LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies

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 plantG and controllerK.


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 G and controller K.
In this setup, u is the control signal and y consists of all signals that will be provided to the controller. The signal w is the performance input and the signal q is the performance output in the sense that the performance requirements are expressed in terms of the "overall gain" (usually measured by the H ∞ or the H 2 norm) of the transfer function from w to q in closed

Chapter 3
loop. In this setup, the aim of the controller design is to satisfy performance requirements expressed as upper bounds on the norm (in case of suboptimal control) or minimize the norm (in optimal control) of the transfer function from w to q. Loosely speaking, a good controller should make the effect of w on q "small" (for suboptimal control) or "as small as possible" (for optimal control). The performance outputs usually consist of weighted versions of the controlled signal, the control error and the control effort. This is achieved by augmenting the original plant with output weighting functions. Good rejection of specific disturbances can be achieved in this framework by using a disturbance model as a weighting function in the transfer path from the performance input w to the performance output q, that is, by modeling the disturbance to be rejected as a weighted version of the performance input. This forces the maximum singular value σ max (G qw (jω)) or, in the single-input single-output case, the amplitude response |G qw (jω)| of the open-loop transfer function to have a very high gain in the frequency regions specified by the disturbance model, or, loosely speaking, enlarges the effect of w on q in certain frequency regions. A reduction of the overall effect of w on q in closed loop will then be mostly achieved by reducing the effect in regions where it is large in open loop. From classical control arguments, it is intuitive that this requires a high loop gain in these frequency regions which in turn usually requires a high controller gain. A high loop gain will give a small sensitivity and in turn a good disturbance rejection (in specified frequency regions).
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 H ∞ design [11] is used to compute a set of controllers. The gain scheduling is then achieved by interpolation between these controllers. For a plant in LPV-LFT form, the design method of Apkarian & Gahinet [1] is used that directly yields a gain-scheduled controller also in LPV-LFT form.
LPV approaches for the rejection of harmonic disturbances have been used by Darengosse [14,15,16] and Heins et al. [12,13]. In the approach of Bohn et al. [5,6], the observer gain is selected from a set of pre-computed gains by switching. In the other approaches of Kinney & de Callafon [16], Heins et al. [13] 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 [14,15] and Heins et al. [12], 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 [2,3] and Shu et al. [18].
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. [12,13], Kinney & de Callafon [14], Du & Shi [8] and Du et al. [9]. An approach based on LPV-LFT control design is used by Ballesteros & Bohn [2,3] and Shu et al. [18].
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. 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

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.

Control design for pLPV systems
A pLPV system is of the form where the system matrix depends affinely on a parameter vector θ, that is Defining The system matrix of a pLPV system A(θ) can be calculated from the M vertices of the polytope Θ by finding the coordinate vector λ that fulfills the conditions of (3) and (4).
Once a representation of a system is obtained in pLPV form, it is possible to find a controller using H ∞ or H 2 techniques for each vertex of the polytope. The controller for a given θ ∈ Θ can be calculated through controllers for the vertex systems. The closed-loop stability is guaranteed even for arbitrarily fast changes of the scheduling parameters if a parameter-independent Lyapunov function is used (for the whole polytope) in the control design. This approach, however, is conservative because fast variations of the scheduling parameters are considered, which might not occur in a practical application. Parameter-dependent Lyapunov functions can be used to include bounds on the rate of change of the parameters, but are not considered here.

Control design for LPV-LFT systems
An LPV system in LFT form is shown in Fig. 2. It consists of a generalized plant G that includes input and output weighting functions and a parametric uncertainty block θ that has been "pulled out" of the system. For this general system, a gain-scheduling controller can be calculated following the method presented in Apkarian & Gahinet [1]. In this method, two sets of linear matrix inequalities (LMIs) are solved. The first set of LMIs determines the feasibility of the problem which means that a bound on the control system performance in the sense of the H ∞ norm can be satisfied. With the second set of LMIs, the controller matrices are calculated from the solution of the first set of LMIs.
As a result of applying this control design method, the gain-scheduling control structure of 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.

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.

Disturbance model
A general model for a harmonic disturbance with n d fixed frequencies is described by A harmonic disturbance can be modeled as the output of an unforced system with system matrix A d and output matrix C d given above in (7) and (10). An input matrix is not required. However, in the generalized plant setup, a performance input is required and the disturbance model acts as an input weighting function on the performance input. This is why the disturbance model above has been given with a nonzero input matrix B d in (9).
The frequency in (8) is fixed and denoted by f i . As in Sec. 4 of the previous chapter, the pLPV disturbance model for n d time-varying frequencies As in Sec. 2.1, (12) can be written in the form of where the matrices A v, i are defined in the same way as A (pLPV) d in (7) and (8), but with a i evaluated for all the vertices of the polytope, with j = 1, 2, . . . , n d . The coordinate vector λ can be calculated using the method described in Sec. 4.4 of the previous chapter.

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

Disturbance model
The state-space representation of a harmonic disturbance for n d fixed frequencies was given by (6)(7)(8)(9)(10). If the frequencies of a harmonic disturbance change between minimal values f i, min and maximal values f i, max , a representation for the variations of the frequencies is given by with and An LPV-LFT model of the disturbance can be written as with and 72 Advances on Analysis and Control of Vibrations -Theory and Applications

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

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 H ∞ techniques are obtained.

Controller synthesis and implementation for pLPV systems
With the generalized plant in pLPV form, an H ∞ -suboptimal controller for each vertex of the polytope can be calculated using standard H ∞ techniques [11]. The steps to obtain them are explained here in detail.

First, two outer factors
and are defined, where null[·] denotes the basis of the null space of a matrix.
Then, the LMIs for feasibility and optimality are solved for X 1 and Y 1 for every A i = A i (θ).
With X 1 and Y 1 , the matrices are calculated. With the matrix is calculated. The matrices and are composed with (59) Finally, the basic LMIs are solved for Ω i for every i.
The state-spaces matrices of the controllers for each vertex can be extracted from The implemented controller is interpolated using the coordinate vector λ in

Controller synthesis and implementation for LPV-LFT systems
In this section, the algorithm for the calculation of the H ∞ -suboptimal gain-scheduling controller from [1] is explained in detail.
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 and With the outer factors, a first set of LMIs corresponding to the feasibility and optimality condition is given as The scalar γ is an upper bound of the maximum singular value, which is given as a constraint. This set of LMIs is solved for R, S, J 3 and L 3 .
The matrices L 1 and L 2 are calculated through and the matrix X (LFT) is computed as with M and N satisfying Then, the basic LMI and is solved for the controller matrix Ω (LFT) . In the last step, the state-space matrices of the controller are extracted from

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. 6.1 and for the LPV-LFT controller on the AVC test bed in Sec. 6.2. Identical hardware setup and sampling frequency as in the previous chapter are used.

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.

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.

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 [10].
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.

G qw
Transfer path between performance input and performance output.
A(θ), B, C, D State-space matrices of a pLPV system.
x k , y k , u k State vector, output and input.
System matrix for the j-th vertex.
w θ , q θ Output and input of the parameter block for the plant in LFT form.
w θ ,q θ Output and input of the parameter block for the controller in LFT form.
n d Number of frequencies of the disturbance.
State-space matrices of the disturbance model for fixed frequencies.  n = n p + 2n d + n W y + n W u Order of matrices X 1 and Y 1 .