A New Approach for Solving Optimal Nonlinear Control Problems Using Decriminalization and Rationalized Haar Functions

This paper presents a numerical method based on quasilinearization and rationalized Haar functions for solving nonlinear optimal control problems including terminal state constraints, state and control inequality constraints. The optimal control problem is converted into a sequence of quadratic programming problems. The rationalized Haar functions with unknown coefficients are used to approximate the control variables and the derivative of the state variables. By adding artificial controls, the number of state and control variables is equal. Then the quasilinearization method is used to change the nonlinear optimal control problems with a sequence of constrained linear-quadratic optimal control problems. To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.


Introduction
A widely used method to solve optimal control problems is the direct method. The direct method converts the optimal control problem into a mathematical programming problem with a large number of parameters and equality constraints [1][2][3][4]. In order to solve these problems, Hussein Jaddu [5] proposed a method to solve the linear-quadratic and the nonlinear optimal control problems by using Chebyshev polynomials to approximate the state variables. Also shifted Legendre polynomials to parameterize the derivative of each of the state variables are introduced to solve the linear-quadratic optimal control problem in [6].
In this paper we present a computational method for solving nonlinear optimal control problems with state and control inequality. The quasilinearization technique is applied to convert the constrained optimal control problems into a sequence of constrained linear-quadratic optimal control problems. Then by applying the Haar wavelets functions to approximate the control variables and the derivative of the state variables, the constrained linear-quadratic optimal control problems are transformed into quadratic programming problems and are well solved.
where r w are given by   T s s

Integration of Haar wavelets
The integration of the ( ) t φ is given by where s s P P × = is the s s × is the s s × operational matrix for integration and is given in [9] as with 1 1 1 1

Problem statement
The problem we are considering is to find the control vector ( ) u τ , and the corresponding state vector ( ) , which minimize (or maximize) the functional

Quasilinearization
Then the optimal control problem (13)-(17) can be replaced with the following sequence of constrained linear-quadratic optimal control problems Minimize ( ) and the performance index is modified to

Problem reformulation
The time transformation is introduced in order to use Haar functions defined on [0,1] t ∈ .
Using this transformation and expressing the optimal control problem (18)-(22) in terms of t , we get Minimize ( ) Approximating each of the system dynamic functions and control variables by Haar series with unknown parameters gives 1 2 , ,, , where I is a n n × dimensional identity matrix, ( ) t φ is 1 s × vector (

Emerging Engineering Approaches and Applications
Also we have where 1 2 , , , .
 is a vector of order . Using the Haar functions' integration operational matrix P , 1 where P is an operational matrix of integration given in Eq. (12). From Eqs.(39) and (43) we obtain The performance index can be approximated by substituting Eqs. ) ( ) Eq.(44) can be simplified to which can be rewritten as 2 ) , ( Then we approximate the inequality constraints of the optimal problem. These constraints can be handled by requiting their satisfaction at a finite number of discrete points, By substituting Eqs. (41) and (42)

Numerical simulations
In this section, we consider the Van der Pol oscillator problem. This example is adapted from [5] and studied by using Chebyshev method. Find the control vector ( ) u t which minimizes and the following terminal state constraints and inequality control constraints ( )   Table.1, we compared the optimal solutions obtained using the proposed method with other solutions in the literature. For 64 k = , the computational results for 1 2 ( ), ( ), ( ) x t x t u t and ( ) z t are given in Figs.1 and 2, respectively.

Conclusion
In this paper, we develop an efficient method for solving nonlinear optimal control problems with terminal constraints, state inequality constraints and inequality control constraints. The technique is based on approximation of both the controls and the derivative of the state into Haar series and converting the optimal control problem into a sequence of linear-quadratic optimal control problems. Illustrative examples are included to demonstrate the effectiveness of the proposed method.