Announcements. Affine dynamics: Example #1. Review: Multiple Lyap. Fcns. Review and Examples: Linear/PWA Quad. Lyapunov Theory

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1 EECE 571M/491M, Spring 2007 Lecture 10 Review and Examples: Linear/PWA Quad. Lyapunov Theory Meeko Oishi, Ph.D. Electrical and Computer Engineering University of British Columbia, BC Announcements Reminder: 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 HW #2 now posted; due Feb. 15 Project proposal due Feb Johansson and Rantzer (1998), DeCarlo et al (2000) 1 2 Review: Multiple Lyap. Fcns. Affine dynamics: Example #1 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 Saturation of a linear system sat(u) V(x) u min u max u 3 4

2 Affine dynamics: Example #1 Piecewise Affine QLF First identify the affine dynamics within each partition First define the sets of indices 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 Then encode the state-space partitions x 2 u = -Kx > u max u = -Kx < u min And the partition boundaries x 1 X 1 X 2 X Piecewise Affine QLF Discontinuous PWA QLF Piecewise Affine Quadratic Lyapunov theorem If there exist symmetric matrices U i, W i, M where U i, W i must have non-negative elements, such that Discontinuous Piecewise Affine Quadratic Lyapunov Thm. If there exist symmetric matrices where U i, W i must have non-negative elements, and vectors such that satisfy the 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 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 7 8

3 Today s lecture Review Piecewise affine quadratic Lyapunov function Discontinuous affine quadratic Lyapunov function Summary of techniques so far Important theorem categories Continuous systems Lyapunov function Quadratic lyapunov function (for differential equations) Discrete quadratic Lyapunov function (for difference equations) Examples Example #1: Switched stability of a VSTOL aircraft M. Oishi and C. Tomlin, IEEE CDC 1999 Example #2: Backing up a truck-trailer C. Altafini, A. Speranzon, K. Johansson, HSCC 2002 C. Altafini, A. Speranzon, B. Wahlberg, IEEE TRA 2001 Example #3: Biological cell dynamics R. Ghosh, C. Tomlin, HSCC 2001 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 9 10 Important system/lf categories Multiple Lyapunov functions theorem Lyapunov function Continuous across modes / Discontinuous across modes Quadratic / Extended quadratic / Other non-quadratic Continuous dynamics Linear / Affine / Other non-linear Differential eqns / Difference eqns All stable / Some unstable Transition function Identity reset map / Non-identity reset map Switching scheme Specified / Irrelevant Lyapunov function Continuous across modes / Discontinuous across modes Quadratic / Extended quadratic / Other non-quadratic Continuous dynamics Linear / Affine / Other non-linear Differential eqns / Difference eqns All stable / Some unstable Transition function Identity reset map / Non-identity reset map Switching scheme Specified / Irrelevant 11 12

4 Common Lyapunov function theorem (we will study this in more detail ) Lyapunov function Continuous across modes / Discontinuous across modes Quadratic / Extended quadratic / Other non-quadratic Continuous dynamics Linear / Affine / Other non-linear Differential eqns / Difference eqns All stable / Some unstable Transition function Identity reset map / Non-identity reset map Switching scheme Specified / Irrelevant Globally quadratic Lyapunov theorem Lyapunov function Continuous across modes / Discontinuous across modes Quadratic / Extended quadratic / Other non-quadratic Continuous dynamics Linear / Affine / Other non-linear Differential eqns / Difference eqns All stable / Some unstable Transition function Identity reset map / Non-identity reset map Switching scheme Specified / Irrelevant Converse globally quadratic Lyapunov theorem Linear quadratic Lyapunov theorem Lyapunov function Continuous across modes / Discontinuous across modes Quadratic / Extended quadratic / Other non-quadratic Continuous dynamics Linear / Affine / Other non-linear Differential eqns / Difference eqns All stable / Some unstable Transition function Identity reset map / Non-identity reset map Switching scheme Specified / Irrelevant Lyapunov function Continuous across modes / Discontinuous across modes Quadratic / Extended quadratic / Other non-quadratic Continuous dynamics Linear / Affine / Other non-linear Differential eqns / Difference eqns All stable / Some unstable Transition function Identity reset map / Non-identity reset map Switching scheme Specified / Irrelevant 15 16

5 Piecewise affine quadratic Lyapunov theorem Lyapunov function Continuous across modes / Discontinuous across modes Quadratic / Extended quadratic / Other non-quadratic Continuous dynamics Linear / Affine / Other non-linear Differential eqns / Difference eqns All stable / Some unstable Transition function Identity reset map / Non-identity reset map Switching scheme Specified / Irrelevant Discontinuous piecewise affine quadratic Lyapunov theorem Lyapunov function Continuous across modes / Discontinuous across modes Quadratic / Extended quadratic / Other non-quadratic Continuous dynamics Linear / Affine / Other non-linear Differential eqns / Difference eqns All stable / Some unstable Transition function Identity reset map / Non-identity reset map Switching scheme Specified / Irrelevant Lyapunov thms yet to come Additional results are possible for specific types of dynamics, switching scheme, state dimension, mode cardinality Lyapunov function Continuous across modes / Discontinuous across modes Quadratic / Extended quadratic / Sum of squares / Other non-quadratic Continuous dynamics Linear / Affine / Polynomial / Other non-linear Differential eqns / Difference eqns All stable / Some unstable Dimension n = 2 / Dimension n > 2 Transition function Identity reset map / Non-identity reset map Switching scheme Specified / Irrelevant Discrete modes Cardinality = 2 / Cardinality > 2 19 Vectored thrust aircraft Three modes of operation Conventional (CTOL) Transition Vertical (VTOL) Nonlinear system approximated by hybrid system Inputs Thrust (T) Pitch acceleration (d 2 /dt 2 ) (through the elevators) Thrust angle acceleration (d 2 "/dt 2 ) (through torque applied to the nozzles) State vector Horizontal position x, horizontal speed dx/dt Vertical position z, vertical speed dz/dt Pitch, pitch rate d/dt Thrust angle ", thrust angle rate d"/dt 20

6 VTOL: Vertical Take-Off/Landing Thrust is 900 from horizontal Nonlinear dynamics CTOL: Conventional Take-Off/Landing Thrust is 00 from horizontal Nonlinear dynamics Closed-loop dynamics linearized by nonlinear control law which tracks desired trajectories in (x, z) (assuming # = 0) Resulting error dynamics satisfy With constants chosen by placing closed-loop poles at 21 With constants chosen to place closed-loop poles at Closed-loop dynamics linearized by nonlinear control law which tracks desired trajectories in (x, z, ") (assuming # = 0) Resulting error dynamics satisfy With constants chosen to place closed-loop poles at 22 TRANSITION: Short Take-Off/Vertical Landing Thrust is from horizontal Nonlinear dynamics Closed-loop dynamics linearized by nonlinear control law which tracks desired trajectories in (x, z) (assuming # = 0) Resulting error dynamics satisfy 23 Switched nonlinear system Placeholder continuous variables in modes with fewer states VTOL TRANS. CTOL Now find a linear quadratic Lyapunov function to guarantee stability of the switched system Easiest to start with a Common Lyapunov function Which we know exists since is stable. 24

7 A Common Lyapunov function guarantees stability for any switching strategy Short Take-off Maneuver: (TRANSITION -> CTOL) A Common Lyapunov function guarantees stability for any switching strategy Vertical Landing Maneuver: (TRANSITION -> VTOL) Experimental truck and trailer 1:16 scale model of commercial vehicles Unstable nonlinear dynamics Nonholonomic system (non-integrable constraints) Goal: back up along a specified path Control not possible with a single, nonlinear controller Hierarchical, hybrid system: Straight-line tracking controller (forwards and backwards) Arc-tracking controller Linearize dynamics around these motions Straight line tracking This mode is open-loop stable in forward motion (v > 0) But open-loop unstable in backwards motion (v < 0) 27 28

8 Arc-tracking Jack-knife configuration (Trailer cannot be pushed backwards) Constraints With a non-zero equilibrium point, The linearized dynamics are Encode constraints into the domain, to ensure system is not allowed to evolve into jack-knife situation Goal: Design a switched controller to move backwards along a desired path, e.g, stabilize tracking error Scenario 1: Two modes Scenario 2: Three modes Switching surfaces C - and C + Switching surfaces C -, C +, $, %, & Results in unacceptably slow convergence of the state 3 (trailer alignment) Choosing these surfaces is part of the control design problem (regions of attraction) Solution: Try adding a third mode, to improve alignment Surfaces must be chosen to satisfy conditions for hybrid stability

9 Summary Initial conditions near the jack-knife scenario Review of the main Lyapunov theorems so far Examples of hybrid systems Stabilizing a VSTOL aircraft Backing up a tractor-trailer Biological circuits Next few classes: Discrete-time piecewise affine Lyapunov theorems Polynomial dynamics and Sum-of-squares Lyapunov functions Special cases of common Lyapunov functions 33 34