Open access peer-reviewed chapter

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

By Pablo Ballesteros, Xinyu Shu, Wiebke Heins and Christian Bohn

Submitted: May 10th 2012Reviewed: May 30th 2012Published: September 5th 2012

DOI: 10.5772/50294

Downloaded: 1143

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 Gand controller K.

Figure 1.

Generalized plant and controller

In this setup, uis the control signal and yconsists of all signals that will be provided to the controller. The signal wis the performance input and the signal qis the performance output in the sense that the performance requirements are expressed in terms of the “overall gain” (usually measured by the Hor the H2norm) of the transfer function from wto qin 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 wto q. Loosely speaking, a good controller should make the effect of won 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 wto 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 ϡmax(Gqw(jϧ))or, in the single-input single-output case, the amplitude response |Gqw(jϧ)|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 won qin certain frequency regions. A reduction of the overall effect of won qin 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 Hdesign [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

xk+11-1yk=A(ϖ)B1-2CDxk1-1uk,E1

where the system matrix depends affinely on a parameter vector ϖ, that is

A(ϖ)=A0+ϖ1A1+ϖ2A2++ϖNAN,E2

with constant matrices Ai. The parameter vector ϖvaries in a polytope ζwith M vertices vjRN. A point ϖζcan be written as a convex combination of vertices, i.e. there exists a coordinate vector ϙ=[ϙ1ϙM]TRMsuch that ϖcan be written as

ϖ=j=1MϙjvjE3

with

ϙj0,j=1Mϙj=1.E4

Defining Av,j=A(vj)for j=1,...,M,the system matrix A(ϖ) can be represented as

A(ϖ)=A(ϙ)=ϙ1Av,1+ϙ2Av,2+...+ϙMAv,M.E5

The system matrix of a pLPV system A(ϖ)can be calculated from the Mvertices of the polytope ζby finding the coordinate vector ϙthat 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 Hor H2techniques for each vertex of the polytope. The controller for a given ϖζcan 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 Gthat includes input and output weighting functions and a parametric uncertainty block ϖ¯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 Hnorm can be satisfied. With the second set of LMIs, the controller matrices are calculated from the solution of the first set of LMIs.

Figure 2.

General LPV-LFT system

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.

Figure 3.

LPV-LFT gain-scheduling control structure

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 ndfixed frequencies is described by

AdBdCd0E6

with

Ad=Ad,100Ad,nd,Ad,i=01-1ai,E7
ai=2cos(2ϞfiT),E8
Bd=Bd,1Bd,nd,Bd,i=11,E9
Cd=Cd,1Cd,ndandCd,i=10.E10

A harmonic disturbance can be modeled as the output of an unforced system with system matrix Adand output matrix Cdgiven 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 Bdin (9).

The frequency in (8) is fixed and denoted by fi. As in Sec. 4 of the previous chapter, the pLPV disturbance model for ndtime-varying frequencies fj,k[fmin,j,fmax,j], j=1,2,,nd, is defined as

Ad(pLPV)(ϖ)Bd(pLPV)Cd(pLPV)0E11

with

Ad(pLPV)(ϖ)=Ad,0+ϖ1Ad,1++ϖndAd,nd.E12

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

Ad(pLPV)(ϖ)=Ad(pLPV)(ϙ)=ϙ1Av,1++ϙMAv,M=i=1MϙiAv,i,E13

where the matrices Av,iare defined in the same way as Ad(pLPV)in (7) and (8), but with aievaluated for all the vertices of the polytope, with j=1,2,,nd. The coordinate vector ϙcan 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

Gp=ApBpCpDpE14

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

Wy(pLPV)=AWy(pLPV)BWy(pLPV)CWy(pLPV)DWy(pLPV),E15
Wu(pLPV)=AWu(pLPV)BWu(pLPV)CWu(pLPV)DWu(pLPV)E16

is illustrated in Fig. 4.

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

xk+1qkyk=Ai(ϖ)Bw(pLPV)Bu(pLPV)Cq(pLPV)Dqw(pLPV)Dqu(pLPV)Cy(pLPV)Dyw(pLPV)Dyu(pLPV)xkwkukE17

where

xk=xp,kTxd,kTxWy,kTxWu,kTT,E18
Ai(ϖ)=ApBpCd(pLPV)000Av,i00BWy(pLPV)Cp0AWy(pLPV)0000AWu(pLPV),E19
Bw(pLPV)Bu(pLPV)=0BpBd(pLPV)0000BWu(pLPV),E20
Cq(pLPV)Cy(pLPV)=DWy(pLPV)Cp0CWy(pLPV)0000CWu(pLPV)Cp000E21
and
Dqw(pLPV)Dqu(pLPV)Dyw(pLPV)Dyu(pLPV)=000DWu(pLPV)00.E22

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

Figure 4.

Plant with pLPV disturbance model and weighting functions

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 ndfixed frequencies was given by (6-10). If the frequencies of a harmonic disturbance change between minimal values fi,minand maximal values fi,max, a representation for the variations of the frequencies is given by

ai(fi)=2cos(2ϞfiT)=a¯i+piϖ¯i,k(fi)E23

with

a¯i=cos(2Ϟfi,maxT)+cos(2Ϟfi,minT),E24
pi=cos(2Ϟfi,maxT)-cos(2Ϟfi,minT)E25
and
ϖ¯i,k[-1,1].E26

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

xd,k+1=Adxd,k+Bd,ϖwϖ,k+Bd,wwd,k,E27
qϖ,k=Cd,ϖxd,k,E28
yd,k=Cd,yxd,k,E29
wϖ,k=ϖ¯kqϖ,kE30

with

Ad=Ad,100Ad,nd,Ad,i=01-1a¯i,E31
Bd,ϖ=Bd,ϖ,100Bd,ϖ,nd,Bd,ϖ,i=0pi,E32
Bd,w=Bd,w,1Bd,w,nd,Bd,w,i=11,E33
Cd,ϖ=Cd,ϖ,100Cd,ϖ,nd,Cd,ϖ,i=01,E34
Cd,y=Cd,y,1Cd,y,nd,Cd,y,i=10E35
and
ϖ¯k=ϖ¯1,k00ϖ¯nd,k.E36

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

Figure 5.

Plant with LPV-LFT disturbance model and weighting functions

xk+1qϖ,kqkyk=ABϖBw(LFT)Bu(LFT)CϖDϖϖDϖwDϖuCq(LFT)DqϖDqw(LFT)Dqu(LFT)Cy(LFT)DyϖDyw(LFT)Dyu(LFT)xkwϖ,kwkukE37

with

xk=xp,kTxd,kTxWy,kTxWu,kTT,E38
A=ApBpCd,y000Ad00BWy(LFT)Cp0AWy(LFT)0000AWu(LFT),E39
BϖBw(LFT)Bu(LFT)=00BpBd,ϖBd,w000000BWu(LFT),E40
CϖCq(LFT)Cy(LFT)=0Cd,ϖ00DWy(LFT)Cp0CWy(LFT)0000CWu(LFT)Cp000E41
and
DϖϖDϖwDϖuDqϖDqw(LFT)Dqu(LFT)DyϖDyw(LFT)Dyu(LFT)=00000000DWu(LFT)000.E42

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 Htechniques are obtained.

5.1. 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 Htechniques [Gahinet Apkarian, 1994]. The steps to obtain them are explained here in detail.

First, two outer factors

NX=nullCy(pLPV)Dyw(pLPV)0E43
and
NY=null(Bu(pLPV))T(Dqu(pLPV))T0E44

are defined, where null[]denotes the basis of the null space of a matrix.

Then, the LMIs

NXTAiTX1Ai-X1AiTX1Bw(pLPV)(Cq(pLPV))T(Bw(pLPV))TX1Ai-ϑ+(Bw(pLPV))TX1Bw(pLPV)(Dqu(pLPV))TCq(pLPV)Dqu(pLPV)-ϑINX<0,E45
NYTAiY1AiT-Y1AiY1(Cq(pLPV))TBw(pLPV)Cq(pLPV)Y1AiT-ϑI+Cq(pLPV)Y1(Cq(pLPV))TDqu(pLPV)(Bw(pLPV))T(Dqu(pLPV))T-ϑNY<0,E46
X1IIY10E47

for feasibility and optimality are solved for X1and Y1for every Ai=Ai(ϖ).

With X1and Y1, the matrices

X1-Y1-1=X2TX2,E48
X(pLPV)=X1X2X2TIE49

are calculated.

With

A¯i=Ai000,E50
B¯=Bw(pLPV)0,E51
C¯=Cq(pLPV)0,E52

the matrix

Ϧi=-(X(pLPV))-1A¯iB¯0A¯iT-X(pLPV)0C¯TB¯T0-ϑ(Dqw(pLPV))T0C¯Dqw(pLPV)-ϑI.E53

is calculated. The matrices

P(pLPV)=B¯T00D¯quTE54
and
Q(pLPV)=0C¯D¯yw0E55

are composed with

-1mmB¯=0Bu(pLPV)I0,E56
-1mmC¯=0ICy(pLPV)0,E57
-1mmD¯qu=0Dqu(pLPV),E58
-1mmD¯yw=0Dyw(pLPV).E59

Finally, the basic LMIs

-1mmϦi+(P(pLPV))TχiQ(pLPV)+(Q(pLPV))TχiP(pLPV)<0E60

are solved for χifor every i.

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

-1mmχi=AKiBKiCKiDKi.E61

The implemented controller is interpolated using the coordinate vector ϙin

χ(pLPV)=ρi=1mϙiχi.E62

5.2. Controller synthesis and implementation for LPV-LFT systems

In this section, the algorithm for the calculation of the H-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

NR=null(Bu(LFT))TDϖu(LFT)(Dqu(LFT))T0E63
and
NS=nullCy(LFT)DyϖDyw(LFT)0.E64

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

NRTARAT-RARCϖTAR(C(LFT))qTBϖBwCϖRAT-ϑJ3+CϖRCϖTCϖR(C(LFT))qTDϖϖDϖwCq(LFT)RATCq(LFT)RCϖTCq(LFT)R(C(LFT))qT-ϑIDqϖDqw(LFT)BϖTDϖϖTDqϖT-ϑL30(Bw(LFT))TDϖwT(Dqw(LFT))T0-ϑNR<0,E65
NSTATSA-SATSBϖATSBw(LFT)CϖT(Cq(LFT))TBϖTSA-ϑL3+BϖTSBϖBϖTSBw(LFT)DϖϖTDqϖT(Bw(LFT))TSA(Bw(LFT))TSBϖ(Bw(LFT))TS(Bw(LFT))-ϑDϖwT(Dqw(LFT))TCϖDϖϖDϖw-ϑJ30Cq(LFT)DqϖDqw(LFT)0-ϑINS<0,E66
RIIS0,E67
L3IIJ30.E68

The scalar ϑis an upper bound of the maximum singular value, which is given as a constraint. This set of LMIs is solved for R, S, J3and L3.

The matrices L1and L2are calculated through

L3-J3-1=L2TL1-1L2,E69

and the matrix X(LFT)is computed as

X(LFT)=SINT0IR0MT,E70

with Mand Nsatisfying

MNT=I-RS.E71

Then, the basic LMI

Ϧ+(Q(LFT))T(χ(LFT))TP(LFT)+(P(LFT))Tχ(LFT)Q(LFT)<0,E72

where

Ϧ=-X-1A0B00A0T-X0C0TB0T0-ϑL0D0T0C0D0-ϑJ0,E73
P(LFT)=B̡T00D̡12T,E74
Q(LFT)=0C̡D̡210,E75
A0=A000,B0=0BϖBw(LFT)000,B̡=0Bu(LFT)0I00,E76
C0=00Cϖ0Cq(LFT)0,C̡=0ICy(LFT)000,D0=0000DϖϖDϖw0DqϖDqw(LFT),E77
D̡12=00I0Dϖu00Dqu(LFT)0,D̡21=0000DyϖDyw(LFT)I00,E78
and
L=L1L2L2TL3,L0=L001,J=L-1,J0=J00I,E79

is solved for the controller matrix χ(LFT). In the last step, the state-space matrices of the controller are extracted from

χ(LFT)=AK(LFT)BK(LFT)CK(LFT)DK(LFT).E80

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.

Figure 6.

Open-loop (gray) and closed-loop (black) amplitude frequency responses for fixed disturbance frequencies of 80, 160, 240 and 320 Hz (left) and of 90, 180, 270 and 360 Hz (right)

Figure 7.

Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured sound pressure (right). The control sequence is off/on/off

Figure 8.

Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured sound pressure (right) in open loop (gray) and closed loop (black)

Figure 9.

Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured sound pressure (right) in open loop (gray) and closed loop (black)

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.

Figure 10.

Open-loop (gray) and closed-loop (black) amplitude frequency responses for fixed disturbance frequencies of 80, 120, 160, 200, 240, 280, 320 and 360 Hz (left) and 100, 140, 180, 220, 260, 300, 340 and 380 Hz (right)

Figure 11.

Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured acceleration (right). The control sequence is off/on/off

Figure 12.

Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured acceleration (right) in open loop (gray) and closed loop (black)

Figure 13.

Results for a disturbance with time-varying frequencies. Variation of the frequencies (left) and measured acceleration (right) in open loop (gray) and closed loop (black)

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)

GGeneralized plant.

KController.

u,yControl input, output signal.

w,qPerformance input, performance output.

ϡmaxMaximum singular value.

GqwTransfer path between performance input and performance output.

A(ϖ),B,C,DState-space matrices of a pLPV system.

xk,yk,ukState vector, output and input.

AiConstant matrices of the polytopic representation of A(ϖ).

ϖParameter vector.

ϖiThe i-th element of the parameter vector.

ζParameter polytope.

vjVertices of the polytope.

MNumber of vertices of the polytope.

NNumber of parameters.

ϙCoordinate vector.

ϙjThe j-th element of the coordinate vector.

Av,j,A(vj)System matrix for the j-th vertex.

ϖ¯Parametric uncertainty block.

wϖ,qϖOutput and input of the parameter block for the plant in LFT form.

w̡ϖ,q̡ϖOutput and input of the parameter block for the controller in LFT form.

ndNumber of frequencies of the disturbance.

Ad(2nd×2nd),State-space matrices of the disturbance model for fixed frequencies.

Bd(2nd×1),

Cd(1×2nd)

Ad,i,Bd,i,Cd,iBlock matrices of Ad,Bdand Cd.

aiScalar parameter for the disturbance model.

T,fiSampling time and the i-th frequency.

Ad(ϖ)(2nd×2nd),State-space matrices of the pLPV disturbance model.

Bd(2nd×1),

Cd(1×2nd)

Ad,iConstant matrices of the polytopic representation of Ad(ϖ).

npOrder of the plant.

GpSystem representation of the plant.

Ap(np×np),State-space matrices of the plant.

Bp(np×1),

Cp(1×np),Dp(1×1)

nWyOrder of the weighting function for y.

Wy,WuSystem representations of the weighting functions.

AWy(nWy×nWy),BWy(nWy×1),State-space matrices of the weighting function for y.

CWy(1×nWy),DWy(1×1)

nWuOrder of the weighting function for u.

AWu(nWu×nWu),BWu(nWu×1),State-space matrices of the weighting function for u.

CWu(1×nWu),DWu(1×1)

xp,k,xd,k,State vectors of plant, disturbance andxWy,k,xWu,kweighting functions.

Ai(ϖ),Bw(pLPV),Bu(pLPV),State-space matrices of the pLPV generalized plant.

Cq(pLPV),Dqw(pLPV),Dqu(pLPV)

Cy(pLPV),Dyw(pLPV),Dqw(pLPV)

0Zero matrix.

Ad,Bd,ϖ,Bd,w,Cd,ϖ,Cd,yState-space matrices of the LFT disturbance model.

wd,ydInput and output of the disturbance model.

up,ypInput and output of the plant.

ại,piScalar parameters for the disturbance model

A,Bϖ,Bw(LFT),Bu(LFT),State-space matrices of the LFT generalized plant.

Cϖ,Dϖϖ,Dϖw,Dϖu,

Cq(LFT),Dqϖ,Dqw(LFT),Dqu(LFT),

Cy(LFT),Dyϖ,Dyw(LFT),Dyu(LFT)

NX((n+3)×(n+2)),NY((n+3)×(n+2))Outer factors to build the LMIs.

X1(n×n),Solutions of the first set of LMIs.

Y1(n×n)

IIdentitiy matrix.

n=np+2nd+nWy+nWuOrder of matrices X1and Y1.

Ϧi(4n+3)×(4n+3)Matrix to build the basic LMI.

X(2n×2n),A¯i(2n×2n),Matrices to build matrix Ϧi.

B¯(2n×1),C¯(2×2n)

P((n+1)×(4n+3)),Q((n+1)×(4n+3))Matrices to build the basic LMI.

B¯(2n×(n+1)),C¯((n+1)×2n),Matrices to obtain P(pLPV)and Q(pLPV).

D¯qu(2×(n+1)),D¯yw((n+1)×1)

χi((n+1)×(n+1))Solution of the basic LMI for the i-th vertex.

AKi(n×n),BKi(n×1),State-space matrices of the controller for theCKi(1×n),DKi(1×1)i-th vertex.

NR((n+2nd+3)×(n+2nd+2)),Outer factors to build the LMIs.

NS((n+2nd+3)×(n+2nd+2))

R(n×n),S(n×n),Solutions of the first set of LMIs.

J3(nd×nd),L3(nd×nd)

ϑUpper bound of the maximum singular value.

M(n×n),N(n×n)Matrices calculated from Rand S.

L1(nd×nd),L2(nd×nd)Matrices to build L.

Ϧ((4n+4nd+3)×(4n+4nd+3))Matrix to build the basic LMI.

X(2n×2n),A0(2n×2n),Matrices needed to build Ϧ.

B0(2n×(2nd+1)),C0((2nd+2)×2n)

D0((2nd+2)×(2nd+1)),J0((2nd+2)×(2nd+2)),

L0((2nd+1)×(2nd+1)),J(2nd×2nd),

L(2nd×2nd)

P((n+nd+1)×(4n+4nd+3)),Matrices to build the basic LMI.

Q((n+nd+1)×(4n+4nd+3))

B̡(2n×(n+nd+1)),C̡((n+nd+1)×2n),Matrices to obtain P(LFT)and Q(LFT).

D̡12((2nd+2)×(n+nd+1)),

D̡21((n+nd+1)×(2nd+1))

χ((n+nd+1)×(n+nd+1))Controller matrix.

AK(n×n),BK(n×(nd+1)),State-space matrices of the controller.

CK((nd+1)×n),DK((nd+1)×(nd+1))

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Pablo Ballesteros, Xinyu Shu, Wiebke Heins and Christian Bohn (September 5th 2012). LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies, Advances on Analysis and Control of Vibrations - Theory and Applications, Mauricio Zapateiro de la Hoz and Francesc Pozo, IntechOpen, DOI: 10.5772/50294. Available from:

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