## 1. Introduction

In this chapter, the same control problem as in the previous chapter is considered, which is the rejection of harmonic disturbances with time-varying frequencies for linear time-invariant (LTI) plants. In the previous chapter, gain-scheduled observer-based state-feedback controllers for this control problem were presented. In the present chapter, two methods for the design of general gain-scheduled output-feedback controllers are presented. As in the previous chapter, the control design is based on a description of the system in linear parameter-varying (LPV) form. One of the design methods presented is based on the polytopic linear parameter-varying (pLPV) system description (which has also been used in the previous chapter) and the other method is based on the description of an LPV system in linear fractional transformation (LPV-LFT) form. The basic idea is to use the well-established norm-optimal control framework based on the generalized plant setup shown in Fig. 1 with the generalized plant

In this setup,

This control design setup is used in this chapter for the rejection of harmonic disturbances with time-varying frequencies. The control design problem is based on a generalized plant obtained through the introduction of a disturbance model that describes the harmonic disturbances and the addition of output weighting functions. Descriptions of the disturbance model in pLPV and in LPV-LFT form are used and lead to generalized plant descriptions that are also in pLPV or LPV-LFT form. Corresponding design methods are then employed to obtain controllers. For a plant in pLPV form, standard

LPV approaches for the rejection of harmonic disturbances have been used by [Darengosse Chevrel, 2000], [Du Shi, 2002, Du et al., 2003], [Bohn et al., 2003, Bohn et al., 2004], [Kinney de Callafon, 2006a, Kinney de Callafon, 2006b, Kinney de Callafon, 2007], [Koro Scherer, 2008], [Witte et al., 2010], [Balini et al., 2011], [Heins et al., 2011, Heins et al., 2012], [Ballesteros Bohn, 2011a, Ballesteros Bohn, 2011b] and [Shu et al., 2011]. [Darengosse Chevrel, 2000], [Du Shi, 2002, Du et al., 2003], [Witte et al., 2010], [Balini et al., 2011] suggested continuous-time LPV approaches. These approaches are tested for a single sinusoidal disturbance by [Darengosse Chevrel, 2000], [Du et al., 2003], [Witte et al., 2010] and [Balini et al., 2011]. Methods based on observer-based state-feedback controllers are presented by [Bohn et al., 2003, Bohn et al., 2004], [Kinney de Callafon, 2006a, Kinney de Callafon, 2006b, Kinney de Callafon, 2007] and [Heins et al., 2011, Heins et al., 2012]. In the approach of [Bohn et al., 2003, Bohn et al., 2004], the observer gain is selected from a set of pre-computed gains by switching. In the other approaches of [Kinney de Callafon, 2007], [Heins et al., 2012] and in the previous chapter, the observer gain is calculated by interpolation. In the other approach presented in the previous chapter, which is also used by [Kinney de Callafon, 2006a, Kinney de Callafon, 2006b] and [Heins et al., 2011], the state-feedback gain is scheduled using interpolation. A general output feedback LPV approach for the rejection of harmonic disturbances is suggested and applied in real time by [Ballesteros Bohn, 2011a, Ballesteros Bohn, 2011b] and [Shu et al., 2011].

The existing LPV approaches can be classified by the control design technique used to obtain the controller. Approaches based on pLPV control design are used by [Heins et al., 2011, Heins et al., 2012], [Kinney de Callafon, 2006a], [Du Shi, 2002] and [Du et al., 2003]. An approach based on LPV-LFT control design is used by [Ballesteros Bohn, 2011a, Ballesteros Bohn, 2011b] and [Shu et al., 2011].

For a practical application, the resulting controller has to be implemented in discrete time. In applications of ANC/AVC, the plant model is often obtained through system identification. This usually gives a discrete-time plant model. If a continuous-time controller is computed, the controller has to be discretized. Since the controller is time varying, this discretization would have to be carried out at each sampling instant. An exact discretization involves the calculation of a matrix exponential, which is computationally too expensive and leads to a distortion of the frequency scale. Usually, this can be tolerated, but not for the suppression of harmonic disturbances. In this context, it is not surprising that the continuous-time design methods of [Darengosse Chevrel, 2000], [Du et al., 2003], [Kinney de Callafon, 2006a] and [Koro Scherer, 2008] are tested only in simulation studies with a very simple system as a plant and a single frequency in the disturbance signal. Exceptions are [Witte et al., 2010] and [Balini et al., 2011], who designed continuous-time controllers which then are approximately discretized. However, [Witte et al., 2010] use a very high sampling frequency of 40 kHz to reject a harmonic disturbance with a frequency up to 48 Hz (in fact, the authors state that they chose “the smallest [sampling time] available by the hardware”) and [Balini et al., 2011] use a maximal sampling frequency of 50 kHz. The control design methods presented in this chapter are realized in discrete time.

The remainder of this chapter is organized as follows. In Sec. 2, pLPV systems and LPV-LFT systems are introduced and the control design for such systems is described. In Sec. 3, it is described how the control problem considered here can be transformed to a generalized plant setup. The required pLPV disturbance model for the harmonic disturbance is introduced in Sec. 3.1 and in Sec. 3.2, it is described how the generalized plant in pLPV form is obtained by combining the disturbance model, the plant and the weighting functions. In Sec. 4, the transformation of the control problem to a generalized plant in LPV-LFT form is treated in essentially the same way, by formulating an LPV-LFT disturbance model (Sec. 4.1) and building a generalized plant in LPV-LFT form (Sec. 4.2). The controller synthesis for both descriptions is described in Sec. 5. Experimental results are presented in Sec. 6 and the chapter finishes with a discussion and some conclusions in Sec. 7.

## 2. Control design setup

In this section, pLPV systems and LPV-LFT systems are introduced and the control design for such systems is described in Sec. 2.1 and 2.2, respectively.

### 2.1. Control design for pLPV systems

A pLPV system is of the form

where the system matrix depends affinely on a parameter vector

with constant matrices * M*vertices

with

Defining

The system matrix of a pLPV system

Once a representation of a system is obtained in pLPV form, it is possible to find a controller using

### 2.2. Control design for LPV-LFT systems

An LPV system in LFT form is shown in Fig. 2. It consists of a generalized plant

As a result of applying this control design method, the gain-scheduling control structure of Fig. 3 is obtained. The time-varying plant parameters are directly used as the gain-scheduling parameters of the controller. This control design method guarantees stability through the small gain theorem. It is often conservative, since the parameter ranges covered are usually larger than the ones that may occur in the real system.

## 3. Generalized plant in pLPV form

As stated in the previous section, to calculate the controller using the pLPV control design method, the generalized plant in pLPV form is needed. In this section, the steps to obtain the generalized plant in pLPV form are discussed. The disturbance model and a representation of the disturbance model in pLPV form are obtained in Sec. 3.1. In Sec. 3.2, the generalized plant is built by combining the plant, the disturbance model in pLPV form and the weighting functions.

### 3.1. Disturbance model

A general model for a harmonic disturbance with

with

A harmonic disturbance can be modeled as the output of an unforced system with system matrix

The frequency in (8) is fixed and denoted by

with

As in Sec. 2.1, (12) can be written in the form of

where the matrices

### 3.2. Generalized plant

A state-space representation of the plant is given by

and it is assumed that the disturbance is acting on the input of the plant.

The block diagram of the generalized plant with the disturbance, the plant and the weighting functions

is illustrated in Fig. 4.

For every vertex of the polytopic system, the generalized plant can be described by

where

Once the generalized plant is obtained, the controller can be calculated using the algorithms in the following section.

## 4. Generalized plant in LPV-LFT form

The same steps as in the previous section are carried out, but in this section the generalized plant in LPV-LFT form is obtained such that the control design method of [Apkarian Gahinet, 1995] can be used. The model of the harmonic disturbance and the generalized plant in LFT form are obtained in Sec. 4.1 and 4.2, respectively. The generalized plant is the result of combining plant, harmonic disturbance and weighting functions.

### 4.1. Disturbance model

The state-space representation of a harmonic disturbance for

with

An LPV-LFT model of the disturbance can be written as

with

### 4.2. Generalized plant

The generalized plant is the result of combining the plant, the harmonic disturbance and the weighting functions and it is shown in Fig. 5. The weighting functions are defined the same way as in (15) and (16). A representation of the generalized plant in LFT form is given by

with

## 5. Controller synthesis and implementation for LPV systems

In this section, algorithms for the calculation of the pLPV and LPV-LFT gain-scheduling controllers are explained in detail. Suboptimal controllers using

### 5.1. Controller synthesis and implementation for pLPV systems

With the generalized plant in pLPV form, an

First, two outer factors

are defined, where

Then, the LMIs

for feasibility and optimality are solved for

With

are calculated.

With

the matrix

is calculated. The matrices

are composed with

Finally, the basic LMIs

are solved for * i*.

The state-spaces matrices of the controllers for each vertex can be extracted from

The implemented controller is interpolated using the coordinate vector

### 5.2. Controller synthesis and implementation for LPV-LFT systems

In this section, the algorithm for the calculation of the

From the state-space representation of the generalized plant the outer factors for the LMIs that have to be solved in the design can be calculated as

With the outer factors, a first set of LMIs corresponding to the feasibility and optimality condition is given as

The scalar

The matrices

and the matrix

with

Then, the basic LMI

where

is solved for the controller matrix

## 6. Experimental results

The gain-scheduled output-feedback controllers obtained through the design procedures presented in this chapter are validated with experimental results. Both controllers have been tested on the ANC and AVC systems. Results are presented for the pLPV gain-scheduled controller on the ANC system in Sec. 9 and for the LPV-LFT controller on the AVC test bed in Sec. 13. Identical hardware setup and sampling frequency as in the previous chapter are used.

### 6.1. Experimental results for the pLPV gain-scheduled controller

The pLPV gain-scheduled controller is validated with experimental results on the ANC headset. The controller is designed to reject a disturbance signal which contains four harmonically related sine signals with fundamental frequency between 80 and 90 Hz. The controller obtained is of 21st order.

Amplitude frequency responses and pressure measured when the fundamental frequency rises suddenly from 80 to 90 Hz are shown in Figs. 6 and 7. An excellent disturbance rejection is achieved even for unrealistically fast variations of the disturbance frequencies. In Fig. 8, results for time-varying frequencies are shown. The performance for fast variations of the fundamental frequency is further studied in Fig. 9. As in the previous chapter, with fast changes of the fundamental frequency the disturbance attenuation performance decreases but the system remains stable.

### 6.2. Experimental results for the LFT gain-scheduled controller

The AVC test bed is used to test the LFT gain-scheduled controller experimentally. The controller is designed to reject a disturbance with eight harmonic components which are selected to be uniformly distributed from 80 to 380 Hz in intervals of 20 Hz. The resulting controller is of 27th order.

Amplitude frequency responses are shown in Fig. 10 and results for an experiment where the frequencies change drastically as a step function in Fig. 11. Results from experiments with time-varying frequencies are shown in Figs. 12 and 13. Excellent disturbance rejection is achieved.

## 7. Discussion and conclusion

Two discrete-time control design methods have been presented in this chapter for the rejection of time-varying frequencies. The output-feedback controllers are obtained through pLPV and LPV-LFT gain-scheduling techniques. The controllers obtained are validated experimentally on an ANC and AVC system. The experimental results show an excellent disturbance rejection even for the case of eight frequency components of the disturbance.

The control design guarantees stability even for arbitrarily fast changes of the disturbance frequencies. This is an advantage over heuristic interpolation methods or adaptive filtering, for which none or only “approximate stability results” are available [Feintuch Bershad, 1993].

To the best of the authors’ knowledge, industrial applications of LPV controllers are rather limited. The results of this chapter show that the implementation of even high-order LPV controllers can be quite straightforward.

### Nomenclature

### Acronyms

ANCActive noise control.

AVCActive vibration control.

LFTLinear fractional transformation.

LMILinear matrix inequality.

LPVLinear parameter varying.

LTILinear time invariant.

pLPVPolytopic linear parameter varying.

### Variables (in order of appearance)

* i*-th element of the parameter vector.

* j*-th element of the coordinate vector.

* j*-th vertex.

* i*-th frequency.

* y*.

* y*.

* u*.

* u*.

* i*-th vertex.

* i*-th vertex.