Robust Hybrid Control Systems Andy Teel with Rafal Goebel, Ricardo Sanfelice, Chaohong Cai Center for Control, Dynamical Systems, and Computation Department of Electrical and Computer Engineering University of California, Santa Barbara
Outline “Hybrid” arcs and systems Robustness: illustrations and general statements “Classical” stability analysis tools in the hybrid setting Control examples Smooth “patchy” control Lyapunov functions Conclusions
“Hybrid” arcs x(j) x(t) Discrete Continuous arc arc P. Collins, MTNS ’04 USCB, NOLCOS ’04 Goebel/T, Automatica ‘06 Not just for bookkeeping or pedagogy. Hybrid arc Provides a natural way of characterizing convergence of solutions: graphical Multiple jumps at Domain convergence (from set-valued analysis). the same “t” allowed.
Robustness from graphical convergence Looking for hybrid systems for which each sequence of solutions has a subsequence converging to a solution. This property comes “for free” for continuous ODEs and difference equations. 15 10 x 1 [m] A ball bouncing from a 5 sequence of decreasing heights. 0 0 0 2 t z3 4 t z2 5 j [jumps] t [ordinary time] 6 t z1 8 10
Hybrid arcs generated by data System data: x may contain variables = ∈ x f x x C � ( ) taking values in a discrete set, timers, etc. + = ∈ x g x x D ( ) “Lateral” evolution 1) ∃ τ > τ ∈ t j t j x if : ( , ), ( , ) dom “Out of the screen” evolution 2) if Now looking for conditions on data to guarantee that each sequence of solutions has a subsequence converging to a solution. For simplicity, f and g are single-valued and continuous…
Bouncing Ball, First Pass � = < h h 0 , 0 ? � � � = − h g � � � + = − γ h h γ ∈ ( 0 , 1 ) { } { } � � = = < � D h h h h = > C h h h : ( , ) : 0 , 0 : ( , ) : 0 1 1
Bouncing Ball, Pass one (cont’d) Ball bouncing from a sequence of decreasing heights: converges to an instantaneous Zeno solution that remains at zero height. But this is not a solution of the BB since the origin does not belong to D . 15 10 x 1 [m] 5 0 0 0 2 t z3 4 t z2 5 j [jumps] t [ordinary time] 6 t z1 8 10
Bouncing Ball, Second Pass � = ≤ h h 0 , 0 ? � � � = − h g � � � + = − γ h h γ ∈ ( 0 , 1 ) { } { } � � = = ≤ � D h h h h = ≥ C h h h : ( , ) : 0 , 0 : ( , ) : 0 2 2 C = D = C D 2 1 2 1 Admits instantaneous Zeno solution at origin.
Conditions on data for convergence property System data: If C and D are closed then each sequence of solutions has a subsequence converging to a solution. Otherwise, no guarantee. This leads to a natural notion of “generalized solutions” (a la Filippov or Krasovskii for discontinuous ODEs) for hybrid systems: = ∈ x f x x C � ( ) Hybrid “Krasovskii” solutions satisfy: + = ∈ x g x x D ( ) . Natural from a control point of view since they agree with “hybrid Hermes” solutions, i.e., the zero noise graphical limit of solutions to = + + ∈ x f x e x e C � ( ) e + = + + ∈ x g x e x e D ( ) .
On “generalized” solutions Biases against (stemming from ODEs): 1) “I don’t work with discontinuous continuous-time systems, so I have no use for generalized solutions.” 2) “The solution notion is flawed because it is too restrictive; e.g., it prevents point stabilization by state feedback for mobile robots.” Rejoinders: 1) For hybrid systems, generalized solutions are relevant even for systems without discontinuities. 2) Every asymptotically controllable nonlinear system (e.g., the mobile robot model) can be stabilized using hybrid feedback with closed flow and jump sets, i.e., using generalized solutions. 3) Value added: Converse Lyapunov theorems, LaSalle’s invariance principle, generic robustness of asymptotic stability become free.
Generalized solutions matter, Example 1 � � 1 1 = C R D = f x x 2 : \ � � ( ) − { } � � 1 1 = = D x x x : ( , ) : 0 . � � 1 2 2 0 0 sequence of = < ε << g x x � � ( ) , 0 1 − ε solutions � � 0 converging to non-solution??? = = D D C R 2 ( , ) Generalized solutions: original solutions plus: and combinations thereof… “Objection! I would implement with ‘zero crossing detection’ (in Simulink.)” { } = ≥ C x x x Or: make D “thick” = x : : 0 � 0 Reply: Put it in 2 3 3 { } = = + D x x = − ε x x perhaps a sector. : : 0 . your model, e.g. sgn( ) 3 1 2 Benefits: Explicit robustness, Lyapunov function exists, LaSalle available.
Generalized solutions matter, Example 2 Target acquisition with Mode 1: drive to obstacle avoidance Mode 2: drive away from = × C R Q on 2 .
Extra generalized solutions = × C R Q 2 Generalized solutions: Simple way to get rid = + D D : inside of circle it describes. new 1 , 1 of extra solutions: = + D D : outside of circle it describes. new (preferred to zero cross 2 , 2 detection or thickening here) = C R D 2 : \ new 1 1 , Benefits as before: = = × ∪ × C R D C C C 2 : \ , ( { 1 }) ( { 2 }) Lyapunov, LaSalle,… new 2 2 , 1 2
LaSalle’s invariance principle Theorem: If ∇ ≤ ≤ ∈ V x f x u x x C ( ), ( ) ( ) 0 c − ≤ ≤ ∈ V g x V x u x x D ( ( )) ( ) ( ) 0 d then each complete, bounded trajectory converges to the largest weakly invariant set contained in ( ) ( ) − − − − = ∩ ∪ ∩ M V r u u g u r 1 1 1 1 : 0 [ ( 0 ) ( ( 0 ))] for some r c d d C Example: � � 0 1 = = = T f x g x x V x x x � � ( ) ( ) , ( ) − � � 1 0 (solutions are not unique) D M Largest weakly invariant set contained in r excludes all points in green except the one on the vertical axis.
A Converse Lyapunov theorem Theorem: If a compact set is asymptotically stable (locally stable + attractive) (using generalized solutions) then there exists a smooth Lyapunov function. ω → R O : Let be ≥ 0 ● positive definite with respect to the compact set, ● proper w.r.t. O , which is an open set related to the basin of attraction, C ∪ D the latter which is open relative to . → α α ∈ V O R K There exists a smooth function : & , : ≥ ∞ 0 1 2 ( ) ( ) α ω ≤ ≤ α ω ∈ x V x x x O 1 ) ( ) ( ) ( ) 1 2 ∇ ≤ − ∈ V x f x V x x C 2 ) ( ), ( ) ( ) ≤ − ∈ V g x V x x D 3 ) ( ( )) exp( 1 ) ( ) Significance: Can be used to show a wide variety of robustness properties. ( ) ( ( ) ) − ω ≤ α − − α ω x t j t j x 1 Note: ( , ) exp( ) ( ( 0 , 0 ) 1 2
Autonomous vs. exogenous perturbations Lyapunov functions can be used = + δ + δ F x f x B B ( ) : co ( ) to show robustness to δ = + δ + δ G x g x B B ( ) : ( ) “autonomous” perturbations, i.e., δ { ( ) } = + δ ∩ ≠ ∅ C x x B C : : inflations of (f,g,C,D): δ { ( ) } = + δ ∩ ≠ ∅ D x x B D : : δ Local separation principles attainable. = + + ∈ x f x e x e C � Solutions with exogenous ( ) disturbances covered by those + = + + ∈ x g x e x e D ( ) . of inflated systems: C e ( 0 , 0 ) However, existence is not > e t t guaranteed for arbitrary signal e : ( , 0 ) 0 D Message: make flow and jump sets overlap, or keep measurements out of the switching conditions, via sample and hold or filtering measurements.
Thermostat control and 2-measure “stability” off mode (q=−1) The set of plant temperatures in the 26 range [20,25] is not forward invariant, 25 ξ (plant temperature) 24 and thus not asymptotically stable. 23 Nevertheless: 22 ( ( ) ) ξ ≤ β ω t j x t j C − ( , ) ( 0 , 0 ) , , 21 2 C − ∩ D − [ 20 , 25 ] β ∈ KLL 20 D − must be positive outside of dashed blue 19 −10 0 10 20 30 40 50 60 70 80 90 100 η (engine temperature) A smooth Lyapunov function ensues: ( ) ( ) on mode (q=+1) α ξ ≤ ≤ α ω V x x 1 ) ( ) ( ) 26 D + 1 2 2 [ 20 , 25 ] 25 C + ∩ D + ∇ ≤ − ∈ V x f x V x x C ξ (plant temperature) 2 ) ( ), ( ) ( ) 24 C + ≤ − ∈ V g x V x x D 23 3 ) ( ( )) exp( 1 ) ( ) 22 and gives several robustness consequences: 21 singular perturbations (actuators or sensors), 20 slowly-varying parameters, sample & hold, 19 −10 0 10 20 30 40 50 60 70 80 90 100 η (engine temperature) small delays, temporal regularization, etc.
“Hybrid handoff” from global to local controller C. Prieur/L. Praly ‘99 Full state feedback: C local : = n D R C \ local local D global Trajectory due to : = n C R D \ global global local controller Extension to output feedback in the presence of norm observers. C. Prieur/T, in preparation
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