Model predictive control (MPC) with its lower request to the mathematical model, excellent control performance, and convenience online calculation has developed into a very important subdiscipline with rich theory foundation and practical application. However, unmeasurable disturbance is widespread in industrial processes, which is difficult to deal with directly at present. In most of the implemented MPC strategies, the method of incorporating a constant output disturbance into the process model is introduced to solve this problem, but it fails to achieve offset-free control once the unmeasured disturbances access the process. Based on the Kalman filter theory, the problem is solved by using a more general disturbance model which is superior to the constant output disturbance model. This paper presents the necessary conditions for offset-free model predictive control based on the model. By applying disturbance model, the unmeasurable disturbance vectors are augmented as the states of control system, and the Kalman filer is used to estimate unmeasurable disturbance and its effect on the output. Then, the dynamic matrix control (DMC) algorithm is improved by utilizing the feed-forward compensation control strategy with the disturbance estimated.

Model predictive control (MPC) is a control algorithm developed in the process of industrial production, for it can deal with physical constraints and linear multivariable systems flexibly in an online optimal way [

Although MPC has many advantages compared to PID control, a weakness still exists: when the control system is influenced by unmeasurable disturbance or modeling errors, and the closed-loop system cannot achieve the expected control performance. In fact, the dynamic matrix control (DMC) has certain limitation. Firstly, the feedback correction is obtained by assuming that there exists a constant output disturbance. Although this method can be done without static error control and is easy to achieve, there is the lack of the advanced estimation for the dynamic characteristics of unmeasurable disturbance, so the effect of disturbance cannot be eased. Secondly, achieving the offset-free control in DMC needs an enough accurate system model even though DMC has the characteristics of multistep prediction and receding optimization. Therefore, the modified DMC makes a great difference in industrial production. The authors in [

In addition to all the methods mentioned above, aiming at the shortcomings of the existing DMC algorithms, we introduce a better performance than the constant output disturbance model which is a more general disturbance model, and an online estimation unmeasurable disturbance Kalman filter is designed. Meanwhile, the offset-free sufficient conditions of modified DMC are shown based on the disturbance model. A simulation example for the continuous stirred tank reactor (CSTR) in industrial production illustrates the effectiveness of the improved approach.

The approach to eliminate the steady-state error is achieved by assuming that the plant is influenced by the step disturbance of constant value and augmenting disturbance into the process model. It is usually assumed that the disturbance is constant, which is estimated based on measurable variables; then the influence of the controlled variable is eliminated through changing the steady-state object of the controller. Although this approach avoids the limitation of the saturation factors, it is required to design interference model and estimate the disturbance [

If the unmeasurable disturbance or modeling error exists in the control system, the closed-loop can never realize offset-free control. It is necessary to find a method to eliminate the steady-state error under the circumstance of unmeasured disturbance, which is the key to achieve the offset-free control.

Consider the linear time-invariant discrete system:

In this framework, the input disturbance can be conveniently described, that is,

The performance of the closed-loop control system directly depends on the accuracy of the disturbance model characterizing the actual system. Therefore, in order to improve the control performance, a disturbance model, which is matched with the entering process, is adopted so as to maximize the accuracy of the disturbance model. The model is shown as follows:

The state disturbance and output disturbance exist in the model simultaneously. If

From the augmented system state estimator in (

Due to the fact that the augmented system is measurable and

With regard to the steady-state control input

Equation (

It can be seen that the controller in steady-state is unconstrained; therefore, the control input can be calculated by the method of infinite horizon controller:

The nominal stability of the feedback gain implies that the matrix

It indicates

The following method will be used to judge whether the augmented matrices

Hantus theorem points out the sufficient and necessary condition of measurable augmented system in [

It only needs to check

the augmented system

DMC is a kind of MPC algorithm based on the step response coefficient model; this paper will present detailed design methods and steps of constrained multivariable DMC algorithm to achieve offset-free control; the improved algorithm not only realizes the offset-free control, but also has the strong ability of disturbance suppression.

From the theory of the DMC algorithm, the estimated

Supposing that only the

Using the feed-forward compensation control strategy with (

If all of the inputs are likely to be affected by the unmeasurable disturbance, then adopting the feed-forward compensation control strategy and considering the

Thus, all the inputs are compensated in the way of feed-forward and all the outputs are considered; each

Consider the

Firstly, let

When the system is influenced by the unmeasurable disturbance, the actual system can be seen as

Consider all of the outputs:

In the control law of (

The above algorithm is based on the uncontrollable input disturbance, and the feed-forward structure of output disturbance can be designed [

Consider that the

If

In this way, the perdition error

Therefore, the design of constrained multivariable offset-free controller is completed. Not only does the above method has good expansibility and is easily applied to control system, but also it significantly improves the performance of the algorithm [

CSTR is widely used in the process of polymer production equipment, and it is the most extensive chemical reactor used in industrial production process.

The dynamic CSTR system can be described as follows:

The operating points are

According to the designed offset-free controller, the parameter values are the length of model

The situation contains two unmeasurable input disturbances:

Figure

True and estimated value of the input disturbance.

Figure

Output of algorithm in constrained condition with input disturbance.

It can be seen that the control increments and inputs in Figures

Figures

Output

Output

The difference between the algorithm and the algorithm in constrained condition is that the calculation method of the control increment; the former is obtained by solving the QP problem, and the latter is obtained by solving extreme conditions. Certainly, the computation burden of unconstrained algorithm is lighter than the constrained algorithm. But, in order to obtain the maximum economic benefits with the constrained condition, the constrained algorithm has greater application value. The DMC algorithm has stronger ability of interference suppression; it can guarantee that the system realizes offset-free control in the presence of unmeasurable disturbance. In addition, the predictive controller is designed in the linear method, which reduces the computation burden and guarantees the system performance.

The authors declare that they have no competing interests.