State-Space Models

State-space representations of LTI models

The representation of a model in state-space is not unique. Coordinate transformation yields state-space models with different matrices but identical dynamics. State coordinate transformation can be useful for achieving minimal realizations of state-space models, or for converting canonical forms for analysis and control design. With the available functionality, you can:

Compute minimal, balanced, modal, and companion forms.
Perform state coordinate and equivalence transformations, and convert descriptor models to explicit form.
Reorder, sort, or eliminate states to simplify models or focus on specific dynamics.
Evaluate system characteristics using controllability and observability matrices and Gramians.
Append states, offsets, or delays to outputs for analyzing internal signals.
Build complex systems by connecting components in series, parallel, feedback, or generalized interconnections.
Scale poorly conditioned models for numerical stability.

Functions

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Realizations

`balreal`	Balanced state-space realization
`modalreal`	Compute modal state-space realization (Since R2023b)
`compreal`	Compute companion state-space realization (Since R2023b)
`minreal`	Minimal realization or pole-zero cancellation
`sminreal`	Eliminates structurally disconnected states, delays, and blocks

Transformations

`ss2ss`	State coordinate transformation for state-space model
`ssequiv`	Equivalence transformation for state-space models (Since R2023b)
`dss2ss`	Convert descriptor state-space model to explicit form (Since R2024a)

State Manipulation

`xperm`	Reorder states in state-space models
`xsort`	Sort states based on state partition
`xelim`	Eliminate states from state-space models (Since R2023b)

Controllability and Observability Analysis

`ctrb`	Controllability of state-space model
`obsv`	Observability of state-space model
`gram`	Controllability and observability Gramians

Model Augmentation

`augstate`	Append state vector to output vector
`augoffset`	Map offset contribution to extra input channel (Since R2024a)
`augdelay`	Append internal delay signal to outputs of state-space model (Since R2026a)

Model Interconnection

`addInterface`	Add interface for physical assembly (Since R2026a)
`assemble`	Assemble components by connecting their physical interfaces (Since R2026a)
`append`	Group models by appending their inputs and outputs
`feedback`	Feedback connection of multiple models
`parallel`	Parallel connection of two models
`series`	Series connection of two models
`connect`	Block diagram interconnections of dynamic systems
`lft`	Generalized feedback interconnection of two models (Redheffer star product)

Scaling and Preprocessing

`prescale`	Optimal scaling of state-space models
`fixInput`	Fix value of some inputs and delete them (Since R2024a)

Topics

State-Space Realizations
A state-space model can be expressed in an infinite number of realizations. Common forms, sometimes called canonical forms, include modal, companion, observable, and controllable forms.
Scaling State-Space Models
When working with state-space models, proper scaling is important for accurate computations.
Scaling State-Space Models to Maximize Accuracy
This example shows that proper scaling of state-space models can be critical for accuracy and provides an overview of automatic and manual rescaling tools.
Use Linearization Offsets to Help Compare Nonlinear and Linearized Responses
Use offsets from linearization to facilitate the comparison of the nonlinear and linearized responses of a Simulink model. (Since R2024a)
Assemble Parts of System Using Coupling Interfaces
Model mass-spring-damper system using assembly of individual components.