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**(s**I** - **A**)-1**B**(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.