# Model Types and Representations (System Identification Toolkit)

You can represent a dynamic system using several types of dynamic system models.

## Model Types

You base the type of a dynamic system model on the properties of the dynamic system that the model represents.

### Linear versus Nonlinear Models

Dynamic system models are either linear or nonlinear. A linear model obeys the principles of superposition and homogeneity. The following equations are true for linear models.

*y*
_{1} = *f *(*u*
_{1})

*y*
_{2} = *f *(*u*
_{2})

*f *(*u*
_{1} + *u*
_{2}) = *f *(*u*
_{1}) + *f *(*u*
_{2}) = *y*
_{1} + *y*
_{2}

*f *(*a*
_{1}
*u*
_{1}) = *a*
_{1}
*f *(*u*
_{1}) = *a*
_{1}
*y*
_{1}

where *u*
_{1} and *u*
_{2} are the system inputs and *y*
_{1} and *y*
_{2} are the system outputs.

Conversely, nonlinear models do not obey the principles of superposition or homogeneity. Nonlinear effects in real-world systems include saturation, dead-zone, friction, backlash, and quantization effects; relays; switches; and rate limiters. Many real-world systems are nonlinear, but you can simulate most real-world systems with linear models to simplify a design or analysis procedure.

### Time-Variant versus Time-Invariant Models

Dynamic system models are either time-variant or time-invariant. The parameters of a time-variant model change with time. For example, you can use a time-variant model to describe an automobile. As fuel burns, the mass of the vehicle changes with time.

Conversely, the parameters of a time-invariant model do not change with time. For an example of a time-invariant model, consider a simple robot. Generally, the dynamic characteristics of robots do not change over short periods of time.

### Continuous versus Discrete Models

Dynamic system models are either continuous or discrete. Both continuous and discrete system models can be linear or nonlinear and time-invariant or time-variant. Continuous models describe how the behavior of a system varies continuously with time, which means you can obtain the properties of a system at any certain moment from the continuous model. Discrete models describe the behavior of a system at separate time instants, which means you cannot obtain the behavior of the system between every two sampling points.

Continuous system models are analog. You derive continuous models of a physical system from differential equations of the system. The coefficients of continuous models have clear physical meanings. For example, you can derive the continuous transfer function of an RC circuit if you know the details of the circuit. The coefficients of the continuous transfer function are the functions of *R* and *C* in the circuit. You use continuous models if you need to match the coefficients of a model to some physical components in the system.

Discrete system models are digital. You derive discrete models of a physical system from difference equations or by converting continuous models to discrete models. In computer-based applications, signals and operations are digital. Thus, you can use discrete models to implement a digital controller or to simulate the behavior of a physical system at discrete instants. You also can use discrete models in the accurate model-based design of a discrete controller for a plant.

## Model Representations

You can use the System Identification Toolkit to represent each model type in any of the following representations:

- Transfer function models
- Zero-pole-gain models
- State-space models

If the model is discrete, you also can use a general-linear polynomial model.