Notes on Linear Robust Control - Robust Control Systems I | MEM 633, Study notes of Mechanical Engineering

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Notes on Linear Robust

Control

MEM 633

October 2, 2002

Professor Harry G. Kwatny

Office: 3-151A

[email protected]

http://www.pages.drexel.edu/faculty/hgk

Contents

• 1 Introduction to the Robust Control Problem.........................................................
• 2 State Space Models
  • 2.1 Solutions of Linear Systems
  • 2.2 The Matrix Exponential
  • 2.3 Controllability
    • More on Invariance
  • 2.4 Observability.......................................................................................................
    • More on Invariance
  • 2.5 Kalman Decomposition
  • 2.6 Thorp-Morse Form..............................................................................................
  • 2.7 Zeros
• 3 Nominal Controller Design: State Space Perspective..........................................
  • 3.1 State Feedback
    • Pole Placement..........................................................................................................
    • The Linear Regulator Problem..................................................................................
  • 3.2 Observers & the Separation Principle...............................................................
  • 3.3 Disturbance Rejection.......................................................................................
• 4 Transfer Function Models......................................................................................
  • 4.1 State Space to Transfer Function
  • 4.2 Frequency Response
  • 4.3 Poles & Zeros of Transfer Functions
  • 4.4 Realizations.......................................................................................................
• 5 Closed Loop Transfer Functions
  • 5.1 Well Posedness
  • 5.2 Closed Loop Transfer Functions.......................................................................
    • Output
    • Error
    • Control
• 6 Performance in the Frequency Domain................................................................
  • 6.1 Sensitivity Functions.........................................................................................
    • A fundamental tradeoff
  • 6.2 Sensitivity Peaks
  • 6.3 Bandwidth
  • 6.4 Limits on Performance......................................................................................
• 7 Nominal Controller Design: Frequency Domain Perspective.............................
  • 7.1Equation Section 1 Full State Feedback Controllers
    • The Quadratic Regulator Problem
    • Min-Max Control
    • Solving the Riccati Equation
  • 7.2 Output Feedback Controllers
    • The Classical H 2 Problem – LQG............................................................................
    • The Modern Paradigm
    • Solution Summary
• 8 Robust Stability & Nyquist Analysis.....................................................................
  • 8.1 SISO Nyquist Analysis
    • Guaranteed Gain and Phase Margin
  • 8.2 MIMO Nyquist Analysis...................................................................................
  • 8.3 M ∆ - Structure..................................................................................................
  • 8.4 Small Gain Theorem
  • 8.5 Robust Stability of the M ∆ - Structure
• 9 Robust Performance
• 10 Some Complex Variable Concepts
  • 10.1 Analytic Functions
    • Residue Theorem
    • Poisson Integrals
    • Bode Waterbed Formula
  • 10.2 Parseval’s Theorem...........................................................................................
• 11 Normed Linear Spaces
• 11.1 Norms and Normed Linear Spaces
  • Examples of normed linear spaces:...........................................................................
• 11.2 System Norms/ Induced Norms
  • Transfer matrix norms...............................................................................................

1 Introduction to the Robust Control Problem

Definition: A state is reachable from x 0 if there exists a finite time t > 0 and a piecewise continuous control u such that.

x 1 ∈ R^ n x(t;x 0 (^) ,u) = x 1 Rx (^) 0 denotes the set of states reachable from x 0.

Let us make a few preliminary observations.

If x 1 is reachable from x 0 in some time , it is reachable in every time t. To see this

simply rescale s :

t 1 (^) > 0

(^11) (^1 11 )

(^1 11 )

0 0

0

st t

st t

t (^) t (^) A(t - ) A(t -s) (^) st (^) t stt

t (^) A(t- ) (^) A(t -t) st (^) t stt

e Bu(s)ds e Bu( )d

e Be u( )d

Thus, we have the replacement u s ( ) → e A(t -t)^1^ u( st t 1 ).

Notice that x 1 is reachable from x 0 if and only if x 1 − e^ At x 0 is reachable from the origin

for 0 < t < ∞, viz

1 0 1 0 0 0

t t x = e At^ x + e A(t-s)^ Bu(s)ds ⇔ x − e At^ x = e A(t-s)Bu(s)ds

As a result, we focus on characterizing the set of states reachable from the origin. Let

U denote the linear vector space of control functions u ( ), τ τ ∈ [0, t 1 ], and X ≅ Rn the

space of states x ( ) t 1. The inner product on U is the usual

1 ˆ , 0 ˆ ( ) (

t (^) T

u u = ∫ u τ u τ) d τ

The mapping A : U →X defined by

(^1) ( 1 ) ( ) 1 0 ( )

t (^) A t

x t = ∫ e −^ τ^ Bu τ d τ (1.1)

defines the state reached from the origin at time when the control u is applied on the

time interval [.

t 1 , t 1 ]

A state x 1 is reachable from the origin over the time interval [0 if and only if the

relation

, t 1 ] A ( ) u = x 1 has a solution u t ( ). Such a solution exists if and only if x 1 (^) ∈ Im A. It is

more convenient to apply the equivalent condition x 1 (^) ∈ Im AA * because.

So let’s us prove this result.

AA^ *^ : X →X

Lemma 1: There exists a solution u of A ( ) u = x 1 if and only if x 1 (^) ∈ Im AA *.

Lemma 1

Proof: Sufficiency ( x 1 (^) ∈ Im AA * ⇒ x 1 ∈ Im A ) is obvious. To prove necessity, assume

x 1 (^) = A u ( ) and Im * x 1 ∉ AA. Since Im

X = A ⊕ker A *,

x 1 has a component in ker A *. It

follows that there exists an x 2 such that AA x * 2 = 0 and x x 1 T 2 (^) ≠ 0. Now,

---

2 2 0 2 0 2 x AA xT = ⇒ A x = ⇒ A x = 0

Then

x x 1 T 2 (^) = Au x , 2 (^) = u A x ,^ * 2 X U =^0

contradicting the condition x x 1 T 2 (^) ≠ 0.

Qed

To use this result, we need to calculate the adjoint mapping A *^ : X →U. It is defined by

x , Au = A x u^ * , X U

Thus,

( )

( )

(^1 1 )

(^1 )

• ( ) 0 0 ( ) 0

T

t T (^) T t A t s

t (^) T A t t T

A x u d x e Bu s ds

B e x u t d

−

∫ t

From this we identify

A *^ ( ) x = B eT AT^ (^ t^1^ − t ) x (1.2)

so that

*^1 (^1 )^ (^1 ) ( ) (^0)

t (^) A t (^) T AT t

AA x = ^ ∫ e −^ τ^ BB e −τ^ d τ x

Consequently, in view of , we have the following result.

Proposition: The set of states reachable from the origin over the time interval [0 , t 1 ]is

*^1 ( 1 ) ( 1 ) Im Im (^0)

AA = ^ t^ e A t^ −^ τ^ BB eT AT^ t −τ^ d τ

Definition: The matrix

(^1) ( 1 ) ( 1 ) ( ) (^1 )

t (^) A t T AT t

GC t = ∫ e −^ τ^ BB e −τ^ d τ

Since GC ( ) t is symmetric ( ⇒ X = Im GC ⊕ker GC ), this is equivalent to

A ker GC ( ), t 0

⊥ B = t >

First show, x ker GC ( ) t x A

⊥ ∈ ⇒ ∈ B. If x ∈ ker GC ( ) t , then x G xT C = 0 so that

( )^2 0 0,^0

t (^) T A t

∫^ x^ e^ −^ τ^ B^ d τ=^ <τ< t

Therefore

xT e AsB = 0, 0 < s < t

Expanding e As and comparing coefficients leads to

xT A Bi = 0, i = 0, … , n − 1

Consequently, xT [ B AB … An −^1 B ] = 0 which implies that x is orthogonal to A B , i.e.

x ∈ A B ⊥.

Now show x ker GC ( ) t x A

⊥ ∈ ⇐ ∈ B. But x A

⊥ ∈ B

0

implies so

reversing the above steps leads to. This is true only if

x T^ [ B AB … An −^1 B ]=

ker (^) C ( )

T x G xC = x ∈ G t.

Qed

Theorem: The system or the matrix pair ( A , B ) is (completely) controllable if and only if R 0 = R^ n , or equivalently rank[ B AB … An^ −^1 B ]= n

Proof:

Qed

We wish to emphasize the geometric aspects of controllability and observability.

To do so fully requires the concept of an invariant subspace.

Definition: A subspace V ⊆ Rn is invariant with respect to A if AV ⊆ V Clearly, every eigenvector defines a one-dimensional invariant subspace. Furthermore,

the set of all vectors h satisfying

Ah = λ h

is called the eigenspace of A associated with the eigenvalue λ. Every eigenspace of A is

an invariant subspace as is every subspace that can be constructed as the sum of

eigenspaces. Perhaps less obvious is the fact that every invariant subspace is the direct

sum of eigenspaces.

) is A -invariant, i.e.,. In fact is the smallest A -invariant

subspace of

R ( 0 AR ( 0 )⊆ R ( 0 ) R ( 0 )

R n containing B. Moreover, if there exists a system of

coordinates in which the state equations take the form:

dim R ( 0 ) = n 1

x x

A A

A

x x

B

(^1) u 2

11 12 22

1 2

1 0 0

L NM

O QP^

= L NM^

O QP

L NM

O QP

• L NM

O QP^

, x 1 ∈Rn^1 , x 2 ∈Rn-n^1

such that the pair ( A 11 , B 1 ) is completely controllable, i.e., A 11 (^) B 1 = Rn , and in fact x 1

are coordinates in R ( ) 0. Hence the restriction of the system to R ( ) 0 ( x 2 =0) results in a

controllable system. Thus, we refer to R ( ) 0 as the controllable subspace.

More on Invariance

Recall that application of the linear control u = K x + v , results in the closed loop system

x^ ^ = ( A + BK ) x + Bv This motivates the following definition:

Definition: A subspace V ⊆ Rn^ is (A,B)–invariant if there exists a state feedback matrix K such that ( A + BK ) V ⊆ V

Now , the following theorem can be established.

Theorem: V ⊆ R n is ( A , B )–invariant if and only if AV ⊆ V + B

2.4 Observability

We briefly review some basic concepts and results for linear autonomous systems

x  = Ax + Bu y = Cx

where. Recall that given an initial state x (0) = x 0 and a control

u ( t ), t > 0, the corresponding trajectory is define by the variations of parameters formula

x ∈ Rn^ , u ∈ Rm , y ∈ R^ p

such that the pair ( C 1 , A 11 ) is completely observable, and x 2 are coordinates in I ( ) 0. We

call I (0)= Nthe unobservable subspace.

More on Invariance

Similar results to those established for controllability can be established for observability.

Definition: A subspace V is (C,A)–invariant if there exists a matrix F such that

⊆ R^ n

( A^ +^ FC V )^ ⊆ V

Note that (A+FC) is the closed loop matrix resulting from output injection.

Theorem: V ⊆ Rn is (C,A)–invariant if and only if

A ( V ∩ ker ( C ))⊆ V

2.5 Kalman Decomposition

2.6 Thorp-Morse Form

2.7 Zeros

Recall for a linear system the output is related to the input and initial state by

Y s ( ) = C sI − A −^1 x (^) 0 + (^) n C sI − A −^1 B + D U s s ( )

G s C sI A B D k n s d s

k s^ z^ s^ z s p s p

m n

= − + = = −^ −

− (^1 ) 1

n s

Y s CN^ s^ x d s

k n s d s s

A partial fraction expansion of the second term shows that this equation can be written in

the form

( ) ( )^0 ( ) ( ) ( )

Y s CN s x^ n s c d s d s s

λ

= ^ + +

 (^) , with c G ( )

λ=^ λ

Moreover, it can be shown that it is always possible to choose x 0 so that the term in

brackets vanishes. If x 0 is so chosen, then

Y s ( ) c s

λ

Moreover, if λ is a system zero, c λ= G ( λ) = 0

0

. Thus, Y s. In summary,

if λ is a system zero, there exists

( ) = 0 ⇒ y t ( ) = 0

x such that x ( ) t = x 0 and u t ( ) = e^ λ t results in.

This, is the essential property of SISO system zeros that we intend to generalize.

y t ( ) = 0

Consider the MIMO system

x  = Ax + Bu y = Cx + Du

Suppose u t ( ) = ge^ λ t , g ∈ Rm. We ask if there exists a solution of the form x ( ) t = x e 0 λ t ,

0 x ∈ R^ n such that y t () ≡0.The assumed solution must satisfy

0 0 (^00)

t t t t

x e Ax e Bge^ t Cx e Dge

λ λ λ λ λ

This leads to

[ ] (^0) 0

I A x Bg Cx Dg

− λ − + =

• =

or

(^0 0) (1.7) I A B x C D g

^ λ −^   −

This represents n + p equations in n + m unknowns. Suppose

rank

I A B

r C D

^ λ − 

Let us consider the following cases. p < m , equation (1.7) always has nontrivial solutions. If the system matrix has maximum

rank, r = n + p , there are m − p independent solutions.

m < p , and r ≥ n + m there are no nontrivial solutions of (1.7). r < n + min( m p , )there are nontrivial solutions.

3 Nominal Controller Design: State Space Perspective

3.1 State Feedback

Pole Placement..........................................................................................................

The Linear Regulator Problem..................................................................................

3.2 Observers & the Separation Principle

3.3 Disturbance Rejection

4 Transfer Function Models

4.1 State Space to Transfer Function

4.2 Frequency Response

4.3 Poles & Zeros of Transfer Functions

4.4 Realizations

5 Closed Loop Transfer Functions

5.1 Well Posedness

5.2 Closed Loop Transfer Functions

K G

r u y

d 1

d 2

Y G U D

U K R Y D

E R Y

1 2

b g b g

Output

Y s I G s K s G s K s R s I G s K s G s D s I G s K s G s K s D s

− − −

1 1 1 1 2

Error

E s I G s K s R s I G s K s G s D s I G s K s G s K s D s

− − −

1 1 1 1 2

Control

U s K s I G s K s R s K s I G s K s G s D s K s I G s K s I G s K s D s

− − −

1 1 1 1 l 2 q 2

6 Performance in the Frequency Domain

6.1 Sensitivity Functions

E s ( ) = I + L −^1 R s ( ) − I + L −^1 GD 1 ( ) s + I + L −^1 LD (^) 2 ( ) s , where L : = GK

Sensitivity function: S := I + L −^1

Complementary sensitivity function: T := I + L −^1 L

L

Consider a scalar system in which is the open loop transfer function and

is the closed loop transfer function. Then compute the (relative) variation

of the closed loop with respect to (relative) variation of the open loop transfer function:

L = GK

T = [ 1 + L ]−^1

dT T dL L

dT dL

L

T

L L L L

L L

L L

L S

− − − −

[ ] [ ]

[ ]

[ ]

[ ]

2 1 1 1

m r

This is Bode’s original reason for the terminology ‘sensitivity function’ for S.

A fundamental tradeoff

Note that I + L −^1 + I + L −^1 L = I

S + T = I T = I + L −^1 L = I + L −^1 −^1 = L I + L −^1

G I + KG −^1 = I + GK −^1 G

GK I + GK −^1 = G I + KG −^1 K = I + GK −^1 GK

E s ( ) = S s R s ( ) ( ) − S s GD ( ) 1 (^) ( ) s + T s D ( ) 2 ( ) s

U s K s I G s K s R s K s I G s K s G s D s K s I G s K s I G s K s D s

− − −

1 1 1 1 l 2 q 2