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Create transfer function model, convert to transfer function model
sys = tf(num,den)
sys = tf(num,den,Ts)
sys = tf(M)
sys = tf(num,den,ltisys)
tfsys =
tf(sys)
tfsys = tf(sys,
'measured')
tfsys = tf(sys, 'noise')
tfsys = tf(sys, 'augmented')
Use tf to create real- or complex-valued transfer function models (TF objects) or to convert state-space or zero-pole-gain models to transfer function form. You can also use tf to create generalized state-space (genss) models or uncertain state-space (uss) models.
sys = tf(num,den) creates a continuous-time transfer function with numerator(s) and denominator(s) specified by num and den. The output sys is:
A tf model object, when num and den are numeric arrays.
A generalized state-space model (genss) when num or den include tunable parameters, such as realp parameters or generalized matrices (genmat).
An uncertain state-space model (uss) when num or den are uncertain (requires Robust Control Toolbox™ software).
In the SISO case, num and den are the real- or complex-valued row vectors of numerator and denominator coefficients ordered in descending powers of s. These two vectors need not have equal length and the transfer function need not be proper. For example, h = tf([1 0],1) specifies the pure derivative h(s) = s.
To create MIMO transfer functions, using one of the following approaches:
Concatenate SISO tf models.
Use the tf command with cell array arguments. In this case, num and den are cell arrays of row vectors with as many rows as outputs and as many columns as inputs. The row vectors num{i,j} and den{i,j} specify the numerator and denominator of the transfer function from input j to output i.
For examples of creating MIMO transfer functions, see Examples and MIMO Transfer Function Model in the Control System Toolbox™ User Guide.
If all SISO entries of a MIMO transfer function have the same denominator, you can set den to the row vector representation of this common denominator. See "Examples" for more details.
sys = tf(num,den,Ts) creates a discrete-time transfer function with sample time Ts (in seconds). Set Ts = -1 to leave the sample time unspecified. The input arguments num and den are as in the continuous-time case and must list the numerator and denominator coefficients in descending powers of z.
sys = tf(M) creates a static gain M (scalar or matrix).
sys = tf(num,den,ltisys) creates a transfer function with properties inherited from the dynamic system model ltisys (including the sample time).
There are several ways to create arrays of transfer functions. To create arrays of SISO or MIMO TF models, either specify the numerator and denominator of each SISO entry using multidimensional cell arrays, or use a for loop to successively assign each TF model in the array. See Model Arrays in the Control System Toolbox User Guide for more information.
Any of the previous syntaxes can be followed by property name/property value pairs
'Property',Value
Each pair specifies a particular property of the model, for example, the input names or the transfer function variable. For information about the properties of tf objects, see Properties. Note that
sys = tf(num,den,'Property1',Value1,...,'PropertyN',ValueN)
is a shortcut for
sys = tf(num,den) set(sys,'Property1',Value1,...,'PropertyN',ValueN)
You can also use real- or complex-valued rational expressions to create a TF model. To do so, first type either:
s = tf('s') to specify a TF model using a rational function in the Laplace variable, s.
z = tf('z',Ts) to specify a TF model with sample time Ts using a rational function in the discrete-time variable, z.
Once you specify either of these variables, you can specify TF models directly as rational expressions in the variable s or z by entering your transfer function as a rational expression in either s or z.
tfsys = tf(sys) converts the dynamic system model sys to transfer function form. The output tfsys is a tf model object representing sys expressed as a transfer function.
If sys is a model with tunable components, such as a genss, genmat, ltiblock.tf, or ltiblock.ss model, the resulting transfer function tfsys takes the current values of the tunable components.
An identified model is represented by an input-output equation of the form y(t) = Gu(t) + He(t), where u(t) is the set of measured input channels and e(t) represents the noise channels. If Λ = LL' represents the covariance of noise e(t), this equation can also be written as: y(t) = Gu(t) + HLv(t), where cov(v(t)) = I.
tfsys = tf(sys), or tfsys = tf(sys, 'measured') converts the measured component of an identified linear model into the transfer function form. sys is a model of type idss, idproc, idtf, idpoly, or idgrey. tfsys represents the relationship between u and y.
tfsys = tf(sys, 'noise') converts the noise component of an identified linear model into the transfer function form. It represents the relationship between the noise input, v(t) and output, y_noise = HL v(t). The noise input channels belong to the InputGroup 'Noise'. The names of the noise input channels are v@yname, where yname is the name of the corresponding output channel. tfsys has as many inputs as outputs.
tfsys = tf(sys, 'augmented') converts both the measured and noise dynamics into a transfer function. tfsys has ny+nu inputs such that the first nu inputs represent the channels u(t) while the remaining by channels represent the noise channels v(t). tfsys.InputGroup contains 2 input groups- 'measured' and 'noise'. tfsys.InputGroup.Measured is set to 1:nu while tfsys.InputGroup.Noise is set to nu+1:nu+ny. tfsys represents the equation y(t) = [G HL] [u; v].
Tip An identified nonlinear model cannot be converted into a transfer function. Use linear approximation functions such as linearize and linapp. |
You can use the syntax:
gensys = tf(num,den)
to create a Generalized state-space (genss) model when one or more of the entries num and den depends on a tunable realp or genmat model. For more information about Generalized state-space models, see Models with Tunable Coefficients.
Transfer Function Model with One-Input Two-Outputs
Create the one-input, two-output transfer function
with input current and outputs torque and ang velocity.
To do this, enter
num = {[1 1] ; 1}; den = {[1 2 2] ; [1 0]}; H = tf(num,den,'inputn','current',... 'outputn',{'torque' 'ang. velocity'},... 'variable','p')
These commands produce the result:
Transfer function from input "current" to output... p + 1 torque: ------------- p^2 + 2 p + 2 1 ang. velocity: - p
Setting the 'variable' property to 'p' causes the result to be displayed as a transfer function of the variable p.
Transfer Function Model Using Rational Expression
To use a rational expression to create a SISO TF model, type
s = tf('s'); H = s/(s^2 + 2*s +10);
This produces the same transfer function as
h = tf([1 0],[1 2 10]);
Multiple-Input Multiple-Output Transfer Function Model
Specify the discrete MIMO transfer function
with common denominator d (z) = z + 0.3 and sample time of 0.2 seconds.
nums = {1 [1 0];[-1 2] 3}; Ts = 0.2; H = tf(nums,[1 0.3],Ts) % Note: row vector for common den. d(z)
Convert State-Space Model to Transfer Function
Compute the transfer function of the state-space model with the following data.
To do this, type
sys = ss([-2 -1;1 -2],[1 1;2 -1],[1 0],[0 1]); tf(sys)
These commands produce the result:
Transfer function from input 1 to output: s - 4.441e-016 -------------- s^2 + 4 s + 5 Transfer function from input 2 to output: s^2 + 5 s + 8 ------------- s^2 + 4 s + 5
Array of Transfer Function Models
You can use a for loop to specify a 10-by-1 array of SISO TF models.
H = tf(zeros(1,1,10)); s = tf('s') for k=1:10, H(:,:,k) = k/(s^2+s+k); end
The first statement pre-allocates the TF array and fills it with zero transfer functions.
Tunable Low-Pass Filter
This example shows how to create the low-pass filter F = a/(s + a) with one tunable parameter a.
You cannot use ltiblock.tf to represent F, because the numerator and denominator coefficients of an ltiblock.tf block are independent. Instead, construct F using the tunable real parameter object realp.
Create a tunable real parameter.
a = realp('a',10);
The realp object a is a tunable parameter with initial value 10.
Use tf to create the tunable filter F:
F = tf(a,[1 a]);
F is a genss object which has the tunable parameter a in its Blocks property. You can connect F with other tunable or numeric models to create more complex models of control systems. For an example, see Control System with Tunable Components.
Extract the measured and noise components of an identified polynomial model into two separate transfer functions. The former (measured component) can serve as a plant model while the latter can serve as a disturbance model for control system design.
load icEngine; z = iddata(y,u,0.04); nb = 2; nf = 2; nc = 1; nd = 3; nk = 3; sys = bj(z, [nb nc nd nf nk]);
sys is a model of the form: y(t) = B/F u(t) + C/D e(t), where B/F represents the measured component and C/D the noise component.
sysMeas = tf(sys, 'measured') sysNoise = tf(sys, 'noise')
Alternatively, use can simply use tf(sys) to extract the measured component.
The control and digital signal processing (DSP) communities tend to use different conventions to specify discrete transfer functions. Most control engineers use the z variable and order the numerator and denominator terms in descending powers of z, for example,
The polynomials z^{2} and z^{2} + 2z + 3 are then specified by the row vectors [1 0 0] and [1 2 3], respectively. By contrast, DSP engineers prefer to write this transfer function as
and specify its numerator as 1 (instead of [1 0 0]) and its denominator as [1 2 3].
tf switches convention based on your choice of variable (value of the 'Variable' property).
Variable | Convention |
---|---|
'z' (default), 'q' | Use the row vector [ak ... a1 a0] to specify the polynomial (coefficients ordered in descending powers of z or q). |
'z^-1' | Use the row vector [b0 b1 ... bk] to specify the polynomial (coefficients in ascending powers of z^{-1}). |
For example,
g = tf([1 1],[1 2 3],0.1);
specifies the discrete transfer function
because z is the default variable. In contrast,
h = tf([1 1],[1 2 3],0.1,'variable','z^-1');
uses the DSP convention and creates
See also filt for direct specification of discrete transfer functions using the DSP convention.
Note that tf stores data so that the numerator and denominator lengths are made equal. Specifically, tf stores the values
num = [0 1 1]; den = [1 2 3];
for g (the numerator is padded with zeros on the left) and the values
num = [1 1 0]; den = [1 2 3];
for h (the numerator is padded with zeros on the right).
filt | frd | genmat | genss | get | ltiblock.tf | realp | set | ss | tfdata | zpk