Skip to main content

Model Predictive Control for Uncertain PWA Systems Based on Robust Observer

Abstract

In this paper, a new robust model predictive control (RMPC) is proposed for uncertain nonlinear systems. The nonlinear behavior is described by uncertain piecewise affine models, where the parametric uncertainties are considered time varying with norm-bounded structure. The proposed control scheme consists of two steps. First, a proportional gain observer is designed. The robustness and the stability of this observer is proved using Lyapunov function and linear matrix inequalities. It is characterized by a simple structure. Second, a RMPC, which denotes a powerful tool for the control of Uncertain PieceWise Affine systems, is established. Simulation of numerical and experimental examples is presented to illustrate the effectiveness of the proposed approach.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

References

  • Alessandri A, Coletta P (2001) Switching observers for continuous-time and discrete-time linear systems. In: American control conference, 2001. Proceedings of the 2001, vol 3. IEEE, pp 2516–2521

  • Atia SB, Messaoud A, Abdennour RB (2015) Supervised model predictive control for discrete-time nonlinear systems with time-varying delay. Int J Comput Appl 114(12):1–8

    Google Scholar 

  • Balluchi A, Benvenuti L, Di Benedetto MD, Vincentelli ALS (2001) A hybrid observer for the driveline dynamics. In: Control conference (ECC), 2001 European. IEEE, pp 618–623

  • Balluchi A, Benvenuti L, Sangiovanni-Vincentelli AL (2002) Observers for hybrid systems with continuous state resets. In: Proceedings of the 10th Mediterranean conference on control and automation-MED

  • Baotić M, Christophersen FJ, Morari M (2003) A new algorithm for constrained finite time optimal control of hybrid systems with a linear performance index. In: 2003 European control conference (ECC). IEEE, pp 3323–3328

  • Bemporad A, Morari M (1999) Control of systems integrating logic, dynamics, and constraints. Automatica 35(3):407–427

    MathSciNet  Article  Google Scholar 

  • Bemporad A, Morari M (1999) Robust model predictive control: a survey. In: Robustness in identification and control. Springer, pp. 207–226

  • Bien Z, Kim JH (1992) A robust stability bound of linear systems with structured uncertainty. IEEE Trans Autom Control 37(10):1549–1551

    MathSciNet  Article  Google Scholar 

  • Birouche A (2006) Contribution sur la synthèse d’observateurs pour les systèmes dynamiques hybrides. Ph.D. thesis, Institut National Polytechnique de Lorraine-INPL

  • Borrelli F, Baotic M, Bemporad A, Morari M (2003) An efficient algorithm for computing the state feedback optimal control law for discrete time hybrid systems. In: Proceedings of the 2003 American control conference, 2003., vol 6. pp 4717–4722

  • Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory, vol 15. Siam, New York

    Book  Google Scholar 

  • Chen H, Scherer CW (2006) Moving horizon h8 control with performance adaptation for constrained linear systems. Automatica 42(6):1033–1040

    MathSciNet  Article  Google Scholar 

  • Chua L, Deng AC (1986) Canonical piecewise-linear modeling. IEEE Trans Circuits Syst 33(5):511–525

    Article  Google Scholar 

  • Daafouz J, Millerioux G, Iung C (2002) A poly-quadratic stability based approach for linear switched systems. Int J Control 75(16–17):1302–1310

    MathSciNet  Article  Google Scholar 

  • Ding B (2010) Constrained robust model predictive control via parameter-dependent dynamic output feedback. Automatica 46(9):1517–1523

    MathSciNet  Article  Google Scholar 

  • Ding B, Xi Y, Cychowski MT, O’Mahony T (2008) A synthesis approach for output feedback robust constrained model predictive control. Automatica 44(1):258–264

  • Ferrari-Trecate G, Muselli M, Liberati D, Morari M (2003) A clustering technique for the identification of piecewise affine systems. Automatica 39(2):205–217

    MathSciNet  Article  Google Scholar 

  • Gao Y, Liu Z, Chen H (2009) Robust h control for constrained discrete-time piecewise affine systems with time-varying parametric uncertainties. IET Control Theory Appl 3(8):1132–1144

    MathSciNet  Article  Google Scholar 

  • Gao Y, Liu Z, Chen H (2012) Robust observer-based control for uncertain discrete-time piecewise affine systems. J Control Theory Appl 10(2):236–243

    MathSciNet  Article  Google Scholar 

  • Gao Z, Cecati C, Ding SX (2015) A survey of fault diagnosis and fault-tolerant techniques-part i: fault diagnosis with model-based and signal-based approaches. IEEE Trans Ind Electron 62(6):3757–3767

    Article  Google Scholar 

  • Gassara H, El Hajjaji A, Chaabane M (2009) Robust control of T–S fuzzy systems with time-varying delay: new approach. In: Proceedings of the 48th IEEE conference on decision and control, 2009 held jointly with the 2009 28th Chinese control conference. CDC/CCC 2009. IEEE, pp 4168–4173

  • Guo SX (2014) Robust reliability based optimal design of h8 control of parametric uncertain systems. J Dyn Syst Meas Control 136(2):024504

    Article  Google Scholar 

  • Hong C, Scherer C (2006) Moving horizon h8 control with performance adaptation for constrained linear systems. Automatica 42(6):1033–1040

    MathSciNet  Article  Google Scholar 

  • Hovakimyan N, Nardi F, Calise A, Kim N (2002) Adaptive output feedback control of uncertain nonlinear systems using single-hidden-layer neural networks. IEEE Trans Neural Netw 13(6):1420–1431

    Article  Google Scholar 

  • Julián P, Jordán M, Desages A (1998) Canonical piecewise-linear approximation of smooth functions. IEEE Trans Circuits Syst I Fund Theory Appl 45(5):567–571

    MathSciNet  Article  Google Scholar 

  • Kerrigan EC, Mayne DQ (2002) Optimal control of constrained, piecewise affine systems with bounded disturbances. In: Proceedings of the 41st IEEE conference on decision and control, 2002., vol 2. IEEE, pp 1552–1557

  • Kim TH, Park JH, Sugie T (2006) Output-feedback model predictive control for lpv systems with input saturation based on quasi-min-max algorithm. In: Proceedings of the 45th IEEE conference on decision and control. IEEE, pp 1454–1459

  • Kothare MV, Balakrishnan V, Morari M (1996) Robust constrained model predictive control using linear matrix inequalities. Automatica 32(10):1361–1379

    MathSciNet  Article  Google Scholar 

  • Lan J, Patton RJ (2016) A new strategy for integration of fault estimation within fault-tolerant control. Automatica 69:48–59

    MathSciNet  Article  Google Scholar 

  • Lee YI, Kouvaritakis B (2001) Receding horizon output feedback control for linear systems with input saturation. IEE Proc Control Theory Appl 148(2):109–115

    Article  Google Scholar 

  • Lee YS, Kwon WH (2002) Delay-dependent robust stabilization of uncertain discrete-time state-delayed systems. In: Preprints of the 15th IFAC world congress. Barcelona, Spain

  • Li D, Xi Y (2010) The feedback robust mpc for lpv systems with bounded rates of parameter changes. IEEE Trans Autom Control 55(2):503–507

    MathSciNet  Article  Google Scholar 

  • Li D, Xi Y, Zheng P (2009) Constrained robust feedback model predictive control for uncertain systems with polytopic description. Int J Control 82(7):1267–1274

    MathSciNet  Article  Google Scholar 

  • Li D, Xi Y, Gao F (2013) Synthesis of dynamic output feedback rmpc with saturated inputs. Automatica 49(4):949–954

    MathSciNet  Article  Google Scholar 

  • Lin JN, Unbehauen R (1992) Canonical piecewise-linear approximations. IEEE Trans Circuits Syst I Fund Theory Appl 39(8):697–699

    MathSciNet  Article  Google Scholar 

  • Mayne DQ, Raković S (2003) Model predictive control of constrained piecewise affine discrete-time systems. Int J Robust Nonlinear Control IFAC Affil J 13(3–4):261–279

    MathSciNet  Article  Google Scholar 

  • Mayne DQ, Raković S, Findeisen R, Allgöwer F (2006) Robust output feedback model predictive control of constrained linear systems. Automatica 42(7):1217–1222

    MathSciNet  Article  Google Scholar 

  • Messaoud A (2010) Sur la représentation et la commande prédictive multimodeles des systemes complexes. Ecole Nationale d’Ingenieurs de Gabes

  • Necoara I, De Schutter B, van den Boom TJ, Hellendoorn J (2004) Model predictive control for perturbed continuous piecewise affine systems with bounded disturbances. In: 43rd IEEE conference on decision and control, 2004. CDC, vol 2. IEEE, pp. 1848–1853

  • Orjuela R, Marx B, Ragot J, Maquin D (2008) Conception d’observateurs robustes pour des systèmes non linéaires incertains: une stratégie multimodèle. In: 5ème Conférence Internationale Francophone d’Automatique, CIFA’2008, p. CDROM

  • Petersen IR (1987) A stabilization algorithm for a class of uncertain linear systems. Syst Control Lett 8(4):351–357

    MathSciNet  Article  Google Scholar 

  • Popescu N, Ivanescu M, Popescu D (2018) A note on observer-based frequency control for a class of systems described by uncertain models. J Dyn Syst Meas Control 140(2):021008

    Article  Google Scholar 

  • Rossiter JA (2003) Model-based predictive control: a practical approach. CRC Press, London

    Google Scholar 

  • Silva MP, Bemporad A, Botto MA, da Costa JS (2003) Optimal control of uncertain piecewise affine/mixed logical dynamical systems. In: European control conference (ECC), 2003. IEEE, pp 1573–1578

  • Thomas J, Olaru S, Buisson J, Dumur D (2006) Robust model predictive control for piecewise affine systems subject to bounded disturbances. IFAC Proc Vol 39(5):329–334

    Article  Google Scholar 

  • Waitman S, Bako L, Massioni P, Scorletti G, Fromion V (2017) Incremental stability of lur’e systems through piecewise-affine approximations. IFAC-PapersOnLine 50(1):1673–1679

  • Waitman S, Massioni P, Bako L, Scorletti G, Fromion V (2016) Incremental l 2-gain analysis of piecewise-affine systems using piecewise quadratic storage functions. In: 2016 IEEE 55th conference on decision and control (CDC). IEEE, pp 1334–1339

  • Wen C, Zhou J, Liu Z, Su H (2011) Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance. IEEE Trans Autom Control 56(7):1672–1678

    MathSciNet  Article  Google Scholar 

  • Yao J, Jiao Z, Ma D (2014) Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE Trans Ind Electron 61(11):6285–6293

    Article  Google Scholar 

  • Zhang J, Tang W (2008) Output feedback h control for uncertain piecewise linear systems. J Dyn Control Syst 14(1):121–144

    MathSciNet  Article  Google Scholar 

  • Zhou J, Wen C (2006) Robust adaptive control of uncertain nonlinear systems in the presence of input saturation. IFAC Proc Vol 39(1):149–154

    Article  Google Scholar 

  • Zhu Y, Li D, Feng G (2005) H controller synthesis of uncertain piecewise continuous-time linear systems. IEE Proc Control Theory Appl 152(5):513–519

    Article  Google Scholar 

  • Zou Y, Li S (2007) Robust model predictive control for piecewise affine systems. Circuits Syst Signal Process 26(3):393–406

    MathSciNet  Article  Google Scholar 

  • Zou Y, Niu Y, Chen B, Jia T (2013) Networked predictive control of constrained linear systems with input quantisation. Int J Syst Sci 44(10):1970–1982

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Olfa Yahya.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yahya, O., Lassoued, Z. & Abderrahim, K. Model Predictive Control for Uncertain PWA Systems Based on Robust Observer. Iran J Sci Technol Trans Electr Eng 46, 109–125 (2022). https://doi.org/10.1007/s40998-021-00446-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40998-021-00446-5

Keywords

  • Nonlinear systems
  • Uncertain piecewise affine systems
  • Robust observer
  • Model predictive control
  • Linear matrix inequalities