Integrator Windup

Transcription

1 Integrator Windup Definition So far we have mainly been concerned with linear behaviour, as is often the case with analysis and design of control systems. There is, however, one nonlinear phenomena that we have to deal with namely saturation of the amplifier. Since the supply voltage is 15 V, it is not possible for the motor amplifier to deliver outputs that are larger than 15 V. The effect of saturation can be quite serious because the feedback loop is effectively broken when the amplifier saturates. In this case, saturation causes the loss of the important feedback information from the system output. The combination of a controller with integral action and a process with saturation is that the integrator may drift when the controller saturates. This is called integrator wind up Windup Protection There are many ways to avoid integrator windup. One possibility is to arrange a feedback that resets the integral when the output saturates. This is illustrated in the block diagram in Figure 3.2. Figure 3.2 PI controller Block Diagram With Protection For Windup For the DCMCT motor, the model of the actuator is simply as follows: u( t )!"! sat 15 ( v( t )) with the saturation function defined as: # -15 x!'! -15 sat 15 ( x )!"! & x -15!(! x and x!(! 15 % $ 15 15!'! x Document Number: 627! Revision: 01! Page: 70

2 The extra feedback loop with the time constant T r is inactive if the the control signal does not saturate because in this case we have u = v. When the controller output saturates, the extra feedback loop drives the saturation error e s to zero which means that the integrator is driven towards a value just at the saturation limit. This means that the control signal will decrease from the saturation limit as soon as the control error goes negative. The windup protection is governed by the parameter a w which ranges between 0 and 1. There is no protection against windup if a w = 0. If a w = 1 the integrator is reset in one sampling period. In discrete time, the PI control law with windup protection can be written as follows: u( k )!"! sat 15 ( v( k )) with: v( k)!"! k p ( b sp r( k )!)! y ( k!)! 1 )) N * -!1!, 0 ( k i ( r( k )!)! y ( k!)! 1 )) h!1! a w ( u ( k!)! 1 )!)! v ( k!)! 1 ) )) / + k!"! 2. where k is the sampling interval. Document Number: 627! Revision: 01! Page: 71

3 Tracking Triangular Signals So far we have investigated tracking of reference signals in the form of square waves. We will now investigate tracking of triangular references. Please answer the following questions. 1. Determine the transfer function, G e,r (s), from reference to control error for a PI control loop characterized by Equation [3.1]. G e,r (s) is defined below: E( s ) G e, r ( s )!"! R( s ) [3.7] where the velocity error is defined as follows: E( s )!"! R( s )!)! 2 m ( s ) [3.8] Document Number: 627! Revision: 01! Page: 72

4 2. When a PI controller is used (k p 3 0 and k i 3 0), apply the final value theorem to calculate the steady-state error, e ss_pi, in response to a ramp reference signal of slope R Using the PI tuning found in Sections Question 10 and Question 11, evaluate the steady-state error e ss_pi for the following ramp slope R 0 : R 0!"! rad 7 s [3.9] Document Number: 627! Revision: 01! Page: 73

5 4. For a system that does not have integral action (k i = 0), apply the final value theorem to calculate the steady-state error, e ss_p, in response to a ramp reference signal of slope R 0. In this configuration, the closed-loop system has a pure proportional controller (k p 3 0, k i = 0, and b sp = 1). Hint: You can obtain e ss_p by first applying the final value theorem for a step input of amplitude R 0 and then integrating. 5. Using k p = 0.1 V.s/rad, evaluate the steady-state error e ss (t) for the ramp slope R 0, as defined in Equation [3.9]. Considering a 0.4 Hz triangular reference signal of slope R 0, calculate the maximum steady-state error, e ss_p_max. Document Number: 627! Revision: 01! Page: 74

6 Response To Load Disturbances Reduction of the effects of load disturbances is a key reason for using control. A torque on the motor axis is a typical example of a load disturbance for a speed control system. In this laboratory we will show the effects of controller tuning on load disturbance response. Please answer the following questions. 1. Considering the regulation problem (for r = 0), determine the closed-loop system block diagram with a disturbance torque input T d applied on the DCMCT inertial load. Assume a PI controller. The block diagram should contain the open-loop transfer function G!,V as formulated in [3.2], and be function of the following system parameters: k p,k i, K, ", and J eq. Hint: Assume V sd = Find the closed-loop transfer function, G!,T (s), from disturbance torque to motor speed, as a function of the following system parameters: k p, k i, K, ", and J eq. Document Number: 627! Revision: 01! Page: 75

7 G!,T (s) is defined below: 2 m ( s ) G :, T ( s )!"! T d ( s ) 3. When a pure integral controller is used (k p " 0 and k i 3 0), apply the final value theorem to calculate the steady-state velocity,! ss_i, in response to a step input disturbance torque of amplitude T d0. Comment. Document Number: 627! Revision: 01! Page: 76

8 4. When a pure proportional controller is used (k p 3 0 and k i = 0), apply the final value theorem to calculate the steady-state velocity,! ss_p, in response to a step input disturbance torque of amplitude T d0. Comment. Document Number: 627! Revision: 01! Page: 77