In this paper, a new design method with performance improvements of multiloop controllers for multivariable systems is proposed. Precise expression is developed to show the relationship between the dynamic- and steady-state characteristics of the multiloop control system and its parameters. First, an equivalent transfer function (ETF) is introduced to decompose the multivariable system, based on which the multiloop controller parameters are calculated. According to the ETF matrix property, an analytical expression for the PI controller for multivariable systems is derived in terms of substituting the ETF matrix for the inverse open-loop transfer function. In the proposed controller design method, no approximation of the inverse of the process model is needed, implying that this method can be applied to some multivariable systems with high dimensions. The simulation results obtained from several examples demonstrate the effectiveness of the proposed method.
In the actual industrial process, especially in multivariable systems such as petroleum, chemical, and biological, the interconnections introduced by the process configuration and control architecture are very common [
Compared to the single loop system, the multiloop control system’s design issues become very complicated as a result of these interconnections, which make it an open topic for many years. There are many reports on this field in the existing literature. From the perspective of controller structure design, these methods can classify into three categories, namely, decoupling plus single loop controller structure, centralized control structure, and multiloop control structure. The classic multiloop PID controller has been widely used in processes with moderate interaction due to some advantages, such as simple control structure, less adjustable parameters, and easy to understand features. For example, a multiloop controller design method is given by Huang et al. [
For the same target, meanwhile, the multiple time delays commonly occur in multivariable processes of high complexity. Particular examples can be found, such as pilot plant distillation columns and high purity distillation columns. The essence of the reason for time delay is the complex interactions between the numerous different pairs of input and output variables. Therefore, approximating the actual model by a reduced-order form by using some model reduction techniques is necessary, for example, the first-order plus dead time (FOPDT) and second-order plus dead time (SOPDT). On the contrary, for systems with time delays, the internal model control (IMC) method is a very practical approach for the design of IMC-PID controllers. Several results have been discussed and reported with some interesting results, extending the IMC-PID of the single-input and single-output (SISO) case to the multivariable case [
Motivated by the above, a novel decoupling method is proposed for multivariable control systems in this paper. By taking the essential effects of the interactions into account, we use the concept of the equivalent transfer function (ETF) to decompose the multivariable system under consideration into multiple univariate loops equivalently. After that, the properties of the ETF matrix and the target closed-loop diagonal transfer function that was specified to the nonminimum phase zeros and inherent time delays are used to derivate the controller parameters. The relationship between proposed controller parameters and the open-loop transfer function is obtained without the prior acknowledge of the specific ETF model. We have established some equations directly to deduce the parameters of the multiloop PI controller. Finally, the proposed approach is applied to different industrial objects to verify its effectiveness.
In the existing literatures, the idea that decomposes a system of the multiloop controller into several signal loops equivalently is the main approach to design controller for the multiloop system. By doing this, the next step is to design the controller for the transfer function obtained with open-loop feature. To give more details, for ETF loop
Block diagram for the concept of the ETF.
The IMC-PID controller tuning method introduced by Lee et al. [
The IMC filter,
To use the above controller as the standard PID form, we need to approximate the feedback controller
The controller stated by equation (
Through the above descriptions, the concept of the ETF is introduced to solve the controllers design problem. In the following section, we will utilize the property of the ETF matrix to find the important relationship between the parameters of the controller and that of the considered model.
For a multivariable system with closed-loop controllers, we can write the forward transfer function to be of the form [
In the above formula,
Due to the right half plane zeros and time delays in multivariable systems, which make an ideal controller that is equal to the inverse of open-loop transfer function matrix unrealizable, the closed-loop transfer function matrix of the IMC system [
According to the properties of the ETF, we have
Comparing equations (
Then,
In literatures [
From equation (
Similar to equations (
For the convenience of presentation, we simplified
Based on the analysis conventional methods to calculate the PI parameters, which decomposed multivariable systems to a set of multiloop control systems, then utilized single-input and single-output design technique, we establish some equations to deduce and solve the parameters of PI directly and omit the intermediate step, i.e., how to calculate ETF parameters. This section derivates a direct relationship between the parameters of the multiloop controller and that of the open-loop transfer function matrix.
To demonstrate the robust stability and also show its distinctive features, a popular and commonly used method of analysing robust stability is used in the presence of other controller design approaches. We use the output multiplication uncertainties to measure the robustness of the target system instead of input uncertainties, due to its advantage of less restrictive [
The error integration criterion is a performance index, which is computed by the integral of the deviation between the actual output and the expected output. The integral absolute error (IAE) index and integrated time absolute error (ITAE) index are widely used because of their good practicability and selectivity. The IAE is introduced as follows:
To analyse the underlying performance, the IAE is first employed to evaluate the control performance of the target closed-loop systems. After that, the ITAE criterion comes in to give a new indicator showing closed-loop performance from the initial point to time infinity:
The system is optimal, which means the IAE and the ITAE get smaller than the relative methods.
In order to illustrate the control performance and robustness of the controller achieved by the proposed method, three typical industrial examples are used for testing. To be specific, the purpose of Example 1 and Example 2 is to show that the proposed method provides better performance than other existing methods for nominal systems. The main target of Example 3 is to demonstrate the performance of the proposed method to the systems of high dimensions.
Consider the VL process [
We compare the multiloop tuning results obtained by the proposed method with those presented by Lee et al. [
Controller parameters.
Methods | Loop | ||||
---|---|---|---|---|---|
Proposed | 1 | −1.7730 | −0.2710 | 1.72 | 0.59 |
2 | 2.9550 | 0.3436 | 0.75 | ||
Lee | 1 | −1.3100 | 0.5800 | — | 0.53 |
2 | 3.9700 | 1.6400 | — | ||
Chen | 1 | 1.2100 | 0.2600 | — | 0.50 |
2 | 3.7400 | 3.4000 | — |
Figure
Closed-loop step responses for Example 1.
To further investigate the robustness of the proposed method, a perturbation uncertainty of
Performance indices.
Methods | ||||
---|---|---|---|---|
Nominal | +50% | Nominal | +50% | |
Proposed | 6.391 | 6.007 | 22.675 | 19.24 |
Lee | 7.102 | 6.899 | 32.369 | 23.99 |
Chen | 7.370 | 7.506 | 27.066 | 25.23 |
Consider a 3 × 3 multivariable system:
According to equations (
Controller parameters.
Methods | Loop | ||||
---|---|---|---|---|---|
Proposed | 1 | 0.9939 | 0.2976 | 8.00 | 0.07 |
2 | −0.1309 | −0.0471 | 12.0 | ||
3 | 7.5314 | 0.7602 | 1.00 | ||
SAT [ | 1 | 2.7100 | 0.3640 | — | 0.01 |
2 | −0.3660 | −0.0350 | — | ||
3 | 4.5600 | 1.4760 | — | ||
Lee [ | 1 | 0.5930 | 0.1730 | — | 0.06 |
2 | −0.1240 | −0.0430 | — | ||
3 | 3.2200 | 0.4210 | — |
Figure
Closed-loop step responses for Example 2.
Similar to the previous method, we also insert a perturbation uncertainty of
Performance indices.
Method | ||||
---|---|---|---|---|
Nominal | +50% | Nominal | +50% | |
Proposed | 180.9597 | 160.7001 | 4431.40 | 3713.8710 |
SAT | 696.4530 | 2233.640 | 40341.5 | 46454.8606 |
Lee | 274.4191 | 223.4985 | 7456.57 | 5722.0498 |
To further illustrate the effectiveness of the proposed method in high-dimensional systems, a system of four dimensions is introduced to verify the control performance and robustness. The corresponding transfer function of A2 system [
Using equations (
Controller parameters.
Methods | Loop | |||||
---|---|---|---|---|---|---|
Proposed | 1 | 2.4881 | 0.0915 | — | 7.0 | 0.15 |
2 | 3.2068 | 0.2241 | — | 2.0 | ||
3 | 4.3252 | 0.1664 | — | 1.0 | ||
4 | 5.7641 | 0.0340 | — | 5.0 | ||
He | 1 | 3.8840 | 0.0940 | 25.75 | — | 0.02 |
2 | 2.5490 | 0.0570 | — | — | ||
3 | 1.3110 | 0.0710 | — | — | ||
4 | 4.2330 | 0.0780 | 23.57 | — | ||
Shen | 1 | 3.0503 | 0.0739 | 20.23 | — | 0.03 |
2 | 2.5020 | 0.0561 | — | — | ||
3 | 1.5450 | 0.0835 | — | — | ||
4 | 1.7530 | 0.0634 | 19.18 | — |
The curves in Figure
Closed-loop step responses for Example 3.
Table
Performance indices.
Methods | ||||
---|---|---|---|---|
Nominal | Nominal | |||
Proposed | 234.4048 | 219.1850 | 16824.34 | 14245.40 |
He | 278.6150 | 447.8400 | 10393.18 | 20731.42 |
Shen | 316.1688 | 323.3098 | 25146.40 | 22231.41 |
In this paper, an effective controller design method for multivariable systems has been developed. Based on the properties of the ETF and the IMC design principle, the decentralized PI controller is established directly by the proposed method. This method analytically derivates the relationships between controller parameters and the open-loop process transfer function. Three different simulation examples have proved the effectiveness of this method and it also can be applied to high-dimensional system. Further work will focus on designing the decentralized controller for nonsquare multivariable systems.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the Natural Science Foundation of China, under Grant no. 61773183.