Solving linear optimal control problems of the time-delayed systems by Adomian decomposition method
Seyed Mehdi Mirhosseini-Alizamini*
AMS subject classifications: Primary 49N05; Secondary 93C05.
Keywords: Multiple time-delay systems; Pontryagin’s maximum principle; Adomian decomposition method.
Introduction
Optimal control of time-delay systems is one of the most challenging mathematical problems in control theory. Indeed, the presence of delay makes analysis and control design much more complicated. Delays occur frequently in mechanics, physics, population dynamics, biological, chemical, electronic and transformation systems . The theory and the application of optimal control for linear time-delay systems have been developed perfectly. However, as for nonlinear systems, synthesis problems that are solved by classic control theory lead to difficult computations. It is well-known that the nonlinear optimal control time-delay systems can be reduced to a TPBVP involing both delay and advance delay terms, implementing the PMP . In general, this TPBVP cannot be solved exactly and most researches have been devoted to find an approximate solution, for nonlinear TPBVP. We briefly review some resent papers that are relevant to the method developed in the current work for time-delay optimal control problem. An averaging approximations for time-delay optimal control problems , The B-spline approximation scheme , the PMP , variational iteration method (VIM) -, a novel feedward-feedback suboptimal control of linear time-delay systems , Haar wavelets approach , hybrid of block-pulse functions and orthonormal basis , composite Chebyshev finite difference method , The Hamilton-Jacobi-Bellman equation , a delay-dependent stability of neutral systems , An iterior-point algorithm and an embedding process that transfers the problem to a new optimal measure problem .
The topic of the ADM has been rapidly growing in recent years. It was first proposed by George Adomian . In this method the solution of functional equations is considered as the sum of an infinite series usually converging to the solution. A lot of research works have been conducted recently in applying this method to a class of linear and nonlinear partial differential equations . The Adomian’s decomposition has many advantages: it does not require any kind of discretization, linearization or perturbation of the variables and of the equation, therefore it does not need any modification of the actual model that could change the solution; is efficient on providing an approximate or even exact solution in a closed form, to linear and nonlinear problems; provides a fast and accurate convergent series and therefore it is only necessary to calculate a few terms of the series in order to obtain a reliable approximate solution; the method depends only on the known function u0(t) and the algorithm is of simple implementation. The method, has been widely applied to solve nonlinear problems, and different modifications are suggested to overcome the demerits arising in the solution procedure .
This paper concerns with a class of nonlinear quadratic optimal control problem with multi-delay systems. Applying the main ideas of the shooting method to the basic and also a ADM. By applying the necessary optimality conditions, we obtained iterative formulas for the ADM. By using the finite-step iteration of algorithm, we can obtain a suboptimal control law. The convergence of the ADM is studied and for illustrate the effectiveness of these methods, some test problems are investigated. Four illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method.
The structure of this paper is arranged as follows: Section 2 is devoted to Pontryagin’s maximum principle used for solving linear time-varying multi-delay system. Section 3 is dedicated to the proposed design approach to solve a close-loop optimal control problem based on the ADM and convergence of the method is demonstrated. Section 4 is devoted to the suboptimal control strategy and algorithm for proposed method. In Section 5 the numerical examples are simulated to show the reasonableness of our theory and demonstrate the performance of our network. Finally, we end this paper with conclusions in Section 6.
Problem statement and optimality conditions
Consider a linear time-varying multi-delay system $$\begin{aligned} \begin{cases}\label{1} \dot x(t)=A(t)x(t)+A_{1}(t)x(t-\tau_{x})+B(t)u(t)+B_{1}(t)u(t-\tau_{u}), \\ x(t)=\phi(t), \hspace{0.7cm}t_{0}-\tau_{x}\leqslant t \leqslant t_{0}, \hspace{2.5cm}\\ u(t)=\psi(t), \hspace{0.7cm}t_{0}-\tau_{u}\leqslant t \leqslant t_{0}, \hspace{2.5cm} \end{cases} \end{aligned}$$ where $x(t)\in \mathbb{R}^n$ and u(t) ∈ ℝm, are the state and control vectors respectively; A(t), A1(t), B(t) and B1(t) are real, piecewise continuous matrices of appropriate dimensions defined on the appropriate intervals; ϕ(t) and ψ(t) are specified initial functions; τx and τu are constant positive scalars. Here, it is assumed that the system [1] is controllable and assume that τu < τx. Find control signal u(t) which minimizes the cost functional: $$\label{2} J=\dfrac{1}{2}x^{T}(t_{f})Q_{f}x(t_{f})+\dfrac{1}{2}\int_{t_{0}}^{t_{f}}\left( x^{T}(t)Q(t)x(t)+u^{T}(t)R(t)u(t)\right)dt,$$ where, the matrix Qf ∈ ℝn × n is symmetric positive semi-definite, Q(t) ∈ ℝn × n and R(t) ∈ ℝm × m are chosen to be positive semi-definite and positive definite matrices respectively. The Hamiltonian function for the problem is $$\begin{aligned} \label{3} \mathcal{H}(x,u,\lambda, t)&=&\dfrac{1}{2}x^{T}(t)Q(t)x(t)+\dfrac{1}{2}u^{T}(t)R(t)u(t)\\&+&\lambda^{T}(t)[A(t)x(t)+A_{1}(t)x(t-\tau_x)+B(t)u(t)+B_{1}(t)u(t-\tau_u)],\nonumber \end{aligned}$$
where λ(t) ∈ ℝn is the vector of the Lagrange multiplier. According to the necessary conditions for optimality, we can obtain the following nonlinear TPBVP : $$\begin{aligned} \dot x(t)= \begin{cases}\label{4} A(t)x(t)+A_{1}(t)x(t-\tau_x)-(S_{1}(t)+S_{2}(t))\lambda(t)\\-S_{3}(t)\lambda(t+\tau_{u})-S_{4}(t)\lambda(t-\tau_{u}),\hspace{0.3cm}t_{0}\leqslant t < t_{f}-\tau_u,\\ A(t)x(t)+A_{1}(t)x(t-\tau_x)-S_1(t)\lambda(t)\\-S_{4}(t)\lambda(t-\tau_{u}),\hspace{0.2cm}t_{f}-\tau_u\leqslant t \leqslant t_{f}, \end{cases} \end{aligned}$$ and $$\begin{aligned} \dot \lambda(t)= \begin{cases}\label{5} -Q(t)x(t)-A^{T}(t)\lambda(t)\\-A^T_{1}(t+\tau_x)\lambda(t+\tau_x), \hspace{0.3cm} t_{0} \leqslant t < t_{f}-\tau_x,\\ -Q(t)x(t)-A^{T}(t)\lambda(t),\hspace{0.3cm} t_{f}-\tau_x\leqslant t \leqslant t_{f}, \end{cases} \end{aligned}$$ with initial conditions $$\begin{aligned} \begin{cases}\label{6} x(t)=\phi(t),\hspace{0.7cm} t_{0}-\tau_{x} \leq t\leqslant t_{0},\\ u(t)=\psi(t), \hspace{0.7cm}t_{0}-\tau_{u}\leqslant t \leqslant t_{0},\\ \lambda(t_{f})=Q_{f}x(t_{f}), \end{cases} \end{aligned}$$ where $$\begin{aligned} S_{1}(t)&=&B(t)R^{-1}(t)B^{T}(t),\\ S_{2}(t)&=&B_{1}(t)R^{-1}(t-\tau_u)B^{T}_{1}(t),\\ S_{3}(t)&=&B(t)R^{-1}(t)B^{T}_{1}(t+\tau_u),\\ S_{4}(t)&=&B_{1}(t)R^{-1}(t-\tau_u)B^{T}(t-\tau_u), \end{aligned}$$x(t−τ) is time-delay term and λ(t+τ) is time-advance term. Also, the optimal control law is obtained by: $$\begin{aligned} u^{\ast}(t)= \begin{cases}\label{7} -R^{-1}(t)B^{T}(t)\lambda(t)\\-R^{-1}(t)B^{T}_{1}(t+\tau_u)\lambda(t+\tau_u), \hspace{0.3cm} t_{0}\leqslant t < t_{f}-\tau_{u},\\ -R^{-1}(t)B^{T}(t)\lambda(t), \hspace{0.3cm} t_{f}-\tau_u\leqslant t \leqslant t_{f}. \end{cases} \end{aligned}$$
The optimal can be implemented as a closed loop optimal if the co-state vector obtained consists of linear function of the states and a nonlinear term which is the adjoint vector sequence, in the form λ(t) = P(t)x(t) + g(t), λ(tf) = Qfx(tf), where P(t) ∈ ℝn × n is unknown positive-semidefinite function matrix, g(t) ∈ ℝn is the adjoint vector.
Substituting [8] into equation [4] yields: $$\begin{aligned} \label{9} \dot x(t)=\left[A(t)-S_{1}(t)P(t)\right]x(t)-S_{1}(t)g(t)+A_{1}(t)x(t-\tau_x)+F(t),\nonumber\\ x(t)=\phi(t),\hspace{0.7cm} t_{0}-\tau \leq t\leqslant t_{0}.\hspace{4.86cm} \end{aligned}$$ where $$\begin{aligned} F(t)= \begin{cases}\label{10} -S_{2}(t)\left[P(t)x(t)+g(t)\right]-S_{3}(t)\left[P(t+\tau_u)x(t+\tau_u)+g(t+\tau_u)\right]\\-S_{4}(t)\left[P(t-\tau_{u})x(t-\tau_{u})+g(t-\tau_{u})\right],\hspace{0.1cm}t_{0}\leqslant t < t_{f}-\tau_{u},\\ -S_{4}(t)\left[P(t-\tau_{u})x(t-\tau_{u})+g(t-\tau_{u})\right],\hspace{0.2cm}t_{f}-\tau_u\leqslant t \leqslant t_{f}. \end{cases} \end{aligned}$$ Computing the derivatives to the both sides with respect to t of equation [8], we have $$\begin{aligned} \label{11} \dot{\lambda}(t)&=&\dot{P}(t)x(t)+P(t)\dot{x}(t)+\dot{g}(t),\hspace{0.5cm}t_{0}\leqslant t \leqslant t_{f}\nonumber\\ &=&\left[\dot{P}(t)+P(t)A(t)-P(t)S_{1}(t)P(t)\right]x(t)-P(t)S_{1}(t)g(t)\nonumber\\&+&P(t)A_{1}(t)x(t-\tau_{x})+P(t)F(t)+\dot{g}(t). \end{aligned}$$ Putting [8] into equation [5], we get $$\begin{aligned} \hspace{-0.45cm} \dot \lambda(t)= \begin{cases}\label{12} -Q(t)x(t)-A^{T}(t)P(t)x(t)-A^{T}(t)g(t)\\-A^{T}_{1}(t+\tau_x)[P(t+\tau_x)x(t+\tau_x)+g(t+\tau_x)], t_{0} \leqslant t < t_{f}-\tau_x,\\ -Q(t)x(t)-A^{T}(t)P(t)x(t)-A^{T}(t)g(t), t_{f}-\tau_x\leqslant t \leqslant t_{f}.\\ \end{cases} \end{aligned}$$
Thus, from [11] and [12], we can obtain the following Riccati matrix differential equation: $$\label{13} -\dot{P}(t)=P(t)A(t)+A^{T}(t)P(t)-P(t)S_{1}(t)P(t)+Q(t),\hspace{0.1cm} P(t_{f})=Q_{f},$$ and adjoint vector differential equation the following: $$\label{14} \dot{g}(t)=-\left[A(t)-S_{1}(t)P(t)\right]^{T}g(t)-P(t)A_{1}(t)x(t-\tau_x)+G(t),\hspace{0.1cm} g(t_{f})=0,$$ where
Substituting [8] into [7] yields: $$\begin{aligned} u^{\ast}(t)= \begin{cases} -R^{-1}(t)B^{T}(t)[P(t)x(t)+g(t)]\nonumber\\-R^{-1}(t)B^{T}_{1}(t+\tau_u)[P(t+\tau_u)x(t+\tau_u) +g(t+\tau_u)], \hspace{0.02cm} t_{0}\leqslant t < t_{f}-\tau_{u},\\ -R^{-1}(t)B^{T}(t)[P(t)x(t)+g(t)], \hspace{0.02cm} t_{f}-\tau_u\leqslant t \leqslant t_{f}. \end{cases} \end{aligned}$$
For the sake of simplicity, let us define the right hand sides of [9] and [14] as follows: f1(t,x,g) = [A(t)−S1(t)P(t)]x(t) − S1(t)g(t) + A1(t)x(t−τx) + F(t),f2(t,x,g) = = − [A(t)−S1(t)P(t)]Tg(t) − P(t)A1(t)x(t−τx) + G(t), where F(t) and G(t) are relations [10] and [15], respectively. Thus the TPBVP in (2.9) and (2.14) changes to: $$\begin{aligned} \begin{cases}\label{18} \dot{x}(t)=f_1(t,x,g),\\ \dot{g}(t)=f_2(t,x,g),\\ x(t_0)=x_0,\hspace{0.2cm}g(t_f)=0.\\ \end{cases} \end{aligned}$$ Note that, relations [18] form a nonlinear TPBVP with time-varying coefficient involving both delay and advance terms. The exact solution of this problem is, in general, extremely difficult, if not impossible. In the next section, we propose another analytic approximate method based on ADM, for this purpose.
Adomian Decomposition Method
In order to illustrate the basic concepts of the ADM, we consider the following equation: ℒ(u) + ℛ(u) + 𝒩(u) = h(t), where u(t) is the unknown function, ℒ is a linear operator which is assumed to be invertible, ℛ is another linear differential operator, 𝒩(u) represents the nonlinear terms, and h is the continuous function. Applying the inverse operator ℒ−1 to both sides of [19], and using the given conditions we obtain u = f − ℒ−1(ℛ(u)) − ℒ−1(𝒩(u)), where the function f(t) represents the terms arising from integrating the function h(t) and using the initial condition.
The standard Adomian method defines the solution u(t) of [19] as a series $$\label{21} u(t)=\sum_{n=0}^{\infty}u_{n}(t),$$ where the components un(t) are usually determined recurrently. Substituting this infinite series into [20] leads to: $$\label{22} \sum_{n=0}^{\infty}u_n(t)=f(t)- \mathcal{L}^{-1}\left(\mathcal{R}(\sum_{n=0}^{\infty}u_n(t))\right)-\mathcal{L}^{-1}\left(\mathcal{N}(\sum_{n=0}^{\infty}u_n(t))\right).$$
The nonlinear term in [21] can be computed by substituting $$\label{23} \mathcal{N}(u)=\sum_{n=0}^{\infty}A_{n}(u_0, u_1, \dots , u_n),$$ where An is the Adomian polynomials, which can be determined by $$\label{24} A_n=\dfrac{1}{n!}\dfrac{\partial ^n}{\partial q^n}\mathcal{N}\left[\sum_{n=0}^{\infty}q^ku_k\right]_{q=0}, \hspace{0.3cm} n=1, 2, 3, \dots ,.$$ Now, substituting [23] into [22] leads to: $$\sum_{n=0}^{\infty}u_n(t)=f(t)- \mathcal{L}^{-1}\left(\mathcal{R}(\sum_{n=0}^{\infty}u_n(t))\right)-\mathcal{L}^{-1}\left(\sum_{n=0}^{\infty}A_n\right).$$ Each term of series [21] is given by the recurrent relation $$\begin{aligned} u_o&=f(t)\\ u_n&=-\mathcal{L}^{-1}(\mathcal{R}(u_{n-1}))-\mathcal{L}^{-1}(A_{n-1}),\hspace{0.3cm} n\geq 1. \end{aligned}$$
Now, we briefly describe how to apply the ADM to systems [18]. For this purpose, we use a shooting method like procedure combine with the ADM for solving TPBVP in [18].
Based on the ADM, we seek the solution {x, g} as follows: $$x=\lim_{N\rightarrow \infty} \sum_{n=0}^{N} x_n, \hspace{1cm} g=\lim_{N\rightarrow \infty} \sum_{n=0}^{N} g_n ,$$ and hence the recursive relationship is found to be $$\begin{aligned} \begin{cases} x_{n+1}=\mathcal{L}^{-1}A_{1,n}, \hspace{0.4cm}n\geq 0,\\ g_{n+1}=\mathcal{L}^{-1}A_{2,n}, \hspace{0.4cm}n\geq 0,\\ x(t_0)=x_0,\hspace{0.2cm}g(t_0)=\alpha,\\ \end{cases} \end{aligned}$$ with inverse ℒ−1(.) = ∫0t(.)dt and $$f_{k}(t,x_n,g_n)=\sum_{n=0}^{\infty}A_{k,n},\hspace{0.3cm} k=1,2,$$ where Ak, n are the Adomian polynomials and are calculated by $$\label{26} A_{k,n}=\dfrac{1}{n!}\dfrac{\partial ^n}{\partial q^n}f_{k}\left(t,\sum_{n=0}^{\infty}q^kx_k, \sum_{n=0}^{\infty}q^kg_k\right)_{q=0}, \hspace{0.3cm} n=0,1, 2, \dots ,.$$ Taking the first n + 1 terms of the nth approximation of x and g as follows: $$\begin{aligned} \begin{cases}\label{27} \Phi_n=x_0+\sum_{i=1}^{n}\mathcal{L}^{-1}(A_{1,i-1}), \\ \Psi_n=g_0+\sum_{i=1}^{n}\mathcal{L}^{-1}(A_{2,i-1}).\\ \end{cases} \end{aligned}$$ Find the sequences Φn = x0 + … + xn and Ψn = g0 + … + gn such that $$\begin{aligned} \begin{cases}\label{28} \Phi_{n}=x_0+\mathcal{L}^{-1}(f_{1}(t,\Phi_{n-1},\Psi_{n-1})), \hspace{0.5cm}n\geq 1,\\ \Psi_{n}=g_0+\mathcal{L}^{-1}(f_{2}(t,\Phi_{n-1},\Psi_{n-1})), \hspace{0.5cm}n\geq 1,\\ \end{cases} \end{aligned}$$ where $x_0(t)=x(t_0)=x_0,\hspace{0.2cm}g_0(t)=g(t_0)=\alpha$.
Suboptimal Control design strategy
Consider the linear time-varying multi-delay system [1] with cost functional [2]. Then, the Nth order suboptimal trajectory-control pair is obtained as follows: $$\begin{aligned} \begin{cases}\label{31} x_{N}(t)=\sum_{k=0}^{N}x_{k}(t)\\ g_{N}(t)=\sum_{k=0}^{N}g_{k}(t), \end{cases} \end{aligned}$$ and $$\begin{aligned} u_{N}(t)= \begin{cases}\label{32} -R^{-1}(t)B^{T}(t)[P(t)x_{N}(t)+g_{N}(t)]\\-R^{-1}(t)B^{T}_{1}(t+\tau_u)[P(t+\tau_u)x_{N}(t+\tau_{u})\\+g_{N}(t+\tau_u)], \hspace{0.3cm} t_{0}\leqslant t < t_{f}-\tau_{u},\\ -R^{-1}(t)B^{T}(t)[P(t)x_{N}(t)+g_{N}(t)], \hspace{0.3cm} t_{f}-\tau_u\leqslant t \leqslant t_{f}. \end{cases} \end{aligned}$$ The integer Nth in [31] and [32] is generally determined according to a concrete control precision. Then, the following cost functional can be calculated:
The Nth order in [31] and [32] has the desirable accuracy, if for given positive constants ϵ > 0, the following condition hold jointly: $$\label{34} \left\vert \dfrac{J_{N}-J_{N-1}}{J_{N}}\right\vert < \epsilon,$$ If the tolerance error bound be chosen small enough, the Nth order suboptimal control law will be very close to u*(t), and thus, the value of cost functional in [33] and its optimal value J* will be almost identical.
Algorithm: Suboptimal control law of system [1]: Step 1: Obtain P(t) from [13]. Let x0(t) = x(t0) = ϕ(t), g0(t) = g(t0) and k = 1. Step 2: Compute xk(t) and gk(t) from [29] and [30]. Step 3: Let N = k and obtain xN(t) and uN(t) from [31] and [32]. Step 4: Calculate JN according to [33]. If $\left\vert \dfrac{J_{N}-J_{N-1}}{J_{N}}\right\vert < \epsilon$, then stop and output uN(t), go to step 5; else, replace k by k + 1 and go to step 2. Step 5: Stop the algorithm; xN(t) and uN(t) are accurate enough.
Numerical examples
In this section, the proposed method is illustrated by some test problems. The calculations are performed using the Matlab software.
FIG. 1.Time difference between events F and A as a function of the length of entering plug for bifurcations B to BXXX, where X stands for 0 or 1.
Conclusion
In this work, the ADM has been successfully applied to find the solution of suboptimal control for linear time-varying systems with multiple state and control delays and with quadratic cost functional is presented. By using the ADM and VIM with the finite-step iteration of algorithm, we can obtain a suboptimal control law. Some numerical examples have been provided to demonstrate the validity and applicability of the proposed method. The method is general and yields very accurate results.
Acknowledgements
Author is grateful to the anonymous referees and the editors for their constructive comments.
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