Active control of frequency varying disturbances in a diesel engine

Control Engineering Practice 20 (2012) 1206–1219

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Active control of frequency varying disturbances in a diesel engine Juha Orivuori a,n, Ilias Zazas b, Steve Daley b a b

Aalto University School of Electrical Engineering, Department of Automation and Systems Technology, PO Box 15500, 00076 AALTO, Finland Institute of Sound and Vibration Research, The University of Southampton, Southampton, SO171BJ, UK

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 April 2011 Accepted 25 June 2012 Available online 15 July 2012

In this paper a method for active vibration isolation of frequency varying tonal disturbances in an engine mounted on a raft is presented. An adaptive nonlinear control algorithm with frequency tracking is introduced to tackle this problem. The studied process is a true MIMO-system with strong crosscouplings and high background noise level. The controller performance is first validated by extensive simulations and then by test bed implementation. It is shown that the proposed method is robust to measurement noise and additional output disturbances, while yielding a high level of vibration suppression with fast convergence rate. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Active control Adaptive control Frequency tracking Optimal control Vibration isolation

1. Introduction Vibration is present in any physical system; most of the time this rather unwanted phenomenon is not unduly problematic, however for a certain class of systems it can cause severe limitations to both the process performance and the quality of the end product. One such class are the processes containing rotating or reciprocating components such as pistons, propeller blades or rotors that act as a natural source for tonal disturbance forces. The tonal disturbances become particularly problematic if they coincide with the natural frequencies of the process, thereby resulting in a significant amplification of the perceived disturbance forces; a situation that is highly likely to arise where process operation speed is variable. Prolonged vibrations cause several problems that can roughly be divided into three categories; namely health risk (Seidel, 1993), process performance and process costs (Nandi, Toliyat, & Xiaodong, 2005). The health risks include severe operator discomfort due to high level process noise and low frequency vibrations causing nausea. The process performance may be limited due to increased safety margins in design, limitations to operating speeds and defective end products. The increased process costs originate from the increased maintenance cost and process downtimes due to wear and tear of supportive structures and critical process components such as bearings. According to the preceding reasons it is clear that attenuation of these periodic

n

Corresponding author. Tel.: þ358 50 409 1840; fax: þ 358 9 470 25208. E-mail addresses: juha.orivuori@aalto.fi, juha.orivuori@tkk.fi (J. Orivuori).

0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2012.06.010

tonal process disturbances would provide significant benefits both in terms of safety and operational profit. Methods for vibration mitigation can be divided into three categories; namely passive, semi-active and active vibration control (Inman, 2006). In this paper active vibration mitigation is considered. The major benefits of this method are the possibility for effective vibration suppression at any frequency, the possibility to redesign the controller without changing the physical process and the availability of all traditional control design methods and analysis tools. The major draws include relatively high design costs, the requirement for actuators and the possibility of closed-loop instability. Although active vibration control as a concept has been around for a number of years it is only in relatively recent times that significant application results have been obtained. The recent increase in the availability of low cost high computation power has made active control methods a viable practical choice. The control designs vary from signal based approaches (Tammi, 2007) to model based approaches with a mixture of feedforward and feedback elements (Bittanti, Lorito, & Strada, 1996; Gupta & Du Val, 1982; Sievers & von Flotow, 1992). Some of the processes on which active vibration control has been applied with success include control of helicopters (Bittanti & Cuzzola, 2002; Knospe, Tamer, & Fittro, (1997);, Bittanti et al., 2002)., vibration isolation of an automotive engine with flexible supports (Bohn, Cortabarria, Hartel, & Kowalczyk, 2004), magnetic bearings (Herzog, Buhler, Gahler, & Larsonneur, 1996; Sun, Krodkiewski, & Cen, 1998), electric ¨ machines (Tammi, H¨atonen, & Daley, 2007; Chiba, Fukao, & Rahman, 2008; Hiromi, Katou, Chiba, Rahman, & Fukao, 2007; Laiho, Tammi, Zenger, & Arkkio, 2008; Laiho, 2009; Orivuori, Zenger, & Sinervo, 2009) and marine applications (Daley, Johnson,

J. Orivuori et al. / Control Engineering Practice 20 (2012) 1206–1219

¨ Pearson, & Dixon, 2004; Daley, Zazas, & H¨atonen, 2008). Most of the preceding methods assume static disturbance frequency, which is usually not a valid assumption, resulting in poor control performance. There exist several different control schemes to tackle the problem of vibration mitigation in the presence of frequency varying disturbances. These methods vary from adaptive gain-scheduled IMC (Kinney & Callafon, 2006, 2007), repetitive and periodic control approaches (Bittanti et al., 2002; Tammi et al., 2007). to robust control formulations, where the disturbance frequency is considered as an unknown parameter (Knospe et al., 1997; Du, Zhang, Lu, & Shi, ¨ glu & Scherer, 2011a; Koro˘ ¨ glu & Scherer, 2011b; 2003; Koro˘ Ballesteros & Bohn, 2011) All of these methods have been successfully implemented in practice, although mostly in SISO-systems, and the disturbance is assumed to be excited at the process input. The synthesis of some of these controllers involves solving multiple LMIs, and in robust control approaches the choice for suitable frequency dependent weighting functions may become a rather tedious process in MIMO-systems. In this paper a problem of vibration isolation from a marine diesel engine mounted on a flexible raft is considered. The engine generates three major tonal disturbances in three dimensions with time varying frequency the effects of which are all to be minimised. The applied control law is a nonlinear controller with frequency tracking obtained through continuous gain scheduling of modified linear quadratic (LQ) optimal controllers. The use of LQ-controllers in vibration control is a rigorously studied approach, for which several effective methods have already been presented. The LQ-design method proposed in this paper has many similarities with the existing methods; however the design structure is modified such that it can be applied to processes with any number of inputs and outputs; also the design procedure is relatively simple, not including any dynamic weighting functions or LMIs. The major difference of the proposed control law to the existing methods is the assumption of the disturbance entering the system at the process output, which is closer to the real scenario and provides better performance in terms of sensitivity to output noise. The disturbance dynamics are included in the model, which is a well-known approach for the conversion of the servo problem into a regulator problem. The model approach enables the direct use of the LQ-regulator design. The major difference in the approach is the introduction of augmented bias states in the disturbance model, which ultimately allows the designer to manipulate the desired robustness margins by changing a single design parameter. This guaranteed robust stability margin enables the use of very low order models, where the process dynamics outside the frequency band of the disturbance can be omitted, while maintaining closed-loop stability. Finally, the control scheme is scalable to processes of any input and output dimensions, without the need to change design structure,

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that is, the performance is still defined by three scalar parameters. The obtained control law can be extended into its nonlinear counterpart with ease by applying gain-scheduling as interpolation of a set of linear controllers designed for a particular frequency. Online frequency tracking is another field that has been rigorously studied and several solutions have already been proposed (Pai, 2010, 2009; Savaresi, Bittanti, & So, 2003). Although there exist several significantly more effective methods, the method utilised in this study, due to its ease of implementation and the fact that it suffices for the given problem, is a standard recursive least squares (RLS) based frequency tracking algorithm (Zazas, Daley, & Pope, 2010) with reconstructed disturbance signal. The combination of the controller and frequency tracking algorithm results in an LQ-based nonlinear adaptive controller with guaranteed robust stability. This paper is structured as follows. In Section 2 the problem formulation and the test process are presented. The modelling and identification is covered in Section 3. The control design is described in detail in Section 4. Simulation results are presented in Section 5 and the test bed results in Section 6. The concluding remarks and discussion are made in Section 7.

2. Problem formulation The process utilised in the study is a marine diesel engine and a generator set attached on a raft, which is supported at each corner by a passive mount as shown in Fig. 1. The reciprocating components within the engine i.e., the pistons act as a natural source for tonal disturbances. In addition wide band process noise is also generated. For most applications the transmission of these low frequency vibrations to surrounding structures is undesirable and therefore needs to be minimised. The major disturbances to be cancelled act at three separate tonal frequencies, namely 52 Hz, 65 Hz and 78 Hz. The tones are harmonically coupled and any variation in the first tone corresponding to the driving frequency of the motor results in same proportional variation in the latter two. Due to the nature of the process, the vibrations cannot be suppressed at their source, hence the control goal is to actively enhance the disturbance attenuation at the mounts of the raft. Although the disturbances are to be minimised in all directions, the emphasis is on the z-direction that is perpendicular to the mounts. Three inertial mass shakers are attached in an angle on the bottom of each mount, providing the required control forces in three dimensions. This arrangement forms part of the patented ‘‘smart-spring’’ concept as described in Daley et al. (2008). The transmitted vibrations in each direction are measured by

Fig. 1. A diesel engine mounted on a raft used as the test process. The disturbance transmissions in three directions are measured by accelerometers attached on the bottom of the mount.

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describes all the phenomena of interest while being of as low order as possible. The following subsections describe the relevant steps taken to obtain such plant model.

acceleration sensors at the bottom of each mount. A schematic diagram of the overall process is given in Fig. 2. The control goal is to minimise the measured accelerations, yielding a process with 12 inputs and 12 outputs. The problem studied in this paper is simplified, without loss of generality, by considering only one corner of the raft, resulting in a 3 by 3 system. The inputs of this system are the shaker control voltages limited to 710 V corresponding to approximately 50 N force. The outputs are the voltage outputs of the sensors directly proportional to the perceived accelerations. Although the operating speed of the engine is set to be constant, in practice it varies within 71 Hz. Because the running frequency of the engine cannot be measured directly without additional instrumentation, it is estimated here using a frequency tracking algorithm. In addition, as the performance of a linear controller is limited to a single predefined frequency and due to the measurement noise, it needs to be designed to be effective in a very narrow frequency band. Hence, even though the frequency variation is relatively small, the linear control law provides insufficient performance for the given problem and therefore a nonlinear frequency adaptive controller is introduced to obtain acceptable control performance.

3.1. Process model The process model in the study describes the dynamics of the control path to the voltage proportional to the acceleration measurements—in essence the actuator response model. Due to the complexity of the system, the model is obtained through data based black box identification by using a prediction error method (PEM), which is a standard iterative method for the identification of linear state-space systems (Ljung, 1999). The modelling data is obtained by feeding a pseudo random signal into each of the actuators one at a time and measuring the resulting voltages that have been filtered with an analogue low-pass filter with the cut-off frequency set at 1 kHz. The sampling rate for the measurements is chosen as 2 kHz. This choice provides adequate information on the process dynamics at the high frequencies while preserving the low frequency dynamics. The original data contains information on the process dynamics on the frequencies that are not relevant to the given problem. Due to these additional dynamics, a suitable model would be of order higher than 100, which is obviously too complex for real-time implementation. Fitting such model for the data is also very sensitive to numerical problems—resulting in poor modelling accuracy. In order to avoid the numerical problems and to obtain a low order model, the data is band-pass filtered in the frequency domain prior to the identification. The filtering is realised by first introducing a DFT, where the elements outside the frequency band are set to zero, given as: 8 0 ,k r klow þ 1 > > > > > þ1 o ko khigh þ 1 F ð k Þ ,k > low < 0 ,khigh þ 1 rk r Nkhigh þ 1 , XðkÞ ¼ ð1Þ > > > þ1 o ko Nk þ1 F ð k Þ Nk > high low > > : 0 Nk þ 1 rk

3. Identification and modelling One of the key concepts for creating a successful model based control scheme is a reliable model representing the process and disturbance dynamics. The model can be defined either by the physical properties of the system, by data based model fitting or a mixture of these. The common approach is to generate a model that

low

PN1

j2pNnk

where FðkÞ ¼ n ¼ 0 xðnÞe ,k ¼ 0,1,    ,N1, is the discrete Fourier-transform (Kamen, 1990), N is the number of samples, x(k) is the signal being filtered, klow ¼ flow(N 1)h and khigh ¼fhigh(N 1) hare the indexes to low and high cut-off frequencies, respectively, h is the sampling interval, flow and fhigh are the low and high cut-off frequencies in Hertz. The filtered time-domain signal is obtained trivially by applying inverse DFT on the filtered signal in Eq. (1), given as: xfilt ðkÞ ¼

1 1 NX 2pnk XðnÞej N ,k ¼ 0,1,. . .,N1 Nn¼0

ð2Þ

In the given problem, the data is band-pass filtered to the frequency range 20 Hz to 180 Hz. The amplitude spectra of the pre-processed data are shown in Fig. 3.

Fig. 2. Illustration of the control scheme.

0.14

0.018

0.12

0.016 0.014 Amplitude

Amplitude

0.1 0.08 0.06 0.04

0.012 0.01 0.008 0.006 0.004

0.02 0

0.002 0

50

100

150

200

Frequency (Hz)

250

300

0

0

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Frequency (Hz)

Fig. 3. Amplitude spectra of the filtered process for a single channel. (a) Output, (b) input.

250

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J. Orivuori et al. / Control Engineering Practice 20 (2012) 1206–1219

where yp(t) is a vector of actuator outputs and u(t) is a vector of control voltages.

After the identification has been carried out the model accuracy is validated in both time and frequency domain. Only the frequency domain validation that is done by comparing the frequency response of the model against the measured frequency response for each channel is presented here. The validation results are shown in Fig. 4. According to the validation results the obtained model provides a reliable representation of the process dynamics within the desired frequency band of interest. It should be noted though that for the x-direction the model estimate is clearly poorer than in the other directions. This inaccuracy in addition to the neglected process dynamics should be taken into account in the control design by ensuring that the controller is sufficiently robust with respect to these known model errors. The balanced realisation of the obtained model can be expressed in continuous time with the following 20th order state-space representation: x_ p ðtÞ ¼ Ap xp ðtÞ þ Bp uðtÞ yp ðtÞ ¼ Cp xp ðtÞ

,

> > :d

Gain (dB)

−20

−10 −20

−20

−40

400

−50

500

0

100

Frequency (Hz)

200

300

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500

−50

−10 −20

−30

−40

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−50

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−20

−10 −20

−40

−40

−40

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400

500

−50

measured modelled

−20 −30

300

500

−10

−30

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0

−30

100

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10

Gain (dB)

Gain (dB)

−10

200

20 measured modelled

0

0

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20 measured modelled

0

−50

0

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20 10

measured modelled

−20

−30

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−10

−30

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0 Gain (dB)

Gain (dB)

−20

300

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0

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10

0

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20 measured modelled

0

0

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20 10

measured modelled

−10

−40 300

ð4Þ

0

−40 200

,

10

−30

100

xddi ðtÞ

20 measured modelled

−30

0

#

where oddi is the disturbance frequency in rad/s, e is a design variable (small number, zero by default), subscript i denotes the tone number. The initial values are given as xddi ð0Þ ¼   0 Addi oddi T , where Addi is the amplitude of the signal.

−30

−50

1



0

−10

0

o2ddi e  ddi ðtÞ ¼ Cddi xddi ðtÞ ¼ 1 0 xddi ðtÞ

10

0 Gain (dB)

" 8 > > < x_ ddi ðtÞ ¼ Addi xddi ðtÞ ¼

20 measured modelled

10

Gain (dB)

The tonal disturbances affecting the process can be described as a sum of sinusoidal disturbances acting at some predefined frequencies. A single self-driven sinusoidal tone is expressed as:

ð3Þ

20

Gain (dB)

3.2. Disturbance model

Gain (dB)

(

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0

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400

500

−50

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Frequency (Hz)

Fig. 4. Validation of the model by comparing the estimated output against the measured output in frequency domain. (a)–(c) Inputs u1-3 to x-direction. (d)–(f) Inputs u1-3 to y-direction. (g)–(i) Inputs u1-3 to z-direction.

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The combined zero mean disturbance with three tones in x,y or z-direction is expressed as:

ð5Þ where dj(t) is the sum of the disturbance tones, subscript j denotes the disturbance direction (x,y or z). The initial values are given as . In practice, the disturbance signal never consists solely of distinct tones, as it is of non-zero mean and includes some random noise. Traditionally, this factor is not taken into account in the disturbance modelling, resulting in an inherently biased estimate for the disturbance. The authors propose the use of augmented states corresponding to these neglected dynamics. These states essentially improve the performance of a stateestimator and, as will be shown later on, will allow direct manipulation of the robustness properties of the controlled closed-loop system. A biased sinusoidal multi-tone tone signal is given as: ^ ¼ sinðxtÞ þ bðtÞ dðtÞ

ð6Þ

Without the loss of generality, the biases can be assumed to be constant deviations from zero (time invariant). This assumption can be made because in the implementation the applied stateestimator compensates the state error resulting from this false assumption at each sample instant. The bias states can now be written as: xbias ðtÞ ¼ eet xbias ð0Þ, ð7Þ   where xbias ð0Þ ¼ b1 b2    bn are the initial deviations from zero. The derivative is given as: x_ bias ðtÞ ¼ exbias ðtÞ,

ð8Þ

where e is a design parameter similar to the one in Eq. (4). The overall non-zero mean disturbance in three directions can be written as:

ð9Þ

3.3. Model composition For control design purposes a model describing the impact of the disturbances to the measured process outputs is required. Such a plant model is obtained by combining the models in Eq. (3) and Eq. (9), yielding the following state-space representation:

ð10Þ

where u(t) is a vector of applied control voltages and y(t) is a vector of actuator outputs subject to tonal disturbances. A schematic diagram of the resulting control scheme is shown in Fig. 5.

Fig. 5. A schematic diagram of the control setup.

4. Control design There exist a wide variety of different nonlinear control schemes for the mitigation of frequency varying disturbances (Knospe et al., 1997; Bittanti et al., 2002; Du et al., 2003; Bohn et al., 2004; Kinney & de Callafon, 2006, 2007; Tammi et al., 2007; ¨ glu et al., 2011a, 2011b; Ballesteros et al., 2011; Knospe et al., Koro˘ 1997). The design approaches for these methods vary from internal model control principles to periodic and robust control schemes, where the disturbance is the unknown parameter. In the most of the approaches, the nonlinear controller is obtained through the direct gain-scheduling, resulting in frequency varying feedback matrices. The performance of each of the methods has been verified in a practical implementation. Although the methods are well applicable, their design includes some relatively involving control theory, which may not be easily applicable by an industrial system specialist implementing the control law. Also the performances of the methods have mostly been assessed in SISO-systems while many of the industrial processes are true MIMO-systems. Another family of controllers commonly used in the suppression of tonal disturbances with static frequencies, which can readily be extended to frequency varying processes are the LMS-based methods, such as convergent control (Tammi, 2007). Unfortunately, they have some severe robustness problems when multiple disturbance tones are suppressed with a slightly damped frequency peak of the underlying process located between the tones. This issue becomes even more severe when MIMO-systems are considered, making the choice of the proper design parameters a very tedious task. These robustness issues are a direct consequence of the fact that the method is not based on the system model (apart from the discrete frequencies). The authors propose a less involving nonlinear model based control design that is based on a set of gain-scheduled linear LQ-controllers, whose design is well known by most control engineers. The proposed controller can be readily extended to mitigate an arbitrary number of frequency varying tonal disturbances in multiple directions with an arbitrary number of control inputs (assuming that adequate computing power is available). The use of LQ-based control design in vibration control is not a novel approach and has been rigorously studied in Sievers, Blackwood, Mercadal, & von Flotow (1991) and Bittanti et al. (1996). Both of these methods provide very high suppression at the given disturbance frequency and have guaranteed stability. However, they have some issues when implemented into MIMOsystems, because a poor choice for controller weighting results in poor stability margins and excessive control usage outside the frequency tones (i.e., they are sensitive to measurement noise). The proposed control approach addresses a solution to this

J. Orivuori et al. / Control Engineering Practice 20 (2012) 1206–1219

problem providing high suppression of tonal disturbances while maintaining low sensitivity to noise and additional disturbances outside the design frequency. The properties of the closed-loop system can be manipulated with three parameters, making the control synthesis a trivial task even for a very complex system. The control law can also be applied on non-square plants, providing the optimal solution for any type of tonal vibration control problem regardless of the system dimensions. For notational convenience, the control design in the following subsections is presented in continuous time although the actual implementation is made in discrete time. This allows the designer to discretise the resulting control law with any method suitable for the process at hand.

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The optimal observer is such that it minimises the cost function: Z 1 T ~ þ uTobs ðtÞRobs uobs ðtÞÞdt, Jobs ¼ ðx~ ðtÞQ obs xðtÞ ð16Þ 0

~ where xðtÞis the relative state-estimation error, uobs(t)is an artificial control signal, Qobs Z0 and Robs 40 are weighting matrices determining the convergence properties of the estimate. The optimal estimator weighting matrix minimising the cost function is obtained by exploiting the duality properties of the system (Rugh, 1996) and is given as: K ¼ PCT R1 obs ,

ð17Þ

where P is the solution of the ARE: T APþ PAT PCR1 obs C P þQ obs ¼ 0

4.1. LQ-control design The optimal linear control law is obtained as state-feedback providing a control signal that minimises the quadratic cost function (Anderson & Moore, 1989): Z 1 J¼ ðzðtÞT Q z zðtÞ þ uðtÞT Rz uðtÞÞdt, ð11Þ 0

where Qz Z0 and Rz 40 are diagonal matrices defining the weighting of plant outputs and control effort of each channel respectively, u(t) is the applied control signal and z(t) is a performance variable. For the given problem the performance variable is chosen such that it expresses the sum of the process output and the disturbance tones that are to be mitigated. The bias states, in essence the disturbances outside the frequency tones, are not to be controlled. The choice of omitting the bias-states is one of the major factors in focusing the control energy on very narrow frequency bands, significantly improving the closed-loop robustness thereof. With these choices, the performance variable is given as: ð12Þ The optimal control effort is given as Anderson et al. (1989): u ðtÞ ¼ LxðtÞ ¼ Rz BT SxðtÞ, n

ð13Þ

where S is the solution of the algebraic Riccati equation: T T AT S þ SASBR1 z B S þ Cz Q z Cz ¼ 0

In Eq. (18) the weighting matrix for estimation error Qobs determines the convergence properties of each state-estimate having therefore a significant impact on the overall closed-loop performance. A poor choice for the weighting matrix leads to very poor control performance. Unfortunately, unlike in the statefeedback design in Eq. (14) there is no obvious way of choosing the weights. There are some rules of thumb for choosing the weights, some of which can be found in Anderson et al. (1989), however none of these are very applicable to the given problem. The authors propose a weighting matrix of the following form for the problems belonging to the class considered herein: Q obs ¼ WGscale ,

ð19Þ

where Gscale is a matrix scaling the squared state variation into unity and W is a weighting matrix given as: n o ð20Þ W ¼ diag a½ 1    1 p b½ 1    1 d g½ 1    1 b , where subscripts p, d and b denote vectors of appropriate dimensions related to the process, disturbance and bias states, respectively. The scalars a, b and g are design parameters determining the convergence properties of the estimate. The determination of the scaling matrix Gscale is a straightforward procedure. First, the output weighting matrix Cp in Eq. (3) is replaced by an identity matrix and the resulting model is converted into a transfer-function matrix G(s), which describes the dynamics of each state, given as:

ð14Þ

It is a well-established fact that in order for the Riccatiequation to have a unique solution, the plant model should not contain any imaginary poles, which is not the case in the system considered herein. This fundamental problem can be evaded by setting the design variable in Eq. (9) a small value, say e ¼10  8. This can be interpreted as the inclusion of a very small damping coefficient in the sinusoidal disturbance. It can be shown that in practice this perturbation has no impact on the obtained feedback gain matrix; although, strictly speaking, this results in a suboptimal solution. 4.2. State-estimator design In order to apply the state-feedback, the unmeasurable process states need to be estimated. For this purpose a deterministic state-observer (Anderson et al., 1989) is used, which is given as: _^ ¼ ðAKCÞxðtÞ ^ þ BuðtÞ þ KyðtÞ, xðtÞ

ð18Þ

ð15Þ

where K is the estimator gain and x^ ðt Þis a vector of estimated states.

ð21Þ

where subscripts m and n denote the number of process states and inputs, respectively. The highest possible variation of each state can now be found as the HN-norm of the related row vectors in the transfer-function matrix. For the given problem, this approach may yield poor results as it is known a priori that the control energy is concentrated mostly on the disturbance frequencies; hence the HN-norm may yield very biased scaling if the operation point is far from the highest amplification of the process. In order to avoid these problems, the scaling is modified such that it only considers the state variation at some pre-defined frequencies. Without loss of generality it can be assumed that the control inputs are scaled to relative unit variance, yielding 9u1 ðjoÞ9 ¼ 9u2 ðjoÞ9 ¼    ¼ 9un ðjoÞ9 ¼ 1. Now the state variation is essentially the row sum of the amplifications of the individual transfer-functions on a frequency, given as: 9xm ðjoÞ9 ¼ 9Gm1 ðjoÞ9 þ9Gm2 ðjoÞ9 þ    þ 9Gmn ðjoÞ9 ¼

n X

9Gmk ðjoÞ9

k¼1

ð22Þ

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In presence of multiple disturbance signals, the scaling factor for a single state is readily given as: gm ¼

n X

9Gmk ðjo1 Þ9 þ

k¼1

¼

p X

n X

9Gmk ðjo2 Þ9 þ    þ

k¼1 n X

n X

9Gmk ðjop Þ9

k¼1

9Gmk ðjor Þ9

ð23Þ

r¼1k¼1

The scaling vector for the process states can now be written as: " T

gp ¼ ½g 1    g m  ¼

p X n X

9G1k ðjor Þ9   

r ¼1k¼1

p X n X

#T 9Gmk ðjor Þ9

4.3. Nonlinear controller The nonlinear controller (Orivuori & Zenger, 2010) is obtained by combining a set of linear controllers designed as described above. In practice the combination is implemented as a continuous gain scheduling by interpolating the static controllers over a predefined frequency range, resulting in a frequency dependent controller expressed as: ( x_ cont ðt,f hz Þ ¼ ðAðf hz ÞBLðf hz ÞÞxcont ðt,f hz Þ þ Kðf hz ÞyðtÞ , ð27Þ uðt,f hz Þ ¼ Lðf hz Þscont ðt,f hz Þ where fhz is the disturbance frequency in Hertz and L(fhz) and K(fhz) are frequency dependent gain matrices.

r ¼1k¼1

ð24Þ With the scaling for the process states determined, all that remains is to define similar scaling for the states related to the disturbance dynamics. The variation of these states is unambiguously defined by the disturbance frequency, under the assumption of unit amplitude. The related scaling vector is given as:

ð25Þ Now, the overall weighting matrix is given as:  h T Gscale ¼ diag gp

gTd

gTb

iT 2

ð26Þ

With the estimation error weighting matrix defined as in Eq. (19), the manipulation of the convergence properties of the estimator becomes an easy task. The application of the scaling matrix makes the weighting problem independent of the underlying system properties, essentially enabling the use of the same weighting parameters defined in Eq. (20) for similar convergence, regardless of the underlying process. The relative ratio of the design parameters a, b and g, related to the process, disturbance and bias properties, explicitly determines the resulting closedloop characteristics. Roughly generalising, the higher the value of a the less robust the closed-loop system is to the modelling errors while the convergence of some initial process states is faster; the higher the value of bthe faster the compensation converges and the higher the obtained steady-state mitigation are; the higher the value of g the higher the closed-loop stability margins are while the convergence speed is slower. In general, the parameters a and b are set first such that some design specifications are met and the parameter g is then used to define the final convergence speed and stability margins. With the proposed design structure the tedious and often heuristic task of choosing the weighting parameters for the control design is reduced to the choice of three parameters determining the closed-loop properties and three diagonal matrices determining the applied control effort. Remark. With the above formulation, the impact of the bias states becomes obvious. As these states are not used in the control feedback, all the energy set to these states on some frequencies increase the closed-loop robustness on these frequencies. Essentially, the bias states can be interpreted as all passfilters, whose gain threshold is set by the g-parameter. The higher the gain on a frequency the less of the signal energy is directed to the process and disturbance states, hence their impact on the realised control effort is minimised. This property can be readily verified by comparing the singular values from the estimator input to the process and disturbance states against the singular values from the estimator input to the bias state while the gparameter is varied.

Remark. Although the nonlinear controller is formed as a composition of optimal linear controllers, the resulting control law is not optimal in general. The design of such optimal controller would require exact a priori information on the disturbance variation, which is not a feasible assumption. Hence, it is rather unlikely that a generic optimal nonlinear controller is ever available. The closed-loop stability for each of the above controllers is inherently guaranteed by LQ-design framework (Kwakernaak & Sivan, 1972) if no parameter variation is present—gain scheduling is frozen. The analytical derivation of the stability under frequency variation would require the analysis of a Lyapunovfunction in terms of both time and frequency, resulting in a high order partial differential equation. This is not a feasible approach to the stability analysis of a general system. Fortunately, it is shown in Desoer (1969), Guo & Rugh (1995), Rugh (1991) that the above controller stabilises the process under smooth variation of the frequency—that is always the case with practical rotating systems. This assumption is further validated by simulations with the process subject to a linear sine sweep (see Section 5). Another study was carried out (the results not presented herein), where the assumption of smooth variation was violated by setting the frequency to switch at every second sample instant between some two discrete values. In this case, the closed-loop system ended in a limit cycle, yet remained BIBO-stable. After the switching was stopped, the amplitude of the perceived disturbances converged to their steady state value. Hence, the proposed control law guarantees closed loop stability if the scheduling variable, the frequency estimate, is bounded within the design range of the nonlinear controller. Naturally a biased frequency estimate may yield poor control performance, explicitly defined by the sensitivity function of the corresponding static controller. However, the process remains stable. The elements of the frequency dependent gain matrices L(fhz) and K(fhz) can be chosen as arbitrary functions of frequency that provide an adequate fit against the point values defined by the feedback matrices for a set of the linear controllers at specific frequencies as shown in Fig. 6 for a single element. It must be emphasised that a poor fit of these functions may result in an unstable closed-loop process, thus care must be taken in choosing them. For the given system the gain matrices are obtained by fitting 4th order polynomial functions of frequency for all individual elements of both matrices over the whole design range. This choice of the functions preserves the structure of the original controller with nonlinear inputs, making the implementation of the controller an easy task. In order for the gain matrices to yield a good generalisation, all of the frequency tones should be varied independently resulting in fitting a function describing a hyper surface spanned by these frequencies. Fortunately, the disturbance tones are closely related to each other and the problem can be simplified into a curve fitting against the

J. Orivuori et al. / Control Engineering Practice 20 (2012) 1206–1219

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0.2 0.15

Desired variation Polynomial fit

Parameter value

0.1 0.05 0

Fig. 7. Reconstruction of the disturbance signal. Gact(z) is the actual control path with output yact(k) and Gp(z) is a model of the control path with output yp(k).

−0.05

expressed as:

−0.1 −0.15 50

eðkÞ ¼ AðkÞsinðod ðkÞkh þ jÞ þnðkÞ, 51

52

53 54 Frequency (Hz)

55

56

ð28Þ

Fig. 6. Variation of an element of the observer gain matrix as a function of frequency.

where A(k) and od(k) are the varying amplitude and frequency, f is an arbitrary constant phase, h is the sampling time and n(k) is the background noise signal. The time varying disturbance frequency od(k) can be expressed as a summation of a constant reference frequency oref and a time varying frequency term oe(k):

frequency of the first tone by defining the higher tones to have some static relation to the first one.

od ðkÞ ¼ oref þ oe ðkÞ

ð29Þ

The estimated disturbance signal can now be expressed as: 4.4. Frequency tracking algorithm For the problem considered herein the speed of the engine varies continuously, and no measurement of the running frequency is available. Hence, an algorithm that estimates the instantaneous value of the disturbance frequency and tunes the parameters of the proposed control algorithm accordingly is required. In this study an RLS-based algorithm is used to extract the frequency information from the process measurement in one direction (Zazas et al., 2010). The implemented estimator algorithm is rather trivial and computationally light, yet it provides adequate accuracy in the given setup and is chosen as the approach thereof. For more complex estimation schemes there exist significantly more effective approaches such as (Pai, 2010, 2009; Savaresi et al., 2003). The major problem with the frequency estimation is the fact that the signal used for the frequency extraction is also the controlled quantity. Hence, when the controller is switched online, the tonal disturbance is suppressed and the measured signal contains less information on the disturbance frequency. This results in a biased frequency estimate, which again results in a poor control effort. This again results in the reappearance of the disturbances in the measured signal and the frequency information related to it. Ultimately, this leads into a cycling system, where the obtained disturbance mitigation varies between significant suppression and no suppression. In order to overcome this problem, the signal fed to the estimator has to be a reconstruction of the plant output rather than the actual measured signal. As the estimated process model is available,the reconstructed signal is obtained as the sum of the true process subject to the disturbance and the negated output of the process model, resulting in a structure (see Fig. 7) similar to those used in IMC. If the control path model is an accurate representation of the true system, for the frequency range of interest, then yp(k) ffiyact(k), hence e(k) ffi d(k). In the case of biased model (a likely occurrence), the reconstructed signal becomes the sum of the residual of the true disturbance and the negated control effort with biased phase and amplitude. Despite these biases the resulting signal still contains the true exact frequency, which can be used for the frequency extraction. The resulting signal is

eðkÞ ¼ AðkÞsinðoref kh þ FðkÞÞ þ nðkÞ,

ð30Þ

where

FðkÞ ¼ oe ðkÞkh þ j

ð31Þ

is the time varying phase of the estimated disturbance signal. Eq. (30) can be expanded and rewritten in a regression form as: eðkÞ ¼ hT ðkÞrðkÞ þ nðkÞ

ð32Þ "

AðkÞcosðFðkÞÞ AðkÞsinðFðkÞÞ hðkÞ ¼ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

where

y1 ðkÞ

y2 ðkÞ

#T and

rðkÞ ¼ ½sinðoref khÞ cosðoref khÞT . The reference vector r(k) is a pair of sine and cosine signals acting on a frequency chosen by the designer. The frequency selection condition is not necessary for the RLS-based frequency tracking algorithm to produce an accurate estimate of the disturbance frequency, however the closer to the disturbance frequency the reference frequency is selected the faster an accurate estimate is obtained. Since Eq. (34) has a standard regression form, the parameter vector h(k) can be estimated with a standard RLS-algorithm such as (Ljung, 2002). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ^ ^ ðkÞ ¼ tan1 It then follows that AðkÞ ¼ y^ 1 ðkÞ þ y^ 2 ðkÞ and F ^ ^ ðy ðkÞÞ=ðy ðkÞÞ. 2

1

The estimated disturbance frequency is then obtained by solving the frequency error in Eq. (29) through differentiation in discrete time and substitution of the result in Eq. (31), yielding: ^ d ðkÞ ¼ oref þ o

^ ðkÞ ð1z1 ÞF h

ð33Þ

5. Simulation study Prior to the test bed implementation, the controller performance is validated by simulations. In the simulations three tonal disturbances are added into each output channel, emulating the real process subject to some unknown disturbance source. The process model is simulated in continuous time, while the controller is updated in discrete time with the sampling rate of 2 kHz, as in the real application. For clarity, the disturbance tones are

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where S(s), T(s) and TI(s) are the sensitivity, complementary sensitivity and input complementary functions, respectively. The obtained vibration mitigation on each frequency is defined by studying the singular values of S(s), given in Fig. 8. According to the results, the controller provides very high vibration mitigation of over 100 dB, while being completely insensitive to disturbances outside to tones that are being controlled. The closed-loop sensitivity to measurement noise, that is whether the controller amplifies the process noise, is determined by studying the singular values of T(s), given in Fig. 9(a). The stability margins of the closed-loop system are defined from the generalised Nyquist diagram Machiejowski (1989), formed by drawing the characteristic loci of the open-loop system over the whole frequency range, given as:

assumed to be in the same phase with the same amplitude. The frequency variation in the true process is small. Hence, in order to prove the applicability of the controller for systems with significant frequency variation, the frequency variation in the simulations is chosen to be a double sided linear sine sweep from 20 Hz to 150 Hz and back in 10 s. The parameters used for the control synthesis in the simulations are: Q z ¼ Robs ¼ I Rz ¼ In 106

a ¼ 103 b ¼ 103

g ¼ 1012

ð34Þ

detðGp ðjoÞGc ðjoÞlðoÞIÞ ¼ 0,8o A 1,1½

In order for the system to be stable the characteristic loci must encircle the point 1 anticlockwise as many times as there are unstable (Smith–McMillan) poles. For the given system this implies no encirclements. The generalised Nyquist diagram is depicted in Fig. 9(b). According to the results, the controlled system is insensitive to measurement noise with average of  30 dB gain in the frequencies outside the controlled tones. The system has high stability margins of approximately 13 dB in every direction. The similar values of the margins are a direct consequence of the scaling and weighting used in the control design. Finally, the robust stability margins of the closed-loop system defining the tolerable modelling errors are determined. In the given problem, where some of the process dynamics were purposefully omitted from the modelling, it is must be verified that the omitted dynamics stay within these limits in order to

5.1. Theoretical control performance The theoretical control performance is assessed in the frequency domain by several different methods. For clarity, the cross-sections of the frequency varying surfaces at 53 Hz basefrequency are considered. It should be noted that for MIMOsystems singular values and characteristic loci are used for the analysis. Hence the results are rather conservative and only reliable for the worst case analysis. The transfer-functions used in the analyses are given as (Skogestad & Postlethwaite,2005): SðsÞ ¼ ðIþ Gp ðsÞGc ðsÞÞ1 TðsÞ ¼ ðI þGp ðsÞGc ðsÞÞ1 Gp ðsÞGc ðsÞ ,

ð35Þ

50

50

0

0 Magnitude (dB)

Magnitude (dB)

TI ðsÞ ¼ Gp ðsÞGc ðsÞðIþ Gp ðsÞGc ðsÞÞ1

−50

−100

−150

−50

−100

−150

σmax

σmax σmin

σmin −200

0

200

400

600

800

ð36Þ

1000

−200 30

40

50

60

70

80

90

100

110

Frequency (Hz)

Frequency (Hz)

Fig. 8. (a) Closed-loop process sensitivity to output disturbances. (b) Close-up of the sensitivity function.

2

10

λ1, Gm=12.36dB

σmax

0

σmin

−20

Imaginary axis

Magnitude (dB)

−10

−30 −40 −50

λ2, Gm=12.83dB

1

λ3, Gm=13.67dB

0.5 0 −0.5 −1

−60

−1.5

−70 −80

1.5

0

200

400

600

Frequency (Hz)

800

1000

−2 −2

−1.5

−1

−0.5

0

0.5

1

1.5

Real axis

Fig. 9. (a) Closed-loop process sensitivity to measurement noise. (b) Generalised Nyquist diagram of the system.

2

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process, namely a change from 20 Hz to 150 Hz and back with the rate of 26 Hz/s. The resulting response of the controller process with respect to the uncontrolled case is shown in Fig. 11(a). The frequency variation of the first disturbance tone is given in Fig. 11(b). According to the results, the controller provides very high vibration damping over the whole frequency band, with no stability issues. Some deviation in the obtained damping can be perceived at certain frequencies. The source of this deviation is either a sudden change in the process dynamics (in the vicinity of the frequency peaks) or the frequency being close to the limits of the scheduled frequency range (the polynomial fit is generally worse in the ends of the curve). Regardless, if the scheduling is frozen at any point, the response converges to amplitudes specified by the linear case at that frequency.

avoid control spill-over and the instability resulting thereof. For MIMO-systems, the robust stability margins are given by two functions (readily derived for example from Machiejowski, 1989), Do(s)¼T  1(s) and DI ðsÞ ¼ T1 I ðsÞ, the output and input error functions, respectively. The singular values of these functions define an upper and lower bounds for the tolerable modelling errors as a function of frequency. The tolerable modelling errors for the given problem are given in Fig. 10. According to the graphs, the system can tolerate approximately 30 dB output modelling errors and practically no input modelling errors. However, as these bounds are given by singular values, they are very conservative thereof. A more accurate description would be obtained by using m-structured singular values; however, in practice the study of the tolerance to output modelling errors only has provided satisfactory accuracy.

6. Test bed implementation 5.2. Time domain control performance After acceptable simulation results are obtained the control algorithm is implemented on the test bed. The process setup is as illustrated in Fig. 2. A 2 kHz sampling rate is used for the process measurements and the control update. Prior to the implementation, the theoretical control limitations are defined in order to verify that the chosen actuators are capable of providing the required control effort.

The time domain control performance is assessed by simulations. These tests provide information on the convergence rate and performance deterioration due to numerical issues – features that cannot be extracted from steady-state analysis. The process is simulated in continuous time, while the controller is implemented in its discrete form with the sampling rate set to 2 kHz (ZOH assumed). The polynomials used in the scheduling are selected to be of the order 15 to cover the dynamics occurring in the extended frequency range. In the simulations, the process is subject to three frequency varying tonal disturbances with the offset of 100. In order to verify the control concept, the frequency variation is chosen significantly higher than that of the true

6.1. Control limitations In practice, the control signal is always limited to some maximum absolute value. This fundamental limitation may cause problems in terms of the control performance and design. The

70

100

60

Magnitude (dB)

Magnitude (dB)

50 50

0

40 30 20 10

σmax −50

0

200

400

600

800

σmax

0

σmin 1000

−10

σmin 0

200

400

Frequency (Hz)

600

800

1000

Frequency (Hz)

Fig. 10. (a) Tolerable input modelling errors. (b) Tolerable output modelling errors.

160 no control z−direction y−direction x−direction

130 120

140

Frequency (Hz)

Amplitude (V)

120 110 100 90

100 80 60

80 40 70 0

1

2

3

4

5 Time (s)

6

7

8

9

10

20

0

2

4

6

8

10

Time (s)

Fig. 11. (a) Response of the process subjected to frequency varying disturbances with and without control. (b) Frequency variation.

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In order to avoid input saturation it is required that: X 1 9Gp ðjoÞWd ðoÞ9 r K,8o A R þ ,9DðjoÞ9 a0,

likelihood of having such problem in the given process can be assessed by studying the required control effort in terms of perfect control (disturbance cancellation). In order to have a clear interpretation, the disturbance signals have to be properly scaled and in the multi-output case, their relative phases have to be taken into account. The characteristics of the disturbance signals can be extracted from the Fourier-transformed process output, in the presence of no control. For the frequency varying case, the analysis would need to be done over the whole frequency range, which is not a feasible approach. Hence, the possible issues are assessed only on a specific frequency (in the given case on 53 Hz) and the decision is based on whether these margins are satisfactory. The frequency characteristics extracted from the process are given as: j0:003705jx ,+167:41 d53Hz ðtÞ ¼ j0:006982jy ,+11:51 j0:03256jz ,+17:81

where K is a vector of the saturation limits of each actuator. For the given process the expected total control effort is: 2 3 0:3476 X 1 6 7 9Gp Wd 9 ¼ 4 0:2446 5 ð40Þ o 0:0703 According to Eq. (42) all shaker inputs are well within the allowable limits while the first two constitute significantly higher effort than the third one, indicating that the perfect control might be possible even with two shakers. It is apparent that in theory the perfect control can be obtained with the current setup.

j0:0099jx ,+681 d66Hz ðtÞ ¼ j0:03102jy ,+132:21 j0:1138jz ,+134:41

Remark. The preceding results assume tonal disturbances; in reality the controller is somewhat sensitive to process noise, hence adding to the realised control effort and lowering the ‘‘gain margin’’ of the actuators. It should also be noted that the possible model errors (especially phase errors) and variation of the disturbance amplitudes all alter the values obtained in Eq. (40).

j0:05152jx ,+781 d79Hz ðtÞ ¼ j0:07752jy ,+165:51 , j0:306jz ,+163:11

ð37Þ

where the phases are relative to t¼0. In terms of the disturbance frequencies, the perfect control is expressed as: þ UðjoÞ ¼ G1 p ðjoÞWd ðoÞDðjoÞ,8o A R ,

6.2. Measured control performance The control performance is evaluated by assessing the vibration mitigation at the process outputs. The control goal is to yield maximal suppression for three tonal disturbances while having no impact on the harmonics of these tones. The time-domain performance of the controller is given in Figs. 12–14. In order to make the control effort more distinguishable the band-pass

ð38Þ

where Wd(o)is a frequency dependent scaling matrix taking the amplitudes and relative phases of the signals into account and D(jo) is a vector of the Fourier components of disturbances at a frequency.

1.5

0.15 without control controlled

1

0.1

0.5

0.05

Amplitude (V)

Amplitude (V)

without control controlled

0

0

−0.5

−0.05

−1

−0.1

−1.5

−0.15 0

2

4

6

8

10

0

2

4

Time (s)

6

8

10

Time (s)

Fig. 12. Process responses in x-direction with and without control, (a) measured, (b) filtered.

1

0.3

without control controlled

0.8

Amplitude (V)

0.4

Amplitude (V)

without control controlled

0.2

0.6

0.2 0 −0.2 −0.4 −0.6

0.1 0 −0.1 −0.2

−0.8 −1

−0.3 0

2

4

6

Time (s)

8

ð39Þ

o

10

0

2

4

6

Time (s)

Fig. 13. Process responses in y-direction with and without control, (a) measured, (b) filtered.

8

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J. Orivuori et al. / Control Engineering Practice 20 (2012) 1206–1219

1.5

1217

0.8 without control controlled

without control controlled

0.6

1

Amplitude (V)

Amplitude (V)

0.4 0.5 0 −0.5

0.2 0 −0.2 −0.4

−1 −1.5

−0.6 0

2

4

6

8

−0.8

10

0

2

4

6

8

10

Time (s)

Time (s)

Fig. 14. Process responses in z-direction with and without control, (a) measured, (b) filtered.

0.4

52.3 Shaker−1 Shaker−2 Shaker−3

0.3

52.25 52.2 Frequency (Hz)

Amplitude (V)

0.2 0.1 0 −0.1

52.15 52.1 52.05

−0.2

52

−0.3

51.95

−0.4

0

2

4

6

8

51.9

10

0

2

4

Time (s)

6

8

10

Time (s)

Fig. 15. (a) Control voltages fed into each shaker. (b) Frequency variation of the 1st disturbance tone.

−25

−25 −30

−30

−35

−35

Amplitude (dB)

Amplitude (dB)

Without control Controlled

−40 −45

−40 −45 −50

−50 −55

Without control Controlled

0

50

100 Frequency (Hz)

150

200

−55

50

55

60

65 70 Frequency (Hz)

75

80

Fig. 16. (a) Frequency spectra of the process output in x-direction with and without control, (b) zoom-in.

filtered results are shown as well. Evidently, the controller provides very high damping in the z-direction while having slightly worse performance in the x- and y-directions. Another important aspect is the applied control voltage; particularly, whether the applied control voltages remain within the allowable limits (710 V). According to the results given in Fig. 15(a), the applied control effort agrees well with the theoretical values derived in Eq. (42) and remains well within the saturation limits. The variation of the disturbance tones is evaluated by studying the output of the frequency estimator, given in Fig. 15(b). According to the graph, the frequency variation is rather small, yet it is sufficient to significantly deteriorate the performance of linear controllers with a very narrow effective frequency band. As the time-domain results were partially inconclusive, the true damping performance is assessed through the analysis of the output power spectra. These spectra make the control impact

clearly distinguishable and help to determine the presence of measurement noise. The spectra for each direction are given in Figs. 16–18. According to the results, nearly perfect suppression is obtained on the frequencies that were to be mitigated while the adjacent frequencies remain unchanged. These results agree well with the results obtained in the theoretical performance analysis in Section 5.1. It is also notable that on the mitigated frequencies the noise floor has been reached, implying that the process is as good as it can be with the given sensors. The overall control performance can be considered highly acceptable, considering the amount of process noise and the fact that the control impact is indeed very good, especially in the z-direction. The adaptation to varying disturbance frequency is clearly working as the running frequency of the engine is constantly altering. The control signals are well within limits and agree with the simulations.

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−15

−15 Without control Controlled

−20

−20 −25 Amplitude (dB)

Amplitude (dB)

−25 −30 −35 −40

−30 −35 −40

−45

−45

−50

−50

−55

Without control Controlled

0

50

100

150

−55 50

200

55

60

Frequency (Hz)

65 70 Frequency (Hz)

75

80

Fig. 17. (a) Frequency spectra of the process output in y-direction with and without control, (b) zoom-in.

−10

−10 Without control Controlled

−15 −20

−20

−25

Amplitude (dB)

Amplitude (dB)

Without control Controlled

−15

−30 −35 −40

−25 −30 −35

−45

−40

−50

−45

−55

−50 0

50

100

150

Frequency (Hz)

200

50

55

60

65

70

75

80

Frequency (Hz)

Fig. 18. (a) Frequency spectra of the process output in z-direction with and without control, (b) zoom-in.

7. Conclusions Blocking of multi tonal disturbances below a passively mounted marine diesel has been shown to be possible using inertial mass shakers driven using a novel nonlinear optimal control law. The obtained disturbance attenuation is exceptionally good and the test bed results agree well with the simulated ones. However, the obtainable control performance is limited to some extent by the process noise and heavy coupling of the actuators with below mount dynamics. A possible enhancement could be the use of different actuator geometry and use of higher sensitivity accelerometers. It was shown that the proposed adaptive control configuration is robust and capable of adapting to varying process conditions, while providing good control performance. Despite a few minor limitations the proposed control scheme has been proved to be highly effective for the class of process studied in the paper.

Acknowledgement The authors are grateful to BAE-Systems for their provision of the experimental hardware and to TEKES (The Finnish Funding Agency for Technology and Innovation) for funding the research visit of the first author to the BAE Systems Centre for Research in Active Control. References Anderson, B. D. O., & Moore, J. B. (1989). Optimal control: Linear quadratic methods. Englewood Cliffs, NJ: Prentice Hall. Ballesteros, P., & Bohn, C. (2011). Disturbance rejection through LPV gainscheduling control with application to active noise cancellation. In Proceedings of the 18th IFAC world congress. Milano, Italy.

Bittanti, S., Lorito, F., & Strada, S. (1996). An LQ approach to active control of vibrations in helicopters. ASME Transactions on Dynamic Systems, Measurement and Control, 118(3), 482–488. Bittanti, S., & Cuzzola, F. A. (2002). Periodic active control of vibrations in helicopters: a gain-scheduled multi-objective approach. Control Engineering Practice, 10(10), 1043–1057. Bohn, C., Cortabarria, A., Hartel, V., & Kowalczyk, K. (2004). Active control of engine induced vibrations in automotive vehicles using disturbance observer gain scheduling. Control Engineering Practice, 12(8), 1029–1039. Chiba, A., Fukao, T., & Rahman, M. A. (2008). Vibration suppression of a flexible shaft with a simplified bearingless induction motor drive. IEEE Transactions on Industry Applications, 44(3), 745. Daley, S., Johnson, F. A., Pearson, J. B., & Dixon, R. (2004). Active control for marine applications. Control Engineering Practice, 12(4), 465–474. ¨ onen, ¨ Daley, S., Zazas, I., & Hat J. (2008). Harmonic control of a ‘smart spring’ machinery vibration isolation system. Journal of Engineering for the Maritime Environment, 222(2), 109–119. Desoer, C. A. (1969). Slowly varying system x_ ¼ AðtÞx. IEEE Transactions on Automatic Control, 14(6), 780–781. Du, H., Zhang, L., Lu, Z., & Shi, X. (2003). LPV technique for the rejection of sinusoidal disturbance with time-varying frequency. IEE Proceedings—Control Theory and Applications, 150(2), 132–138. Guo, D., & Rugh, W. J. (1995). A stability result for linear parameter-varying systems. Systems & Control Letters, 24(1), 1–5. Gupta, N. K., & Du Val, R. W. (1982). A new approach for active control of Rotorcraft vibration pages. Journal of Guidance, Control and Dynamics, 5(2), 143–150. Herzog, R., Buhler, P., Gahler, C., & Larsonneur, R. (1996). Unbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings. IEEE Transactions on Control Systems Technology, 4(5), 580–586. Hiromi, T., Katou, T., Chiba, A., Rahman, M., & Fukao, T. (2007). A novel magnetic suspension-force compensation in bearingless inductionmotor drive with squirrel-cage rotor. IEEE Transactions on Industry Applications, 43(1), 66–76. Inman, D. J. (2006). Vibration with control. Hoboken, NJ: Wiley. Kamen, E. W. (1990). Introduction to signals and systems (2nd ed.). New York: Macmillan. Kinney, C. E., & de Callafon, R. A. (2006). An adaptive Internal Model-based Controller for Periodic Disturbance Rejection. In Proceedings of the 14th IFAC Symposium on System Identification. Newcastle, Australia. Kinney, C. E., & de Callafon, R. A. (2007). A comparison of fixed point designs and time-varying observers for scheduling repetitive controllers. In Proceedings of the 46th IEEE conference on decision and control. New Orleans, USA.

J. Orivuori et al. / Control Engineering Practice 20 (2012) 1206–1219

Knospe, C. R., Tamer, S. M., & Fittro, R. (1997). Rotor synchronous response control: approaches for addressing speed dependence. Journal of Vibration and Control, 3(4), 435–458. ¨ glu, H., & Scherer, C. W. (2011a). Robust generalized asymptotic regulation Koro˘ against non-stationary sinusoidal disturbances with uncertain frequencies. International Journal of Robust and Nonlinear Control, 21(8), 883–903. ¨ glu, H, & Scherer, CW. (2011b). Scheduled control for robust attenuation of Koro˘ non-stationary sinusoidal disturbances with measurable frequencies. Automatica, 47(3), 504–514. Kwakernaak, H., & Sivan, R. (1972). Linear optimal control systems. New York: Wiley-Interscience. Laiho A. (2009). Electromechanical modelling and active control of flexural rotor vibration in cage rotor electrical machines. Doctoral thesis, VTT Publications: 712. Espoo: Otamedia. Laiho, A., Tammi, K., Zenger, K., & Arkkio, A. (2008). A model-based flexural rotor vibration control in cage induction machines by a builtin force actuator. Electrical Engineering (Archiv f¨ ur Elektrotechnik), 90(6), 407–423. Ljung, L. (1999). System identification: Theory for the user (2nd ed.). Upper Saddle River, NJ: Prentice Hall. Ljung, L. (2002). Recursive identification algorithms. Circuits Systems Signal Processing, 21(1), 57–68. Machiejowski, J. M. (1989). Multivariable feedback design. Wokingham: AddisonWesley. Nandi, S., Toliyat, H. A., & Xiaodong, L. (2005). Condition monitoring and fault diagnosis of electrical motors—a review. IEEE Transactions on Energy Conversion, 20(4), 719–729. Orivuori, J., & Zenger, K. (2010). Active control of vibrations in a rolling process by nonlinear optimal controller. In: Proceedings of the 10th international conference on motion and vibration (MOVIC2010). Tokyo, Japan. Orivuori, J., Zenger, K., & Sinervo, A. (2009). Active control of rotor vibrations by advanced control methods. In Proceedings of ICSV16 in recent developments in acoustics, noise and vibration (ICSV16). Krako´w, Poland. Pai, P. F. (2009). Three-point frequency tracking method. Strutural Health Monitoring, 8(6), 425–442.

1219

Pai, P. F. (2010). Online tracking of instantaneous frequency and amplitude of dynamical system response. Mechanical Systems and Signal Processing, 24(4), 1007–1024. Rugh, W. J. (1991). Analytical framework for gain scheduling. IEEE Control Systems Magazine, 11(1), 79–84. Rugh, W. J. (1996). Linear system theory. Upper Saddle River, NJ: Prentice Hall. Savaresi, S. M., Bittanti, S., & So, H. C. (2003). Closed-form unbiased frequency estimation of a noisy sinusoid using notch filters. IEEE Transactions on Automatic Control, 48(8), 1285–1291. Seidel, H. (1993). Selected health risks caused by long-term, whole-body vibration. American Journal of Industrial Medicine, 23, 589–604. Sievers, L. A., Blackwood, G. H., Mercadal, M., & von Flotow, A. H. (1991). MIMO narrowband isturbance rejection using frequency shaping of cost functionals. In Proceedings of American control conference. Boston, MA, USA. Sievers, L. A., & von Flotow, A. H. (1992). Comparison and extensions of control methods for narrow-band disturbance rejection. IEEE Transactions on Signal Processing, 40(10), 2377–2391. Skogestad, S., & Postlethwaite, I. (2005). Multivariable feedback control: Analysis and design (2nd ed.). Chichester: Wiley. Sun, L., Krodkiewski, J. M., & Cen, Y. (1998). Self-tuning adaptive control of forced vibration in rotor systems using an active journal bearing. Journal of Sound and Vibration, 213(1), 1–14. Tammi K. (2007). Active control of radial rotor vibrations: Identification, feedback, feedforward, and repetitive control methods. Doctoral thesis, VTT Publications: 634. Espoo: Otamedia. ¨ onen, ¨ Tammi, K., Hat J., & Daley, S. (2007). Novel adaptive repetitive algorithm for active vibration control of a variable-speed rotor. Journal of Mechanical Science and Technology, 21(6), 855–859. Zazas, I., Daley, S., & Pope, S. A. (2010). Marine diesel vibration isolation using an RLS based harmonic control algorithm with frequency tracking. In Proceedings of 19th international congress and exposition on noise control engineering (INTERNOISE 2010). Lisbon, Portugal.