This paper proposes a multiinput and multioutput (MIMO) quasi-autoregressive eXogenous (ARX) model and a multivariable-decoupling proportional integral differential (PID) controller for MIMO nonlinear systems based on the proposed model. The proposed MIMO quasi-ARX model improves the performance of ordinary quasi-ARX model. The proposed controller consists of a traditional PID controller with a decoupling compensator and a feed-forward compensator for the nonlinear dynamics based on the MIMO quasi-ARX model. Then an adaptive control algorithm is presented using the MIMO quasi-ARX radial basis function network (RBFN) prediction model and some stability analysis of control system is shown. Simulation results show the effectiveness of the proposed control method.
Nonlinear system control has become a considerable topic in the field of control engineering [
Due to the complexity of nonlinear Multi-Input and Multi-Output (MIMO) systems, most of the control techniques developed for SISO systems cannot be extended directly for MIMO systems. One of the main difficulties in MIMO nonlinear system control is coupling problem. As such, it is important to investigate the realization of decoupling control. Many adaptive decoupling control algorithms have been proposed to deal with coupling in nonlinear system based on linear methods and nonlinear networks [
In our previous work, a quasi-autoregressive exogenous (ARX) model with an ARX-like macromodel part and a kernel part was proposed, and a controller was designed for SISO systems [
The SISO model and control methods based on quasi-ARX model cannot directly be applied to MIMO nonlinear systems. Motivated by the above discussions, an MIMO quasi-ARX model is first proposed for MIMO nonlinear systems and then a nonlinear multivariable decoupling PID controller is proposed based on the MIMO quasi-ARX model, which consists of a traditional PID controller with a decoupling compensator and a feed-forward compensator for the nonlinear dynamics based on the MIMO quasi-ARX model. Then an adaptive controller is presented using the MIMO quasi-ARX RBFN prediction model. The parameters of such controller are selected based on the generalized minimum control variance. In this paper, quasi-ARX RBFN model is divided into two parts: the linear part is used to guarantee the stability and decoupling, and the nonlinear part is used to improve the accuracy.
The paper is organized as follows: in Section
Consider an MIMO nonlinear dynamical system with input-output relation as
Under the continuous condition, the unknown nonlinear function
However, we need to get
Now, introduce two polynomial matrices
By (
The coefficients matrices
Similar with [
Then, the linear and nonlinear expression of system (
Now, we will propose an MIMO quasi-ARX RBFN model. However, the
Introduce the following performance index:
The optimal control law minimizing (
The controller (
The multivariable decoupling PID control system based on MIMO quasi-ARX model.
A velocity-type form of the PID controller is given:
Now let us initialize
If the process is known,
Parameter vector
The parameter
Comparing with [
There are some assumption made.
(i)
For the MIMO nonlinear (
The nonlinear part estimation error vector can be described by
We can see that, if the nonlinear decoupling PID controller (
From (
Then, the boundness of all the signals in the closed-loop system is got.
The tracking error of the system is obtained as
In order to study the behavior of the proposed control method, a numerical simulation is described in this section. The MIMO nonlinear system to be controlled is described by
In this example, a system disturbance appears when
In this example, the proposed control method in Sections
Under the same simulation conditions and with the same parameters value, the control output results by a typical PID controller are given for comparison, where the PID controller has neither the decoupling compensator nor the nonlinear part. The control outputs are shown in Figure
Control outputs.
Corresponding control inputs of the PID control method.
Corresponding control inputs of the proposed control method.
Table
Comparison results of errors based on two control method.
Mean of errors | Variance of errors | |
---|---|---|
0.0132 | 0.1350 | |
: proposed method | −0.0090 | 0.0668 |
−0.0067 | 0.0157 | |
: proposed method | −0.0039 | 0.0098 |
In this paper, an MIMO quasi-ARX model is first introduced, and a nonlinear multivariable decoupling PID controller is proposed based on the proposed model for MIMO nonlinear systems. The proposed controller consists of a traditional PID controller with a decoupling compensator and a feed-forward compensator for the nonlinear dynamics based on the MIMO quasi-ARX model. And an adaptive control system is presented using the MIMO quasi-ARX RBFN prediction model. The parameters of such controller are selected based on the generalized minimum control variance. The proposed control method has more simplicity structures and better control performance. The nonlinear part is not a black box whose parameters can be determined by