Numerical Approximation For Optimal Control Based On Wavelet Method

Optimal control is derived from classical calculus of variations. The motivations include the development of aeronautie technic and the demands of engineering and industry. All of them promotes a breakthrough in the theory of control engineerina. Modern control theory become an independent field. Optimal control is a major branch of control theory, which concern how to establish the mathematical model for some objects. Optimal control seeks a permission control to make the state of a system transfer from a given initial value to a destination in a terminate time, while maximize the performance index.Many mathematicians pay much attention to optimal control for its conspicuous and rigors represent formulation and the need from engineering. In the last decade, wavelet method is introduced as a new tool for solving numerical system in optimal control. The Multi-Resolution-Analysis of wavelet have shown some advantages in numerical approaches.This paper is organized as follow:In the first chapter, some basic theory of optimal control is introduced. We formulate the mathematical description of the optimal control. Optimal control seeks a permission set u(t)∈Ωto make the state of a system transfer from a given initial value x₀ to a destination set A in a terminate time t₁(?t₀). while maximize the performance index J(u).Direct and indirect methods are use to solve the problems in optimal control. With the rapid development of the science and technique, the demand for new method for optimal control is raised. Some approaches appear for exploring the new methods. Wavelet method is a successful one among them.In the second chapter, the definition and basic theory of wavelet are introduced. Denoteψ(x)∈L²(R),ψ_j,k(x)= 2₂¹Ïˆ(2^jx-k). If {ψ_j,k}_j,k∈z is a standard orthogonal basis of L²(R).thenψis a orthogonal wavelet, {ψ_j,k}_j,k∈z is orthogonal basis, W_j = span{ψ_j,k(x)}_k∈z is a subspace of wavelet.From the definition of the wavelet, the wavelet basis is designed by a Multi-Resolution-Analysis (MRA) framework. From the special spatial decomposition of MRA, we can devise wavelet basis which have some remarkable advantages. Multi-Resolution-Analysis is one of them. With the Multi-Resolution-Analysis ability, some optimal control can be simple in numerical aspect.In the third chapter, we convert a integral question into an ordinary algebra system, which can be solved easily. We discuss three optimal control problem with wavelet method.The first is a optimal control problem in linear-quadratic differential saddle-Point game.The system state equation is:E(?)(t) = Ax(t) + Bu(t) + Cv(t), t∈[0.t_f], Ex(0) = Ex₀. (1)The performance index is defined as follows:whereE-rank{E)≤n, and |sE - A|≠0,x(t)∈Rⁿ,u(t)∈R₃^Ï„,v(t)∈R₂^Ï„,v(t)∈Г₁^a×Г₂^a(?)Ω₁×Ω₂.The permission set is chosen as:whereQ≥0, R₁ > 0, R₂ > 0, A, B₁, B₂, Q_fis a matrix with proper dimension. problem of the saddle-point strategy for linear-quadratic differential game. find u(t)∈Γ₁^ato make L(u,v)to minimum ;at the same time find v(t)∈Γ₂^ato make L(u,v) maximum. So find (u*,v*)to make L(u*,v) < L(u*,v*)≤L(u,v*).As it is difficult to find a analytical solution, we present a numerical method with the MRA and orthogonal basis of wavelet.The first step is to chose Daubechies wavelet basis; The second step is to span the functions in the state equation and performance index with the Daubechies wavelet basis;The third step is to convert the integral problem to a matrix algebra optimal with some tools, such as operation matrix of integrate , product and coefficient matrices and fast wavelet transform.The fourth step is to solve the algebra problem, then an approximation solution for optimal control is obtained.Then we consider a optimal control for a linear time variable system.The system state equation is:(?)(t)=A(t)x(t)+B(t)u(t),x(0)=x₀ (4)The performance index is defined as follows:J(x,t)=(1/2)integral from n=0 to tf [x^T(t)Q(t)x(t)+u^TR(t)u(t)]dt (5)where x{t)∈C_n¹[0,t_f],u(t)∈L_T²[0,t_f],t_f,A(t)∈R^n×n,B(t)∈R^n×r. and the element in them is a continuous function of t, Q{t)∈R^n×n is a half positive definite matrices, R(t)∈R^r×r is a positive definite matrices.To get the optimal solution u*(t,x), it is necessary to get the solution of a Riccati equation with additional constrain of a integral equation. As it is very difficulty in numerical aspect, we discuss this problem with wavelet.The first step is to chose Daubechies wavelet basis; The second step is to span the functions in the state equation and performance index with the Daubechies wavelet basis; The third step is to convert the integral problem to a algebra quadratic constrained programming with some tools, such as operation matrix of integrate , product and coefficient matrices and fast wavelet transform.The fourth step is to solve the algebra quadratic constrained programming problem, then an approximation solution for optimal control is obtained.We consider the optimal control for a distributed parameter systems.Some optimal control problem is formulated as a one order ordinary differential equation system. For a system control by a pritial differential equation. we get the approximation solution with wavelet. Let us consider a optimal control of a system of parallel beams.The system state equation is:where the parameters in the system isK_i=E_iI_i,m_i=p_iA_ithe second in (6) represents the coupling between the two beams.The boundary value is:uⁱ(0,t)=uⁱ(l,t)= 0(?)_x²uⁱ(0,t)=(?)_x²uⁱ(l,t)= 0,i = 1.2.t∈Ω_t. (7)And the initial value is:uⁱ(x,0) = u₀ⁱ(x) (?)_tu²(x,0)=u₁ⁱ(x ),i =1,2,t∈Ω_x. (8) Consider the set of admissible distributions:We introduce the following performance index:where (?) (t) = (f₁₁(t),f₁₂(t),…,f_1n1(t),f₂₁(t),f₂₂(t),…,f_2n2(t)).μ_i≥0 andμ_i≠0, i = 1…6.The optimal control of the double beam can now be expressed aswhere uⁱ(x,t) is subject to (7)-(9).The first step is to convert the control problem in modal-space, thus the question with partial differential equation is transform as an ordinary differential equation question.The second step is to chose Legendre wavelet basis; The third step is to span the functions in the state equation and performance index with the Legendre wavelet basis;The fourth step is to convert the integral problem to a linear algebra with some tools, such as operation matrix of integrate , product and coefficient matrices and fast wavelet transform.The last step is to solve the linear algebra problem, then an approximation solution for optimal control is obtained.With three examples above, we can solve the optimal control with MRA and orthogonal bases of wavelet. Numerical simulations show that wavelet method can get a precision solution easily.