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Proceedings ArticleDOI

Discretization of Linear Fractional Representations of LPV systems

01 Dec 2009-pp 7424-7429

TL;DR: The proposed and existing methods are compared and analyzed in terms of approximation error, considering ideal zero-order hold actuation and sampling, and criteria to choose appropriate sampling times with respect to the investigated methods are presented.

AbstractCommonly, controllers for Linear Parameter- Varying (LPV) systems are designed in continuous-time using a Linear Fractional Representation (LFR) of the plant. However, the resulting controllers are implemented on digital hardware. Furthermore, discrete-time LPV synthesis approaches require a discrete-time model of the plant which is often derived from continuous-time first-principle models. Existing discretization approaches for LFRs suffer from disadvantages like alternation of dynamics, complexity, etc. To overcome the disadvantages, novel discretization methods are derived. These approaches are compared to existing techniques and analyzed in terms of approximation error, considering ideal zero-order hold actuation and sampling.

Topics: Discretization (58%), Linear system (55%)

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Citations
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Journal ArticleDOI

[...]

08 Dec 2001-BMJ
TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

30,199 citations


Journal ArticleDOI
Abstract: Discretisation of linear parameter-varying (LPV) systems is a relevant, but insufficiently investigated problem of both LPV control design and system identification. In this contribution, existing results on the discretisation of LPV state-space models with static dependence (without memory) on the scheduling signal are surveyed and new methods are introduced. These approaches are analysed in terms of approximation error, considering ideal zero-order hold actuation and sampling of the input-output signals and scheduling variables of the system. Criteria to choose appropriate sampling periods with respect to the investigated methods are also presented. The application of the considered approaches on state-space representations with dynamic dependence (with memory) on the scheduling is investigated in a higher-order hold sense.

54 citations


Journal ArticleDOI
Abstract: Controllers in the linear parameter-varying (LPV) framework are commonly designed in continuous time (CT) requiring accurate and low-order CT models of the system. However, identification of CT-LPV models is largely unsolved, representing a gap between the available LPV identification methods and the needs of control synthesis. In order to bridge this gap, direct identification of CT-LPV systems in an input-output setting is investigated, focusing on the case when the noise part of the data generating system is an additive discrete-time (DT) coloured noise process. To provide consistent model parameter estimates in this setting, a refined instrumental variable (IV) approach is proposed and its properties are analysed based on the prediction-error framework. The benefits of the introduced direct CT-IV approach over identification in the DT case are demonstrated through a representative simulation example inspired by the Rao-Garnier benchmark.

27 citations


Cites background from "Discretization of Linear Fractional..."

  • ...Unfortunately, transformation of DT-LPV models to CT-LPV mo dels is more complicated than in the LTI case and despite recent advances in LPV discretiza tion theory (see [13], [14]) the theory of CT realization of DT models is still in an immature st ate....

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Journal ArticleDOI
TL;DR: The proposed and existing methods are compared and analyzed in terms of approximation error, considering ideal zero-order hold actuation and sampling, and criteria to choose appropriate sampling times with respect to the investigated methods are presented.
Abstract: Commonly, controllers for linear parameter-varying (LPV) systems are designed in continuous time using a linear fractional representation (LFR) of the plant. However, the resulting controllers are implemented on digital hardware. Furthermore, discrete-time LPV synthesis approaches require a discrete-time model of the plant which is often derived from a continuous-time first-principle model. Existing discretization approaches for LFRs describing LPV systems suffer from disadvantages like the possibility of serious approximation errors, issues of complexity, etc. To explore the disadvantages, existing discretization methods are reviewed and novel approaches are derived to overcome them. The proposed and existing methods are compared and analyzed in terms of approximation error, considering ideal zero-order hold actuation and sampling. Criteria to choose appropriate sampling times with respect to the investigated methods are also presented. The proposed discretization methods are tested and compared both on a simulation example and on the electronic throttle control problem of a race motorcycle.

22 citations


Posted Content
TL;DR: A Kalman-style realization theory for discrete-time affine LPV systems is formulated and it is shown that an input-output map has a realization by an affineLPV system if and only if it satisfies certain types of input- output equations.
Abstract: We formulate a Kalman-style realization theory for discrete-time affine LPV systems. By an affine LPV system we mean an LPV system whose matrices are affine functions of the scheduling parameter. In this paper we characterize those input-output behaviors which exactly correspond to affine LPV systems. In addition, we characterize minimal affine LPV systems which realize a given input-output behavior. Furthermore, we explain the relationship between Markov-parameters, Hankel-matrices, existence of an affine LPV realization and minimality. The results are derived by reducing the problem to the realization problem for linear switched systems. In this way, as a secondary contribution, we formally demonstrate the close relationship between LPV systems and linear switched systems. In addition we show that an input-output map has a realization by an affine LPV system if and only if it satisfies certain types of input-output equations.

19 citations


Cites background from "Discretization of Linear Fractional..."

  • ...It is well known that there is a correspondence between LPVs and LFT representations [19], [17]....

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References
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Journal ArticleDOI

[...]

08 Dec 2001-BMJ
TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

30,199 citations


"Discretization of Linear Fractional..." refers methods in this paper

  • ...Full zero-order hold approaches A commonly used approach, like in [4], [5], is to apply ZOHs and sampling on all signals of (1a-b) (see Figure 1b)....

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  • ...Basically the available methods use Zero-Order Hold (ZOH) and First-Order Hold (1OH) approaches to restrict the variations of the signals of the LFR in the sample interval which results in a DT description of the dynamics [4], [5], [6], [7], [8], [9], [10], [11]....

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Book
17 Aug 1995
Abstract: This paper will very briefly review the history of the relationship between modern optimal control and robust control. The latter is commonly viewed as having arisen in reaction to certain perceived inadequacies of the former. More recently, the distinction has effectively disappeared. Once-controversial notions of robust control have become thoroughly mainstream, and optimal control methods permeate robust control theory. This has been especially true in H-infinity theory, the primary focus of this paper.

6,941 citations


"Discretization of Linear Fractional..." refers background in this paper

  • ..., [3])...

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  • ...The main reason is that stability and performance requirements can be more conveniently expressed in CT, like in a mixed sensitivity setting [3]....

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  • ..., [1]–[3]), often require LPV models in a linear fractional representation (LFR), as depicted in Fig....

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Journal Article

4,497 citations


01 Jan 2002

4,051 citations


"Discretization of Linear Fractional..." refers methods in this paper

  • ...In Section VI a numerical example is given for the comparison of the approaches....

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Book
01 Jan 1991
Abstract: The subject of this book is the solution of stiff differential equations and of differential-algebraic systems (differential equations with constraints). The book is divided into four chapters. The beginning of each chapter is of introductory nature, followed by practical applications, the discussion of numerical results, theoretical investigations on the order and accuracy, linear and nonlinear stability, convergence and asymptotic expansions. Stiff and differential-algebraic problems arise everywhere in scientific computations (e.g., in physics, chemistry, biology, control engineering, electrical network analysis, mechanical systems). Many applications as well as computer programs are presented.

4,038 citations