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State Definition

In view of the above discussion, it is perhaps plausible that the state $ \underline{x}(t) = [x_1(t), \ldots, x_N(t)]^T$ of a physical system at time $ t$ can be defined as a collection of state variables $ x_i(t)$, wherein each state variable $ x_i(t)$ is a physical amplitude (pressure, velocity, position, $ \ldots$) corresponding to a degree of freedom of the system. We define a degree of freedom as a single dimension of energy storage. The net result is that it is possible to compute the stored energy in any degree of freedom (the system's ``memory'') from its corresponding state-variable amplitude.

For example, an ideal mass $ m$ can store only kinetic energy $ E_m \eqsp \frac{1}{2}m\, v^2$, where $ v={\dot x}$ denotes the mass's velocity along the $ x$ axis. Therefore, velocity is the natural choice of state variable for an ideal point-mass. Coincidentally, we reached this conclusion independently above by writing $ f=ma$ in state-space form $ \dot{v}=(1/m)f$. Note that a point mass that can move freely in 3D space has three degrees of freedom and therefore needs three state variables $ (v_x,v_y,v_z)$ in its physical model. In typical models from musical acoustics (e.g., for the piano hammer), masses are allowed only one degree of freedom, corresponding to being constrained to move along a 1D line, like an ideal spring. We'll study the ideal mass further in §7.1.2.

Another state-variable example is provided by an ideal spring described by Hooke's law $ f=kx$B.1.3), where $ k$ denotes the spring constant, and $ x$ denotes the spring displacement from rest. Springs thus contribute a force proportional to displacement in Newtonian ODEs. Such a spring can only store the physical work (force times distance), expended to displace, it in the form of potential energy $ E_k \eqsp \frac{1}{2}k\, x^2$. More about ideal springs will be discussed in §7.1.3. Thus, spring displacement is the most natural choice of state variable for a spring.

In so-called RLC electrical circuits (consisting of resistors $ R_i$, inductors $ L_i$, and capacitors $ C_i$), the state variables are typically defined as all of the capacitor voltages (or charges) and inductor currents. We will discuss RLC electrical circuits further below.

There is no state variable for each resistor current in an RLC circuit because a resistor dissipates energy but does not store it--it has no ``memory'' like capacitors and inductors. The state (current $ I$, say) of a resistor $ R$ is determined by the voltage $ V$ across it, according to Ohm's law $ V=IR$, and that voltage is supplied by the capacitors, inductors, and voltage-sources, etc., to which it is connected. Analogous remarks apply to the dashpot, which is the mechanical analog of the resistor--we do not assign state variables to dashpots. (If we do, such as by mistake, then we will obtain state variables that are linearly dependent on other state variables, and the order of the system appears to be larger than it really is. This does not normally cause problems, and there are many numerical ways to later ``prune'' the state down to its proper order.)

Masses, springs, dashpots, inductors, capacitors, and resistors are examples of so-called lumped elements. Perhaps the simplest distributed element is the continuous ideal delay line. Because it carries a continuum of independent amplitudes, the order (number of state variables) is infinity for a continuous delay line of any length! However, in practice, we often work with sampled, bandlimited systems, and in this domain, delay lines have a finite number of state variables (one for each delay element). Networks of lumped elements yield finite-order state-space models, while even one distributed element jumps the order to infinity until it is bandlimited and sampled.

In summary, a state variable may be defined as a physical amplitude for some energy-storing degree of freedom. In models of mechanical systems, a state variable is needed for each ideal spring and point mass (times the number of dimensions in which it can move). For RLC electric circuits, a state variable is needed for each capacitor and inductor. If there are any switches, their state is also needed in the state vector (e.g., as boolean variables). In discrete-time systems such as digital filters, each unit-sample delay element contributes one (continuous) state variable to the model.

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Impulse Response of State Space Models
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Numerical Integration of General State-Space Models