TUTORIAL on LOGIC-BASED CONTROL Part I: SWITCHED CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign MED ’02, Lisbon

OUTLINE Switched Control Systems Stability of Switched Systems Questions, Break

OUTLINE Switched Control Systems Stability of Switched Systems Questions, Break

SWITCHED and HYBRID SYSTEMS Switching can be: State-dependent or Time-dependent Hybrid systems: interaction of continuous and discrete dynamics Switched systems: continuous systems with discrete switchings emphasis on properties of continuous state and is a switching signal where is a family of systems Autonomous or Controlled

SWITCHING CONTROL Classical continuous feedback paradigm: u y P C u y P Plant: But logical decisions are often necessary: The closed-loop system is hybrid u y C1C1 C2C2 l o g i c P

REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above

REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above

PARKING PROBLEM Nonholonomic constraint: wheels do not slip

OBSTRUCTION to STABILIZATION Solution: move away first ?

REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above

OUTPUT FEEDBACK switched system Example: harmonic oscillator

QUANTIZED FEEDBACK PLANT QUANTIZER CONTROLLER u x q(x) x sensitivity M values

OBSTRUCTION to STABILIZATION Assume: fixed Asymptotic stabilization is impossible

MOTIVATING EXAMPLES 1. Temperature sensor normal too low too high 2. Camera with zoom Tracking a golf ball 3. Coding and decoding

VARYING the SENSITIVITY zoom out zoom in Why switch ? More realistic Easier to design and analyze Robust to time delays

LINEAR SYSTEMS is GAS:Assume: Along solutions of quantization error Then can achieve GAS we have for some s.t.

SWITCHING POLICY We have level sets of V.

NONLINEAR SYSTEMS is GAS: Assume: s.t. Need: along solutions of wh er e is pos. def., increasing, and unbounded (this is input-to-state stability wrt measurement error) Then can achieve GAS quantization error

EXTENSIONS and APPLICATIONS Arbitrary quantization regions Active probing for information Output and input quantization Relaxing the assumptions Performance-based design Application to visual servoing

REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above

MODEL UNCERTAINTY parametric uncertainty unmodeled dynamics Also, noise and disturbances Adaptive control (continuous tuning) vs. supervisory control (switching)

SUPERVISORY CONTROL switching signal Plant Supervisor Controller 2 Controller m Controller 1 u1u1 u2u2 umum y u candidate controllers......

STABILITY of SWITCHED SYSTEMS switching stops in finite time Stable if: unstable slow switching (on the average) “locally confined” switching common Lyapunov function

REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above

PARKING PROBLEM under UNCERTAINTY Unknown parameters correspond to the radius of rear wheels and distance between them p 1 p 2 p 1

SIMULATION

OUTLINE Switched Control Systems Stability of Switched Systems Questions, Break

OUTLINE Switched Control Systems Stability of Switched Systems Questions, Break

TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching

UNIFORM STABILITY is a piecewise constant switching signal where is a family of GAS systems Want GUAS w.r.t. GUES: :

COMMON LYAPUNOV FUNCTION is GUAS if and only if s.t. is not enough Example: if Usually we take P compact and f p continuous Corollary: is GAS

SWITCHED LINEAR SYSTEMS LAS for every GUES but not necessarily quadratic common Lyapunov function

COMMUTING STABLE MATRICES => GUES quadratic common Lyap fcn: … t …

LIE ALGEBRAS and STABILITY Lie algebra: Lie bracket: g is nilpotent if s.t. g is solvable if s.t.

SOLVABLE LIE ALGEBRA => GUES Lie’s Theorem: triangular form is solvable quadratic common Lyap fcn: D diagonal

SOLVABLE + COMPACT => GUES Levi decomposition: radical (max solvable ideal) is compact => GUES quadratic common Lyap fcn

SOLVABLE + NONCOMPACT => GUES is not compact a set of stable generators for that gives GUES a set of stable generators for that leads to an unstable system Lie algebra doesn’t provide enough information

NONLINEAR SYSTEMS Commuting systems => GUAS Linearization ???

REMARKS on LIE-ALGEBRAIC CRITERIA Checkable conditions In terms of the original data Independent of representation Not robust to small perturbations

SYSTEMS with SPECIAL STRUCTURE Triangular systems Linear => GUES Nonlinear: need ISS conditions Passive: => GUAS Small gain: => GUES 2D systems convex combs of stable quadratic common Lyap fcn Feedback systems u y -

MULTIPLE LYAPUNOV FUNCTIONS GAS respective Lyapunov functions t is GAS Very useful for analysis of state-dependent switching

MULTIPLE LYAPUNOV FUNCTIONS t is GAS decreasing sequence

DWELL TIME The switching times satisfy dwell time GES respective Lyapunov functions t Need: must be

AVERAGE DWELL TIME # of switches on average dwell time dwell time: cannot switch twice if no switching: cannot switch if => is GAS if

SWITCHED LINEAR SYSTEMS GUES over all with large enough Finite induced norms for The case when some subsystems are unstable

STABILIZATION by SWITCHING both unstable Assume: stable for some So for each : either or

UNSTABLE CONVEX COMBINATIONS Can also use multiple Lyapunov functions LMIs