Arithmetic Operations on Matrix Signals

This example shows how to perform arithmetic operations on signals carrying matrix and vector data.

In this example, the model performs a series of matrix operations on the input matrices and vectors to transform a system with state-space representation to its controllable canonical form, also known as phase variable form. The phase variable form provides a simplified representation of the system that eases the controllability analysis and helps in model reduction.

The canonicalform model contains the following components that show different mathematical operations.

• Input data

• Multiplication

• Inverse

• Canonical Form

Open the model.

open_system('canonicalform.slx')

Input Data

The Input data component represents the input matrix and vectors in state-space form.

$$\underset{}{\overset{\dot{}}{x}}=Ax+Bu$$

$$y=Cx+Du$$

In this example, $$A$$ is 3-by-3 matrix that represents the system, $$B$$ is 3-by-1 input vector, $$C$$ is 3-by-1 output vector, and the model assumes $$D$$ to be 0.

Multiplication

The controllability matrix for the state-space system is defined as:

$$Q=[BAB{A}^{2}B.....A^{n-1}B]$$, where n represents the number of states.

The Multiplication component computes the controllability matrix Q from the state-space system.

Controllable Canonical Form

The model modifies the state-space equation using the substitution $$x=P^{-1}z$$ and $$D=0$$ as:

$$\underset{}{\overset{\dot{}}{z}}=PA{P}^{-1}z+PBu$$

$$y=C{P}^{-1}z$$

In these equations, $$P=[P_1{P}_{1}A{P}_1{A}^2]$$, where $$P_1$$ is the last row of $$Q_{inv}$$. The Inverse component computes the inverse of the controllability matrix Q and transformation matrix P.

The equation is further simplified as:

$$\underset{}{\overset{\dot{}}{z}}=A_0+B_{0}u$$

$$y=C_{0}z$$

where $$A_0=PA{P}^{-1}$$, $$B_0=PB$$, and $$C_0=C{P}^{-1}$$ .

The Canonical Form component computes the canonical form of the state-space system.

Simulate the model and visualize $$A_0$$, $$B_0$$, $$C_0$$ using Display blocks.

sim('canonicalform.slx');