LinearAlgebra - SHUYOU YU

Robust Control ��

��

��

v3.2, 2013�3��

�� Linear Algebra

21

• Linear subspaces

• Eigenvalues and Eigenvectors

• Matrix inversion formulas

• Invariant subspaces

• Vector norms and matrix norms


22

Vector norm

A real-valued function ‖ · ‖ defined on a vector space X is said to be a

norm if it satisfies

• ‖x‖ ≥ 0 (positivity);

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

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

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

for any x, y ∈ X

�� , ��


23

p-norm: x ∈ Cn

‖x‖p :=(

n∑i=1

|xi|p)1/p

, 1 ≤ p ≤ ∞

1− norm: ‖x‖1 :=n∑

i=1

|xi|

2− norm: ‖x‖2 :=√√√√ n∑

i=1

|xi|2

∞− norm: ‖x‖∞ := max1≤i≤n

|xi|


24

Matrix norms A = [aij] ∈ Cm×n

induced p-norms: matrix norm induced by a vector p-norm

‖A‖p := supx �=0

‖Ax‖p‖x‖p , 1 ≤ p ≤ ∞

Interpretation: input u = x, output y = Ax, ‖A‖p gives the

input/output amplification gain

example: 2-norm: ‖A‖2 :=√

λmax(A∗A)

Frobenius norm: is not an induced norm

‖A‖F :=√

trace(A∗A) =

√√√√ m∑i=1

n∑j=1

|aij|2

Matlab commands: norm, normest


25

Unitary matrix U ∈ Fm×n

A unitary matrix is an complex matrix satisfying

U∗U = UU∗ = I

Some properties of unitary matrices:

• U−1 = U∗

• < Ux, Uy >=< x, y >

• the columns of U form an orthonormal basis of Cn

• the rows of U form an orthonormal basis of Cn

• U preserves length: ‖Ux‖2 = ‖x‖2• U is a normal matrix with eigenvalues lying on the unit circle

Unitary matrices ��/�� , ��

�(length, distance, angle, energy)


26

Singular value decomposition A ∈ Fm×n

There exist unitary matrices U = [u1, u2, . . . , um] ∈ Fm×m and V =

[v1, v2, . . . , vn] ∈ Fn×n such that

A = UΣV ∗, Σ =

[Σ1 0

0 0

], Σ1 =

⎡⎢⎢⎢⎢⎣

σ1 0 · · · 0

0 σ2 0 · · · 0...

.... . .

...

0 0 0 σp

⎤⎥⎥⎥⎥⎦

σ1 ≥ σ2 ≥ · · · ≥ σp, p = minm,n

Hence

A =

p∑i=1

σiuiv∗i

σi is the ith singular value of A

�� Linear Algebra — Singular value decomposition

27

ui and vj are the ith left singular vector and the jth right singular vector

Avi = σiui

A∗ui = σivi

Sketch of proof

Av1 = UΣV ∗v1 = UΣ

⎡⎢⎢⎢⎢⎣

v∗1v∗2...

v∗n

⎤⎥⎥⎥⎥⎦ v1 = UΣ

⎡⎢⎢⎢⎢⎣

1

0...

0

⎤⎥⎥⎥⎥⎦ = U

⎡⎢⎢⎢⎢⎣

σ1

0...

0

⎤⎥⎥⎥⎥⎦

=[u1 u2 · · · um

]⎡⎢⎢⎢⎢⎣

σ1

0...

0

⎤⎥⎥⎥⎥⎦ = σ1u1


28

σ2i is an eigenvalue of AA∗ or A∗A

A = UΣV ∗, AA∗ = UΣV ∗V ΣU∗ = UΣ2U∗

eig(AA∗) = eig(UΣ2U∗) = {σ21, σ

22, · · · , σ2

p}ui is an eigenvector of AA∗ and vi is an eigenvector of A∗A

A∗Avi = A∗σiui = σiA∗ui = σ2

i vi

AA∗ui = Aσivi = σiAvi = σ2i ui

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

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

If define y = Ax =⇒ an interpretation of singular values


29

��v1 is the direction in which ‖y‖ is largest for all ‖x‖ = 1,

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

Example 1: A = UΣV ∗

A =

(cos θ1 − sin θ1

sin θ1 cos θ1

)(σ1 0

0 σ2

)(cos θ2 − sin θ2

sin θ2 cos θ2

)

A maps a unit circle to an ellipsoid with semiaxes of σ1 and σ2


30

��:

��: v1 is the highest gain input (control) direction, vp is

the lowest gain input direction Av1 = σ1u1, Avp = σpup

��: u1 is the highest gain output (observing) direction, up

is the lowest gain output direction A∗u1 = σ1v1, A∗up = σpvp

σ(A) := max‖x‖=1

‖Ax‖ = maxx �=0

‖Ax‖‖x‖

σ(A) := min‖x‖=1

‖Ax‖ = minx�=0

‖Ax‖‖x‖


31

Example 2: �� A ∈ Rm×n ��

A = UΣV ∗, Σ =

[Σ1 0

0 0

],

Σ1 = diag(

σ1 · · · σl σl+1 · · · σp,), σl+1 � σl

Σ1 ∈ Rl×l ��=⇒ ��

Example 3: Model reduction A ∈ Fm×n ��

�� Linear Systems – ��

32

[x

y

]=

[A B

C D

][x

u

]

Controllability: The dynamical system or the pair (A,B) is

said to be controllable if, for any initial state x(0) = x0, t1 > 0

and final state x1, there exists a (piecewise continuous) input

u(·) such that the solution of the system satisfies x(t1) = x1.

(A,B) �� F �� A + BF ��

��

Stabilizability: A ��

(A,B) �� F �� A+BF ��

�� Linear Systems – ��

33

[x

y

]=

[A B

C D

][x

u

]

Observability: The dynamical system or the pair (C,A) is

said to be observable if, for any t1 > 0, the initial state

x(0) = x0 can be determined from the time history of the

input u(t) and the output y(t) in the interval [0, t1].

(C,A) �� L �� A+LC ��

Detectability: A ��

(C,A) �� L �� A+ LC ��

�� Linear Systems: Pole-Zero Cancelation

34

y u11s

11

ss

�� λ1 = 1 � λ2 = −1 ��: et � e−t

G(s) =1

s− 1

s− 1

s+ 1=

1

s+ 1; �� et ��

�� x1 = y � x2 = u− ξ[x1

x2

]=

[1 −1

0 −1

][x1

x2

]+

[1

2

]u, y =

[1 0

] [ x1

x2

]

��: (A,B) ��, (C,A) ��

��: �� et � u ��( ξ ��, � ξ ��

��), �� y ��


35

y u11s

11

ss

�� λ1 = 1 λ2 = −1 ��: et e−t

G(s) =s− 1

s+ 1

1

s− 1=

1

s+ 1; �� et �

�� x1 = y x2 = ξ[x1

x2

]=

[−1 1

0 1

][x1

x2

]+

[1

1

]u, y =

[1 0

] [ x1

x2

]

��: (A,B) ��, (C,A) ��

��: �� et � u ��, �� y �

�(�� ξ ��, � ξ ��)


36

G(s) =

[1

s+11

s+22

s+21

s+1

], K(s) =

[s+2s−√

2− s+1

s−√2

0 1

]

K(s)G(s) =

[− s+

√2

(s+1)(s+2)0

2s+2

1s+1

]

G(s) must have an unstable zero√2 that cancels the unstable pole

of K(s) !!!

MIMO system poles and zeros

�� H2 and H∞ Spaces

37

��: ��; ��(�� )

Let f(s) be a complex-valued function defined on S ⊂ C

f(s) is said to be analytic at a point z0 in S if it is differentiable

at z0 and also at each point in some neighborhood of z0

Examples

• G(s) = 1s+1

is analytic on jR and in the open right-half plane

• G(s) = 1sis not analytic on jR

• G(s) = 1s−1

is analytic in the open left-half plane

• �� G(s) � ��

��: s = σ+ jω ��, s�, jω ��(��), σ > 0 ��

��(��), σ < 0 ��(��)

�� H2 and H∞ Spaces — ��

38

�� s ��

• � jR ��: �� 1s�

• � jR �� rhp ��: �� 1s� 1

s−a�(a > 0)

• �� lhp ��: �� 1s+a�(a > 0)

�� =⇒

• � jR ��: �� 1s�

• � jR �� rhp ��: ��

� 1s� 1

s−a�(a > 0)

• �� lhp ��: �� 1s+a

�(a > 0)

�� H2 and H∞ Spaces — ��(��)��

39

�� =⇒ ��(��)��

• �� L2 ��(��)

‖G‖2 :=√

< G,G > =

√1

2π

∫ +∞

−∞trace{G∗(jω)G(jω)}dω

=⇒

- � jR ��, �� L2 ��

=⇒ L2 ��

- � jR �� rhp ��, �� L2 ��

=⇒ H2 ��: a subspace of L2

- �� lhp ��, �� L2 ��

=⇒ H⊥2 ��: the orthogonal complement of H2 in L2

�� H2 and H∞ Spaces — ��(��)��

40

�� =⇒ ��(��)��

• �� L∞ ��

‖G‖∞ := ess supω∈R

σ[G(jω)]

=⇒

- � jR ��, �� L∞ ��

=⇒ L∞ ��

- � jR �� rhp ��, �� L∞ ��

=⇒ H∞ ��: a subspace of L∞- �� lhp ��, �� L∞ ��

=⇒ H⊥∞ ��: the orthogonal complement of H∞ in L∞

�� H2 and H∞ Spaces — L2 ��L2 ��

41

H2

L2(jR)

H⊥2L2(−∞, 0]

L2(−∞,∞)

L2[0,∞)

P−

P+

Laplace Transform

Inverse Transform

Inverse Transform

Laplace Transform

Inverse Transform

Laplace Transform

P−

P+

� �

��

� �

� �

� �

�� H2 and H∞ Spaces — H2 ��

42

The H2 norm

‖G‖2 :=√

1

2π

∫ +∞

−∞trace{G∗(jω)G(jω)}dω

=

√1

2π

∫ +∞

−∞trace{G(jω)G∗(jω)}dω

= ‖g‖2

=

√∫ +∞

−∞trace{g∗(t)g(t)}dt =

√∫ +∞

−∞trace{g(t)g∗(t)}dt

trace(A) :=n∑

i=1

aii


43

Computing H2 norms with the observability Gramian

D = 0 and A stable

G(s) =

[A B

C 0

]:= C(SI − A)−1B

g(t) = L−1(G(s)) =

{CeAtB, t ≥ 0

0, t < 0

‖G‖22 =∫ ∞

0

trace(gT (t)g(t)

)dt =

∫ ∞

0

trace(BT eA

T tCTCeAtB)dt

= trace

(BT

∫ ∞

0

eAT tCTCeAtdt

)B = trace

(BTQB

)Observability Gramian Q =

∫∞0

eAT tCTCeAtdt satisfies

ATQ+QA+ CCT = 0


44

Computing H2 norms with the controllability Gramian

D = 0 and A stable

g(t) = L−1(G(s)) =

{CeAtB, t ≥ 0

0, t < 0

‖G‖22 =∫ ∞

0

trace(g(t)gT (t)

)dt =

∫ ∞

0

trace(CeAtBBT eA

T tCT)dt

= trace

(C

∫ ∞

0

eAT tBBT eAtdt

)CT = trace

(CPCT

)Controllability Gramian P =

∫∞0

eAT tBBT eAtdt satisfies

AP + PAT +BBT = 0


45

Computing H2 norms with the controllability Gramian

D = 0 and A unstable

Decompose G(s) into stable and antistable parts

G(s) = Gs(s) +Gu(s)

then

‖G(s)‖22 = ‖Gs(s)‖22 + ‖Gu(s)‖22where (note that Gu(−s) is stable!)

‖Gu(s)‖22 = ‖Gu(−s)‖22 = ‖GTu (−s)‖22


46

Computing H2 norms: Example 4.1

G =

[3(s+3)

(s−1)(s+2)2

s−1s+1

(s+2)(s+3)1

s−4

]= Gs +Gu

sdecomp(G) ⇒ Gs =

⎡⎢⎢⎢⎢⎣

−2 0 −1 0

0 −3 2 0

1 0 0 0

1 1 0 0

⎤⎥⎥⎥⎥⎦ , Gu =

⎡⎢⎢⎢⎢⎣

1 0 4 2

0 4 0 1

1 0 0 0

0 1 0 0

⎤⎥⎥⎥⎥⎦

h2norm(Gs) ⇒ ‖Gs‖2 = 0.6055

h2norm(cjt(Gu) ⇒ ‖Gu‖2 = 3.182

‖G‖2 =√

‖Gs‖22 + ‖Gu‖22 = 3.2393


47

�� H∞ ��(��)��, �� L∞ ��(��)

�� L∞ space is a Banach space of matrix-valued functions that are

(essentially) bounded on jR (��), with L∞ norm

‖F‖∞ := ess supω∈R

σ[F (jω)]

RL∞ space is the rational subspace of L∞

�� RL∞ ��”��”��? proper, real rational, no

poles on jR (∵ bounded on jR)

G(s) ∈ RL∞ �� G(s) ��, ��


48

H∞ space is a (closed) subspace of L∞ with functions that are

analytic and bounded in the open right-half plane, with H∞ norm

‖F‖∞ := supRe(s)>0

σ[F (s)] = supω∈R

σ[F (jω)]

RH∞ space is the rational subspace of H∞

�� RH∞ ��”��”��? proper, real rational, no

poles on open right-half plane (∵ bounded in the open rhp)

G(s) ∈ RH∞ �� G(s) ��, ��


49

Computing L∞ and H∞ norms: the largest value of the

singular value plot over ω

G(s) ∈ RL∞ , ‖G(s)‖∞ := supω∈R

σ[G(jω)]

G(s) ∈ RH∞ , ‖G(s)‖∞ := supω∈R

σ[G(jω)]

��: To get an estimate, set up a fine grid of frequency points

{ω1, ω2, · · · , ωN}an estimate for ‖G‖∞ is given by

‖G‖∞,estimate = max1≤k≤N

σ[G(jωk)]


50

Computing H∞ norms: the largest value of the singular value

plot over ω

SISO ��

�� u(t) = u0 sin(ω0t+ φ0)

�� y(t) = u0|G(jω0)| sin(ω0t+ φ0 + ∠G(jω0))

‖G(s)‖∞ = supω0

|G(jω0)|

H∞ ��!


51

Computing H∞ norms: the largest value of the singular value

plot over ω.

MIMO ��

�� u(t) =

⎡⎢⎢⎢⎢⎣

u1 sin(ω0t+ φ1)

u2 sin(ω0t+ φ2)...

uq sin(ω0t+ φq)

⎤⎥⎥⎥⎥⎦ , u =

⎡⎢⎢⎢⎢⎣

u1

u2...

uq

⎤⎥⎥⎥⎥⎦

�� y(t) =

⎡⎢⎢⎢⎢⎣

y1 sin(ω0t+ θ1)

y2 sin(ω0t+ θ2)...

yp sin(ω0t+ θp)

⎤⎥⎥⎥⎥⎦ , y =

⎡⎢⎢⎢⎢⎣

y1

y2...

yp

⎤⎥⎥⎥⎥⎦

‖G(s)‖∞ = supφi,ω0,u

‖y‖‖u‖


52

Example 4.2

hinfnorm(G,0.0001) ⇒ ‖G(s)‖∞ = 11.47, ωmax = 0.8483

�� 0.8483 �� 11.47 (��)��!

Internal Stability Feedback Stucture

53

Standard feedback configurey: ��

n: ��

r: ��

d: (��)��

di: (��)��

Transfer function realization

G(s) =

[A B

C D

]:= C(SI − A)−1B +D

Internal Stability Well-Posedness

54

Definition: A feedback system is said to be well-posed if all

closed-loop transfer matrices are well-defined and proper.

Well-posedness ⇔

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

I − K(∞)P (∞) is invertible

I − P (∞)K(∞) is invertible[I −K(∞)

−P (∞) I

]is invertible

Internal Stability Well-Posedness

55

The well-posedness condition is simple to state in terms of

state-space realizations.

P =

[A B

C D

], K =

[A B

C D

].

=⇒ P (∞) = D, K(∞) = D

Well-posedness ⇔

⎧⎪⎨⎪⎩[

I −D

−D I

]is invertible

I − DD is invertible

Internal Stability Internal Stability

56

Definition: The system is said to be internally stable if all transfer

functions from

[w1

w2

]to

[e1

e2

]belongs to RH∞

Example

P =s− 1

s+ 1, K = − 1

s− 1[e1

e2

]=

[s+1s+2

− s+1(s−1)(s+2)

s−1s+2

s+1s+2

][w1

w2

]⇒ not internally stable


57

Internally stable ⇐⇒ Eq.(5.4)[I −K

−P I

]−1

=

[(I − KP )−1 K(I − PK)−1

P (I − KP )−1 (I − KP )−1

]∈ RH∞

=

[I + K(I − PK)−1P K(I − PK)−1

(I − PK)−1P (I − PK)−1

]∈ RH∞

�� (5.4) ��

e1 = w1 + Ke2 ⇔ w1 = e1 − Ke2

e2 = Pe1 + w2 ⇔ w2 = −Pe1 + e2

⇒ e1 = (I − KP )−1w1 + (I − KP )−1Kw2

Or ⇒ e2 = (I − PK)−1Pw1 + (I − PK)−1Pw2


58

⇒ e1 = (I − KP )−1w1 + (I − KP )−1Kw2

e2 = P (I − KP )−1w1 + (I + P (I − KP )−1K)w2

Or ⇒ e2 = (I − PK)−1Pw1 + (I − PK)−1Pw2

e1 = (I + K(I − PK)−1P )w1 + K(I − PK)−1Pw2

[e1

e2

]=

[I −K

−P I

]−1 [w1

w2

]

=

[(I − KP )−1 K(I − PK)−1

P (I − KP )−1 (I − PK)−1

][w1

w2

]

Or =

[I + K(I − PK)−1P K(I − PK)−1

(I − PK)−1P (I − PK)−1

][w1

w2

]


59

• Suppose K ∈ RH∞. Internal stability ⇔P (I − KP )−1 ∈ RH∞

• Suppose P ∈ RH∞. Internal stability ⇔K(I − PK)−1 ∈ RH∞

• Suppose P, K ∈ RH∞. Internal stability ⇔(I−PK)−1 ∈ RH∞

• Suppose no unstable pole-zero cancellation in PK.

Internal stability ⇔(I − P (s)K(s))−1 ∈ RH∞

In general, the system is internally stable ⇐⇒ it is well-posed and

• the number of open rhp poles of P (s)K(s) = nk + np

⇔ no unstable pole-zero cancellation

• (I − P (s)K(s))−1 is stable

nk : the number of open right-half plane (rhp) poles of K(s)

np : the number of open right-half plane (rhp) poles of P (s)


60

Example 5.1

P =

[1

s−10

0 1s+1

], K =

[− s−1

s+1−1

0 −1

]

• it is well-posed

√nk + np = 1 ��

× (I − PK)−1 =

[s+1s+2

(s+1)2

(s+2)2(s−1)

0 s+1s+2

]unstable

⇒ The closed-loop system is not stable, even though

det(I − PK) =(s+ 2)2

(s+ 1)2has no unstable zero

��!

LinearAlgebra - SHUYOU YU

Documents

Transcript of LinearAlgebra - SHUYOU YU