Haar Wavelet-Based Optimal Control of Time-Varying State-Delayed Systems: A Computational Method H. R. Karimil P. Jabehdar Maralani' B. Moshiri' B. Lohmann2 'Control & Intelligent Processing Center of Excellence, ECE, Faculty of Engineering, University of Tehran, P. 0. Box: 14395-515, Tehran, Iran. E-mail: hrkarimigut.ac.ir 2Institute of Automatic Control, Technical University of Munich, Munich, Germany Abstract Using Haar wavelets, a computational method is of OFs can be divided into three classes such as piecewise presented to determine the piecewise constant feedback constant basis functions (PCBFs) like HFs, WFs and BPFs; controls for a finite-time linear optimal control problem of a orthogonal polynomials like Laguerre, Legendre and time-varying state-delayed system. The method is simple and Chebyshev as well as sine-cosine functions in Fourier computationally advantageous. The approximated optimal series [16]. trajectory and optimal control are calculated using Haar We extend the results in [11, 12, 13] to finite-time wavelet integral operational matrix, Haar wavelet product operational matrix and llaar wavelet delay operational eoptmal control problem of Thme-varying systems with time matrix. An illustrative example is included to demonstrate the delay in the state vector. The properties of Haar wavelets validity and applicability of the technique. rwavelet integral operational matrix , Haar wavelet product operational matrix and Haar wavelet delay operational matrix are given and are utilized to provide a I. INTRODUCTION systematic computational framework to find the Ti sare often present in engineering systems, approximated optimal trajectory and finite-time optimal Time delays~~~~~~~ control of the time-varying state-delayed system with which have been generally regarded as a main source on instability and poor.performance.There stabilization respect to a quadratic cost function by solving only the instablity and poor performance. Therefore, linelzho and optimization of state-delayed systems are fields of linear algebraic equations instead of solving the differential intense research [8, 14, 15] equations. One of the main advantages is solving linear Wavelet theory is a relatively new and an emerging area algebraic equations instead of solving nonlinear Riccati in mathematical research [3]. It has been applied in a wide equation to optimize the control problem of the time- disciplines such as signal varying state-delayed systems. We demonstrate the range of engineering d1cple uha 1nlprocessing, pattern recognition and computational graphics. Recently, applicability of the technique in an illustrative example. **o* ~~~~Notations. some of the attempts are made in solving surface integral equations, improving the finite difference time domain A matrix A with dimension r x method, solving linear differential equations and nonlinear Ir identity matrix with dimension r x r; partial differential equations and modelling nonlinear °r zero matrix with dimension r x r; semiconductor devices [5, 9, 11, 12, 13, 17, 19, 21] °rxs zero matrix with dimension r x s; On the other hand, orthogonal functions (OFs) like Haar & Kronecker product; functions (HFs) [9, 11], Walsh functions (WFs) [6], block vec(X) the vector obtained by putting matrix x into one pulse functions (BPFs) [18], Laguerre polynomials [10], column. Legendre polynomials [4], Chebyshev functions [8] and Fourier series [20], often used to represent an arbitrary time II. PROPERTIES OF HAAR WAVELETS functions, have received considerable attention in dealing A. Haar Functions (HEs) with various problems of dynamic systems. The main The oldest and most basic of the wavelet systems is characteristic of this technique is that it reduces these named Haar wavelet [7] that is a group of square waves problems to those of solving a system of algebraic with magnitude of ±1 in the interval [o, i). In other words, equations for the solution of problems described by the HFs are defind on the interval [o, i) as differential equations, such as analysis of linear time- invariant, time-varying systems, model reduction, optimal control and system identification. Thus, the solution, I0(t) tfr[O, l), identification and optimisation procedure are either greatly 2 ({=l for (1) reduced or much simplified accordingly. The available sets 1 for t E [± 1), 1-4244-0020-1/05/$20.OO a2005 IEEE 1 of 6