Switched systems: stability

Transcription

1 Switched systems: stability

2 OUTLINE Switched Systems Stability of Switched Systems

3 OUTLINE Switched Systems Stability of Switched Systems

4 a family of systems SWITCHED SYSTEMS

5 SWITCHED SYSTEMS a family of systems a signal that orchestrates the switching between them s

6 GRAPH REPRESENTATION

7 SWITCHED LINEAR SYSTEMS a family of linear systems a signal that orchestrates the switching between them s

8 SWITCHING time-dependent versus state-dependent switching autonomous versus controlled switching

9 SWITCHING time-dependent versus state-dependent switching autonomous versus controlled switching

10 TIME-DEPENDENT SWITCHING s s piecewise constant function of time (exogenous) switching signal s(t) specifies the system that is active at time t

11 STATE-DEPENDENT SWITCHING Q(x) s (endogenous) switching signal the state space X is partitioned into operating regions, each one associated to a system s(x) specifies the system that is active when the state is x

12 STATE-DEPENDENT SWITCHING

13 SWITCHING time-dependent versus state-dependent switching autonomous versus controlled switching

14 AUTONOMOUS SWITCHING Switching events are triggered by an external mechanism over which we do not have control Examples: unpredictable environmental factors component failures

15 CONTROLLED SWITCHING Switching are imposed so as to achieve a desired behavior of the resulting system switched control systems

16 SWITCHING CONTROL

17 SWITCHING CONTROL The closed-loop system is a switched system

18 Reasons for switching: SWITCHING CONTROL nature of the control problem (system with different operation phases)

19 Reasons for switching: SWITCHING CONTROL nature of the control problem (system with different operation phases) Example: flight control system

20 Reasons for switching: large modeling uncertainty SWITCHING CONTROL

21 SWITCHING CONTROL Reasons for switching: large modeling uncertainty Example: adaptive switching control P = admissible model set C 1 C 2 C 3 C 4 controller cover

22 Reasons for switching: sensor/actuator limitations SWITCHING CONTROL

23 SWITCHING CONTROL Reasons for switching: sensor/actuator limitations Example: quantized control u PLANT x q(x) QUANTIZER q(x) x CONTROLLER

24 EXAMPLE: THERMOSTAT Temperature in a room controlled by a thermostat switching a heater on and off Dynamics of the temperature x (in C): heater on: heater off: Goal: regulate the temperature around 20 C Strategy: turn the heater from OFF to ON as soon as x 18 turn the heater from ON to OFF as soon as x 22 ON x OFF hysteretic behavior

25 EXAMPLE: THERMOSTAT evolution of the temperature x starting from the initial condition x(0) = 5 with the heater ON evolution of the heater status starting from the initial condition x(0) = 5 with the heater ON

26 EXAMPLE: THERMOSTAT Continuous dynamics linear ODEs describing the temperature evolution Discrete dynamics finite automaton describing the behavior of the thermostat Q = {ON,OFF} ON = (OFF,e 1 ) e 1 = [x 18] OFF = (ON,e 2 ) e 2 = [x 22]

27 EXAMPLE: THERMOSTAT evolution of the temperature x starting from the initial condition x(0) = 5 with the heater ON x generates the events causing a transition from OFF to ON x 18 from ON to OFF x 22 evolution of the heater status starting from the initial condition x(0) = 5 with the heater ON

28 EXAMPLE: THERMOSTAT evolution of the temperature x starting from the initial condition x(0) = 5 with the heater ON A heater transition causes a switch in the dynamics governing x evolution of the heater status starting from the initial condition x(0) = 5 with the heater ON

29 EXAMPLE: THERMOSTAT interface ODE dx/dt = x + u continuous systems controlled by a discrete logic

30 EXAMPLE: THERMOSTAT interface ODE dx/dt = x + u quantized control input (ON heating power u = 6 OFF heating power u= 0) continuous systems controlled by a discrete logic

31 SWITCHING CONTROL Reasons for switching: nature of the control problem (system with different operation phases) large modeling uncertainty sensor/actuator limitations

32 OUTLINE Switched Systems Stability of Switched Systems

33 SWITCHED SYSTEMS: EQUILIBRIUM family of systems with

34 SWITCHED SYSTEMS: EQUILIBRIUM family of systems with x = 0 is an equilibrium of the switched system Stability of the equilibrium x=0?

35 SWITCHING BETWEEN AS. STABLE LINEAR SYSTEMS

36 SWITCHING BETWEEN AS. STABLE LINEAR SYSTEMS x = 0 is unstable!

37 SWITCHING BETWEEN AS. STABLE LINEAR SYSTEMS x = 0 is unstable! overshoots sum up

38 SWITCHING BETWEEN AS. STABLE LINEAR SYSTEMS x = 0 is unstable! Problem: find conditions that guarantee asymptotic stability under arbitrary switching

39 SWITCHING BETWEEN AS. STABLE LINEAR SYSTEMS x = 0 is stable!

40 SWITCHING BETWEEN AS. STABLE LINEAR SYSTEMS x = 0 is stable! Problem: identify those switching signals that preserve asymptotic stability

41 SWITCHING BETWEEN UNSTABLE LINEAR SYSTEMS

42 SWITCHING BETWEEN UNSTABLE LINEAR SYSTEMS x = 0 is stable! Problem: identify those switching signals that ensure asymptotic stability

43 Stability for arbitrary switching Stability for constrained switching

44 Stability for arbitrary switching Stability for constrained switching

45 GLOBAL UNIFORM ASYMPTOTIC STABILITY (GUAS) The equilibrium x e =0 is GUAS if it is globally asymptotically stable, uniformly with respect to the switching signals s

46 GLOBAL UNIFORM ASYMPTOTIC STABILITY (GUAS) Assumption: family of systems with GAS equilibrium in x=0 Remark: if the equilibrium x=0 is not GAS for one of the systems, then it cannot be GUAS for the switched system

47 Def. The family of systems COMMON LYAPUNOV FUNCTION share a radially unbounded common Lyapunov function at x=0 if there exists a continuously differentiable function V: X! < such that

48 COMMON LYAPUNOV FUNCTION If all systems in the family share a radially unbounded common Lyapunov function V: X! < at x = 0, then, the equilibrium x = 0 is GUAS. Proof. Same reasoning as in standard Lyapunov theorem

49 COMMON LYAPUNOV FUNCTION V ( x( t)) s = 1 s = 2 s = 1 s = 2 t

50 SWITCHED LINEAR SYSTEMS family of systems time-dependent switching rule

51 COMMON QUADRATIC LYAPUNOV FUNCTION If there exists such that then, the equilibrium x = 0 is GUAS. Proof. is a radially unbounded common Lyapunov function at x = 0. Remark: A set of LMIs to solve. This problem can be reformulated as a convex optimization problem. Efficient solvers exist.

52 GLOBALLY QUADRATIC LYAPUNOV FUNCTION The existence of a globally quadratic Lyapunov function is not necessary for x =0 to be GUAS Example: x=0 is GUAS but there is no common quadratic Lyapunov function

53 SWITCHED LINEAR SYSTEMS WITH A SPECIAL STRUCTURE Hurwitz matrices commute are upper (or lower) triangular

54 COMMUTING HURWITZ MATRICES GUAS A 1 A 2 A 2 A 1 s 1 s 2 s 1 s 2 t s1 t1 2 2 s t A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A2 ( t... t1 ) A1 ( s... s1 ) e k e k x(0) 0

55 COMMUTING HURWITZ MATRICES GUAS 1 A 2 A 2 A 1 quadratic common Lyapunov function: A

56 COMMUTING HURWITZ MATRICES GUAS 1 A 2 A 2 A 1 quadratic common Lyapunov function: A

57 TRIANGULAR HURWITZ MATRICES GUAS

58 TRIANGULAR HURWITZ MATRICES GUAS

59 TRIANGULAR HURWITZ MATRICES GUAS exponentially stable system exponentially decaying perturbation

60 TRIANGULAR HURWITZ MATRICES => GUAS quadratic common Lyapunov function with P diagonal

61 SWITCHED SYSTEMS WITH A SPECIAL STRUCTURE Hurwitz matrices commute are upper (or lower) triangular can be transformed to upper (or lower) triangular form by a common similarity transformation

62 Stability for arbitrary switching Stability for constrained switching

63 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 t s1 t1 2 2 The switching intervals satisfy s t t i, s i dwell time D

64 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 t s1 t1 2 2 The switching intervals satisfy s t, i s i t D A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0)

65 STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (exponential stability): Let the equilibrium point x e =0 be asymptotically stable. Then, the rate of convergence to x e =0 is exponential: for all x(0) = x 0 2 < n, where 0 2 (0, min i Re{ i (A)} ) and >0 is an appropriate constant. Remark:

66 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 t s1 t1 2 2 The switching intervals satisfy s t, i s i t D A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) t t 0 e A i e 0 e e D 0 D log slowest decay rate so that the inequality holds 8 i

67 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 t s1 t1 2 2 The switching intervals satisfy s t, i s i t D A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A t 0 D log D e i e e 1 (0, 0) D log 0

68 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 t s1 t1 2 2 A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) s t 0

69 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 t s1 t1 2 2 A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) s t

70 STABILITY UNDER STATE-DEPENDENT SWITCHING s: X Q : s(x) = i if x 2 X i

71 STATE-DEPENDENT COMMON LYAPUNOV FUNCTIONS If V: < n! < is a C 1 radially unbounded function such that V(x) state-dependent common Lyapunov function then, x = 0 is GAS for Remarks: need that only when s is equal to q, i.e. on X q matrices A q are not required to be Hurwitz

72 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A A ( 1 A 1 ) 2 Hurwitz for some (0,1)

73 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A A ( 1 A 1 ) 2 Hurwitz for some (0,1) A T P PA 0

74 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A A ( 1 A 1 ) 2 Hurwitz for some (0,1) T ( A1 P PA1 ) (1 )( A2 P PA2 ) T 0

75 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A A ( 1 A 1 ) 2 Hurwitz for some (0,1) So for each : either x T T ( A1 P PA1 ) (1 )( A2 P PA2 ) x 0 T 1 x ( A P PA1) 0 or x T T T 2 x ( A P PA2) 0 0

76 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A A ( 1 A 1 ) 2 Hurwitz for some (0,1) So for each : either x T T ( A1 P PA1 ) (1 )( A2 P PA2 ) x 0 T 1 x ( A P PA1) 0 or x T T T 2 x ( A P PA2) 0 0 Region where system 1 is active for the system Region where system 2 is active is a Lyapunov function at x =0 => GAS

77 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A A ( 1 A 1 ) 2 Hurwitz for some (0,1) So for each : either x T T ( A1 P PA1 ) (1 )( A2 P PA2 ) x 0 T 1 x ( A P PA1) 0 or x T T T 2 x ( A P PA2) 0 0 Region where system 1 is active Region where system 2 is active

78 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A A ( 1 A 1 ) 2 Hurwitz for some (0,1) So for each : either x T T ( A1 P PA1 ) (1 )( A2 P PA2 ) x 0 T 1 x ( A P PA1) 0 or x T T T 2 x ( A P PA2) 0 0

79 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable

80 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable

81 Main source: Switching in Systems and Control Daniel Liberzon, Birkhauser, 2003.