State variable system representation

State variables, typically denoted \(x_i\), are members of a minimal set of variables that completely expresses the state (or status) of a system. All variables in the system can be expressed algebraically in terms of state variables and input variables, typically denoted \(u_i\).

A state-determined system model is a system for which

1. a mathematical description in terms of \(n\) state variables \(x_i\),

2. initial conditions \(x_i(t_0)\), and

3. inputs \(u_i(t)\) for \(t\ge t_0\)

are sufficient conditions to determine \(x_i(t)\) for all \(t\ge t_0\). We call \(n\) the system order.

The state, input, and output variables are all functions of time. Typically, we construct vector-valued functions of time for each. The so-called state vector \(\bm{x}\) is actually a vector-valued function of time \(\bm{x}: \mathbb{R}\rightarrow\mathbb{R}^n\). The \(i\)th value of \(\bm{x}\) is a state variable denoted \(x_i\).

Similarly, the so-called input vector \(\bm{u}\) is actually a vector-valued function of time \(\bm{u}: \mathbb{R}\rightarrow\mathbb{R}^r\), where \(r\) is the number of inputs. The \(i\)th value of \(\bm{u}\) is an input variable denoted \(u_i\).

Finally, the so-called output vector \(\bm{y}\) is actually a vector-valued function of time \(\bm{y}: \mathbb{R}\rightarrow\mathbb{R}^m\), where \(m\) is the number of outputs. The \(i\)th value of \(\bm{y}\) is an output variable denoted \(y_i\).

Most systems encountered in engineering practice can be modeled as state-determined. For these systems, the number of state variables \(n\) is equal to the number of independent energy storage elements.

Since to know the state vector \(\bm{x}\) is to know everything about the state, the energy stored in each element can be determined from \(\bm{x}\). Therefore, the time-derivative \(d\bm{x}/dt\) describes the power flow.

The choice of state variables represented by \(\bm{x}\) is not unique. In fact, any basis transformation yields another valid state vector. This is because, despite a vector’s components changing when its basis is changed, a “symmetric” change also occurs to its basis vectors. This means a vector is a coordinate-independent object, and the same goes for vector-valued functions. This is not to say that there aren’t invalid choices for a state vector. There are. But if a valid state vector is given in one basis, any basis transformation yields a valid state vector.

One aspect of the state vector is invariant, however: it must always be a vector-valued function in \(\mathbb{R}^n\). Our method of analysis will yield a special basis for our state vectors. Some methods yield rather unnatural state variables (e.g. the third time-derivative of the voltage across a capacitor), but ours will yield natural state variables (e.g. the voltage across a capacitor or the force through a spring).