State Space Modelling¶
Representation of a system that replaces an \(n\)th order differential equation with a single first order matrix differential equation.
The state space representation is given through 2 equations
| State | \(\dot q(t) = Aq(t) + Bx(t)\) |
| Output | \(y(t) = Cq(t) + Dx(t)\) |
where
| Dimension | |||
|---|---|---|---|
| \(q\) | State Vector | \(n \times 1\) | Constant |
| \(A\) | State Matrix | \(n \times n\) | Constant |
| \(B\) | Input Matrix | \(n \times r\) | Constant |
| \(x\) | Input | \(r \times 1\) | Function of time |
| \(C\) | Output matrix | \(m \times n\) | Constant |
| \(D\) | Direct transition matrix | \(m \times r\) | Constant |
| \(y\) | Output | \(m \times 1\) | Function of time |
Advantages¶
- Concise notation: Even large systems can be represented using 2 simple equations
- Easy to develop general techniques to solve systems, as all systems are represented by the same notation
- Computers easily simulate first-order equations
Example¶
\[ 2 \dfrac{d^3 y}{dt^3} + 4 \dfrac{d^2 y}{dt^2} + 6 \dfrac{dy}{dt} + 8y = 10 u(t) \\ \implies 2 y''' + 4 y'' + 6 y' + 8y = 10 u(t) \]
Since DE is of 3rd order, there are 3 state variables $$ x_1 = y, x_2 = \dot y, x_3 = \ddot y \ \implies 2 \dot x_3 + 2 x_3 + 6 x_2 + 8x_1 = 10u(t) $$
Equation Representation¶
\[ \begin{aligned} \dot x_1 &= x_2 \\ \dot x_2 &= x_3 \\ \dot x_3 &= -4x_1 - 3x_2 - 2x_3 + 5u(t) \end{aligned} \]
Matrix Representation¶
\[ \begin{aligned} \begin{bmatrix} \dot x_1 \\ \dot x_2 \\ \dot x_3 \end{bmatrix} &= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -4 & -3 & -2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 5 \end{bmatrix} u(t) \\ y &= \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \end{aligned} \]
Representation of Kalman Filter¶
\[ x_k = A x_{k-1} + B u_{k-1} + w_{k-1} \]
\[ z_k = H x_k + v_k \]
