Linear control of time-domain constrained systems

This paper presents a general framework for the design of linear controllers for linear systems subject to time-domain constraints. The design framework exploits sums-of-squares techniques to incorporate the time-domain constraints on closed-loop signals and leads to conditions in terms of linear matrix inequalities (LMIs). This control design framework offers, in addition to constraint satisfaction, also the possibility of including an optimization objective that can be used to minimize steady state (tracking) errors, to decrease the settling time, to reduce overshoot and so on. The effectiveness of the framework is shown via a numerical example.

[1]  Bart De Schutter,et al.  On hybrid systems and closed-loop MPC systems , 2002, IEEE Trans. Autom. Control..

[2]  Mato Baotic,et al.  Stabilizing low complexity feedback control of constrained piecewise affine systems , 2005, Autom..

[3]  Alberto Bemporad,et al.  An algorithm for multi-parametric quadratic programming and explicit MPC solutions , 2003, Autom..

[4]  Gene F. Franklin,et al.  Feedback Control of Dynamic Systems , 1986 .

[5]  Semyon M. Meerkov,et al.  Simultaneous Design of Controllers and Instrumentation: ILQR/ILQG , 2010, IEEE Transactions on Automatic Control.

[6]  Manfred Morari,et al.  Model predictive control: Theory and practice - A survey , 1989, Autom..

[7]  V. Kučera,et al.  Proper solutions of polynomial equations , 1999 .

[8]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[9]  David L Powers,et al.  Boundary Value Problems and Partial Differential Equations Ed. 6 , 2009 .

[10]  Vladimír Kucera.  The pole placement equation. A survey , 1994, Kybernetika.

[11]  Tor Arne Johansen,et al.  Hardware Synthesis of Explicit Model Predictive Controllers , 2007, IEEE Transactions on Control Systems Technology.

[12]  Sophie Tarbouriech,et al.  Control of linear systems subject to time-domain constraints with polynomial pole placement and LMIs , 2005, IEEE Transactions on Automatic Control.

[13]  Sophie Tarbouriech,et al.  Advanced strategies in control systems with input and output constraints , 2007 .

[14]  Ali Saberi,et al.  Constrained stabilization problems for linear plants , 2002, Autom..

[15]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[16]  Frank J. Christophersen,et al.  Polynomial Approximation of Closed-form MPC for Piecewise Affine Systems , 2008 .

[17]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[18]  Anton A. Stoorvogel,et al.  Linear quadratic regulator problem with positive controls , 1998 .

[19]  N. Asmar,et al.  Partial Differential Equations with Fourier Series and Boundary Value Problems , 2004 .

[20]  Ilya V. Kolmanovsky,et al.  Nonlinear tracking control in the presence of state and control constraints: a generalized reference governor , 2002, Autom..

[21]  Maxim V. Subbotin,et al.  Continuous Time Linear Quadratic Regulator With Control Constraints via Convex Duality , 2007, IEEE Transactions on Automatic Control.

[22]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[23]  Semyon M. Meerkov,et al.  An LQR/LQG theory for systems with saturating actuators , 2001, IEEE Trans. Autom. Control..

[24]  Didier Henrion,et al.  GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi , 2003, TOMS.

[25]  Bruce A. Francis,et al.  Feedback Control Theory , 1992 .

[26]  Pablo A. Parrilo,et al.  Introducing SOSTOOLS: a general purpose sum of squares programming solver , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[27]  E. Mosca,et al.  Nonlinear control of constrained linear systems via predictive reference management , 1997, IEEE Trans. Autom. Control..

[28]  Wpmh Maurice Heemels,et al.  Input‐to‐state stabilizing sub‐optimal NMPC with an application to DC–DC converters , 2008 .

[29]  S. Joe Qin,et al.  A survey of industrial model predictive control technology , 2003 .

[30]  M. Laurent.  Sums of Squares, Moment Matrices and Optimization Over Polynomials , 2009 .

[31]  S. Tarbouriech,et al.  Anti-windup design: an overview of some recent advances and open problems , 2009 .

[32]  W. P. M. H. Heemels,et al.  Linear control of time-domain constrained systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[33]  R. Goebel,et al.  Continuous time constrained linear quadratic regulator - convex duality approach , 2005, Proceedings of the 2005, American Control Conference, 2005..

[34]  Alberto Bemporad,et al.  The explicit linear quadratic regulator for constrained systems , 2003, Autom..

[35]  Ali Saberi,et al.  Control of Linear Systems with Regulation and Input Constraints , 2000 .