Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 380609, 9 pages

http://dx.doi.org/10.1155/2015/380609

## LPV Observer-Based Strategy for Rejection of Periodic Disturbances with Time-Varying Frequency

^{1}Departamento de Ingeniería Eléctrica y Electrónica, Universidad Nacional de Colombia, Bogotá, Colombia^{2}Institut d’Organització i Control de Sistemes Industrials, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain

Received 21 December 2014; Revised 29 April 2015; Accepted 4 May 2015

Academic Editor: Peter Dabnichki

Copyright © 2015 G. A. Ramos 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

Rejection of periodic disturbances is an important issue in control theory and engineering applications. Conventional strategies like repetitive control and resonant control can deal adequately with this problem but they fail when the frequency of the disturbance varies with time. This paper proposes a Linear Parameter Varying (LPV) resonant observer-based control for periodic signal rejection which is able to deal with the changes in frequency of the disturbance signal. The observer includes, in an embedded way, an internal model of the disturbance that is based on its harmonic decomposition. In this way, the frequency of the disturbance signal constitutes a parameter that can be adjusted according to the variations of the signal. The resulting disturbance estimation is then used by a control law that cancels the periodic disturbance term while controlling a specified tracking task. The proposed scheme lets the control designer address the disturbance estimation and tracking problems separately. Experimental results, on a mechatronic test bed, show that the proposed LPV resonant observer-based control successfully rejects periodic disturbances under varying frequency conditions.

#### 1. Introduction

Rejection of periodic disturbances has been a subject of great interest in control theory and engineering applications. Periodic disturbances are present in many applications like robotics [1, 2], power inverters [3], power active filters [4], and wind turbines [5], among others.

The most common control strategies used for rejection of periodic disturbances are repetitive control (RC) [6, 7] and resonant control [8]. RC constitutes a very efficient methodology in control applications that require tracking and/or rejection of periodic signals (see [9]). It is based on the internal model principle (IMP), thus requiring the inclusion of a periodic signal model in the control loop. However, one of the main drawbacks of RC appears when the frequency of the signals is uncertain or varies with time. In these cases, the traditional RC suffers from a significant performance loss [10]. In order to solve this problem, different strategies have been reported in the literature: a variable structure RC has been proposed in [11], where the frequency of the internal model is adapted to follow the exogenous signal changes; a High Order Repetitive Controller (HORC) is presented in [12] which is robust against frequency variations and [13, 14] propose a varying sampling controller for which the frequency discrete representation remains invariant. Similarly, based on the IMP, the resonant control [15] is dedicated to the tracking/rejection of selected harmonics present in a given signal.

Alternatively, this problem can be treated using an observer-based control scheme. Under this approach, the observer is in charge of obtaining an estimate of the disturbance that is then used by the control law to reject the real disturbance. A review of disturbance observers design can be found in [16].

In this paper an LPV observer-based strategy is proposed aimed at rejecting periodic disturbances under variable frequency conditions. The proposed observer is formulated such that it includes an internal model of a periodic signal. This internal model is built from the decomposition of the periodic signal on its harmonic components. Thus, the observer is able to estimate the states of the plant and each of the selected frequency components. To overcome the frequency variation problem, the frequency of the signal, which is structurally embedded in the observer, is changed according to the exogenous variations. Furthermore, the observer gains are reaccommodated according to the different operating points caused by the varying frequency. The tuning of the system is a combined methodology that uses the pole placement technique for reference tracking and optimal Kalman-Bucy approach for the configuration of the resonant observer. Finally, stability analysis can be formulated in an LPV systems framework. In this way, since the closed-loop system is affine with respect to the varying frequency parameter a simple condition to establish the stability can be stated.

The experimental validation of the proposal is carried out in a mechatronic platform. This is based on a DC motor exposed to a rotating periodic disturbance torque. Experimental results show that the proposed approach exhibits very high performance, reducing effectively the effect of the disturbances over the angular speed. It is also shown that the performance is preserved under variations of the disturbance frequency. Unlike classic control schemes for handling periodic signals, resonant and repetitive control, the proposed architecture offers the advantage of independently designing the disturbance rejection and tracking reference signals.

This paper is organized as follows. Section 2 describes the architecture of the proposed controller, Section 3 describes the platform and the experimental results, and finally conclusions and future work are proposed in Section 4.

#### 2. Structure of Resonant Observer-Based Control

This section describes the controller architecture of the proposed LPV observer-based strategy. The controller is composed by a disturbance observer, a state feedback control, and the reference internal model. The disturbance estimation is used to compensate the disturbance signal using the Active Disturbance Rejection Control (ADRC) philosophy. A complete stability analysis and some tuning criteria are provided.

##### 2.1. Plant Model

Consider the following state-space linear plant model:where is the state vector, is the control action, is the disturbance signal, and is the system output. Similarly, is the state transition matrix, is the input vector, and is the output vector. The system defined by is assumed to be a minimal representation being both controllable and observable.

##### 2.2. Disturbance Model

In this work we are dealing with disturbances that can be written aswithwhere is the frequency of each component and and are assumed unknown. In this work it is assumed that and are constant or piecewise constant. Under this hypothesis the frequency content of is locally concentrated around .

Although the values of might be arbitrarily assigned, a particular case with great relevance is when , where . This case will be assumed from now in this work. Consequently is defined as and . In case is constant, is a -periodic signal and are the different harmonic frequency components.

Each sinusoidal term can be thought as generated by the following system, with appropriate initial conditions:where is the fundamental frequency which is assumed measurable (or known). In state-space this can be written as with , , and .

Therefore, the disturbance signal, , admits the following state-space representation:where and

##### 2.3. Augmented System

We can extend the plant model to include the disturbance signal using ; thus we obtain the following augmented model:where This system, with appropriate initial conditions, is equivalent to (1) subject to (2) and (3). It is important to notice that (8) has no disturbance input and it is an observable system but it is not completely controllable system. The controllable subsystem corresponds to the plant while the noncontrollable subsystem corresponds to the disturbance model. Note that the disturbance is an exogenous signal and consequently it cannot be modified through the control action.

##### 2.4. Resonant Observer

In order to observe the state of (8) a Luenberger observer is proposed:where is the augmented system state estimation and are the observer gains.

The estimation error is defined as follows: . Therefore, using (8) and (10) the estimation error evolution can be written as

###### 2.4.1. Stability Analysis

The stability of system (11) depends on the matrix:In order to prove the stability of system (11), a Lyapunov function can be formulated:Consequently in order to guarantee closed-loop stability the following inequality must be fulfilled:so it is necessary that Defining where and are two symmetric matrices, the stability condition can be stated as Using LPV theory [17, 18], this condition can be checked in terms of an LMI which must be evaluated in four points defined by and .

###### 2.4.2. Tuning Procedure

In Section 2.4.1 observer stability conditions have been established. These conditions do not uniquely determine the observer gain, . An approach which provides a simple and convenient framework is optimal estimation.

Although a theory for optimal estimation for time-varying systems exists [19], it implies difficult implementation and it is not easy to apply in practice. In Linear Time Invariant framework, it is well-known that the Kalman-Bucy filter constitutes the optimal Luenberger observer where the estimation error covariance is minimized [19]. Thus, the optimal observer gain is defined bywhere is the unique positive-semidefinite solution of the algebraic Riccati equation: where and are the spectral density matrices of the measurement and process noise, respectively. In this work an optimal tuning, based on the Kalman filter, is proposed. The observer gain is proposed to be linear varying aswhere and are obtained by forcing and . Thus, and are obtained from (18) for and , respectively.

Proposed approach guarantees stability at the extreme values of . Stability at intermediate points must be checked through conditions established in Section 2.4.1.

##### 2.5. Closed-Loop System

In this section, the closed-loop stability of the system obtained using the observer estimation to close the loop is analyzed. Based on the state estimation, a state feedback control law is used; taking this into account the complete system takes the following form:where corresponds to the state feedback gain.

In order to simplify the analysis, these equations are written using the estimation error:obtainingAs it can be seen the well-known separation principle can also be applied to this LPV system. The dynamics of depends onwhich can be decomposed on the periodic signal generator dynamics and the closed-loop plant dynamics. Note that the closed-loop plant dynamics is described by an LTI system dynamics and only depends on , which can be stabilized with a suitable selection of the constant gain .

##### 2.6. Closed-Loop System including Reference Internal Model

The controller which has been introduced in the previous section will be useful if we are interested in stabilizing the origin, but in most cases we are interested in tracking a reference. In order to guarantee this, the reference internal model will be introduced in the controller (see Figure 3): with this new control law the complete closed-loop system is defined byThe stability of this system can be analyzed in terms of the observer dynamics, **,** and the matrix,Note that this matrix is time invariant so it can be analyzed using regular methods, and and can be tuned using LTI methods.

#### 3. Case Study

The system used for the experimental validation of the proposed control strategy is a mechatronic system affected by nonlinear periodic disturbances. It consists of a Pulse Width Modulation (PWM) electronic amplifier, a DC motor, a Pulses Per Revolution (PPR) incremental encoder, and a magnetic setup that generates a periodic torque under constant angular speed, . This disturbance torque applied to the plant is a nonlinear function of the angular position, . Hence, the control objective is to regulate the angular speed of the motor to a desired value despite the periodic torque disturbance. A detailed scheme of the mechatronic system, control loop, and the experimental setup can be observed in Figure 1. The reader is encouraged to read [20] for more detailed explanation of this system (roto-magnet plant).