COMMUTATION RELATIONS and

STABILITY of SWITCHED SYSTEMS

Daniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

COMMUTING STABLE MATRICES => GUES

For subsystems – similarly

(commuting Hurwitz matrices)

...

quadratic common Lyapunov function[Narendra–Balakrishnan ’94]

COMMUTING STABLE MATRICES => GUES

Alternative proof:

is a common Lyapunov function

Nilpotent means sufficiently high-order Lie brackets are 0

NILPOTENT LIE ALGEBRA => GUES

Lie algebra:

Lie bracket:

Hence still GUES [Gurvits ’95]

For example:

(2nd-order nilpotent)

In 2nd-order nilpotent case

Recall: in commuting case

SOLVABLE LIE ALGEBRA => GUES

Example:

quadratic common Lyap fcn diagonal

exponentially fast

[Kutepov ’82, L–Hespanha–Morse ’99]

Larger class containing all nilpotent Lie algebras

Suff. high-order brackets with certain structure are 0

exp fast

Lie’s Theorem: is solvable triangular form

MORE GENERAL LIE ALGEBRAS

Levi decomposition:

radical (max solvable ideal)

There exists one set of stable generators for which

gives rise to a GUES switched system, and another

which gives an unstable one

[Agrachev–L ’01]

• is compact (purely imaginary eigenvalues) GUES,

quadratic common Lyap fcn

• is not compact not enough info in Lie algebra:

SUMMARY: LINEAR CASE

Lie algebra w.r.t.

Assuming GES of all modes, GUES is guaranteed for:

• commuting subsystems:

• nilpotent Lie algebras (suff. high-order Lie brackets are 0)e.g.

• solvable Lie algebras (triangular up to coord. transf.)

• solvable + compact (purely imaginary eigenvalues)

Further extension based only on Lie algebra is not possible

Quadratic common Lyapunov function exists in all these cases

SWITCHED NONLINEAR SYSTEMS

Lie bracket of nonlinear vector fields:

Reduces to earlier notion for linear vector fields(modulo the sign)

SWITCHED NONLINEAR SYSTEMS

• Linearization (Lyapunov’s indirect method)

Can prove by trajectory analysis [Mancilla-Aguilar ’00]

or common Lyapunov function [Shim et al. ’98, Vu–L ’05]

• Global results beyond commuting case – ?

[Unsolved Problems in Math. Systems & Control Theory ’04]

• Commuting systems

GUAS

SPECIAL CASE

globally asymptotically stable

Want to show: is GUAS

Will show: differential inclusion

is GAS

OPTIMAL CONTROL APPROACH

Associated control system:

where

(original switched system )

Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot]:

fix and small enough

MAXIMUM PRINCIPLE

is linear in

at most 1 switch

(unless )

GAS

Optimal control:(along optimal trajectory)

GENERAL CASE

GAS

Want: polynomial of degree

(proof – by induction on )

bang-bang with switches

THEOREM

Suppose:

• GAS, backward complete, analytic

• s.t.

and

Then differential inclusion is GAS,

and switched system is GUAS [Margaliot–L ’06]

Further work in [Sharon–Margaliot ’07]

REMARKS on LIE-ALGEBRAIC CRITERIA

• Checkable

conditions• In terms of the original

data• Independent of representation

• Not robust to small perturbations

In any neighborhood of any pair of matricesthere exists a pair of matrices generating the entire Lie algebra [Agrachev–L ’01]

How to measure closeness to a “nice” Lie algebra?

ŁOJASIEWICZ INEQUALITY

f : Rn ! R real analytic function

Z := f x 2 Rn : f (x) = 0g – zero set of f

K ½RnThen for every compact, s.t.9C;®> 0

Cjf (x)j ¸ dist(x;Z)® 8x 2 K

Meaning: if thenjf (x)j · " dist(x;Z) · ±:= (C")1=®

When is a maximum of finitely many polynomials, explicit

bounds on the exponent can be obtained

f®

ŁOJASIEWICZ APPLIED TO OUR SET – UP f A1;A2; : : : ;AN g – finite set of Hurwitz matrices

Suppose k[A i;A j ]kF · " 8 i; j

where is Frobenius normkAkF :=ptrAAT

Let f (A1;A2; : : : ;AN ) := maxi;j k[A i;A j ]k2FThis is max of polynomials of degree 4 in

coefficients of (or can use sum to get a single polynomial)

N (N ¡ 1)=2A i

Let be distance from to nearest -tuple

of pairwise commuting matrices (Frobenius norm of difference between stacked matrices)

± (A1; : : : ;AN ) N(B1; : : : ;BN )

Łojasiewicz inequality gives Cf (A1;A2; : : : ;AN ) ¸ ±®

i.e., withA i = B i + ¢ i k¢ ikF · (C"2)1=®

Higher-order commutators (near nilpotent / solvable) – similar

PERTURBATION ANALYSISA i = B i + ¢ i; k¢ ikF · (C"2)1=®

B i commute (or generate nilpotent or solvable Lie algebra)

P B i + B Ti P = ¡ Qi < 0 8i

) 9 common Lyapunov function V (x) = xTP x:

which is still negative definite if k¢ ikF < ¸min(Qi)2¸max(P )

P A i + ATi P = ¡ Qi + P ¢ i + ¢ Ti PFor original matrices:

For commuting and solvable cases, specific known constructions of

can be used to estimate the right-hand side [Baryshnikov–L, CDC ’13]

P

C;® depend on (# of matrices) and on compact set where they liveNdeg f = 4 ) ® can be explicitly estimated

estimating is a bit more difficult [Ji–Kollar–Shiffman]C

stability of is ensured if _x = A¾x (C"2)1=®< ¸min(Qi)2¸max(P )

)

DISCRETE TIME: BOUNDS on COMMUTATORS

inductionbasis

contraction small

– Schur stable

GUES:

Let

GUES

Idea of proof: take , , consider