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83
New results in linear filtering and prediction theory
 TRANS. ASME, SER. D, J. BASIC ENG
, 1961
"... A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this "variance equation " completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary sta ..."
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Cited by 585 (0 self)
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A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this "variance equation " completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed sidebyside. Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.
Contributions to the Theory of Optimal Control
, 1960
"... This paper was in fact the first to introduce the RDE as an algorithm for computing the state feedback gain of the optimal controller for a general linear system with a quadratic performance criterion. RDE had emerged earlier in the study of the second variations in the calculus of variations, but i ..."
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Cited by 112 (1 self)
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This paper was in fact the first to introduce the RDE as an algorithm for computing the state feedback gain of the optimal controller for a general linear system with a quadratic performance criterion. RDE had emerged earlier in the study of the second variations in the calculus of variations, but its use in general linear systems, where the optimal trajectory needs to be generated by a control input, was new. The analysis throughout the paper concentrates on timevarying systems, and uses the HamiltonJacobi theory to arrive at RDE and to deduce optimality of the LQ control gain. We now know, however, that an alternative way to prove optimality in least squares is by showing how RDE allows one to "complete the square" (see, e.g., [5], [18]).
Issues in the convergence of control with communication and computation
 Proceedings of the 43rd IEEE Conference on Decision and Control
, 2004
"... Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the United States ..."
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Cited by 32 (6 self)
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Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the United States
A probabilistic algorithm to test local algebraic observability in polynomial time
, 2001
"... The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the inputoutput behavior of an experiment? How many of the remaining variables should we assume to be known in order to determi ..."
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Cited by 24 (2 self)
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The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the inputoutput behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the local algebraic observability problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a probabilistic seminumerical algorithm that proposes a solution to this problem in polynomial time. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the complexity of evaluation of the model and in the number of variables. Furthermore, we show that the size of the integers involved in the computations is polynomial in the number of variables and in the degree of the system. Last, we estimate the probability of success of our algorithm.
Conewise linear systems: nonZenoness 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 15 (4 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 Pproperty 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: shorttime and Ttime (or finitetime) local/global observability. For the shorttime observability notions, constructive, finitely verifiable algebraic (both sufficient and necessary) conditions are derived. Due to their longtime modetransitional behavior, which is very difficult to predict, only partial results are obtained for the Ttime observable states. Nevertheless, we completely resolve the Ttime local observability for the bimodal conewise linear system, for finite T, and provide numerical examples to illustrate the difficulty associated with the longtime observability.
On the controllability of bimodal piecewise linear systems
 IN:ALUR R, PAPPAS GJ (EDS.) HYBRID SYSTEMS: COMPUTATIONAND CONTROL LNCS 2993
"... This paper studies controllability of bimodal systems that consist of two linear dynamics on each side of a given hyperplane. We show that the controllability properties of these systems can be inferred from those of linear systems for which the inputs are constrained in a certain way. Inspired by ..."
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Cited by 15 (4 self)
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This paper studies controllability of bimodal systems that consist of two linear dynamics on each side of a given hyperplane. We show that the controllability properties of these systems can be inferred from those of linear systems for which the inputs are constrained in a certain way. Inspired by the earlier work on constrained controllability of linear systems, we derive necessary and sufficient conditions for a bimodal piecewise linear system to be controllable.
A New Approach to Linear Filtering . . .
"... AN IMPORTANT class of theoretical and practical problems in communication and control is of a statistical nature. Such problems are: (i) Prediction of random signals; (ii) separation of random signals from random noise; (iii) detection of ..."
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Cited by 10 (0 self)
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AN IMPORTANT class of theoretical and practical problems in communication and control is of a statistical nature. Such problems are: (i) Prediction of random signals; (ii) separation of random signals from random noise; (iii) detection of
Positivity and reachability of fractional electrical circuits
 ACTA MECHANICA ET AUTOMATICA 5(2): 42–51
"... Conditions for the positivity of fractional linear electrical circuits composed of resistors, coils, condensators and voltage (current) sources are established. It is shown that: 1) the fractional electrical circuit composed of resistors, coils and voltage source is positive for any values of their ..."
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Cited by 10 (6 self)
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Conditions for the positivity of fractional linear electrical circuits composed of resistors, coils, condensators and voltage (current) sources are established. It is shown that: 1) the fractional electrical circuit composed of resistors, coils and voltage source is positive for any values of their resistances, inductances and source voltages if and only if the number of coils is less or equal to the number of its linearly independent meshes, 2) the fractional electrical circuit is not positive for any values of its resistances, capacitances and source voltages if each its branch contains resistor, capacitor and voltage source, It is also shown that the fractional positive electrical circuits of R, C, e type are reachable if and only if the conductances between their nodes are zero and the fractional positive electrical circuits of R, L, e type are reachable if and only if the resistances belonging to two meshes are zero.
Algebraic necessary and sufficient conditions for the controllability of conewise linear systems
 IEEE TRANS. ON AUTOMATIC CONTROL
, 2008
"... The problem of checking certain controllability properties of even very simple piecewise linear systems is known to be undecidable. This paper focuses on conewise linear systems, i.e., systems for which the state space is partitioned into conical regions and a linear dynamics is active on each of t ..."
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Cited by 10 (3 self)
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The problem of checking certain controllability properties of even very simple piecewise linear systems is known to be undecidable. This paper focuses on conewise linear systems, i.e., systems for which the state space is partitioned into conical regions and a linear dynamics is active on each of these regions. For this class of systems, we present algebraic necessary and sufficient conditions for controllability. We also show that the classical results of controllability of linear systems and inputconstrained linear systems can be recovered from our main result. Our treatment employs tools both from geometric control theory and mathematical programming.
AlgebraCoalgebra Duality in Brzozowski’s Minimization Algorithm
"... We give a new presentation of Brzozowski’s algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simpl ..."
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Cited by 7 (2 self)
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We give a new presentation of Brzozowski’s algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations. Notably, we derive algorithms to obtain minimal, language equivalent automata from Moore, nondeterministic and weighted automata.