Modern Control

Normally speaking, we know much about classical control, in the form of:


\dot{x}(t) = ax(t) + bu(t) \longleftrightarrow sX(s) - x(0) = aX(S) + bU(s)

With the left part being a derivative equation in continuous time, while the right being its tranformation in the complex domain field.

Note
\dot{x}(t) = ax(t) + bu(t) \longleftrightarrow x(k+1) = ax(k) + bu(k)
These are equivalent, but the latter one is in discrete time.

A brief recap over Classical Control

Be Y(s) our output variable in classical control and U(s) our input variable. The associated transfer function G(s) is:


G(s) = \frac{Y(s)}{U(s)}

Root Locus

Bode Diagram

Nyquist Diagram

State Space Representation

State Matrices

A state space representation has 4 Matrices: A, B, C, D with coefficients in \R:

A: State Matrix [x_rows, x_columns]
B: Input Matrix [x_rows, u_columns]
C: Output Matrix [y_rows, x_columns]
D: Direct Coupling Matrix [y_rows, u_columns]


\begin{cases}
\dot{x}(t) = Ax(t) + Bu(t) \;\;\;\; \text{Dynamic of the system}\\
      y(t) = C{x}(t) + Du(t) \;\;\;\; \text{Static of the outputs}
\end{cases}

This can be represented with the following diagrams:

Continuous Time:

Discrete time:

State Vector

This is a state vector [x_rows, 1]:


x(t) =  \begin{bmatrix}
            x_1(t)\\
            \dots\\
            x_x(t)
        \end{bmatrix}

\text{or} \:

x(k) =  \begin{bmatrix}
            x_1(k)\\
            \dots\\
            x_x(k)
        \end{bmatrix}

Basically, from this we can know each next step of the state vector, represented as:


x(k + 1) = f\left( 
    x(k), u(k)
\right) = Ax(k) + Bu(k)

Case Studies

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Congestion Control
Video Player Control
Deep Learning

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