# Computation of Stabilizing PI and PID parameters for multivariable system with time delays

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2 approach was proposed to approximate the decoupled system by SOPDT models. We present in section 4 and 5 the ( ) Hermite Biehler theorem that is useful for PI and PID controllers synthesis. In section 6 the proposed method is applied to the Wood & Berry binary distillation column plant [7] to show its effectiveness and applicability. Finally ( ) conclusion remarks are made. And is the smallest time delay in each column of. The decoupled process can be then written as follows: II. SMITH PREDICTOR FOR MIMO SYSTEMS WITH DELAYS For a SISO delayed, Smith proposed a compensation scheme which can eliminate the delay of the characteristic equation closed loop which facilitates its control and greatly improves its impulse response [12]. This type of control can be applied to MIMO systems with multiple delays. To eliminate the coupling between the various loops, we will then present the design of decoupling control Smith proposed by Wang et al [13]. Therefore, the resulting decoupled system will be reduced to a SOPDT plant which seems to be an useful tool for solving the problem of control of MIMO systems with multiple delays. The structure of multivariable control by Smith predictor [13] is illustrated in Fig. 1: III. MODEL REDUCTION After decoupling a MIMO system interactions are eliminated but the resulting decoupled process given by equation (4) is complex. In order to simplify the system and ensure an efficient control, the model is approximated by a SOPDT plant in this work. In order to find an approximation by a SOPDT model for, these unknown parameters, namely,, and (i=1, 2,, n) should be determined. The standard recursive least square approach [14] is adopted here. The resulting L (s) then obtained can be written as follows: IV. PRELIMINARY RESULTS FOR ANALYZING TIME DELAY SYSTEM Several problems in process control engineering are related to the presence of delays. These delays intervene in dynamic models whose characteristic equations are of the following form [5, 25]: Fig.1 Multivariable Smith predictor structure. δ(s)=d(s)+ (s)+ (s)+ + (s) (6) With R, Y, C, D and G represent respectively the input, output, controller, decoupler, and the process which is a stable nonsingular matrix ( det(g(0) 0)). And (s) is the same as H(s) except with no delay. Consider the multivariable system with the transfer matrix: [ ] The transfer function of the closed loop system is given by: In the case of multivariable control by Smith Predictor, the decoupler D (s) is determined in a way that the matrix of the transfer function H(s) is diagonal and is expressed as follows: : d(s) and (s) are polynomials with real coefficients and represent time delays. These characteristic equations are recognized as quasi-polynomials. Under the following assumptions: Deg (d(s))= n and deg ( (s)) < n for i=1,2,,m (7) One can consider the quasi-polynomials described by: = δ(s) The zeros of δ(s) are identical to those of since does not have any finite zeros in the complex plan. However, the quasi-polynomial has a principal term since the coefficient of the term containing the highest powers of s and is nonzero. If does not have a principal term, then it has an infinity roots with positive real parts [5]. 2 ISSN:

3 The stability of the system with characteristic equation (6) is The proposed method leads to an efficient calculation of the equivalent to the condition that all the zeros of must be proportional and integral gains and achieving stability. in the open left half of the complex plan. We said that The characteristic equation of the closed-loop system is given is Hurwitz or is stable. The following theorem gives a by: necessary and sufficient condition for the stability of. δ (s) =K( s) + s (12) Theorem 1 (Hermite Biehler) [5] We deduce the quasi-polynomial: Let be given by (8), and write: δ(s)=k( s) +s (13) = (ω) + j (ω) (9) where (ω) and (ω) represent respectively the real and imaginary parts of. Under conditions and, is stable if and only if: 1: (ω) and (ω) have only simple, real roots and these interlace, 2: > 0 for some in [,+ ]. and (ω) denote the first derivative with respect to ω of (ω) and (ω), respectively. A crucial stage in the application of the precedent theorem is to verify that and have only real roots. Such a property can be checked while using the following theorem. Theorem 2 [5] Let M and N designate the highest powers of s and which appear in. Let η be an appropriate constant such that the coefficient of terms of highest degree in (ω) and (ω) do not vanish at ω = η. Then a necessary and sufficient condition that (ω) and (ω) have only real roots is that in each of the intervals 2lπ+η < ω < 2lπ+η, l =, +1, (ω) or (ω) have exactly 4lN + M real roots for a sufficiently large. V. PI CONTROL FOR SECOND ORDER DELAY SYSTEM A second order system with delay can be mathematically expressed by a transfer function having the following form: (10) K is the static gain of the plant, L is the time delay and and are the plant parameters. The parameters are always positive. In this section, the stabilization of the plant is assured by the PI controller designed as follow: (11) by replacing s by jω, we get: = (ω) + j (ω) With:, - Clearly, the parameter only affects the real part of whereas the parameter affects the imaginary part. Let s put z = L, we get:, - From the previous expressions, the real part depends on the controller parameter whereas the imaginary part depends on. In order to determine the range of, Silva et al. (2005) and Farkh et al. (2009a) use the following theorem [17, 18]: ( ) is the solution of the equation tan= the interval [0,π]. in For values outside this range, there are no stabilizing PID controllers. All of the values of the parameter given in (16) verify the first condition of Hermite Biehler theorem, which required that the roots of (z) and (z) are simple and interlaced. Applying the condition, > 0, we can compute the set of parameter controller gains that verifies the interlace property of the roots. Thus we can rewrite as follows:, - (17),. / - (18) Fig.2 The closed-loop system with controller Let s put, j = 1, 2, 3... the roots of and a. Interlacing the roots of and is equivalent to (since > 0),,... We can 3 ISSN:

4 use the interlacing property and the fact that and has only real roots to establish that possess real roots too. From the previous equations we get the following inequalities:, - We notice that have the same expression as in (15) then to determine the range of we will use the same theorem. ; (19) All of the values of the parameter given in (16) verify the first condition of Hermite Biehler theorem, which required that the roots of (z) and (z) are simple and interlaced. Applying the condition, > 0, we can From these inequalities, it is clear that the odd bounds must compute the set of and gains that verifies the interlace be strictly positive; however the even bounds are negative property of the roots. in order to find a feasible range of. From which we have: Thus we can rewrite as follows: * + (20) Algorithm for determining PI parameters [17] 1) Choose in the interval suggested in (16) and initialize j = 1, 2) find the roots of, 3) Compute the parameter associated with the previously founded, 4) Determine the lower and the upper bounds for as follows: * + 5) Go to step 1. VI. PID CONTROL FOR SECOND ORDER DELAY SYSTEM Considering the same system expressed by equation (10) shown in Fig.2, we attempt to achieve stabilization with PID controller presented by: (21) The characteristic equation of the closed-loop system is given by: δ(s)=k( s+ ) + s (22) we deduce the quasi-polynomial: K( s+ )+s (23) by replacing s by jω, we get: = (ω) + j (ω) [ ] And by putting z = L, we get:, - (26), - Let s put, j = 1, 2, 3... the roots of, m and b. By using the interlacing property and the fact that and has only real roots to establish that possess real roots too we get the following inequalities: In order to determine the cross-section of the stabilizing region in the space for each value of the parameter given in (16), the theorem of Farkh et al. [19] is used. It is defined as: 1) A trapezoid T if >, 2) A triangle if > [19]. Algorithm for determining PID parameters [19] 1) Choose in the interval suggested in (16) and initialize j = 1, 2) Find the roots of, 3) Compute the parameter, associated with the for j = 1, 2, 3 founded, 4) Determine the stability region in the plane using Farkh et al. Theorem [19]. 5) Go to step 1. VII. SIMULATION RESULTS ISSN:

5 To show the effectiveness of the proposed design methods, we consider the following binary distillation column: [ ] The decoupler is designed using Eq. (3): [ ] The resulting diagonal decoupled system is: ( ) Fig.3 PI controller stability region for In order to determine PI and PID controllers, and should be expressed as SOPDT processes. Using the standard recursive least square, SOPDT models for and are determined as follow: In order to determine values stabilizing and, we look for α in interval [0, π] satisfying Eq. (16).we find respectively the following results: = range is given by: 0.97< < = 1.89 range is given by: 1< Fig.4 PI controller stability region for By sweeping over, a stability region is defined in the plane. A three dimensional curve is then obtained as shown in Fig.5 and Fig.6. < The systems stability regions, obtained in (, presented in Fig.3 and Fig.4. ) plane are Fig.5 PID controller stability domain for 5 ISSN:

6 the process static gains do not affect decoupling regulation of the output responses. By using Fig. 5 and Fig. 6, we choose with the following PID controllers: [ ] Fig.6 PID controller stability domain for By using Fig. 3 and Fig. 4, we start with the following PI controllers: [ ] Fig.9 First output Step responses with PID Controller ( = ; = and = ). Fig.7 First output Step responses with PI Controller ( = and = ). Fig.10 Second output Step responses with PID Controller ( = 0.12; = and =0). A unit step disturbance has been applied to the process input at t=850s. The simulation results shown in Fig. 9 and Fig. 10 reveal that the proposed control strategy gives robust system performance. The obtained results show the effectiveness of the proposed approach. Fig.8 Second output Step responses with PI Controller ( = 1.5 and = 0.03). Output responses to unit step function in the first input and the second input are shown in Fig. 7 and Fig. 8 for the simulation, an unit step disturbance has been applied to the process input at t=850 s. It is clearly seen that perturbations of VIII. CONCLUSION This paper proposed a simple method to tune decentralized PID controllers for MIMO plants with multiple delays. The MIMO was first decoupled using the design of decoupling Smith control. A model reduction was proposed to find a suitable (SOPDT) model for each element of the resulting diagonal process. An extension of Hermit-Biehler theorem is used to find stabilizing PI and PID sets for the reduced system This approach is finally investigated through 6 ISSN:

7 its application to the Wood & Berry binary distillation column plant. REFERENCES [1] GJomez, G. I., & Goodwin, G. C. (1996). Integral constraints on sensitivity vectors for multivariable linear systems. Automatica, 32(4), [2] HORWITZ, I. : Synthesis of Feedback Control Systems, Chapter 5, New York Academic Press, [3] H. Takatsu, T. Itoh, and M. Araki, Future needs for the control theory in industries report and topics of the control technology survey in Japanese industry, Journal of Process Control, vol. 8, no. 5-6, pp , [4] GOODWIN, G. GRAEBE, S. F. SALGADO,M. E. : Control System Design, Prentice Hall, [5] ASTR OM, K. WITTENMARK, B. : Computer Controlled Systems, Theory and Design, Prentice Hall, [6] H. Shu, X. Guo, and H. Shu, PID neural networks in multivariable systems, in Proceedings of the IEEE International Symposium on Intelligent Control, pp , [7] DOMINIQUE N G, PATRICK L, ALAIN O. Robust control design for multivariable plants with time-delays [J]. Chemical Engineering Journal, 2009, 146(3): [8] LUYBEN W L. Simple method for tuning SISO controllers in multivariable system [J]. Industrial and Engineering Chemistry Process Design and Development, 1986, 25(3): [9] JIETAE L, THOMAD F E. Multiloop PI/PID control system improvement via adjusting the dominant pole or the peal amplitude ratio [J]. Chemical Engineering Science, 2006, 61(5): [10] [10] ASTROM K J, JOHANSSON K H, WANG Qing-guo. Design of decoupled PI controllers for two-by-two systems [J]. IEE Proceeding Control Theory & Applications, 2002, 149(1): [11] SAEED T, IAN G, PETER J F. Tuning of decentralized PI (PID) controllers for TITO processes [J]. Control Systems Engineering, 2006, 14(9): [12] Holt, B. R., & Morari, M. (1985a). Design of resilient processing plants VI: The effect of right-half-plane zeros on dynamic resilience. Chemical Engineering Science, 40, [13] HUANG C, GUI W, YANG C, XIE Y. Design of decoupling Smith control for multivariable system with time delays. J. Cent. South Univ. Technol. 18: , [14] S.Skogestad.Simple analytical rules for model reduction and PID controller tuning, Journal of Process Control, vol. 13, pp ,2003. [15] G. J. Silva, A. Datta, and S. P. Bhattacharyya. PID Controllers for Time-Delay Systems, Control Engineering, Birkh auser, Boston, Mass, USA, [16] S. P. Bhattacharyya, H. Chapellat, and L. H. Keel. Robust Control: The Parametric Approach, Prentice-Hall, Upper Saddle River, NJ, USA, [17] R.Farkh, K.Laabidi and M.Ksouri (2009a),PI control for second order delay system with tuning parameter optimization. International Journal of Electrical and Electronics Engineering 3: 1. [18] GJ. Silva, A. Datta and SP. Bhattacharyya(2005). PID Controllers fortime Delay Systems. London: Springer. [19] R.Farkh, K.Laabidi and M.Ksouri (2009a) Computation of all stabilizing PID gain for second-order delay system. Mathematical Problems in Engineering. doi: /2009/ ISSN:

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