S.P. Bhattacharyya, H. Chapellat, L.H. Keel, Robust Control - The Parametric Approach. Prentice Hall, Upper Saddle River, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....0 Use the zero exclusion principle to test graphically the robust stability of the given uncertain polynomial. First, let s transform the given polynomial into the centered form 3 p(s,q) 0.5 s 2s 2 4S 3 qis i i=0 with the uncertainty bound unchanged. Now type pO = 0. 5 s 2 s 2 4 s 3; W=[2,5,3,1]; r = 1; omega = 0: 05:2; The graphical output is generated by spherplot (pO, omega, r, W) Figure 2 Value sets for a polynomial family with spherical uncertainty bounding set. It can be clearly seen that the origin is included in the value sets, so we conclude that the given polynomial is not ....

....sets for a polynomial family with spherical uncertainty bounding set. It can be clearly seen that the origin is included in the value sets, so we conclude that the given polynomial is not robustly stable. 4. ROBUSTNESS MARGIN WITH POLYNOMIAL TOOLBOX Example 2 Consider the nominal polynomial [3], pp. 147: pO = 433.5 667.25 S 502.25 S 2 251.25 S 3 14 S 5 S 6; The diagonal entries of the weighting matrix are given weight = 43.5, 33.36, 25.137, 15.075, 5.6175, The vector of frequencies is chosen as omega = 0:0.01:5; To compute the maximum norm on the uncertainty vector just ....

Bhattacharyya, S.P., Chapellat, H., Keel, L.H. (1995). Robust Control: The Parametric Approach, Prentice Hall.

....The main reason is that control systems are in face of a changing environment, model errors, approximations, perturbations, disturbances, and so on. A summary of robust control may be found in reprint volumes and some excellent books (such as Dorato [5 7] Barmish [3] Bhatacharyya et al. [4], Zhou et al. 20] and others) It is known by experiments that the flight control system with its poles staying in a specific approximate region bounded by two arcs and two lines in the left half complex plane provides a good ride quality for flight [8, 10] This region is called a Good Ride ....

....about the robust pole clustering in GRQR for control systems with state feedback, output feedback and observer will be followed in other papers. Concerning robust pole clustering, the research can be carried out via the closed loop system characteristic polynomial in the frequency domain, e.g. [2 4] and the closed loop system matrix in the time domain [9, 11, 14 19, 21] For example, there is research work [3, 4] via Karitonov s theorem [12] and edge theorem [2] in frequency domain. In time domain, most work is via generalized Lyapunov theorem [11, 16, 19] initially developed by Gutman and ....

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S. P. Bhattacharyya, H. Chapellat and L. H. Keel, Robust Control: The Parametric Approach, Prentice Hall, Upper Saddle River, NJ, 1995.

....the solutions of this polynomial system. This method has been extensively studied, developed and has been implemented on many Computer Algebra Systems. Robust control refers to the control of a plant with significant uncertaintyby using a fixed controller and is a main branchofmoderncontrol theory [4,6,18]. Stability robustness analysis is a basic and fundamental topic in robust control. Robustness margin, kM , is a measure of the stability analysis of a feedback system. However, it is a known result that robustness margin problem is NP hard. Thus, pursuing the exact calculation of stability ....

....analysis of a feedback system. However, it is a known result that robustness margin problem is NP hard. Thus, pursuing the exact calculation of stability robustness margin is a tough job for control engineers. Grid method is a popular way to compute kM which be considered as Exact computation [6]. However, the dimension of the uncertainty space in robust stability analysis is normally high, for example, 20. If wechoose 20 sampling points in each uncertainty, it means that 20 20 evaluations must be made. This task is beyond the abilityofany current supercomputer. Therefore, the ....

S. Bhattacharyya, H. Chapellat and L. Keel, Robust Control - The Parametric Approach, Prentice-Hall, New York, 1995.

....effectiveness of the overall methodology, despite the inherent difficulty of the addressed problem. 1 Introduction Many works, mainly devoted to robust performance analysis, have appeared in the literature on the subject of robust regulation of linear systems affected by parametric uncertainties [1,2,3]. However, for the case of nonlinear uncertainties, effectivetechniques to synthesize robust controllers achieving predefined performances are apparently not available. Aiming to provide new tools for robust synthesis, in this paper wefocus our efforts on the synthesis of a feedforward feedback ....

S.P. Bhattacharyya, H. Chapellat, L.H. Keel, "Robust Control: The parametric Approach", Prentice Hall, 1995.

....problem of robust stability under structured and unstructured uncertainties and the quantitative theory of robust adaptive control. The problem can be transformed to be the computation problem of the maximal H1 norm or shifted H1 norm of a weighted transfer function with parametric uncertainties [1,2], which is also called parametric H1 or shifted H1 problems. The standard H1 norm of a stable transfer function P (s) is defined as kP (s)k 1 : sup 2R jP (j )j, and the shifted H1 norm of a H ffi stable transfer function P (s) is defined as kP (s)k ffi 1 : kP (s ; ffi)k 1 = sup ....

S.P. Bhattacharyya, H. Chapellat and L.H. Keel, "Robust Control, The Parametric Approach ", Prentice Hall PTR, 1995.

....very complicated uncertainty structures with several independent parameters. The case of simpler structures was described elsewhere (Sebek, Pejchova, 1999) The underlying methods as well as other solutions that can also be built from the Polynomial Toolbox macros are described in Barmish (1996) Bhattacharyya, Chapellat and Keel (1995) and other textbooks. VALUE SET AND ZERO EXCLUSION CONDITION A family of polynomials discussed in this paper is a set PpqqQ=# ( of polynomials depending on one or more independent parameters from a parameter bounding set Q . The parameters are described by the vector q. The set Q is ....

Bhattacharyya S. P., Chapellat H. and Keel L.H. (1995). Robust Control: The Parametric Approach.

....a polynomial belong to some region of the complex plane is of fundamental importance in control theory, because it is closely related to stability of dynamical systems. Standard methods for analyzing stability of polynomials are comprehensively described in [Gantmacher, 1959] or more recently in [Bhattacharyya et al. 1995]. The most popular stability analysis techniques are probably the determinantal Routh Hurwitz or Schur Cohn criteria, and Lyapunov s second approach. Lyapunov like techniques gradually gained a lot of interest among control people because of the development of efficient numerical procedures to ....

S. P. Bhattacharyya, H. Chapellat and L. H. Keel "Robust Control: The Parametric Approach", Prentice Hall, Upper Saddle River, New Jersey, 1995.

....The problem of robust stabilization of a plant affected by parametric uncertainty is of fundamental importance in control. Even though significant progress has been made recently in the realm of parametric robust control and robust stability analysis, a very few design algorithms are available [3]. One of the main hindrance behind the development of an efficient, systematic robust design tool is the well known fact that the stability region in the space of polynomial coefficients is non convex in general [1] Since most of the design problems can be formulated as optimization problems in ....

....the merit of being tractable. This is the approach we pursue in this paper. The choice of the uncertainty model is crucial in the design procedure. It is now admitted that interval uncertainty is more suitable for robust stability analysis (based on the theorem of Kharitonov and its variations [3]) than for robust design. The assumption that each plant parameter is constrained to an uncertainty range that is independent of all other parameters usually proves overly conservative. For this reason, ellipsoidal parametric uncertainty [4, 10] may be viewed as an interesting alternative ....

S. P. Bhattacharyya, H. Chapellat and L. H. Keel, "Robust Control: The Parametric Approach", Prentice Hall, Upper Saddle River, New Jersey, 1995.

.... specifically, we specialize to the periodic setting the integral logarithmic criterion for stability of linear time varying systems given in [Dasgupta et al. 1994] Exploiting such a criterion together with a technique for computing the real parametric stability margin of a family of polynomials [Bhattacharyya et al. 1995], we obtain a lower bound of the degree of exponential stability of periodic solutions. Two di#erent application examples are presented to illustrate the reliability of the obtained lower bound as well as its usefulness in controlling bifurcations and chaos. The remainder of the paper is organized ....

.... of exponential stability consists in finding a region of the complex plane containing all the roots of the family of characteristic polynomials det[sI A Bk # C] as k ranges over [k , k ] Problems of this kind have been recently investigated in the context of robust stability analysis [Bhattacharyya et al. 1995], eds. Garulli et al. 1999] Hereafter, we recall the concept of real parametric stability margin of a family of polynomials. Consider the following family of polynomials P # : # P (s) P 0 (s) n q # i=1 q i P i (s) #q# # # # # where P 0 (s) P 1 (s) P n q (s) are ....

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Bhattacharyya, S., Chapellat, H. & Keel, L. [1995] Robust Control: The Parametric Approach (Prentice Hall PTR, Upper Saddle River, NJ).

....of physical parameters. As far as polynomial methods are concerned, this leads to polynomial matrices with uncertain coefficients. Several analysis and design tools have emerged to cope with these uncertainties. They were coined out as robust control techniques, see e.g. the recent textbooks [2, 5, 49]. It must be however pointed out that most the results available so far concern robust stability of constant matrices or scalar polynomials. A very few works deal with robust stability of polynomial matrices. Some first results in this direction were obtained in [25] where stability of polynomial ....

....works deal with robust stability of polynomial matrices. Some first results in this direction were obtained in [25] where stability of polynomial matrix polytopes is studied. Polytopic uncertainty is probably the most general representation of the parametric uncertainty affecting a linear system [2, 5]. However, this generality has recently led to the somehow disappointing conclusion that most of the stability analysis problems involving polytopic uncertainty cannot be solved efficiently [7] The aim of the authors in [25] was then to show that sufficient robust stability conditions can however ....

S. P. Bhattacharyya, H. Chapellat, L. H. Keel "Robust Control: The Parametric Approach", Prentice Hall, Upper Saddle River, New Jersey, 1995.

....project No. LN00B096. 1 (half plane, disk) of the complex plane has strong implications in stability theory [13] The deep study of relationships existing between the roots of a polynomial and its coefficients has led to the development of the so called parametric approach to systems control [2, 4]. The main hindrance to the development of efficient analysis and design tools based on parametric methods is the well known fact that the stability region in the space of polynomial coefficients is non convex in general. For example, when the stability region in the space of polynomial roots is ....

S. P. Bhattacharyya, H. Chapellat and L. H. Keel, "Robust Control: The Parametric Approach", Prentice Hall, Upper Saddle River, New Jersey, 1995.

....or variation of the system parameters. 1 The zeros of a polynomial matrix are the roots of its determinant. 2 A lot of research efforts has been recently devoted to the investigation of this problem, which has been coined out as the robust stability analysis problem, see e.g. the textbooks [2, 3, 29] and references therein. Quite naturally, the problem of checking robust stability of uncertain linear systems amounts to checking robust stability of uncertain polynomial matrices. A very few works have been devoted so far to the study of robust stability of polynomial matrices, probably because ....

S. P. Bhattacharyya, H. Chapellat and L. H. Keel, "Robust Control: The Parametric Approach", Prentice Hall, Upper Saddle River, New Jersey, 1995.

....the analysis and design of uncertain systems. This motivates the computation of the value set of uncertain polynomials. Considerable attention has recently been attracted and many significant results have been presented including Kharitonov theorem [6] Edge Theorem [1] and ZeroExclusion principle [2]. In Quantitative Feedback Theory (QFT) and other frequency domain robust design methods, it is essential to describe the uncertain plant including parameter uncertainties, unstructured uncertainties and mixed uncertainties in the frequency domain. In addition, it is also useful to be able to ....

S. P. Bhattacharyya, H. Chapellat, and L. H. Keel. Robust Control: The Parametric Approach. Prentice Hall Inc., 1995.

.... timeinvariant systems with uncertain parameters are related to the problem of robust stability of segments of polynomials and quasipolynomials [7] 9] Research on systems with uncertain parameters has been very active in recent years and a number of useful results have been obtained, see [2] [4] and the references therein. The problem under which conditions a complex (real) polynomial q has the property that for all complex (real) Hurwitz stable polynomials p with deg p deg q, the Hurwitz stability of p q implies the Hurwitz stability of the whole segment [p; p q] is called the ....

S.P. Bhattacharyya, H. Chapellat, L.H. Keel, Robust Control - The Parametric Approach. Prentice Hall, Upper Saddle River, 1995.

....four polynomials are fixed to x Gamma i or x i . As an immediate important consequence, stability of an interval polynomial family can be checked efficiently with a polynomial time algorithm. Many researchers attempted to extend Kharitonov s result to other polynomial families, see [3] and [6] for a good overview. The most successful generalization is probably the so called Edge Theorem [4] valid for an affine polynomial family fp 0 (s) x 1 p 1 (s) x 2 p 2 (s) Delta Delta Delta x n p n (s) x Gamma i x i x i ; i = 1; ng (1) where each p i (s) is a given ....

....Components x i represent unknown compensator parameters. For clarity, we underline that this problem is distinct from the (even more challenging) design issue of stabilizing with a fixed compensator a family of plants whose scalar parameters are restricted to lie within given intervals [6, 28, 14]. In turn, this latter problem is dual to non fragile design [20] where 1 In this paper, a stable polynomial has its zeros in the open left half plane. a given plant must be stabilized by any fixed order compensator whose parameters lie within given intervals. Note however that in this paper ....

S. P. Bhattacharyya, H. Chapellat and L. H. Keel, "Robust Control: The Parametric Approach", Prentice Hall, Upper Saddle River, New Jersey, 1995.

....of vectors q that preserve stability. It has been shown that for the transfer function model, using Kharitonov theory, that some problems of this type can be reduced to the problem of simultaneous stabilization of a finite collection of plants. See, for example chapter 11 in (Barmish, 1988) and (Bhattacharyya et al. 1995). 2.0.2 Performance In general, stability is but one of many requirements of a closed loop control system. We will focus here on performance measures specified in the frequency domain, since measures of this type fit the QE problem formulation very nicely. Problem 4: Frequency Domain ....

Bhattacharyya, S., Chapellat, H., and Keel, L. (1995). Robust Control: The Parametric Approach. Prentice-Hall PTR, Upper Saddle River, NJ, 1st edition.

....stability [11, 9] Quite naturally, the problem of checking robust stability of uncertain linear systems amounts to checking robust stability of uncertain polynomial matrices. Polytopic uncertainty is probably the most general representation of the parametric uncertainty affecting a linear system [2, 3]. However, this generality has recently led to the somehow disappointing conclusion that most of the stability analysis problems involving polytopic uncertainty are NP hard [4] Following the hierarchy described in [2] we now briefly review these results. In increasing order of complexity, we can ....

....of complexity, we can distinguish between stability of ffl Interval polynomials, where each coefficient of the polynomial varies independently in a given interval. Kharitonov showed that stability of an interval family of continuous time polynomials can be checked efficiently in polynomial time [2, 3]. Unfortunately, there is no broad generalization of Kharitonov s result to other stability regions [4] ffl Polytopes of polynomials, sometimes also referred to as affine polynomial families, which are linear combinations of a set of given polynomials. These families have value sets which are ....

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S. P. Bhattacharyya, H. Chapellat and L. H. Keel, "Robust Control: The Parametric Approach", Prentice Hall, Upper Saddle River, New Jersey, 1995.

.... in the design procedure, it will produce huge burden in the subsequent computation of the robust stability and robust performance bounds [7] In view of this, several approaches have been proposed to simplify the procedure of template generation for plants with special uncertain structures (see [6] and [4] Bailey and Hui [2] consided the case where the uncertain parameters in the numerator and denominator are independent and affine. When the uncertain parameters in the numerator and denominator are dependent and affine, Fu [9] and Bartlett [5] claim the boundary of the templates are only ....

....of Kharitonov polynomial and Kharitonov segment, Tesi and Vicino [14] and Keel and Bhattacharyya [11, 12] show that the boundary of the frequency response can be produced by the Kharitonov polynomials and Kharitonov segments. This approach is further extended to the multilinear interval plants in [6] but the resultant templates are conservative as shown in this paper. Template generation for a structured uncertain plant is considered in this paper. There is no restriction on the structure of uncertain parameters. A general procedure for template generation of uncertain plant including ....

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S. P. Bhattacharyya, H. Chapellat, and L. H. Keel. Robust Control: The Parametric Approach. Prentice Hall Inc., 1995.

....cost. 1 Introduction We focus on the problem of robust stabilization of an uncertain single input single output plant whose parameters belong to given real intervals. Even though significant progress has been made recently in the realm of parametric robust control see textbooks [3] and [6] for a good overview this is a largely unsolved problem for which a very few results are Corresponding author. E mail henrion laas.fr. FAX 33 5 61 33 69 69. available. Indeed, Bhattacharyya and co workers [6, p. xii] point out that a significant deficiency of control theory at the present ....

....obtained by setting r i (s) k j i (s) for some j = 1; 2; 3; 4 and every fixed integer i between 1 and t. There are at most 4 t distinct elements in K. With these notations, the following result can be considered as a particular case of the Generalized Kharitonov s Theorem described in [6]. Theorem 2 Consider a t tuple of fixed polynomials fff 1 (s) ff t (s)g of the form ff i (s) s f i (g i s h i ) where f i 0 is an arbitrary integer and g i ; h i are arbitrary real numbers. Then, the interval polynomial ff 1 (s)r 1 (s) Delta Delta Delta ff t (s)r t (s) ....

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S. P. Bhattacharyya, H. Chapellat and L. H. Keel, "Robust Control: The Parametric Approach", Prentice Hall, Upper Saddle River, New Jersey, 1995.

....some performance requirements. P P D K Figure 1: Robust Control Scheme In this paper we will consider structured uncertainties in the plant, to represent the effect of (generally) slowly time varying parameters whose exact values are unknown but which are known to belong to a given set [1]. Most control algorithms proposed in the literature do not consider the problems introduced by implementing uncertain controllers. We first remark that it is reasonable to consider only structured uncertainties in the controller since by design, one can choose its exact structure even though the ....

S. Bhattacharyya, H. Chapellat, and L. Keel, Robust Control: The Parametric Approach. Upper Saddle River, NJ: Prentice-Hall PTR, 1st ed., 1995.

....can be cast into the two following categories: ffl Kharitonov like results, for systems with parametric or structured uncertainty. Launched by the seminal Kharitonov theorem for polynomials with interval uncertainty, these results generally apply to systems described in a transfer function setting [3, 5]. ffl Lyapunov like results, for systems with frequency domain or unstructured uncertainty. Originating from the work of Lyapunov on the stability of motion, these results generally apply to systems described in a state space setting [6, 21, 20] 1 This work was supported by the Barrande ....

.... an uncertain parameter belonging to a real symmetric interval, i.e. jrj r max : 1) Several kinds of stability can be defined with respect to uncertain polynomial p(z; r) In this section, we first focus on the definition most frequently encountered in the parametric approach to control systems [3, 5]. Definition 1 Uncertain polynomial p(z; r) is robustly stable if it is stable for any fixed value of r such that (1) holds. Definition 2 The real stability radius of p(z; r) is the smallest absolute value of r that destabilizes p(z; r) i.e. r R = min r max s.t. p(z; r) is not Schur. There ....

S. P. Bhattacharyya, H. Chapellat and L. H. Keel, Robust Control: The Parametric Approach, Prentice Hall, Englewood Cliffs, New Jersey, 1995.

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S.P. Bhattacharyya, H. Chapellat, L.H. Keel, Robust Control - The Parametric Approach. Prentice Hall, Upper Saddle River, 1995.

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Bhattacharyya, S. P., H. Chapellat, and L. H. Keel (1995). Robust Control: The Parametric Approach, Prentice Hall, Upper Saddle River, New Jersey.

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S.P. Bhattacharyya, H. Chapellat, and L.H. Keel. Robust Control: The Parametric Approach. Prentice Hall, 1995.

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S.P. Bhattacharyya, H. Chapellat, and L.H. Keel. Robust Control: The Parametric Approach. Prentice Hall, 1995.

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Bhattacharyya, S. P., H. Chapellat, L. H. Keel, Robust Control : The Parametric Approach,Prentice Hall PTR, 1995.

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S. P. Bhattacharyya, H. Chapellat and L. H. Keel. Robust Control: The Parametric Approach. Prentice Hall, Upper Saddle River, New Jersey, 1995.

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Bhattacharyya, S.P., Chapellat, H., Keel, L.H. (1995). Robust control { The parametric approach. Prentice-Hall, Inc. Upper Saddle River.

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