Open Access is an initiative that aims to make scientific research freely available to all. To date our community has made over 100 million downloads. It’s based on principles of collaboration, unobstructed discovery, and, most importantly, scientific progression. As PhD students, we found it difficult to access the research we needed, so we decided to create a new Open Access publisher that levels the playing field for scientists across the world. How? By making research easy to access, and puts the academic needs of the researchers before the business interests of publishers.
We are a community of more than 103,000 authors and editors from 3,291 institutions spanning 160 countries, including Nobel Prize winners and some of the world’s most-cited researchers. Publishing on IntechOpen allows authors to earn citations and find new collaborators, meaning more people see your work not only from your own field of study, but from other related fields too.
To purchase hard copies of this book, please contact the representative in India:
CBS Publishers & Distributors Pvt. Ltd.
www.cbspd.com
|
customercare@cbspd.com
Institute of Electrical Information Technology, Clausthal University of Technology, Clausthal-Zellerfeld, Germany
Xinyu Shu
Institute of Electrical Information Technology, Clausthal University of Technology, Clausthal-Zellerfeld, Germany
Wiebke Heins
Institute of Electrical Information Technology, Clausthal University of Technology, Clausthal-Zellerfeld, Germany
Christian Bohn
Institute of Electrical Information Technology, Clausthal University of Technology, Clausthal-Zellerfeld, Germany
*Address all correspondence to:
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 G and controller K.
Figure 1.
Generalized plant and controller
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 H2 norm) of the transfer function from w to q 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 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 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 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 [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.
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.
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 ϖ varies in a polytope ζ with M vertices vj∇RN. A point ϖ∇ζ can be written as a convex combination of vertices, i.e. there exists a coordinate vector ϙ=[ϙ1⋮ϙM]T∇RM such that ϖ can be written as
ϖ=∐j=1MϙjvjE3
with
ϙj≤0,∐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 M vertices of the polytope ζ by finding the coordinate vector ϙ 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 H2 techniques for each vertex of the polytope. The controller for a given ϖ∇ζ 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 G that 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 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.
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.
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 nd fixed frequencies is described by
AdBdCd0E6
with
Ad=Ad,1⋮0⋭⋰⋭0⋮Ad,nd,Ad,i=01-1ai,E7
ai=2cos(2ϞfiT),E8
Bd=Bd,1⋭Bd,nd,Bd,i=11,E9
Cd=Cd,1⋮Cd,ndandCd,i=10.E10
A harmonic disturbance can be modeled as the output of an unforced system with system matrix Ad and output matrix Cd 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 Bd in (9).
The frequency in (8) is fixed and denoted by fi. As in Sec. 4 of the previous chapter, the pLPV disturbance model for nd time-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)=A d,0+ϖ1A d,1+⋮+ϖndAd,nd.E12
As in Sec. 2.1, (12) can be written in the form of
where the matrices Av,i are defined in the same way as Ad(pLPV) in (7) and (8), but with ai evaluated for all the vertices of the polytope, with j=1,2,‧,nd. The coordinate vector ϙ 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
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
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 nd fixed frequencies was given by (6-10). If the frequencies of a harmonic disturbance change between minimal values fi, min and 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,1⋮0⋭⋰⋭0⋮Ad,nd,Ad,i=01-1a¯i,E31
Bd,ϖ=Bd,ϖ,1⋮0⋭⋰⋭0⋮Bd,ϖ,nd,Bd,ϖ,i=0pi,E32
Bd,w=Bd,w,1⋭Bd,w,nd,Bd,w,i=11,E33
Cd,ϖ=Cd,ϖ,1⋮0⋭⋰⋭0⋮Cd,ϖ,nd,Cd,ϖ,i=01,E34
Cd,y=Cd,y,1⋮Cd,y,nd,Cd,y,i=10E35
and
ϖ¯k=ϖ¯1,k⋮0⋭⋰⋭0⋮ϖ¯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
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 H∝ techniques 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 H∝ techniques [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.
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 ϙ in
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
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)
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)
A d,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))
References
1.ApkarianP.GahinetP.1995A convex charachterization of gain-scheduled H∞ controllers. IEEE Transactions on Automatic Control 4085364
2.BallesterosP.BohnC.LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequenciesrol. Proceedings of the American Control ConferenceSan Francisco, June 2011. 134045
3.BallesterosP.BohnC.2011bDisturbance rejection through LPV gain-scheduling control with application to active noise cancellation. Proceedings of the IFAC World Congress. Milan, August 2011. 7897902
4.BaliniH. M. N. K.SchererC. W.WitteJ.2011Performance enhancement for AMB systems using unstable H∞ controllers. IEEE Transactions on Control Systems Technology 19147992
5.BohnC.CortabarriaA.HärtelV.KowalczykK.2003LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequenciesngs of the SPIE’s 10th Annual International Symposium on Smart Structures and Materials. San Diego, March 2003. Paper 5049-68504968
6.BohnC.CortabarriaA.HärtelV.KowalczykK.2004LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying FrequenciesControl Engineering Practice12102939
7.DarengosseC.ChevrelP.2000Linear parameter-varying controller design for active power filters. Proceedings of the IFAC Control Systems Design. Bratislava, June 2000. 6570
8.DuH.ShiX.2002LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying FrequenciesAnchorageAmerican Control Conference. Anchorage, May 2002. 466869
9.DuH.ZhangL.ShiX.2003LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying FrequenciesIEE Proceedings on Control Theory and Applications 15013238
10.FeintuchP. L.BershadN. J.LoA. K.1993A frequency-domain model for filtered LMS algorithms- Stability analysis, design, and elimination of the training mode. IEEE Transactions on Signal Processing 41151831
11.GahinetP.ApkarianP.1994A linear matrix inequality approach to H∞ control. International Journal of Robust and Nonlinear Control 442148
12.HeinsW.BallesterosP.BohnC.2011Gain-scheduled state-feedback control for active cancellation of multisine disturbances with time-varying frequencies. Presented at the 10th MARDiH Conference on Active Noise and Vibration Control Methods. Krakow-Wojanow, Poland, June 2011.
13.HeinsW.BallesterosP.BohnC.2012Experimental evaluation of an LPV-gain-scheduled observer for rejecting multisine disturbances with time-varying frequencies. Proceedings of the American Control Conference. Montreal, June 2012. Accepted for publication.
14.KinneyC. E.de CallafonR. A.2006aScheduling control for periodic disturbance attenuation. Proceedings of the American Control Conference. Minneapolis, June 2006. 478893
15.KinneyC. E.de CallafonR. A.2006bAn adaptive internal model-based controller for periodic disturbance rejection. Proceedings of the 14th IFAC Symposium on System Identification. Newcastle, Australia, March 2006. 27378
16.KinneyC. E.de CallafonR. A.2007LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying FrequenciesProceedings of the 46th IEEE Conference on Decision and Control. New Orleans, December 2007. 284449
17.KöroğluH.SchererC. W.2008LPV control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies. Proceedings of the 17th IFAC World Congress. Korea, July 2008. 492833
18.ShuX.BallesterosP.BohnC.2011Active vibration control for harmonic disturbances with time-varying frequencies through LPV gain scheduling. Proceedings of the 23rd Chinese Control and Decision Conference. Mianyang, China, May 2011. 72833
19.WitteJ.BaliniH. M. N. K.SchererC. W.2010LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying FrequenciesProceedings of the IEEE International Conference on Control ApplicationsYokohama, September 2010. 95055
Written By
Pablo Ballesteros, Xinyu Shu, Wiebke Heins and Christian Bohn