Descriptor State-Space

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

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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 your 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.

• E — n-by-n, where n is the number of states in the system.

• A — n-by-n, where n is the number of states in the system.

• B — n-by-m, where:

  • n is the number of states in the system.

  • m is the number of system inputs.

• C — r-by-n, where:

  • r is the number of system outputs.

  • n is the number of states in the system.

• D — r-by-m, where:

  • r is the number of system outputs.

  • m is the number of system inputs.

Examples

Model a Series RLC Circuit

Model a series RLC circuit while avoiding algebraic loops by using the Descriptor State-Space block.

• Open Model

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

Input 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

Output 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 | 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 — System B 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 — System C 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 — System D 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.

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

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

Alternatively, use X0.

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 whether the system has direct feedthrough 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.

• 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 — Representation of tunable parameters in simulation

Auto (default) | Optimized | Unconstrained

To model time variant systems, you can tune the system matrix parameters during simulation. This parameter specifies how the software represents the system matrix parameters in the compiled model or the simulation target for accelerator and rapid accelerator simulations.

• Auto — The software determines how to represent tunable parameters by analyzing the system matrices.

• Optimized — The software optimizes the parameter representation for simulation execution.

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

• Unconstrained — The software does not optimize the parameter representation for simulation execution.

  This option places the fewest restrictions on the values you can specify when you tune the 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 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) | 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 — Use the absolute tolerance specified by the Absolute tolerance configuration parameter.

• Real, positive, scalar — Use the specified value as the 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. To open the Model Discretizer, in the Apps tab, under Control Systems, click Model Discretizer.

Version History

Introduced in R2018b

R2026a: Model time variant DAE systems

Model time variant linear 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.