Here, we define the unified framework we are working with, and the problem we wish to solve. This will be the basis upon the theory in the next sections are developed upon.
State space models are numerically efficient to solve, can handle complex systems and allow for a more geometric understanding of dynamic systems, and form the basis of much of modern control theory
High Level
<aside> 📖 A state space model represents a system by a series of first order differential state equations and algebraic output equations
General Definition
<aside> 📖 State space models have the following form for state equations:
$$ x_1' =f_1(x_1, \dots, x_n, u_1, \dots, u_m) \\ \vdots \\ x_n' = f_n(x_1, \dots, x_n, u_1, \dots, u_m) $$
Note these $f_n$ may be nonlinear
Output equations generally have the following form:
$$ y_1 = h_1(x_1, \dots, x_n, u_1, \dots, u_m) \\ \vdots \\ y_p = h_p(x_1, \dots, x_n, u_1, \dots, u_m) $$
</aside>
Vectorized Definition
<aside> 📖 For linear systems, i.e. $f_n, h_p$ are linear, then we can write these systems of equations as matrices:
$$ \mathbf x' = \mathbf {Ax} + \mathbf {Bu} \\ \mathbf y' = \mathbf {Cx} + \mathbf {Du} $$
</aside>