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Lasso-MPC – Predictive Control with ℓ1-Regularised Least Squares

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Abstract

Denote the set of integers (reals) as \(\mathbb {I}\) (\(\mathbb {R}\)), the set of non-negative integers (reals) as \(\mathbb {I}_{\ge 0}\) (\(\mathbb {R}_{\ge 0}\)) and the set of positive integers (reals) as \(\mathbb {I}_{>0}\) (\(\mathbb {R}_{>0}\)). The integers from 0 to N are denoted as \(\mathbb {I}_{[0,N]}\). Given a square matrix A and the scalar \(\lambda _i\) denoting the ith eigenvalue of A, then \(\lambda _{\max }(A)=\max _i|\lambda _i|\), \(\lambda _{\min }(A)=\min _i|\lambda _i|\).

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Notes

  1. 1.

    In the worst case a complexity of \(\mathcal {O}(N^3m^3)\), where N is the MPC prediction horizon length and m is the number of control inputs.

  2. 2.

    Equality constraints can be, for instance, accommodated as back-to-back inequalities, \([h(\chi )^T,\ -h(\chi )^T]^T\le 0\). This is generally not done in practise, as it makes the optimisation problem primal degenerate and more difficult to solve.

  3. 3.

    Uniform continuity is implied by continuity if all the involved sets are bounded, e.g. C-sets, (see Theorem 2.3.3).

  4. 4.

    The terminal controller is never applied to the plant.

  5. 5.

    Redundant inequalities can be removed at a preliminary stage.

  6. 6.

    This includes a corrigendum of the ISS gain provided in Theorem III.7 of Gallieri and Maciejowski (2013a).

References

  • Alessio A, Bemporad A (2009) A survey on explicit model predictive control. Nonlinear model predictive control. Lecture notes in control and information sciences. Springer, Berlin, pp 345–369

    Google Scholar 

  • Astrom KJ (1970) Introduction to stochastic control theory. Academic Press, New York

    MATH  Google Scholar 

  • Bemporad A, Morari M, Dua V, Pistikopoulos EN (2002) The explicit Linear quadratic regulator for constrained systems. Automatica 38:3–20

    Google Scholar 

  • Blanchini F (1999) Set invariance in control. Automatica 35:1747–1767

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Borrelli F, Baotic M, Bemporad A, Morari M (2008) Efficient on-line computation of explicit model predictive control. SIAM J Control Optim 47:2470–2489

    Article  MathSciNet  MATH  Google Scholar 

  • Borrelli F, Baotić M, Pekar J, Stewart G (2009) On the complexity of explicit MPC laws. In: Proceedings of the European control conference (ECC), pp 2408–2413

    Google Scholar 

  • Boyd S, Vandenberghe L (2004) Convex Optimization. Cambridge University Press, Cambridge

    Google Scholar 

  • Boyd S, Vandenberghe L (2006–2007) Lecture notes of EE364b, Stanford University, Winter, Subgradients

    Google Scholar 

  • Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Bravo JM, Limon D, Alamo T, Camacho EF (2005) On the computation of invariant sets for constrained nonlinear systems: an interval arithmetic approach. Automatica 41:1583–1589

    Article  MathSciNet  MATH  Google Scholar 

  • Chen H, Allgöwer F (1998) A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34(10):1205–1217

    Article  MathSciNet  MATH  Google Scholar 

  • Chisci L, Rossiter JA, Zappa G (2001) Systems with persistent disturbances: predictive control with restricted constraints. Automatica 37:1019–1028

    Article  MathSciNet  MATH  Google Scholar 

  • Chmielewski D, Manousiouthakis V (1996) On constrained infinite-time linear quadratic optimal control. Syst Control Lett 29:121–129

    Article  MathSciNet  MATH  Google Scholar 

  • Di Cairano S, Liang W, Kolmanovsky IV, Kuang ML, Phillips AM (2013) Power smoothing energy management and its application to a series hybrid powertrain. IEEE Trans Control Syst Technol 21(6):2091–2103

    Article  Google Scholar 

  • Ferramosca A, Limon D, Alvarado I, Alamo T, Camacho EF (2008) MPC for tracking with optimal closed-loop performance. In: Proceedings of the conference on decision and control (CDC), pp 4055–4060

    Google Scholar 

  • Ferramosca A, Limon D, Alvarado I, Alamo T, Camacho EF (2009a) MPC for tracking of constrained nonlinear systems. In: Proceedings of the conference on decision and control (CDC), pp 7978–7983

    Google Scholar 

  • Ferramosca A, Limon D, Alvarado I, Alamo T, Camacho EF (2009b) MPC for tracking with optimal closed-loop performance. Automatica 45:1975–1978

    Article  MathSciNet  MATH  Google Scholar 

  • Fiacchini M, Alamo T, Camacho EF (2012) Invariant sets computation for convex difference inclusions systems. Syst Control Lett 8:819–826

    Article  MathSciNet  MATH  Google Scholar 

  • Fletcher R (1987) Practical methods of optimization, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Gallieri M, Maciejowski J (2013a) Soft-constrained lasso-MPC for robust LTI tracking: enlarged feasible region and an ISS gain estimate. In: Proceedings of the conference on decision and control (CDC), pp 7101–7106

    Google Scholar 

  • Gallieri M, Maciejowski JM (2013b) Stabilising terminal cost and terminal controller for \(\ell _{asso}\)-MPC: enhanced optimality and region of attraction. In: Proceedings of the European control conference (ECC), pp 524–529

    Google Scholar 

  • Goodwin G, Seron MM, De Doná JA (2005) Constrained control and estimation. An optimisation approach. Springer, London

    Book  MATH  Google Scholar 

  • Goulart PJ, Kerrigan EC, Maciejowski JM (2006) Optimization over state feedback policies for robust control with constraints. Automatica 42(4):523–533

    Article  MathSciNet  MATH  Google Scholar 

  • Grammatico S, Pannocchia, G (2013) Achieving a large domain of attraction with short-horizon linear MPC via polyhedral Lyapunov functions. In: Proceedings of the European control conference (ECC), pp 1059–1064

    Google Scholar 

  • Grimm G, Messina MJ, Tuna SE, Teel AR (2004) Examples when nonlinear model predictive control is nonrobust. Automatica 40(10):1729–1738

    Article  MathSciNet  MATH  Google Scholar 

  • Grüne L (2011) Economic receding horizon control without terminal constraints. Research Report, University of Bayreuth

    Google Scholar 

  • Grüne L, Pannek J (2011) Nonlinear model predictive control: theory and algorithms, 1st edn. Springer, London

    Book  MATH  Google Scholar 

  • Hindi H (2004) A tutorial on convex optimization. In: Proceedings of the American control conference (ACC), pp 3252–3265

    Google Scholar 

  • Hindi H (2006) A tutorial on convex optimization II: duality and interior point methods. In: Proceedings of the American control conference (ACC), pp 686–696

    Google Scholar 

  • Horn RA, Johnson CR (2010) Matrix analysis. Cambridge University Press, Cambridge

    Google Scholar 

  • IBM (2013) ILOG CPLEX v. 12.2.0.2

    Google Scholar 

  • Jiang Z-P, Wang Y (2001) Input-to-state stability for discrete-time nonlinear systems. Automatica 37(6):857–869

    Article  MathSciNet  MATH  Google Scholar 

  • Johansen TA, Grancharova A, Grancharova R (2003) Approximate explicit constrained linear model predictive control via orthogonal search tree. IEEE Trans Autom Control 48:810–815

    Article  MathSciNet  MATH  Google Scholar 

  • Kakade SM, Shalev-Shwartz S, Tewari A (2012) Regularization techniques for learning with matrices. J Mach Learn Res 13(1):1865–1890

    MathSciNet  MATH  Google Scholar 

  • Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395

    Article  MathSciNet  MATH  Google Scholar 

  • Kerrigan EC (2000) Robust constraint satisfaction : invariant sets and predictive control. Ph.D. thesis, University of Cambridge - St. John’s college

    Google Scholar 

  • Kerrigan EC, Maciejowski JM (2000a) Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control. In: Proceedings of the conference on decision and control (CDC), pp 4951–4956

    Google Scholar 

  • Kerrigan EC, Maciejowski JM (2000b) Soft constraints and exact penalty functions in model predictive control. In: Proceedings of the UKACC international conference (Control 2000)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Kuwata Y, Richards A, How J (2007) Robust receding horizon control using generalized constraint tightening. In: Proceedings of the American control conference (ACC), pp 4482–4487

    Google Scholar 

  • Lazar M (2006) Model predictive control of hybrid systems: stability and robustness. Ph.D. thesis, Technische Universiteit Eindhoven

    Google Scholar 

  • Lazar M, Heemels WPMH (2009) Predictive control of hybrid systems: input-to-state stability results for sub-optimal solutions. Automatica 45(1):180–185

    Article  MathSciNet  MATH  Google Scholar 

  • Lazar M, Heemels WPMH, Weiland S, Bemporad A (2006) Stabilizing model predictive control of hybrid systems. IEEE Trans Autom Control 51(11):1813–1818

    Article  MathSciNet  Google Scholar 

  • Lazar M, Heemels WPMH, Muñoz de la Peña D, Alamo T (2009) further results on robust MPC using linear matrix inequalities. In: Magni L, Raimondo DM, Allgöwer F (eds) Nonlinear model predictive control, vol 384. Lecture notes in control and information sciences. Springer, Berlin

    Google Scholar 

  • Lerner D (2010) Introduction to mathematical analysis, vol 500. Lecture notes of Math. University of Kansas

    Google Scholar 

  • Limón D, Alamo T, Salas F, Camacho EF (2006) Input to state stability of min-max MPC controllers for nonlinear systems with bounded uncertainties. Automatica 42(5):797–803

    Article  MathSciNet  MATH  Google Scholar 

  • Limon D, Alamo T, Salas F, Camacho EF (2006) On the stability of constrained MPC without terminal constraints. IEEE Trans Autom Control 51(6):832–836

    Article  MathSciNet  Google Scholar 

  • Limon D, Alvarado I, Alamo T, Camacho EF (2008) MPC for tracking piecewise constant references for constrained linear systems. Automatica 44(9):2382–2387

    Article  MathSciNet  MATH  Google Scholar 

  • Limon D, Alamo T, Raimondo DM, Muñoz de la Peña D, Bravo JM, Ferramosca A, Camacho EF (2009) Input-to-state stability: a unifying framework for robust model predictive control. Nonlinear Model Predict Control 384:1–26

    Article  MATH  Google Scholar 

  • Maciejowski JM (2002) Predictive Control with Constraints. Pearson Education, Prentice Hall

    Google Scholar 

  • Mariéthoz S, Domahidi A, Morari M (2012) High-bandwidth explicit model predictive control of electrical drives. IEEE Trans Ind Appl 48(6):1980–1992

    Article  Google Scholar 

  • Mayne DQ, Rawlings J, Rao C, Scokaert P (2000) Constrained model predictive control: stability and optimality. Automatica 36(6):789–814

    Article  MathSciNet  MATH  Google Scholar 

  • Nesterov Y (2009) Primal-dual subgradient methods for convex problems. Math Program Ser B 120(1):221–259

    Article  MathSciNet  MATH  Google Scholar 

  • Nocedal J, Wright SJ (1999) Numerical optimisation. Operations research series. Springer, New York

    Google Scholar 

  • Osborne MR, Presnell B, Turlach A (2000) On the LASSO and its dual. J Comput Graph Stat 9(2):319–337

    MathSciNet  Google Scholar 

  • Pannocchia G, Rawlings JB, Wright SJ (2011) Conditions under which suboptimal nonlinear MPC is inherently robust. Syst Control Lett 60(9):747–755

    Article  MathSciNet  MATH  Google Scholar 

  • Rao CV, Rawlings JB (2000) Linear programming and model predictive control. J Process Control 10(2–3):283–289

    Article  Google Scholar 

  • Rawlings JB, Mayne DQ (2009) Model predictive control theory and design. Nob Hill Pub Llc, Madison

    Google Scholar 

  • Richards A, How J (2006) Robust stable model predictive control with constraint tightening. In: Proceedings of the American control conference (ACC), pp 1558–1562

    Google Scholar 

  • Richter S, Jones CN, Morari M (2013) Certification aspects of the fast gradient method for solving the dual of parametric convex programs. Math Methods Oper Res 77(3):305–321

    Article  MathSciNet  MATH  Google Scholar 

  • Robbins JW (n.d) Advanced calculus, vol 521. Lecture notes of Math. University of Wisconsin

    Google Scholar 

  • Scokaert POM, Mayne DQ, Rawlings JB (1999) Suboptimal model predictive control (feasibility implies stability). IEEE Trans Autom Control 44(3):648–654

    Article  MathSciNet  MATH  Google Scholar 

  • Shekhar RC, Maciejowski, JM (2012) Optimal constraint tightening policies for robust variable horizon model predictive control. In: Proceedings of the conference on decision and control (CDC), pp 5170–5175

    Google Scholar 

  • Summers S, Jones C, Lygeros J, Morari M (2011) A multiresolution approximation method for fast explicit model predictive control. IEEE Trans Autom Control 56(11):2530–2541

    Article  MathSciNet  Google Scholar 

  • Tondel P, Johansen TA, Bemporad A (2003) Further results on multiparametric quadratic programming. In: Proceedings of the conference on decision and control (CDC), pp 3173–3178

    Google Scholar 

  • Trench WF (2013) Introduction to real analysis. Pearson Education, Prentice Hall

    MATH  Google Scholar 

  • Vidyasagar M (1993) Nonlinear systems analysis, 2nd edn. Classics in applied mathematics. SIAM: Society for Industrial and Applied Mathematics, New Jersey

    Google Scholar 

  • Vinnicombe G (2000) Uncertainty and feedback, \(H_\infty \) loop shaping and the \(\nu \) gap metric. Imperial College press, London

    Book  Google Scholar 

  • Vinter R (2000) Optimal control. Birkhäuser, Boston

    MATH  Google Scholar 

  • Walker M (n.d.) Lecture notes of Economics, vol. 519. University of Arizona

    Google Scholar 

  • Wang Y, Boyd S (2010) Fast model predictive control using online optimization. IEEE Trans Control Syst Technol 18(2):267–278

    Article  Google Scholar 

  • Zeilinger MN, Jones CN, Morari M (2010) Robust stability properties of soft constrained MPC. In: Proceedings of the conference on decision and control (CDC), pp 5276–5282

    Google Scholar 

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  1. Analytics, McLaren Applied Technologies, Woking, UK

    Marco Gallieri

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Gallieri, M. (2016). Background. In: Lasso-MPC – Predictive Control with ℓ1-Regularised Least Squares. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-27963-3_2

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