Hybrid Systems Course Lyapunov stability
|
|
- Isaac Harvey
- 5 years ago
- Views:
Transcription
1 Hybrid Systems Course Lyapunov stability OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata
2 OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata ORDINARY DIFFERENTIAL EQUATIONS An ordinary differential equation is a mathematical model of a continuous state continuous time system: X = < n state space f: < n! < n vector field (assigns a velocity vector to each x) Given an initial value x 0 2 X, an execution (solution in the sense of Caratheodory) over the time interval [0,T) is a function x: [0,T)! < n such that: x(0) = x 0 x is continuous and piecewise differentiable
3 ODE SOLUTION: WELL-POSEDNESS Theorem [global existence and uniqueness non-blocking, deterministic, non-zeno] If f: < n! < n is globally Lipschitz continuous, then 8 x 0 there exists a single solution with x(0)=x 0 defined on [0,1). STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous Definition (equilibrium): x e 2 < n for which f(x e )=0 Remark: {x e } is an invariant set
4 Definition (stable equilibrium): Graphically: perturbed motion execution starting from x(0)=x 0 d x e equilibrium motion small perturbations lead to small changes in behavior Definition (asymptotically stable equilibrium): and d can be chosen so that Graphically: perturbed motion d x e equilibrium motion small perturbations lead to small changes in behavior and are re-absorbed, in the long run
5 Definition (asymptotically stable equilibrium): and d can be chosen so that Graphically: d xe small perturbations lead to small changes in behavior and are re-absorbed, in the long run EXAMPLE: PENDULUM l friction coefficient (a) m
6 m EXAMPLE: PENDULUM unstable equilibrium EXAMPLE: PENDULUM m as. stable equilibrium
7 EXAMPLE: PENDULUM l m Let x e be asymptotically stable. Definition (domain of attraction): The domain of attraction of x e is the set of x 0 such that execution starting from x(0)=x 0 Definition (globally asymptotically stable equilibrium): x e is globally asymptotically stable (GAS) if its domain of attraction is the whole state space < n
8 m EXAMPLE: PENDULUM m as. stable equilibrium small perturbations are absorbed, not all perturbations not GAS Let x e be asymptotically stable. Definition (exponential stability): x e is exponentially stable if 9 a, d, >0 such that
9 STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous Definition (equilibrium): x e 2 < n for which f(x e )=0 Without loss of generality we suppose that x e = 0 if not, then z := x -x e! dz/dt = g(z), g(z) := f(z+x e ) (g(0) = 0) STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous How to prove stability? find a function V: < n! < such that V(0) = 0 and V(x) >0, for all x 0 V(x) is decreasing along the executions of the system V(x) = 3 V(x) = 2 x(t)
10 STABILITY OF CONTINUOUS SYSTEMS execution x(t) behavior of V along the execution x(t): V(t): = V(x(t)) candidate function V(x) Advantage with respect to exhaustive check of all executions? STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous V: < n! < continuously differentiable (C 1 ) function Rate of change of V along the execution of the ODE system: (Lie derivative of V with respect to f) gradient vector
11 STABILITY OF CONTINUOUS SYSTEMS with f: < n! < n globally Lipschitz continuous V: < n! < continuously differentiable (C 1 ) function Rate of change of V along the execution of the ODE system: (Lie derivative of V with respect to f) gradient vector No need to solve the ODE for evaluating if V(x) decreases along the executions of the system LYAPUNOV STABILITY Theorem (Lyapunov stability Theorem): Let x e = 0 be an equilibrium for the system and D½ < n an open set containing x e = 0. If V: D! < is a C 1 function such that V positive definite on D Then, x e is stable. V non increasing along system executions in D (negative semidefinite)
12 EXAMPLE: PENDULUM l friction coefficient (a) m energy function x e stable LYAPUNOV STABILITY Theorem (Lyapunov stability Theorem): Let x e = 0 be an equilibrium for the system and D½ < n an open set containing x e = 0. If V: D! < is a C 1 function such that Then, x e is stable. If it holds also that Then, x e is asymptotically stable (AS)
13 LYAPUNOV GAS THEOREM Theorem (Barbashin-Krasovski Theorem): Let x e = 0 be an equilibrium for the system. If V: < n! < is a C 1 function such that V positive definite on < n V decreasing along system executions in < n (negative definite) V radially unbounded Then, x e is globally asymptotically stable (GAS). STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is an equilibrium for the system the elements of matrix e At are linear combinations of e i(a)t, i=1,2,,n
14 STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is an equilibrium for the system x e =0 is asymptotically stable if and only if A is Hurwitz (all eigenvalues with real part <0) asymptotic stability GAS STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is an equilibrium for the system x e =0 is asymptotically stable if and only if A is Hurwitz (all eigenvalues with real part <0) asymptotic stability GAS Alternative characterization
15 STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Lyapunov equation STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Lyapunov equation Remarks: Q positive definite (Q>0) iff x T Qx >0 for all x 0 Q positive semidefinite (Q 0) iff x T Qx 0 for all x and x T Q x = 0 for some x 0
16 STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Proof. (if) V(x) =x T P x is a Lyapunov function Lyapunov equation STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Proof. (only if) Consider Lyapunov equation
17 STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point x e =0 is asymptotically stable if and only if for all matrices Q = Q T positive definite (Q>0) the A T P+PA = -Q has a unique solution P=P T >0. Proof. (only if) Consider P = P T and P>0 easy to show P unique can be proven by contradiction Lyapunov equation STABILITY OF LINEAR CONTINUOUS SYSTEMS Remarks: for a linear system existence of a (quadratic) Lyapunov function V(x) =x T P x is a necessary and sufficient condition for asymptotic stability it is easy to compute a Lyapunov function since the Lyapunov equation is a linear algebraic equation A T P+PA = -Q
18 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. 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. Im eigenvalues of A o o o o Re
19 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: STABILITY OF LINEAR CONTINUOUS SYSTEMS Proof (exponential stability): A + 0 I is Hurwitz (eigenvalues are equal to (A) + 0 ) Then, there exists P = P T >0 such that (A + 0 I) T P + P (A + 0 I) <0 which leads to x(t) T [A T P + P A]x(t) < x(t) T P x(t) Define V(x) = x T P x, then from which
20 STABILITY OF LINEAR CONTINUOUS SYSTEMS (cont d) Proof (exponential stability): thus finally leading to 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:
21 STABILITY OF LINEAR CONTINUOUS SYSTEMS x e = 0 is an equilibrium for the system x e =0 is asymptotically stable if and only if A is Hurwitz (all eigenvalues with real part <0) asymptotic stability GAS exponential stability GES OUTLINE Focus: stability of an equilibrium point continuous systems decribed by ordinary differential equations (brief review) hybrid automata
22 HYBRID AUTOMATA: FORMAL DEFINITION A hybrid automaton H is a collection H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1,q 2, } is a set of discrete states (modes) X = < n is the continuous state space f: Q X! < n is a set of vector fields on X Init µ Q X is a set of initial states Dom: Q! 2 X assigns to each q2 Q a domain Dom(q) of X E µ Q Q is a set of transitions (edges) G: E! 2 X is a set of guards (guard condition) R: E X! 2 X is a set of reset maps q = q 1 q = q 2
23 HYBRID TIME SET A hybrid time set is a finite or infinite sequence of intervals = {I i, i=0,1,, M } such that I i = [ i, i ] for i < M I M = [ M, M ] or I M = [ M, M ) if M<1 i = i+1 i i I = [ I 1 1 ] t 2 1 I t 1 [ 0 ] 0 0 t 1 Á t 2 Á t 3 Á t 4 t 4 I time instants in are 3 3 linearly ordered [ ] 3 t 3 HYBRID TIME SET: LENGTH Two notions of length for a hybrid time set = {I i, i=0,1,, M }: Discrete extent: < > = M+1 Continuous extent: = i=0,1,..,m i - i number of discrete transitions total duration of intervals in < > = 4 = 3-0 [ ] 0 I 0 0 I 3 [ ] 3 3 I = [ I 1 ] 1
24 HYBRID TIME SET: CLASSIFICATION A hybrid time set = {I i, i=0,1,, M } is Finite: if < > is finite and I M = [ M, M ] Infinite: if < > is infinite or is infinite Zeno: if < > is infinite but is finite finite infinite infinite Zeno Zeno HYBRID TRAJECTORY A hybrid trajectory (, q, x) consists of: A hybrid time set = {I i, i=0,1,, M } Two sequences of functions q = {q i ( ), i=0,1,, M } and x = {x i ( ), i=0,1,, M } such that q i : I i! Q x i : I i! X
25 HYBRID AUTOMATA: EXECUTION A hybrid trajectory (, q, x) is an execution (solution) of the hybrid automaton H = (Q,X,f,Init,Dom,E,G,R) if it satisfies the following conditions: Initial condition: (q 0 ( 0 ), x 0 ( 0 )) 2 Init Continuous evolution: for all i such that i < i q i : I i! Q is constant x i :I i! X is the solution to the ODE associated with q i ( i ) x i (t) 2 Dom(q i ( i )), t2 [ i, i ) Discrete evolution: (q i ( i ),q i+1 ( i+1 )) 2 E transition is feasible x i ( i ) 2 G((q i ( i ),q i+1 ( i+1 ))) guard condition satisfied x i+1 ( i+1 ) 2 R((q i ( i ),q i+1 ( i+1 )),x i ( i )) reset condition satisfied Well-posedness? HYBRID AUTOMATA: EXECUTION Problems due the hybrid nature: for some initial state (q,x) infinite execution of finite duration Zeno no infinite execution blocking multiple executions non-deterministic We denote by H (q,x) the set of (maximal) executions of H starting from (q,x) H (q,x) 1 the set of infinite executions of H starting from (q,x)
26 STABILITY OF HYBRID AUTOMATA Definition (equilibrium): H = (Q,X,f,Init,Dom,E,G,R) x e =0 2 X is an equilibrium point of H if: f(q,0) = 0 for all q 2 Q if ((q,q )2 E) Æ (02 G((q,q )) ) R((q,q ),0) = {0} Remarks: discrete transitions are allowed out of (q,0) but only to (q,0) if (q,0) 2 Init and (, q, x) is an execution of H starting from (q,0), then x(t) = 0 for all t2 EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1, q 2 } X = < 2 f(q 1,x) = A 1 x and f(q 2,x) = A 2 x with: Init = Q {x2 X: x >0} Dom(q 1 ) = {x2 X: x 1 x 2 0} Dom(q 2 ) = {x2 X: x 1 x 2 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x2 X: x 1 x 2 0} G((q 2,q 1 )) = {x2 X: x 1 x 2 0} R((q 1,q 2 ),x) = R((q 2,q 1 ),x) = {x}
27 EXAMPLE: SWITCHED LINEAR SYSTEM x 2 x 1 EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1, q 2 } X = < 2 f(q 1,x) = A 1 x and f(q 2,x) = A 2 x with: Init = Q {x2 X: x >0} Dom(q 1 ) = {x2 X: x 1 x 2 0} Dom(q 2 ) = {x2 X: x 1 x 2 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x2 X: x 1 x 2 0} G((q 2,q 1 )) = {x2 X: x 1 x 2 0} R((q 1,q 2 ),x) = R((q 2,q 1 ),x) = {x} x e = 0 is an equilibrium: f(q,0) = 0 & R((q,q ),0) = {0}
28 STABILITY OF HYBRID AUTOMATA H = (Q,X,f,Init,Dom,E,G,R) Definition (stable equilibrium): Let x e = 0 2 X be an equilibrium point of H. x e = 0 is stable if set of (maximal) executions starting from (q 0, x 0 ) 2 Init STABILITY OF HYBRID AUTOMATA H = (Q,X,f,Init,Dom,E,G,R) Definition (stable equilibrium): Let x e = 0 2 X be an equilibrium point of H. x e = 0 is stable if set of (maximal) executions starting from (q 0, x 0 ) 2 Init Remark: Stability does not imply convergence To analyse convergence, only infinite executions should be considered
29 STABILITY OF HYBRID AUTOMATA H = (Q,X,f,Init,Dom,E,G,R) Definition (stable equilibrium): Let x e = 0 2 X be an equilibrium point of H. x e = 0 is stable if Definition (asymptotically stable equilibrium): set of (maximal) executions starting from (q 0, x 0 ) 2 Init Let x e = 0 2 X be an equilibrium point of H. x e = 0 is asymptotically stable if it is stable and d>0 that can be chosen so that set of infinite executions starting from (q 0, x 0 ) 2 Init 1 := i ( i - i ) continuous extent 1 < 1 if Zeno STABILITY OF HYBRID AUTOMATA H = (Q,X,f,Init,Dom,E,G,R) Definition (stable equilibrium): Let x e = 0 2 X be an equilibrium point of H. x e = 0 is stable if Question: x e = 0 stable equilibrium for each continuous system dx/dt = f(q,x) implies that x e = 0 stable equilibrium for H?
30 EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1, q 2 } X = < 2 f(q 1,x) = A 1 x and f(q 2,x) = A 2 x with: Init = Q {x2 X: x >0} Dom(q 1 ) = {x2 X: x 1 x 2 0} Dom(q 2 ) = {x2 X: x 1 x 2 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x2 X: x 1 x 2 0} G((q 2,q 1 )) = {x2 X: x 1 x 2 0} R((q 1,q 2 ),x) = R((q 2,q 1 ),x) = {x} x e = 0 is an equilibrium: f(q,0) = 0 & R((q,q ),0) = {0} EXAMPLE: SWITCHED LINEAR SYSTEM x 2 x 1
31 EXAMPLE: SWITCHED LINEAR SYSTEM Swiching between asymptotically stable linear systems. EXAMPLE: SWITCHED LINEAR SYSTEM q 1 q 2 q 2 q 1 q 1 : quadrants 2 and 4 q 2 : quadrants 1 and 3 Switching between asymptotically stable linear systems, but x e = 0 unstable equilibrium of H
32 EXAMPLE: SWITCHED LINEAR SYSTEM x( i+1 ) > x( i ) overshoots sum up EXAMPLE: SWITCHED LINEAR SYSTEM q 2 q 1 q 1 q 2 q 1 : quadrants 1 and 3 q 2 : quadrants 2 and 4
33 EXAMPLE: SWITCHED LINEAR SYSTEM x( i+1 ) < x( i ) Theorem (Lyapunov stability): LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. Consider V: Q D! < is a C 1 function in x such that for all q 2 Q: If for all (, q, x) 2 H (q0,x 0 ) with (q 0,x 0 ) 2 Init Å (Q D), and all q 2 Q, the sequence {V(q( i ),x( i )): q( i ) =q } is non-increasing (or empty), then, x e = 0 is a stable equilibrium of H.
34 Theorem (Lyapunov stability): LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. Consider V: Q D! < is a C 1 function in x such that for all q 2 Q: V(q,x) Lyapunov function for continuous system q ) x e =0 is stable equilibrium for system q If for all (, q, x) 2 H (q0,x 0 ) with (q 0,x 0 ) 2 Init Å (Q D), and all q 2 Q, the sequence {V(q( i ),x( i )): q( i ) =q } is non-increasing (or empty), then, x e = 0 is a stable equilibrium of H. Theorem (Lyapunov stability): LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. Consider V: Q D! < is a C 1 function in x such that for all q 2 Q: V(q,x) Lyapunov function for continuous system q ) x e =0 is stable equilibrium for system q If for all (, q, x) 2 H (q0,x 0 ) with (q 0,x 0 ) 2 Init Å (Q D), and all q 2 Q, the sequence {V(q( i ),x( i )): q( i ) =q } is non-increasing (or empty), then, x e = 0 is a stable equilibrium of H.
35 Sketch of the proof. LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) V(q(t),x(t)) V(q 1,x(t)) Lyapunov function for system q 1! decreases when q(t) = q 1, but can increase when q(t) q 1 V(q 2,x(t)) q(t)= q 1 q(t)= q 1 [ ][ ][ ][ 0 0 = 1 1 = 2 2 = 3 LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Sketch of the proof. V(q(t),x(t)) V(q 1,x(t)) {V(q 1,x( i ))} non-increasing q(t)= q 1 q(t)= q 1 [ ][ ][ ][ 0 0 = 1 1 = 2 2 = 3
36 Sketch of the proof. LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) V(q(t),x(t)) V(q 1,x(t)) q(t)= q 1 q(t)= q 1 [ ][ ][ ][ 0 0 = 1 1 = 2 2 = 3 LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) V(q(t),x(t)) Lyapunov-like function
37 EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) Q = {q 1, q 2 } X = < 2 f(q 1,x) = A 1 x and f(q 2,x) = A 2 x with: Init = Q {x2 X: x >0} Dom(q 1 ) = {x2 X: Cx 0} Dom(q 2 ) = {x2 X: Cx 0} E = {(q 1,q 2 ),(q 2,q 1 )} G((q 1,q 2 )) = {x2 X: Cx 0} G((q 2,q 1 )) = {x2 X: Cx 0}, C T 2 < 2 R((q 1,q 2 ),x) = R((q 2,q 1 ),x) = {x} EXAMPLE: SWITCHED LINEAR SYSTEM H = (Q,X,f,Init,Dom,E,G,R) q 1 q 2
38 EXAMPLE: SWITCHED LINEAR SYSTEM x 2 Cx = 0 x 1 EXAMPLE: SWITCHED LINEAR SYSTEM Proof that x e = 0 is a stable equilibrium of H for any C T 2 <2 : x e = 0 is an equilibrium: f(q 1,0) = f(q 2,0) = 0 R((q 1,q 2 ),0) = R((q 2,q 1 ),0) = {0}
39 EXAMPLE: SWITCHED LINEAR SYSTEM Proof that x e = 0 is a stable equilibrium of H for any C T 2 <2 : x e = 0 is an equilibrium: f(q 1,0) = f(q 2,0) = 0 x e = 0 is stable: R((q 1,q 2 ),0) = R((q 2,q 1 ),0) = {0} consider the candidate Lyapunov-like function: V(q i,x) = x T P i x, where P i =P i T >0 solution to A i T P i + P i A i = - I (V(q i,x) is a Lyapunov function for the asymptotically stable linear system q i ) In each discrete state, the continuous system is as. stable. EXAMPLE: SWITCHED LINEAR SYSTEM Proof that x e = 0 is a stable equilibrium of H for any C T 2 <2: x e = 0 is an equilibrium: f(q 1,0) = f(q 2,0) = 0 R((q 1,q 2 ),0) = R((q 2,q 1 ),0) = {0} x e = 0 is stable: consider the candidate Lyapunov-like function: V(q i,x) = x T P i x, where P i =P T i >0 solution to A T i P i + P i A i = - I
40 EXAMPLE: SWITCHED LINEAR SYSTEM Test for non-increasing sequence condition Let q( i )=q 1 and x( i )=z. EXAMPLE: SWITCHED LINEAR SYSTEM Test for non-increasing sequence condition Since V(q 1,x(t)) is not increasing during [ i, i ], then, when x(t) intersects the switching line at i, it does at a z with a 2 (0,1], hence x( i+1 ) = x( i ) x( i ). Let q( i+1 )=q 2 -z Cx = 0 i = i+1 i z
41 EXAMPLE: SWITCHED LINEAR SYSTEM Test for non-increasing sequence condition Since V(q 2,x(t)) is decreasing during [ i+1, i+1 ], then, when x(t) intersects the switching line at i+1, x( i+2 ) = x( i+1 ) x( i+1 ) x( i ) -z Cx = 0 i = i+1 i z i+2 EXAMPLE: SWITCHED LINEAR SYSTEM Test for non-increasing sequence condition From this, it follows that V(q 1,x( i+2 )) V(q 1,x( i )) -z Cx = 0 i = i+1 i z i+2
42 LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Drawbacks of the approach based on Lyapunov-like functions: In general, it is hard to find a Lyapunov-like function The sequence {V(q( i ),x( i )): q( i ) =q } must be evaluated, which may require solving the ODEs LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Corollary (common Lyapunov function): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. If V: D! < is a C 1 function such that for all q 2 Q: then, x e = 0 is a stable equilibrium of H.
43 LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Corollary (common Lyapunov function): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. If V: D! < is a C 1 function such that for all q 2 Q: independent of q then, x e = 0 is a stable equilibrium of H. LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Corollary (common Lyapunov function): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. If V: D! < is a C 1 function such that for all q 2 Q: V(x) common Lyapunov function for all systems q then, x e = 0 is a stable equilibrium of H.
44 LYAPUNOV STABILITY H = (Q,X,f,Init,Dom,E,G,R) Corollary (common Lyapunov function): Let x e = 0 be an equilibrium for H with R((q,q ),x) = {x}, 8 (q,q )2 E, and D½ X=< n an open set containing x e = 0. If V: D! < is a C 1 function such that for all q 2 Q: V(x) common Lyapunov function for all systems q then, x e = 0 is a stable equilibrium of H. Proof: Define W(q,x) = V(x), 8 q 2 Q and apply the previous theorem V ( x( t)) same V function + identity reset map t
45 COMPUTATIONAL LYAPUNOV METHODS H PL = (Q,X,f,Init,Dom,E,G,R) non-zeno and such that for all q k 2 Q: f(q k,x) = A k x (linear vector fields) Init ½ [ q2 Q {q } Dom(q) (initialization within the domain) for all x2 X, the set Jump(q k,x):= {(q,x ): (q k,q )2 E, x2g((q k,q )), x 2R((q k,q ),x)} has cardinality 1 if x 2 Dom(q k ), 0 otherwise (discrete transitions occur only from the boundary of the domains) (q,x ) 2 Jump(q k,x)! x 2 Dom(q ) and x = x (trivial reset for x) COMPUTATIONAL LYAPUNOV METHODS H PL = (Q,X,f,Init,Dom,E,G,R) non-zeno and such that for all q k 2 Q: f(q k,x) = A k x (linear vector fields) Init ½ [ q2 Q {q } Dom(q) (initialization within the domain) for all x2 X, the set Jump(q k,x):= {(q,x ): (q k,q )2 E, x2g((q k,q )), x 2R((q k,q ),x)} has cardinality 1 if x 2 Dom(q k ), 0 otherwise (discrete transitions occur only from the boundary of the domains) (q,x ) 2 Jump(q k,x)! x 2 Dom(q ) and x = x (trivial reset for x) For this class of Piecewise Linear hybrid automata computationally attractive methods exist to construct the Lyapunov-like function
46 GLOBALLY QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (globally quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists P=P T >0 such that A T k P+ PA k < 0, 8 k Then, x e = 0 is asymptotically stable. Remark: V(x)=x T Px is a common Lyapunov function x e = 0 is stable GLOBALLY QUADRATIC LYAPUNOV FUNCTION Proof (showing exponential stability): There exists >0 such that A k T P+ PA k + I 0, 8 k There exists a unique, infinite, non-zeno execution (,q,x) for every initial state with x:! < n satisfying where k :! [0,1] is such that k k (t)=1, t2 [ i, i ]. Let V(x) = x T Px. Then, for t2 [ i, i ).
47 GLOBALLY QUADRATIC LYAPUNOV FUNCTION Proof. (cont d) min and max eigenvalues of P Since min x 2 V(x) max x 2, then and, hence, which leads to Then, Since 1 =1 (non-zeno), then x(t) goes to zero exponentially as t! 1 GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 conditions of the theorem satisfied with P = I
48 GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 GLOBALLY QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (globally quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists P=P T >0 such that A T k P+ PA k < 0, 8 k Then, x e = 0 is asymptotically stable. Remark: A set of LMIs to solve. This problem can be reformulated as a convex optimization problem. Efficient solvers exist.
49 GLOBALLY QUADRATIC LYAPUNOV FUNCTION Suppose that A k, k=1,2,,n, are Hurwitz matrices. Then, the set of linear matrix inequalities A k T P+ PA k < 0, k=1,2,,n, where P is positive definite symmetric is not feasible if and only if there exist positive definite symmetric matrices R k, k=1,2,,n, such that Proof. Based on results in convex analysis GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 stable node stable focus
50 GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 stable node stable focus no globally quadratic Lyapunov function exists although x e = 0 stable equilibrium PIECEWISE QUADRATIC LYAPUNOV FUNCTION Idea: consider different quadratic Lyapunov functions on each domain and glue them together so as to provide a (nonquadratic) Lyapunov function for H that is continuous at the switching times
51 PIECEWISE QUADRATIC LYAPUNOV FUNCTION Idea: consider different quadratic Lyapunov functions on each domain and glue them together so as to provide a (nonquadratic) Lyapunov function for H that is continuous at the switching times Developed for piecewise linear systems with structured domain and reset LMIs characterization COMPUTATIONAL LYAPUNOV METHODS H PL = (Q,X,f,Init,Dom,E,G,R) non-zeno and such that for all q k 2 Q: f(q k,x) = A k x (linear vector fields) Init ½ [ q2 Q {q } Dom(q) (initialization within the domain) for all x2 X, the set Jump(q k,x):= {(q,x ): (q k,q )2 E, x2g((q k,q )), x 2R((q k,q ),x)} has cardinality 1 if x 2 Dom(q), 0 otherwise (discrete transitions occur only from the boundary of the domains) (q,x ) 2 Jump(q k,x)! x 2 Dom(q ) and x = x (trivial reset for x)
52 PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) satisfies the following additional assumptions: Dom(q) = {x 2 X: E q1 x 0, E q2 x 0,, E qn x 0} (each domain is a polygon) E q = [E q1 T E q2 T E qnt ] T 2 < n n defines the domain. (q,x ) 2 Jump(q,x) F q x = F q x, q q, x =x (matching condition at the boundaries of dom(q) and dom(q )) PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 P k E kt W k E k > 0 Then, x e = 0 is asymptotically stable.
53 PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 P k E kt W k E k > 0 Then, x e = 0 is asymptotically stable. Proof based on the fact that V(x)=x T P k x, x Dom(q k ) is a Lyapunov-like function for H PL, strictly decreasing along its executions PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 P k E kt W k E k > 0 P k positive definite within Dom(k) Then, x e = 0 is asymptotically stable. Proof based on the fact that V(x)=x T P k x, x Dom(q k ) is a Lyapunov-like function for H PL, strictly decreasing along its executions
54 PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 A T k P k + PA k < 0 within Dom(k) P k E kt W k E k > 0 P k positive definite within Dom(k) Then, x e = 0 is asymptotically stable. Proof based on the fact that V(x)=x T P k x, x Dom(q k ) is a Lyapunov-like function for H PL, strictly decreasing along its executions PIECEWISE QUADRATIC LYAPUNOV FUNCTION H PL = (Q,X,f,Init,Dom,E,G,R) Theorem (piecewise quadratic Lyapunov function): Let x e = 0 be an equilibrium for H PL. If there exists U k =U kt, W k =W kt, with all non-negative elements and M=M T, such that P k =F kt MF k satisfies A T k P k + PA k + E kt U k E k < 0 continuitity of V(x) P k E kt W k E k > 0 Then, x e = 0 is asymptotically stable. Proof based on the fact that V(x)=x T P k x, x Dom(q k ) is a Lyapunov-like function for H PL, strictly decreasing along its executions
55 GLOBALLY QUADRATIC LYAPUNOV FUNCTION q 1 q 2 level curves of the piecewise quadratic Lyapunov function (red lines) phase plot of some continuous state trajectories (blue lines) REFERENCES H.K. Khalil. Nonlinear Systems. Prentice Hall, S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, M. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. on Automatic Control, 43(4): , H. Ye, A. Michel, and L. Hou. Stability theory for hybrid dynamical systems. IEEE Transactions on Automatic Control, 43(4): , M. Johansson and A. Rantzer. Computation of piecewise quadratic Lyapunov function for hybrid systems. IEEE Transactions on Automatic Control, 43(4): , R.A. Decarlo, M.S. Branicky, S. Petterson, and B. Lennartson. Perspectives and results on the stability and stabilization of hybrid systems. Proceedings of the IEEE, 88(7): , 2000.
56 Hybrid Systems Course Switched systems & stability OUTLINE Switched Systems Stability of Switched Systems
57 OUTLINE Switched Systems Stability of Switched Systems SWITCHED SYSTEMS a family of systems
58 SWITCHED SYSTEMS a family of systems a signal that orchestrates the switching between them s Note: The value of x is preserved when a switching occurs SWITCHED SYSTEMS AS HYBRID SYSTEMS
59 SWITCHED SYSTEMS vs. HYBRID AUTOMATA switched systems can be seen as a higher-level abstraction of hybrid automata (details of the discrete behavior neglected) simpler to describe but with more solutions than the original hybrid automata (conservative analysis results) SWITCHED SYSTEMS vs. HYBRID AUTOMATA switched systems can be seen as a higher-level abstraction of hybrid automata (details of the discrete behavior neglected) simpler to describe but with more solutions than the original hybrid automata (conservative analysis results) Switched systems are of interest in their own right
60 SWITCHING time-dependent versus state-dependent switching autonomous versus controlled switching SWITCHING time-dependent versus state-dependent switching autonomous versus controlled switching
61 TIME-DEPENDENT SWITCHING s s (exogenous) switching signal piecewise constant function of time s(t) specifies the system that is active at time t SWITCHED LINEAR SYSTEMS family of systems time-dependent switching rule
62 STATE-DEPENDENT SWITCHING s(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 STATE-DEPENDENT SWITCHING
63 SWITCHING time-dependent versus state-dependent switching autonomous versus controlled switching AUTONOMOUS SWITCHING Switching events are triggered by an external mechanism over which we do not have control Examples: unpredictable environmental factors component failures
64 CONTROLLED SWITCHING Switching are imposed so as to achieve a desired behavior of the resulting system switched control systems SWITCHING CONTROL
65 SWITCHING CONTROL The closed-loop system is a switched system Reasons for switching: SWITCHING CONTROL nature of the control problem (system with different operation phases)
66 Reasons for switching: SWITCHING CONTROL nature of the control problem (system with different operation phases) Example: flight control system Reasons for switching: large modeling uncertainty SWITCHING CONTROL
67 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 Reasons for switching: sensor/actuator limitations SWITCHING CONTROL
68 SWITCHING CONTROL Reasons for switching: sensor/actuator limitations Example: quantized control u PLANT x q(x) QUANTIZER q(x) x CONTROLLER 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
69 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] 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
70 CONTROLLED SWITCHING Switching are imposed so as to achieve a desired behavior of the resulting system switched control systems Reasons for switching: large modeling uncertainty nature of the control problem (phase systems) sensor/actuator limitations OUTLINE Switched Systems Stability of Switched Systems
71 SWITCHED SYSTEMS: TIME-DEPENDENT SWITCHING family of systems with piecewise constant switching signal SWITCHED SYSTEMS: TIME-DEPENDENT SWITCHING family of systems with piecewise constant switching signal Stability of the equilibrium x e =0?
72 SWITCHING BETWEEN AS. STABLE SYSTEMS SWITCHING BETWEEN AS. STABLE SYSTEMS unstable!
73 SWITCHING BETWEEN AS. STABLE SYSTEMS unstable! Problem: find conditions that guarantee asymptotic stability under arbitrary switching SWITCHING BETWEEN UNSTABLE SYSTEMS
74 SWITCHING BETWEEN UNSTABLE SYSTEMS stable! SWITCHING BETWEEN UNSTABLE SYSTEMS stable! Problem: identify those switching signals that ensure asymptotic stability
75 Stability for arbitrary switching Stability for constrained switching Stability for arbitrary switching Stability for constrained switching
76 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 GLOBAL UNIFORM ASYMPTOTIC STABILITY (GUAS) Assumption: family of systems with GAS equilibrium in x=0 Remark: if the equilibrium x e =0 is not GAS for one of the systems, then it cannot be GUAS for the switched system
77 COMMON LYAPUNOV FUNCTION The family of systems share a radially unbounded common Lyapunov function at x e =0 if there exists a continuously differentiable function V such that COMMON LYAPUNOV FUNCTION If all systems in the family share a radially unbounded common Lyapunov function at x e =0, then, the equilibrium x e =0 is GUAS. Proof. Same reasoning as for more general hybrid systems
78 GLOBALLY QUADRATIC LYAPUNOV FUNCTION If there exists such that then, the equilibrium x e =0 is GUAS. Proof. is a radially unbounded common Lyapunov function at x e =0. GLOBALLY QUADRATIC LYAPUNOV FUNCTION The existence of a globally quadratic Lyapunov function is not necessary for x e =0 to be GUAS Example: x e =0 is GUAS but there is no common quadratic Lyapunov function
79 SWITCHED SYSTEMS WITH A SPECIAL STRUCTURE Hurwitz matrices commute are upper (or lower) triangular COMMUTING HURWITZ MATRICES => GUAS A 1 A 2 A 2 A 1 s 1 s 2 s 1 s 2 s t s1 t1 2 2 t A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A e t t A s 2 ( k... 1 ) 1 (... 1 ) e k x(0) s 0
80 COMMUTING HURWITZ MATRICES => GUAS 1 A 2 A 2 A 1 quadratic common Lyapunov function: A COMMUTING HURWITZ MATRICES => GUAS 1 A 2 A 2 A 1 quadratic common Lyapunov function: A
81 TRIANGULAR HURWITZ MATRICES => GUAS TRIANGULAR HURWITZ MATRICES => GUAS
82 TRIANGULAR HURWITZ MATRICES => GUAS exponentially stable system exponentially decaying perturbation TRIANGULAR HURWITZ MATRICES => GUAS quadratic common Lyapunov function with P diagonal
83 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 Stability for arbitrary switching Stability for constrained switching
84 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t t i, s i dwell time D STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) t D
85 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: STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A 0 D 0 0 e i t t e e e D D t log slowest decay rate so that the inequality holds 8 i
86 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A 0 D log e i t e t D STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A 0 D log e i t e D t log D 0 (0, 0)
87 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 The switching intervals satisfy t, i s i A t x (t) e 2 k A s e 1 k e A 2 t 1 e A1s1 x(0) A 0 D log D e i t e e 1 D t log D (0, log 0) D 0 D 0 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s 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) t 0
88 STABILITY UNDER SLOW SWITCHING Hurwitz matrices s 1 s 2 s 1 s 2 s t s1 t1 2 2 A t e 2 k A s e 1 k e A 2 t 1 A1s1 x (t) e x(0) t DWELL TIME: EXTENSIONS adaptive version: the dwell time is selected based on matrix A i so as to make the dynamics of the system contract by some 2 (0,1) during the dwell time average dwell time N s ( T, t) N 0 T t AD ( t, T # of switches on ) average dwell time N0 0 no switching: cannot switch if T t AD N 1 dwell time: cannot switch twice if T t AD 0 Same bound on AD as in the dwell time case. Larger values of x(t) in finite time because of N 0
89 STABILITY UNDER STATE-DEPENDENT SWITCHING s: X Q : s(x) = i if x 2 X i 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 e = 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
90 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 A T P PA 0
91 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 T T 1 a( A P PA1 ) (1 a)( A2 P PA2 ) 0 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 So for each : either x a( A P PA1 ) (1 a)( A2 P PA2 ) T T T 1 x 0 T 1 ( A P PA1) x 0 or x T 0 T 2 ( A P PA2) x 0
92 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 So for each : either x T T T 1 a( A P PA1 ) (1 a)( A2 P PA2 ) x 0 T 1 ( A P PA1) x 0 or x T 0 T 2 ( A P PA2) x 0 Region where system 1 is active for the system Region where system 2 is active is a Lyapunov function at x e =0 => GAS STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 So for each : either x T T T 1 a( A P PA1 ) (1 a)( A2 P PA2 ) x 0 T 1 ( A P PA1) x 0 or x T 0 T 2 ( A P PA2) x 0 Region where system 1 is active Region where system 2 is active
93 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Assume: A aa ( 1 a A stable for some a (0,1 ) 1 ) 2 So for each : either x T T T 1 a( A P PA1 ) (1 a)( A2 P PA2 ) x 0 T 1 ( A P PA1) x 0 or x T 0 T 2 ( A P PA2) x 0 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Theorem: If the matrices A 1 and A 2 have a Hurwitz combination, then, there exists a state dependent switching strategy such that the switching system x = A σ x is GAS
94 STABILIZATION BY SWITCHING x A1 x, x A2 x both unstable Theorem: If the matrices A 1 and A 2 have a Hurwitz combination, then, there exists a state dependent switching strategy such that the switching system x = A σ x is GAS Extensions to the m>2 matrices case: two matrices A i and A j have a Hurwitz combination more than 2 matrices have a Hurwitz combination Main source: Switching in Systems and Control Daniel Liberzon, Birkhauser, 2003.
Hybrid Systems - Lecture n. 3 Lyapunov stability
OUTLINE Focus: stability of equilibrium point Hybrid Systems - Lecture n. 3 Lyapunov stability Maria Prandini DEI - Politecnico di Milano E-mail: prandini@elet.polimi.it continuous systems decribed by
More informationSwitched systems: stability
Switched systems: stability OUTLINE Switched Systems Stability of Switched Systems OUTLINE Switched Systems Stability of Switched Systems a family of systems SWITCHED SYSTEMS SWITCHED SYSTEMS a family
More informationLyapunov stability ORDINARY DIFFERENTIAL EQUATIONS
Lyapunov stability ORDINARY DIFFERENTIAL EQUATIONS An ordinary differential equation is a mathematical model of a continuous state continuous time system: X = < n state space f: < n! < n vector field (assigns
More informationCourse on Hybrid Systems
Course on Hybrid Systems Maria Prandini Politecnico di Milano, Italy Organizer and lecturer: Maria Prandini Politecnico di Milano, Italy maria.prandini@polimi.it Additional lecturers: CONTACT INFO Goran
More informationHybrid Control and Switched Systems. Lecture #11 Stability of switched system: Arbitrary switching
Hybrid Control and Switched Systems Lecture #11 Stability of switched system: Arbitrary switching João P. Hespanha University of California at Santa Barbara Stability under arbitrary switching Instability
More informationEE291E Lecture Notes 3 Autonomous Hybrid Automata
EE9E Lecture Notes 3 Autonomous Hybrid Automata Claire J. Tomlin January, 8 The lecture notes for this course are based on the first draft of a research monograph: Hybrid Systems. The monograph is copyright
More informationModeling & Control of Hybrid Systems Chapter 4 Stability
Modeling & Control of Hybrid Systems Chapter 4 Stability Overview 1. Switched systems 2. Lyapunov theory for smooth and linear systems 3. Stability for any switching signal 4. Stability for given switching
More informationGLOBAL ANALYSIS OF PIECEWISE LINEAR SYSTEMS USING IMPACT MAPS AND QUADRATIC SURFACE LYAPUNOV FUNCTIONS
GLOBAL ANALYSIS OF PIECEWISE LINEAR SYSTEMS USING IMPACT MAPS AND QUADRATIC SURFACE LYAPUNOV FUNCTIONS Jorge M. Gonçalves, Alexandre Megretski y, Munther A. Dahleh y California Institute of Technology
More informationHybrid Systems Techniques for Convergence of Solutions to Switching Systems
Hybrid Systems Techniques for Convergence of Solutions to Switching Systems Rafal Goebel, Ricardo G. Sanfelice, and Andrew R. Teel Abstract Invariance principles for hybrid systems are used to derive invariance
More informationThe servo problem for piecewise linear systems
The servo problem for piecewise linear systems Stefan Solyom and Anders Rantzer Department of Automatic Control Lund Institute of Technology Box 8, S-22 Lund Sweden {stefan rantzer}@control.lth.se Abstract
More informationEN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015
EN530.678 Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 Prof: Marin Kobilarov 0.1 Model prerequisites Consider ẋ = f(t, x). We will make the following basic assumptions
More informationAnnouncements. Review. Announcements. Piecewise Affine Quadratic Lyapunov Theory. EECE 571M/491M, Spring 2007 Lecture 9
EECE 571M/491M, Spring 2007 Lecture 9 Piecewise Affine Quadratic Lyapunov Theory Meeko Oishi, Ph.D. Electrical and Computer Engineering University of British Columbia, BC Announcements Lecture review examples
More informationUniversity of California. Berkeley, CA fzhangjun johans lygeros Abstract
Dynamical Systems Revisited: Hybrid Systems with Zeno Executions Jun Zhang, Karl Henrik Johansson y, John Lygeros, and Shankar Sastry Department of Electrical Engineering and Computer Sciences University
More informationStability lectures. Stability of Linear Systems. Stability of Linear Systems. Stability of Continuous Systems. EECE 571M/491M, Spring 2008 Lecture 5
EECE 571M/491M, Spring 2008 Lecture 5 Stability of Continuous Systems http://courses.ece.ubc.ca/491m moishi@ece.ubc.ca Dr. Meeko Oishi Electrical and Computer Engineering University of British Columbia,
More informationNonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems
Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Time-varying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin
More informationConvergence Rate of Nonlinear Switched Systems
Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the
More informationLecture 4. Chapter 4: Lyapunov Stability. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.
Lecture 4 Chapter 4: Lyapunov Stability Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 4 p. 1/86 Autonomous Systems Consider the autonomous system ẋ
More informationLyapunov Stability Theory
Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous
More informationStability of Deterministic Finite State Machines
2005 American Control Conference June 8-10, 2005. Portland, OR, USA FrA17.3 Stability of Deterministic Finite State Machines Danielle C. Tarraf 1 Munther A. Dahleh 2 Alexandre Megretski 3 Abstract We approach
More informationGramians based model reduction for hybrid switched systems
Gramians based model reduction for hybrid switched systems Y. Chahlaoui Younes.Chahlaoui@manchester.ac.uk Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) School of Mathematics
More informationConverse Lyapunov theorem and Input-to-State Stability
Converse Lyapunov theorem and Input-to-State Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts
More informationStability Analysis of a Proportional with Intermittent Integral Control System
American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, ThB4. Stability Analysis of a Proportional with Intermittent Integral Control System Jin Lu and Lyndon J. Brown Abstract
More informationSolution of Linear State-space Systems
Solution of Linear State-space Systems Homogeneous (u=0) LTV systems first Theorem (Peano-Baker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state
More informationAnalysis and design of switched normal systems
Nonlinear Analysis 65 (2006) 2248 2259 www.elsevier.com/locate/na Analysis and design of switched normal systems Guisheng Zhai a,, Xuping Xu b, Hai Lin c, Anthony N. Michel c a Department of Mechanical
More informationAnnouncements. Affine dynamics: Example #1. Review: Multiple Lyap. Fcns. Review and Examples: Linear/PWA Quad. Lyapunov Theory
EECE 571M/491M, Spring 2007 Lecture 10 Review and Examples: Linear/PWA Quad. Lyapunov Theory Meeko Oishi, Ph.D. Electrical and Computer Engineering University of British Columbia, BC Announcements Reminder:
More informationSolution of Additional Exercises for Chapter 4
1 1. (1) Try V (x) = 1 (x 1 + x ). Solution of Additional Exercises for Chapter 4 V (x) = x 1 ( x 1 + x ) x = x 1 x + x 1 x In the neighborhood of the origin, the term (x 1 + x ) dominates. Hence, the
More informationHybrid Control and Switched Systems. Lecture #7 Stability and convergence of ODEs
Hybrid Control and Switched Systems Lecture #7 Stability and convergence of ODEs João P. Hespanha University of California at Santa Barbara Summary Lyapunov stability of ODEs epsilon-delta and beta-function
More informationADAPTIVE control of uncertain time-varying plants is a
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 1, JANUARY 2011 27 Supervisory Control of Uncertain Linear Time-Varying Systems Linh Vu, Member, IEEE, Daniel Liberzon, Senior Member, IEEE Abstract
More informationDisturbance Attenuation Properties for Discrete-Time Uncertain Switched Linear Systems
Disturbance Attenuation Properties for Discrete-Time Uncertain Switched Linear Systems Hai Lin Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556, USA Panos J. Antsaklis
More informationSTABILIZATION THROUGH HYBRID CONTROL
STABILIZATION THROUGH HYBRID CONTROL João P. Hespanha, Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106-9560, USA. Keywords: Hybrid Systems; Switched
More informationGlobal Analysis of Piecewise Linear Systems Using Impact Maps and Surface Lyapunov Functions
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 12, DECEMBER 2003 2089 Global Analysis of Piecewise Linear Systems Using Impact Maps and Surface Lyapunov Functions Jorge M Gonçalves, Alexandre Megretski,
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Lyapunov Stability - I Hanz Richter Mechanical Engineering Department Cleveland State University Definition of Stability - Lyapunov Sense Lyapunov
More informationL 2 -induced Gains of Switched Systems and Classes of Switching Signals
L 2 -induced Gains of Switched Systems and Classes of Switching Signals Kenji Hirata and João P. Hespanha Abstract This paper addresses the L 2-induced gain analysis for switched linear systems. We exploit
More information2 Lyapunov Stability. x(0) x 0 < δ x(t) x 0 < ɛ
1 2 Lyapunov Stability Whereas I/O stability is concerned with the effect of inputs on outputs, Lyapunov stability deals with unforced systems: ẋ = f(x, t) (1) where x R n, t R +, and f : R n R + R n.
More informationNonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems
Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Time-varying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin
More informationChapter III. Stability of Linear Systems
1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,
More informationBenchmark problems in stability and design of. switched systems. Daniel Liberzon and A. Stephen Morse. Department of Electrical Engineering
Benchmark problems in stability and design of switched systems Daniel Liberzon and A. Stephen Morse Department of Electrical Engineering Yale University New Haven, CT 06520-8267 fliberzon, morseg@sysc.eng.yale.edu
More informationModeling and Analysis of Dynamic Systems
Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System - Stability
More informationTopic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis
Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of
More informationNonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1
Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium
More informationEE C128 / ME C134 Feedback Control Systems
EE C128 / ME C134 Feedback Control Systems Lecture Additional Material Introduction to Model Predictive Control Maximilian Balandat Department of Electrical Engineering & Computer Science University of
More informationMulti-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures
Preprints of the 19th World Congress The International Federation of Automatic Control Multi-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures Eric Peterson Harry G.
More informationIntroduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems
p. 1/5 Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems p. 2/5 Time-varying Systems ẋ = f(t, x) f(t, x) is piecewise continuous in t and locally Lipschitz in x for all t
More informationOn one Application of Newton s Method to Stability Problem
Journal of Multidciplinary Engineering Science Technology (JMEST) ISSN: 359-0040 Vol. Issue 5, December - 204 On one Application of Newton s Method to Stability Problem Şerife Yılmaz Department of Mathematics,
More informationPractical Stabilization of Integrator Switched Systems
Practical Stabilization of Integrator Switched Systems Xuping Xu and Panos J. Antsaklis 2 Department of Electrical and Computer Engineering Penn State Erie, Erie, PA 6563, USA. E-mail: Xuping-Xu@psu.edu
More informationOn optimal quadratic Lyapunov functions for polynomial systems
On optimal quadratic Lyapunov functions for polynomial systems G. Chesi 1,A.Tesi 2, A. Vicino 1 1 Dipartimento di Ingegneria dell Informazione, Università disiena Via Roma 56, 53100 Siena, Italy 2 Dipartimento
More informationNonlinear Control Lecture 5: Stability Analysis II
Nonlinear Control Lecture 5: Stability Analysis II Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 5 1/41
More informationNonlinear Control. Nonlinear Control Lecture # 3 Stability of Equilibrium Points
Nonlinear Control Lecture # 3 Stability of Equilibrium Points The Invariance Principle Definitions Let x(t) be a solution of ẋ = f(x) A point p is a positive limit point of x(t) if there is a sequence
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 20: LMI/SOS Tools for the Study of Hybrid Systems Stability Concepts There are several classes of problems for
More informationNonlinear Control Design for Linear Differential Inclusions via Convex Hull Quadratic Lyapunov Functions
Nonlinear Control Design for Linear Differential Inclusions via Convex Hull Quadratic Lyapunov Functions Tingshu Hu Abstract This paper presents a nonlinear control design method for robust stabilization
More informationDOMAIN OF ATTRACTION: ESTIMATES FOR NON-POLYNOMIAL SYSTEMS VIA LMIS. Graziano Chesi
DOMAIN OF ATTRACTION: ESTIMATES FOR NON-POLYNOMIAL SYSTEMS VIA LMIS Graziano Chesi Dipartimento di Ingegneria dell Informazione Università di Siena Email: chesi@dii.unisi.it Abstract: Estimating the Domain
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. XII - Lyapunov Stability - Hassan K. Khalil
LYAPUNO STABILITY Hassan K. Khalil Department of Electrical and Computer Enigneering, Michigan State University, USA. Keywords: Asymptotic stability, Autonomous systems, Exponential stability, Global asymptotic
More informationMCE/EEC 647/747: Robot Dynamics and Control. Lecture 8: Basic Lyapunov Stability Theory
MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 p.1/17 Stability in the sense of Lyapunov A
More informationUsing Lyapunov Theory I
Lecture 33 Stability Analysis of Nonlinear Systems Using Lyapunov heory I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Outline Motivation Definitions
More informationLinear Matrix Inequalities in Robust Control. Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University MTNS 2002
Linear Matrix Inequalities in Robust Control Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University MTNS 2002 Objective A brief introduction to LMI techniques for Robust Control Emphasis on
More informationI. D. Landau, A. Karimi: A Course on Adaptive Control Adaptive Control. Part 9: Adaptive Control with Multiple Models and Switching
I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 1 Adaptive Control Part 9: Adaptive Control with Multiple Models and Switching I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 2 Outline
More informationStabilization of constrained linear systems via smoothed truncated ellipsoids
Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Stabilization of constrained linear systems via smoothed truncated ellipsoids A. Balestrino, E. Crisostomi, S. Grammatico,
More informationStability in the sense of Lyapunov
CHAPTER 5 Stability in the sense of Lyapunov Stability is one of the most important properties characterizing a system s qualitative behavior. There are a number of stability concepts used in the study
More informationTechnical Notes and Correspondence
1108 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY 2002 echnical Notes and Correspondence Stability Analysis of Piecewise Discrete-ime Linear Systems Gang Feng Abstract his note presents a stability
More informationOn common quadratic Lyapunov functions for stable discrete-time LTI systems
On common quadratic Lyapunov functions for stable discrete-time LTI systems Oliver Mason, Hamilton Institute, NUI Maynooth, Maynooth, Co. Kildare, Ireland. (oliver.mason@may.ie) Robert Shorten 1, Hamilton
More informationDynamical Systems Revisited: Hybrid Systems with Zeno Executions
Dynamical Systems Revisited: Hybrid Systems with Zeno Executions Jun Zhang, Karl Henrik Johansson, John Lygeros, and Shankar Sastry Department of Electrical Engineering and Computer Sciences University
More informationMinimum-Phase Property of Nonlinear Systems in Terms of a Dissipation Inequality
Minimum-Phase Property of Nonlinear Systems in Terms of a Dissipation Inequality Christian Ebenbauer Institute for Systems Theory in Engineering, University of Stuttgart, 70550 Stuttgart, Germany ce@ist.uni-stuttgart.de
More informationGlobal Analysis of Piecewise Linear Systems Using Impact Maps and Quadratic Surface Lyapunov Functions
Global Analysis of Piecewise Linear Systems Using Impact Maps and Quadratic Surface Lyapunov Functions Jorge M. Gonçalves, Alexandre Megretski, Munther A. Dahleh Department of EECS, Room 35-41 MIT, Cambridge,
More informationStability and Stabilizability of Switched Linear Systems: A Short Survey of Recent Results
Proceedings of the 2005 IEEE International Symposium on Intelligent Control Limassol, Cyprus, June 27-29, 2005 MoA01-5 Stability and Stabilizability of Switched Linear Systems: A Short Survey of Recent
More informationRank-one LMIs and Lyapunov's Inequality. Gjerrit Meinsma 4. Abstract. We describe a new proof of the well-known Lyapunov's matrix inequality about
Rank-one LMIs and Lyapunov's Inequality Didier Henrion 1;; Gjerrit Meinsma Abstract We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix
More informationNonlinear Control. Nonlinear Control Lecture # 2 Stability of Equilibrium Points
Nonlinear Control Lecture # 2 Stability of Equilibrium Points Basic Concepts ẋ = f(x) f is locally Lipschitz over a domain D R n Suppose x D is an equilibrium point; that is, f( x) = 0 Characterize and
More informationOutline. Input to state Stability. Nonlinear Realization. Recall: _ Space. _ Space: Space of all piecewise continuous functions
Outline Input to state Stability Motivation for Input to State Stability (ISS) ISS Lyapunov function. Stability theorems. M. Sami Fadali Professor EBME University of Nevada, Reno 1 2 Recall: _ Space _
More informationHybrid Control and Switched Systems. Lecture #1 Hybrid systems are everywhere: Examples
Hybrid Control and Switched Systems Lecture #1 Hybrid systems are everywhere: Examples João P. Hespanha University of California at Santa Barbara Summary Examples of hybrid systems 1. Bouncing ball 2.
More informationLyapunov Stability Analysis: Open Loop
Copyright F.L. Lewis 008 All rights reserved Updated: hursday, August 8, 008 Lyapunov Stability Analysis: Open Loop We know that the stability of linear time-invariant (LI) dynamical systems can be determined
More informationWE CONSIDER linear systems subject to input saturation
440 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 3, MARCH 2003 Composite Quadratic Lyapunov Functions for Constrained Control Systems Tingshu Hu, Senior Member, IEEE, Zongli Lin, Senior Member, IEEE
More informationModels for Control and Verification
Outline Models for Control and Verification Ian Mitchell Department of Computer Science The University of British Columbia Classes of models Well-posed models Difference Equations Nonlinear Ordinary Differential
More informationHybrid Control and Switched Systems. Lecture #9 Analysis tools for hybrid systems: Impact maps
Hybrid Control and Switched Systems Lecture #9 Analysis tools for hybrid systems: Impact maps João P. Hespanha University of California at Santa Barbara Summary Analysis tools for hybrid systems Impact
More informationOn common linear/quadratic Lyapunov functions for switched linear systems
On common linear/quadratic Lyapunov functions for switched linear systems M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland, USA
More informationStability of Hybrid Control Systems Based on Time-State Control Forms
Stability of Hybrid Control Systems Based on Time-State Control Forms Yoshikatsu HOSHI, Mitsuji SAMPEI, Shigeki NAKAURA Department of Mechanical and Control Engineering Tokyo Institute of Technology 2
More informationNonlinear System Analysis
Nonlinear System Analysis Lyapunov Based Approach Lecture 4 Module 1 Dr. Laxmidhar Behera Department of Electrical Engineering, Indian Institute of Technology, Kanpur. January 4, 2003 Intelligent Control
More informationDynamical Systems & Lyapunov Stability
Dynamical Systems & Lyapunov Stability Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Ordinary Differential Equations Existence & uniqueness Continuous dependence
More informationStability of switched block upper-triangular linear systems with switching delay: Application to large distributed systems
American Control Conference on O'Farrell Street, San Francisco, CA, USA June 9 - July, Stability of switched block upper-triangular linear systems with switching delay: Application to large distributed
More informationEE363 homework 7 solutions
EE363 Prof. S. Boyd EE363 homework 7 solutions 1. Gain margin for a linear quadratic regulator. Let K be the optimal state feedback gain for the LQR problem with system ẋ = Ax + Bu, state cost matrix Q,
More informationAsymptotic convergence of constrained primal-dual dynamics
Asymptotic convergence of constrained primal-dual dynamics Ashish Cherukuri a, Enrique Mallada b, Jorge Cortés a a Department of Mechanical and Aerospace Engineering, University of California, San Diego,
More informationEML5311 Lyapunov Stability & Robust Control Design
EML5311 Lyapunov Stability & Robust Control Design 1 Lyapunov Stability criterion In Robust control design of nonlinear uncertain systems, stability theory plays an important role in engineering systems.
More informationAQUANTIZER is a device that converts a real-valued
830 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 4, APRIL 2012 Input to State Stabilizing Controller for Systems With Coarse Quantization Yoav Sharon, Member, IEEE, Daniel Liberzon, Senior Member,
More informationESC794: Special Topics: Model Predictive Control
ESC794: Special Topics: Model Predictive Control Discrete-Time Systems Hanz Richter, Professor Mechanical Engineering Department Cleveland State University Discrete-Time vs. Sampled-Data Systems A continuous-time
More information7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system
7 Stability 7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system ẋ(t) = A x(t), x(0) = x 0, A R n n, x 0 R n. (14) The origin x = 0 is a globally asymptotically
More informationLecture 6 Verification of Hybrid Systems
Lecture 6 Verification of Hybrid Systems Ufuk Topcu Nok Wongpiromsarn Richard M. Murray AFRL, 25 April 2012 Outline: A hybrid system model Finite-state abstractions and use of model checking Deductive
More informationOn Dwell Time Minimization for Switched Delay Systems: Free-Weighting Matrices Method
On Dwell Time Minimization for Switched Delay Systems: Free-Weighting Matrices Method Ahmet Taha Koru Akın Delibaşı and Hitay Özbay Abstract In this paper we present a quasi-convex minimization method
More informationHYBRID AND SWITCHED SYSTEMS ECE229 WINTER 2004
HYBRID AND SWITCHED SYSTEMS ECE229 WINTER 2004 Course description As computers, digital networks, and embedded systems become ubiquitous and increasingly complex, one needs to understand the coupling between
More informationAverage-Consensus of Multi-Agent Systems with Direct Topology Based on Event-Triggered Control
Outline Background Preliminaries Consensus Numerical simulations Conclusions Average-Consensus of Multi-Agent Systems with Direct Topology Based on Event-Triggered Control Email: lzhx@nankai.edu.cn, chenzq@nankai.edu.cn
More informationSwitched Systems: Mixing Logic with Differential Equations
research supported by NSF Switched Systems: Mixing Logic with Differential Equations João P. Hespanha Center for Control Dynamical Systems and Computation Outline Logic-based switched systems framework
More informationSATURATION is an ubiquitous nonlinearity in engineering
1770 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 11, NOVEMBER 2006 Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions Tingshu Hu, Andrew R Teel, Fellow,
More informationTo appear in IEEE Control Systems Magazine 1. Basic Problems in Stability and Design of. Switched Systems. Yale University. New Haven, CT
To appear in IEEE Control Systems Magazine 1 Basic Problems in Stability and Design of Switched Systems Daniel Liberzon and A. Stephen Morse Department of Electrical Engineering Yale University New Haven,
More informationStability of Nonlinear Systems An Introduction
Stability of Nonlinear Systems An Introduction Michael Baldea Department of Chemical Engineering The University of Texas at Austin April 3, 2012 The Concept of Stability Consider the generic nonlinear
More informationComplex Dynamic Systems: Qualitative vs Quantitative analysis
Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems Chiara Mocenni Department of Information Engineering and Mathematics University of Siena (mocenni@diism.unisi.it) Dynamic
More informationNonlinear systems. Lyapunov stability theory. G. Ferrari Trecate
Nonlinear systems Lyapunov stability theory G. Ferrari Trecate Dipartimento di Ingegneria Industriale e dell Informazione Università degli Studi di Pavia Advanced automation and control Ferrari Trecate
More informationSTABILITY ANALYSIS OF DYNAMIC SYSTEMS
C. Melchiorri (DEI) Automatic Control & System Theory 1 AUTOMATIC CONTROL AND SYSTEM THEORY STABILITY ANALYSIS OF DYNAMIC SYSTEMS Claudio Melchiorri Dipartimento di Ingegneria dell Energia Elettrica e
More informationEstimating the Region of Attraction of Ordinary Differential Equations by Quantified Constraint Solving
Estimating the Region of Attraction of Ordinary Differential Equations by Quantified Constraint Solving Henning Burchardt and Stefan Ratschan October 31, 2007 Abstract We formulate the problem of estimating
More informationH State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 11, NO 2, APRIL 2003 271 H State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions Doo Jin Choi and PooGyeon
More informationFeedback Control CONTROL THEORY FUNDAMENTALS. Feedback Control: A History. Feedback Control: A History (contd.) Anuradha Annaswamy
Feedback Control CONTROL THEORY FUNDAMENTALS Actuator Sensor + Anuradha Annaswamy Active adaptive Control Laboratory Massachusetts Institute of Technology must follow with» Speed» Accuracy Feeback: Measure
More information1 Introduction In this paper, we study the stability of continuous and discrete-time switched linear systems using piecewise linear Lyapunov functions
Design of Stabilizing Switching Control Laws for Discrete and Continuous-Time Linear Systems Using Piecewise-Linear Lyapunov Functions Λ Xenofon D. Koutsoukos Xerox Palo Alto Research Center Coyote Hill
More informationRobust Observer for Uncertain T S model of a Synchronous Machine
Recent Advances in Circuits Communications Signal Processing Robust Observer for Uncertain T S model of a Synchronous Machine OUAALINE Najat ELALAMI Noureddine Laboratory of Automation Computer Engineering
More informationAsymptotic Disturbance Attenuation Properties for Continuous-Time Uncertain Switched Linear Systems
Proceedings of the 17th World Congress The International Federation of Automatic Control Asymptotic Disturbance Attenuation Properties for Continuous-Time Uncertain Switched Linear Systems Hai Lin Panos
More information