Page 61 - MATLAB an introduction with applications

46 ——— MATLAB: An Introduction with Applications

to the controller-canonical form state-space equations
x′ = Ax + Bu
y = Cx + Du
The function has four output matrices:
[A, B, C, D] = tf2ss(num,den) Determines the matrices A, B, C and D of the controller-canonical form state-
space equations. The input arguments num and den contain the coefficients,
in descending powers of s, of the numerator and denominator polynomials of
the transfer function that is to be converted.
tf 2zp Function: The tf 2zp function converts the polynomial transfer function

bs + b s n− 1 + + +
n
... b s b
() =
Hs 0 1 n− 1 n
m
+
as + a s m− 1 + + m− 1 s a m
... a
0
1
to the zero-pole-gain transfer function
(s − z )(s z )...(s − z )
−
() =
Hs k 1 2 n
(s − p 1 )(s − p 2 )...(s − p m )
The function has three output matrices:
[z, p, k] = tf2zp(num, den) Determines the zeros (z), poles (p) and associated gain (k) of the zero-pole-
gain transfer function using the coefficients, in descending powers of s, of
the numerator and denominator of the polynomial transfer function that is to
be converted.
zp2tf Function: The zp2tf function converts the zero-pole-gain transfer function
−
(s − z )(s z )...(s − z )
() =
Hs k 1 2 n
(s − p 1 )(s − p 2 )...(s − p m )
to the polynomial transfer function
n
bs + b s n− 1 + + +
... b s b
() =
Hs 0 1 n− 1 n
m
+
as + a s m− 1 + + m− 1 s a m
... a
1
0
The function has two output matrices:
[num, den] = zp2tf(z, p, k) Determines the vectors num and den containing the coefficients, in descending
powers of s, of the numerator and denominator of the polynomial transfer
function. p is a column vector of the pole locations of the zero-pole-gain transfer
function, z is a matrix of the corresponding zero locations, having one column
for each output of a multi-output system, k is the gain of the zero-pole-gain
transfer function. In the case of a single-output system, z is a column vector of
the zero locations corresponding to the pole locations of vector p.
zp2ss Function: The zp2ss function converts the zero-pole-gain transfer function
−
(s − z )(s z )...(s − z )
() =
Hs k 1 2 n
(s − p 1 )(s − p 2 )...(s − p m )
to the controller-canonical form state-space equations

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