Block Diagrams - Section 4.4 - Explain open-loop and closed-loop transfer functions

Oct 26, 2026

Section 4.4 Block Diagrams explains how Laplace transform properties extend transfer functions to interconnected linear systems. It derives the open-loop transfer function as the ratio of output and input Laplace transforms under zero initial conditions, then discusses how block diagrams represent the resulting relationships. It further introduces cascade and parallel connections and shows how feedback (unity and nonunity) forms closed-loop configurations, deriving corresponding transfer functions and noting the role of loop and forward/feedback paths.

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4.4 Block Diagrams

Open-loop transfer function via Laplace transforms

Cascade and parallel system connections

Unity and nonunity feedback configurations

Closed-loop transfer function for unity feedback

Nonunity feedback, loop transfer function, and complex configurations

4.4 Block Diagrams Using the Laplace transform linearity and convolution properties we can easily extend the concept of the transfer function to conﬁgurations of several connected linear systems. In that way we will ﬁnd the equivalent transfer functions for cascade and parallel connections of systems, introduce the feedback (closed-loop) conﬁguration, and deﬁne the corresponding feedback system transfer function. In Section 4.3.1 we have deﬁned the transfer function of a linear time invariant continuous-time system. The system transfer function is the ratio of the Laplace transform of the system output and the Laplace transform of the system input under the assumption that the system initial conditions are zero. This transfer function infact represents the open-loop continuous-time system transfer function. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall 2003. Prepared by Professor Zoran Gajic 4–91 For the system at rest, the system input produces on the system output the signal 􀀀 ✁ given by 􀀀 ✁ which is symbolically represented in Figure 4.6 using a block diagram. F(s) Y(s)=H(s)F(s) H(s) Figure 4.6: An open-loop transfer function Note that the above block diagram can also be used in the case when the system initial conditions are different from zero. In such a case, an additive component coming from the system initial conditions should be added to the system output. For that reason in all block diagrams presented in this section we will denote the system output by ✄ ☎ ✄ ✆ . The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall 2003. Prepared by Professor Zoran Gajic 4–92 The open-loop transfer function is derived from the differential equation that is obtained using known physical laws (mathematical modeling procedure) . The accuracy of coefﬁcients that appear in the open-loop system transfer function depends on the accuracy of the system coefﬁcients (inductors, resistors, masses, friction coefﬁcients, and so on) . These coefﬁcients are not always perfectly known. Even more, the coefﬁcients change either due to aging or due to internal and/or external system disturbances. Due to changes in the system coefﬁcients (system parameters), it can happen that the actual system output (in the open-loop system conﬁguration) is pretty different from the one obtained analytically. The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall 2003. Prepared by Professor Zoran Gajic 4–93 A way to cope with the system parameter changes, and a way to reduce the impact of those changes on the system output, is to form a closed-loop system conﬁguration, also known as the system feedback conﬁguration. We can feed back the system output and form the closed-loop around the system as presented in Figure 4.7a. The directed path (as indicated by the arrows) from to is called the forward path and the directed path from to is called the feedback path. Such a feedback loop is called a unity feedback loop. In general, we can put a dynamic element in the feedback loop (another open-loop transfer function) as presented in Figure 4.7b. For notational convenience, we will denote the transfer function in the feedback path by and the transfer function in the forward path by . Sometimes, in the feedback path, we put a static element equal to a constant, that is, . The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall 2003. Prepared by Professor Zoran Gajic 4–94 (a) (b) Figure 4.7: Closed-loop system conﬁgurations: (a) unity feedback; (b) nonunity feedback The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall 2003. Prepared by Professor Zoran Gajic 4–95 In the feedback conﬁguration presented in Figure 4.7a, the output signal is fed back and compared to the input signal, and the differe

FAQs

How is the open-loop transfer function defined for a continuous-time linear time-invariant system?

It is defined as the ratio of the Laplace transform of the system output to the Laplace transform of the system input, assuming zero initial conditions. This represents the open-loop continuous-time system transfer function.

What block diagram structure corresponds to a unity feedback loop?

The system output is fed back and compared to the input signal, using the difference as a new input. The feedback path is unity, and the feedback signal is typically taken with a negative sign to avoid instability.

How does nonunity feedback differ from unity feedback in closed-loop transfer function derivation?

With nonunity feedback, another transfer function is placed in the feedback loop. The closed-loop transfer function is defined similarly, using the forward path and feedback transfer functions, and the product forms the loop transfer function.

Oct 26, 2026

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