Descriptor State-Space

R2026b

Model continuous linear system as system of implicit state-space equations

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Descriptor State-Space block

To add a block to a model, double-click the canvas and start typing the block name. Then, select the block from the list.

Libraries:
Simulink / Continuous

Description

The Descriptor State-Space block models a continuous linear system as a system of differential equations expressed in the implicit form:

$$\begin{array}{c}E\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

where:

• x is the state vector.

• u is the input vector.

• y is the output vector.

• The matrices E, A, B, C, and D contain the coefficients of the terms in the system equations.

You can use the Descriptor State-Space block to model both time invariant and time variant linear implicit systems.

• To model time-invariant systems, specify the E, A, B, C, and D parameters as constant matrices.

• To model time-variant systems, tune the A, B, C, D, and E parameters during simulation. (since R2026a)

Specify System Matrices

To model a system, specify the system matrices using the E, A, B, C, and D parameters. You can specify either sparse or full matrices. By default, MATLAB^® creates full matrices.

• To convert a full matrix to a sparse matrix, use the sparse function.

• To convert a sparse matrix to a full matrix, use the full function.

• To determine whether a matrix is sparse, use the issparse function.

For more information, see Constructing Sparse Matrices.

The number of states, inputs, and outputs in the system determine the dimensions of the system matrices. For a system with n states, m inputs, and r outputs, the dimensions are:

• E — n-by-n

• A — n-by-n

• B — n-by-m

• C — r-by-n

• D — r-by-m

Examples

Create Low-Order LPV Model of CPU and Heat Sink Model

Obtain a low-order linear parameter-varying (LPV) surrogate for the high-fidelity finite element model of the heat sink.

(Control System Toolbox)

• Since R2024a

Reduced-Order Modeling Technique for System-Level Simulation of Aircraft Wing Spar

Run system-level simulations by using the Descriptor State-Space block to implement a reduced-order model of the I-beam in an aircraft wing spar.

(Partial Differential Equation Toolbox)

Ports

Input

Port_1 — System inputs

scalar | vector

Input values must be real. The dimensions of the input signal must match the number of system inputs.

• When the system has one input, connect a scalar signal.

• When the system has multiple inputs, connect a vector signal that contains an element for each system input.

This port has direct feedthrough when:

• You set the Direct Feedthrough parameter to True.

• You set the Direct Feedthrough parameter to Auto and the software determines that the block has direct feedthrough.

Data Types: double

Output

Port_1 — System outputs

scalar | vector

The dimensions of the output signal match the number of system outputs.

• When the system has one output, the block produces a scalar signal.

• When the system has multiple outputs, the block produces a vector signal that contains an element for each system output.

Data Types: double

Parameters

E — Mass matrix

'1' (default) | scalar | matrix

Specify a matrix that:

• Contains only real values.

• Has dimensions n-by-n, where n is the number of states in the system.

The mass matrix can be singular or nonsingular.

• When the mass matrix is singular, model the system using the Descriptor State-Space block. Because the mass matrix is not invertible, you cannot express the system equations in the explicit form.

• When the mass matrix is nonsingular, you can model the system using either the Descriptor State-Space block or the State-Space block. Because the mass matrix is invertible, you can express the system equations in the explicit form, $$\dot{x}=E^{-1}Ax+E^{-1}Bu$$.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	E
Values:	'1' (default) | scalar | matrix
Data Types:	string | char

Example: set_param("MyModel/Descriptor State-Space",E="[1 2;3 4]")

A — System A matrix

'1' (default) | scalar | vector | matrix

Specify a matrix that:

• Contains only real values.

• Has dimensions n-by-n, where n is the number of states in the system.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	A
Values:	'1' (default) | scalar | vector | matrix
Data Types:	string | char

Example: set_param("MyModel/Descriptor State-Space",A="[1 2;3 4]")

B — Input-to-state matrix

'1' (default) | scalar | vector | matrix

Specify a matrix that:

• Contains only real values.

• Has dimensions n-by-m, where n is the number of states in the system, and m is the number of system inputs.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	B
Values:	'1' (default) | scalar | vector | matrix
Data Types:	string | char

Example: set_param("MyModel/Descriptor State-Space",B="[1 2]")

C — State-to-output matrix

'1' (default) | scalar | vector | matrix

Specify a matrix that:

• Contains only real values.

• Has dimensions r-by-n, where r is the number of system outputs, and n is the number of states in the system.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	C
Values:	'1' (default) | scalar | vector | matrix
Data Types:	string | char

Example: set_param("MyModel/Descriptor State-Space",C="[3 4]")

D — Feedthrough matrix

'1' (default) | scalar | vector | matrix

Specify a matrix that:

• Contains only real values.

• Has dimensions r-by-m, where r is the number of system outputs, and m is the number of system inputs.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	D
Values:	'1' (default) | scalar | vector | matrix
Data Types:	string | char

Example: set_param("MyModel/Descriptor State-Space",D="5")

Initial conditions — Initial values of system states

0 (default) | scalar | vector | matrix

To initialize the system, specify the initial values of states in the system. States cannot have Inf or NaN values.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

This parameter has two names: InitialCondition and X0. You can use either name to set or get the parameter value.

Parameter:	InitialCondition
Values:	'0' (default) | scalar | vector | matrix
Data Types:	string | char

Example: set_param("MyModel/Descriptor State-Space",InitialCondition="[2 3]")

Example: set_param("MyModel/Descriptor State-Space",X0="[2 3])

Direct feedthrough — Option to specify whether system has direct feedthrough

True (default) | False | Auto

When the block implements a system with more than 500 continuous states, specifying this parameter can speed up block execution during simulation. A system has direct feedthrough if the value of at least one output signal directly depends on at least one input signal value. By default, the software assumes that the system has direct feedthrough.

• True — The system has direct feedthrough.

• False — The system does not have direct feedthrough.

• Auto — The software determines whether the system has direct feedthrough based on the characteristics of the specified matrices.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	DirectFeedthrough
Values:	'True' (default) | "False" | "Auto"

Example: set_param("MyModel/Descriptor State-Space",DirectFeedthrough="False")

Linearize to sparse model — Option to linearize system to sparse model during linearization

on (default) | off

This parameter specifies how the software linearizes the system when you linearize the model that contains the block.

• on — System linearized to sparse system

• off — System linearized to nonsparse explicit state-space system

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	LinearizeToSparse
Values:	'off' (default) | "on"

Example: set_param("MyModel/Descriptor State-Space",LinearizeToSparse="on")

Parameter tunability — Level of parameter tunability during simulation

Auto (default) | Optimized | Unconstrained

To model time-variant systems, you can tune the system matrix parameters during simulation. This parameter specifies the level of parameter tunability during simulation. The parameter value determines how the software represents the system matrix parameters in the compiled model for normal mode simulations and the simulation target for accelerator and rapid accelerator simulations. The choice involves a tradeoff between flexibility and performance.

• Auto — The software chooses the level of parameter tunability by analyzing the system matrices.

• Optimized — This option improves performance but places additional restrictions on the values you can specify when you tune the system matrix parameters during simulation.

• Unconstrained — This option provides the most flexibility, with the fewest restrictions on the values you can specify when you tune the system matrix parameters during simulation.

In all cases, the pattern of the mass matrix must remain fixed. When you tune the E parameter during simulation, only elements that have nonzero values in the initial matrix are tunable.

The values you can specify when you tune the A, B, C, and D parameters depend on whether you specify the initial value as a sparse or full matrix and the value of this parameter.

Initial Matrix Parameter Value	Optimized Parameter Tunability	Unconstrained Parameter Tunability
Sparse matrix	Matrix pattern is fixed. Only elements with nonzero values in the initial matrix are tunable.	The matrix pattern can change, but the number of nonzero elements in the matrix cannot increase.
Full matrix	The matrix pattern can change, but the number of nonzero elements in the matrix cannot increase.	All elements of the matrix can change.

Dependencies

To tune the system matrices using the Parameter Writer block, specify the parameters as full matrices.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	ParameterTunability
Values:	'Auto' (default) | "Optimized" | "Unconstrained"

Example: set_param("MyModel/Descriptor State-Space",ParameterTunability="Optimized")

Absolute tolerance — Absolute tolerance for block states

auto (default) | -1 | scalar | vector

Variable-step solvers use absolute and relative tolerances to determine whether the error in state calculations is acceptable when choosing the step size. This parameter specifies the absolute tolerance to use for states in the system implemented by this block.

• auto or -1 — Inherit the absolute tolerance from the Absolute tolerance configuration parameter.

• Real, positive scalar — Specify an absolute tolerance for all states.

• Real, positive vector — Specify an absolute tolerance for each state. The number of elements in the vector must be the same as the number of states in the system.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	AbsoluteTolerance
Values:	'auto' (default) | "-1" | positive real scalar | positive real vector
Data Types:	string | char

Example: set_param("MyModel/Descriptor State-Space",AbsoluteTolerance="-1")

State Name (e.g., 'position') — Option to specify state names

'' (default) | character vector | cell array of character vectors | ...

If you do not specify a custom name for every state in the system:

• Each specified name is used for more than one state.

• The number of states must be evenly divisible by the number of specified names.

For example, if the system has four states, you must specify one, two, or four names. If you specify two names, the first two states use the first name and the last two states use the second name.

To specify a single custom state name using the Property Inspector or the Block Parameters dialog box, include single quotes ('') around the name. For example, to specify the name position, enter 'position'. If you omit the single quotes, the software interprets the value as the name of a MATLAB variable.

To use default state names, specify this parameter as an empty character vector ('').

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	ContinuousStateAttributes
Values:	'' (default)
Data Types:	char | cell

Example: set_param("MyModel/Descriptor State-Space",ContinuousStateAttributes={'position','velocity'})

Block Characteristics

Data Types	double
Direct Feedthrough	noa
Multidimensional Signals	no
Variable-Size Signals	no
Zero-Crossing Detection	no
a Direct feedthrough characteristics for this block depend on block parameter values.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using Simulink® Coder™.

Not recommended for production code generation.

Before R2024b: The Descriptor State-Space block supports only C code generation and does not support C++ code generation.

Consider using the Model Discretizer to map continuous blocks, such as the Descriptor State-Space block, to discrete equivalents that support code generation. For more information, see Model Discretizer.

Version History

Introduced in R2018b

R2026a: Model time variant DAE systems

Model time-variant linear differential-algebraic equation (DAE) systems by tuning the A, B, C, D, and E parameters during simulation.

The pattern of the mass matrix must remain fixed. When you tune the E parameter during simulation, only elements that have nonzero values in the initial matrix are tunable.

The values you can specify when you tune the A, B, C, and D parameters depend on whether you specify the initial value as a sparse or full matrix and the value of the Parameter tunability parameter.

Initial Matrix Parameter Value	Optimized Parameter Tunability	Unconstrained Parameter Tunability
Sparse matrix	Matrix pattern is fixed. Only elements with nonzero values in the initial matrix are tunable.	The matrix pattern can change, but the number of nonzero elements in the matrix cannot increase.
Full matrix	The matrix pattern can change, but the number of nonzero elements in the matrix cannot increase.	All elements of the matrix can change.

R2024b: C++ code generation support

You can generate C++ code for implicit systems that have a mass matrix, such as those implemented using the Descriptor State-Space block. Use the Language (Simulink Coder) parameter to specify the language for code generation.