IMT Institutional Repository: No conditions. Results ordered -Date Deposited. 2024-03-29T13:48:29ZEPrintshttp://eprints.imtlucca.it/images/logowhite.pnghttp://eprints.imtlucca.it/2014-07-16T12:09:17Z2014-12-03T13:05:43Zhttp://eprints.imtlucca.it/id/eprint/2260This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/22602014-07-16T12:09:17ZStabilizing dynamic controllers for hybrid systems: a hybrid control Lyapunov function approach
This paper proposes a dynamic controller structure and a systematic design procedure for stabilizing discrete-time hybrid systems. The proposed approach is based on the concept of control Lyapunov functions (CLFs), which, when available, can be used to design a stabilizing state-feedback control law. In general, the construction of a CLF for hybrid dynamical systems involving both continuous and discrete states is extremely complicated, especially in the presence of non-trivial discrete dynamics. Therefore, we introduce the novel concept of a hybrid control Lyapunov function, which allows the compositional design of a discrete and a continuous part of the CLF, and we formally prove that the existence of a hybrid CLF guarantees the existence of a classical CLF. A constructive procedure is provided to synthesize a hybrid CLF, by expanding the dynamics of the hybrid system with a specific controller dynamics. We show that this synthesis procedure leads to a dynamic controller that can be implemented by a receding horizon control strategy, and that the associated optimization problem is numerically tractable for a fairly general class of hybrid systems, useful in real world applications. Compared to classical hybrid receding horizon control algorithms, the proposed approach typically requires a shorter prediction horizon to guarantee asymptotic stability of the closed-loop system, which yields a reduction of the computational burden, as illustrated through two examples.Stefano Di CairanoW.P.M.H. HeemelsMircea LazarAlberto Bemporadalberto.bemporad@imtlucca.it2014-07-08T14:21:55Z2014-07-08T14:21:55Zhttp://eprints.imtlucca.it/id/eprint/2253This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/22532014-07-08T14:21:55ZStability of hybrid model predictive controlIn this paper we investigate the stability of hybrid systems in closed-loop 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 1-norm based
MPC cost, they are obtained as 1-norm inequalities. It is shown that by using 1-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 on-line
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.Mircea LazarW.P.M.H. HeemelsSiep WeilandAlberto Bemporadalberto.bemporad@imtlucca.it2014-07-08T13:48:09Z2014-07-08T13:48:09Zhttp://eprints.imtlucca.it/id/eprint/2252This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/22522014-07-08T13:48:09ZNon-smooth model predictive control: stability and applications to hybrid systems In this report we investigate the stability of hybrid systems in closed-loop 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 on-line 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.Mircea LazarW.P.M.H. HeemelsSiep WeilandAlberto Bemporadalberto.bemporad@imtlucca.it2014-07-01T12:35:16Z2014-07-01T12:35:16Zhttp://eprints.imtlucca.it/id/eprint/2228This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/22282014-07-01T12:35:16ZHybrid control lyapunov functions for the stabilization of hybridsystems
The design of stabilizing controllers for hybrid systems is
particularly challenging due to the heterogeneity present
within the system itself. In this paper we propose a constructive
procedure to design stabilizing dynamic controllers for
a fairly general class of hybrid systems. The proposed technique
is based on the concept of a hybrid control Lyapunov
function (hybrid CLF) that was previously introduced by the
authors. In this paper we generalize the concept of hybrid
control Lyapunov function, and we show that the existence
of a hybrid CLF guarantees the existence of a standard control
Lyapunov function (CLF) for the hybrid system. We
provide a constructive procedure to design a hybrid CLF
and the corresponding dynamic control law, which is stabilizing
because of the established connection to a standard
CLF that becomes a Lyapunov function for the closed-loop
system. The obtained control law can be conveniently implemented
by constrained predictive control in the form of
a receding horizon control strategy. A numerical example
highlighting the features of the proposed approach is presented.Stefano Di CairanoW.P.M.H. HeemelsMircea LazarAlberto Bemporadalberto.bemporad@imtlucca.it2012-04-26T10:50:14Z2012-07-06T12:20:13Zhttp://eprints.imtlucca.it/id/eprint/1264This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/12642012-04-26T10:50:14ZAssessment of non-centralised model predictive control techniques for electrical power networks Model predictive control (MPC) is one of the few advanced control methodologies that have proven to be very successful in real-life applications. An attractive feature of MPC is its capability of explicitly taking state and input constraints into account. Recently, there has been an increasing interest in the usage of MPC schemes to control electrical power networks. The major obstacle for implementation lies in the large scale of these systems, which is prohibitive for a centralised approach. In this article, we therefore assess and compare the suitability of several non-centralised predictive control schemes for power balancing, to provide valuable insights that can contribute to the successful implementation of non-centralised MPC in the real-life electrical power system. Ralph M. HermansAndrej JokicMircea LazarAlessandro AlessioPaul Van den boschIan HiskensAlberto Bemporadalberto.bemporad@imtlucca.it2011-07-27T08:54:00Z2011-08-05T13:56:12Zhttp://eprints.imtlucca.it/id/eprint/562This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/5622011-07-27T08:54:00ZStabilizing receding horizon control of piecewise linear systems: An LMI approachReceding horizon control has recently been used for regulating discrete-time Piecewise Affine (PWA) systems. One of the obstructions for implementation consists in guaranteeing closed-loop 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 set-up 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.Mircea LazarW.P.M.H. HeemelsSiep WeilandAlberto Bemporadalberto.bemporad@imtlucca.it2011-07-27T08:47:36Z2011-08-05T13:54:10Zhttp://eprints.imtlucca.it/id/eprint/521This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/5212011-07-27T08:47:36ZStabilization conditions for model predictive control of constrained PWA systemsModel predictive control (MPC) has recently been applied to several relevant classes of hybrid systems with promising results. These developments generated an increasing interest towards issues such as stability and computational problems that arise in hybrid MPC. Stability aspects have been addressed only marginally. In this paper we present an extension of the terminal cost and constraint set method for guaranteeing stability in MPC to the class of constrained piecewise affine systems. Semidefinite programming is used to calculate the employed terminal weight matrix that ensures stability for quadratic cost based MPC. A procedure for computing a robust positively invariant set for piecewise linear systems is also developed. The implementation of the proposed method is illustrated by an example.Mircea LazarW.P.M.H. HeemelsSiep WeilandAlberto Bemporadalberto.bemporad@imtlucca.it2011-07-27T08:47:24Z2011-08-05T13:52:14Zhttp://eprints.imtlucca.it/id/eprint/526This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/5262011-07-27T08:47:24ZInfinity norms as Lyapunov functions for model predictive control of constrained PWA systemsIn this paper we develop a priori stabilization conditions for infinity norm based hybrid MPC in the terminal cost and constraint set fashion. Closed-loop 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. Mircea LazarW.P.M.H. HeemelsSiep WeilandAlberto Bemporadalberto.bemporad@imtlucca.itOctavian Pastravanu2011-07-27T08:47:22Z2011-08-05T13:51:39Zhttp://eprints.imtlucca.it/id/eprint/530This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/5302011-07-27T08:47:22ZOn the stability and robustness of non-smooth nonlinear MPCThis paper considers discrete-time nonlinear, possibly discontinuous, systems in closed-loop with Model Predictive
Controllers (MPC). The aim of the paper is to provide a priori sufficient conditions for asymptotic stability in
the Lyapunov sense and robust stability, while allowing for both the system dynamics and the value function of the MPC cost (the usual candidate Lyapunov function in MPC) to be discontinuous functions of the state. The motivation for this work lies in the recent development of MPC for hybrid systems, which are inherently discontinuous and nonlinear systems. As an application of the general theory, it is shown that Lyapunov stability is achieved in hybrid MPC. For a particular class of piecewise affine systems, a modified MPC set-up is proposed, which is proven to be robust to small additive disturbances via an input-to-state stability argument.Mircea LazarW.P.M.H. HeemelsAlberto Bemporadalberto.bemporad@imtlucca.itSiep Weiland2011-07-27T08:45:16Z2011-08-05T13:48:46Zhttp://eprints.imtlucca.it/id/eprint/528This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/5282011-07-27T08:45:16ZOn the stability of quadratic forms based model predictive control of constrained PWA systemsIn this paper we investigate the stability of discrete-time PWA systems in closed-loop 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
closed-loop 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 S-procedure technique
is employed to reduce conservativeness of the stabilization
conditions and a linear matrix inequalities set-up 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 on-line optimization problem associated with MPC leads to a mixed integer quadratic programming problem, which can be solved by standard optimization tools.Mircea LazarW.P.M.H. HeemelsSiep WeilandAlberto Bemporadalberto.bemporad@imtlucca.it2011-07-27T08:43:43Z2011-08-05T13:38:38Zhttp://eprints.imtlucca.it/id/eprint/538This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/5382011-07-27T08:43:43ZSquaring the circle: An algorithm for generating polyhedral invariant sets from ellipsoidal onesThis 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 beta-contractive ellipsoidal set E, the key idea is to construct a polyhedral set that lies between the ellipsoidal sets betaE 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 costsMircea LazarAlessandro AlessioAlberto Bemporadalberto.bemporad@imtlucca.itW.P.M.H. Heemels2011-07-27T08:43:40Z2011-08-05T13:37:53Zhttp://eprints.imtlucca.it/id/eprint/543This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/5432011-07-27T08:43:40ZConvex Polyhedral Invariant Sets for Closed-Loop Linear MPC Systems Given an asymptotically stabilizing linear MPC controller, this paper proposes an algorithm to construct invariant polyhedral sets for the closed-loop system. Rather than exploiting an explicit form of the MPC controller, 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) les gamma of the MPC value function (implicitly defined by the quadratic programming problem associated with MPC or explicitly computed via multiparametric quadratic programming), while the outer convex set is the level set of a the value function of a modified multiparametric quadratic program (implicitly or explicitly defined). The level gamma acts as a tuning parameter for deciding the size of the polyhedral invariant containing the inner set, ranging from the origin (gamma = 0) to the maximum invariant set around the origin where the solution to the unconstrained MPC problem remains feasible, up to the whole domain of definition of the controller (possibly the whole state space Ropfn) (gamma = inf). Potential applications of the technique include reachability analysis of MPC systems and generation of constraints to supervisory decision algorithms on top of MPC loopsAlessandro AlessioAlberto Bemporadalberto.bemporad@imtlucca.itW.P.M.H. HeemelsMircea Lazar2011-07-27T08:40:33Z2011-08-05T13:19:18Zhttp://eprints.imtlucca.it/id/eprint/493This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/4932011-07-27T08:40:33ZStabilizing model predictive control of hybrid systemsIn this note, we investigate the stability of hybrid systems in closed-loop with model predictive controllers (MPC). A priori sufficient conditions for Lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion, while allowing for discontinuous system dynamics and discontinuous MPC value functions. For constrained piecewise affine (PWA) systems as prediction models, we present novel techniques for computing a terminal cost and a terminal constraint set that satisfy the developed stabilization conditions. For quadratic MPC costs, these conditions translate into a linear matrix inequality while, for MPC costs based on 1, infin-norms, they are obtained as norm inequalities. New ways for calculating low complexity piecewise polyhedral positively invariant sets for PWA systems are also presented. An example illustrates the developed theoryMircea LazarW.P.M.H. HeemelsSiep WeilandAlberto Bemporadalberto.bemporad@imtlucca.it2011-07-27T08:39:18Z2011-08-05T13:17:43Zhttp://eprints.imtlucca.it/id/eprint/516This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/5162011-07-27T08:39:18ZDiscrete-time non-smooth nonlinear MPC: Stability and robustnessThis paper considers discrete-time nonlinear, possibly discontinuous, systems in closed-loop with model predictive controllers (MPC). The aim of the paper is to provide a priori sufficient conditions for asymptotic stability in the Lyapunov sense and input-to-state stability (ISS), while allowing for both the system dynamics and the value function of the MPC cost to be discontinuous functions of the state. The motivation for this work lies in the recent development of MPC for hybrid systems, which are inherently discontinuous and nonlinear. For a particular class of discontinuous piecewise affine systems, a new MPC set-up based on infinity norms is proposed, which is proven to be ISS to bounded additive disturbances. This ISS result does not require continuity of the system dynamics nor of the MPC value function. Mircea LazarW.P.M.H. HeemelsAlberto Bemporadalberto.bemporad@imtlucca.itSiep Weiland2011-07-27T08:36:50Z2011-08-05T13:03:30Zhttp://eprints.imtlucca.it/id/eprint/479This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/4792011-07-27T08:36:50ZSquaring the circle: an algorithm for generating polyhedral invariant sets from ellipsoidal onesThis paper presents a new (geometrical) approach to the computation of polyhedral (robustly) positively invariant (PI) sets for general (possibly discontinuous) nonlinear discrete-time systems possibly affected by disturbances. Given a β-contractive ellipsoidal set View the MathML source, the key idea is to construct a polyhedral set that lies between the ellipsoidal sets View the MathML source and View the MathML source. A proof that the resulting polyhedral set is contractive and thus, PI, is given, and a new algorithm is developed to construct the desired polyhedral set. 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.
Alessandro AlessioMircea LazarAlberto Bemporadalberto.bemporad@imtlucca.itW.P.M.H. Heemels2011-07-27T08:36:24Z2011-08-05T12:57:10Zhttp://eprints.imtlucca.it/id/eprint/514This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/5142011-07-27T08:36:24ZA control Lyapunov approach to predictive control of hybrid systemsIn this paper we consider the stabilization of hybrid systems with both continuous and discrete dynamics via predictive control. To deal with the presence of discrete dynamics we adopt a “hybrid” control Lyapunov function approach, which consists of using two different functions.
A Lyapunov-like function is designed to ensure finite-time convergence of the discrete state to a target value, while asymptotic stability of the continuous state is guaranteed via a classical local control Lyapunov function. We show that by combining these two functions in a proper manner it is no longer necessary that the control Lyapunov function for the continuous dynamics decreases at each time step. This leads to a significant reduction of conservativeness in contrast with classical Lyapunov based predictive control. Furthermore, the proposed approach also leads
to a reduction of the horizon length needed for recursive feasibility with respect to standard predictive control approaches.Stefano Di CairanoMircea LazarAlberto Bemporadalberto.bemporad@imtlucca.itW.P.M.H. Heemels2011-07-27T08:36:19Z2012-04-26T10:50:02Zhttp://eprints.imtlucca.it/id/eprint/616This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/6162011-07-27T08:36:19ZAssessment of decentralized model predictive control techniques for power networksModel predictive control (MPC) is one of the few advanced control methodologies that have proven to be very successful in real-life control applications. MPC has the capability to guarantee optimality with respect to a de- sired performance cost function, while explicitly taking con- straints into account. Recently, there has been an increas- ing interest in the usage of MPC schemes to control power networks. The major obstacle for implementation lies in the large scale of power networks, which is prohibitive for a centralized approach. In this paper we critically assess and compare the suitability of three model predictive control schemes for controlling power networks. These techniques are analyzed with respect to the following relevant characteristics: the performance of the closed-loop system, which is evaluated and compared to the performance achieved with the classical automatic generation control (AGC) structure; the decentralized implementation, which is investigated in terms of size of the models used for prediction, required measurements and data communication, type of cost function and the computational time required by each algorithm to obtain the control action. Based on the investigated properties mentioned above, the study presented in this paper provides valuable insights that can contribute to the successful decentralized implementation of MPC in real-life electrical power networks.Armand DamoiseauxAndrej JokicMircea LazarAlessandro AlessioPaul Van den boschIan HiskensAlberto Bemporadalberto.bemporad@imtlucca.it2011-07-27T08:30:07Z2011-08-04T07:29:06Zhttp://eprints.imtlucca.it/id/eprint/426This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/4262011-07-27T08:30:07ZOn the synthesis of piecewise affine control lawsPiecewise affine (PWA) control laws offer an attractive solution to real-time control of linear, nonlinear and hybrid systems. In this paper we provide a compact exposition of the existing state-of-the-art methods for the synthesis of PWA control laws using optimization-based methods.Alberto Bemporadalberto.bemporad@imtlucca.itW.P.M.H. HeemelsMircea Lazar