Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 10: Singular Perturbations and Averaging This lecture presents results which describe local behavior of parameter-dependent OdE models in cases when dependence on a parameter is not continuous in the usual sense 10.1 Singularly perturbed ODE In this section we consider parameter-dependent systems of equations i(t)=f(ar(t),y(t),t), g(x(t),y(t),t), 10.1 where e E[0, Eo] is a small positive parameter. When e>0,(10.1)is an ODE model For e=0,(10.1)is a combination of algebraic and differential equations. Models such as(10. 1), where y represents a set of less relevant, fast changing parameters, are fre- "classicall"approach to dealing with uncertainty, complexity, and nonlineari ons is the quently studied in physics and mechanics. One can say that singular perturbatie 10.1.1 The Tikhonov's heorem A typical question asked about the singularly perturbed system(10. 1)is whether its solutions with e >0 converge to the solutions of(10.1)with e =0 as E-0. A suffi cient condition for such convergence is that the Jacobian of g with respect to its second argument should be a hurwitz matrix in the region of interest Theorem 10.1 Let o: to, t1]H+ R", yo: [to, til brm be continuous functions tisfying equations io(t)=f(aro(t), yo(t), t),0=g(ao(t), yo(t), t) Version of October 15. 2003
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 10: Singular Perturbations and Averaging1 This lecture presents results which describe local behavior of parameter-dependent ODE models in cases when dependence on a parameter is not continuous in the usual sense. 10.1 Singularly perturbed ODE In this section we consider parameter-dependent systems of equations x˙ (t) = f(x(t), y(t), t), (10.1) �y˙ = g(x(t), y(t), t), where � → [0, �0] is a small positive parameter. When � > 0, (10.1) is an ODE model. For � = 0, (10.1) is a combination of algebraic and differential equations. Models such as (10.1), where y represents a set of less relevant, fast changing parameters, are frequently studied in physics and mechanics. One can say that singular perturbations is the “classical” approach to dealing with uncertainty, complexity, and nonlinearity. 10.1.1 The Tikhonov’s Theorem A typical question asked about the singularly perturbed system (10.1) is whether its solutions with � > 0 converge to the solutions of (10.1) with � = 0 as � � 0. A suffi cient condition for such convergence is that the Jacobian of g with respect to its second argument should be a Hurwitz matrix in the region of interest. Theorem 10.1 Let x0 : [t0, t1] ∞� Rn, y0 : [t0, t1] ∞� Rm be continuous functions satisfying equations x˙ 0(t) = f(x0(t), y0(t), t), 0 = g(x0(t), y0(t), t), 1Version of October 15, 2003
where f:R"×Rn×R→ R" and g:R"×R"×R→ R are continuous functions. Assume that f, g are continuously differentiable with respect to their first two arguments in a neigborhood of the trajectory co(t), yo(t), and that the derivative ()=92(xo(t),o(t) is a Hurwitz matrin for all t e to, t1. Then for every t2 E(to, ti) there erists d>0 and C>0 such that inequalities ao(t)-c(t)l s Ce for all e [ to, t, and lyo(t)-y(t)lsce for all tE [t2, ti] for all solutions of (10. 1)with lr(to)-ro(to)l0,a= a(t) can be considered a constant when predicting the behavior of y. From this viewpoint, for a given t E(to, t1), one can expect that y(t +e where y1: 0, oo) is the solution of the "fast motion"ODE i1(r)=9(x0(),y(7),1(0)=y(D) Since yo(t) is an equilibrium of the ODE, and the standard linearization around this equilibrium yields 6(7)≈A(E)6() where 8(1)=y1(r)-yo(t), one can expect that y1(T)- yo() exponentially as T-+oo whenever A(t)is a Hurwitz matrix and ly(b-yo (b)l is small enough. Hence, when E>0 is small enough, one can expect that y(t) a yo(t) 10.1.2 Proof of Theorem 10.1 First, let us show that the interval [to, til can be subdivided into subintervals Ak 7k-1,n], where k∈{1,2 and to= To 0 for which Pk A(t)+A(t)Pk0 such that P(t)A(t)+A(tP(t) Since A depends continuously on t, there exists an open interval A(t) such that tE A( P(t)A(7)+A()P(t)<-Ir∈△(t)
2 where f : Rn × Rm × R ∞� Rn and g : Rn × Rm × R ∞� Rm are continuous functions. Assume that f, g are continuously differentiable with respect to their first two arguments in a neigborhood of the trajectory x0(t), y0(t), and that the derivative A(t) = g2 � (x0(t), y0(t), t) is a Hurwitz matrix for all t → [t0, t1]. Then for every t2 → (t0, t1) there exists d > 0 and C > 0 such that inequalities |x0(t) − x(t)| ≈ C� for all t → [t0, t1] and |y0(t) − y(t)| ≈ C� for all t → [t2, t1] for all solutions of (10.1) with |x(t0) − x0(t0)| ≈ �, |y(t0) − y0(t0)| ≈ d, and � → (0, d). The theorem was originally proven by A. Tikhonov in 1930-s. It expresses a simple principle, which suggests that, for small � > 0, x = x(t) can be considered a constant when predicting the behavior of y. From this viewpoint, for a given t ¯ → (t0, t1), one can expect that y(t ¯+ �� ) � y1(� ), where y1 : [0,∀) is the solution of the “fast motion” ODE y˙1(� ) = g(x0(t ¯), y1(� )), y1(0) = y(t ¯). Since y0(t ¯) is an equilibrium of the ODE, and the standard linearization around this equilibrium yields � ˙(� ) � A(t ¯)�(� ) where �(� ) = y1(� ) − y0(t ¯), one can expect that y1(� ) � y0(t ¯) exponentially as � � ∀ whenever A(t ¯) is a Hurwitz matrix and |y(t ¯) − y0(t ¯)| is small enough. Hence, when � > 0 is small enough, one can expect that y(t) � y0(t). 10.1.2 Proof of Theorem 10.1 First, let us show that the interval [t0, t1] can be subdivided into subintervals �k = [�k−1, �k], where k → {1, 2, . . . , N} and t0 = �0 0 for which PkA(t) + A(t) � Pk 0 such that P(t)A(t) + A(t) � P(t) < −I. Since A depends continuously on t, there exists an open interval �(t) such that t → �(t) and P(t)A(� ) + A(� ) � P(t) < −I � � → �(t)
Now the open intervals A(t)with t E [to, t, cover the whole closed bounded interval to, ti, and taking a finite number of tk, k=1, . m such that [to, ti is completely covered by A(tk) yields the desired partition subdivision of to, til exist cod, o t h th due to the coutinuous differentiablity of g, for every u> o therd Jf(xo0(t)+6-,3o(t)+y,1)-f(o(t),3(1,t)≤C(|62|+|) and lg(xo(1)+6x,y0(t)+6y,t)-A(t)by≤C|bx+列 for all t∈R,bz∈Rn,by∈ R satisfying t∈{to,t1],|1x-x0()≤r,1,-3(t)≤r Fort∈△klet lk=(2P5)/2 Then. for 6(t)=x(t)-x0(t),y(t)=y(t)-o(t), we have x|≤C1(1x|+|6y|k) e|6k≤-ql6yk+C1|6+C1 (10.2) as long as 8, du are sufficiently small, where C1, g are positive constants which do not depend on k. Combining these two derivative bounds yield 18=|+(EC1/q)15, D)<C218-+EC2 for some constant C2 independent of k. Hence 162(7k-1+r)|≤er(62(7k-1)+(eC1/q)(7k-1))+C3 for T E0, Tk-Tk-1. With the aid of this bound for the growth of 8=I, inequality(10. 2) yields a bound for 8,li 6(7k-1+7)≤exp(-qr/e)|b(xk-1)+C4(|62(k-1)+(eC1/q)b(k-1))+C46, which in turn yields the result of Theorem 10.1
3 Now the open intervals �(t) with t → [t0, t1] cover the whole closed bounded interval [t0, t1], and taking a finite number of t ¯k, k = 1, . . . , m such that [t0, t1] is completely covered by �(t ¯k) yields the desired partition subdivision of [t0, t1]. Second, note that, due to the continuous differentiability of f, g, for every µ > 0 there exist C, r > 0 such that ¯ ¯ ¯ ¯ |f(x0(t) + �x, y0(t) + �y, t) − f(x0(t), y0(t), t)| ≈ C(|�x| + |�y|) and ¯ ¯ ¯ ¯ ¯ |g(x0(t) + �x, y0(t) + �y, t) − A(t)�y| ≈ C|�x| + µ|�y| ¯ for all t → R, � ¯ x → Rn, �y → Rm satisfying ¯ ¯ t → [t0, t1], |�x − x0(t)| ≈ r, |�y − y0(t)| ≈ r. For t → �k let |�y|k = (�y � Pk�y) 1/2 . Then, for �x(t) = x(t) − x0(t), �y(t) = y(t) − y0(t), we have |� ˙ x| ≈ C1(|�x| + |�y|k), �|� ˙ y|k ≈ −q|�y|k + C1|�x| + �C1 (10.2) as long as �x, �y are sufficiently small, where C1, q are positive constants which do not depend on k. Combining these two derivative bounds yields d (|�x| + (�C1/q)|�y|) ≈ C2|�x| + �C2 dt for some constant C2 independent of k. Hence |�x(�k−1 + � )| ≈ eC3� (|�x(�k−1)| + (�C1/q)|�y(�k−1)|) + C3� for � → [0, �k − �k−1]. With the aid of this bound for the growth of |�x|, inequality (10.2) yields a bound for |�y|k: |�y(�k−1 + � )| ≈ exp(−q�/�)|�y(�k−1)| + C4(|�x(�k−1)| + (�C1/q)|�y(�k−1)|) + C4�, which in turn yields the result of Theorem 10.1
10.2 Averaging Another case of "potentially discontinuous"dependence on parameters is covered by the following"averaging" result Theorem 10.2 Let f: R"XRxRHR be a continuous function which is T-periodic with respect to its second argument t, and continuously differentiable with respect to its first argument. Let toE r be such that f(Co, t, e)=0 for all t, E. ForiEr define ∫(z,∈) ∫(正,t,e) If df /d xl2=0 e-o is a Hurwitz matric, then, for sufficiently small e>0, the equilibrium x≡0 of the system i(t)=∈f(x,t,) is exponentially stable Though the parameter dependence in Theorem 10.2 is continuous, the question asked about the behavior at t= oo, which makes system behavior for e =0 not a valid indicator of what will occur for e>0 being sufficiently small.(Indeed, for e=0 the quilibrium io is not asymptotically stable. To prove Theorem 10.2, consider the function S: R"XRH R which maps z(0), to r(o)=s((0), e), where a( ) is a solution of (10.3). It is sufficient to show that the derivative (Jacobian)S(I, e) of S with respect to its first argument, evaluated at i=To and e>0 sufficiently small, is a Schur matrix. Note first that, according to the rules on differentiating with respect to initial conditions, S(0, e)=A(T, e), where d△(t,e)df (0,t,∈)△(t,e),△(0,∈)=I Consider D(t, e) defined by d△(t,e)df (0,t,0)(t,e),△(0,)=1 Let S(t)be the derivative of A(t, e) with respect to e at e=0. According to the rule for differentiating solutions of ODE with respect to parameters 6(t) ,1,O)d1 Hence d(r=df/ d
4 10.2 Averaging Another case of “potentially discontinuous” dependence on parameters is covered by the following “averaging” result. Theorem 10.2 Let f : Rn × R × R ∞� Rn be a continuous function which is � -periodic with respect to its second argument t, and continuously differentiable with respect to its first argument. Let x¯0 → Rn be such that f(¯x0,t,�) = 0 for all t,�. For x¯ → Rn define � � ¯f(¯x, �) = f(¯x,t,�). 0 ¯ If df/dx|x=0,�=0 is a Hurwitz matrix, then, for sufficiently small � > 0, the equilibrium x ≤ 0 of the system x˙ (t) = �f(x,t,�) (10.3) is exponentially stable. Though the parameter dependence in Theorem 10.2 is continuous, the question asked is about the behavior at t = ∀, which makes system behavior for � = 0 not a valid indicator of what will occur for � > 0 being sufficiently small. (Indeed, for � = 0 the equilibrium x¯0 is not asymptotically stable.) To prove Theorem 10.2, consider the function S : Rn × R ∞� Rn which maps x(0),� to x(� ) = S(x(0),�), where x(·) is a solution of (10.3). It is sufficient to show that the derivative (Jacobian) S x, �) of S˙ with respect to its first argument, evaluated at ¯ x0 ˙(¯ x = ¯ and � > 0 sufficiently small, is a Schur matrix. Note first that, according to the rules on differentiating with respect to initial conditions, S˙(¯x0,�) = �(�, �), where d�(t,�) df = � (0, t,�)�(t,�), �(0, �) = I. dt dx Consider D¯(t,�) defined by d�( ¯ t,�) df = ¯ ¯ � (0, t, 0)�(t, �), �(0,�) = I. dt dx Let ¯ �(t) be the derivative of �(t, �) with respect to � at � = 0. According to the rule for differentiating solutions of ODE with respect to parameters, � t df �(t) = (0,t1, 0)dt1. 0 dx Hence ¯ �(� ) = df/dx|x=0,�=0
is by assumption a hurwitz matrix. On the other hand △(r,)-△(7,e)=o(e Combining this with △(r,e)=I+6(7)e+o(∈) yields △(r,e)=I+6(7)e+o() Since S()is a Hurwitz matrix, this implies that all eigenvalues of A(T, e) have absolute value strictly less than one for all sufficiently small E>0
5 is by assumption a Hurwitz matrix. On the other hand, �( ¯ �, �) − �(�, �) = o(�). Combining this with �( ¯ �, �) = I + �(� )� + o(�) yields �(�, �) = I + �(� )� + o(�). Since �(� ) is a Hurwitz matrix, this implies that all eigenvalues of �(�, �) have absolute value strictly less than one for all sufficiently small � > 0