Modeling, Analysis and Control Radu Grosu Vienna University of - PowerPoint PPT Presentation

Hybrid Systems Modeling, Analysis and Control Radu Grosu Vienna University of Technology Lecture 6

Continuous AND Discrete Systems Control Theory Computer Science Continuous systems Discrete systems approximation, stability abstraction, composition control, robustness concurrency, verification Hybrid Systems Software controlled systems Embedded real-time systems Multi-agent systems

Models and Tools Dynamic systems with continuous & discrete state variables Continuous Part Discrete Part Differential equations, Automata, Petri nets, Models transfer functions, Statecharts, Lyapunov functions, Boolean algebra, formal Analytical Tools eigenvalue analysis, logics, verification, Matlab, Matrix x , Statemate, Rational Software Tools VisSim, Rose, SMV,

Modeling a Hybrid System Model of System Model of Model of Physics Software continuous dynamics discrete dynamics

Hybrid Automaton (HA) locations or modes guard (discrete states) edge n x  inv ( n ) e : guard ( x )  0  m action ( x, x ) x  inv ( m ) x  init ( m ) jump transformation invariant: HA may remain in initial m as long as x  inv ( m ) condition continuous dynamics

Example: Bouncing Ball Ball has mass m and position x Ball initially at position x 0 and at rest Ball bounces when hitting ground at x  0

Bouncing Ball: Free Fall Condition for free fall: x  0 Differential equations: First order

Bouncing Ball: Bouncing Condition for bouncing: x  0 Action for bouncing:  v   c v Coefficient c: deformation, friction

Bouncing Ball: Hybrid Automaton x  x 0 , v  0 initial condition location freeFall x  0 invariant discrete transition flow label bounce: guard  v   c v action

Bouncing Ball: Associated Program initial condition location freeFall invariant discrete transition flow label bounce: guard  v   c  v action

Execution of Bouncing Ball x (t) 0 Position ( x ) x (t) 1 x (t) 2 x (t) x (t) 3 4 T T T 3 T T Time ( t ) 0 1 2 4 Velocity ( v ) v (t) v (t) 4 v (t) 3 v (t) 2 v (t) 1 0 T T T T 3 T Time ( t ) 0 1 2 4

Boost DC-DC Converter i L  i 0 v c  v 0 s 0 s  0

Boost DC-DC Converter i L  i 0 v c  v 0 s  1 s 0 s 1 s  1 s  0 s  0

Boost DC-DC Converter i L  i 0 v c  v 0 s  1 s 0 s 1 s  0 s  1 s  0 float i L  i 0 , v c  v 0 , d  d 0 ; bool s  0; while true { while ( s  0) { R L L  i L  1 i L  i L  ( L  v S )  d v C  v C  1 1 C   v C  d R C  R 0 read( s ) }

Execution of Boost DC-DC Converter Capacitor Voltage and Inductor Current 16 200 Parameters : Current i L , mA , V U s = 20 V 12 150 Voltage u C þ L = 1 mH 8 100 C = 50 nF 4 50 L = 1 k W 0 0 R 0 1 2 3 C = 10 W R Time t , ms 0 = 10 k W R Load Voltage dt = 200 ns 16 200 Current i L , mA Voltage u R 0 , V U max = 16 V 12 150 U min = 14 V 8 100 4 50 0 0 0 1 2 3 Time t , ms

Hybrid Automaton H Variables: Continuous variables x  [ x 1 ,..., x n ] Control Graph: Finite directed multigraph ( V , E ) Finite set V of control modes Finite set E of control switches Vertex labeling functions: for each v  V Initial states: init( v )( x ) defines initial region Invariant: inv( v )( x ) defines invariant region Edge labeling functions: for each e  E Guard: guard( e )( x ) defines enabling region  Update: action( e )( x, x ) defines the reset region Synchronization labels: label( e ) defines communication

Executions of a Hybrid Automaton State: ( m , x ) such that x  inv( m ) Initialization: ( m , x ) such that x  init( m ) Two types of state updates: Discrete switches: ( m , x )  a ( m ,   x ) if e  ( m,m')  E  label( e )  a  guard( e )( x )  0  action(e)( x ,  x ) Continuous flows: ( m , x )  f ( m ,  x ) if  f : [0,T]  R n . f (0)  x  f (T)   x