Author Topic: Robust Impulse Eliminating Internal Model Control of Singular Systems  (Read 2450 times)

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Robust Impulse Eliminating Internal Model Control of Singular Systems: A Robust Control Approach
Author : M.M. Share Pasand, H.D. Taghirad
International Journal of Scientific & Engineering Research, IJSER - Volume 2, Issue 4, April-2011
ISSN 2229-5518
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Abstract— The problem of model based internal control of singular systems is investigated and the limitations of directly extending the control schemes for normal systems to singular ones were analyzed in this paper. A robust approach is proposed in order to establish the control scheme for singular systems, and moreover, to present a framework for robust control of singular systems in presence of modeling uncertainties. The theory is developed through a number of theorems, and several simulation examples are included and their physical inter-pretations are given to verify the proof of concept.

Index Terms— singular systems, Impulsive behavior, internal model control, Model based control, robust control, tracking problem, impulse elimination.

Singular systems represent a general framework for linear systems [1]. A singular model is an appropriate model for describing large scale interconnected systems, constrained robots and other differential algebraic systems with linear algebraic constraints [2]. Also singular models can be utilized to model a system when the dependent variable is displacement rather than time [3]. Since the first time they were introduced [4], several efforts have been made to control singular systems [5-9]. As the singular systems were firstly introduced in the state space form representation [4], they were usually studied in time do-main. In [5] the problem of finite mode pole placement is studied, while simultaneous impulse elimination and robust stabilization problem is considered in [6], robust Eigen-structure assignment of finite modes is studied in [7]. In [8] strict impulse elimination is studied using time derivative feedback of the states and [9] investigated the output feedback control using a compensator. In fact most of the existing methods are extensions of the control schemes for standard systems [5],[6],[10]. In the singular system control context the control objectives are more complicated due to the existing obstacles such as algebraic loop phenomenon, impulsive behavior [11] , and regularity of the closed loop [8,9] . Unlike the time domain methods, there are very few works on the frequency domain control of singular systems. In the frequency domain, the tracking problem, robust control problem and impulse elimination can be treated more conveniently. Specifically the so called Internal Model Control (IMC) method provides a very interesting framework for analyzing the algebraic loop, regularity of the closed loop and impulse elimination problems of singular systems. Furthermore, most of existing methods in robust control of singular systems are limited to study a special form of uncertainty. They as-sumed matrix E to be exactly known [6, 7 and 10]. This assumption is more restrictive than it appears, because it limits the system to be impulsive while some uncertainties may exist which lead to a strictly proper system for a singular model. Therefore this paper suggests a new concept for robust control. While previous works on robust control focuses on robust stability and robust performance, as it comes to descriptor systems, robust properness of closed loop should be studied. The internal model framework for controlling singular systems provides a more logical uncertainty model and release the restrictive assumptions made in the existing state space methods for robust control of singular systems. Also it provides offset free tracking capability of the closed loop as well as being able to well treat delayed systems. The main obstacle which arises in the internal model control of singular systems is that the internal model cannot be implemeted easily, because it is generally improper. Even in computer aided control systems it is not easy to simulate a singular system, since the discrete model needs future input data to determine the system state vector at the present time [1]. This problem results in an inevitable mismatch between the plant and the parallel model used in IMC.

Notice that, defining the disk shaped multiplicative uncertainty leads to an unbounded uncertainty profile which is not suitable in robust design of singular systems. This paper provides solution to the latter problem by introducing the singular internal model filter in series with the conventional internal model filter. The aforementioned filter eases the design procedure, bounds the uncertainty profile. Also it makes the closed loop strictly proper and eliminates impulsive modes by smoothing the control action as much as needed. Another role of the introduced filter is to make it possible to design robust controller in the conventional context.  The paper is organized as fol-lows. In the next section backgrounds are discussed and the obstacles in control of singular systems are presented, and some major limitations of the direct extension of IMC are explained. In the third section the proposed method is studied and the filter design procedure is illustrated. In the fourth section several examples and simulations are given to examine the algorithm both in terms of robustness properties and closed loop performance. Finally, the concluding remarks are given in last section. 

Definitions and Singular Systems Characteristics
As Descriptor models are a straight extension of standard state space models [1], control problem for these systems has a wider range of objectives. A control system for a standard plant is designed such that the closed loop is stable and has a predefined performance and acceptable robustness properties. A singular control system, on the other hand, should be designed such that it is impulse free, regular and doesn’t include any algebraic loops in addition to the aforementioned properties. These control objectives combined with the standard objectives make the control of singular systems more challenging. Robust control of singular systems is the most challenging issue because it requires robustness not only in the stability and per-formance but also in regularity and properness.   State space robust control schemes require robust observers in order to work properly and do not guarantee strict properness of the closed loop also they usually result in more complicated derivation algorithms. The main advantage to use internal model control scheme in here, is that the IMC provides an effective tool in frequency domain without introducing complicated methods in evaluation of closed loop performance and stability. There-fore, IMC can be regarded as a proper alternative for existing state space methods. Also IMC provides a simple framework for algebraic loop and properness analysis of singular control systems which is much simpler than that in state space methods or other frequency domain schemes.

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