reader is referred to [51, 19], where the projection of the initialand final profile onto the basis spanned by power series of the underlying function space is


0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

t

ξω=1.1

ω=2.0

Figure 2.1: Trajectory ξ∗(t ) defined in (2.18) with (2.19), (2.20) for ξ∗0 = 0, ξ∗T = 1, T = 1, and ω ∈ 1.1,2.

proposed. Another projection technique is suggested in [63] based on the parametrizationapproach introduced in Section 2.4 below.

2.3.1.3 Simulation results

To illustrate the theoretical concepts simulation results are presented for the feedforward control ofthe heat equation (2.9). The boundary input is determined by evaluating the input parametrization in(2.13) with the desired trajectory ξ∗(t) for the basic output constructed as described before, i.e., bymaking use of (2.18). The series for u∗(t ) is cut off after 21 addends. Numerical results are shown inFigure 2.2 for ξ∗(t ) of Gevrey order α= 2 and in Figure 2.3 when reducing the Gevrey order to α= 1.5.The transition time is assigned as T = 1. The numerical evaluation reveals that a reduction in α resultsin an increase of the slope of ξ∗(t ) and thus in an increase in input amplitude.

0

0.5

1

0

0.5

1

1.50

0.5

1

1.5

2

zt

x

0 0.5 1 1.50

0.5

1

1.5

2

t

u*

Figure 2.2: Feedforward boundary control of the heat equation (2.9) for Gevrey order α= 2 and k = 1: Stateevolution x(z , t ) (left), feedforward control u∗(t ) (right).

2.3.2 Flatness–based trajectory planning for the linear wave equation

The wave equation is the prototype of so–called hyperbolic PDEs, that exhibit wave dynamics with fi-nite speed of propagation. This classification is based on the so–called characteristics or characteristiccurves (the reader is here referred to, e.g., [43, 124]).


0

0.5

1

0

0.5

1

1.50

1

2

3

4

zt

x

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

t

u*

Figure 2.3: Feedforward boundary control of the heat equation (2.9) for Gevrey order α= 1.5 and k = 1: Stateevolution x(z , t ) (left), feedforward control u∗(t ) (right).

In the following the implications of the finite speed of propagation are illustrated by developingflatness–based trajectory planning for the linear wave equation with boundary control, i.e.,

∂2t x(z , t ) = c2∂2

z x(z , t ), z ∈ (0,1), t > 0 (2.21a)

∂z x(0, t ) = u(t ), ∂z x(1, t ) = 0, t > 0 (2.21b)

x(z ,0) = x0(z ), ∂t x(z ,0) = x1(z ) z ∈ [0,1] (2.21c)

with c denoting the phase velocity. Assuming zero initial conditions application of the Laplacetransform transfers (2.21) into a boundary value problem, whose solution in the operator domain canbe determined as

x(z , s) =−c cosh(µ(s)(1− z ))

s sinhµ(s)u(s) (2.22)

with µ(s) = s/c. Proceeding as in Section 2.3.1 the basic output ξ(s) = x(1, s) can be introduced in theoperational domain which yields the state and input parametrizations

x(z , s) = cosh(µ(s)(1− z ))ξ(s) (2.23)

u(s) =− s

csinh(µ(s))ξ(s). (2.24)

These equations of the inverse system can be transferred into the time domain by taking into accountthe shifting property which yields

x(z , t ) = 1

2

[ξ

(t + 1− z

c

)σ

(t + 1− z

c

)+ξ

(t − 1− z

c

)σ

(t − 1− z

c

)], (2.25a)

u(t ) =− 1

2c

[ξ

(t + 1

c

)σ

(t + 1

c

)− ξ

(t − 1

c

)σ

(t − 1

c

)]. (2.25b)

Differing from the heat equation example delayed and advanced arguments arise, that also need toenter the assignment of admissible trajectories ξ∗(t) for the basic output ξ(t). In addition it shouldbe mentioned, that it is sufficient to assign ξ∗(t) ∈ C 0(R) which, by (2.25b), implies a (piecewise)continuous input u∗(t ).


ξ∗(t )

ξ∗(t )

t

1a

1

− 1a

01c

1c +a 1

c +2a

Figure 2.4: Triangular–shaped desired trajectory for the basic output ξ∗(t ) and its derivative ξ∗(t ).

To illustrate this consider the desired triangular–shaped trajectory ξ∗(t ) of width 2a depicted in Figure2.4. Causality requires ξ∗(t ≤ 1/c) = 0. The corresponding feedforward control u∗(t ) determined from(2.25b) is shown in 2.5 for a pulse width of 2a < 2/c (top) and 2a > 2/c (bottom). The effect of the

0

0

u∗(t )

u∗(t )

t

t

1c

a 2c

2a 2c +a

2c +2a

a 1c

2a 2c

2c +a

2c +2a

− 12ac

12ac

1ac

− 12ac

12ac

Figure 2.5: Feedforward control u∗(t) for the desired trajectory ξ∗(t) of Figure (2.4) for a < 1/c (top) anda > 1/c (bottom).

feedforward control u∗(t) on the spatial–temporal dynamics of the wave equation is illustrated inFigure 2.6 in the (z , t , x)–plane. The first contribution of the feedforward control (2.25b), i.e.,

u∗(1)(t ) =− 1

2cξ

(t + 1

c

)σ

(t + 1

c

)(2.26)

induces a triangular impulse, that propagates along the characteristic curves in the (z , t)–plane toright border z = 1. The time of arrival of the first signal is t (1) = 1/c. Reflection at the free end impliesamplitude doubling at z = 1 and traveling of an image of the initial impulse to the left border z = 0.


0

x(z, t )

z1

t

a

2a

2c

2c +a

2c +2a

1c

1c +a

1c +2a

12

1

Reflected wave

Control ∂z x(0, t ) = u∗(t )

Incoming waveinduced by control

Extinction by wavesuperposition

Figure 2.6: Feedforward boundary control for the wave equation (2.21) to realize trajectory tracking x(1, t ) =ξ(t ) → ξ∗(t ) with ξ∗(t ) from Figure 2.4 and pulse width a > 1/c.

The time of arrive of the first backward–traveling signal is t (2) = 2/c. This wave superposes with the(second) contribution imposed by the feedforward control u∗(t ), i.e.,

u∗(2)(t ) = 1

2cξ

(t − 1

c

)σ

(t − 1

c

)(2.27)

which results in the extinction of the incoming wave.

Summarizing these results it has to be pointed out, that trajectory planning for hyperbolic PDEs has totake into account the finite speed of propagation. This corresponds to a minimal control time which,e.g., does not arise for the parabolic heat equation, whose speed of propagation is infinite.

2.4 Riesz spectral operators

The spectral analysis of a finite– or infinite–dimensional linear operator is a well–established andpowerful mathematical tool for stability analysis and feedback control design. The dynamic systemproperties are thereby determined based on the eigenvalue distribution and the respective set ofeigenvectors. For infinite–dimensional systems governed by PDEs certain restrictions apply that arein particular related to the possible existence of continuous spectra.

However, a wide class of physically important systems including, e.g., diffusion–convection–reaction,wave, Euler–Bernoulli, and Timoshenko beam equations, yields so–called Riesz spectral operators.These have the favorable property of a discrete eigenvalues distribution with the respective eigenvec-tors and adjoint eigenvectors spanning an orthogonal basis for the underlying function space. On the


one hand this significantly simplifies the analysis of structural properties such as controllability andobservability which can be performed rather similar to the finite–dimensional case [17]. On the otherhand Riesz spectral operators satisfy the so–called spectrum determined growth assumption whichimplies, that stability can be directly evaluated based on the eigenvalue distribution [17, 57].

2.4.1 Fundamental concepts

To motivate the notion of a Riesz basis and to determine properties of the important class of (Riesz)spectral operators subsequently some preliminary results on normed linear spaces and spectralanalysis are summarized according to the exposition in [63].

Let X denote a Hilbert space with inner product ⟨·, ·⟩X and induced norm ‖x‖X =√⟨x , x⟩X . Given abasis (ek )k∈N of a Hilbert space X , then the sequence ( f k )k∈N biorthogonal to a basis (ek )k∈N of aHilbert space X is also a basis of X [27]. Thus, any x ∈ X can be uniquely expanded into the series

x = ∑k∈N

⟨x , f k⟩X ek , (2.28)

which is convergent in the norm ‖ ·‖X .

Remark 2.3

We remark the following:

• If not stated otherwise throughout this chapter the natural numbersN are considered asthe index set. However, all results hold similarly if another countable index set is used, e.g.,Z. The choice depends on the particular system under consideration.

• As is common in the functional analytic or operator theoretic setting the reference to theindependent coordinates, in particular to the spatial coordinate z , is omitted subsequently.

With this, the following theorem can be introduced, which provides a procedure to deduce that agiven sequence indeed represents a Riesz basis.

Theorem 2.1

The following assertions are equivalent:

1. The sequence (ek )k∈N forms a basis of the Hilbert space X equivalent to an orthonormalbasis, i.e. (ek )k∈N is a Riesz basis.

2. The sequence (ek )k∈N becomes an orthonormal basis of the space X by an appropriate re-placement of the inner product ⟨·, ·⟩X by some topologically equivalent new inner product⟨·, ·⟩1, i.e., ∃c1, c2 > 0 such that for all x ∈ X the induced norms satisfy c1⟨x , x⟩X ≤ ⟨x , x⟩1 ≤c2⟨x , x⟩X .

3. The sequence (ek )k∈N is complete in X and there exists positive constants m, M suchthat for any positive integer k ′ and arbitrary αk , k = 1, . . . ,k ′, one has m

∑k ′k=1 |αk |2 ≤

‖∑k ′k=1αk ek‖ ≤ M

∑k ′k=1 |αk |2.

4. The sequence (ek )k∈N is complete in X , there exists a complete biorthogonal sequence( f k )k∈N, and for any x ∈ X one has

∑k∈N |⟨x ,ek⟩X |2 <∞ and

∑k∈N |⟨x , f k⟩X |2 <∞.

2.4 Riesz spectral operators 33

5. The sequence (ek )k∈N is complete in X and the Gramian matrix given by [⟨ek ,e j ⟩X ]k, j∈Ngenerates a bounded invertible operator on `2.

The equivalences 1–4 are also known as the Bari theorem [27, Theorem. VI.2.1] or [126, Theorem.9] with the latter pointing out the equivalence with 5. The Bari theorem implies that any x ∈ X canbe represented as a linear combination of the individual ek , k ∈N, even if these are not mutuallyorthogonal but form a Riesz basis [17, 29, 118].

Corollary 2.1. Let (ek )k∈N be a Riesz basis. Then

1. there exists a sequence ( f k )k∈N biorthogonal to (ek )k∈N, which forms a Riesz basis for X ;

2. every x ∈ X can be uniquely expressed as

x = ∑k∈N

⟨x , f k⟩X ek

and there exist constants m, M > 0 such that

m∑

k∈N|⟨x , f k⟩X |2 ≤ ‖x‖2

X ≤ M∑

k∈N|⟨x , f k⟩X |2.

With these preparations so–called (Riesz) spectral operators on Hilbert spaces can be introduced [45,29, 106].

Definition 2.2: Scalar operator

Let (ek )k∈N be a Riesz basis in a Hilbert space X , let ( f k )k∈N be the Riesz basis biorthogonal to(ek )k∈N, and let (λk )k∈N be a sequence inC. The linear operator

Ax = ∑k∈N

λk⟨x , f k⟩X ek

in X with domain

D(A) =

x ∈ X :∑

k∈N|λk |2|⟨x , f k⟩X |2 <∞

is called a scalar operator.

Definition 2.3: Spectral operator

An operator A in X is called a spectral operator if it can be represented in the form

A=S+N (2.29)

with S a scalar operator and N a bounded finite rank nilpotent operator commuting with S.

By imposing a restriction on the sequence (λk )k∈N the following Lemma can be verified [118].

Lemma 2.1

Let (ek )k∈N and ( f k )k∈N be orthonormal Riesz bases and let (λk )k∈N be a sequence inC. Thenthe following statements are equivalent:


1. The sequence (λk )k∈N is bounded.

2. The series Ax = ∑k∈Nλk⟨x , f k⟩X ek is convergent for every x ∈ X and the thus defined

operator A is bounded on X .

If the above statements are true, then supk∈N |λk | ≤ ‖A‖ ≤pM/m supk∈N |λk |, where m, M as

introduced in Theorem 2.13.

Given a bounded operator A ∈L (X ), the last statement implies that λk ∈σp (A), where σp (A) denotesthe point spectrum of the operator A [118, Proposition 2.2.10]. By restricting the analysis to operatorswith a pure point spectrum, the relationship between the spectral properties of the operator and theproperties of the Riesz basis of the Hilbert space X can be further exploited by analyzing the Rieszbasis properties of the root vectors or generalized eigenvectors, respectively, of a linear operator A, i.e.the sequence of its eigenvectors and associated vectors.

Remark 2.4

In the following only spectral operators with mutually disjoint discrete eigenvalues are consid-ered and the interested reader is referred to [63] for the general situation. This assumption inparticular implies, that N in Definition 2.3 reduces to the zero operator so that A=S and A isdiagonalizable. Also recall, that the spectrum σ(A) of the closed linear operator A contains alleigenvalues, i.e. all λ ∈C for which the equation(

A−λI)φ = 0

has at least one nonzero solutionφ ∈ X . In this caseφ is called an eigenvector.

The subsequently considered class of (Riesz) spectral operators can be formulated as follows (see also[29, Theorems. 2.9, 2.12] and [106, Theorem. 4]).

Theorem 2.2

Let A be a closed linear operator with isolated (point) spectrum σp (A) = (λk )k∈N and σp (A)being totally disconnected5. Assume that the set of eigenvectors (φk )k∈N forms a Riesz basis forX . Then

1. the set of eigenvectors (ψk )k∈N of the adjoint operator A∗ forms a Riesz basis for X , whichis biorthogonal to (φk )k∈N;

2. A is a (Riesz) spectral operator according to Definition 2.3 with

Ax= ∑k∈N

λk⟨x ,ψk⟩Xφk (2.30)

for all x ∈D(A), where

D(A) =

x ∈ X :∑

k∈N|λk |2|⟨x ,ψk⟩X |2 <∞

. (2.31)

In view of Corollary 2.1 the above result yields, that if the set of eigenvectors (φk )k∈N forms a Rieszbasis for X , then every x ∈ X can be uniquely expressed in terms of the Fourier series

5Any two elements of σp (A) cannot be connected by a segment lying entirely in the closure σp (A).


x = ∑k∈N

⟨x ,ψk⟩Xφk . (2.32)

There are counterexamples which show, that the (generalized) eigenvectors of an operator A notnecessarily form a Riesz basis for X [29]. This makes it necessary to use one of the criteria provided inTheorem 2.1 to verify the Riesz basis property.

Theorem 2.3

Let A be a Riesz spectral operator. Then

1. λ ∈ ρ(A) if and only if infk∈N |λ−λk | > 0. In this case the resolvent can be expressed as

(λI−A)−1x = ∑k∈N

1

λ−λk⟨x ,ψk⟩Xφk . (2.33)

2. The operator A generates a C0–semigroup T(t) if and only if supk∈Nℜλk <∞. In thiscase,

T(t )x = ∑k∈N

eλk t ⟨x ,ψk⟩Xφk (2.34)

and A or T(t ), respectively, satisfy the spectrum determined growth condition.

The proof of Theorem 2.3 can be found in [17] or [29] for A being a spectral operator according toDefinition 2.3.

Remark 2.5

Once the Riesz basis of eigenvectors is chosen, it is particularly convenient to exploit the equiv-alence of both the operator A as well as the resolvent (λI−A)−1 with infinite–dimensionalmatrices in the space `2. Recalling Theorem 2.2 we know, that any x ∈ X can be represented bythe Fourier series (2.32) with (⟨x ,ψk⟩X )k∈N ∈ `2. In other words X is isometric isomorph to `2.

2.4.2 Flatness–based state and input parametrization

In the following, distributed parameter systems in abstract form governed by

∂t x(t ) =Ax(t )+Bu(t ) (2.35a)

x(0) = x0 ∈D(A) (2.35b)

are considered for x(t ) ∈ X . Moreover, we assume

1. A is a (Riesz) spectral operator according to Definition 2.3 with non–zero eigenvalues λk , k ∈N;

2. the initial state x0 is a steady state satisfying Ax0 = 0, x0 ∈ D(A) such that without loss ofgenerality we can consider x0 = 0;

In addition, in view of the following examples we


3. restrict the analysis to input operators of the form

Bu(t ) =m∑

l=1bl ul (t ) (2.36)

with the spatial input characteristics bl = bl (z ) and refer to [63] for a general treatise and

4. assume that (2.35) is approximately controllable, i.e.,

rk[⟨b1,ψk⟩X . . . ⟨bm ,ψk⟩X

]= 1

for all k ∈N.

Remark 2.6

As already pointed out in Remark 2.3 we omit to explicitly provide the dependency of thearising variables on the spatial coordinate. In particular we have x(s) ≡ x(z , s),φk ≡φk (z ) andψk ≡ψk (z ).

2.4.2.1 Construction of a basic output in the operational domain

The resolvent operator corresponds to the Laplace transform of the C0–semigroup generated by A,i.e.,

x(s) = (sI−A

)−1Bu(s), s ∈ ρ(A) with s > sup

k∈Nℜλk ,

where s ∈C denotes the Laplace variable and x(s) and u(s) the Laplace transforms of x(t) and u(t).By recalling (2.33) this yields for all s ∈ ρ(A) with s > supk∈Nℜλk , that

x(s) = ∑k∈N

1

s −λk⟨Bu(s),ψk⟩Xφk =− ∑

k∈N

1

λk

1

1− sλk

⟨Bu(s),ψk⟩Xφk . (2.37)

The above expression serves as basis for the construction of a basic output for the linear system (2.35).For this, we re–write the resolvent by sufficiently extending numerator and denominator

x(s) =− ∑k∈N

1

λk

∏j∈N, j 6=k

(1− s

λ j

)∏

j∈N(1− s

λ j

) ⟨Bu(s),ψk⟩Xφk

and observe, that in view of (2.36) we have

⟨Bu(s),ψk⟩X =m∑

l=1⟨bl ,ψk⟩X ul (s).

Substitution into the previous expression yields

x(s) =− ∑k∈N

1

λk

m∑l=1

⟨bl ,ψk⟩Xφk

∏j∈N, j 6=k

(1− s

λ j

) ul (s)∏j∈N

(1− s

λ j

) .

Formally introducing the variables ξl (s), l = 1, . . . ,m by


ul (s) = ∏j∈N

(1− s

λ j

)︸︷︷︸

Du(s)

ξl (s) (2.38a)

provides

x(s) =− ∑k∈N

1

λk

m∑l=1

⟨bl ,ψk⟩Xφk

∏j∈N, j 6=k

(1− s

λ j

)︸︷︷︸

Dxk (s)

ξl (s). (2.38b)

Expressions (2.38) can be interpreted as state and input parametrizations in the operational (Laplace)domain in terms of ξl (s). Hence, in accordance with the previous results ξl (s), l = 1, . . . ,m is called abasic output in the operational domain.

2.4.2.2 Time domain representation and convergence analysis

The state and input parametrizations (2.38) are so far only formal since their (uniform) convergencehas to be ensured. Convergence obviously relies on the properties of the basic output and as suchreduces to a problem of assigning suitable trajectories to ξl (t ). In addition it is decisive to note, thatthe introduced operators Du(s) and Dx

k (s) define entire functions in the complex domain s ∈C. Withthis, the main result reads as follows.

Theorem 2.4

Let (λk )k∈N be the sequence of disjoint eigenvalues of the Riesz spectral operator A. Assumethat (λk )k∈N is of convergence exponent γ and genus g s . Then Du(s) is an entire functions oforder %= γ and admits a MacLaurin series expansion

Du(s) = ∑n∈N

cn sn−1, c1 = 1. (2.39)

If Du(s) is in addition of finite type τ, then

f (t ) = Du(∂t )ξ(t ) = ∑n∈N

cn∂n−1t ξ(t ) (2.40)

convergences uniformly for ξ(t ) ∈GD,α(R) with α≤ 1/% and ‖ f (t )‖∞ is bounded.

For a proof of Theorem 2.4 consult [63]. Identical properties can be deduced for Dxk (s) based on the

results for Du(s). As a result of the previous analysis the time–domain representation of (2.38) followsimmediately by taking into account the MacLaurin series expansion, i.e.,

ul (t ) = Du(∂t )ξl (t ) = ∑n∈N

cn∂nt ξ(t ) (2.41a)

x(t ) =− ∑k∈N

1

λkDx

k (∂t )ξl (t )m∑

l=1⟨bl ,ψk⟩Xφk . (2.41b)


Remark 2.7

The convergence of (2.41b) imposes an additional condition, that is omitted but is derived in[63]. This is due to the summation over k ∈Nweighting the coefficients of the basis functionsφk so that the L2–convergence of this Fourier series has to be ensured.

In view of the convergence result, the assignment of suitable desired trajectories ξl ,∗(t ) for the basicoutput ξl (t ) follows along the lines of Section 2.3.1.2.

To apply these results obviously certain notions from the theory of entire functions are required,that are summarized in the following remark (see [63] for a comprehensive overview of the requirednotions and properties in view of flatness–based methods).

Remark 2.8

The so–called maximal modulus M(η) of an entire function f (s) is defined as

M(η) = max|s|=η

| f (s)|. (2.42)

Obviously, M(η) enables to characterize the growth of the entire function f (s). Thereby, twoproperties are essential, namely type and order:

• The entire function f (s) is of finite order if M(η) <as exp(ηk ) for some k > 0. The order % isthe infimum of those k for which the asymptotic inequality <as is fulfilled. With this, wehave

eη%−ε <n M(η) <as eη

%+ε

and by taking the logarithm twice we conclude

%= limsupη→∞

loglog M(η)

logη.

• The function f (s) has a finite type if for some A > 0 the inequality M(η) <as exp(Aη%) holds.The type τ is the infimum of those A for which the asymptotic inequality <as is satisfied.Moreover, this implies the inequalities

e(τ−ε)η% <n M(η) <as e(τ+ε)η%

and hence

τ= limsupη→∞

log M(η)

η%. (2.43)

If for a given % the type of f (s) is infinite, then the function is of maximal type. If 0 < τ<∞,then the type is normal while for τ= 0 the type is minimal.

Given a non–decreasing sequence (an)n∈N, an ∈C the so–called counting function N (η) is

N (η) = #an , n ∈N : |an | ≤ η. (2.44)


and its order %1 [12, Theorem 2.5.8] can be defined as

%1 = limsupη→∞

logN (η)

logη. (2.45)

The counting function is a particularly useful tool to deduce properties of entire functions.

Given a sequence (an)n∈N, an ∈ C with an 6= 0, limn→∞ an → ∞ the convergence exponent isdefined as the infimum of positive numbers γ for which the series

∑n∈N

1

|an |γ(2.46)

converges. The relationship between the convergence exponent of a sequence and its countingfunction is given in the following lemma [53, Section 3.2].

Lemma 2.2

Given a sequence (an)n∈N, an ∈Cwith an 6= 0, limn→∞ an →∞, then γ= %1.

Denote by g s +1 the smallest positive integer γ for which (2.46) converges. Then the integer g s

is called the genus of the sequence (an)n∈N. The genus g s is not necessarily equal to the genus ofthe entire function f (s) but there is a class of entire functions, where equality holds. Assumethat the sequence (an)n∈N is of genus g s . With this, consider now the infinite product

Π(s) = ∏n∈N

G

(s

an, g s

), (2.47)

with the so–called Weierstrass primary factors G (s, g s) defined as

G (s, g s) =

1− s, g s = 0

(1− s)expF (s, g s), g s > 0, F (s, g s) =

g s∑i=1

si

i. (2.48)

The infinite product (2.47) converges absolutely and uniformly in every disk s ∈C : |s| ≤ R <∞[53, Section 4.1] andΠ(s) is called the Weierstrass canonical product of genus g s . Following Boas[12, Theorem 2.6.5],Π(s) defines an entire function of order equal to the convergence exponent ofits zeros.

With these preliminary considerations one of the main theorems in the theory of entire functionscan formulated, which provides a general representation formula for entire functions of finiteorder [53, Section 4.2].

Theorem 2.5: Hadamard theorem

An entire function f (s) of finite order % may be represented in the form

f (s) = smePq (s)∞∏

n=1G

(s

an, g s

), (2.49)

where the sequence (an)n∈N of genus g s includes all nonzero roots of the function f (s),g s ≤ %, Pq (s) is a polynomial in s of degree q ≤ %, and m is the multiplicity of the root at theorigin.


The product representation is useful to connect the growth of an entire function and the distri-bution of its zeros.

Theorem 2.6

The convergence exponent γ of the zero set of an entire function f (s) of non–integer orderis equal to the order of growth % of f (s).

For a proof, see, e.g., Levin [53, Section 5.1]. If the entire function is of integer order, then nosuch simple result is available since the order might be larger than the distribution of its zerosmight indicate. As an example consider exp(s), which has no (finite) zeros but is of order 1. Leta% denote the coefficient of s% in the polynomial Pq (s) in the Hadamard representation (2.49).

2.4.3 Application to the linear heat and wave equation with in–domain control

In the following, we consider the application of the spectral design approach for the linear heat andwave equation defined on the line z = [0,1] with Dirichlet boundary conditions only, i.e., x(0, t) =x(1, t ) = 0. In–domain control in terms of b(z1)u(t ) is assumed with the spatial characteristic

b(z ) =σ(z −a)−σ(z −b)

for 0 < a < b < 1. As is shown in Section 1.3 both equations, i.e., (1.24) for the heat equation and (1.28)for the wave equation, can be put into the form of an abstract Cauchy problem defined in a suitableHilbert space.

It is a straightforward task to determine the eigenvalues and eigenvectors of the respective (self–adjoint) operators A. For the heat equation, we obtain

λ(heat )k =−(kπ)2 φ(heat )

k =ψ(heat )k =

p2sin(kπz ), k ∈N (2.50)

while for the wave equation eigenvalues and eigenvectors follow as

λ(w ave)k = ıkπ φ(w ave)

k =ψ(w ave)k =

[1

λ(w ave)k

]Fk sin(kπz ) k ∈Z\ 0 (2.51)

with Fk = 1/(kπ). Both sets of eigenvectors φ(heat )k k∈N and φ(w ave)

k k∈Z\0 form orthonormal bases

for the respective spaces X (heat ) = L2(0,1) and X (w ave) = H 10 (0,1)×L2(0,1) and hence Riesz bases.

Thus the operators A(heat ) and A(w ave) are Riesz spectral operators or scalar operators in the sense ofDefinition 2.2 and the flatness–based state and input parametrizations can be directly obtained fromthe results above.

2.4.3.1 Heat equation

The evaluation of (2.38) yields

Du(s) = ∏n∈N

(1− s

λ(heat )n

)= sinh(

ps)p

s(2.52)

Dxk (s) = ∏

n∈N,n 6=k

(1− s

λ(heat )n

)=−

λ(heat )k

s −λ(heat )k

sinh(p

s)ps

. (2.53)


0 0.2 0.4 0.60

10

20

30

40

50

t

u∗ξ∗

00.5

1 00.2

0.40.6

0

0.2

0.4

0.6

0.8

1

tz1

x

x0

xT

Figure 2.7: Feedforward control u∗(t ) and basic output ξ∗(t ) (left); numerical solution of the heat equationwith in–domain control when applying u∗(t ) (right). ©2012, Springer

Moreover, we have bk = ⟨b,φ(heat )k ⟩X (heat ) =p

2/(kπ)(cos(akπ)−cos(bkπ)) and we assume, that a, bare chosen so that bk 6= 0 for all k ∈Nwhich guarantees the approximate controllability.

It can be rather easily verified, that Du(s) is an entire function of finite type and finite order %= 1/2.With Theorem 2.4 the convergence of the formal parametrizations for the heat equation followsimmediately provided, that ξ(t ) is a Gevrey class function of order α< 2. The input u(t ) = Du(∂t )ξ(t )is determined using (2.52) and results in the series

u(t ) =∞∑

n=0

∂nt ξ(t )

(2n +1)!. (2.54)

The basic output is subsequently chosen to realize a finite time transition between steady states whichare governed by

xs(us) =(z1

∫ 1

0

∫ η

0b(ζ)dζdη−

∫ z1

0

∫ η

0b(ζ)dζdη

)us . (2.55)

Noting us = ξs in steady state conditions (see (2.52)) the desired trajectory ξ∗(t ) for the basic output isassigned as

ξ∗(t ) = ξs,0 + (ξs,T −ξs,0)Θω,T (t )

withΘω,T (t ) from (2.19) for ω> 1. Substitution into (2.54) provides the feedforward control u∗(t ) re-quired to achieve the transition starting at the steady state xs(z ;ξs,0) to the final steady state xs(z ;ξs,T )within the time interval t ∈ [0,T ]. Consistency with the zero initial state implies ξs,0 = 0.

Simulation results are shown in Figure 2.7 for ξs,T = 19.5, T = 0.5, and ω= 2 and the spatial character-istic being restricted to z ∈ (1/2,3/4).

2.4.3.2 Wave equation

Evaluating (2.38) for the wave equation provides

Du(s) = ∏n∈Z\0

(1− s

λ(w ave)n

)= ∏

n∈N

(1+ s2

λ(w ave)n λ(w ave)

k

)= ∏

n∈N

(1+ s2

(kπ)2

)= sinh(s)

s(2.56)

Dxk (s) = ∏

n∈Z\0,n 6=k

(1− s

λ(w ave)n

)=−

λ(w ave)k

s −λ(w ave)k

sinh(s)

s. (2.57)


0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

t

u∗

ξ∗

00.5

1 01

2

0

0.2

0.4

0.6

0.8

1

tz1

x

x0

xT

Figure 2.8: Feedforward control u∗(t ) and basic output ξ∗(t ) (left); numerical solution of the wave equationwith in–domain control when applying u∗(t ) (right). ©2012, Springer

The following analysis holds true if bk = ∫ 10 b(z )ψ(w ave)

2,k (z )dz = Fkλ(w ave)k /(kπ)(cos(akπ)−cos(bkπ)) 6=

0 for all k ∈Nwhich implies the approximate controllability of the system.

We can rather easily verify, that Du(s) as introduced in (2.57) is entire, of finite type and of order %= 1.Convergence according to Theorem 2.4 requires the basic output ξ(t) to be of Gevrey order α ≤ 1.However, as is shown in Section 2.3.1.2, this does not allow to realize finite time transitions betweensteady states. To resolve this issue note

Du(s)ξ(s) = sinh(s)

sξ(s) = e s −e−s

2sξ(s)

1

2

(ξ(t +1)σ(t +1)− ξ(t −1)σ(t −1)

)= u(t ), (2.58)

with6 ξ(t) = ∫ t0− ξ(p)dp. Obviously, the parametrization exactly recovers the wave dynamics with

a finite speed of wave propagation so that the feedforward control involves advanced and delayedarguments. The wave dynamics also implies the existence of a minimal transition time, here Tmin = 2,which corresponds to twice the wave speed.

Since the operator (2.58) does not induce any differentiability requirement the realization of a finitetime transition between an initial zero steady state and a final non–zero steady state can be achievedfor the basic output trajectory being even a discontinuous function of time. For simulation, we makeuse of

ξ∗(t ) = ξs,0δ(t )+ (ξs,T −ξs,0)σ(t −1) (2.59)

or

ξ∗(t ) = ξs,0 + (ξs,T −ξs,0)(t −1)σ(t −1),

respectively. The shift t −1 is introduced for causality purposes to guarantee u∗(t ) = 0 for t < 0.

Simulation results are shown in Figure 2.8 for ξs,0 = 0, ξs,T = 19.5 and T = 2 with the spatial character-istic being restricted to z ∈ (1/2,3/4).

2.5 Extension to nonlinear problems

To conclude this part on flatness–based trajectory planning for PDE systems we give a brief outlookto a rather general design approach based on formal integration. This technique has been proposedin [72] for scalar linear diffusion–convection–reaction systems with higher–dimensional domain.

6Since a basic output is not necessarily unique either ξ(t ) or ξ(t ) can be considered here.

2.5 Extension to nonlinear problems 43

Recently, the extension of this approach to general classes of semilinear PDEs has been achieved[99].

For the sake of simplicity we subsequently restrict ourselves to scalar semilinear PDEs of the form

∂t x(z , t ) = ∂2z x(z , t )+ f (x(z , t )), z ∈ (0,1), t > 0 (2.60a)

∂z x(0, t ) = 0, x(1, t ) = u(t ), t > 0 (2.60b)

x(z ,0) = x0(z ), z ∈ [0,1] (2.60c)

with the nonlinear function f (·) being locally Lipschitz continuous. Re–writing (2.60) in the form

∂2z x(z , t ) = ∂t x(z , t )− f (x(z , t ))

and integrating formally twice with respect to z yields

x(z , t ) = x(0, t )+ z∂z x(0, t )+∫ z

0

∫ r

0

(∂t x(s, t )− f (x(s, t ))

)dsdr. (2.61)

For the determination of x(0, t ) and ∂z x(0, t ) consider the boundary conditions (2.60b) which provide∂z x(0, t ) = 0 but leave the value x(0, t) free. Hence, ξ(t) = x(0, t) may serve as candidate for a basicoutput since

x(z , t ) = ξ(t )+∫ z

0

∫ r

0

(∂t x(s, t )− f (x(s, t ))

)dsdr (2.62)

u(t ) = ξ(t )+∫ 1

0

∫ r

0

(∂t x(s, t )− f (x(s, t ))

)dsdr = x(1, t ) (2.63)

can be considered as an implicit state and input parametrization in terms of ξ(t). To render theparametrization explicit the following iterative scheme

x j+1(z , t ) = ξ(t )+∫ z

0

∫ r

0

(∂t x j (s, t )− f (x j (s, t ))

)dsdr, j = 0,1, . . . , x0(z , t ) = ξ(t ) (2.64)

can be utilized, that allows for a successive evaluation depending on ξ(t ). The main challenge in thisapproach is given by the verification, that the limit

x(z , t ;ξ(t )) = limj→∞

x j (z , t ) (2.65)

exists and fulfills (2.60). This requires to consider the problem in so–called scales of Banach spacesin Gevrey classes, in principle generalizing the well–known Cauchy–Kowalevski theorem to Gevreyclass functions. The analysis of this problem is, however, beyond the scope of this treatise and theinterested reader is referred to [98].

Numerical results for the feedforward control based on formal integration for the semilinear problem(2.60) are summarized in Figure 2.9 for different functions f (·). The desired trajectory for the basicoutput is determined similar to the exposition in Section 2.3.1.2 with the difference, that also threesteady state profiles are connected by the temporal path.


0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

t

ξ∗ ,

x(0,t)

Des.Sim.

0 0.1 0.2 0.3 0.4 0.5−0.5

0

0.5

1

1.5

2

2.5

3

3.5

t

u∗

0

0.5

1

00.10.20.30.4

−1

0

1

2

3

4

zt

x

(a) Polynomial nonlinearity f (x) = x3.

0 0.1 0.2 0.3 0.4 0.50.5

0.6

0.7

0.8

0.9

1

1.1

t

ξ∗ ,

u(0,t)

Des.Sim.

0 0.1 0.2 0.3 0.4 0.50.5

1

1.5

2

t

u∗

0

0.5

1

00.10.20.30.4

0.5

1

1.5

2

zt

x

(b) Gevrey class 2 nonlinearity f (x) = exp(−90/x) for x(z , t ) > 0, f (0) = 0.

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

t

ξ∗ ,

u(0,t)

Des.Sim.

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

t

u∗

0

0.5

1

00.5

11.5

0

1

2

3

zt

x

(c) Real analytic nonlinearity f (x) = exp(x/(1+x/20)).

Figure 2.9: Numerical results for the scalar semilinear PDE (2.60) with nonlinearities f (x) as indicated above.Left column: comparison of x(0, t) and ξ∗(t); middle column: feedforward control u∗(t); rightcolumn: evolving profile x(z , t ) for u∗(t ). ©2013, IEEE

References







References 45




















References 47



















References 49





















References 51

3Feedback stabilization and observer designusing backstepping

This section summarizes some basic results for the design of backstepping controllers for PDE systems.For finite–dimensional nonlinear control systems backstepping is a well–established Lyapunov–basedmethod to determine stabilizing feedback control [49, 104, 112]. The extension of the backstepping ap-proach to certain classes of infinite–dimensional systems is exemplarily presented for two distributedparameter systems involving diffusion, convection and reaction. It is thereby also shown, that thecombination of backstepping and flatness allows to determine stabilizing tracking controllers evenfor semilinear PDEs.

3.1 Introduction to Lyapunov’s stability theory

For the subsequent introduction of the backstepping concept some basic knowledge of stability theoryfor PDEs is required. For this brief introduction consider the following linear diffusion–reactionsystem

∂t x(z, t ) = ∂2z x(z, t )−βx(z, t ), z ∈ (0,1), t > 0 (3.1a)

x(0, t ) = 0, x(1, t ) = 0, t > 0 (3.1b)

x(z,0) = x0(z), z ∈ [0,1]. (3.1c)

We want to answer the question if this system is exponentially stable in the sense of an L2–norm of thestate x(z, t ). Recall that exponential stability in this sense implies the existence of constants M ,ω> 0such that

‖x‖L2 (t ) ≤ Me−ωt‖x‖L2 (0). (3.2)

Remark 3.1: Relationship to finite dimensional linear systems

For finite dimensional linear systems in the form

x(t ) = Ax(t ), t > 0 (3.3a)

x(0) = x0 ∈Rn (3.3b)

this question can be answered in different ways by, e.g., analyzing the eigenvalues of the systemmatrix A and verifying, that these have only strictly negative real part. An alternative test is givenby the following criteria which states, that the linear system is exponentially stable if and only iffor any positive definite matrix Q ∈Rn×n there exists a positive definite matrix P = P T ∈Rn×n

such that

PA+ AT P =−Q. (3.4)

53

This result is based on the analysis of the Lyapunov function V (t ) =V (x(t )) = xT (t )P x(t ) and itsrate of change along a solution trajectory of (3.3), i.e.,

∂t V (t ) = xT (t )AT P x(t )+xT (t )PAx(t ) = xT (t )(PA+ AT P

)x(t )

!=−xT (t )Qx(t ) (3.5)

which is negative definite. Hence, the stability analysis reduces to finding the solution P of theLyapunov equation (3.4).

Lyapunov equations can be similarly defined for PDEs by making use of the abstract formulation(1.17). However, their practical use is very limited due to the necessity to solve an operator equa-tion [82]. This is further complicated by the fact, that norm equivalence is lost. In other wordswhile all norms in finite dimensions are equivalent this property is lost in infinite dimensionalfunction spaces so that stability essentially depends on the chosen norm. As such the applicationof Lyapunov’s stability analysis for PDEs requires to gain experience with deriving estimates inthe arising norms.

Remark 3.2: Lyapunov stability

It should be recalled that Lyapunov stability refers to the stability of a rest position or steadystate, respectively, of the differential equation.

To analyze the stability of (3.1) consider the Lyapunov functional1

V (t ) = 1

2‖x‖2

L2 (t ) = 1

2

∫ 1

0x2(z, t )dz. (3.6)

Its rate of change along a solution of (3.1) evaluates to

∂t V (t ) =∫ 1

0x(z, t )∂t x(z, t )dz =

∫ 1

0x(z, t )∂2

z x(z, t )dz −β∫ 1

0x2(z, t )dz

= [x(z, t )∂z x(z, t )

]10 −

∫ 1

0

(∂z x(z, t )

)2dz −β‖x‖2L2 (t )

=−∫ 1

0

(∂z x(z, t )

)2dz −β‖x‖2L2 (t ) (3.7)

This implies that V (t) is bounded. For the further analysis certain inequalities are useful which aresummarized below:

• Minkowski inequality: Let x(z), y(z) ∈ L2(0,1), then

‖x + y‖L2 ≤ ‖x‖L2 +‖y‖L2 (3.8)

This inequality is the triangle inequality for functions in L2(0,1).

• Young’s inequality: For a,b,α ∈Rwe have

ab ≤ α

2a2 + 1

2αb2. (3.9)

The proof of Young’s inequality follows directly from the binomial theorem.

1For PDEs one has to consider functionals, that map elements of the function space, i.e., functions to the real or complexnumbers.

54 Chapter 3 Feedback stabilization and observer design using backstepping

• Cauchy–Schwarz inequality: Let x(z), y(z) ∈ L2(0,1), then

[∫ 1

0x(z)y(z)dz

]2≤

∫ 1

0x2(z)dz

∫ 1

0y2(z)dz = ‖x‖2

L2‖y‖2L2 . (3.10)

• Poincaré inequality: Let x(z) ∈ H 1(0,1), then

∫ 1

0x2(z)dz ≤ 2x2(0)+4

∫ 1

0

(∂z x(z)

)2 dz (3.11)∫ 1

0x2(z)dz ≤ 2x2(1)+4

∫ 1

0

(∂z x(z)

)2 dz (3.12)

• Agmon inequality: Let x(z) ∈ H 1(0,1), then

maxz∈[0,1]

|x(z)|2 ≤ x2(0)+2(∫ 1

0x2(z)dz

) 12(∫ 1

0

(∂z x(z)

)2 dz) 1

2(3.13)

maxz∈[0,1]

|x(z)|2 ≤ x2(1)+2(∫ 1

0x2(z)dz

) 12(∫ 1

0

(∂z x(z)

)2 dz) 1

2(3.14)

Hence, with the Poincaré inequality evaluated for the considered boundary conditions equation (3.7)can be bounded from above according to

∂t V (t ) ≤−(β+ 1

4

)‖x‖2

L2 (t ) ≤−2(β+ 1

4

)V (t ).

Solving this equation for V (t ) provides

V (t ) ≤ e−2(β+ 14 )t V (0)

and hence

‖x‖L2 (t ) ≤ e−(β+ 14 )t‖x0‖L2

with ‖x0‖L2 = ‖x‖L2 (0) denoting the L2–norm of the initial profile x0(z). As a result, the system (3.1) isexponentially stable in the L2–norm if β+ 1

4 > 0.

In particular it is possible to verify the following result, that generalizes the above analysis to coupledlinear systems of PDEs [57].

Theorem 3.1

The linear system

x(t ) =Ax(t ), t > 0

x(0) = x0 ∈D(A) ⊂ X

is exponentially stable, if there exists a Lyapunov functional V (t ) =V (x(t )) such that

(i) α‖x‖2X (t ) ≤V (x(t )) ≤β‖x‖2

X (t ) with α,β> 0 and

(ii) ∂t V (x(t )) ≤−γ‖x‖2X (t ) for some γ> 0.

3.1 Introduction to Lyapunov’s stability theory 55

The theorem implies

V (x(t )) ≤−γ‖x‖2X (t ) ≤−γ

βV (x(t )) =−µV (x(t ))

so that V (x(t )) ≤V (x(0))e−µt . The latter motivates the estimate

‖x‖X (t ) ≤√β

α‖x‖X (0)e−

µ

2 t =√β

α‖x0‖X e−

µ

2 t

and hence exponential stability of the linear system.

Note that different other norms can be used in the stability analysis assuming higher regularity. Inparticular even pointwise exponential stability can be analyzed by making use of Agmon’s inequalityintroduced above (see, e.g., [50]).

3.2 Feedback control for a linear diffusion–reaction system usingbackstepping

In the following an introduction to backstepping–based stabilization of PDEs is given by considering alinear diffusion–reaction system.

3.2.1 Stabilization problem

We consider the stabilization of the diffusion–reaction system

∂t x(z, t ) = ∂2z x(z, t )+αx(z, t ), z ∈ (0,1), t > 0 (3.15a)

x(0, t ) = 0, x(1, t ) = u(t ), t > 0 (3.15b)

x(z,0) = x0(z), z ∈ [0,1] (3.15c)

which is unstable for α > π2 by means of properly designing a state–feedback controller using theboundary input u(t ). The main idea of the backstepping design approach is to use the Volterra integraltransformation

w(z, t ) = x(z, t )−∫ z

0k(z,ζ)x(ζ, t )dζ (3.16)

to transfer (3.15) into the target system

∂t w(z, t ) = ∂2z w(z, t )−βw(z, t ), z ∈ (0,1), t > 0 (3.17a)

w(0, t ) = 0, w(1, t ) = 0, t > 0 (3.17b)

w(z,0) = w0(z), z ∈ [0,1]. (3.17c)

As is shown in the previous section the target system (3.17) is exponentially stable in the L2–norm ifβ+ 1

4 > 0. To achieve the transformation into this target system it is at first necessary to determine theintegral kernel k(z,ζ).


3.2.2 Kernel computation

To derive of the equations governing the kernel k(z,ζ) substitute the target system (3.17) into thetransformation (3.16) and take into account the equations of the original system (3.15) for evaluation.For this, successive differentiation of (3.16) is needed which yields

∂z w(z, t ) = ∂z x(z, t )−k(z, z)x(z, t )−∫ z

0∂z k(z,ζ)x(ζ, t )dζ

∂2z w(z, t ) = ∂2

z x(z, t )−dz k(z, z)x(z, t )−k(z, z)∂z x(z, t )−∂z k(z, z)x(z, t )−∫ z

0∂2

z k(z,ζ)x(ζ, t )dζ

with the total differential dz k(z, z) = ∂z k(z, z)+∂ζk(z, z) and

∂t w(z, t ) = ∂t x(z, t )−∫ z

0k(z,ζ)∂t x(ζ, t )dζ

= ∂2z x(z, t )+αx(z, t )−

∫ z

0k(z,ζ)

(∂2ζx(ζ, t )+αx(ζ, t )

)dζ

= ∂2z x(z, t )+αx(z, t )− [

k(z,ζ)∂ζx(ζ, t )−∂ζk(z,ζ)x(ζ, t )]ζ=zζ=0

−∫ z

0

(∂2ζk(z,ζ)+αk(z,ζ)

)x(ζ, t )dζ.

Substitution of these expressions into (3.17a) results in

0 = ∂t w(z, t )−∂2z w(z, t )+βw(z, t )

= x(z, t )(µ+2dz k(z, z)

)−x(0, t )∂ζk(z,0)+∂z x(0, t )k(z,0)

+∫ z

0x(ζ, t )

(∂2

z k(z,ζ)−∂2ζk(z,ζ)− (α+β)k(z,ζ)

)dζ.

In view of the boundary conditions (3.15b) this provides the so–called kernel PDE

∂2z k(z,ζ)−∂2

ζk(z,ζ) =µk(z,ζ), ζ ∈ (0,1), z ∈ (ζ,1) (3.18a)

2dz k(z, z)+µ= 0 (3.18b)

k(z,0) = 0 (3.18c)

with µ=α+β. The triangular domain ζ ∈ (0,1), z ∈ (ζ,1) is shown in Figure 3.1 (left). The boundaryconditions (3.18b), (3.18c) are equivalent to

k(z, z) =−µ2

z. (3.19)

3.2.3 Solution of the kernel PDE

The solution of the kernel PDE (3.18) makes use of the fact, that the PDE (3.18a) is similar to the waveequation (compare with (1.10)). Hence, solution techniques developed for the wave equation can bedirectly applied to determine k(z,ζ). For this, the so–called method of characteristics is applied whichexploits a suitable change of the independent coordinates according to

η= z +ζ, χ= z −ζ. (3.20)

3.2 Feedback control for a linear diffusion–reaction system using backstepping 57

0

z

1

ζ1

0

η

1

2

χ1

Figure 3.1: Domain of kernel PDE: (left) in the (z,ζ)–plane with ζ ∈ (0,1), z ∈ (ζ,1) and (right) in the (χ,η)–planewith χ ∈ (0,1), η ∈ (χ,2−χ).

Introducing k(η,χ) = k(z,ζ) allows to transfer (3.18) into the normal form

4∂η∂χk(η,χ) =µk(η,χ), χ ∈ (0,1), η ∈ (χ,2−χ) (3.21a)

k(η,0) =−µ4η (3.21b)

k(η,η) = k(χ,χ) = 0. (3.21c)

The transformed (still triangular) domain in the (χ,η)–plane is depicted in Figure 3.1 (right). With this,the solution of (3.21) can be obtained by formally integrating the PDE with respect to χ (form 0 to χ),i.e.,

∂ηk(η,χ) = ∂ηk(η,0)+ µ

4

∫ χ

0k(η, q)dq

(3.21b)= −µ4

(1−

∫ χ

0k(η, q)dq

)followed by an integration with respect to η (from χ to η, see Figure 3.1 (right)), i.e.,

k(η,χ) =−µ4

(η−χ−

∫ η

χ

∫ χ

0k(p, q)dqdp

). (3.22)

The resulting implicit solution in terms of an integral equation can be made explicit by consideringthe so–called method of successive approximation. For this, introduce the series

k(η,χ) =∞∑

n=0kn(η,χ), (3.23)

whose coefficients are computed recursively2 according to

k0(η,χ) =−µ4

(η−χ)

kn(η,χ) = µ

4

∫ η

χ

∫ χ

0kn−1(p, q)dqdp, n ≥ 1.

2For the determination of the recursive rule substitute (3.23) into (3.22) and appropriately sort the arising terms.


It can be rather easily shown using induction, that the series coefficients fulfill

kn(η,χ) =−(µ

4

)n+1 (η−χ)ηnχn

n!(n +1)!. (3.24)

Evaluation of the series ansatz (3.23) with series coefficients (3.24) enables to determine a closed–formexpression taking into account the modified Bessel functions of the first kind I1(·), i.e.,

k(η,χ) =−µ2

(η−χ)I1(

pµηχ)

pµηχ

,

Reverting the coordinate transformation (3.20) thus results in the kernel

k(z,ζ) =−µζ I1(√µ(z2 −ζ2))√µ(z2 −ζ2)

. (3.25)

3.2.4 Inverse transformation

To establish a one–to–one correspondence between the original and the target system it is necessaryto verify invertibility of the Volterra transformation (3.16). Consider

x(z, t ) = w(z, t )+∫ z

0g (z,ζ)w(ζ, t )dζ (3.26)

and proceed as before, i.e. differentiate (3.26) with respect to t , twice with respect to z, and substitutethe obtained expressions into the equations (3.15) for the diffusion–reaction system in x(z, t ). Aftersome intermediate but standard computations it can be shown, that the equations governing theinverse transformation g (z,ζ) are identical to those of the original kernel except for the coefficient µin the kernel PDE, who has to be replaced by −µ, i.e.,

∂2z g (z,ζ)−∂2

ζg (z,ζ) =−µg (z,ζ), ζ ∈ (0,1), z ∈ (ζ,1) (3.27a)

2dz g (z, z)+µ= 0, g (z,0) = 0. (3.27b)

Thus the same solution procedure can be applied to explicitly determine g (z,ζ). However, as is shownsubsequently, the desired stabilization assertion is obtained immediately without computing g (z,ζ)but by verifying, that g (z,ζ) is bounded in z and ζ.

3.2.5 State–feedback controller and closed–loop stability

Having computed the integral kernel k(z,ζ) allows to determine the state–feedback controller, that isrequired to realize the (invertible) transformation from the original diffusion–reaction system (3.15)into the exponentially stable target system (3.17). Evaluating the boundary condition (3.15b) at z = 1with (3.16) provides in view of (3.17b) at z = 1 the state–feedback controller

u(t ) =∫ 1

0k(1,ζ)x(ζ, t )dζ=−µ

∫ 1

0ζ

I1(√µ(1−ζ2))√µ(1−ζ2)

x(ζ, t )dζ. (3.28)

The notion state–feedback controller becomes immediately apparent since the complete spatial–temporal evolution of the state variable x(z, t ) is required for evaluation. Hence, the implementationof (3.28) relies on amending the control loop by state–observer which can be similarly achieved usingbackstepping (see Exercise 3).

3.2 Feedback control for a linear diffusion–reaction system using backstepping 59

For the verification, that the state–feedback controller (3.28) does indeed stabilize the originaldiffusion–reaction system (3.15) recall, that the target system is exponentially stable so that its solutionsatisfies

‖w‖L2 (t ) ≤ e−λt‖w0‖L2 (3.29)

for λ=β+ 14 > 0. In addition, (3.16) implies the bound

‖w‖L2 (t ) ≤ ‖x‖L2 (t )+∥∥∥∥∫ z

0k(z,ζ)x(ζ, t )dζ

∥∥∥∥L2

.

The last term can be further estimated as follows∥∥∥∫ z


∥∥∥2

L2=

∫ 1

0

∣∣∣∫ z


∣∣∣2dz ≤

∫ 1

0

(∫ 1

0|k(z,ζ)||x(ζ, t )|dζ

)2dz

≤∫ 1

0‖k‖2

L2 (z)‖x‖2L2 (t )dz (Cauchy–Schwarz inequality)

≤C 2‖x‖2L2 (t ), (boundedness of k(z,ζ))

for a sufficiently large constant C > 0. As a result, we have

‖w‖L2 (t ) ≤ (1+C )‖x‖L2 (t ). (3.30)

Applying a similar sequence of estimates on the inverse kernel (3.26) provides

‖x‖L2 (t ) ≤ (1+D)‖w‖L2 (t )

for a sufficiently large constant D > 0. Combining both bounds finally yields

‖x‖L2 (t )(3.29)≤ (1+D)e−λt‖w0‖L2

(3.30)≤ (1+C )(1+D)︸︷︷︸=M

e−λt‖x0‖L2 , (3.31)

and thus the exponential stability of the closed–loop control system consisting of (3.15) with state–feedback controller (3.28).

3.3 Tracking control for a semilinear diffusion–reaction system usingflatness and backstepping

In the following, the introductory example of backstepping is extended in several directions. For this,we consider a tracking control problem given a semilinear diffusion–convection–reaction system.Here, flatness–based trajectory planning and feedback stabilization using backstepping are combinedto determine an exponentially stabilizing tracking controller for

∂t x(z, t ) =λ∂2z x(z, t )+ν∂z x(z, t )+ f (x(z, t )), z ∈ (0,1) t > 0. (3.32a)

The parameters are assumed to satisfy λ,ν> 0, i.e. convection takes place in the negative z-direction.The PDE (3.32a) can be considered as the simplified model of a tubular reactor with reaction rategiven by f (x(z, t)). For consistency, a no–flux boundary condition (BC) and a Danckwert’s BC areimposed at the outlet (z = 0) and inlet (z = 1), respectively, i.e.

∂z x(0, t ) = 0, t > 0 (3.32b)


λ

ν∂z x(1, t ) = u(t )−x(1, t ), t > 0. (3.32c)

For example, if x(z, t ) represents a concentration, then the input u(t ) represents changes in the inflowconcentration. The initial state

x(0, z) = x0(z) = 0, z ∈ [0,1] (3.32d)

represents a stationary profile for (3.32a)-(3.32c). The controlled output is chosen as the state value atthe outlet, i.e.

y(t ) = x(0, t ), t ≥ 0. (3.32e)

The considered tracking control problem concerns the design of a boundary controller u(t ) = u∗(t )+ue (t ) consisting of a feedforward part u∗(t ) and a feedback part ue (t ) to ensure exponentially stabletracking of a suitably chosen output trajectory t 7→ y∗(t) for the realization of the transition fromthe initial stationary profile x0(z) to the new operating profile xT (z) within the finite time intervalt ∈ (0,T ], T ¿∞. The proposed solution follows the lines of [70, 73, 71] and proceeds as follows:

• At first, flatness–based methods are used to solve the trajectory planning problem, i.e., to de-termine the feedforward control u∗(t ), that is required to achieve the desired spatial–temporaltransition. For this, formal power series combined with resummation techniques to also incorpo-rate divergent state and input parametrizations can be applied (see, e.g., [70, 73]). Alternatively,formal integration can be used as is elaborated in Section 2.5 following [99, 98].

• Since the flatness–based state and input parametrizations solve the diffusion–convection–reaction system they furthermore enable to introduce the distributed parameter tracking errorsystem. Assuming only small deviations between the actual and the desired state the lineariza-tion around the reference profile x∗(z, t ) can be exploited for the feedback control design.

Remark 3.3

The solution of the trajectory planning problem using flatness is rather involving and requiresseveral steps when addressing also the convergence or divergence problem, respectively. Sincethese are outside the scope of this introductory course computational details are subsequentlyskipped and the reader is referred to [70, 73, 75]. Hence, we assume, that a (differential) stateand input parametrization in terms of a basic output is available in the form

x(z, t ) =ψ1(ξ(t ), ξ(t ), . . . ,ξ(β)(t ), . . .

)u(t ) =ψ2

(ξ(t ), ξ(t ), . . . ,ξ(β+1)(t ), . . .

).

(3.33)

It is thereby rather straightforward to show that ξ(t ) = x(0, t ) is a basic output for (3.32).

3.3.1 Flatness–based trajectory planning and feedforward control

For the numerical evaluation we make use of a polynomial nonlinearity

f (x(z, t )) =3∑

j=1p j (x(z, t )) j . (3.34)

The desired trajectory ξ∗(t ) for the basic output ξ(t ) is determined according to the analysis in Section2.3.1.2 to realize the desired transition between steady states. The feedforward control u∗(t ) obtained

3.3 Tracking control for a semilinear diffusion–reaction system using flatness and backstepping 61

0 1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

t

u*

kSum

pSum

(a)

0

0.5

1

0

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

1.2

zt

x*

(b)

y(t )

xT (z)

x0(z)

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

t

y

Des.

Sim.

(c)

0

0.5

1

01

23

45

67

0

0.2

0.4

0.6

0.8

1

1.2

zt

x

(d)

Figure 3.2: Simulation results for flatness–based feedforward control for (3.32) with λ = 1, ν = p1 = p3 = 3,p2 = 0. (a) Feedforward control u∗(t ); (b) Desired profile x∗(z, t ); (c) Comparison of desired andobtained output trajectories y∗(t ) and y(t ) = x(0, t ) when applying u∗(t ) from (a) to a simulationmodel of (3.32); (d) Simulated profile x(z, t ).

by evaluating (3.33) is shown in Figure 3.2 (a). Here, a resummation technique (label ’ksum’) is used todeal with the arising divergent behavior and to extract a meaningful limit from the parametrization(see also [65]). The system parameters are chosen as λ= 1, ν= p1 = p3 = 3, and p2 = 0. The respectivespatially and temporally varying desired profile x∗(z, t) is shown in Figure 3.2 (b). This illustratesthe solution of the tracking control task, i.e., the realization of a finite time transition between aninitial stationary profile x0(z) and a final stationary profile xT (z) within a prescribed time intervalt ∈ (0,T ] along the output trajectory y∗(t ). However, making use of the thus determined feedforwardcontrol u∗(t ) as the input to a simulation model of the diffusion–convection–reaction system (3.32)reveals, that the chosen final profile xT (z) corresponds to an unstable steady state. Figures 3.2 (c)and (d) illustrate, that the system converges to a new stationary state. This is due to the existenceof multiple steady states for nonlinear DPSs such that up to three stationary profiles with differingstability properties exist for the considered cubic nonlinearity when solving the respective boundaryvalue problem for (3.32) with a constant boundary input u(t) = u. The stability of the stationaryprofiles can be determined by considering the linearization of (3.32) around the stationary profilesx0(z) and xT (z). For λ= 1, ν= p1 = p3 = 3, and p2 = 0 in the first case the dominating eigenvalues arelocated at −1.63,−14.28,−44.36 while in the second case the dominating eigenvalues are obtained at2.83,−10.39,−40.39. Obviously, to realize the desired transition and to stabilize the unstable finalprofile the feedforward control has to be amended by a stabilizing feedback controller.


3.3.2 Stabilization problem

Due to the available inverse system representation (3.33) in terms of the basic output ξ(t) = x(0, t)the distributed parameter tracking error system can be directly determined. Since the desired statex∗(z, t ) satisfies the PDE (3.32) with u(t ) and y(t ) replaced by the respective feedforward control u∗(t )and the desired trajectory y∗(t ), it follows that

∂t xe (z, t ) =λ∂2z xe (z, t )+ν∂z xe (z, t )+ f (xe (z, t )+x∗(z, t ))− f (x∗(z, t )), z ∈ (0,1), t > 0 (3.35a)

∂z xe (0, t ) = 0, t > 0 (3.35b)

λ

ν∂z xe (1, t ) = ue (t )−xe (1, t ), t > 0 (3.35c)

xe (0, z) = xe,0(z), z ∈ [0,1], (3.35d)

where xe,0(z) = x0(z)−x∗(0, z), ue (t ) = u(t )−u∗(t ). Assuming, that the deviation between x(z, t ) andx∗(z, t ) is small the PDE (3.35a) can be linearized with respect to the desired spatial–temporal statex∗(z, t ). Here, we make use of

f (xe (z, t )+x∗(z, t ))− f (x∗(z, t )) ≈ ∂x f (x∗(z, t ))xe (z, t ) (3.36)

which transfers (3.35a) into a linear PDE with spatially and temporally varying coefficients dependingnonlinearly on the parametrized desired profile x∗(z, t ). By introducing the classical transformation

xe (z, t ) = exp(− ν

2λz)xe (z, t ) (3.37)

the convective term in (3.35a) can be eliminated such that (3.35) can be re–written as

∂t xe (z, t ) =λ∂2z xe (z, t )+α(z, t )xe (z, t ), z ∈ (0,1), t > 0 (3.38a)

∂z xe (0, t ) = ν

2λxe (0, t ), t > 0 (3.38b)

λ

ν∂z xe (1, t ) = ue (t )−µxe (1, t ), t > 0 (3.38c)

xe (0, z) = xe,0(z), z ∈ [0,1], (3.38d)

which is independent of the first order derivative with respect to z. Here, the coefficients are given byα(z, t ) = ∂x f (x∗(z, t ))−ν2/(4λ), and µ= 1/2. In addition, the transformed boundary input and initialcondition follow as ue (t ) = exp(ν/(2λ))ue (t ) and xe,0(z) = exp(ν/(2λ)z)xe,0(z), respectively.

The linear distributed parameter tracking error system with spatially and temporally varying parameter(3.38) serves as the basis for the design of an exponentially stabilizing state-feedback controller. Similarto the introductory example in Section 3.2 a time varying Volterra integral transformation

w(z, t ) = xe (z, t )−∫ z

0k(z,ζ, t )xe (ζ, t )dζ (3.39)

is used to transform the tracking error system (3.38) into the exponentially stable target system

∂t w(z, t ) =λ∂2z w(z, t )−βw(z, t ), z ∈ (0,1), t > 0 (3.40a)

∂z w(0, t ) = κw(0, t ), t > 0 (3.40b)

∂z w(1, t ) =−µw(1, t ), t > 0 (3.40c)

w(0, z) = xe,0(z), z ∈ [0,1] (3.40d)


with β> 0. Note that the kernel k(z,ζ, t ) has to be chosen time variant in order to deal with the timeand spatially varying parameter α(z, t ) in the PDE (3.38a).

Exercise 3.1. Use a Lyapunov approach to determine the parameters β, κ, µ to ensure exponentialstability of (3.40) in the L2–norm.

The state–feedback control which is required to realize the desired transformation follows immediatelyfrom the evaluation of (3.38c) with (3.39), i.e.

ue (t ) =∫ 1

0

[λ

ν∂z k(1,ζ, t )+µk(1,ζ, t )

]xe (ζ, t )dζ+ λ

νk(1,1, t )xe (1, t ). (3.41)

3.3.3 Kernel computation

The computation of the kernel k(z,ζ, t ) requires the evaluation of the equations for the target system(3.40) using the transformation (3.39) followed by the substitution of the equations for the trackingerror system (3.38) and the respective integration by parts. This yields

∂t w(z, t ) = ∂t xe (z, t )−∫ z

0

(∂t k(z,ζ, t )xe (ζ, t )+k(z,ζ, t )∂t xe (z, t )

)dζ

=λ∂2z xe (z, t )+α(z, t )xe (z, t )

−∫ z

0

(∂t k(z,ζ, t )xe (ζ, t )+k(z,ζ, t )

[λ∂2

ζ xe (ζ, t )+α(ζ, t )xe (ζ, t )])

dζ

=λ∂2z xe (z, t )+α(z, t )xe (z, t )−λ(

k(z, z, t )∂z xe (z, t )−∂ζk(z, z, t )xe (z, t ))

+λ(k(z,0, t )∂z xe (0, t )−∂ζk(z,0, t )xe (0, t )

)−

∫ z

0xe (ζ, t )

(∂t k(z,ζ, t )+λ∂2

ζk(z,ζ, t )+α(ζ, t )k(z,ζ, t ))dζ

and

∂2z w(z, t ) = ∂2

z xe (z, t )−dz k(z, z, t )xe (z, t )−k(z, z, t )∂z xe (z, t )−∂z k(z, z, t )xe (z, t )

−∫ z

0∂2

z k(z, t ,ζ)xe (ζ, t )dζ,

where dz k(z, z, t) = ∂z k(z, z, t)+∂ζk(z, z, t). Hence, the evaluation of (3.40) together with the BCs

(3.38b), (3.38c) of the tracking error system provides the PDE for the backstepping kernel

∂t k(z,ζ, t ) =λ[∂2

z k(z,ζ, t )−∂2ζk(z,ζ, t )

]−γ(ζ, t )k(z,ζ, t ), ζ ∈ (0,1), z ∈ (ζ,1) (3.42)

with γ(ζ, t ) =β+α(ζ, t ) and the BCs

2λdz k(z, z, t ) =−γ(z, t ), k(0,0, t ) = ν

2λ−κ (3.43a)

−∂ζk(z,0, t )+ ν

2λk(z,0, t ) = 0. (3.43b)

The domain of the kernel PDE is given by the triangle depicted in Figure 3.1 (left). The initial conditionfor k(z,ζ, t ) follows from (3.38d) and (3.40d) such that

k(z,ζ,0) = 0. (3.43c)


3.3.4 Solution of the kernel PDE

Compared with the kernel PDE (3.21a) for the time invariant case (3.42) is a rather unusual type ofPDE, since the operator on the right–hand side is hyperbolic such that classical analytical or numericalapproximation schemes, e.g., finite–differences cannot be directly applied. Note that closed–formsolutions exist if α(z, t ) is constant or only time–varying (see, e.g., [111]).

For the determination of an approximate solution to (3.43) in the following the approach of [15] isused which is based on a successive approximation of k(z,ζ, t). For this, introduce the change ofcoordinates η= z +ζ, χ= z −ζ and let k(η,χ, t ) = k(z,ζ, t ) which in view of (3.42) yields

∂η∂χk(η,χ, t ) = 1

4λ

[∂t k(η,χ, t )+γ

(η−χ2

, t)k(η,χ, t )

]. (3.44a)

This PDE has to be solved on the domain χ ∈ (0,1), η ∈ (χ,2−χ) which is depicted in Figure 3.1 (right).The respective boundary conditions follow from (3.43a), (3.43b), i.e.,

∂ηk(η,0, t ) =−γ(η2 , t )

4λ, k(0,0, t ) = ν

2λ−κ, (3.44b)

∂χk(η,η, t )−∂ηk(η,η, t )+ ν

2λk(η,η, t ) = 0. (3.44c)

As before, formally integrating (3.44) with respect to χ and η yields the implicit solution

k(η,χ, t ) = A(χ, t )+B(η,χ, t )+ 1

4λ

∫ η

χ

∫ χ

0

[∂t k(p, q, t )+γ

( p −q

2, t

)k(p, q, t )

]dqdp (3.45)

where

A(χ, t ) =∫ χ

0e−

ν2λ (χ−p)

[a(t , p)− 1

2λγ( p

2, t

)]dp

a(t , p) = 1

2λ

∫ p

0

[∂t k(p, q, t )+γ

( p −q

2, t

)k(p, q, t )

]dq

B(η,χ, t ) =− 1

4λ

∫ η

χγ( p

2, t

)dp +

( ν2λ

−κ)e−

ν2λχ.

Following [15] we look for a solution of (3.45) in the form of the series

k(η,χ, t ) =∞∑

n=0kn(η,χ, t ), (3.46)

where the series coefficients are determined recursively according to

k0(η,χ, t ) = B(η,χ, t )− 1

4λ

∫ χ

0e−

ν2λ (χ−p)γ

( p

2, t

)dp

and

kn(η,χ, t ) =∫ χ

0e−

ν2λ (χ−p)an−1(p, t )dp + 1

4λ

∫ η

χ

∫ χ

0

[∂t kn−1(p, q, t )+γ

( p −q

2, t

)kn−1(p, q, t )

]dqdp

with an−1(p, t ) denoting a(p, t ) defined above for k(p, q, t ) replaced by kn−1(p, q, t ).

The convergence analysis of the series (3.46) will not be presented here. However, note that for ananalytic coefficient α(z, t ) convergence can be deduced from the results of [15]. Moreover, a completeconvergence analysis for general cases is provided in [71, 63].


0

1

2

0

0.5

1

−10

−5

0

5

ηχ

k

t=0 t=2 t=4 0

1

2

0

0.5

1

−20

−15

−10

−5

0

5

ηχ

k

t=0 t=2 t=4

Figure 3.3: Kernel k(η,χ, t) determined by successive approximation (3.46) using 10 addends in the (η,χ)–plane for times t ∈ 0,2,4 with κ = 0 and β = 5 (left) and β = 10 (right) in (3.40a). The systemparameters are chosen as λ= 1, ν= p1 = p3 = 3, p2 = 0.

Numerical results for the kernel are depicted in Figure 3.3. The nonlinearity f (x(z, t)) is assignedas defined in (3.34) in terms of a polynomial of degree three. Here, the kernel k(η,χ, t) determinedfrom the successive approximation procedure with 10 addends in (3.46) is shown at times t ∈ 0,2,4for the parameters λ= 1, ν= p1 = p3 = 3, p2 = 0 in the tracking error system (3.35a) correspondingto α(z, t) = 3(1+3(x∗(z, t))2)−9/4 in the transformed PDE (3.38a). The parameters for the targetsystem (3.40) are chosen as κ= 0 and β= 5 (left) and β= 10 (right). The desired profile x∗(z, t) forthe computation of α(z, t) is obtained from the parametrization (3.33) evaluated using the desiredtrajectory ξ∗(t ) defined in (2.18) with T = 4 and ω= 1. This in particular yields that the kernel satisfiesk(z,ζ, t ) = k(z,ζ, t ) for t ≥ T .

3.3.5 Simulation results

Based on the previous analysis simulation results are presented for the tracking control of (3.32) torealize the transition from the stationary initial profile x0(z) = 0 to the final stationary profile xT (z)within the prescribed finite time interval t ∈ (0,4] along the desired trajectory ξ∗(t ) for the basic outputξ(t ). Note that xT (z) corresponds to the solution of the boundary-value problem consisting of (3.32a)for t → ∞ or respectively ∂t x(z, t ) = 0, (3.32b) and (3.32e) with xT (0) = ξ∗ = 1 due to the flatnessproperty. The system parameters are chosen identical to those above, i.e., λ= 1, ν= p1 = p3 = 3, p2 = 0with the cubic nonlinearity (3.34).

Simulation results are depicted in Figure 3.4. In particular Figure 3.4 (a) shows the evolution of thestate x(z, t) in the (z, t)–domain when controlled by u(t) = u∗(t)+ue (t). Here, u∗(t) denotes theflatness–based feedforward control introduced in (3.33). The term ue (t ) represents the backsteppingstate–feedback controller ue (t) = exp(−ν/(2λ))ue (t) with ue (t) as defined in (3.41) for the targetsystem (3.40) with β= 10 and κ= 0. As indicated, the desired transition along the prescribed pathis realized in a highly accurate way, which is confirmed in Figure 3.4 (b), where the obtained outputy(t) = x(0, t) is compared with the desired trajectory y∗(t) for β ∈ 5,10, κ= 0 in the target system(3.40). As is illustrated in Figure 3.4 (c) for the time evolution of the L2–norm of xe (z, t) an increasein β allows a further reduction of the tracking error. In addition, the comparison of the magnitudesof u∗(t) and ue (t) for β ∈ 5,10 in Figure 3.4 (d) clearly illustrates that the feedback part is severalmagnitudes smaller than the feedforward part which is typical for the proposed control approach,where the feedforward control ensures the desired tracking behavior while the feedback part accountsfor the stabilization of the tracking error.


0

0.5

1

0

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

1.2

zt

x(a)

y(t )

xT (z)

x0(z)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

t

y

Des.

β=5

β=10

(b)

0 1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

u*

t

0 1 2 3 4 5 6 7−2

0

2

4

6x 10

−3

ue

u*(t)

β=5

β=10

(d)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

x 10−3

t

‖xe‖

β=5

β=10

(c)

Figure 3.4: Simulation results for combined flatness–based feedforward control and backstepping errorfeedback control (3.41) for (3.32) with λ = 1, ν = p1 = p3 = 3, p2 = 0. The target system (3.40)is parametrized with β ∈ 5,10 and κ = 0. (a) obtained state profile x(z, t) in the (z, t)–domainillustrating the achieved transition between x0(z) and xT (z) within t ∈ (0,4] along y∗(t ); (b) outputy(t) compared to desired output y∗(t); (c) time evolution of the L2–norm of the tracking errorxe (z, t ) = x(z, t )−x∗(z, t ); (d) applied feedforward control u∗(t ) and error feedback control ue (t ).

3.4 Extensions of PDE backstepping

Various generalizations of the backstepping approach for state–feedback control and state–observerdesign are available. This comprises:

• Tracking control for linear parabolic PDEs with spatially and time varying parameters takinginto account trajectory planning for the target system [71, 64];

• Feedback stabilization of certain classes of semilinear parabolic PDEs [121, 122];

• Tracking control for certain classes of semilinear parabolic PDEs [73, 64];

• Feedback stabilization of anti–stable linear wave equations [48];

• State–observer design for parabolic PDEs in single and higher–dimensional domains [110, 50,41, 39, 64, 40, 67];

• Feedback stabilization and tracking control for parabolic PDEs with higher–dimensional spatialdomain using multi–linear Volterra integral transformations [64];

3.4 Extensions of PDE backstepping 67

• Extensions to coupled systems of linear diffusion–reaction systems [5] and coupled systems ofhyperbolic PDEs [16, 35].

For a comprehensive overview on backstepping techniques for distributed parameter systems includ-ing also adaptive concepts the reader is referred to the monograph [50].


References





















References 69





















References 71




















References 73















References 75

4Lyapunov–based feedback stabilization andobserver design

In the following the design of stabilizing feedback controllers for PDEs is addressed using Lyapunov’sstability theory. Differing from the presentation in Section 3 we will focus on asymptotic stabilizationso that certain preliminaries have to be introduced leading to LaSalle’s invariance principle. Theseconcepts are utilized to achieve vibration suppression in a flexible beam by means of embeddedpiezoelectric patch actuators. Here, a dynamic feedback controller is designed, that makes use of adistributed parameter state–observer . Experimental data is provided to verify the applicability of thisdesign approach.

4.1 Extensions to Lyapunov’s stability theory

Based on the introduction to Lyapunov’s stability theory in Section 3.1 in the following certain exten-sions are provided, that are required for stability analysis and feedback stabilization. Our startingpoint is the abstract formulation of an autonomous PDE system in the form

x(t ) =Ax(t ), t > 0 (4.1a)

x(0) = x0 ∈D(A). (4.1b)

As before the state x(t ) ∈ X is defined in some Hilbert space X with A referring to a linear differentialoperator, that maps elements of its domain D(A) to X . We assume that A is the so–called infinitesimalgenerator of a C0–semigroup T(t). For an introduction to this concept the reader is referred, e.g.,to [17, 118]. Subsequently, it suffices to observe the implication of the generator property, namelyx(t) = T(t)x0 solves (4.1). With this, asymptotic stability and exponential stability as well as otherstability concepts such as weak or strong stability can be defined [57, Chapter 3])

Definition 4.1

The C0–semigroup T(t ) is called

1. asymptotically stable if ∀x ∈ X : ‖T(t )x‖X → 0 as t →∞;

2. exponentially stable if ∃M , ω> 0 such that1

‖T‖(t ) ≤ Me−ωt for t ≥ 0. (4.2)

The constant ω is called the decay rate.

The treatise in Section 3.1 addresses exponential stability and essentially relies on verifying theconditions of Theorem 3.1, i.e., (i) the Lyapunov functional V (x(t )) has to be bounded from below andfrom above by the norm of x(t) and (ii) the rate of change ∂t V (x(t)) can be bounded by −µV (x(t))

1The norm in (4.2) is the operator norm.

77

for non–negative µ. While in applications condition (i) is often immediately fulfilled the secondcondition (ii) proves to strongly limit the usability of Lyapunov’s direct method. In particular, infinite–dimensional function spaces are not compact which is a crucial preliminary for Lyapunov’s stabilitytheory given finite–dimensional nonlinear systems. As a consequence, for PDE systems we can nolonger conclude stability from ∂t V (x(t )) < 0 or ∂t V (x(t )) ≤ 0. This issue can be addressed by LaSalle’sinvariance principle, whose formulation requires to introduce some additional preliminaries.

Let T(t ) denote a continuous (nonlinear) contraction2 semigroup defined on closed subset D(T) of areal Banach space. Let x ∈D(T), then

• γ(x) =⋃t≥0T(t )x denotes the orbit through x and

• $(x) = y ∈ D(T) : y = limn→∞T(t n)x with t n < t n+1 and t n → ∞ as n → ∞ is the possiblyempty $–limit set of x .

A sufficient condition for $(x) to be non–empty is provided below [57].

Theorem 4.1

Let γ(x) be precompact3 for x ∈D(T), i.e., γ(x) is compact4. Then $(x) is non–empty, compact,connected5 and

limt→∞dist(T(t )x ,$(x)) = 0.

Here, dist(y ,Ω) for y ∈ Y andΩ⊂ Y denotes the distance of y toΩ, i.e.,

dist(y ,Ω) = infw∈Ω

‖y −w‖Y .

With this, LaSalle’s invariance principle reads as follows [30, 109, 55, 56].

Theorem 4.2: LaSalle’s invariance principle

Let V (x) be a Lyapunov function(al), i.e., V (x) > 0 and ∂t V (x) ≤ 0, in D(T) and let E be the largestpositively invariant subset of x ∈D(T) : ∂t V (x) = 0. If x ∈D(T) and γ(x) is precompact, then

limt→∞dist(T(t )x ,E ) = 0.

Invariance of E under T(t ) implies, that T(t )E = E for all t ≥ 0.

Theorems 4.1 and 4.2 provide a criteria to prove, that the distance between the trajectory T(t )x andthe $–limit set or E , respectively, approaches zero as t →∞ for all x ∈ D(T). If E = 0 this yieldsasymptotic stability. While theoretically appealing the verification of relative compactness of the orbitγ(x) restricts applicability. To address this, consider the following result which is a consequence ofthe so–called Crandell–Liggett theorem [18, 57].

2The semigroup T(t ) is a contraction semigroup if ‖T‖(t ) ≤ 1, ∀t ≥ 0.3A set S in a normed linear space is precompact or relatively compact if its closure is compact. The closure S of S is obtained

by adding to S all limit points of sequences in S [17].4A set S in a normed linear space is compact if every sequence in S contains a convergent subsequence with its limit point

in S [17].5This implies that $(x) cannot be separated into two disjoint, non–empty, open sets.

78 Chapter 4 Lyapunov–based feedback stabilization and observer design

Theorem 4.3

Let A be a dissipative operator6 defined in the Banach space X with

D(A) ⊂ ran(I−λA)

for sufficiently small λ and let T(t ) be the contraction semigroup defined by

T(t )x = limn→∞

(I− t

nA

)−nx .

Assume also, that 0 ∈ ran(A) and that (I−λA)−1 is compact7 for one λ> 0. Then γ(x) is precom-pact for all x ∈D(A).

Subsequently an application of this theory is applied for the vibration suppression for a flexible beamwith embedded piezoelectric actuation.

4.2 Vibration suppression for a flexible beam

The design of a feedback controller for the flexible beam structure already introduced in Section 1.2.3is considered with the desire to suppress unwanted oscillations in the weakly damped system. Theconfiguration is shown in Figure 4.1. We remark, that this section is based on [103, 100].

zz

z

Lc

bc

hc

zp,1 zp,2

mtm , Itm

bp

Lp

x(Lc , t )

Figure 4.1: Cantilever beam with pairs of patches.

The equations of motion have been already presented in Section 1.2.3 with their abstract formulationintroduced in Example 1.3. For the sake of completeness the equations are recalled below, i.e.,

x(t ) =Ax(t )+Bu(t ), t > 0 (4.3a)

x(0) = x0 =[x0(z) v0(z) v0(Lc ) ∂z v0(Lc )

]T ∈D(A) (4.3b)

with

Ax =

x2

− 1µ(z)

(γe (z)x2 +∂2

z

(E I (z)∂2

z x1))

1mtm

∂z

(E I (z)∂2

z x1) |z=Lc

− 1Itm

(E I (z)∂2

z x1) |z=Lc

, B=− 1

µ(z)

0 . . . 0

Γ1(z) . . . Γm(z)0 . . . 00 . . . 0

, (4.3c)

6Let A be a linear operator with domain D(A) dense in the Hilbert space X . Then the operator A is called dissipative ifℜ⟨Ax , x⟩ ≤ 0 for all x ∈D(A).

7Let X and Y be normed linear spaces and let T be a linear bounded operator from X to Y . Then T is a compact operator ifit maps bounded sets of X into precompact sets of Y [17].

4.2 Vibration suppression for a flexible beam 79

and domain

D(A) = x ∈ X : x1 ∈ (H 4(0,Lc )∩H 20 (0,Lc )), x2 ∈ H 2

0 (0,Lc ), x3 ∈R, x4 ∈Rwith

x3 = x2(Lc ), x4 = ∂z x2(Lc ). (4.3d)

The state vector is thereby defined as

x(t ) =

x1(t )x2(t )x3(t )x4(t )

=

x(·, t )∂t x(·, t )∂t x(Lc , t

∂t∂z x(Lc , t )

∈ X (4.3e)

for X = H 20 (0,Lc )×L2(0,Lc )×R2 with H 2

0 (0,Lc ) = H 2(0,Lc ) : x(0) = ∂z x(0) = 0. The space X is aHilbert space when equipped with the inner product

⟨x , y⟩X =∫ Lc

0

(µx2 y2 +E I∂2

z x1∂2z y1

)dz +mtm x3 y3 + Itm x4 y4 (4.3f)

and the norm induced according to ‖x‖X =√⟨x , x⟩X . This norm is also called energy norm. In additionrecall the definition of the spatial input characteristics

Γk (z) = Γp,k∂2zΛ

εk (z) (4.3g)

with

Λεk (z) = (%ε(z − zp,k )−%ε(z − zp,k −Lp )

), (4.3h)

and %ε(z) a (possibly smooth) transition function from %ε(z) = 0 for z <−ε/2 to %ε(z) = 1 for z > ε/2.

4.2.1 Well–posedness of the model

By making use of the Lumer–Phillips theorem, see, e.g., [57], we can verify that the operator A is theinfinitesimal generator of a C0–semigroup of contractions. This requires to prove, that A is dissipativeand there exists λ0 > 0 such that the range of (λ0I−A) is X with I denoting the identity operator.

Proposition 4.1. The operator A is dissipative.

Proof. Consider the total energy E (t ) = 12‖x‖2

X (t ) = 12 ⟨x , x⟩X of the free (u(t ) = 0) beam system and its

rate of change along a solution trajectory, i.e.,

∂t E(t ) = 1

2

(⟨∂t x , x⟩X +⟨x ,∂t x⟩X)= 1

2

(⟨Ax , x⟩X +⟨x ,Ax⟩X)

= 1

2

(⟨Ax , x⟩X +⟨A∗x , x⟩X)=ℜ⟨Ax , x⟩X

(4.3c)= −∫ Lc

0γe (z)|x2(z, t )|2dz ≤ 0.

(4.4)

Here, A∗ denotes the so–called adjoint operator. Obviously, dissipativity of A can be deduced fromthe last line, where some intermediate but straightforward computations using integration by partsare skipped.

The verification, that there is a λ0 > 0 implying that (λ0I−A) : X → X is onto is rather involving.Luckily [56, Theorem 1.2.4] implies, that this is equivalent to prove existence and boundedness of theinverse operator A−1.

Proposition 4.2. The operator A−1 exists and is bounded.


Proof. To verify this claim we explicitly compute the inverse operator. For this, solve Ax = ζ withx ∈D(A) for given ζ ∈ X which yields

x1 =∫ z

0

∫ s

0

1

E I (r )

∫ Lc

r

∫ Lc

qh1(p)dpdqdr ds −

∫ z

0

∫ s

0

1

E I (r )(h2(Lc − r )+h3)dr ds

x2 = ζ1

x3 = ζ1|z=Lc

x4 = ∂zζ1|z=Lc ,

(4.5)

with h1(z) = −µ(z)ζ2(z) − γe (z)ζ1(z), h2 = mtmζ3, and h3 = Itmζ4. This implies the existence ofA−1. Note also, that given ζ ∈ X , i.e., ζ1(z) ∈ H 2

0 (0,Lc ), ζ2(z) ∈ L2(0,Lc ) and ζ3, ζ4 ∈ R we havex ∈ (H 4(0,Lc )∩H 2

0 )×H 20 (0,Lc )×R2 = D(A). In addition, by estimating the norm it can be shown, that

the boundedness of ζ implies the boundedness of x such that A−1 is a linear bounded operator fromX to D(A), i.e., A−1 ∈L (X ,D(A)).

Lemma 4.1

The operator A is the infinitesimal generator of a C0–semigroup of contractions T(t ).

Proof. This is a direct consequence of the Lumer–Phillips theorem in view of Proposition 4.1 and 4.2or [56, Theorem 1.2.4].

Finally note if B ∈ L (Rm , X ) is an admissible input operator, then by [118, Proposition 4.2.5 andRemark 4.1.3] the initial value problem (4.3) has a unique so–called mild solution in X in the form

x(t ) =T(t )x0 +∫ t

0T(t −τ)Bu(τ)dτ (4.6)

for u(t ) ∈ L2loc([0,∞);Rm) and x0 ∈D(A) ⊂ X .

4.2.2 Lyapunov–based feedback control design

The feedback control design for the considered Euler–Bernoulli beam model exploits the fact, that thetotal energy E (t ) = 1

2‖x‖2X (t ) = 1

2 ⟨x , x(t )⟩X is positive definite and hence is a candidate for a Lyapunovfunctional. Taking into account (4.4) for the rate of change of E(t ) along a solution trajectory of (4.3)for the free case with u(t ) = 0 yields for the non–autonomous system

∂t E(t ) = 1

2

(⟨∂t x , x⟩X +⟨x ,∂t x⟩X)= 1

2

(⟨Ax , x⟩X +⟨x ,Ax⟩X)+ 1

2

(⟨Bu, x⟩X +⟨x ,Bu⟩X)

=ℜ⟨Ax , x⟩X + 1

2

(⟨Bu, x⟩X +⟨x ,Bu⟩X)

(4.3c)= −∫ Lc

0

(γe (z)|x2(z, t )|2 +

m∑k=1

Γk (z)x2(z, t )uk (t ))dz.

(4.7)

Obviously the choice of the feedback control

uk (t ) = κk

∫ Lc

0Γk (z)x2(z, t )dz = κk

∫ Lc

0Γp,k (∂2

zΛεk (z))x2(z, t )dz︸︷︷︸

= Mk (t )

, κk > 0 (4.8)


renders ∂t E (t ) negative semi–definite. Hence, the closed–loop control system consisting of (4.3) with(4.8) is given by

∂t x = Ax , t > 0 (4.9a)

x(0) = x0 ∈D(A) (4.9b)

with the operator

Ax =

x2

− 1µ(z)

(γe (z)x2 +∂2

z

(E I∂2

z x1))− 1

µ(z)

∑mk=1Γk (z)κk Mk

1mtm

∂z(E I∂2

z x1) |z=Lc

− 1Itm

(E I∂2

z x1) |z=Lc

(4.10)

defined on the domain D(A) = D(A).

Theorem 4.4

The operator A is the infinitesimal generator T(t ) of a C0–semigroup of contractions and T(t ) isasymptotically stable.

Proof. The proof, that A generates a C0–semigroup of contractions is almost identical to the proof ofPropositions 4.1 and 4.2 exploiting the Lumer–Phillips theorem or [56, Theorem 1.2.4]. Note, that thedissipativity of A is guaranteed by construction if κk > 0.

To verify asymptotic stability of the C0–semigroup we make use of LaSalle’s invariance principle asis introduced in Theorem 4.2. According to this principle all solutions of the closed–loop controlsystem (4.9) asymptotically tend to the maximal invariant subset of the set E = x(t ) ∈ X : ∂t E(t ) = 0provided, that the solution trajectories are precompact in X . Precompactness can be analyzed, e.g., bymaking use of Theorem 4.3. We have already seen (proof of A being the generator of a C0–semigroupof contractions), that A−1 exists and is bounded. Moreover, the embedding of D(A) into X is compactwhich is a consequence of the so–called Sobolev embedding theorem [56, 2]. This implies, that A−1 is acompact operator in X . The operator A is also closed by [17, Theorem A.3.46]. Hence, [45, Chapter 3,Theorem 6.29] yields, that the resolvent R(λ,A) = (λI− A)−1 is compact for any λ in the resolvent setof A. Since it can be easily seen from the definition of A, that 0 is in the range of A precompactness ofthe solution trajectories is ensured by Theorem 4.3 so that the invariance principle can be applied.

Investigation of (4.7) provides the set

E = x(t ) ∈ X : ∂t E(t ) = 0 = x(t ) ∈ X : ∂t x(·, t ) = x2(t ) = 0.

Hence, the maximal invariant set of E consists of the solutions x(t ) ∈ X satisfying the equations

0 =− 1

µ(z)∂2

z

(E I∂2

z x1)

0 = 1

mtm∂z

(E I∂2

z x1) |z=Lc

0 =− 1

Itm

(E I∂2

z x1) |z=Lc .

Since x(t) ∈ X implies x1(0) = ∂z x1(0) = 0 the only element of E is the zero element x(t) = 0. As aresult asymptotic stability of the closed–loop control system (4.9) follows from LaSalle’s invarianceprinciple.


The evaluation of the determined control law (4.8) relies on the availability of the velocity profilex2(t ) = ∂t x(·, t ) which is hardly completely accessible for measurement. Hence, the control–loop hasto be amended by state–observer.

4.2.3 Lyapunov–based state–observer design

Based on the equations of motion (4.3) a Luenberger–type state–observer is designed to reconstructthe velocity profile x2(t ) = ∂t x(·, t ) from the measurement of the beam’s tip deflection

y(t ) = x1(Lc , t ). (4.11)

The state–observer in the state variable x(z, t ) is set up as a copy of the original system

µ(z)∂2t x(z, t )+γe (z)∂t x(z, t )+∂2

z

(E I∂2

z x(z, t ))=−

m∑k=1

Γk (z)uk (t ), z ∈ (0,Lc ), t > 0 (4.12a)

with the boundary conditions extended by the functions l1(t ) and l2(t ) in the form

x(z, t ) = 0,∂z x(z, t ) = 0, z = 0 (4.12b)

E I (z)∂2z x(z, t )+ Itm∂

2t ∂z x(z, t ) = l1(t ),

∂z

(E I (z)∂2

z x(z, t ))−mtm∂

2t x(z, t ) = l2(t ).

z = Lc (4.12c)


x(z,0) = x0(z), ∂t x(z,0) = v0(z). (4.12d)

Introducing the observer error x(z, t ) = x(z, t )−x(z, t ) yields the observer error dynamics

µ(z)∂2t x(z, t )+γe (z)∂t x(z, t )+∂2

z

(E I∂2

z x(z, t ))= 0, z ∈ (0,Lc ), t > 0 (4.13a)

with the boundary conditions

x(z, t ) = 0,∂z x(z, t ) = 0, z = 0 (4.13b)

E I (z)∂2z x(z, t )+ Itm∂

2t ∂z x(z, t ) = l1(t ),

∂z

(E I (z)∂2

z x(z, t ))−mtm∂

2t x(z, t ) = l2(t ).

z = Lc (4.13c)


x(z,0) = x0(z), ∂t x(z,0) = v0(z). (4.13d)

For the determination of the correction terms l1(t), l2(t) we, similar to the control design, considerthe rate of change of the energy of the observer error system (for the sake of simplicity the energy termis written in the state variable x(z, t )) defined by

E(t ) = 1

2

(∫ Lc

0

(µ(z)(∂t x(z, t ))2 +E I (z)(∂2

z x(z, t ))2)dz +mtm(∂t x(Lc , t ))2 + Itm(∂t∂z x(z, t ))2

)which yields after some intermediate computations using integration by parts

∂t E(t ) =−∫ Lc

0γe (z)(∂t x)2dz − l2(t )∂t x(Lc , t )+ l1(t )∂t∂z x(Lc , t )


≤−l2(t )∂t x(Lc , t )+ l1(t )∂t∂z x(Lc , t ). (4.14)

Choosing the degrees–of–freedom as

l1(t ) =−α1∂t∂z x(Lc , t ), l2(t ) =α2∂t x(Lc , t ), α1, α2 > 0 (4.15)

renders ∂t E(t ) negative semi–definite.

Remark 4.1

The boundary conditions (4.13c) illustrate, that the functions l1(t ) and l2(t ) can be interpretedas a bending moment and a force acting at the position z = Lc . This also confirms, that(l1(t ),∂t∂z x(Lc , t )) and (l2(t ),∂t x(Lc , t )) represent collocated power pairings enabling to adjustthe energy stored in the observer error system.

Remark 4.2

Since the correction terms l1(t), l2(t) are proportional to velocities, static deviations betweenthe estimated and the measured signals cannot be considered. From a practical point of view itis hence recommended to introduce an extended Lyapunov functional, i.e.,

V (t ) = E(t )+α3

2(x(Lc , t ))2 + α4

2(∂z x(Lc , t ))2

with α3, α4 > 0. Evaluation of the rate of change of V (t ) along a solution trajectory x(z, t ) resultsin

∂t V (t ) = ∂t E + (α3x(Lc , t )∂t x(Lc , t )+α4∂z x(Lc , t )∂t∂z x(Lc , t )

)≤ ∂t x(Lc , t )

[α3x(Lc , t )− l2(t )

]+∂t∂z x(Lc , t )[α4∂z x(Lc , t )+ l1(t )

]. (4.16)

Hence, the choice

l1(t ) =−(α1∂t∂z x(Lc , t )+α4∂z x(Lc , t )

), l2(t ) =α2∂t x(Lc , t )+α3x(Lc , t ), α1α2 > 0

(4.17)

results in a dissipative observer error system with ∂t V (t ) ≤ 0.

For the asymptotic stability analysis of the observer error system (4.13) with (4.17) we introduce itsabstract formulation

˙x(t ) = Ax , t > 0 (4.18a)

x(0) = x0 ∈D(A) (4.18b)

with the state vector

x(t ) =

x(·, t )∂t x(·, t )∂t x(Lc , t )∂t∂z x(Lc , t )

(4.18c)


and the operator

Ax =

x2

− 1µ(z)

(γe (z)x2 +∂2

z

(E I∂2

z x1))

1mtm

∂z(E I∂2

z x1) |z=Lc − 1

mtml2

− 1Itm

E I∂2z x1|z=Lc + 1

Itml1

. (4.18d)

The domain is defined as D(A) = (H 4(0,Lc )∩H 20 (0,Lc )×H 2

0 (0,Lc )×R2) and is dense in X = H 20 (0,Lc )×

L2(0,Lc )×R2 which is a Hilbert space when equipped with the inner product

⟨x , y⟩X =∫ Lc

0

(µ(z)x2 y2 +E I (z)∂2

z x1∂2z y1

)dz +mtm x3 y3 + Itm x4 y4

+α3

(x1 y1

)z=Lc

+α4

(∂z x1∂z y1

)z=Lc

(4.19)

for all x , y ∈ X .

Theorem 4.5

The operator A is the infinitesimal generator T(t ) of a C0–semigroup of contractions and T(t ) isasymptotically stable.

Proof. The proof is almost identical to the proof of Theorem 4.4 with the difference, that the Lyapunovfunctional V (t ) = 1

2 ⟨x , x⟩X has to be considered instead of E(t ).

The main implication of this section is, that an estimation of the spatial–temporal evolution of thestate variable can be obtained by means of the state–observer (4.12) solely from the measurementsof (∂t x(Lc , t)), ∂t∂z x(Lc , t), x(Lc , t), and ∂z x(Lc , t). The velocity signals can be thereby obtained byappropriate filtering of the position signals.

4.2.4 Stability of the composite system

For finite–dimensional linear time invariant systems the separation principle guarantees, that feed-back control and state–observer can be independently designed. This property is not directly ob-vious for distributed parameter systems so that subsequently the composite system consisting ofthe controlled cantilever beam and the observer error system is considered in the extended statexext (t ) = [xT (t ), xT (t )]T ∈ X × X . In particular we have

xext (t ) =Aext xext , t > 0 (4.20a)

xext (0) = xext0 ∈D(Aext ) (4.20b)

with the operator

Aext =[A P

0 A

](4.20c)


Figure 4.2: Picture of the cantilevered beam with two pairs of MFC actuators and tip mass.

defined on the domain D(Aext ) =D(A)×D(A). Under the assumptions introduced above it can beshown, that P is a linear bounded operator, i.e., P ∈L (X , X ), given by

Px =

0

− 1µ(z)

∑mk=1Γk (z)κk

∫ Lc0 Γk (z)x2dz

00

. (4.21)

By making use of the separation principle introduced in [69, Theorem 3] for the composition ofgenerators of asymptotically stable C0–semigroups coupled according to Aext the asymptotic stabilityof the composite system can be deduced.

4.2.5 Experimental results

The experimental evaluation of the developed control concept involving the state–observer is per-formed for a cantilevered beam actuated by macro–fiber composite (MFC) patch actuators as isdepicted in Figure 4.2. The beam consists of a fiber reinforced composite material with dimensionsLc = 0.406m, bc = 0.045m, and hc = 0.75×10−3 m and an end mass of mtm = 0.0126kg. The rotationalinertia Itm of the end mass is supposed to be zero. Two pairs of MFC patches with an active area ofdimension Lp = 85×10−3 m, bp = 57×10−3 m, hp = 3×10−4 m are bonded to the beam at distanceszp,1 = 0.031m and zp,2 = 0.246m from the clamped edge. Two laser sensors are used to measurethe beam deflection at zm,1 = 0.88Lc m and zm,2 = 0.95Lc m. Assuming negligible bending betweenz = zm,1 and z = Lc the deflection and the angle at the beam’s tip are approximated by extrapolation.

For the realization of the state–observer Galerkin’s method is used to determine a finite–dimensionalapproximation of the distributed parameter system (4.12). In particular the approach suggested in[101] is applied with the basis function chosen as the first 5 eigenmodes of a uniform cantilever beam.

Remark 4.3

To simplify implementation of the control law (4.8) the spatial patch characteristics is assumeddiscontinuous, i.e.,Λ0

k (z) = limε→0Λεk (z) =σ(z−zp,k )−σ(z−zp,k−Lp ), whereσ(·) is the Heaviside


function. Since (4.8) now involves the spatial derivative of the Dirac delta function δ(z) due to∂2

zσ(z) = ∂zδ(z) integration by parts yields the control law in the form

uk (t ) = κk

∫ Lc

0Γk (z)x2(z, t )dz = κkΓp,k

(∂t∂z x(zp,k +Lp,k , t )−∂t∂z x(zp,k , t )

). (4.22)

In addition, the effect of the voltages uk (t) can be interpreted as a pair of pointwise bendingmoments located at the boundary of the patch actuators. It has to be mentioned, that the arisingdiscontinuities require to analyze the weak form the equations of motion as is done, e.g., in [7,100, 102].

4.2.5.1 Model validation

The parameters of the beam model are identified by minimizing the mean squares error betweenmeasured and simulated step responses based on the finite–dimensional approximation. Due tothe inherent actuator nonlinearities a hysteresis and creep compensation is utilized for driving thepiezoelectric patches. The step response of the beam compared to the numerical simulation is shownin Figure 4.3 confirming excellent agreement.

0 5 10 15 20

0

0.005

0.01

0.015

0.02

0.025

t (s)

y 1(m

)

y1ysim1

Figure 4.3: Measured and simulated beam’s tip deflection due to a voltage step of u1 =−500V at the patchpair located at zp,1 at t = 1s followed by a voltage step of u2 = 500V at the patch pair at zp,2 att = 12s. ©2011, IEEE

4.2.5.2 Vibration suppression

To evaluate the control performance experimental results for vibration suppression are presented,when exciting the beam by an impact hammer which is positioned at z = 0.37 m.

As is shown in Figure 4.4 for the uncontrolled case this results in weakly damped vibrations originatingfrom high flexibility of the beam. Contrary, in the controlled case the oscillations are immediatelydamped out as depicted in Figure 4.5 (a) with Figure 4.5 (b) illustrating the applied voltage signal to theMFC patch pairs. The control signal results from the evaluation of (4.22) with estimates obtained fromthe observer and subsequent processing by the hysteresis and creep compensator. To illustrate theadmissibility of the observer quantities Figure 4.5 (c) presents snapshots of the estimated deflectionprofile of the beam at different instances of time.


0 1 2 3 4 5 6 7 8

−0.05

0

0.05

t (s)

y 1(m

)

Figure 4.4: Vibrations generated by an impact hammer positioned near the tip of the beam.

0 1 2 3 4 5 6 7 8

−0.05

0

0.05

t (s)

y 1(m

)

(a)

0 1 2 3 4 5 6 7 8

0

200

400

600

800

t (s)

ufs(V

)

ufs1ufs2

(b)

0 0.1 0.2 0.3 0.4

−0.05

0

0.05

x

w(x,t)

0 0.1 0.2 0.3 0.4

−0.05

0

0.05

x

w(x,t)

t=0.07t=0.59t=0.84t=1.07t=1.32t=1.58t=2.18

(c)

Figure 4.5: Vibration control in case of an impact near the tip of the beam: (a) deflection y1(t) = x(zm,2, t);(b) voltages applied to the MFC patch pairs (only signal applied to the front–side, superscriptfs,is shown); (c) estimated deflection profile at instances of time corresponding to minimal andmaximal values of the beam’s tip deflection. ©2011, IEEE


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motion planning and tracking control for distributed

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