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 and controller .
In this setup, is the control signal and consists of all signals that will be provided to the controller. The signal is the performance input and the signal is the performance output in the sense that the performance requirements are expressed in terms of the “overall gain” (usually measured by the or the norm) of the transfer function from to in closed 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 to . Loosely speaking, a good controller should make the effect of on “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 to the performance output , that is, by modeling the disturbance to be rejected as a weighted version of the performance input. This forces the maximum singular value or, in the single-input single-output case, the amplitude response 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 on in certain frequency regions. A reduction of the overall effect of on 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 design [Gahinet Apkarian, 1994] 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, 1995] 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 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 , that is
with constant matrices . The parameter vector varies in a polytope with M vertices . A point can be written as a convex combination of vertices, i.e. there exists a coordinate vector such that can be written as
Defining for the system matrix () can be represented as
Once a representation of a system is obtained in pLPV form, it is possible to find a controller using or techniques for each vertex of the polytope. The controller for a given 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.
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 that includes input and output weighting functions and a parametric uncertainty block 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, 1995]. 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 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 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 fixed frequencies is described by
A harmonic disturbance can be modeled as the output of an unforced system with system matrix and output matrix 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 in (9).
The frequency in (8) is fixed and denoted by . As in Sec. 4 of the previous chapter, the pLPV disturbance model for time-varying frequencies , , is defined as
As in Sec. 2.1, (12) can be written in the form of
where the matrices are defined in the same way as in (7) and (8), but with evaluated for all the vertices of the polytope, with . The coordinate vector can be calculated using the method described in Sec. 4.4 of the previous chapter.
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
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 fixed frequencies was given by (6-10). If the frequencies of a harmonic disturbance change between minimal values and maximal values , a representation for the variations of the frequencies is given by
An LPV-LFT model of the disturbance can be written as
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
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 techniques are obtained.
5.1. Controller synthesis and implementation for pLPV systems
With the generalized plant in pLPV form, an -suboptimal controller for each vertex of the polytope can be calculated using standard techniques [Gahinet Apkarian, 1994]. The steps to obtain them are explained here in detail.
First, two outer factors
are defined, where denotes the basis of the null space of a matrix.
Then, the LMIs
for feasibility and optimality are solved for and for every .
With and , the matrices
is calculated. The matrices
are composed with
Finally, the basic LMIs
are solved for 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 in
5.2. Controller synthesis and implementation for LPV-LFT systems
In this section, the algorithm for the calculation of the -suboptimal gain-scheduling controller from [Apkarian Gahinet, 1995] 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
With the outer factors, a first set of LMIs corresponding to the feasibility and optimality condition is given as
The scalar is an upper bound of the maximum singular value, which is given as a constraint. This set of LMIs is solved for , , and .
The matrices and are calculated through
and the matrix is computed as
with and satisfying
Then, the basic LMI
is solved for the controller matrix . In the last step, the state-space matrices of the controller are extracted from
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.
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)
Control input, output signal.
Performance input, performance output.
Maximum singular value.
Transfer path between performance input and performance output.
State-space matrices of a pLPV system.
State vector, output and input.
Constant matrices of the polytopic representation of .
The i-th element of the parameter vector.
Vertices of the polytope.
Number of vertices of the polytope.
Number of parameters.
The j-th element of the coordinate vector.
System matrix for the j-th vertex.
Parametric uncertainty block.
Output and input of the parameter block for the plant in LFT form.
Output and input of the parameter block for the controller in LFT form.
Number of frequencies of the disturbance.
State-space matrices of the disturbance model for fixed frequencies.
Block matrices of and .
Scalar parameter for the disturbance model.
Sampling time and the i-th frequency.
State-space matrices of the pLPV disturbance model.
Constant matrices of the polytopic representation of .
Order of the plant.
System representation of the plant.
State-space matrices of the plant.
Order of the weighting function for y.
System representations of the weighting functions.
State-space matrices of the weighting function for y.
Order of the weighting function for u.
State-space matrices of the weighting function for u.
State vectors of plant, disturbance andweighting functions.
State-space matrices of the pLPV generalized plant.
State-space matrices of the LFT disturbance model.
Input and output of the disturbance model.
Input and output of the plant.
Scalar parameters for the disturbance model
State-space matrices of the LFT generalized plant.
Outer factors to build the LMIs.
Solutions of the first set of LMIs.
Order of matrices and .
Matrix to build the basic LMI.
Matrices to build matrix .
Matrices to build the basic LMI.
Matrices to obtain and .
Solution of the basic LMI for the i-th vertex.
State-space matrices of the controller for thei-th vertex.
Outer factors to build the LMIs.
Solutions of the first set of LMIs.
Upper bound of the maximum singular value.
Matrices calculated from and .
Matrices to build .
Matrix to build the basic LMI.
Matrices needed to build .
Matrices to build the basic LMI.
Matrices to obtain and .
State-space matrices of the controller.