State-Space

Implement linear state-space system

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Libraries:
Simulink / Continuous

Description

The State-Space block implements a system whose behavior you define as

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\\ {x|}_{t=t_0}=x_0,\end{array}$$

where x is the state vector, u is the input vector, y is the output vector, and x0 is the initial condition of the state vector. The A, B, C, and D matrices can be specified as either sparse matrices or dense matrices. The matrix coefficients must have these characteristics:

• A must be an n-by-n matrix, where n is the number of states.

• B must be an n-by-m matrix, where m is the number of inputs.

• C must be an r-by-n matrix, where r is the number of outputs.

• D must be an r-by-m matrix.

  

In general, the block has one input port and one output port. The number of rows in C or D matrix is the same as the width of the output port. The number of columns in the B or D matrix are the same as the width of the input port. If you want to model an autonomous linear system with no inputs, set the B and D matrices to empty. In this case, the block acts as a source block with no input port and one output port, and implements the following system:

$$\begin{array}{l}\dot{x}=Ax\\ y=Cx\\ {x|}_{t=t_0}=x_0.\end{array}$$

Simulink^® software converts a matrix containing zeros to a sparse matrix for efficient multiplication.

Examples

Double Spring Mass System

Model a double spring-mass-damper system with a periodically varying forcing function. The model uses an S-Function block to animate the mass system during simulation. In the system, the only sensor is attached to the mass on the left, and the actuator is attached to the mass on the left. The example uses state estimation and linear-quadratic regulator (LQR) control.

• Open Model

Ports

Input

Port_1 — Input signal

scalar | vector

Real-valued input vector of type double, where the width equals the number of columns in the B and D matrices. For more information, see Description.

Data Types: double

Output

Port_1 — Output vector

scalar | vector

Real-valued output vector of data type double, with width equal to the number of rows in the C and D matrices. For more information, see Description.

Data Types: double

Parameters

A — Matrix coefficient, A

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

Specify the matrix coefficient A, as a real-valued n-by-n matrix, where n is the number of states. For more information on the matrix coefficients, see Description.

Programmatic Use

Block Parameter: A

Type: character vector, string

Values: scalar | vector | matrix | sparse matrix

Default: '1'

B — Matrix coefficient, B

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

Specify the matrix coefficient B, as a real-valued n-by-m matrix, where n is the number of states and m is the number of inputs. For more information on the matrix coefficients, see Description.

Programmatic Use

Block Parameter: B

Type: character vector, string

Values: scalar | vector | matrix | sparse matrix

Default: '1'

C — Matrix coefficient, C

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

Specify the matrix coefficient C as a real-valued r-by-n matrix, where r is the number of outputs and n is the number of states. For more information on the matrix coefficients, see Description.

Programmatic Use

Block Parameter: C

Type: character vector, string

Values: scalar | vector | matrix | sparse matrix

Default: '1'

D — Matrix coefficient, D

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

Specify the matrix coefficient D as a real-valued r-by-m matrix, where r is the number of outputs and m is the number of inputs. For more information on the matrix coefficients, see Description.

Programmatic Use

Block Parameter: D

Type: character vector, string

Values: scalar | vector | matrix | sparse matrix

Default: '1'

Initial conditions — Initial state vector

0 (default) | scalar | vector

Specify the initial state vector.

Limitations

The initial conditions of this block cannot be inf or NaN.

Programmatic Use

Block Parameter: InitialCondition

Type: character vector, string

Values: scalar | vector

Default: '0'

Alternatively, use X0.

Parameter tunability — Choose tunable representation of block parameters

Auto (default) | Optimized | Unconstrained

Tunability level of the state-space matrices (A,B,C, and D ) for accelerated simulation modes and deployed simulations using the Simulink Compiler™. When set to Auto, Simulink chooses the appropriate parameter tunability level.

For sparse matrix coefficients, set the parameter to Optimized to allow tunability of non-zero elements while keeping the pattern and number of non-zero elements constant. Set this parameter to Unconstrained to allow all elements to be tunable, so long as the number of non-zero elements is kept constant, that is, you can change the pattern of the sparse matrix.

For dense matrix coefficients, select Optimized to allow tunability of all matrix elements, provided the number of non-zero elements initially specified in the matrix is kept constant. Set this parameter to Unconstrained to allow full tunability of all matrix elements.

Note

To tune the D matrix of the block when D = 0, you must enable the Allow non-zero values for D matrix initially specified as zero parameter.

Programmatic Use

Block Parameter: ParameterTunability

Type: character vector | string

Values: 'Auto' | 'Optimized' | 'Unconstrained'

Default: 'Auto'

Allow non-zero values for D matrix initially specified as zero — Allow tunability of D matrix when D = 0

off (default) | on

Enable this parameter to support tunability of D even when D = 0.

Note

Enabling this parameter enables direct feedthrough for the State-Space block.

Programmatic Use

Block Parameter: AllowTunableDMatrix

Type: character vector | string

Values: 'off' | 'on'

Default: 'off'

Absolute tolerance — Absolute tolerance for computing block states

auto (default) | scalar | vector

Variable-step solvers use absolute and relative tolerances when choosing the step size to determine whether the error in state calculations is acceptable.

To inherit the absolute tolerance from the Absolute tolerance configuration parameter, specify this parameter value as auto or -1.

To specify an absolute tolerance for this block that overrides the value specified for the Absolute tolerance configuration parameter:

• Enter a real, positive scalar value to use to compute all block states.

• Enter a real vector with dimensions that match the dimensions of the continuous states for the block.

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 number | vector of positive real scalar numbers
Data Types:	char | string

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

State Name (e.g., 'position') — Option to assign unique names to states

'' (default) | character vector | cell array of character vectors | MATLAB^® variable | ...

Use this parameter to optionally assign names to the states of this block. The names you assign apply only to the states of this block.

• To use default state names, leave this field blank ('').

• To assign a single name to a single state, enter the name between quotes. For example, to name a single state position, enter 'position'.

• To assign names to multiple states, specify this parameter value as a cell array of character vectors. Each name in the cell array must be unique. For example, to assign the names a, b, and c, enter {'a','b','c'}.

• To specify the names using a MATLAB variable, enter the name of the variable without quotes. For example, to use the variable names to specify the state names, enter names.

You can specify a number of names that is less than the number of states in the block. In this case, the state names are used for multiple states, and the number of states must divide evenly into the number of state names. For example, when you specify two names for a block that has four states, the first name is used for the first two states, and the second name is used for the last two states.

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) | valid MATLAB variable name
Data Types:	char | string | cell

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

Block Characteristics

Data Types	double
Direct Feedthrough	yes
Multidimensional Signals	no
Variable-Size Signals	no
Zero-Crossing Detection	no

Extended Capabilities

C/C++ Code Generation

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

Not recommended for production-quality code. Relates to resource limits and restrictions on speed and memory often found in embedded systems. The code generated can contain dynamic allocation and freeing of memory, recursion, additional memory overhead, and widely-varying execution times. While the code is functionally valid and generally acceptable in resource-rich environments, smaller embedded targets often cannot support such code.

In general, consider using the Simulink Model Discretizer to map continuous blocks into discrete equivalents that support production code generation. To start the Model Discretizer, in the Simulink Editor, on the Apps tab, under Apps, under Control Systems, click Model Discretizer. One exception is the Second-Order Integrator block because, for this block, the Model Discretizer produces an approximate discretization.

Version History

Introduced before R2006a