Using Lagrange multipliers method for constrained optimal control problems

Математика / 4. Прикладная математика

Dr. Murzbekov Z.N., PhD Aipanov Sh.A.

Kazakh National University, Almaty Technological University

Almaty, Kazakhstan

USING LAGRANGE MULTIPLIERS METHOD

FOR CONSTRAINED OPTIMAL CONTROL PROBLEMS

1. Introduction. There are various examples of mathematical statement and methods for solving the optimal control problems (OCP) in papers devoted to dynamic systems modeling and automatic control. The problem is to find an optimal programmed control depending on running time and an initial state of a system or to find an optimal synthesized control as a function of a system’s current state. For solving the OCP in the first statement we can apply the Pontryagin’s maximum principle [1] (a problem is reduced to a boundary problem for a system of ordinary differential equations) and in the second case we can use the dynamic programming method (a problem is reduced to the Bellman’s equation [2]). The analytical solution of the linear-quadratic OCP with fixed endpoints of trajectories (without constraints on control values) was suggested in [3, 4].

Development of the fast and constructive algorithms for building the controls possessing various necessary properties is an actual problem in modern information technologies applications. In the given article we consider the OCP with fixed initial and final states of the system, taking into account constraints on control values, and present a constructive algorithm for designing closed-loop controls. The OCP with fixed endpoints of trajectories arise, for instance, in exploring dynamics of flight vehicles, robotic and electric power systems, chemical and nuclear reactors, etc. In simple cases a dynamics of investigated system can be described by linear differential equations and as a criterion of control quality can be used a quadratic functional.

2. Problem statement. We consider a control system described by the differential equation of the form

(1)

with the given initial state

(2)

and final state

(3)

with constraints on control values

(4)

Here is a -vector of the object state; is a -vector of piecewise continuous control action; , are matrices of dimensions , respectively (entries of these matrices are continuous functions); is a ‑vector of continuous functions; , are ‑vectors with components, which are piecewise continuous functions. A system dynamics is considered within the time interval , where and are the preset initial and final time moments. It is supposed that the system (1) is fully controllable at time moment .

A quality of control is described by the functional

(5)

where , , are matrices of dimensions , , respectively; , are vectors of dimensions , respectively; is a scalar function. The prime symbol (') denotes the transpose of a matrix or a vector.

The problem is to find a synthesizing control , subject to constraints (4), that transfers the system (1) from the initial state (2) to the final state (3) (coordinate origin) within the specified time interval and minimize the functional (5).

If a final time moment isn’t specified beforehand and it is assumed that in the functional (5), then we obtain a time optimal control problem; let’s assume that it has a solution and the optimal value of the functional is equal to . In (1)-(5) we suppose that the final time moment is fixed and satisfies the inequality ; it is necessary for a solution existence of the OCP under consideration. When the final condition (3) is given as , we can use the substitution for which we obtain the problem of the form (1)-(5) too. In practical problems we often have , , , , but here we consider the functional of the form (5), because such statement of the problem may be useful in implementation of some numerical algorithms for solving the OCP of more general form.

3. Algorithm for solving the problem. Let’s consider a symmetric -matrix , subject to the Riccati differential equation

(6)

where , , .

We use to denote a symmetric -matrix of the form

Here is a -matrix; is a fundamental matrix solution for the differential equation of the form , where .

Let’s consider a -vector-function , satisfying the differential equation

(7)

where

(8)

Using the above denotations the solution of the OCP under consideration (1)-(5) can be formulated as the following theorem.

T h e o r e m . Let be a positive definite matrix with nonnegative entries in interval and be a positive semi-definite matrix. It is assumed that the system (1) is fully controllable at time moment and also that . Then

1) the optimal trajectory of system movement , for OPC (1)-(5) is determined from differential equation

(9)

where matrix and vector satisfy differential equations (6) and (7) respectively;

2) the optimal control is of the form

(10)

where values of the vector-functions and are determined by formula (8).

The theorem can be proved on the base of sufficient conditions of optimality for dynamic systems with fixed endpoints of trajectories, which obtained in [5] by using Lagrange multipliers of special form. A vector can be chosen to provide a fulfillment of the final condition (3) for the solution of differential equation (9).

We can find optimal trajectory and optimal control for OCP (1)-(5) applying the following algorithm.

S t a g e 1 . Integrate the system of differential equations over the interval :

(11)

(12)

where is an arbitrary positive semi-definite symmetric matrix. As the result of integrating the system (11), (12) we determine the matrices and .

S t a g e 2 . Integrate the system of differential equations over the interval :

(13)

and calculate

. (14)

In (13) the symbol denotes a -dimensional identity matrix.

S t a g e 3. Integrate the system of differential equations over the interval :

(15)

The obtained solution corresponds to the required optimal movement trajectory , . During the integrating process we calculate the values of functions , , , by formula (8) and therefore we can find the optimal control values , using (10).

If then we have , so when these inequalities are hold there is no need to calculate an inverse matrix in right parts of differential equations (15). Furthermore, matrices and in (11)-(13), (15) are symmetric, so we can consider only upper triangular parts of these matrices (with diagonal elements) when solving the system of differential equations by numerical methods; it allows to reduce the amount of computations.

4. Example. We consider the following optimal control problem: minimize the functional

under conditions

The Riccati differential equation (11) will have the solution , when setting the final condition . Then the required optimal control can be written in the form (10), where

For the example under consideration we have

then by formula (14) we calculate .

According to (10) we find the optimal control of the form

where a switching time moment is . The optimal trajectories , in time interval can be found from the system of differential equations (15)

Here functions and are elements of the matrix , where . Note that there is no need to calculate the entries of matrix in the interval as we have in this interval. Plots of , and are represented in Fig. 1. As can be seen from this figure, the required control provides the final condition fulfillment with high accuracy (in numerical calculations we obtained , ).

Fig. 1. Optimal control and optimal trajectories

5. Conclusions. In this article we consider linear nonstationary systems with quadratic functional and present the method for building the synthesizing control moving the system from the initial state to the desired final state within the given time interval in the presence of constraints on control values.

The problem is solved using the Lagrange multipliers depending on phase coordinates and time. Due to choosing of we managed to build an optimal control based on principles of feedback loops; and were chosen to satisfy so called complementary slackness conditions in Lagrange multipliers method.

Due to choosing of the condition we can obtain various representation forms for the required optimal control with different and , . The optimal control and optimal trajectory can be found using the algorithm represented in Section 3.

The algorithm suggested for solving the linear-quadratic OCP with fixed endpoints of trajectories and constraints on control values was implemented in FORTRAN and approved for an example model.

References:

1. Понтрягин Л.С., Болтянский В.Г., Гамкрелидзе Р.В., Мищенко Е.Ф. Математическая теория оптимальных процессов. – М.: Наука, 1976. – 392 с.

2. Bellman R. Dynamic programming. – New York: Dover Publications, Inc., 2003. – 366 p.

3. Aipanov Sh.A., Murzabekov Z.N. Optimal control of linear systems with fixed ends of trajectories and with quadratic functional // Journal of Computer and Systems Sciences International. – 1996. – Vol. 34, No. 2. – P. 166-172.

4. Aipanov Sh.A., Murzabekov Z.N. The solution of a linear-quadratic optimal control problem for fixed endpoints of system // Differential Equations. – 1996. – Vol. 32, No. 6. – P. 848-849.

5. Мурзабеков З.Н. Достаточные условия оптимальности динамических систем управления с закрепленными концами // Математический журнал. – 2004. – Т. 4, № 2 (12). – С. 52-59.