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Numeric LTI models are the basic representation of linear systems or components of linear systems whose coefficients are fixed numeric values.
You can use Numeric LTI models to represent block diagram components such as plant or sensor dynamics. By connecting Numeric LTI models together, you can derive Numeric LTI models of block diagrams. Use Numeric LTI models for most modeling, analysis, and control design tasks, including:
Analyzing linear system dynamics using analysis commands such as bode, step, or impulse.
Designing controllers for linear systems using SISO Design Tool or the PID Tuner GUI.
Designing controllers using control design commands such as pidtune, rlocus, or lqr/lqg.
Control System Toolbox™ includes the following types of numeric LTI models:
Control System Toolbox software supports transfer functions that are continuoustime or discretetime, and SISO or MIMO. You can also have time delays in your transfer function representation.
A SISO transfer function is expressed as the ratio:
$$G\left(s\right)=\frac{N\left(s\right)}{D\left(s\right)},$$
of polynomials N(s) and D(s), called the numerator and denominator polynomials, respectively.
You can represent linear systems as transfer functions in polynomial or factorized (zeropolegain) form. For example, the polynomialform transfer function:
$$G\left(s\right)=\frac{{s}^{2}3s4}{{s}^{2}+5s+6}$$
can be rewritten in factorized form as:
$$G\left(s\right)=\frac{\left(s+1\right)\left(s4\right)}{\left(s+2\right)\left(s+3\right)}.$$
The tf model object represents transfer functions in polynomial form. The zpk model object represents transfer functions in factorized form.
MIMO transfer functions are arrays of SISO transfer functions. For example:
$$G\left(s\right)=\left[\begin{array}{c}\frac{s3}{s+4}\\ \frac{s+1}{s+2}\end{array}\right]$$
is a oneinput, two output transfer function.
Use the commands described in the following table to create transfer functions.
Command  Description 

tf  Create tf objects representing continuoustime or discretetime transfer functions in polynomial form. 
zpk  Create zpk objects representing continuoustime or discretetime transfer functions in zeropolegain (factorized) form. 
filt  Create tf objects representing discretetime transfer functions using digital signal processing (DSP) convention. 
Statespace models rely on linear differential equations or difference equations to describe system dynamics. Control System Toolbox software supports SISO or MIMO statespace models in continuous or discrete time. Statespace models can include time delays. You can represent statespace models in either explicit or descriptor (implicit) form.
Statespace models can result from:
Linearizing a set of ordinary differential equations that represent a physical model of the system.
Statespace model identification using System Identification Toolbox™ software.
Statespace realization of transfer functions. (See Conversion Between Model Types for more information.)
Use ss model objects to represent statespace models.
Explicit continuoustime statespace models have the following form:
$$\begin{array}{c}\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du\end{array}$$
where x is the state vector. u is the input vector, and y is the output vector. A, B, C, and D are the statespace matrices that express the system dynamics.
A discretetime explicit statespace model takes the following form:
$$\begin{array}{c}x\left[n+1\right]=Ax\left[n\right]+Bu\left[n\right]\\ y\left[n\right]=Cx\left[n\right]+Du\left[n\right]\end{array}$$
where the vectors x[n], u[n], and y[n] are the state, input, and output vectors for the nth sample.
A descriptor statespace model is a generalized form of statespace model. In continuous time, a descriptor statespace model takes the following form:
$$\begin{array}{c}E\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du\end{array}$$
where x is the state vector. u is the input vector, and y is the output vector. A, B, C, D, and E are the statespace matrices.
Use the commands described in the following table to create statespace models.
Command  Description 

ss  Create explicit statespace model. 
dss  Create descriptor (implicit) statespace model. 
delayss  Create statespace models with specified time delays. 
In the Control System Toolbox software, you can use frd models to store, manipulate, and analyze frequency response data. An frd model stores a vector of frequency points with the corresponding complex frequency response data you obtain either through simulations or experimentally.
For example, suppose you measure frequency response data for the SISO system you want to model. You can measure such data by driving the system with a sine wave at a set of frequencies ω_{1}, ω_{2}, ,...,ω_{n}, as shown:
At steady state, the measured response y_{i}(t) to the driving signal at each frequency ω_{i} takes the following form:
$${y}_{i}\left(t\right)=a\mathrm{sin}\left({\omega}_{i}t+b\right),\text{\hspace{1em}}i=1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}n.$$
The measurement yields the complex frequency response G at each input frequency:
$$G\left(j{\omega}_{i}\right)=a{e}^{jb},\text{\hspace{1em}}i=1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}n.$$
You can do most frequencydomain analysis tasks on frd models, but you cannot perform timedomain simulations with them. For information on frequency response analysis of linear systems, see Chapter 8 of [1].
Use the following commands to create FRD models.
Command  Description 

frd  Create frd objects from frequency response data. 
frestimate  Create frd objects by estimating the frequency response of a Simulink^{®} model. This approach requires Simulink Control Design™ software. See Frequency Response Estimation in the Simulink Control Design documentation for more information. 
You can represent continuoustime ProportionalIntegralDerivative (PID) controllers in either parallel or standard form. The two forms differ in the parameters used to express the proportional, integral, and derivative actions and the filter on the derivative term, as shown in the following table.
Form  Formula 

Parallel  $$C={K}_{p}+\frac{{K}_{i}}{s}+\frac{{K}_{d}s}{{T}_{f}s+1},$$ where:

Standard  $$C={K}_{p}\left(1+\frac{1}{{T}_{i}s}+\frac{{T}_{d}s}{\frac{{T}_{d}}{N}s+1}\right),$$ where:

Use a controller form that is convenient for your application. For example, if you want to express the integrator and derivative actions in terms of time constants, use Standard form.
Discretetime PID controllers are expressed by the following formulas.
Form  Formula 

Parallel  $$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)},$$ where:

Standard  $$C={K}_{p}\left(1+\frac{1}{{T}_{i}}IF\left(z\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\right),$$ where:

IF(z) and DF(z) are the discrete integrator formulas for the integrator and derivative filter, respectively. Use the IFormula and DFormula properties of the pid or pidstd model objects to set the IF(z) and DF(z) formulas. The next table shows available formulas for IF(z) and DF(z). T_{s} is the sample time.
IFormula or DFormula  IF(z) or DF(z) 

ForwardEuler (default)  $$\frac{{T}_{s}}{z1}$$ 
BackwardEuler  $$\frac{{T}_{s}z}{z1}$$ 
Trapezoidal  $$\frac{{T}_{s}}{2}\frac{z+1}{z1}$$ 
If you do not specify a value for IFormula, DFormula, or both, ForwardEuler is used by default.