2. DISCRETE STATE SPACE MODELS FOR SYSTEM IDENTIFICATION
So far we have studied several model equations for structural system identification, which are continuous in time. In modern measurement practice, discrete sampling has become the standard methodology. In other words, the sensor output as well as the force input are recorded in discrete numbers. Therefore, an appropriate model for system identification must be cast into a discrete form that reflects the measurement environment. The derivation of a discrete state space model begins with the solution of the continuous state space model which is recalled here for convenience:

3. DISCRETE STATE SPACE MODELS FOR SYSTEM IDENTIFICATION
- cont’d
A simple discrete approximation of the above continuous solution is to employ sampling with a zero-order hold. By zero-order hold it is meant to hold the excitation input u constant during each sampling step:

4. Discrete solution form of the internal variable (x):
DISCRETE STATE SPACE MODELS FOR SYSTEM IDENTIFICATION
- cont’d
Discrete solution form of the internal variable (x):

5. Discrete form of the state space model:
DISCRETE STATE SPACE MODELS FOR SYSTEM IDENTIFICATION
- cont’d
Discrete form of the state space model:
It is noted that C and D have not been affected by the discrete form.
Why?

6. Output Sequence of Discrete State Space Model
Consider an initially-relaxed system, namely, the system with the trivial initial condition (x(0) = 0). For this case the output sequence is given by

7. (Discrete Impulse Response Functions(DIRFs) )
Markov Parameters
(Discrete Impulse Response Functions(DIRFs) )
When u(0) = 1, and
u(1) = u(2) = … = u(k) =0, the excitation (input) is called
the discrete impulse excitation. The corresponding discrete
Solution sequence is thus the discrete impulse response functions,
or is referred to Markov parameters defined by:
For a general train of excitations, the response is given in terms of the Markov parameters as

8. STATE OBSERVER MODEL
The identification models discussed so far in the preceding sections are based on the physical models that are derivable from analytical deterministic modeling process. It will be seen in the subsequent chapters that the length of the input data used for system identification assumes that the impulse response h(t) decays to a sufficiently small amplitude within that time duration.
There are many instances where the system to be identified exhibits very little damping such as composite structures and many light-weight space structures, often contaminated with noises. For such systems, one may inject an artificial damping by augmenting the output into the state space model by modifying the model equation or embed a filter to deal with noises.

9. STATE OBSERVER MODEL - cont’d
The state observer model equation begins with
stated as

10. STATE OBSERVER MODEL - cont’d

11. STATE OBSERVER MODEL - cont’d

12. Continuous transfer function:
TRANSFER FUNCTION FOR DISCRETE STATEE SPACE MODEL
Continuous transfer function:
Discrete transfer function:

13. Relation between the two-sided Laplace Transform
and z-Transform:

14. Matlab Toolbox Functions

16. Luckily, Matlab has the conversion routine!
>> num=[1/ /2]
>> den =[1 -2 1];
>> [A, B, C, D] = tf2ss(num,den)
A =[ ];
B =[ 1
0];
C = [ ]
D =