Dynamic systems and first-order models · Fundamentals of Control Engineering

Prediction

Two identical tanks receive the same inflow. One has twice the cross-sectional area. Which level changes faster?

Do not reach for a formula yet. Decide what the tank stores and what resists the change.

Dynamics begin with accumulation

A static model says what output corresponds to an input after everything settles. A dynamic model also describes the path between the two. For a tank with cross-sectional area $A$ , inflow $q_\mathrm{in}$ , and linear outflow $q_\mathrm{out}=h/R$ , conservation gives

A\frac{dh}{dt}=q_\mathrm{in}-\frac{h}{R}.

The left side is storage; the right side is the imbalance that changes the stored quantity. Rearranging produces a standard first-order model:

RA\frac{dh}{dt}+h=R q_\mathrm{in}.

So $T=RA$ is the time constant and $K=R$ is the static gain from inflow to level.

Concept check

If A doubles while R stays fixed, what changes?

Reactive experiment: separate speed from steady-state gain

Worked example: a thermal room model

Let $C$ be the room’s thermal capacitance, $R$ the thermal resistance to outside, $T_r$ room temperature, $T_o$ outdoor temperature, and $Q$ heating power. The energy balance is

C\dot T_r = Q - \frac{T_r-T_o}{R}.

The structure is the same as the tank: stored energy changes because inflows and outflows do not balance. Changing symbols does not change the dynamics.

Derivation

Deviation variables remove the operating point

Choose an equilibrium where 0 = Q₀ - (Tᵣ₀-Tₒ₀)/R. Define deviations y=Tᵣ-Tᵣ₀, u=Q-Q₀, and hold outdoor temperature fixed. Subtracting the equilibrium equation gives RC ẏ + y = R u. The model now describes changes around the chosen operating point.

Unofficial TUM ACE exam prep

Recognize the PT1 normal form

Practise moving from the physical balance to T ẏ + y = K u and state the units of both parameters. Do not memorize a parameter mapping without checking dimensions.

Guided exercise

A sensor behaves as $0.4\dot y+y=2u$ . State the gain and time constant, then predict the final output and relative speed after a unit step.

Hint

Compare coefficients directly with T ẏ + y = K u.

Summary

A dynamic model describes accumulation, not only equilibrium.
Many physical systems reduce to storage plus resistance.
In a PT1 model, $K$ controls the final scale and $T$ controls the time scale.

Retrieve it later

Starting from a tank balance, derive T and K and explain their units.

Answer from memory tomorrow, then return to check your reasoning.