Relay Approach for tuning of PID controller - International Journal of ...

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Amar G Khalore et al ,Int.J.Computer Technology & Applications,Vol 3 (3), 1237-1242 ISSN:2229-6093 therefore be classified according to this experimental procedure. The Tuning methods can be broadly classified into the following categories where the classification is based on the availability of the process model and the model type. 2.1.1 Limit cycle method: This method was proposed by Niederlinski (1971) which is a more natural extension of the ZN tuning procedure of the MIMO case. It is based on replacing the controllers by gains and identifying a critical point consisting of n scalar critical gains and the critical frequency. The main departure from the SISO case is that MIMO systems have infinitely many critical points. The collection of these points defines a hyper surface in the gains space called the stability limit. Consequently, one has to prespecify the desired critical point, e.g. equal loop gains. The choice of the desired critical point depends on the relative importance of the various loops, which is commonly expressed through weighting factors. Once the parameters of the critical point have been determined, the controllers are tuned in a fashion similar to the classical ZN rules, possibly with some modifications. This method is briefly described below: (i) Choose n weighing factors ci (i = 1, 2 . . . n) for the relative control quality of the n controlled variables. (ii) Using the best input output pairing bring the P - controlled system to the stability limit keeping the following relations between loop gains as K . G C K . G C ci i, i(0) i ci 1 i 1, i 1(0) i 1 --------------------------- (2.1) Where Kc, i is the gain of the P controller in the ith loop and Gi,i are the diagonal process gains. (iii) Determine the critical frequency wci from the oscillation period Tu and the critical gain Kui when the oscillations just commences with Kci at Kui (iv) Determine the controller parameters using the Generalised Ziegler Nichols formulae listed in table 2.1 where the choice of coefficients αi depends on the ratio i c ci . Such that Ωc is the critical frequency when all the P- controlled loops are active, whereas wci is the critical frequency of the ith loop. (v) Check whether the control quality is satisfactory. If not change αi appropriately and return to step (ii). “Table 1: Generalised Ziegler Nichols Tuning Rules” where, Controller P Parameters K c T i T d α 1 K u PI α 2 K u 0.8T u PID α 3 K u 0.5T u 0.12T u 0.5 ≤ α 1 ≤ √0.5, 0.45 ≤ α 2 ≤ √0.45 and 0.6 ≤ α 3 ≤ √0.3 2.1.2 Biggest log modulus method: The BLT (biggest log modulus tuning) method also aims at tuning decentralized PID controllers. First, the settings for the individual controllers are determined via ZN rules, ignoring interactions, and then all settings are detuned by a common factor that is determined via a Nyquist-like plot of the closedloop characteristic polynomial and some distance criterion. This method is an extension of the classical Nyquist stability criterion method for SISO systems. The farther away the Nyquist plot of the loop transfer function is from the (−1, 0) point the more stable the system is. One commonly used measure of the distance of G(jw)H(jw) contour from the (−1, 0) point is the maximum log modulus GH (L c ) max where Lc 20log 1 GH For multivariable system this measure becomes the closed loop log modulus given by L cm W ( jw) 20log 1 W ( jw ) ----------------------- (2.2) The proposed tuning method on varying a factor F until the “Biggest log modulus” (L c ) max is equal to some reasonable number and hence the name “Biggest Log Modulus Tuning” (BLT). IJCTA | MAY-JUNE 2012 Available online@www.ijcta.com 1238
Amar G Khalore et al ,Int.J.Computer Technology & Applications,Vol 3 (3), 1237-1242 ISSN:2229-6093 This method can be outlined as in the following steps: (i) Compute the Ziegler Nichols PI tuning parameters for each individual loop, based on the ultimate gain and ultimate period information. (ii) Choose a factor F between 2 to 5 and compute the proportional gain Kc and the integral time τi for each loop using the relationships K c K F ZN ----------------------------------------- (2.3) (iii) Calculate the function in equation (2.2) over appropriate frequency range. (iv) Compute the closed loop log modulus equation (2.3) and keep adjusting F till the value of (L c ) max = 2n, where n is the order of the system. In the improved BLT method, the modeling is accomplished under certain structural assumptions by two relay experiments for each function of the process transfer matrix. Both the BLT method and the improved one are thus off-line methods that require good analytical models. 2.1.3 Relay feedback method: The PID relay auto-tuner of Astrom and Hagglund is one of the simplest and most robust auto-tuning techniques for process controllers and has been successfully applied to industry for more than 15 years. In recent years, relay feedback method have found a new lease of life in the automatic tuning of PID controllers and in the initialization of other sophisticated adaptive controllers. This tuner is based on the approximate estimation of the critical point on the process frequency response from relay oscillations. A continuous cycling of the controlled variable is generated from a relay feedback experiment and the important process information, ultimate gain and ultimate period can be extracted directly from the experiment. This is a very efficient way, i.e., a one shot solution, to generate a sustained oscillations. The success of this auto tuner is due to the fact that the identification and tuning mechanism is so simple that process operators understand how it works. Moreover, it works well even in slow and non-linear processes. To understand the relay feedback system it is vital to understand the describing function (DF) analysis. Describing Function Analysis: The describing function (DF) of a nonlinear element is defined as the complex ratio of the fundamental component of the output to the sinusoidal input. A more general definition says it is the covariance of the given input signal and the output divided by the variance of the input. The DF is a quasilinear representation of the non-linear element subjected to usually a sinusoidal input, and its use in the analysis of a non-linear system is thus based on the assumption that the nonlinear element has a sinusoidal input. Assume that the non-linear element has a sinusoidal input. x( t) a cos 2 t T ----------------------------- (2.4) For two periodic signals x(t) and u(t) of period T, the cross correlation function Rxu(ŧ ) is Defined by, T 1 Rxu ( ) x( t). u( t ) dt ------------------ (2.5) T 0 Where is the time delay. Also, the covariance of two signals is the value of their cross correlation function, with zero delay. Hence, the definition of the describing function N(a) of the non-linear element becomes, Na ( ) R R xu xx (0) (0) ----------------------------------- (2.6) Let x(t) be the input to the non-linear element and u(t) its output. As assumed earlier the input is sinusoidal and the relay output which would be a rectangular wave could be written as a Fourier sum as shown below, x( t) acos( t ) u( t) a cos( s t ) s 1 s Thus, the cross- correlation function between the input x(t) and the output u(t) is computed tobe, 2 1 aa1 xu ( ) cos( ) ( ) cos( ) 2 2 0 R t a t u t d t ------------------------------------- (2.7) IJCTA | MAY-JUNE 2012 Available online@www.ijcta.com 1239
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