Invertibility of linear time-invariant dynamical systems

Sain, M.K.; Massey, James L.

doi:10.1109/tac.1969.1099133

Cited by 457 publications

(207 citation statements)

References 12 publications

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“…Of course, our construction of an optimal estimator is intimately related to the construction of a left inverse. The version presented here seems to be new in that it uses a differentiating and an integrating part in parallel rather than in series as in [23] and [39] or in a hybrid form as in [37]. It remains to prove iii).…”

Section: Proof Of Theorem 54mentioning

confidence: 98%

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A geometric approach to the singular filtering problem

Schumacher¹

1985

IEEE Trans. Automat. Contr.

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Abstract-We consider the least-squares filtering problem for a stationary Gaussian process when the observation is not fuJly corrupted by white noise, the so-called "singular" case. An optimal estimator is constructed consisting of an integrating part, which is, as in the regular case, computed from a spectral factorization or an equivalent matrix problem, and a differentiating part whose parameters are computed from a single matrix equation. This improves on older results which either work under restrictive assumptions, or describe the solution only as the result of some nested algorithm.

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Section: Proof Of Theorem 54mentioning

confidence: 98%

“…To give another characterization of the sequences { Vk (:E)} and { Tk(L)}, consider the following definitions from [23]. For a given system :E (A, B, C, D), write (fork ;;;i: 1)…”

Section: Signal Model Equivalence (Sme)mentioning

confidence: 99%

A geometric approach to the singular filtering problem

Schumacher¹

1985

IEEE Trans. Automat. Contr.

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“…The reader is referred to (Sain and Massey 1969) for related results. Our concern next is an algorithm for data-driven output matching control.…”

Section: Proposition 14 (Model-based Output Matching Control) If a Cmentioning

confidence: 99%

Data-driven simulation and control

Markovsky¹,

Rapisarda²

2008

International Journal of Control

287

285

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Classical linear time-invariant system simulation methods are based on a transfer function, impulse response, or input/state/output representation. We present a method for computing the response of a system to a given input and initial conditions directly from a trajectory of the system, without explicitly identifying the system from the data. Similarly to the classical approach for simulation, the classical approach for control is model-based: first a model representation is derived from given data of the plant and then a control law is synthesised using the model and the control specifications. We present an approach for computing a linear quadratic tracking control signal that circumvents the identification step. The results are derived assuming exact data and the simulated response or control input is constructed off-line.

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“…Such a problem has attracted the interest of the control community since it has direct applications in many domains, such as data secure transmission where the unknown input is the message, and fault detection and isolation where the fault is the unknown input. In fact, left invertibility problem has been studied since at least forty five years ago in linear control theory [33], [34] and thirty five year ago in nonlinear control theory [15], [35]. Most of those works are in the context of nonlinear systems without time delays, or linear systems with commensurate delays [2], [41].…”

Section: Introductionmentioning

confidence: 99%

“…Such a problem has attracted the interest of the control community since it has direct applications in many domains, such as data secure transmission where the unknown input is the message, and fault detection and isolation where the fault is the unknown input. In fact, left invertibility problem has been studied since at least forty five years ago in linear control theory [33] In the literature, an important tool based on non-commutative ring, proposed in [36], is used to analyze nonlinear time delay systems in algebraic framework. Using this framework, many notions are extended to the case of nonlinear time * Corresponding author.…”

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confidence: 99%

Finite-time and asymptotic left inversion of nonlinear time-delay systems

2018

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In this paper we investigate the left invertibility problem for a class of nonlinear time-delay systems. In both cases of time delay systems with and without internal dynamics the invertibility conditions are given. A new approach based on the use of higher order sliding mode observer is developed for finite-time left invertibility and for asymptotic left inversion. Causal and non-causal estimation of the unknown inputs are respectively discussed. The results are illustrated by numerical examples in order to show the efficiency of the method and its limits.Key words: Left inversion, nonlinear system, nonlinear time-delay system, asymptotic left inversion, internal dynamics. IntroductionTime delay systems represent one of the most studied class of systems in control theory. Since the 60', many different problems are studied such as stability and stabilization of time delay systems, observation and observer design, parameter identification, etc. In the present work we are interested in the left invertibility problem of time delay systems with internal dynamics. The problem of unknown inputs recovering from the outputs is crucial. Such a problem has attracted the interest of the control community since it has direct applications in many domains, such as data secure transmission where the unknown input is the message, and fault detection and isolation where the fault is the unknown input. In fact, left invertibility problem has been studied since at least forty five years ago in linear control theory [33] In the literature, an important tool based on non-commutative ring, proposed in [36], is used to analyze nonlinear time delay systems in algebraic framework. Using this framework, many notions are extended to the case of nonlinear time * Corresponding author.Email address: zohra.kader@inria.fr (Zohra Kader). delay systems and many results have already been obtained and published [40], [37]. In the context of constant time delay, the notions of Lie derivatives and relative degree are defined, and the differences between causal and non-causal invertibility are clarified in [38]. The canonical form of invertibility is also given in [39], and in [42] a method for estimating the unknown inputs is proposed. However, the algorithm for left invertibility proposed in [39] was only for system without internal dynamics (see also [4] and [13] and their references). For system with or without time-delay, the main difficulty when the internal dynamics appears is to estimate the state of such dynamics. One interesting solution in order to overcome such a difficulty is to allow the derivative of the unknown inputs [31] with a geometrical approach and with an algebraic one. If, however, the input derivatives are not possible, then it is necessary to compute and analyze the internal dynamics.For nonlinear systems without delay, if the vector fields associated to the inputs verify involutivity property, then the internal dynamics does not depend on the unknown input. However, this rule is not valid for nonlinear systems with...

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Invertibility of linear time-invariant dynamical systems

Cited by 457 publications

References 12 publications

A geometric approach to the singular filtering problem

A geometric approach to the singular filtering problem

Data-driven simulation and control

Finite-time and asymptotic left inversion of nonlinear time-delay systems

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