Bilinear system is a class of systems that is derived by introducing the interactive product term of the state variable and the control variable in the linear state equations. It is a rather special nonlinear system, which exists widely in the engineering community of electric, mechanical, biological, chemical and other fields. The study of the bilinear system starts in the 50’s, then becomes an important branch of nonlinear system and made a series of research results. Fang proposed a method of research the stability of MIMO bilinear system [1]; the output-feedback control for bilinear system was studied by Sasaki [2]; a global feedback stability analysis method of bilinear system was presented by Jerbi [3]. In the real Industrial process control, time-delay is ubiquity and the mathematical models put forward from engineering technology, physical, chemical and biomedical had obvious delay amount, which can not be neglected in some accurate control systems. Tang had studied optimal disturbance rejection problem for time-delay system in recent years [4]. Meanwhile, various forms of external disturbances exist on the control system, such as sinusoidal disturbance, periodic perturbation, step disturbance, etc. So it has important actual meanings to study time-delay bilinear system affected by external disturbance [5].

This paper proposed an optimal control design method for time-delay bilinear systems affected by sinusoidal disturbances based on state-feedback linearization [6,7]. Based on the differential homeomorphism, the model of the system that is researched is changed to a time-delay pseudo linear system through the coordinate transition, then through delay-free transform, the time-delay pseudo linear system is converted to an easy pseudo linear system. At last, an optimal control law which is used to eliminate the influence of the disturbances is derived from a Riccati equation and Matrix equations [8,9]. It is shown that the method is easy to realize and has a good convergence.

Consider time-delay bilinear system affected by sinusoidal disturbances

where, is the state vector; is the control vector; is the external disturbances vector; is the output vector; A, B, D, N_j are scalar matrixes of appropriate dimensions; x_j is the j-th component of state vector; is the bilinear term; is the scalar function of x.

Assumption 1 The external disturbances can be expressed as

which is an m-dimensional sinusoidal vector with known frequencies, amplitudes and phases are measurable. By transformation, time-delay bilinear system could be changed as

Change the bilinear system to the general expression of nonlinear system

where, , and j, g are continuously differentiable functions.

Assumption 2 The relation degree of the nonlinear system is r, that is

Through exact delay-free transformation, the nonlinear system (4) can be converted to an easy pseudo linear system (6)   

is the new state vector.

Then based on the theory of linear quadratic optimal control, an optimal control law which is used to eliminate the influence of the disturbances is derived.

The time-delay bilinear system (1) is changed to the equivalent delay-free linear system affected by disturbances

where, is the new state vector; is the new control vector; A is state coefficient matrix; B is control coefficient matrix; the system (6) is completely controllable.

When select the quadratic performance index of system (6) as

where Q is a positive semi-definite matrix, R is a positive definite symmetric matrix.

Theorem: Consider the optimal control problem of system (6) with the quadratic performance index (14). Suppose the system is completely controllable and observable, the optimal disturbance rejection control law is unique existence and can be expressed as

P is the unique positive definite solution of the matrix Equation (16)

P₁ is the unique solution of the matrix Equation (17)

P₂ is the unique solution of the matrix Equation (18)

where

Proof: According to the necessary conditions of the optimal control problem based on maximum principle, the optimal control law of system (6) with the quadratic performance index (14) can be expressed as

where is the solution of the following two-point boundary value problem

In order to obtain the solution to the problem in (14), let

where, are both undetermined matrix.

From expression (2), the vector function satisfies the following vector differential equation

Taking the derivatives to the sides in (21) and substituting (22) into it get (23)

Add (23) into the first expression of (20), obtain (24)

Considering for any and, Equation (24) is all hold, so we can get matrix Equations (16)-(18). The optimal control law of the bilinear systems affected by sinusoidal disturbances is as follow

Consider a time-delay bilinear system with sinusoidal disturbances, system parameters as follows

According to calculation, through local linearization, we can get a two-step linear system affected by sinusoidal disturbances

where

The form of the disturbances described as follows

Using model transformation, change the system with time-delay to a linear controllable system without delay

where,.

Select, , we get

The simulation results of, and are presented in Figures 1 and 2.

It is shown that the optimal control design method for the system affected by sinusoidal disturbances based on state-feedback linearization is easily realized, and has a good convergence through the simulation results.

This paper concentrates on the solution of the optimal control problem for time-delay bilinear system affected by sinusoidal disturbances with known frequency and measurable amplitude and phase based on state feedback, a method of optimal control law is given. Simulation re-

sults show that the designed control law is accurate and easy to implement.