Control Synthesis and Reconfiguration for Hybrid Systems

1. Control Synthesis and Reconfiguration for Hybrid Systems October 2001 Sherif Abdelwahed ISIS Vanderbilt University

2. Introduction Hybrid systemscombine discrete-event or mode switching dynamics with continuous evolution according to differential equations or inclusions. States take values over real and discrete sets. discrete Continuous e.g. task, activity, count, status, operation modes e.g. location, velocity, temperature, pressure variables Variables changing in discrete steps; dynamics modeled by finite state structure (automata) Variables changing continuously with time; dynamics modeled by differential equations dynamics Modeling domain can be the abstraction of: continuous systems with phased operation (Robots, diodes), continuous systems controlled by discrete inputs (switches, valves), Coordinating processes (multi-agent systems), etc.

3. Research Directions • Modeling and Simulation • Classify discreet phenomena; existence (non-blocking), uniqueness • (determinism), non-zenoness (liveness) [Branicky, Astrom] • Composition and abstraction operations [Henzenger, Lynch, Sifackis, Varia] • Analysis and Verification • Stability analysis, Lyapunov techniques [Branicky, Michel] • Model checking [Alur, Humdinger Sifakis] • Deductive techniques [Lynch, Manna, Pnueli] • Controller Synthesis • Optimal control [Branicky-Borkar-Mitter] • Supervisory control [Lemmon-Koutsoukos-Antsaklis] • Algorithm synthesis [Maler et.al, Wong-Toi] • Diagnosis • Output/Control observation [Hashtrudi-Zad] • Failure analysis, control reconfiguration [ISIS, Parisini]

4. Basic Hybrid System • Q finite set of discrete states or control modes • G  Q  Q, control graph or discrete transitions x G1,2 q2 22 x =f2(x,v) x Inv2 q1 x =f1(x,v) x Inv1 x R1,2 • For each discrete mode q  Q, a dynamical system: • Xq  nstate space for mode q; possible values • x = (x1, …, xn) of real variables when in mode q. • V  m an input vector of real values • fq= Xq  V +  Xqis the continuous flow of a • vector field (system of o.d.e) on Xq; model how the • system continuously evolves over time t + • Invq  Xq Vset of invariant states, or the domain • of permitted evolution in q. Typically enforce a time • bound on how long the system can stay in mode q. • Initq  Xq set of initial states for mode q. x G2,3 x R3,1 q3 x =f3(x,v) x Inv3 x R2,3 x G3,1 x R3,3 x G3,3 • For each discrete transition (q,q’)  G, • Gq,q’  Xqis theguard set of “trigger event” for the • discrete transition (q,q’) • Rq,q’ : Xq  V Xqis the reset map: state x non- • deterministically reset to some x’ Rq,q’(x); e.g. • identify relation on some variables. Hybrid State Space: XH := q Q ({q}  Xq)

5. q3 q1 q2 to=0 t1 t2 t3 Hybrid System Control Problems • Control Patterns: • Feedback map restricting H to another • hybrid system H’ where L(H’)  L(H) • Discrete-event supervision observing • finite outputs from and supply finite • inputs to H • Game theory extensions to hybrid systems; • balance the effect of disturbance • Mixed-integer quadratic programming: • discrete transitions captured as variables over • finite set of integers • A trajectory of a HS, H from a state • (qo,xo) is a map  : T XH; T is a • connected time sequences. • In every time sequence the discrete state • is assumed fixed • L(H) is the set of all trajectories starting • from any state (q,x) in H • Control Problems: • Safety: Given a region R  Q  Xq find a control strategy to keep the system within R • Reachability: Given two regions S, T  Q Xq find a control strategy to derive the system from any state in S to a state in T • Optimization: Find a control strategy to optimize a given cost function

6. Discrete Abstraction of Hybrid Systems - Approaches The idea to transform a hybrid systems into a finite transitions structure that is equivalent to the initial system with respect to the given problem. Abstraction approaches • Equivalence based • 1. Language • H’ preserves the language generated by • H; L(H’) = L(H) • Based on congruence relation over the set • of strings generated by the system • Very restrictive • 2. Reachability • H’ preserves the reachability properties of • H; f(Pre(q,x)) = Pre(f(q,x)) • Based on bisimulation equivalence • Proved to be decidable for rectangular, • O- minimal hybrid systems. • Even if decidable, the abstracted state • space is typically huge. • Conservative abstractions • H’ extends the behavior of H; L(H’)  L(H) • or H’ partially preserves the reachability relation in H • Can be formulated for PL-constrained and finite input PL-constrained regions • Another approach assumed finite inputs and observations (outputs). The abstracted state space is formed of (finite) sequences of inputs and outputs. • Another approach approximates the reachability set by orthogonal polyhedral. • Can offer different levels of abstraction (by providing more inputs/outputs).

7. Switching Controller of Hybrid Systems - Introduction Control Problem: Given a hybrid system H = (Q, X, Inv, Init, G, R) and a set F  Q  Xq: Find a restricting non-blocking hybrid system H’ such that for every   L(H’) and for every t T, (t) R . • Algorithm: • Po := F  {qQ Invq); k=0 • Repeat • Pk+1 = PkPre(Pk) • k := k+1 • UntilPk = Pk-1 • P* = Pk Controller: The automata H’ = (Q, X, Inv’, Init’, G’, R’) is the solution to the control problem; where Inv’q = Invq P*, G’q,q’ = Gq,q’ P* P*, etc. • Approach: • Restricts the system dynamics: PL systems, xq(t+1) = Aq x(t) • PL-contstrained inputs • Approximate Pk to a finite union of hybercubes, or orthogonal (griddy) polyhedral. • Implementation Issues: • Effective computation of Pre, , and checking equivalence (=). • The algorithm may not terminate even these operations are computable and even for very simple class of systems (PCD).

8. Switching Controller of Hybrid Systems - Implementation • Approximation Algorithm: • Po := F; Vo=V; • Repeat • Vk+1 = Postr(Vk); • Dk = conv(Vk-1  Vk); • Dk = bloat(Dk) • Dk = griddy(Dk) • Dk = Pk-1  Dk • UntilPK+1 = PK • Approximate Solution: • Identify the set F as the convex hull of a set of • points V. • Compute D = conv(V  Postr(V)). This set is • an approximation of Post[0,r](V). • Push the faces of D outward to obtain a bloated • polyhedron D’ which contains the required set • Overapproximating D’ by a griddy polyhedron Post23(x1) Post2r(F) Post2r(x2) Post[0,2r](F) Postr(x1) Postr(F) Postr(x2) Post[0,r](F) x1 F x2

9. Discrete Abstraction of Hybrid Systems – IO approach Behavior abstraction: System behavior: c  (Ud, Yd)T The abstract system behavior l satisfies c  l Therefore for a specification spec and a controller cont we get l cont  spec  c cont spec The system: x(k+1) = f(x(k), w(k),ud(k)), yd(k) = q(x(k)), where, w(k) W:= {w | w  r, ||w|| 1} ud(k)  Ud = {ud1, …, udn} Yd(k)  Yd = {yd1, …, ydm} • Abstraction behaviors: • The abstraction is done by restricting the observation horizon to a finite level l. • The approximate behavior of the system is defined recursively as • l,o = c,o • l,k+1 ={ [bk, (ydj,udi)] | bk  l,k, udi Ud, ydj Yl(bk,l)} • Where • Yl(bk,l) is the set of all possible outputs generated at time k+1 given the string bk,l The abstract system: States: xd(k) = [yd(0)] if k=0 = [yd[0 …k], ud[0..k-1]] if k l = [yd[k-l … k], ud[k-l …k-1]] if k >l Transitions: (xd, ud, x’d) is a transition in Alif it is consistent with the state discription and compatible with continuous the system dynamics

10. Control Reconfiguration of Hybrid Systems - Model • Problem Setting • The System • A hybrid system H with: • Linear cont. dynamics: fq = Aqx+Bqu • Piecewise-linear (PL) discrete • constraints: Invq, Initq, Gq,q’are PL • The specification • the system has to remain in a given safe region defined by a set of PL constraints. • detects faulty components • provides the current value of • the system parameters • provides enough information to • observe the current state Piecewise Linear Hybrid System Diagnoser Sensors Alarms Samplers measurements of variables, Observer components • compute the current system state • adjust the controller for the new • system parameters • assumes finite control policies • provide stable and efficient • transitions between controllers states parameters update Configuration engine Switches Valves Regulators control Controller input

11. Control Reconfiguration of Hybrid Systems - Approach Discrete and continuous diagnoser Current systems data Controller Synthesis • Discrete Abstraction • Divide the state space into finite • set of regions • In any region, the system can be • driven to the adjacent regions Hybrid model parameters Global discrete observer • Supervisory Control • based on the abstract state • machine obtained by the partition • it is required to move the system • from current region to safe region • movement is global abstract control current discrete state Hybrid System discrete input Local continuous observer • Continuous Control • based on the discrete supervisor • continuous controller is established • for each region • drive the system from a region to • the guard of the next one. continuous input local detailed control current continuous state