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Stability Analysis of Linear Switched Systems: An Optimal Control Approach

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## Stability Analysis of Linear Switched Systems: An Optimal Control Approach

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**Part 1**Stability Analysis of LinearSwitched Systems:An Optimal Control Approach Michael Margaliot School of Elec. Eng. Tel Aviv University, Israel Joint work with: Gideon Langholz (TAU), Daniel Liberzon (UIUC), Michael S. Branicky (CWRU), Joao Hespanha (UCSB).**Overview**• Switched systems • Global asymptotic stability • The edge of stability • Stability analysis: • An optimal control approach • A geometric approach • An integrated approach • Conclusions**Switched Systems**Systems that can switch between several possible modes of operation. Mode 1 Mode 2**Example 1**server**linear filter**Example 2 Switched power converter 50v 100v**Example 3**A multi-controller scheme plant + controller1 controller2 switching logic Switched controllers are stronger than “regular” controllers.**More Examples**• Air traffic control • Biological switches • Turbo-decoding • ……**Synthesis of Switched Systems**Driving: use mode 1 (wheels) Braking: use mode 2 (legs) The advantage: no compromise**Linear Systems**Solution: Definition: The system is globallyasymptotically stable if Theorem: stability A is called a Hurwitzmatrix.**Linear Switched Systems**A system that can switch between them: Two (or more) linear systems: 10**Stability**Linear switched system: Definition: Globally uniformly asymptotically stable (GUAS): In other words, for any AKA, “stability under arbitrary switching”. 11**A Necessary Condition for GUAS**The switching law yields Thus, a necessary condition for GUAS is that both are Hurwitz. Then instability can only arise due to repeated switching. 12**Why is the GUAS problem difficult?**Answer 1: The number of possible switching laws is huge. 13**Why is the GUAS problem difficult?**Answer 2: Even if each linear subsystem is stable, the switched system may not be GUAS. 14**Why is the GUAS problem difficult?**Answer 2: Even if each linear subsystem is stable, the switched system may not be GUAS. 15**Stability of Each Subsystem is Not Enough**A multi-controller scheme plant + controller1 controller2 switching logic Even when each closed-loop is stable, the switched system may not be GUAS.**Easy Case #1**A trajectory of the switched system: Suppose that the matrices commute: Then and since both matrices are Hurwitz, the switched system is GUAS. 17**Easy Case #2**Suppose that both matrices are upper triangular: Then so Now so This proves GUAS. 18**Optimal Control Approach**Basic idea: (1) A relaxation: linear switched system → bilinear control system (2) characterize the “most destabilizing” control (3) the switched system is GUAS iff Pioneered by E. S. Pyatnitsky (1970s). 19**where is the set of measurable functions**taking values in [0,1]. Optimal Control Approach Relaxation: the switched system: → a bilinear control system:**Optimal Control Approach**The bilinear control system (BCS) is globally asymptotically stable (GAS) if: TheoremThe BCS is GAS if and only if the linear switched system is GUAS.**Optimal Control Approach**The most destabilizing control: Fix a final time . Let Optimal control problem: find a control that maximizes Intuition: maximize the distance to the origin.**Optimal Control Approach and Stability**Theorem The BCS is GAS iff**Edge of Stability**GAS The BCS: Consider original BCS GAS 24**Edge of Stability**GAS The BCS: Consider original BCS GAS Definition: k* is the minimal value of k>0 such that GAS is lost. 25**Edge of Stability**The BCS: Consider Definition: k* is the minimal value of k>0 such that GAS is lost. The system is said to be on the edge of stability. 26**Edge of Stability**The BCS: Consider Definition: k* is the minimal value of k>0 such that GAS is lost. k k 0 1 k* 0 1 k* Proposition: our original BCS is GAS iff k*>1. 27**Edge of Stability**The BCS: Consider Proposition: our original BCS is GAS iff k*>1. → we can always reduce the problem of analyzing GUAS to the problem of determining the edge of stability. 28**Edge of Stability When n=2**Consider The trajectory x* corresponding to u*: A closed periodic trajectory 29**Solving Optimal Control Problems**is a functional: Two approaches: • The Hamilton-Jacobi-Bellman (HJB) equation. • The Maximum Principle.**Solving Optimal Control Problems**1. The HJB equation. Intuition: there exists a function and V can only decrease on any other trajectory of the system.**The HJB Equation**Find such that Integrating: or An upper bound for , obtained for the maximizing (HJB).**The HJB for a BCS:**Hence, In general, finding is difficult. Note: u* depends on only.**The Maximum Principle**Let Then, Differentiating we get A differential equation for with a boundary condition at**Summarizing,**The WCSL is the maximizing that is, We can simulate the optimal solution backwards in time.**Result #1 (Margaliot & Langholz, 2003)**An explicit solution for the HJB equation, when n=2, and {A,B} is on the “edge of stability”. This yields an easily verifiable necessary and sufficient condition for stability of second-order switched linear systems.**Basic Idea**The HJB eq. is: Thus, Let be a first integral of that is, Then is a concatenation of two first integrals and**Example:**where and**Nonlinear Switched Systems**with GAS. Problem: Find a sufficient condition guaranteeing GAS of (NLDI).**Lie-Algebraic Approach**For the sake of simplicity, we present the approach for LDIs, that is, and**Commutation and GAS**Suppose that A and B commute, AB=BA, then Definition: The Lie bracketof Ax and Bx is [Ax,Bx]:=ABx-BAx. Hence, [Ax,Bx]=0 implies GAS.**Lie Brackets and Geometry**Consider A calculation yields:**Geometry of Car Parking**This is why we can park our car. The term is the reason it takes so long.**Nilpotency**We saw that [A,B]=0 implies GAS. What if [A,[A,B]]=[B,[A,B]]=0? Definition: k’th order nilpotency - all Lie brackets involving k terms vanish. [A,B]=0 → 1st order nil. [A,[A,B]]=[B,[A,B]]=0 → 2nd order nil.**Nilpotency and Stability**We saw that 1st order nilpotency Implies GAS. A natural question: Does k’th order nilpotency imply GAS?**Some Known Results**Switched linear systems: • k=2 implies GAS (Gurvits,1995). • k order nilpotency implies GAS (Liberzon, Hespanha, and Morse, 1999).(The proof is based on Lie’s Theorem) Switched nonlinear systems: • k=1 implies GAS. • An open problem: higher orders of k? (Liberzon, 2003)**A Partial Answer**Result #2 (Margaliot & Liberzon, 2004) 3rd order nilpotency implies GAS. Proof: Consider the WCSL Define the switching function**Differentiating m(t) yields**2nd order nilpotency no switching in the WCSL! Differentiating again, we get 3rd order nilpotency up to a single switching in the WCSL.**Singular Arcs**If m(t)0, then the Maximum Principle provides no direct information. Singularity can be ruled out using the auxiliary system.**Summary**• Parking cars is an underpaid job. • Switched systems and differential inclusions are important in various scientific fields, and pose interesting theoretical questions. • Stability analysis is difficult. A natural and useful idea is to consider the worst-case trajectory.