To model a system, we want “the simplest mathematical description that adequately predicts the response of the system to all anticipated inputs.”

We restrict our discussion to systems described by ordinary differential equations in state variable form. Wikipedia has a decent overview of state space here, and the notation I’ll be using is pretty similar.

If \( x_1(t), x_2(t), …, x_n(t) \) are the state variables (or the states) of the process at time \( t \), and \( u_1(t), u_2(t), …, u_m(t) \) are control inputs to the process at time \( t \), then we can describe the system by \( n \) first-order differential equations thus: \[ \begin{aligned} \dot{x_1}(t) &= a_1(x_1(t), x_2(t), …, x_n(t), u_1(t), u_2(t), …, u_m(t), t) \\ \dot{x_2}(t) &= a_2(x_1(t), x_2(t), …, x_n(t), u_1(t), u_2(t), …, u_m(t), t) \\ {} & ⋮ \\ \dot{x_n}(t) &= a_n(x_1(t), x_2(t), …, x_n(t), u_1(t), u_2(t), …, u_m(t), t). \end{aligned} \]

Now define the state and control vectors \[ \def\avec{\boldsymbol a} \def\uvec{\boldsymbol u} \def\xvec{\boldsymbol x} \xvec(t) := \begin{bmatrix} x_1(t) \\ x_2(t) \\ ⋮ \\ x_n(t) \end{bmatrix} \quad\text{and}\quad \uvec(t) := \begin{bmatrix} u_1(t) \\ u_2(t) \\ ⋮ \\ u_n(t) \end{bmatrix}. \] The state equation can now be rewritten as \[ \dot\xvec = \avec(\xvec(t), \uvec(t), t) \] where the definition of \( \avec \) is apparent.

Next time, a word on system & control constraints.