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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 581 (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.
Stabilizing Model Predictive Control of Hybrid Systems
"... Abstract—In this note, we investigate the stability of hybrid systems in closedloop 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 dis ..."
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Cited by 46 (31 self)
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Abstract—In this note, we investigate the stability of hybrid systems in closedloop 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,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 theory. Index Terms—Hybrid systems, Lyapunov stability, model predictive control (MPC), piecewise affine systems. I.
Asymptotic behaviour of nonlinear systems
 The American Mathematical Monthly
, 2004
"... processes are of great importance in the applications of differential equations, dynam ..."
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Cited by 14 (1 self)
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processes are of great importance in the applications of differential equations, dynam
On the stability and robustness of nonsmooth nonlinear model predictive control
 in Workshop on Assessment and Future Directions of NMPC, FreudenstadtLauterbad
, 2005
"... This paper considers discretetime nonlinear, possibly discontinuous, systems in closedloop 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 sys ..."
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Cited by 8 (6 self)
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This paper considers discretetime nonlinear, possibly discontinuous, systems in closedloop 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 setup is proposed, which is proven to be robust to small additive disturbances via an inputtostate stability argument. 1 An introductory survey One of the problems in Model Predictive Control (MPC) that has received an increased attention over the years consists in guaranteeing closedloop stability for the controlled system. The usual approach to ensure stability in MPC is to consider the value function of the MPC cost as a candidate Lyapunov function. Then, if the system dynamics and the MPC value function are continuous, the classical Lyapunov stability theory [1] can be used to prove that the MPC control law is stabilizing [2]. The first results that weaken the requirement that the MPC value function must be continuous in the state were presented in [3,4], which consider terminal equality constraint MPC.
Lyapunov functions, stability and inputtostate stability subtleties for discretetime discontinuous systems
 IEEE Transactions on Automatic Control
, 2009
"... Abstract—In this note we consider stability analysis of discretetime discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discretetime discontinuous dynamics. Also, we indicate that a particular type of Ly ..."
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Abstract—In this note we consider stability analysis of discretetime discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discretetime discontinuous dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discretetime discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discretetime hybrid systems. In contrast to existing results based on smooth Lyapunov functions, we develop several inputtostate stability tests that explicitly employ an available discontinuous Lyapunov function.
Theory, stochastic stability and applications of stochastic delay differential equations: A survey of results
 Differential Equations Dynam. Systems
"... This paper surveys some results in stochastic differential delay equations beginning with ”On stationary solutions of a stochastic differential equations ” by K. Ito and M. Nisio, 1964, and also some results in stochastic stability beginning with the ”Stability of positive supermartingales ” by R. B ..."
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This paper surveys some results in stochastic differential delay equations beginning with ”On stationary solutions of a stochastic differential equations ” by K. Ito and M. Nisio, 1964, and also some results in stochastic stability beginning with the ”Stability of positive supermartingales ” by R. Bucy, 1964. The problems discussed in this survey are the existence and uniqueness of solutions of stochastic differential delay equations (or stochastic differential functional equations, or stochastic affine hereditary systems), Markov property of solutions of SDDE’s, stochastic stability, elements of ergodic theory, numerical approximation, parameter estimation, applications in biology and finance. 1 Theory of Stochastic Delay Differential Equations 1.1. The first paper of this review on the SDDEs is [12] (Ito, Nisio, 1964). Let x(t), t ∈ (−∞, +∞), be a stochastic process, Fuv(x) be a minimal σalgebra (Borel), with respect to (w.r.t.) which x(t) is measurable for every t ∈ [u, v]. Let w(t) be a Wiener process, t ∈ (−∞, +∞), w(0) = 0, and let Fuv(dw) be a minimal Borel σalgebra, w.r.t. which w(s) − w(t) is measurable for every (t, s) with u ≤ t < s ≤ v. Minimal σalgebra, which contains F1, F2,..., is determined by F1 ∨ F2 ∨... ∨ Fk. Let s be arbitrary but fixed number. A stochastic process y(θ): = x(s + θ), θ ≤ 0, (1.1) is called a past (or memory) of the process x at the moment s and denoted by πsx. C− denotes a space of all continuous functions defined on the negative semiaxis (−∞, 0]. C− is a metric space with the metric ρ−(f, g):= k=1 2 −k f − gk/(1 + f − gk), (1.2) where hk: = max−k≤t≤0 h(t). Let a(f) and b(f) be continuous functionals defined on C − with metric ρ − given by (1.2). Stochastic process x(t) is called a solution of the SDDE t ∈ (−∞, +∞), if dx(t) = a(πtx)dt + b(πtx)dw(t), (1.3)
MODELING SHORTEST PATH GAMES WITH PETRI NETS: A LYAPUNOV BASED THEORY
"... In this paper we introduce a new modeling paradigm for shortest path games representation with Petri nets. Whereas previous works have restricted attention to tracking the net using Bellman’s equation as a utility function, this work uses a Lyapunovlike function. In this sense, we change the tradit ..."
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In this paper we introduce a new modeling paradigm for shortest path games representation with Petri nets. Whereas previous works have restricted attention to tracking the net using Bellman’s equation as a utility function, this work uses a Lyapunovlike function. In this sense, we change the traditional cost function by a trajectorytracking function which is also an optimal costtotarget function. This makes a significant difference in the conceptualization of the problem domain, allowing the replacement of the Nash equilibrium point by the Lyapunov equilibrium point in game theory. We show that the Lyapunov equilibrium point coincides with the Nash equilibrium point. As a consequence, all properties of equilibrium and stability are preserved in game theory. This is the most important contribution of this work. The potential of this approach remains in its formal proof simplicity for the existence of an equilibrium point.
Iterative learning control of hysteresis in piezobased nanopositioners: theory and application in atomic force microscopes
, 2003
"... This is to certify that I have examined this copy of a doctoral dissertation by ..."
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Cited by 5 (0 self)
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This is to certify that I have examined this copy of a doctoral dissertation by
COLORED DECISION PROCESS PETRI NETS: MODELING, ANALYSIS AND STABILITY
"... In this paper we introduce a new modeling paradigm for developing a decision process representation called the Colored Decision Process Petri Net (CDPPN). It extends the Colored Petri Net (CPN) theoretic approach including Markov decision processes. CPNs are used for process representation taking ad ..."
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In this paper we introduce a new modeling paradigm for developing a decision process representation called the Colored Decision Process Petri Net (CDPPN). It extends the Colored Petri Net (CPN) theoretic approach including Markov decision processes. CPNs are used for process representation taking advantage of the formal semantic and the graphical display. A Markov decision process is utilized as a tool for trajectory planning via a utility function. The main point of the CDPPN is its ability to represent the markdynamic and trajectorydynamic properties of a decision process. Within the markdynamic properties framework we show that CDPPN theoretic notions of equilibrium and stability are those of the CPN. In the trajectorydynamic properties framework, we optimize the utility function used for trajectory planning in the CDPPN by a Lyapunovlike function, obtaining as a result new characterizations for final decision points (optimum point) and stability. Moreover, we show that CDPPN markdynamic and Lyapunov trajectorydynamic properties of equilibrium, stability and final decision points converge under certain restrictions. We propose an algorithm for optimum trajectory planning that makes use of the graphical representation (CPN) and the utility function. Moreover, we consider some results and discuss possible directions for further research.
On infinity norms as Lyapunov functions: Alternative necessary and sufficient conditions
 in Proceedings 49th IEEE Conf. on Dec. and Contr
, 2010
"... Abstract — This paper considers the synthesis of infinity norm Lyapunov functions for discretetime linear systems. A proper conic partition of the statespace is employed to construct a finite set of linear inequalities in the elements of the Lyapunov weight matrix. Under typical assumptions, it is ..."
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Abstract — This paper considers the synthesis of infinity norm Lyapunov functions for discretetime linear systems. A proper conic partition of the statespace is employed to construct a finite set of linear inequalities in the elements of the Lyapunov weight matrix. Under typical assumptions, it is proven that the feasibility of the derived set of linear inequalities is equivalent with the existence of an infinity norm Lyapunov function. Furthermore, it is shown that the developed solution extends naturally to several relevant classes of discretetime nonlinear systems. I.