lecture notes: week 1a ece/mae 7360 optimal and robust control · control systems area fall'03...

Lecture Notes: Week 1a

ECE/MAE 7360Optimal and Robust Control

( Fall 2003 Offering)

Instructor: Dr YangQuan Chen, CSOIS, ECE Dept.,Tel. : (435)797-0148.E-mail: [email protected] or, [email protected]

Control Systems AreaFall'03 Course Offering

ECE/MAE 7360 Optimal and Robust Control. Advanced methods of control systemanalysis and design. Operator approaches to optimal control, including LQR/LQG/LTR,mu-analysis, H-infinity loop shaping and gap metric etc. Prerequisite: ECE 6320 orinstructor approval. (3 cr) (alternate Fall).

Day/Time/Venue : MW 2:30-3:45 PM. EL-112 (Control Lab)

Instructor: Dr YangQuan Chen, CSOIS, ECE Dept., (435)797-0148.

Text: Kemin Zhou, with John Doyle, Essentials of Robust Control, Prentice-Hall, 1998.

Course Description: Robust control is concerned with the problem of designing controlsystems when there is uncertainty about the model of the system to be controlled or whenthere are (possibly uncertain) external disturbances influencing the behavior of thesystem. Optimal control is concerned with the design of control systems to achieve aprescribed performance (e.g., to find a controller for a given linear system that minimizesa quadratic cost function). While optimal control theory was originally derived using thetechniques of calculus of variation, most robust control methodologies have beendeveloped from an operator-theoretic perspective. In this course we will mainly use anoperator approach to study the basic results in robust control that have been developedover the last fifteen years. However, mathematical programming based techniques forsolving optimal control problems will also be briefly covered. This course provides aunified treatment of multivariable control system design for systems subject touncertainty and performance requirements.

Course Topics:

1. Review of multivariable linear control theory and balanced modelrealization/reduction.

2. Signal/system norms and ∞H / 2H spaces and internal stability.3. Performance specification and limitations.4. Modeling uncertainty and robustness.5. LFT and mu synthesis.6. Parameterization of controllers.7. 2H -optimal control (LQR/Kalman Filter /LQG/LTR.)8. ∞H -optimal control (for unstructured perturbations).9. Gap metric10. Solving optimal control problems numerically.

ECE/MAE 7360: Optimal and Robust Control Course Syllabus - Fall 2003

From http://www.ece.usu.edu/academics/graduate_courses.html ***7360. Optimal and Robust Control. Advanced methods of control system analysis and design. Operator approaches to optimal control, including LQR, LQG, and L1 optimization techniques. Robust control theory, including QFT, H-infinity, and interval polynomial approaches. Prerequisite: ECE/MAE 6320 or instructor approval. Also taught as MAE 7360. (3 cr) (Sp) Instructor: YangQuan Chen, Center for Self-Organizing and Intelligent Systems Department of Electrical and Computer Engineering, Utah State University Room EL152; Tel.(435)797-0148, [email protected] Lecture Day/Time/Venue: MW 2:30-3:45 PM. EL-112 (Control Lab) Ofice Hours: MW 1:15-2:30 PM. Text: Kemin Zhou, with John Doyle, Essentials of Robust Control, Prentice-Hall, 1998. References: Will be give by the Instructor via email/web/ftp. Software: (1) MATLAB Control Systems Toolbox (2) MATLAB mu-Synthesis Toolbox (3) RIOTS_95: MATLAB Toolbox for solving general optimal control problems. Course Requirements: Homework 40 points Mid-term take home exam 10 points Focused Individual Study Project/presentation 10 points Design project 40 points Notes: 1. The course will follow the outline on the next page. 2. The course will cover material from most chapters of the text as well as some materials taken

from the instructor's notes. 3. The course will be conducted as follows:

a) There will be lectures by the instructor on most Mondays/Wednesdays b) Homework or project assignments will be given, via e-mail, on the weekly basis

normally on Wednesday. The due is by the end of the next Wednesday. c) There will be a midterm take-home exam. d) For each student, a focused individual study project (FISP) is to be done with a

literature survey and a class presentation. Topics can be chosen by the individual student, subject to the approval of the Instructor.

e) There are totally 4 design projects using MATLAB Simulink/Control Systems Toolbox/mu-Synthesis Toolbox/RIOTS_95 Toolbox. The Instructor will provide a free student edition of RIOTS_95 (worth $99.00) for solving general optimal control problems.

f) There is no final exam.

Course Description: Robust control is concerned with the problem of designing control systems when there is uncertainty about the model of the system to be controlled or when there are (possibly uncertain) external disturbances influencing the behavior of the system. Optimal control is concerned with the design of control systems to achieve a prescribed performance (e.g., to find a controller for a given linear system that minimizes a quadratic cost function). While optimal control theory was originally derived using the techniques of calculus of variation, most robust control methodologies have been developed from an operator-theoretic perspective. In this course we will mainly use an operator approach to study the basic results in robust control that have been developed over the last fifteen years. However, mathematical programming based techniques for solving optimal control problems will also be briefly covered. This course provides a unified treatment of multivariable control system design for systems subject to uncertainty and performance requirements. Course Topics and Approximate Schedule: Course Topics: 1. Review of multivariable linear control theory and balanced model realization/reduction. 2. Signal/system norms and / spaces and internal stability. 3. Performance specification and limitations. 4. Modeling uncertainty and robustness. 5. LFT and mu synthesis. 6. Parameterization of controllers. 7. -optimal control (LQR/Kalman Filter /LQG/LTR.) 8. -optimal control (for unstructured perturbations). 9. Gap metric 10. Solving optimal control problems numerically. wk #

Mondays Wednesdays Homework/Project

1 Aug. 25 – Chapter 1 Introduction/linear algebra

Aug. 27 – Chapter 2,3 Review /linear system theory

HW#1

2 Sept. 1 – No class Labor Day

Sept. 3 – No class (ASME DETC03 Conference)

3 Sept. 8 -- Chapter 4, 5 Norms, Stability

Sept. 10 – Chapter 6 Performance Specs/Limitation

HW#2

4 Sept. 15 – Chapter 6 More on performance limitations.

Sept. 17 – Chapter 7 Balanced Model Reduction

Proj.#1: Inverted Pendulum control revisited

5 Sept. 22 -- Chapter 8 Modeling Uncertainty

Sept. 24 – Chapter 9 LFT: Linear Fractional Transform

HW#3

6 Sept. 29 – Chapter 10 mu and mu synthesis

Oct. 1 – Chapter 10 More on mu

HW#4

7 Oct. 6 – Chapter 11 Controller parameterization (Youla-paramterization)

Oct. 8 – Chapter 12,13 LQR/H2 control

Project#2: Space-shuttle robustness analysis (stability and performance)

8 Oct. 13 – Lecturer's Notes LQG/LTR

Oct. 15 – Chapter 14 H-infinity Control

HW#5

9 Oct. 20 -- Chapter 14

H-infinity Control Oct. 22 – Chapter 14 H-infinity Control

mid-term take home exam

10 Oct. 27 -- Chapter 15 H-infinity Controller order-reduction

Oct. 29 – Chapter 16 H-infinity loop shaping

HW#6

11 Nov. 3 – Chapter 16 H-infinity loop shaping

Nov. 5 – – Chapter 16 H-infinity loop shaping

Project#3: H-infinity control (performance) design of high-maneuvering airplane

12 Nov. 10 – Chapter 17 Gap metric

Nov. 12 – Chapter 17 nu-Gap metric

HW#7

13 Nov. 17 – Instructor's notes Mathematical foundation of RIOTS_95

Nov. 19 – Instructor's notes Sample applications of RIOTS_95

HW#8

14 Nov. 24 – FISP presentations (3 students)

Nov. 26 – No class. Thanksgiving

Project #4: Solving optimal control problems (you define your own OCP!) using RIOTS_95

15 Dec. 1 – FISP presentations (2 students)

Dec. 3 - FISP presentations (2 students)

16 Dec. 8 - No class (IEEE CDC'03 Conference)

Dec. 10 – No class (IEEE CDC'03 Conference)

Email exit interview Due: Dec. 15.

No Final Exam Everything due on Dec. 12, 12:00PM.

Possible Topics for FISP (not limited to the following, students may propose their own topic of interest subject to the Instructor’s approval) 1. 1l - and ∞l -optimal control (for rejection of unknown but bounded disturbances) 2. Structured perturbations, Kharitonov's Theorem 3. Quantitative feedback theory (QFT ). 4. Linear matrix inequalities (LMI ). 5. and many more …

3

Classical control in the 1930’s and 1940’s

Bode, Nyquist, Nichols, . . .

• Feedback amplifier design

• Single input, single output (SISO)

• Frequency domain

• Graphical techniques

• Emphasized design tradeoffs

– Effects of uncertainty

– Nonminimum phase systems

– Performance vs. robustness

Problems with classical control

Overwhelmed by complex systems:

• Highly coupled multiple input, multiple output systems

• Nonlinear systems

• Time-domain performance specifications

4

The origins of modern control theory

Early years

• Wiener (1930’s - 1950’s) Generalized harmonic analysis, cybernetics,

filtering, prediction, smoothing

• Kolmogorov (1940’s) Stochastic processes

• Linear and nonlinear programming (1940’s - )

Optimal control

• Bellman’s Dynamic Programming (1950’s)

• Pontryagin’s Maximum Principle (1950’s)

• Linear optimal control (late 1950’s and 1960’s)

– Kalman Filtering

– Linear-Quadratic (LQ) regulator problem

– Stochastic optimal control (LQG)

5

The diversification of modern control

in the 1960’s and 1970’s

• Applications of Maximum Principle and Optimization

– Zoom maneuver for time-to-climb

– Spacecraft guidance (e.g. Apollo)

– Scheduling, resource management, etc.

• Linear optimal control

• Linear systems theory

– Controllability, observability, realization theory

– Geometric theory, disturbance decoupling

– Pole assignment

– Algebraic systems theory

• Nonlinear extensions

– Nonlinear stability theory, small gain, Lyapunov

– Geometric theory

– Nonlinear filtering

• Extension of LQ theory to infinite-dimensional systems

• Adaptive control

6

Modern control application: Shuttle reentry

The problem is to control the reentry of the shuttle, from orbit to

landing. The modern control approach is to break the problem into two

pieces:

• Trajectory optimization

• Flight control

• Trajectory optimization: tremendous use of modern control principles

– State estimation (filtering) for navigation

– Bang-bang control of thrusters

– Digital autopilot

– Nonlinear optimal trajectory selection

• Flight control: primarily used classical methods with lots of simulation

– Gain scheduled linear designs

– Uncertainty studied with ad-hoc methods

Modern control has had little impact on feedback design because it

neglects fundamental feedback tradeoffs and the role of plant uncertainty.

7

The 1970’s and the return of the frequency domain

Motivated by the inadequacies of modern control, many researchers

returned to the frequency domain for methods for MIMO feedback control.

• British school

– Inverse Nyquist Array

– Characteristic Loci

• Singular values

– MIMO generalization of Bode gain plots

– MIMO generalization of Bode design

– Crude MIMO representations of uncertainty

• Multivariable loopshaping and LQG/LTR

– Attempt to reconcile modern and classical methods

– Popular, but hopelessly flawed

– Too crude a representation of uncertainty

While these methods allowed modern and classical methods to be blended

to handle many MIMO design problems, it became clear that fundamen-

tally new methods needed to be developed to handle complex, uncertain,

interconnected MIMO systems.

8

Postmodern Control

• Mostly for fun. Sick of “modern control,” but wanted a name equally

pretentious and self-absorbed.

• Other possible names are inadequate:

– Robust ( too narrow, sounds too macho)

– Neoclassical (boring, sounds vaguely fascist )

– Cyberpunk ( too nihilistic )

• Analogy with postmodern movement in art, architecture, literature,

social criticism, philosophy of science, feminism, etc. ( talk about

pretentious ).

The tenets of postmodern control theory

• Theories don’t design control systems, engineers do.

• The application of any methodology to real problems will require some

leap of faith on the part of the engineer (and some ad hoc fixes).

• The goal of the theoretician should be to make this leap smaller and

the ad hoc fixes less dominant.

9

Issues in postmodern control theory

• More connection with data

• Modeling

– Flexible signal representation and performance objectives

– Flexible uncertainty representations

– Nonlinear nominal models

– Uncertainty modeling in specific domains

• Analysis

• System Identification

– Nonprobabilistic theory

– System ID with plant uncertainty

– Resolving ambiguity; “uncertainty about uncertainty”

– Attributing residuals to perturbations, not just noise

– Interaction with modeling and system design

• Optimal control and filtering

– H∞ optimal control

– More general optimal control with mixed norms

– Robust performance for complex systems with structured uncer-

tainty

10

Chapter 2: Linear Algebra

• linear subspaces

• eigenvalues and eigenvectors

• matrix inversion formulas

• invariant subspaces

• vector norms and matrix norms

• singular value decomposition

• generalized inverses

• semidefinite matrices

11

Linear Subspaces

• linear combination:

α1x1 + . . . + αkxk, xi ∈ Fn, α ∈ F

span{x1, x2, . . . , xk} := {x = α1x1 + . . .+ αkxk : αi ∈ F}.

• x1, x2, . . . , xk ∈ Fn linearly dependent if there exists α1, . . . , αk ∈ F

not all zero such that α1x2 + . . . + αkxk = 0; otherwise they are

linearly independent.

• {x1, x2, . . . , xk} ∈ S is a basis for S if x1, x2, . . . , xk are linearly

independent and S = span{x1, x2, . . . , xk}.

• {x1, x2, . . . , xk} in Fn are mutually orthogonal if x∗i xj = 0 for all

i 6= j and orthonormal if x∗i xj = δij.

• orthogonal complement of a subspace S ⊂ Fn:

S⊥ := {y ∈ Fn : y∗x = 0 for all x ∈ S}.

• linear transformation

A : Fn 7−→ Fm.

• kernel or null space

KerA = N(A) := {x ∈ Fn : Ax = 0},

and the image or range of A is

ImA = R(A) := {y ∈ Fm : y = Ax, x ∈ Fn}.

Let ai, i = 1, 2, . . . , n denote the columns of a matrix A ∈ Fm×n,

then

ImA = span{a1, a2, . . . , an}.

12

• The rank of a matrix A is defined by

rank(A) = dim(ImA).

rank(A) = rank(A∗). A ∈ Fm×n is full row rank if m ≤ n and

rank(A) = m. A is full column rank if n ≤ m and rank(A) = n.

• unitary matrix U∗U = I = UU∗.

• Let D ∈ Fn×k (n > k) be such that D∗D = I. Then there exists a

matrix D⊥ ∈ Fn×(n−k) such that[D D⊥

]is a unitary matrix.

• Sylvester equation

AX +XB = C

with A ∈ Fn×n, B ∈ Fm×m, and C ∈ Fn×m has a unique solution

X ∈ Fn×m if and only if λi(A) + λj(B) 6= 0, ∀i = 1, 2, . . . , n and

j = 1, 2, . . . ,m.

“Lyapunov Equation”: B = A∗.

• Let A ∈ Fm×n and B ∈ Fn×k. Then

rank (A) + rank(B)− n ≤ rank(AB) ≤ min{rank (A), rank(B)}.

• the trace of A = [aij] ∈ Cn×n

Trace(A) :=n∑i=1aii.

Trace has the following properties:

Trace(αA) = αTrace(A), ∀α ∈ C, A ∈ Cn×n

Trace(A +B) = Trace(A) + Trace(B), ∀A, B ∈ Cn×n

Trace(AB) = Trace(BA), ∀A ∈ Cn×m, B ∈ Cm×n.

13

Eigenvalues and Eigenvectors

• The eigenvalues and eigenvectors of A ∈ Cn×n: λ, x ∈ Cn

Ax = λx

x is a right eigenvector

y is a left eigenvector:

y∗A = λy∗.

• eigenvalues: the roots of det(λI −A).

• the spectral radius: ρ(A) := max1≤i≤n |λi|

• Jordan canonical form: A ∈ Cn×n, ∃ T

A = TJT−1

where

J = diag{J1, J2, . . . , Jl}Ji = diag{Ji1, Ji2, . . . , Jimi

}

Jij =

λi 1

λi 1. . . . . .

λi 1

λi

∈ Cnij×nij

The transformation T has the following form:

T =[T1 T2 . . . Tl

]

Ti =[Ti1 Ti2 . . . Timi

]

Tij =[tij1 tij2 . . . tijnij

]

14

where tij1 are the eigenvectors of A,

Atij1 = λitij1,

and tijk 6= 0 defined by the following linear equations for k ≥ 2

(A− λiI)tijk = tij(k−1)

are called the generalized eigenvectors of A.

A ∈ Rn×n with distinct eigenvalues can be diagonalized:

A[x1 x2 · · · xn

]=

[x1 x2 · · · xn

]

λ1

λ2. . .

λn

.

and has the following spectral decomposition:

A =n∑i=1λixiy

∗i

where yi ∈ Cn is given by

y∗1y∗2...

y∗n

=

[x1 x2 · · · xn

]−1.

• A ∈ Rn×n with real eigenvalue λ ∈ R ⇒ real eigenvector x ∈ Rn.

• A is Hermitian, i.e., A = A∗ ⇒ ∃ unitary U such that A = UΛU∗

and Λ = diag{λ1, λ2, . . . , λn} is real.

15

Matrix Inversion Formulas

• A11 A12

A21 A22

=

I 0

A21A−111 I

A11 0

0 ∆

I A−1

11 A12

0 I

∆ := A22 − A21A

−111 A12

• A11 A12

A21 A22

=

I A12A−122

0 I

∆̂ 0

0 A22

I 0

A−122 A21 I

∆̂ := A11 − A12A

−122 A21

• A11 A12

A21 A22

−1

=

A−111 + A−1

11 A12∆−1A21A−111 −A−1

11 A12∆−1

−∆−1A21A−111 ∆−1

and A11 A12

A21 A22

−1

=

∆̂−1 −∆̂−1A12A−122

−A−122 A21∆̂−1 A−1

22 +A−122 A21∆̂−1A12A

−122

. A11 0

A21 A22

−1

=

A−111 0

−A−122 A21A

−111 A−1

22

A11 A12

0 A22

−1

=

A−111 −A−1

11 A12A−122

0 A−122

.• detA = detA11 det(A22−A21A

−111 A12) = detA22 det(A11−A12A

−122 A21).

In particular, for any B ∈ Cm×n and C ∈ Cn×m, we have

det

Im B

−C In

= det(In + CB) = det(Im +BC)

and for x, y ∈ Cn det(In + xy∗) = 1 + y∗x.

• matrix inversion lemma:

(A11−A12A−122 A21)−1 = A−1

11 +A−111 A12(A22−A21A

−111 A12)−1A21A

−111 .

16

Invariant Subspaces

• a subspace S ⊂ Cn is an A-invariant subspace if Ax ∈ S for every

x ∈ S.

For example, {0}, Cn, and KerA are all A-invariant subspaces.

Let λ and x be an eigenvalue and a corresponding eigenvector of

A ∈ Cn×n. Then S := span{x} is an A-invariant subspace since

Ax = λx ∈ S.

In general, let λ1, . . . , λk (not necessarily distinct) and xi be a set of

eigenvalues and a set of corresponding eigenvectors and the generalized

eigenvectors. Then S = span{x1, . . . , xk} is an A-invariant subspace

provided that all the lower rank generalized eigenvectors are included.

• An A-invariant subspace S ⊂ Cn is called a stable invariant subspace

if all the eigenvalues of A constrained to S have negative real parts.

Stable invariant subspaces are used to compute the stabilizing solu-

tions of the algebraic Riccati equations

• Example

A[x1 x2 x3 x4

]=

[x1 x2 x3 x4

]

λ1 1

λ1

λ3

λ4

with Reλ1 < 0, λ3 < 0, and λ4 > 0. Then it is easy to verify that

S1 = span{x1} S12 = span{x1, x2} S123 = span{x1, x2, x3}S3 = span{x3} S13 = span{x1, x3} S124 = span{x1, x2, x4}S4 = span{x4} S14 = span{x1, x4} S34 = span{x3, x4}

are all A-invariant subspaces. Moreover, S1, S3, S12, S13, and S123 are

stable A-invariant subspaces.

17

However, the subspaces

S2 = span{x2}, S23 = span{x2, x3}

S24 = span{x2, x4}, S234 = span{x2, x3, x4}are not A-invariant subspaces since the lower rank generalized eigen-

vector x1 of x2 is not in these subspaces.

To illustrate, consider the subspace S23. It is an A-invariant subspace

if Ax2 ∈ S23. Since

Ax2 = λx2 + x1,

Ax2 ∈ S23 would require that x1 be a linear combination of x2 and

x3, but this is impossible since x1 is independent of x2 and x3.

18

Vector Norms and Matrix Norms

X a vector space. ‖·‖ is a norm if

(i) ‖x‖ ≥ 0 (positivity);

(ii) ‖x‖ = 0 if and only if x = 0 (positive definiteness);

(iii) ‖αx‖ = |α| ‖x‖, for any scalar α (homogeneity);

(iv) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ (triangle inequality)

for any x ∈ X and y ∈ X .

Let x ∈ Cn. Then we define the vector p-norm of x as

‖x‖p :=

n∑i=1|xi|p

1/p

, for 1 ≤ p ≤ ∞.

In particular, when p = 1, 2,∞ we have

‖x‖1 :=n∑i=1|xi|;

‖x‖2 :=

√√√√√ n∑i=1|xi|2;

‖x‖∞ := max1≤i≤n

|xi|.

the matrix norm induced by a vector p-norm is defined as

‖A‖p := supx6=0

‖Ax‖p‖x‖p

.

In particular, for p = 1, 2,∞, the corresponding induced matrix norm can

be computed as

‖A‖1 = max1≤j≤n

m∑i=1|aij| (column sum) ;

‖A‖2 =√λmax(A∗A) ;

19

‖A‖∞ = max1≤i≤m

n∑j=1|aij| (row sum) .

The Euclidean 2-norm has some very nice properties:

Let x ∈ Fn and y ∈ Fm.

1. Suppose n ≥ m. Then ‖x‖ = ‖y‖ iff there is a matrix U ∈ Fn×m

such that x = Uy and U∗U = I.

2. Suppose n = m. Then |x∗y| ≤ ‖x‖ ‖y‖. Moreover, the equality

holds iff x = αy for some α ∈ F or y = 0.

3. ‖x‖ ≤ ‖y‖ iff there is a matrix ∆ ∈ Fn×m with ‖∆‖ ≤ 1 such that

x = ∆y. Furthermore, ‖x‖ < ‖y‖ iff ‖∆‖ < 1.

4. ‖Ux‖ = ‖x‖ for any appropriately dimensioned unitary matrices U .

Frobenius norm

‖A‖F :=√

Trace(A∗A) =

√√√√√ m∑i=1

n∑j=1|aij|2 .

Let A and B be any matrices with appropriate dimensions. Then

1. ρ(A) ≤ ‖A‖ (This is also true for F norm and any induced matrix

norm).

2. ‖AB‖ ≤ ‖A‖ ‖B‖. In particular, this gives∥∥∥A−1

∥∥∥ ≥ ‖A‖−1 if A is

invertible. (This is also true for any induced matrix norm.)

3. ‖UAV ‖ = ‖A‖, and ‖UAV ‖F = ‖A‖F , for any appropriately di-

mensioned unitary matrices U and V .

4. ‖AB‖F ≤ ‖A‖ ‖B‖F and ‖AB‖F ≤ ‖B‖ ‖A‖F .

20

Singular Value Decomposition

Let A ∈ Fm×n. There exist unitary matrices

U = [u1, u2, . . . , um] ∈ Fm×m

V = [v1, v2, . . . , vn] ∈ Fn×n

such that

A = UΣV ∗, Σ =

Σ1 0

0 0

where

Σ1 =

σ1 0 · · · 0

0 σ2 · · · 0... ... . . . ...

0 0 · · · σp

and

σ1 ≥ σ2 ≥ · · · ≥ σp ≥ 0, p = min{m,n}.Singular values are good measures of the “size” of a matrix

Singular vectors are good indications of strong/weak input or output

directions.

Note that

Avi = σiui

A∗ui = σivi.

A∗Avi = σ2i vi

AA∗ui = σ2i ui.

σ(A) = σmax(A) = σ1 = the largest singular value of A;

and

σ(A) = σmin(A) = σp = the smallest singular value of A .

21

Geometrically, the singular values of a matrix A are precisely the lengths

of the semi-axes of the hyper-ellipsoid E defined by

E = {y : y = Ax, x ∈ Cn, ‖x‖ = 1}.

Thus v1 is the direction in which ‖y‖ is the largest for all ‖x‖ = 1; while

vn is the direction in which ‖y‖ is the smallest for all ‖x‖ = 1.

v1 (vn) is the highest (lowest) gain input direction

u1 (um) is the highest (lowest) gain observing direction

e.g.,

A =

cos θ1 − sin θ1

sin θ1 cos θ1

σ1

σ2

cos θ2 − sin θ2

sin θ2 cos θ2

.A maps a unit disk to an ellipsoid with semi-axes of σ1 and σ2.

alternative definitions:

σ(A) := max‖x‖=1

‖Ax‖

and for the smallest singular value σ of a tall matrix:

σ(A) := min‖x‖=1

‖Ax‖ .

Suppose A and ∆ are square matrices. Then

(i) |σ(A + ∆)− σ(A)| ≤ σ(∆);

(ii) σ(A∆) ≥ σ(A)σ(∆);

(iii) σ(A−1) =1

σ(A)if A is invertible.

22

Some useful properties

Let A ∈ Fm×n and

σ1 ≥ σ2 ≥ · · · ≥ σr > σr+1 = · · · = 0, r ≤ min{m,n}.

Then

1. rank(A) = r;

2. KerA = span{vr+1, . . . , vn} and (KerA)⊥ = span{v1, . . . , vr};

3. ImA = span{u1, . . . , ur} and (ImA)⊥ = span{ur+1, . . . , um};

4. A ∈ Fm×n has a dyadic expansion:

A =r∑i=1σiuiv

∗i = UrΣrV

∗r

whereUr = [u1, . . . , ur], Vr = [v1, . . . , vr], and Σr = diag (σ1, . . . , σr);

5. ‖A‖2F = σ2

1 + σ22 + · · · + σ2

r ;

6. ‖A‖ = σ1;

7. σi(U0AV0) = σi(A), i = 1, . . . , p for any appropriately dimensioned

unitary matrices U0 and V0;

8. Let k < r = rank(A) and Ak :=∑ki=1 σiuiv

∗i , then

minrank(B)≤k

‖A−B‖ = ‖A− Ak‖ = σk+1.

23

Generalized Inverses

Let A ∈ Cm×n. X ∈ Cn×m is a right inverse if AX = I. one of the

right inverses is given by X = A∗(AA∗)−1.

Y A = I then Y is a left inverse of A.

pseudo-inverseor Moore-Penrose inverse A+:

(i) AA+A = A;

(ii) A+AA+ = A+;

(iii) (AA+)∗ = AA+;

(iv) (A+A)∗ = A+A.

pseudo-inverse is unique.

A = BC

B has full column rank and C has full row rank. Then

A+ = C∗(CC∗)−1(B∗B)−1B∗.

or

A = UΣV ∗

with

Σ =

Σr 0

0 0

, Σr > 0.

Then A+ = V Σ+U∗ with

Σ+ =

Σ−1r 0

0 0

.

24

Semidefinite Matrices

• A = A∗ is positive definite (semi-definite) denoted by A > 0 (≥ 0),

if x∗Ax > 0 (≥ 0) for all x 6= 0.

• A ∈ Fn×n and A = A∗ ≥ 0, ∃ B ∈ Fn×r with r ≥ rank(A) such that

A = BB∗.

• Let B ∈ Fm×n and C ∈ Fk×n. Suppose m ≥ k and B∗B = C∗C.

∃ U ∈ Fm×k such that U∗U = I and B = UC.

• square root for a positive semi-definite matrix A, A1/2 = (A1/2)∗ ≥ 0,

by

A = A1/2A1/2.

Clearly, A1/2 can be computed by using spectral decomposition or

SVD: let A = UΛU∗, then

A1/2 = UΛ1/2U∗

where

Λ = diag{λ1, . . . , λn}, Λ1/2 = diag{√λ1, . . . ,

√λn}.

• A = A∗ > 0 and B = B∗ ≥ 0. Then A > B iff ρ(BA−1) < 1.

• Let X = X∗ ≥ 0 be partitioned as

X =

X11 X12

X∗12 X22

.Then KerX22 ⊂ KerX12. Consequently, if X+

22 is the pseudo-inverse

of X22, then Y = X12X+22 solves

Y X22 = X12

and X11 X12

X∗12 X22

=

I X12X+22

0 I

X11 −X12X

+22X

∗12 0

0 X22

I 0

X+22X

∗12 I

.

3

Reference Textbooks

• G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic

Systems, 3rd Edition, Addison Wesley, New York, 1994.

• B. D. O. Anderson and J. B. Moore, Optimal Control, Prentice Hall, London, 1989.

• F. L. Lewis, Applied Optimal Control and Estimation, Prentice Hall, Englewood Cliffs,

New Jersey, 1992.

• A. Saberi, B. M. Chen and P. Sannuti, Loop Transfer Recovery: Analysis and Design,

Springer, London, 1993.

• A. Saberi, P. Sannuti and B. M. Chen, H2 Optimal Control, Prentice Hall, London, 1995.

• B. M. Chen, Robust and H∞ Control, Springer, London, 2000.

Prepared by Ben M. Chen

6

Revision: Basic Concepts


7

What is a control system?

System to be controlledController

Desired

performance:

REFERENCE

INPUT

to the

system

Information

about the

system:

OUTPUT

+–

Difference:

ERROR

Objective: To make the system OUTPUT and the desired REFERENCE as close

as possible, i.e., to make the ERROR as small as possible.

Key Issues: 1) How to describe the system to be controlled? (Modelling)

2) How to design the controller? (Control)

aircraft, missiles,

economic systems,

cars, etc


8

Some Control Systems Examples:

System to be controlledController+

–

OUTPUTINPUTREFERENCE

Economic SystemDesired

PerformanceGovernment

Policies


9

A Live Demonstration on Control of a Coupled-Tank System through Internet Based

Virtual Laboratory Developed by NUS

The objective is to control the flow levels of two coupled tanks. It is a reduced-scale

model of some commonly used chemical plants.


10m

uv

m

bv =+&

Modelling of Some Physical Systems

A simple mechanical system:

By the well-known Newton’s Law of motion: f = m a, where f is the total force applied to an

object with a mass m and a is the acceleration, we have

A cruise-controlsystem

force u

friction

force bx&

x displacement

accelerationx&&

mass

m

m

ux

m

bxxmxbu =+⇔=− &&&&&&

This a 2nd order Ordinary Differential Equation with respect to displacement x. It can be

written as a 1st order ODE with respect to speed v = :x&

← model of the cruise control system, u is input force, v is output.


11

Controller+

–


A cruise-control system:

?+

–

speed vu90 km/h

m

uv

m

bv =+&


12

Basic electrical systems:

v

i

R

resistor

Riv =

capacitor

Cv (t)

i (t)

dt

dvCi =

inductor

Lv (t)

i (t)

dt

diLv =

Kirchhoff’s Voltage Law (KVL):

The sum of voltage drops around anyclose loop in a circuit is 0.

v5

v1

v4

v3

v2

054321 =++++ vvvvv

Kirchhoff’s Current Law (KCL):

The sum of currents entering/leaving anote/closed surface is 0.

i i

i

ii

1

23

4

5i i

i

ii

1

23

4

5

054321 =++++ iiiii


13

Modelling of a simple electrical system:

i

vi

RC vo

To find out relationship between the input (vi) and the output (vo) for the circuit:

dt

dvRCRivR

o==

dt

dvCi o=

By KVL, we have 0io =−+ vvv R

0io

oio =−+=−+ vdt

dvRCvvvv R

iooioo vvvRCvv

dt

dvRC =+⇔=+ & A dynamic model

of the circuit


14

Controller+

–


Control the output voltage of the electrical system:

?+

–

vovi230 Volts

viR

C vo

ioo vvvRC =+&


15

Ordinary Differential Equations

Many real life problems can be modelled as an ODE of the following form:

This is called a 2nd order ODE as the highest order derivative in the equation is 2. The ODE

is said to be homogeneous if u(t) = 0. In fact, many systems can be modelled or

approximated as a 1st order ODE, i.e.,

)()()()( 01 tutyatyaty =++ &&&

An ODE is also called the time-domain model of the system, because it can be seen the above

equations that y(t) and u(t) are functions of time t. The key issue associated with ODE is: how

to find its solution? That is: how to find an explicit expression for y(t) from the given equation?

)()()( 0 tutyaty =+&


16

State Space Representation

Recall that many real life problems can be modelled as an ODE of the following form:

)()()()( 01 tutyatyaty =++ &&&

If we define so-called state variables,

yx

yx

&==

2

1

uxaxauyayayx

xyx

+−−=+−−====

1021012

21

&&&&

&&

[ ]

==

+

−−

=

2

1

1

2

1

102

1 01,1

010

x

xxyu

x

x

aax

x

&

&

We can rewrite these equations in a more compact (matrix) form,

This is called the state space representation of the ODE or the dynamic systems.


17

Laplace Transform and Inverse Laplace Transform

Let us first examine the following time-domain functions:

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

TIME (Second)

Mag

nitu

de

A cosine function with a frequency f = 0.2 Hz.

Note that it has a period T = 5 seconds.

( ) ( ) ( )ttttx πππ 6.1cos8.0sin4.0cos)( +=

What are frequencies of this function?

0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

TIME (Second)

Mag

nitu

deLaplace transform is a tool to convert a time-domain function into a frequency-domain one

in which information about frequencies of the function can be captured. It is often much

easier to solve problems in frequency-domain with the help of Laplace transform.


18

Laplace Transform:

Given a time-domain function f (t), its Laplace transform is defined as follows:

{ } ∫∞

−==0

)()()( dtetftfLsF st

Example 1: Find the Laplace transform of a constant function f (t) = 1.

0)(,1

11

01111

)()( 0

000

>=

⋅−−⋅−=

−−−=−=== ∞−

∞

−∞

−∞

− ∫∫ ssss

es

es

es

dtedtetfsF ststst Re

Example 2: Find the Laplace transform of an exponential function f (t) = e – a t.

( ) ( ) asas

eas

dtedteedtetfsF tastasstatst −>+

=+

−====∞

+−∞

+−∞

−−∞

− ∫∫∫ )(,11

)()(0000

Re


19

Inverse Laplace Transform

Given a frequency-domain function F(s), the inverse Laplace transform is to convert it back

to its original time-domain function f (t).

( )2

2

1

1

1

11

)()(

aste

ase

st

s

sFtf

at

at

+⇔

+⇔

⇔

⇔

⇔

−

−

( )

( ) 22

22

22

22

cos

sin

cos

sin

)()(

bas

asbte

bas

bbte

as

sat

as

aat

sFtf

at

at

+++

⇔

++⇔

+⇔

+⇔

⇔

−

−

Here are some very useful Laplace and inverse Laplace transform pairs:


20

Some useful properties of Laplace transform:

{ } { } { } )()()()()()( 221122112211 sFasFatfLatfLatfatfaL +=+=+

1. Superposition:

2. Differentiation: Assume that f (0) = 0.

{ } { } )()()()(

ssFtfsLtfLdt

tdfL ===

&

{ } { } )()()()( 22

2

2

sFstfLstfLdt

tfdL ===

&&

3. Integration:

( ) { } )(1

)(1

0

sFs

tfLs

dfLt

==

∫ ζζ


21

Re-express ODE Models using Laplace Transform (Transfer Function)

Recall that the mechanical system in the cruise-control problem with m = 1 can be

represented by an ODE:

ubvv =+&

Taking Laplace transform on both sides of the equation, we obtain

{ } { } { } { } { }uLbvLvLuLbvvL =+⇒=+ &&

{ } { } { } )()()( sUsbVssVuLvbLvsL =+⇒=+⇒

( )bssU

sVsUsVbs

+=⇒=+⇒

1

)(

)()()(

This is called the transfer function of the system model

)(sG=


22

Controller+

–


A cruise-control system in frequency domain:

driver? auto?+

–

speed V (s)U (s)R (s)

bssG

+=

1)(


23

In general, a feedback control system can be represented by the following block diagram:

+

U (s)R (s))(sG)(sK

Y (s)

–

E (s)

Given a system represented by G(s) and a reference R(s), the objective of control system

design is to find a control law (or controller) K(s) such that the resulting output Y(s) is as

close to reference R(s) as possible, or the error E(s) = R(s) –Y(s) is as small as possible.

However, many other factors of life have to be carefully considered when dealing with real-

life problems. These factors include:

R (s)

+ U (s))(sG)(sK

Y (s)–

E (s)

disturbances noisesuncertainties

nonlinearities


24

Control Techniques – A Brief View:

There are tons of research published in the literature on how to design control laws for various

purposes. These can be roughly classified as the following:

♦ Classical control: Proportional-integral-derivative (PID) control, developed in 1940s and used

for control of industrial processes. Examples: chemical plants, commercial aeroplanes.

♦ Optimal control: Linear quadratic regulator control, Kalman filter, H2 control, developed in

1960s to achieve certain optimal performance and boomed by NASA Apollo Project.

♦ Robust control: H∞ control, developed in 1980s & 90s to handle systems with uncertainties

and disturbances and with high performances. Example: military systems.

♦ Nonlinear control: Currently hot research topics, developed to handle nonlinear systems

with high performances. Examples: military systems such as aircraft, missiles.

♦ Intelligent control: Knowledge-based control, adaptive control, neural and fuzzy control, etc.,

researched heavily in 1990s, developed to handle systems with unknown models.

Examples: economic systems, social systems, human systems.Prepared by Ben M. Chen

25

Classical Control

Let us examine the following block diagram of control system:

+

U (s)R (s))(sG)(sK

Y (s)

–

E (s)

Recall that the objective of control system design is trying to match the output Y(s) to the

reference R(s). Thus, it is important to find the relationship between them. Recall that

)()()()()(

)( sUsGsYsU

sYsG =⇒=

Similarly, we have , and .)()()( sEsKsU = )()()( sYsRsE −= Thus,

[ ])()()()()()()()()()( sYsRsKsGsEsKsGsUsGsY −===

[ ] )()()()()()(1)()()()()()()( sRsKsGsYsKsGsYsKsGsRsKsGsY =+⇒−=

)()(1)()(

)()(

)(sKsG

sKsG

sR

sYsH

+==⇒ Closed-loop transfer function from R to Y.


26

as

bsG

+=)(

s

ksk

s

kksK ipi

p

+=+=)(

We’ll focus on control system design of some first order systems with a

proportional-integral (PI) controller, . This implies

Thus, the block diagram of the control system can be simplified as,

)()(1)()(

)(sKsG

sKsGsH

+=

R (s) Y (s)

The whole control problem becomes how to choose an appropriate K(s) such that the

resulting H(s) would yield desired properties between R and Y.

ip

ip

bksbkas

bksbk

sKsG

sKsGsH

++++

=+

=)()()(1

)()()( 2

The closed-loop system H(s) is a second order system as its denominator is a polynomial s

of degree 2.


27

Stability of Control Systems

Example 1: Consider a closed-loop system with,

11

)( 2 −=

ssH

R (s) = 1 Y (s)

We have

1

5.0

1

5.0

)1)(1(

1

1

1)()()( 2 +

−−

=−+

=−

==sssss

sRsHsY

Using the Laplace transform table, we obtain

ase at

+⇔− 1

15.0

5.0+

⇔−

se t

15.0

5.0−

⇔s

et

)(5.0)( tt eety −−=

This system is said to be unstable because the

output response y(t) goes to infinity as time t is

getting larger and large. This happens because

the denominator of H(s) has one positive root at

s = 1.0 2 4 6 8 10

0

2000

4000

6000

8000

10000

12000

Time (seconds)

)(ty

&


28

Example 2: Consider a closed-loop system with,

23

1)( 2 ++

=ss

sHR (s) = 1 Y (s)

We have

2

1

1

1

)2)(1(

1

23

1)()()( 2 +

−+

=++

=++

==ssssss

sRsHsY

Using the Laplace transform table, we obtain tt eety 2)( −− −=

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

Time (seconds)

)(ty

This system is said to be stable because

the output response y(t) goes to 0 as time

t is getting larger and large. This happens

because the denominator of H(s) has no

positive roots.


29

We consider a general 2nd order system,

The system is stable if the denominator of the system, i.e., , has no

positive roots. It is unstable if it has positive roots. In particular,

22

2

2)(

nn

n

sssH

ωζωω

++=

R (s) = 0 Y (s)

02 22 =++ nn ss ωζω

Marginally Stable

Unstable

Stable


30

Stability in the State Space Representation

Consider a general linear system characterized by a state space form,

Then,

1. It is stable if and only if all the eigenvalues of A are in the open left-half plane.

2. It is marginally stable if and only if A has eigenvalues are in the closed left-half

plane with some (simple) on the imaginary axis.

3. It is unstable if and only if A has at least one eigenvalue in the right-half plane.

u

u

D

B

x

x

C

A

y

x

++

==

&

L.H.P.

Stable Region

R.H.P.

Unstable Region


31

Lyapunov Stability

Consider a general dynamic system, . If there exists a so-called Lyapunov function

V(x), which satisfies the following conditions:

1. V(x) is continuous in x and V(0) = 0;

2. V(x) > 0 (positive definite);

3. (negative definite),

then we can say that the system is asymptotically stable at x = 0. If in addition,

then we can say that the system is globally asymptotically stable at x = 0. In this case, the

stability is independent of the initial condition x(0).

)(xfx =&

0)()( <∂∂= xfxVxV&

∞→∞→ xxV as,)(


32

Lyapunov Stability for Linear Systems

Consider a linear system, . The system is asymptotically stable (i.e., the eigenvalues

of matrix A are all in the open RHP) if for any given appropriate dimensional real positive

definite matrix Q = QT > 0, there exists a real positive definite solution P = PT > 0 for the

following Lyapunov equation:

Proof. Define a Lyapunov function . Obviously, the first and second conditions

on the previous page are satisfied. Now consider

Hence, the third condition is also satisfied. The result follows.

Note that the condition, Q = QT > 0, can be replaced by Q = QT ≥ 0 and being

detectable.

xAx =&

QPAPA −=+T

xPxxV T=)(

( ) 0)()( <−=+=+=+= QxxxPAPAxxAPxxPxAxPxxPxxV TTTTTTT &&&

2

1, QA


33

Behavior of Second Order Systems with a Step Inputs

Again, consider the following block diagram with a standard 2nd order system,

The behavior of the system is as follows:

22

2

2)(

nn

n

sssH

ωζωω

++=

R (s) = 1/s Y (s)

r = 1

The behavior of the system is

fully characterized by ζ ,

which is called the damping

ratio, and ωn , which is called

the natural frequency.


34

Control System Design with Time-domain Specifications

1% settling time

overshoot

rise time

strt

pM

22

2

2)(

nn

n

sssH

ωζωω

++=

R (s) = 1/s Y (s)

r = 1

tn

rt ω8.1

≅

nst ζω

6.4≅


35

ip

ip

bksbkas

bksbk

sKsG

sKsG

sR

sYsH

++++

=+

==)()()(1

)()()()(

)(2

+

U (s)R (s))(sG)(sK

Y (s)

–

E (s)

PID Design Technique:

s

ksk

s

kksK ipi

p

+=+=)(with and results a closed-loop system:

as

bsG

+=)(

The key issue now is to choose parameters kp and ki such that the above resulting system

has desired properties, such as prescribed settling time and overshoot.

Compare this with the standard 2nd order system:

22

2

2)(

nn

n

sssH

ωζωω

++=

in

pn

bk

bka

=

+=2

2

ω

ζω

bk

b

ak

ni

np

2

2

ω

ζω

=

−=


36

To achieve an overshoot less than 25%, we obtain

from the figure on the right that 4.0>ζ

xTo achieve a settling time of 10 s, we use

767.0106.0

6.46.46.4=

×==⇒=

s

n

n

s tt

ζω

ζω

6.0=ζTo be safe, we choose

Cruise-Control System Design

Recall the model for the cruise-control system, i.e., . Assume that the

mass of the car is 3000 kg and the friction coefficient b = 1. Design a PI controller for it

such that the speed of the car will reach the desired speed 90 km/h in 10 seconds (i.e., the

settling time is 10 s) and the maximum overshoot is less than 25%.

mbsm

sU

sV

+=

1

)()(


37

The transfer function of the cruise-control system,

000333.030001

300013000

11

)()(

)( ===⇒+

=+

== basm

bsm

sU

sYsG

bk

b

ak

ni

np

2

2

ω

ζω

=

−=

Again, using the formulae derived,

17653000/1767.0

27603000/1

3000/1767.06.022

22

===

=−××

=−

=

bk

b

ak

ni

np

ω

ζω

The final cruise-control system:

+–

SpeedReference90 km/h

s

17652760 +


38

Simulation Result:

The resulting

overshoot is

less than 25%

and the settling

time is about 10

seconds.

Thus, our

design goal is

achieved.0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

0

2 0

4 0

6 0

8 0

100

120

Tim e in S e c o n d s

Sp

ee

d i

n km

/h


39

+

r)(sG)(sK

y

–

e

Bode Plots

Consider the following feedback control system,

+

r)()( sGsK

y

–

e

Bode Plots are the the magnitude and phase responses of the open-loop transfer function,

i.e., K(s) G(s), with s being replaced by jω. For example, for the ball and beam system we

considered earlier, we have

( )222

3.27.33.27.31023.037.0)()(

ωω

ωωω −

+=+=+===

=

j

s

s

sssGsK

jsjsjs

o1807.3

3.2tan)()(,

)3.2(7.3)()( 1

2

22

−

=∠

+= − ω

ωωω

ωωω jGjKjGjK


40

10-1

100

101

-20

0

20

40

60

Frequency (rad/sec)

Mag

nitu

de (

dB)

10-1

100

101

-180

-160

-140

-120

-100

-80

Frequency (rad/sec)

Ph

ase

(deg

rees

)

Bode magnitude and phase plots of the ball and beam system:


41

10-1

100

101

-60

-40

-20

0

20

Frequency (rad/sec)

Mag

nitu

de (

dB)

10-1

100

101

-250

-200

-150

-100

-50

0

Frequency (rad/sec)

Ph

ase

(deg

rees

)

gaincrossoverfrequency phase

crossoverfrequency

gainmargin

phasemargin

Gain and phase margins


42-0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

Real Axis

Imag

Axi

s

Nyquist Plot

Instead of separating into magnitude and phase diagrams as in Bode plots, Nyquist plot

maps the open-loop transfer function K(s) G(s) directly onto a complex plane, e.g.,


43

–1

PM

GM1

Gain and phase margins

The gain margin and phase margin can also be found from the Nyquist plot by zooming in

the region in the neighbourhood of the origin.

o180)()(,)()(

1 =∠= ppp

pp

jGjKjGjK

ωωωωω

that such is whereGM

Mathematically,

1)()( such that is where,180)()( PM =+∠= ggggg jGjKjGjK ωωωωω o

Remark: Gain margin is the maximum

additional gain you can apply to the

closed-loop system such that it will still

remain stable. Similarly, phase margin

is the maximum phase you can tolerate

to the closed-loop system such that it

will still remain stable.


44

10-1

100

101

-20

0

20

40

60

Frequency (rad/sec)

Mag

nitu

de (

dB)

10-1

100

101

-180

-160

-140

-120

-100

-80

Frequency (rad/sec)

Ph

ase

(deg

rees

)

Example: Gain and phase margins of the ball and beam system: PM = 58°, GM = ∞


lecture notes: week 1a ece/mae 7360 optimal and robust control · control systems area fall'03...

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