Lecture #13 Stability under slow switching & state-dependent switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched.

Lecture #13 Stability under slow switching & state-dependent switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched Systems

Summary Stability under slow switching Dwell-time switching Average dwell-time Stability under brief instabilities Stability under state-dependent switching State-dependent common Lyapunov function Stabilization through switching Multiple Lyapunov functions LaSalle’s invariance principle

Switched system parameterized family of vector fields ´ f p : R n ! R n p 2  parameter set switching signal ´ piecewise constant signal  : [0, 1 ) !   ´ set of admissible pairs ( , x) with  a switching signal and x a signal in R n t  = 1  = 3  = 2  = 1 switching times A solution to the switched system is a pair (  x) 2  for which 1.on every open interval on which  is constant, x is a solution to 2.at every switching time t, x(t) =  (  (t),  – (t), x – (t) ) time-varying ODE

Three notions of stability Definition (class  function definition): The equilibrium point x eq is stable if 9  2  : ||x(t) – x eq || ·  (||x(t 0 ) – x eq ||) 8 t ¸ t 0 ¸ 0, ||x(t 0 ) – x eq || · c along any solution (  x) 2  to the switched system Definition: The equilibrium point x eq 2 R n is asymptotically stable if it is Lyapunov stable and for every solution that exists on [0, 1 ) x(t) ! x eq as t !1. Definition (class  function definition): The equilibrium point x eq 2 R n is uniformly asymptotically stable if 9  2  : ||x(t) – x eq || ·  (||x(t 0 ) – x eq ||,t – t 0 ) 8 t ¸ t 0 ¸ 0 along any solution (  x) 2  to the switched system  is independent of x(t 0 ) and   is independent of x(t 0 ) and  exponential stability when  (s,t) = c e - t s with c, > 0

Stability under slow switching  all ´ set of all pairs ( , x) with  piecewise constant and x piecewise continuous  (p, q, x) = x 8 p,q 2 , x 2 R n no resets So far … any switching signal is admissible Slow switching:  dwell [  D ] ´ switching signals with “dwell-time”  D > 0, i.e., interval between consecutive discontinuities larger or equal to  D Slow switching on the average:  ave [  D, N 0 ] ´ switching signals with “average dwell-time”  D > 0 and “chatter-bound” N 0 > 0, i.e., # of discontinuities of  in the open interval ( ,t)  [  D ] =  ave [  D,1] Why? Now… switched linear systems

Stability under slow switching switched linear systems  [  D ] ´ switching signals with “dwell-time”  D > 0, i.e., interval between consecutive discontinuities larger or equal to  D Theorem: (  finite) If all A q, q 2  are asymptotically stable, there exists a dwell-time  D such that the switched system is uniformly (exponentially) asymptotically stable over  dwell [  D ] Why? 1 st For a switched linear system state-transition matrix (  -dependent) t 1, t 2, t 3, …, t k ´ switching times of  in the interval [t,  )

Stability under slow switching switched linear systems  [  D ] ´ switching signals with “dwell-time”  D > 0, i.e., interval between consecutive discontinuities larger or equal to  D Theorem: (  finite) If all A q, q 2  are asymptotically stable, there exists a dwell-time  D such that the switched system is uniformly (exponentially) asymptotically stable over  dwell [  D ] Why? 2 st Since all the A q, q 2  are asymptotically stable: 9 c,   > 0 ||e A q t || · c e –  t 3 nd Taking norms of the state-transition matrix…

Stability under slow switching switched linear systems  [  D ] ´ switching signals with “dwell-time”  D > 0, i.e., interval between consecutive discontinuities larger or equal to  D Theorem: (  finite) If all A q, q 2  are asymptotically stable, there exists a dwell-time  D such that the switched system is uniformly (exponentially) asymptotically stable over  dwell [  D ] Why? 3 nd 4 th Pick  D > 0, 2 (0, 0 ) such that Always possible? yes: can pick

Stability under slow switching switched linear systems  [  D ] ´ switching signals with “dwell-time”  D > 0, i.e., interval between consecutive discontinuities larger or equal to  D Theorem: (  finite) If all A q, q 2  are asymptotically stable, there exists a dwell-time  D such that the switched system is uniformly (exponentially) asymptotically stable over  dwell [  D ] Why? 3 nd 4 th 5 th Then exponential convergence to zero (with rate independent of  )

Stability under slow switching switched linear systems  [  D ] ´ switching signals with “dwell-time”  D > 0, i.e., interval between consecutive discontinuities larger or equal to  D Theorem: (  infinite) Assuming the sets {A q : q 2  } & { R p,q : p, q 2  } are compact. If all A q, q 2  are asymptotically stable, there exists a dwell-time  D such that the switched system is uniformly (exponentially) asymptotically stable over  dwell [  D ]

 ave [  D, N 0 ] ´ switching signals with “average dwell-time”  D > 0 and “chatter-bound” N 0 > 0, i.e., Stability under slow switching on the average switched linear systems Theorem: (  finite) If all the A q, q 2  are asymptotically stable, there exists an average dwell-time  D such that for every chatter-bound N 0 the switched system is uniformly (exponentially) asymptotically stable over  ave [  D, N 0 ] Why? 1 st As before … 2 nd But k is the number of switchings in [t,  ) so exponential decrease as long as (w.l.g we assume r c > 1) # of switchings in ( ,t)

Stability under slow switching on the average switched linear systems Theorem: (  infinite) Assuming the sets {A q : q 2  } & { R p,q : p, q 2  } are compact. If all the A q, q 2  are asymptotically stable, there exists an average dwell-time  D such that for every chatter-bound N 0 the switched system is uniformly (exponentially) asymptotically stable over  ave [  D, N 0 ]  ave [  D, N 0 ] ´ switching signals with “average dwell-time”  D > 0 and “chatter-bound” N 0 > 0, i.e., 1.Same results would hold for any subset of  ave [  D, N 0 ] 2.Some versions of these results also exist for nonlinear systems 3.One may still have stability if some of the A q are unstable, provided that  does not “dwell” on these values for a long time (switching under brief instabilities) # of switchings in ( ,t)

So far… state-independent switching  all ´ set of all pairs ( , x) with  piecewise constant and x piecewise continuous Slow switching:  [  D ] ´ switching signals with “dwell-time”  D > 0, i.e., interval between consecutive discontinuities larger or equal to  D Slow switching on the average:  ave [  D, N 0 ] ´ switching signals with “average dwell-time”  D > 0 and “chatter-bound” N 0 > 0, i.e., # of discontinuities of  in the open interval ( ,t) switched linear systems no resets Arbitrary switching:

Current-state dependent switching no resets  [  ] ´ set of all pairs ( , x) with  piecewise constant and x piecewise continuous such that 8 t,  (t) = q is allowed only if x(t) 2  q Current-state dependent switching   q 2 R n : q 2  } ´ (not necessarily disjoint) covering of R n, i.e., [ q 2   q = R n 11 22  = 1  = 2  = 1 or 2 Thus ( , x) 2  [  ] if and only if x(t) 2   (t) 8 t

Common Lyapunov function for arbitrary switching Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n ! R such that Then for arbitrary switching  all 1.the equilibrium point x eq is Lyapunov stable 2.if W(z) = 0 only for z = x eq then x eq is (glob) uniformly asymptotically stable. Why? (for simplicity consider x eq = 0) 1 st Take an arbitrary solution ( , x) and define v(t) V( x(t) ) 8 t ¸ 0 2 nd Therefore V( x(t) ) is always bounded…

Common Lyapunov function for current-state dep. switching Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n ! R such that Then for current-state dependent switching  [  ] 1.the equilibrium point x eq is Lyapunov stable 2.if W(z) = 0 only for z = x eq then x eq is (glob) uniformly asymptotically stable. Why? (for simplicity consider x eq = 0) 1 st Take an arbitrary solution ( , x) and define v(t) V( x(t) ) 8 t ¸ 0 2 nd Therefore still holds because x(t) 2   (t) Same conclusions as before …

Note that: Same conclusion would hold for any subset of  [  ] Some (or all) the unswitched systems may not be stable This theorem does not guarantee existence of solutions (as opposed to the usual Lyapunov Theorem and the ones for state independent switching)… Common Lyapunov function for current-state dep. switching Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n ! R such that Then for current-state dependent switching  [  ] 1.the equilibrium point x eq is Lyapunov stable 2.if W(z) = 0 only for z = x eq then x eq is (glob) uniformly asymptotically stable.

Common Lyapunov function for current-state dep. switching Theorem: Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n ! R such that Then for current-state dependent switching  [  ] 1.the equilibrium point x eq is Lyapunov stable 2.if W(z) = 0 only for z = x eq then x eq is (glob) uniformly asymptotically stable. E.g.,  {–1, +1},  –1 [0, 1 ),  +1 (– 1,0) For x eq = 0 is an equilibrium point and for V(z) z 2 no solutions exists

Stabilization through switching Given a family of unstable vector fields f q, q 2  Is there a covering  for which the current-state dependent set of switching signals  [  ] results in stability? Theorem: If there exists a set of constants q ¸ 0, q 2  such that  q q =1 and x eq is an (asymptotically) stable equilibrium point of the ODE then there is a current-state dependent set of switching signals  [  ] for which x eq is an (asymptotically) stable equilibrium point of the switched system. Why? 1 st Since the convex combination is asymptotically stable, it has a Lyapunov function V: since all the q ¸ 0, for every z, at least one of the terms must be · 0 convex combination of the f q

Stabilization through switching Given a family of unstable vector fields f q, q 2  Theorem: If there exists a set of constants q ¸ 0, q 2  such that  q q =1 and x eq is an (asymptotically) stable equilibrium point of the ODE then there is a current-state dependent set of switching signals  [  ] for which x eq is an (asymptotically) stable equilibrium point of the switched system. convex combination of the f q Why? 2 nd Define 1.every point in R n belongs to one of the  q )  {  q : q 2  } form a covering 2. V is a common Lyapunov function for current-state dep. switching Is there a covering  for which the current-state dependent set of switching signals  [  ] results in stability?

Stabilization through switching Given a family of unstable vector fields f q, q 2  Theorem: If there exists a set of constants q ¸ 0, q 2  such that  q q =1 and x eq is an (asymptotically) stable equilibrium point of the ODE then there is a current-state dependent set of switching signals  [  ] for which x eq is an (asymptotically) stable equilibrium point of the switched system. convex combination of the f q But these covers may lead to non-existence of solution (Zeno) Is there a covering  for which the current-state dependent set of switching signals  [  ] results in stability?

Example  = 2  = 1 The two regions actually intersect. One can use this to prevent Zeno (e.g., through hysteresis)…

Multiple Lyapunov functions Given a solution ( , x) and defining v(t) V  (t) ( x(t) ) 8 t ¸ 0 1.On an interval [ , t) where  = q (constant) V q : R n ! R, q 2  ´ family of Lyapunov functions (cont. dif., pos. def., rad. unb.) 2.But at a switching time t, where  – (t) = p   (t) = q, v decreases v may be discontinuous (even without reset)

Multiple Lyapunov functions Given a solution ( , x) and defining v(t) V  (t) ( x(t) ) 8 t ¸ 0 1.On an interval [ , t) where  = q (constant) V q : R n ! R, q 2  ´ family of Lyapunov functions (cont. dif., pos. def., rad. unb.) 2.But at a switching time t, where  – (t) = p   (t) = q, v decreases  = 1  = 2  = 1 t v=V1(x)v=V1(x) v=V2(x)v=V2(x) v=V1(x)v=V1(x)  = 2  = 1 t v=V1(x)v=V1(x) v=V2(x)v=V2(x) v=V1(x)v=V1(x) we would be okay if v would not increase at switching times

Multiple Lyapunov functions Theorem: (  finite) Suppose there exists a family of continuously differentiable, positive definite, radially unbounded functions V q : R n ! R, q 2  such that Then 1.the equilibrium point x eq is Lyapunov stable 2.if W(z) = 0 only for z = x eq then x eq is (glob) uniformly asymptotically stable. and at any z 2 R n where a switching signal in  can jump from p to q Why? (for simplicity consider x eq = 0) 1 st Take an arbitrary solution ( , x) and define v(t) V  ( x(t) ) 8 t ¸ 0 while  is constant: and, at points of discontinuity of  : v – (t) ¸ v(t) does not increase from now on same as before …

Multiple Lyapunov functions Theorem: (  finite) Suppose there exists a family of continuously differentiable, positive definite, radially unbounded functions V q : R n ! R, q 2  such that Then 1.the equilibrium point x eq is Lyapunov stable 2.if W(z) = 0 only for z = x eq then x eq is (glob) uniformly asymptotically stable. and at any z 2 R n where a switching signal in  can jump from p to q Why? (for simplicity consider x eq = 0) 2 nd Since 9  1,  2 2  1 :  1 (||x||) · V q (x) ·  2 (||x||) 3 rd If 9  3 :W(x) · –  3 (||x||) class  function independent of 

Multiple Lyapunov functions The V q ’s need not be positive definite and radially unbounded “everywhere” It is enough that 9  1,  2 2  1 :  1 (||z||) · V q (z) ·  2 (||z||) 8 q 2 , z 2  q Theorem: (  finite) Suppose there exists a family of continuously differentiable, positive definite, radially unbounded functions V q : R n ! R, q 2  such that Then 1.the equilibrium point x eq is Lyapunov stable 2.if W(z) = 0 only for z = x eq then x eq is (glob) uniformly asymptotically stable. and at any z 2 R n where a switching signal in  can jump from p to q

LaSalle’s Invariance Principle (ODE) Theorem (LaSalle Invariance Principle): Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: R n ! R such that Then x eq is a Lyapunov stable equilibrium and the solution always exists globally. Moreover, x(t) converges to the largest invariant set M contained in E { z 2 R n : W(z) = 0 } M 2 R n is an invariant set ´ x(t 0 ) 2 M ) x(t) 2 M 8 t ¸ t 0 Note that: 1.When W(z) = 0 only for z = x eq then E = {x eq }. Since M ½ E, M = {x eq } and therefore x(t) ! x eq ) asympt. stability 2.Even when E is larger then {x eq } we often have M = {x eq } and can conclude asymptotic stability.

LaSalle’s Invariance Principle (linear system) Theorem (LaSalle Invariance Principle–linear system, quadratic V): Suppose there exists a positive definite matrix P A’ P + P A · – Q · 0 Then the system is stable. Moreover, x(t) converges to the largest invariant set M contained in E { z 2 R n : Q z = 0 } Note that: 1.Since Q ¸ 0 we can always write Q = C’ C … M 2 R n is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0

LaSalle’s Invariance Principle (linear system) Theorem (LaSalle Invariance Principle–linear system, quadratic V): Suppose there exists a positive definite matrix P A’ P + P A · – C’C · 0 Then the system is stable. Moreover, x(t) converges to the largest invariant set M contained in E { z 2 R n : C z = 0 } M 2 R n is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0 Note that: 2.When Q > 0 then E = {0}. Since M ½ E, M = {0} and therefore x(t) ! 0 ) asympt. stability 3.Even when E is larger then {0} we often have M = {0} and can conclude asymptotic stability. When does this happen ? Why? show that C’Cz = 0 ) Cz = 0

Asymptotic stability from LaSalle’s IP M 2 R n is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0 M ´ largest invariant set contained in E { z 2 R n : C z = 0 } x 0 2 M if and only if x(t) e A t x 0 2 M ½ E 8 t ¸ 0 m (Why?)

Asymptotic stability from LaSalle’s IP M 2 R n is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0 M ´ largest invariant set contained in E { z 2 R n : C z = 0 } x 0 2 M if and only if x(t) e A t x 0 2 M ½ E 8 t ¸ 0 (check that this is indeed an invariant set …)

LaSalle’s Invariance Principle (linear system) Theorem (LaSalle Invariance Principle–linear system, quadratic V): Suppose there exists a positive definite matrix P A’ P + P A · – C’C · 0 Then the system is stable. Moreover, x(t) converges to M 2 R n is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0 When O is nonsingular, we have asymptotic stability (pair (C,A) is said to be observable) observability matrix of the pair (C,A)

Back to switched linear systems… Theorem: (  finite) Suppose there exist positive definite matrices P q 2 R n £ n, q 2  such that A q ’ P q + P q A q · – C q ’C q · 0 8 q 2  and at any z 2 R n where a switching signal in  [  ] can jump from p to q z’ P p z ¸ z’ R’ q p P q R q p z Then the switched system is stable. Moreover, if every pair (C q,A q ), q 2  is observable then 1.if  ½  weak-dwell then it is asymptotically stable 2.if  ½  p-dwell [  D,T] then it is uniformly asymptotically stable. from general theorem

Sets of switching signals  dwell [  D ] ´ switching signals with “dwell-time”  D > 0, i.e., interval between consecutive discontinuities larger or equal to  D  ave [  D, N 0 ] ´ switching signals with “average dwell-time”  D > 0 and “chatter-bound” N 0 > 0, i.e.,  p-dwell [  D,T] ´ switching signals with “persistent dwell-time”  D > 0 and “period of persistency” T > 0, i.e., 9 infinitely many intervals of length ¸  D on which sigma is constant & consecutive intervals with this property are separated by no more than T  weak-dwell [  D > 0  p-dwell [  D,+ 1 ] ´ each  has persistent dwell-time > 0 ¸D¸D ¸D¸D · T ¸D¸D  dwell [  D ] ½  ave [  D, N 0 ] ½  p-dwell [   D,T] ½  weak-dwell ½  all

LaSalle’s IP for switched systems Theorem: (  finite) Suppose there exist positive definite matrices P q 2 R n £ n, q 2  such that A q ’ P q + P q A q · – C q ’C q · 0 8 q 2  and at any z 2 R n where a switching signal in  [  ] can jump from p to q V p (z) ¸ V q (R q p z) Then the switched system is stable. Moreover, if every pair (C q,A q ), q 2  is observable then 1.if  ½  weak-dwell then it is asymptotically stable 2.if  ½  p-dwell [  D,T] then it is uniformly asymptotically stable. from general theorem  p-dwell [  D,T] ´ switching signals with “persistent dwell-time”  D > 0 and “period of persistency” T > 0, i.e., 9 infinitely many intervals of length ¸  D on which sigma is constant & consecutive intervals with this property are separated by no more than T  weak-dwell [  D > 0  p-dwell [  D,+ 1 ] ´ each  has persistent dwell-time > 0  dwell [  D ] ½  ave [  D, N 0 ] ½  p-dwell [   D,T] ½  weak-dwell ½  all

Example Choosing P 1 = P 2 = I common Lyapunov function 1.One can find   weak-dwell for which we do not have asymptotic stability 2.Stability is not uniform on  weak-dwell, because one can find  2  weak-dwell for which convergence is “arbitrarily slow” (problems, e.g., close to the x 2 =0 axis) nonsingular (observable)

LaSalle’s IP for switched systems Theorem: (  finite) Suppose there exist positive definite matrices P q 2 R n £ n, q 2  such that A q ’ P q + P q A q · – C q ’C q · 0 8 q 2  and at any z 2 R n where a switching signal in  [  ] can jump from p to q V p (z) ¸ V q (R q p z) Then the switched system is stable. Moreover, if every pair (C q,A q ), q 2  is observable then 1.if  ½  weak-dwell then it is asymptotically stable 2.if  ½  p-dwell [  D,T] then it is uniformly asymptotically stable. from general theorem a)Finiteness of  could be replaced by compactness b)In some cases it is sufficient for all pairs (C q,A q ), q 2  to be detectable (e.g., when A q = A + B F q ) c)When the pairs (C q,A q ), q 2  are not observable x converges to the smallest subspace  that is invariant for all unswitched system and contains the kernels of all O q d)There are nonlinear versions of this result (no uniformity?)

Next lecture… Computational methods to construct multiple Lyapunov functions—Linear Matrix Inequalities (LMIs) Applications (vision-based control)

Lecture #13 Stability under slow switching & state-dependent switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched.

Similar presentations

Presentation on theme: "Lecture #13 Stability under slow switching & state-dependent switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched."— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Lecture #13 Stability under slow switching & state-dependent switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched.

Similar presentations

Presentation on theme: "Lecture #13 Stability under slow switching & state-dependent switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched."— Presentation transcript:

Similar presentations

About project

Feedback