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.
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 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.
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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 …
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TL;DR: In this article, a survey of state-space discretisation of linear parameter-varying (LPV) systems with static and dynamic dependence on the scheduling signal is presented.
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.
62 citations
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.
28 citations
TL;DR: In this paper, the authors proposed a direct identification of CT-LPV systems in an input-output setting, focusing on the case when the noise part of the data generating system is an additive discrete-time (DT) coloured noise process.
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.
28 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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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.
20 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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Book•
02 Jun 1994TL;DR: The design and analysis of linear control systems and matrix equations problems, (Lyapunov equations, Sylvester equations, the algebraic Riccati equations), the pole-placement problems, stability problems, and frequency response problems, have been studied.
Abstract: The design and analysis of linear control systems:
$$\dot x(t) = Ax(t) + Bu(t),\,\,\,\,\,\,\,\,\,\,y = Cx(t)$$
(1.1)
and
$$x_{k + 1} = Ax_k + Bu_k ,\,\,\,\,\,yk = Cx_k$$
(1.2)
give rise to many interesting linear algebra problems. Some of the important ones are: controllability and observability problems, the problem of computing the exponential matrix e At ,the matrix equations problems: (Lyapunov equations, Sylvester equations, the algebraic Riccati equations), the pole-placement problems, stability problems, and frequency response problems. These problems have been very widely studied in the literature. There exists a voluminous work both on theory and computations. Theory is very rich. Unfortunately, the same can not be remarked about computations. Many of the earlier methods were developed before the computer era, and are not based on numerically sound techniques. Fortunately, in the last twenty years or so, numerically effective techniques have been developed for most of these problems, and numerical analysis aspects of these methods (e.g. study of stability by round-off error analysis) and of the problems themselves (e.g. study of sensitivity) have been studied.
281 citations
TL;DR: In this paper, a solution of the mixed H 2 =H 1 problem with reduced order controller for time-varying systems in terms of solvability of diierential linear matrix inequalities and rank conditions is provided, including a detailed discussion of how to construct a controller.
Abstract: This paper provides a solution of the mixed H 2 =H 1 problem with reduced order controller for time-varying systems in terms of solvability of diierential linear matrix inequalities and rank conditions, including a detailed discussion of how to construct a controller. Immediate specialization lead to a solution of the full order problem and the mixed H 2 =H 1 problem for linear systems whose description depend on an unknown but in real-time measurable time-varying parameter. As known for the H 1 problem, we completely resolve the quadratic mixed H 2 =H 1 problem (under certain hypotheses on the parameter dependence) by reducing it to the solution of nitely many algebraic linear matrix inequalities. However, we also point out directions how to overcome the conservatism introduced by using constant solutions of the diierential inequalities. Finally, we clarify that a simple specialization leads to a linear matrix inequality approach to the pure H 2 problem for general LTI systems, and we reveal how to compute the optimal value. 1 Notation R n is equipped with the Euclidean norm, and R nm with the corresponding induced norm, both denoted as k:k. L p denotes the signal space L n p 0; 1) (for an appropriate n) and is equipped with the standard norm k:k p (deened with the Euclidean spatial norm). Functions are tacitly assumed to be continuous and bounded, and smooth functions are, in addition, continuously diierentiable. Time functions are functions deened on 0; 1). For a symmetric valued function X(s) deened on a set S, X is said to be strictly positive (X 0) if there exists an > 0 such that X(s) I for all s 2 S. For the system or input output mapping _ x = Ax + Bu; y = Cx + Du with x(0) = 0 we use the notation \" A B C D #. If A B C D ! is a constant matrix, the system is called LTI, if it is a time function, the system is called LTV. The time function A is exponentially stable if the system _ x = Ax has this property.
158 citations
"Discretization of Linear Fractional..." refers background in this paper
...…2, 2628 CD, Delft, The Netherlands, email: {r.toth,p.s.c.heuberger,p.m.j.vandenhof}@tudelft.nl. M. Lovera is with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci 20133, Milano, Italy, email: lovera@elet.polimi.it. has not been analyzed so far....
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TL;DR: This paper reviews the issues of digital control implementation, from algorithms through current hardware up to the various problems arising with non-ideal behaviour of digital controllers.
158 citations
"Discretization of Linear Fractional..." refers methods in this paper
...The paper is organized as follows: First, in Section II, LFRs of LPV systems are defined....
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...In Section III existing approaches of LFR discretization are investigated pointing out the need for improvement....
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...In Section V properties of the introduced methods are presented in terms of discretization error and preservation of stability....
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...In Section VI a numerical example is given for the comparison of the approaches....
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...Using an exact discretization setting in Section IV, popular discretization methods of the LTI framework are extended to LFRs....
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TL;DR: In this paper, a linear parameter-varying (LPV) control technique was used for the missile pitch-axis autopilot design, and the controller gain-scheduling function was constructed as affine matrix-valued function in the polytopic co-ordinates of the scheduled parameter.
Abstract: In this paper, the missile pitch-axis autopilot design is revisited using a new and recently available linear parameter-varying (LPV) control technique. The missile plant model is characterized by a linear fractional transformation (LFT) representation. The synthesis task is conducted by exploiting new capabilities of the LPV method: firstly, a set of H2/H∞ criteria defined channel-wise is considered; secondly, different Lyapunov and scaling variables are used for each channel/specification which is known to reduce conserva tism; and finally, the controller gain-scheduling function is constructed as affine matrix-valued function in the polytopic co-ordinates of the scheduled parameter. All these features are examined and evaluated in turn for the missile control problem. The method is shown to provide additional flexibility to tradeoff conflicting and demanding performance and robustness specifications for the missile while preserving the practical advantage of previous single-objective LPV methods. Finally, the method is shown to perform very satisfactorily for the missile autopilot design over a wide range of operating conditions. Copyright © 2001 John Wiley & Sons, Ltd.
116 citations
TL;DR: In this article, the authors revisited the missile pitch-axis autopilot design using a new recently available Linear Parameter-Varying (LPV) control technique, which is characterized by a linear fractional transformation (LFT) representation.
82 citations