State Space Models

Equations of motion for any physical system may be conveniently formulated in terms of the state of the system [333]:

$$\underline{{\dot x}}(t) = f_t[\underline{x}(t),\underline{u}(t)] \protect$$ (2.6)

Here, $\underline{x}(t)$ denotes the state of the system at time $t$ , $\underline{u}(t)$ is a vector of external inputs (typically forces), and the general vector function $f_t$ specifies how the current state $\underline{x}(t)$ and inputs $\underline{u}(t)$ cause a change in the state at time $t$ by affecting its time derivative $\underline{{\dot x}}(t)$ . Note that the function $f_t$ may itself be time varying in general. The model of Eq.(1.6) is extremely general for causal physical systems. Even the functionality of the human brain is well cast in such a form.

Equation (1.6) is diagrammed in Fig.1.4.

Figure 1.4: Continuous-time state-space model $\underline {{\dot x}}(t) = f_t[\underline {x}(t),\underline {u}(t)]$ .

The key property of the state vector $\underline{x}(t)$ in this formulation is that it completely determines the system at time $t$ , so that future states depend only on the current state and on any inputs at time $t$ and beyond.^2.8 In particular, all past states and the entire input history are ``summarized'' by the current state $\underline{x}(t)$ . Thus, $\underline{x}(t)$ must include all ``memory'' of the system.