Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 6: Storage Functions And Stability Analysis This lecture presents results describing the relation between existence of Lyapunov or storage functions and stability of dynamical systems 6.1 Stability of an equilibria n this section we consider ode models i(t=a(r(t)), where a: XHR is a continuous function defined on an open subset X of R". Remem- ber that a point io E X is an equilibrium of (6. 1)if a(io)=0, i.e. if r(t)=To is a solution of (6. 1). Depending on the behavior of other solutions of(6.1)(they may stay close to Co, or converge to Io as t-00, or satisfy some other specifications) the equilibrium may be called stable, asymptotically stable, etc. Various types of stability of equilibria can be derived using storage functions. On the other hand, in many cases existence of storage functions with certain properties is impled by stability of equilibria. 6.1.1 Locally stable equilibr Remember that a point to E X is called a(locally) stable equilibrium of ODE(6. 1) if for every e>0 there exists 8>0 such that all maximal solutions =a(t)of (6.1)with (0)- ol s are deinfed for all t>0, and satisfy lz(t)-Tol e for all t20 The statement below uses the notion of a lower semicontinuity: a function f: Y+R, defined on a subset y of r" is called lower semicontinuous if lim f,f(x)≥f()Vz,∈Y 0,r>0正∈Y:|一正,< Version of September 24, 2003
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 6: Storage Functions And Stability Analysis1 This lecture presents results describing the relation between existence of Lyapunov or storage functions and stability of dynamical systems. 6.1 Stability of an equilibria In this section we consider ODE models x˙ (t) = a(x(t)), (6.1) where a : X �� Rn is a continuous function defined on an open subset X of Rn. Remember that a point x¯0 ≤ X is an equilibrium of (6.1) if a(¯x0) = 0, i.e. if x(t) ≥ x¯0 is a solution of (6.1). Depending on the behavior of other solutions of (6.1) (they may stay close to x¯0, or converge to x¯0 as t � →, or satisfy some other specifications) the equilibrium may be called stable, asymptotically stable, etc. Various types of stability of equilibria can be derived using storage functions. On the other hand, in many cases existence of storage functions with certain properties is impled by stability of equilibria. 6.1.1 Locally stable equilibria Remember that a point x¯0 ≤ X is called a (locally) stable equilibrium of ODE (6.1) if for every � > 0 there exists � > 0 such that all maximal solutions x = x(t) of (6.1) with |x(0) − x¯0| � � are deinfed for all t ∀ 0, and satisfy |x(t) − x¯0| 0 x¯→Y : |x¯−x¯�|<r 1Version of September 24, 2003
Theorem 6.1 to E X is a locally stable equilibrium of (6.1) if and only if there exist c>0 and a lower semicontinuous function V: Bio)H+R, defined on B2(0)={z:‖x-l‖l0, one can find d>0 such that V(min{e,c/2})>V(z)z:|-o<6. Hence a solution =a(t)of (6.1)with an initial condition such that l r(0)-iol<8(and hence V(z(o))<v(minE, c/2))cannot cross the sphere i-iol= mine, c/21 To prove that(i)implies(ii), define V by V(5)=suppl z(t)-Tol: t20,x(0)=I, a( satisfies(6.1)) Since by assumption, solutions starting close enough to io never leave a given disc cen- tered at io, V is well defined in a neigborhood Xo of o. Then, by its very definition V((t)) is not increasing for every solution of(6. 1)starting in Xo. Since V is a supremum, it is lower semicontinuous(actually, here we use the fact, not mentioned before, that if Tk=Tk(t) are solutions of( 6. 1)such that Tk (to)-i0o and Tk(t1)-ilo then there exists a solution of (6.1) with a(to)=i0 and a(t1)=ioo). Moreover, V is continuous at o because of stability of the equilibrium o One can ask whether existence of a Lyapunov function from a better class(say,con- tinuous functions) is possible. The answer, in general, is negative, as demonstrated by the following example Example 6.1 The equilibrium To=0 of the first order ODE Let a: R+R be defined erp(-1/i)sgn(i)sin2(E),I+0 a(T 7=0 Then a is arbitrary number of times differentiable and the equilibrium Io=0 of (6. 1)is locally stable. However, every continuous function V: R+R which does not increase along system trajectories will achieve a marimum at Io=0
� 2 Theorem 6.1 x¯0 ≤ X is a locally stable equilibrium of (6.1) if and only if there exist c > 0 and a lower semicontinuous function V : Bc(¯x0) �� R, defined on Bc(¯x0) = {x¯ : ∞x − x¯0∞ 0, one can find � > 0 such that V x) � ¯ x − ¯ ˆ (min{�, c/2}) > V (¯ x : |¯ x0| < �. Hence a solution x = x(t) of (6.1) with an initial condition such that |x(0) − x¯0| < � (and hence V (x(0)) < Vˆ (min{�, c/2}) cannot cross the sphere |x¯ − x¯0| = min{�, c/2}. To prove that (i) implies (ii), define V by V (¯x) = sup{∞x(t) − x¯0∞ : t ∀ 0, x(0) = x, ¯ x(·) satisfies (6.1) }. (6.2) Since, by assumption, solutions starting close enough to x¯0 never leave a given disc centered at x¯0, V is well defined in a neigborhood X0 of x0. Then, by its very definition, V (x(t)) is not increasing for every solution of (6.1) starting in X0. Since V is a supremum, it is lower semicontinuous (actually, here we use the fact, not mentioned before, that if 0 � and xk(t1) � x¯1 xk = xk(t) are solutions of (6.1) such that xk(t0) � x¯ � then there exists 0 � and x(t1) = x¯1 a solution of (6.1) with x(t0) = x¯ �). Moreover, V is continuous at x0, because of stability of the equilibrium x0. One can ask whether existence of a Lyapunov function from a better class (say, continuous functions) is possible. The answer, in general, is negative, as demonstrated by the following example. Example 6.1 The equilibrium x¯0 = 0 of the first order ODE Let a : R �� R be defined by x2 exp(−1/¯ )sgn(¯x)sin x), x ≡ 2 (¯ ¯ = 0, a(¯x) = 0, x¯ = 0. Then a is arbitrary number of times differentialble and the equilibrium x¯0 = 0 of (6.1) is locally stable. However, every continuous function V : R �� R which does not increase along system trajectories will achieve a maximum at x¯0 = 0
For the case of a linear system, however, local stability of equilibrium Io=0 implies existence of a Lyapunov function which is a positive definite quadratic form Theorem 6.2 Ifa: R"HR" is defined by a(i)=At here A is a given n-by-n matric, then equilibrium To=0 of(6. 1)is locally stable if and only if there erists a matrix Q=Q>0 such that V(r(t))=r(t)'Qa(t) is monotonically non-increasing along the solutions of(6.1) The proof of this theorem, which can be based on considering a Jordan form of A, is usually a part of a standard linear systems class 6.1.2 Locally asymptotically stable equilibria A point To is called a(locally) asymptotically stable equilibrium of (6. 1) if it is a stable quilibria, and, in addition, there exists eo>0 such that every solution of (6. 1)with (0)-iol Eo converges to To as t-0o Theorem 6.3 IfV: X HR is a continuous function such that V(x0)<V(z)V∈X/{fo}, and V(r(t)) is strictly monotonically decreasing for every solation of (6.1)encept r(t) To then To is a locally asymptotically stable equilibrium of(6.1) From Theorem 6.1, To is a locally stable equilibrium. It is sufficient to show that every solution =a(t) of(6. 1) starting sufficiently close to To will converge to Co as t-00. Assume the contrary. Then (t) is bounded, and hence will have at least one limit point I, which is not Io. In addition, the limit V of V(a(t)) will exist Consider a solution a*=a(t) starting from that point. By continuous dependence on initial conditions we conclude that V(,(t))=V is constant along this solution, which contradicts the assumptions A similar theorem deriving existence of a smooth Lyapunov function is also valid Theorem 6.4 If to is an asymptotically stable equilibrium of system(6. 1)where a: XH r" is a continuously differentiable function defined on an open subset X ofr" the erists a continuously differentiable function V: B(io)+R such that V(io)<v(i)for all≠ to and Vv(i)a(a)<0 VIE B(Co)/Col
3 For the case of a linear system, however, local stability of equilibrium x¯0 = 0 implies existence of a Lyapunov function which is a positive definite quadratic form. Theorem 6.2 If a : Rn �� Rn is defined by a(¯) x = Ax¯ where A is a given n- by-n matrix, then equilibrium x¯0 = 0 of (6.1) is locally stable if and ∗ only if there exists a matrix Q = Q∗ > 0 such that V (x(t)) = x(t) Qx(t) is monotonically non-increasing along the solutions of (6.1). The proof of this theorem, which can be based on considering a Jordan form of A, is usually a part of a standard linear systems class. 6.1.2 Locally asymptotically stable equilibria A point x¯0 is called a (locally) asymptotically stable equilibrium of (6.1) if it is a stable equilibria, and, in addition, there exists e0 > 0 such that every solution of (6.1) with |x(0) − x¯0| < �0 converges to x¯0 as t � →. Theorem 6.3 If V : X �� R is a continuous function such that V (¯x0) < V (¯x) � x¯ ≤ X/{x¯0}, and V (x(t)) is strictly monotonically decreasing for every solution of (6.1) except x(t) ≥ x¯0 then x¯0 is a locally asymptotically stable equilibrium of (6.1). Proof From Theorem 6.1, x¯0 is a locally stable equilibrium. It is sufficient to show that every solution x = x(t) of (6.1) starting sufficiently close to x¯0 will converge to x¯0 as t � →. Assume the contrary. Then x(t) is bounded, and hence will have at x ¯ least one limit point ¯� which is not x¯0. In addition, the limit V of V (x(t)) will exist. Consider a solution x� = x�(t) starting from that point. By continuous dependence on ¯ initial conditions we conclude that V (x�(t)) = V is constant along this solution, which contradicts the assumptions. A similar theorem deriving existence of a smooth Lyapunov function is also valid. Theorem 6.4 If x¯0 is an asymptotically stable equilibrium of system (6.1) where a : X �� Rn is a continuously differentiable function defined on an open subset X of Rn then there exists a continuously differentiable function V : B�(¯x0) �� R such that V (¯x x 0) < V (¯) for all x¯ = ¯ ≡ x0 and ∈V (¯x)a(¯x) < 0 � x¯ ≤ B�(¯x0)/{x¯0}
Proof Define v by V(x(0)= p(lr(t)l)dt where P: [ 0, oo)H0, oo) is positive for positive arguments and continuously differen- tiable. If V is correctly defined and differentiable, differentiation of v((t)) with respect to t at t=0 yields v(x(O)a(x(O)=-p(x(0)2) which proves the theorem. To make the integral convergent and continuously differen- tiable, it is sufficient to make p(y) converging to zero quickly enough as y-0 For the case of a linear system, a classical lyapunov theorem shows that local stability of equilibrium to =0 implies existence of a strict Lyapunov function which is a positive definite quadratic form Theorem 6.5 Ifa: R"H+R is defined by (z)=A where A is a given n-by-n matrin, then equilibrium Io=0 of(6.1)is locally asymptotically stable if and only if there erists a matric Q=Q>0 such that, for v(r)=i'QD VV(2)A=-22 6.1.3 Globally asymptotically stable equilibria Here we consider the case when a: R"bR in defined for all vectors. An equilibrium Co of (6.1) is called globally asymptotically stable if it is locally stable and every solution of(6.1) converges to o as t→∞ Theorem 6.6 If function V: R"HR has a unique minimum at io, is strictly mono- tonically decreasing along every trajectory of (6. 1)except r(t)=io, and has bounded level sets then io is a globally asymptotically stable equilibrium of (6.1) The proof of the theorem follows the lines of the proof of Theorem 6.4. Note that the assumption that the level sets of V are bounded is critically important: without it, some solutions of(6.1)may converge to infinity instead of To
4 Proof Define V by � V (x(0)) = �(|x(t)| 2 )dt, 0 where � : [0,→) �� [0,→) is positive for positive arguments and continuously differentiable. If V is correctly defined and differentiable, differentiation of V (x(t)) with respect to t at t = 0 yields ∈V (x(0))a(x(0)) = −�(|x(0)| 2 ), which proves the theorem. To make the integral convergent and continuously differentiable, it is sufficient to make �(y) converging to zero quickly enough as y � 0. For the case of a linear system, a classical Lyapunov theorem shows that local stability of equilibrium x¯0 = 0 implies existence of a strict Lyapunov function which is a positive definite quadratic form. Theorem 6.5 If a : Rn �� Rn is defined by a(¯x) = Ax¯ where A is a given n- by-n matrix, then equilibrium x¯0 = 0 of (6.1) is locally asymptotically stable if and only if there exists a matrix Q = Q∗ > 0 such that, for V (¯x) = ¯x x, ∗ Q¯ ∈V (¯x)Ax¯ = −|x¯| 2 . 6.1.3 Globally asymptotically stable equilibria Here we consider the case when a : Rn �� Rn in defined for all vectors. An equilibrium x¯0 of (6.1) is called globally asymptotically stable if it is locally stable and every solution of (6.1) converges to x¯0 as t � →. Theorem 6.6 If function V : R ¯ n �� R has a unique minimum at x0, is strictly monotonically decreasing along every trajectory of (6.1)except x(t) ≥ x¯0, and has bounded level sets then x¯0 is a globally asymptotically stable equilibrium of (6.1). The proof of the theorem follows the lines of the proof of Theorem 6.4. Note that the assumption that the level sets of V are bounded is critically important: without it, some solutions of (6.1) may converge to infinity instead of x¯0