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

2. Introduction discrete e.g. task, activity, count, status, operation modes Variables changing in discrete steps; dynamics modeled by finite state structure (automata) Continuous e.g. location, velocity, temperature, pressure Variables changing continuously with time; dynamics modeled by differential equations variables dynamics Hybrid systems combine discrete-event or mode switching dynamics with continuous evolution according to differential equations or inclusions. States take values over real and discrete sets. 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 For each discrete mode q  Q, a dynamical system: X q   n state space for mode q; possible values x = (x 1, …, x n ) of real variables when in mode q. V   m an input vector of real values f q = X q  V   +  X q is the continuous flow of a vector field (system of o.d.e) on X q ; model how the system continuously evolves over time t   + Inv q  X q  V set 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. Init q  X q set of initial states for mode q. For each discrete transition (q,q’)  G, G q,q’  X q is the guard set of “trigger event” for the discrete transition (q,q’) R q,q’ : X q  V  X q is the reset map: state x non- deterministically reset to some x’  R q,q’ (x); e.g. identify relation on some variables. q 1 x  =f 1 (x,v) x  Inv 1 q 2 22 x  =f 2 (x,v) x  Inv 2 q 3 x  =f 3 (x,v) x  Inv 3 x  G 1,2 x  G 3,1 x  G 2,3 x  G 3,3 x  R 2,3 x  R 1,2 x  R 3,1 x  R 3,3 Hybrid State Space: X H :=  q  Q ({q}  Xq)

5. Hybrid System Control Problems A trajectory of a HS, H from a state (q o,x o ) is a map  : T  X H ; 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 t o =0 q1q1 q2q2 q3q3 t1t1 t2t2 t3t3 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 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

6. Discrete Abstraction of Hybrid Systems - Approaches 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). 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. Abstraction 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.

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  X q : Find a restricting non- blocking hybrid system H’ such that for every   L(H’) and for every t  T,  (t)  R. Algorithm: P o := F  {  q  Q Inv q ); k=0 Repeat P k+1 = P k  Pre(P k ) k := k+1 Until P k = P k-1 P * = P k Controller: The automata H’ = (Q, X, Inv’, Init’, G’, R’) is the solution to the control problem; where Inv’ q = Inv q  P*, G’ q,q’ = G q,q’  P*  P*, etc. 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). Approach: Restricts the system dynamics: PL systems, x q (t+1) = A q x(t) PL-contstrained inputs Approximate P k to a finite union of hybercubes, or orthogonal (griddy) polyhedral.

8. Switching Controller of Hybrid Systems - Implementation Approximate Solution: Identify the set F as the convex hull of a set of points V. Compute D = conv(V  Post r (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 Approximation Algorithm: P o := F; V o =V; Repeat V k+1 = Post r (V k ); D k = conv(V k-1  V k ); D k = bloat(D k ) D k = griddy(D k ) D k = P k-1  D k Until P K+1 = P K x1 x2 Post r (x1) Post r (x2 ) F Post r (F) Post 2r (F) Post 23 (x1) Post 2r (x2 ) Post [0,r] (F) Post [0,2r] (F)

9. Discrete Abstraction of Hybrid Systems – IO approach The system: x(k+1) = f(x(k), w(k),u d (k)), y d (k) = q(x(k)), where, w(k)  W:= {w | w   r, ||w||   1} u d (k)  U d = {u d1, …, u dn } Y d (k)  Y d = {y d1, …, y dm } Behavior abstraction: System behavior:  c  (U d, Y d ) 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 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 ={ [b k, (y dj,u di )] | b k   l,k, u di  U d, y dj  Y l (b k,l )} Where Y l (b k,l ) is the set of all possible outputs generated at time k+1 given the string b k,l The abstract system: States: x d (k) = [y d (0)] if k=0 = [y d [0 …k], u d [0..k-1]] if k  l = [y d [k-l … k], u d [k-l …k-1]] if k >l Transitions: (x d, u d, x’ d ) is a transition in A l if 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: f q = A q x+B q u Piecewise-linear (PL) discrete constraints: Inv q, Init q, G q,q’ are PL The specification the system has to remain in a given safe region defined by a set of PL constraints. Piecewise Linear Hybrid System Configuration engine Diagnoser Observer detects faulty components provides the current value of the system parameters provides enough information to observe the current state Controller compute the current system state adjust the controller for the new system parameters assumes finite control policies provide stable and efficient transitions between controllers components measurements of variables, states parameters update control input Sensors Alarms Samplers Switches Valves Regulators

11. Control Reconfiguration of Hybrid Systems - Approach Current systems data Hybrid System 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 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 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. Hybrid model parameters current discrete state current continuous state Global discrete observer Local continuous observer discrete input continuous input global abstract control local detailed control Discrete and continuous diagnoser