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44
An Algebraic Geometric Approach to the Identification of a Class of Linear Hybrid Systems
- In Proc. of IEEE Conference on Decision and Control
, 2003
"... We propose an algebraic geometric solution to the identification of a class of linear hybrid systems. We show that the identification of the model parameters can be decoupled from the inference of the hybrid state and the switching mechanism generating the transitions, hence we do not constraint the ..."
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Cited by 60 (15 self)
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We propose an algebraic geometric solution to the identification of a class of linear hybrid systems. We show that the identification of the model parameters can be decoupled from the inference of the hybrid state and the switching mechanism generating the transitions, hence we do not constraint the switches to be separated by a minimum dwell time. The decoupling is obtained from the so-called hybrid decoupling constraint, which establishes a connection between linear hybrid system identification, polynomial factorization and hyperplane clustering. In essence, we represent the number of discrete states n as the degree of a homogeneous polynomial p and the model parameters as factors of p. We then show that one can estimate n from a rank constraint on the data, the coe#cients of p from a linear system, and the model parameters from the derivatives of p. The solution is closed form if and only if n 4. Once the model parameters have been identified, the estimation of the hybrid state becomes a simpler problem. Although our algorithm is designed for noiseless data, we also present simulation results with noisy data. 1
Analysis and synthesis of switched linear control systems
, 2005
"... Switched linear systems have a long history of interest in the control community, and have attracted considerable attention recently because they are not only practically relevant, but also tangible with the rich results in the linear system theory. Rapid progress in the field has generated many new ..."
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Cited by 54 (3 self)
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Switched linear systems have a long history of interest in the control community, and have attracted considerable attention recently because they are not only practically relevant, but also tangible with the rich results in the linear system theory. Rapid progress in the field has generated many new ideas and powerful tools. This paper provides a concise and timely survey on analysis and synthesis of switched linear control systems, and presents the basic concepts and main properties of switched linear systems in a systematic manner. The fundamental topics include (i) controllability and observability, (ii) system structural decomposition, (iii) feedback controller design for stabilization, and (iv) optimal control.
Observability of piecewise-affine hybrid systems
- Hybrid Systems: Computation and Control, 7th International Workshop, HSCC 2004,, volume 2993 of Lecture Notes in Computer Science
, 2004
"... Abstract. We consider observability for a class of piecewise-affine hy-brid systems without inputs. The aim is to give verifiable conditions for observability in terms of linear equations and inequalities. We first dis-cuss a number of important concepts, such as discrete-event detectability and tra ..."
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Cited by 23 (1 self)
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Abstract. We consider observability for a class of piecewise-affine hy-brid systems without inputs. The aim is to give verifiable conditions for observability in terms of linear equations and inequalities. We first dis-cuss a number of important concepts, such as discrete-event detectability and trajectory observability. We give sufficient conditions for observabil-ity, observability in infinitesimal time, and observability after a single discrete event. The former conditions are used to construct an observer for the system, the latter are applied to deduce observability for an ex-ample system. 1
Conewise linear systems: non-Zenoness and observability
- SIAM J. Control Optim
"... Abstract. Conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation. This class of dynamical systems represents a large numbe ..."
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Cited by 18 (7 self)
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Abstract. Conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation. This class of dynamical systems represents a large number of piecewise linear systems, most notably, linear complementarity systems with the P-property and their generalizations to affine variational systems, which have many applications in engineering systems and dynamic optimization. The challenges of dealing with this type of hybrid system are due to two major characteristics: mode switchings are triggered by state evolution, and states are constrained in each mode. In this paper, we first establish the absence of Zeno states in such a system. Based on this fundamental result, we then investigate and relate several state observability notions: short-time and T-time (or finite-time) local/global observability. For the short-time observability notions, constructive, finitely verifiable algebraic (both sufficient and necessary) conditions are derived. Due to their long-time mode-transitional behavior, which is very difficult to predict, only partial results are obtained for the T-time observable states. Nevertheless, we completely resolve the T-time local observability for the bimodal conewise linear system, for finite T, and provide numerical examples to illustrate the difficulty associated with the long-time observability.
Observability criteria and estimator design for stochastic linear hybrid systems
- in Proceedings of the IEE European Control Conference
, 2003
"... systems A stochastic linear hybrid system is said to be observable if the hybrid state of the system can be uniquely determined from its output. In this paper, we derive conditions for the observability of stochastic linear hybrid systems by exploiting the information obtained from system noise char ..."
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Cited by 14 (2 self)
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systems A stochastic linear hybrid system is said to be observable if the hybrid state of the system can be uniquely determined from its output. In this paper, we derive conditions for the observability of stochastic linear hybrid systems by exploiting the information obtained from system noise characteristics. Having established the necessary criteria for observability, we study the effect of these conditions on estimator design, and also find bounds on the switching times of the system to achieve guaranteed estimator performance. We then apply these results to the estimation of a two-mode aircraft trajectory. 1
Invertibility of Switched linear systems
"... We address the invertibility problem for switched systems, which is the problem of recovering the switching signal and the input uniquely given an output and an initial state. In the context of hybrid systems, this corresponds to recovering the discrete state and the input from partial measurements ..."
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Cited by 13 (1 self)
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We address the invertibility problem for switched systems, which is the problem of recovering the switching signal and the input uniquely given an output and an initial state. In the context of hybrid systems, this corresponds to recovering the discrete state and the input from partial measurements of the continuous state. In solving the invertibility problem, we introduce the concept of singular pairs for two systems. We give a necessary and sufficient condition for a switched system to be invertible, which says that the individual subsystems should be invertible and there should be no singular pairs. When the individual subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included.
Identification of PWARX Hybrid Models with Unknown and Possibly Different Orders
- In Proceedings of IEEE American Control Conference
, 2004
"... We consider the problem of identifying the orders and the model parameters of PWARX hybrid models from noiseless input/output data. We cast the identification problem in an algebraic geometric framework in which the number of discrete states corresponds to the degree of a multivariate polynomial p a ..."
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Cited by 11 (4 self)
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We consider the problem of identifying the orders and the model parameters of PWARX hybrid models from noiseless input/output data. We cast the identification problem in an algebraic geometric framework in which the number of discrete states corresponds to the degree of a multivariate polynomial p and the orders and the model parameters are encoded on the factors of p. We derive a rank constraint on the input/output data from which one can estimate the coefficients of p. Given p, we show that one can estimate the orders and the parameters of each ARX model from the derivatives of p at a collection of regressors that minimize a certain objective function. Our solution does not require previous knowledge about the orders of the ARX models (only an upper bound is needed), nor does it constraint the orders to be equal. Also the switching mechanism can be arbitrary, hence the switches need not be separated by a minimum dwell time. We illustrate our approach with an algebraic example of a switching circuit and with simulation results in the presence of noisy data.
Discrete state estimators for systems on a lattice
- Automatica
, 2006
"... Abstract. We address the problem of estimating discrete variables in a class of deterministic transition systems where the continuous variables are available for measurement. This simplified scenario has practical interest, for example, in the case of decentralized multi-robot systems. In these syst ..."
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Cited by 10 (5 self)
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Abstract. We address the problem of estimating discrete variables in a class of deterministic transition systems where the continuous variables are available for measurement. This simplified scenario has practical interest, for example, in the case of decentralized multi-robot systems. In these systems, the continuous variables represent physical quantities such as the position and velocity of a robot, while discrete variables may represent the state of the logical system that is used for control and coordination. We propose a novel approach to the estimation of discrete variables using basic lattice theory that overcomes some of the severe complexity issues encountered in previous work. We show how to construct the proposed estimator for a multi-robot system performing a cooperative assignment task.
Continuization of Timed Petri Nets: From Performance Evaluation to Observation and Control
- In Proc. of the 26th Int. Conf. On Application and Theory of Petri Nets and Other Models of Concurrency
, 2005
"... Abstract. State explosion is a fundamental problem in the analysis and synthesis of discrete event systems. Continuous Petri nets can be seen as a relaxation of discrete models allowing more efficient (in some cases polynomial time) analysis and synthesis algorithms. Nevertheless computational costs ..."
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Cited by 6 (2 self)
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Abstract. State explosion is a fundamental problem in the analysis and synthesis of discrete event systems. Continuous Petri nets can be seen as a relaxation of discrete models allowing more efficient (in some cases polynomial time) analysis and synthesis algorithms. Nevertheless computational costs can be reduced at the expense of the analyzability of some properties. Even more, some net systems do not allow any kind of continuization. The present work first considers these aspects and some of the alternative formalisms usable for continuous relaxations of discrete systems. Particular emphasis is done later on the presentation of some results concerning performance evaluation, parametric design and marking (i.e., state) observation and control. Even if a significant amount of results are available today for continuous net systems, many essential issues are still not solved. A list of some of these are given in the introduction as an invitation to work on them. 1
