Response of Linear/Invariant Systems

For linear/invariant systems there are four basic single-input, single-response cases, illustrated in Fig. 7.2. They are characterized by the inputs, which are, respectively:

1. a unit impulse at t = 0

2. a unit step at / = 0

3. a sinusoid of unit amplitude and frequency /

4. white noise

In the first two the system is specified to be quiescent for t < 0 and to be subjected to a control or disturbance input at t = 0. In the last two cases the input is presumed to have been present for a very long time. In these two the system is assumed to be sta­ble, so that any initial transients have died out. Thus in case 3 the response is also a steady sinusoid and in case 4 it is a statistically steady state. We discuss the first three of these cases in the following, but the fourth, involving the theory of random processes, is outside the scope of this text. The interested reader will find a full ac­count of that topic in Etkin, 1972.



Figure 7.2 The four basic response problems. (1) Impulse response. (2) Step response. (3) Frequency response. (4) Response to white noise.


A central and indispensable concept for response analysis is the transfer function that relates a particular input to a particular response. The transfer function, almost uni­versally denoted G(s), is the ratio of the Laplace transform of the response to that of the input for the special case when the system is quiescent for t < 0. A system with n state variables xt and m controls c, would therefore have a matrix of nm transfer func­tions Gfs).

The Laplace transform of (7.1,4) is.

sx — Ax – f – Be



(si – A)x = Be


x = Gc




G(s) = (si – A) ‘B


is the matrix of transfer functions. The response of the ith state variable is then given by

Ф) = ^С, р)ф) (7.2,3)


For a single-input single-response system with transfer function G(s) we have simply

Systems in series.


When two systems are in series, so that the response of the first is the input to the second, as in Fig. 7.3, the overall transfer function is seen to be the product of the two. That is,

Xi(s) = Gfs)c(s)

and x2(s) = G2(s)xl(s) = G2(s)G,(s)c(s)

Thus the overall transfer function is

G(s) = x2(s)/c(s) = G,0)G20) (7.2,5)

Similarly for n systems in series the overall transfer function is

G(s) = G,(*)G2(s)-G„(s) (7.2,6)


High-order linear/invariant systems, such as those that occur in aerospace practise, can always be represented by a chain of subsystems like (7.2,6). This is important, because the elemental building blocks that make up the chain are each of a simple kind—either first-order or second-order. To prove this we note from the definition of an inverse matrix (Appendix A. l) that

adj (si – A)

det (si — A)

We saw in Sec. 6.1 that det (A — si) is the characteristic polynomial of the system. We also have, from the definition of the adjoint matrix as the transpose of the matrix of cofactors, that each element of the numerator of the right side of (7.2,7) is also a polynomial in s. (See Exercise 7.1.) Thus it follows from (1.2,2b), on noting that В is a matrix of constants, that each element of G is a ratio of two polynomials, which can be written as



in which /(s) is the characteristic polynomial. It is seen that all the transfer functions of the system have the same denominator and differ from one another only in the dif­ferent numerators. Since f(s) has the roots A, . . . A,„ the denominator can be factored to give


" (s – A,)C? – A2) ••• (s – A„)

Figure 7.4 High-order systems as a “chain.”

Now some of the eigenvalues Ar are real, but others occur in complex pairs, so to ob­tain a product of factors containing only real numbers we rewrite the denominator thus

m 1/2 (n+m)

f(s)=r[(s-Ar) П (s2 + a^ + br) (7.2,10)

r= 1 r=m+ 1

Here Ar are the m real roots of f(s) and the quadratic factors with real coefficients a, and br produce the (n — m) complex roots. It is then clearly evident that the transfer function (7.2,9) is also the overall transfer function of the fictitious system made up of the series of elements shown in Fig. 7.4. The leading component Nfs) is of course particular to the system, but all the remaining ones are of one or other of two simple kinds. These two, first-order components and second-order components, may there­fore be regarded as the basic building blocks of linear/invariant systems. It is for this reason that it is important to understand their characteristics well—the properties of all higher-order systems can be inferred directly from those of these two basic ele­ments.

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