CRITICAL CASES IN SLOW/FAST CONTROL PROBLEMS

Abstract

The aim of the paper is to describe the main critical cases in the theory of singularly perturbed optimal control problems and to give examples which are typical for slow/fast systems. The theory has traditionally dealt only with perturbation problems near normally hyperbolic manifold of singularities and this manifold is supposed to isolated. We reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory. We illustrate our approach by several examples of control problems.

Keywords

• integral manifolds
• singular perturbations
• optimal control

Introduction

Consider singularly perturbed di®erential systems of the type dx dy

= f (x; y; t; "); " = g(x; y; t; ") ( 1 ) dt dt with vector variables x and y, and a small positive parameter ". Such systems play an important role as mathematical models of numerous nonlinear phenomena in di®erent ¯elds (see e.g. [1{7]).

A usual approach in the qualitative study of ( 1 ) is to consider ¯rst the so called degenerate system dx

= f (x; y; t; 0); 0 = g(x; y; t; 0) ( 2 ) dt and then to draw conclusions for the qualitative behavior of the full system ( 1 ) for su±ciently small ". In order to recall a basic result of the geometric theory of singularly perturbed systems we introduce the following notation and assumptions for su±ciently small positive "0, 0 · " · "0. (A1). Functions f and g are su±ciently smooth and uniformly bounded together with all their derivatives. (A2). There are some region G 2 Rm and a function h(x; t; ") of the same smoothness as g such that g(x; h(x; t); t; 0) ´ 0

8(x; t) 2 G £ R: (A3). The spectrum of the Jacobian matrix B(x; t) = gy(x; h(x; t); t; 0) is uniformly separated from the imaginary axis for all (x; t) 2 G £ R, i.e. the eigenvalues ¸i(x; t)(i = 1; : : : ; n) of the matrix B(x; t) satisfy the inequality jRe¸i(x; t)j ¸ ° for some positive number °.

Then the following result is valid (see e.g. [8, 9]): Proposition 1. Under the assumptions (A1) ¡ (A3) there is a su±ciently small positive "1, "1 · "0, such that for " 2 I1 system ( 1 ) has a smooth integral manifold M" ( slow integral manifold) with the representation M" := ©(x; y; t) 2 Rn+m+1 : y = Ã(x; t; "); (x; t) 2 G £ Rª and with the asymptotic expansion Ã(x; t; ") = h(x; t) + "Ã1(x; t) + : : : : The motion on this manifold is described by the slow di®erential equation ( 3 ) x_ = f (x; Ã(x; t; "); t; "): Remark 1. The global boundedness assumption in (A1) with respect to (x; y) can be relaxed by modifying f and g outside some bounded region of Rn £ Rm. Remark 2. In applications it is usually assumed that the spectrum of the Jacobian matrix gy(x; h(x; t); t; 0) is located in the left half plane. Under this additional hypothesis the manifold M" is exponentially attracting for " 2 I1. The case that assumption (A3) is violated is called critical. We distinguish three subcases: 1. The Jacobian matrix gy(x; y; t; 0) is singular on some subspace of Rm £ Rn £ R. In that case, system ( 1 ) is referred to as a singular singularly perturbed system [10]. This subcase has been treated in [3, 4, 7, 10, 11]. 2. The Jacobian matrix gy(x; y; t; 0) has eigenvalues on the imaginary axis with nonvanishing imaginary parts. A similar case has been investigated in [3, 4, 12]. 3. The Jacobian matrix gy(x; y; t; 0) is singular on the set M0 := f(x; y; t) 2 Rm £ Rn £ R : y = h(x; t); (x; t) 2 G £ Rg. In that case, y = h(x; t) is generically an isolated root of g = 0 but not a simple one. Other critical cases were considered, for example, in [3, 4, 13{21]. The critical case (i) is considered in Section 2 as applied to the high-gain control problem, the case (ii) is considered in Section 3 as applied to the manipulator control, the case (iii) is considered in Section 4 as applied to the partially cheap control problem. Two critical cases, (i) and (ii) are combined in the the optimal control problem which is analyzed in Section 5, and therefore it is possible to say that this Section is devoted to the consideration of the twice critical case. It is not inconceivable that combinations of other pairs of critical cases and even triple critical case are of interest as well and possibly they will be considered later.

Singular singularly perturbed systems

For a better idea of the problems we wish to examine, and to gain some insight into why the term in the title is used, we initially consider the following di®erential system At ¯rst glance it would seem that there are two fast variables z1 and z2 and we apply the proposed approach to the analysis of this system. Setting " equal to zero we obtain the linear algebraic system Apart from the trivial solution this system possesses an one-parameter family of solutions z1 = s; z2 = 2s, where s is a real parameter. Thus there is no isolated solution to the degenerate system. The reason is that the matrix is singular, i.e. det A(0) = 0 and the singularly perturbed system in this case is called a singular singularly perturbed system. In fact, in this particular system it is possible to extract a slow variable and obtain a system with one slow and one fast variable. Taking into account that the rows of matrix A are proportional for " = 0 (proportionality constant is equal 3), we introduce a new variable x = z2 ¡ 3z1; and obtain the following di®erential equation for the slow variable x_ = ¡x + z2: To obtain the full solution, it is possible to use either of the two equations for z1 or z2 as a fast equation. If we choose the equation for z2, then, after taking into account x ¡ z2 = ¡3z1; we obtain the singularly perturbed equation "z_2 = ¡(2 + ")x ¡ (1 ¡ ")z2: As a result we obtain the system x_ = ¡x + z2;

"z_2 = ¡(2 + ")x ¡ (1 ¡ ")z2; which has the form ( 1 ). It is easy to check by direct substitution into the invariance equation that this last system possesses the one-dimensional attractive slow invariant manifold z2 = kx. On substituting for z2 the above equations imply "k(¡1 + k)x = ¡(2 + ")x ¡ (1 ¡ ")kx; and this implies "k2 + (1 ¡ 2")k + 2 + " = 0: Setting in the last equation k = k0 + "k1 + O("2) and equating the powers of ", we obtain k0 = ¡2, k1 = ¡9. Thus, the invariant manifold has the form z2 = ¡(2 + 9" + O("2))x = ¡(2 + 9" + O("2))(z2 ¡ 3z1); or, in equivalent form z2 = (2 + 3" + O("2))z1: ( 4 )

High-gain control

Consider the control system x_ = ³(x) + B(x)u; x(0) = x0; where x 2 Rn, u 2 Rr and t ¸ 0. The vector function ³ and the matrix function B are taken to be su±ciently smooth and bounded. The control vector u is to be selected in such a way as to transfer the vector x from x = x0 to a su±ciently small neighborhood of a smooth m{dimensional surface S(x) = 0: A commonly employed feedback control is where K is a constant r £ m{matrix and " is a small positive parameter, see [6] and references therein.

Suppose that we can choose the matrix K in such a way that the matrix ¡N (x; t) = ¡GBK is stable, i.e., all eigenvalues have strictly negative real parts, and its inverse matrix is bounded, and introduce the additional variable y = S(x), then x and y satisfy the system "x_ = "³(x) ¡ B(x)Ky;

x(0) = x0; "y_ = "G(x)³(x) ¡ G(x)B(x)Ky;

y(0) = y0 = S(x0); where G(x) = @S=@x. The reduced (" = 0) algebraic problem possesses an n{parameter family of solutions x = v; y = 0. The role of A is played by the singular matrix The latter singular singularly perturbed di®erential system possesses an n{ dimensional slow integral manifold ( 5 ) x = v;

y = "N ¡1(v; t)G(v)³(v) + O("2): The °ow on the manifold is governed by v_ = [I ¡ B(v)KN ¡1(v)G(v)]³(v) + O("): Introduce the new variable y = z + "N ¡1(x)G(x)³(x): Then for z we obtain the equations "z_ = ¡N (x)z + O("): y = N ¡1G³ + O(") It is now clear that the representations are valid for for all t > 0. Thus, under the control law ( 5 ) the trajectory very quickly attains the "{neighborhood of S(x) = 0.

( 6 ) ¡N (x) = ¡GBK. Under this control for the variable x Let us introduce the modi¯ed control

1 u = ¡ " K £S(x) ¡ "N ¡1(x)G(x)³(x)¤ ; with the stable matrix we obtain the equation "x_ = " £I ¡ B(x)K(GBK)¡1G(x)¤ ³(x) ¡ B(x)KS(x); and for the variable y = S(x) we obtain the equation "y_ = ¡N (x)y; y = O ³e¡º"¡1t´ ; º > 0; t > 0; " ! 0 for some positive º. Note that the modi¯ed control law turns y ´ 0 invariant and exponentially attractive. This means that the modi¯ed control law ( 6 ) is more preferable than the usually used law ( 5 ). Surprisingly, there is no need for using of asymptotic expansion to design the control law since the modi¯ed control law gives the exact result.

Weakly attractive integral manifolds

In this section we consider the system ( 1 ) when the matrix B = gy(x; Á(x; t); t; 0) has eigenvalues on the imaginary axis with nonvanishing imaginary parts. If the eigenvalues at " = 0 are pure imaginary but after taking into account the perturbations of higher order they move to the complex left half-plane, then the system under consideration has stable slow integral manifolds. It seems reasonable to say that this kind of problem for gyroscopic systems was previously investigated by the integral manifolds method, see, for example, [12]. Note that in the eigenvalues of the matrix of the linearized fast subsystem in such problems have the form ¡® § i¯=" with positive ® and the existence problem for slow integral manifolds has not any connections with the bifurcation problems when the real parts of the eigenvalues change their sign. Some problems of the mechanics of manipulators with high-frequency and weakly damped transient regimes are now discussed in this context. More results along this line can be found in [3, 4, 12].

Control of a one rigid-link °exible-joint manipulator Consider a simple model of a rigid-link °exible joint manipulator [22, 23], where Jm is the motor inertia, J1 is the link inertia, M is the link mass, l is the link length, c is the damping coe±cient, k is the sti®ness. The model is described by the equations: J1qÄ1 + M gl sin q1 + c(q_1 ¡ q_m) + k(q1 ¡ qm) = 0; JmqÄm ¡ c(q_1 ¡ q_m) ¡ k(q1 ¡ qm) = u: Here q1 is the link angle, qm is the rotor angle, and u is the torque input which is the controller.

The control problem under consideration consists of a tracking problem in which it is desired that the link coordinate q1 follows a time-varying smooth and bounded desired trajectory qd(t) so that jqd(t) ¡ q1(t)j ! 0 as t ! 1 [22, 23]. If we rewrite the original system in the form J1qÄ1 + JmqÄm + M gl sin q1 = u; qÄ1 ¡ qÄm + +c then the use of the small parameter " = 1=pk and new variables Note that neglecting all terms of order O("2) in the r.h.s. of the last equation we obtain the independent subsystem

( 7 ) ( 8 ) ( 9 ) ( 10 ) solutions of which are characterized by high frequency ¼ p(1=J1 + 1=Jm)=" and relatively slow decay c(1=J1 + 1=Jm)=2, since this di®erential system has the characteristic polynomial which possesses complex zeros

c µ 1 ¸1;2 = ¡ 2 J1 + 1 ¶ Jm

i sµ 1 § " J1 + 1 ¶ Jm ¡ "2 c2 µ 1 4

J1 + 1 ¶2 Jm : Since the real part of these numbers is negative, for the analysis of the manipulator model under consideration it is possible to use the slow invariant manifold noting that the reducibility principle holds for this manifold (the exact statement may be found in [3]). The terms of O("2) of the subsystem ( 9 ) lead us to conclude that the slow invariant manifold may be found in the form y1 = "2Y + O("3) and y2 = O("3), where Y = ¡ · M gl

J1 sin(x1) + u0 ¸ µ 1 Jm

+ J1

Jm 1 ¶¡1 : q1 = x1 + "2

Jm This allows us to rewrite the system ( 11 ) on the slow invariant manifold using the original variable q1 instead x1 in the form

The case of multiple root of the degenerate equation

We consider system ( 1 ) under the assumptions (A1) and (A2). Instead of hypothesis (A3) we suppose det gy(x; h(x; t); t; 0) ´ 0

8(x; t) 2 G £ R; that is, y = h(x; t) is not a simple root of the degenerate equation g(x; y; t; 0) = 0: Under this assumption we cannot apply Proposition 1.1 to system ( 1 ) in order to establish the existence of a slow integral manifold near M0 for small ". Our goal is to derive conditions which imply that for su±ciently small " system ( 1 ) has at least one integral manifold M" with the representation y = Ãi(x; t; ") = h(x; t) + "qi h1;i(x; t) + "2qi h2;i(x; t) + : : : : where qi; 0 < qi < 1; is a rational number.

The key idea to solve this problem consists in looking for scalings and transformations of the type

Here we used the representation u = u0 + "2u1 + O("3). Thus, the °ow on this manifold is described by equations " = ¹r; y = y~(¹; z; x; t); t = t~(¹; ¿ ) such that system ( 1 ) can be reduced to a system dz d¿ dx d¿ = f (x; z; ¿; ¹); ¹

= g(x; z; ¿; ¹) to which Proposition 1.1 can be applied.

Thrice critical case

Consider the control system "x_ = A(t; ")x + "B(t; ")u; x 2 Rn+m; x(0) = x0 ( 12 ) ( 13 ) ( 14 ) ( 15 ) ( 16 ) ( 17 ) (xT (t)Q(t)x(t) + "uT (t)R(t)u(t))dt: ( 18 ) where A; F1; Q are (n £ n)-matrices, B is (n £ m)-matrix, and R is (m £ m)matrix. Suppose that all these matrices have the following asymptotic presentations with respect to ": with the cost functional J = 1 xT ( 1 )F x( 1 ) + 2

1 1 Z 2

0 A(t; ") =

X "j Aj(t); B(t; ") =

X "j Bj (t); j¸0 j¸0 j¸0 X "j Qj (t); X "j Rj (t); j¸0 X "j Fj j¸0 Q(t; ") = R(t; ") = F (") = with smooth on t matrix coe±cients, t 2 [0; 1].

The solution of this problem is the optimal linear feedback control law u = ¡"¡1R¡1BT P (t; ")x; where P satis¯es the di®erential matrix Riccati equation "P_ = ¡P A ¡ AT P + P SP ¡ "Q;

P (1; ") = F: Setting " = 0 we obtain the matrix algebraic equation ¡M A0 ¡ A0T M + M S0M = 0; LX = XA0 + A0T X: where S0 = B0R0¡1B0T : It is clear that the main role plays the linear operator For this class of systems the eigenvalues of A0 are pure imaginary and the spectrum of the linear operator L has a nontrivial kernel, since sums (¸i(t) + ¸j (t)); i; j = 1; : : : ; n; form its spectrum. This means that the Riccati equation is singular singularly perturbed. Thus, the problem under consideration is critical in this sense. Moreover, under taking into account that zero eigenvalues are multiple and all other, nonzero eigenvalues of L, are pure imaginary, it is possible to say that this problem is thrice critical. First, we need to separate it into a slow and a fast subsystem. At ¯rst glance, all three equations are singularly perturbed. However, setting " = 0 we obtain p1 = p2 = p3 = 0, and we should consider the matrix of leading terms on the right hand side of the system, which has the form 0 1 1 A : 0 Obviously, this matrix has a zero eigenvalue and two pure imaginary eigenvalues, i.e. the problem under consideration is twice critical.Moreover, the trivial solution is multiple. This means that we have thrice critical case. Let " = ¹2. Introducing the new variables

Example

Let A = S = µ ¡" ¡1 1 ¶ ¡" and then s = q1 + q3, we obtain the di®erential system ¹s_ = 2q3 + ¹q2 + 2¹s + ¹q22 + ¹q32 + 4 + ¹=4; ¹2q_2 = ¡s + 2¹2q2 + 2q3 + ¹q2 + ¹2q2q3 + ¹=2 + ¹2; ¹2q_3 = ¡2q2 + 2¹q3 + 2¹2q3 + ¹2q32 + 2¹ with the slow variable s and two fast variables q2; q3.

The last system possesses one-dimensional slow invariant manifold which is weakly attractive with respect to argument 1 ¡ t because the main matrix of the fast subsystem is µ ¹ ¡2 2 ¶ 2¹

: Thus, the dimension of the system of Riccati di®erential equations can be reduced from three to one. Let us construct the slow integral manifold using the fact that it can be asymptotically expanded in powers of the small parameter. Setting q2 = '(s; ¹) = ¹'1(s) + ¹2 : : : ;

( 19 ) ( 20 ) we obtain Critical cases for singularly perturbed di®erential systems are studied in the paper. We have considered singularly perturbed control problems as applications. It has been shown that the reduction of dimensions of these problems can be done by means of the integral manifold method. The slow integral manifolds for the matrix Riccati equation of linear-quadratic control problem are constructed and it is shown that the method of integral manifolds allows us to reduce the dimension of control problems. This approach was used for the investigation of optimal ¯ltering problems in [28, 29].

Acknowledgements

This work is supported in part by the Ministry of education and science of the Russian Federation in the framework of the implementation of Program of increasing the competitiveness of SSAU for 2013{2020 years.

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