Constraints and robustness

An actuator can apply only 100% heating even when the controller requests 120%. If integration ignores the difference, error can accumulate internally and delay recovery after the target becomes feasible. This chapter connects PID with frequency-domain analysis.

Requested versus applied action

Let $u_c=u_b+K_{p}e+z$ be the unclipped request. Applied action is

$$u=\mathrm{clip}(u_c,0,1).$$

Only $u$ enters the thermal plant. With ambient 20 °C, heat loss 20 W/K and maximum heating 1000 W, the highest equilibrium is 70 °C. A target of 80 °C cannot be reached by increasing integral gain.

With $\dot{z}=K_{i}e$, positive error keeps increasing $z$ during saturation. After the target drops to 40 °C, even negative error may take time to unwind it. Clipping the output does not stop the internal state growing.

Start at 40 °C, request 80 °C, then return to 40 °C at 200 s. Compare temperature, unclipped request, applied action and integral contribution with and without back-calculation. Anti-windup does not make the original infeasible target feasible.

Back-calculation and conditional integration

Back-calculation feeds the applied-minus-requested difference into the integral state:

$$\dot{z}=K_{i}e+\frac{u-u_c}{T_t},T_t>0.$$

At upper saturation, the correction is negative. With $e=10$ K, $K_i=0.001$, $u_c=1.4$, $u=1$, $T_t=20$ s, $\dot{z}=0.01-0.02=-0.01{\mathrm{s}}^{-1}$, rather than +0.01 without correction. Smaller $T_t$ speeds this internal tracking loop but can introduce excessively fast dynamics.

Conditional integration instead pauses integration when saturation and error push farther into the limit. Integration that moves the request back into range should remain possible; do not freeze permanently whenever saturation occurs.

Feedforward, other controllers or rate limits require an actual-action signal measured at the appropriate location. Manual-to-automatic transitions also need a consistent integral state to avoid jumps. MathWorks examples distinguish back-calculation, clamping and external tracking.

Identify the sensitivity paths

For unity negative feedback, output disturbance $d_o$, measurement $y+n$ and loop $L=PC$, rearrange $y=PC(r-y-n)+d_o$:

$$S=\frac{1}{1+L},T=\frac{L}{1+L},y=Tr+S{d}_o-Tn.$$

$T$ maps reference to output and, with a minus sign, sensor noise to output. $S$ maps this output disturbance to output. Meanwhile $u=CSr-CS{d}_o-CSn$: small output noise can coexist with large control activity.

A disturbance at the plant input instead has path $PS$. Extra thermal power uses the power-to-temperature plant, not the duty-to-temperature gain. Draw the injection point before assigning a transfer.

$S+T=1$ is a complex identity, not $|S|+|T|=1$. Both cannot be arbitrarily small at the same frequency, but full cross-frequency “waterbed” integral statements require additional assumptions.

Define the uncertainty before claiming robustness

Keep the controller fixed and vary thermal capacitance, heat-loss coefficient and delay separately. Compare tracking, disturbance recovery, peak action and saturation duration. Capacitance mainly changes dynamics; heat loss also changes static gain; delay reduces phase. A single unspecified “percentage error” hides these differences.

More formally, write $P=P_0(1+W\Delta )$, where $W$ bounds frequency-dependent multiplicative uncertainty and stable $\Delta$ satisfies $\Vert \Delta {\Vert }_{\infty }\le 1$. With nominal internal stability and a well-posed interconnection, $\Vert W{T}_0{\Vert }_{\infty }<1$ is a small-gain sufficient condition for robust stability. A few parameter samples do not prove stability over the entire set.

Check your understanding

At upper saturation, should integration pause after error becomes negative?

Why can an actuator move frequently while measured output looks smooth?

Further reading

The python-control robust SISO example uses frequency weights to express performance requirements. Physical validation must still include model validity, actuator limits and measurement quality.