Difference between four matrices of state space?

I already know that there are two common types of representations used for mathematical modelling. One is transfer function and the other is state space.

When dealing with state space representation, we see 4 different matrices A (system matrix), B (input matrix), C (Output matrix) and D (feedforward matrix)?

What is the difference between all four of these matrices, especially 'A' and 'D' matrix? As the other two 'B' and 'C' somehow indicate their function/application that they might store input and output.

• What happens if $\dot x = x + B u$? What happens if $\dot x = Ax$? How are they different? What happens if $y = C x$? What happens if $y = D u$? How are they different? How are all of these different from $\dot x = Ax + Bu,\,y = Cx + Du$? – TimWescott Sep 17 at 15:13

All four of the matrices interact with the system at different points. I think the best way to understand them is to look at the formulae while referring to the general diagram:

When you look at the above diagram we can see that the four matrices are in various points that correlate with their general names. $$\B\$$ is at the input, $$\C\$$ is at the output, $$\D\$$ feeds the input forward to the output (feed-forward), and $$\A\$$ is in the heart of the system mixing with the integrator.

The input signal $$\u\$$ is fed through the input matric $$\B\$$. Following that conversion, it is added to the signal $$\x\$$ passing through the system matrix $$\A\$$:

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

The input $$\u\$$ is also "fed-forward", by-passing the system, through matrix $$\D\$$ and it is added together with the output of the inner system $$\x\$$ multiplied by our output matrix $$\C\$$:

$$y = Cx + Du$$

This is the general diagram. The values of the actual matrices change how the block diagram would look. For example, if $$\D = 0 \$$ there would be no feed-forward portion on the upper part of the diagram.

The state space representation looks like this mathematically

$$\ \dot{x} =A x + Bu\$$

$$\ y = C x +Du\$$

This is what it looks like in block diagram form:

The reason for the integrator is this:

$$\\dot{x} =A x \$$

$$\ x s =A x \$$

$$\ x =A \frac{x}{s} \$$

• Ax + Bu and Cx + Du – Chu Sep 17 at 15:57
• why there is integrator(1/s)?? – abt Sep 17 at 16:53
• @abt $\int \dot x\: dt=x$ – Chu Sep 17 at 19:11