Performance of Suboptimal Controllers for Affine-Quadratic Problems

In this article, affine-quadratic control problems are studied. Error bounds are derived for the difference between the performance indices corresponding to the optimal and a class of suboptimal controls. In particular, it is shown that the performance of these suboptimal controls is close to that of the optimal control whenever the error in estimating the costate initial condition is small.


Introduction
One of the most active areas in control theory is optimal control and methods to find them [1]- [3].It has a wide range of practical applications in engineering (Aerospace, Chemical, Mechanical, Electrical), science (Physics, Biology), and economics (see e.g.[4]- [7]).Optimal control theory has been developed for linear systems ( [1] [2] [8]) and explicit formulae for computing optimal control inputs are available.However, control of nonlinear systems is much more challenging and obtaining formulae for optimal controls seems in general not possible.This motivated researchers to study various classes of nonlinear control problems separately, and affine-qudratic problems is one such class.In a recent paper [9], the optimal control for affine-quadratic problems is obtained in terms of the associated costate.But, in practice, it is difficult to compute the costate (at each time t ) as the knowledge of its terminal condition is required.
In this article, we study the affine-quadratic control problem given by ((1), ( 2)).We note that a method for finding the initial condition for the costate is recently proposed [10].This allows one to compute the initial costate (at 0 t = ) exactly or approximately.This approximation of the initial costate and the explicit formula for optimal control (as in [11]) are shown, in this article, which give rise to suboptimal controls of practical importance.More precisely, our main theorem (Theorem 2) provides an upper bound for the difference in performance between these suboptimal and optimal control.
The article is organized as follows.In Section 2, the affine-quadratic control problem is described.We also explain how to obtain the optimal control in terms of costate.The main (Theorem 2) is proved in Section 3.This theorem provides a method to obtain the costate (without the knowledge of its terminal value) which results in an explicit formula and performance bounds for a class of suboptimal controls.Notation: For ( ) , , , ∈  , and 1 p ≤ < ∞ , we use the notation ( )

Problem Description
We consider the affine control system with the quadratic cost functional Here ( ) n x t ∈  is the state vector, ( ) m u t ∈  is the control vector, : , , , : , and ' denotes transposition.
Throughout this paper, it is assumed that , S Q are positive semidefinite, R is positive definite, the functions , f g are continuously differentiable with bounded derivatives, the control input ( ) Under these assumptions, for each admissible control ( ) u ⋅ ∈  there exist a unique solution (trajectory) of the control system (1) denoted by The value function of the control problem given by ( 1), (2), is defined as Similarly a control input ( ) Given 0 x , the optimal control problem is to find a control ( ) The Hamiltonian associated with the optimal control problem (1), ( 2), is given as where n λ ∈  is the adjoint vector.
To derive an expression for the optimal control ( ) u * ⋅ (for ( ) 0 x ), it is convenient to introduce the adjoint system: Here . We now state the Pontryagin's Minimum Principle (PMP) for the affine-quadratic control system (1), ( 2), which provides a set of necessary conditions for ( ) u * ⋅ to be optimal [12].x , as by the Equation (4).Then the optimal control (for ( ) Proof.The proof follows immediately from the above theorem.□ Now to obtain ( ) (in ( 5)) in terms of ( ) x t * , we solve the coupled systems given in ( 1) and ( 4) together with the initial conditions In general, solving this coupled system and finding a closed form solution ( ) is very difficult.However it may be easier to find 0 λ approximately.Such an approximation 0 λ will lead to the associated adjoint state ( ) . In the next section, we provide bounds for the difference between the performance indices corresponding to ( ) u * ⋅ and ( ) û ⋅ .

Performance of Suboptimal Controllers
In this section, we prove the main result.⋅ be the solution of the coupled system ((1), ( 4)) with initial condition ( ) The constant C depends only on the matrix function g and the constant K depends only on its gra- dient.
Proof.Note that  6), we first consider the term using Cauchy-Schwarz inequality From R.H.S. of (6), we next consider the term ( ) ( ) ( ) ( ) In a similar manner (as for ( 7)), we have ( ) ( ) ( ) ( ) From R.H.S. of ( 6), we next consider the term   Now using assumption on the matrix function g , we have that the matrix function h is continuously differentiable and has bounded derivatives.Therefore , where sup , max Using this and following the procedure as for the inequality (7), we get  This implies that ( ) ( ) ( ) ( ) ⋅ ⋅ is a good suboptimal control when 0 λ is a good approximation of 0 λ .We emphasize the fact that ( ) ˆt λ (and hence ( ) û t ) can be computed at each time t as 0 λ is known.

Theorem 2
Consider the affine-quadratic control problem (1),(2) ) It follows from the previous theorem that