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| R2012a Documentation → Control System Toolbox | |
Learn more about Control System Toolbox |
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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 continuous-time or discrete-time, and SISO or MIMO. You can also have time delays in your transfer function representation.
A SISO transfer function is expressed as the ratio:
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 (zero-pole-gain) form. For example, the polynomial-form transfer function:
can be rewritten in factorized form as:
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:
is a one-input, two output transfer function.
Use the commands described in the following table to create transfer functions.
Command | Description |
|---|---|
| tf | Create tf objects representing continuous-time or discrete-time transfer functions in polynomial form. |
| zpk | Create zpk objects representing continuous-time or discrete-time transfer functions in zero-pole-gain (factorized) form. |
| filt | Create tf objects representing discrete-time transfer functions using digital signal processing (DSP) convention. |
For examples of using these commands, see Model Creation and the reference pages for each command.
State-space models rely on linear differential equations or difference equations to describe system dynamics. Control System Toolbox software supports SISO or MIMO state-space models in continuous or discrete time. State-space models can include time delays. You can represent state-space models in either explicit or descriptor (implicit) form.
State-space models can result from:
Linearizing a set of ordinary differential equations that represent a physical model of the system.
State-space model identification using System Identification Toolbox software.
State-space realization of transfer functions. (See Conversion Between Model Types for more information.)
Use ss model objects to represent state-space models. For examples of creating state-space models, see Model Creation.
Explicit continuous-time state-space models have the following form:
where x is the state vector. u is the input vector, and y is the output vector. A, B, C, and D are the state-space matrices that express the system dynamics.
A discrete-time explicit state-space model takes the following form:
where the vectors x[n], u[n], and y[n] are the state, input, and output vectors for the nth sample.
A descriptor state-space model is a generalized form of state-space model. In continuous time, a descriptor state-space model takes the following form:
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 state-space matrices.
Use the commands described in the following table to create state-space models.
| Command | Description |
|---|---|
| ss | Create explicit state-space model. |
| dss | Create descriptor (implicit) state-space model. |
| delayss | Create state-space models with specified time delays. |
For examples of using these commands, see Model Creation and the reference pages for each command.
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 ω1, ω2, ,...,ωn, as shown:
At steady state, the measured response yi(t) to the driving signal at each frequency ωi takes the following form:
The measurement yields the complex frequency response G at each input frequency:
You can do most frequency-domain analysis tasks on frd models, but you cannot perform time-domain 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 User's Guide for more information. |
For examples creating FRD models, see Model Creation and the frd reference page.
You can represent continuous-time Proportional-Integral-Derivative (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 |
where:
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| Standard |
where:
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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.
For examples of creating continuous-time PID Controllers, see Model Creation and the pid and pidstd reference pages.
Discrete-time PID controllers are expressed by the following formulas.
| Form | Formula |
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| Parallel |
where:
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| Standard |
where:
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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). Ts is the sample time.
| IFormula or DFormula | IF(z) or DF(z) |
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| ForwardEuler (default) |
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| BackwardEuler |
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| Trapezoidal |
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If you do not specify a value for IFormula, DFormula, or both, ForwardEuler is used by default.
For examples of creating discrete-time PID Controllers, see Model Creation and the pid and pidstd reference pages.
 | Generalized Models | Models with Tunable Coefficients |  |
Learn more about resources for designing, testing, and implementing control systems.
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