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In this paper, a MIMO PI design procedure is proposed for linear time invariant (LTI) systems with multiple time delays. The controller tuning is established in two stages and guarantees performances for set-point changes, disturbance variations, and parametric uncertainties. In the first stage, an iterative linear matrix inequality (ILMI) approach is extended to design PI controllers for systems with multiple time delays without performance guarantee, a priori. The second stage is devoted to improve the closed-loop performances by minimizing sensitivity functions. Simulations results carried out on the unstable distillation column, the stable industrial scale polymerization (ISP) reactor, and the non-minimum phase 4-tank benchmark prove the efficiency of the proposed approach. A comparative analysis with the conventional internal model control (IMC) approach, a multiloop IMC-PI approach, and a previous ILMI PID approach proves the superiority of the proposed approach compared to the related ones.

PID controllers have been at the heart of control engineering practice for several decades [

On the other hand, iterative linear matrix inequalities (ILMIs) are known to be powerful tools to solve multivariable control problems. Particularly, ILMI approaches were already used to design PID controllers for LTI systems without delays [

As Loop Shaping (LS) techniques [

To illustrate the effectiveness and the performances of the proposed approach, three examples of multiple time delay systems including unstable, stable, and non-minimum systems are considered. A comparative analysis with related approaches is also given to prove the superiority of the proposed approach.

The paper is organized as follows: The problem formulation is stated in Section

Consider a nominal multivariable LTI system with multiple time delays described by

The objective is to design a finite dimensional PI controller described by

Let

For such PI controller there are

The last control problem is very complex since system (

In this stage, the infinite dimensional system (

The set-point vector

The PI controller (

The finite dimensional closed-loop dynamics

To this end, the PI design procedure is proposed in Section

The objective of the subproblem 2 is to design a shaped controller described by

The most crucial part of the design procedure is to find the appropriate weighting matrices

Each delayed variable can be modeled as a distributed parameter system described by a partial differential equation as follows [

Modeling a delayed variable via a distributed parameter system.

For numerical simulation or control design purposes, an infinite dimensional system is generally reduced to a finite dimensional system by using an approximation method. Within the framework of weighted residuals methods, the orthogonal collocation method is applied in this paper to approximate the partial differential equations described by relation (

The principle of the orthogonal collocation method is to search a finite dimensional approximation for the distributed parameter variable

By applying delay variable approximation on each delayed variable of the vectors

Let consider Cauchy’s formula for the interpolation error defined by Lefèvre et al. [

Hence, we try to choose the interior collocation points

By considering the case study of the Chebyshev polynomials belonging to the family of Jacobi polynomials, and, corresponding to the values of the parameters

In the following, the SOF transformation of the PI controller of the delayed system (

The multivariable LTI system with multiple time delays (

Due to the term

The following is a constructive ILMI algorithm for PI control of LTI MIMO with multiple delays systems, and the explanations are given in Remark

OP1: Minimize

OP2: Minimize

Due to the bad performances generated by initial data selection in ILMI algorithms [

In this section, a modified approach of Loop Shaping technique [

Block diagram of the controlled system.

Let us define the input loop transfer matrix,

Block diagram of a controlled shaped system.

Loop Shaping design procedure.

If

Note that the final PI controller designed in Algorithm

It must be noted that there are severe limitations when the conventional Loop Shaping design procedure is used for MIMO systems as discussed in [

In this section, simulation results will be performed using three typical examples: the distillation column (unstable system), the ISP reactor (stable system), and the 4-tank process (non-minimum phase system). Furthermore, we will illustrate the superiority of the proposed approach over related ones for set-point tracking, disturbance rejection, and parametric uncertainties scenarios. The comparative study will be established between the following approaches:

The proposed PI controller designed via the Algorithms

The PI controller designed via Algorithm

The PID controller designed in [

IMC-PI controller approach [

The conventional IMC-PID approach [

For the different simulations, unit step changes in the set-points and disturbances are made to the 1st and 2nd loops. Furthermore, the robustness of the controller is evaluated by considering a perturbation uncertainty of ±10% in the important parameters, particularity, gains, and delays of the process.

Just for the second example, we will prove that the most conventional multiloop IMC-PID control approach proposed by Economou and Morari [

For systems with given transfer matrix, the passage from the matrix transfer to a minimal state space model is established using Gilbert method detailed in [

For the orthogonal collocation method, optimal parameters are chosen such as

To prove the validity of the transformation between the state space representation and the corresponding transfer matrix and the approximation of the delayed system, let us introduce the following errors: let

Due the bad performances obtained via the PID controller designed via the approach given in [

Consider the typical example of the distillation column described in [

Using the orthogonal collocation method, the following matrices are obtained for model (

Models validation of the distillation column.

Figure

Singular values of

The resulting performance indices for the proposed multivariable controller and the one computed by ILMI method for the nominal and perturbed system cases are summarized in Tables

Comparative analysis of controller’s performances: the distillation column case study.

Tuning method | Set-point | Disturbance | ||
---|---|---|---|---|

IAE | TV | IAE | TV | |

Proposed | 175.16 | 19.82 | 453.14 | 19.77 |

PI controller | 494.05 | 1.810 | 1366.10 | 6.69 |

Robustness analysis under ±10% in the gain and delay: the distillation column with input delays.

Tuning method | The distillation column with input delays (+10%) | The distillation column with input delays (−10%) | ||||||
---|---|---|---|---|---|---|---|---|

Set-point | Disturbance | Set-point | Disturbance | |||||

IAE | TV | IAE | TV | IAE | TV | IAE | TV | |

Proposed | 166.17 | 60.42 | 453.14 | 19.77 | 189.34 | 11.97 | 371.23 | 15.83 |

PI controller | 457.45 | 1.82 | 1366.10 | 6.69 | 537.84 | 1.80 | 1118 | 5.18 |

Consider the ISP reactor system described by its transfer matrix given by Chien et al. [

Let us first test for the previous system the conventional IMC-PID approach proposed by Economou & Morari [

Interaction measures of the ISP reactor via the IMC-PID approach [

Pairing | IMC interaction measure | |
---|---|---|

| | |

| | |

| | |

| | |

| | |

Let us now apply the extended IMC-PI controller approach proposed by Vu and Lee [

Multiloop PI controller design of the ISP reactor via the IMC-PI approach [

Loop | | |
---|---|---|

1 | 0.4211 | 0.1068 |

2 | 0.1320 | 0.1121 |

For

Comparative analysis of the controller’s performances: the ISP reactor case study.

Tuning method | Set-point | Disturbance | ||
---|---|---|---|---|

IAE | TV | IAE | TV | |

Proposed | 20.24 | 10.74 | 123.08 | 2.66 |

PI | 174.42 | 1.97 | 224.96 | 1.33 |

PID [ | 440.57 | 4.93 | 417.66 | 1.48 |

IMC-PI [ | 3.97 | 1.98 | 49.05 | 9.13 |

Robustness analysis under ±10% parametric uncertainties: The ISP reactor case study.

Tuning method | ISP (+10%) | ISP (−10%) | ||||||
---|---|---|---|---|---|---|---|---|

Set-point | Disturbance | Set-point | Disturbance | |||||

IAE | TV | IAE | TV | IAE | TV | IAE | TV | |

Proposed | 19.75 | 8.72 | 120.67 | 2.90 | 21.28 | 9.27 | 127.46 | 2.54 |

PI | 158.68 | 1.95 | 224.06 | 1.33 | 193.50 | 1.94 | 225.85 | 1.33 |

PID [ | 404.24 | 4.82 | 415.56 | 1.28 | 482.68 | 4.94 | 418.84 | 1.52 |

IMC-PI [ | 4.36 | 2.05 | 48.33 | 8.26 | 3.58 | 1.92 | 49.95 | 10.32 |

Consider the quadruple-tank process for which one of the two transmission-zeros of the linearized system dynamics can be moved between the positive and negative real axis [

To investigate the validity of the 4-tank process, the different model errors are depicted in Figure

Comparative analysis of controller’s performances. The non-minimum phase 4-tank process.

Tuning method | Set-point | Disturbance | ||
---|---|---|---|---|

IAE | TV | IAE | TV | |

Proposed | 479.04 | 119.47 | 98.24 | 15.20 |

PI | 4005.50 | 39.10 | 105.91 | 11.48 |

PID [ | 4100.80 | 338.06 | 105.22 | 6.75 |

Robustness analysis under ±10% in the gain and delay: the non-minimum phase 4-tank process case study.

Tuning method | 4-tank process (+10%) | 4-tank process (−10%) | ||||||
---|---|---|---|---|---|---|---|---|

Set-point | Disturbance | Set-point | Disturbance | |||||

IAE | TV | IAE | TV | IAE | TV | IAE | TV | |

Proposed | 418.26 | 120.51 | 99.43 | 15.84 | 510.06 | 98.66 | 94.85 | 13.21 |

PI | 3678.10 | 37.14 | 106.96 | 11.99 | 4383.80 | 41.88 | 102.31 | 11.58 |

PID [ | 3768.90 | 333.93 | 106.24 | 7.04 | 4482.50 | 339.06 | 101.65 | 5.76 |

Furthermore, noise rejection in high frequencies is also known as an important requirement in a control system design. In order to evaluate the effect of such noise on the closed-loop performances of the most complex examples considered in this paper, simulation results have been conducted taking into account of White Gaussian Noise Measurements (WGNM) with a variation of 0.01 V and zero mean. It is apparent from Figure

Equations (

Input and output sensitivity matrices of the shaped model for the distillation column.

Closed-loop responses and controller output responses to set-point changes for the distillation column.

Closed-loop responses and controller output responses to unit step changes in the disturbance for the distillation column.

IMC interaction measures for the ISP reactor: fail of the conventional IMC-PID approach.

Models validation of the ISP reactor.

Singular values of

Input and output sensitivity matrices of the shaped model for the ISP reactor.

Closed-loop responses and controller output responses to set-point changes for the ISP reactor.

Closed-loop responses and controller output responses to unit step changes in the disturbance for the ISP reactor.

Models validation of the 4-tank process.

Singular values of

Input and output sensitivity matrices of the shaped model for the 4-tank process.

Closed-loop responses and controller output responses to set-point changes for the 4-tank process.

Closed-loop responses and controller output responses to unit step changes in the disturbance for the 4-tank process.

Closed-loop responses and controller output responses without WGNM and under WGNM.

Previous results can be summarized as follows:

Even more the controller design procedure has not considered decoupling principle of multivariable systems; the proposed approach provides generally superior performances by the smallest total IAE for the set-point changes, disturbance changes and parametric uncertainties, over related approaches, for the unstable distillation column, the ISP reactor, and the 4-tank process as summarized by Tables

The proposed method succeeds to synthesize a MIMO PI controller for the ISP reactor when the IMC-PID approach proposed by Economou and Morari [

The proposed method is applicable to the 4-tank process where the IMC-PI approach [

The proposed method is applicable to the unstable distillation column with input delays where the IMC-PI approach proposed by Vu and Lee [

For the 4-tank process, some large IAE and TV values listed are due to non-minimum phase system characteristics; it is obvious that non-minimum zero dynamics cause performance deterioration of the closed-loop system responses (initial undershoot, overshoot and zero crossings) and then increase IAE and TV performances indices.

This paper presents a MIMO PI controller design procedure for LTI MIMO systems with multiple time delays by means of ILMI and sensitivity functions. The proposed Loop Shaping design procedure with minimizing sensibility functions yields to optimized closed-loop system responses. The distillation column, the ISP reactor, and the 4-tank process as benchmarks of unstable, stable, and non-minimum phase systems are provided to illustrate the validity, effectiveness, and robustness of the proposed method. Considering different case studies (set-point tracking, disturbance rejection, and parametric uncertainties), a comparative analysis between the proposed method and related ones showed that the proposed method afforded the superior performances both in the nominal and in the perturbed case studies.

The authors declare that there is no conflict of interests regarding the publication of this paper.