State, Sensing, and Actuation

A bridge page showing how state variables, sensors, and control inputs become the basic objects of real dynamical systems.
Modified

April 26, 2026

Keywords

state, sensing, actuation, state-space, control

1 Application Snapshot

A large fraction of systems work can be summarized in one sentence:

something evolves over time, you only measure part of it, and you try to influence it through inputs

That sentence already contains the three basic objects:

  • state
  • sensing
  • actuation

This page is the shortest bridge from the site’s math modules into the language used in control, robotics, navigation, and sequential decision-making.

2 Problem Setting

A system is usually described by:

  • a state \(x\), which stores the information needed to predict future evolution
  • an input \(u\), which represents what we can command or apply
  • an output \(y\), which is what sensors actually report

In continuous time, a common model is

\[ \dot{x}(t) = f(x(t), u(t)), \qquad y(t) = h(x(t)). \]

In discrete time, the same idea becomes

\[ x_{t+1} = f(x_t, u_t), \qquad y_t = h(x_t). \]

If the world is noisy, we often add process and measurement noise:

\[ x_{t+1} = f(x_t, u_t, w_t), \qquad y_t = h(x_t) + v_t. \]

The key point is that the output need not reveal the full state.

3 Why This Math Appears

This language reuses several math layers already on the site:

  • Linear Algebra: states, inputs, and outputs are often vectors; system models are often matrices or linear maps
  • ODEs and Dynamical Systems: the state evolves according to differential or difference equations
  • Control and Dynamics: feedback laws act on the state or its estimate
  • Signal Processing and Estimation: sensors are noisy, delayed, partial, or filtered
  • Stochastic Control and Dynamic Programming: decisions are made repeatedly under uncertainty

So state-space language is not a side topic. It is the common translation layer between math and real systems.

4 Math Objects In Use

  • state vector \(x\)
  • input or control \(u\)
  • output or measurement \(y\)
  • dynamics law \(f\)
  • observation law \(h\)
  • sometimes disturbance or process noise \(w\)
  • sometimes measurement noise \(v\)

In linear time-invariant form, these objects often become

\[ \dot{x} = Ax + Bu, \qquad y = Cx + Du \]

or the discrete-time analog

\[ x_{t+1} = Ax_t + Bu_t, \qquad y_t = Cx_t + Du_t. \]

5 A Small Worked Walkthrough

Consider a simple vertical-motion model for a drone:

\[ \dot{h} = v, \qquad \dot{v} = -g + \alpha u + d(t), \qquad y = h + \eta. \]

Here:

  • \(h\) is height
  • \(v\) is vertical velocity
  • \(u\) is commanded thrust
  • \(d(t)\) is an unmodeled disturbance such as wind
  • \(y\) is a noisy height measurement

The natural state is

\[ x = \begin{bmatrix} h \\ v \end{bmatrix}. \]

This example makes the roles clear:

  • state: height and velocity together determine future motion
  • actuation: thrust changes the acceleration
  • sensing: the sensor only reports a noisy version of height

So even if height is observed, velocity may still be hidden and must be estimated or inferred.

That is exactly why state estimation and feedback appear so quickly after the first state-space model.

6 Implementation or Computation Note

The main practical questions start immediately after the model is written:

  1. Feedback How should the input depend on the current state or estimated state?

  2. Estimation If only noisy outputs are measured, how do we reconstruct the hidden state?

  3. Sampling If the controller runs on a computer, how does the continuous model become a discrete update?

Use these as the strongest follow-on pages:

7 Failure Modes

  • treating the sensor output as if it were the full state
  • forgetting that hidden variables can still drive future evolution
  • confusing disturbances with control inputs
  • writing down a feedback law before deciding what is actually measured
  • ignoring sampling and actuator limits when jumping from theory to implementation

8 Paper Bridge

9 Sources and Further Reading

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