Minimal realizations of nonlinear systems

Kotta, Ülle; Moog, Claude H.; Tõnso, Maris

doi:10.1016/j.automatica.2018.05.007

“…On the theoretical side, we prove that if a DAE of order ℎ can be realized by a system in the state-space form, then it can be realized by a system of dimension ℎ (that is, by a locally observable one). This result is related to a theorem by Sussmann [19] and its analogues for rational realizations [11] (see also [6,20]) which state that, for a realization problem (analytic or rational), if a realization exists, it can always be taken to be observable at the expense of allowing a realization to be defined not on an affine space but on an arbitrary variety. We achieve only local observability but guarantee the existence of a realization defined on an affine space.…”

Section: Introductionmentioning

confidence: 82%

“…In the nonlinear case, there are several versions of the problem depending on where f and g are sought. Two popular classes considered in this paper are rational functions and input-affine rational functions as in [11,12,17], but one could also consider algebraic, analytic, or smooth functions [14,19,20,22]. From the constructive standpoint, the case of single-output-no-input systems (for which rational and inputaffine rational functions coincide) has been considered by Forsman [4].…”

Section: Introductionmentioning

confidence: 99%

On realizing differential-algebraic equations by rational dynamical systems

Pavlov,

Pogudin

2022

Preprint

0

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Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs of the system can typically be related by more complicated differential-algebraic equations (DAEs). Therefore, a natural question (referred to as the realizability problem) is: given a differential-algebraic equation (say, fitted from data), does it come from a partially observed dynamical system? A special case in which the functions involved in the dynamical system are rational is of particular interest. For a single differential-algebraic equation in a single output variable, Forsman has shown that it is realizable by a rational dynamical system if and only if the corresponding hypersurface is unirational, and he turned this into an algorithm in the first-order case.In this paper, we study a more general case of single-input-singleoutput equations. We show that if a realization by a rational dynamical system exists, the system can be taken to have the dimension equal to the order of the DAE. We provide a complete algorithm for first-order DAEs. We also show that the same approach can be used for higher-order DAEs using several examples from the literature. CCS CONCEPTS• Computing methodologies → Symbolic calculus algorithms.

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“…The linear quadratic regulator is the extension of pole placement technique that tends to find the control input so as to place the poles of the system at a desired optimal position. The main idea in linear quadratic regulator control design is to minimize the quadratic cost function of J given in (9) [24], [25].…”

Section: Systemanalysis and Controller Designmentioning

confidence: 99%

“…The LQR should minimize this cost function (performance index) while obtaining the state feedback gains K that drives the system to the desired operating point. It turns out that regardless of the values of Q and R, the cost function has a unique minimum that can be obtained by solving the following Algebraic Riccati equation [25].…”

Section: Systemanalysis and Controller Designmentioning

confidence: 99%

Optimization of automobile active suspension system using minimal order

Finecomes

¹

,

Gebre²,

Mesene

³

et al. 2022

IJECE

1

0

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<p><span>This paper presents an analysis and design of linear quadratic regulator for reduced order full car suspension model incorporating the dynamics of the actuator to improve system performance, aims at benefiting: Ride comfort, long life of vehicle, and stability of vehicle. Vehicle’s road holding or handling and braking for good active safety and driving pleasure, and keeping vehicle occupants comfortable and reasonably well isolated from road noise, bumps, and vibrations are become a key research area conducted by many researchers around the globe. Different researchers were tested effectiveness of different controllers for different vehicle model without considering the actuator dynamics. In this paper full vehicle model was reduced to a minimal order using minimal realization technique. The entire system responses were simulated in MATLAB/Simulink environment. The effectiveness of linear quadratic regulator controller was compared for the system model with and without actuator dynamics for different road profiles. The simulation results were indicated that percentage reduction in the peak value of vertical and horizontal velocity for the linear quadratic regulator with actuator dynamics relative to linear quadratic regulator without actuator dynamics was 28.57%. Overall simulation results were demonstrated that proposed control scheme has able to improve the effectiveness of the car model for both ride comfort and stability.</span></p>

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“…In the nonlinear case, there are several versions of the problem depending on where f and g are sought. Two popular classes considered in this paper are rational functions and input-affine rational functions as in [12,13,21], but one could also consider algebraic, analytic, or smooth functions [18,23,24,26]. From the constructive standpoint, the case of single-output-no-input systems (for which rational and input-affine rational functions coincide) has been considered by Forsman [4].…”

Section: Introductionmentioning

confidence: 99%

On Realizing Differential-Algebraic Equations by Rational Dynamical Systems

Павлов

¹

,

Pogudin

²

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs of the system can typically be related by more complicated differential-algebraic equations (DAEs). Therefore, a natural question (referred to as the realizability problem) is: given a differential-algebraic equation (say, fitted from data), does it come from a partially observed dynamical system? A special case in which the functions involved in the dynamical system are rational is of particular interest. For a single differential-algebraic equation in a single output variable, Forsman has shown that it is realizable by a rational dynamical system if and only if the corresponding hypersurface is unirational, and he turned this into an algorithm in the first-order case.In this paper, we study a more general case of single-input-singleoutput equations. We show that if a realization by a rational dynamical system exists, the system can be taken to have the dimension equal to the order of the DAE. We provide a complete algorithm for first-order DAEs. We also show that the same approach can be used for higher-order DAEs using several examples from the literature. CCS CONCEPTS• Computing methodologies → Symbolic calculus algorithms.

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Minimal realizations of nonlinear systems

Cited by 11 publications

References 7 publications

On realizing differential-algebraic equations by rational dynamical systems

On realizing differential-algebraic equations by rational dynamical systems

Optimization of automobile active suspension system using minimal order

On Realizing Differential-Algebraic Equations by Rational Dynamical Systems

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