Applied Mathematical Sciences, Vol. 8, 24, no. 2, 6-68 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ams.24.4863 Numerical Investigation of the Time Invariant Optimal Control of Singular Systems Using Adomian Decomposition Method S. Sekar Department of Mathematics Government Arts College (Autonomous) Salem 636 7 Tamil Nadu, India A. Kavitha Department of Mathematics J.K.K. Nataraja College of Arts and Science Komarapalayam, Namakkal 638 83 Tamil Nadu, India Copyright c 24 S. Sekar and A. Kavitha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper deals with the implementation of Adomian Decomposition Method (ADM) to solve the optimal control problem of linear time invariant singular systems with a quadratic cost functional. The idea is that the states and the input are expressed in terms of ADM. The method simplifies the system of state equations into a set of algebraic equations which can be solved using a digital computer. Illustrative example is included to demonstrate the validity and applicability of the technique. Mathematics Subject Classification: 65L5, 65L8 Keywords: Optimal control, Singular Systems, Adomian Decomposition Method, Fourth order Runge-Kutta Method
62 S. Sekar and A. Kavitha Introduction Mathematical models in the form of optimal singular systems are essential in describing mechanical multi-body systems and many electrical circuits. They are particularly useful when a non-linear system subjected to algebraic constraints is linear along a nominal trajectory. Because these systems may exhibit behaviors so different from those of the regular linear systems, theories well established for regular systems without generalization. In view of this situation, numerous research articles have been dedicated to singular systems in the past three decades. Finding an optimal control subject to linear quadratic cost is crucial in many control problems 4. For the linear systems in regular state-space, the quadratic optimal control problem for singular systems, however, remained unexplored until the 98s. Two prominent works addressing this issue in the time-invariant setting is Cobb 7. The time-optimal control of singular systems was analyzed by Wansheng Tang et al. 5 Many models that enter into this framework can be found in practice and, in particular, in the existing literature. Among these we can mention: Chen and Hsiao 5, Chen and Shih 6, applied Walsh series to study the problem of optimal control of time-invariant and time-varying linear systems. It is to be noted that from the study of past literature that Cobb 7 and Pandolfi 3 seems to have been the first authors to consider the optimal regulator problem of continuous time singular systems. Both of them used state feed backs and their results were derived by the aid of Ricatti-type matrix equations. K. Balachandran and K. Murugesan 3 derived optimal control of singular systems via single-term Walsh series. C.F. Chen and C.H. Hsiao 5 Walsh series analysis in optimal control in detail. W.L. Chen and Y.P. Shih 6 analysis and optimum control of time varying linear systems via Walsh functions. H. Maurer gave numerical solution of singular control problems using multiple shooting techniques. H.J. Oberle gave the numerical computation of singular control functions in trajectory optimization problems. H.J. Pesch 2 highlight a practical guide to the solution of real-life optimal control problems. M. Razzaghi and H. Marzban 4 derived optimal control of singular systems via piecewise linear polynomial functions. The above all the systems investigated in the cited articles. The paper is organized as follows: In section 2 we describe the formula for the ADM. Section 3 is devoted to the formulation of the optimal control problem. In section 4 formulation of time invariant optimal control of singular systems and in section 5, example of time invariant optimal control of singular systems and we report our numerical findings and demonstrate the accuracy of the proposed method.
Numerical investigation of the time invariant optimal control 63 2 Adomian Decomposition Method Suppose k is a positive integer f, f 2,...f k and k are real continuous functions defined on some domain G. To obtain k differentiable functions y, y 2,...y k defined on the interval I such that (t, y (t), y 2 (t),...y k (t)) G for t I. Let us consider the problems in the following system of ordinary differential equations: dy i (t) = f i (t, y (t), y 2 (t),...y k (t)) () dt where β i is a specified constant vector, y i (t) is the solution vector for i =, 2,...k. In the decomposition method, () is approximated by the operators in the form: Ly i (t) = f i (t, y (t), y 2 (t),...y k (t)) where L is the first order operator defined by L = d and i =, 2,...k. Assuming the inverse operator dt of L is L which is invertible and denoted by L (.) = t t (.)dt, then applying L to Ly i (t) yields L Ly i (t) = L f i (t, y (t), y 2 (t),...y k (t)) where i =, 2,...k. Thus y i (t) = y i (t ) + L f i (t, y (t), y 2 (t),...y k (t)). Hence the decomposition method consists of representing y i (t) in the decomposition series form given by y i (t) = f i,n (t, y (t), y 2 (t),...y k (t)) n= where the components y i,n, n and i =, 2,...k can be computed readily in a recursive manner. Then the series solution is obtained as y i (t) = y i, (t) + L f i,n (t, y (t), y 2 (t),...y k (t)) n= For a detailed explanation of decomposition method and a general formula of Adomian polynomials, we refer reader to Adomian -2. 3 Optimal control of singular systems The linear time-invariant singular system represented in 3 is considered. Assuming that K det(s K A), B =, K = I p K 2 T In p Where K = Now the problem can be stated as follows: Given the initial state x( ) = x find a control vector u(t) that generates a state x(t)
64 S. Sekar and A. Kavitha such that x(t f ) = x f, where t f is a prescribed time and x f is a fixed vector, and minimizes the cost functional J = tf L(x, u)dt where L = 2 (xt Qx + u T Ru), Q and R denote given real symmetric constant matrices. In case the initial state x( ) is not known, the method developed by E-Tohami et al. 8 may be used to reconstruct the state. It has been proved by Lovass-Nagy et al. 9, which the problem of finding an optimal control reduces to the solution of a two-point boundary value problem. Let x = x x 2 x is (n p) and x 2 is p, K 2 = and K 22 is p p A = A A 2, A = A A 2 K2 K 22 T, A 2 = T where K 2 is p (n p) A2 where A, A 2, A 2, A 22 are respectively (n p) (n p), (n p) p, p (n p), p p. Further take Q = Q Q 2 A 22 where Q and Q 2 are (n p) n and p n respectively. Then we have the following equations (Lovass-Nagy et al. 9) where p = p p 2 dx dt = A x + A 2 x 2 (2) dx K 2 dt + K dx 22 dt A 2x A 22 x 2 = u (3) dp dt = AT p + K2R T du dt + AT 2Ru Q x (4) A T 2p = K22R T du dt + AT 22Ru Q 2 x (5) is the co-state vector corresponding to time invariant singular systems. From the equations (2)-(5), optimal state and optimal control can be calculated. The governing equations for determining u(t) and x(t) for the time-invariant and time-varying optimal control problem can be obtained using the set of equations (2) to (5), which have been mentioned earlier. It is to be noted that the above governing equations may not suit all types of time-invariant optimal control problems. Hence, it is necessary to investigate further to derive the governing equations exclusively (a generalized form) for the time-invariant optimal control problem of the form in 3. T
Numerical investigation of the time invariant optimal control 65 4 Formulation of time invariant optimal control of singular systems Rearranging Equations (2)-(5), we have the following system. K 2 K 22 K22 T K2 T x x 2 u p which can be written in the form = A A 2 A 2 A 22 Q T 2 Q 22 A 22 R Q Q 2 A T 2 A T 2 A T Ky (t) = My(t) x x 2 u p Where M = K = K 2 K 22 K22 T K2 T A A 2 A 2 A 22 Q T 2 Q 22 A 22 R Q Q 2 A T 2 A T 2 A T and y = x x 2 u p T where the matrix K is singular so that it is called as time-invariant singular systems and it can not be written in the standard form. 5 Example for the time invariant optimal control of singular systems The linear singular system 7, 9 is considered x with initial condition x() = The performance index x 2 = 2 x x 2 + u J = 2 tf (x T x + u 2 )dt
66 S. Sekar and A. Kavitha Let z = x. Then we obtain z z 2 = z z 2 + u (6) is obtained, since 2 (xt x+u 2 ) = 2 (zt z +u 2 ) minimization of L(z, u) = 2 (zt z + u 2 ) with respect to z and u will yield the same result as minimization of L(x, u) with respect to x and u. The exact solution of the system (6) is and the optimal control is z (t) = 2 exp( 2t) z 2 (t) = exp( 2t) (7) u(t) = 2 exp( 2t) (8) using ADM, RK method and equation (7) the discrete time solutions and the exact solutions of z(t) are calculated and is presented in Table and Table 2 along with the solution obtained by Balachandran and Murugesan 3 using Single Term Walsh Series technique. The corresponding optimal control u(t) is calculated by using ADM, RK method and equation (8) and the results are presented in Tables -2. Table : Discrete solutions for z (t) and z 2 (t) Exact Solution RK Solution ADM Solution t z (t) z 2 (t) z (t) z 2 (t) z (t) z 2 (t). -.63856.86823 -.63858.86825 -.63857.86824.2 -.53293.753638 -.53297.753642 -.53295.75364.3 -.462625.65425 -.46263.654257 -.462628.654254.4 -.466.56797 -.4624.567979 -.462.567975.5 -.348652.49369 -.348662.49379 -.348657.49374.6 -.32673.42844 -.32685.42856 -.32679.4285.7 -.262758.37595 -.262772.3769 -.262765.3762.8 -.2286.32259 -.22822.32267 -.2284.322599.9 -.9824.2849 -.9842.2867 -.9833.2858. -.799.2437 -.7929.24337 -.799.24327
Numerical investigation of the time invariant optimal control 67 Table 2: Discrete solutions for u(t) Exact Solution RK Solution ADM Solution t u(t) u(t) u(t)..63856.63858.63857.2.53293.53297.53295.3.462625.46263.462628.4.466.4624.462.5.348652.348662.348657.6.32673.32685.32679.7.262758.262772.262765.8.2286.22822.2284.9.9824.984.9833..799.7929.799 6 Conclusion The obtained results of the time-invariant optimal control of linear singular systems with quadratic performance index show that the ADM works well for finding the state vector and the control input vector. From the Tables 2, it can be observed that for most of the time intervals, the absolute error is less in ADM when compared to the classical fourth order RK method, which yields a little error, along with the exact solutions of the problem. Illustrative example has shown the validity and applicability of the proposed method. Acknowledgements. The authors gratefully acknowledge the Dr. A. Murugesan, Assistant Professor, Department of Mathematics, Government Arts College (Autonomous), Cherry Road, Salem - 636 7, for his encouragement and support. The authors also thank Dr. S. Mehar Banu, Assistant Professor, Department of Mathematics, Government Arts College for Women (Autonomous), Salem - 636 8, Tamil Nadu, India, for her kind help and suggestions. References G. Adomian, Solving Frontier Problems of Physics: Decomposition method, Kluwer, Boston, MA, (994). 2 G. Adomian, A Review of the Decomposition Method in Applied Mathematics, J. Math. Anal. Appl., 35 (988), 5-544.
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