Mathematical Problems in Engineering

Volume 2018, Article ID 4095473, 12 pages

https://doi.org/10.1155/2018/4095473

## Robust Linear Output Regulation Using Extended State Observer

^{1}Faculty of Mechanical Engineering, University of Tabriz, 29 Bahman, Tabriz, Iran^{2}Institute of Computational Engineering, Ruhr‐Universität Bochum, Universitätsstr. 150, D‐44801 Bochum, Germany

Correspondence should be addressed to Atta Oveisi; ed.bur@isievo.atta

Received 16 February 2018; Revised 31 May 2018; Accepted 5 June 2018; Published 12 August 2018

Academic Editor: Francesco Riganti-Fulginei

Copyright © 2018 Mehran Hosseini-Pishrobat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a disturbance rejection-based solution to the problem of robust output regulation. The mismatch between the underlying plant and its nominal mathematical model is formulated by two disturbance classes. The first class is assumed to be generated by an autonomous linear system while for the second class no specific dynamical structure is considered. Accordingly, the robustness of the closed-loop system against the first disturbance class is achieved by following the internal model principle. On the other hand, in the framework of disturbance rejection control, an extended state observer (ESO) is designed to approximate and compensate for the second class, i.e., unstructured disturbances. As a result, the proposed output regulation method can deal with a vast range of uncertainties. Finally, the stability of the closed-loop system based on the proposed compound controller is carried out via Lyapunov and center manifold analyses, and some results on the robust output regulation are drawn. A representative simulation example is also presented to show the effectiveness of the control method.

#### 1. Introduction

The theory of output regulation in its essence deals with the problem of tracking/disturbance rejection of a class of signals generated by autonomous dynamic systems while guaranteeing the closed-loop stability. For linear systems, this theory is well-established, and the solution of the associated control problem is obtained by the* internal model principle *[1, 2]. The principle is also instrumental in addressing the problem of robust output regulation. On this basis, robust design methods have been proposed in the literature for linear systems with uncertain parameters [3–6]. These methods consider the effect of external disturbances by an exogenous signal which belongs to the solution space of a particular differential equation. According to such a differential equation, the steady-state behavior of the underlying plant should be considered to examine the solvability of the output regulation problem. After determining the steady-state forms of the plant trajectories that satisfy the tracking/disturbance rejection requirements, the internal model-based design can be pursued [7]. By this method, a stabilizing controller that incorporates a suitable internal model of the class of the desired signals offers guaranteed asymptotic rejection/tracking of any signal from that class.

In this paper, we depart from the conventional formulation of the output regulation problem by considering the effect of disturbances that cannot be modeled by a process signal of a differential equation. As a matter of fact, in many practical applications, such modeling assumption on the disturbances is restrictive. In many systems, the disturbances are not only caused by exogenous factors, but also from endogenous factors such as structural variations, parametric uncertainty, and unmodeled dynamics. The latter factors usually appear as state-dependent disturbances in the system’s mathematical model. For example, in many structural vibration control problems, the main source of the disturbance is the unmodeled higher order modes. Even though the dominant higher order modes can be obtained via suitable system identification methods, a comprehensive dynamic model for the disturbances is not feasible [8]. As another example, in many mechanical and electromechanical systems, parametric uncertainty, friction, and nonlinear effects bring about state-dependent disturbances that are not guaranteed to belong to the solution space of any particular dynamic system [9]. On this basis, our goal in this paper is to extend the applicability of the output regulation method to such examples. To this end, we recast the output regulation problem into the active disturbance rejection framework [10, 11]. Active disturbance rejection control (ADRC) is a robust control method for systems with large uncertainty and disturbances [10, 12–14]. The key idea of the ADRC is to treat a robust control problem as an estimation-rejection problem. More specifically, in the first step, all sorts of discrepancies between the physical system and its nominal mathematical model including parameter variations, unknown nonlinearities, and external disturbances/noises are lumped into a* total disturbance* term [14]. Next, an estimator referred to as extended state observer (ESO) is applied to estimate the total disturbance and then reject it in a closed-loop framework. The active disturbance rejection control offers several promising advantages over the conventional control methods including the following: ADRC is an active robustification method that in comparison to classical robust techniques (based on worst-case analysis) minimizes the conservatism in the design; ADRC can handle large uncertainty/disturbances from both internal and external sources; ADRC does not require involved, accurate system modeling; this feature enables a disturbance-oriented design instead of a model-oriented one [12–14]. More details about the ADRC and its various engineering applications can be found in [10, 14] and the recent book [13].

In light of the above discussion, active disturbance rejection provides a suitable framework to generalize the applicability of the output regulation to the systems whose disturbances do not necessarily satisfy a differential equation. The main contribution of this paper is reported in terms of combining the merits of output regulation and disturbance rejection control techniques to achieve a better tracking performance in the presence of a general class of disturbance/uncertainty. The class of considered systems is fairly general, which includes linear systems subjected to mismatched disturbances as well as state-dependent uncertainties. Following the methodology of ADRC, we lump all disturbances/uncertainties of the underlying system into a total disturbance. The total disturbance term is decomposed into two parts. The first part can be classically modeled as the solution of a differential equation while the second part has no specific dynamical structure. The modeled component of the total disturbance is handled via standard method of the output regulation theory, which entails embedding an internal model into the closed-loop system. This approach will also guarantee output tracking of the reference signals that can be generated by the internal model. The unmodeled component of the total disturbance is handled by the method of active disturbance rejection. To this end, the system dynamics is extended by appending an integrator channel that transmits the uncertainty of the unmodeled total disturbance to its time derivative. Meanwhile, an ESO is designed that continuously monitors input-output signals of the extended system to estimate the unmodeled disturbance component. By combining this estimated value with the nominal output regulation control, the mismatch disturbance rejection is achieved. We note that since the ADRC uses an integral action to model the dynamical behavior of the disturbance, the method can be construed as an approximate output regulation technique [16]. This point of view is also explained in [17], where active disturbance rejection and output regulation are combined together for an improved robust vibration control of a MEMS gyroscope. As a novelty in the disturbance rejection aspect, a new ESO design method is proposed based on the convex programming. To this end, the ESO error dynamics is presented as a Lur'e system and its stability condition is phrased in terms of linear matrix inequalities (LMIs). Additionally, the proposed ESO applies nonlinear gain approach which enhances its immunity to the measurement noises.

We note that various methods have been reported in the literature for the general problem of disturbance rejection/attenuation of control systems; for example, LMIs-based disturbance attenuation for nonlinear systems with input delays [18], operator-based disturbance attenuation/rejection [19], disturbance observer-based disturbance attenuation for stochastic systems [20], disturbance rejection based on the equivalent-input-disturbance approach [21], ESO-based disturbance rejection controller for fully actuated Euler-Lagrange systems [22], and sliding mode observer/controller hybrid control structure [15]. With respect to these methods, the main advantages of our method lie in combining the strong points of the output regulation and the disturbance rejection techniques. The output regulation component enables a guaranteed asymptotic rejection/tracking of the disturbances/reference signals with known dynamics. Moreover, for such disturbances/reference signals, no assumptions on the energy boundedness nor control matching are required. For disturbances without known dynamic behavior, the ESO-based disturbance rejection engages to fortify the output regulation. In other words, our method attempts to get the most out of the available information about the disturbances with the primary aim of disturbance rejection rather than disturbance attenuation.

The remainder of this paper is organized as follows. In Section 2, the control problem is formulated in terms of the output regulation theory. In Section 3, a nominal regulator is designed to handle the tracking problem as well as the rejection of the disturbance components with known dynamic characteristics. In Section 4, to compensate for the disturbances without known characteristics, an ESO is developed. A convex optimization-based design method for the ESO is proposed in Section 5. The stability of the closed-loop system is investigated in Section 6. In Section 7, a simulation example based on the plotter system is elaborated. Finally, the paper is concluded by Section 8.

#### 2. Formulation of the Control Problem

Consider the following square state space model: where is the state, is the control input, denotes the measured performance output, and is a sufficiently smooth (differentiable) nonlinear function representing the total disturbance. , , and are real matrices of appropriate dimensions satisfying the following assumption.

*Assumption 1. *The pair is controllable and is observable.

The control goal is to asymptotically track a given reference signal, i.e., , while boundedness of the all other signals is guaranteed in the presence of the total disturbance . The reference signal is considered to be generated by the solution of a fixed linear autonomous dynamic system of the formwhere and and are real-valued matrices with proper dimensions. The following assumption is taken to guarantee the persistence and boundedness of the reference signal [23].

*Assumption 2. *All eigenvalues of the matrix have zero real parts with multiplicity one in the minimal polynomial.

By setting , the system in (1) can be rewritten asThe systems in (1) and (2) are trajectory equivalent in the sense that, under the same initial values, their state trajectories coincide with each other [24, 25]. Therefore, we carry out the controller design and analysis on the basis of (2).

The conventional output regulation theory is based on the assumption that the perturbing disturbances belong to the solution space of the dynamic system given in (2). Here, we generalize this assumption by including disturbances which do not necessarily comply with this particular dynamics.

*Assumption 3. *The total disturbance signal satisfieswhere is a given matrix with the pair being observable. The signal is bounded and sufficiently differentiable such that is bounded as well.

*Remark 4. *The rationale behind Assumption 3 can be explained as follows. In many engineering systems, only the dominant frequencies of the disturbances can be obtained via time and/or frequency domain identification methods (see, for example, [8]). In such systems, one can embed the known dominant frequencies into the system (2) and model the corresponding disturbance approximation error into the term, . As another example, consider the tracking control problems where the dominant dynamics of the underlying systems are linear. Assuming that the norm of the state-dependent disturbances is smaller than the convergence rate of the linear part, it is conceivable that, for a successful tracking, the frequency spectrum of the disturbances will contain the modes of the reference signals. Accordingly, (4) forms a suitable basis for modeling of the disturbances. An example of such modeling can be found in [9].

Let us define the tracking performance of the control system asNote that here and after the dependence on time variable () is dropped (unless necessary) in the formulation for the sake of readability. The following composite system is proposed regarding the underlying output regulation control problemIn order to examine the solvability of the output regulation problem, first, we consider a steady-state condition in which the output tracking is achieved. In such a steady-state condition, the trajectories of the underlying system should be well-defined to ensure that the solution of the output regulation problem exists. Denoting , , , and , as the steady-state vectors of the state, control, unstructured, and the structured disturbances, respectively, it is easy to obtainSetting the solutions as and , for some and , we obtain the following Sylvester-type matrix equations from (7)

Theorem 5. *For any given matrices and , there exist unique matrices and satisfying (8) if and only if the following assumption holds.*

*Assumption 6. *For all ( represents the vector of eigenvalues of the matrix (.));

*Proof. *See [26] for the proof.

From a geometrical point of view, solutions of the matrix equations (8) —known as the* regulator equations*—corresponds to a zero-error controlled invariant manifold defined by for the composite system (6). In other terms, for an initial condition of the form , under the control input , the trajectory will evolve on for all and the tracking error (5) will be identically equal to zero. From this standpoint, solving the formulated output regulation problem amounts to rendering the manifold globally attractive by a suitable control function [7]. Since exact dynamic characterization of the total disturbance signal is not available, we pursue* practical* stabilization of the manifold for the composite system (6). More specifically, trajectories of the closed-loop system all should converge to a small compact set around by approximately recovering the solutions of the regulator equations. To this end, we propose a disturbance rejection-based, two-degrees-of-freedom control structure. The control system is composed of a nominal output regulator equipped with an ESO for disturbance rejection. Accordingly, the main design stages of this control system are as follows.

*Stage 1. *The nominal controller solves the output regulation problem for the given dynamical system in (5) with the difference that the state equation is simplified toNote that the corresponding control input is designed to drive the nominal closed-loop trajectories toward by recovering the nominal part of the steady-state control signal defined by . This corresponds to the feedforward control input required for tracking as well as compensating for .

*Stage 2. *On the other hand, the disturbance rejection loop utilizes an ESO that continuously monitors the input-output signals of the plant to provide an estimate of the disturbance, denoted by . The obtained estimated value is then integrated with the nominal control input to cancel out the disturbance signal. Accordingly, the overall control input is in the formThe configuration of the proposed control system is schematically illustrated by the block diagram in Figure 1.