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# Dynamical behaviour of a closed loop system

Prepared with an understanding of models, transfer functions and basics about control loops, we now consider control systems and the types of feedback design principles available. Figure 7.1 shows a block diagram of a closed loop system

with the four classical components: controller, actuator, plant and measurement device. It is often convenient to combine the controller and actuator into one controller component, while the measurement device is often assigned to the plant. Usually a set of disturbances may occur, each of them can enter the plant at different locations. The transition behaviour of the plant and of the parts of the plant between disturbance input and plant output, respectively, is denoted by . From this a block diagram of the closed loop system is obtained according to Figure 7.2.

For linear plants all the disturbances can be combined into one single cumulative disturbance

according to Figure 7.2. This cumulative disturbance will act at the plant output (see Figure 7.3). Furthermore, by a suitable choice of it can be shown that the structure from Figure 7.2 is also valid for disturbances entering at other locations in the closed loop.

The transition behaviour of this control loop is specified according to the two inputs (command and disturbance) either command behaviour or disturbance behaviour. The transfer function of the controller elements - briefly called in the following only controller - is and those of the plant . From Figure 7.3 the controlled variable of the closed loop is

Rearranging, then it follows

 (7.1)

Using this equation, the control system tasks already mentioned in section 1.3 can be formulated more precisely as follows:

a)
For the transfer function of the closed loop for disturbance behaviour the disturbance transfer function

 (7.2)

is obtained.
b)
Similarly for the transfer function of the closed loop for command behaviour is the command transfer function

 (7.3)

Both transfer functions and contain the dynamical control factor

 (7.4)

with

 (7.5)

Opening the closed loop for and according to Figure 7.4 at an arbitrary location and defining with respect to the route of the transfer elements the input as and the output as , the transfer function of the open loop

 (7.6)

is obtained.

If can be described by a rational fraction, by setting the denominator of Eq. (7.2) or Eq. (7.3) to zero one obtains for the closed loop the condition

 (7.7)

analogous to Eq. (3.12) for the characteristic equation in the form

 (7.8)

The overall goal in designing a control system is to use the principle of feedback to cause the controlled variable to follow a desired command variable accurately regardless of the command variable's path and to minimise the effect of any external disturbances or changes in the dynamics of the plant. Reaching this goal economically the standard structure of Figure 7.3 is a relatively complex task if one must meet the basic requirements listed below:

a)
The minimum requirement is that the closed loop is stable.

b)
Disturbances should be rejected or they must have a small influence on the controlled variable .

c)
The controlled variable should track the command input as precisely and as fast as possible.

d)
The closed loop should be as insensitive as possible with respect to changes in the plant parameters.

In order to fulfil the requirements in the ideal case, the command transfer function must be according to requirement c)

 (7.9)

and the disturbance transfer function according to requirement b)

 (7.10)

A rigorous realisation of these requirements is not possible for physical and technical reasons. The problem will be illustrated using the following simple example.

Example 7.1.1   A common actuator in control systems is the DC motor. It provides rotary motion for a current input. The dynamical behaviour between current and speed is described by the simplified transfer function

 (7.11)

In order to compensate the plant dynamics, a candidate controller may be

 (7.12)

The open-loop transfer function is

 (7.13)

which shows a proportional behaviour. On step inputs to the controller the speed will jump, which is physically not possible due to the inertia of the motor. According to section 3.3 the controller in Eq.(7.12) is not realisable. Adding a pole in the controller transfer function to the left in the plane at will cure this problem, but with a delayed speed response. Figure 7.5a shows the controlled speed
for a unit step in the command input. The time constant of the closed loop system changes as the feedback gain increases. Increasing the controller gain will speed-up the behaviour and reduce the steady-state error, but will also increase the control effort as shown in Figure 7.5b. As the current of the motor is limited for physical reasons the manipulated variable is also limited. Increasing the controller gain to an arbitrary high value is not suitable. During the design of a controller such limitations have to be taken into account.

It is often true that closed-loop systems have a faster response as the feedback gain is increased, and if there are no other effects, this is generally desirable. However, systems typically also become less well damped and even unstable as the gain increases. This is shown when we mount the same DC motor on a robotic manipulator and control the speed of the manipulator arm using the same type of controller. In this case the speed of the arm movement is the controlled variable . The transfer function between the current of the DC motor and the speed of the arm is

 (7.14)

Figure 7.6 shows the step response of this control
system.

From the example given above it can be seen that a definite limit exists on how high we can make the gain. But there is a design tradeoff between gain and steady-state error. Attempts to resolve the conflict between small steady-state errors and good transient or dynamic responses must be undertaken. These two essential aspects of performance are considered when a control system is designed: the transient performance and the steady-state performance. The following sections deal with these aspects in more detail.

Next: Static properties of the Up: Behaviour of linear continuous-time Previous: Behaviour of linear continuous-time   Contents
Christian Schmid 2005-05-09