The stability of the nonlinear equations can be determined by finding the following state-space form
and determining the eigenvalues of the A (state-space)
The nonlinear dynamic state equations (1a) and (2a) are
let the state, and input variables be defined in deviation variable
5.1 Stability Analysis
Performing the linearization, we obtain the following elements
where we define the following parameters for more compact representation
From the analysis presented above, the state-space A matrix is
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 +
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
»[amat,lambda] = cstr_amat(8.564,311.2)
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)
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)
-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.