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.

`tf` |
Create transfer function model, convert to transfer function model |

`zpk` |
Create zero-pole-gain model; convert to zero-pole-gain model |

`ss` |
Create state-space model, convert to state-space model |

`frd` |
Create frequency-response data model, convert to frequency-response data model |

`filt` |
Specify discrete transfer functions in DSP format |

`dss` |
Create descriptor state-space models |

`pid` |
Create PID controller in parallel form, convert to parallel-form PID controller |

`pidstd` |
Create a PID controller in standard form, convert to standard-form PID controller |

`pid2` |
Create 2-DOF PID controller in parallel form, convert to parallel-form 2-DOF PID controller |

`pidstd2` |
Create 2-DOF PID controller in standard form, convert to standard-form 2-DOF PID controller |

LTI System | Use linear system model object in Simulink |

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

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.

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.

**Simulink Block for LTI Systems**

Use the LTI System block to import Control System Toolbox™ model
objects into a Simulink^{®} model.

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

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.

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

**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.

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

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

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