Before a drone banks.
Before a controller commands.
Before a trajectory bends across the sky—
there must be structure.
There must be a formulation:
a way of expressing the system that holds all its motion, forces, and intent—
not yet in motion,
but ready for it.
This is the role of General Formulation:
a foundational step in system design where physics, mathematics, and engineering align.
It’s where we ask:
– What is this system?
– What governs its behavior?
– What can we control—and what can’t we?
What Is General Formulation?
A general formulation defines a system’s dynamics in a unified mathematical structure—before simplifications, approximations, or special cases.
It doesn’t assume linearity.
It doesn’t assume time-invariance.
It assumes only that the system is governed by laws,
and that those laws must be captured cleanly.
Typically, a general formulation consists of:
- State Variables (x)
Describe the internal configuration of the system (e.g., position, velocity, attitude, actuator states). - Input Variables (u)
Describe external influences we can control (e.g., thrust, torque, flap deflections). - Output Variables (y)
Describe what we observe or measure (e.g., altitude, pitch angle, velocity). - Dynamics Equation
A (possibly nonlinear) differential equation:
ẋ = f(x, u, t)
Describes how the system evolves over time. - Output Equation
y = h(x, u, t)
Maps internal states to observable quantities.
This is the canvas from which LTI models, nonlinear control, adaptive laws, or estimation filters all emerge.
Why General Formulation Matters
Because every control strategy, every estimator, every planner
must begin with a correct understanding of how the system behaves.
A poor formulation means:
– Controllers won’t stabilize
– Estimators will drift
– Simulations will lie
– Real systems will misbehave
But a strong, general formulation becomes the truth beneath all behavior.
It allows you to:
– Simulate real dynamics
– Linearize around trim points
– Derive optimal controls
– Estimate unobservable states
– Predict system response under various inputs
Where It’s Used
– Aircraft dynamics: combining 6-DOF motion, actuator models, aerodynamics, and environmental forces
– Autonomous vehicles: modeling chassis, sensors, and delay-affected control paths
– Robotic arms: capturing joint dynamics, payloads, and friction
– Spacecraft and satellites: incorporating orbital dynamics and attitude control
– Swarm coordination: building common models to describe interaction and individual response
In each, the general formulation gives the system a body—and a voice.
From General to Specific
Once the general formulation is in place, it can be:
– Linearized for small-signal analysis (yielding LTI models)
– Parameter-varying to account for operating condition shifts (yielding LPV systems)
– Extended to include noise and uncertainty (for robust or stochastic control)
– Reduced for real-time applications or onboard computing constraints
But these are all refinements.
The formulation comes first.
Why It Still Matters
In modern autonomy, we chase speed, adaptation, intelligence.
But speed without structure collapses.
Adaptation without foundations misleads.
And intelligence without formulation is guesswork.
General formulation is the blueprint before the blueprint—
the silent act of naming what a system is,
so that everything else it becomes can stand on solid ground.
Because before motion,
there must be meaning.
And in the space between math and machine,
formulation is where understanding begins.