Lecture Note 7: Switching Stabilization via Control-Lyapunov Function

ECE7850: Hybrid Systems:Theory and Applications Lecture Note 7: Switching Stabilization via Control-Lyapunov Function Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State University, Columbu, Ohio, USA Spring 2017 Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 1 / 28

Outline Classical Control-Lyapunov Function Approach Switching Stabilization Problem Switching Stabilization via Control Lyapunov Function Special Case: Quadratic Switching Stabilization Special Case: Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 2 / 28

Stabilization Problem I Consider general nonlinear control system: ẋ = f(x, u) - f : X U X, where X R n is state space and U R m is control space - assume f is locally Lipschitz in (x, u) - for simplicity and without loss of generality, we assume zero control input will result in equilibrium at origin, i.e., f(0, 0) = 0 Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 3 / 28

Stabilization Problem II Roughly speaking, there are two classes of stabilization questions: - Open-loop: whether there is a control input signal u : R + U such that the time-varying system ẋ(t) = f(x(t), u(t)) is asymptotically stable? This is often referred to as the Asymptotic Controllability problem. - Feedback: whether there is a state-feedback control law µ : X U under which the closed-loop system ẋ(t) = f(x(t), µ(x(t))) is asymptotically stable? Precise definitions of asymptotic controllability and feedback stabilizability can be quite involved and depend on many additional assumptions. We will not get into those technical details. (See [Son89; SS95; PND99]) Under some conditions, these two types of stabilization problems are equivalent Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 4 / 28

Control-Lyapunov Function I Feedback stabilization is concerned with constructing control law to ensure closed-loop stability. Lyapunov function is the most important tool for stability analysis. Under a given feedback law µ(x(t)), the closed-loop system is stable if a PD C 1 function V such that (V x (x)) T f(x, µ(x)) < 0, x 0 By the converse Lyapunov function theorems, if the closed-loop system is stable, it must have certain kind of Lypuanov functions V Stabilization can be thought of as finding the control law µ that minimizes the Lie derivative of some Lypunov function of the closed-loop system Control-Lyapunov Function Approach Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 5 / 28

Control-Lyapunov Function II Definition 1 ((Smooth) Control-Lyapunov Function). A C 1 PD function V : R n R + is called a Control Lyapunov Function (CLF) if a PD function W such that inf u U ( V (x))t f(x, u) < W (x), x Control Lyapunov function V can be used to generate stabilizing control law: } µ (x) = argmin u U {l(u) : ( V (x)) T f(x, u) < W (x) (1) when l(u) = u, the resulting controller is called pointwise min-norm controller [PND99] Classical results on control-lyapunov functions are mostly based on input affine system: f(x, u) = f(x) + g(x)u, for which general formula (Sontag s formula) of µ exists which does not involve solving optimization problems [Son89] Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 6 / 28

Control-Lyapunov Function III Theorem 1. Assume (i) ẋ = f(x, u) has a control Lyapunov function V ; and (ii) the corresponding µ defined in (1) makes f(x, µ (x)) locally Lipschitz. Then µ asymptotically stabilize the system. This theorem follows immediately by applying the standard Lypapunov stability theorem to the closed-loop system The result can be easily extended to obtain stronger stability result. - e.g.: If β 1 x α V (x) β 2 x α and W can be chosen as cv (x) with c > 0, then µ exponentially stabilizes the system. Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 7 / 28

Switching Stabilization Problem I Consider a switched nonlinear system ẋ(t) = f σ(t) (x(t)) (2) σ(t) Q, where Q is finite f i locally Lipschitz, for i Q origin is a common equilibrium, i.e. f i (0) = 0, i Q different classes of admissible switching signals: - S m: set of all measurable switching signals - S p: set of all piecewise constant switching signals - S p[τ]: set of switching signals with dwell time no smaller than τ Switching Stabilization Problem Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 8 / 28

Switching Stabilization Problem II Switching Stabilization Problem: view σ(t) in (2) as the control input and design the switching control to stabilize the continuous state x(t). Conceptually, this is similar to the classical nonlinear stabilization problem. But there are some key differences: the control set Q is discrete, and f σ (x) is not (locally) Lipschitz in (x, u). Most existing results in classical nonlinear stabilization cannot be directly used. Open-Loop Switching Stabilization: Find an admissible switching signal σ so that the solution to ẋ(t) = f σ(t) (x(t)) is asymptotically stable - depends on the assumption on admissible switching signals, e.g. S m or S p or S p[τ] for some τ > 0 - this is similar to the asymptotic controllability problem for classical nonlinear system Switching Stabilization Problem Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 9 / 28

Switching Stabilization Problem III Feedback Switching Stabilization: Design a (state-dependent) switching law ν : R n Q so that the cl-system ẋ = f ν(x) (x) is asymp. (or exp.) stable. - Note: Even when all the subsystem vector fields f i is smooth, the closed-loop vector field is typically discontinuous - The stabilization problem relies crucially on the solution notion adopted for the closed-loop discontinuous system - Feedback stabilization in Fillipov Sense: Filippov solution notion is adopted for the discontinuous CL-system - Feedback stabilization in sample-and-hold sense: sample-and-hold solution notion is adopted for the discontinuous CL-system Switching Stabilization Problem Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 10 / 28

Switching Stabilizability I When is a given switched system stabilizable? Switching Stabilizable with S m (or S p ): if σ S m (or S p ) so that ẋ(t) = f σ(t) (x(t)) asymp. stable. Feedback Stabilizable in Filippov Sense: if a switching law ν : R n Q so that all Filippov solutions of the cl-system ẋ = f ν(x) (x) is asymp. stable. Feedback Stabilizable in Sample-and-Hold Sense: if a switching law ν : R n Q so that all sample-and-hold solutions of the cl-system ẋ = f ν(x) (x) is asymp. stable. These stabilizability concepts are different. Their relations are unknown in general except for switching linear systems [LZ16] Switching Stabilization Problem Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 11 / 28

Switching Stabilizability II Theorem 2 (Switching Stabilizability Theorem for SLS [LZ16]). For switched linear system: ẋ(t) = f σ(t) (x(t)) with finite subsystems. The following statements are equivalent open-loop switching stabilizable with measurable switching input (i.e. σ S m) open-loop switching stabilizable with piecewise constant switching input (i.e. σ S p) feedback switching stabilizable in Filippov sense feedback switching stabilizable in sample-and-hold sense Switching Stabilization Problem Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 12 / 28

Switching Stabilization via Control Lyapunov Function I We will focus on feedback stabilization in Filippov sense via a control-lyapunov function approach Question: can we use the classical control Lyapunov function approach to design switching control? - Yes, but there are additional challenges - Challenge 1: smooth control Lyapunov function is too restrictive for switched/hybrid systems need to extend the framework to enable the use of nonsmooth control Lyapunov functions - Challenge 2: the generated switching law ν (even with a smooth control Lyapunov function) will make the cl-system vector field f ν (x)(x) discontinuous; need to analyze possible sliding motions Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 13 / 28

Piecewise Smooth Control-Lyapunov Function I Focus on a large class of nonsmooth control Lyapunov functions: piecewise smooth functions. A function g : R n R is called piecewise smooth if it is continuous and there exists a finite collection of disjoint and open sets Ω 1,..., Ω m R n, such that (i) j Ωj = R n ; (ii) g is C 1 on Ω j ; (iii) Ω j is a differentiable manifold Important properties of piecewise smooth function - directional derivative exists everywhere - nonsmooth surface Ω j is of measure 0 One-sided directional derivative of V along η: V (x; η) = lim t 0 V (x + tη) V (x) t (3) Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 14 / 28

Piecewise Smooth Control-Lyapunov Function II Derivative along vector field direction in mode i Q: V (x; fi (x)) If V is differentiable, then V (x; η) = ( V (x)) T η and V (x; fi (x)) = V (x) T f i (x) - NOTE: in this case, the V (x; η) is linear w.r.t η, which may not hold for nonsmooth V More general definition of Control Lyapunov Function: a PD function V (not necessarily C 1 ) is called a control Lypapunov function for switched system (2) if a PD function W s.t. { V (x; fi (x)) exists, x, i (4) min i Q V (x; fi (x)) < W (x), x Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 15 / 28

Piecewise Smooth Control-Lyapunov Function III Switching law generation: ν (x) = argmin i Q V (x; fi (x)) (5) If V is a control-lyapunov function, then under the switching law ν, all classical solutions (excluding sliding motion) to the cl-system f ν (x)(x) is asymp. stable When W in (4) can be chosen as cv (x) for some c > 0, then exponential stability (excluding sliding motions) can be concluded. Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 16 / 28

Piecewise Smooth Control-Lyapunov Function IV What happens if there are sliding motions? - suppose x(t) involves sliding motion during [t 1, t 2]. - For any t [t 1, t 2], ẋ(t) = i I sm α if i(x(t)) with i I sm α i = 1 - We should require ( ) V x(t); α if i(x(t) W (x(t)) (6) i I sm Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 17 / 28

Piecewise Smooth Control-Lyapunov Function V - If V smooth, then (4) (6) - If V nonsmooth, (6) is not easy to check/guarantee in general CLF-based switching stabilization approach: In many cases, control-lyapunov function conditions (4) can be translated into LMIs or BMIs, so one can search for CLF by solving LMIs or BMIs. Once a CLF is found, the switching law (5) ensures Filippov solution of CL-system asymp. stable. If needed, sliding motion stability can be guaranteed if the CLF also satisfies (6) General results about piecewise smooth CLF can be found in [LZ17]. Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 18 / 28

Special Case: Quadratic Switching Stabilization I We now focus on a simple special case for which the CLF is chosen to be of quadratic form. Definition 2. The system is called quadratically stabilizable if there exists a quadratic control Lyapunov function The most extensively studied class of switching stabilization problems A quadratic control Lyapunov function: V (x) = x T P x needs to satisfy P 0 and min i Q { V (x)t f i (x)} < 0 (7) The corresponding switching law: ν (x) = argmin i Q { V (x) T f i (x)} Special Case: Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 19 / 28

Special Case: Quadratic Switching Stabilization II Theorem 3. For switched nonlinear system (2), if P satisfying (7), then cl-trajectory (including sliding motion) under ν is stable. Stable sliding motion is automatically guaranteed If exponential stability is desired: then should require min { V i Q (x)t f i (x)} < αv (x) Special Case: Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 20 / 28

Special Case: Quadratic Switching Stabilization III For switched linear systems ẋ = A σ x, quadratically switching stabilizable requires min i Q xt ( A T i P + P A i ) x < 0, x - Checking the condition is NP-hard - A sufficient condition is the existence of convex combination i αiai that is stable proof: stable convex combination means ( l ) T ( l ) α i A i P + P α i A i 0 (8) i=1 i=1 (8) is a bilinear matrix inequality; still NP-hard to solve but good numerical algorithms exist: e.g. path-following method (see HHB99) Special Case: Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 21 / 28

Special Case: Quadratic Switching Stabilization IV Extension to co-design of switching and continuous controls - Model: ẋ = A ix + B iu, i Q - Assume u = K ix for mode i - BMI (8) becomes: ( l ) T ( i=1 αi (Ai + BKi) l ) P + P i=1 αi (Ai + BiKi) 0 - change of variable: X = P 1 and Y i = K ip 1 [ ] α i XA T i + Yi T Bi T + A ix + B iy i 0 (9) i - numerical algorithm for (8) can be used to solve (9) as well. Special Case: Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 22 / 28

Special Case: Piecewise Quadratic Switching Stabilization I Even for switched linear systems, quadratic stabilizability is much weaker than switching stabilizability Piecewise quadratic control Lyapunov function is a natural extension The most important class of such nonsmooth control Lyapunov functions is obtained by taking pointwise minimum of a finite number of quadratic functions: V min (x) = min j J xt P j x (10) - # of quadratic functions (i.e. J ) is different from # of subsystems (i.e. Q ) in general Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 23 / 28

Special Case: Piecewise Quadratic Switching Stabilization II V min is nonconvex and nonsmooth (piecewise smooth). Level set of V min : Switching strategy is simply: ν = argmin i Q V min (x; f i (x)) It can be verified that: for switched linear systems {A i } and exponential stability (i.e. W (x) = cv min (x) for some c > 0). The CLF conditions (4) can guaratneed by the following BMIs { P, c > 0, β jk 0, α ij [0, 1], i αij = 1, for each j, s.t. ( ) T ( ) i Q αijai P j + P j i Q αijai k J β jk(p j P k ) cp j, for each j J The conditions are only sufficient and can be conservative. Derivations and insights can be found in [HML08; LZ17] Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 24 / 28

Special Case: Piecewise Quadratic Switching Stabilization III Why considering pointwise minimum functions? Theorem 4 (pointwise minimum CLF automatically guarantee stable sliding motions). For switched nonlinear system (2), let V min(x) = min j J V j(x), where V j is a smooth function (not necessarily quadratic). If V min(x) is a CLF (i.e. it satisfies condition (4)), then the CL-sys under switching law (5) is asymp. stable including sliding motions. Therefore, if system has a pointwise-min CLF, then stable sliding motion is automatically guaranteed (no need to further check condition (6)) Proof can be found in [LZ17] Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 25 / 28

Special Case: Piecewise Quadratic Switching Stabilization IV Why is pointwise minimum piecewise quadratic CLF important? Theorem 5 (Converse CLF for switched linear system). A switched linear system is switching stabilziable (in any open-loop or closed-loop stabilizability sense) if and only if there exists a piecewise quadratic CLF of form (10). To study switching stabilization of switched linear systems (regardless of open-loop or feedback based control), it suffices to consider piecewise quadratic CLFs that can be written as a pointwise minimum of a finite number of quadratic functions. Proof can be found in [LZ16]. Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 26 / 28

Conclusion Control-Lyapunov function (CLF) is an effective approach to design stabilizing control law. Once a CLF is found, stabilizing law can be constructed by minimizing the Lie derivative of CLF over admissible control inputs For switched systems, one needs to consider nonsmooth CLF and needs to analyze sliding motion Two important cases: if a switched nonlinear system has a quadratic or pointwise-min CLF, then stable sliding motion is automatically guaranteed. For switched linear systems, open-loop and feedback switching stabilizability (in Filippov or sample-and-hold sense) are all equivalent to the existence of a pointwise-min piecewise quadratic CLF. Further reading: [LZ16; LZ17; HML08] Next Lecture: Discret-time Optimal Control Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 27 / 28

References [Son89] [SS95] [PND99] [HML08] [LZ16] [LZ17] Eduardo D Sontag. A fffdfffdfffduniversalfffdfffdfffdconstruction of Artstein s theorem on nonlinear stabilization. In: Systems & control letters 13.2 (1989). Eduardo Sontag and Héctor J Sussmann. Nonsmooth control-lyapunov functions. In: Decision and Control, 1995., Proceedings of the 34th IEEE Conference on. Vol. 3. IEEE. 1995. James A Primbs, Vesna Nevistić, and John C Doyle. Nonlinear optimal control: A control Lyapunov function and receding horizon perspective. In: Asian Journal of Control 1.1 (1999). Tingshu Hu, Liqiang Ma, and Zongli Lin. Stabilization of switched systems via composite quadratic functions. In: IEEE Transactions on Automatic Control 53.11 (2008). Yueyun Lu and Wei Zhang. On switching stabilizability for continuous-time switched linear systems. In: IEEE Transactions on Automatic Control 61.11 (2016). Yueyun Lu and Wei Zhang. A piecewise smooth control-lyapunov function framework for switching stabilization. In: Automatica 76 (2017). Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 28 / 28