Servo denetim sistemlerinin incelenmesi

Abstract

Zaman içinde değişen sistem parametreleri ve bozucu etkilere rağmen, verilen bir referans işaretini mümkün olan en az hata ile takip edecek denetim algorit malarının tasarımı, uzun süredir araştırma konusu olmaya devam etmektedir. Bu çalışmada, bu algoritmalardan bazıları incelenmiş ve simülasyonlar yapılarak, per formansları karşılaştırılmıştır. Servo denetim yapısı olarak, ileri beslemeli iki serbestlik dereceli yapı alınmıştır. Bu yapı içinde geri besleme işaretini üretmek için orantı türev integral (PID), doğrusal ikincil Gauss dağılımlı (LQG) ve kendini ayarlayan (STR) denetim yasaları kullanılarak gerekli simülasyonlar yapılmış ve sonuçlar karşılaştırmalı olarak ver ilmiştir.

Control algorithms capable of following a given reference signal, inspite of dis turbances and system parameter variations are long beeing investigated. In this thesis, some of these algorithms are discussed and their performances are illustrated by the help of computer simulations. Regulator type control systems try to keep the output and the states of the system fixed at a given set point. In the servo control problem, the objective is to make the states and outputs of the system to follow a desired trajectory. Practical systems often have specifications that involve both servo and regulation properties. This is traditionally solved by using a two degrees of freedom structure. Block diagram of two degrees of freedom structure is shown in Figure 1. Gil ^y Plant G/b Figure 1 Block diagram of two degrees of freedom controller. The feedback controller G/j is first designed to obtain a closed loop system that is insensitive to load disturbances and to plant uncertainty. The feedforward compensator is then designed to obtain desired servo properties. To calculate the feedforward signal, the one step ahead value of the reference is used. If the reference signal is not available, it have to be predicted using an appropriate model and a predictor. Let the closed loop systems transfer function, H(k), be given in discrete time as y(k) «(Jb) = H(k) = Alg + «2 q2 + hq + b2 (1) Where, y(k) is the measured system output, u(k) is the control signal, and aj, a2, b\ and &2 are system parameters in discrete time, q is the forward shift operator in discrete time with the following properties: qnf(kh) = f(kh + nh) qnf(k) = /(* + l) Then, we can rearrange (1) to give y(k + 1) + biy{k) + b2y(k - 1) = aiu(k) + a2u(k - 1) (2) In the servo problem, the objective is to make y(* + l) = r(fc + l) (3) where r(k + 1) is the reference signal at time k + 1. If we substitute (3) in (1), for um(k) we obtain nm{k) = - [r(k + 1) + hy{k) + b2y{k - 1) - a2u{k - 1)] (4) Here um is the feedforward compensator, and a\, a2, b\ and b2 are parameters of the closed loop system. In this study we used 3 different types of regulators to obtain a robust and stable closed loop. First of them is the PID regulator, described as ufb(t) = K e(t) + ^j\(T)dr + Td^ (5) Here e is the difference between the reference signal and measured system output, v,fb(t) is the feedback control signal, T{ is the integration time constant, Td is the differentiation time constant and K is the gain of the controller. Secondly, we used LQG control to obtain feedback signal. In LQG control, the system is assumed to be linear but it may be time varying. The objective is to choose the control signal in such a way that, a given quadratic loss function of control signal and system outputs is minimum. The obtained control rule is optimum if disturbance distributions are Gaussian random processes. The solution is given as follows: Let the discrete time system be described in state space by x(kh + h) = $x(kh) + Tu(kh) + v(kh) y(kh) = Cx(kh) + e(kh) ^ Here v and u are random variables with zero mean value and Gaussian distribution. Let's define R\ = Variance (v(kh)v (kh)) R\2 - Variance (v(kh)e(kh)) R2 - Variance (e2(kh)) VI Let's use Kalman filter, in (6), to estimate the one step ahead value of the system state. Kalman filter is then given by (7) x(k+l) = $x(k) + Tu(k) + K(k)[y(k)-Cx(k)] (8) K{k) = $P(k)CT(R2 + CP(fc)CT)-1 (9) P(k + l) = $P{k)$T + R1-$P{k)CT(R2 + CP{k)CT)-lCP{k)$T Then there is a unique control strategy that minimizes the loss function given by (5.12). This control strategy is given by : u(Jfe) = -L(k)x(k) (10) Here x is the estimated state, and L is the time varying gain matrix, given by (5.18). The last regulator used to obtain feedback signal is the Self Tuning Regulator. STR is an adaptive control in the sense that it tunes its own parameters. A block diagram of STR is shown in Figure 2. For an STR to be realized, first, plant parameters have to be estimated using an appropriate estimation algorithm, like Least Squares Estimator or Kalman Filter. Secondly, a controller design procedure have to be given. Figure 2 Block diagram of Self Tuning Regulator. vıı The organization of this thesis is as follows. In Chapter 2, system and noise models are given. These models are used in the simulations. The equations of the harmonic oscillator and the double integrator are given as linear system models. Harmonic oscillator model will be used in the self tuning regulator problem. The given double integrator model, with time varying moment of inertia, will be the controlled system model in the simulations. The measurement noise is modelled as discretization noise. Least squares parameter estimation method is given in Chapter 3. Here, the parameter estimation problem is formulated as an optimization problem, where the best model is the one that best fits the data according to a given criterion. For the least squares parameter estimation problem, the criterion is to minimize the discrete time loss function of : m = !>(«(*)) fe=l where, e is the difference between the measured output and the computed output of the identification model. Solution of identification problem is given in Chapter 3.3. Then, this solution is rearranged in such a way that the results obtained for jV observations could be used in order to get the estimates for İV + 1 observations. This procedure is referred as recursive least squares identifications. To illustrate the parameter estimation performance of the recursive least squares algorithm, two simulations are made. In the first simulation, parameters of a linear plant have been estimated and shown. In the second simulation, plant parameters are made to change with time. The simulation results of this time varying parameter estimation problem are given at the end of Chapter 3. In Chapter 4, PID regulator equations are derived. Properties of discrete time PID regulator are discussed to some degree. Simulations are made with constant and time varying process models. Linear quadratic optimal control problem is formulated in Chapter 5. In LQ problem, the process to be controlled is assumed to be linear, and a quadratic loss function is minimized by choosing appropriate control signals. The solution of the problem is obtained as a state feedback controller with time varying gain. The case with noise in. the system is considered next. For the Gaussian random distributed noise, the problem is named LQG regulator problem, and the optimal solution that minimizes the quadratic loss function is given without proof. Simulation results indicate that LQ controller gives good results for deterministic cases, but its perfor mance decreases for the noisy processes. LQG controller gave good performance for both deterministic case and noisy process. In Chapter 6, self tuning regulator is discussed. A self tuner algorithm, with vm pole placement design is given. System parameters are estimated by recursive least squares method, given in Chapter 3. In Chapter 7, PID, LQG and STR regulators are used to generate feedback signals in a two degrees of freedom servo controller structure. Simulation results of the servo controller with STR regulator have shown supe rior model following properties when compared to the servo controllers with LQG and PID feedback, but the control signal is remarked to be oscillating. Even though the produced control signals are different, using LQG and PID feedback in two degrees of freedom structure give similar reference following errors.