1.6: State Variable Models

State Variable Models 

State variable models are time-domain models that express system behavior as time derivatives of state variables, i.e., the variables to express the process state. The state variables are typically selected as the natural variables associated with the energy storage elements in the process, but alternate variables can also be used. The state equations of the system describe the time derivatives of the state variables. When the state equations are linear, they are expressed in a vector-matrix form.

In the case of electrical networks, capacitor voltages and inductor currents serve as natural state variables. In the case of mechanical systems, positions and velocities of inertial masses serve as natural state variables. In thermal systems, heat flow is a natural state variable. In hydraulic systems, the head (height of the liquid in the reservoir) is a natural state variable.

The number of state variables determines the system order, however, the choice of state variables for a system model is not unique. For example, in a mechanical system model, position and momentum can serve as state variables in place of position and velocity.

Example \(\PageIndex{1}\)

A series RLC circuit driven by a constant voltage source contains two energy storage elements, an inductor and a capacitor. Accordingly, let the inductor current, \(i(t)\), and the capacitor voltage, \(v_ c (t)\), serve as state variables. Then, the circuit behavior is represented by the following equations:

\[C\frac{ dv_ c }{ dt} =i,\, \, \, L\frac{ di}{ dt} =V_ s -v_ c -Ri \nonumber \]

In vector-matrix form, the state equations are represented as:

\[\frac d{ dt} \left[\begin{array}{c} {v_ c } \\ {i} \end{array}\right]=\left[\begin{array}{cc} {0} & {1/C} \\ {-1/L} & {-R/L} \end{array}\right]\left[\begin{array}{c} {v_ c } \\ {i} \end{array}\right]+\left[\begin{array}{c} {0} \\ {1/L} \end{array}\right]V_ s \nonumber \]

Let \(v_ c\) denote the circuit output; then, the output equation is formed as:

\[v_ c =\left[\begin{array}{cc} {1} & {0} \end{array}\right]\left[\begin{array}{c} {v_ c } \\ {i} \end{array}\right] \nonumber \]

Example \(\PageIndex{2}\)

The dynamic equation of the mass–spring–damper system is given as:

\[m\frac{ d^{2} x(t)}{ dt^{2} } +b\frac{ dx(t)}{ dt} +kx(t)=f(t) \nonumber \]

Let the position, \(x(t)\), and velocity, \(v(t)=\dot{x}(t)\) serve as the state variables, and let \(x(t)\) represent the output; then, the state and output equations for the model are given as:

\[\frac {d}{ dt} \left[\begin{array}{c} {x} \\ {v} \end{array}\right]=\left[\begin{array}{cc} {0} & {1} \\ {-k/m} & {-b/m} \end{array}\right]\left[\begin{array}{c} {x} \\ {v} \end{array}\right]+\left[\begin{array}{c} {0} \\ {1/m} \end{array}\right] f \nonumber \]

\[x=\left[\begin{array}{cc} {1} & {0} \end{array}\right] \left[\begin{array}{c} {x} \\ {v} \end{array}\right] \nonumber \]

Example \(\PageIndex{3}\)

The dynamic equations for the DC motor are given as:

\[L\frac{ di_{a} (t)}{ dt} +Ri_a (t)+k_b \omega (t)=V_a (t) \nonumber \]

\[J\frac{ d\omega (t)}{ dt} +b\omega (t)-k_t i_a (t)=0 \nonumber \]

Let \(i_a (t),\; \omega (t)\) serve as the state variables, and let \(\omega (t)\) represent the output; then, the state variable model of the DC motor is given as:

\[\frac {d}{ dt} \left[\begin{array}{c} {i_a} \\ {\omega} \end{array}\right] = \left[\begin{array}{cc} {-R/L} & {-k_b /L} \\ {k_{t} /J} & {-b/J} \end{array}\right] \left[\begin{array}{c} {i_a} \\ {\omega} \end{array}\right]+\left[\begin{array}{c} {1/L} \\ {0} \end{array}\right] V_{a} \nonumber \]

\[\omega =\left[\begin{array}{cc} {0} & {1} \end{array}\right]\left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right] \nonumber \]

For a small DC motor, let the following parameter values be assumed:\(R=1\Omega ,\; L=1\; mH,\; \; J=0.01\; kg \cdot m^2 ,\; b=0.1\; \frac{{ N}\cdot s}{rad} ,\; k_t = k_b =0.05\). Then, the state variable model of the motor is given as:

\[\frac {d}{ dt} \left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right]=\left[\begin{array}{cc} {-100} & {-5} \\ {5} & {-10} \end{array}\right]\left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right]+\left[\begin{array}{c} {100} \\ {0} \end{array}\right] V_{ a} \nonumber \]

\[\omega =\left[\begin{array}{cc} {0} & {1} \end{array}\right]\left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right] \nonumber \]

State variable models are covered in more detail in Chapters 8-10.