Common models of linear systems, such as transfer
functions and state-space models

Numeric linear-time-invariant (LTI) models are the basic building
blocks that you use to represent linear systems. Numeric LTI model
objects let you store dynamic systems in commonly-used representations.
For example, `tf`

models represent transfer functions
in terms of the coefficients of their numerator and denominator polynomials,
and `ss`

models represent LTI systems in terms
of their state-space matrices. There are also LTI model types specialized
for representing PID controllers in terms of their proportional, integral,
and derivative coefficients.

Build up a more complex model of a control system by representing individual components as LTI models and connecting the components to model your control architecture. For an example, see Control System Modeling with Model Objects.

LTI System | Use linear time invariant system model object in Simulink |

LPV System | Simulate Linear Parameter-Varying (LPV) systems |

**Control System Modeling with Model Objects**

Model objects can represent components such as the plant, actuators, sensors, or controllers. You connect model objects to build aggregate models that represent the combined response of multiple elements.

Model objects represent linear systems as specialized data containers that encapsulate model data and attributes in a structured way.

Ways to use model objects include linear analysis, compensator design, and control system tuning.

**Creating Continuous-Time Models**

This example shows how to create continuous-time linear models using the `tf`

, `zpk`

, `ss`

, and `frd`

commands.

Represent transfer functions in terms of numerator and denominator coefficients or zeros, poles, and gain.

Represent state-space models in terms of the state-space matrices.

**Frequency Response Data (FRD) Models**

Represent dynamic systems in terms of the magnitude and phase of their responses at various frequencies.

**Proportional-Integral-Derivative (PID) Controllers**

Represent PID controllers in terms of controller gains or time constants.

**Two-Degree-of-Freedom PID Controllers**

2-DOF PID controllers can achieve faster disturbance rejection without significant increase of overshoot in setpoint tracking.

**Using the Right Model Representation**

This example shows some best practices for working with LTI models.

This example shows how to create discrete-time linear models using the `tf`

, `zpk`

, `ss`

, and `frd`

commands.

Represent discrete-time numeric models by specifying a sample time when you create the model object.

**Discrete-Time Proportional-Integral-Derivative (PID) Controllers**

The integrator and filter terms in discrete-time PID controllers can be represented by several different formulas.

Create MIMO transfer functions by concatenating SISO transfer functions or by specifying coefficient sets for each I/O channel.

These examples show how to represent MIMO systems as state-space models.

**MIMO Frequency Response Data Models**

Use frequency-response data from multiple I/O pairs in a system to create a MIMO frequency response model.

**Select Input/Output Pairs in MIMO Models**

Extract particular I/O channels from a MIMO dynamic system model.

**Import LTI Model Objects into Simulink**

Use the LTI System block to import linear system model objects
into Simulink^{®}.

Model object types include numeric models, for representing systems with fixed coefficients, and generalized models for systems with tunable or uncertain coefficients.

Represent systems that have internal dynamics or memory of past states, such as integrators, delays, transfer functions, and state-space models.

Numeric LTI Models represent dynamic elements, such as transfer functions or state-space models, with fixed coefficients.

Represent static input/output relationships, including tunable or uncertain parameters and arrays.