Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 8: Local Behavior at eqilibria This lecture presents results which describe local behavior of autonomous systems in terms of Taylor series expansions of system equations in a neigborhood of an equilibrium 8. 1 First order conditions This section describes the relation between eigenvalues of a Jacobian a'(io) and behavior of ODE t)=a(a(t)) or a difference equation r(t+1)=a(x(t) in a neigborhood of equilibrium i In the statements below it is assumed that a X hr is a continuous function defined on an open subset X C R". It is further assumed that Io E X, and there exists an n-by-n matrix A such that la(io +d-a(io)-A8l 6 If derivatives dak/dri of each component ak of a with respect to each component i of r exist at io, A is the matrix with coefficients dak/dri, i.e. the Jacobian of the system However, differentiability at a single point io does not guarantee that( 8.3)holds. On the other hand,(8.3) follows from continuous differentiability of a in a neigborhood of I Version of October 3. 2003
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 8: Local Behavior at Eqilibria1 This lecture presents results which describe local behavior of autonomous systems in terms of Taylor series expansions of system equations in a neigborhood of an equilibrium. 8.1 First order conditions This section describes the relation between eigenvalues of a Jacobian a� (¯x0) and behavior of ODE x˙ (t) = a(x(t)) (8.1) or a difference equation x(t + 1) = a(x(t)) (8.2) in a neigborhood of equilibrium x¯0. In the statements below, it is assumed that a : X ≥� Rn is a continuous function defined on an open subset X � Rn. It is further assumed that x¯0 ⊂ X, and there exists an n-by-n matrix A such that |a(¯x0 + λ) − a(¯x0) − Aλ| � 0 as |λ| � 0. (8.3) |λ| If derivatives dak/dxi of each component ak of a with respect to each cpomponent xi of x exist at x¯0, A is the matrix with coefficients dak/dxi, i.e. the Jacobian of the system. However, differentiability at a single point x¯0 does not guarantee that (8.3) holds. On the other hand, (8.3) follows from continuous differentiability of a in a neigborhood of x¯0. 1Version of October 3, 2003
Example 8.1 Function a: R bR. defined by 一(正-)2「亚1 2 子+(一)2L亚2 for i #0, and by a(0)=0, is differentiable with respect to I1 and I2 at every point IE R, and its Jacobian a'(0)= A equals minus identity matrix. However, condition (8.3)is not satisfied (note that a'(i)is not continuous at I=0) 8.1.1 The continuous time case Let us call an equilibrium to of(8.1)exponentially stable if there exist positive real num- bers e, r, C such that every solution x: 0, T]b X with Iz(0)-Iol e satisfies r(t)-o|≤Ce"r(0)-oyt≥0. The following theorem can be attributed directly to Lyapunov Theorem 8.1 Assume that a( o)=0 and condition(8.) is satisfied. Then (a)if A= a'(To) is a Hurwitz matric(i.e. if all eigenvalues of a have negative real part) then o is a (locally exponentially stable equilibrium of (8.1) (b)if A= a() has an eigenvalue with a non-negative real part then io is not an exponentially stable equilibrium of (8.1); (c) if A=a(to) has an eigenvalue with a positive real part then io is not a stable equilibrium of (8.1) Note that Theorem 8.1 does not cover all possible cases: if A is not a Hurwitz matrix and does not have eigenvalues with positive real part then the statement says very little and for a good reason: the equilibr rulun ima Ly turn out to be asymptotically stable or unstable. Note also that the equilibrium i=0 from Example 8. 1(where a is differentiable but does not satisfy(8.3)is not stable, despite the fact that A=-I has all eigenvalues at -1 Example 8.2 The equilibrium i=0 of the ODE t)=a(t)+B2(t)3 is asympotically stable when a 0(due to Theorem 8.1), but also when a=0 and B>0. In addition, the equilibrium is stable but not asymptotically stable when a=B=0
�� � 2 Example 8.1 Function a : R2 ≥� R2 , defined by ¯ ¯2 ¯2 − (¯2 x2)2 � x1 x1x2 x1 − ¯ ¯ 2 x1 � a = ¯2 ¯2 x2 x¯2 x ¯ 1x2 + (¯2 − ¯2) x 2 x2 1 for x¯ = 0, ≤ and by a(0) = 0, is differentiable with respect to x¯1 and x¯2 at every point x¯ ⊂ R2 , and its Jacobian a� (0) = A equals minus identity matrix. However, condition (8.3) is not satisfied (note that a� (¯x) is not continuous at x¯ = 0). 8.1.1 The continuous time case Let us call an equilibrium x¯0 of (8.1) exponentially stable if there exist positive real numbers σ, r, C such that every solution x : [0, T] ≥� X with |x(0) − x¯0| 0 (due to Theorem 8.1), but also when � = 0 and � > 0. In addition, the equilibrium is stable but not asymptotically stable when � = � = 0
8.1.2 Proof of theorem 8.1 The proof of (a)can be viewed as an excercise in"storage function construction"outlined n the previous lecture. Indeed, assuming, for simplicity, that To =0,(8.1)can be re- written as i(t)=A.x(t)+w(t), w(t)=a(a(t))-Ax(t) Here the linear part has standard storage functions VITI(I)=IPC, P=P with supply rates Lr(z,)=27P(A7+m) In addition, due to(8.3), for every 8>0 there exists e>0 such that the nonlinear component w(t) satisfies the sector constraint dNL(x(t),(t)=6(t)|2-|(t)|2≥0, as long as z(t) E. Since A is a Hurwitx matrix, P=P can be chosen positive definite and such that PA+AP=-I (z,)=0Lr(,)+ToNL(,) 1)22+2xP-l2≤(6-1)2-2P·|l|-r2 where Pl is the largest singular value of P, is a supply rate for the storage function V= ViTI for every constant T≥0. When T=16‖P‖and=0.25/r, we have (Z,)≤-0.522 which proves that, for a(t)<e, the inequality 1 V(x(1)≤-0.5(t)|2≤ 2|P v((t)) (x(0)t≥0, where d=1/2 P-l, as long as r(t) <e. Since P|·|x(t)≥v(x()≥‖P-l-1x(t)l, this implies(a) The proofs of(b)and(c)are more involved, based on showing that solutions which start at To+du, where v is an eigenvector of A corresponding to an eigenvalue with a non- negative(strictly positive) real part, cannot converge to Co quickly enough(respectively, diverge from io
3 8.1.2 Proof of Theorem 8.1 The proof of (a) can be viewed as an excercise in “storage function construction” outlined in the previous lecture. Indeed, assuming, for simplicity, that x¯0 = 0, (8.1) can be rewritten as x˙ (t) = Ax(t) + w(t), w(t) = a(x(t)) − Ax(t). Here the linear part has standard storage functions VLT I (¯ x� x) = ¯ Px, ¯ P = P� with supply rates αLT I (¯ w) = 2¯� x, ¯ x x P(A¯ ¯ + w). In addition, due to (8.3), for every λ > 0 there exists σ > 0 such that the nonlinear component w(t) satisfies the sector constraint 2 αNL(x(t), w(t)) = λ|x(t)| 2 − |w(t)| → 0, as long as |x(t)| < σ. Since A is a Hurwitx matrix, P = P � can be chosen positive definite and such that P A + A� P = −I. Then α(¯x, w¯) = αLT I (¯x, w¯) + δαNL(¯x, w¯) x| 2 + 2δ ¯ w − δ | ¯ ¯ ¯ w| − δ |w| 2 = (δ λ − 1)|¯ x , � P ¯ w| 2 ∀ (δ λ − 1)|x| 2 − 2∈P∈ · |x| · | ¯ ¯ where ∈P∈ is the largest singular value of P, is a supply rate for the storage function V = VLT I for every constant δ → 0. When δ = 16∈P∈ and λ = 0.25/δ , we have α(¯x, w¯) ∀ −0.5|x¯| 2 , which proves that, for |x(t)| < σ, the inequality 1 V (x(t)) ∀ −0.5|x(t)| 2 ∀ − V (x(t)). 2∈P −1∈ Hence V (x(t)) ∀ e−dtV (x(0)) � t → 0, where d = 1/2∈P −1∈, as long as |x(t)| < σ. Since −1 ∈P∈ · |x(t)| → V (x(t)) → ∈P −1 ∈ · |x(t)|, this implies (a). The proofs of (b) and (c) are more involved, based on showing that solutions which start at x¯0 +λv, where v is an eigenvector of A corresponding to an eigenvalue with a nonnegative (strictly positive) real part, cannot converge to x¯0 quickly enough (respectively, diverge from x¯0)
To prove(b), take a real number d E(0, r/2)such that no two eigenvalues of A sum up to-2d. Then P= P be the unique solution of the Lyapunov equation P(A+dD+(A'+dIP Note that P is non-singular: otherwise, if Pu=0 for some v#0, it follows that 12=u(P(A+d)+(4+dD)P)=(Pv)(+dD)+v(4+dn)(P)=0. In addition, P=P is not positive semidefinite: since, by assumption, A + dI has an eigenvector ufo which corresponds to an eigenvalue A with a positive real part, we have 2Re(a)up hence uPu0 be small enough so that zPu≤0.5 By assumption, there exists 8>0 such that a(z)-A≤ffor同≤6. Then dcea c(t)'pr(t) e2a(2dr(yPr(t)+2r(t)PAx()+2r(t)'P(a(x(t)-Ar(t) ≤-0.5e24|r()2 as long as a (t)is a solution of(8. 1)and lz(t)l 2d The proof of(c)is similar to that of (a) 8.1.3 The discrete time case The results for the discrete time case are similar to Theorem 8.1, with the real parts of the eigenvalues being replaced by the difference between their absolute values and 1 Let us call an equilibrium io of(8.2) exponentially stable if there exist positive real numbers 6, r, C such that every solution z: 0, T]H X with r(0)-Tol e satisfies (t)-iol s Ce(0)-iol Vt=0, 1, 2, Theorem 8.2 Assume that a(io)=0 and condition(8.3) is satisfied. Then
� 4 To prove (b), take a real number d ⊂ (0, r/2) such that no two eigenvalues of A sum up to −2d. Then P = P� be the unique solution of the Lyapunov equation P(A + dI) + (A� + dI)P = −I. Note that P is non-singular: otherwise, if P v = 0 for some v =≤ 0, it follows that � � (A� −|v| 2 = v (P(A + dI) + (A� + dI)P)v = (P v) � (A + dI)v + v + dI)(P v) = 0. In addition, P = P� is not positive semidefinite: since, by assumption, A + dI has an eigenvector u =≤ 0 which corresponds to an eigenvalue � with a positive real part, we have −|u| 2 = −2Re(�)u P u, hence u� P u 0 be small enough so that 2¯ ¯ 2 x ¯ � P w ∀ 0.5|x| for |w| ∀ σ|x|. By assumption, there exists λ > 0 such that |a(¯x) − Ax¯ ¯ ¯ | ∀ σ|x| for |x| ∀ λ. Then d (e2dt 2dt x(t) � P x(t)) = e (2dx(t) � P x(t) + 2x(t) � P Ax(t) + 2x(t) � P(a(x(t)) − Ax(t))) dt 2 ∀ −0.5e2dt|x(t)| as long as x(t) is a solution of (8.1) and |x(t)| ∀ λ. In particular, this means that if x(0)� P x(0) ∀ −R 2d. The proof of (c) is similar to that of (a). 8.1.3 The discrete time case The results for the discrete time case are similar to Theorem 8.1, with the real parts of the eigenvalues being replaced by the difference between their absolute values and 1. Let us call an equilibrium x¯0 of (8.2) exponentially stable if there exist positive real numbers σ, r, C such that every solution x : [0, T] ≥� X with |x(0) − x¯0| < σ satisfies x0| ∀ Ce−rt |x(t) − ¯ |x(0) − x¯0| � t = 0, 1, 2, . . . . Theorem 8.2 Assume that a(¯x0) = 0 and condition (8.3) is satisfied. Then
(a)if A=a'(io) is a Schur matrit(i.e. if all eigenvalues of a have absolute value less than one) then To is a(locally exponentially stable equilibrium of (8.2) (b)if A=a(lo) has an eigenvalue with absolute value greater than 1 then io is not an exponentially stable equilibrium of (8.2 (c) if A=aio) has an eigenvalue with absolute value strictly larger than 1 then To is not a stable equilibrium of(8.2) 8.2 Higher order conditions When the Jacobian A=a'(Io)of( 8.1)evaluated at the equilibrium To has no eigenvalues with positive real part, but has some eigenvalues on the imaginary axis, local stability analysis becomes much more complicated. Based on the proof of Theorem 8.1, it is natural to expect that system states corresponding to strictly stable eigenvalues will behave in a predictably stable fashion, and hence the behavior of system states corresponding to the eigenvalues on the imaginary axis will determine local stability or instability of the equilibrium. 8.2.1 A Center Manifold Theorem n this subsection we assume for simplicity that io=0 is the studied equilibrium of (8.1) e. a(0)=0. Assume also that a is k times continuously differentiable in a neigborhood of To=0, where k 1, and that A= a(O)has no eigenvalues with positive real part, but has eigenvalues on the imaginary axis, as well as in the open left half plane Re(s)2 times continuously differentiable in a neigbor- hood of Io=0. Assume that a(0)=0 and a(0)=A 0 A where As is a Hurwitz p-by-p matric, and all eigenvalues of the q-by-q matric Ac have zero real part. Then (a) there erists e>0 and a function h: RHR, k-l times continuously differ entiable in a neigborhood of the origin, such that h(0)=0, h(0)=0, and every solution r(t)=[ct); Is(t) of(8.1)with Ts(0)=h(rc(o)) and with arc(o)l<e satisfies s(t)=h(ao(t))for as long as c()<E;
� � � � � 5 (a) if A = a� (¯x0) is a Schur matrix (i.e. if all eigenvalues of A have absolute value less than one) then x¯0 is a (locally) exponentially stable equilibrium of (8.2); (b) if A = a� (¯x0) has an eigenvalue with absolute value greater than 1 then x¯0 is not an exponentially stable equilibrium of (8.2); (c) if A = a� (¯x0) has an eigenvalue with absolute value strictly larger than 1 then x¯0 is not a stable equilibrium of (8.2). 8.2 Higher order conditions When the Jacobian A = a� (¯x0) of (8.1) evaluated at the equilibrium x¯0 has no eigenvalues with positive real part, but has some eigenvalues on the imaginary axis, local stability analysis becomes much more complicated. Based on the proof of Theorem 8.1, it is natural to expect that system states corresponding to strictly stable eigenvalues will behave in a predictably stable fashion, and hence the behavior of system states corresponding to the eigenvalues on the imaginary axis will determine local stability or instability of the equilibrium. 8.2.1 A Center Manifold Theorem In this subsection we assume for simplicity that x¯0 = 0 is the studied equilibrium of (8.1), i.e. a(0) = 0. Assume also that a is k times continuously differentiable in a neigborhood of x¯0 = 0, where k → 1, and that A = a� (0) has no eigenvalues with positive real part, but has eigenvalues on the imaginary axis, as well as in the open left half plane Re(s) 0 and a function h : Rq ≥� Rp , k − 1 times continuously differentiable in a neigborhood of the origin, such that h(0) = 0, h� (0) = 0, and every solution x(t) = [xc(t); xs(t)] of (8.1) with xs(0) = h(xc(0)) and with |xc(0)| < σ satisfies xs(t) = h(x0(t)) for as long as |xc(t)| < σ;
(b) for every function h from (a), the equilibrium io =0 of(8.1) is locally stable (asymptotically stable) /unstable if and only if the equilibrium Ic=0 of the ODE dot e(t)=a(ze(t); h(ac(t)) (8.4) is locally stable(asymptotically stable)/unstable/; (c) if the equilibrium Ic=0 of (8.4) is stable then there exist constants r>0,y>0 such that for every solution x= a(t)of (8.1)with z(o) r there exists a solution Tc=rc(t)of(8.4) such that lim eta(t)-lrc(t); h(a(t))I=0 The set of points M={2=[c;h():||0 is small enough, is called the central manifold of (8.1). Theorem 8.3, called frequently the center manifold theorem, allows one to reduce the dimension of the system to be analyzed from n to g, as long as the function h defining the central manifold can be alculated exactly or to a sufficient degree of accuracy to judge local stability of 8.4 Example 8.3 This example is taken from Sastry, p. 312. Consider system i1(t)=-x1(t)+kr2(t)2, i2(t)=x1(t)r2(t), where k is a real parameter. In this case n= 2, p=q=1, Ac=0, As=-1, and k can be arbitrarily large. According to Theorem 8.3, there exists a k times differentiable function h: R HR such that 1= h(2)is an invariant manifold of the OdE (at least in a neigborhood of the origin). Hence ky=h(y)+h(gh(y)y for all sufficiently small y. For the 4th order Taylor series expansion h(y)=h2y2+h3y3+h4y4+o(y4),h(y)=2h2y+ h4y2+o(y3) comparing the coefficients on both sides of the ODE for h yields h2= k, h3=0, h4 2k2. Hence the center manifols ODE has the form (t)=kxe(t)3+o(x(t)3) which means stability for k 0
6 (b) for every function h from (a), the equilibrium x¯0 = 0 of (8.1) is locally stable (asymptotically stable) [unstable] if and only if the equilibrium x¯c = 0 of the ODE dotxc(t) = a([xc(t); h(xc(t))]) (8.4) is locally stable (asymptotically stable) [unstable]; (c) if the equilibrium x¯c = 0 of (8.4) is stable then there exist constants r > 0, β > 0 such that for every solution x = x(t) of (8.1) with |x(0)| 0 is small enough, is called the central manifold of (8.1). Theorem 8.3, called frequently the center manifold theorem, allows one to reduce the dimension of the system to be analyzed from n to q, as long as the function h defining the central manifold can be calculated exactly or to a sufficient degree of accuracy to judge local stability of (8.4). Example 8.3 This example is taken from Sastry, p. 312. Consider system x˙ 1(t) = −x1(t) + kx2(t) 2 , x˙ 2(t) = x1(t)x2(t), where k is a real parameter. In this case n = 2, p = q = 1, Ac = 0, As = −1, and k can be arbitrarily large. According to Theorem 8.3, there exists a k times differentiable function h : R ≥� R such that x1 = h(x2) is an invariant manifold of the ODE (at least, in a neigborhood of the origin). Hence ky2 = h(y) + h˙(y)h(y)y for all sufficiently small y. For the 4th order Taylor series expansion 3 4 ˙ 4 h(y) = h2y2 + h3y + h4y4 + o(y ), h(y) = 2h2y + 3h3y2 + 4h4y + o(y3 ), comparing the coefficients on both sides of the ODE for h yields h2 = k, h3 = 0, h4 = −2k2. Hence the center manifols ODE has the form x˙ c(t) = kxc(t) 3 + o(xc(t) 3 ), which means stability for k 0