An INTRODUCTION to SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign Based on joint work with J.P. Hespanha (UCSB) and A.S. Morse (Yale) UTC-IASE online seminar, July 21-22, 2014 1 of 34
SWITCHING CONTROL y u Plant: P y u P Classical continuous feedback paradigm: C y u P But logical decisions are often necessary: C 1 C 2 l o g i c 2 of 34
REASONS for SWITCHING • Nature of the control problem • Sensor or actuator limitations • Large modeling uncertainty • Combinations of the above 3 of 34
REASONS for SWITCHING • Nature of the control problem • Sensor or actuator limitations • Large modeling uncertainty • Combinations of the above 4 of 34
MODELING UNCERTAINTY unmodeled dynamics parametric 0 uncertainty Also, noise and disturbance Adaptive control (continuous tuning) vs. supervisory control (switching) 5 of 34
EXAMPLE Scalar system: , otherwise unknown (purely parametric uncertainty) stable not implementable Controller family: Could also take controller index set 6 of 34
SUPERVISORY CONTROL ARCHITECTURE Supervisor candidate controllers u 1 Controller y u u 2 Plant Controller . . . u m Controller . . . – switching signal, takes values in – switching controller 7 of 34
TYPES of SUPERVISION • Prescheduled (prerouted) • Performance-based (direct) • Estimator-based (indirect) 8 of 34
TYPES of SUPERVISION • Prescheduled (prerouted) • Performance-based (direct) • Estimator-based (indirect) 9 of 34
OUTLINE • Basic components of supervisor • Design objectives and general analysis • Achieving the design objectives (highlights) 10 of 34
OUTLINE • Basic components of supervisor • Design objectives and general analysis • Achieving the design objectives (highlights) 11 of 34
SUPERVISOR y 1 e 1 y Multi- y 2 e 2 estimation errors: . . . Estimator e p y p u . . . . . . Want to be small Then small indicates likely 12 of 34
EXAMPLE Multi-estimator: exp fast => 13 of 34
EXAMPLE disturbance Multi-estimator: exp fast => 14 of 34
STATE SHARING Bad! Not implementable if is infinite The system produces the same signals 15 of 34
SUPERVISOR y 1 e 1 y 1 Monitoring Multi- y 2 e 2 2 . . Signals . Estimator e p y p u Generator p . . . . . . . . . Examples: 16 of 34
EXAMPLE Multi-estimator: – can use state sharing 17 of 34
SUPERVISOR y 1 e 1 y 1 Monitoring Multi- y 2 e 2 Switching 2 . . Signals . Estimator Logic e p y p u Generator p . . . . . . . . . Basic idea: , controllers: Justification? Plant gives stable small => small => plant likely in => closed-loop system (“certainty equivalence”) 18 of 34
SUPERVISOR y 1 e 1 y 1 Monitoring Multi- y 2 e 2 Switching 2 . . Signals . Estimator Logic e p y p u Generator p . . . . . . . . . Basic idea: , controllers: Justification? Plant gives stable small => small => plant likely in => closed-loop system only know converse! small => Need: gives stable closed-loop system This is detectability w.r.t. 19 of 34
DETECTABILITY Linear case: plant in closed loop with view as output Want this system to be detectable “output injection” matrix 9 L q : A q ¡ L q C q is Hurwitz x = (A q ¡ L q C q )x + L q e q _ asympt. stable 20 of 34
SUPERVISOR y 1 e 1 y 1 Monitoring Multi- y 2 e 2 Switching 2 . . Signals . Estimator Logic e p y p u Generator p . . . . . . . . . We know: is small Switching logic (roughly): This (hopefully) guarantees that is small small => stable closed-loop switched system Need: This is switched detectability 21 of 34
DETECTABILITY under SWITCHING plant in closed Switched system: loop with view as output Want this system to be detectable: Assumed detectable for each frozen value of Output injection: need this to be asympt. stable Thus needs to be “ non-destabilizing ” : • switching stops in finite time • slow switching (on the average) 22 of 34
SUMMARY of BASIC PROPERTIES Multi-estimator: 1. At least one estimation error ( ) is small • when • is bounded for bounded & Candidate controllers: 2. For each , closed-loop system is detectable w.r.t. Switching logic: 3. is bounded in terms of the smallest 4. Switched closed-loop system is detectable w.r.t. provided this is true for every frozen value of : for 3, want to switch to conflicting for 4, want to switch slowly or stop 23 of 34
SUMMARY of BASIC PROPERTIES Multi-estimator: 1. At least one estimation error ( ) is small • when • is bounded for bounded & Candidate controllers: 2. For each , closed-loop system is detectable w.r.t. Switching logic: 3. is bounded in terms of the smallest 4. Switched closed-loop system is detectable w.r.t. provided this is true for every frozen value of 1 + 3 => is small Analysis: => state is small 2 + 4 => detectability w.r.t. 24 of 34
OUTLINE • Basic components of supervisor • Design objectives and general analysis • Achieving the design objectives (highlights) 25 of 34
CANDIDATE CONTROLLERS y fixed Plant u Controller y e Multi- q q estimator y 26 of 34
CANDIDATE CONTROLLERS y fixed Plant P u Controller y e Multi- q q estimator C y E Linear: overall system is detectable w.r.t. if i. system inside the box is stable ii. plant is detectable => , , Need to show: C E P => => => => C , P E ii i 27 of 34
CANDIDATE CONTROLLERS y fixed Plant P u Controller y e Multi- q q estimator C y E Linear: overall system is detectable w.r.t. if i. system inside the box is stable ii. plant is detectable Nonlinear: same result holds if stability and detectability are interpreted in the ISS / OSS sense: external signal 28 of 34
CANDIDATE CONTROLLERS y fixed Plant P u Controller y e Multi- q q estimator C y E Linear: overall system is detectable w.r.t. if i. system inside the box is stable ii. plant is detectable Nonlinear: same result holds if stability and detectability are interpreted in the integral-ISS/OSS sense: 29 of 34
CANDIDATE CONTROLLERS y fixed Plant P u Controller y e Multi- q q estimator C y E Linear: overall system is detectable w.r.t. if i. system inside the box is stable ii. plant is detectable For minimum-phase plants, it is enough to ask that the system inside the box be output-stabilized 30 of 34
SWITCHING LOGIC: DWELL-TIME – dwell time – monitoring signals Initialize Wait time units Find ? no yes Detectability is preserved if is large enough Obtaining a bound on in terms of is harder Not suitable for nonlinear systems (finite escape) 31 of 34
SWITCHING LOGIC: HYSTERESIS – hysteresis constant – monitoring signals Initialize Find ? yes no or (scale-independent) 32 of 34
SWITCHING LOGIC: HYSTERESIS Initialize Find ? yes no finite, bounded switching stops in finite time => This applies to exp fast, Linear, bounded average dwell time => 33 of 34
TOY EXAMPLE: PARKING PROBLEM x 2 x cos p 1 w 1 1 x w sin p 1 2 1 w p 2 2 x 1 Unknown parameters correspond to the 1 , p p 2 radius of rear wheels and distance between them 34 of 34
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