Announcements. Review. Announcements. Piecewise Affine Quadratic Lyapunov Theory. EECE 571M/491M, Spring 2007 Lecture 9

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1 EECE 571M/491M, Spring 2007 Lecture 9 Piecewise Affine Quadratic Lyapunov Theory Meeko Oishi, Ph.D. Electrical and Computer Engineering University of British Columbia, BC Announcements Lecture review examples Demonstrate main concepts from previous lecture 2 each Sign-up sheet minutes, informal Homework #2 posted, due Thurs Feb 15 Project proposals (1-2 paragraphs) due Thurs Feb 15 moishi@ece.ubc.ca Tomlin LN 6, Johansson and Rantzer (1998), DeCarlo et al (2000) 1 2 Announcements Review IEEE Control Systems Society, Vancouver chapter seminar Explicit robust model predictive control in the recursive closed-loop prediction Friday, Feb. 9, 1pm, Kaiser 2020 Dr. Danlei Chu, Research Scientist, Honeywell Canada Global Quadratic Lyapunov functions Easiest to start with Irrespective of switching strategy Common Lyapunov function Solved by LMIs Piecewise Quadratic Lyapunov functions When no global quadratic Lyapunov function exists Solved by LMIs Switched linear dynamics 3 4

2 Review: Multiple Lyap. Fcns. Review: GQLF Theorem Consider a Lyapunov-like function V(q,x): When the system is evolving in mode q, V(q,x) must decrease or maintain the same value Every time mode q is re-visited, the value V(q,x) must be lower than it was last time the system entered mode q. When the system switches into a new mode q, V may jump in value For inactive modes p, V(p,x) may increase Requires solving for V directly Theorem: Globally Quadratic Lyapunov Function Consider a hybrid automaton H = (Q, X, f, R, Init, Dom) with equilibrium x* = 0. Assume that V(x) If there exists a matrix such that then x* = 0 is exponentially stable. 5 6 Review: GQLF Theorem Review: GQLF Example Theorem: Converse Globally Quadratic Lyapunov Function Consider a hybrid automaton H = (Q, X, f, R, Init, Dom) with equilibrium x* = 0. Assume that Phase plane plot If there exists matrices such that then no globally quadratic Lyapunov function exists. A1 A2 7 8

3 Review: Piecewise-linear QLF Review: Piecewise-linear QLF When no common Lyapunov function exists Different Lyapunov-like functions in each mode Disjoint partition of the state-space Described by intersections of hyperplanes Domain is set of convex polyhedra Linear dynamics in each mode Goal: Find P i such that P i > 0 for x in X i S-procedure transforms the requirements for a piecewise-linear quadratic lyapunov function into something that be solved through LMI techniques Introduce new symmetric matrices U i, M where U i must have non-negative elements To find a Lyapunov function V(q,x) solve the following LMI for the new matrices U i, M: P i + E i W i E i > 0 for x in R n E i x 0 for x in X i A T P i +P i A + E i U i E i < 0 for x in R n F i x = F k x for x on bdry P i = F i M F of X i, X k i A T P i +P i A < 0 for x in X i and V(p,x) maintains continuity across modes 9 These three conditions assure that V(q,x) is positive definitn the active mode, d/dt ( V(q,x) ) is negative definitn the active mode, and V(q,x) is continuous across transitions 10 Today s lecture Affine dynamics Review Global linear quadratic Lyapunov function Piecewise linear quadratic Lyapunov function Example Affine dynamics: Equilibrium at x* = - A \ b Piecewise affine quadratic Lyapunov function (Johansson and Rantzer, 1998) LMI Toolbox Relaxed PWA QLF (QLF with discontinuities) Let Sliding modes 11 12

4 Affine dynamics: Example #1 Affine dynamics: Example #1 Saturation of a linear system Will the system remain stable despite saturation? Standard stability analysis assumes unbounded control input Not possibln physical systems Piecewise affine stability analysis can answer this question sat(u) x 2 u = -Kx > u max u = -Kx < u min u min u max u x Consider the case when State-spacs partitioned into disjoint regions, only some of which may contain the origin Reset map for x is thdentity Discontinuities in extended state (e.g., in the last element of ) Dynamics are affine or linear and stabln each mode Only one transition is possible from each mode For modes with affine dynamics, consider the Lyapunov function Lyapunov function depends on the mode, and whether or not the associated domain in that mode contains the origin Sufficient condition for stability For modes with linear dynamics, let Q i =0 and R i =0 such that 15 16

5 First define the sets of indices Then encode the state-space partitions And the partition boundaries Cell boundaries vs. domains To restrict x in X i to a linear (as opposed to affine) function in x for parts of the computation Algorithm to find [E i ] : Encode domains by E i x + 0 For i in I 0 (domain includes the origin), eliminate all domain boundaries that do not go through the origin. Replace these rows with 0. For i in I 1 (domain does not include the origin) If thers only one boundary, augment [E i ] with [0 1xn 1] Otherwise do not change [E i ] This is equivalent to encoding domain directly into [E i ] [x; 1] 0 Assuming that [E i ] are found in this way Goal: Find a Lyapunov function V(q,x) which assures that V(q,x) is positive definitn the active mode, d/dt ( V(q,x) ) is negative definitn the active mode, and V(q,x) is continuous across transitions under piecewise affine dynamics over polyhedral sets S-procedure As with the piecewise linear QLF theory Allows use of LMI solvers Piecewise Affine Quadratic Lyapunov theorem If there exist symmetric matrices U i, M where U i must have non-negative elements, such that satisfy the linear matrix inequalities Eliminates direct reference to polyhedral sets X i Encodes state partitions into linear matrix inequalities Then the piecewise affine dynamical system dx/dt = A i x + b i, x 0 = x(0), will have piecewise continuous trajectories x(t) which tend to zero exponentially 19 20

6 Affine dynamics: Example #1 Relaxing First identify the affine dynamics within each partition Then express the partitions in terms of the matrices E i and the switching boundaries in terms of F i Formulate the LMI to be solved Consider systems for which X i are not disjoint Transitions may allow a system to evolve under different dynamics in the same region of the statespace, depending on the switching logic x 2 Transitions can be enabled and forced u = -Kx > u max u = -Kx < u min Discontinuities in V(q,x) are allowed so long as V(q,x) decreases across mode transitions x 1 X 1 X 2 X Relaxing Relaxing General switched system with piecewise affine dynamics Note that the hyperplane g i,i = 0 For mode q i, the domain (and its boundary) are encoded into matrices E i (as for standard PWA system) And let Discontinuities in V(q,x) are allowed so long as V(q,x) decreases across mode transitions: Switching surfacs encoded in new hyperplane g i,k such that Expressed as an LMI in terms of P i, P k, z i,k 23 24

7 Discontinuous PWA QLF Example #2 Discontinuous Piecewise Affine Quadratic Lyapunov Thm. If there exist symmetric matrices where U i must have non-negative elements, and vectors such that Consider the dynamical system Then the dynamical system dx/dt = A i x + b i, q k = R(q i, x), x 0 = x(0), will have piecewise continuous trajectories x(t) which tend to zero exponentially From M. Johansson, PhD thesis, 1999, Lund University Example #2 Example #2 Define the matrices To compute the positive definite matrices From M. Johansson, PhD thesis, 1999, Lund University From M. Johansson, PhD thesis, 1999, Lund University 27 28

8 Sliding Modes Sliding Modes Implicit assumption that piecewise-affine dynamics do not chatter However, attractive sliding modes are possible Proving stability of a sliding mods more difficult, but still possible Can also be formulated as LMI constraint Detection of sliding modes Requires extension of solution in the sense of Filippov (Solution lies within convex hull of dynamics) Define a sliding surface of width -> 0 Show decreasing Lyapunov function along sliding surface Example #3: x 1 = Summary Piecewise Affine Quadratic Lyapunov functions Switched affine dynamics Continuous Lyapunov function Solved by LMIs Discontinuous Piecewise Quadratic Lyapunov functions Switched affine dynamics Discontinuous Lyapunov function Solved by LMIs Sliding modes in piecewise linear/affine systems 31