Ch_ 1

1.4 Stability 31

Apparently, the stable subspace of this system is spanned by the vectors e, and e,,, while the unstable subspace is spanned by e, and e,.

Example 1.11. ~1ver . t~dpen~1111~itl101ftfr1111~ ~ .ictior1.

In Example 1.5 (Section 1.3.4), we discussed the Jordan normal form of the A matrix of the inverted pendulum with negligible friction. There we found a double characteristic value 0 and the single characteristic values Jmand -mThe. null space corresponding to the characteristic value 0 is spanned by the first two columns of the transformation matrix T, that is, by

These two column vectors, together with the characteristic vector corre-

span the unstable subspace of the system. The stable subspace is spanned by the remaining characteristic vector

1.4.4" Investigation of the Stability of Nonlinear Systems through Linearization

Most of the material o r this book is concerned with the design of linear control systems. One major goal in the design of suc11 systems is stability. In

32 Elements of Linear System Theory

later chapters very powerful techniques for finding stable linear feedback control systems are developed. As we have seen, however, actual systems are never linear, and the linear models used are obtained by linearization.

This means that we design systems whose linearized models possess good properties. The question now is: What remains of these properties when the actual nonlinear system is implemented? Here the following result is helpful.

Suppose tltat the systenz has an equilibri~nnstate x, and that the j'is~ctionf possessespartial deriuatiues with respect to the conlponents of s at xa. Suppose that the linearized state d~rerentialequation about x, is

w11ere the constant rnatrix A is the Jacobiarl o f f at s,. T11etr ifA is asy~ttptotically stable, tlre solutiotl x(t) = z, is an as)~npoticallJJstable soh~tionof

1-148.

Fo r a proof we refer the reader to Roseau (1966). Note that of course we cannot conclude anything about stability in the large from the linearized state differential equation.

This theorem leads to a reassuring conclusion. Suppose that we are confronted with an initially unstable system, and that we use linearized equations to find a controller that makes the linearized system stable. Then it can he shown from the theorem that the actual nonlinear system with this controller will at least be asymptotically stable for small deviations from the equilibrium state.

Note, however, that the theorem is reassuring only when the system contains "smooth" nonlinearities. If discontinuous elements occur (dead zones, stiction) this theory is of no help.

We conclude by noting that if some of the characteristic values of A have zero real parts while all the other characteristic values have strictly negative real parts no conclusions about the stability of x, can be drawn from the linearized analysis. If A has some characteristic values with positive real parts, however, x, is not stable in any sense (Roseau, 1966).

An example of the application of this theorem is given in Chapter 2 (Example 2.6, Section 2.4).

1.5 Trnnsforrn Analysis 33

1 . 5 T R A N S F O R M A N A L Y S I S O F T I M E - I N V A R I A N T S Y S T E M S

1.5.1Solution of the State Differential Equation through Laplace Transformation

Often it is helpful t o analyze time-invariant linear systems through Laplace transformation. We define the Laplace transform of a time-varying vector z(t) as follows

where s is a complex variable. A boldface capital indicates the Laplace transform of the corresponding lowercase time function. The Laplace transform is defined for those values of s for which 1-150 converges. We see that the Laplace transform of a time-varying vector z(t) is simply a vector whose components are the Laplace transforms of the components of z(t).

Let us first consider the homogeneous state differential equation

where A is a constant matrix. Laplace transformation yields

since all the usual rules of Laplace tiansformations for scalar expressions carry over to the vector case (Polak and Wong, 1970). Solution for X(s) yields

The Laplace transform of a time-varying matrix is obtained by transforming each of its elements. Theorem 1.17 is particularly convenient for obtaining the explicit form of the transition matrix as long as n is not too large, irrespective of whether o r not A is diagonalizable.

The matrix function ( s l - A)-' is called the resolue~~tof A. The following result is useful (Zadeh and Desoer, 1963; Bass and Gura, 1965).

where C i s constant. Laplace transformation and substitution of 1-165 yields

Y ( s )= CX(s) = C(sI - A)-'~(0)+ C(sI - A)-'BU(s), 1-167

which is the equivalent in the Laplace transform domain of the time domain expression 1-70 with to = 0 :

0

For x(0) = 0 the expression 1-167 reduces to

where

The matrix H(s) is called the transfer matrix of the system. If H(s) and U ( s ) are known, the zero initial state response of the system can he found by inverse Laplace transformation of 1-169.

By Theorem 1.17 it follows immediately from 1-170 that the transfer matrix H(s) is the Laplace transform of the matrix function K(t) = Cex p (At)B, t 2 0. I t is seen from 1-168 that K(t - .r), t 2 T, is precisely the impulse response matrix of the system.

From Theorem 1.18 we note that the transfer matrix can be written in the form

where P(s) is a matrix whose elements are polynomials in s. The elements of the transfer matrix H(s) are therefore rational functions of s. The common denominator of the elements of H(s)is det (sI - A ) , onless cancellation occurs of factors of the form s - A,, where ',is a characteristic value of A , in all the elements of H(s).

We call the roots of the common denominator of H(s) the poles of the trmwfer i v a t r i H(s). I f no cancellation occurs, the poles of the transfer matrix are precisely the poles of the system, that is, the characteristic values of A.

If the input rr(t) and the output variable ~ ( tare) both one-dimensional, the transfer matrix reduces to a scalar transfer j'ifirnction. Fo r multiinput multioutput systems, each element H,,(s) of the transfer matrix H(s) is the transfer function from the j-th component of the input to the i-th component of the output.

The system has the transfer matrix

The impulse response matrix 1-75 of the system follows immediately by inverse Laplace transformation of 1-178.

1.5.2 Frequency Response

In this section we study the frequency response of time-invariant systems, that is, the response to an input of the form

where I,,,is a constant vector. We express the solution of the state differential equation

in terms of the solution of the homogeneous equation plus a particular solution. Let us first try to find a particular solution of the form

where x,,, is a constant vector to be determined. I t is easily found that this particular solution is given by

where a is an arbitrary constant vector. The general solution of the inhomogeneous equation 1-180 is therefore

The constant vector a can be determined from the initial conditions. If the system 1-180 is asy~i~ptoticallj~stable, the first term of the solution will eventually vanish as t increases, and the second term represents the steadystate response of the state to the input 1-179. The corresponding steady-state

We note that in this expression the transfer matrix H(s) appears with s replaced by jw. We call H(jw) thefieqttencj~response iiiatris of the system.

Once we have obtained the response to complex periodic inputs of the type 1-179, the steady-state response to real, sinusoidal inputs is easily found. Suppose that the ic-th component p,(t) of the input u ( f )is given as follows

Assume that all other components of the input are identically zero. Then the steady-state response of the i-th component ili(t)of the output y(t) is given by

A convenient manner of representing scalar frequency response functions is through asymptotic Bode plots (D'Auo and Houpis, 1966). Melsa (1970) gives a FORTRAN computer program for plotting the modulus and the

. argument of a scalar frequency response function.

In conclusion, we remark that it follows from the results of this section that the steady-state response of an asymptotically stable system with frequency response matrix H ( j o ) to a constant input

~ ( t=) I!,,,

is given by

~ ( t=) H(o)~t,,,.

Example 1.14. Stirred tank.

The stirred tank of Example 1.2 has the transfer matrix (Example 1.13)

The system is asymptotically stable so that it makes sense to consider the frequency response matrix. With the numerical data of Example 1.2, we have

1.5.3 Zeroes of Transfer Matrices

Let us consider the single-input single-output system

where p(t) and q ( t ) are the scalar input and output variable, respectively, b is a column vector, and c a row vector. The transfer matrix of this system reduces to a transfer function which is given by

Denote the characteristic polynomial of A as

Then H(s) can be written as

where, if A is an 11 x n matrix, then $(s) is a polynomial of degree n and v ( s ) apolynomial of degree 11 - 1 o r less. The roots of y ( s ) we call the zeroes of the sJUten1 1-194. Note that we determine the zeroes before cancelling any common factors of y(s ) and The zeroes of H(s) that remain after cancellation we call the zer'oes of the transfer f~mcfiol~ .

I n the case of a multiinput multioutput system, H(s) is a matrix. Each entry of H(s) is a transfer function which has its own zeroes. I t is not obvious how to define "the zeroes of H(s)" in this case. I n the remainder of this section we give a definition that is motivated by the results of Section 3.8. Only square transfer matrices are considered.

First we have the following result (Haley, 1967).

Theorem 1.19. Consider the sj~stenl