LAB 7 PID CONTROL 1. LAB OBJECTIVE The objective of this lab is to design and implement a PID (Proportional-Integral- Derivative) controller and to study the effect of integral and derivative control actions on the system. We will also learn about the importance of the bandwidth of a controller. 2. BACKGROUND 2.1 Closed-loop bandwidth The frequency range of input signals that a closed-loop system can track successfully without significant attenuation is called the closed-loop bandwidth of the system. It is defined as the frequency at which output amplitude becomes 2/1 (or -3 dB) Example: Consider the proportional control system shown below. Let KBmB = 5.5, TBmB = 0.13, KBpB = 10. Figure 1: Block diagram of closed-loop proportional control system that controls the shaft position of a DC servomotor. The closed loop transfer function is given by mpmmpKKsTsKKsRs2)()( (1) The bode plot of the above transfer function is shown in Figure 2. We can see from the Bode plot that the –3dB point occurs around 30 rad/s. This is considered to be the bandwidth of the closed-loop system. In other words, this controller can track signals that vary with frequencies up to 30 rad/s (4.7Hz) without significant attenuation.2.2 Experimental determination of bandwidth We can determine the bandwidth experimentally by observing how well the plant tracks sinusoidal input signals of different frequencies. Figure 2: Bode plot of the closed loop transfer function of the control system shown in Figure 1 2.3 Position control Figure 3: Position control loop.Figure 3 shows a position control loop for the lab DC servomotor. A variety of control strategies, represented by the Laplace function GBCB(s), can be used to control the shaft position. We have already seen proportional (P) control. In this section we will consider three other types of control that use not only the error of the shaft position, but also the integral and/or the derivative of this error. The three additional types of control to be considered include PI, PD, and PID control. 2.4 PI controller sKKsGipc)( (2) Proportional-integral (PI) control considers both the magnitude of the system error and the integral of this error. For the DC servomotor, by integrating the error of the shaft position over time, scaling the integral, and adding this term to the control voltage, steady-state errors in position can be eliminated that P control alone may not be able to cancel. This is the primary reason to add integral control action, to eliminate steady state error. The drawback of adding integral action is that it may have a destabilizing influence on the system response if KBiB is not properly selected. Implementation error = (reference position – actual position) integrated_error = integrated_error + error x TBsB (TBsB is the sampling period) control voltage = KBp Bx error + KBiB x integrated_error The integrated_error term is calculated by summing the error at each time instant. In C the statement integrated_error += error*Ts; does this. It is necessary to initialize integrated_error to zero before the start of the integration process. 2.5 PD controller sKKsGdpc)( (3) Proportional-derivative (PD) control considers both the magnitude of the system error and the derivative of this error. Derivative control has the effect of adding damping to a system, and, thus, has a stabilizing influence on the system response.Implementation error = (reference position – actual position) error_derivative = (error – previous_error)/TBs control voltage = KBpB x error + KBdB x e rror_derivative 2.6 PID controller sKsKKsGdipc)( (4) Proportional-integral-derivative control (PID) combines the stabilizing influence of the derivative term and the reduction in steady-state error from the integral term. Implementation error = (reference position – actual position) integrated_error = integrated_error + error x TBsB error_derivative = (error – previous_error)/TBsB control voltage = KBpB x error + KBiB x integrated_error + KBdB x error_derivative 2.7 Ziegler-Nichols rule for tuning a PID controller The Ziegler-Nichols tuning rule is widely used to tune PID controller gains in process control systems. For a PID controller of the form  sTsTKsGdipc11)( (4) The parameters pK, iT and dT are calculated using the following design criteria. Ziegler-Nichols design rule Type of controller pK iT dT P 0.5 x scrK - - PI 0.45 x scrK scrP /1.2 - PID 0.6 x scrK 0.5 x scrP 0.125 x scrP scrK and scrPare obtained by experiment. scrK is the critical gain where the shaft exhibits sustained oscillations (marginal stability). scrP designates the period of these oscillations.For the PID control law given in Section 8.2.3.3, KBpB = KBpB, KBiB = KBpB/TBiB, and KBdB = KBpBTBdB. 3. PRELAB a) Modify the program you wrote in the Lab 6 ( P r o p o r t i o n a l C o n t r o l ) prelab (question 1) so that the reference displacement signal is a sine wave instead of a step input. Use a magnitude of 250 counts and a frequency of 5 Hz. Use Kp of 0.004. You will vary the frequency of the reference signal and observe tracking performance. As before, store encoder data in an array variable for plotting with print_into_matlab. b) Determine the gain values for a PI controller and a PID controller using the Ziegler-Nichols tuning rules. Use the values for scrK and scrPthat you experimentally determined in the previous lab. c) Write a program to implement PI position control. Use gain values obtained from Question 2. Use a step input of 500 counts. d) Write a program to implement PID position control. Use gain values obtained from Question 2. Use a step input of 500 counts. 4. LAB PROCEDURE Exercise 1 Implement Prelab Question 1. Vary the reference signal frequency according to the following table. Plot