Ideal Sliding Mode in the Problems of Convex Optimization ...

Automation and Remote Control, Vol. 63, No. 7, 2002, pp. 1199–1203. Translated from Avtomatika i Telemekhanika, No. 7, 2002, pp. 179–183. c 2002 by Venets. Original Russian Text Copyright

NOTES

Ideal Sliding Mode in the Problems of Convex Optimization V. I. Venets Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia Received February 14, 2002

Abstract—The characteristics of the sliding mode that appears with using the continuous convex-programming algorithms based on the exact penalty functions were discussed. For the case under study, the ideal sliding mode was shown to occur in the absence of infinite number of switchings.

1. INTRODUCTION The sliding modes are widely used to study various applications of control. They enable one to design systems featuring the desired characteristics where application of the ordinary smooth controls is difficult or impossible at all. By the ideal sliding mode is usually [1] meant the motion along the discontinuity surface that occurs if the phase trajectories in the neighborhood of this surface are directed “toward” each other. The authors mostly believe (see [2–5], for example) that an infinite number of switchings corresponds to the ideal sliding modes, but sometimes they assert [6] that these (ideal) modes exist only as theoretical constructions. It turns out, however, that the ideal sliding modes do not necessarily have an infinite number of switchings. This fact may be illustrated by the optimization problems that are often used as examples in the study of the sliding modes [1]. 2. CONTINUOUS ALGORITHMS OF CONVEX OPTIMIZATION In what follows, we employ the terminology and notation of nonlinear programming, but one could pass with no trouble to the notation used by the experts in the field of sliding modes (for example, in [1]). Let us consider a rather general formulation of the problem of convex programming formulated as follows: f (x) → min, x ∈ G = {x ∈ Rn : gi (x) ≤ 0, i = 1, . . . , m},

(1)

where all functions f, gi , i = 1, . . . , n, are the proper convex functions that are definite and continuously differentiable everywhere on Rn . The requirement on continuous differentiability is not a must and was introduced here just to simplify the presentation. We assume that a solution of problem (1) exists on a nonempty set X ∗ ⊂ G. To solve the original problem (1), we use the method of penalty functions or, to be more correct, exact penalty functions. In doing so, the original problem is replaced by that of determining the unconditional minimum of the nonsmooth convex function F (x, q) = f (x) + q

m X

gi+ (x),

i=1

c 2002 MAIK “Nauka/Interperiodica” 0005-1179/02/6307-1199$27.00

(2)

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where

(

gi+ (x)

=

0 gi (x)

if if

gi (x) ≤ 0 gi (x) > 0,

and q > 0 is the penalty coefficient. The existence conditions for the critical penalty coefficient q0 < ∞ that make problems (1) and (2) equivalent for all q > q0 are well known [7]. To determine the unconditional minimum of the convex function (2), we employ the continuous subgradient algorithm [8–9] and consider the system of differential inclusions dx ∈ −∂x F (x, q), dt

x(t0 ) = x0 ∈ Rn ,

(3)

where ∂x F (x, q) is the partial subdifferential of function (2) that is convex in the argument x ∈ Rn . In the case under consideration, ∂x F (x, q) is a convex bounded closed set for all x ∈ Rn and q > 0, and ∂x F (x, q) is the maximal monotone operator. Bresis proved that that for any x(t0 ) = x0 ∈ Rn there exists a unique solution x(t) of (3) defined for all t ≥ t0 [10]. Moreover, x(t) is an absolute continuous function for which the right derivative D+ x(t) is defined for all t ≥ t0 and D+ x(t) = −∂x F 0 (x, q),

(4)

where ∂x F 0 (x, q) is the canonical restriction [10] on the multivalued operator ∂x F (x, q). 3. IDEAL SLIDING MODE IN THE PROBLEMS OF CONVEX OPTIMIZATION The approach to the problems of convex optimization as based on the system of differential inclusions (3) is in agreement with the sliding mode-based method of solving the optimization problems [1]. The conditions formulated in [1] as those for existence of sliding modes arise in fact on some surfaces gi (x) = 0, i = 1, . . . , m, and/or their intersections. At the same time, the solutions of (3) have some interesting characteristics allowing one to assert that they correspond to the ideal sliding mode. The present author introduced the notion of sharp minimum [11]. For the case at hand, function (3) will be said to have the sharp minimum if the following condition is satisfied: inf

min

hy, ∂x F 0 (x, q)i ≥ ν > 0,

(5)

x∈Rn /X ∗ y∈∂x F (x,q)

where h·, ·i stands for the scalar product and X ∗ is the set of the minimum points of (2) which, under an appropriate choice of the penalty coefficient q coincides with the set of the optimal solutions of the original problem (1); we assume here without loss of generality that the existence conditions for the critical penalty coefficient are met. As follows from the theorem of [11], any solution x(t) of the system of differential inclusions (3) reaches the points of the set X ∗ in a finite time depending on the choice of initial approximation x0 and ν in (5). Therefore, beginning from some finite time instant T ∗ (x0 , ν), the solution x(t) belongs to the set X ∗ where the right side of system (3) is identically zero. Since the solution x(t) is an absolutely continuous function, the function x(t) has the derivative— and not only the right derivative (4)—almost everywhere [12]. This assertion suggests that the motion in system (3) indeed takes place on the corresponding surfaces or their intersections, that is, is the ideal sliding mode. Therefore, the ideal sliding modes exist (we do not discuss here technical feasibility of such systems), and motion in such modes does not consist of an infinite number of switchings. A similar result is obtained if the system of differential inclusions [9] having in its right side the operation projecting the subdifferential ∂x F (x, q) on the admissible set G is used instead of (3). AUTOMATION AND REMOTE CONTROL

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4. EXAMPLE The following simple example (see figure) is considered by way of illustration. Let the original problem lie in determining the minimum of the function f (x) = x2

(6)

on the admissible set G made up by the system of linear inequalities g1 (x) = −x1 − x2 + 2 ≤ 0, g2 (x) = x1 − x2 − 2 ≤ 0 ≤ 0.

(7)

Solution of problem (6), (7) is obtained at a single point x∗ = (2, 0) where both constraints are satisfied as equalities. In the case under consideration, solution of the system of differential inclusions (3) presents no difficulties. For problem (6), (7), the existence conditions for the critical penalty coefficient are satisfied, and for simplicity in (2) one can assume that q = 1. Let at time t0 = 0 the initial approximation x(t0 ) = x0 = (0, 4), that is, let it belong to the admissible set G. At this point, the penalty function (2) is differentiable and (3) is representable as the system of ordinary differential equations dx1 = 0, x01 = 0, dt dx2 = −1, x02 = 4. dt

(8)

Its solution x1 (t) = x01 , x2 (t) = x02 − t to which the trajectory shown in the Figure by ⇓ corresponds reaches at time t = t1 = 2 the point x1 = (0, 2) where the first constraint of system (7) is satisfied as equality; further motion obeys

x2 6 4 @ ⇓ ⇓ @ ⇓ @ 3 ⇓ ⇓ @ G ⇓ g1 (x) = 0@2 ⇓ g2 (x) = 0 @& @&& 1 @&& @&& @ x1 −1 1 2@ 3 4 5 @ −1 x∗ = (2, 0) @ @ −2 @

Figure.

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another system and not the system of differential equations (8). At the point x1 = (0, 2), the penalty function (2) becomes as follows: F (x, 1) = x2 + (−x1 − x2 + 2)+ ,

(9)

where the superscript + is as above. Further motion in system (3) proceeds along the surface g1 (x) = 0. Indeed, the canonical restriction of the multivalued operator ∂x F (x, q) is represented here by the vector ∂x F (x, 1) = (−0.5; 0.5), and system (3) assumes the following simple form: dx1 = 0.5, x11 = 0, dt dx2 = −0.5, x12 = 2. dt

(10)

Its solution x1 (t) = x11 + 0.5t, x2 (t) = x12 − 0.5t to which the trajectory shown in the Figure by & corresponds at the time instant t2 = 4 hits the optimal point x∗ = (2, 0) and remains there. Therefore, system (3) realizes in fact the ideal sliding mode enabling one to reach the optimal point in a finite time by moving partially on the discontinuity surface. It can be mentioned that in the case under study the numerical realization of this algorithm also provides ideal sliding and the optimal solution in a finite number of steps. Indeed, let the ith constraint of system (7) be satisfied for the numerical realization as equality to within some given accuracy ε if kgi (x)k ≤ ε. Then, the neighborhood of the optimal solution x∗ where both constraints are satisfied with the above accuracy will be reached in two conditional steps. At the first conditional step, the value of x2 will be reduced under constant x1 = 0 until the first constraint of system (7) is satisfied to within the given accuracy ε, and at the second conditional step, motion, that is, sliding, will proceed in parallel to the plane g1 (x) = 0 until the second constraint of system (7) is satisfied to within the given accuracy ε. The number of actual steps of the iterative process will depend on the particular chosen algorithm and given accuracy ε, but the number of commutations in the case of this sliding will be other than infinite. REFERENCES 1. Utkin, V.I., Skol’zyashchie rezhimy i ikh primenenie v sistemakh s peremennoi strukturoi (Sliding Modes in Variable-structure Systems), Moscow: Nauka, 1974. 2. Moura, J.T. and Olgac, N., Robust Lyapunov Control with Perturbation Estimation, Proc. Am. Control Conf., Seattle, 1995, vol. 4, pp. 3458–3462. 3. Garcia, P.F.D., Cortizo, P.C., de Menezes, B.R., et al., Sliding Mode Control for Current Distribution in Parallel Connected DC-to-DC Converters, IEE-EPA, 1998, vol. 145, no. 4, pp. 333–338. 4. Levant, A., Controlling Output Variables via Higher Order Sliding Modes, Proc. Eur. Control Conf., Karlsruhe, 1999. 5. Levant, A., Universal SISO Sliding-Mode Controllers with Finite-Time Convergence, IEEE Trans. Autom. Control, 2001, vol. 46, no. 9, pp. 1447–1451. 6. Korondi, P. and Hashimoto, H., Park Vector Based Sliding Mode Control of UPS with Unbalanced and Nonlinear Load, Periodica Polytechnica, Ser. El. Eng., 1999, vol. 43, no. 1, pp. 65–79. AUTOMATION AND REMOTE CONTROL

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7. Venets, V.I., The Saddle Point of the Lagrange Function and Nonsmooth Penalty Functions in Convex Programming, Avtom. Telemekh., 1974, no. 8, pp. 109–118. 8. Venets, V.I., Stability of Continuous Subgradient Algorithms, Avtom. Telemekh., 1983, no. 1, pp. 26–32. 9. Venets, V.I., Continuous Algorithms for Solution of Convex Optimization Problems and Finding Saddle Points of Convex-Concave Functions with the Use of Projection, Optimization, 1985, vol. 16, no. 1, pp. 519–533. 10. Bresis, H., Op´erateurs maximaux monotones et semigroupes de contraction dan les espaces de Hilbert , Amsterdam: Elsevier, 1973. 11. Venets, V.I., Finite Convergence of Continuous Subgradient Optimization Algorithms, in Jahrestagung “Mathematische Optimierung,” Rostok, 1986, pp. 79–82. 12. Grauert, H., Lieb, I., and Fisher, W. Differential- und Integralrechnung, Berlin: Springer-Verlag, 1967. Translated under the title Differentsial’noe i integral’noe ischislenie, Moscow: Mir, 1971.

This paper was recommended for publication by N.A. Bobylev, a member of the Editorial Board

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