Frequency Response of Nonlinear Systems A. B. Ritter and J. M. Douglas' Department of Chemical Engineering, Tiniversity of Rochester, Rochester, A'. P.
The equations used to describe the dynamic operation of most chemical processes either are nonlinear or have variable coefficients. Previous discussions of the dynamic response of these systems normally have been based on a linearization of the state equations. A procedure which can be used to determine whether or not the linearized analysis is valid f o r both lumped and distributed parameter process is presented. It can be used to assess the potential adva ntages of periodic operation of a stable process.
CHE~IICAL
ESGISIWRS have displayed a great interest in process dynamics and control theory during the last decade. The current approach is to use the conservation equationsi.e., the conservation of mass, energy, and monientiimto derive models describing the dynamic characteristics of processes. Since most of these models are nonlinear or the input to the system appears as a variable coefficient, it' is very difficult to determine the dynamic response of the plant even for simple inputs. Thus, it' has been coninion practice t,o linearize the equations around some steady-st'ate operat'ing condition of interest and then find the response of the linearized system. The linearization is based on a Taylor series expansion, arid it is knowii that for sufficiently small fluctuations this model will give valid estimate$. However, the problem of establishing a region where the approximation is satisfactory ha5 received litt,le attention in the literature. Most investigators assume that linear models will be applicable over the whole range of engineering interest and then attempt to confirm this assumption by comparing experiniental data with the predictions obtained from the theory. Of course, this met,hod runs into difficulty when an exact confirmation is not observed, since one is never certain whether the linearization or some other assumpt'ion introduced to simplify the model is the cause of the discrepancy. This paper shows that' classical perturbation theory (Minorsky) 1962) can be used t o determine the validity of the linearization technique. I n addition, the results provide a qualitative insight into the different kinds of behavior of linear and nonlinear systems. Specifically, the approxiinate solutions indicate that the frequency responses of nonlinear systems contain higher harmonics, as well as t'he fundamental component. and that the time average output of the plant, will be different from that predicted frorii.steady-st*'1t e considerations. This latter result implies that dynamic might have :i performance better than the optimum steadystate design. Before presenting some cxamples which illustrate the superiority of periodic processing, it may be worth while to review the essential abpects of the aiialysis.
where x1 ,xn are the n state or dependent variables; ul, ,up are the p inputs which include both the control variables and the diqturbanceq; and j t represents some arbitrary function of these quantities. Often it is advantageous to qimplify the equations by expanding any complicated nonlinear functions which appear on the right-hand sides of Equation 1 in a Taylor series around some steady-state operating point
where a repeated index indicates summat,ion. The normal linearization procedure is based on the argument that if me consider small enough changes in the inputs, (uj - ~ j , ~ the deviations of the state variables from t'heir steady-state values, ( x j - xjs), will be sufficiently small so that all of the quadratic and higher-order terms in the expansion will be negligible in comparison with these linear terms. Thus, the system equations become a set of linear equations having constant coefficient,s, and we know that it always is possible to find analytical solutions for the dynamic response. As an additional consideration, however, we would like to be able to ascertain the range of system parameters and the magnitudes of the input changes where the quadratic terms begin to become ap1,rrciable. This can lie aeconiplished by associating an artificial parameter, p , with the quadratic and higher-order tcrins
Perturbation Theory
-\e first consider a lumped parameter plant, rlescrihed by the set of equations
where
Piewit addrezs, University of JIassachiisett s, Amherst, JIahs. 01002
and the term in Fkpiation 2 has been US
[(1-;sin$-cos-
Ft
:OC
)
cos - - 1 cos wt
2vsw
Outlet temperature fluctuations
+
+ (wg c o s ; -wzs i n - -
w z us
APPROXIMATE SOLUTION 0 :NUMERICAL INTEGRATION
1.
POINTS FIRST ORDER CORRECTION TERMS
'03
Figure 9.
(31)
where eo, el, e2, . . . are functions of position in the exchanger and periodic functions of time which we must determine. Following the procedure described earlier, we find that
Frequency Response of a Steam-Heated Exchanger
.05
+
C 0
\
- 03 -04-
26
l&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
\
3
4
5
i
/-LINEAR FREQUENCY RESPONSE
.-' /
6 8 9 OIMENSIONLESS TIME
IO
,'I
I2
I3
Id
.03 -
w =22.0FLOWRATE
.01 -
ONLY
-.03
,-CONDENSATE TEMPERATURE ONLY 4
8
12 16 20 24 28 32 FREQUENCY, w ( C Y C L E S / M I N )
36
40
B
cos US
wz
cos 0s
US
+ B sin 2az + 21 cos w
-
~
vs
Thus, we obtain an explicit expression for the output. T h e first term in square brackets is the conventional linear frequency response, while the second is the first-order correction function. This solution makes it apparent that if
A d
vsw
