Sparse Second Order

R2026b

Represent sparse second-order models in Simulink

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Sparse Second Order 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:
Control System Toolbox / Linear Time Invariant

Description

The Sparse Second Order block lets you represent second-order sparse state-space models, in Simulink^®. Such sparse models arise from finite element analysis (FEA) and are useful in fields like structural analysis, fluid flow, heat transfer and electromagnetics. The resultant matrices from this type of modeling are quite large with a sparse pattern. In continuous time, a second-order sparse mass-spring-damper state-space model is represented in the following form:

$$\begin{array}{r}\text{M }\ddot{q}(t)+\text{C }\dot{q}(t)+\text{K }q(t)\text{ = B }u(t)\\ y(t)\text{ = F }q(t)+\text{G }\dot{q}(t)+\text{D }u(t)\end{array}$$

Here, the full state vector is given by $$\left[q,\dot{q}\right]$$, where $$q$$ and $$\dot{q}$$ are the displacement and velocity vectors. u and y represent the inputs and outputs, respectively. M, C, and K represent the mass, damping and stiffness matrices, respectively. B is the input matrix, while F and G are the output matrices resulting from the two components of the state vector. D is the input-to-output matrix.

Examples

Linearize Simulink Model to a Sparse Second-Order Model Object

Linearize to a sparse second-order model object from your Simulink model using Simulink Control Design™ software.

• Open Live Script

Ports

Input

Input 1 — Input signal

scalar | vector

Real-valued input vector of type double whose size is equal the number of columns in the B and D matrices.

Data Types: double

Output

Output 1 — Output vector

scalar | vector

Real-valued output vector of type double whose size is equal to the number of rows in the F, G and D matrices.

Data Types: double

Parameters

M (mass) — Mass matrix

1 (default) | scalar | Nq-by-Nq sparse matrix

Mass matrix, specified as an Nq-by-Nq sparse matrix where, Nq is the number of nodes.

Programmatic Use

Block Parameter: M

Type: scalar, square sparse matrix

Default: 1

C (damping) — Damping matrix

0 (default) | scalar | Nq-by-Nq sparse matrix

Damping matrix, specified as an Nq-by-Nq sparse matrix where, Nq is the number of nodes.

Programmatic Use

Block Parameter: C

Type: scalar, square sparse matrix

Default: 0

K (stiffness) — Stiffness matrix

1 (default) | scalar | Nq-by-Nq sparse matrix

Stiffness matrix, specified as an Nq-by-Nq sparse matrix where, Nq is the number of nodes.

Programmatic Use

Block Parameter: K

Type: scalar, square sparse matrix

Default: 1

B — Input-to-state matrix

1 (default) | scalar | Nq-by-Nu sparse matrix

Input-to-state matrix, specified as an Nq-by-Nu sparse matrix where, Nq is the number of nodes and Nu is the number of inputs.

Programmatic Use

Block Parameter: B

Type: scalar, sparse matrix

Default: 1

F — Displacement-to-output matrix

1 (default) | scalar | Ny-by-Nq sparse matrix

Displacement-to-output matrix, specified as an Ny-by-Nq sparse matrix where, Nq is the number of nodes and Ny is the number of outputs.

Programmatic Use

Block Parameter: F

Type: scalar, sparse matrix

Default: 1

G — Velocity-to-output matrix

0 (default) | scalar | Ny-by-Nq sparse matrix

Velocity-to-output matrix, specified as an Ny-by-Nq sparse matrix where, Nq is the number of nodes and Ny is the number of outputs.

Programmatic Use

Block Parameter: G

Type: scalar, sparse matrix

Default: 0

D — Input-to-output matrix

0 (default) | scalar | Ny-by-Nu sparse matrix

Input-to-output matrix, specified as an Ny-by-Nu sparse matrix where, Ny is the number of outputs and Nu is the number of inputs.

Programmatic Use

Block Parameter: D

Type: scalar, sparse matrix

Default: 0

Initial position q(0) — Initial values for displacement vector

0 (default) | scalar | vector of doubles

Initial values for displacement vector $$q$$, specified as a vector of doubles. $$q$$ and $$\dot{q}$$ are the displacement and velocity vectors that make up the state vector.

Programmatic Use

Block Parameter: q0

Type: scalar, vector of doubles

Default: 0

Initial velocity q'(0) — Initial values for velocity vector

0 (default) | scalar | vector of doubles

Initial values for velocity vector $$\dot{q}$$, specified as a vector of doubles. $$q$$ and $$\dot{q}$$ are the displacement and velocity vectors that make up the state vector.

Programmatic Use

Block Parameter: dq0

Type: scalar, vector of doubles

Default: 0

Note

For linearization with Simulink Control Design, the linearized model is a mechss model object when the Sparse Second Order block is present in your Simulink model.

For more information, see Sparse Model Basics.

For an example, see Linearize Simulink Model to a Sparse Second-Order Model Object.

Programmatic Use: Add Block

To add the block programmatically, use the add_block function with one of these library block paths as the source.

"cstblocks/Linear Time Invariant/Sparse Second Order"

Example: add_block("cstblocks/Linear Time Invariant/Sparse Second Order","myModel/myBlock",MakeNameUnique="on")

Extended Capabilities

C/C++ Code Generation

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

Not recommended for production code.

Version History

Introduced in R2020b