Topics to discuss: Computing H -Norm for Stable Linear Systems€¦ · Topics to discuss: • Computing H2-Norm for Stable ... January 26, 2010. Optimal Control for Linear Systems

Lecture 3: How to Compute H2 and H∞-Norms

Topics to discuss:

• Computing H2-Norm for Stable Linear Systems

c© A.Shiriaev/L.Freidovich. January 26, 2010. Optimal Control for Linear Systems: Lecture 3 – p. 1/19


Topics to discuss:


• Computing H∞-Norm for Stable Linear Systems


Computing H2-Norm:

H2 norm is the energy of the output of the system

X = AX +[B1, . . . , Bm

]

w1

· · ·

wm

,

y1

· · ·

yp

=

C1

· · ·

Cp

X

obtained in response to a vector impulse functions w, where

• y(t) is a p vector, y(t) =[y1(t), . . . , yp(t)

]T

• w is an m vector, w(t) =[δ1(t), . . . , δm(t)

]T

• G(s) =[C1, . . . , Cp

]T

(s In×n − A)−1 [B1, . . . , Bm

]is

the p × m transfer matrix


Computing H2-Norm:


X = AX +[B1, . . . , Bm

]

w1

· · ·

wm

,

y1

· · ·

yp

=

C1

· · ·

Cp

X

obtained in response to a vector impulse functions w, where

• y(t) is a p vector, y(t) =[y1(t), . . . , yp(t)

]T

• w is an m vector, w(t) =[δ1(t), . . . , δm(t)

]T

• G(s) =[C1, . . . , Cp

]T

(s In×n − A)−1 [B1, . . . , Bm

]is

the p × m transfer matrix

The yi(t)-component to the impulse in wj-channel is

yij(t) =

{

CieAtBj, t > 0

0, t ≤ 0


Computing H2-Norm:


X = AX +[B1, . . . , Bm

]

w1

· · ·

wm

,

y1

· · ·

yp

=

C1

· · ·

Cp

X

obtained in response to w(t) =[δ1(t), . . . , δm(t)

]T .

The yi(t)-component to the impulse in wj-channel is

yij(t) =

{

CieAtBj, t > 0

0, t ≤ 0g(t) :=

y11(t) · · · y1m(t)

· · · · · ·

yp1(t) · · · ypm(t)

The H2-norm of the system is the sum of the energies of yij(t)

∣∣G(·)

∣∣2:=

√√√√√

+∞∫

0

p∑

i=1

m∑

j=1

y2ij(t)dt =

√√√√√

+∞∫

0

trace [gT (t)g(t)] dt


Computing H2-Norm:

Given a stable MIMO linear system

x = Ax + Bw, y = C x

there are several ways to compute its H2-norm:

1. Directly evaluate+∞∫

0

trace [gT (t)g(t)] dt numerically!


Computing H2-Norm:





0


2. Use Laplace transform and Parceval relation

• Lg(t)(s) =

∫ +∞

0g(t)e−stdt := G(s)

•

+∞∫

0

trace [gT (t)g(t)] dt =1

2π

+∞∫

−∞

trace [G∗(jω)G(jω)] dω


Computing H2-Norm:





0


2. Use Laplace transform and Parceval relation

• Lg(t)(s) =

∫ +∞

0g(t)e−stdt := G(s)

•

+∞∫

0

trace [gT (t)g(t)] dt =1

2π

+∞∫

−∞

trace [G∗(jω)G(jω)] dω

3. Solve a Lyapunov equation without computing integrals!


Computing H2-Norm:

‖G(·)‖2 =

√√√√√

+∞∫

0



Computing H2-Norm:

‖G(·)‖2 =

√√√√√

+∞∫

0


∥∥G(·)

∥∥2=

√√√√√

+∞∫

0

trace[(CeAtB

)T

CeAtB]dt


Computing H2-Norm:

‖G(·)‖2 =

√√√√√

+∞∫

0


∥∥G(·)

∥∥2=

√√√√√

+∞∫

0

trace[BTeA

T tCTCeAtB]dt


Computing H2-Norm:

‖G(·)‖2 =

√√√√√

+∞∫

0


∥∥G(·)

∥∥2=

√√√√√√

+∞∫

0

trace[

BT eAT tCTCeAt B

]

dt


Computing H2-Norm:

‖G(·)‖2 =

√√√√√

+∞∫

0


∥∥G(·)

∥∥2=

√√√√√√trace

BT

+∞∫

0

eAT tCTCeAt dt B


Computing H2-Norm:

‖G(·)‖2 =

√√√√√

+∞∫

0


∥∥G(·)

∥∥2=

√

trace[BT QB

], Q =

+∞∫

0

eAT tCTCeAt dt


Computing H2-Norm:

‖G(·)‖2 =

√√√√√

+∞∫

0


∥∥G(·)

∥∥2=

√

trace[BT QB

], Q =

+∞∫

0

eAT tCTCeAt dt

Important Observation:

The observability Grammian Q can be computed withoutintegration as a solution of the Lyapunov equation

QA + ATQ + CTC = 0


Example 1Consider the system

x = −x + u + w, u is control variable, w is disturbance

To stabilize the position x = 0, we have chosen P-controller

u = −10x





u = −10x

Then the closed-loop system is

x = −10x − x + w





u = −10x


x = −10x − x + w

The state-space model of the system is

d

dt

[

x

x

]

=

[

0 1

−10 −1

]

︸︷︷︸

= A

[

x

x

]

+

[

0

1

]

︸︷︷︸

= B

w, y =[1, 0

]

︸︷︷︸

= C

[

x

x

]





u = −10x


x = −10x − x + w

The state-space model of the system is

d

dt

[

x

x

]

=

[

0 1

−10 −1

]

︸︷︷︸

= A

[

x

x

]

+

[

0

1

]

︸︷︷︸

= B

w, y =[1, 0

]

︸︷︷︸

= C

[

x

x

]

How to access the quality of the feedback design?


Example 1 (Cont’d)To compute H2-norm, we need to solve the Lyapunov equation

QA + ATQ + CTC = 0

that is[

q11 q12

q12 q22

] [

0 1

−10 −1

]

+

[

0 1

−10 −1

]T[

q11 q12

q12 q22

]

=

= −[1, 0

]T[1, 0

]



QA + ATQ + CTC = 0

that is[

q11 q12

q12 q22

] [

0 1

−10 −1

]

+

[

0 −10

1 −1

] [

q11 q12

q12 q22

]

= −

[

1 0

0 0

]



QA + ATQ + CTC = 0

that is[

q11 q12

q12 q22

] [

0 1

−10 −1

]

+

[

0 −10

1 −1

] [

q11 q12

q12 q22

]

= −

[

1 0

0 0

]

[

−10q12 q11 − q12

−10q22 q12 − q22

]

+

[

−10q12 −10q22

q11 − q12 q12 − q22

]

= −

[

1 0

0 0

]



QA + ATQ + CTC = 0

that is[

q11 q12

q12 q22

] [

0 1

−10 −1

]

+

[

0 −10

1 −1

] [

q11 q12

q12 q22

]

= −

[

1 0

0 0

]

[

−10q12 q11 − q12

−10q22 q12 − q22

]

+

[

−10q12 −10q22

q11 − q12 q12 − q22

]

= −

[

1 0

0 0

]

−20q12 = −1, q11 − q12 − 10q22 = 0, q12 − q22 = 0



QA + ATQ + CTC = 0

that is[

q11 q12

q12 q22

] [

0 1

−10 −1

]

+

[

0 −10

1 −1

] [

q11 q12

q12 q22

]

= −

[

1 0

0 0

]

[

−10q12 q11 − q12

−10q22 q12 − q22

]

+

[

−10q12 −10q22

q11 − q12 q12 − q22

]

= −

[

1 0

0 0

]

−20q12 = −1, q11 − q12 − 10q22 = 0, q12 − q22 = 0

q12 =1

20, q22 =

1

20, q11 =

11

20


Example 1 (Cont’d)

To compute H2-norm, we need to solve the Lyapunov equationQA + ATQ + CTC = 0

and the solution is

Q =1

20

[

11 1

1 1

]




and the solution is

Q =1

20

[

11 1

1 1

]

The H2-norm of our linear system (squared) is then

∥∥G(·)

∥∥2

2= trace [BTQB] =

1

20[0, 1]

[

11 1

1 1

] [

0

1

]

=1

20= 0.05




and the solution is

Q =1

20

[

11 1

1 1

]

The H2-norm of our linear system (squared) is then

∥∥G(·)

∥∥2

2= trace [BTQB] =

1

20[0, 1]

[

11 1

1 1

] [

0

1

]

=1

20= 0.05

A=[0, 1; -10, -1]; B=[0; 1]; C=[1, 0]; D=0;plant=ss(A,B,C,D);

% computing H2-normH2=norm(plant,2), H2 2


Alternative Way of Computing H2-Norm:

Important Observation: A solution of the linear equation

AP + PAT + BBT = 0

i.e. the controllability Grammian, has the form

P =

∫ +∞

0eAtBBTeA

T tdt




AP + PAT + BBT = 0


P =

∫ +∞

0eAtBBTeA

T tdt

∫ +∞

0g(t) gT (t)dt =

∫ +∞

0CeAtB

[CeAtB

]T

dt




AP + PAT + BBT = 0


P =

∫ +∞

0eAtBBTeA

T tdt

∫ +∞

0g(t) gT (t)dt =

∫ +∞

0CeAtB

[CeAtB

]T

dt

=

∫ +∞

0CeAtBBTeA

T tCTdt




AP + PAT + BBT = 0


P =

∫ +∞

0eAtBBTeA

T tdt

∫ +∞

0g(t) gT (t)dt =

∫ +∞

0CeAtB

[CeAtB

]T

dt

=

∫ +∞

0CeAtBBTeA

T tCTdt

= C

[∫ +∞

0eAtBBTeA

T tdt

]

CT




AP + PAT + BBT = 0


P =

∫ +∞

0eAtBBTeA

T tdt

∫ +∞

0g(t) gT (t)dt =

∫ +∞

0CeAtB

[CeAtB

]T

dt

=

∫ +∞

0CeAtBBTeA

T tCTdt

= C

[∫ +∞

0eAtBBTeA

T tdt

]

CT = CPCT




AP + PAT + BBT = 0


P =

∫ +∞

0eAtBBTeA

T tdt

∫ +∞

0g(t) gT (t)dt = CPCT

∥∥G(·)

∥∥2

2=

∫ +∞

0trace [gT (t) g(t)] dt =

∫ +∞

0trace [g(t) gT (t)] dt




AP + PAT + BBT = 0


P =

∫ +∞

0eAtBBTeA

T tdt

∫ +∞

0g(t) gT (t)dt = CPCT

∥∥G(·)

∥∥2

2=

∫ +∞

0trace [g(t) gT (t)] dt = trace [CPCT ]




• Computing H∞-Norm for Stable Linear Systems


H∞-norm for SISO stable systemsThe steady-state output of the SISO stable system

X = AX + Bw, y = C X

to the input function

w(t) = a sin(ω t + φ)

with unknown a, ω, φ ∈ R1, a 6= 0, is

yss(t) =∣∣G(jω)

∣∣ · a · sin

(ω t + φ + argG(jω)

)

where G(s) = C(sI − A)−1B ⇐ Y (s) = G(s)W (s).


H∞-norm for SISO stable systemsThe steady-state output of the SISO stable system

X = AX + Bw, y = C X

to the input function

w(t) = a sin(ω t + φ)

with unknown a, ω, φ ∈ R1, a 6= 0, is

yss(t) =∣∣G(jω)

∣∣ · a · sin

(ω t + φ + argG(jω)

)

where G(s) = C(sI − A)−1B ⇐ Y (s) = G(s)W (s).

The H∞-norm is the maximal possible amplification, i.e.

∥∥G(·)

∥∥∞

=

supω∈R1

|Y (jω)|

supω∈R1

|W (jω)|= sup

ω∈R1

∣∣G(jω)

∣∣



Our closed-loop system is

d

dt

[

x

x

]

=

[

0 1

−10 −1

]

︸︷︷︸

= A

[

x

x

]

+

[

0

1

]

︸︷︷︸

= B

w, y =[1, 0

]

︸︷︷︸

= C

[

x

x

]



Our closed-loop system is

d

dt

[

x

x

]

=

[

0 1

−10 −1

]

︸︷︷︸

= A

[

x

x

]

+

[

0

1

]

︸︷︷︸

= B

w, y =[1, 0

]

︸︷︷︸

= C

[

x

x

]

% computing Hinf-normHinf=norm(plant,inf)

ga=Hinf;[mag,phase,w] = bode(plant,{0.01,10}), grid;magnitudes=mag(:);

figure(1),ga line=ga*ones(size(w));plot(w,magnitudes,w,ga line,’r--’), gridxlabel(’frequency’), ylabel(’magnitude’)


H∞-norm for MIMO stable systemsThe steady-state k-th output of the general stable system

X = AX +

m∑

i=1

Bi wi,

y1

· · ·

yp

=

C1 X

· · ·

Cp X

to the input functions wi(t) = ai sin(ω t + φi),

yk(t) =

m∑

i=1

∣∣Gki(jω)

∣∣ · ai · sin

(ω t + φi + argGki(jω)

)

where Gki(s) = Ck(sI − A)−1Bi ⇐ Yk(s) = Gki(s)Wi(s).


H∞-norm for MIMO stable systemsThe steady-state k-th output of the general stable system

X = AX +

m∑

i=1

Bi wi,

y1

· · ·

yp

=

C1 X

· · ·

Cp X

to the input functions wi(t) = ai sin(ω t + φi),

yk(t) =

m∑

i=1

∣∣Gki(jω)

∣∣ · ai · sin

(ω t + φi + argGki(jω)

)

where Gki(s) = Ck(sI − A)−1Bi ⇐ Yk(s) = Gki(s)Wi(s).

The H∞-norm is the maximal possible vector amplification

∥∥G(·)

∥∥∞

=

supω∈R1

(p∑

k=1

|Yk(jω)|2)

supω∈R1

(m∑

i=1

|Wi(jω)|2) = sup

ω∈R1

σ{G(jω)

}


Computing an Approximation for H∞-Norm

Given a stable linear system

x = Ax+Bw, y = Cx+Dw,{G(s) = D + C(sIn − A)−1B

},

its H∞-norm equals the largest singular value

∥∥G(·)

∥∥∞

= supω∈R1

σ{G(jω)

}= sup

ω∈R1

√

max1≤i≤p

λi {G∗(jω)G(jω)}


Computing an Approximation for H∞-Norm

Given a stable linear system


},

its H∞-norm equals the largest singular value

∥∥G(·)

∥∥∞

= supω∈R1

σ{G(jω)

}= sup

ω∈R1

√

max1≤i≤p

λi {G∗(jω)G(jω)}

To approximate∥∥G(·)

∥∥∞

we can choose a set of frequencies

{ω1, ω2, . . . , ωN}

and search for

max1≤k≤N

σ {G(jωk)} ≈ ‖G(·)‖∞


Computing a Bound on H∞-Norm if D = 0

Given a number γ > 0 and a stable linear system

x = Ax + Bw, y = C x,{G(s) = C(sIn − A)−1B

}

Then∥∥G(·)

∥∥∞

< γ if and only if





}

Then∥∥G(·)

∥∥∞

< γ if and only if the following 2n× 2n matrix

H(A,B,C, γ) =

A 1

γ2BBT

−CT C −AT

has no imaginary eigenvalues.





}

Then∥∥G(·)

∥∥∞


H(A,B,C, γ) =

A 1

γ2BBT

−CT C −AT


The proof is based on defining

Φ(s) := γ2 I−GT (−s)G(s), Φ−1 =

A 1

γ2B BT 1

γ2B

−CT C −AT 0

0 1

γ2BT 1

γ2I





}

Then∥∥G(·)

∥∥∞


H(A,B,C, γ) =

A 1

γ2BBT

−CT C −AT


The proof is based on defining

Φ(s) := γ2 I − GT (−s)G(s)

and using the fact that∥∥G(·)

∥∥∞

< γ ⇔ Φ(jω) > 0 (positive

definite) for ω ∈ R1 ⇔ Φ−1(jω) has no imaginary poles.


Computing a Lower Bound on H∞-Norm



}

Then |G(·)|∞ < γ if and only if





}


• σ{D}< γ;





}


• σ{D}< γ;

• The following 2n × 2n matrix H = H(A,B,C,D, γ) hasno imaginary eigenvalues





}


• σ{D}< γ;

• The following 2n × 2n matrix H = H(A,B,C,D, γ) hasno imaginary eigenvalues

H =

A + BR−1DTC BR−1BT

−CT

(I + DR−1DT

)C −

(A + BR−1DTC

)T

R = γIn − DTD


Example 2

m1x1 = −k1(x1 − x2) − b1(x1 − x2) + u1

m2x2 = k1(x1 − x2) + b1(x1 − x2) − k2x2 − b2x2 + u2



The dynamical system

m1x1 = −k1(x1 − x2) − b1(x1 − x2) + u1

m2x2 = k1(x1 − x2) + b1(x1 − x2) − k2x2 − b2x2 + u2

can be written in the state-space from as follows

d

dt

z1

z2

z3

z4

=

0 0 1 0

0 0 0 1

− k1

m1

k1

m1

− b1

m1

b1

m1

k1

m2

−k1+k2

m2

b1

m2

−b1+b2

m2

z1

z2

z3

z4

+

0 0

0 01

m1

0

0 1m2

[

u1

u2

]

[

y1

y2

]

=

[

1 0 0 0

0 1 0 0

]

z1

z2

z3

z4



k1=1; k2=4; b1=0.2; b2=0.1; m1=1; m2=2;

A=[0, 0, 1, 0; 0, 0, 0, 1;...-k1/m1, k1/m1, -b1/m1, b1/m1;k1/m2, -(k2+k1)/m2, b1/m2, -(b1+b2)/m2];

B=[0, 0; 0, 0; 1/m1, 0; 0, 1/m2];

C=[1, 0, 0, 0; 0, 1, 0, 0];

D=zeros(2);

plant=ss(A,B,C,D);

% computing H2-normH2=norm(plant,2), h2norm(pck(A,B,C,D)),

% computing Hinf-normHinf=norm(plant,inf), hinfnorm(pck(A,B,C,D),0.00001),

figure(2), sigma(plant,0.1,10); grid ondisp(20*log10(Hinf)),


Next Lecture / Assignments:

Next meeting: January 27, 13:15-15:00, in A205,

Next lecture (January 29, 10:15-12:00, in A206):“Well-Posedness and Internal Stability”.


Next Lecture / Assignments:

Next meeting: January 27, 13:15-15:00, in A205,

Next lecture (January 29, 10:15-12:00, in A206):“Well-Posedness and Internal Stability”.

Problem: Let G(s) =1

(s2 + 2 ζ s + 1)(s + 1).

Compute∥∥G(·)

∥∥2

and∥∥G(·)

∥∥∞

using Bode plots and

state-space algorithms, respectively for ζ ∈{1, 0.1, 0.0001

}

and compare the results.

Verify your solutions using Matlab. Show all your work.The assignment is due at 13:15 on February 2, 2010.


Topics to discuss: Computing H -Norm for Stable Linear Systems€¦ · Topics to discuss: • Computing H2-Norm for Stable ... January 26, 2010. Optimal Control for Linear Systems

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