Open- and Closed-Loop Systems

A control system was defined in Section 1.1 as an interconnection of components forming a system configuration that will provide a desired system response. An open control system is shown in Fig.2.15. For contrast, a closed-loop negative feedback control system is shown in Fig.2.16.

Fig. 2.15. Open-loop system. Fig. 2.16. Closed-loop system.

The prime difference between the open- and closed-loop systems is the generation and utilization of the error signal. The closed-loop system, when operating correctly, operates so that the error will be reduced to a minimum value. The signal e(t) is the measure of the error of the system and is equal to the error e(t)=x(t)-y(t) when . The utilization of this signal to control the process results in a closed-loop sequence of operations that is called a feedback system.

Let us consider the closed-loop system shown in Fig. 2.17. in order to determine the transfer functions of the closed-loop system.

Fig.2.17. Closed-loop control system.

The open-loop transfer function (feed-forward transfer function) is expressed by the equation

. (2.52)

The open-loop transfer function for the output signal, , (forward transfer function) concerning the input signal, , when the disturbance action z(t)=0, is

, (2.53)

and

. (2.54)

When x(t)=0, we have the open-loop transfer function for the output (forward transfer function), relatively to the disturbance action as follows

, (2.55)

then the output is

. (2.56)

The close-loop transfer function for the output signal, , when the disturbance action is assumed to be equal zero, z(t)=0, is

(2.57)

then the output is characterized by the equation

. (2.58)

The close-loop transfer function for the output, , concerning the disturbance action z(t), when the input x(t)=0, is

, (2.59)

then the equation for the output is

. (2.60)

When the control system is subjected to the control action, x(t), and to the disturbance action, z(t), we have

, (2.61)

In the case of unity negative feedback system shown in Fig.2.18 , then the forward transfer function concerning the error, when z(t)=0, can be written as

. (2.62)

Fig. 2.18. Closed-loop system with unity negative feedback.

The transfer function concerning the error when x(t)=0 and is

. (2.63)

Terms and Concepts

Block diagrams.Consist of unidirectional, operational blocks that represent the transfer functions of the elements of the system.

Characteristic equation. The relation formed by equating to zero the denominator of the transfer function.

Disturbance signal.An unwanted input signal which affects the system’s output signal.

Error signal. The difference between the desired output and the actual output.

Laplace transform.A transformation of a function f(t) from the time domain into the complex frequency domain yielding F(s).

Linear approximation. An approximate model that results in a linear relationship between the output and the input of the device.

Mathematical models. Descriptions of the behavior of a system using mathematics.

Transfer function.The transfer function in the operational form is the ratio of the forcing operator to the characteristic operator.The transfer function in the Laplace form is the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all initial conditions assumed to be zero.

The Instantaneous Element

An instantaneous (proportional) element is described by the linear relationship

(3.30)

or by the transfer function

, (3.31)

where k is the gain of the element (for k>1 we have the amplification of the input signal, and for k<1 we obtain the attenuation of the input).

The instantaneous element behavior is represented by the equation

h(t)=k. (3.32)

The frequency response of this element is characterized by the following expressions

;

; ; ; (3.33)

; .

The transient response to a step input for instantaneous element is shown in Fig.3.11(a), the log-magnitude and phase diagram is shown in Fig. 3.11(b).

(a) (b)

Fig. 3.11. Transient response to a step unit input (a) and log-magnitude and phase diagram (b) for an instantaneous (proportional) element.

The block diagram representation for this element is shown in Fig.3.12.

(a) (b)

Fig. 3.12. Block diagram representation of an instantaneous (proportional) element.

A gear train, an amplifier, a tachometer are an examples of the instantaneous elements.

Lagging Elements