Hussein Jaddu

Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials

Abstract In this paper, a numerical method to solve nonlinear optimal control problems with terminal state constraints, control inequality constraints and simple bounds on the state variables, is presented. The method converts the optimal control problem into a sequence of quadratic programming problems. To this end, the quasilinearization method is used to replace the nonlinear optimal control problem with a sequence of constrained linear-quadratic optimal control problems, then each of the state variables is approximated by a finite length Chebyshev series with unknown parameters. The method gives the information of the quadratic programming problem explicitly (The Hessian, the gradient of the cost function and the Jacobian of the constraints). To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.

Spectral method for constrained linear-quadratic optimal control

A computational method based on Chebyshev spectral method is presented to solve the linear-quadratic optimal control problem subject to terminal state equality constraints and state-control inequality constraints. The method approximates each of the system state variables and each of the control variables by a finite Chebyshev series of unknown parameters. The method converts the optimal control problem into a quadratic programming problem which can be solved more easily than the original problem. This paper gives explicit results that simplify the implementation of the method. To show the numerical behavior of the proposed method, the simulation results of an example are presented.

Computation of optimal control trajectories using Chebyshev polynomials: parameterization, and quadratic programming

An algorithm is proposed to solve the optimal control problem for linear and nonlinear systems with quadratic performance index. The method is based on parameterizing the state variables by Chebyshev series. The control variables are obtained from the system state equations as a function of the approximated state variables. In this method, there is no need to integrate the system state equations, and the performance index is evaluated by an algorithm which is also proposed in this paper. This converts the optimal control problem into a small size parameter optimization problem which is quadratic in the unknown parameters, therefore the optimal value of these parameters can be obtained by using quadratic programming results. Some numerical examples are presented to show the usefulness of the proposed algorithm. Copyright

Computational method based on state parametrization for solving constrained nonlinear optimal control problems

In this paper, we present a method for solving nonlinear optimal control problems subject to terminal state constraints and control saturation constraints. This method is based on using the quasilinearization and the state variables parametrization by using Chebyshev polynomials. In this method, there is no need to integrate the system state equations or the costate equations. By virtue of the quasilinearization and the state parametrization, the difficult constrained nonlinear optimal control problem is approximated by a sequence of small quadratic programming problems which can be solved easily. To show the effectiveness of the proposed method, the simulation results of a numerical example are shown.

Computation of optimal feedback gains for time-varying LQ optimal control

A computational method is proposed to compute the optimal feedback control law of time-varying linear quadratic optimal control problem. The idea of the method is to use Chebyshev polynomials of the first type and their differentiation operational matrix to solve the matrix Riccati equation. To show the effectiveness of the proposed method, the simulation result of an example is shown.

Optimal closed loop control for nonlinear systems using Chebyshev polynomials

A method is proposed to construct the optimal feedback control law of a nonlinear optimal control problem. The method is based on two steps: the first step is to determine the open loop optimal control and trajectories, by using the quasilinearization and the state variables parameterization via Chebyshev polynomials of the first type; the second step is to use the results of the last quasilinearization iteration, when acceptable convergence error is achieved, to obtain the optimal feedback control law. To show the effectiveness of the proposed method, the simulation results of a nonlinear optimal control problem are shown.

Construction of optimal feedback control for nonlinear systems via Chebyshev polynomials

A method is proposed to determine the optimal feedback control law of a class of nonlinear optimal control problems. The method is based on two steps. The first step is to determine the open-hop optimal control and trajectories, by using the quasilinearization and the state variables parametrization via Chebyshev polynomials of the first type. Therefore the nonlinear optimal control problem is replaced by a sequence of small quadratic programming problems which can easily be solved. The second step is to use the results of the last quasilinearization iteration, when an acceptable convergence error is achieved, to obtain the optimal feedback control law. To this end, the matrix Riccati equation and another n linear differential equations are solved using the Chebyshev polynomials of the first type. Moreover, the differentiation operational matrix of Chebyshev polynomials is introduced. To show the effectiveness of the proposed method, the simulation results of a nonlinear optimal control problem are shown.

Closed Form Solution of Nonlinear-Quadratic Optimal Control Problem by State-Control Parameterization using Chebyshev Polynomials

In this paper the quasilinearization technique along with the Chebyshev polynomials of the first type are used to solve the nonlinear-quadratic optimal control problem with terminal state constraints. The quasilinearization is used to convert the nonlinear quadratic optimal control problem into sequence of linear quadratic optimal control problems. Then by approximating the state and control variables using Chebyshev polynomials, the optimal control problem can be approximated by a sequence of quadratic programming problems. The paper presents a closed form solution that relates the parameters of each of the quadratic programming problems to the original problem parameters. To illustrate the numerical behavior of the proposed method, the solution of the Van der Pol oscillator problem with and without terminal state constraints is presented.

Computational algorithm based on state parametrization for constrained nonlinear optimal control problem

An algorithm, based on quasi-linearization and state variable parametrization by using Chebyshev polynomials, is proposed to solve the nonlinear optimal control problem subject to terminal state constraints and control saturation constraints. By the virtue of the second method of quasi-linearization and state parametrization, the difficult nonlinear optimal control problem is converted into a sequence of standard quadratic programming problems. An example is given to show the effectiveness of the proposed algorithm.