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Research Article | Open Access
Wajdi Belhaj, Olfa Boubaker, "MIMO PI Controllers for LTI Systems with Multiple Time Delays Based on ILMIs and Sensitivity Functions", Mathematical Problems in Engineering, vol. 2017, Article ID 1241545, 25 pages, 2017. https://doi.org/10.1155/2017/1241545
MIMO PI Controllers for LTI Systems with Multiple Time Delays Based on ILMIs and Sensitivity Functions
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 [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  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. , 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 , 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 :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.
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 (9) augmented by boundary conditions (10)-(11) for its simplicity since it avoids integration .
The principle of the orthogonal collocation method is to search a finite dimensional approximation for the distributed parameter variable in the following form :where denotes the approximation of ; is the order reduction; are unknown time-varying coefficients chosen such that the approximated solution is the exact one at the collocation points such that and are the th order Lagrange interpolation polynomials; that is:where are the collocation points of the method. In this paper, the internal collocation points are considered as the zeros of the th order Jacobi polynomial defined for by Lefèvre et al.  as follows:with and where coefficients and are defined as follows:where and are two constant parameters affecting the position of the collocation points.
By applying delay variable approximation on each delayed variable of the vectors , , and , the following finite dimensional equations can be then obtained from the partial differential equation (9), : augmented by the following outputs:where for , are computed as given in .
Let consider Cauchy’s formula for the interpolation error defined by Lefèvre et al.  as follows:and assume that the unknown solution is sufficiently continuously differentiable; we have thenwhere and .
Hence, we try to choose the interior collocation points that minimize the interpolation error (19). Without any a priori knowledge on the behavior of the exact solution, this problem reduces to finding such that is minimal.
By considering the case study of the Chebyshev polynomials belonging to the family of Jacobi polynomials, and, corresponding to the values of the parameters , the corresponding minimal norm is given by Lefèvre et al.  as follows: Through the result (21), we demonstrate that the interpolation error for a variable delay approximation is always bounded for the parameters .
4. Main Results
4.1. PI Controller Design via ILMIs
In the following, the SOF transformation of the PI controller of the delayed system (1) will be detailed. Using (18), the system (1) can be written as follows:Let now where:and let:where:The state space of a new augmented system controlled via an SOF controller is then deduced as follows:where:Taking into account (27), the control law (2) under Assumption 1 can be written as follows: On the other hand, we have from (27) the following:We can deduce that once the matrix is designed the closed-loop system (4)–(6) is asymptotically stable. Considering analogy between (29) and (30), the original PI gains can be recovered as follows:
Theorem 5. The multivariable LTI system with multiple time delays (1) is stabilizable via SOF if and only if there exist a constant matrix and a symmetric positive definite matrix , satisfying the following matrix inequality: such that
Sufficiency. Note thatFrom Lyapunov’s theory, the closed-loop system is then asymptotically stable.
Necessity. Suppose that is asymptotically stable for some . Then there exists such thatIt is easy to find that there exists a scalar such thatthat is,Obviously, (37) is equivalent to By substituting with in (38), we obtain inequality (32). Condition (33) is already proved in (31).
Due to the term , (32) cannot be simplified to an LMI. Similarly to SOF control problem described in [14–16], an ILMI algorithm can be addressed to solve the Bilinear Matrix Inequality (BMI) in (32). To accommodate the term, an additional design variable is introduced. Because for any and for the same dimension, we obtainBy combining (32) and (39), a sufficient condition for the existence of SOF gain matrix is obtained such thatConsidering the stabilizability via SOF concept , if the matrix inequality (40) has a solution , then there exist a real number and a fixed matrix such thatBased on the idea that all eigenvalues of are shifted progressively towards the left-half-plane through the reduction of , we may close in on the feasibility of (32) . Using Schur complement, inequality (41) is equivalent to the following matrix inequality:The previous inequality (42) points to an iterative approach to solve and ; namely, if is fixed in (42), then it reduces to an LMI problem in the unknowns , and .
The following is a constructive ILMI algorithm for PI control of LTI MIMO with multiple delays systems, and the explanations are given in Remark 7.
Step 1. Define the orthogonal collocation method parameters , , and .
Step 2. Transform the infinite dimensional system (1) to a finite dimensional system (17)-(18) by computing matrices , , , , , , , , and .
Step 3. Design the SOF transformation to give the system’s state space realization (). If it does proceed to Step 4.
Step 4. Set and choose where .
Step 5. Solve the following optimization problem for , , and :
OP1: Minimize subject to the following LMI constraints:where . Denote by the minimized value of .
Step 6. If , the feedback matrix gains are and . Stop. Otherwise, go to Step 7.
Step 7. Solve the following optimization problem for and .
OP2: Minimize subject to LMI constraints (43) with , where stands for the trace of a square matrix. Denote by the optimal . The feedback matrix gains are and .
Step 8. If , where is a prescribed tolerance, go to Step 9; otherwise, set , and go to Step 5.
Step 9. It cannot be decided by this algorithm whether SOF problem is solvable. Stop.
4.2. Improving Closed-Loop Performances
In this section, a modified approach of Loop Shaping technique  will be introduced to design the multiloop PI controller as described in subproblem 2. Figure 2 shows the block diagram of the controlled system where and denote the set-point vector, the output vector, the control signal vector, the process control signal vector, the error vector, the output disturbance vector, and the input disturbance vector, respectively.
Let us define the input loop transfer matrix, , and the output transfer matrix, , respectively, as follows :The input sensitivity matrix is defined as the transfer matrix from to such asand the output sensitivity matrix is defined as the transfer matrix from to such thatThe input and output complementary sensitivity matrices are defined as follows: It is easy to see that the closed-loop system, if it is internally stable, satisfies the following equations:Equation (48) shows that the effects of the disturbance on the plant output can be made “small” by making the output sensitivity function small, as is fixed. Similarly, (50) shows the effect of the set-point to the error by making as small as possible. The notion of smallness for a transfer matrix in a certain range of frequencies can be made explicit using frequency dependent singular values particularly minimizing where is the maximum singular value. Similarly to the conventional Loop Shaping design [19, 20], shaping the open-loop nominal system corresponds to shaping the loop gain using the pre- and postcompensators and . The new shaped control system is shown in Figure 3. It is obvious that a well designed control system should meet, at least, the following requirements: () stability, () set-point tracking, and () output disturbance attenuation. Let us then define the input loop transfer matrix, , and the output transfer matrix, , asThe input sensitivity matrix is defined as the transfer matrix from to as follows:and the output sensitivity matrix is defined as the transfer matrix from to as follows:The input and output complementary sensitivity matrices are defined as It is easy to see that the closed-loop system, if it is internally stable, satisfies the following equations:As is fixed, from (56) and (58), good set-point tracking and output disturbance attenuation would require the maximum singular value of the output sensitivity matrix of the shaped system be made small such asIt should be indicated that improving the closed-loop shaped system performances over those of the nominal system would require be made smaller than , particularly in the low frequency range where is usually significant. As and are fixed by subproblem 1, and play a key role in the Loop Shaping design procedure. Thus, synthesis of the shaped controller is reduced to choose an appropriate and in order to guarantee closed-loop performances, under the following:such thatFigure 4 synthesizes the Loop Shaping design procedure proposed in this paper where the proposed Loop Shaping design procedure is stated below:
Step 1. Consider the PI controller designed via Algorithm 6. Assume that the closed-loop system performances are not well performed and define the control objectives for the desired closed-loop system responses (good disturbance rejection, steady state error minimization).
Step 2. Choose a precompensator and a postcompensator such that the singular values of the nominal plant are shaped to give a desired open-loop shape (high low frequency gain and low high frequency gain).
Step 3. For the shaped plant , if the control objectives and constraint (62) are satisfied then go to Step 4. Else adjust and and go to Step 2.
Step 4. Synthesize a final feedback controller for the nominal plant bywhere is given in (7). For tuning purpose, we always choose ; thenStep 5. Verify that the desired closed-loop system responses are met. If yes stop. Else adjust and and go to Step 2.
Remark 9. If is a nonsquare matrix such as or , then it is obvious that the proposed algorithm does not hold. Some minor modifications are required to tackle this problem. In fact, compatible dimensions for the shaped plant and the shaped controller require taking , , , and with the same dimension . Thereby, the dimensions of and are chosen to be equal to and the proposed Loop Shaping design procedure still holds.
Remark 10. Note that the final PI controller designed in Algorithm 8 is related to the PI controller designed in Algorithm 6. Indeed, , given by relation (64), is designed, on one hand, using the full MIMO PI controller given by relation (3) computed via Algorithm 6, and, on the other hand, using weighting functions properly designed following Algorithm 8.
Remark 11. It must be noted that there are severe limitations when the conventional Loop Shaping design procedure is used for MIMO systems as discussed in . Among these limitations, it may still be much harder to find a stabilizing if for non-minimum phase or unstable systems. However, this paper succeeds in overcoming these limitations thanks to the first stage of the design procedure that guarantees internally stable closed-loop.
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:(i)The proposed PI controller designed via the Algorithms 6 and 8.(ii)The PI controller designed via Algorithm 6.(iii)The PID controller designed in .(iv)IMC-PI controller approach (v)The conventional IMC-PID approach Sedumi and Yalmip Toolbox  are used to solve ILMIs. To evaluate the closed-loop performances, the Integral Absolute Error (IAE) and the Total Variation (TV) criteria are considered. They are defined, respectively, by Vu and Lee  as follows:where is finite time chosen for the integral approach steady state value and is defined as the total error between the set-points and the outputs.
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  fails and that the multiloop IMC-PI proposed by Vu and Lee  has less TV performances. This last approach will not be tested on the third example since it is only appropriate for first-order systems.
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 and . The different performances singular values are plotted by means of and by noting that is equal to .
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 be the error between the unit step response to the state space representation (1) and the corresponding transfer matrix of the LTI MIMO with multiple time delays . is defined as the total error between the delayed system (1) outputs and the approximated ones by model (4)-(5) using the orthogonal collocation method.
Remark 12. Due the bad performances obtained via the PID controller designed via the approach given in , an additive filter is joined to the derivative action to attenuate noises. Thus, the transfer matrix of the PID controller with filter considered is described by It should be noted that the PID controller with filter (66) is applied to stable and non-minimum phase systems. Due to bad simulation results performed for the unstable system, the PID controller with filter is not considered.
5.1. Example 1: The Distillation Column System
Consider the typical example of the distillation column described in [6, 32, 33] belonging to the class of MIMO unstable plants with input delays; its transfer matrix model is described by Mete et al.  as follows: Applying a column decomposition method  for (67), the state space representation (1) can be deduced as follows:for h and h.
Using the orthogonal collocation method, the following matrices are obtained for model (4)-(5):By solving Algorithm 6 with and yielding to , the following PI gains are obtained:Thus, the PI controller transfer matrix is given byFigure 5 illustrates the validity of the passage between the transfer matrix model (67) and the state space representation (1) and also the validity of the approximated model (4)-(5) obtained via the orthogonal collocation method.
Figure 6 proves that the PI design procedure satisfies the desired specifications for a precompensator and postcompensator chosen as follows:To boost the low frequency gain and give almost zero steady state error, is chosen accordingly as an approximated PI precompensator.
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 1 and 2. The proposed controller affords better performances especially for the second output and better disturbance rejection over PI controller as shown by Figures 8 and 9. As listed in Tables 1 and 2, the controller settings of the proposed method provide superior performances by the smallest total IAE for both case studies: set-point changes, disturbances changes, and parametric uncertainties. Acceptable TV indices are also shown by the proposed method for this process.
5.2. Example 2: The Industrial Scale Polymerization (ISP) Reactor
Let us first test for the previous system the conventional IMC-PID approach proposed by Economou & Morari . In this approach, the IMC interaction measure surfaces are practical tools to assess the potential value of the multiloop design. By their definitions, they must vary between 0 and 1 such that for the th input/output pair of a particular system configuration, the Row IMC interaction measure is the quantity defined by whereas, for the same input/output pair and configuration, the complementary quantity is defined byThe following scenarios are expected:(i), corresponds to significant interactions between the multiple loops and an overall poor performances of the multiloop structure are expected.(ii), corresponds to good pairing and SISO controllers can be designed for each loop and granting good performances.For the ISP reactor, the transfer function matrix (73) can be written as follows:From (75) and (76), we compute the IMC interactions measure for each pairing as summarized in Table 3. Figure 10 shows the IMC interaction measure for controlling and controlling for the original pairing whereas reverse pairing is shown as the IMC interaction measure for controlling and controlling . As it can be observed by the IMC measure interaction, the original pairing as the reverse pairing cannot guarantee good performances that is why such an approach fails.
Let us now apply the extended IMC-PI controller approach proposed by Vu and Lee  for the class of TITO multidelay processes with first-order plus delay time (FOPDT) systems. From the ISP reactor and referring to , the following data are deduced: The steady state relative gain array of the ISP reactor is , which proves that the closed-loop gain is greater than the open-loop gain. The ISP reactor system does not then exhibit open-loop diagonal dominance. The diagonal PI-multi-loop controller parameters and for each loop , are then designed and given by Table 4. We have also verified that the two outputs converge to the set-points in response to a unit step.
For and , the reduced model (4)-(5) is obtained via the collocation method. Figure 11 proves the validity of models (1) and (4)-(5). By solving Algorithm 6 using the parameters + 0.0020 and and yielding , the following PI matrix gains are given:The transfer matrix of the related computed PI controller is given byBy solving algorithm in , the PID feedback matrix gains are given byFigure 12 proves that PI design procedure satisfies the desired specifications for a precompensator and postcompensator chosen as follows:For a sequential unit step change in the set-points at and h, one can see that the proposed controller has the faster rising time and settling response over other ILMI approaches as shown by Figure 14. The disturbance model is taken as as in . Unit step changes in the disturbance were also made to the 1st and 2nd loops, respectively, as shown by Figure 15. The resulting performance indices for the nominal and perturbed system cases for various tuning methods are given in Tables 5 and 6. The proposed MIMO PI controller provides superior performances over PI controller and PID controller designed via ILMI approaches by means of the smallest total IAE. Acceptable TV values are also listed by the proposed method.
5.3. Example 3: The 4-Tank Process
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 . The corresponding model with multiple delays is described in  by taking into account transport delays between valves and tanks. Applying the numerical values corresponding to the non-minimum phase model found in , system (1) is given byThe measurement level signals and are and where V/cm. The output matrix is then given by