arX

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0293

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Oct

201

7

A Laguerre homotopy method for optimal control of nonlinear

systems in semi-infinite interval

Haijun Yu 1,2, Hassan Saberi Nik †

1 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China2NCMIS & LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing,

Academy of Mathematics and Systems Science, Beijing 100190, China

email: hyu@lsec.cc.ac.cn†Department of Mathematics, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran

email: saberi hssn@yahoo.com

July 20, 2018

Abstract

This paper presents a Laguerre homotopy method for optimal control problems in semi-infiniteintervals (LaHOC), with particular interests given to nonlinear interconnected large-scale dynamicsystems. In LaHOC, spectral homotopy analysis method is used to derive an iterative solver forthe nonlinear two-point boundary value problem derived from Pontryagins maximum principle.A proof of local convergence of the LaHOC is provided. Numerical comparisons are made be-tween the LaHOC, Matlab BVP5C generated results and results from literature for two nonlinearoptimal control problems. The results show that LaHOC is superior in both accuracy and efficiency.

Keywords: Laguerre method; collocation method; optimal control problems; spectral homotopyanalysis method; semi-infinite interval.

1 Introduction

Large-scale systems are found in many practical applications, such as power systems and physicalplants. During the past several years, the problem of analysis and synthesis for dynamic large-scalesystems has received considerable attention. Based on the characteristics of large-scale systemsmany results have been proposed, such as modelling, stability, robust control, decentralized, andso on [1, 2, 3, 4, 5, 6].

The optimal control problem (OCP) of nonlinear large-scale systems has been widely investi-gated in recent decades. For instance, a new successive approximation approach (SAA) was pro-posed in [7]. In this approach, instead of directly solving the nonlinear large-scale two-point bound-ary value problem (TPBVP), derived from the maximum principle, a sequence of non-homogeneouslinear time-varying TPBVPs is solved iteratively. Also, in [9] a new technique, called the modalseries method, has been has been extended to solve a class of infinite horizon OCPs of nonlinearinterconnected large-scale dynamic systems, where the cost function is assumed to be quadratic

1

http://arxiv.org/abs/1710.02935v1

and decoupled. This method provides the solution of autonomous nonlinear systems in terms offundamental and interacting modes. Conventional methods of optimal control are generally im-practical for many nonlinear large-scale systems because of the dimensionality problem and highcomplexity in calculations. One example is the state-dependent Riccati equation (SDRE) method[8]. Although this scheme has been widely used in many applications, its major limitation is thatit needs to solve a sequence of matrix Riccati algebraic equations at each sample state along thetrajectory. This property may take a long computing time and large memory space. Therefore,developing new methods is necessary for solving nonlinear large-scale optimal control problems [10].

The use of spectral methods for optimal control problem usually leads to more efficient methodthan finite element or finite different approaches. Chebyshev and Legendre method are commonlyused for problems in finite intervals [11, 12]. For infinite or semi-infinite intervals, there are severalchoices for the approximation bases: Hermite polynomials/functions [13], Laguerre polynomials/functions [43], mapped Jacobi bases [14, 15, 16]. Furthormore, one class of very important appli-cations of OCP in unbounded intervals is the minimum action method (MAM) [17] used in findingthe most probable transition path in phase transition phenomena. Using MAM to study spatialextended transitions, such as fluid instability transition is usually equivalent to solve a large-scalednonlinear optimal control problem [18, 19].

The homotopy analysis method is an analytical technique for solving nonlinear differentialequations. The HAM [20, 21] was first proposed by Liao in 1992 to solve lots of nonlinear problems.This method has been successfully applied to many nonlinear problems, such as physical modelswith an infinite number of singularities [22], nonlinear eigenvalue problems [23], fractional Sturm-Liouville problems [24], optimal control problems [25, 26], Cahn-Hilliard initial value problem [27],semi-linear elliptic boundary value problems [28] and so on [29]. The HAM contains a certainauxiliary parameter ~ which provides us with a simple way to adjust and control the convergenceregion and rate of convergence of the series solution. Moreover, by means of the so-called ~-curve, itis easy to determine the valid regions of ~ to gain a convergent series solution. The HAM howeversuffers from a number of restrictive measures, such as the requirement that the solution soughtought to conform to the so-called rule of solution expression and the rule of coefficient ergodicity.These HAM requirements are meant to ensure that the implementation of the method results in aseries of differential equations which can be solved analytically.

Recently, Motsa et al. [30, 31, 32] proposed a spectral modification of the homotopy analysismethod, the spectral-homotopy analysis method (SHAM). The SHAM approach imports some ofthe ideas of the HAM such as the use of the convergence controlling auxiliary parameter. Inthe implementation of the SHAM, the sequence of the so-called deformation differential equationsare converted into a matrix system by applying Chebyshev or Legendre pseudospectral method[32]. But so far, to our knowledge, there is no work concerning the combination of Laguerrepolynomails [33] with the HAM. This paper presents a spectral homotopy analysis method basedon modified Laguerre-Radau interpolation to solve nonlinear large-scale optimal control problems.This process has several advantages. First, it possesses the spectral accuracy [34, 35]. Next, itis easier to be implemented, especially for nonlinear systems. Furthermore, it is applicable tolong-time calculations.

The paper is organized as follows. The nonlinear interconnected OCP and optimality conditionsis described in section 2. In Section 3, we propose the new algorithm by using the modified Laguerrepolynomials. The convergence of the proposed method is proved in section 4. We present numericalresults in Section 5, which demonstrate the spectral accuracy of proposed methods. The final section

2

is for concluding remarks.

2 The nonlinear interconnected OCP

Consider a nonlinear interconnected large-scale dynamic system which can be decomposed into Ninterconnected subsystems. The ith subsystem for i = 1, 2, · · · , N is described by:

ẋi(t) = Aixi(t) +Biui(t) + fi(x(t)), t > t0,xi(t0) = xi0 ,

(2.1)

with xi ∈ Rni denoting the state vector, ui ∈ R

mi the control vector of the ith subsystem,

respectively, x = (xT1 , xT2 , · · · x

TN )

T ,N∑

i=1

ni = n, Fi : Rn → Rni is a nonlinear analytic vector function

where Fi(0) = 0, and xi0 ∈ Rni is the initial state vector. Also, Ai and Bi are constant matrices of

appropriate dimensions such that the pair (Ai, Bi) is completely controllable [9]. Furthermore, theinfinite horizon quadratic cost function to be minimized is given by:

J =1

2

K∑

i=1

{∫ ∞

t0

(xTi (t)Qxi(t) + uTi (t)Riui(t))dt

}

(2.2)

where Qi ∈ Rni×ni and Ri ∈ R

mi×mi are positive semidefinite and positive definite matrices, respec-tively. Note that the quadratic cost function (2.2) is assumed to be decoupled as a superpositionof the cost functions of the subsystems.

According to Pontryagin’s maximum principle, the optimality conditions are obtained as thefollowing nonlinear TPBVP:

ẋi(t) = Aixi(t)−BiR−1i B

Ti λi(t) + fi(x(t)), t > t0,

λ̇i(t) = −Qixi(t)−ATi λi(t)−Ψi(x(t), λ(t)), t > t0,

xi(t0) = xi0 , λi(∞) = 0,i = 1, 2, · · · ,K,

(2.3)

where λi(t) ∈ Rni is the co-state vector, λ = (λT1 , λ

T2 , · · · λ

TK)

T , and

Ψi(x(t), λ(t)) =

K∑

j=1

∂fj(x(t))

∂xi(t)λj(t). Also the optimal control law of the ith subsystem is given by

u∗i (t) = −R−1i B

Ti λi(t), t > t0, i = 1, 2, · · ·K. (2.4)

Unfortunately, problem (2.3) is a nonlinear largescale TPBVP which is decomposed into Ninterconnected subproblems. In general, it is extremely difficult to solve this problem analyticallyor even numerically, except in a few simple cases. In order to overcome this difficulty, we willpresented the LaHOC method in the next section.

3 Laguerre polynomials and spectral homotopy analysis method

In this section, we give a brief description of the basic idea of the Laguerre homotopy method forsolving nonlinear boundary value problems. At first, we take into account the following propertiesof the modified Laguerre polynomials.

3

3.1 Properties of the modified Laguerre polynomials

Let ωβ(t) = e−βt, β > 0, and define the weighted space L2ωβ(0,∞) as usual, with the following inner

product and norm, [36]:

(u, v)ωβ =

∫ ∞

0u(t)v(t)ωβ(t)dt, ||v||ωβ = (v, v)ωβ (3.1)

The modified Laguerre polynomial of degree l is defined by :

Lβl (t) =1

l!eβt

dl

dtl(tle−βt), l ≥ 0. (3.2)

They satisfy the recurrence relation

d

dtLβl (t) =

d

dtLβl−1(t)− βL

βl−1(t), l ≥ 1. (3.3)

The set of Laguerre polynomials is a complete L2ωβ (0,∞)-orthogonal system, namely,

(Lβl ,Lβm)ωβ =

1

βδl,m, (3.4)

where δl,m is the Kronecker symbol. Thus, for any v ∈ L2ωβ(0,∞),

v(t) =

∞∑

j=0

v̂lLβl (t), (3.5)

where the coefficients v̂l are given by

v̂l = β(v,Lβl )ωβ . (3.6)

Now, let N be any positive integer, and PN (0,∞) the set of all algebraic polynomials of degreeat most N . We denote by tNβ,j, 0 ≤ j ≤ N the nodes of modified Laguerre-Radau interpolation.

Indeed, tNβ,0 = 0 and tNβ,j, 1 ≤ j ≤ N are the distinct zeros of

ddtLβN+1(t) By using (3.3), the

corresponding Christoffel numbers are as follows:

ωNβ,0 =1

β(N + 1), ωNβ,j =

1

β(N + 1)LβN (tNβ,j) L

βN+1(t

Nβ,j)

, (3.7)

For any Φ ∈ P2N (0,∞),N∑

j=0

Φ(tNβ,j)ωNβ,j =

∫ ∞

0Φ(t)ωβ(t)dt. (3.8)

Next, we define the following discrete inner product and norm,

(u, v)ωβ ,N =

N∑

j=0

u(tNβ,j)v(tNβ,j)ω

Nβ,j, ||v||ωβ ,N = (v, v)

12ωβ ,N

. (3.9)

For any Φ, ψ ∈ PN (0,∞),

(Φ, ψ)ωβ = (Φ, ψ)ωβ ,N , ||v||ωβ = ||v||ωβ ,N . (3.10)

4

3.2 Spectral homotopy analysis method

In this section, we give a description of the SHAM with the Laguerre polynomials basis. This willbe followed by a description of the new version of the SHAM algorithm [30]. To this end, considera general n dimensional initial value problem described as

ż(t) = f(t, z(t)), z(t0) = z0, (3.11)

z : R → Rn, f : R× Rn → Rn (3.12)

We make the usual assumption that f is sufficiently smooth for linearization techniques to be valid.If z = (z1, z2, . . . , zn) we can apply the SHAM by rewriting equation (3.11) as

żr +

n∑

k=1

σr,kzk + gr(z1, z2, . . . , zn) = 0, (3.13)

subject to the initial conditionszr(0) = z

0r . (3.14)

where z0r are the given initial conditions, σr,k are known constant parameters and gr is the nonlinearcomponent of the rth equation.

The SHAM approach imports the conventional ideas of the standard homotopy analysis methodby defining the following zeroth-order deformation equations

(1− q)Lr [z̃r(t; q)− zr,0(t)] = q~rNr[z̃(t; q)], (3.15)

where q ∈ [0, 1] is an embedding parameter, z̃r(t; q) are unknown functions, ~r is a convergencecontrolling parameter. The operators Lr and Nr are defined as

Lr[z̃r(t; q)] =∂z̃r∂t

+

n∑

k=1

σr,kz̃k, (3.16)

Nr[z̃(t; q)] = Lr[z̃r(t; q)] + gr[z̃1(t; q), z̃2(t; q), . . . , z̃n(t; q)]. (3.17)

Using the ideas of the standard HAM approach [21], we differentiate the zeroth-order equations(3.15) m times with respect to q and then set q = 0 and finally divide the resulting equationsby m! to obtain the following equations, which are referred to as the mth order (or higher order)deformation equations,

Lr[zr,m(t)− χmzr,m−1(t)] = ~rRr,m−1, m ≥ 1, (3.18)

subject tozr,m(0) = 0, (3.19)

where

Rr,m−1 =1

(m− 1)!

∂m−1Nr[z̃(t; q)]

∂qm−1

∣

∣

∣

∣

q=0

, (3.20)

and

χm =

{

0, m 6 1,1, m > 1.

(3.21)

5

After obtaining solutions for equations (3.18), the approximate solution for each zr(t) is deter-mined as the series solution

zr(t) = zr,0(t) + zr,1(t) + zr,2(t) + . . . (3.22)

A HAM solution is said to be of order M if the above series is truncated at m =M , that is, if

zr(t) =M∑

m=0

zr,m(t). (3.23)

A suitable initial guess to start off the SHAM algorithm is obtained by solving the linear part of(3.13) subject to the given initial conditions, that is, we solve

Lr[zr,0(t)] = φr(t), zr,0(0) = z0r . (3.24)

If equation (3.24) cannot be solved exactly, the spectral collocation method is used as a meansof solution. The solution zr,0(t) of equation (3.24) is then fed to (3.18) which is iteratively solvedfor zr,m(t) (for m = 1, 2, 3 . . . ,M).

In this paper, we use the Laguerre pseudo-spectral method to solve equations (3.18-3.20). Thepseudo-spectral derivative DN (z) of a continuous function z is defined by:

DN (z) = D[IN (z)], (3.25)

that is, DN (z) is the derivative of the interpolating polynomial of z.Moreover, DN can be expressedin terms of a matrix, the pseudo-spectral derivation matrix Dβ :

Dβ = [(dβ)ij ]i,j=0,1,··· ,N .

Indeed, given the nodes {x(β)j }

Nj=0, an approximation z ∈ P

(β)N of an unknown function and {(hβ)j},

the Lagrange interpolation polynomials associated to the points xj , differentiating m times theexpression

zβ(x) =

N∑

j=0

zβ(xj)(hβ)j(x),

yields:

z(m)β (xk) =

N∑

j=0

(hβ)(m)j (xk)zβ(xj), 0 ≤ k ≤ N.

If we define:

z(m)β =

(

z(m)β (x0), z

(m)β (x1), · · · , z

(m)β (xN )

)T

, zβ = z(0)β ,

D(m)β =

[

(dβ)(m)ij = (hβ)

(m)j (xi)

]

0≤i,j≤N,

(dβ)(m)ij = (hβ)

(m)j (xi),

then:Dβ = D

(1)β , (dβ)ij = (dβ)

(1)ij .

We now state two important results. The first ensures that it is sufficient to compute the first orderdifferentiation matrix, the second gives the general expression of its entries.

6

Lemma 3.1 [33]

D(m)β = Dβ .Dβ · · ·Dβ = D

mβ , m ≥ 1. (3.26)

Let {x(β)j }

Nj=0 be the Gauss-Laguerre (GL) or Gauss-Laguerre-Radau (GLR) nodes and z ∈ P

(β)N .

Let {(hβ)j(x)}Nj=0 be the Lagrange interpolation polynomials relative to {x

(β)j }

Nj=0. From Lemma

3.1, we have:

z(m)β = D

mβ zβ, m ≥ 1.

Next we have:

Lemma 3.2 [33] The entries of the differentiation matrix Dβ associated to the GL and GLR points

{x(β)j }

Nj=0 have the following form:

• GL points: {x(β)j }

Nj=0 are the zeros of L

(β)N+1(x),

dij =

L(β)N

(

x(β)i

)

(

x(β)i −x

(β)j

)

L(β)N

(

x(β)j

) if i 6= j,

βx(β)i −N−2

2x(β)i

if i = j,

(3.27)

• GLR points: x0 = 0, {x(β)j }

Nj=1 are the zeros of

∂∂x

L(β)N+1(x),

dij =

L(β)N+1

(

x(β)i

)

(

x(β)i −x

(β)j

)

L(β)N+1

(

x(β)j

) if i 6= j,

β2 if i = j 6= 0,

−βN2 if i = j = 0,

(3.28)

Applying the the Laguerre spectral collocation method in equations (3.18-3.20) gives

A [Wm − χmWm−1] = ~rRm−1, Wm(τ0) = 0, Wm(τN ) = 0, (3.29)

where Rm−1 is an (N + 1)n× 1 vector corresponding to Rr,m−1 when evaluated at the collocationpoints and Wm = [z̃1,m; z̃2,m; . . . ; z̃n,m].

The matrix A is an (N + 1)n × (N + 1)n matrix that is derived from transforming the linearoperator Lr using the derivative matrix Dβ (we omit subscipt β for simplicity) and is defined as

A =

A11 A12 · · · A1nA21 A22 · · · A2n...

. . ....

An1 An2 · · · Ann

, with Apq =

{

D+ σpqI, p = q,σpqI, p 6= q,

(3.30)

where I is an identity matrix of order N + 1.Thus, starting from the initial approximation, the recurrence formula (3.29) can be used to

obtain the solution zr(t).

7

4 Convergence analysis of LaHOC

To analysis the convergence of LaHOC, we first recall the mth order (or higher order) deformationequation,

L[zm(t)− χmzm−1(t)] = ~H(t)Rm−1, (4.1)

subject to the initial conditionzm,1:n(t0) = 0, (4.2)

where H(t) 6= 0 is an auxiliary function,

Rm−1 = L[zm−1] +Nm−1[z0, z1, · · · , zm−1]− (1− χm)φ(t). (4.3)

where zr,m, Lr andNr in (3.18) are the rth components of zm−1 and operators L andN , respectively.Let us define the nonlinear operator N and the sequence {Zm}

∞m=0 as,

N [z(t)] =∞∑

k=0

Nk(z0, z1, · · · , zk), (4.4)

Z0 = z0,Z1 = z0 + z1,...Zm = z0 + z1 + z2 + · · ·+ zm.

(4.5)

Therefore, we have

L[zm(t)] = ~H(t){m−1∑

k=0

L[zk] +m−1∑

k=0

Nk − φ(t)}, (4.6)

from (4.5) we have

L[Zm(t)− Zm−1(t)] = ~H(t){L[Zm−1] +N [Zm−1]− φ(t)}, (4.7)

subject to the initial condition

Zm,1:n(t0) = 0. (4.8)

Consequently, the collocation method is based on a solution ZN (t) ∈ PN+1(0,∞), for (4.7) suchthat

L[ZNm (tNβ,k)− Z

Nm−1(t

Nβ,k)] = ~H

N (tNβ,k){L[ZNm−1(t

Nβ,k)] +N [Z

Nm−1(t

Nβ,k)]− φ

N (tNβ,k)}, (4.9)

subject to the initial condition

ZNm,1:n(t0) = 0. (4.10)

From (4.9) we have

L[ZNm (tNβ,k)] = (1 + ~H

N (tNβ,k))L[ZNm−1(t

Nβ,k)] + ~H

N (tNβ,k){N [ZNm−1(t

Nβ,k)]− φ

N (tNβ,k)},

0 ≤ k ≤ N, m ≥ 1, (4.11)

ZNm,1:n(t0) = 0,m ≥ 0.

8

Now, we choose L[Z(t)] = ddtZ +α(t)Z, N [Z(t)] = −α(t)Z − f(t, Z) and φ(t) ≡ 0 where α(t) is anarbitrary analytic function.Let Z̃Nm (t) = Z

Nm (t)− Z

Nm−1(t), then we have from (4.11) that

L[Z̃Nm (tNβ,k)] = (1 + ~H(t

Nβ,k))L[Z

Nm−1(t

Nβ,k)− Z

Nm−2(t

Nβ,k)] + ~H(t

NT,k)

{N [ZNm−1(tNβ,k)]−N [Z

Nm−2(t

Nβ,k)]},

0 ≤ k ≤ N,m ≥ 1, (4.12)

or according to the definitions of L[Z(t)] and N [Z(t)],

d

dt[Z̃Nm (t

Nβ,k)] + α(t

Nβ,k)Z̃

Nm = (1 + ~H(t

Nβ,k))

d

dt[Z̃Nm−1(t

Nβ,k)] + α(t

Nβ,k)Z̃

Nm−1

− ~H(tNβ,k){f(tNβ,k, Z

Nm−1(t

Nβ,k))− f(t

Nβ,k, Z

Nm−2(t

Nβ,k))},

0 ≤ k ≤ N,m ≥ 1, (4.13)

Theorem 4.1 Assume that for any k = 0, 1, ..., N,Zk = {ZNm (t

Nβ,k)}

∞0 is the LaHOC sequence

produced by (4.11). Furthermore, assume α0 = min t∈[0,∞) α(t), α1 = max t∈[0,∞) |α(t)| and H =max t∈[0,∞) |H(t)| and

||f(., ZNm )− f(., ZNm−1)||ωβ ,N ≤ Lf ||Z

Nm − Z

Nm−1||ωβ ,N . (4.14)

for some constant Lf > 0. Then for any initial n-vector ZN0 (t

Nβ,k), Zk converges to some Ẑ(t

Nβ,k)

which is the exact solution of (3.13), at any GLR point, tNβ,k, if

γ =N |1 + ~H|+ α1 + |~|HLf

β/2 + α0< 1. (4.15)

Proof. 1. Using (3.1) and integrating by parts yield that

(

Z̃Nm ,d

dtZ̃Nm

)

ωβ ,N

=

(

Z̃Nm ,d

dtZ̃Nm

)

ωβ

=1

2

[

e−βt(Z̃Nm )2 |∞0 +

∫ ∞

0βe−βt(Z̃Nm )

2dt

]

, (4.16)

then, we have

2

(

Z̃Nm ,d

dtZ̃Nm

)

ωβ ,N

= β‖Z̃Nm‖2ωβ, ‖Z̃Nm‖ωβ ,N = ‖Z̃

Nm‖ωβ , (4.17)

by (4.17) and from the Cauchy inequality we obtain that

β‖Z̃Nm‖2ωβ

≤ 2‖Z̃Nm‖ωβ ,N‖d

dt(Z̃Nm )‖ωβ ,N , (4.18)

from where

‖Z̃Nm‖ωβ ≤2

β‖d

dt(Z̃Nm )‖ωβ , (4.19)

9

2. Taking discrete weighted inner product of (4.13) with Z̃Nm (tNβ,k), we have

(

d

dtZ̃Nm + α(t)Z̃

Nm , Z̃

Nm

)

ωβ ,N

=

(

(1 + ~H)d

dtZ̃Nm−1 + α(t)Z̃

Nm−1, Z̃

Nm

)

ωβ ,N

− ~(

H(t)[f(tNβ,k, ZNm−1 − f(t

Nβ,k, Z

Nm−2)], Z̃

Nm

)

ωβ ,N

0 ≤ k ≤ N,m ≥ 1, (4.20)

Therefore, a combination with Cauchy inequality and (4.17) leads to

(β

2+ α0)‖Z̃

Nm‖ωβ ≤ |1 + ~H|‖

d

dtZ̃Nm−1‖ωβ + α1||Z̃

Nm−1||ωβ

+|~|H‖f(tNβ,k, ZNm−1 − f(t

Nβ,k, Z

Nm−2)‖ωβ ,N (4.21)

Then by using inverse inequality of Laguerre polynomial and (4.14), we get

(β

2+ α0)‖Z̃

Nm‖ωβ ≤ (N |1 + ~H|+ α1 + |~|HLf )‖Z̃

Nm−1‖ωβ , (4.22)

which is∥

∥

∥Z̃Nm

∥

∥

∥

ωβ≤N |1 + ~H|+ α1 + |~|HLf

β/2 + α0

∥

∥

∥Z̃Nm−1

∥

∥

∥

ωβ= γ

∥

∥

∥Z̃Nm−1

∥

∥

∥

ωβ. (4.23)

Hence, we have∥

∥

∥Z̃Nm

∥

∥

∥

ωβ≤ γ

∥

∥

∥Z̃Nm−1

∥

∥

∥

ωβ≤ · · · ≤ γm

∥

∥

∥Z̃N0

∥

∥

∥

ωβ. (4.24)

Then for any m′ ≥ m ≥ 1,

∥

∥ZNm′ − ZNm

∥

∥

ωβ≤

m′∑

i=m+1

∥

∥

∥Z̃Ni

∥

∥

∥

ωβ≤

m′∑

i=m+1

γi∥

∥

∥Z̃N0

∥

∥

∥

ωβ≤γm+1

1− γ

∥

∥

∥Z̃N0

∥

∥

∥

ωβ. (4.25)

Since γ ∈ [0, 1),∥

∥ZNm′ − ZNm

∥

∥

ωβ→ 0 as m,m′ → ∞. Thus Zk is a Cauchy sequence; and since R

n

is a Banach space, Zk has a limit Ẑ(tNβ,k). Taking limit m→ ∞ in (4.9), yields

L[Ẑ(tNβ,k)− Ẑ(tNβ,k)] = 0 = ~H(t

Nβ,k){L[Ẑ(t

Nβ,k)] +N [Ẑ(t

Nβ,k)]− φ

N (tNβ,k)},

Ẑ(0) = z0.

Thus, Ẑ(tNβ,k) is the exact solution of (3.13) at any GLR point tNβ,k. Also, by noticing the definition

of ẐN(t), it is easy to verify ẐN(tNβ,k) = Ẑ(tNβ,k) and the proof is completed.

5 Numerical experiments

To demonstrate the applicability of the LaHOC algorithm as an appropriate tool for solving in-finite horizon optimal control for nonlinear large-scale dynamical systems, we apply the proposedalgorithm to several test problems.

10

Test problem 3.1. Consider the two-order nonlinear composite system described by [7]:

ẋ1(t) = x1(t) + u1(t)− x31(t) + x

22(t), (5.1)

ẋ2(t) = −x2(t) + u2(t) + x1(t)x2(t) + x32(t), (5.2)

x1(0) = 0, x2(0) = 0.8. (5.3)

The quadratic cost functional to be minimized is given by:

J =1

2

2∑

i=1

∫ ∞

0(x2i (t) + u

2i (t))dt, (5.4)

In this example, we have A1 = B1 = B2 = 1, A2 = −1, Q1 = Q2 = R1 = R2 = 1, f1(x) =−x31(t) + x

22(t), f2(x) = x1(t)x2(t) + x

32(t).

Then, according to the optimal control theory (2.3), the optimality conditions can be written as:

ẋ1(t) = x1(t)− λ1(t)− x31(t) + x

22(t), (5.5)

ẋ2(t) = −x2(t)− λ2(t) + x1(t)x2(t) + x32(t), (5.6)

λ̇1(t) = −x1(t)− λ1(t) + 3x21(t)λ1(t)− x2(t)λ2(t), (5.7)

λ̇2(t) = −x2(t) + λ2(t)− 2x2(t)λ1(t)− x1(t)λ2(t)− 3x22(t)λ2(t), (5.8)

x1(0) = 0, x2(0) = 0.8, λ1(∞) = 0, λ2(∞) = 0. (5.9)

Also the optimal control laws are u1(t) = −λ1, u2(t) = −λ2.In this example, the parameters used in the LaHOC algorithms are

Lr =

ddt

− 1 0 1 00 d

dt+ 1 0 1

1 0 ddt

+ 1 0

0 0 0 ddt

− 1

, A =

D− I O I OO D+ I O II O D+ I OO O O D− I

, (5.10)

11

Fr =

x31 − x22

−x1x2 − x32

−3x21λ1 + x2λ22x2λ1 + x1λ2 + 3x

22λ2

, φ =

00000

, (5.11)

Rr,m−1 = Lr[xr,m−1] +Qr,m−1, (5.12)

Qr,m−1 =

−m−1∑

j=0

Z1,m−1−j(t)

j∑

k=0

Z1,j(t)Z1,j−k(t) +m−1∑

j=0

Z2,jZ2,m−1−j

m−1∑

j=0

Z1,j(t)Z2,m−1−j(t) +

m−1∑

j=0

Z2,m−1−j(t)

j∑

k=0

Z2,j(t)Z2,j−k(t)

3m−1∑

j=0

Z3,m−1−j(t)

j∑

k=0

Z1,j(t)Z1,j−k(t)−m−1∑

j=0

Z2,j(t)Z4,m−1−j(t)

−2m−1∑

j=0

Z2,j(t)Z3,m−1−j(t)−m−1∑

j=0

Z1,j(t)Z4,m−1−j(t)− 3m−1∑

j=0

Z4,m−1−j(t)

j∑

k=0

Z2,j(t)Z2,j−k(t)

(5.13)

With these definitions, the LaHOC algorithm gives

Xr,m = (χm + ~r)Xr,m−1 + ~rA−1Qr,m−1, (5.14)

Because the right hand side of equation (5.14) is known, the solution can easily be obtained byusing methods for solving linear system of equations.

Table 1 gives a comparison between the present LaHOC results for N = 100 and ~ = −0.6 andthe numerically generated BVP5C [42], at selected values of time t. It can be seen from the table thatthere is in good agreement between the two results. Moreover, our calculations show the betteraccuracy of LaHOC. In comparison with the BVP5C, it is noteworthy that the LaHOC controls theerror bounds while preserving the CPU time. The CPU time of LaHOC is 0.606532 s, and BVP5Cis 1.109817 s.

Figure 1 and Figure 2 show the suboptimal states and control for m = 19 iterations of LaHOC,compared to MATLAB built-in function BVP5C. The convergence of LaHOC ieteration is depictedin Figure 3. Also, Figure 4 presents that the minimum objective functional |Jj − JN | converges to0, where j = 20, 30, . . . , 110 and N = 120.

The results obtained with the present method are in good agreement with results of the succes-sive approximation method used by Tang and Sun [7].

12

Table 1: Comparison between the LaHOC solution when N = 100 and ~ = −0.6 and BVP5Csolution.

x1(t) x2(t) λ1(t) λ2(t)

t LaHOC BV P5C LaHOC BV P5C LaHOC BV P5C LaHOC BV P5C

0.113 0.013872 0.013872 0.689067 0.689067 0.388387 0.388387 0.557556 0.5575560.494 0.031434 0.031434 0.412872 0.412872 0.195039 0.195039 0.236820 0.2368201.152 0.021573 0.021573 0.164529 0.164529 0.070317 0.070317 0.075704 0.0757042.107 0.006800 0.006800 0.042594 0.042594 0.017627 0.017627 0.018077 0.0180773.389 0.001168 0.001168 0.006943 0.006943 0.002852 0.002852 0.002887 0.0028875.047 0.000113 0.000113 0.000666 0.000666 0.000273 0.000273 0.000276 0.000276

0 5 10 15 20t

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x1

-4

-2

0

2

4

6

8

10

12

14

LaHOC-B

VP5C

×10-7

LaHOCBVP5CLaHOC-BVP5C

0 5 10 15 20t

-0.5

0

0.5

1

x2

-5

0

5

10

LaHOC-B

VP5C

×10-6

LaHOCBVP5CLaHOC-BVP5C

Fig. 1. The amplitudes of optimal state variables.

0 5 10 15 20t

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

u1

-12

-10

-8

-6

-4

-2

0

2

LaHOC-B

VP5C

×10-6

LaHOCBVP5CLaHOC-BVP5C

0 5 10 15 20t

-1

-0.5

0

u2

-2

0

2

LaHOC-B

VP5C

×10-5

LaHOCBVP5CLaHOC-BVP5C

Fig. 2. The amplitudes of optimal control variables.

13

20 30 40 50 60 70 80 90 100 110N

10-3

10-2

10-1

cost

func

tion

erro

r

The minimum cost converges to 0.027605

Fig. 3. The minimum cost onvergence.

0 10 20 30 40 50 60 70Numer of iteration

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

102

|xm|

20 30 40 50 60 70 80 90 100 110 120N

67

67.5

68

68.5

69

69.5

70

itera

tion

num

ber

Fig. 4. Convergence of LaHOC iteration: Left) the error reduction in each iteration when N = 120;Right) number of iterations needed when error reduction threshold is 10−12.

Test problem 3.2. Consider the Euler dynamics and kinematics of a rigid body related to controllaws to regulate the attitude of spacecraft and aircraft [7]:

{

ρ̇(t) = 12 (I − S(ρ(t)) + ρ(t)ρT (t))ω(t),

ω̇(t) = J−1S(ω(t))J ω(t) + J−1u(t),(5.15)

where J = diag(10, 6.3, 8.5), ρ = (ρ1, ρ2, ρ3)T ∈ R3 is the vector of Rodrigues parameters, ω =

(ω1, ω2, ω3)T ∈ R3, is the angular velocity, and u = (u1, u2, u3)

T ∈ R3, is the control torque. Thesymbol S(.) is a skew symmetric matrix of the form

S(ω) =

0 ω3 −ω2−ω3 0 ω1ω2 −ω1 0

, (5.16)

In addition, the initial conditions are ρ(0) = (0.3735, 0.4115, 0.2521)T and ω(0) = (0, 0, 0)T .

14

Then, according to the optimal control theory (2.3), the optimality conditions can be written as:

ρ̇1(t) =1

2ω1(t) +

1

2ω1(t)ρ

21(t) +

1

2ω2(t)ρ1(t)ρ2(t) +

1

2ω3(t)ρ1(t)ρ3(t), (5.17)

ρ̇2(t) =1

2ω2(t) +

1

2ω2(t)ρ

22(t) +

1

2ω1(t)ρ1(t)ρ2(t) +

1

2ω3(t)ρ2(t)ρ3(t), (5.18)

ρ̇3(t) =1

2ω3(t) +

1

2ω3(t)ρ

23(t) +

1

2ω1(t)ρ1(t)ρ3(t) +

1

2ω2(t)ρ2(t)ρ3(t), (5.19)

ω̇1(t) = −11

50ω2(t)ω3(t)−

1

100λ4(t), (5.20)

ω̇2(t) = −5

21ω1(t)ω3(t)−

100

3969λ5(t), (5.21)

ω̇3(t) =37

85ω1(t)ω2(t)−

4

289λ6(t), (5.22)

λ̇1(t) = −λ1(t)ω1(t)ρ1(t)− ρ1(t)−1

2λ1(t)ω2(t)ρ2(t)−

1

2λ1(t)ω3(t)ρ3(t)

−1

2λ2(t)ω1(t)ρ2(t)−

1

2λ3(t)ω1(t)ρ3(t), (5.23)

λ̇2(t) = −λ2(t)ω2(t)ρ2(t)− ρ2(t)−1

2λ1(t)ω2(t)ρ1(t)−

1

2λ2(t)ω1(t)ρ1(t)

−1

2λ2(t)ω3(t)ρ3(t)−

1

2λ3(t)ω2(t)ρ3(t), (5.24)

λ̇3(t) = −λ3(t)ω3(t)ρ3(t)− ρ3(t)−1

2λ1(t)ω3(t)ρ1(t)−

1

2λ2(t)ω3(t)ρ2(t)

−1

2λ3(t)ω1(t)ρ1(t)−

1

2λ3(t)ω2(t)ρ2(t), (5.25)

λ̇4(t) = −37

85λ6(t)ω2(t) +

5

21λ5(t)ω3(t)−

1

2λ1(t)ρ

21(t)−

1

2λ2(t)ρ1(t)ρ2(t)

−1

2λ3(t)ρ1(t)ρ3(t)−

1

2λ1(t)− ω1(t), (5.26)

λ̇5(t) =11

50λ4(t)ω3(t)−

37

85λ6(t)ω1(t)−

1

2λ2(t)ρ

22(t)−

1

2λ1(t)ρ1(t)ρ2(t)

−1

2λ3(t)ρ2(t)ρ3(t)−

1

2λ2(t)− ω2(t), (5.27)

λ̇6(t) = −11

50λ4(t)ω2(t) +

5

21λ5(t)ω1(t)−

1

2λ3(t)ρ

23(t)−

1

2λ1(t)ρ1(t)ρ3(t)

−1

2λ2(t)ρ2(t)ρ3(t)−

1

2λ3(t)− ω3(t), (5.28)

ρ1(0) = 0.3735, ρ2(0) = 0.4115, ρ3(0) = 0.2521, ω1(0) = 0, ω2(0) = 0, ω3(0) = 0,

and the optimal control laws are u1(t) = −110λ4, u2(t) = −

1063λ5, u3(t) = −

217λ6.

In this example, the parameters used in the LaHOC algorithms are

15

Lr =

ddt

0 0 −12 0 0 0 0 0 0 0 00 d

dt0 0 −12 0 0 0 0 0 0 0

0 0 ddt

0 0 −12 0 0 0 0 0 00 0 0 d

dt0 0 0 0 0 1100 0 0

0 0 0 0 ddt

0 0 0 0 0 1003969 0

0 0 0 0 0 ddt

0 0 0 0 0 42891 0 0 0 0 0 d

dt0 0 0 0 0

0 1 0 0 0 0 0 ddt

0 0 0 0

0 0 1 0 0 0 0 0 ddt

0 0 0

0 0 0 1 0 0 12 0 0ddt

0 0

0 0 0 0 1 0 0 12 0 0ddt

0

0 0 0 0 0 1 0 0 12 0 0ddt

, (5.29)

A =

D O O −12I O O O O O O O OO D O O −12I I O O O O O OO O D O O −12I O O O O O OO O O D O O O O O 1100I O OO O O O D O O O O O 1003969I OO O O O O D O O O O O 4289II O O O O O D O O O O OO I O O O O O D O O O OO O I O O O O O D O O OO O O I O O 12I O O D O OO O O O I O O 12I O O D OO O O O O I O O 12I O O D

, (5.30)

Rr,m−1 = Lr[xr,m−1] +Qr,m−1, (5.31)

Qr,m−1 =

12

m−1∑

j=0

Z4,m−1−j

j∑

k=0

Z1,jZ1,j−k +m−1∑

j=0

Z5,m−1−j

j∑

k=0

Z1,jZ2,j−k

+

m−1∑

j=0

Z6,m−1−j

j∑

k=0

Z1,jZ3,j−k,

12

m−1∑

j=0

Z5,m−1−j

j∑

k=0

Z2,jZ2,j−k +m−1∑

j=0

Z4,m−1−j

j∑

k=0

Z1,jZ2,j−k

+

m−1∑

j=0

Z6,m−1−j

j∑

k=0

Z2,jZ3,j−k,

12

m−1∑

j=0

Z6,m−1−j

j∑

k=0

Z3,jZ3,j−k +m−1∑

j=0

Z4,m−1−j

j∑

k=0

Z1,jZ3,j−k

+

m−1∑

j=0

Z5,m−1−j

j∑

k=0

Z2,jZ3,j−k,

, r = 1, 2, 3, (5.32)

16

Qr,m−1 =

−1150

m−1∑

j=0

Z5,jZ6,m−1−j ,

− 521

m−1∑

j=0

Z4,jZ6,m−1−j ,

3785

m−1∑

j=0

Z4,jZ5,m−1−j ,

−m−1∑

j=0

Z7,m−1−j

j∑

k=0

Z4,jZ1,j−k −1

2

m−1∑

j=0

Z7,m−1−j

j∑

k=0

Z5,jZ2,j−k −1

2

m−1∑

j=0

Z7,m−1−j

j∑

k=0

Z6,jZ3,j−k −1

2

m−1∑

j=0

Z8,m−1−j

j∑

k=0

Z4,jZ2,j−k −1

2

m−1∑

j=0

Z9,m−1−j

j∑

k=0

Z4,jZ3,j−k,

−m−1∑

j=0

Z8,m−1−j

j∑

k=0

Z5,jZ2,j−k −1

2

m−1∑

j=0

Z7,m−1−j

j∑

k=0

Z5,jZ1,j−k −1

2

m−1∑

j=0

Z8,m−1−j

j∑

k=0

Z4,jZ1,j−k −1

2

m−1∑

j=0

Z8,m−1−j

j∑

k=0

Z6,jZ3,j−k −1

2

m−1∑

j=0

Z8,m−1−j

j∑

k=0

Z5,jZ3,j−k,

−m−1∑

j=0

Z9,m−1−j

j∑

k=0

Z6,jZ3,j−k −1

2

m−1∑

j=0

Z7,m−1−j

j∑

k=0

Z6,jZ1,j−k −1

2

m−1∑

j=0

Z8,m−1−j

j∑

k=0

Z6,jZ2,j−k −1

2

m−1∑

j=0

Z9,m−1−j

j∑

k=0

Z4,jZ1,j−k −1

2

m−1∑

j=0

Z9,m−1−j

j∑

k=0

Z5,jZ2,j−k,

−3785

m−1∑

j=0

Z12,jZ5,m−1−j +5

21

m−1∑

j=0

Z11,jZ6,m−1−j −1

2

m−1∑

j=0

Z7,m−1−j

j∑

k=0

Z1,jZ1,j−k

−12

m−1∑

j=0

Z8,m−1−j

j∑

k=0

Z1,jZ2,j−k −1

2

m−1∑

j=0

Z9,m−1−j

j∑

k=0

Z1,jZ3,j−k,

1150

m−1∑

j=0

Z10,jZ6,m−1−j −37

85

m−1∑

j=0

Z12,jZ4,m−1−j −1

2

m−1∑

j=0

Z8,m−1−j

j∑

k=0

Z2,jZ2,j−k

−12

m−1∑

j=0

Z7,m−1−j

j∑

k=0

Z1,jZ2,j−k −1

2

m−1∑

j=0

Z9,m−1−j

j∑

k=0

Z2,jZ3,j−k,

−1150

m−1∑

j=0

Z10,jZ5,m−1−j +5

21

m−1∑

j=0

Z11,jZ4,m−1−j −1

2

m−1∑

j=0

Z9,m−1−j

j∑

k=0

Z3,jZ3,j−k

−12

m−1∑

j=0

Z7,m−1−j

j∑

k=0

Z1,jZ3,j−k −1

2

m−1∑

j=0

Z8,m−1−j

j∑

k=0

Z2,jZ3,j−k,

(5.33)

(5.34)

With these definitions, the LaHOC algorithm gives

Xr,m = (χm + ~r)Xr,m−1 + ~rA−1Qr,m−1, (5.35)

17

Because the right hand side of equation (5.35) is known, the solution can easily be obtained byusing methods for solving linear system of equations.

Tables 2 and 3, give a comparison between the present LaHOC results for N = 50 and ~ = −1and the numerically generated BVP5C at selected values of time t. It can be seen from the tablesthat there is in good agreement between the two results. Moreover, our calculations show that theaccuracy of LaHOC is faster. In comparison with the BVP5C, it is noteworthy that the LaHOCcontrols the error bounds while preserving the CPU time. The CPU time of LaHOC is 1.009860 s,and BVP5C is 4.514071 s.

Figurs. 5-9 show the suboptimal states and control for m = 20 iterations of LaHOC, comparedto MATLAB built-in function BVP5C. The convergence of Laguerre-LaHOC ieteration is depictedin Figure 10.

The obtained optimal trajectories and optimal controls are almost identical to those obtainedby Jajarmi et al. [9].

Table 2: Comparison between the LaHOC solution when N = 50 and ~ = −1 and BVP5C solution.

ρ1(t) ρ2(t) ρ3(t)

t LaHOC BV P5C LaHOC BV P5C LaHOC BV P5C

0.409 0.371513 0.371389 0.408328 0.408146 0.251619 0.2504031.950 0.337343 0.335574 0.355885 0.353367 0.241531 0.2219424.663 0.237026 0.232989 0.215281 0.210198 0.198296 0.1445248.597 0.107445 0.103268 0.066722 0.062265 0.112940 0.05755420.488 -0.010891 -0.011225 -0.006986 -0.007030 -0.003736 -0.00537838.855 0.000248 0.000274 0.000140 0.000138 0.000053 0.000159

Table 3: Comparison between the LaHOC solution when N = 50 and ~ = −1 and BVP5C solution.

ω1(t) ω2(t) ω3(t)

t LaHOC BV P5C LaHOC BV P5C LaHOC BV P5C

0.409 -0.013313 -0.013421 -0.023872 -0.024420 -0.000899 -0.0116411.950 -0.047399 -0.047871 -0.077873 -0.079134 -0.009730 -0.0398064.663 -0.066195 -0.067190 -0.090208 -0.090958 -0.032073 -0.0493968.597 -0.051563 -0.051458 -0.050159 -0.049573 -0.040398 -0.03182920.488 -0.000271 0.000040 0.002386 0.002574 -0.002575 0.00038638.855 0.000147 0.000132 -0.000059 -0.000062 0.000127 0.000009

18

0 20 40 60 80 100 120t

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4ρ1

LaHOCBVP5C

0 20 40 60 80 100 120t

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

ρ2

LaHOCBVP5C

Fig. 5. The amplitudes of optimal state variables.

0 20 40 60 80 100 120−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

t

ρ3

LaHOCBVP5C

0 20 40 60 80 100 120t

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

ω1

LaHOCBVP5C

Fig. 6. The amplitudes of optimal state variables.

0 20 40 60 80 100 120t

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

ω2

LaHOCBVP5C

0 20 40 60 80 100 120−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

t

ω3

LaHOCBVP5C

Fig. 7. The amplitudes of optimal state variables.

19

0 20 40 60 80 100 120t

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1u1

LaHOCBVP5C

0 20 40 60 80 100 120t

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

u2

LaHOCBVP5C

Fig. 8. The amplitudes of optimal control variables.

0 20 40 60 80 100 120t

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

u3

LaHOCBVP5C

Fig. 9. The amplitudes of optimal control variables.

20 25 30 35 40 45 50N

10-2

10-1

100

min

imum

cos

t ->

0.15

301

The minimum cost converges to 0.15301

Fig. 10. The minimum cost onvergence.

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6 Conclusion

In this paper, an effective method based upon the spectral homotopy method with Laguerre basis(LaHOC) is proposed for finding the numerical solutions of the infinite horizon optimal controlproblem of nonlinear interconnected large-scale dynamic systems. Modified Laguerre method isused to discretize the equation of optimal condition, while homotopy method is used to constructan iterative scheme. Two illustrative examples demonstrated that LaHOC has spectral accuracyand very good efficiency, which is comparable to well established numerical methods such as theMATLAB BVP5C solver. The second example shows when the multi-components have differenttime and amplitude scales, one need to use adaptive rescaling technique in the Laguerre bases toimprove accuracy, which deserves a further study.

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1 Introduction2 The nonlinear interconnected OCP3 Laguerre polynomials and spectral homotopy analysis method 3.1 Properties of the modified Laguerre polynomials3.2 Spectral homotopy analysis method

4 Convergence analysis of LaHOC5 Numerical experiments6 Conclusion