Fall 2011 Lab38a2r.doc ET 438a Automatic Control Systems Technology Laboratory 2 Proportional Control Action Objective: Construct a proportional controller using OP AMP circuits and measure its steady-state and transient response. View the response of a first order process to proportional control action. Theoretical Background Automatic control has two principle functions: to maintain control output as close as possible to the desired setpoint value while the system is subject to outside disturbances (regulator action), and to respond as quickly as possible to changes in the setpoint value. Figure 1. Feedback Control System. Figure 1 shows block diagram of a typical control system. This is a negative feedback control system because the error signal is produced by subtracting the output from the setpoint value at the summing node. Another characteristic of this type of feedback is that the output is measured after the changes are made in the system under control, also called the plant. Sensors and signal conditioning may be used to transform a mechanical variable (e.g. flow, speed, pressure, temperature) into electrical signals. In some cases such as OP AMPs, the signals may already be electrical quantities such as voltage or current. An error signal is produced by taking the difference between the desired value and the measured value. This is the control error signal. Ideally this signal should be zero after the setpoint has been changed or an external disturbance has occurred on the process. If the control was initially in a balanced state the system will come into a new balanced condition after the changes. The controller block takes the error signal and modifies itFall 2011 Lab38a2r.doc 2 to produce an output signal called the manipulated variable. This output signal is used to control some element in the final process. The final control element is the actual apparatus that modifies the process. An example of this is a dc motor speed controller where: Control variable = motor speed Sensor/signal conditioning = speed transducer Set-point = desired speed Final control element = variable dc power supply Control variable = motor terminal voltage Manipulated variable = power supply voltage The controller can take several forms. The simplest mode of control is to amplify the error signal with a constant gain, Kp, and use this as the manipulated variable. This is called proportional control. Figure 2 shows the input output response of a practical proportional controller. All practical devices have maximum and minimum limits on output. The final control elements, such as valves and heaters, have limits on flow and temperature. The proportional controller will also have limited output. The controllable range will be between these limits. This range is called the proportional band. An OP AMP circuit will be used to design the proportional controller for this lab. For OP AMPs, controller limits will be the saturation voltage of the IC used in circuit construction. The proportional control band is inversely proportional to the gain of the proportional controller. The gain of the controller is defined as Figure 2. Proportional Controller Input/Output characteristic.Fall 2011 Lab38a2r.doc 3 tMeasuremen - SPOutput = Kp∆ ( 1) Where, Kp = proportional controller gain SP = setpoint value ∆Output = change in controller output. The difference between the setpoint and the measurement is the controller error signal, so the gain formula for the proportional controller simplifies to eOutput = Kp∆∆ ( 2) Where, ∆e = the change in error value. The proportional band as a percentage of the controller output is given by, 100% K1 = %PBp• ( 3) The effects of increasing the proportional gain on the proportional band are shown in Figure 3. In this figure, K1 > K2 >K3. The proportional band decreases as the gain increases which makes the overall control more sensitive. It also reduces the controllable range of the overall system since a small error signal may cause the controller to reach a limit. Figure 3. Effects of Increased Proportional Gain.Fall 2011 Lab38a2r.doc 4 Proportional control has two components, the bias and the amplified error. The bias is also called the control offset. The bias is the control output when the error signal is zero. This is usually set at 50% of the controller output. This is the point where the setpoint and sensor signals are equal. When the setpoint is moved from this value an error signal is produced and the controller must increase or decrease its output to maintain the system in balance. Figure 4 shows this action graphically. The proportional controller output can be described mathematically as an equation for a line. C + e K = Cbpo ( 4) Where Co = the controller output Cb = the controller bias value e = the control input error Kp = the controller proportional gain The error signal is the difference between the desired setpoint value and the measured value of the control output. This relationship can be represented by b - r = e ( 5) Where, r = the setpoint value b = the measured indication of the control output. Given the desired ranges for input error and controller output the values of Kp and Cb in Figure 4. Effects of Imbalance on Proportional Controller Output.Fall 2011 Lab38a2r.doc 5 Equation 4 can be found using the point-slope formula for a straight line. This relationship is shown below. )e - (e e - eC - C = ) C- C (minminmaxminmaxmino ( 6) Figure 5 shows a block diagram of a proportional controller. The error signal,e is multiplied by the proportional gain, Kp. The result is then added to the bias signal Cb to form the output Co. Figure 5. Block Diagram Equivalent of a Proportional Controller. An inverting summing amplifier can implement this relationship. Figure 6 shows a schematic for an OP AMP implementation of a proportional controller. The performance of this circuit was investigated in Lab 1. The input/output relationships for the inverting summing amplifier are given by: eVC = V RRK = RRC + e K = CV RR+ V RR - = V1bb2fp1fbpob2f11fc=•••• In this formulation, the bias value is determined by a constant input voltage and a gain set by the values of the resistors Rf and R2. The proportional gain is set by the ratio