Stability Proofs for Hybrid Systems Jens Oehlerking - PowerPoint PPT Presentation

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Stability Proofs for Hybrid Systems Jens Oehlerking jens.oehlerking@informatik.uni-oldenburg.de Abteilung Systemsoftware und verteilte Systeme Department f¨ ur Informatik Carl von Ossietzky Universit¨ at Oldenburg March 18, 2010 Jens Oehlerking 1/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Goal of this Talk This talk will give an introduction to: • stability theory for hybrid systems • Lyapunov function based results that allow verification • concrete verification methods based on nonlinear optimization • decompositional techniques to ease the proof obligations Jens Oehlerking 2/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Outline 1 Hybrid Systems and Stability 2 Lyapunov Functions 3 Lyapunov Function Computation 4 Decomposition 5 Conclusion Jens Oehlerking 3/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Outline 1 Hybrid Systems and Stability 2 Lyapunov Functions 3 Lyapunov Function Computation 4 Decomposition 5 Conclusion Jens Oehlerking 4/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Closed-loop Contol Classical application for hybrid systems: closed-loop control 4 2 0 −2 −4 −6 −8 −10 −12 −14 −16 0 100 200 300 400 500 600 700 800 900 1000 • system variables must be driven toward a designated target (equilibrium state x e ∈ R n ) • small disturbances should only cause small deviations from this equilibrium Jens Oehlerking 5/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Stability Any given bound on the deviation from the equilibrium x e should be respected, if we choose the initial state close enough. ∀ ǫ > 0 ∃ δ > 0 : || x (0) − x e || < δ = ⇒ ∀ t : || x ( t ) − x e || < ǫ In temporal logic: ∀ ǫ > 0 ∃ δ > 0 : || x − x e || < δ = ⇒ � ( || x − x e || < ǫ ) Here: initial state represents result of a transient disturbance, or initial error after changing the set point of the system. Intuitively: no chaotic behavior, where small disturbances cause huge changes in behavior. Jens Oehlerking 6/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Convergence Intuitively: A (hybrid) system is convergent, if, from any initial state, the system state converges to equilibrium x e . ∀ ǫ > 0 ∃ t 0 > 0 ∀ t > t 0 : || x ( t ) − x e || < ǫ x e In temporal logic: q ∀ ǫ > 0 : ♦� ( || x − x e || < ǫ ) ⇒ conjunction of infinitely many “finally globally” proper- ties Jens Oehlerking 7/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Global Asymptotic Stability Global asymptotic stability is the conjunction of these two properties: A system is globally asymptotically stable (GAS), if for all trajectories x ( t ): ∀ ǫ > 0 ∃ δ > 0 : || x − x e || < δ )) = ⇒ � ( || x − x e || < ǫ ) and ∀ ǫ > 0 : ♦� ( || x − x e || < ǫ ) It is not inherently clear • how fast convergence will be • how far the system can stray from x e But this can be derived in many cases! Jens Oehlerking 8/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Hybrid Automata A hybrid automaton is a finite automaton where • each node (=mode) has a differential equation x = 1 ˙ x = − 1 ˙ Jens Oehlerking 9/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Hybrid Automata A hybrid automaton is a finite automaton where • each node (=mode) has a differential equation • optionally, each node has an invariant predicate x = 1 ˙ x = − 1 ˙ x < 10 x > 10 Jens Oehlerking 9/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Hybrid Automata A hybrid automaton is a finite automaton where • each node (=mode) has a differential equation • optionally, each node has an invariant predicate • each edge (= mode transition) has an associated guard predicate x = 1 ˙ x = − 1 ˙ x = 10 x < 10 x > 10 Jens Oehlerking 9/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Hybrid Automata A hybrid automaton is a finite automaton where • each node (=mode) has a differential equation • optionally, each node has an invariant predicate • each edge (= mode transition) has an associated guard predicate • optionally, each edge has a discrete update function x = 10 ∧ x + = x + 1 x = 1 ˙ x = − 1 ˙ x < 10 x > 10 Jens Oehlerking 9/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Runs of Hybrid automata x = f 1 ( x ) ˙ I 1 Jens Oehlerking 10/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Runs of Hybrid automata g 1 ∧ x + = u 1 ( x ) x = f 1 ( x ) ˙ x = f 2 ( x ) ˙ I 1 I 2 Jens Oehlerking 10/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Runs of Hybrid automata g 1 ∧ x + = u 1 ( x ) x = f 1 ( x ) ˙ x = f 2 ( x ) ˙ I 1 I 2 Jens Oehlerking 10/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Runs of Hybrid automata g 2 ∧ g 1 ∧ x + = u 2 ( x ) x + = u 1 ( x ) x = f 3 ( x ) ˙ x = f 1 ( x ) ˙ x = f 2 ( x ) ˙ I 3 I 1 I 2 Jens Oehlerking 10/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Proving Stability Proving stability requires different techniques than proving safety properties, because • we need to prove that sets of states will definitely be reached • this includes disproving oscillations in the state space • predicate abstraction techniques will result in loops in the abstraction • we need to guarantee some sort of progress toward the equilibrium in all situations • proof decomposition is not easy (see next slide) Jens Oehlerking 11/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Compositionality • very desirable: Break down stability proofs of large systems into smaller sub-proofs • this works well for safety proofs, but is very difficult for stability! • individual modes of a stable hybrid system can be unstable • (all) individual modes of an unstable hybrid system can be stable • therefore, we cannot easily use “divide and conquer” strategies on hybrid automata! Jens Oehlerking 12/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Unstable System with Stable Modes 2 10 2 1.5 1.5 5 1 1 0.5 0.5 0 0 0 −0.5 −5 −0.5 −1 −1 −1.5 −10 −1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 0 5 10 15 20 25 30 35 Jens Oehlerking 13/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Stable System with Unstable Modes 5 6 4 1 4 3 0.8 2 2 1 0.6 0 0 0.4 −1 −2 −2 0.2 −3 −4 −4 0 −6 −5 −0.5 0 0.5 −8 −6 −4 −2 0 2 4 6 8 −3 −2 −1 0 1 2 3 Jens Oehlerking 14/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Outline 1 Hybrid Systems and Stability 2 Lyapunov Functions 3 Lyapunov Function Computation 4 Decomposition 5 Conclusion Jens Oehlerking 15/64 Stability Proofs for Hybrid Systems

HS/Stability Lyapunov Functions LF Computation Decomposition Conclusion References Lyapunov Functions – Intuition We need a way of arguing about progress toward x e . Basic Idea (Lyapunov, 1907): Measure the “energy” of the system. The energy must be: • at its minimum at x e • strictly decreasing over time along any valid run of the system, unless we are already at x e • radially unbounded Jens Oehlerking 16/64 Stability Proofs for Hybrid Systems