Congestion Control for Nonlinear TD-SCDMA Discrete Networks Based on TCP/IP

—A successive approximation approach (SAA) is developed to obtain a new congestion controller for the nonlinear TD-SCDMA network control systems based on TCP/IP. By using the successive approximation approach, the original optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term that is the limit of the solution sequence of the adjoint vector differential equations. By using the finite-time iteration of nonlinear compensation term of optimal solution sequence, we can obtain a suboptimal control law for TD-SCDMA network control systems based on TCP/IP.


I. INTRODUCTION
It is well known that the insertion of the TD-SCDMA network based on TCP/IP in the feedback control loop makes the analysis and design of a network control systems complex because the network imposes an undetermined communication delay [1] . Therefore, conventional control theories with many ideal assumptions must be revaluated before they can be applied to network control systems. For instance, the stochastic optimal controller and the optimal state estimator of a network control system whose network induced delay is shorter than a sampling period have been proposed by Nilsson [2] . In Ref. [3] a model-based network control system was introduced. This control architecture has as main objective the reduction of the data packets transmitted over the network by a networked control system.
TD-SCDMA Network control systems based on TCP/IP can be described by nonlinear systems [4] . An amount of literature related to the analysis and controller design of such systems has been developed over the past decades. The stability region estimation and controller design for nonlinear systems with uncertainties are considered [5] . While for the quadratic cost functional in the state and control, the optimal state feedback control problem often leads to solving a Hamilton-Jacobi-Bellman (HJB) equation or a nonlinear two-point boundary value (TPBV) problem. But for the general regulation problem of nonlinear systems, with the exception of simplest case, there is no analytic optimal control in explicit feedback form. This has spirited up researchers to develop many methods to obtain an approximate solution to the HJB equations or the nonlinear TPBV problems as well as obtain a suboptimal feedback control [6][7] .
Since TD-SCDMA network control systems based on TCP/IP are an integrated research area, which is not only concerned about control, but also relevant to communication, we must combine the knowledge of control and communication together to improve the system performance. Following this direction, in this paper, we address a novel scheme that integrates control technology with communication technology for a class of nonlinear network control systems [8] .  We consider the TD-SCDMA networked control systems based on TCP/IP consisting of a collection of nonlinear plants whose feedback control loops are closed via a shared network link, as illustrated in Figure 1. All sample values of plant states are transmitted in one package [9] .The k-th plant is given by

II. PROBLEM FORMULATION
where x is an n -dimensional real state vector, u an rdimensional real control vector, B an r n ! constant matrix, 0 x a known initial state vector. Assume that Nonlinear function sequence g may be expanded into the series form where f is the nonlinear term whose order size is larger system (1) may be rewritten as The control objective, in an optimal control sense, is to find a control law ) ( * k u , which may make the quadratic performance index where R is an r r ! positive-definite matrix and Q Q f , are n n ! semi-positive-definite matrices.

III. PRELIMINARIES
As we know, we may get the optimal control law of the quadratic performance index (4) if and only if the system in (1) satisfies the following two-point boundary value problem: with the boundary conditions: Since (5) is a nonlinear two-point boundary problem, in a general way, it is difficult to get the solution whether the exact solution or the numerical solution.
We will propose a sensitivity approach to simplify the two-point boundary value problem in (5) and help get the optimal control law. Construct the following twopoint boundary value problem, in which a sensitive with boundary conditions . Obviously, when 1 = ! , the two-point boundary value problem in (6) (i denotes the i th-order derivative of the series with respect to ! when 0 = ! . In order to guarantee convergence for the series in According to the same reasoning process, we may also get the conclusion that Substituting (7), (11), and (12) into the two sides of (6), we may obtain ! " Substituting (17) into (16), we obtain the 0th-order optimal control law as follows: are known functions which are the solutions obtained in the (i-1)th iteration, two-point boundary problem in (20) is a linear nonhomogeneous one. In order to solve this problem, let V. AN ILLUSTRATED EXAMPLE Consider the optimal control problem for a bilinear model of a TD-SCDMA networked control system based on TCP/IP described by (1) and (3) The state variables 1