5. Linearization of Dynamic Equations

The stability of the nonlinear equations can be determined by finding the following state-space form

(11)

and determining the eigenvalues of the A (state-space) matrix.

The nonlinear dynamic state equations (1a) and (2a) are

(1a)

(2a)

let the state, and input variables be defined in deviation variable form

5.1 Stability Analysis

Performing the linearization, we obtain the following elements for A

where we define the following parameters for more compact representation

or

From the analysis presented above, the state-space A matrix is

(12)

The stability characteristics are determined by the eigenvalues of A, which are obtained by solving det (l I - A) = 0.

det (l I - A)=(l - A11)(l - A22) - A12A21

=l2 - (A11+ A22) l + A11A22 - A12A21

=l2 - (tr A) l + det (A)

the eigenvalues are the solution to the second-order polynomial

l2 - (tr A) l + det (A) =0(13)

The stability of a particular operating point is determined by finding the A matrix for that particular operating point, and finding the eigenvalues of the A matrix.

Here we show the eigenvalues for each of the three case 2 steady-state operating points. We use the function routine cstr_amat.m shown in appendix 3 to find the A matrices.

Operating point 1

The concentration and temperature are 8.564 kgmol/m3 and 311.2 K, respectively.

»[amat,lambda] = cstr_amat(8.564,311.2)

amat =

-1.1680 -0.0886

2.0030 -0.2443

lambda =

-0.8957

-0.5166

Both of the eigenvalues are negative, indicating that the point is stable, which is consistent with the results of Figure 2.

Operating point 2

The concentration and temperature are 5.518 and 339.1, respectively.

»[amat,lambda] = cstr_amat(5.518,339.1)

amat =

-1.8124 -0.2324

9.6837 1.4697

lambda =

-0.8369

0.4942

One of the eigenvalues is positive, indicating that the point is unstable. This is consistent with the responses presented in Figure 3.

Operating point 3

»[amat,lambda] = cstr_amat(2.359,368.1)

amat =

-4.2445 -0.3367

38.6748 2.7132

lambda =

-0.7657 + 0.9584i

-0.7657 - 0.9584i

The real portion of each eigenvalue is negative, indicating that the point is stable; again, this is consistent with the responses in Figure 4.

5.2 Input/Output Transfer Function Analysis

The input-output transfer functions can be found from

G(s)= C(sI - A)-1B(14)

where the elements of the B matrix corresponding to the first input (u1 = Tj - Tjs) are

the reader should find the elements of the B matrix that correspond to the second and third input variables (see exercise 8)

Here we show only the transfer functions for the low temperature steady-state for case 2. The input/output transfer function relating jacket temperature to reactor concentration (state 1) is

and the input/output transfer function relating jacket temperature to reactor temperature (state 2) is

Notice that the transfer function for concentration is a pure second-order system (no numerator polynomial) while the transfer function for temperature has a first-order numerator and second-order denominator. This indicates that there is a greater ìlagî between jacket temperature and concentration than between jacket temperature and reactor temperature. This makes physical sense, because a change in jacket temperature must first affect the reactor temperature before affecting the reactor concentration.