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发表于 6-6-2005 07:00 AM
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设定方程式
Defining Transfer Functions in Matlab
There are two forms of transfer function representation in Matlab. The most obvious is the polynomial form where
G(s) = b(s) a(s)
= s2 + 2 s + 3 s3 + 4 s2 + 5 s + 6
is entered as two row vectors with the polynomial coefficients entered in the order of ascending powers of s.
>> b = [1, 2, 3];
>> a = [1, 4, 5, 6];
Missing coefficients, must be entered as zero: so q(s) = s2 + 2s and r(s) = s4 + s2 + 1 are entered as
>> q = [1, 2, 0];
>> r = [1, 0, 2, 0, 1];
An alternative form of representation for transfer functions is the factored polynomial, for example
G(s) = (s + 1)(s + 3) s(s + 2)(s + 4)
The advantage of this formulation is that the zeros of the numerator and denominator polynomials are obvious by inspection. So it is often used in the preliminary analysis of the performance of a dynamic system. The poles of this transfer function are s = 0, -2, -4 and the zeros are s = -1, -3. In Matlab, this form of transfer function is specified by a column vector of the zeros and a column vector of the poles:
>> z = [-1; -3];
>> p = [0; -2; -4];
A third parameter, the overall gain K, completes the definition of the so called pole-zero-gain form of transfer function. In this case K = 1
>> K = 1;
The Linear Time Invariant System Object
Starting from version 4 of the Control System Toolbox (distributed with Matlab version 5), the Mathworks have introduced a new data object for the creation and manipulation of system transfer functions. This object is called the Linear Time Invariant (LTI) System Object. It is used to gather the components of a transfer function (or state-space model) into a single variable which can then easily be combined with other LTI system objects.
To create a LTI system object representing a factored transfer function the following command is issued:
>> G = zpk(z,p,K)
Zero/pole/gain:
(s+1) (s+3)
-------------
s (s+2) (s+4)
The expanded numerator and denominator form of the transfer function is readily obtained by using a ``data extraction'' function.
>> [num,den]=tfdata(G,'v')
num =
0 1 4 3
den =
1 6 8 0
LTI system objects can also be created from the expanded form of a transfer function directly:
>> G2=tf(num,den)
Transfer function:
s^2 + 4 s + 3
-----------------
s^3 + 6 s^2 + 8 s
and the zeros and poles similarly extracted:
>> [zeros,poles,gain]=zpkdata(G2,'v')
zeros =
-3
-1
poles =
0
-4
-2
gain =
1
Setting LTI Properties
Numerous options are available to document the LTI system objects that you create. For example, suppose the transfer function G represents a servomechanism with input ``Voltage'' and output ``Angular Position''. We can add this information to the LTI system as follows:
>> set(G,'inputname','Voltage','ouputname','Angular Position')
>> G
Zero/pole/gain from input "Voltage" to output "Angular Position":
(s+1) (s+3)
-------------
s (s+2) (s+4)
Such documentary information is probably best added when the LTI system object is created, for example as:
>> G3=zpk(z,p,K,'inputname','Armature Voltage (V)',...
'outputname','Load Shaft Position (rad)',...
'notes','An armature voltage controlled servomechanism')
Zero/pole/gain from input "Armature Voltage (V)"
to output "Load Shaft Position (rad)":
(s+1) (s+3)
-------------
s (s+2) (s+4)
Once the LTI object has been documented, the documentation can be extracted using commands like:
>> get(G3,'notes')
ans =
'An armature voltage controlled servomechanism'
One can also access the documentation using an ``object reference'' notation
>> in=G3.inputname, out=G3.outputname
in =
'Armature Voltage (V)'
out =
'Load Shaft Position (rad)'
All the documentation available on an LTI system object may be extracted with a single command:
>> get(G3)
z = {1x1 cell}
p = {1x1 cell}
k = 1
Variable = 's'
Ts = 0
Td = 0
InputName = {'Armature Voltage (V)'}
OutputName = {'Load Shaft Position (rad)'}
Notes = {[1x45 char]}
UserData = []
There are numerous other documentation features provided for LTI system objects. Please consult the on-line help for full details.
System Transformations
Matlab supports the easy transformation of LTI system objects between expanded and factored forms1. For example to convert a transfer function from `expanded' form to pole-zero-gain form the following command is used:
>> G4 = zpk(G2)
Transfer function:
s^2 + 4 s + 3
-----------------
s^3 + 6 s^2 + 8 s
To convert from zero-pole-gain form to expanded form we use the function tf:
>> G5 = tf(G)
Transfer function from input "Voltage" to output "Angular Position":
s^2 + 4 s + 3
-----------------
s^3 + 6 s^2 + 8 s
Please note that these transformations are merely a convenience that allow you to work with your preferred form of representation. Most of the tools that deal with LTI system objects will work with any form2. Furthermore, you can always use the data extraction functions zpdata and tfdata to extract the zero-pole-gain and numerator-denominator parameters from a LTI system, no matter in which form it was originally defined, without the need for an explicit conversion.
Combining LTI System Objects
A powerful feature of the LTI system object representation is the ease with which LTI objects can be combined. For example, suppose we have two transfer functions
G1(s) = s+1 s+3
and
G2(s) = 10 s(s+2)
then the series combination of the two transfer functions Gs(s) = G1(s)G2(s) is obtained using the ``*'' (multiplication) operator:
>> G1=tf([1 1],[1 3]);
>> G2=tf(10,conv([1 0],[1 2])); % conv is polynomial multiplication
>> Gs=G1*G2 % series connection of two sys objects
Transfer function:
10 s + 10
-----------------
s^3 + 5 s^2 + 6 s
>> [zeros,poles,K]=zpkdata(Gs,'v')
zeros =
-1
poles =
0
-3
-2
K =
10
The parallel connection of two LTI system objects corresponds to addition Gp = G1(s)+G2(s):
>> Gp = G1 + G2
Transfer function:
s^3 + 3 s^2 + 12 s + 30
-----------------------
s^3 + 5 s^2 + 6 s
The feedback connection of two LTI system objects is also supported. The function feedback is used for this. Let
G(s) = 2s2 + 5s +1 s2 + 2s + 3
be the forward transfer function of a closed-loop system and
H(s) = 5(s+2) (s+10)
be the feedback network. Then the closed-loop transfer function3 is
Gc(s) = G(s) 1 + G(s)H(s)
.
In matlab:
>> G = tf([2 5 1],[1 2 3],'inputname','torque',...
'outputname','velocity');
>> H = zpk(-2,-10,5);
>> Gc = feedback(G,H) % negative feedback assumed
Zero/pole/gain from input "torque" to output "velocity":
0.18182 (s+10) (s+2.281) (s+0.2192)
-----------------------------------
(s+3.419) (s^2 + 1.763s + 1.064)
The Analysis of LTI System Objects
Matlab uses the LTI system objects as parameters for the analysis tools such as impulse, step, nyquist, bode and rlocus. As an example of their use try:
>> rlocus(G*H) % root locus
>> bode(G*H)% open-loop frequency response
>> step(Gc) % closed-loop step response
>> bode(Gc) % closed-loop frequency response
Matlab also provides two interactive graphical tools that work with LTI system objects.
* ltiview is a graphical tool that can be used to analyze systems defined by LTI objects. It provides easy access to LTI objects and time and frequency response analysis tools.
* rltool is an interactive tool for designing controllers using the root locus method.
You are encouraged to experiment with these tools.
Partial Fraction Expansions
Matlab provides a command called residue that returns the partial fraction expansion of a transfer function. That is, given
G(s) = sm +bm-1sm-1 + ¼+ b1 s + b0 sn + an-1sn-1 + ¼+a1 s + a0
it returns
R1 s + p1
+ R2 s + p2
+ ¼ Rn s + pn
+ K(s)
where pi are the poles of the transfer function, Ri are the coefficients of the partial fraction terms (called the residues of the poles) and K(s) is a remainder polynomial which is usually empty. To use this, the starting point must (rather perversely) be the expanded form of the transfer function in polynomial form. Thus given
C(s) = 5(s + 2) s(s + 3)(s + 10)
we obtain the partial fraction expansion using the Matlab command sequence:
>> k = 5; z = [-2]; p = [0; -3; -10]; % zero-pole-gain form
>> C = zpk(z,p,k);
>> [num,den] = tfdata(C,'v')
num =
0 0 5 10
den =
1 13 30 0
(Note that the leading terms in num are zero).
>> [r,p,k] = residue(num,den)
r =
-0.5714
0.2381
0.3333
p =
-10
-3
0
k =
[]
which we interpret to mean
C(s) = 0.3333 s
+ 0.2381 s + 3
- 0.5714 s + 5
.
If C(s) represents the step response of the system
G(s) = 5(s + 2) (s + 3)(s + 10)
then the step response is, by inspection,
c(t) = 0.3333h(t) + 0.2381 e-3t - 0.5714 e-10t.
You can check this with the command:
>> newC = tf([5, 10],[1, 13, 30])
>> step(newC)
(where the 1/s term has been eliminated because step provides the forcing function itself). This should give exactly the same results as:
t = 0:.05:1.5; % time vector
c = 0.3333 + 0.2381 * exp(-3*t) - 0.5714 * exp(-10*t);
plot(t,c)
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