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Infinity norms as Lyapunov functions for model predictive control of constrained PWA systems
 IN: HYBRID SYSTEMS: COMPUTATION AND CONTROL. VOLUME 3414 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... In this paper we develop a priori stabilization conditions for infinity norm based hybrid MPC in the terminal cost and constraint set fashion. Closedloop stability is achieved using infinity norm inequalities that guarantee that the value function corresponding to the MPC cost is a Lyapunov funct ..."
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Cited by 9 (6 self)
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In this paper we develop a priori stabilization conditions for infinity norm based hybrid MPC in the terminal cost and constraint set fashion. Closedloop stability is achieved using infinity norm inequalities that guarantee that the value function corresponding to the MPC cost is a Lyapunov function of the controlled system. We show that Lyapunov asymptotic stability can be achieved even though the MPC value function may be discontinuous. One of the advantages of this hybrid MPC scheme is that the terminal constraint set can be directly obtained as a sublevel set of the calculated terminal cost, which is also a local piecewise linear Lyapunov function. This yields a new method to obtain positively invariant sets for PWA systems.
Squaring the Circle: An Algorithm for Generating Polyhedral Invariant Sets from Ellipsoidal Ones
, 2006
"... This paper presents a new (geometrical) approach to the computation of polyhedral positively invariant sets for general (possibly discontinuous) nonlinear systems, possibly affected by disturbances. Given a βcontractive ellipsoidal set E, the key idea is to construct a polyhedral set that lies betw ..."
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Cited by 4 (2 self)
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This paper presents a new (geometrical) approach to the computation of polyhedral positively invariant sets for general (possibly discontinuous) nonlinear systems, possibly affected by disturbances. Given a βcontractive ellipsoidal set E, the key idea is to construct a polyhedral set that lies between the ellipsoidal sets βE and E. A proof that the resulting polyhedral set is positively invariant (and contractive under an additional assumption) is given, and a new algorithm is developed to construct the desired polyhedral set. An advantage of the proposed method is that the problem of computing polyhedral invariant sets is formulated as a number of Quadratic Programming (QP) problems. The number of QP problems is guaranteed to be finite and therefore, the algorithm has finite termination. An important application of the proposed algorithm is the computation of polyhedral terminal constraint sets for model predictive control based on quadratic costs.
On the stability of quadratic forms based Model Predictive Control of constrained PWA systems
, 2005
"... In this paper we investigate the stability of discretetime PWA systems in closedloop with quadratic cost based Model Predictive Controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability. We prove that Lyapunov stability can be achieved for the closedloop s ..."
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Cited by 3 (3 self)
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In this paper we investigate the stability of discretetime PWA systems in closedloop with quadratic cost based Model Predictive Controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability. We prove that Lyapunov stability can be achieved for the closedloop system even though the considered Lyapunov function and the system dynamics may be discontinuous. The stabilization conditions are derived using a terminal cost and constraint set method. An Sprocedure technique is employed to reduce conservativeness of the stabilization conditions and a linear matrix inequalities setup is developed in order to calculate the terminal cost. A new algorithm for computing piecewise polyhedral positively invariant sets for PWA systems is also presented. In this manner, the online optimization problem associated with MPC leads to a mixed integer quadratic programming problem, which can be solved by standard optimization tools.
Stabilizing receding horizon control of piecewise linear systems: An LMI approach
 in Proceedings of the 16th Symposium on Mathematical Theory of Networks and Systems
, 2004
"... Abstract: Receding horizon control has recently been used for regulating discretetime Piecewise Affine (PWA) systems. One of the obstructions for implementation consists in guaranteeing closedloop stability a priori. This is an issue that has only been addressed marginally in the literature. In th ..."
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Cited by 2 (2 self)
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Abstract: Receding horizon control has recently been used for regulating discretetime Piecewise Affine (PWA) systems. One of the obstructions for implementation consists in guaranteeing closedloop stability a priori. This is an issue that has only been addressed marginally in the literature. In this paper we present an extension of the terminal cost method for guaranteeing stability in receding horizon control to the class of unconstrained Piecewise Linear (PWL) systems. A linear matrix inequalities setup is developed to calculate the terminal weight matrix and the auxiliary feedback gains that ensure stability for quadratic cost based receding horizon control. It is shown that the PWL statefeedback control law employed in the stability proof globally asymptotically stabilizes the origin of the PWL system. The additional conditions needed to extend these results to constrained PWA systems are also pointed out. The implementation of the proposed method is illustrated by an example.
Robustly stabilizing MPC for perturbed PWL systems: Extended report
, 2005
"... If you want to cite this report, please use the following reference instead: ..."
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Cited by 2 (1 self)
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If you want to cite this report, please use the following reference instead:
An Algorithm for the Computation of Polyhedral Invariant Sets for ClosedLoop Linear MPC Systems
"... Abstract — Given an asymptotically stabilizing linear MPC controller, this paper proposes an algorithm to construct invariant polyhedral sets for the closedloop system. The approach exploits a recently developed DC (Difference of Convex functions) programming technique developed by the authors to c ..."
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Abstract — Given an asymptotically stabilizing linear MPC controller, this paper proposes an algorithm to construct invariant polyhedral sets for the closedloop system. The approach exploits a recently developed DC (Difference of Convex functions) programming technique developed by the authors to construct a polyhedral set in between two convex sets. Here, the inner convex set is any given level set V (x) = γ of the MPC value function, while the outer convex set is the level set of a the value function of a modified multiparametric quadratic program. The level gamma acts as a tuning knob for deciding the size of the polyhedral invariant containing the inner set, ranging from the origin (γ = 0) to the maximum invariant set around the origin where the unconstrained MPC solution remains feasible, up to the feasibility domain (possibly the whole state space R n) provided by the multiparametric QP solver (γ = inf). Potential applications of the technique include reachability analysis of MPC systems and generation of constraints to supervisory decision algorithms on top of MPC loops. During the interactive session the numerical behavior of the proposed algorithm in Matlab will be demonstrated, based on a software implementation on top of the Hybrid Toolbox for MPC setup, simulation, and multiparametric programming solutions. Index Terms — Linear systems, Model predictive control, Polyhedral invariant sets, Stability.
Stability of Hybrid Model . . .
, 2005
"... In this paper we investigate the stability of hybrid systems in closedloop with Model Predictive Controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability and exponential stability. A general theory is presented which proves that Lyapunov stability is achieved ..."
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In this paper we investigate the stability of hybrid systems in closedloop with Model Predictive Controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability and exponential stability. A general theory is presented which proves that Lyapunov stability is achieved for both terminal cost and constraint set and terminal equality constraint hybrid MPC, even though the considered Lyapunov function and the system dynamics may be discontinuous. For particular choices of MPC criteria and constrained Piecewise Affine (PWA) systems as the prediction models we develop novel algorithms for computing the terminal cost and the terminal constraint set. For a quadratic MPC cost, the stabilization conditions translate into a linear matrix inequality while, for an ∞norm based MPC cost, they are obtained as ∞norm inequalities. It is shown that by using ∞norms, the terminal constraint set is automatically obtained as a polyhedron or a finite union of polyhedra by taking a sublevel set of the calculated terminal cost function. New algorithms are developed for calculating polyhedral or piecewise polyhedral positively invariant sets for PWA systems. In this manner, the online optimization problem leads to a mixed integer quadratic programming problem or to a mixed integer linear programming problem, which can be solved by standard optimization tools. Several examples illustrate the effectiveness of the developed methodology.
CONTROL OF PWA SYSTEMS USING A STABLE RECEDING HORIZON METHOD
, 2005
"... In this paper we derive stabilization conditions for the class of piecewise affine (PWA) systems using the linear matrix inequality (LMI) framework. We take into account the piecewise structure of the system and therefore the matrix inequalities that we solve are less conservative. Using the upper ..."
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In this paper we derive stabilization conditions for the class of piecewise affine (PWA) systems using the linear matrix inequality (LMI) framework. We take into account the piecewise structure of the system and therefore the matrix inequalities that we solve are less conservative. Using the upper bound of the infinitehorizon quadratic cost as a terminal cost and constructing also a convex terminal set we show that the receding horizon control stabilizes the PWA system. We derive also an algorithm for enlarging the terminal set based on a backward procedure; therefore, the prediction horizon can be chosen shorter, removing some computations offline.
DOI: 10.1109/CCA.2007.4389435 Adaptive Model Predictive Control of the Hybrid Dynamics of a Fuel
, 2008
"... Abstract — In this paper, an adaptive control scheme for the safe operation of a fuel cell system is presented. The aim of the control design is to guarantee that the oxygen ratio do not reach dangerous values. A first level of control is given by a feedforward control. An improved behavior is obtai ..."
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Abstract — In this paper, an adaptive control scheme for the safe operation of a fuel cell system is presented. The aim of the control design is to guarantee that the oxygen ratio do not reach dangerous values. A first level of control is given by a feedforward control. An improved behavior is obtained using an adaptive predictive controller to determine the voltage to be applied to the air compressor. An admissible robust control invariant set for the PWA model of the system is computed. The control action of the predictive controller is obtained in such a way that the state is always included in the safe region characterized by the admissible robust control invariant set. This guarantees that the proposed controller always provides safe evolutions of the system. I.
STABILIZING RECEDING HORIZON CONTROL OF PIECEWISE LINEAR SYSTEMS: AN LMI APPROACH1
"... Abstract: Receding horizon control has recently been used for regulating discretetime Piecewise Affine (PWA) systems. One of the obstructions for implementation consists in guaranteeing closedloop stability a priori. This is an issue that has only been addressed marginally in the literature. In th ..."
Abstract
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Abstract: Receding horizon control has recently been used for regulating discretetime Piecewise Affine (PWA) systems. One of the obstructions for implementation consists in guaranteeing closedloop stability a priori. This is an issue that has only been addressed marginally in the literature. In this paper we present an extension of the terminal cost method for guaranteeing stability in receding horizon control to the class of unconstrained Piecewise Linear (PWL) systems. A linear matrix inequalities setup is developed to calculate the terminal weight matrix and the auxiliary feedback gains that ensure stability for quadratic cost based receding horizon control. It is shown that the PWL statefeedback control law employed in the stability proof globally asymptotically stabilizes the origin of the PWL system. The additional conditions needed to extend these results to constrained PWA systems are also pointed out. The implementation of the proposed method is illustrated by an example.