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Published in S.P. Bhattacharyya, L.H. Keel, of UNCERTAIN DYNAMIC SYSTEMS, 2020
This key theorem unlocked the door leading to the development of a large number of interesting results in the area of real parametric uncertainty. However Kharitonov’s theorem itself is of somewhat limited applicability in control problems. To explain this consider the control system shown below in Figure 1. Let F(s):=F1(s)F2(s)G(s):=N(s)D(s)
Robust Control
Published in P.N. Paraskevopoulos, Modern Control Engineering, 2017
According to Kharitonov’s theorem, in order to guarantee robust stability of the given interval polynomial, it is sufficient to test the stability of the above four Kharitonov polynomials. This can be accomplished by using an algebraic stability criterion, e.g., the Routh criterion. Applying the Routh criterion in the four Kharitonov’s polynomials, we obtain the following four Routh tables: Polynomial p1(s): Polynomial p2(s): Polynomial p3(s): Polynomial p4(s):
K
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
Kharitonov theorem the most popular necessary and sufficient condition of robust stability for characteristic interval polynomials. It states that robust stability of uncertain linear time-invariant system with characteristic interval polynomial p(s, a) is stable if and only if the following four polynomials have all roots endowed with negative parts:
An Integrated Electromechanical Model of the Fixed-Speed Induction Generator for Turbine-Grid Interactions Analysis
Published in Electric Power Components and Systems, 2018
Da Xie, Wangping Wu, Xitian Wang, Chenghong Gu, Yanchi Zhang, Furong Li
For the interval system described above, it is not easy to determine the stability simply through calculating the eigenvalue or applying Routh Hurwitz stability criterion. The Kharitonov theorem is effective in the stability analysis of the interval system, and it is convenient and quick to determine the stability of the interval polynomial.
Model Order Reduction of LTI Interval Systems Using Differentiation Method Based on Kharitonov’s Theorem
Published in IETE Journal of Research, 2022
Sudharsana Rao Potturu, Rajendra Prasad
The aforementioned reduction methods are developed for fixed coefficient transfer functions or state space models. It is a fact that designing a controller based on the fixed coefficient transfer function or state space model is often unrealistic because the practical system parameters vary within the certain interval. Kharitonov [10] proposed a pioneer technique to deal with interval systems. The Kharitonov’s theorem is useful for finding the stability of interval systems by separating the interval polynomial into its four Kharitonov fixed coefficient polynomials; if these four Kharitonov polynomials meet the requirement of the stability criterion, then the interval system is said to be stable. Bandyopadhyay et al. [11] extended the fixed parameter reduction methods to deal with interval systems. In the study by Bandyopadhyay et al. [11], the reduction method is found by using the Routh-Pade approximation technique to deal with interval systems. Later, Hwang and Yang [12] said that the Routh approximation method may loss its stability preservation property due to irreversibility of interval arithmetic operation. Later, Dolgin and Zeheb [13] and Yang [14] proposed the modified Routh-Pade approximation method to avoid limitations of [12]. Later, Bandyopadhyay et al. [15] proposed the Routh approximation method to deal with higher order interval systems (HOIS). This technique preserves the initial time moments of the OHOS in its ROM by constructing tables. To overcome the limitation mentioned in Ref. [12], Bandyopadhyay et al. [15] presented a method by constructing γ-δ table formation of the Routh approximation method using Kharitonov polynomials. This technique provides stable ROIMs. Later, Sastry et al. [16] presented the reduced interval method by using the modified Routh approximation method. This method requires only table formation to obtain stable ROIM. But still, this method has some limitations, i.e. this method always provides some steady-state errors. After that, many interval reduction methods were developed for finding stable and better approximation of ROIMs. Recently, Siva Kumar et al. [17] proposed a reduction method to deal with HOIS, and the reduced-order numerator and denominators are achieved by using the Routh approximation method and Kharitonov’s polynomials. This technique preserves the dominant impulse response properties of the original system in its ROM. Later, Siva and Gulshad [18] presented order simplification of HOIS based on the stability equation method and integral square error (ISE). More recently, Vijay Anand et al. [19] proposed the reduction method using the soft computing technique, and the ROIMs are obtained based on particle swarm optimization (PSO) of ISE and impulse response error (IRE).