Robust Strict Positive Realness Research Articles

Fixed‐Structure ℋ∞ Controller Design for Systems with Polytopic Uncertainty: An LMI Solution

AbstractThis note provides a new method for fixed‐structure H∞ controller design for discrete‐time single input single output (SISO) systems with polytopic uncertainty. New conditions are derived based on the concept of robust strict positive realness (SPRness) of an uncertain polynomial with respect to a parameter‐dependent polynomial. The quality of this approximation depends on this SPR‐maker. A procedure is proposed for choosing the parameter‐dependent SPR‐maker that guarantees the improvement of the performance for the designed controller in comparison with the traditional approaches employed fixed SPR‐makers. The proposed conditions are given in terms of solutions to a set of linear matrix inequalities. The effectiveness of the proposed approach is demonstrated by comparing with the existing results.

Fixed-Order H∞ Controller Design for Systems with Polytopic Uncertainty

AbstractThis note provides sufficient conditions for fixed-order H∞ controller design for SISO systems with polytopic uncertainty. New conditions are derived based on the concept of robust strict positive realness of an uncertain polynomial with respect to a parameter dependent polynomial. The main feature of the proposed method is that the non-convex set of fixed-order stabilizing controllers is approximated with an inner set which is not necessarily a convex set. The quality of this approximation depends on an initial stabilizing controller. The proposed conditions are in terms of solution to a set of linear matrix inequalities. The effectiveness of the proposed approach is demonstrated by comparing with the existing results.

Properties of strictly positive real functions: products and compositions

In this work necessary conditions on one side are sufficient conditions on another side given such that the product of strictly positive real (SPR) functions to be an SPR0 function. Also, sufficient conditions are given for the composition of SPR functions to be an SPR function. As a consequence, a result on robust SPR transfer functions, under non-linearly correlated perturbations in the coefficients, induced by the composition and product with SPR functions is presented. Based on these facts, we give an application on preservation of robust and simultaneous stabilisation of single-input single-output (SISO), linear time-invariant (LTI) systems. The applicability of the results is shown through two examples.

Synthesis of robust strictly positive real discrete-time systems with l/sub 2/ parametric perturbations

The author addresses the robust strict-positive realness problem for families of discrete-time polynomials where the uncertainty is described via a l/sub 2/ ball in coefficient space. It is shown constructively that under the only assumption that all the polynomials of the family are Schur, the sought filter can be provided in closed form as a polynomial or rational function with an a-priori bounded degree. The proposed synthesis procedure is based on the solution of a simple polynomial factorization problem.

Algebraic solution to the robust SPR problem for two polynomials

In this paper we provide an algebraic method for the design of an appropriate transfer function making simultaneously two polynomials strictly positive real (SPR), in the so-called robust SPR problem for two polynomials. This problem arises in the design of recursive estimation algorithms used for identification purposes and in active noise control (ANC) applications. In the former a given transfer function involving the poles of the unknown transfer function must be made SPR; in the latter, the phase of a transfer function must be matched within a range of (− π /2, π /2) . Robust SPR design methods try to fulfill the phase requirements for all possible transfer functions belonging to a given uncertainty description. Here we focus on uncertainties described by two polynomials.

Algebraic solution to the robust SPR problem for two polynomials

In this paper we provide an algebraic method for the design of an appropriate transfer function making simultaneously two polynomials strictly positive real (SPR), in the so-called robust SPR problem for two polynomials. This problem arises in the design of recursive estimation algorithms used for identification purposes and in active noise control (ANC) applications. In the former a given transfer function involving the poles of the unknown transfer function must be made SPR; in the latter, the phase of a transfer function must be matched within a range of (− π/2, π/2) . Robust SPR design methods try to fulfill the phase requirements for all possible transfer functions belonging to a given uncertainty description. Here we focus on uncertainties described by two polynomials.

On the strengthened robust SPR problem for discrete-time systems

The well-known necessary and sufficient condition to find a solution to the robust strictly positive real (SPR) problem, namely, that the phase difference of the polynomials involved must be less than π, was found to be also necessary and sufficient in the strengthened robust SPR problem by Anderson and Landau in a recent work, only for the continuous-time case. In this brief communication, we derive a necessary condition in the discrete-time case, which cannot be formulated in phase terms, and which is more stringent than the condition of the robust SPR problem, thereby establishing a fundamental difference between both continuous-time and discrete-time domains.

Composite Interval Control Systems: Some Strong Kharitonov-Like Properties

The interval model plays an important role in parametric robust control. This paper deals with the strict positive realness of a class of composite interval control systems, and establishes a strong Kharitonov-like extreme point criterion. We also discuss the robust strict positive realness of the shifted interval plant-controller family, and improve some previous results in the literature. Furthermore, for the collection of Popov plots of an interval transfer function family, we prove that a large portion of its outer boundary comes from the sixteen Kharitonov transfer functions. Finally, we study the beta boundedness of interval systems. A strong Kharitonov-like vertex result is established, namely, that all members in the interval system family are beta bounded if and only if the sixteen Kharitonov critical vertices are beta bounded.

The robust SPR problem: Design algorithms and new applications

Practical methods to solve the robust strictly positive real (SPR) problem are presented in this paper. Passivity conditions, in the form of an SPR requirement, are sufficient for the convergence of the hyperstable family of adaptive IIR algorithms. The robust SPR problem tries to get the satisfaction of the mentioned passivity condition for an uncertain set describing the unknown system. We show how this robust problem can be practically solved, considerably extending previous works on the topic. We also find another important area for the robust SPR problem in the active noise control problem. Adaptive algorithms for active noise control are more challenging than those used in classical scenarios; the error cannot be accessed, but a filtered version of it. This imposes a certain knowledge of the frequency response which filters the error, in such a way that its phase must be estimated with an error in the range (− π/2, π/2) . The robust SPR problem can take care of different transfer functions corresponding to varying operating conditions, thus yielding robust adaptive algorithms.

Performance Evaluation of Composite Interval Systems

Interval model plays an important role in parametric robust control. This paper deals with the strict positive realness of a class of composite interval systems, and establishes a strong Kharitonov-like extreme point criterion. We also discuss the robust strict positive realness of the shifted interval plant-controller family, and improve some previous results in literature. Finally, for the collection of Popov plots of interval transfer function family, we prove that a large portion of its outer boundary comes from the sixteen Kharitonov transfer functions.

On strict positive realness of multilinearly parametrized interval systems

The strict positive realness of a class of multilinearly parametrized interval systems is discussed, and a strong Kharitonov-like theorem is established. This result is applied to the absolute stability study of a class of nonlinear feedback systems with both parameter uncertainty and unstructured uncertainty, and an extreme point criterion for robust absolute stability is obtained. The robust strict positive realness of the shifted interval plant-controller family is discussed, and some previous results in literature are improved.

Finite test of robust strict positive realness

The paper is concerned with the problem of testing the robust strict positive realness (SPRness) of a family of rational functions with both the numerator and the denominator dependent on the same set of parameters. We show that this problem can be solved by using a series of Routh tables. In other words, the robust SPRness of the whole family can be tested by performing only a finite number of elementary operations (arithmetic operations, logical operations and sign tests).

On the analysis and synthesis of robust strict positive real systems

As the first main result, this paper presents a criterion which is very easy to be tested for robust strict positive realness of perturbed systems. Then, it considers such two problems as follows: 1) When does there exist a real polynomial, or more generally, a real transfer function B(s), such that B(s)/D(s) is strict positive real (SPR) for all D(s)/spl isin/D, D is a convex polytopic polynomial family? 2) When does there exist a compensator C(s) such that N(s)C(s)/D(s) is SPR for all N(s)/spl isin/N, D(s)/spl isin/D, N and D are both convex polytopic polynomial families? Necessary and sufficient conditions for such B(s) and C(s) to exist are given by this paper. Further the methods for synthesizing such B(s) and C(s) are also given. The results of this paper only involve computation of the extremes of N and D, and adopt diagrammatic methods like Bode plot; therefore, they are very simple and convenient to apply.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust strict positive realness design of interval systems

The paper is devoted to the study of robust strict positive realness design of interval systems, i.e. considering the question of designing a single controller which both robustly stabilizes an interval denominator system by unity feedback and makes the closed-loop transfer function robustly strictly positive real. By establishing a new computationally checkable criterion for the strict positive realness of a rational function, the extreme point result for a class of interval denominator systems is obtained, which significantly simplies the controller design procedure.

Design criteria for robust strict positive realness in adaptive schemes

Design of filters ensuring convergence of recursive estimation algorithms in the presence of uncertainty in the plant model is a key problem in the area of identification and adaptive control. This paper addresses the problem of designing filters ensuring strict positive realness of a family of uncertain polynomials over an assigned region of the complex plane. The uncertainty is assumed to be both structured and unstructured. When the structured uncertainty of the family is represented through regions of root location of a certain shape (for instance, circles centered on the real axis), an optimal solution of the problem is provided. For more general uncertainty regions, a simple procedure for constructing sub-optimal solutions to the filter design problem is proposed. A numerical example is fully developed to show the effectiveness of the proposed approach.

Convexity properties of polynomials with assigned root location

This paper presents some geometrical results on the domain of (generalized) stability for a family of nth order polynomials. Regions of root location such that the convex hull of the corresponding domain of stability in the coefficient space is a polyhedron are investigated, and specific regions for which the convex hull is an n+1 vertex polyhedron are derived. The discrete-time stability domain falls in the latter class of regions. Implications of the results for the design of filters solving the robust strict positive realness for families of rational transfer functions with uncertainty in the numerator are also developed. >

004 Design criteria for robust strict positive realness in adaptive schemes

004 Design criteria for robust strict positive realness in adaptive schemes

Robust strict positive realness: new results for interval plant-controller families

A frequency domain approach is used to derive several robust strict positive realness results for interval plants and interval plant plus controller families of transfer functions. Based on simple frequency domain properties of transfer functions, the approach provides a framework for obtaining new results and constructing easy proofs of several important existing results on robust strict positive realness. The main new result states that the minimum of the real part of the transfer functions belonging to an interval plant controller family is achieved on one of the 32 Kharitonov segments of the interval plant. The argument used in the proof is of wider interest and suggests easy ways of proving that robustness of other different frequency properties of interval plant plus controller families of transfer functions, such as robust stability of H/sub infinity /-norm computation, can be deduced from a fixed number of segments of transfer functions of the family.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust stability of polynomial families and robust strict positive realness of rational function families

Dual to the interval model, the diamond model proves to be an alternative powerful device for taking into account the variation of parameters in prescribed ranges. This paper deals with the robust stability of certain diamond polynomial families and the strict positive realness of certain diamond rational function families. By exploiting the geometric characterizations of their value sets, we show that, for the strict positive realness of the rational function family with the coefficients of numerator and denominator each varying in diamonds, the same property of only 24 specially selected extreme rational functions will suffice. An interesting result concerning the stability of the so-called simplex polynomial family is provided, namely, the stability of 4 extreme points and 2 exposed edges in this family will ensure the stability of the whole family. We also extend this result to the strict positive realness of the simplex rational function family where 30 extreme points and 2 exposed edges need to be c...

Robust strict positive realness: characterization and construction

Let P be a convex set of real polynomials. Consideration is given to the question of when there exists a real polynomial b(s), or more generally, a real transfer function b(s), such that p(s)/b(s) is strictly positive real for all p(s) in P. Necessary and sufficient conditions are found for the transfer function b(s) case, and when the degree of the polynomials in P is restricted, such conditions are also found for the polynomial b(s) case. Closely related results are also obtained for a z-transform version of the problem. The results have application in adaptive systems. >