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A Nash game approach to mixed \({\mathcal {H}}_{2}/{\mathcal {H}}_{\infty }\) model predictive control: part 2—stability and robustness

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Abstract

This paper is a note on the stability and robustness of the Nash Game (Vamvoudakis et al. in Adaptive optimal control algorithm for zero-sum nash games with integral reinforcement learning, 2012; Ning et al. in Optim Control Appl Methods. doi:10.1002/oca.2042, 2012; Bouyer et al. in Concurrent games with ordered objectives, 2012) based mixed \({\mathcal {H}}_{2}/{\mathcal {H}}_{\infty }\) Model Predictive Controllers (Aadaleesan and Saha in Mixed \(\mathcal {H}_{2}/\mathcal {H}_{\infty }\) Model Predictive Control for Unstable and Non-Minimum Constrained Processes, 2008; Aadaleesan in Nash Game based Mixed \(\mathcal {H}_{2}/\mathcal {H}_{\infty }\) Model Predictive Control applied with Laguerre-Wavelet Network Model, 2011) for linear state feedback systems addressed in Part 1 (Aadaleesan and Saha in Int J Dyn Control, 2016) of this series. The mixed \({\mathcal {H}}_{2}/{\mathcal {H}}_{\infty }\) MPC proposed in Part 1 (Aadaleesan and Saha in Int J Dyn Control, 2016) and that developed by Orukpe et al. (Model predictive control based on mixed \(\mathcal {H}_2/\mathcal {H}_{\infty }\) control approach, 2007) are compared in this Part 2. The issues of stability and robustness of the multi-criterion optimal control are dealt in this paper using set theoretic concepts.

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Notes

  1. A set-valued map \(F:\varXi \leadsto \varLambda \) can also be regarded as a single-valued map \(F:\varXi \rightarrow 2^{\varLambda }\) (where \(2^{\varLambda }\) denotes the power set of all the subsets of \(\varLambda \)).

  2. The notation \(|\cdot |_{{\mathcal {T}}}\) represents the Euclidean point-to-set distance function, that is, \(|\cdot |_{{\mathcal {T}}}:=d(\cdot ,{\mathcal {T}})\).

  3. \({sat}(\cdot )\) represents the saturation nonlinearity. Note, in the present context of the paper, where input constraint is considered, \({sat}(\cdot )\equiv \partial {\mathcal {U}}\).

  4. Recall the use of slack variable \(\{\epsilon _{k}^{1},\epsilon _{k}^{2}\}\).

  5. A C-Set is a convex and compact set containing the origin [32].

  6. \({\mathcal {P}}\) represents the set of all solution pair \(\left\{ P_{k}^{I},P_{k}^{II}\right\} \) that satisfies cAREs, for the given system conditions and controller design objectives.

  7. The control input \(u_{k}\in \partial {\mathcal {U}}\) which is an admissible control such that the response \(x_{k}\) of the system is in Ext(\({\mathcal {R}}_{k}\)) is called the Extremal Control.

References

  1. Vamvoudakis KG, Vrabie D, Lewis FL (2012) Adaptive optimal control algorithm for zero-sum nash games with integral reinforcement learning. In: Proceesings of AIAA Guidance, Navigation, and Control Conference. Minneapolis, Minnesota, p 4773

  2. Ning G, Tie L, Cai KY, Wu XY (2012) Nash equilibrium solutions of tracking game for bilinear systems with exponential reference signals. Optim Control Appl Methods. doi:10.1002/oca.2042

    MATH  Google Scholar 

  3. Bouyer P, Brenguier R, Markey N, Ummels M (2012) Concurrent games with ordered objectives. In: Proceedings of the 15th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS’12). Tallinn, Estonia, pp 301–315

  4. Aadaleesan P, Saha P (2008) Mixed \({\cal {H}}_{2}/\cal H\cal _{\infty }\) Model Predictive Control for Unstable and Non-Minimum Constrained Processes. In: Proceedings of Annual Meeting of American Institute of Chemical Engineers. Philadelphia

  5. Aadaleesan P (2011) Nash Game based Mixed \({\cal {H}}_{2}/\cal H\cal _{\infty }\) Model Predictive Control applied with Laguerre-Wavelet Network Model. PhD Dissertation, IIT Guwahati

  6. Aadaleesan P, Saha P (2016) A Nash game approach to mixed \({\cal {H}}_{2}/\cal H\cal _{\infty }\) model predictive control : part 1 - state feedback linear system. Int J Dyn Control. doi:10.1007/s40435-016-0261-y

  7. Orukpe PE, Jaimoukha IM, El-Zobaidi HMH (2007) Model predictive control based on mixed \({\cal {H}}_2/\cal H\cal _{\infty }\) control approach. In: Proceedings of American Control Conference. New York, pp 6147–6150

  8. Bertsekas DP, Rhodes IB (1973) Sufficiently informative functions and the minimax feedback control of uncertain dynamic systems. IEEE Trans Autom Control 18:117–124

    Article  MATH  MathSciNet  Google Scholar 

  9. Blanchini F (1999) Set invariance in control: a survey. Automatica 35(11):1747–1767

    Article  MATH  MathSciNet  Google Scholar 

  10. Blanchini F, Miani S (2008) Set-theoretic methods in control. Birkhäuser, Boston

    MATH  Google Scholar 

  11. Cwikel M, Gutman PO (1986) Convergence of an algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states. IEEE Trans Autom Control 31(5):457–459

    Article  MATH  Google Scholar 

  12. Delfour M, Mitter S (1969) Reachability of perturbed systems and min sup problems. SIAM J Control 7(4):521–533

    Article  MATH  MathSciNet  Google Scholar 

  13. Gutman PO, Cwikel M (1986) Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states. IEEE Trans Autom Control 31(4):373–376

    Article  MATH  MathSciNet  Google Scholar 

  14. Kerrigan EC (2000) Robust constraint satisfaction : Invariant sets and predictive control. Ph.D. thesis, Univ. Cambridge, Cambridge

  15. Raković SV, Kerrigan EC, Kouramas KI, Mayne DQ Invariant approximations of robustly postively invariant sets for linear constrained linear discrete-time systems subject to bounded disturbances. Tech. rep., University of Cambridge, UK

  16. Witsenhausen HS (1968) A minimax control problem for sampled linear systems. IEEE Trans Autom Control 13(1):5–21

    Article  MathSciNet  Google Scholar 

  17. Gutman PO, Cwikel M (1987) An algorithm to find maximal state constriant sets for discrete-time linear dynamical systems with bounded controls and states. IEEE Trans Autom Control 32(3):251–254

  18. Gilbert EG, Tan KT (1991) Linear systems with state and control constraints: the theory and applications of maximal output admissible sets. IEEE Trans Autom Control 36(9):1008–1020

    Article  MATH  MathSciNet  Google Scholar 

  19. Blanchini F (1991) Ultimate boundedness control for uncertain discrete-time uncertain systems via set-induced lyapunov functions. In: Proceedings of 30th IEEE Conference on Decision and Control. Brighton, pp 1755–1760

  20. Hu T, Lin Z (2001) Control systems with actuator saturation. Birkhäuser, Boston

  21. Jiang ZP, Sontag E, Wang Y (1999) Input-to-state stability for discrete-time nonlinear systems. In: Proceedings of the 14th triennial IFAC world congress

  22. Freeman RA, Kokotović PV (2008) Robust nonlinear control design: state-space and Lyapunov techniques. Birkhäuser, Boston

  23. Grimm G, Messina MJ, Tuna SE, Teel AR (2004) Examples when nonlinear model predictive control is nonrobust. Automatica 40:1729–1738

    Article  MATH  MathSciNet  Google Scholar 

  24. Kerrigan EC, Maciejowski JM (2000) Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control. In: Proceedings of the 39th IEEE Conference on Decision and Control. Sydney, pp 4951–4956

  25. Kerrigan EC, Maciejowski JM (2001) Robust feasibility in model predictive control : Necessary and sufficient conditions. In: Proceedings of the 40th IEEE Conference on Decision and Control. Florida, pp 728–733

  26. Sontag ED (1990) Further facts about input to state stabilization. IEEE Trans Autom Control 35:473–476

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Kim J, Yoon T, Jadbabaie A, De Persis C (2004) Input-state stabilizing MPC for naturally stable linear systems subject to input constraints. In: Proceedings of 43th- IEEE Conference on Decision and Control. Paradise Island, Bahamas, pp 5041–5046

  29. Goulart PJ, Kerrigan EC, Maciejowski JM (2006) Optimization over state feedback policies for robust control with constraints. Automatica 42:523–533

    Article  MATH  MathSciNet  Google Scholar 

  30. Jiang ZP, Wang Y (2001) Input-to-state stability for discrete-time nonlinear systems. Automatica 37:857–869

    Article  MATH  MathSciNet  Google Scholar 

  31. Goulart PJ, Kerrigan EC, Maciejowski JM (2005) State feedback policies for robust receding horizon control : Uniqueness, Continuity and Stability. In: Proceedings of 44th- IEEE Conference on Decision and Control. Seville, pp 3753–3758

  32. Blanchini F (1994) Ultimate boundedness control for uncertain discrete-time systems via set-induced lyapunov functions. IEEE Trans Autom Control 39(2):428–433

    Article  MATH  MathSciNet  Google Scholar 

  33. Iwasaki T, Fu M (2002) Actuator saturation control. Control engineering. In: Kapila V, Grigoriadis KM (eds) Regional \({\cal {H}}_2\) performance synthesis. Marcel Dekker Inc., New York

    Google Scholar 

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Author notes

  1. Pakkiriswamy Aadaleesan

    Present address: School of Electrical and Electronics Engineering, SASTRA University, Thanjavur, TN, 613 402, India

Authors and Affiliations

  1. Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

    Pakkiriswamy Aadaleesan & Prabirkumar Saha

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  1. Pakkiriswamy Aadaleesan
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  2. Prabirkumar Saha
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Correspondence to Prabirkumar Saha.

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Aadaleesan, P., Saha, P. A Nash game approach to mixed \({\mathcal {H}}_{2}/{\mathcal {H}}_{\infty }\) model predictive control: part 2—stability and robustness. Int. J. Dynam. Control 5, 1073–1088 (2017). https://doi.org/10.1007/s40435-016-0259-5

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