Announcements. Review: Lyap. thms so far. Review: Multiple Lyap. Fcns. Discrete-time PWL/PWA Quadratic Lyapunov Theory

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1 EECE 571M/491M, Spring 2007 Lecture 11 Discrete-time PWL/PWA Quadratic Lyapunov Theory Meeko Oishi, Ph.D. Electrical and Computer Engineering University of British Columbia, BC Johansson and Rantzer (1998); Feng (2006); Trecate, Cuzzola, Mignone, Morari (2002). 1 Announcements! Abbreviated class, Thursday Feb. 15 (class will still meet!)! Upcoming seminar:! Prof. Deborah Estrin, UCLA! Wireless sensing systems: Observing the previously unobservable! 4-5pm, Hugh Dempster Pavilion 310! HW #2 due! Project proposal due -- will be included in final project grade! Last homework will be posted after break! will include Feb. 27 lecture! Midterm, March 1! Open note, in-class, 1.5 hr exam! 20% of final grade! Will not need computers/calculators! Covers everything through Feb. 15 lecture. 2 Review: Lyap. thms so far Review: Multiple Lyap. Fcns.! Important theorem categories! Continuous systems! Quadratic lyapunov function (for differential equations)! Discrete quadratic Lyapunov function (for difference equations)! Hybrid systems! Multiple Lyapunov functions! Common Lyapunov function! Globally quadratic Lyapunov theorem! Converse globally quadratic Lyapunov theorem! Linear quadratic Lyapunov theorem! Piecewise affine quadratic Lyapunov theorem! Discontinuous piecewise affine quadratic Lyapunov theorem V(x) 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 3 4

2 Review: Lyap. thms so far Review: Lyap. thms so far! Multiple Lyapunov functions theorem! Common Lyapunov function theorem (we will study this in more detail ) 5 6 Review: Lyap. thms so far Review: Lyap. thms so far! Globally quadratic Lyapunov theorem! Converse globally quadratic Lyapunov theorem 7 8

3 Review: Lyap. thms so far Today s lecture! Linear quadratic Lyapunov theorem! Review! Summary of techniques so far! Modeling Discrete-time PWA/PWL systems! Discrete-time PWL stability! Global quadratic Lyapunov theorem! Converse global quadratic Lyapunov theorem! Piecewise linear quadratic Lyapunov theorem! Discrete-time PWA stability! Piecewise affine quadratic Lyapunov function 9 10! Discrete-time dynamics! Note that! where! Sets X i partition the state-space R n into closed polyhedral regions! I is the set of indices of the regions! the set of all possible transitions from one region to another is defined as can also be expressed as the linear system X 1 X 3! Note that when a i = 0, the system is discrete-time piecewise linear (not affine) X

4 ! As with continuous-time systems, first consider a common Lyapunov function! Consider the switched system whose switching scheme is unspecified (e.g. arbitrary switching)! Stability of such a system can be proven with the existence of a common Lyapunov function! One type of common Lyapunov function is the global quadratic Lyapunov function! The existence of a common Lyapunov function also proves stability of the piecewise linear system Global Quadratic Lyapunov Theorem (Discretetime)! If there exists a symmetric, positive definite matrix P such that! Then the discrete-time piecewise linear system is exponentially stable Converse Lyapunov Theorem (Discrete-time)! Consider the discrete-time piecewise linear system! If there exists symmetric, positive definite matrices R i such that! Then the discrete-time piecewise linear system! With matrices does not have a global quadratic Lyapunov function

5 ! 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! Consider the case when all state-space partitions contain the origin! The polyhedral sets can be written as inequalities in the state! Where E i can represent either domains or cells where continuous evolution is allowed within each mode.! Goal: Find P i such that! P i > 0 for x in X i! A i T P i A i - P i < 0 for x in X i! A i T P j A i - P i < 0 for x k in X i, x k+1 in X j! Note that V(x) may not be continuous across modes 17 18! Transitions may not occur exactly at known boundaries! Therefore discontinuities in V may occur across mode transitions! V must still decrease across mode transitions: Piecewise Linear Quadratic Lyapunov Theorem (Discrete-time)! If there exists symmetric, positive-definite matrices P i and symmetric, non-negative matrices U i, W i, Q ij, such that! where X i X3 X j! Then the discrete-time piecewise linear system! and x k-1 x k x k+1 is exponentially stable

6 ! Recall! Find a piecewise quadratic Lyapunov function! where! Therefore the domains of polyhedral sets are encoded by! such that 21 22! To solve this problem, find the symmetric matrices! Using the LMI Matlab toolbox, the solution is with non-negative elements! which fulfill the LMIs! Since a piecewise quadratic Lyapunov function exists, the piecewise linear system is exponentially stable

7 Example #2: Piecewise Linear System! Consider the discrete-time piecewise linear system Example #2: Piecewise Linear System! No global quadratic Lyapunov function exists for this system, yet simulations appear to converge to 0.! with stable system matrices (eigenvalues within the unit circle) Example #2: Piecewise Linear System! Define the sets X 1, X 2, X 3, X 4 in terms of the matrices! Now reconsider the case when a i! 0 for some modes (affine dynamics) and X i does not include the origin in some modes.! First define the sets of indices! And solve the corresponding LMI for symmetric, positive definite P i, and symmetric U i, W i, Q ij, with non-negative elements.! Computed solution:! Then encode the state-space partitions! Therefore since a piecewise quadratic Lyapunov function exists, the piecewise linear system is stable.! Note that! e i = 0 for polyhedral sets whose boundaries go through the origin! a i = 0 for modes whose polyhedral sets contain the origin 27 28

8 ! Consider a Lyapunov function with the form! Where the extended state is! Given! Different Lyapunov-like functions in each mode! Disjoint partition of the state-space! Affine or linear dynamics in each mode! Find such that! Furthermore, define or Piecewise Affine Quadratic Lyapunov Theorem (G. Feng)! If there exist symmetric matrices where U i, W i, Q ij have non-negative elements, such that Example #3: Piecewise Affine System! Consider the discrete-time piecewise affine system! With system matrices and vectors X 1 X 2 X 3 X 4! Then the dynamical system dx/dt = A i x + a i, x 0 = x(0), will have piecewise continuous trajectories x(t) which tend to zero exponentially 31! and 32

9 Example #3: Piecewise Affine System! Define the extended-state matrices X 1 X 2 X 3 X 4 Example #3: Piecewise Affine System! Define the system cells through the matrices X 1 X 2 X 3 X 4! Define the set of possible mode transitions! Algorithm to find [E i e i ] : (Same as for continuous-time PWA systems)! Encode domains by E i x +e i " 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 there is only one boundary, augment [E i e i ] with [0 1xn 1]! Otherwise do not change [E i e i ] Summary Example #3: Piecewise Affine System! Computed Lyapunov-like function with! Review of the main Lyapunov theorems so far! Discrete-time PWL systems! Global quadratic Lyapunov theorem! Converse global quadratic Lyapunov theorem! Piecewise quadratic Lyapunov theorem! Discrete-time PWA systems! Piecewise quadratic Lyapunov theorem! Assures exponential stability of the discrete-time PWA system! Next classes:! Feb. 15, abbreviated class! Feb. 27, Hybrid stability with polynomial continuous dynamics! Mar. 1, In-class midterm 35 36