Choosing State Variables And Observables

Why the Choice of Variables Matters

A model is only as useful as the information its variables preserve. A quantity that changes over time may be central to the system’s evolution, while another quantity is measured because it is useful to observe or report. Those roles are related, but they are not automatically the same.

For example, a reservoir’s water volume changes as water enters and leaves. Its water level can be measured and may be the quantity people care about. The level, however, depends on both the volume and the reservoir’s shape. Choosing carefully lets us write the evolution in the variable directly governed by the balance, then calculate the measured output from that variable.

This builds on stock-rate-total model architecture: a state variable often represents stored or accumulated quantity, and its rate describes how that quantity changes. The additional question here is how to distinguish such a state from an observable derived from it.

States and Observables

In a differential-equation model, the state is commonly written as \(\mathbf{x}(t)\), with its evolution described by an equation of the form

Here \(\mathbf{u}\) represents known inputs, such as a prescribed control or an externally supplied flow, and \(\mathbf{p}\) represents fixed model parameters. An observable can be written as \(y(t)=G(t,\mathbf{x}(t),\mathbf{u}(t),\mathbf{p})\). This notation expresses two different jobs: the state equation describes how the model evolves, while the output relation describes what is reported or measured.

A measured quantity is not automatically a state variable. It may be a state, but it may instead be calculated from the state. Conversely, a quantity can be a state even when it is not directly measured, if it is needed to represent the system’s evolution. The appropriate choice depends on the model: states must retain the information needed for its evolution, while observables can be chosen for practical or scientific interest.

Worked Example: Reservoir Volume as the State and Level as the Observable

Why the Chain Rule Connects State and Output

The reservoir calculation is an instance of a general rule. Suppose a scalar state \(x(t)\) determines an observable through \(y(t)=g(x(t))\), where \(g\) is differentiable at the relevant state value. The chain rule gives

This says that the output’s rate depends both on how quickly the state changes and on how sensitive the output is to changes in that state. It also explains why one cannot generally copy a rate from a state variable to an observable: the factor \(g'(x(t))\) may not be \(1\).

If the relation is instead written directly as \(y=g(x)\), the observable’s derivative can be found from that formula when it is differentiable. A useful modeling check is to make sure the state-to-output relation is appropriate over the range being considered. If the same state value could correspond to several different output values, a single-valued function \(y=g(x)\) would not describe the observable without adding further information.

Worked Example: A Derived Pressure Reading

When More Than One State Is Needed

A single observable can conceal distinctions that matter to a model. If separate parts of a system can change independently, one state variable may not preserve enough information to describe their evolution. A total can be a useful output while failing to reveal how that total is distributed.

A Practical Selection Test

When choosing variables for a model, ask what information must be retained to describe the system’s changing condition. Then separate that information from quantities calculated for observation or reporting.

This test is a guide, not a rule that every model must use one particular set of states. Different state choices may describe the same system, but each choice must support a coherent evolution model and a clear connection to the quantities of interest. In the reservoir example, volume is convenient because the net flow directly specifies its rate. Level is still essential as an observable, and it could be used as the evolving variable if its rate equation were obtained from the volume-level relation.

A Common Mistake: Treating a Level Rate as a Volume Rate

A tempting but incorrect calculation is to say that a net inflow of \(2\text{ m}^3/\text{min}\) means the reservoir level rises at \(2\text{ m/min}\). This confuses two different quantities. In the example, the water surface area at \(h=2\text{ m}\) is \(12\text{ m}^2\), so a one-metre rise would correspond locally to about \(12\text{ m}^3\) of added water, not \(1\text{ m}^3\).

The incorrect value fails a direct check: substituting it into \(\frac{dV}{dt}=(8+2h)\frac{dh}{dt}\) at \(h=2\) gives \(12(2)=24\text{ m}^3/\text{min}\), not the specified \(2\text{ m}^3/\text{min}\). The corrected value gives \(12(1/6)=2\text{ m}^3/\text{min}\). Checking the rate against the state-to-output relation catches the mistake.

Why the Distinction Is Useful

Separating states from observables makes a model easier to construct, interpret, and check. The state variables express the evolving condition that the model must track. Observables connect that condition to measurements, decisions, or reported quantities. One state can support several observables, as reservoir volume supports both level and pressure; several states can also be summarized by a single output, as two reservoir volumes are summarized by their total.

The distinction also helps identify missing information. If an output is being used as the only state, ask whether the same output value could represent different underlying conditions that behave differently. If so, the output alone is not a sufficient state description. Choosing variables carefully at the start prevents a model from losing distinctions it needs later.