Proceedingsof the American Control Conference San Diego, Cafifomia ● June 1999
Adaptive Real-Time Control of a Christmas Tkee Using a Novel Signal Processing Approach J. Hu and H.R. Wu School
C.T. Chou, Control
Faculty of Information
P.O. Box 5031,2600
The LMS algorithm forms the core of a number of adaptive control schemes. It has on one hand provided certain degree of robustness to these algorithms but on the other hand its convergence rate is notoriously slow. In this paper we propose a novel method to initialize the LMS algorithm which dramatically shorten its adaptation period. We also prove that under certain condition this initialization can be computed without explicit knowledge of eitherthe reference or noise spectrum. Although the work is illustrated using the filtered-X scheme, this novel idea can also be applied to other similar schemes. Two successful experiments starting from black-box identification to real-time control have verified the effectiveness of this novel idea.
1 Introduction For durability and fatigue evaluations, it is hoped thatreal or synthesized target time histories can be reproduced in laboratory testrigs. When the field dataand testing rig are available, the most important thing is to find a technique that reproduces these field data as accurately as possible. From the control point of view this is exactly a tracking problem integratedwith plant modeling and/or identification. A similar situationcan be found in the field of active noise control and active vibration control, where the system tries to reproduce the primary input in order to cancel the noise or vibration, The Least Mean Square (LMS) algorithm [1, 2] plays a fundamental role in these areas. The power of the LMS algorithm lies in its simplicity in implementation as well as its robustness. In practical applications, the slow convergence rate of the LMS algorithm is notorious, which hinders its great potential. Although much progress has been made e.g. [3, 4, 5], the problem remains inherently serious. In this paper we propose what we call the FastConvergence Method (FCM) to speed up the convergence rateof the LMS algorithm. FCM initializes the LMS algorithm by using the solution to an optimal Hz model matching problem. Under certain condition, this solution can be computed without explicit knowledge of either the reference or noise spectrum,
@ 1999 AACC
D. Westwick, and Systems,
G. Nijsse Delft University
GA Delft, The Netherlands
which is of great advantage. Although the work is illustratedusing the filtered-X scheme, this novel idea can also be applied to other similar schemes, e.g. filter-e. To a certain extent, FCM bears certain similarities to the preview method and to the work of . However, FCM differs from these methods in the sense that it does not simply compute an optimal controller using model matching but it goes further to incorporate this controller as an initial startingpoint for the LMS algorithm, which is of fundamental importance in shorteningthe adaptationperiod of LMS algorithm in the filtered-X algorithm. The organization of this paper is as follows: in section 2 the main results of the FCM will be presented. In section 3, the experimental results from two real-time experiments will be given. The first experiment is taken from  and it shows that FCM improves convergence rate tremendously. The second experiment is performed on a mechatronic device provided by Leuven Measurement System Int., Belgium. Finally, the conclusions will be given in section 4.
2 Main Results For simplicity, we first consider the basic adaptive control scheme using the LMS algorithm as shown in figure 1 where r is the reference signal, y is the output, e is the error signal and u is an external disturbance which is uncorrelated with r. Both the plant P and reference model M are assumed to be discrete-time stable rationalmatrix transferfunctions; we further assume thatP has more outputs than inputs, has no transmissionzeros on the unit circle and has full rank. The key idea of this setting is to adjust the controller C such that y is close to the reference trajectory Mr, or in other words, the following cost function is to be minimized: (1) Let cD, denote the spectrum of r and under certain regularity conditions a spectral factorization of Or exists and is given by: 0,= ylyI* (2)
where v is square, stable and minimum phase. Since the external disturbance n is not correlated with r, the minimization problem (1) is therefore independent of the spectrum of n; in fact, it can readily be shown that the solution to problem (1) is equivalent to thatof following Hz model matching problem (3) We have the following result:
Theorem 1 Let P = PiPObe an inner-outer factorization  of P with Pi inner and PO outec then the solution to the model matching problem (3) is uniquely given by c
where * and [ ]+ denote respectively complex conjugate transpose and the projection operator onto H2.. Note that the assumptions that P is tall, hus no transmission zeros on the unit circle and has fill rank guarantee that the inverse of POexists and belongs to Hz [111.
Note thata special case to the solution (4) is when P is stable and minimum phase. In this case, Pi is identity and C is simply given by P- lM which is independent of both the external reference and noise spectra. However, in the nonminimum case, the solution to the model matching problem generally depends on the input spectrum which may not be readily available. The following derivation shows how this problem can be circumvented.
Proposition 1 Let us write M as the product of a pure delay part z-d and a causal part M, i.e. M
Zfd is su#iciently large, the solution to the Hz model matching problem is approximately given by c
which is independent of both the reference input and noise spectra.
Proofi (Sketch) Since P~My.ris an h function, the theory of 2-sided z-transform guaranteesthatit can be written as a Laurent power series with both positive and negative powers of z (corresponding to the anticausal and causal parts respectively.) Furthermore, the coefficients of extreme positive and negative powers must decay to zero. Thus, if d is sufficiently large, the anticausal part of PfiklyI is negligible and we have [P~Mw]+ % P~MyI. Thus, we have C z P–lA4. A It is important to point out here that the introduction of a large number of delays in the reference model does not
contradict our control objective. Let us first recall that our fundamental objective is to replicate the reference signals which have been collected during field test at the outputs of a test-rig. There is no restriction thatthe signals have to be reproduced instantaneously. The introduction of delays in the reference model simply means that the reference signal will be reproduced later and this is perfectly acceptable for durability and fatigue evaluations. Although the above derivation is based on the basic adaptive control scheme of figure 1, the same machinery can be applied to other adaptive algorithms such as filtered-X and filtered-E. The standardfiltered-X algorithm is depicted in figure 2 where in addition to the basic scheme, a plant model ~ and a noise canceler Cn are used. This scheme is found to be very robust and has been used extensively in many active noise and vibration control applications. However, as we mentioned before, a severe bottleneck of these algorithms is their slow convergence rate. We suggest to overcome this problem by the Fast Convergence Method (FCM): First, solve the H2 model matching problem as in Proposition 1 using an identified plant model F(z) and the controller so obtained can then be used to derive the noise canceler off-line. Second, compute a FIR approximation of the controller obtained in the first step by truncation. Third, use these controller coefficients to initialize the transversal filter of the LMS algorithm and startthe adaptationprocess. The main thrustof FCM is to initialize the controller from a good startingpoint before the adaptation begins, instead of startingit from either zero or an arbitraryinitial condition. This is the reason why it gives fast convergence. But, why is the H2 model matching solution a good guess? It is well known thatthe optimal LMS solution for a given adaptation rate p >0 differs from the optimal Hz model matching solution but these two solutions are sufficiently close to each other if p is small. Since a small adaptationrate is normally used, FCM therefore provides a reasonably good starting point for fast adaptation. The filtered-X algorithm has been found to be very robust, both in theory and in practice, in the sense that LMS still converges even though the plant model ~ is inaccurate  and this is the reason why this algorithm has been so widely applied. For FCM, a poorly identified plant model may give rise to a low quality model matching solution; however due to robustness of filtered-X, this solution may still be well within the convergence region ! In general it can also cope with wide eigenvalue spread of the input without having to use Discrete Cosine Transformation (DCT) .
3 Real-time experiments Two real-time experiments were conducted to verify the FCM described in Section 2. Transversal filters are used. Because of the constraints on the existing hardware and
software, a noise canceler was not included in the real-time experiment though it worked perfectly in a non-real-time simulation environment. The real-time operating system used is version 5.3 of VxWork.srunning on a Power PC 604 equipped with 16-bit A/D and D/A channels. The Real-time workshop of Simulink was used in the Matlab 5 environment to generate the real-time target code. 3.1 Experiment I: an example from  To give a convincing comparison, this example is taken from pp. 188-194 of  where the filtered-X algorithm was found to be still far from convergence even after 106 adaptations. Moreover, the input signal to the controller has an eigenvalue spread of 30,000 to 1. This example therefore poses an excellent benchmark for the FCM. At a sampling rate of 10HZ, the discretised plant is O.1O32(Z– 1.0513 )(z+0.8609) ‘(z)
(Z- 0.9048) 2(z - 0.8187)
Note thatthe plant is non-minimum phase. We implemented the original analogue plant on a circuit board and plugged it into the VxWork real-time operating system. For this example, we chose to identify a FIR model using the LMS algorithm. This identified model was used with the filtered-X algorithm where the controller had 150 weights and 50 delays were used in the reference model. By using the FCM, the optimal model matching solution was obtained in FIR form without utilizing any knowledge of external signal, The convolution of the estimated plant and the controller was found to give an almost perfect unit impulse at the 50th sample, see figure 3. This means that an approximate inverse of the plant had been obtained. The model matching solution was then used to initialize the filtered-X algorithm with an adaptationrate of 10–6. The tracking performance with a first order Markov process as the reference is shown in figure 4 where almost perfect tracking performance is observed immediately after the 50 delays imposed by the reference model. This is an amazing result compared with the case of 106 adaptation steps with still no convergence in sight. Equally well tracking performance was observed after 1000 samples (see figure 5) and beyond (not shown here.) 3.2 Experiment II: Christmas tree After successful completion of the first experiment, furtherexperiments were conducted on a mechatronic structure known as a Christmas tree, see figure 6. This was done in collaboration with Leuven Measurement System Int., Belgium, where this structureis used as a benchmark internally in the company to test its signal replication algorithms. At the base of the structure is an electric motor which shakes the Christmas tree and the sensor picks up the acceleration signal at the top of the tree. Due to high frequency vibration, the system exhibits flexibility. The blockage function of the net makes first principle modeling of the process nontrivial. The ultimate goal of the experiment is to replicate an acceleration signal provided by the company.
Previous testswithin the company shown thatthe Christmas tree has a bandwidth of more than 1 kHz and the sampling rate should therefore be at least 2 kHz. Due to hardware limitation we could only use a sampling rate of up to 1 kHz and a significant mismatch between the identified model and the plant was to be expected. A black box identification experiment was carried out and the model was identified using subspace identification algorithm  from the SMI toolbox . The experiment was carried out with 50 delays in the reference model and an adaptation rate of 10–4. Altogether 2000 samples were recorded to illustrate the performance. Figure 7 shows the tracking performance in the first 200 samples and almost perfect tracking can be observed right after the 50 delays specified by the reference model. An objective measure of the performance is given by the variance accounted for (VAF) which is defined as:
~ _ Variance(y – j)
where y and j are respectively the the target signal and the measured output. Two closely matched signals will give a VAF close to 100%. For the first 200 samples, the VAF between the target output and the measured output is 88.1%. This is improved to 90.4% in the next 200 samples due to the adaptation process, see figure 8 for a time plot. The VAF for the entire 2000 samples is 88.7%, which is considered to be rather high. The tracking performance over frequency domain is shown in figure 9. As mentioned before, a noise canceler was not used in the experiment due to hardware limitation; otherwise, a better performance could be expected.
In this paper we propose a method to speed up the slow convergence rate of LMS algorithm by initializing it with a model matching solution. This initialization, though not optimal itself, can be sufficiently close to the truly optimal solution to allow fast convergence to take place and can be computed without explicit knowledge of either the external reference signal or noise spectrum. The effectiveness of this method is demonstrated by two real-time experiments where almost instantaneousconvergence of the LMS algorithm was observed. The authors would like to thank Acknowledgments: Messrs. Coppens, Debille and De Cuyper of Leuven Measurement System Int., Belgium, for their support and cooperation in carrying out the Christmas tree experiment as well as to Mr. Van Geest of the Control Laboratory for his assistanceon using VxWorks.
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Figure 1: Basic adaptive control scheme,
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Figure 2: Filtered-X LMS algorithm scheme.
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Figure 3: Convolution of the model matching controller with the identified plant model.
Figure 4: Trackingperformancein the first 200 samplesincluding 50 delayunitssetby thereferencemodel.
Figure 7: ‘llme domaintrackingperformanceover the first200 samples.
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Time domain tracking performance over the second 200 samdes.
Figure 5: Trackingperformanceafter 1000 samplesof adaptation.
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