Uyarlamalı Model Kontrol

Abstract

Klasik kapalı çevrimli kontrol sistemleri, farklı çevre şartlarında sistem parametrelerinde değişiklik yaratmadığı sürece tasarımı kolay ve düşük maliyetinden dolayı tercih edilir. Ancak değişken çevre şartlarının sistem parametrelerini büyük ölçüde değiştirdiği ve giriş değerlerinin önceden kestirilemediği uygulamalarda klasik kontrol tamamen yetersiz kalmaktadır. Özellikle robotik, radar, iletişim ve biyomedikal cihazlar gibi ileri teknoloji uygulamalarında son yıllarda tercih edilen uyarlamalı kontrol sistemleri üzerinde sürekli çalışmalar yapılmakta ve son zamanlarda yapay sinir ağlarıyla entegrasyonu konusunda da ciddi ilerlemeler kaydedilmektedir. Canlı organizmaların değişken çevre şartlarında dengelerini korumaları prensibini kendisine çıkış noktası olan bu sistem yapı itibariyle de bir öğrenen sistemdir. Bu tez çalışmasında uyarlamalı sistemlerin genel tanıtımı yapılmış, bu alan içinde önemli bir yere sahip olan uyarlamalı modelleme ve buna bağlı olarak uyarlamalı model kontrol detaylı olarak incelenmiştir. İkinci dereceden bir sistem, MATLAB programlama dili kullanılarak önce modellemesi yapılmış ve rasgele verilen giriş değerlerine karşılık sistemin ve modelin performansları değerlendirilmiştir. Modellemede, gerçek sistem çıkışı ile model çıkışı arasındaki farkı minimum yapabilmek için LMS (en küçük kareler) algoritması kullanılmıştır. Tasarım sırasında bazı parametreler değiştirilerek modelin üzerindeki etkisi incelenmiştir. Kontrol için, sistem direkt programlama yöntemiyle yeniden yapılandırılmış ve ileri zamanlı hesaplama yöntemiyle sistemin ve modelin kontrolü sağlanmıştır. Sistemin uyarlamalı modellemesi ve kontrolüyle ilgili yapılan bütün simülasyonlar grafiklerle verilmiştir.

Adaptive plant modeling or plant identification is an important function for all adaptive control systems. In the practical world, the plant to be controlled may be unknown and possibly time variable. Firstly, it is assumed that the plant is continuous, stable or stabilized, linear and time invariant. A discrete-time adaptive modeling system samples the plant input and output and automatically adjusts its internal parameters to produce a sampled output which is a close match to the samples of the plant output when the samples of the plant input are used as the input to the adaptive model. When the plant and its model produce similar output signals, the adaptive impulse response is a good representation of the plant impulse response. In fact, the discrete-time adaptive model is a model of the samples of the impulse response of the plant. The basic idea is illustrated in Fig.l, where all signals and systems are considered to be sampled. [1,2,5] Plant, output 7 Error M Input 9k + /*\ dk -*- Unknown plant *¦{ S 1 A Adaptive model ~EZL Plant output - Error (b) Fig. 1 Adaptive Modeling [3] An adaptive filter (transversal filter) used in modeling,that is, imitating the behaviour of physical dynamic systems which may be regarded as unknown. The form of the adaptive filter to be considered as a tapped delay line, variable weights whose input signals are the signals at the delay-line taps, a summer to add the weighted signals, and an adaptation process that automatically seeks an optimal impulse response by adjusting the weights. Fig. 2 illustrates the adaptive filter which is casual and has a finite impulse response (FIR) [5]. Output signal (continuous) Fig.2 Modeling a plant by transversal filter [5] There is an input signal vector with elements xo,xi,,xl, a corresponding set of adjustable weights, w0,Wi,,wL,a summing unit, and a single output signal, y. Input output relation is formulated as follows. yk = E wik xk.i 1=0 (1.1) Wk = [ w0k wik wLk ] T yk=xkTWk = WkTxk Adaptive algorithm is, ek = dk -yk=dk - XkT W = dk- WT Xk Wk+i= Wk+2uek Xk (1.2) (1.3) (1.4) (1.5) The output signal, yk, is simply subtracted from the desired signal, dk, to produce the error signal, 8k.The objective of the modeling process to minimize the error signal 8k so that for the same input signals the model and the plant generate the same outputs. In order to minimize the error signal, LMS (least-mean-squares) algorithm is used which use a special estimate of the gradient that is valid for adaptive filter. Formula (1.5) is obtained from this algorithm by using steepest descent method which is used for gradient searching. With a stationary input and a stationary plant, the LMS algorithm is known to be convergent when the convergence factor |X is chosen in the range, lA,max>l/trR> u>0 (1.6) where R is the input autocorrelation matrix, A.max is its largest eigenvalue. System to be modeled is chosen as a second order simple system and its transfer function is given below. 1 (1.7) s2 + s + 1 The sampling period is determined according to settling time of the impulse response of discretized plant. Ts equals to 0.1 second which is small enough. As the settling time is about 10 seconds the number of weights is calculated as 10/0.1=100. Therefore the weight vector with 100 elements used as an adaptive filter in this study. By using MATLAB programming language which is very common in control applications, the above system is modeled and for each 200 iteration cycle three different step input values is entered both the plant ant the model and their outputs are illustrated in Fig. 3. Fig. 3 Comparison of model output with plant output over 600 iteration cyle with several step input values, u=0.01 For the smaller convergence factor, convergent would not be satisfactory. In adaptive control systems, the input signals usually can not be foreseen. Therefore, a random input signal generator is programmed in this study. The random signals are firstly passed through an second order filter in order to soften their sharp peaks. After this filtered random output is entered to the plant and the model. Similar comparison between the model and the plant output is illustrated in Fig. 4. It is noticed that the model can follow the plant perfectly from the beginning as the |j. is chosen correctly. If it had been chosen smaller, there would be lack of following the plant at the beginning but it XI would follow after a few hundreds steps. For the bigger value of u=0.18 it is observed that model becomes unstable and for the smaller value the error increases in predictable manner. 1 - :Random Input 2 - : Plant Output 3 : Model Output 400 600 No. of samples 800 1000 Fig.4 Comparison of model output with plant output with random input signals, u=0.08 A block diagram of the adaptive control system is shown in Fig. 5. The functions labelled "forward-time calculation" is the inverse of the model in reality. This box generates Xk from the reference command input rk and from the weight vector Wk and the input vector Xk. Fig.5 Block Diagram of Adaptive Model Control System [3] XII Assuming that E[e2] has been driven close to zero by the adaptive process, the objective of this box is to derive Xk from r^ such that rk and yk are equal. If rk and yk are equal, then (with Sk2) the plant output gk(equal to desired response dk) will be close to rk, which is the ultimate goal for this study. If it is considered that there are delays in the plant such as transport lag, the Xk can be written as follows, with yk equal to rk L Xk= 1 /W2k [rk- W0k' " S WnkXk+2-n] (1.8) n=3 Before using this formula in MATLAB program, the plant is programmed in direct form. 500 1000 1500 No.of samples 2000 2500 1 - : Input 2 - : Plant output Fig.6 Tracking performance: A comparison of the plant output and the model output (Control starts at 500.step)