# Discrete-time predictive control with overparameterized delay-plant models and an identified cancellation order

Zdzisław Kowalczuk; Piotr Suchomski

International Journal of Applied Mathematics and Computer Science (2005)

- Volume: 15, Issue: 1, page 5-34
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topKowalczuk, Zdzisław, and Suchomski, Piotr. "Discrete-time predictive control with overparameterized delay-plant models and an identified cancellation order." International Journal of Applied Mathematics and Computer Science 15.1 (2005): 5-34. <http://eudml.org/doc/207728>.

@article{Kowalczuk2005,

abstract = {The paper presents several solutions to the discrete-time generalized predictive (GPC) controller problem, including an anticipative filtration mechanism, which are suitable for plants with nonzero transportation delays. Necessary modifications of the GPC design procedure required for controlling plants based on their non-minimal models are discussed in detail. Although inevitably invoking the troublesome pole-zero cancellation problem, such models can be used in adaptive systems as a remedy for the uncertainty or variability of the model order. The purpose of this paper is to present a complete GPC controller design for delay plants that is robust to the overparameterization of the plant model. Refined conditions for the existence and stability of GPC control solutions in terms of pertinent design parameters are given, and explicit forms of closed-loop characteristic polynomials are provided. The issue of identifying the model cancellation order is also considered, and practical solutions are proposed. The presented ideas are illustrated numerically.},

author = {Kowalczuk, Zdzisław, Suchomski, Piotr},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {delay plants; robust control synthesis; system design; adaptive control; overparameterization; predictive control; overparametrization; anticipative filtering; pole-zero cancellation},

language = {eng},

number = {1},

pages = {5-34},

title = {Discrete-time predictive control with overparameterized delay-plant models and an identified cancellation order},

url = {http://eudml.org/doc/207728},

volume = {15},

year = {2005},

}

TY - JOUR

AU - Kowalczuk, Zdzisław

AU - Suchomski, Piotr

TI - Discrete-time predictive control with overparameterized delay-plant models and an identified cancellation order

JO - International Journal of Applied Mathematics and Computer Science

PY - 2005

VL - 15

IS - 1

SP - 5

EP - 34

AB - The paper presents several solutions to the discrete-time generalized predictive (GPC) controller problem, including an anticipative filtration mechanism, which are suitable for plants with nonzero transportation delays. Necessary modifications of the GPC design procedure required for controlling plants based on their non-minimal models are discussed in detail. Although inevitably invoking the troublesome pole-zero cancellation problem, such models can be used in adaptive systems as a remedy for the uncertainty or variability of the model order. The purpose of this paper is to present a complete GPC controller design for delay plants that is robust to the overparameterization of the plant model. Refined conditions for the existence and stability of GPC control solutions in terms of pertinent design parameters are given, and explicit forms of closed-loop characteristic polynomials are provided. The issue of identifying the model cancellation order is also considered, and practical solutions are proposed. The presented ideas are illustrated numerically.

LA - eng

KW - delay plants; robust control synthesis; system design; adaptive control; overparameterization; predictive control; overparametrization; anticipative filtering; pole-zero cancellation

UR - http://eudml.org/doc/207728

ER -

## References

top- Arent K., Mareels I.M.Y. and Polderman J.W. (1995): Adaptive control of linear systems based on approximate models. - Proc. 3rd European Control Conf., Rome, Italy, Vol. 2, pp. 868-873.
- Arent K., Mareels I.M.Y. and Polderman J.W. (1998): The pole-zero cancellation problem in adaptive control: A solution for minimum phase systems by approximate models. - Europ. J. Contr., Vol. 4, No. 3, pp. 320-332. Zbl0926.93041
- Banerjee P. and Shah S.L. (1992): Tuning guidelines for robust generalized predictive control. - Proc. 31st IEEE Conf. Decision Control, Tucson, USA, pp. 3233-3234.
- Berlin F. and Frank P.M. (1992): Advanced optimal predictive control. - Proc. American Control Conf., Chicago, USA, (TM8) pp. 2027-2031.
- Bjorck A. (1996): Numerical Methods for Least Squares Problems. - Philadelphia: SIAM. Zbl0847.65023
- van den Boom A.J.W. and van den Enden A.W.M. (1974): The determination of the orders of process- and noise dynamics. - Automatica, Vol. 10, No. 3, pp. 245-256. Zbl0307.93038
- Boullion T.L. and Odell P.L. (1971): Generalized Inverse Matrices. - New York: Wiley. Zbl0223.15002
- Bunch J.R. and Nielsen Ch.P. (1978): Updating the singular value decomposition. - Numer. Math., Vol. 31, No. 2, pp. 111-129. Zbl0421.65028
- Camacho E.F. and Bordons C. (1999): Model predictive control. - Berlin: Springer. Zbl1223.93037
- Clarke D.W. (1988): Application of generalized predictive control to industrial processes. - IEEE Contr. Syst. Mag., Vol. 8, No. 2, pp. 49-55.
- Clarke D.W. and Mohtadi C. (1989): Properties of generalized predictive control. - Automatica, Vol. 25, No. 6, pp. 859-876. Zbl0699.93028
- Clarke D.W., Mohtadi C. and Tuffs P.S. (1987): Generalized predictive control - Part I. The basic algorithm, Part II. Extensions and interpretations. - Automatica, Vol. 23, No. 2, pp. 137-148, 149-160. Zbl0621.93032
- Clarke D.W. and Scattolini R. (1991): Constrained receding horizon predictive control. - IEE Proc. D Contr. Theory Applic., Vol. 138, No. 4,pp. 347-354. Zbl0743.93063
- Cullen C.G. and Hall C.A. (1971): On determining whether two polynomials are relatively prime. - IEEE Trans. Automat. Contr., Vol. 16, No. 4, pp. 369-370.
- Cutler C.R. and Ramaker B.L. (1980): Dynamic matrix control -A computer control algorithm. - Proc. Joint American Control Conf., San Francisco, USA, (WP5-B).
- Demircioglu H. and Clarke D.W. (1993): Generalised predictive control with end-point state weighting. - IEE Proc. D Contr. Theory Applic., Vol. 140, No. 4, pp. 275-282. Zbl0786.93002
- Elshafei A.L., Dumont G. and Elnaggar A. (1991): Perturbation analysis of GPC with one step control horizon. - Automatica, Vol. 27, No. 4, pp. 725-728.
- Elshafei A.-L., Dumont G. and Elnaggar A. (1995): Stability and convergence analyses of an adaptive GPC based on state-space modelling. - Int. J. Contr., Vol. 61, No. 1, pp. 193-210. Zbl0825.93358
- Filali S. and Wertz V. (2001): Using genetic algorithms to optimize the design parameters of generalized predictive controllers. - Int. J. Syst.Sci., Vol. 32, No. 4, pp. 503-512. Zbl1002.93510
- Gawthrop P.J. (1987): Continuous-time Self-tuning Control. Vol. 1: Design. - Letchworth, UK: Research Studies Press. Zbl0666.93044
- Gawthrop P.J., Jones R.W. and Sbarbaro D.G. (1996): Emulator-based control and internal model control: complementary approaches to robust control design. - Automatica, Vol. 32, No. 8, pp. 1223-1227. Zbl0850.93261
- Georgiou A., Georgakis C. and Luyben W.L. (1988): Nonlinear dynamic matrix control for high-purity distillation columns. - AIChE J., Vol. 34, pp. 1287-1298.
- Grimble M.J. (1990): LQG predictive optimal control for adaptive applications. - Automatica, Vol. 26, No. 6, pp. 949-961. Zbl0717.93062
- Grimble M.J. (1992): Generalized predictive control: an introduction to the advantages and limitations. - Int. J. Syst. Sci., Vol. 23, No. 1, pp. 85-98. Zbl0777.93089
- Grimble M.J. (1993): Long range predictive optimal control law with guaranteed stability for process control applications. - Trans. ASME J. Dyn.Syst. Measur. Contr., Vol. 115, No. 4, pp. 600-610. Zbl0796.93105
- Golub G.H. and van Loan C.F. (1996): Matrix Computations. - Baltimore: The Johns Hopkins University Press.
- Gorez R., Wertz V. and Zhu K.Y. (1987): On a generalized predictive control algorithm. - Syst. Contr. Lett., Vol. 9, pp. 369-377. Zbl0624.93037
- Halpern M.E. (1988): Modified pole-assignment controller for plant models with exact or near pole-zero cancellation. - IEE Proc. D Contr. Theory Applic., Vol. 135, No. 3, pp. 189-195. Zbl0642.93029
- Hangstrup M.E., Ordys A.W. and Grimble M.J. (1997): Multivariable state space LQGPC. - Techn. Rep., Industrial Control Centre, University of Strathclyde, Glasgow (Scotland), UK, (R-1997-4187).
- Higham N.J. (1996): Accuracy and Stability of Numerical Algorithms. - Philadelphia: SIAM. Zbl0847.65010
- Hinde Jr.R.F. and Cooper D.J. (1994): A pattern-based approach to excitation diagnostics for adaptive process control. - Chem. Engng. Sci., Vol. 49, pp. 1403-1415.
- Ho B.L. and Kalman R.E. (1966): Effective construction of linear state-variable models from inputoutput functions. - Regelungstechnik, Vol. 14, No. 12,pp. 545-548. Zbl0145.12701
- Jolly T. and Bentsman J. (1993): Generalized predictive control algorithms with guaranteed frozen-time stability and bounded tracking error. - Proc.s American Control Conf., San Francisco (CA), USA, (WM7), pp. 384-388.
- de Keyser R., and van Cauwenberghe A. (1981): A self-tuning multistep predictor application. - Automatica, Vol. 17, No. 2, pp. 167-174.
- Kleinman D.L. (1974): Stabilizing a discrete, constant, linear system with application to iterative methods for solving the Riccati equation. - IEEE Trans. Automat. Contr., Vol. 19, No. 3, pp. 252-254. Zbl0297.93046
- van der Kooji D.J. and Polderman J.W. (1993): Adaptive control of a restricted class of minimum phase systems on the basis of approximate models. - Proc. 2nd Europ. Contr. Conf., Groningen, Netherlands, Vol. 1, pp. 199-201.
- Kouvaritakis B. and Rossiter J.A. (1993): Multivariable stable generalised predictive control. - IEE Proc. D Contr. Theory Applic., Vol. 140, No. 5, pp. 364-372. Zbl0786.93056
- Kouvaritakis B., Rossiter J.A. and Chang A.O.T. (1992): Stable generalised predictive control: an algorithm with guaranteed stability. - IEE Proc. D Contr. Theory Applic., Vol. 139, No. 4, pp. 349-362. Zbl0765.93065
- Kouvaritakis B., Rossiter J.A. and Gossner J.R. (1997): Improved algorithm for multivariable stable generalised predictive control. - IEE Proc. D Contr. Theory Applic., Vol. 144, No. 4, pp. 309-312. Zbl0900.93087
- Kowalczuk Z., Suchomski P. and Marcinczyk A. (1996): Discrete-time and continuous-time generalised predictive control with anticipated filtration: tuning rules. - Int. J. Appl. Math. Comput. Sci., Vol. 6, No. 4, pp. 707-732. Zbl0867.93032
- Kowalczuk Z. and Suchomski P. (1997): Generalised predictive control of delay systems. - Proc. 4th Europ. Control Conf., Brussels, Belgium, Vol. 2, (CD-ROM: FR-M, B1) pp. 1-6. Zbl0926.93028
- Kowalczuk Z. and Suchomski P. (1998): Two-degree-of-freedom stable GPC design.- Proc. IFAC Workshop Adaptive Systems in Control and Signal Processing, Glasgow, Scotland, U.K., pp. 243-248.
- Kowalczuk Z. and Suchomski P. (1999): Analytical design of stable continuous-time generalised predictive control. - Int. J. Appl. Math. Comput. Sci., Vol. 9, No. 1, pp. 53-100. Zbl0926.93028
- Kowalczuk Z. and Suchomski P. (2001): Discrete-time predictive control design applied to overparameterized delay-plant models. - Faculty Report, Electronics Telecom. Computer Sci., Gdańsk University of Technology, No. 5, pp. 1-33.
- Kowalczuk Z. and Suchomski P. (2002): Simple stable discrete-time generalised predictive control with anticipated filtration of control error. - Contr.Cybern., Vol. 31, No. 1, pp. 1-22. Zbl1027.93014
- Kowalczuk Z. and Suchomski P. (2004): Robust continuous-time controller design via structural Youla-Kučera parameterization with application to predictive control. - Optim. Contr. Applic. Meth., Vol. 24, pp. 1-28. Zbl1073.93024
- Kr''amer K. and Unbehauen H. (1992): Aspects in selecting predictive adaptive control algorithms. - Proc. American Control Conf., Chicago, USA, (TP8) pp. 2396-2401.
- Kreisselmeier G. (1986): A robust indirect adaptive control approach. - Int. J. Contr., Vol. 43, No. 1, pp. 161-175. Zbl0589.93039
- Kung S.Y. (1978): A new identification and model reduction algorithm via singular decomposition. - Proc. 12th Asilomar Conf. Circuits, Systems and Computers, Pacific Grove (CA), USA, pp. 705-714.
- Kwon W.H. and Byun D.G. (1989): Receding horizon tracking control as a predictive control and its stability properties. - Int. J. Contr., Vol. 50, No. 5, pp. 1807-1824. Zbl0688.93020
- Kwon W.H., Noh S. and Lee Y. (1992a): Stability guaranteed generalized predictive controls and its equivalence to receding horizon tracking control. - Proc. American Control Conf., Chicago, USA, (TM8) pp. 2037-2041.
- Kwon W.H. and Pearson A.E. (1975): On the stabilization of a discrete constant linear system. - IEEE Trans. Automat. Contr., Vol. 20, No. 6, pp. 800-801. Zbl0319.93040
- Kwon W.H. and Pearson A.E. (1978): On feedback stabilization of time-varying discrete linear systems. - IEEE Trans. Automat. Contr., Vol. 23, No. 3, pp. 479-481. Zbl0378.93038
- Kwon W.H., Young Y.I. and Noh S. (1992b): Partition of GPC into a state observer and a state feedback controller. - Proc. American Control Conf., Chicago, USA, (TM8) pp. 2032-2036.
- Landau I.D., Lozano R. and M'Saad M. (1998): Adaptive Control. - Berlin: Springer.
- Lee J.H., Morari M. and Garcia C.E. (1994): State-space interpretation of model predictive control. - Automatica, Vol. 30, No. 4, pp. 707-717. Zbl0800.93025
- Li S., Lim K. and Fisher D. (1989): A state-space formulation for model predictive control. - AIChE J., Vol. 35, pp. 241-249.
- Lim K.W., Ho W.K., Lee T.H., Ling K.V. and Xu, W. (1998): Generalised predictive controller with pole restriction. - IEE Proc. D Contr. Theory Applic., Vol. 145, No. 2, pp. 219-225.
- Longchamp R. (1983): Singular perturbation analysis of a receding horizon controller. - Automatica, Vol. 19, No. 3, pp. 303-308. Zbl0509.93031
- Lozano-Leal R. (1989): Robust adaptive regulation without persistent excitation. - IEEE Trans. Automat. Contr., Vol. 34, No. 12, pp. 1260-1266. Zbl0689.93038
- Lozano-Leal R. and Goodwin G. (1985): A globally convergent adaptive pole placement algorithm without a persistency of excitation requirement. - IEEE Trans. Automat. Contr., Vol. 30, No. 8, pp. 795-798. Zbl0565.93028
- Lozano-Leal R. and Zhao X.H. (1994): Adaptive pole placement without excitation probing signals. - IEEE Trans. Automat. Contr., Vol. 39, No. 1, pp. 47-58. Zbl0796.93068
- Maciejowski J.M. (2002): Predictive control with constraints. - Harlow: Prentice Hall. Zbl0978.93002
- McIntosh A.R., Shah S.L. and Fisher D.G. (1991): Analysis and tuning of adaptive generalized predictive control. - Canad. J. Chem. Eng., Vol. 69, No. 2, pp. 97-110.
- Middleton R.H. and Goodwin G.C. (1990): Digital Control and Estimation. - Englewood Cliffs, N.J.: Prentice Hall.
- Mohtadi C. and Clarke D.W. (1986): Generalized Predictive Control, LQ or Pole Placement: A Unified Approach. - Proc. 25th IEEE Conf. Decision and Control, Athens, Greece, pp. 1536-1541.
- Mosca E. and Zhang J. (1992): Stable redesign of predictive control. - Automatica, Vol. 28, No. 6, pp. 1229-1233. Zbl0775.93056
- Ogata K. (1995): Discrete-Time Control Systems. - Englewood Cliffs, N.J.: Prentice Hall.
- Ordys A.W. and Clarke D.W. (1993): A state-space description for GPC controllers. - Int. Journ. Systems Science, Vol. 24, No. 9, pp. 1727-1744. Zbl0788.93008
- Ossman K.A. and Kamen E.D. (1987): Adaptive regulation of MIMO linear discrete-time systems without requiring a persistent excitation. - IEEE Trans. Automat. Contr., Vol. 32, No. 5, pp. 397-404. Zbl0617.93036
- van Overschee P., de Moor B. and Fovereel W. (1997): Numerical algorithms for subspace state-space system identification. - Proc. ASME Design Engineering Technical Conf., Sacramento (CA), USA, (DETEC97VIB4254).
- Peng L., Fisher D.G. and Shah S.L. (1992): Tuning generalized predictive control using a pole-placement criterion. - Proc. American Control Conf., Chicago, USA, (TP8) pp. 2391-2395.
- Peng L., Fisher D.G. and Shah S.L. (1993): On-line tuning of GPC using a pole placement criterion. - Proc. American Control Conf., San Francisco, USA, (FM11) pp. 2637-2641.
- Peng Y. and Hanus R. (1991): A tuning strategy for generalized predictive control. - Contr. Theory Adv. Technol., Vol. 7, No. 1, pp. 153-166.
- Rani Y.K. and Unbehauen H. (1996): Tuning and auto-tuning in predictive control. - Proc. 13th Triennial World Congress of IFAC, San Francisco (CA), USA, Vol. C, (3b-04-1) pp. 109-114.
- Rani Y.K. and Unbehauen H. (1997): Study of predictive controller tuning methods. - Automatica, Vol. 33, No. 12, pp. 2243-2248. Zbl0900.93084
- Richalet J., Rault A., Testud J.L. and Papon J. (1978): Model predictive heuristic control: Applications to industrial processes. - Automatica, Vol. 14, No. 5, pp. 413-428.
- Ricker N. L. (1990): Model predictive control with state estimation. - Ind. Eng. Chem. Res., Vol. 29, No. 3, pp. 374-382.
- Rissanen J. (1971): Recursive identification of linear systems. - SIAM J. Contr. Optim., Vol. 9, No. 3, pp. 420-430. Zbl0204.46202
- Rouhani R. and Mehra R.K. (1982): Model algorithmic control (MAC). Basic theoretical properties. - Automatica, Vol. 18, No. 4, pp. 401-414. Zbl0483.93045
- Rossiter J.A. (1997): Predictive controllers with guaranteed stability and mean-level controllers for unstable plants. - Europ. J. Contr., Vol. 3, No. 2, pp. 292-303. Zbl0895.93018
- Rossiter J.A. (2003): Model-based predictive control: a practical approach. - Boca Raton: CRC Press.
- Rossiter J.A., Chisci L. and Lombardi A. (1997): Stabilizing predictive control algorithms in the presence of common factors. - Proc. 4th European Control Conf., Brussels, Belgium, (CD-ROM: TH-E, B5) pp. 1-6.
- Rossiter J.A. and Kouvaritakis B. (1994): Numerical robustness and efficiency of generalised predictive control algorithms with guaranteed stability. - IEE Proc. D Contr. Theory Applic., Vol. 141, No. 3, pp. 154-162. Zbl0799.93049
- Rossiter J.A., Kouvaritakis B. and Rice M.J. (1998): A numerically robust state-space approach to stable-predictive control strategies. - Automatica, Vol. 34, No. 1, pp. 65-73. Zbl0913.93022
- Scattolini R. and Bittanti S. (1990): On the choice of the horizon in long-range predictive control: Some simple criteria. - Automatica, Vol. 26, No. 5,pp. 915-917. Zbl0701.93033
- Scokaert P.O.M. and Clarke D.W. (1994): Stabilising properties of constrained predictive control. - Proc. IEE-D Contr. Theory Applic., Vol. 141, No. 5, pp. 295-304. Zbl0925.93555
- Shah S.L., Mohtadi C. and Clarke D.W. (1987): Multivariable adaptive control without prior knowledge of the delay matrix. - Syst. Contr. Lett., Vol. 9,pp. 295-306. Zbl0623.93048
- Silverman L.M. (1971): Realization of linear dynamical systems. - IEEE Trans. Automat. Contr., Vol. 16, No. 6, pp. 554-567.
- Soderstrom T. (1974): Convergence properties of the generalized least squares identification method. - Automatica, Vol. 10, No. 6, pp. 617-626. Zbl0293.93025
- Söderström T. (1975): Test of pole-zero cancellation in estimated models. - Automatica, Vol. 11, No. 5, pp. 537-541. Zbl0316.93013
- Söderström T. and Stoica P. (1989): System Identification. - Hemel Hemstead, U.K.: Prentice Hall.
- Stewart G.W. (1998): Matrix Algorithms. Vol. I: Basic Decompositions. - Philadelphia: SIAM.
- Stoica P. and Söderström T. (1996): Common factor detection and estimation. - Techn. Rep. Systems Control Group, Dept. Technology, Uppsala University, Uppsala, Sweden.
- Suchomski P. and Kowalczuk Z. (2002a): Pre-arrangement of solvability, complexity, stability and quality of GPC systems. - Int. J. Adapt. Contr. Signal Proces., Vol. 16, No. 3, pp. 177-191. Zbl1006.93024
- Suchomski P. and Kowalczuk Z. (2002b): Analytical design of stable delta-domain generalized predictive control. - Optim. Contr. Applic. Meth., Vol. 23, pp. 239-273. Zbl1072.93510
- Taube B. (1991): Clearness of tuning possibilities in long-range predictive control (LRPC). - Proc. 30th IEEE Conf. Decision and Control, Brighton, UK, (F2-10) pp. 2726-2728.
- Taube B. and Lampe B. (1992): LQGPC - a predictive design as tradeoff between LQG and GPC. - Proc. 31st Conf. Decision and Control, Tucson, USA, (FP6) pp. 3579-3581.
- Tether A.J. (1970): Construction of minimal linear state-variable models from finite input-output data. - IEEE Trans. Automat. Contr., Vol. 15, No. 4,pp. 427-436.
- Unbehauen H. and Göhring B. (1974): Tests for determining model order in parameter estimation. - Automatica, Vol. 10, No. 3, pp. 233-244. Zbl0281.62092
- Vogt W.G. and Bose N.K. (1970): A method to determine whether two polynomials are relatively prime. - IEEE Trans. Autom. Contr., Vol. 15, No. 3, pp. 379-380.
- Yoon T.W. and Clarke D.W. (1993): Receding-horizon predictive control with exponential weighting. - Int. J. Syst. Sci., Vol. 24, No. 9,pp. 1745-1757. Zbl0785.93008
- Wellstead P.E. and Zarrop M.B. (1991): Self-tuning Systems. Control and Signal Processing. - Chichester: Wiley. Zbl0505.93047
- Xi Y. (1989): Minimal form of a predictive controller based on the step-response model. - Int. J. Contr., Vol. 49, No. 1, pp. 57-64. Zbl0664.93011
- Ydstie B.E. (1984): Extended horizon adaptive control. - Proc. 9ths Triennal World Congress of IFAC, Budapest, Hungary, Vol. VII, (14.4E-4) pp. 133-137.
- Ydstie B.E., Kershenbaum L.S. and Sargent R.W.H. (1985): Theory and application of the extended horizon self-tuning controller. - AIChE J., Vol. 31, No. 11, pp. 1771-1780.
- Zeiger H.P. and McEwen A.J. (1974): Approximate linear realizations of given dimension via Ho's algorithm. - IEEE Trans. Automat. Contr., Vol. 19, No. 2,pp. 153. Zbl0301.93017
- Zhang J. (1996): Property analysis of GPC based on coefficient mapping. - Proc. 13th Triennial World Congress of IFAC, San Francisco, USA, Vol. C, (2a-13-5)pp. 457-462.
- Zhang J. and Xi Y. (1998): Some GPC stability results. - Int. J. Contr., Vol. 70, No. 5, pp. 831-840. Zbl0930.93035

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.