robust and lpv control of mimo systems part 3: linear ...o.sename/docs/lpv...robust and lpv control...

Robust and LPV control of MIMO systemsPart 3: Linear Parameter Varying systems: from modelling to control

Olivier Sename

GIPSA-Lab

Tecnologico de Monterrey, July 2016

Olivier Sename (GIPSA-Lab) Robust and LPV control - part 3 Tecnologico de Monterrey, July 2016 1 / 64

1. What is a Linear Parameter Varying systems?

2. Classes (models) of LPV systems

3. How to approximate a nonlinear system by an LPV one ?

4. Identification of LPV systems

5. Some properties of LPV systems

6. Stability of LPV systems

7. LPV Control & ObservationThe Static State feedback caseThe Dynamic Output feedback caseLPV observer design

8. Summary of LPV approach interests

O. Sename [GIPSA-lab] 2/64

What is a Linear Parameter Varying systems?

LPV systems

Definition of an Linear Parameter Varying system

Σ(ρ) :

xzy

=

A(ρ) B1(ρ) B2(ρ)C1(ρ) D11(ρ) D12(ρ)C2(ρ) D21(ρ) D22(ρ)

xwu

x(t) ∈ Rn, ...., ρ = (ρ1(t), ρ2(t), . . . , ρN (t)) ∈ Ω, is a vector of time-varying parameters (Ωconvex set), assumed to be known ∀t



LPV systems


Σ(ρ) :

xzy

=


xwu


System (ρ)

ExogeneousInputs

Controlledoutputs

Measuredoutputs

z

ControlInputs

w

u y

(Scherer, ACC Tutorial 2012)

10/60

Ma

them

ati

cal

Sys

tem

sT

heo

ryExample

Dampened mass-spring system:

p+ c _p+ k(t) p = u; y = x

First-order state-space representation:

d

dt

0@ p

_p

1A =

0@ 0 1

k(t) c

1A0@ p

_p

1A+

0@ 0

1

1Au;

y =1 0

0@ p

_p

1A

Only parameter is k(t)

System matrix depends affinely on this parameter

Could view c as another parameter - keep it simple for now ...



LPV systems


Σ(ρ) :

xzy

=


xwu


The frozen Bode plots forc = 1 and k ∈ [1, 3]

(Scherer, ACC Tutorial 2012)

10/60

Ma

them

ati

cal

Sys

tem

sT

heo

ryExample

Dampened mass-spring system:

p+ c _p+ k(t) p = u; y = x

First-order state-space representation:

d

dt

0@ p

_p

1A =

0@ 0 1

k(t) c

1A0@ p

_p

1A+

0@ 0

1

1Au;

y =1 0

0@ p

_p

1A

Only parameter is k(t)

System matrix depends affinely on this parameter

Could view c as another parameter - keep it simple for now ...



LPV systems (2)

Let the LPV system be:

Σ(ρ) :

xzy

=


xwu

x(t) ∈ Rn, ...., ρ = (ρ1(t), ρ2(t), . . . , ρN (t)) ∈ Uρ, is a vector of time-varying parameters (Uρconvex set)• ρ(.) varies in the set of continously differentiable parameter curves ρ : [0,∞)→ RN .

It is assumed to be known or measurable.• The parameters ρ are always assumed to be bounded:

ρ ∈ Uρ ⊂ RN and Uρ compact (1)

defined by the minimal ρi, and maximal ρi values of ρi(t)

ρi(t) ∈ [ρi, ρi], ∀i

• The system matrices A(.) .... are continuous on Uρ



LPV systems (3): about the parameters

• Parameters are exogenous if they are external variables. The system is in that case nonstationary.See the previous damped mass-sping system.

• Parameters are endogenous if they are function of the state variables, ρ = ρ(x(t), t), and, inthat case, the LPV system is referred to as a quasi-LPV system.This case is encountered when approximating Nonlinear systems.For instance:

x(t) = x2(t) = ρ(t)x(t)

with ρ(t) = x(t).• It is sometimes required that the derivative of the parameters are bounded, i.e:

ρ ∈ Uρ ⊂ RN and Uρ compact (2)

defined by the minimal νi, and maximal νi values of ρi(t)

ρi(t) ∈ [νi, νi], ∀i

This corresponds to the case of slow varying parameters• Other representations can be considered if ρ is piecewise-constant, or varies in a finite set of

elements (ρ(t) ∈ 0, 1 for switching systems)

Next, several classes of LPV models are presented, and some ways to go from one class toanother are given.



Some comments

• LPV systems can model uncertain systems (ρ fixed but unknown) or parameter-varyingmodels (ρ(t))

LPV=linear or nonlinear?

• What is often referred to as gain-scheduling control, corresponds to Jacobian linearization ofthe nonlinear plant about a family of equilibrium points Shamma (90), Rugh & Shamma (2000)In terms of control design this means, lineraization around operating conditiosn, design (ateach operating points) of a LTI controller, and interpolation of the LTI controllers in betweenoperating conditions (often used in Aerospace and Automotive industries).Pros: Simplicity of design for a non linear systemCons: No a priori guarantee of stability nor robustness

• But: this differs from quasi-LPV representations where nonlinearities are hiddden in someparameter descriptions (as seen later in the course)

LPV=LTV

• Theoretical analysis of LPV system properties (stability, controllability, observability), oftenfalls into the framework of LTV systems or of nonlinear ones (for quasi-LPV representations),see (Blanchini).



Some references

Those not to be ignored

• Modelling, identification : (Bruzelius, Bamieh, Lovera, Toth) + 2011 TCST Special Issue on"Applied LPV modelling and identification"

• Control (Shamma, Apkarian & Gahinet, Adams, Packard, Beker, Seiler, Grigoriadis ...)• Stability, stabilization (Scherer, Wu, Blanchini ...)• Geometric analysis (Bokor & Balas)• Survey paper: Hoffmann & Werner, 2015• Fault tolerant control: special issues by

• Balas, 2012: in International Journal of Adaptive Control and Signal Processing• Casavola, Rodrigues & Theilliol, 2015: in International Journal of Robust and Nonlinear Control

Some recent books

• R. Toth, Modeling and identification of linear parameter-varying systems, Springer 2010• J. Mohammadpour, C. Scherer, (Eds), Control of Linear Parameter Varying Systems with

Applications, Springer-Verlag New York, 2012• O. Sename, P. Gaspar, J. Bokor (Eds), Robust Control and Linear Parameter Varying

Approaches: Application to Vehicle Dynamics, Springer, 2013


Classes (models) of LPV systems

Outline











Class 1: Affine parameter dependence

In this case, the system matrices are such that:

A(ρ) = A0 +A1ρ1 + ...+ANρN

This is the case of the damped mass-sping system:

p+ cp+ k(t)p = u

considering the state space representationx(t) = A(k(t))x(t) +Bu(t),

y(t) = Cx(t) +Du(t)(3)

with x(t) ∈ R2 and

A(k(t)) =

(0 1−k(t) −c

), B =

(01

), C =

(1 0

), D = 0

Denoting the varying parameter ρ(t) = k(t), we get:

A0 =

(0 10 −c

), A1 =

(0 0−1 0

)



Class 2: Polynomial parameter dependence


A(ρ) = A0 +A1ρ+A2ρ2 + ...+ASρ

S

Example 1

Let consider for instance the following example in (Briat, 2015):

x(t) = x3(t)

which can be written asx(t) = −ρ(t)2x(t),with ρ(t) = x(t)

Example 2

Another example is the sampling-dependent discrete-time system representation (Robert et al,2010):

Gd :

xk+1 = Ad(h) xk +Bd(h) ukyk = C(h) xk +D uk

(4)

where Ad(h) = eAh is approximated using Taylor expansions, such as:

Ad(h) ≈ I +N∑i=1

Ai

i!hi



Class 3: Rational parameter dependence


A(ρ) = [An0 +An1ρn1 + ...+AnNρnN ][I +Ad1ρd1 + ...+AdNρdN ]−1

The bicycle model

To study lateral dynamics the so-called bicycle model considers the dynamics of the ya rate ψ andof the vehicle sideslip angle β, as follows:

[ψ

β

]=

−l2fCf+l2rCr

Izv

lrCr−lfCf

Iz

1 +lrCr−lfCf

mv2−Cf+Cr

mv

[ ψβ

]+

[lfCf

IzCf

mv

]δ+ (5)

δ+ being the controlled steering angle.Considering the vehicle longitudinal velocity v as a varying parameter, i.e ρ = v, this model can berecast into an LPV system with rational dependency on ρ.



Class 4: Polytopic models

A polytopic system is represented as

Σ(ρ) =Z∑k=1

αk(ρ)

[Ak BkCk Dk

], with

2N∑k=1

αk(ρ) = 1 , αk(ρ) > 0

where[Ak BkCk Dk

]are LTI systems.

This representation is often used to rewrite an affine LPV system. Indeed, assuming that theparameters are bounded (ρi ∈

[ρi

ρi]), the vector of parameters evolves inside a polytope

represented by Z = 2N vertices ωi, as

ρ ∈ Coω1, . . . , ωZ (6)

It is then written as the convex combination:

ρ =Z∑i=1

αiωi, αi ≥ 0,Z∑i=1

αi = 1 (7)

where the vertices are defined by a vector ωi = [νi1, . . . , νiN ] where νij equals ρj or ρj .

The LTI system[Ak BkCk Dk

]here corresponds to the LPV system frozen at the vertex k.



From a generic affine LPV systems to a poytopic model

For a LPV system with 2 parameters, boundend[ρ

1,2ρ1,2

], the corresponding polytope

onws 4 vertices as:

Pρ =

(ρ1, ρ

2), (ρ

1, ρ2), (ρ1, ρ2

), (ρ1, ρ2)

(8)

The polytopic coordinates are (αi) are obtained as:

ω1 = (ρ1, ρ

2), α1 =

(ρ1 − ρ1ρ1 − ρ1

)×(ρ2 − ρ2ρ2 − ρ2

)

ω2 = (ρ1, ρ2), α2 =

(ρ1 − ρ1ρ1 − ρ1

)×(ρ2 − ρ2ρ2 − ρ2

)

ω3 = (ρ1, ρ2), α3 =

(ρ1 − ρ1ρ1 − ρ1

)×(ρ2 − ρ2ρ2 − ρ2

)

ω4 = (ρ1, ρ2), α4 =

(ρ1 − ρ1ρ1 − ρ1

)×(ρ2 − ρ2ρ2 − ρ2

)(9)

where ρ1 and ρ2 are the instantaneous values of the parameters (ρ(k)i in the implementation

step).The LPV system is then rewritten under the polytopic representation:(

A(ρ1,2) B(ρ1,2)C(ρ1,2) D(ρ1,2)

)= α1

(A(ω1) B(ω1)C(ω1) D(ω1)

)+ α2

(A(ω2) B(ω2)C(ω2) D(ω2)

)+ α3

(A(ω3) B(ω3)C(ω3) D(ω3)

)+ α4

(A(ω4) B(ω4)C(ω4) D(ω4)

)(10)



Class 5: LFT (or LFR) models

z∆w∆

zw P

∆

u y

Figure: System under LFT form

∆ is defined such as: z∆ = ∆(.)w∆.It represents the parameter variations.

∆(.) is a linear function of theparameter vector.

The equation of the system under LFR representation is asfollows:

xz∆zy

=

A B∆ B1 B2

C∆ D∆∆ D∆1 D∆2

C1 D1∆ D11 D12

C2 D2∆ D21 D22

xw∆

wu

Denoting the transfer matrix N(s) as:[z∆z

]=

[N11(s) N12(s)N21(s) N22(s)

] [w∆

w

]The Linear Fractional Representation (LFR) gives then thetransfer matrix from w to z, and is referred to as the upperLinear Fractional Transformation (LFT) :

Fu(N,∆) = N22 +N21∆(I −N11∆)−1N12

This LFT exists and iswell-posed if (I −N11∆)−1 is invertible



Few facts on LPV models under Linear Fractional Representations

LFT modelling

• Such a representation is the direct extension of LFT forms (originally used for uncertainties inthe context of µ-analysis), to model systems with time-varying parameters.

• It can be used to model nonlinear systems (with an adhoc definition of the parameters), andalso linear system scheduled by the operating conditions (i.e. with a varying equilibrium)

The "gain scheduling" approach

• One interest is that it allows a systematic synthesis of robust controllers, in the framework ofthe integral quadratic constraint (IQC) approach (Megretski & Rantzer, 1997), (Scherer,2001), (Veenman & Scherer, 2014).Moreover, through the defintion of particular IQC-multipliers, it allows to model a large verietyof problems including time-varying parametric uncertainties, passive nonlinearities,Norm-bounded uncertainties/nonlinearities, sector bounded nonlinearities, delays...

• This framework allows to design (robust) gain-scheduled controllers/observers, in particular inthe context of L2 -induced performances



About LFT definition

Parameter properties

• ∆(ρ) (L2 → L2 ) is bounded (i.e ∆(ρ)T∆(ρ) ≤ 1) and causal.• it can be always chosen as ∆(ρ) = diag(ρ1In1, . . . , ρN InN ), where ni is the number of

occurrences of parameter ρi, where |ρi| < 1, ∀i

LFT vs LPV

Let consider the LPV part of the global LFR system,

[xz∆

]=

[A B∆

C∆ D∆∆

] [xw∆

]with w∆ = ∆(ρ)z∆. If the system is well-posed, i.e if (I −∆(ρ)D∆∆) is non singular ∀ρ, then theLFT form equals:

x = (A+B∆∆(ρ)(I −∆(ρ)D∆∆)−1C∆)x(t)

which is nothing else than the original LPV system:

x = A(ρ)x



An academic example

Let us consider the following nonlinear system:

G :

x1 = −x1 + x2sin(x1)x2 = −x2 + wz = x1

(11)

In order to use an LFT, let us define the input:

u∆ = ∆(ρ)x2 := ρx2, with ρ = sin(x1)


Then the previous system can berewritten into the following LFR:

∆

∆zw

uy

where ∆ and y∆ are given as:

∆(ρ) =[ρ], y∆ =

(x2

)and N given by the state space representation:

N :

x1

x2

==

−x1 + u∆

−x2 + wy∆ = x2

z = x1

(12)


The Autonomous Underwater Vehicle case

Considered the NL model with 2 state variables (the pitch angle θ and velocity q):

θ =cos(φ)q − sin(φ)r

Mq =− p× r(Ix − Iz)−m[Zg(q × w − r × v)]− (Zqm− ZfµV )gsin(θ)

− (Xqm−XfµV )gcos(θ)cos(φ) +Mwqw|q|+Mqqq|q|+ Ffins (13)

where M interia matrix, m AUV mass, V volume and µ mass density. The other parameters(Ix, Iz , Zg , Zq , Zf , Xq , Xf ,Mwq) appear in the dynamical and hydrodynamical functions.Ffins : control input forces and moments due to the fins.

Tangent linearization around Xeq = [0 ueq 0 0 0 0 0 0 θeq 0 0 0]:

˙θ = q

M ˙q = [−(Zgm− ZfµV )gcos(θeq) + (Xgm−XfµV )gsin(θeq)]θ + Ffinseq

where θ and q are the variations of θ and q.



A LFR model

An LFR model of the vehicle described in the form:

z∆w∆

zkwk P (z)

∆(ρ)

uk yk

The ∆ block contains the varying part of the model, which depends on the linearization point (θeq).The LFR form (without external inputs w and controlled output z) is defined by:

xk+1 = Axk +[B∆ B1

] (w∆

u

)(z∆y

)=

[C∆C1

]xk +

[D∆ 00 D1

](w∆

u

) (14)



A LFR model

with

A =

[0 10 0

], B∆ =

[0 0

−(Zqm− ZfµV )g (Xqm−XfµV )g

]B1 =

[0

Ffins

], C∆ =

[1 10 0

], C1 =

[1 00 1

]D∆ =

[0 00 0

], D1 =

[0 00 0

](15)

z∆ =

[θ

θ

]; w∆ = ∆z∆; ∆ =

[ρ1 00 ρ2

](16)

with: ρ1 = cos(θeq)ρ2 = sin(θeq)

(17)


How to approximate a nonlinear system by an LPV one ?

Outline











Nonlinear vs Linear Differential Inclusion

See (Boyd et al, 1994).Let consider the nonlinear system

ΣNL :

x = f(x(t), w(t))z = g(x(t), w(t))

(18)

Suppose that, for each x, w and t, there is a matrix G(x,w, t) ∈ Ω s.t.:[f(x,w)g(x,w)

]= G(x,w, t)

[xw

](19)

where Ω ∈ R(nx+nz)×(nx+nu).As said in (Boyd et al, 1994):"Then of course every trajectory of the nonlinear system (18) is also a trajectory of the LDI definedby (19). If we can prove that every trajectory of the LDI defined by (19) has some property (e.g.,converges to zero), then a fortiori we have proved that every trajectory of the nonlinear system(18) has this property."



LPV modelling of a quarter car vehicle suspension model


Figure: Simple quarter vehicle model for semi-active suspension control

Quarter vehicle dynamics

mszs = −kszdef − Fdampermuszus = kszdef + Fdamper − kt (zus − zr)

(20)

zdef = zs − zus : damper deflection, zdef = zs − zus : deflection velocity.

• The damper’s characteristics : Force-Deflection-Deflection Velocity relation

Fdamper = g(zdef , zdef

)(21)

where g can be linear or nonlinear.


LPV modelling of a quarter car vehicle suspension model (cont.)

A Semi-active nonlinear MR damper model [Gu et al., 2006, Nino-Juarez et al., 2008]

Fdamper = c0zdef + k0zdef + fItanh(c1zdef + k1zdef

)(22)

• The tanh function allows to model the bi-viscous behavior.• 5c0, k0, c1, k1): constant parameters. k0, k1 dedicated to the hysteresis behavior.• fI is a controllable force and depends on input current I.

LPV model

Choosing ρ = tanh(c1zdef + k1zdef

), and denoting u = fI the control input, the quarter car

model can be represented as: x(t) = Ax(t) +B(ρ)u(t),

y(t) = Cx(t) +Du(t)(23)



Example of a one-tank model - MATLAB session 1

Usually the hydraulic equation is non linear and of the form

SdH

dt= Qe −Qs

where H is the tank height, S the tank surface, Qe the input flow, and Qs the output flow definedby Qs = kt

√H.

Definition the state space model

The system is represented by an Ordinary Differential Equation whose solution depends on H(t0)and Qe. Clearly H is the system state, Qe is the input, and the system can be represented as:

x(t) = f(x(t), u(t)), x(0) = x0 (24)

with x = H, f = − ktS

√x+ 1

Su

In steady state : Qe = Q0, H = H0



Example of a one-tank model - MATLAB session 1 (cont.)

Consider the variations qe, qs, h around the steady state as:Qe = Q0 + qe; Qs = Q0 + qs; H = H0 + h.This leads to the equation :

Sdh

dt= qe − kt(

√H0 + h−

√H0)

Denoting the state variable x = h, the control input u = qe, the output y = h, the tangentlinearization gives

x = Ax+Bu (25)

y = Cx (26)

with A = − kt2S√H0

, B = 1S

and C = 1.Now implement 3 types of models and compare with the nonlinear one:• The linear model around a fixed operating point• A LPV model obtained defining an adequate LDI• A LPV model defined in the LFR framework


Identification of LPV systems

Outline











Models definition

Identification often requires discrete-time models than can be of 2 forms:

Discrete-time state space model

ΣLPV

x(k + 1) = A(ρ(k))x(k) +B(ρ(k))u(k),y(k) = C(ρ(k))x(k) +D(ρ(k))u(k)

(27)

Input-output models

y(k) = −na∑i=1

ai(ρ(k))y(k − i) +

nb∑j=1

bi(ρ(k))u(k − j) (28)

where ai’s and bi’s can be affine, polynomial, rational, of LFR, functions of theparamteter vector ρ



Derivation of LPV models (Lovera, Grenoble summer school 2011)

How to derive control-oriented LPV models?

From non linear models (see before)

Analytical/numerical approaches based on a non linear model or simulator - see (Marcos andBalas 2004) for an overview

Identification from input/output data

Two broad classes of methods can be defined:

• Global approaches:• a single experiment→ the parameter is also excited• a parameter-dependent model is directly obtained• Do exist for input/output and state space models

• Local approaches:• multiple experiments→ constant parameter values• many LTI models are obtained, which have to be interpolated



About model transformation

Many approaches need a conversion from input-output to state space models (to go throughcanonical forms), which is easy for LTI systems....

for LTI systems

Both systems (A,B,C,D) and (T−1AT, T−1B,CT,D) have equivalent properties (stability,observability, controllability, transfer functions...).

But for LPV systems

The state transformation may depend on time, i.e. T (ρ(k)).

In that case, the transformation x(k) = T (ρ(k))xnew(k) applied to

ΣLPV

x(k + 1) = A(ρ(k))x(k) +B(ρ(k))u(k),y(k) = C(ρ(k))x(k) +D(ρ(k))u(k)

(29)

leads to (denoting for simplicity all matrices parameter dependency Mk for M(ρ(k))):

ΣnewLPV

xnewk+1 = (T−1

k+1AkTk)xnewk + (T−1k+1Bk)uk,

yk = (CkTk)xnewk + (Dk)uk(30)

The change of basis changes locally according to the parameter values and variations !In particular T (ρ(k)) must be invertitble ∀k.



Some literature

Global approaches - input/output models

(Bamieh & Giarre 99, 02): characterisation of persistency of excitation conditions for input-outputLPV models

Previdi & Lovera 03, 04): NLPV model class (LFT feeded by a neural network model for thescheduling policy)

(Toth, 07 + book 2010): An LPV system can be viewed as a collection of "local" behaviours(associated with constant parameter values

Global approaches - state space models

(Lee & Poolla): maximum likelihood (ML) algorithm for the identification of MIMO LPV-LFTmodels (PEM algorithm)

(Verhaegen et al, 02, 07, 09...): Supspace methods

Local approaches

Interpolation of locally identified LTI models... need to pay attentin to:• Input/output form (Toth, 07 + book 2010): interpolating transfer function coefficients• State space form (Steinbuch et al, 03): consistency of state space basis


Some properties of LPV systems

Outline











LPV systems properties

Let consider the LPV syetsm

Σρ

x(t) = A(ρ(t))x(t) +B(ρ(t))u(t), x(0) = x0

y(t) = C(ρ(t))x(t) +D(ρ(t))u(t)(31)

What kind of properties we should pay attention to?

When ρ is fixed (constant) the previous system is LTI and• controllability, observability, stability, are uniquely defined• controllability⇔ reachabillity, observability⇔ reconstructibility• these properties are equivalent by a state change of basis.

But when ρ(t) is time varying .....

• these facts may not be true (asymptotic and exponential stabilty may differ)• need to study properies of Linear Time-Varying systems.• A generalization of the exp(At) is needed, defining the state transition matrix Φ(t, t0, ρ(t))

• For a change of basis T (t) with x(t) = T (t)xnew(t) then, x(t) = T (t)xnew(t) + T (t)xnew(t)



A brief insight on observability

Let us consider the system

Σρ

x(t) = A(ρ(t))x(t), x(0) = x0

y(t) = C(ρ(t))x(t)(32)

Observability refers to the ability to estimate the initial state variable.

Definition

The system (32) is completely observable if, given the control and the output over the intervalt0 ≤ t ≤ T , one can determine any initial state x(t0).

Following some studies on observability for Linear Time-Varying Systems (LTV) (Silverman &Meadows, SIAM 67), to characterize the observability we need to define the fundamental matrixΨ, satisfying Ψ(t) = A(ρ(t))Ψ(t) (with Ψ(t0) = In) allowing to state that the solution of the stateequation can be deduced as:

x(t) = Ψ(t, t0, ρ(t))x(t0)

which is not easy to compute. Note that, for LTI systems, Ψ(t, t0) = eA(t−t0)



A brief insight on observability (2)

In an analogous way the unobservability property is defined as : a state x(t) is not observable ifthe corresponding output vanishes, i.e. if the following holds: y(t) = y(t) = y(t) = . . . = 0. In thecase of LTV systems it corresponds to:

Definition

The LPV system (32) is completely observable if rankO = n ∀t, where

O =[oT1 oT2 . . . oTn

]Twhere o1 = C(ρ) and oi+1 = oiA+ oi, i > 1 (for instance o2 = ρ

∂C(ρ)∂ρ

+ C(ρ)A(ρ)).


A weaker notion of observability can bedefined for the LPV systems (32) in thefunctional sense O function of ρ(t).

Definition

The LPV system (32) is structurallyobservable if rankO = n

This does not guarantee that O is invertible∀t and for all parameter values.

Finally the above notion differ from the directextension of the observability matrix for LTI

systems, i.e O =

C(ρ)

C(ρ)A(ρ)...

C(ρ)An−1(ρ)

.This definition is ONLY valid if ρ is constant,i.e. it corresponds to the observability of theLTI systems frozen at the values of theconstant parameter vector ρ.


A few examples


Σ1(ρ) :

x(t) = A(ρ)x(t)y(t) = C(ρ)x(t)

with

A =

(1 1ρ(t) 2

), C =

(ρ(t) 1

)Observability matrix with ρf is a frozen valueof ρ(t):

O =

(ρf 12ρf ρf + 2

)which is of rank 2 apart for ρf = 0.Therefore the LTI frozen systems areobservable.

However the observaility matrix of theconsidered time-varying system is given by:

O =

(ρ(t) 1

ρ(t) + 2ρ(t) ρ(t) + 2

)which is of rank 2 in the functional sense.Therefore the structural rank of Σ1(ρ) is 2.However it is of rank 1 if ρ satisfiesρ(t) = ρ(t)2. The system is then notcompletely observable.Therefore, for some specific parameterdefinitions, the parameter variations maytherefore induce a loss of observability.


A few examples (2)

Let us consider now


Σ(ρ) :

x(t) = A(ρ)x(t)y(t) = C(ρ)x(t)

with

A =

(1 0ρ(t) 2

), C =

(ρ(t) 1

)Observability matrix with ρf is a frozen valueof ρ(t):

O =

(ρf 12ρf 2

)which is of rank 1.Therefore the LTI frozen systems are notobservable.

However the observaility matrix of theconsidered time-varying system is given by:

O =

(ρ(t) 1

ρ(t) + 2ρ(t) 2

)which is of rank 2 in the functional sense(apart if ρ(t) 6= 0). Therefore the structuralrank of Σ1(ρ) is 2.In that case the time-variations of ρ(t)provide a persistent excitation on the systemoutput.

Stability of LPV systems

Outline











Problem stament and facts

Recall

For LTI systems all notions of stability are equivalent: global/local, asymptotic/exponential,time-domain (Lyapunov)/frequency-domain (Bode, poles...).

Why stability analysis fo LPV systems is not an easy task?

Let consider x = A(ρ(t))x. Stability analysis is more involved (as for LTV systems) since:• there is a set of solutions for a given x0 (family of systems from ρ variations)• the system may be stable for frozen parameter values and unstable for varying parameters

(as for switching systems)• asymptotic and exponential stability are no more equivalent and cannot be characterized by

the eigenvalues of A(ρ(t)).• In term of design, we will often rely on the notion of quadratic stability (using quadratic

Lyapunov function V (x) = xTPX) which is stronger but easier to check for stability andsimpler to use for control and observer design, see (Wu, PhD 95)

Robust or LPV? (Blanchini,00 & 07)

• Robust analysis and control: dedicated to LTI systems subject to time-varying uncertainties• LPV (or gain-scheduling) analysis and control: dedicated to LTV systems or to linearizations

of non linear systems along the trajectory of ρO. Sename [GIPSA-lab] 39/64


Recall: robust stability with time-invariant uncertainties

This concept is very useful for the stability analysis of uncertain systems.Let us consider an uncertain system

x = A(δ)x

where δ is an parameter vector that belongs to an uncertainty set ∆.

Problem statement

Is the system asymptotically stable for all δ in ∆?

Definition

The considered system is said to be quadratically stable for all uncertainties δ ∈ ∆ if there exists a(single) Lyapunov function V (x) = xTPx with P = PT > 0 s.t

A(δ)TP + PA(δ) < 0, for all δ ∈ ∆ (33)

Computation

For polytopic uncertaities (convex set), i.e. if ρ ∈ Coω1, . . . , ωZ, then, the problem becomesfeasible since it remains to find P = PT > 0 such that:

A(ωi)TP + PA(ωi) < 0, i = 1, . . . , Z



Quadratic stability for time-varying parameters

Let us consider the LPV systemx = A(ρ(t))x

where ρ(t) is an time-varying parameter vector that belongs to an uncertainty set Ω.

Use of a single Lyapunov function

If there exists P = PT > 0 such that:

A(ρ(t))TP + PA(ρ(t)) < 0, ∀ρ(t) ∈ Ω

then the system is stable for arbitrarily fast time-varying uncertainties

Remarks

• Quadractic stability imples exponential stability (Wu, 95)• It is an infinite dimension problem (can be relaxed for polytopic uncertainties)• It could be conservative since stability is checked for any variation of the parameters !

Pay attentation in what follows: LPV system means TIME-VARYING parameters so a polytopicLPV system is not an uncertain polytopic system (in the latter case the coefficient αi of thepolytopic description are constant even if unknown)



Parameter Varying Lyapunov functions

Let consider now a parameter dependent Lyapunov function Vρ(x(t)) = x(t)TP (ρ)x(t) > 0 forevery x 6= 0 and V (0) = 0.

Uncertain systems (ρ is time-invariant)

The uncertain system x = A(ρ)x is exponentially stable if there exists Vρ such that (classicalapproach for polytopic uncertain systems):

A(ρ)TP (ρ) + P (ρ)A(ρ) < 0, ∀ ρ ∈ Ω

LPV systems (ρ is time-varying)

The uncertain system x = A(ρ(t))x is exponentially stable if there exists Vρ such that:

A(ρ)TP (ρ) + P (ρ)A(ρ) +

N∑i=1

ρi∂P (ρ)

∂ρi< 0 ∀ ρ(t) ∈ Ω

which, in addition to bounded parameters, needs to consider rate-bounded parameter variations.Such a condition is more complex since:• It involves the partial differentiation of P• it has to be checked for all ρ(t) ∈ Ω

• It implies to choose a parametrization of P (ρ): from affine to polynomial



Stability of LFR models of LPV systems

See Scherer, Apkarian & Gahinet).In that case, the method corresponds to the generalization of Robust stability tools for LFR modelsto time-varying parameters block ∆(ρ(t)), which is often referred to as the Scaled Small GainTheorem.It is generally assumed, in the LPV framework, that the matrix has a diagonal structure

∆(ρ) = diag(In1 ⊗ ρ1, . . . , Inp ⊗ ρp) (34)

Let us introduce the set of D-scalings defined by

L∆ := L > 0 : L∆(ρ) = ∆(ρ)L, ∀∆ (35)

where n = ||col(n1, . . . , np)||1 and

Lemma

The LFR system (A,B∆, C∆, D∆∆) is asymptotically stable and||L1/2(D∆∆ + C∆(sIn −A)−1B∆)L−1/2||∞ if there exist P = PT 0 and L ∈ L∆ such thatATP + PA PB∆ CT∆L

? −L DT∆∆L? ? −L

≺ 0 (36)



L2 stability of LPV systems (Wu, 95)

Definition

Given a parametrically dependent stable LPV system Σρ = (A(ρ), B(ρ), C(ρ), D(ρ)) for zeroinitial conditions x0. The induced L2 norm is defined as:

||Σρ||i,2 = supρ(t)∈Ω

supw(t) 6=0∈L2

‖y‖2‖u‖2

which is often referred to as (by abuse of langage) the H∞ gain ||Σρ||∞ of the LPV system.

Theorem

A sufficient condition for the L2 stability of system Σρ is the generalized BRL, using parameterdependent Lyapunov functions, i.e assuming |ρi| < νi, ∀i, if there exists P (ρ) > 0, ∀ρ s.t

A(ρ)TP (ρ) + P (ρ)A(ρ) +∑Ni=1 νi

∂P (ρ)

∂ρiP (ρ) B(ρ) C(ρ)T

B(ρ)T P (ρ) −γ I D(ρ)T

C(ρ) D(ρ) −γ I

< 0, ∀i. (37)

then ||Σρ||i,2 ≤ γ


LPV Control & Observation

Outline










LPV Control & Observation

Towards LPV control

The "gain scheduling" approach

System (ρ)Controller (ρ)

Adaptationstrategy

Measured or estimatedParameters

References ControlInputs

Outputs

Externalparameters

Some references

• Modelling, identification : (Bruzelius, Bamieh, Lovera, Toth)• Control (Shamma, Apkarian & Gahinet, Adams, Packard, Beker ...)• Stability, stabilization (Scherer, Wu, Blanchini ...)• Geometric analysis (Bokor & Balas)


LPV Control & Observation The Static State feedback case

State feedback control design: pole placement (1)

Let consider the system Σ(ρ):

x(t) = A(ρ)x(t) +B(ρ)u(t) (38)

y(t) = C(ρ)x(t) +D(ρ)u(t)

The objective is to find a state feedback control law u = −F (ρ)x+G(ρ)r, where r is thereference signal s.t:• the closed-loop system is stable• the output y tracks the reference r (unit closed-loop gain y/r)

The closed-loop system is

x(t) = (A(ρ)−B(ρ)F (ρ))x(t) +B(ρ)G(ρ)r (39)

y(t) = (C(ρ)x(t)−D(ρ)F (ρ))x(t) +D(ρ)G(ρ)r

Then we must consider the following issues:• What controllability property shall we consider?• What parameter dependency should we define for (F (ρ), G(ρ))?• How to choose the poles (dynamics) of the closed-loop system?



State feedback control design: pole placement (2)

Here we can tackle several problems:

Robust SF control:• Design a nominal state feedback control (F,G) (for Σ(ρnominal))• Check quadractic stability and performances of the closed-loop system, with

(A(ρ)−B(ρ)F ) ...

LPV SF control with fixed performances:• Choose some desired poles (p1, p2, . . . , pn) for the CL system• Design F (ρ) such that eig(A(ρ)−B(ρ)F (ρ)) = (p1, p2, . . . , pn). Could be

analytic design or ’frozen-type’.• Check stability (quadractic?) of the CL system when ρ is time-varying

LPV SF control with varying (adaptive) performances:• Choose some desired poles (p1(ρ), p2(ρ), . . . , pn(ρ)) for the CL system• Design F (ρ) such that eig(A(ρ)−B(ρ)F (ρ)) = (p1(ρ), p2(ρ), . . . , pn(ρ)).

Could be analytic design or ’frozen-type’.• Check stability (quadractic?) of the CL system when ρ is time-varying• This latter case allows to schedule the performances according to the

parameter changes, so to handle the trade-off, function of the parametervariations, between closed-loop system transient dynamics and control costlimitations



The H∞ state feedback control problem

Let consider the system:

x(t) = A(ρ)x(t) +B1(ρ)w(t) +B2(ρ)u(t) (40)

y(t) = C(ρ)x(t) +D11(ρ)w(t) +D12(ρ)u(t)

The objective is to find a state feedback control law u = −K(ρ)x s.t:

‖Tyw(s)‖∞ ≤ γ

The method consists in applying the Bounded Real Lemma to the closed-loop system, and thentry to obtain some convex solutions (LMI formulation).Following the framework of quadratic stabilty, this is achieved if and only is there exists a positivedefinite symmetric matrix P (i.e P = PT > 0 s.t (A(ρ)− B2(ρ)K(ρ))T P + P (A(ρ)− B2(ρ)K(ρ)) P B1(ρ) (C(ρ)−D12(ρ)K(ρ))T

? −γ I DT11(ρ)

? ? −γ I

< 0.

(41)



Solution of the state feedback control problem

Use of change of variables

First, left and right multiplication by diag(P−1, In, In), and use Q = P−1 and Y(ρ) = −K(ρ)P−1.It leads to Q > 0 and A(ρ)Q + B2(ρ)Y(ρ) + QAT (ρ) + Y(ρ)TBT

2 (ρ) B1(ρ) QCT (ρ)− YT (ρ)DT12(ρ)

? −γ I DT11(ρ)

? ? −γ I

< 0. (42)

The state feedback controller is then:

K(ρ) = −Y(ρ)Q−1

How to solve (42)?

• It is indeed not an LMI in ρ due to B2(ρ)Y(ρ) and to D12(ρ)Y(ρ), and still infinite dimensional• For polytopic systems, a solution exists if either we choose Y time-invariant or if B2 and D12

are parameter independent• In the general case this requires to impose some parameter dependency on Y(ρ) (affine,

polynomial ...) and to solve the problems trying to linearize it (or using gridding techniques(Balasz, Packard, Seiler)).


LPV Control & Observation The Dynamic Output feedback case

The H∞/LPV control problem

Definition

Find a LPV controller C(ρ) s.t the closed-loop system is stable and for γ∞ > 0, sup ‖z‖2‖w‖2< γ∞,

• Unbounded set of LMIs (Linear Matrix Inequalities) to be solved (ρ ∈ Ω)• Some approaches: polytopic, LFT, gridding. See Arzelier [HDR, 2005], Bruzelius [Thesis,

2004], Apkarian et al. [TAC, 1995]...

A solution: The "polytopic" approach [C. Scherer et al. 1997]

• Problem solved off line for each vertex of a polytope (convex optimisation) (using here asingle Lyapunov function i.e. quadratic stabilization).

• On-line the controller is computed as the convex combination of local linear controllers

C(ρ) =

2N∑k=1

αk(ρ)

[Ac(ωk) Bc(ωk)Cc(ωk) Dc(ωk)

],

2N∑k=1

αk(ρ) = 1 , αk(ρ) > 0

6

-

ρ2

ρ1ρ1

ρ2

ρ2

ρ1

C(ω1)

C(ω2) C(ω4)

C(ω3)

C(ρ)

• Easy implementation !!



LPV control design

Generalized plant

w z

y u

Controller S (ρ)

∑ (ρ)

Problem on standard form

CL (ρ)

w: exagenous input u: control input

z: output to minimize y: measurement

Dynamical LPV generalized plant:

Σ(ρ) :

xzy

=


xwu

(43)

LPV controller structure:

S(ρ) :

[xc

u

]=

[Ac(ρ) Bc(ρ)Cc(ρ) Dc(ρ)

] [xc

y

](44)

LPV closed-loop system:

CL(ρ) :

[ξz

]=

[A(ρ) B(ρ)C(ρ) D(ρ)

] [ξw

](45)



LPV control design

H∞ criteria Apkarian et al. [TAC, 1995]

Stabilize system CL(ρ) (find K > 0) while minimizing γ∞. A(ρ)TK +KA(ρ) KB∞(ρ) C∞(ρ)T

B∞(ρ)TK −γ2∞I D∞(ρ)T

C∞(ρ) D∞(ρ) −I

< 0

Infinite set of LMIs to solve (ρ ∈ Ω) (Ω is convex)

LPV control designs Arzelier [HDR, 2005], Bruzelius [Thesis, 2004]

LFT, Gridding, Polytopic



LPV control design

Polytopic approach

Solve the LMIs at each vertex of the polytope formed by the extremum values of each varyingparameter, with a common K Lyapunov function.

C(ρ) =2N∑k=1

αk(ρ)


]where,

αk(ρ) =

∏Nj=1 |ρj − Cc(ωk)j |∏N

j=1(ρj − ρj),

where Cc(ωk)j = ρj if (ωk)j = ρj

or ρj otherwise.

2N∑k=1

αk(ρ) = 1 , αk(ρ) > 0



LPV control design

Polytopic approach

Solve the LMIs at each vertex of the polytope formed by the extremum values of each varyingparameter, with a common K Lyapunov function.

C(ρ) =2N∑k=1

αk(ρ)


]

6

-

ρ2

ρ1ρ1

ρ2

ρ2

ρ1

C(ω1)

C(ω2) C(ω4)

C(ω3)

C(ρ)



LPV/H∞ control synthesis

Proposition - feasibility (brief) Scherer et al. (1997)

Solve the following problem at each vertices of the parametrized points (illustration with 2parameters):

γ∗ = min γs.t. (47) |ρ1,ρ2s.t. (47) |ρ1,ρ2s.t. (47) |ρ1,ρ2s.t. (47) |ρ1,ρ2

(46)

AX +B2C(ρ1, ρ2) + (?)T (?)T (?)T (?)T

A(ρ1, ρ2) +AT YA+ B(ρ1, ρ2)C2 + (?)T (?)T (?)T

BT1 BT1 Y +DT21B(ρ1, ρ2)T −γI (?)T

C1X +D12C(ρ1, ρ2) C1 D11 −γI

≺ 0

[X II Y

] 0

(47)



LPV/H∞ control synthesis

Proposition - reconstruction (brief) Scherer et al. (1997)

Reconstruct the controllers as,

solve (49) |ρ1,ρ2(49) |ρ1,ρ2(49) |ρ1,ρ2(49) |ρ1,ρ2

(48)

Cc(ρ1, ρ2) = C(ρ1, ρ2)M−T

Bc(ρ1, ρ2) = N−1B(ρ1, ρ2)

Ac(ρ1, ρ2) = N−1(A(ρ1, ρ2)− Y AX −NBc(ρ1, ρ2)C2X

− Y B2Cc(ρ1, ρ2)MT)M−T

(49)

where M and N are defined such that MNT = I −XY which may be chosen by applying asingular value decomposition and a Cholesky factorization.


LPV Control & Observation LPV observer design

Definition LPV observers

Definition

Let consider the LPV system:

x(t) = A(ρ)x(t) +B(ρ)u(t)y(t) = C(ρ)x(t)

(50)

The following LPV state space representation

˙x(t) = A(ρ)x(t) +B(ρ)u(t) + L(ρ)(y(t)− C(ρ)x(t))x0to be defined

(51)

is said to be an observer for (50) if

limt→∞

(x(t)− x(t))→ 0 ∀ρ(t) ∈ Ω

where x(t) ∈ Rn is the estimated state of x(t) and L(ρ) is the n× p observer gain matrix to bedesigned.



Some issues for LPV observer design

The estimated error, e(t) := x(t)− x(t), satisfies:

e(t) = (A− LC)(ρ)e(t) (52)

The two main problems to be handle are then• What observability property shall we consider?• What parameter dependency should we define for L(ρ)?

Quadractic detectability (Wu, 95)

A simple solution is to consider a single Lyapunov function in order to guarantte the quadraticdetectability, i.e:

(A(ρ)− L(ρ)C(ρ))T P + P (A(ρ)− L(ρ)C(ρ)) < 0

Some remarks:• The previous problem can be solved using a polytopic approach only if C(ρ) = C, a constant

matrix• If this is not solvable, one can try using Parameter dependent Lyapunov functions, but the

coupling between L(ρ) and P(ρ) will lead to soved non affine LMIs (a polynomial or a griddingapproach is then needed).



Some issues for LPV observer design (2)

On key issue in observer implementation concerns the knownledge of ρ(t). While previously theresult is valid if ρ(t) is perfectly known, such a following observer description must be used if ρ(t)is estimated:

˙x(t) = A(ρ)x(t) +B(ρ)u(t) + L(ρ)(y(t)− C(ρ)x(t)) (53)

Denoting ∆A = A(ρ)− = A(ρ), ∆B = B(ρ)−B(ρ), ∆C = C(ρ)− C(ρ), and∆L = L(ρ)− L(ρ), this leads for the estimation error equation:

e(t) = (A− LC)(ρ)e(t) + (∆A+ L(ρ).∆C)x+ ∆Bu(t) (54)

If C(ρ) = C and B(ρ) are constant matrices, then we get the uncertain estimated error system

e(t) = (A(ρ)− L(ρ)C)e(t) + ∆Ax(t) (55)

The stability analysis is indeed more involved due to the state vector x (see (Daafouz et al, 2010)for the discrete-time case). Either ∆Ax(t) should be considered as a disturbance, or a stateaugmentation approach is to be used (which has to be done in closed-loop control).

Observer-based control

For control design in the latter case, the following state feedback should be used:

u(t) = −F (ρ)x(t)


Summary of LPV approach interests

Outline











Interest of the LPV approach

LPV is a key tool to the control of complex systems.

Some examples :

Modelling of complex systems (non linear)

• Use of a quasi-LPV representation to include non linearities in a linear state space model(even delays)

• Transformation of constraints (e.g. saturation) into an ’external’ parameter• Modelling of LTV, hybrid (e.g. switching control)

BUT :

A q-LPV system is not equivalent to the non linear one:• stability: ρ = ρ(x(t), t) is assumed to be bounded... so are the state trajectories• controllability: some non controllable modes of a non linear system may vanish according to

the LPV representation• observability: unobservability may occur for some specific parameter variations



Interest of the LPV approach

Some of works using LPV approaches - former PhD students

Gain-scheduled control

• Account for various operating conditions using a variable "equilibrium point": (Gauthier 2007)• Control with real-time performance adaptation using parameter dependent weighting

functions from endogenous or exogenous parameters (Poussot 2008, Do 2011)• Analysis and control of LPV Time-Delay Systems: delay-scheduled control Briat 2008• Control under computation constraints: H∞ variable sampling rate controller with sampling

dependent performances (Robert 2007, Roche 2011, Robert et al., IEEE TCST 2010))

Coordination of several actuators for MIMO systems

• An LPV structure for control allocation Poussot et al. (CEP 2011)• Selection of a specific parameter for the control activation (of each actuator) Poussot et al.

(VSD 2011), Doumiati et al (EJC 2013), Fergani et al (IEEE TVT 2015)

Incorporate fault-(diagnosis, accomodation, tolerant control) properties

• LPV fault-scheduling control: see Sename et al (Systol 2013, ICSTCC 2015).



Some Grenoble PhD students on LPV control

• Waleed Nwesaty, " LPV/H∞ control design of on-board energy management systems for electrical vehicles", PhD GIPSA-lab, Univertisté GrenobleAlpes, 2015.

• Soheib Fergani, "Robust LPV/H∞ MIMO control for vehicle dynamics, PhD GIPSA-lab, Univertisté Grenoble Alpes, 2014.

• Maria Rivas, "Modeling and Control of a Spark Ignited Engine for Euro 6 European Normative", PhD, GIPSA-lab / RENAULT, Grenoble INP, 2012.

• Ahn-Lam Do, "LPV Approach for Semi-active Suspension Control & Joint Improvement of Comfort and Security", PhD, GIPSA-lab, Grenoble INP,2011.

• David Hernandez, "Robust control of hybrid electro-chemical generators", PhD, GIPSA-lab / G2Elab, Grenoble INP, 2011.

• Emilie Roche, "Commande Linéaire à Paramètres Variants discrète à échantillonnage variable : application à un sous-marin autonome", PhD,GIPSA-lab, Grenoble INP, 2011.

• Sébastien Aubouet, "Semi-active SOBEN suspensions modelling and control", PhD, GIPSA-lab / SOBEN, INP Grenoble, 2010.

• Charles Poussot-Vassal, "Robust LPV Multivariable Global Chassis Control", PhD , GIPSA-lab, INP Grenoble, 2008.

• Corentin Briat, "Robust control and observation of LPV time-delay systems", PhD, GIPSA-lab, INP Grenoble, 2008.

• Christophe Gauthier, "Commande multivariable de la pression d’injection dans un moteur Diesel Common Rail", PhD, LAG / DELPHI, Grenoble INP,2007.

• David Robert, "Contribution à l’interaction commande/ordonnacement", PhD, LAG, Grenoble INP, 2007.

• Alessandro ZIN, "Sur la commande robuste de suspensions automobiles en vue du contrôle global de châssis", PhD, LAG / Grenoble INP, 2005.



Some references

• W. Nwesaty, A.I. Bratcu, O. Sename, "Power sources coordination through multivariable LPV/H∞ control with application to multi-source electricvehicles", to appear in IET Control theory and Applications, 2016;

• S. Fergani, O. Sename and L. Dugard, "An LPV/H∞ Integrated Vehicle Dynamic Controller," in IEEE Transactions on Vehicular Technology, vol.65, no. 4, pp. 1880-1889, April 2016.

• O. Sename, "The LPV approach: the key to controlling vehicle dynamics ?", pleanry talk at 19th International Conference on System Theory, Controland Computing (ICSTCC), Cheile Gradistei, Romania, 2015.

• Doumiati, M., Sename, O., Dugard, L., Martinez-Molina, J.-J., Gaspar, P., Szabo, Z., "Integrated vehicle dynamics control via coordination of activefront steering and rear braking", (2013) European Journal of Control, 19 (2), pp. 121-143.;Among the 3 Most Cited Articles published since 2011, extracted from Scopus.

• O. Sename, J. C. Tudon-Martinez and S. Fergani, "LPV methods for fault-tolerant vehicle dynamic control," plenary paper, 2013 Conference onControl and Fault-Tolerant Systems (SysTol), Nice, 2013, pp. 116-130.

• C. Poussot-Vassal, O. Sename, L. Dugard, P. Gaspar, Z. Szabo & J. Bokor, "Attitude and Handling Improvements Through Gain-scheduledSuspensions and Brakes Control", Control Engineering Practice (CEP), Vol. 19(3), March, 2011, pp. 252-263.

• C. Poussot-Vassal, O. Sename, L. Dugard, S.M. Savaresi, "Vehicle Dynamic Stability Improvements Through Gain-Scheduled Steering and BrakingControl", Vehicle System Dynamics (VSD), vol 49, Nb 10, pp 1597-1621, 2011

• Briat, C.; Sename, O., "Design of LPV observers for LPV time-delay systems: an algebraic approach", International Journal of Control, vol 84, nb 9,pp 1533-1542, 2011

• Robert, D., Sename, O., and Simon, D. (2010). An H∞ LPV design for sampling varying controllers: experimentation with a T inverted pendulum.IEEE Transactions on Control Systems Technology, 18(3):741–749.

• C. Briat, O. Sename, and J.-F. Lafay, ""Memory-resilient gain-scheduled state-feedback control of uncertain LTI/LPV systems with time-varying delays"Systems & Control Let., vol. 59, pp. 451-459, 2010.

• C. Briat, O. Sename, and J. Lafay, "Hinf delay-scheduled control of linear systems with time-varying delays," IEEE Transactions in Automatic Control,vol. 42, no. 8, pp. 2255-2260, 2009.

• Poussot-Vassal, C.; Sename, O.; Dugard, L.; Gaspar, P.; Szabo, Z. & Bokor, J. "A New Semi-active Suspension Control Strategy Through LPVTechnique", Control Engineering Practice, 2008, 16, 1519-1534

• Zin A., Sename O., Gaspar P., Dugard L., Bokor J. "Robust LPV - Hinf Control for Active Suspensions with Performance Adaptation in view of GlobalChassis Control", Vehicle System Dynamics, Vol. 46, No. 10, 889-912, October 2008

• Gauthier C., Sename O., Dugard L., Meissonnier G. "An LFT approach to Hinf control design for diesel engine common rail injection system", Oil &Gas Science and Technology, vol 62, nb 4, pp. 513-522 (2007)


robust and lpv control of mimo systems part 3: linear ...o.sename/docs/lpv...robust and lpv control...

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