SGS22/150/OHK2/3T/12 “Mechatronics and adaptronics 202

Bukovský, I.

Pawlik, V.

Steinbaue, P.

DSpace at University of West Bohemia

2022 November 7 - 9, 2022Srní37   conference with international participation  thOnline identification using linear neural unitwith guaranteed weights convergenceV. Pawlik, P. Steinbauer, I. BukovskýFaculty of Mechanical Engineering, CTU in Prague, Technická 4, 160 00 Praha 6, Czech RepublicFor many modern control strategies, a model of the controlled plant must be known. Thismodel does not need to be known a priori and can be found online using an adaptive process.Online identification is very useful, especially if the plant’s properties change during the run, forexample, in context of vibration testing. This model can be a high order neural unit (HONU),such as a linear neural unit (LNU) with structureym =n∑i=0wixi = wTx, (1)where x is the input vector and w is the neural weights vector [1].The general HONU output formula isym = wT colx, (2)where w and colx are both column vectors. The neural weights w can be adapted using a gra-dient descent algorithm, for example, normalised least mean squares (NLMS), which is derivedby optimising a criterionJ =12e(k)2, (3)where the error e is the difference between the desired and the current outpute(k) = y(k)− ym(k). (4)The gradient of the criterion J with respect to w, which is the steepest direction, is∂J∂w= e(k)(∂J∂wy(k)− ∂J∂wym(k))= e(k) (0− colx)) = −e(k)colx. (5)The weights are then adjusted toward the minimum of the criterionw(k + 1) = w(k)− µk∂J∂w= w(k) + µne(k)colx(k), (6)using a learning rate µn = µ 1ε+||colx(k)|| , where ε ≈ 1e−5 and the normalisation improves con-vergence and µ ∈ (0, 2).To assess the performance of HONU, the sum of squared errors (SSE) over a certain horizonNe might be usedSSE =k∑i=k−Nee(i)2. (7)To ensure convergence of the weights [2], the learning rate µ might be chosen iterativelysuch that||A(k)|| = ||I− µncolx(k)colx(k)T || ≤ 1, (8)where I is the identity matrix and A(k) is the matrix of update dynamicsw(k + 1) = A(k)w(k) + B(k)u(k) =(I− µncolx(k)colx(k)T)w(k) + µny(k)colx(k). (9)At the start of the adaptation process the value µ = 2 is chosen and the decreased µ(k + 1) =0.9µ(k), until the condition (8) is satisfied. Then the weights are updated using the update rule(6). If the condition is violated during the run, then the learning rate µ is lowered until a valuethat satisfies the condition is reached again.A plant with a transfer functionY (s) =30530s+ 3.765e06s4 + 204.2s3 + 15278s2 + 415450s+ 588660U(s) (10)was used as a test system with square wave input signal. The input signal and the system outputare shown in Fig. 1 with time step ∆T = 0.01 s.0 2 4 6 8 10 12 14 16 18 20-0.500.5u0 2 4 6 8 10 12 14 16 18 20t [s]-1012yFig. 1. The input and output of the testing plantThe input vector x consist of bias and a number of samples of the input and output of theplantx(k + 1) =1u(k + 1)...u(k − nu + 1)y(k)...y(k − ny), (11)where nu and ny are chosen such that the best performance is achieved. The performance ofvarious choices is shown in Fig. 2. Performance is measured with Ne = 100.0 2 4 6 8 10 12 14 16 18 20-3-2-10123nu = 10, ny = 0, mu = 0.697, SSEmin = 0.86ymy0 2 4 6 8 10 12 14 16 18 20100102SSE0 2 4 6 8 10 12 14 16 18 2011.5270 2 4 6 8 10 12 14 16 18 20t [s]012;0 2 4 6 8 10 12 14 16 18 20-1-0.500.511.52nu = 10, ny = 10, mu = 0.458, SSEmin = 0.0094ymy0 2 4 6 8 10 12 14 16 18 20100SSE0 2 4 6 8 10 12 14 16 18 200.511.5270 2 4 6 8 10 12 14 16 18 20t [s]012;Fig. 2. Performance of the LNU for identification with various choices of nu and nu. The best perfor-mance is achieved with nu = 10 and ny = 2The linear neural unit shows great performance in the identification of the plant. The plantis sufficiently approximated for a wide selection of nu and ny, which is enabled by the selectionof µk that guarantees the convergence of the weights. In future work, this approach will be usedto automatically identify a model for an adaptive controller.AcknowledgementThe work has been supported by the project SGS22/150/OHK2/3T/12 “Mechatronics and adap-tronics 2022” of Czech Technical University in Prague.0 2 4 6 8 10 12 14 16 18 20-1-0.500.511.52nu = 10, ny = 2, mu = 0.628, SSEmin = 0.00023ymy0 2 4 6 8 10 12 14 16 18 20100SSE0 2 4 6 8 10 12 14 16 18 2011.5270 2 4 6 8 10 12 14 16 18 20t [s]012;0 2 4 6 8 10 12 14 16 18 20-1-0.500.511.52nu = 0, ny = 10, mu = 0.565, SSEmin = 0.003ymy0 2 4 6 8 10 12 14 16 18 20100SSE0 2 4 6 8 10 12 14 16 18 2011.5270 2 4 6 8 10 12 14 16 18 20t [s]00.51;Fig. 2. (Continued)References[1] Bukovsky, I., Voracek, J., Ichiji, K., Noriyasu, H., Higher order neural units for efficient adaptivecontrol of weakly nonlinear systems, Proceedings of the 9th International Joint Conference onComputational Intelligence, Funchal, Madeira, Portugal, 2017, pp. 149-157.[2] Bukovsky, I., Dohnal, G., Benes, P. M., Ichiji, K., Homma, N., Letter on convergence of in-parameter-linear nonlinear neural architectures with gradient learnings, IEEE Transactions onNeural Networks and Learning Systems, 2021, doi: 10.1109/TNNLS.2021.3123533. (in press)

Online identification using linear neural unit with guaranteed weights convergence

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Online identification using linear neural unit with guaranteed weights convergence

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