Computing the distance between a nonlinear model and its linear approximation: an approach

Kihas, Dejan; Marquez, Horacio J.

doi:10.1016/j.compchemeng.2004.08.002

Cited by 12 publications

(8 citation statements)

References 10 publications

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“…Then (13) Proof: Follows immediately from Theorem 4. The expression on the right hand side of (13) is actually equal to the AE-NLM of the memoryless operator associated with .…”

Section: Nonlinearity Measures Of Memoryless Systems and Steady-mentioning

confidence: 94%

“…The most common setup for nonlinearity measures [1], [3], [24] and linear modeling for nonlinear systems [12], [6], [13], [18], [19] is to consider the distance between the nonlinear system and a (linear) model as measured by the gain of the error system . 2 If the error system is defined this way, it readily follows that the nonlinear system can be represented as the parallel connection of the nominal linear model and the error system , , see Fig.…”

Section: Model Quality Indices Nonlinearity Measures and Best mentioning

confidence: 99%

“…Mäkilä and Partington [6] give the best linear models for discrete-time bi-gain systems and prove the existence of a best linear model for nonlinear finite impulse response filters. In [13], a procedure is proposed to approximately compute the -gain of the difference between a nonlinear system and its state space linearization (both in continuous time). Nikolaou and Hanagandi introduce an inner product on operators in order to derive a linear model for nonlinear systems [14].…”

Section: Introductionmentioning

confidence: 99%

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On System Gains, Nonlinearity Measures, and Linear Models for Nonlinear Systems

Schweickhardt

Allgöwer

2009

IEEE Trans. Automat. Contr.

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This article is concerned with the assessment and derivation of (best) linear models for nonlinear systems. We introduce a general framework reminiscent of uncertainty descriptions in linear robust control theory in order to define a family of six model quality indices for linear models of nonlinear systems. Each quality index corresponds to the gain of an error system, where the nonlinear system can be represented as the interconnection of a nominal linear model and this error system. For an analysis in a certain region of operation, these gains are considered over a specified signal set. A best linear model is a linear model that minimizes the quality index and the minimal quality index serves at the same time as a measure of nonlinearity of the nonlinear system. We show that the model quality indices and nonlinearity measures can be defined under very weak assumptions, and we give conditions that guarantee the existence of optimal linear models. We discuss the structure of the problem of computing the nonlinearity measures. We show that the local linearization of a state space system is a best linear model for the limiting case of a vanishing operating range for one of the measures. We show the relation between the steady-state behavior of a system and its gain and nonlinearity measures. Furthermore we give linear models and upper bounds for nonlinearity measures for systems composed of a linear dynamic and a nonlinear static part. In the case of scalar (SISO) systems, these bounds are given by the sector bounds of the nonlinearity. Finally, it is shown that a lower bound on the nonlinearity measures can be derived using harmonic analysis. Several small examples serve to illustrate the results and the underlying ideas.

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“…Then (13) Proof: Follows immediately from Theorem 4. The expression on the right hand side of (13) is actually equal to the AE-NLM of the memoryless operator associated with .…”

Section: Nonlinearity Measures Of Memoryless Systems and Steady-mentioning

confidence: 94%

Section: Model Quality Indices Nonlinearity Measures and Best mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On System Gains, Nonlinearity Measures, and Linear Models for Nonlinear Systems

Schweickhardt

Allgöwer

2009

IEEE Trans. Automat. Contr.

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“…Few results exist in the open literature that allow relating the time responses of a nonlinear system to those of its linearization by quantitative means (e.g. [11], [12]). These approaches conservatively bound from above a certain measure of the trajectories distance by maximizing some nonlinear time-varying function over a vector space.…”

Section: Evaluation Of Nonlinear Trajectories Approximation Errormentioning

confidence: 99%

A hybrid approach to robustness analyses of flight control laws in re-entry applications

et al. 2010

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The present paper aims at improving the efficiency of the robustness analyses of flight control laws with respect to conventional techniques, especially when applied to vehicles following time-varying reference trajectories, such as in an atmospheric re-entry. A nonlinear robustness criterion is proposed, stemming from the practical stability framework, which allows dealing effectively with such cases. A novel approach is presented, which exploits the convexity of linear time varying systems, coupled to an approximate description of the original nonlinear system by a certain number of its time-varying linearizations. The suitability of the approximating systems is evaluated in a probabilistic fashion making use of the unscented transformation technique. The effectiveness and potentials of the method are ascertained by application to the robustness analysis of the longitudinal flight control laws of the Italian Aerospace Research Center (CIRA) experimental vehicle USV

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“…In [5], the best linear model for discrete-time bi-gain systems is given with respect to the l 1 -norm and the existence of a best linear model for nonlinear finite impulse response filters is proved. In [6], a procedure is proposed to approximately compute the L 2 -gain of the error system defined as the difference between a nonlinear system and its local linearization.…”

Section: Introductionmentioning

confidence: 99%

A robustness approach to linear control of mildly nonlinear processes

Schweickhardt

Allgöwer

2007

Intl J Robust & Nonlinear

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SUMMARYWe present a novel approach toward linear control of nonlinear systems. Combining robust control theory and nonlinearity measures, we derive a method to (i) assess the nonlinearity of a given control system, (ii) derive a suitable linear model (not necessarily equivalent to the local linearization), and (iii) design a linear controller that guarantees stability of the closed loop containing the nonlinear process. A distinctive feature of the approach is that the nonlinearity analysis, linear model derivation and linear controller synthesis can be done on an operating regime specified by the designer. Examples are given to illustrate the approach.

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Computing the distance between a nonlinear model and its linear approximation: an approach

Cited by 12 publications

References 10 publications

On System Gains, Nonlinearity Measures, and Linear Models for Nonlinear Systems

On System Gains, Nonlinearity Measures, and Linear Models for Nonlinear Systems

A hybrid approach to robustness analyses of flight control laws in re-entry applications

A robustness approach to linear control of mildly nonlinear processes

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