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Engineering & Technology · Control, Automation & Mechatronics

Dynamic models, inputs, outputs and disturbances

The same motor command can produce different speeds when a load changes. A model becomes useful when it explains what can change, what information it needs and which observations would challenge its prediction. A formula connecting two steady readings alone cannot describe the entire movement between them.

Build on Dynamics & Vibrations and Instrumentation & Embedded Integration. All model coefficients, response curves, trajectories, test observations and thresholds here are original teaching inputs. They explain reasoning rather than supply equipment settings, safety ratings or operating procedures.

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Identify a dynamic model’s inputs, outputs, state, disturbance and intended operating range.

01

Assign the variables a role

The plant is the process being influenced. The manipulated input is the quantity the controller can change; the output is the quantity selected for observation or regulation. A disturbance affects the plant without being chosen to achieve the control objective. These roles depend on the chosen boundary. Supply voltage may be a commanded input in one study and an unwanted variation in another.

A state records enough information about the model’s present condition to predict its future when subsequent inputs are supplied. Position alone does not specify a moving mass: velocity matters too. A temperature model may need several internal temperatures. A state is not necessarily measured directly, and choosing fewer states is an approximation whose consequences should be checked.

02

Read a rate equation before solving it

Consider dv/dt=(u−v)/τ+d. Here v is speed in m/s, u a speed-equivalent manipulated input in m/s, τ a positive time constant in seconds and d an acceleration disturbance in m/s². The input is deliberately expressed in model units; a real voltage-to-force actuator needs another relationship. Every term has units of acceleration.

For constant u and d, equilibrium requires zero derivative, giving v_eq=u+τd. A displaced speed approaches that equilibrium exponentially with time constant τ. This model neglects saturation, delay and speed-dependent resistance. It is linear with constant coefficients. A local linear approximation to a nonlinear process may hold near one operating condition while failing during startup or a large disturbance.

A model needs the disturbance and initial stateTwo original first-order predictions have identical command and initial speed but different constant disturbances.A disturbance changes the prediction for the same commandSpeed / (m/s)Elapsed time / s02468012345Solid: d=0; dashed: d=−0.5 m/s²; both v(0)=0Constructed model: dv/dt = (u−v)/2 + d, with u=4 m/s
Time 0–8 s; speed 0–5 m/s. For dv/dt=(u−v)/2+d, u=4 m/s and v(0)=0, the d=0 prediction tends to 4 m/s; d=−.5 m/s² tends to 3 m/s. At 2 s the predictions are 2.5285 and 1.8964 m/s. Both have 2 s time constant. Commands are speed-equivalent model inputs, not specified motor voltages.
03

Separate identification from checking a prediction

Parameters can come from physical balances, measured response or both. Record the input history and initial condition as well as the output. Otherwise a slow start can be mistaken for a smaller steady gain. Use additional observations to check the fitted model; reusing every fitting point as its only validation hides possible overfitting. Examine residual patterns rather than demanding a visually smooth curve.

One command, two load conditions

Supply τ=2 s, u=4 m/s and v(0)=0. Without disturbance, v(t)=4(1−exp(−t/2)) m/s. With a constant opposing d=−0.5 m/s² applied from the start, equilibrium is 4+2(−.5)=3 m/s and v(t)=3(1−exp(−t/2)) m/s. At 2 s these predictions are approximately 2.529 and 1.896 m/s.

Both curves share the 2 s time constant while tending to different final speeds. This is an original illustrative plant, not motor performance data. A measured curve that continues accelerating after the predicted settling period would challenge its assumptions or supplied inputs.

Try the reasoning

  1. If the initial speed is already 3 m/s in the disturbed case, what happens?

    Show the answer and reasoning

    It remains 3 m/s in this ideal model: the derivative is zero because (4−3)/2−.5=0.

  2. Can this model predict required electrical power?

    Show the answer and reasoning

    No. The speed-equivalent input lacks an electrical conversion and loss model; speed response alone does not identify watts.

  3. A single observed speed fits two parameter sets. Has the model been identified uniquely?

    Show the answer and reasoning

    No. Different dynamics can share one value. A varied, recorded input and time history are needed to distinguish the candidates.

What to carry forward

A model’s boundary, units, state and neglected effects determine what its predictions can support.

References

Teaching resources and research supporting this lesson’s explanations and examples.

  1. Jonathan P. How and Emilio Frazzoli · MIT16.302 sources
    1. State-space models, lecture5 (2010), selected foundational sectionsJonathan P. How and Emilio Frazzoli · MIT16.30Primary teaching or institutional technical source · PDFocw.mit.eduRead source
    2. Introduction, lecture1 (2010), objectives and model uncertaintyJonathan P. How and Emilio Frazzoli · MIT16.30Primary teaching or institutional technical source · PDFocw.mit.eduRead source