STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign IAAC Workshop, Herzliya, Israel, June 1, 2009

TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching

MULTIPLE LYAPUNOV FUNCTIONS – GAS – respective Lyapunov functions is GAS No GUAS => no common Lyapunov function Useful for analysis of state-dependent switching

MULTIPLE LYAPUNOV FUNCTIONS decreasing sequence decreasing sequence [DeCarlo, Branicky] GAS Need to have information about solutions

DWELL TIME The switching times satisfy – GES dwell time – respective Lyapunov functions System is stable if switching is slow enough (dwell time large enough) Continuous counterpart – stability of slowly time-varying systems

DWELL TIME The switching times satisfy – GES Need: Of course need same thing for V_2

DWELL TIME The switching times satisfy – GES Need: Here is the calculation

DWELL TIME The switching times satisfy – GES Need: must be From this we get a bound on dwell time (just take the log) must be

AVERAGE DWELL TIME – dwell time: cannot switch twice if # of switches on average dwell time – dwell time: cannot switch twice if Motivate ADT

AVERAGE DWELL TIME Theorem: [Hespanha ‘99] Switched system is GAS if # of switches on average dwell time Theorem: [Hespanha ‘99] Switched system is GAS if Lyapunov functions s.t. . No proof for now, but will see it later (for the case of inputs) Useful for analysis of hysteresis-based switching logics

MULTIPLE WEAK LYAPUNOV FUNCTIONS Theorem: is GAS if . observable for each s.t. there are infinitely many switching intervals of length – milder than ADT For every pair of switching times s.t. have Extends to nonlinear switched systems as before

APPLICATION: FEEDBACK SYSTEMS (Popov criterion) linear system observable positive real Weak Lyapunov functions: Corollary: switched system is GAS if s.t. infinitely many switching intervals of length For every pair of switching times at which we have Last condition quite restrictive for control design purposes, but we don’t need equality See also invariance principles for switched systems in: [Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel]

STATE-DEPENDENT SWITCHING Switched system unstable for some no common But switched system is stable for (many) other switch on the axes (Multiple) Lyapunov functions are useful for this But not a common one! Only decreases in appropriate regions I’m just giving examples here, but can search for MLFs systematically by LMIs is a Lyapunov function

STATE-DEPENDENT SWITCHING Switched system unstable for some no common But switched system is stable for (many) other Switch on y-axis level sets of For this switching signal GAS is pretty clear: with every half-rotation get closer to 0 Need MLFs to prove this GAS

STABILIZATION by SWITCHING – both unstable Assume: stable for some

STABILIZATION by SWITCHING – both unstable Assume: stable for some So for each either or Can verbally mention epsilon-robustification So stable convex combination implies quadratic stability (single Lyapunov function) [Wicks et al. ’98]

UNSTABLE CONVEX COMBINATIONS Motivate: stable conv comb difficult to find, may not exist Difference with previous examples: both systems unstable Single Lyapunov function – weak one (Feron’s result, see notes) Reference: my book Can also use multiple Lyapunov functions Linear matrix inequalities

SWITCHED SYSTEMS with INPUTS and OUTPUTS Outline: Background Input-to-state stability (ISS) Main results ISS under ADT switching Invertibility of switched systems

INPUT-TO-STATE STABILITY (ISS) class Nonlinear gain functions: ISS [Sontag ’89]: class , e.g. class Equivalent Lyapunov characterization [Sontag–Wang]: Think of u as disturbance, not control Pause to interpret ISS in terms of BIBS and CICS (means: pos. def., rad. unbdd.) without loss of generality, can replace by

ISS under ADT SWITCHING Suppose functions . class functions and constants such that : each subsystem is ISS Explain the meaning of \alpha_{1,2}, that lambda is no loss of generality, and discuss \mu Note independence of p (OK if P is finite or compact) The proof will be different from the paper (it works better for ISS but not for iISS) If has average dwell time then switched system is ISS [Vu–Chatterjee–L, Automatica, Apr 2007]

SKETCH of PROOF 1 2 1 3 Let be switching times on Consider Recall ADT definition: Norm of u is up to infinity or up to t, by causality doesn’t matter Then there will be stage 4 and so on Will now analyze each stage separately

SKETCH of PROOF 1 2 2 1 3 3 – ISS Special cases: GAS when Common ISS-Lyapunov function Last case: don’t exit the level set of V Special cases: GAS when ISS without switching (single ) ISS under arbitrary switching if (common )

VARIANTS Output-to-state stability (OSS) [M. Müller] Integral ISS: finds application in switching adaptive control Output-to-state stability (OSS) [M. Müller] Stochastic versions of ISS for randomly switched systems [D. Chatterjee] First is in the same paper, also mention stability margins Some subsystems not ISS [Müller, Chatterjee]

SWITCHED SYSTEMS with INPUTS and OUTPUTS Outline: Background Input-to-state stability (ISS) Main results ISS under ADT switching Invertibility of switched systems [Vu–L, Automatica, Apr 2008; Tanwani–L, CDC 2008]

PROBLEM FORMULATION Invertibility problem: recover uniquely from for given If this is possible, we say that the system is invertible Mention variants: strong invertibility around x_0, strong invertibility Sundaram, Daafouz: know sigma , recover u, discrete time – and only for linear systems Vidal, Babaali, De Santis: switched observability/mode identification, recover sigma and x (no u or known u). Also studied discernibility for control – again, only for linear Desirable: fault detection (in power systems) Undesirable: security (in multi-agent networked systems) Related work: [Sundaram–Hadjicostis, Millerioux–Daafouz]; [Vidal et al., Babaali et al., De Santis et al.]

MOTIVATING EXAMPLE Guess: because Will check this guess/conjecture later Guess:

INVERTIBILITY of NON-SWITCHED SYSTEMS Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham] Background slide: invertibility problem has been well-studied

INVERTIBILITY of NON-SWITCHED SYSTEMS Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham] Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh]

INVERTIBILITY of NON-SWITCHED SYSTEMS Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham] Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh] SISO nonlinear system affine in control: Suppose it has relative degree at : Then we can solve for : I’m dropping time arguments, stress that these are functions of time Inversion works on some time interval Important: the role of the inverse system is to give the input but also the state produced by it Abuse of notation: in the paper, the state of the system is z. Emphasize that need to initialize with the same state. Inverse system

BACK to the EXAMPLE We can check that each subsystem is invertible – similar Can mention reduced inverse For MIMO systems, can use nonlinear structure algorithm

SWITCH-SINGULAR PAIRS Consider two subsystems and is a switch-singular pair if such that ||| Invertibility of subsystems was the first ingredient for switched invertibility, this will be the second ingredient Say “x_0 and y form a switch-singular pair” Abuse of notation: p, q in the formula stand for sigma identically equal to p, q (in Linh’s prelim this is written via H_{p,x_0}, not sure if it’s better)

FUNCTIONAL REPRODUCIBILITY SISO system affine in control with relative degree at : For given and , that produces this output if and only if Slide transition: to check for switch-singular pairs, we need to characterize the set of ouputs that each subsystem can produce from a given initial condition (and then check their intersection) Reason: (i) terms that we have no control over have to match, (ii) then choose u (state feedback)p to have y^{(r)}(t) \equiv y_d^{(r)}(t), and initial conditions match => integrate to get the result

CHECKING for SWITCH-SINGULAR PAIRS is a switch-singular pair for SISO subsystems with relative degrees if and only if For linear systems, this can be characterized by a matrix rank condition MIMO systems – via nonlinear structure algorithm Existence of switch-singular pairs is difficult to check in general

MAIN RESULT Theorem: Switched system is invertible at over output set if and only if each subsystem is invertible at and there are no switched-singular pairs Idea of proof: no switch-singular pairs can recover Details: output set (differentiability), subintervals (finite escape), strong invertibility subsystems are invertible can recover The devil is in the details

BACK to the EXAMPLE Let us check for switched singular pairs: Stop here because relative degree For every , and with form a switch-singular pair Switched system is not invertible on the diagonal

OUTPUT GENERATION Given and , find (if exist) s. t. may be unique for some but not all Recall our example again: A slightly different twist: switched system not necessarily invertible, want an algorithm to find all sigma, u Tacitly assuming that subsystems are invertible, otherwise there will be infinitely many inputs Will use the example one last time to illustrate how the algorithm works

OUTPUT GENERATION Given and , find (if exist) s. t. may be unique for some but not all Recall our example again: Solution from : switch-singular pair

OUTPUT GENERATION Given and , find (if exist) s. t. may be unique for some but not all Recall our example again: Solution from : For sigma=1 can continue up to \pi because we know there are no more singular pairs before \pi For sigma=2 will have to check for singular pairs switch-singular pair

OUTPUT GENERATION Given and , find (if exist) s. t. may be unique for some but not all Recall our example again: Case 1: no switch at Then up to Must switch when y is discontinuous At , must switch to 2 But then won’t match the given output

OUTPUT GENERATION Given and , find (if exist) s. t. may be unique for some but not all Recall our example again: Case 2: switch at No more switch-singular pairs Still need to investigate Case 2, because we don’t know if this output is generatable at all

OUTPUT GENERATION Given and , find (if exist) s. t. may be unique for some but not all Recall our example again: Case 2: switch at No more switch-singular pairs

OUTPUT GENERATION Given and , find (if exist) s. t. may be unique for some but not all Recall our example again: Case 2: switch at No more switch-singular pairs Didn’t give the formulas for u just to save space, they are easy We see how one switch can help recover an earlier “hidden” switch We also obtain from