Control system with anti-windup measure

In practical applications the manipulated variable must not exceed given extreme values. This is the case due to either the bounded power of the actuator or to the physical constraints of the plant. In most cases these hard limitations of the manipulated variable must be respected. This means that the modulus of the manipulated variable must not exceed given bounds

For control system design using linear methods it is difficult to cope with this problem and to abide by the bounds on the manipulated variable. When the design is performed such that the amplitudes of the manipulated variable are small and do not reach the bounds, the actuator is not fully exploited and, thus, the control response is slow. On the other hand, when the bounds are exceeded for a reasonable period of time undesirable control behaviour may be obtained.

In order to discuss the problem, the bounds are described by a saturation element, as shown in Figure 11.10.

Figure 11.10: Block diagram of a control system with a bounded manipulated variable

The variable is the manipulated variable obtained from the controller and the manipulated variable acting on the plant, which is determined from

When the bounds are exceeded the nonlinear saturation characteristic will take effect and influence the dynamical behaviour. In some cases the closed-loop system may also become unstable or show an oscillating behaviour.

This undesired phenomenon, called the windup effect, occurs in all control systems where an integrator is used in the controller. This integrator is necessary to have a zero steady-state control error. In order to demonstrate this effect, the example from section 9.3 is taken. The plant is given by Eq. (9.41) and the controller by Eq. (9.51). The step response of the closed loop without saturation is shown in Figure 11.11, where the response of the controlled variable is the same as in Figure 9.20. If the manipulated variable is bounded by (with saturation) the rise time increases due to the smaller values of the manipulated value in the period from 0.1s to 2s. The increased maximum overshoot and settling time reflect a worse control behaviour. The reason for this is the following: From the beginning, the control error decreases and changes sign at . As is very large at this time ( ), the manipulated variable cannot be reduced despite the negative control error. This will only occur when falls below at 2s. The problem is obviously that the controller continues to integrate though the manipulated variable has already reached its bound. As the controller output further grows unnecessarily, this is called the windup effect.

The goal of an anti-windup measure is to counteract the integration of the controller. This can be performed by feeding back the difference to the controller. Figure 11.12 shows a simple approach for an anti-windup measure, where the difference is weighted by the factor and fed into the controller. Figure 11.11 shows the improvement of the behaviour for . The settling time is close to the case without saturation, but the maximum overshoot is half of that without saturation.

Figure 11.11: Step response of the closed loop system with and without anti-windup measure, (a) manipulated variable and controller output , (b) controlled variable

Figure 11.12: Block diagram of a control system with an anti-windup measure