Models for Hybrid Systems Bart De Schutter TU Delft, The - PDF document

st 1 HYCON PhD School on Hybrid Systems www.ist-hycon.org www.unisi.it Models for Hybrid Systems Bart De Schutter TU Delft, The Netherlands b.deschutter@its.tudelft.nl scimanyd suounitnoc enibmoc smetsys dirbyH lacipyt (snoitauqe ecnereffid ro laitnereffid) scimanyd etercsid dna stnalp lacisyhp fo fo lacipyt (snoitidnoc lacigol dna atamotua) fo senilpicsid gninibmoc yB .cigol lortnoc ,yroeht lortnoc dna smetsys dna ecneics retupmoc dilos a edivorp smetsys dirbyh no hcraeser ,sisylana eht rof sloot lanoitatupmoc dna yroeht fo ngised lortnoc dna ,noitacifirev ,noitalumis egral a ni desu era dna ,''smetsys deddebme`` ria ,smetsys evitomotua) snoitacilppa fo yteirav ssecorp ,smetsys lacigoloib ,tnemeganam ciffart .(srehto ynam dna ,seirtsudni HYSCOM IEEE CSS Technical Committee on Hybrid Systems 4 Siena, July 1 9-22, 2005 - Rectorate of the University of Siena

Models for Hybrid Systems 1st HYCON Summer School on Hybrid Systems Overview Siena, Italy, July 13–18, 2005 1. Hybrid systems — Definition, examples & challenges Models for Hybrid Systems 2. Hybrid system models — Overview & issues 3. Models for event-driven systems — Automata 4. Hybrid automata Bart De Schutter (b.deschutter@dcsc.tudelft.nl) Delft Center for Systems and Control 5. PWA systems and related model classes (MLD, LC, MMPS) Delft University of Technology 6. Timed automata 7. (Timed Petri nets) 8. Summary 1. Hybrid systems 1.2 More formal definition 1.1 Informal definition • System can be in one of several modes • Hybrid = combination of continuous and discrete dynamics • In each mode: behavior described by system of difference or differential equations • Temperature control system: • Mode switches due to occurrence of “events” T > T upp x 2 = f 2 ( x 2 , u ) ˙ on mode off mode y = g 2 ( x 2 , u ) ˙ ˙ T = f on ( T , w ) T = f off ( T , w ) T < T low x 1 = f 1 ( x 1 , u ) ˙ x 3 = f 3 ( x 3 , u ) ˙ y = g 1 ( x 1 , u ) y = g 3 ( x 3 , u ) hs models.1 hs models.2 1.2 More formal definition (continued) 1.3 Examples • At switching time instant: • Hierarchical control in process industry → possible state reset or state dimension change • Telecommunication systems • Mode transitions may be caused by • Manufacturing systems – external control signal • Air traffic coordination and control – internal control signal • Batch processes – dynamics of system itself (crossing of boundary in state space) whirlpool (e.g. beer brewing) mashing boiling wort separation water water malt holding vessel cooling maturation/ air conditioning fermentation filtration packaging hs models.3 hs models.4 Human intervention in smooth systems → hybrid

1.3 Examples (continued) 1.4 Challenges • Traffic control • Analysis and control • Automatic platooning • Nowadays: • Evolution of rigid bodies – often still heuristic & ad-hoc (contact/no contact) – often focus still exclusively on either continuous or discrete dynamics • Electrical networks (switching, diodes) → structured approach necessary • Fermentation process (lag, growth, stationary, inactivation) • Consider hybrid nature of systems • Saturation, hysteresis • Combination of systems & control, computer science, mathematics, and simulation • Actuator and sensor failures Switching between dynamical regimes → hybrid hs models.5 hs models.6 2. Hybrid system models 2.2 Models for time-driven systems 2.1 Introduction • Continuous-time time-driven systems: • Continuous-state / discrete-state x ( t ) = f ( x ( t ) , u ( t )) ˙ y ( t ) = g ( x ( t ) , u ( t )) • Continuous-time / discrete-time • Time-driven / event-driven – time-driven → state changes as time progresses, • Discrete-time (or sampled) time-driven systems: i.e., continuously (for CT), or at every tick of clock (for DT) x ( k + 1 ) = f ( x ( k ) , u ( k )) – event-driven → state changes due to occurrence of event y ( k ) = g ( x ( k ) , u ( k ) event: ∗ start or end of an activity ∗ asynchronous (occurrence times not necessarily equidistant) Combinations → “hybrid” hs models.7 hs models.8 2.3 Models for event-driven systems 2.3 Models for event-driven systems (continued) Z • No general framework (similar situation for hybrid systems) • Automata Z • Basic trade-off: • Petri nets • (max,+) algebraic models modeling power ↔ decision power • Markov chains / Markov processes ⇒ application-specific • Extended state machines • Generalized semi-Markov processes • Networks of waiting queues • . . . ⇒ no general framework Note: see also lecture on “Discrete-event modeling and diagnosis of quantized systems” hs models.9 hs models.10

2.4 Models for hybrid systems 2.4 Models for hybrid systems (continued) Z • Timed or hybrid Petri nets • Computer simulation models • Differential automata • Predicate calculus Z Z • Hybrid automata • Piecewise-affine models Z • Brockett’s model • Timed automata Z • Mixed logical dynamic models • . . . • Duration calculus Note: focus in this lecture is on non-stochastic models • Real-time temporal logics (see also lectures on “Stochastic Hybrid Systems”) • Timed communicating sequential processes • Switched bond graphs • ../.. hs models.11 hs models.12 2.5 Models for hybrid systems — Issues Intermezzo: Undecidable and NP-hard problems ⇒ no general modeling & analysis framework • Undecidable problems → no algorithm at all can be given for solving the modeling power ↔ decision power problem in general • NP-complete and NP-hard problems + computational complexity (NP-hard, undecidable) ⇒ special subclasses – decision problem : solution is either “yes” or “no” e.g., traveling salesman decision problem: hierarchical / modular approach Given a network of cities, intercity distances, and a number B , does there exist a tour with length � B ? – search problem e.g., traveling salesman problem: Given a network of cities, intercity distances, what is the shortest tour? hs models.13 intermezzo.1 P and NP-complete decision problems P and NP-complete decision problems • Each problem in NP can be solved in exponential time: T ( n ) � 2 n k • time complexity function T ( n ) : largest amount of time needed to solve problem instance of size n ( worst case! ) • NP-complete problems: “hardest” class in NP: • polynomial time algorithm: – any NP-complete problem solvable in polynomial time T ( n ) � | p ( n ) | for some polynomial p ⇒ every problem in NP solvable in polynomial time – any problem in NP intractable → class P: solvable by polynomial time algorithm ⇒ NP-complete problems also intractable • nondeterministic computer: NP – guessing stage (tour) NP-complete – checking stage (compute length of tour + compare it with B ) P → class NP: “ nondeterministically polynomial ” i.e., time complexity of checking stage is polynomial if P � = NP intermezzo.2 intermezzo.3

NP-hard problems Examples of NP-hard and undecidable problems • decision problem is NP-complete ⇒ search problem is NP-hard • Consider simple hybrid system: � • NP-hard problems: at least as hard as NP-complete problems if c T x ( k ) � 0 A 1 x ( k ) x ( k + 1 ) = if c T x ( k ) < 0 – NP-complete (decision problem) A 2 x ( k ) → solvable in polynomial time if and only if P = NP → deciding whether system is stable or not is NP-hard – NP-hard (search problem) • Given two Petri nets, do they have the same reachability set? → cannot be solved in polynomial time unless P = NP → undecidable intermezzo.4 intermezzo.5 Back to the main topic — Hybrid system models Hybrid system models (continued) • Many modeling frameworks for hybrid systems • Computer simulation and verification tools: Modelica, HyTech, ⇒ trade-off: modeling power ↔ decision power, tractability KRONOS, Chi, 20-sim, UPPAAL, . . . + simulation models can represent plant with high degree of • Hybrid automata: detail (high modeling power) – very general, high modeling power, but low decision power - computationally very demanding for large systems – analysis and control → computationally hard - difficult to understand from simulation how behavior depends (NP-hard, undecidable problems) on model parameters • In this lecture: special classes of hybrid systems for which tractable analysis and control design techniques are available ( → see next lectures) hs models.14 hs models.15 3. Models for event-driven systems 3.1 Automata (continued) 3.1 Automata Evolution of automaton Automaton • Given state q ∈ Q and discrete input symbol u ∈ U , Automaton is defined by triple Σ = ( Q , U , φ ) with transition function φ defines collection of next possible states: φ ( q , u ) ⊆ Q • Q : finite or countable set of discrete states • If each set of next states has 0 or 1 element: • U : finite or countable set of discrete inputs (“input alphabet”) → “deterministic” automaton • φ : Q × U �→ P ( Q ) : partial transition function . • If some set of next states has more than 1 element: where P ( Q ) is power set of Q (set of all subsets) → “non-deterministic” automaton Finite automaton: Q and U finite hs models.16 hs models.17