Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1241545, 25 pages

https://doi.org/10.1155/2017/1241545

## MIMO PI Controllers for LTI Systems with Multiple Time Delays Based on ILMIs and Sensitivity Functions

National Institute of Applied Sciences and Technology (INSAT), Centre Urbain Nord, BP 676, 1080 Tunis Cedex, Tunisia

Correspondence should be addressed to Olfa Boubaker

Received 12 August 2016; Revised 9 November 2016; Accepted 5 December 2016; Published 20 February 2017

Academic Editor: Tamas Kalmar-Nagy

Copyright © 2017 Wajdi Belhaj and Olfa Boubaker. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

PID controllers have been at the heart of control engineering practice for several decades [1, 2]. They are widely used in industrial applications as no other controllers match simple control structure, fewer tuning parameters, and robustness against uncertainties. However, until now, a high percentage of PID control systems seem to be badly tuned and many difficulties occur essentially when the multi-input multi-output systems are considered [3–5]. One major reason may be explained by coupling interactions between the different loops and mainly the negligence of uncertain and immeasurable dead times. Tuning multiloop PID controllers for LTI systems with multiple time delays [6–9] is then considered until now as a challenging problem in control theory. In this framework, the internal model control (IMC) method is considered as the most conventional and effective approach for PID controller design while taking into account time delays [10–12]. The design of MIMO IMC-PID controllers is based on a series of SISO controllers using IMC interaction measures between the different loops. This method becomes very hard when the number of inputs/outputs increases. Even more, its implementation may fail when the interaction measures between the different loops are so high. The last difficulty represents the main disadvantage of this method and an alternative solution was proposed by Vu and Lee [13] to solve such a problem. Unfortunately, this result remains applicable only when the first-order MIMO systems are considered.

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 [14–18]. The basic idea was based on transforming the PID controller into an equivalent static output feedback (SOF) stabilization one by augmenting, using some new state variables, the dimension of the controlled system. Unfortunately, such controllers are known by their bad performances compared to those designed via IMC approaches (when applicable).

As Loop Shaping (LS) techniques [19, 20] are well known for their abilities to improve the closed-loop system performances by minimizing the signal transmission from load disturbances and measurement noise to input and output process or in terms of requirements on the sensitivity functions and/or complementary sensitivity functions [21–23], this paper suggests using this concept for improving the MIMO PI controller performances computed via ILMIs. The proposed approach overcomes the problems introduced by the well known IMC method when the fully cross-coupled multivariable systems are considered. Its implementation requires two steps: in the first step, the ILMI method proposed by Zheng et al. [14], appropriate for systems without delays, is extended for the design of PI controller for multiple time delay systems. As such approach generally gives bad performances, a second stage is then launched in order to improve the performances of the closed-loop system by shaping the already designed PI controller by minimizing the sensitivity function of the system.

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 2. Model reduction of the MIMO system with multiple time delays is detailed in Section 3. Section 4 is devoted to the main results. Section 5 shows the validity of the proposed approach where a comparative study with related approaches using typical examples for set-point tracking, disturbance rejection, and parametric uncertainties scenarios is considered.

#### 2. Problem Statement

Consider a nominal multivariable LTI system with multiple time delays described bywhere , , and are the state vector, the control vector, and the output vector, respectively. , , , , and are known constant matrices. , , and are time delays.

The objective is to design a finite dimensional PI controller described by where , , is the set-point vector and , are proportional and time integral gain matrices, respectively, that stabilize the system (1) to the set-point vector.

Let be the general transfer matrix of the delayed system (1), computed as described in [24], and the transfer matrix of the PI controller given bywhere is the th element of the transfer matrix , is the proportional gain of the th element of , and is the integral gain of the th element of ,

For such PI controller there are parameters to be tuned for a plant with inputs and outputs.

The last control problem is very complex since system (1) is a MIMO infinite dimensional system. To be relaxed, the control problem will be organized in two subproblems.

##### 2.1. Subproblem 1: Design a Finite Dimensional PI Controller for Just SOF Stabilization

In this stage, the infinite dimensional system (1) will be reduced to the finite dimensional system (4)-(5) whereas the PI controller (2) will be transformed into the SOF controller (6) described, respectively, bywhere , , and are the sate vector, the control vector, and the output vector of the approximated system, respectively. , , and are matrices related to the approximated system to be computed using the approximation method and the SOF transformation and and is the SOF feedback gain matrix, to be designed under the following assumptions.

*Assumption 1. *The set-point vector in (2) is assumed to be null .

*Assumption 2. *, , and are assumed to be uncertain but constant delays.

*Assumption 3. *The PI controller (2) is well-posed.

*Assumption 4. *The finite dimensional closed-loop dynamics with a realization is stabilizable via SOF controller.

To this end, the PI design procedure is proposed in Section 4.1.

##### 2.2. Subproblem 2: Set-Point Stabilization and Output Disturbance Attenuation by Minimizing Sensitivity Functions

The objective of the subproblem 2 is to design a shaped controller described bythat improves the closed-loop response considering a shaped system described by where and are a pre- and postcompensators to be chosen in order to satisfy, in closed-loop, performance specifications such as set-point stabilization and load disturbance rejection.

The most crucial part of the design procedure is to find the appropriate weighting matrices and . Note that the shape of the weights is determined by the closed-loop design specifications. Once the desired loop shape is achieved, the final controller to be applied to the nominal transfer matrix is then constructed. To this end, a Loop Shaping design procedure is proposed in Section 4.2.

#### 3. Model Reduction of the MIMO System with Multiple Time Delays

Each delayed variable can be modeled as a distributed parameter system described by a partial differential equation as follows [25]:with the boundary condition and the output equations: where and are time and pseudospace variables, respectively. As shown by Figure 1, , , and are the input, the state variable, and the output of the delay block, respectively. is a constant time delay.