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Abstract: This study presents the design of both H∞ Loop Shaping Control (HLSC) and Internal Model Control (IMC) strategies for linear time delay systems. For first order time delay system, a systematic approach for weight selection based on the sensitivity function was proposed, then compared to the internal model control strategy. For both methods, the synthesis was based on the Padéapproximation. Two cases are considered for time delay: upper or lower than system time constant. Simulation results for the proposed approaches are acceptable ever in presence of disturbances and model mismatches.
Key words: H∞ loop shaping control, internal model control, sensitivity function, constant delay, Padé approximation, model mismatches.
1. Introduction
Robust control is a branch of control theory. It has been developed as a technique to make machine do what the model predicted. This theory deals with uncertainty in controller design. Robust methods are designed to achieve robust performance and/or stability in presence of bounded modeling errors. There are many techniques such as Adaptive Control, Fuzzy Control, Predictive Control, Internal Model Control, H∞ Loop Shaping Control, etc. The H∞ Loop Shaping Control (HLSC) design is an efficient technique for constructing robust controllers. It has been successfully used in a wide variety of applications [1-3]. This structure incorporates loop shaping methods to obtain robust stability/performance tradeoffs, a particular H∞ optimization problem to guarantee closed loop stability and a level of robust stability at all frequencies.
However, weighting function selection is not an easy task to pass and the order of the final controller is usually high. Loop shaping weighting functions are created in two steps. In the first, the desired loop shape is determined. In the second, the designer chooses loop shaping weights [4-9].
Gain and phase margins are used in loop shaping controller design as a measure of robustness. But, this robustness is a hard task for the loop shaping designer searching an acceptable balance between gain, phase margins and tracking performance.
As well as the HLSC, the Internal Model Control(IMC) strategy was proposed by Garcia and Morari in 1982 [10] and finished by a set of articles by the same authors [11-13]. This method has realized an increasing interest in the last few years. The stability of the IMC depends only on the stability of both nominal plant and controller.
The purpose of this paper is to compare two robust control methods for linear time delay systems. This paper is organized as follows: In section 2, the description of the H∞ Loop Shaping Control and the proposed weight selection procedure which is based on sensitivity function. In section 3, the Internal Model Control strategy is presented. Section 4 is devoted to Padé approximation of constant time delay systems. In section 5, a first-order time delay system is chosen as an application example. In which, the delay is
approximated by the first and the second order Padéapproximation. Both strategies were investigated. Finally, section 6 is given a conclusion.
2. H∞ Loop Shaping Control 2.1 H∞ Loop Shaping Control Design
The H∞ Loop Shaping Control method is derived from the H∞ robust stabilization combined with the classical Loop Shaping. The “loop shape” refers to the magnitude of the loop transfer function .LK G= as a frequency function [5, 7]. It is essentially a two-step procedure; in the first step, the singular values of the open loop plant are shaped by pre and post compensators. In the second, the resulting shaped plant is robustly stabilized.
The robust stability H∞ problem is to find min
After that, the gain and the phase margins of the shaped plant will be calculated. In practice, if the gain margin (GM) is about ± 6 dB and the phase margin(PM) is between 40 degrees a response with adequate margins of stability would be found; or else, W1 will be readjusted until better margins values would be obtained [6].
3. Internal Model Control
The Internal Model Control, shown in Fig. 3, was first proposed by Garcia and Morari in 1982 [10] and finished by a set of articles by the same authors [11-13]. As its name indicates, the IMC incorporates a simulation of the process by an internal model M in its control structure, where r is the setpoint and d is the disturbance. The control u obtained is applied simultaneously to the process G and to its model M.
This method has acquired an increasing interest in the last few years because of the inherent robustness of its structure. It was the subject of several publications
4. Delay Approximation
Time delay systems exist in many engineering fields. They are essentially generated by transporting mass,
energy or information or processing time. Constant delays are sometimes structured by places of their genesis: input time delay, dead time of the system and output time delay.
Control systems with dead time are difficult to analyse and simulate, because their closed-loop are infinite dimensional systems. In some situations, as in frequency response based analysis of control systems containing a time delay, it is necessary to substitute?with an approximation in form of a rational transfer function. The most common approximation is the Padé approximations [18-19]. This method has an important feature that there are no restrictions on the degrees of both polynomials.
Padé approximations introduce unstable zeros into the transfer functions. It is recommended to use it for 10n≤only. Their poles are stable for all practically usable orders and have tendency to group to each other [19].
5. Time Delay System Control
In this paper, a first-order stable system with pure delay is studied with the two structures IMC and HLSC. The constant delay is approximated using Padé where polynomials have the same order which is 1 and 2(5 iτ<).
Two cases will be studied system time constant upper or lower than delay:
Then, the delays expressions are replaced by their approximations, and the obtained transfer functions are as follows:
The gain and phase margins for the four transfer functions and the value of gamma optimum are given in Table 1.
The HLSC system responses with the Padéapproximations for both cases are given in Figs. 6-7 where a step disturbance with 0.5 as magnitude is applied at 30 s.
All responses are stable, with overshoots but without static errors. A good disturbance rejection is achieved. In addition, the responses for the first order Padéapproximation are faster than the second order.
5.3 Comparison
The IMC responses are faster than the HLSC. But, for the cases 1 and 2, both structures showed acceptable responses quite comparable.
5.4 Model Mismatch
Let’s consider a model mismatch at the nominal transfer function (16) after the computing of the controllers:
Then the IMC and the HLSC step responses are studied in presence of disturbances in Figs. 8-11.
Despite the model mismatch, IMC system responses(Figs. 8-9) are stables with acceptable disturbance rejection. The responses for the first order Padéapproximation are faster than the second order.
As in the IMC, the HLSC illustrate stable responses with acceptable disturbance rejection in presence of model mismatches.
6. Conclusions
In this paper, time delay systems with pure delay were studied with the Internal Model Control and the H∞ Loop Shaping Control strategies where the synthesis was based on first and second order Padéapproximation of the delay. The weight selection in the H∞ loop shaping control is focused on the sensitivity function of the open loop shape function of systems. Both strategies were studied in presence of disturbances and model mismatches and acceptable results were shown.
References
[1] F. Tado, O.P. Lopez, T. Alvarez, Control of neutralization processes by robust loopshaping, IEEE Transactions on Control Systems Technology 8 (2) (2000) 236-246.
[2] M. Boukhnifer, A. Ferreira, J.G. Fontaine, Loop shaping robust control for scaled teleoperation systems, in: Proceeding of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Spain, December 2005, pp. 7894-7899.
[3] S. Kaitwanidvilai, P. Olranthichachat, M. Parnichkun, Fixed structure robust loop shaping controller for a buck-boost converter using genetic algorithm, in: Proceeding of the International MultiConference of Engineers and Computer Scientists, Hong Kong, March 2008.
[4] D. McFarlane, K. Glover, A loop shaping design procedure using H∞ synthesis, IEEE Transactions on Automatic Control 37 (6) (1992) 759-769.
[5] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, Wiley, January 1997.
[6] E. Prempain, Statistic H∞ loop shaping control, in: Proceeding of the International Conference on Control, University of Bath, September 2004.
[7] K. Zhou, J. C. Doyle, Essentials of Robust Control, Prenctice Hall, May 1999.
[8] M. Ejaz, M.N. Arbab, S.W. Slah, Weight selection in H∞loop shaping using lead/lag compensators, in: 2nd
International Conference on Emerging Technologies, Pakistan, November 13-14, 2006.
[9] M. Ejaz, M.N. Arbab. Automatic weight selection in H∞loop shaping using genetic algorithm, in: 2nd International Conference on Emerging Technologies, Pakistan, November 13-14, 2006.
[10] M. Morari, C.E. Gar?ia, Internal model control: 1. Unifying review and some new results, Ind. Eng. Chem. Process Des. Dev. 21 (2) (1982) 308-323.
[11] M. Morari, C.E. Gar?ia, Internal model control: 2. Design procedure for multivariables systems. Ind. Eng. Chem. Process Des. Dev. 24 (2) (1985) 472-484.
[12] M. Morari, C.E. Gar?ia, Internal model control: 3. Multivariables control law computation and tuning guidelines, Ind. Eng. Chem. Process Des. Dev. 24 (2)(1985) 448-494.
[13] C.G. Economou, M. Morari, B.O. Palsson, Internal model control: 5. Extension to nonlinear systems, Ind. Eng. Chem. Process Des. Dev. 25 (2) (1986) 403-411.
[14] S. Farhati, S.B.H.A. Naoui, M.N. Abdelkrim, Internal model control of time delay systems, in: 10th International Conference on Sciences and Techniques of Automatic Control & Computer Engineering, Tunisia, December 20-22, 2009.
[15] S.B.H. Ali, Sur la commande robuste par modèle interne, Ph.D. Thesis, National Engineering School of Tunis, Tunisia, 2003.
[16] J.P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (2003) 1667-1694.
[17] M. Morari, E. Zafiriou, Robust Process Control, Prentice Hall, 1989.
[18] M. Vajta, Some remarks on Padé-approximations, in: 3rd TEMPUS-INTCOM Symposium, Hungary, September 9-14, 2000.
[19] V. Hanta, A. Prochazka, Rational approximation of time delay, in: International Conference on Technical Computing, Prague, 2009.
Key words: H∞ loop shaping control, internal model control, sensitivity function, constant delay, Padé approximation, model mismatches.
1. Introduction
Robust control is a branch of control theory. It has been developed as a technique to make machine do what the model predicted. This theory deals with uncertainty in controller design. Robust methods are designed to achieve robust performance and/or stability in presence of bounded modeling errors. There are many techniques such as Adaptive Control, Fuzzy Control, Predictive Control, Internal Model Control, H∞ Loop Shaping Control, etc. The H∞ Loop Shaping Control (HLSC) design is an efficient technique for constructing robust controllers. It has been successfully used in a wide variety of applications [1-3]. This structure incorporates loop shaping methods to obtain robust stability/performance tradeoffs, a particular H∞ optimization problem to guarantee closed loop stability and a level of robust stability at all frequencies.
However, weighting function selection is not an easy task to pass and the order of the final controller is usually high. Loop shaping weighting functions are created in two steps. In the first, the desired loop shape is determined. In the second, the designer chooses loop shaping weights [4-9].
Gain and phase margins are used in loop shaping controller design as a measure of robustness. But, this robustness is a hard task for the loop shaping designer searching an acceptable balance between gain, phase margins and tracking performance.
As well as the HLSC, the Internal Model Control(IMC) strategy was proposed by Garcia and Morari in 1982 [10] and finished by a set of articles by the same authors [11-13]. This method has realized an increasing interest in the last few years. The stability of the IMC depends only on the stability of both nominal plant and controller.
The purpose of this paper is to compare two robust control methods for linear time delay systems. This paper is organized as follows: In section 2, the description of the H∞ Loop Shaping Control and the proposed weight selection procedure which is based on sensitivity function. In section 3, the Internal Model Control strategy is presented. Section 4 is devoted to Padé approximation of constant time delay systems. In section 5, a first-order time delay system is chosen as an application example. In which, the delay is
approximated by the first and the second order Padéapproximation. Both strategies were investigated. Finally, section 6 is given a conclusion.
2. H∞ Loop Shaping Control 2.1 H∞ Loop Shaping Control Design
The H∞ Loop Shaping Control method is derived from the H∞ robust stabilization combined with the classical Loop Shaping. The “loop shape” refers to the magnitude of the loop transfer function .LK G= as a frequency function [5, 7]. It is essentially a two-step procedure; in the first step, the singular values of the open loop plant are shaped by pre and post compensators. In the second, the resulting shaped plant is robustly stabilized.
The robust stability H∞ problem is to find min
After that, the gain and the phase margins of the shaped plant will be calculated. In practice, if the gain margin (GM) is about ± 6 dB and the phase margin(PM) is between 40 degrees a response with adequate margins of stability would be found; or else, W1 will be readjusted until better margins values would be obtained [6].
3. Internal Model Control
The Internal Model Control, shown in Fig. 3, was first proposed by Garcia and Morari in 1982 [10] and finished by a set of articles by the same authors [11-13]. As its name indicates, the IMC incorporates a simulation of the process by an internal model M in its control structure, where r is the setpoint and d is the disturbance. The control u obtained is applied simultaneously to the process G and to its model M.
This method has acquired an increasing interest in the last few years because of the inherent robustness of its structure. It was the subject of several publications
4. Delay Approximation
Time delay systems exist in many engineering fields. They are essentially generated by transporting mass,
energy or information or processing time. Constant delays are sometimes structured by places of their genesis: input time delay, dead time of the system and output time delay.
Control systems with dead time are difficult to analyse and simulate, because their closed-loop are infinite dimensional systems. In some situations, as in frequency response based analysis of control systems containing a time delay, it is necessary to substitute?with an approximation in form of a rational transfer function. The most common approximation is the Padé approximations [18-19]. This method has an important feature that there are no restrictions on the degrees of both polynomials.
Padé approximations introduce unstable zeros into the transfer functions. It is recommended to use it for 10n≤only. Their poles are stable for all practically usable orders and have tendency to group to each other [19].
5. Time Delay System Control
In this paper, a first-order stable system with pure delay is studied with the two structures IMC and HLSC. The constant delay is approximated using Padé where polynomials have the same order which is 1 and 2(5 iτ<).
Two cases will be studied system time constant upper or lower than delay:
Then, the delays expressions are replaced by their approximations, and the obtained transfer functions are as follows:
The gain and phase margins for the four transfer functions and the value of gamma optimum are given in Table 1.
The HLSC system responses with the Padéapproximations for both cases are given in Figs. 6-7 where a step disturbance with 0.5 as magnitude is applied at 30 s.
All responses are stable, with overshoots but without static errors. A good disturbance rejection is achieved. In addition, the responses for the first order Padéapproximation are faster than the second order.
5.3 Comparison
The IMC responses are faster than the HLSC. But, for the cases 1 and 2, both structures showed acceptable responses quite comparable.
5.4 Model Mismatch
Let’s consider a model mismatch at the nominal transfer function (16) after the computing of the controllers:
Then the IMC and the HLSC step responses are studied in presence of disturbances in Figs. 8-11.
Despite the model mismatch, IMC system responses(Figs. 8-9) are stables with acceptable disturbance rejection. The responses for the first order Padéapproximation are faster than the second order.
As in the IMC, the HLSC illustrate stable responses with acceptable disturbance rejection in presence of model mismatches.
6. Conclusions
In this paper, time delay systems with pure delay were studied with the Internal Model Control and the H∞ Loop Shaping Control strategies where the synthesis was based on first and second order Padéapproximation of the delay. The weight selection in the H∞ loop shaping control is focused on the sensitivity function of the open loop shape function of systems. Both strategies were studied in presence of disturbances and model mismatches and acceptable results were shown.
References
[1] F. Tado, O.P. Lopez, T. Alvarez, Control of neutralization processes by robust loopshaping, IEEE Transactions on Control Systems Technology 8 (2) (2000) 236-246.
[2] M. Boukhnifer, A. Ferreira, J.G. Fontaine, Loop shaping robust control for scaled teleoperation systems, in: Proceeding of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Spain, December 2005, pp. 7894-7899.
[3] S. Kaitwanidvilai, P. Olranthichachat, M. Parnichkun, Fixed structure robust loop shaping controller for a buck-boost converter using genetic algorithm, in: Proceeding of the International MultiConference of Engineers and Computer Scientists, Hong Kong, March 2008.
[4] D. McFarlane, K. Glover, A loop shaping design procedure using H∞ synthesis, IEEE Transactions on Automatic Control 37 (6) (1992) 759-769.
[5] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, Wiley, January 1997.
[6] E. Prempain, Statistic H∞ loop shaping control, in: Proceeding of the International Conference on Control, University of Bath, September 2004.
[7] K. Zhou, J. C. Doyle, Essentials of Robust Control, Prenctice Hall, May 1999.
[8] M. Ejaz, M.N. Arbab, S.W. Slah, Weight selection in H∞loop shaping using lead/lag compensators, in: 2nd
International Conference on Emerging Technologies, Pakistan, November 13-14, 2006.
[9] M. Ejaz, M.N. Arbab. Automatic weight selection in H∞loop shaping using genetic algorithm, in: 2nd International Conference on Emerging Technologies, Pakistan, November 13-14, 2006.
[10] M. Morari, C.E. Gar?ia, Internal model control: 1. Unifying review and some new results, Ind. Eng. Chem. Process Des. Dev. 21 (2) (1982) 308-323.
[11] M. Morari, C.E. Gar?ia, Internal model control: 2. Design procedure for multivariables systems. Ind. Eng. Chem. Process Des. Dev. 24 (2) (1985) 472-484.
[12] M. Morari, C.E. Gar?ia, Internal model control: 3. Multivariables control law computation and tuning guidelines, Ind. Eng. Chem. Process Des. Dev. 24 (2)(1985) 448-494.
[13] C.G. Economou, M. Morari, B.O. Palsson, Internal model control: 5. Extension to nonlinear systems, Ind. Eng. Chem. Process Des. Dev. 25 (2) (1986) 403-411.
[14] S. Farhati, S.B.H.A. Naoui, M.N. Abdelkrim, Internal model control of time delay systems, in: 10th International Conference on Sciences and Techniques of Automatic Control & Computer Engineering, Tunisia, December 20-22, 2009.
[15] S.B.H. Ali, Sur la commande robuste par modèle interne, Ph.D. Thesis, National Engineering School of Tunis, Tunisia, 2003.
[16] J.P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (2003) 1667-1694.
[17] M. Morari, E. Zafiriou, Robust Process Control, Prentice Hall, 1989.
[18] M. Vajta, Some remarks on Padé-approximations, in: 3rd TEMPUS-INTCOM Symposium, Hungary, September 9-14, 2000.
[19] V. Hanta, A. Prochazka, Rational approximation of time delay, in: International Conference on Technical Computing, Prague, 2009.