When you discretize the system, the state matrices have different meanings:
- x(k+1) = A*x(k) + B*u(k)
- y(k) = C*x(k) + D*u(k)
That is, eq(1) shows how the system can be represented as a sum of the values at previous time steps, rather than defining a system of differential equations that describes system behavior. That naturally changes to the structure of the A matrix, and explains why you cannot simply replace their current values with different ones. By default, "c2d" uses a zero-order hold for the conversion, though you could use a different method depending on your application and desired outputs. You might find this page in the documentation on conversion methods to be relevant.
That said, you can perform a similarity transformation to get a different system representation. For example, you can use MATLAB's "canon" function to transform the system into modal or companion form. These forms are based on transformations of the A matrix, so while they will not have the same meaning that they have with continuous systems, they are still valid conversions that will obtain nicer representations, e.g. for the following second-order system:
>> A = [0 1; -0.5 -1]; B = [0; 1]; C = [1 0]; D = 0;
>> sys = ss(A,B,C,D);
>> sysd = c2d(sys,0.01)
sysd =
A =
x1 x2
x1 1 0.00995
x2 -0.004975 0.99
B =
u1
x1 4.983e-05
x2 0.00995
C =
x1 x2
y1 1 0
D =
u1
y1 0
Sample time: 0.01 seconds
Discrete-time state-space model.
>> [sysd2,T] = canon(sysd,'companion')
sysd2 =
A =
x1 x2
x1 0 -0.99
x2 1 1.99
B =
u1
x1 1
x2 0
C =
x1 x2
y1 4.983e-05 0.0001488
D =
u1
y1 0
Sample time: 0.01 seconds
Discrete-time state-space model.
T =
1.0e+04 *
-0.9950 0.0150
1.0050 -0.0050
Alternatively, if there is a form that you would like to transform to and you have the corresponding transformation matrix T, you can use "ss2ss" to perform the similarity transformation and obtain an equivalent state representation.
