responses [2]. For the grey – box models is to compute the coefficients of ordinary differential and difference equations for systems modelled from first principles.
Figure 2.1 : Block Diagram of Model Identification Method[2]
2.3.1 Transfer Function
The definition of transfer function is multiplying the factor in the equation for transform of output to the transform of the input [3]. There are two
types of transfer function which are continuous system transfer function and discrete-time system transfer function. The difference of both transfer function
is the domain use which is s-domain for continuous and z-domain for discrete- time. There are listed a few properties of both continuous system and discrete-
time transfer function.
There are six properties of continuous system transfer function stated
in [3] which are: 1. The Laplace transform of its impulse response is У
δ
t, t ≥ 0. So that, the transform of the output Ps if the system input with transfer function is an
impulse and all initial values of zero.
2. By taking the Laplace transform and ignoring all terms come up from initial values, the transfer function of the system can be determined from the
system differential equation. The transfer function is
Y s P s
U s
2.1
3. By replacing the s-variable with the differential operator D which is D ≡ ddt, the differential equation of the system can be obtained from the
transfer function.
4. Characteristic equation can be used in determining the stability of a time- invariant linear system. The denominator of the transfer function is the
characteristic polynomial. So, in continuous systems, it is stable if all roots of the denominator have negative real parts.
5. The denominator root gives the poles of the system while the numerator roots give the zeros of the system. By specifying the poles and zero, the
transfer function can be specified to within a constant, K gain factor. A pole-zero maps in the s-plane can represent the poles and zero system.
6. The system is considered minimum phase if the transfer functions has no poles or zero with positive real parts.
There are also six properties of discrete-time system transfer function as listed below[3]:
1. Pz is the z-transform of its Kronecker delta response У
δ
k, k = 0,1, … 2. Pz can be used in obtaining the difference equation of the system by
replacing the z variable with the shift operator Z defined for any integers k and n by
[ ]
n
Z y k y k n
2.2
3. The Pz denominator is the characteristic polynomial of the system. So, the system is stable if all the denominator roots located in the z-plane unit
circle.
4. The denominator indicates the poles and the numerator indicate the zeros of the system. The pole-zero map in the z-plane can be used to represent the
system poles and zeros. The poles-zero map of Pz used to construct the output response by including the poles and zeros of the input Uz. in
specifying the Pz, the system poles and zeros and the gain factor K are specified. The Pz equation is as shows below:
1 2
1 2
... ...
n n
K z z z z z z
P z z p z p
z p
2.3
5. The order of the denominator polynomial must be greater or equal to the order of the numerator polynomial of the transfer function of a causal
physically realizable discrete-time system.
6. The steady state response of a discrete-time system to a unit step input is called the d.c gain
The block diagram below shows the transfer function with the left is the input, the right is the output and inside the block is the system transfer
function. The denominator of the transfer function is identifical to the characteristic polynomial of the differential equation.
Figure 2.2 : Block diagram of the transfer function
1 1
1 1
... ...
m m
m m
n n
n n
b s b s
b a s
a s a
2.1 Rs
Cs
2.4 MODEL PREDICTIVE CONTROLLER
