Open access peer-reviewed chapter

LPV Gain-Scheduled Observer-Based State Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies

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

Submitted: December 6th 2011Reviewed: May 30th 2012Published: September 5th 2012

DOI: 10.5772/50293

Downloaded: 1682

1. Introduction

The design of controllers for the rejection of multisine disturbances with time-varying frequencies is considered. The frequencies are assumed to be known. Such a control problem frequently arises in active noise and vibration control (ANC/AVC) applications where the disturbances are caused by imbalances due to rotating or oscillating masses or periodically fluctuating excitations, for example the torque of a combustion engine, and the rotational speed is measured. Application examples are automobiles and aircrafts.

For the rejection of disturbances with time-varying frequencies, time-varying controllers that are automatically adjusted to the disturbance frequencies are usually used. Although time-invariant controllers might be sufficient in some applications [Koro Scherer, 2008a], time-varying controllers usually result in a much better performance, particularly if the disturbance frequencies vary over fairly wide ranges. Such a controller can be constructed in several ways (see Sec. 2.1). Two observer-based state-feedback controllers are presented in this chapter (see Sec. 3). General output-feedback controllers are treated in the next chapter. The approaches presented in this chapter use state augmentation in order to achieve disturbance rejection. One consists of a time-invariant plant observer and a time-varying state-feedback gain for the state-augmented plant, where the state augmenation is based on a time-varying error filter, as proposed by [Kinney de Callafon, 2006a]. The other controller approach is based on the disturbance observer of [Bohn et al., 2003], where the plant is augmented with a time-varying disturbance model. A time-varying observer for the overall system and a time-invariant state-feedback gain are used to track and reject the disturbance.

The remainder of this chapter is organized as follows. In Sec. 2, existing approaches to the problem are classified and some general control design considerations are discussed. The state-augmented observer-based state-feedback approaches are described in Sec. 3. In Sec. 4, the calculation of stabilizing state-feedback gains for time-varying systems in polytopic linear parameter-varying (pLPV) form is discussed. The application of this method for the rejection of harmonic disturbances is treated in Sec. 5. Real-time results are presented in Sec. 6. A discussion and conclusions are given in Sec. 7.

2. Controller design: Overview, stability and implementation aspects

In this section, an overview of control approaches for the rejection of harmonic disturbances with time-varying frequencies is given (Sec. 2.1) and stability and implementation aspects are discussed (Secs. 2.2 and 2.3, respectively).

2.1. Overview and classification of control approaches

A common approach in ANC/AVC is adaptive filtering, where the filter weights are usually updated with the FxLMS algorithm [Kuo Morgan, 1996]. In most cases, disturbance feedforward is used, although it is possible to use an adaptive filter in feedback control with a technique called “secondary path neutralization” [Kuo Morgan, 1996], which is equivalent to internal model control [Morari Zafiriou, 1989].

Another approach is to use gain scheduling, where the scheduling parameters are calculated from the disturbance frequencies. This can be further subdivided into indirect and direct scheduling methods. In indirect scheduling, the controller, or part of it, for example a state-feedback or observer gain, is determined from a set of pre-computed quantities through interpolation or switching. For example, for linear parameter-varying (LPV) systems, where the uncertain parameters are contained in a polytope, one controller is calculated for each vertex and the resulting controller is obtained from interpolation [Apkarian et al., 1995]. For the rejection of harmonic disturbances, continuous-time LPV approaches have been suggested by [Darengosse Chevrel, 2000], [Du Shi, 2002, Du et al., 2003], [Witte et al., 2010], [Balini et al., 2011] and 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 this chapter, the observer gain is calculated by interpolation. In the other approach presented here, 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.

In direct scheduling, the dependence of the controller on the scheduling parameter does not correspond to a simple interpolation or switching law [Apkarian et al., 1995, Kinney de Callafon, 2007, Koro Scherer, 2008b, Ballesteros Bohn, 2011a, Ballesteros Bohn, 2011b, Shu et al., 2011]. For example, for LPV systems where the parameter dependence is expressed as a linear fractional transformation (LFT), the uncertain parameters also enter the controller through an LFT [Apkarian et al., 1995]. For harmonic disturbances, an LPV-LFT approach has been suggested and applied in real time by [Ballesteros Bohn, 2011a, Ballesteros Bohn, 2011b] and [Shu et al., 2011]. Another example for direct scheduling is a controller based on a time-varying state estimator, for example a Kalman filter, where the scheduling parameters enter the controller through the recursive equations for the state estimate and the error covariance matrix. Such a controller is presented and compared to an indirect (interpolation) approach by [Kinney de Callafon, 2007].

2.2. Stability considerations

The existing design approaches can be classified as approaches that take stability into consideration and such that do not. In indirect scheduling, for example, the controllers or gains are sometimes pre-computed for fixed operating points and then interpolated in an ad-hoc fashion [Bohn et al., 2003, Bohn et al., 2004, Kinney de Callafon, 2006b]. Stability is then not guaranteed, although it might be expected that the system is stable for slow variations of the scheduling parameter. For the adaptive filtering approaches, only approximate stability results seem available to date [Feintuch Bershad, 1993, Kuo Morgan, 1996]. To take stability into consideration, it is attractive to model the control problem as an LPV system and then use suitable gain-scheduling techniques [Darengosse Chevrel, 2000, Kinney de Callafon, 2006a, Koro Scherer, 2008b, Witte et al., 2010, Balini et al., 2011, Ballesteros Bohn, 2011a, Ballesteros Bohn, 2011b, Shu et al., 2011, Heins et al., 2011, Heins et al., 2012]. If parameter-independent Lyapunov functions are used, this guarantees closed-loop stability even for arbitrarily fast changes of the scheduling parameters at the expense of conservatism. The methods presented in this chapter guarantee closed-loop stability. In designs based on parameter-dependent Lyapunov functions, limits on the rate of change of the parameters can be incorporated.

2.3. Implementation aspects

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. It is therefore most natural to carry out the whole design in discrete time. If a continuous-time controller is computed, the controller has to be discretized. Since the controllers considered here are time-varying, the 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. Particularly in LPV gain-scheduling control, an approximate discretization is proposed by [Apkarian, 1997]. However, this leads to a distortion of the frequency scale. Usually, this can be tolerated, but not for the suppression of harmonic disturbances. It is therefore not surprising that the continuous-time controllers of [Darengosse Chevrel, 2000], [Du et al., 2003], [Kinney de Callafon, 2006a] and [Koro Scherer, 2008b] are only tested in simulation studies with a very simple system as a plant and a single frequency in the disturbance signal. The design methods that are tested in real time are usually formulated in discrete time [Bohn et al., 2003, Bohn et al., 2004, Kinney de Callafon, 2006b, Kinney de Callafon, 2007, Ballesteros Bohn, 2011a, Ballesteros Bohn, 2011b, Shu et al., 2011, Heins et al., 2011, Heins et al., 2012]. 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. It seems more natural to directly carry out the design in discrete time to avoid discretization issues.

3. State-augmented observer-based state-feedback control

Usually, disturbances act somewhere on the plant under consideration. The control objective is then to reject this disturbance. For the design methods considered in this paper, it is assumed that the disturbance acts on the plant input. For linear systems, a disturbance acting somewhere in the plant can be represented as a disturbance acting on the plant input under very mild assumptions. Therefore, assuming an input disturbance does not mean that the true disturbance has to act on the plant input (in fact, it usually does not). The design method used in this chapter is based on the internal model principle [Francis Wonham, 1976]. Applying this principle results in controllers with high gain in the frequency ranges where disturbances are assumed. These disturbances are then rejected no matter where they act upon the plant.

A discrete-time model for the plant with a disturbance yd, kacting on the input is given by

xp, k+1=Apxp, k+Bp(up, k+yd, k),yp, k=Cpxp, k.E1

In the approaches presented in the following, the disturbance rejection will be achieved with observer-based state-feedback controllers. The design methods presented here use state augmentation to add certain desired dynamics to the controller. In the time-invariant case, such a state augmentation can be used to prescribe controller poles. It can easily be verified that controller poles show up as zeros in the closed-loop disturbance transfer functions. Therefore, controller poles can be chosen to correspond to disturbances that are to be suppressed.

This argumentation, of course, only holds in the time-invariant case for which transfer functions and poles are defined (and only for controller poles that are not cancelled by plant zeros). If a time-varying internal model is used, it is not easy to interpret what happens when the internal model changes. Conceptually, the argument that the controller has high (infinite) gain at the disturbance frequencies that are to be rejected should still hold. It is confirmed by experiments (see Sec. 6) that even for fairly fast changes of the disturbance frequencies, excellent disturbance rejection is achieved.

3.1. State augmentation through an error filter

In this approach, disturbance rejection is achieved by including additional dynamics in a state-feedback controller through error filtering. A general error filter in state-space representation is given by

xM, k+1=AMxM, k+BMrk-yp, kE2

and interconnected with the plant model (without disturbance input)

xp, k+1=Apxp, k+Bpup, k,yp, k=Cpxp, k.E3

as shown in Fig. 1. Usually, the controller error e=r-ypis used as input for the error filter. In the application considered here, however, no set-point rwill be given and the error filter basically acts as an output filter with input-yp. Nonetheless it will be referred to as error filter in the following.

Figure 1.

Interconnection of plant model and error filter

The dynamics of the error filter can be used to include additional desired dynamics in a state-feedback controller for the overall system given by the plant and the filter. The additional dynamics can be chosen such that they describe the disturbances that the controller should reject, which corresponds to the internal model principle [Francis Wonham, 1976]. The additional dynamics can therefore be referred to as the “internal (disturbance) model.” If, as in the control problem considered in this chapter, the disturbance characteristics change over time, a time-varying internal model

xM, k+1=AM, kxM, k-BMyp, kE4

can be used.

In order to design a state-feedback controller for the overall system, first the overall system model is formed by combining the plant model (3) and the error filter dynamics (4) through introduction of an augmented state vector. This yields

xM, k+1xp, k+1=AM, k-BMCp0ApxM, kxp, k+0Bpup, k,yp, k=0CpxM, kxp, k.E5

Then, a state-feedback gain Kkfor the overall system (5) can be designed. Due to the time-varying dynamics of the error filter, the state-feedback gain may be time-varying.

Usually the plant states are not available for feedback. Therefore, an identity observer

x^p, k+1=(Ap-Lp, kCp)x^p, k+Bpup, k+Lp, kyp, kE6

is used to obtain an estimate x^pof the plant statesxp. The estimate is then used for feedback instead of the unknown plant states.

For an overall state-feedback gain

Kk=KM, k    Kp, k,E7

where KM, kand Kp, kdenote the parts which feed back the error filter states xMand the plant states xp(or in this case, the estimated plant statesx^p), respectively, the state-feedback law is given by

up, k=-KM, kxM, k-Kp, kx^p, k.E8

The structure of the overall closed-loop system is shown in Fig. 2. The plant is affected by a disturbance ydat the plant input. The plant output ypis used as input for the error filter (with a sign reversal), which outputs are its statesxM, and for the identity observer for the plant. The latter gives an estimate x^pof the plant states computed from ypandup. The states are then combined and fed back with the overall state-feedback gain consisting of KM, kandKp, k.

Figure 2.

Structure of the resulting controller based on error filtering

For stability analysis of the closed-loop system it is convenient to separate the state-space representation of the overall closed-loop dynamics including plant, observer, error filter, state-feedback and disturbance input in several parts as derived in the following. First, the controlled plant under disturbance at the input can be derived by combining (1), (6) and (8) to

xp, k+1=(Ap-BpKp, k)xp, k-BpKM, kxM, k-BpKp, k(x^p, k-xp, k)+Bpyd, k.E9

It is driven by the error filter statesxM, the observer error x^p-xpand the real unknown disturbanceyd.

The dynamics of the identity observer for the plant states are described by (6). Since the observer does not include a model of the disturbance, the dynamics of the observer error, from now on denoted byx~p, are given as

x~p, k+1=(Ap-Lp, kCp)x~p, k-Bpyd, k,E10

which means that the observer is driven by the unknown disturbanceyd. With the error filter dynamics given by (4), the overall system can be described as

x~p, k+1xM, k+1xp, k+1=Ap-Lp, kCp000AM, k-BMCp-BpKp, k-BpKM, kAp-BpKp, kx~p, kxM, kxp, k+-Bp0Bpyd, k.E11

The overall system behavior is shown in Fig. 3. From this representation it can be seen that the time-varying dynamics of the augmented plant under state feedback are driven by the time-varying dynamics of the observer error for the plant states. The augmented plant under state feedback and the dynamics of the observer error for the plant states can therefore be interpreted as two distinct systems connected in series. From this it follows that choosing an observer gain Lp, ksuch that Ap-Lp, kCpis stable and a state-feedback gain such that the time-varying dynamics of the augmented plant under state feedback are stable guarantees overall stability of the closed-loop system.

Figure 3.

Dynamics of the overall closed-loop system for stability analysis

3.2. State augmentation through a disturbance model

Figure 4.

System model for the disturbance observer approach

In this approach, the disturbance ydacting on the plant input upis explicitly modeled as the output of an unforced linear time-varying system

xd, k+1=Ad, kxd, k,yd, k=Cdxd, kE12

as shown in Fig. 4. An overall linear time-varying system model is then given by the combination of the plant model (1), which is affected by the disturbance yd, kat the plant input, and disturbance through

xd, k+1xp, k+1=Ad, k0BpCdApxd, kxp, k+0Bpup, k,yp, k=0Cpxd, kxp, k.E13

With

xk=xd, kxp, k,Ak=Ad, k0BpCdAp,B=0Bp,C=0Cp,E14

the overall system can be written as

xk+1=Akxk+Bup, k,yp,  k=Cxk.E15

Estimates for the states of the disturbance and the plant model can then be obtained by application of a linear time-varying identity observer with time-varying observer gain Lk

x^k+1=(Ak-LkC)x^k+Bup, k+Lkyp, k.E16

A state-feedback control for the overall system is given by

up, k=-Kkx^k.E17

Figure 5.

Observer-based state-feedback control in the disturbance observer approach

Figure 6.

Detailed illustration of observer structure

This leads to the typical structure of an observer-based state-feedback controller, but due to the time-varying disturbance model, the observer gain as well as state-feedback gain might be time-varying. Fig. 5 shows the general structure. The idea behind the disturbance observer is to use the estimate of the disturbance states (with a sign reversal) as control input. This should lead to perfect disturbance cancellation if the disturbance model is chosen appropriately. For a more detailed analysis of this aspect, in the following observer gains and state-feedback gains are seperated into a part that feeds back the disturbance states via Ld, kand Kd, kand a part that feeds back the plant states or their estimates via Lp, kandKp, k. The estimated disturbance states can be described by

x^d, k+1=Ad, kx^d, k-Ld, kCpx^p, k+Ld, kyp, k,E18

and the estimated plant states are given through

x^p, k+1=(Ap-Lp, kCp)x^p, k+BpCdx^d, k+Lp, kyp, k+Bpup, k,E19

where

Lk=Ld, k    Lp, k.E20

In Fig. 6, this more detailed representation is illustrated.

For stability analysis of the overall system including overall observer dynamics and state feedback, the dynamics of the observer error for the plant states

x^p, k+1-xp, k+1=(Ap-Lp, kCp)(x^p, k-xp, k)+BpCdx^d, k-Bpyd, kE21

are considered.

The dynamics of the states of the plant under state feedback are given by

xp, k+1=(Ap-BpKp, k)xp, k-BpKd, kx^d, k+BpKp, k(x^p, k-xp, k)+Bpyd, k.E22

This yields the overall closed-loop system dynamics given by

x^d, k+1x~p, k+1xp, k+1=Ad, k-Ld, kCp0BpCdAp-Lp, kCp0-BpKd, kBpKp, kAp-BpKp, kx^d, kx~p, kxp, k+0-BpBpyd, k.E23

Fig. 7 illustrates the structure of the overall system dynamics. This representation shows, similiar to the error filtering approach, that the dynamics of the plant under state feedback are driven by the time-varying dynamics of the observer for the augmented system, where in this case the outputs are given by the estimate of the disturbance states and the observer error for the plant states. The plant under state feedback and the observer of the augmented system can therefore be considered as two distinct systems in series connection. From this it follows that as long as a stabilizing state-feedback gain Kp, kfor the linear time-invariant plant and a stabilizing observer gain for the augmented system model is chosen, the overall closed-loop system is stable. It is also evident that the choice of the disturbance model does not influence the overall system stability, as long as the disturbance model itself is stable. Also, the choice of Kd, khas no effect on the overall system stability (as long as Kd, kx^d, kremains bounded). Therefore, the overall state-feedback gain Kdoes not have to be time-varying since only the linear time-invariant plant must be stabilized by Kpand the choice of Kdis not important for system stability. Furthermore, it can be seen that, assuming a perfect disturbance model, complete disturbance cancellation will be achieved after some transient ifKd=Cd.

Figure 7.

Dynamics of the overall closed-loop system for stability analysis

4. Gain-scheduled state-feedback and observer design for pLPV systems

In this section, design methods for time-varying state-feedback and observer gains for pLPV systems based on quadratic stability theory are reviewed.

An LPV system is a linear system described by

xk+1=A( θ )xk+B( θ )ukyk=C( θ )xk+D( θ )ukE24

withxkRn, where the state-space matrices depend on a possibly time-varying parameter vector

 θ =θ1θ2θNT Θ RN.E25

In the following, only unforced systems of the form

xk+1=A( θ )xkE26

are considered. In general, an LPV system is called a pLPV system, if the matrices depend affinely on the varying parameter. For the unforced system given in (26), this means that it is a pLPV system, if the system matrix A()depends affinely on θ , that is

A( θ )=A0+θ1A1+θ2A2++θNAN,E27

where AiRn×nfor all i=0,1,...,Nare constant matrices and  Θ is a convex polytope in RNwith a finite set of verticesV={v1,v2,...,vM}RN. A point  θ  Θ can be written as a convex combination of vertices, which means there exists a coordinate vector  λ =λ1 λ2  λMTRMsuch that  θ can be written as

 θ =j=1Mλjvj,    λj0    j=1,...,M,    j=1Mλj=1.E28

Defining Av,j=A(vj)forj=1,,M, A( θ )can be represented as

A( θ )=A( λ )=λ1Av,1+λ2Av,2++λMAv,M.E29

The matrices Av,jcan be considered as system matrices of linear time-invariant vertex systems

xk+1=Av,jxk,    j=1,, M,E30

and the current system (26) is a convex combination of the vertex systems in (30).

The pLPV system (26) is called quadratically stable in  Θ if and only if there exists a positive definite matrix Psuch that for all  θ  Θ 

AT( θ )PA( θ )-P<0,E31

or equivalently via the Schur complement [Horn Johnson, 1985]

PPA( θ )AT( θ )PP>0,E32

where “>0” and “<0” indicate positive and negative definiteness, respectively.

Since (32) has to hold for every θ , this constitutes an infinite set of linear matrix inequalities (LMIs), which is computationally intractable. By the following result, it is possible to use a finite set of inequalities. It holds that a pLPV system in the form of (26) is quadratically stable for every  θ  Θ if and only if there exists a symmetric positive definite matrix Psuch that

Av, jTPAv, j-P<0,E33

or equivalently

PPAv, jAv, jTPP>0,E34

for allj=1,, M. This result is based on one single quadratic Lyapunov function that assures stability for the whole parameter space Θ . Therefore,  θ is allowed to change arbitrarily fast with time and  θ and  λ can explicitly be assumed time-varying and denoted as  θ kand λ k, respectively. A proof is given in [Amato, 2006].

These results can be applied to the design of time-varying state-feedback and observer gains, as reviewed in the following sections.

4.1. State-feedback design for pLPV systems based on quadratic stability

A system of the form

xk+1=A( θ k)xk+BukE35

is considered and the objective is to find a state-feedback gain K( θ k)that quadratically stabilizes the closed-loop system

xk+1=(A( θ k)-BK( θ k))xk.E36

From the above result it follows that it suffices to find a single symmetric positive definite matrix Pand a finite set of matrices Kv, jsuch that

PP(Av, j-BKv, j)(Av, j-BKv, j)TPP>0,    j=1,, M.E37

Then, K( θ k)has to be chosen as

K( θ k)=j=1Mλj, kKv, j=K( λ k),E38

because then it holds that

A( θ k)-BK( θ k)=j=1Mλj, kAv, j-Bj=1Mλj, kKv, j=j=1Mλj, k(Av, j-BKv, j),E39

and therefore quadratic stability of (36) is implied due to the results presented above.

4.2. Computation of state-feedback gains for the vertex systems

In this section, sufficient conditions that guarantee closed-loop stability and a certain H2performance level for the controlled vertex systems that can be used to calculate state-feedback gains for the vertex systems are derived. To use an H2performance level, an additional performance input is introduced that enters the state update equation for each vertex system of the pLPV system in (35). This input can be interpreted as process noise. For everyj=1,,M, this yields

xk+1=Av,jxk+Buk+wk.E40

As a performance output, the artificial signal

zk=Q12xkR12ukE41

is defined that weighs the states and the control signal with matrices Q12andR12, respectively. The objective then is to find state-feedback gains Kv, jthat stabilize the system and minimize the H2norm of the transfer path from wktozk, when the control signal ukin (40) and (41) is chosen as

uk=-Kv,jxk.E42

This objective can be stated as the minimization of the H2norm of the system

z=GwE43

which has the state-space representation

xk+1=A~xk+B~wk,zk=C~xk,E44

with

A~=Av,j-BKv,j,    B~=I,    C~=Q12-R12Kv,j.E45

If the system is stable, the H2norm of this discrete-time linear time-invariant system is given by

G22=traceC~WcC~T,E46

where the controllability gramian Wcsatisfies the discrete-time Lyapunov equation

A~WcA~T-Wc+B~B~T=0.E47

Therefore, if there exist symmetric positive definite matrices Pand Wsuch that

A~PA~T-P+B~B~T<0,E48
W-C~PC~T>0,E49
traceW<γ2,E50

then it follows that G2<γfor aγ>0. Through the Schur complement, (48) and (49) can be transformed to

PPA~TA~PP-B~B~T>0,E51
WC~PPC~TP>0.E52

Introducing

Q~=Q120,    R~=0R12,    Yv, j=Kv, jP,E53

it follows that if solutions for the matrix variablesP, WandYv, j, j=1,,M, can be found that satisfy the 2M+1LMIs

PAv, jP-BYv, jTAv, jP-BYv, jP-I>0,    j=1,,M,E54
WQ~P-R~Yv, jQ~P-R~Yv, jTP>0,    j=1,,M,E55
traceW<γ2,E56

then the system Ghas an H2norm bounded byγ. From the solutions PandYv, j, the state-feedback gain for each vertex system can be calculated as

Kv, j=Yv, jP-1.E57

Quadratic stability is then implied for each closed-loop vertex system because of (48). In order to guarantee quadratic stability for the whole parameter space, solutions for the matrix variables Pand Win (54)-(56) have to be the same for all vertex systems. Therefore, if solutions are found, also the performance bound γis guaranteed for every fixed  θ in the parameter space Θ , since it depends only on PandW.

Once state-feedback gains are found for the vertex systems, in any instant of time of a realization of the pLPV system (35) with θ k Θ , a state-feedback gain can be found via interpolation according to (38). Quadratic stability of the closed-loop system (36) is guaranteed. The question of how to compute the coordinates  λ kwith the properties described in (28) depends on the specific polytope Θ . The case of an N-dimensional hyper box will be considered in detail in Sec. 4.4.

4.3. Observer design for pLPV systems based on quadratic stability

Although the observer design is dual to the controller design, the LMIs for observer design are shortly presented here for completeness.

For observer design, a system of the form

xk+1=A( θ k)xk+Buk,yk=CxkE58

is considered and the objective is to find an observer gain L( θ k)that quadratically (and therefore, asymptotically) stabilizes the dynamics of the observer error

x~k+1=(A( θ k)-L( θ k)C)x~k.E59

From the above result it follows that it suffices to find a single symmetric positive definite matrix Pand a finite set of matrices Lv, jsuch that

PP(Av, j-Lv, jC)(Av, j-Lv, jC)TPP>0,j=1,, M,E60

and L( θ k)has to be chosen as

L( θ k)=j=1Mλj, kLv, j=L( λ k).E61

As in the design of the state-feedback gain in the previous section, sufficient conditions for quadratic stability of (59) and a certain H2performance level can be derived. It is assumed that white noise of unit covariance, given by w1and w2and weighted by matrices Q12and R12respectively, affects the states and the outputs of the vertex systems of (58). This gives

xk+1=A( θ k)xk+Cuk+Q12w1, kyk=Cxk+R12w2, k.E62

The H2-norm of the transfer path from this noise input to the observer error is chosen as the performance measure. Defining

wk=w1, kw2, k,E63

this transfer path is described by

x~k+1=Av, j-Lv, kCx~k+Q12-Lv, kR12wk.E64

Therefore the objective is to minimize the H2-norm of the system z=Gwwith state-space representation

xk+1=A~xk+B~wk,zk=C~xk,E65

with

A~=Av,j-Lv,jC,    B~=Q12-Lv, kR12,    C~=I.E66

If the system is stable, the H2norm of this discrete-time linear time-invariant system is given by

G22=traceB~TWoB~,E67

where the observability gramian Wosatisfies the discrete-time Lyapunov equation

A~WoA~T-Wo+C~TC~=0.E68

For the same arguments as in Sec. 4.1 and with the introduction of

Q~=QT20,    R~=0RT2,    Yv, j=Lv, jTP,    j=1,...,M,E69

it follows that if solutions for the matrix variablesP, WandYv, j, j=1,,M, can be found that satisfy the 2M+1LMIs

PPAv, j-Yv, jTC(PAv, j-Yv, jTC)TP-I>0,    j=1,...,M,E70
WQ~P-R~Yv, j(Q~P-R~Yv, j)TP>0,    j=1,...,M,E71
traceW<γ2,E72

then the system Ghas an H2norm bounded byγ. From the solutions PandYv, j, the observer gain for each vertex system can be calculated as

Lv, j=P-TYv, jT.E73

In order to guarantee quadratic stability for the whole parameter space, solutions for the matrix variables Pand Win (70)-(72) have to be the same for all vertex systems. Therefore, if solutions are found, also the performance bound γis guaranteed for every fixed  θ in the parameter space Θ .

Once observer gains are found for the vertex systems, in any instant of time an observer gain for the realization of the pLPV system (58) with  θ k Θ can be found via interpolation according to (61).

4.4. Computation of the coordinates for interpolation

The computation of the coordinates  λ kRMrequired for the interpolation depends on the specific polytope  Θ RNthat is used to cover the parameter space. In general, for a parameter vector θ k Θ , where  Θ denotes a polytope with Mverticesv1,v2,,vMRN, the coordinate vector  λ kas required for the interpolation is given by the solution to the constrained system of linear equations

v1v2vM111 λ k= θ k1,    0λj, k1,    j=1,,M.E74

Existence of the solution is guaranteed as long as θ k Θ . If the polytope considered has less than or exactly N+1vertices, the constrained linear equation system admits exactly one solution. If the polytope is given by more than N+1vertices, the system is underdetermined and more than one solution is possible. Since the controller design is not a linear operation, different choices of coordinates might lead to different properties of the resulting controller.

In most cases, ranges of components of the parameter vector are given by closed intervals in Rdetermined by upper and lower bounds, such that

θi, kθi, min,θi, max,i=1,,N.E75

If no further information on the relations between parameters are known, the polytope  Θ has to be chosen as the N-dimensional hyper box, also referred to as the parameter box, given by

 Θ =θ1, min,θ1, max×θ2, min,θ2, max××θN, min,θN, max.E76

One way to compute a set of coordinates  λ kfor this case is introduced by [Apkarian et al., 1995] and generalized and implemented for an arbitrary number of vertices in the LMI Control Toolbox for Matlab [Gahinet et al., 1995]. Another implementation of this approach is proposed here that is suitable for real-time implementation purposes where variables have to be pre-initialized with fixed dimensions. [Daafouz et al., 2000] presented a compact way of writing the calculation scheme, on which the scheme proposed here and in [Heins et al., 2012] is based. If the order of vertices is not changed, any of the mentioned approaches leads to the same coordinates.

The i-th entry vj, iof a vertex jof the parameter box  Θ is either the lower bound θi, minor the upper bound θi, maxofθi, k. Now, 2Nvectors

bimin=bimin, 1bimin, M,bimax=bimax, 1bimax, ME77

are pre-computed such that

bimax, j=1θi, max-θi, min,if  vj, i=θi, max,    0,if  vj, i=θi, min,E78
bimin, j=1θi, max-θi, min,if  vj, i=θi, min,    0,if  vj, i=θi, max.E79

The following steps are then carried out in every sampling instant:

1.    θi, k=cos(2πfi, kT), i=1,...,N,E80
2.    cimax, k=θi, k-θi, min,    cimin, k=θi, max-θi, k, i=1,...,N,E81
3.    λj, k=i=1N(bimax, jcimax, k+bimin, jcimin, k),j=1,...,M.E82
(82)

5. Application for the rejection of harmonic disturbances

In this section, the methods described in Sec. 4 are applied to the case of a harmonic multisine disturbance. Specific system models and the transformation that leads to a pLPV system are presented.

5.1. Internal model for harmonic disturbances

The disturbance is assumed to be a multisine. Let Nbe the number of components of the disturbance and fi, kthe frequency of the i-th component at sampling instantk. The frequencies are assumed to vary in intervalsfi, min, fi, max0, 0.5fs, where fsdenotes the sampling frequency.

As discussed in Sec. 3, an error filter or a disturbance model can be used to include desired dynamics in the controller according to the internal model principle [Francis Wonham, 1976]. Since the following relations are used for either AM, kor Ad, kdepending on the control approach, the notation A M/d, kis employed. The dynamics for a multisine disturbance with time-varying frequency are given by a discrete-time state-space model with system matrix

A M/d, k=AM1/d1, k00AMN/dN, k,E83

where

AMi/di, k=cos(2πfi, kT)sin(2πfi, kT)-sin(2πfi, kT)cos(2πfi, kT)E84

with sampling timeT=1/fs. This time-varying matrix might therefore be used as system matrix AM, kof the error filter described in Sec. 3.1 or Ad, kof the disturbance model described in Sec. 3.2. In order to find a representation of A M/d, kthat depends affinely on a parameter vector θ , one approach could be to choose

θi, k=cos(2πfi, kT),  i=1,...,N.E85

Unfortunately, through

sin(2πfi, kT)=1-θi, k2,  i=1,...,N,E86

a non-affine dependence is introduced into the model. To circumvent this, additional parameters

θi, k=sin(2πfi, kT), i=N+1,...,2NE87

could be introduced. The use of this internal model leads to a large dimension of the parameter space and a polytope with many vertices. It can be expected that controllers based on this model are quite conservative, if there exists a solution to the LMIs at all. Therefore, a simplified disturbance model that leads to less conservative conditions is used for the controller design.

In the case of constant frequencies, instead of (84), the matrices

AMi/di=01-12cos(2πfi,T), i=1,...,NE88

can be used in the system matrix of the internal model (83). This leads to a time-invariant overall system model. As discussed in Sec. 3, the internal model then is a way to determine controller poles (that show up as zeros in the disturbance transfer functions). With the above approach, for a frequencyf, a complex conjugate controller pole pair is placed at exp(j2πfT)andexp(-j2πfT), which causes complete asymptotic rejection of harmonic disturbances with this frequency.

Although this model does not take into account the rate of change of the disturbance frequencies and the argumentation based on controller poles is valid only in the time-invariant case, this simpler model is used here even for the case of time-varying frequencies, since it reduces the dimension of the parameter space. Thus, the matrices

AMi/di, k=01-12cos(2πfi, kT), i=1,...,NE89

are used for the time-varying disturbance model. This might have an effect on the disturbance attenuation for fast changes of the frequencies, but it is expected that other effects are more dominant than the “incorrect” disturbance model. There will always be delay between the measured frequency used in the controller and the true frequency. Also, if the disturbance and the control signal do not enter at exactly the same point and the disturbance frequency varies, it is unclear which frequency “is present” at the point where the control signal enters the plant at a certain time. As shown in Sec. 3, closed-loop stability is not affected by the choice of the internal model.

5.2. Transformation to pLPV form

If the internal model can be written in the form of a pLPV system, it is possible to find a representation of the system matrix Akof the overall system of the error filter approach (5) and the disturbance observer approach (15) in the form of a pLPV system as well. Then, the controller design and the gain-scheduling procedure can be carried out based on quadratic stability theory of pLPV systems, as has been presented in Sec. 4.

If the sampling theorem applies (which should be the case for any practical application), it holds that 02πfi, kTπfor alli=1,...,N. Since the function fcos(2πfT)is monotonically decreasing on0, π, the frequencies under consideration have to fulfill 02πfi, kTπfor all i=1,...,Nin order to guarantee a unique mapping between frequencies and parameters. This can usually be achieved by choosing a sufficiently small value forT. Then, parameters  θ ias defined in (85) are bounded by

θi, min=cos(2πfi, maxT), θi, max=cos(2πfi, minT), i=1,...,N,E90

respectively. Defining

 Θ =θ1, min, θ1, max×θ2, min, θ2, max× ... ×θN, min, θN, maxRNE91

with vertices v1,v2,...,vMRNit follows that the parameter vector

 θ k=θ1, kθ2, kθN, kRNE92

is always inside the hyper box Θ . Then, the system matrix A M/d, kof the internal model can be written as

A M/d, k=A M/d( θ k)=A M0/d0+θ1, kA M1/d1+θ2, kA M2/d2+ ... +θN, kA MN/dNE93

with

A M0/d0=A M0/d0, 100A M0/d0, N,E94

where

A M0/d0, i=01-10, i=1,...,N,E95

and A Mi/dimatrices with zero entries only except for

A Mi/di(2i,2i)=2, i=1,...,N.E96

Therefore, the system matrix A M/d, kof the internal model used for the error filter from Sec. 3.1 and the disturbance model in Sec. 3.2 can be written in pLPV form based on the N-dimensional hyper box  Θ with verticesv1,v2,...,vMRN. The system matrices Akof (5) in the error filter approach can then be written as

Ak=A(  θ k)=A 0+θ1, kA 1+...+θN, kA NE97

with

A 0=A M0BMCp0Ap,E98
A i=A Mi000, i=1,...,N,E99

and analogously in (15) for the disturbance observer approach

Ak=A( θ k)=A 0+θ1, kA 1+...+θN, kA NE100

with

A 0=ApBpCd0A d0,E101
A i=000A di, i=1,...,N.E102

Since  Θ is a convex polytope, in every instant there exist coordinatesλj, k, j=1,...,M, such that

 θ k=j=1Mλj, kvj.E103

Since Akdepends affinely on  θ kin both approaches, it follows that the system matrix can be expressed as

A( θ k)=j=1Mλj, kAv, j=A( λ k),E104

where

Av, j=A(vj),j=1,,M.E105

The required time-varying state-feedback and observer gains for the overall systems (5) and (15), respectively, can therefore be obtained by application of the design methods based on quadratic stability theory as reviewed in Sec. 4.

5.3. Controller implementation

Once the pLPV representation of the considered system is found, the required time-varying state-feedback and observer gains for the overall systems (5) and (15), respectively, can be obtained by application of the design methods based on quadratic stability theory as reviewed in Sec. 4.

The time-invariant vertex state-feedback or vertex observer gains are calculated in advance. Since the plant itself is linear time-invariant, also the time-invariant observer required for the estimation of the plant states in the error filter approach as well as the time-invariant state-feedback gain of the disturbance observer approach can be calculated off-line with a standard method (e.g. LQR or pole placement). The controllers are then implemented according to the structures given in Sec. 3 and shown in Figs. 8 and 9 with the interpolation. Wherever possible, system matrices are updated in every sampling instant directly with the measured frequencies, while the time-varying state-feedback and observer gains are obtained by interpolation with the coordinates obtained by the scheme given in Sec. 4.4.

Figure 8.

Control structure and interpolation for the error filter approach

Figure 9.

Control structure and interpolation the disturbance model approach

6. Real-time results

The gain-scheduled observer-based state-feedback controllers obtained through the design procedures presented in this chapter are validated with experimental results. Both controllers have been tested on an AVC and an ANC system and were found to work well. Results are presented for the test of the error filter with time-varying state-feedback gain on an ANC headset and the controller based on a disturbance observer with time-varying observer gain on an AVC test bed.

6.1. Real-time results for state-feedback gain-scheduling (error filtering) approach

The controller based on a disturbance observer with time-varying observer gain is tested experimentally on a Sennheiser PXC 300 headset. The experimental setup is shown schematically in Fig. 10. An external loudspeaker is used to generate a harmonic disturbance. The headset has one microphone on each loudspeaker. The objective is to cancel the disturbance with the loudspeakers of the headset. An anti-aliasing filter is applied to the output signal and a reconstruction filter to the control input. The control algorithm is implemented on a rapid control prototyping unit (dSpace MicroAutoBox). A sampling frequency of 1 kHz was used.

Figure 10.

Block diagramm (left) and photograph (right) of the ANC system

Standard black-box system identification techniques were used to obtain the transfer function between output and input of the control unit. The identified transfer function is of 12th order and the resulting controller of 20th order. As a disturbance signal, a sum of four harmonically related sine signals with fundamental frequency between 90 Hz and 100 Hz is used. The control algorithm has been implemented on both sides and results for the right side are shown.

Fig. 11 shows the amplitude frequency responses for constant fundamental frequencies of 90 Hz and 100 Hz. The amplitude frequency response plots show that amplification takes place in some frequency ranges, which is the known “waterbed” effect. It might cause problems in practical applications where significant background noise, e.g. broadband stochastic disturbances, is present. In Fig. 12, results are shown for a case where the fundamental frequency suddenly jumps from 90 Hz to 100 Hz. The controller remains stable even for such drastic variations of the fundamental frequency. The transient spike might be undesirable, particularly in ANC applications where it could be audible. However, a step change in the frequency does not commonly occur in real applications.

Figure 11.

Open-loop (gray) and closed-loop (black) amplitude frequency responses for fixed disturbance frequencies of 90, 180, 270 and 360 Hz (left) and of 100, 200, 300 and 400 Hz (right)

Figure 12.

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

In Fig. 13, results for a gradually changing fundamental frequency are shown. The measurements show that excellent disturbance attenuation is achieved. The performance of the controller is tested with fast variations of the disturbance frequencies. Results for this experiment are shown in Fig. 14. It can be observed that the attenuation performance decreases, but the system remains stable.

Figure 13.

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 14.

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. Real-time results for observer gain-scheduling (disturbance model) approach

The scheme and a photograph of the AVC test bed are shown in Fig. 15. Two shakers (inertia mass actuators) are attached to a steel cantilever beam. One shaker acts as the disturbance source and the other shaker is driven by the control signal to counteract this disturbance. An accelerometer is used to measure the output signal. An anti-aliasing filter is applied to the output signal and a reconstruction filter to the control input.

Figure 15.

Schematic representation (left) and photograph (right) of the AVC system

A multisine test signal is used to identify the transfer function between output and input of the control unit using standard black-box system identification techniques. The controller is implemented on a rapid control prototyping unit (dSpace MicroAutoBox). A sampling frequency of 1 kHz was chosen. The identified system is of 10th order and the controller of 16th order.

Figure 16.

Open-loop (gray) and closed-loop (black) amplitude frequency responses for fixed disturbance frequencies of 110, 220 and 330 Hz (left) and of 120, 240 and 360 Hz (right)

Figure 17.

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

As a disturbance signal, a sum of three harmonically related sine signals with fundamental frequency between 110 and 120 Hz is used. The amplitude frequency responses of the open-loop and closed-loop systems for fixed disturbance frequencies of 110 Hz, 220 Hz and 330 Hz and 120 Hz, 240 Hz and 360 Hz are shown in Fig. 16. The resonance frequencies are damped by a factor 0.9 using pole placement with the state-feedback gainKp. In Fig. 17, results are shown for a (rather unrealistic) case where the fundamental frequency suddenly jumps from 110 Hz to 120 Hz. After a short transient spike the disturbance rejection resumes quickly. The transient spike might be undesirable but a step change in the frequency does not commonly occur in real applications. The measurements confirm the excellent disturbance rejection that can be expected from the amplitude frequency responses. The amplitude response plots show that amplification takes place in frequency ranges between the rejected frequencies. This is due to Bode’s sensitivity integral (“waterbed” effect). Whether this is tolerable or not depends on the application. It is possible to limit the maximum disturbance amplification by a suitable overall controller design (which need not be observer based), but at the expense of either not fully suppressing the harmonic disturbances (that is, the notches in the frequency response would not get to zero) or requiring the frequency measurements to be very exact (that is, making the notches narrower) or worsening the transient behavior (that is, it would take longer before a harmonic disturbance is suppressed to a certain level). However, in this chapter only harmonic disturbances are considered, so these issues are not addressed in the design.

In Fig. 18, results for a gradually changing fundamental frequency are shown. It decreases linearly from 120 Hz to 110 Hz, remains constant for a while and then rises back to 120 Hz. This is repeated four times, every time in a shorter time interval. At the end the fourth harmonic rises from 330 Hz to 360 Hz and decreases back to 330 Hz in less than two seconds. The effect of fast variations of the disturbance frequencies has been further investigated in another experiment. The results are shown in Fig. 19. The disturbance frequency varies sinusoidally between the minimum and the maximum value with a period that decays from 10 to 0.5 seconds. It is seen that for very fast frequency variations, the attenuation performance decreases slightly (but the system remains stable).

Figure 18.

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 19.

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 LPV observer-based state-feedback controllers for the rejection of harmonic disturbances with time-varying frequencies are presented. The control design methods are based on quadratic stability theory for pLPV systems. They guarantee stability of the closed-loop system also for arbitrarily fast changes of the disturbance frequencies. This is an advantage over other approaches such as adaptive filtering or heuristic gain scheduling. The experimental results show that an excellent disturbance rejection is achieved and that the proposed controllers can be applied in a real-time setup.

However, some degree of conservatism is present in these approaches. Using a single parameter independent Lyapunov function limits the range of admissible disturbance frequencies that can be covered with the resulting controller. Also, the polytope that contains the uncertain parameters could be chosen much smaller and with fewer vertices than the cuboid applied here, if information on the relations between the disturbance frequencies is given (as, for example, in the case of harmonically related frequencies). Thus, feasibility of the LMIs and the upper bound on the system performance could be improved as well as the computation time for the coordinates required for the on-line interpolation. This might be important for applications where many harmonics have to be cancelled, a wide frequency range has to be covered and the computational resources are limited, for example in automotive applications [Bohn et al., 2003]. This aspect as well as a direct comparison of the proposed methods with conventional algorithms (such as the FxLMS) will be subject of future research.

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

Nomenclature

Acronyms

ANCActive noise control.

AVCActive vibration control.

FxLMSFiltered-x least mean squares.

LFTLinear fractional transformation.

LMILinear matrix inequality.

LPVLinear parameter varying.

pLPVPolytopic linear parameter varying.

Variables (in order of appearance)

Ap,Bp,CpState-space matrices of the plant.

xp, kState vector of the plant.

yp, k,up, kOutput signal and control input of the plant.

yd, kDisturbance signal.

AM,BMState-space matrices of the error filter.

xM, kState-space vector of the error filter.

rk,ekReference and error signal.

AM, kTime-varying system matrix of the error filter.

0Zero matrix.

IIdentity matrix.

LpObserver gain for the plant.

x^p, kEstimated state vector of the plant.

KkTime-varying state-feedback gain for the overall system.

KM, k,Kp, kTime-varying state-feedback gains for the error filter and the plant.

x~p, kObserver error of the plant.

Ad, k,CdState-space matrices of the disturbance model.

xd, kState vector of the disturbance model.

Ak,B,CState-space matrices of the overall system.

LkTime-varying observer gain of the overall system.

Ld, k,Lp, kTime-varying observer gains for the disturbance model and the plant.

x^d, kEstimated state vector of the disturbance model.

Kd, k,Kp, kTime-varying state-feedback gains for the disturbance model and the plant.

A( θ ),B( θ ),State-space matrices of an LPV system.

C( θ ),D( θ )E106

 θ =θ1    θ2    θNTParameter vector.

NSize of the parameter vector.

 Θ Parameter polytope.

A iConstant matrices of a polytopic representation.

VSet of vertices of the parameter polytope.

vjVertex j of the parameter polytope.

 λ =λ1    λ2    λMTCoordinate vector.

MSize of the coordinate vector.

Av,jSystem matrices for the j-th vertex.

PSymmetric positive definite matrix.

θkVector of time-varying parameters.

 λ k=λ1, k    λ2, k    λM, kTTime-varying coordinate vector.

Kv,jState-feedback gain for the j-th vertex.

wk,zk,ukProcess noise, performance output and control input.

Q,RWeighting matrices.

GTransfer function between wkandzk.

A~,B~,C~State-space matrices of the transfer path from wktozk.

Q~,R~Matrices to build the LMIs.

WcControllability gramian.

WoObservability gramian.

WSymmetric positive definite matrix.

Yv, jSolution of the LMIs for the j-th vertex.

Lv,jObserver gain for the j-th vertex.

bimin,bimaxVectors for the computation of the coordinate vector.

cimin, k,cimax, k,Scalars for the computation of the coordinate vector.

bimin, j,bimax, jE107

AM/d, kSystem matrix of a general time-varying internal disturbance model.

© 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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Wiebke Heins, Pablo Ballesteros, Xinyu Shu and Christian Bohn (September 5th 2012). LPV Gain-Scheduled Observer-Based State 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: 10.5772/50293. Available from:

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