In this paper we deal with the problem of estimating the marking of a labeled Petri net system ba... more In this paper we deal with the problem of estimating the marking of a labeled Petri net system based on the observation of transitions labels. In particular, we assume that a certain number of transitions are labeled with the empty string ε, while a different label taken from a given alphabet is assigned to all the other transitions. Transitions labeled with the empty string are called silent because their firing cannot be observed. Under some technical assumptions on the structure of the T ε −induced subnet, where T ε denotes the set of silent transitions, we formally prove that the set of markings consistent with the observed word can be represented by a linear system with a fixed structure that does not depend on the length of the observed word.
In this paper we consider a quantized discrete-time linear quadratic regulator (DLQR) problem, na... more In this paper we consider a quantized discrete-time linear quadratic regulator (DLQR) problem, namely a DLQR problem where the input u may only take values in a given finite set U. Based on our previous results on the optimal control of hybrid systems we show that the optimal control law for the quantized DLQR problem takes the form of a feedback control law, that can be obtained from a partition of the state space C, computed off-line. The numerical simulations carried out enabled us to observe a particular structure of C, related to the solution of the non-quantized DLQR problem.
In this contribution we address an optimal control problem for a class of discretetime hybrid aut... more In this contribution we address an optimal control problem for a class of discretetime hybrid automata under safety and liveness constraints. The solution is based on a hierarchical decomposition of the problem, where the low-level controller enforces safety and liveness constraints while the high-level controller exploits the remaining degrees of freedom for performance optimization. Lower-level control is based on a discrete abstraction of the continuous dynamics. The action of low-level control can be interpreted as restricting invariants in the hybrid automaton representing the plant model. A state feedback solution for the high-level control is provided, based on the off-line construction of an appropriate partition of the state space.
The optimal control of switched linear autonomous systems with quadratic performance index over i... more The optimal control of switched linear autonomous systems with quadratic performance index over infinite time horizon is considered. The decision variables are the switching times and the mode sequence, under the following constraints: the mode sequence has finite length; an automaton restricts the possible switches within adjacent locations, with a cost associated to each switch; the time interval between two consecutive switches is greater than a fixed quantity. A state-feedback solution is computed off-line through a numerical procedure. The method is extended to the case of infinite number of switches.
In this contribution we address an optimal control problem for a class of discretetime hybrid aut... more In this contribution we address an optimal control problem for a class of discretetime hybrid automata under safety and liveness constraints. The solution is based on a hierarchical decomposition of the problem, where the low-level controller enforces safety and liveness constraints while the high-level controller exploits the remaining degrees of freedom for performance optimization. Lower-level control is based on a discrete abstraction of the continuous dynamics. The action of low-level control can be interpreted as restricting invariants in the hybrid automaton representing the plant model. A state feedback solution for the high-level control is provided, based on the off-line construction of an appropriate partition of the state space.
Copyright G IFAC Analysis and Design of Hybrid Systems, Brittany, France, 2003 IFAC Publications ... more Copyright G IFAC Analysis and Design of Hybrid Systems, Brittany, France, 2003 IFAC Publications www. elseviercom/locate'ifac OPTIMAL STATE-FEEDBACK QUADRATIC REGULATION OF LINEAR HYBRID AUTOMATA Alberto Bemporad* Daniele Corona** Alessandro Giua** ...
In this contribution, we propose a two-level hierarchical control strategy to solve the problem o... more In this contribution, we propose a two-level hierarchical control strategy to solve the problem of minimising a cost function for a switched affine system under safety and liveness constraints. The constraints are handled on the lower level by a discrete supervisory controller designed using l-complete approximation. Its action can be represented as state invariants added to the plant model. In a bottom-up strategy, we can then synthesise a high-level controller, which computes a set of switching tables using the remaining degrees of freedom to optimise a quadratic performance index.
In this paper we deal with the problem of estimating the marking of a labeled Petri net with nond... more In this paper we deal with the problem of estimating the marking of a labeled Petri net with nondeterministic transitions. In particular, we consider the case in which nondeterminism is due to the presence of transitions that share the same label and that can be simultaneously enabled. Under the assumption that: the structure of the net is known, the initial marking is known, the transition labels can be observed, the nondeterministic transitions are contact-free, we present a technique for characterizing the set of markings that are consistent with the actual observation. More precisely, we show that the set of markings consistent with an observed word can be represented by a linear system with a fixed structure that does not depend on the length of the observed word.
An adaptive cruise controller (ACC) is a device used in modern automotive applications that aims ... more An adaptive cruise controller (ACC) is a device used in modern automotive applications that aims to achieve the tracking of a leading vehicle, allowing safety, comfort driving and overall improvement of traffic streams. Through appropriate modeling and design it permits also to address other tasks such as energy saving, environmental protection and reduction of the mechanical stress. For a type of small car, a SMART, we propose the design of an ACC using control methods based on tuned proportional-integral (PI) action and on model predictive control (MPC), developed for linear and piecewise affine (PWA) systems. We implement these controllers and compare the main properties and strong/weak points of each method, collecting the comparison keys in a table. The trade-off between complexity and accuracy of the solutions is also discussed.
* Dip. Ing. Elettrica ed Elettronica, U. di Cagliari, Italy Email: {daniele . corona , giua , sea... more * Dip. Ing. Elettrica ed Elettronica, U. di Cagliari, Italy Email: {daniele . corona , giua , seatzu} diee . unica . it Abstract: In this paper we consider switched systems composed of LTI non Hurwitz dynamics and we deal with the problem of computing an appropriate switching law such ...
We consider the optimal control of switched linear autonomous systems whose switching sequence is... more We consider the optimal control of switched linear autonomous systems whose switching sequence is determined by a controlled automaton and where the objective is to minimize a quadratic performance index over an infinite time horizon. The quantities to be optimized are the sequence of switching times and the sequence of modes (or "locations"), under the following constraints: the sequence of modes has a finite length; the discrete dynamics of the automaton restricts the possible switches from a given location to an adjacent location, with a cost associated to each switch; the time interval between two consecutive switching times is greater than a fixed quantity. We first show how a state-feedback solution can be computed off-line through a numerical procedure. Then we show how the proposed procedure can be extended to the case of an infinite number of switches.
In this paper we investigate the possibility of applying the hybrid Model Predictive Control (MPC... more In this paper we investigate the possibility of applying the hybrid Model Predictive Control (MPC) framework to solve a control problem regarding tracking of a moving vehicle. The study originates from the design of an adaptive cruise controller (ACC) of a Smart car, that aims to closely follow a reference trajectory transmitted by a leading vehicle. The physical behavior of the Smart and the constraints arising from the specifications related to safety and security issues make the hybrid MPC framework suitable for this task. An adaptation of the terminal cost and constraint set MPC approach, which is commonly used for fixed set-point regulation, is employed in order to achieve good tracking of a timevarying reference trajectory. The simulation results indicate the effectiveness of the developed hybrid MPC algorithm and the industrial feasibility with respect to on-line computation restrictions.
This paper deals with optimal control of switched piecewise affine autonomous systems, where the ... more This paper deals with optimal control of switched piecewise affine autonomous systems, where the objective is to minimize a performance index over an infinite time horizon. We assume that the switching sequence has a finite length, and that the decision variables are the switching instants and the sequence of operating modes. We present two different approaches for solving such an
In this paper we extend a technique developed to design a feedback stabilizing control law for a ... more In this paper we extend a technique developed to design a feedback stabilizing control law for a class of autonomous switched system. More specifically we extend the switching table procedure to a particular class of switched systems the dynamics of which either do not have an equilibrium point or, if they do, it is not common. This study is motivated by the application of the DC-DC buck-boost converter. The design of the control law is based on dynamic programming arguments and it is a partition of the state space into switching look-up tables. A comparison with a Lyapunov based technique is also discussed. D. Corona and B. De Schutter are with the
In this paper we deal with the problem of estimating the marking of a labeled Petri net with nond... more In this paper we deal with the problem of estimating the marking of a labeled Petri net with nondeterministic transitions. In particular, we consider the case in which nondeterminism is due to the presence of transitions that share the same label and that can be simultaneously enabled. Under the assumption that: the structure of the net is known, the initial marking is known, the transition labels can be observed, the nondeterministic transitions are contact-free, we present a technique for characterizing the set of markings that are consistent with the actual observation. More precisely, we show that the set of markings consistent with an observed word can be represented by a linear system with a fixed structure that does not depend on the length of the observed word.
In this paper we extend a technique developed to design a feedback stabilizing control law for au... more In this paper we extend a technique developed to design a feedback stabilizing control law for autonomous switched systems all modes of which are unstable. More specifically, we extend the switching table procedure to the class of affine switched systems, the dynamics of which either do not have an equilibrium point or, if they do, it is not common. This method is then applied to the DC-DC buck-boost converter. The design of the control law is based on dynamic programming and it results in a partition of the state space into switching look-up tables. A comparison with a Lyapunov based technique is also discussed.
In this paper we deal with the problem of estimating the marking of a labeled Petri net system ba... more In this paper we deal with the problem of estimating the marking of a labeled Petri net system based on the observation of transitions labels. In particular, we assume that a certain number of transitions are labeled with the empty string ε, while a different label taken from a given alphabet is assigned to all the other transitions. Transitions labeled with the empty string are called silent because their firing cannot be observed. Under some technical assumptions on the structure of the T ε −induced subnet, where T ε denotes the set of silent transitions, we formally prove that the set of markings consistent with the observed word can be represented by a linear system with a fixed structure that does not depend on the length of the observed word.
We define a new class of hybrid systems called Autonomous Hybrid Automata that can be seen as a g... more We define a new class of hybrid systems called Autonomous Hybrid Automata that can be seen as a generalization of the class of switched systems we have considered in previous works. In this new model there are two types of edges: a controllable edge represents a mode switch that can be triggered by the controller; an autonomous edge represents a mode switch that is triggered by the continuous state of the system as it reaches a given threshold. We show how to solve an infinite time horizon quadratic optimization problem with a numerically viable procedure for such a class of Hybrid Automata; the optimal control law is a state-feedback.
In this paper we deal with the problem of estimating the marking of a labeled Petri net system ba... more In this paper we deal with the problem of estimating the marking of a labeled Petri net system based on the observation of transitions labels. In particular, we assume that a certain number of transitions are labeled with the empty string ε, while a different label taken from a given alphabet is assigned to all the other transitions. Transitions labeled with the empty string are called silent because their firing cannot be observed. Under some technical assumptions on the structure of the T ε −induced subnet, where T ε denotes the set of silent transitions, we formally prove that the set of markings consistent with the observed word can be represented by a linear system with a fixed structure that does not depend on the length of the observed word.
In this paper we consider a quantized discrete-time linear quadratic regulator (DLQR) problem, na... more In this paper we consider a quantized discrete-time linear quadratic regulator (DLQR) problem, namely a DLQR problem where the input u may only take values in a given finite set U. Based on our previous results on the optimal control of hybrid systems we show that the optimal control law for the quantized DLQR problem takes the form of a feedback control law, that can be obtained from a partition of the state space C, computed off-line. The numerical simulations carried out enabled us to observe a particular structure of C, related to the solution of the non-quantized DLQR problem.
In this contribution we address an optimal control problem for a class of discretetime hybrid aut... more In this contribution we address an optimal control problem for a class of discretetime hybrid automata under safety and liveness constraints. The solution is based on a hierarchical decomposition of the problem, where the low-level controller enforces safety and liveness constraints while the high-level controller exploits the remaining degrees of freedom for performance optimization. Lower-level control is based on a discrete abstraction of the continuous dynamics. The action of low-level control can be interpreted as restricting invariants in the hybrid automaton representing the plant model. A state feedback solution for the high-level control is provided, based on the off-line construction of an appropriate partition of the state space.
The optimal control of switched linear autonomous systems with quadratic performance index over i... more The optimal control of switched linear autonomous systems with quadratic performance index over infinite time horizon is considered. The decision variables are the switching times and the mode sequence, under the following constraints: the mode sequence has finite length; an automaton restricts the possible switches within adjacent locations, with a cost associated to each switch; the time interval between two consecutive switches is greater than a fixed quantity. A state-feedback solution is computed off-line through a numerical procedure. The method is extended to the case of infinite number of switches.
In this contribution we address an optimal control problem for a class of discretetime hybrid aut... more In this contribution we address an optimal control problem for a class of discretetime hybrid automata under safety and liveness constraints. The solution is based on a hierarchical decomposition of the problem, where the low-level controller enforces safety and liveness constraints while the high-level controller exploits the remaining degrees of freedom for performance optimization. Lower-level control is based on a discrete abstraction of the continuous dynamics. The action of low-level control can be interpreted as restricting invariants in the hybrid automaton representing the plant model. A state feedback solution for the high-level control is provided, based on the off-line construction of an appropriate partition of the state space.
Copyright G IFAC Analysis and Design of Hybrid Systems, Brittany, France, 2003 IFAC Publications ... more Copyright G IFAC Analysis and Design of Hybrid Systems, Brittany, France, 2003 IFAC Publications www. elseviercom/locate'ifac OPTIMAL STATE-FEEDBACK QUADRATIC REGULATION OF LINEAR HYBRID AUTOMATA Alberto Bemporad* Daniele Corona** Alessandro Giua** ...
In this contribution, we propose a two-level hierarchical control strategy to solve the problem o... more In this contribution, we propose a two-level hierarchical control strategy to solve the problem of minimising a cost function for a switched affine system under safety and liveness constraints. The constraints are handled on the lower level by a discrete supervisory controller designed using l-complete approximation. Its action can be represented as state invariants added to the plant model. In a bottom-up strategy, we can then synthesise a high-level controller, which computes a set of switching tables using the remaining degrees of freedom to optimise a quadratic performance index.
In this paper we deal with the problem of estimating the marking of a labeled Petri net with nond... more In this paper we deal with the problem of estimating the marking of a labeled Petri net with nondeterministic transitions. In particular, we consider the case in which nondeterminism is due to the presence of transitions that share the same label and that can be simultaneously enabled. Under the assumption that: the structure of the net is known, the initial marking is known, the transition labels can be observed, the nondeterministic transitions are contact-free, we present a technique for characterizing the set of markings that are consistent with the actual observation. More precisely, we show that the set of markings consistent with an observed word can be represented by a linear system with a fixed structure that does not depend on the length of the observed word.
An adaptive cruise controller (ACC) is a device used in modern automotive applications that aims ... more An adaptive cruise controller (ACC) is a device used in modern automotive applications that aims to achieve the tracking of a leading vehicle, allowing safety, comfort driving and overall improvement of traffic streams. Through appropriate modeling and design it permits also to address other tasks such as energy saving, environmental protection and reduction of the mechanical stress. For a type of small car, a SMART, we propose the design of an ACC using control methods based on tuned proportional-integral (PI) action and on model predictive control (MPC), developed for linear and piecewise affine (PWA) systems. We implement these controllers and compare the main properties and strong/weak points of each method, collecting the comparison keys in a table. The trade-off between complexity and accuracy of the solutions is also discussed.
* Dip. Ing. Elettrica ed Elettronica, U. di Cagliari, Italy Email: {daniele . corona , giua , sea... more * Dip. Ing. Elettrica ed Elettronica, U. di Cagliari, Italy Email: {daniele . corona , giua , seatzu} diee . unica . it Abstract: In this paper we consider switched systems composed of LTI non Hurwitz dynamics and we deal with the problem of computing an appropriate switching law such ...
We consider the optimal control of switched linear autonomous systems whose switching sequence is... more We consider the optimal control of switched linear autonomous systems whose switching sequence is determined by a controlled automaton and where the objective is to minimize a quadratic performance index over an infinite time horizon. The quantities to be optimized are the sequence of switching times and the sequence of modes (or "locations"), under the following constraints: the sequence of modes has a finite length; the discrete dynamics of the automaton restricts the possible switches from a given location to an adjacent location, with a cost associated to each switch; the time interval between two consecutive switching times is greater than a fixed quantity. We first show how a state-feedback solution can be computed off-line through a numerical procedure. Then we show how the proposed procedure can be extended to the case of an infinite number of switches.
In this paper we investigate the possibility of applying the hybrid Model Predictive Control (MPC... more In this paper we investigate the possibility of applying the hybrid Model Predictive Control (MPC) framework to solve a control problem regarding tracking of a moving vehicle. The study originates from the design of an adaptive cruise controller (ACC) of a Smart car, that aims to closely follow a reference trajectory transmitted by a leading vehicle. The physical behavior of the Smart and the constraints arising from the specifications related to safety and security issues make the hybrid MPC framework suitable for this task. An adaptation of the terminal cost and constraint set MPC approach, which is commonly used for fixed set-point regulation, is employed in order to achieve good tracking of a timevarying reference trajectory. The simulation results indicate the effectiveness of the developed hybrid MPC algorithm and the industrial feasibility with respect to on-line computation restrictions.
This paper deals with optimal control of switched piecewise affine autonomous systems, where the ... more This paper deals with optimal control of switched piecewise affine autonomous systems, where the objective is to minimize a performance index over an infinite time horizon. We assume that the switching sequence has a finite length, and that the decision variables are the switching instants and the sequence of operating modes. We present two different approaches for solving such an
In this paper we extend a technique developed to design a feedback stabilizing control law for a ... more In this paper we extend a technique developed to design a feedback stabilizing control law for a class of autonomous switched system. More specifically we extend the switching table procedure to a particular class of switched systems the dynamics of which either do not have an equilibrium point or, if they do, it is not common. This study is motivated by the application of the DC-DC buck-boost converter. The design of the control law is based on dynamic programming arguments and it is a partition of the state space into switching look-up tables. A comparison with a Lyapunov based technique is also discussed. D. Corona and B. De Schutter are with the
In this paper we deal with the problem of estimating the marking of a labeled Petri net with nond... more In this paper we deal with the problem of estimating the marking of a labeled Petri net with nondeterministic transitions. In particular, we consider the case in which nondeterminism is due to the presence of transitions that share the same label and that can be simultaneously enabled. Under the assumption that: the structure of the net is known, the initial marking is known, the transition labels can be observed, the nondeterministic transitions are contact-free, we present a technique for characterizing the set of markings that are consistent with the actual observation. More precisely, we show that the set of markings consistent with an observed word can be represented by a linear system with a fixed structure that does not depend on the length of the observed word.
In this paper we extend a technique developed to design a feedback stabilizing control law for au... more In this paper we extend a technique developed to design a feedback stabilizing control law for autonomous switched systems all modes of which are unstable. More specifically, we extend the switching table procedure to the class of affine switched systems, the dynamics of which either do not have an equilibrium point or, if they do, it is not common. This method is then applied to the DC-DC buck-boost converter. The design of the control law is based on dynamic programming and it results in a partition of the state space into switching look-up tables. A comparison with a Lyapunov based technique is also discussed.
In this paper we deal with the problem of estimating the marking of a labeled Petri net system ba... more In this paper we deal with the problem of estimating the marking of a labeled Petri net system based on the observation of transitions labels. In particular, we assume that a certain number of transitions are labeled with the empty string ε, while a different label taken from a given alphabet is assigned to all the other transitions. Transitions labeled with the empty string are called silent because their firing cannot be observed. Under some technical assumptions on the structure of the T ε −induced subnet, where T ε denotes the set of silent transitions, we formally prove that the set of markings consistent with the observed word can be represented by a linear system with a fixed structure that does not depend on the length of the observed word.
We define a new class of hybrid systems called Autonomous Hybrid Automata that can be seen as a g... more We define a new class of hybrid systems called Autonomous Hybrid Automata that can be seen as a generalization of the class of switched systems we have considered in previous works. In this new model there are two types of edges: a controllable edge represents a mode switch that can be triggered by the controller; an autonomous edge represents a mode switch that is triggered by the continuous state of the system as it reaches a given threshold. We show how to solve an infinite time horizon quadratic optimization problem with a numerically viable procedure for such a class of Hybrid Automata; the optimal control law is a state-feedback.
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