Robust Tracking of Linear Systems for Matched Uncertainty using LQR Approach

ISSN: 2347-6087 © 2014 | Published by The Standard International Journals (The SIJ) 1 Abstract—Robust control is derived for general linear uncertain systems with scalar control. In this paper we have shown way to solve a robust control problem of a uncertain system (matched uncertainty) by transforming it into an optimal control problem which becomes Linear Quadratic Regulator problem (LQR) for linear systems and can be solved by using algebraic Riccati equation. We assume that the control objective is not only to drive the states to zero but also to track a non-zero reference signal which is to be assumed as polynomial function of time. We also emphasized on finding the largest value of uncertain parameter for which given system is on the verge of instability, which has not been mentioned in the previous work. The control is derived by using LQR method and has state negative feedback form. The performance of proposed controller is examined by simulating linear system model in MATLAB environment.


INTRODUCTION
HE early methods of Bode and others in control theory were fairly robust. The state space models invented in 1960s and 1920s were sometimes found to lack of robustness, prompting to improve them which were the start of the robust theory. Robust control is a limb of control hypothesis that expressly manages uncertainty in its approach to controller design. Robust methods aim to achieve robust performance or stability in the presence of bounded modelling errors. In real life, the systems to be controlled have many uncertainties that can deviate the system performance from the desired. So designing effective robust controllers for such systems becomes significant. In the past, there have been many works on robust control design of uncertain systems, but the work focused on the robust regulator problem.
In Barmish (1985), necessary and sufficient conditions for quadratic stability of unforced system and then controlled system are determined for the state and the output feedback case. In that linear time varying systems subjected to norm bounded, structured uncertainties considered. Barmish et al., (1983) emphasized on the problem of designing a stabilizing controller for a class of uncertain dynamical systems. The stability and control of uncertain systems using co-positive linear Lyapunov function along with dissipativity theory is done by Corentin Briat. The Integral Linear Constraints (ILC"s) are used for the robust analysis. For the class of robust stabilization problems the construction of universal controller is discussed by Battilotti (1997). General theorems on the construction of the universal controllers were discussed, which requires that a certain non-linear inequality is solvable or equivalently that a robust control Lyapunov function does exist. A type of observer called the Proportional-Integral (PI) observer proposed for the design of the robust controller by Beale & Shafani (1989). In Amato et al., (1996), linear time-varying systems subject to norm bounded, one block, structured uncertainties presented. This observer contrasts from the ordinary one by an integration path which gives extra degree of freedom [Beale & Shafani, 1989]. Whereas augmented systems approach is used for the measurement based robust tracking problem in Benson & Schmitendorf (1997). The robust optimal control of linear uncertain system with 2 -bounded uncertainties is discussed in Chen (1993). The robust tracking control problem for a class of partial information MIMO LTV systems using multilinear model approximation approach is employed because only a finite number of discrete characteristic time points For decentralized robust control of a class of large-scale uncertain dynamical system using deterministic design approach proposed by Chen (1988). The two types of robust control algorithm are proposed, namely local control and the global control. The local control utilizes the local state of each subsystem as feedback while global control uses local state as well as states of neighboring subsystems as the feedback. In Dawson et al., (1990), the stability of a PD controller is examined for the trajectory tracking problem of robot manipulator. A Lyapunov"s second method is used to derive a uniform boundness for the PD controller. The likelihood of deciding a linear control law when the full state can't be measured and observers are designed to estimate the state is recognized in Faryar & Schmitendorf (1991). The proposed control law and the observer are designed using two Riccati equations. Focus was on systems which satisfied matched uncertainty condition. A class of non-linear uncertain systems is considered for the stabilization using polynomial robust control in which the bound of uncertainty does not need to be known [Han & Chen, 1992]. Leitmann (1979) discusses a class of linear dynamical systems in which the system and system input matrices, and additionally the input, are uncertain. Uncertainties are assumed to be measurable functions of time who may range in given compact sets.
In this paper, we presented an optimal control approach for robust tracking of linear systems and given largest value of uncertainty parameter for which system under consideration remains stable. The presented work is an extension of results presented in Haihua Tan et al., (2009) for matched uncertainty case. The simulation results clearly show improved robustness for given linear system for different values of uncertainty parameter. The proposed approach is very simple to implement in MATLAB context. The paper is organized as follows: section 2, discusses the foundation of robust control design, methodology for robust tracking of linear systems for matched uncertainty. In section 3, case study of simple linear system with uncertainty parameter is reviewed, and finally some brief conclusions are given in section 4.

II. ROBUST TRACKING USING STATE FEEDBACK
We first present results obtained in Haihua Tan et al., (2009), that will form the theoretical foundation of our approach.

Background
For linear systems, the approach considers the following system, are state variables, ∈ are control inputs, and ∈ are uncertain parameters. If the uncertainty of ( ) is in the range of , that is, the uncertainty in ( ) can be written as for some ( 0 ), where 0 ∈ is nominal value of , then we say that the matching condition is satisfied.

Methodology
An optical approach is proposed that translates robust control problem to an optimal control problem and if the system under consideration is linear system then optimal control problem will become Linear Quadratic problem. This LQR problem then can be solved by solving an algebraic Ricatti equation.

Robust Tracking using State Feedback for Matched Uncertainty
The linear system is described by Our objective is to design a feedback control law "u' under which the output asymptotically tracks the reference signal with no error, that is, → as → ∞. In determining the control law, we assume that the following information is available for control: Where is an upper bond on uncertainty ; that is, for all ∈ , ≤ 2 . The upper bound on the uncertainty = ( ) 0 0 … 0 is denoted by . We further assume that 0 , is controllable, otherwise, the solution, may not exist.

Riccati Equation:
[Donald E. Kirk, 2004] Let the system equation given by And if the performance measure is given by The HJB equation is given by 0 Where * is minimized Hamiltonian function given by, (1) and (6) we get, Plug in equation (12) in (11) to obtain * * = + + − 1 2 * − 1 2 * The one of the solution of the form is given by, This is known as Ricatti equation. To solve the LQR problem, we first solve the algebraic Ricatti equation for . Then the solution to the LQR problem is given by So the control law is given by (17), We now show that the system, (16) is asymptotically stable for all p ϵ P. Now, let us consider the following Lyapunov function candidate: = Clearly, In particular,

Example 1 [Haihua Tan et al., 2009]:
Let us consider the following second-order system, where ∈ 0,10 is the uncertainty. We would like to design a state feedback control law to ensure that the output y is tracking a reference signal = 1 for all ∈ 0,10 . We first construct the augmented system as follows  To see the performance of the closed-loop system, we conduct simulations, whose results are shown in Figure2-5, for = 1 and = 11 respectively. Here X1 and X2 are the states of system. From the simulation results of figure: 2-3, we see that the system response is not fast. This is because the poles of the controlled system are not far away from the imaginary axis, that is, the stability margin is not large. Also from figure for p=11 it can be seen that response is under damped response. This implies that as p increases system moves towards instability. Thus we can conclude that system is less robust. If we want a large stability margin and fast response, we may want to place the poles of the controlled system at least distance away from the imaginary axis. In other words, we want to solve the following problem [Haihua Tan et al., 2009].

Robust Tracking Problem with Guaranteed Stability Margin
For an arbitrary positive real number , find a feedback control law = such that the controlled system has all its poles on the left of − and → for all possible ∈ . The solution to the above problem is given as follows.
Assume that the matching condition is satisfied and there exist a nominal value 0 ∈ such that 0 , is controllable. Then the solution to the robust tracking problem with guaranteed stability margin exists and is given by = = − , where is the solution to the algebraic Riccati equation