An Improved Approach for Stability Analysis of Discrete System
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An Improved Approach for Stability Analysis of Discrete System Vilas H. Gaidhane1 · Yogesh V. Hote2 Received: 29 September 2017 / Revised: 13 June 2018 / Accepted: 3 July 2018 © Brazilian Society for Automatics--SBA 2018
Abstract This paper presents an improved technique to analyze the stability margin of the discrete systems. The presented approach is based on the reduced conservatism of eigenvalues and scaling of Gerschgorin circles. The mathematical proof and sufficient condition with suitable scaling is stated for the conservatism of eigenvalues. Moreover, the presented technique is demonstrated with the numerical examples. Further, the improved results are compared with the existing methods. Keywords Discrete system · Gerschgorin theorem · Conservatism of eigenvalues · Scaling · Stability margin
1 Introduction The mathematical modeling and analysis of a physical system in engineering application result in the complex and nonlinear problem. This also leads to several difficulties in the analysis and synthesis of the system performance. Therefore, an efficient and simple technique for the analysis of the systems has remained a challenge for researchers. One of the most concerns in the performance of the continuous as well as the discrete system is a stability. A system is said to be stable if all the roots of system lie in the left half of the s-plane (Choudhury 1973). Recently, a numerous work has been carried out on stability analysis of delay-independent linear time-varying systems (Li et al. 2016; Gao and Li 2011). Moreover, Li and Gao (2011) presented a new model transformation of discrete-time systems, including time-varying delay for stability calculations. In discrete stable system, all the roots must lie within the unit circle of the z-plane. The system becomes unstable if any pole of the system moves outside the unit circle (Choudhury
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Vilas H. Gaidhane [email protected] Yogesh V. Hote [email protected]
1
Department of Electrical and Electronics Engineering, Birla Institute of Technology and Science, Pilani, Dubai Campus, International Academic City, Dubai, UAE
2
Department of Electrical Engineering, Indian Institute of Technology (IIT), Roorkee, India
2005). It can be represented easily by transforming s-plane into the z-plane by replacing the variable as s α ± jβ, z esτ eατ e± jβτ eατ (cosβτ + jsinβτ )
(1)
In above Eq. (1), α > 0 represent the right half of the splane (unstable system) which is analogous to the circle with radius larger than unity in z-plane. Thus, for a stable system, all the roots of the discrete system must lie within the unity circle (Phillips and Nagle 2007). The stability calculation is of paramount importance in system modeling and analysis and become essential in the design of electronic circuit such as filters, oscillators, feedback amplifiers and feedback control systems (Meng and Chen 2014; Lu 1998; Yang and Xu 2007). In the past, the linear matrix inequality (LMI)-based techniques were presented for global asymptotic stability of discrete time del
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