SOLUTION OF NONLINEAR DELAY OPTIMAL

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 9, Number 5, September 2010

doi:10.3934/cpaa.2010.9.1379 pp. 1379–1389

SOLUTION OF NONLINEAR DELAY OPTIMAL CONTROL PROBLEMS USING A COMPOSITE PSEUDOSPECTRAL COLLOCATION METHOD

Hamid Reza Marzban Department of Mathematical Sciences, Isfahan University of Technology Isfahan, 8415683111, Iran

Hamid Reza Tabrizidooz Department of Mathematics, Faculty of Science, University of Kashan Kashan, 8731751167, Iran Abstract. We develop a composite collocation approximation scheme for the numerical solution of nonlinear delay optimal control problems. For this purpose, we present an extension and also modification for the Gauss pseudospectral method using the hybrid of block-pulse functions and Lagrange polynomials based on the Legendre-Gauss points. In this respect, we derive the corresponding operational matrix of derivative according to the weak representation of derivative operator. In order to demonstrate the applicability, efficiency and accuracy of the proposed method, we examine two illustrative examples.

1. Introduction and preliminaries. The control of systems with time-delay has been of considerable concern. Delays occur frequently in biological, chemical, electronic and transportation systems [9]. Time-delay systems are therefore a very important class of systems which their control and optimization have been of interest to many investigators. The application of Pontryagin’s maximum principle to the optimization of control systems with time-delays, as outlined by Kharatishvili [10], results in a system of coupled two point boundary value problem involving both delayed and advanced terms whose exact solution, except in very special cases, is very difficult. Therefore, the main object of all computational aspects of optimal time-delay systems has been to devise a methodology to avoid the solution of the mentioned coupled two point boundary value problem. Banks and Burns [3] considered a particular approximation scheme which can be employed to solve optimal control problems governed by linear functional differential equations. Their approximation scheme involves approximation of linear functional differential equations by systems of high-order ordinary differential equations using known results from linear semigroup theory. In [17], a method were used to solve a class of optimal control problems involving nonlinear hereditary systems with linear control constraints. Lee [12] considered an approximation scheme for solving timedelay optimal control problems with terminal inequality constraints. Elnagar and Kazemi [7] developed a pseudospectral approximation scheme for solving the class 2000 Mathematics Subject Classification. Primary: 49M25; Secondary: 65D25, 65M70. Key words and phrases. Delay optimal control problems, Lagrange polynomials, hybrid functions, pseudospectral methods.

1379

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HAMID REZA MARZBAN AND HAMID REZA TABRIZIDOOZ

of time-delayed functional differential equation control systems. Employing various approximation schemes, optimal time-delay systems were solved with different degrees of accuracy. The application of hybrid of block-pulse functions plus orthogonal polynomials to the discretization of linear delay systems has generated excellent results [14, 15]. Accordingly, the hybrid of block-pulse functions and Lagrange polynomials based on the Legendre-Gauss points were used to present the composite interpolation method for which the corresponding error of approximation is superior to that for the classical spectral methods [16]. In this paper, using the hybrid of blockpulse functions and Lagrange polynomials based on the Legendre-Gauss points, we present a composite collocation method for solving delay optimal control problems. In this respect, according to the weak representation of the derivative operator, we derive the corresponding operational matrix of derivative. The proposed method can be considered as an extension and also modification for the Gauss pseudospectral method introduced in [4, 8]. In order to demonstrate the applicability and efficiency of the method, we examine two nonlinear delay optimal control problems. Here we introduce the preliminaries of the composite interpolation method required for our subsequent development. The reader is referred to [5, 6] for details. Let t0 , t1 , . . . , tM be the Legendre-Gauss points, i.e. the zeros of the Legendre polynomial of degree M + 1, and w0 , w1 , . . . , wM be the corresponding quadrature weights. For m = 0, 1, . . . , M , let Lm (t) denote the Lagrange polynomial of degree M corresponding to the point tm , defined by Lm (t) =

M Y

i=0, i6=m



t − ti tm − ti



.

For the rest, we let N ≥ 1 and M ≥ 0 to be integers and also confine our attention to the functions defined on the interval [0, 1]. We let PN M denote the set of all functions whose restriction to each interval n−1 n ( N , N ), n = 1, 2, . . . , N , is a polynomial of degree at most M , and P C(N ) denote n the set of all functions whose restriction to each interval ( n−1 N , N ), n = 1, 2, . . . , N , n−1 n has a continuous extension to the interval [ N , N ]. According to these notations, it is readily verified that P C(N ) is a linear subspace of L2 (0, 1), the space of square Lebesgue integrable functions, and PN M is a linear subspace of P C(N ). The hybrid of block-pulse functions and Lagrange polynomials φnm , n = 1, 2, . . . , N, m = 0, 1, . . . , M , are defined on the interval [0, 1) as  n Lm (2N t − 2n + 1) if n−1 N ≤ t< N , φnm (t) = 0 otherwise where n and m are the orders of the block-pulse function and the Lagrange polynomial, respectively. It can be checked that the set of hybrid functions {φnm : n = 1, 2, . . . , N, m = 0, 1, . . . , M } forms an orthogonal basis for PN M. In order to have our fundamental definition, for n = 1, 2, . . . , N and m = 0, 1, . . . , M , we set 1 tnm = (tm + 2n − 1), 2N n i.e. the corresponding point of tm in the interval [ n−1 N , N ] by means of the natural linear transformation.

SOLUTION OF NONLINEAR DELAY OPTIMAL

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Definition 1.1. Let v ∈ P C(N ), then the composite interpolation formula of v N denoted by IM (v) is a function in PN M defined by N IM (v)(t) =

N X M X

v(tnm ) φnm (t).

n=1 m=0 N According to this, IM (v)(tnm ) = v(tnm ), n = 1, 2, . . . , N , m = 0, 1, . . . , M . Note that in the case N = 1, the composite interpolation formula coincides with the transformed classical interpolation formula based on the Legendre-Gauss points.

2. Function approximation. In this section, we explain our method for approximating a function, its integration and its derivation. We also present the operational matrix of derivative based on the weak representation of the derivative operator. For a function v ∈ P C(N ), we approximate v by the composite interpolation N formula IM (v), as N v(t) ≃ IM (v)(t).

(1)

Accordingly, we approximate the integration of the function v over the interval N [0, 1] by the integration of IM (v) over the interval [0, 1]. Then, using the LegendreGauss quadrature formula, Z 1 Z 1 N M 1 XX N v(t) dt ≃ IM (v)(t) dt = v(tnm ) wm . (2) 2N n=1 m=0 0 0 N We also approximate the derivative of v by the derivative of IM (v), i.e.

d N d v(t) ≃ I (v)(t). dt dt M

(3)

For the sequence, we let N IM (v)(t)

=

N X M X

v(tnm ) φnm (t) = Φ(t)T V

(4)

n=1 m=0

and d N I (v)(t) dt M where V

=

and S



=

N X M X

snm φnm (t) = Φ(t)T S,

(5)

n=1 m=0

T |v(t10 ), . . . , v(t1M )|, . . . , |v(tN 0 ), . . . , v(tN M )|

=



T |s10 , . . . , s1M |, . . . , |sN 0 , . . . , sN M |

are the vectors of coefficients and  T Φ(t) = |φ10 (t), . . . , φ1M (t)|, . . . , |φN 0 (t), . . . , φN M (t)|

is the vector of hybrid functions. The relation between N (M + 1) × 1 vector V introduced in Eq. (4) and N (M + 1) × 1 vector S introduced in Eq. (5) is expressed by S = D V, where N (M + 1) × N (M + 1) matrix D is the operational matrix of derivative.

(6)

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HAMID REZA MARZBAN AND HAMID REZA TABRIZIDOOZ

In order to construct the operational matrix of derivative D, we use the approach of weak representation of the derivative operator in which the derivative operator d n−1 n dt act on each interval ( N , N ) in the weak sense [1]. The advantage of using this approach is that the information in the neighboring subintervals are simultaneously used and thus additional linking constraints (e.g. continuity) are not needed. In Eq. (5), by multiplying φnm (t), n = 1, 2, . . . , N , m = 0, 1, . . . , M , and then integrating over the interval [0, 1], we can get snm , n = 1, 2, . . . , N , m = 0, 1, . . . , M , as Z Nn d N 2N snm = IM (v)(t) φnm (t) dt. wm n−1 dt N Then, using integration by parts, we obtain !   X M n n − 1 2N N N ) Lm (−1) − qmm′ v(tnm′ ) , snm = IM (v)( ) Lm (1) − IM (v)( wm N N m′ =0 (7) where using Legendre-Gauss quadrature formula, Z 1 d d ′ qmm = Lm′ (t) Lm (t) dt = wm′ Lm (tm′ ). dt dt −1 N N (v)(1), using Eq. (4), we have For evaluating IM (v)(0) and IM N IM (v)(0) =

M X

v(t1m′ ) Lm′ (−1)

(8)

M X

v(tN m′ ) Lm′ (1).

(9)

m′ =0

and N IM (v)(1)

=

m′ =0 n N Also in the case N ≥ 2, for evaluating IM (v)( N ), n = 1, 2, . . . , N − 1, we use N IM (v)(

M M n 1 X 1 X )= v(tnm′ ) Lm′ (1) + v(t(n+1)m′ ) Lm′ (−1). N 2 ′ 2 ′ m =0

(10)

m =0

Substituting Eqs. (8)–(10) in Eq. (7), we can express the operational matrix of derivative D as follows: In the case N = 1, D = 2 R,

(11)

in the case N = 2, D=4



R1 Q

H R3



,

and in the case N ≥ 3, 

R1  Q   D = 2N   

H R2 .. .

H .. . Q

..

. R2 Q



   ,  H  R3

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where, for m, m′ = 0, 1, . . . , M , we set  1  [R]mm′ = Lm (1) Lm′ (1) − Lm (−1) Lm′ (−1) − qmm′ , wm  1 1 Lm (1) Lm′ (1) − Lm (−1) Lm′ (−1) − qmm′ , [R1 ]mm′ = wm 2  1 1 1 [R2 ]mm′ = Lm (1) Lm′ (1) − Lm (−1) Lm′ (−1) − qmm′ , wm 2 2  1  1 Lm (1) Lm′ (1) − Lm (−1) Lm′ (−1) − qmm′ , [R3 ]mm′ = wm 2  1 1 [H]mm′ = Lm (1) Lm′ (−1) , wm 2  1  1 − Lm (−1) Lm′ (1) . [Q]mm′ = wm 2 As it is seen, for the case N ≥ 3, the matrix D is a block tridiagonal matrix. For the case N = 1, it is verified that the operational matrix of derivative given in Eq. (11) coincides with the transformed operational matrix of derivative obtained in the Gauss pseudospectral method. 3. Approximation errors. In this section, we discuss some error estimations concerning the approximations used in the previous section. We state these results in terms of the Sobolev norms. For this purpose, we introduce some notations. For integer s ≥ 0, we let P H s (N ) denote the set of all functions whose restriction n s n−1 n to each interval ( n−1 N , N ), n = 1, 2, . . . , N , is in H ( N , N ), the Sobolev space of n−1 n order s on the interval ( N , N ). Here by Sobolev norm of a function v of integer order s on the interval (a, b), we mean !1/2 !1/2 s Z b s X X (k) 2 (k) 2 ||v||H s (a,b) = |v (t)| dt = ||v ||L2 (a,b) , a

k=0

k=0

(k)

where v denotes the distributional derivative of v of order k. According to the notation, it is readily verified that P H s (N ) is a linear subspaces of L2 (0, 1) and also, in the case s ≥ 1, is contained in P C(N ). Theorem 3.1. [16] Suppose v ∈ P H s (N ) with s ≥ 1. Then N ||v − IM (v)||L2 (0,1) ≤ C M −s |v|H 0;s;M ;N (0,1) ,

(12)

and for 1 ≤ r ≤ s, 1

N ||v − IM (v)||H r (0,1) ≤ C M 2r− 2 −s |v|H r;s;M ;N (0,1) .

(13)

Here we set 

|v|H r;s;M ;N (0,1) = 

s X

N 2r−2k

N X

n=1

k=min(s,M+1)

1/2

||vn(k) ||2L2 ( n−1 , n )  N

N

,

n where vn , n = 1, 2, . . . , N , denote the restriction of v to the interval ( n−1 N , N ), n = 1, 2, . . . , N , and also !1/2 N M X X N 2 ||v − IM (v)||H r (0,1) = ||vn − v(tnm )φnm ||H r ( n−1 , n ) , N

n=1

m=0

N

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HAMID REZA MARZBAN AND HAMID REZA TABRIZIDOOZ

and C denotes a positive constant that depends upon the type of the norms, but which is independent of the function v and the integers N and M . Corollary 1. By setting M ≥ s − 1 in Eqs. (12) and (13), we get !1/2 N X N −s −s (s) 2 ||v − IM (v)||L2 (0,1) ≤ C M N ||vn ||L2 ( n−1 , n ) , N

(14)

N

n=1

and for 1 ≤ r ≤ s,

N ||v − IM (v)||H r (0,1) ≤ C M

2r− 21 −s

N r−s

N X

||vn(s) ||2L2 ( n−1 , n )

n=1

N

N

!1/2

.

(15)

Note that for v ∈ H s (0, 1) with s ≥ 0, it is checked that !1/2 N X (s) 2 = ||v (s) ||L2 (0,1) , ||vn ||L2 ( n−1 , n ) N

N

n=1

therefore in the case that v is infinitely smooth, i.e. v ∈ H s (0, 1) for every s ≥ 0, N Eqs. (12)–(15) show that the rate of convergence of IM (v) to the function v is faster 1 1 than N to the power of M + 1 − r and any power of M . 4. The problem and its discretization. In this section, we consider a general nonlinear delay optimal control problem defined on the interval [0, 1] and explain our method to discretize the problem into a nonlinear programming problem. Consider the nonlinear time-varying delay system

l

˙ x(t)

= f (x(t), x(t − td ), u(t), t) ,

x(t)

= Ψ(t),

−td ≤ t ≤ 0,

l

0 ≤ t ≤ 1,

(16) (17)

q

where x(t) ∈ R , Ψ(t) ∈ R , u(t) ∈ R . Moreover Ψ(t) is an arbitrary known function and td is the delay time. The problem is to find the optimal control u(t) and the corresponding state trajectory x(t), 0 ≤ t ≤ 1, satisfying Eqs. (16) and (17), while minimizing the functional Z 1 J = h (x(1)) + g (x(t), u(t), t) dt. (18) 0

The vector function f and the scalar functions h and g are generally nonlinear, and assumed to be continuously differentiable with respect to their arguments. It is assumed the problem (16)–(18) has a unique solution. Let  T x(t) = x1 (t), x2 (t), . . . , xl (t) ,  T u(t) = u1 (t), u2 (t), . . . , uq (t) .

Using Eqs. (1) and (4), we expand each of xi (t) and uj (t), i = 1, 2, . . . , l, j = 1, 2, . . . , q, in terms of hybrid functions as xi (t)

≃ ΦT (t) Xi ,

uj (t)

≃ ΦT (t) Uj ,

where Xi and Uj , respectively, consist the values of xi (t) and uj (t) at the points tnm , n = 1, 2, . . . , N , m = 0, 1, . . . , M .

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For a compact notation form, we set ˆ Φ(t) ˆ 1 (t) Φ

= Il ⊗ Φ(t), = Iq ⊗ Φ(t),

where Il and Iq are the l- and q-dimensional identity matrices and ⊗ denotes the Kronecker product [11]. Then, we can write x(t)

≃

u(t)

≃

ˆ T (t) X, Φ ˆ T (t) U, Φ 1

(19) (20)

where X and U are vectors of order lN (M + 1) × 1 and qN (M + 1) × 1, respectively, given by  T X = X1T , X2T , . . . , XlT ,  T U1T , U2T , . . . , UqT . U = Substituting Eqs. (19)–(20) in Eq. (18) and using Eq. (2), we obtain

where

N X M     X ˆ T (1)X + 1 J ≃h Φ g x(tnm ), u(tnm ), tnm wm , 2N n=1 m=0

x(tnm ) = u(tnm ) =

(21)

 T ˆ T (tnm ) X, x1 (tnm ), x2 (tnm ), . . . , xl (tnm ) =Φ  T ˆ T1 (tnm ) U. u1 (tnm ), u2 (tnm ), . . . , uq (tnm ) =Φ

To discretize the system dynamics, we consider the points tnm , n = 1, 2, . . . , N , m = 0, 1, . . . , M , as the collocation points and substitute them in Eq. (16). Then using Eqs. (3) and (5), we get N M X X

n′ =1 m′ =0

  Dn:m , n′ :m′ x(tn′ m′ ) − f x(tnm ), x(tnm − td ), u(tnm ), tnm = 0,

n = 1, 2, . . . , N,

m = 0, 1, . . . , M,

(22)

where n : m = (n − 1)M + m + 1 and n′ : m′ = (n′ − 1)M + m′ + 1 and Dn:m , n′ :m′ stands for (n : m , n′ : m′ )-th component of the matrix D introduced in Eq. (6). Also, by setting t = 0 in Eq. (19) and using Eq. (17), we obtain ˆ T (0)X − Ψ(0) = 0. Φ

(23)

The benefit of the proposed method appears when we describe the expression L x(tnm − td ), where td = K (L and K are positive integers which L < K and their greatest common divisor is 1), in terms of the elements of X. In this case, by choosing N = KN0 (N0 is a positive integer), we obtain  Ψ(tnm − td ) n = 1, . . . , LN0 x(tnm − td ) = . x(t(n−LN0 )m ) n = LN0 + 1, . . . , KN0 L Note that by setting N = KN0 , for n = 1, . . . , LN0 , we have tnm −td = tnm − K