Graphical computation of gain and phase margin specifications-oriented robust PID controllers for uncertain systems with time-varying delay

This paper proposes a novel graphical method to compute all feasible gain and phase margin specifications-oriented robust PID controllers to stabilize uncertain control systems with time-varying delay. A virtual gain-phase margin tester compensator is incorporated to guarantee the concerned system with certain robust safety margins. The complex Kharitonov theorem is used to characterize the parametric uncertainties of the considered system and is exploited as a stability criterion for the Hurwitz property of a family of polynomials with complex coefficients varying within given intervals. The coefficients of the characteristic equation are overbounded and eight vertex Kharitonov polynomials are derived to perform stability analysis. The stability equation method and the parameter plane method are exploited to portray constant gain margin and phase margin boundaries. The feasible controllers stabilizing every one of the eight vertex polynomials are identified in the parameter plane by taking the overlapped region of the plotted boundaries. The overlapped region of the useful region of each vertex polynomial is the Kharitonov region, which represents all the feasible specifications-oriented robust PID controller gain sets. Variations of the Kharitonov region with respect to variations of the derivative gain are extensively studied. The way to select representative points from the Kharitonov region for designing robust controllers is suggested. Finally, three illustrative examples with computer simulations are provided to demonstrate the effectiveness and confirm the validity of the proposed methodology. Based on the pre-specified gain and phase margin specifications, a non-conservative Kharitonov region can be graphically identified directly in the parameter plane for designing robust PID controllers.     

Robust stability of a class of polynomials with coefficientsdepending multilinearly on perturbations

Necessary and sufficient conditions are given for robust stability of a family of polynomials. Each polynomial is obtained by a multilinearity perturbation structure. Restrictions on the multilinearity are involved, but, in contrast to existing literature, these restrictions are derived from physical considerations stemming from analysis of a closed-loop interval feedback system. The main result indicates that all polynomials in the family of polynomials have their zeros in the strict left half-plane if and only if two requirements are satisfied at each frequency. The first requirement is the zero exclusion condition involving four Kharitonov rectangles. The second requirement is that a specially constructed &thetas;0-parameterized set of 16 intervals must cover the positive reals for each &thetas; ε[0,2π]     

A new extension to Kharitonov's theorem

An extension to a well-known theorem due to Kharitonov is presented, Kharitonov's theorem gives a necessary and sufficient condition for all polynomials in a given family to be Hurwitz stable. In Kharitonov's theorem, the family of polynomials considered is obtained by allowing each of the polynomial coefficients to vary independently within an interval. Kharitonov's theorem shows that stability of this family of polynomials can be determined by looking at the stability of four specially constructed vertex polynomials. Kharitonov's theorem is extended to allow for more general families of polynomials and to allow a given margin of stability to be guaranteed for the family of polynomials     

Robustness analysis for linear dynamical systems with linearlycorrelated parametric uncertainties

The authors propose an approach for robust pole location analysis of linear dynamical systems with parametric uncertainties. Linear control systems with characteristic polynomials whose coefficients are affine in a vector of uncertain physical parameters are considered. A design region in complex plane for system pole placement and a nominal parameter vector generating a characteristic polynomial with roots in that region are given. The proposed method allows the computation of maximal domains bounded by linear inequalities and centered at the nominal point in system parameter space, preserving system poles in the given region. The solution of this problem is shown to also solve the problem of testing robot location of a given polytope of polynomials in parameter space. It is proved that for stability problems for continuous-time systems with independent perturbations on polynomial coefficients, this method generates the four extreme Kharitonov polynomials     

Robust stability of a special class of polynomial matrices with control applications

The robust stability problem of uncertain continuous-time systems described by higher-order dynamic equations is considered in this paper. Previous results on robust stability of Metzlerian matrices are extended to matrix polynomials, with the coefficient matrices having exactly the same Metzlerian structure. After defining the structured uncertainty for this class of polynomial matrices, we provide an explicit expression for the real stability radius and derive simplified formulae for several special cases. We also report on alternative approaches for investigating robust Hurwitz stability and strong stability of polynomial matrices. Several illustrative examples throughout the paper support the theoretical development. Moreover, an application example is included to demonstrate uncertainty modeling and robust stability analysis used in control design.     

Some results on robust stability of discrete-time systems

The theorem of Kharitonov on the Hurwitz property of interval families of polynomials cannot be extended, in genera, to obtain sufficient conditions for the stability of families of characteristic polynomials of discrete-time systems. Necessary and sufficient conditions for this stability problem are given. Such conditions naturally give rise to a computationally efficient stability test which requires the solution of a one-parameter optimization problem and which can be considered as a counterpart to the Kharitonov test for continuous-time systems. At the same time, the method used to derive the stability conditions provides a procedure for solving another stability robustness problem, i.e. the estimation of the largest domain of stability with a rectangular box shape around given nominal values of the polynomial coefficients     

A Kharitonov‐like theorem for robust stability independent of delay of interval quasipolynomials

In this paper, a Kharitonov‐like theorem is proved for testing robust stability independent of delay of interval quasipolynomials, p(s)+∑eq_k(s), where p and q_k's are interval polynomials with uncertain coefficients. It is shown that the robust stability test of the quasipolynomial basically reduces to the stability test of a set of Kharitonov‐like vertex quasipolynomials, where stability is interpreted as stability independent of delay. As discovered in (IEEE Trans. Autom. Control 2008; 53 :1219–1234), the well‐known vertex‐type robust stability result reported in (IMA J. Math. Contr. Info. 1988; 5 :117–123) (See also (IEEE Trans. Circ. Syst. 1990; 37 (7):969–972; Proc. 34th IEEE Conf. Decision Contr., New Orleans, LA, December 1995; 392–394) does contain a flaw. An alternative approach is proposed in (IEEE Trans. Autom. Control 2008; 53 :1219–1234), and both frequency sweeping and vertex type robust stability tests are developed for quasipolynomials with polytopic coefficient uncertainties. Under a specific assumption, it is shown in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) that robust stability independent of delay of an interval quasipolynomial can be reduced to stability independent of delay of a set of Kharitonov‐like vertex quasipolynomials. In this paper, we show that the assumption made in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) is redundant, and the Kharitonov‐like result reported in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) is true without any additional assumption, and can be applied to all quasipolynomials. The key idea used in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) was the equivalence of Hurwitz stability and ?_‐o‐stability for interval polynomials with constant term never equal to zero. This simple observation implies that the well‐known Kharitonov theorem for Hurwitz stability can be applied for ?_‐o‐stability, provided that the constant term of the interval polynomial never vanishes. However, this line of approach is based on a specific assumption, which we call the CNF‐assumption. In this paper, we follow a different approach: First, robust ?_‐o‐stability problem is studied in a more general framework, including the cases where degree drop is allowed, and the constant term as well as other higher‐orders terms can vanish. Then, generalized Kharitonov‐like theorems are proved for ?_‐o‐stability, and inspired by the techniques used in (IEEE Trans. Autom. Control 2008; 53 :1219–1234), it is shown that robust stability independent of delay of an interval quasipolynomial can be reduced to stability independent of delay of a set of Kharitonov‐like vertex quasipolynomials, even if the assumption adopted in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) is not satisfied. Copyright © 2009 John Wiley & Sons, Ltd.     

On the thirty-two virtual polynomials to stabilize an intervalplant

This paper studies the conservatism of the 32 virtual polynomials to stabilize an interval plant. It is shown that working with the 32 virtual vertices is generally less conservative than with the Kharitonov polynomials of the smallest interval polynomial containing the characteristic polynomial polytope. By means of the former, it is possible to find all the controllers such that the value set of the polytope of characteristic polynomials is applied in two quadrants as a maximum for each ω; while using the latter, only some of them can be found. The cases in which both methods coincide are also analyzed, and the conditions on the numerator and denominator of the controller are developed. Thus, this coincidence can be known a priori from the characteristics of the coefficients of the numerator and denominator of the controller. It is shown that these conditions are satisfied by the first-order controllers     

To stabilize an interval plant family it suffices to simultaneouslystabilize sixty-four polynomials

It is shown that stability of three specific polynomial families can be deduced from the stability of a finite number of polynomials. These polynomial families are the characteristic polynomials of unity feedback loops with the controller in the forward path, and where the plant includes a specific form of parameter uncertainty. For the first polynomial family, the plant has parameter uncertainty in the even or odd terms of the numerator or denominator polynomial. For the second polynomial family the plant has a numerator or denominator which is an interval polynomial. For the third polynomial family, the plant is interval. Because of the structure of these results it is shown that they lead to robust stabilization results. Two examples are included. The approach employed here was developed for plants with affine uncertainty. It is demonstrated that considerable simplification results if the plants under investigation are interval     

Robust stability of a family of disc polynomials

In his well-known theorem, V. L. Kharitonov established that Hurwitz stability of a set f1 of interval polynomials with complex coefficients (polynomials where each coefficient varies in an arbitrary but prescribed rectangle of the complex plane) is equivalent to the Hurwitz stability of only eight polynomials in this set. In this paper we consider an alternative but equally meaningful model of uncertainty by introducing a set fD of disc polynomials, characterized by the fact that each coefficient of a typical element P(s) in fD can be any complex number in an arbitrary but fixed disc of the complex plane. Our result shows that the entire set is Hurwitz stable if and only if the ‘center’ polynomial is stable, and the H _∞-norms of two specific stable rational functions are less than one. Our result can be readily extended to deal with the Schur stability problem and the resulting condition is equally simple.     

2‐D ALGEBRAIC TEST FOR ROBUST STABILITY OF QUASIPOLYNOMIALS WITH INTERVAL PARAMETERS

Determining the robust stability of interval quasipolynomials leads to a NP problem: an enormous number of testing edge polynomials. This paper develops an efficient approach to reducing the number of testing edge polynomials. This paper solves the stability test problem of interval quasipolynomials by transforming interval quasipolynomials into two‐dimensional (2‐D) interval polynomials. It is shown that the robust stability of an interval 2‐D polynomial can ensure the stability of the quasipolynomial, and the algebraic test algorithm for 2‐D s‐z interval polynomials is provided. The stability of 2‐D s‐z vertex polynomials and 2‐D s‐z edge polynomials were tested by using a Schur Table of complex polynomials.     

Systems of linear difference equations with bounded and robustly bounded solutions

In this paper, we investigate higher-order systems of linear difference equations where the associated characteristic matrix polynomial is self-inversive. We consider classes of equations with bounded solutions. It is known that stability properties of higher-order systems of linear difference equations are determined by the characteristic values of the corresponding matrix polynomials. All solutions are bounded (in both time directions) if the spectrum of the corresponding matrix polynomial lies on the unit circle, and moreover if the characteristic values of modulus 1 are semisimple. If the corresponding matrix polynomial is self-inversive, then one can use the inner radius of the numerical range to obtain a criterion for boundedness of solutions. We show that all solutions are bounded if the inner radius is greater than 1. In the case of matrix polynomials with positive definite coefficient matrices, we derive a computable lower bound for the inner radius and we obtain a criterion for robust boundedness.     

An alternative proof of Kharitonov's theorem

An alternative proof is presented of Kharitonov's theorem for real polynomials. The proof shows that if an unstable root exists in the interval family, then another unstable root must also show up in what is called the Kharitonov plane, which is delimited by the four Kharitonov polynomials. This fact is proved by using a simple lemma dealing with convex combinations of polynomials. Then a well-known result is utilized to prove that when the four Kharitonov polynomials are stable, the Kharitonov plane must also be stable, and this contradiction proves the theorem     

On robust control of a class of nonlinear mechanical systems with parametric uncertainty

This article presents a method to design a robust controller for a class of nonlinear mechanical systems. The parameters of the system are assumed uncertain in the sense that they are known only within intervals of known lower and upper limits. The method involves using Kharitonov stability theory of interval polynomials to design a robust stabilising controller for the considered class of systems. To formulate the interval parameter problem in joint space, the dynamic equation, which is derived in Cartesian space for parallel kinematic machines, is transformed to joint space using the Jacobian matrix and its time derivative. The nonlinear joint space model is linearised yielding an interval linear model and a robust controller with interval parameter gains is then found. A simulation study on a dynamic model of Stewart platform based parallel kinematics machine demonstrates the design procedure and shows its effectiveness.     

Frequency domain conditions for the stability of perturbed polynomials

New necessary and sufficient conditions for the stability of perturbed polynomials of continuous systems are given in the frequency domain. The conditions are equivalent and in some respects more powerful than the well-known Kharitonov conditions. The new conditions allow considerable freedom in distributing the available uncertainty margin among the different coefficients of a polynomial and provide an indication as to whether the maximum allowable margin of uncertainty for a given polynomial has been reached.     

区间分数阶系统的鲁棒稳定性判别准则:0 < α < 1

针对同元阶次在0和1之间的区间分数阶系统,提出了类似Kharitonov定理的鲁棒稳定性判别准则. 研究了区间分数阶系统分母的主分支函数值集不包含原点所需满足的条件.根据除零原理, 给出了区间分数阶系统鲁棒稳定的顶点和棱边条件. 定义了由分母函数系数构成的矩阵,通过检验矩阵是否在负实轴上存在特征值来检验棱边条件. 最后,通过对两个数值算例的分析说明了这种方法的有效性.     

Necessary and sufficient conditions for robust oscillatory stability

The problem of robust oscillatory stability of uncertain systems is investigated in this article. For the uncertain systems, whose characteristic polynomial sets belong to the interval polynomial family or diamond polynomial family, sufficient and necessary conditions are given based on the stability and/or oscillation properties of some special extreme point polynomials. A systematic approach exploiting Yang's complete discrmination system is proposed to check the robust oscillatory stability of such uncertain systems. The proposed method is efficient in computation and can be easily implemented.     

On a root distribution criterion for interval polynomials

H. Kokame and T. Mori (1991) and C.B. Soh (1990) derived conditions under which an interval polynomial has a given number of roots in the open left-half plane and the other roots in the open right-half plane. However, the one-shot-test approach using Sylvester's resultant matrices and Bezoutian matrices implies that the implemented conditions are only sufficient (not necessary) for an interval polynomial to have at least one root in the open left-half plane and open right-half plane. Alternative necessary and sufficient conditions, which only require the root locations of four polynomials to check the root distribution of an interval polynomial, are presented     

A class of stability regions for which a Kharitonov-like theoremholds

Families of complex polynomials whose coefficients lie within given intervals are discussed. In particular, the problem of determining if all polynomials in a family have the property that all of their roots lie within a given region is discussed. Towards this end, a notion of a Kharitonov region is defined. Roughly speaking, a Kharitonov region is a region in the complex plane with the following property: given any suitable family of polynomials, in order to determine if all polynomials in the family have all of their roots in the region, it suffices to check only the vertex polynomials of the family. The main result is a sufficient condition for a given region to be a Kharitonov region     

Kharitonov regions: it suffices to check a subset of vertexpolynomials

The authors deal with the D-stability property of interval polynomials. In particular, they show that certain D-domains are Kharitonov regions. That is, the D-stability of interval polynomials is implied by the D-stability of all its vertex polynomials. They then proceed to show that it suffices to check the D-stability of a subset of the vertex polynomials