(a) Analytical Development

The transfer function is defined as the Laplace transform of the deviation of a linear system output from its equilibrium value divided by the Laplace transform of the deviation of an input from its value at equilibrium. It is usually written as

Example 2.3.1. Determine the transfer function <50(s)/<5/(s) for the following lumped-parameter equation:

dO/dt = — ЗО + 41

Since this is a linear system, the equation for the deviation from equilibrium (involving <50 and SI) is exactly the same as the equation for absolute values of the variables (involving 0 and I). Thus we may write

dSOjdt = -3S0 + 431

Now Laplace transform:

s <50(s) — <50(0) = -3 <50(s) + 4 <50(s)

The value of <50(0) is zero because the initial state is the equilibrium state. The transfer function is

<50(s) _ 4

<5/(s) s + 3 ^

Transfer functions may also be developed for distributed-parameter systems.

Example 2.3.2. Develop a transfer function that relates internal temperatures to a change in the surface temperature applied simultaneously at each sur­face of a one-dimensional slab.

The equation (in terms of deviation from equilibrium) is

d2ST _ 8 ST
a 8z2 8t

where a is a constant, T the temperature, and z the position. Laplace trans­form to obtain

82 <5T(s) s

The solution of this ordinary differential equation is

<5T(s) = A exp[(s/a)1/2z] + В exp[(s/a)1/2z]

Now use appropriate boundary conditions:

(1) 3T(L, s) = 36(s)

(в is the slab surface temperature, and L the half thickness of the slab.)

(2) ^(0,s) = 0

The result is

3T(z, s) cosh[(s/a)1/2z]

<50(s) cosh[(s/a)1/2L] ®

In general, the transfer function for a lumped-parameter system will be a ratio of polynomials in s with a finite number of poles. The transfer function for a distributed-parameter system will be a transcendental function of s with an infinite number of poles.

(b) Stability

The stability of a linear system is determined by the values of the poles of the transfer function. If all of the poles have negative real parts, the system is stable. If any pole has a positive real part, the system is unstable. If a complex conjugate pair of poles lies on the imaginary axis, the system will have an undamped oscillatory response, and the system is on the stability- instability boundary. System stability can be determined by a direct calcula­tion of the poles or by use of stability criteria (2) that provide an indication of stability versus instability without a calculation of all the poles. Further­more, some of these criteria also provide a measure of relative stability so that the stability margin may be evaluated. A description of the various stability criteria and relative stability measures is beyond the scope of this book. However, we are interested in relative stability measures that can be determined experimentally. The Nyquist stability criterion provides a suitable relative stability measure and is described in Section 2.4.

(c) Block Diagrams

Transfer functions provide input-output relations that may be defined for each part of a system and combined to give complete input-output relations for the whole system. This involves the use of block diagrams and

their combination using block-diagram algebra. An input-output relation is indicated by a block and input and output lines as shown in Fig. 2.1.

Total system models are obtained by combining subsystems, which may be arranged in series, parallel, or feedback configurations as shown in Fig. 2.2.

Fig. 2.2. Block diagram combinations.

	G,
? ‘
	H

c. Feedback Combination

The overall system transfer function G for a series arrangement is obtained multiplying each of the serially connected transfer functions

G = GlG2G3 ■ ■ • (2.3.2)

The overall transfer function for a parallel arrangement is obtained by adding each of the parallel transfer functions:

G = G, + G2 + G3 + ••• (2.3.3)

The overall transfer function for the feedback arrangement shown in Fig. 2.2 is

G = G,/( 1 + GlH)

Note that the sign of the feedback term at the summing junction is negative, indicating that the feedback is subtracted at the summing junction. This is the most common form, but the sign of the feedback signal may also be positive, giving

G = G,/( 1 — GlH) (2.3.5)

For feedback systems, a distinction is made between the closed-loop transfer function and the open-loop transfer function. For the feedback system of Fig. 2.2, the closed-loop transfer function is the overall input — output relation,

G = GJ( + GlH) (2.3.6)

The open-loop transfer function is defined as GtH. It gives the input-output relation that would describe a system obtained by breaking the feedback loop at some point, inserting an input downstream of the break, and measuring the output upstream of the break. For example, Fig. 2.3 shows an open-loop system. The transfer function GXH gives the input-output

Fig. 2.3. Feedback combination for the open-loop transfer function.

relation — b/c. Since the open-loop transfer function includes all of the sub­system transfer functions that occur in the closed-loop transfer function, it is possible to obtain information about closed-loop system performance by studying the open-loop transfer function. In particular, the Nyquist stability criterion described in the next section uses the open-loop transfer function to determine the stability of the closed-loop system.