This paper tackles the problem of simultaneous estimation of the state and the unknown disturbance of an MIMO disturbed system and designs the disturbance rejection controller according to the estimation information. Through a series of transformations, we can transform the original system into two subsystems and then propose a sliding mode observer and a descriptor system form observer, respectively. Our algorithm can simultaneously estimate the state and the unknown disturbance. The estimation error is shown to be bounded within a small region. Moreover, the controller algorithm developed in this paper can effectively avoid the peaking phenomenon. Finally, the feasibility and the performance using the proposed method are analyzed and demonstrated with two simulated examples.

Disturbances coming from the environment often constitute an annoyance in the operation of dynamic systems. Disturbance rejection control (DRC), in which the controller is designed to suppress the disturbance’s effect, is the major concern in the design of feedback control systems. Since external disturbances are usually not accessible for measurement, in the early development of disturbance rejection control, high gain control is used to suppress the unknown disturbance. Examples of high-gain controllers include the geometric control algorithm [

There is another DRC method in which an observer algorithm is first proposed to estimate the unknown disturbance and then cancel the disturbance’s effect by the control input. The advantage of this approach is that the disturbance canceling control does not need to be high gain. There are various formulations related to the unknown disturbance estimation. According to the transfer function approach, the disturbance observer is known to be very effective in compensating disturbances [

For the linear system having both unknown disturbance and measurement noise, we develop an observer design method, which can be successfully implemented in systems with unstable invariant zeros. With reference to [

In the next section, a class of linear systems having both unknown disturbances and sensor noises is first introduced with three assumptions in relation to the system matrices. Section

Consider an MIMO system with both unknown disturbance and measurement noise

The pair

System matrices

The unknown disturbance and the measurement noise have the upper bound.

For linear systems with unknown inputs in the state and output equations, previous studies [

When system (

If system (

Since system (

Following the above procedures, we know that system (

Consider system (

First, we show that the pair

We know from Theorem

Since the pair

For subsystem (

Recall that most disturbance rejection controllers are of high-gain control design, which has the disadvantage of peaking phenomenon in control input. An important merit of the proposed method is that it can avoid the peaking phenomenon when applied to DRC. In high-gain control design, the magnitude of the disturbance estimate

In order to deal with nonminimum phase systems, Chen [

First, it follows from Section

If the pair

Following the aforementioned transformations, we have

Let

Consider system (

We show that the pair

Since the pair

To demonstrate the effectiveness of the proposed method, the sensor noise and the unknown disturbance are introduced into the system and its state space form is given by

True and estimation values of

True and estimation values of disturbance for Example

Responses of system states

Control input for Example

To demonstrate the proposed observer design method in Section

True and estimation values of

True and estimation values of

System outputs

System inputs

True and estimation values of

True and estimation values of

For a class of MIMO systems with the unknown disturbances and the measurement noises, this paper has developed an observer design method consisting of the sliding mode observer and the descriptor system transformation. The proposed estimation method can simultaneously reconstruct the system state and the unknown disturbance even though the system is nonminimum phase with respect to the relation between the output and the unknown disturbance. The robust stability of the estimation dynamics can be guaranteed and the estimation error is shown to be bounded within a small region. Compared with conventional high-gain disturbance rejection controllers, our controller can avoid the peaking phenomenon and does not require the derivative of output. Simulation results demonstrate that the proposed control scheme exhibits reasonably good system performance.

The author acknowledges the financial support provided by the National Science Council of Taiwan, Taiwan under Grant NSC 99-2221-E-161-007.