Hybrid functions approach for nonlinear constrained optimal control problems

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Abstract

In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrix of integration is introduced. This matrix is then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Highlights

► Hybrid functions consisting of the combination of block-pulse with polynomial series have been shown to be a powerful tool. ► The hybrid of block-pulse and Bernoulli polynomials is introduced. We then solve the constrained optimal control problems. ► The operational matrices of integration P and the integration of the cross product of two hybrid functions D are given. ► The matrices P and D for this hybrid functions have many zero elements. This reduces the CPU time and computer memory.

Introduction

In the classical development, it is well-known that the variational method of optimal control theory, which typically consists of the calculus of variations and Pontryagin’s methods [1], can be used to derive a set of necessary conditions that must be satisfied by an optimal control law and its associated state-control equations. These necessary conditions of optimality lead to a (generally nonlinear) two-point boundary-value problem (2PBVP) that must be solved to determine the explicit expression for the optimal control. Except in some special cases, the solution of this 2PBVP is difficult, and in some cases not practical to obtain. Various alternative computational techniques for optimal control problems have been developed in the literature (see for example [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and the references therein).

The available sets of orthogonal functions can be divided into three classes. The first class includes sets of piecewise constant basis functions (PCBF’s) (e.g., block-pulse, Haar, Walsh, etc.). The second class consists of sets of orthogonal polynomials (e.g., Chebyshev, Laguerre, Legendre, etc.). The third class is the set of sine-cosine functions in the Fourier series. Orthogonal functions have been used when dealing with various problems of the dynamical systems. The approach is based on converting the underlying differential equation into an integral equation through integration, approximating various signals involved in the equation by truncated orthogonal functions, and using the operational matrix of integration P to eliminate the integral operations. The matrix P can be uniquely determined based on the particular orthogonal functions.

Among PCBF’s, block-pulse functions are found to be very attractive, in view of their properties of simplicity and disjointedness and among orthogonal polynomials, the shifted Legendre polynomials Lm(t), m = 0, 1, 2,  , where 0  t  1, is computationally more effective [16]. The Bernoulli polynomials and Taylor series are not based on orthogonal functions, nevertheless, they possesses the operational matrices of integration. However, since the integration of the cross product of two Taylor series vectors is given in terms of a Hilbert matrix [17], which are known to be ill conditioned, the applications of Taylor series are limited.

For approximating an arbitrary time function the advantages of Bernoulli polynomials βm(t), m = 0, 1, 2,  , where 0  t  1, over shifted Legendre polynomials are:

  • (a)

    The operational matrix P, in Bernoulli polynomials has less errors than P for shifted Legendre polynomials for 1 < m < 10. This is because for P in βm(t) we ignore the term βm+1(t)m+1 while for P in Lm(t) we ignore the term Lm+1(t)2(2m+1);

  • (b)

    The Bernoulli polynomials have less terms than shifted Legendre polynomials. For example β6(t), has 5 terms while L6(t), has 7 terms, and this difference will increase by increasing m. Hence for approximating an arbitrary function we use less CPU time by applying Bernoulli polynomials as compared to shifted Legendre polynomials;

  • (c)

    The coefficient of individual terms in Bernoulli polynomials βm(t), are smaller than the coefficient of individual terms in the shifted Legendre polynomials Lm(t). Since the computational errors in the product are related to the coefficients of individual terms, the computational errors are less by using Bernoulli polynomials.

In recent years the hybrid functions consisting of the combination of block-pulse functions with Chebyshev polynomials [18], [19], [20], Legendre polynomials [21], [22], [23], or Taylor series [24], [25] have been shown to be a mathematical power tool for discretization of selected problems. Among these three hybrid functions, the hybrid functions of block-pulse and Legendre polynomials have shown to be computationally more effective.

Most of the computing techniques for the solution of optimal control problems successfully solve the unconstrained problems, but the presence of inequality constraints often results in both analytical and computational difficulties. Theoretical aspects of trajectory inequality constraints have been studied in [3], [4]. Mehra and Davis [4] noted that difficulties arising from handling trajectory inequality constraints are due to the exclusive use of control variables as independent variables, and they presented the so-called generalized gradient technique. The approach proposed in [9], [10] for solving optimal control problems with constraints, requires that the state and control variables, the system dynamics, the boundary conditions, the constraints, and the performance index be expanded in Chebyshev series with unknown coefficients. The resulting system of nonlinear equations have to be solved for these unknowns by some kind of iterative method. The unknowns which evolve from the Chebyshev series expansion of the performance index also have to be calculated at each step of the iteration. A Fourier-based state parameterization approach for solving linear quadratic optimal control problems is proposed in [11]. The approach proposed in [11], is based on approximating the state variable by the sum of a third-order polynomial and a finite-term Fourier-type series. Their method requires that the influence matrix multiplied by the control vector in the system dynamics is invertible, otherwise a penalty function technique is imposed to produce another invertible influence matrix.

In the present paper we present a new direct computational method to solve the nonlinear constrained quadratic optimal control problems. Here, we approximate the optimization problem by a nonlinear programming (NLP) one. The approach draws upon the power of well-developed NLP techniques and computer codes (see, e.g., [12], [26], [27], [28], [29], [30], [31]). Our method consists of reducing the optimal control problem to a NLP one by first expanding the state rate x˙(t) and the control vector u(t) as a hybrid function with unknown coefficients. These hybrid functions, which consist of block-pulse functions and Bernoulli polynomials are introduced. The operational matrix of integration P and the integration of the cross product of two hybrid functions of block-pulse and Bernoulli polynomials D are given.

The paper is organized as follows: In Section 2 we describe the basic formulation of the hybrid functions required for our subsequent development. Section 3 is devoted to the formulation of optimal control problems. Section 4 summarizes the application of this method to the optimal control problems, and in Section 5, we report our numerical finding and demonstrate the accuracy of the proposed method.

Section snippets

Hybrid of block-pulse and Bernoulli polynomials

Hybrid functions bnm(t), n = 1, 2,  , N, m = 0, 1,  , M are defined on the interval [0, tf) asbnm(t)=βmNtft-n+1,tn-1Ntf,nNtf),0,otherwise,where n and m are the order of block-pulse functions and Bernoulli polynomials, respectively. In Eq. (1), βm(t), m = 0, 1, 2, … are the Bernoulli polynomials of order m, which can be defined by [32] asβm(t)=k=0mmkαktm-kwhere αk, k = 0, 1,  , m are Bernoulli numbers. These numbers are a sequence of signed rational numbers which arise in the series expansions of trigonometric

Problem statement

Consider the following class of nonlinear systems with inequality constraints,x˙(t)=F(t,x(t),u(t))Sj(t,x(t),u(t))0,j=1,2,,w,t[0,1]x(0)=x0where x(t) and u(t) are K1 × 1 and K2 × 1 state and control vectors respectively. The problem is to find the optimal control u(t) and the corresponding state trajectory x(t), 0  t  1 satisfying Eqs. (17), (18), (19) while minimizing (or maximizing) the quadratic performance indexJ=12xT(1)Zx(1)+1201(xT(t)Q(t)x(t)+uT(t)R(t)u(t))dt,where T denotes transposition, Z,

The numerical method

By expanding each of the derivatives of the components of the state vector and each of the components of the control vector in the hybrid of block-pulse and Bernoulli polynomials, we getx˙q(t)=m=0Mn=1Nanmqbnm(t)=AqTB(t),q=1,2,,K1us(t)=m=0Mn=1Nhnmsbnm(t)=HsTB(t),s=1,2,,K2whereAq=a10q,a20q,,aN0q,a11q,a21q,,aN1q,,a1Mq,a2Mq,,aNMqT,q=1,2,,K1Hs=h10s,h20s,,hN0s,h11s,h21s,,hN1s,,h1Ms,h2Ms,,hNMsT,s=1,2,,K2.Using Eqs. (21), (22), (23), (24) the derivative of the state variable x˙(t) and

Illustrative examples

In this section four examples are given to demonstrate the applicability and accuracy of our method. In all examples the package of Mathematica (7.0) has been used to solve the test problems considered in this paper.

Conclusion

In the present work the hybrid of block-pulse and Bernoulli polynomials are used to solve nonlinear constrained optimal control problems. The problem has been reduced to a problem of solving a nonlinear programming one to which existing well-developed algorithms may be applied. The matrices D, and P in Eqs. (8), (9) have large numbers of zero elements and they are sparse, hence the present method is very attractive and reduces the CPU time and the computer memory. Illustrative examples are

Acknowledgment

The authors wish to express their sincere thanks to an anonymous referee for valuable suggestions that improved the final manuscript.

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