Co 12

Convex Optimization and SystemTheory

Lecturer : Prof.dr. Anton A. Stoorvogel

E-mail: [email protected]

1/27 Electrical Engineering, Mathematics and Computing ScienceEEMCS

Consider a system:

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

(∗)

The system with supply function s is said to be dissipative if

there exists a function V such that:

V(x(t0))+t1∫

t0

s(w(t), z(t))dt � V(x(t1))

for all t0 � t1 and all (w,x, z) satisfying (∗).


Consider a system:


(∗)

The system with supply function s is said to be conservative if

there exists a function V such that:

V(x(t0))+t1∫

t0

s(w(t), z(t))dt = V(x(t1))



First law of thermodynamics

E(x(t0))+t1∫

t0

Q(t)+W(t)dt = E(x(t1))

E Internal energy

S Entropy

Q Rate of heating

T Temperature

W Rate of work


Second law of thermodynamics

S(x(t0))+t1∫

t0

−Q(t)T(t)

dt = S(x(t1))

E Internal energy

S Entropy

Q Rate of heating

T Temperature

W Rate of work


Consider a system:


(∗)

The system with supply function s is said to be strictly

dissipative if there exists a function V and an ε > 0 such that:

V(x(t0))+t1∫

t0

s(w(t), z(t))dt − ε2

t1∫

t0

‖w(t)‖2 dt � V(x(t1))



x∗ ∈ X is a fixed reference point in the state space. We consider

normalized storage functions which satisfy

V(x(t0))+t1∫

t0

s(w(t), z(t))dt � V(x(t1))

and additionally

V(x∗) = 0

We assume that we can reach all points from x∗ via an

appropriate choice for w and conversely.


Available storage

Vav(x0) := supw,t1

−t1∫

0

s(w(t), z(t))dt

such that (w,x, z) satisfy (∗) while t1 � 0, x(0) = x0 and

x(t1) = x∗.


Required storage

Vreq(x0) := infw,t−1

0∫

t−1

s(w(t), z(t))dt

such that (w,x, z) satisfy (∗) while t1 � 0, x(t−1) = x∗ and

x(0) = x0.


The system is dissipative if and only if the available storage is

well-defined and finite for all initial conditions

The system is dissipative if and only if the required storage is

well-defined and finite for all end conditions


Linear systems with quadratic supply rate

Consider a system:

x = Ax + Ewz = Cx +Dw

(∗)

and

s(w, z) =⎛⎜⎝wz

⎞⎟⎠

T⎛⎜⎝Q S

ST R

⎞⎟⎠⎛⎜⎝wz

⎞⎟⎠


A dissipative system with a quadratic supply rate has a

quadratic storage function.

V(x) = xTPx

Moreover available and required storage are also both quadratic.


Consider a linear system with a quadratic supply rate

We define V by:

V(x) = xTPx

The system is dissipative if and only if there exists a symmetric

matrix P such that:

⎛⎜⎝ A

TP + PA+ CTRC PE +DTRC + SCETP + CTST + CTRD Q+ SD +DTST +DTRD

⎞⎟⎠ � 0

In that case V is a storage function.


Consider a linear system with a quadratic supply rate

We define V by:

V(x) = xTPx

The system is strictly dissipative if and only if there exists a

symmetric matrix P such that:

⎛⎜⎝ A

TP + PA+ CTRC PE +DTRC + SCETP + CTST + CTRD Q+ SD +DTST +DTRD

⎞⎟⎠ ≺ 0

In that case V is a storage function.


Positive real lemma

A system is dissipative with respect to the quadratic supply rate

s(w,z) = zTw if and only if there exists a symmetric matrix Psuch that: ⎛

⎜⎝ATP + PA PE − CT

ETP − CT −D −DT

⎞⎟⎠ � 0


Bounded real lemma

A system is dissipative with respect to the quadratic supply rate

s(w,z) = wTw − zTz if and only if there exists a symmetric

matrix P such that:

⎛⎜⎝A

TP + PA+ CTC PE + CTD

ETP +DTC DTD − I

⎞⎟⎠ � 0


Stability

Consider a nonlinear system:

x = f(x)

x0 is called an equilibrium point if f(x0) = 0



x = f(x), x(0) = x0

The equilibrium point x∗ is called stable if for all ε > 0 there

exists δ > 0 such that

‖x∗ − x0‖ < δ �⇒ ‖x(t)− x∗‖ � ε for all t � 0



x = f(x), x(0) = x0

The equilibrium point x∗ is called asymptotically stable if

• the equilibrium point is stable,

• there exists δ > 0 such that

‖x∗ − x0‖ < δ �⇒ limt→∞

x(t) = 0


Lyapunov stability


x = f(x), x(0) = x0

If there exists a positive definite, continuously differentiable

function V with V(x∗) = 0 and

ddtV(x(t)) � 0

for all initial conditions x0 and t > 0 then the equilibrium x∗ is

stable.


Lyapunov stability


x = f(x), x(0) = x0

If there exists a positive definite, continuously differentiable

function V with V(x∗) = 0 and

ddtV(x(t)) < 0

for all initial conditions x0 and t > 0 then the equilibrium x∗ is

asymptotically stable.


Consider a linear system

x = Ax

If there exists a matrix P 0 such that

ATP + PA � 0

then the equilibrium 0 is stable.


Consider a linear system

x = Ax

If there exists a matrix P 0 such that

ATP + PA ≺ 0

then the equilibrium 0 is asymptotically stable.


H∞ performance

Consider a system:

x = Ax + Ew, x(0) = 0

z = Cx +Dw(∗)

supw∈L2w≠0

‖z‖‖w‖


The H∞ performance is less than γ if and only if there exists P∞such that

⎛⎜⎝A

TP∞ + P∞A+ CTC P∞E + CTD

ETP∞ +DTC DTD − γ2I

⎞⎟⎠ � 0


H2 performance

Consider a system:

x = Ax + Ew, x(0) = 0

z = Cx +Dw(∗)

p∑i=1

‖zi‖2

where zi is the output of the system with input wi = δ(t)ej .Here w(t) ∈ �p and e1, . . . ep is a basis of �p while δ(t) is the

Dirac impulse.


The H2 performance is equal to

traceEP2ET

where

ATP2 + P2A+ CTC = 0


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