Research Article: Linear systems with unstructured multiplicative uncertainty: Modeling and robust stability analysis

Date Published: July 12, 2017

Publisher: Public Library of Science

Author(s): Radek Matušů, Bilal Şenol, Celaleddin Yeroğlu, Xiaosong Hu.


This article deals with continuous-time Linear Time-Invariant (LTI) Single-Input Single-Output (SISO) systems affected by unstructured multiplicative uncertainty. More specifically, its aim is to present an approach to the construction of uncertain models based on the appropriate selection of a nominal system and a weight function and to apply the fundamentals of robust stability investigation for considered sort of systems. The initial theoretical parts are followed by three extensive illustrative examples in which the first order time-delay, second order and third order plants with parametric uncertainty are modeled as systems with unstructured multiplicative uncertainty and subsequently, the robust stability of selected feedback loops containing constructed models and chosen controllers is analyzed and obtained results are discussed.

Partial Text

The robustness of control systems represents an attractive research topic whose necessity is boosted by everyday control engineering practice [1–5]. The principal problem is that the mathematical model of a controlled plant almost never matches real plant behavior exactly due to the understandable effort to build and use a simple enough LTI model in which the potential complex features are simplified or neglected and furthermore, since the real physical parameters of a system can vary owing to a number of reasons. The popular and effective approach as to how to systematically study the influence of uncertainty and overcome this discrepancy in control tasks is provided by robust control. The problem of designing a robust controller typically means ensuring the robust stability and robust performance [6–8].

The first and fundamental step in robust control is to respect the difference between the true behavior of a control loop and its mathematical description by means of exploiting the uncertain model. Roughly speaking, one fixed “nominal” model is replaced by a whole family of models represented by some neighborhood of the nominal one. This neighborhood can be quantified essentially by means of two main approaches.

Under the assumption of multiplicative uncertainty, the closed-loop system is robustly stable if and only if [14, 15]:
‖WM(s)T0(s)‖∞<1(5) where T0(s) represents a complementary sensitivity function defined by: T0(s)=L0(s)1+L0(s)(6) and where L0(s) is the open-loop frequency transfer function: L0(s)=C(s)G0(s)(7) From the fundamental inequality (5), the following relation can be derived: |WM(jω)L0(jω)1+L0(jω)|<1∀ω⇒|WM(jω)L0(jω)|<|L0(jω)−(−1)|∀ω(8) This expression means that the closed-loop system is robustly stable if and only if the envelope of Nyquist diagrams with a radius of |WM(jω)L0(jω)| and center L0(jω) does not include the critical point [-1, 0j]. The depiction of this condition is shown in Fig 2. This key section presents examples of the possible construction of multiplicative uncertainty model for the commonly used forms of controlled plants, i.e. for the first order time-delay plant, second order plant and third order plant. Moreover, robust stability is analyzed for a closed loop with the second or third order plant model and selected feedback controllers. This article focused on the modeling and robust stability analysis of continuous-time LTI SISO systems with unstructured multiplicative uncertainty. The examples presented herein have shown the techniques for the construction of multiplicative uncertainty models from systems with parametric uncertainty via the selection of suitable nominal models and weight functions. Moreover, the robust stability of the feedback control loops that contain multiplicative uncertainty plants was analyzed and their conservatism in comparison with the usage of “original” parametric uncertainty plants was discussed.   Source:


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