PID Controller Tuning Methods

Everything in Parts 3 and 4 began from a transfer function. Give me \(G(s)\) and I will give you a root locus, a Bode plot, a gain margin, and a controller designed against any of them. In a heat exchanger, a distillation column, or a paper machine, nobody has \(G(s)\). What exists is a valve, a sensor, and a plant whose dynamics are the accumulated result of pipes, mixing, thermal mass and transport delay. Writing down the differential equations of Chapter 2 for such a system is possible in principle and hopeless in practice.

A tuning rule closes that gap. It is a formula that takes two or three numbers measured directly from the plant and returns \(K_p\), \(T_i\) and \(T_d\) — no model, no locus, no plot. The price of that convenience is that the rule must assume a plant shape, and it will behave well only on plants that actually have that shape. Understanding which shape each rule assumes is the difference between using a tuning table and being used by it.

There are two ways to get the numbers. An open-loop test breaks the loop, steps the actuator, and records the response — this gives the reaction curve of Section 20-3. A closed-loop test leaves the loop closed and pushes it deliberately to the edge of instability — this gives the ultimate cycle of Section 20-4. Everything else in this chapter is a refinement of one of those two experiments.

Open-loop step responses of industrial plants look remarkably alike: nothing happens for a while, then the output rises along an S-shaped curve and levels off. Three numbers describe that shape well enough for control purposes, and the model that carries them is the first-order-plus-dead-time (FOPDT) plant.

The compression is legitimate because of what happens to a chain of lags. A cascade of \(n\) identical first-order lags has, at high frequency, a phase that falls without bound and a magnitude that rolls off steeply — behaviour indistinguishable, over the one decade around crossover that actually decides the loop, from a single lag followed by a delay. The FOPDT model is therefore not a claim about the plant's physics; it is a claim that the plant's phase near crossover is captured by one lag and one delay. That is all a controller needs.

The three parameters are read off a process reaction curve: put the controller in manual, wait for steady state, apply a step \(\Delta u\) to the actuator, and record the output. Draw the tangent at the point of steepest slope. Where the tangent meets the initial level defines \(L\); where it meets the final level defines \(L+T\); the total output change \(\Delta c\) gives \(K=\Delta c/\Delta u\).

The ratio \(L/T\) is called the controllability ratio, and it decides how hard the loop can be pushed. A plant with \(L/T\) below about \(0.1\) is lag-dominant: the delay is negligible, the phase lag near crossover comes from the lag alone, and a high-gain controller is perfectly safe. A plant with \(L/T\) above \(1\) is delay-dominant: the delay contributes phase without contributing attenuation, which is exactly the situation Chapter 16 identified as dangerous, and such a loop must be detuned severely. The classical rules were calibrated in the middle of that range, roughly \(0.1\le L/T\le 1\), and it is there that they work.

Ziegler and Nichols asked a blunt question: given only \(L\) and the initial slope of the reaction curve, what gains give a response that a plant operator would accept? Their answer, arrived at by extensive simulation on an analogue computer and then verified on real loops, was the first tuning table in the subject's history, and it is still the first one every engineer learns.

Write the slope of the tangent as \(R=K/T\), the rate at which the output climbs per unit step of input. The delay \(L\) is the time during which the controller acts blind: whatever the controller does at time \(t\), the effect does not appear until \(t+L\), and during that window the error keeps growing at the rate \(R\) times the current control effort. The product \(RL\) therefore measures how far the plant can drift before the controller sees any consequence, and the proportional gain must scale as its reciprocal.

The constants of proportionality are what Ziegler and Nichols supplied, along with the observation that \(T_i\) and \(T_d\) should both be measured in units of \(L\), since \(L\) is the only time scale the loop's stability actually cares about.

Read the table as a sentence rather than three rows. Adding integral action forces the gain down by ten per cent, because the extra \(90^\circ\) of low-frequency phase lag identified in Section 19-4 eats into the margin. Adding derivative action then allows the gain to go back up, by a third relative to PI, because the derivative zero returns phase near crossover — precisely the trade recorded in the last table of Chapter 19, now with numbers attached. The integral time falls from \(3.33L\) to \(2L\) for the same reason: the derivative term is paying for the extra integral aggression.

The second Ziegler–Nichols method never opens the loop. Switch the controller to pure proportional — set \(T_i=\infty\) and \(T_d=0\) — and raise \(K_p\) from a small value while watching the output. The response becomes progressively more oscillatory, and at one particular gain the oscillation neither grows nor decays. That gain is the ultimate gain \(K_u\), and the period of the sustained oscillation is the ultimate period \(P_u\).

Chapters 11, 16 and 17 already told us exactly what this experiment measures. Sustained oscillation means a pair of closed-loop poles sitting on the imaginary axis: the root locus of Chapter 12 is crossing the \(j\omega\) axis, the Nyquist plot of Chapter 16 is passing through \(-1\), and the Bode plot of Chapter 17 has unity gain at the phase-crossover frequency. All three statements are the same statement.

The proportional entry is the easiest to justify. Setting \(K_p=0.5K_u\) leaves a gain margin of exactly \(2\), or \(6\) dB — the lower end of the range Chapter 17 called acceptable. Everything else in the table is then a correction to that starting point, and the PID row hides a piece of structure worth uncovering.

Ziegler and Nichols tuned for a quarter-amplitude decay ratio: each overshoot should be a quarter of the one before it. On a 1940s chart recorder that criterion is easy to judge by eye, which is why it was chosen. Chapter 8 lets us translate it into the language of damping, and the translation is unflattering.

The second overshoot is one quarter of the first because the first is one half of the step: quarter decay is fifty per cent overshoot, exactly, with no room for negotiation. A damping ratio of \(0.215\) is far below the \(0.5\) to \(0.7\) that Chapter 13 treated as a reasonable design target, and it leaves a phase margin in the region of \(20^\circ\). Ziegler–Nichols settings are aggressive by construction, not by accident.

They are also fragile. A loop with \(\zeta=0.215\) has almost no margin left for the plant gain to drift, and process gains drift constantly with throughput, feed composition and equipment fouling. The settings are best regarded as a first pass — a place from which to detune — and the refined tables below are simply the detuning written down in advance.

The Ziegler–Nichols reaction-curve rules use \(L\) and the slope \(R\) but discard \(T\) once the product \(RL\) has been formed. That is acceptable while \(L/T\) is small. When the delay becomes comparable with the time constant, two plants with the same \(RL\) but very different \(L/T\) behave quite differently, and the rules start to give loops that are far too fast.

Cohen and Coon addressed this in 1953 by fitting their rules to the FOPDT model over the whole range of \(L/T\), targeting the same quarter-decay criterion but also demanding a minimum offset for proportional-only control. The result keeps the \(T/KL\) skeleton of Ziegler–Nichols and adds a correction term in \(L/T\).

Letting \(L/T\to 0\) in the PID row recovers \(K_p=1.33\,T/KL\), \(T_i=32L/13=2.46L\) and \(T_d=4L/11=0.36L\) — recognisably the Ziegler–Nichols PID row with slightly more gain and a slightly longer integral time. The correction terms only bite when \(L/T\) is appreciable, which is precisely the situation Ziegler–Nichols handles badly.

Cohen–Coon inherits the quarter-decay target, so it inherits the aggression as well; it is a better-conditioned version of the same philosophy, not a more conservative one. For genuinely delay-dominant plants the honest answer is not a table at all, but the model-based approach of the next section, in which the desired closed-loop speed is chosen explicitly rather than inherited from a 1940s chart recorder.

Every rule so far answers the question "what gains will produce a response of a certain shape?" Direct synthesis reverses the question: state the closed-loop transfer function you want, and solve for the controller that delivers it. If the loop is \(L(s)=G_c(s)G(s)\) and the closed-loop response is \(M(s)=L/(1+L)\), then the controller is determined by inversion.

The choice of \(M(s)\) is not free. It must contain the plant's delay — no controller can make an output move before the delay has elapsed — and it must be no faster than the actuator can support. The standard choice for an FOPDT plant is therefore a first-order response with the same delay and one adjustable time constant \(\lambda\).

Two features make this family different from everything before it. First, \(\lambda\) is a genuine design knob with physical meaning — it is the closed-loop time constant — so a specification such as "settle in thirty seconds" translates directly into a number instead of into trial and error. Second, the derivation shows why the rule is safe: with \(\lambda=L\) the gain is \(K_p=T/2KL\), a little over half the Ziegler–Nichols PI value of \(0.9T/KL\), and the integral time is \(T\) rather than \(3.33L\), typically several times longer. The IMC rule is the Tyreus–Luyben correction arrived at from first principles rather than from experience.

Retaining a second-order term in the delay expansion — the Padé approximation \(e^{-Ls}\approx(1-Ls/2)/(1+Ls/2)\) — produces a PID instead of a PI, with

The ultimate-cycle experiment has an obvious defect: it asks an engineer to drive a working plant to the edge of instability, by trial and error, while hoping the oscillation does not grow. Åström and Hägglund's relay method obtains the same two numbers without ever going unstable, and it is the technique behind the "auto-tune" button on essentially every industrial controller.

Replace the controller temporarily by an on–off relay of amplitude \(\pm d\). The relay drives the plant hard in whichever direction reduces the error, the plant output crosses the setpoint, the relay switches, and the loop settles into a self-sustaining limit cycle. Its period is measured directly and its amplitude \(a\) is recorded at the output.

Why the limit cycle appears, and what its amplitude means, follows from the describing function — the subject of Chapter 29, borrowed here for one result. The relay's output is a square wave; expand it in a Fourier series and keep only the fundamental. A square wave of amplitude \(d\) has a fundamental of amplitude \(4d/\pi\), so a relay driven by a sinusoid of amplitude \(a\) behaves like an equivalent gain

A sustained oscillation requires the loop gain to be exactly \(-1\) at the oscillation frequency, so \(N(a)G(j\omega)=-1\). Since \(N(a)\) is real and positive, the frequency must be the one at which \(\angle G=-180^\circ\) — the ultimate frequency — and the magnitude condition then fixes the amplitude.

A tuning exercise is not finished when the gains are computed. Three checks belong to every loop before it is left running.

The first is a margin check. Compute the gain and phase margins of the tuned loop by the methods of Chapter 17; anything below \(6\) dB or \(30^\circ\) should be regarded as provisional. The second is a disturbance test. Tuning rules are quoted for setpoint response, but most loops spend their lives rejecting disturbances, and the two responses are different — a controller with a long \(T_i\) may track a setpoint beautifully and reject a load change slowly. The third is anti-windup. Section 19-8 described what happens when the integrator keeps accumulating against a saturated actuator; an aggressively tuned loop reaches saturation more often, so the more aggressive the rule, the more essential the limiter.

Identify which experiment supplies the data before reaching for a table, and state the units of every gain you compute. Difficulty rises down the list.