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