System Identification
for Control.

§ 01 Foundations & Workflow

Before any algebra, a mental model of the loop you're trying to close.

System identification is the art of turning experimental data into dynamic models. You give a physical plant some carefully chosen input, record its reaction, and fit an equation that predicts what it will do next time. The fit is never perfect — but a model that is simple, validated, and honest about its uncertainty is worth more to a controller designer than any block diagram drawn from first principles alone.

The three boxes: white, gray, black

The canonical workflow

• Design the experiment. Pick input signal, amplitude, sampling rate, duration. This is where 80% of success is won or lost.
• Collect and pre-process. Detrend, remove means, decimate if oversampled, apply anti-alias filtering.
• Choose a model structure. First-order? ARX? State-space of order 4? Start simple.
• Estimate parameters. Least squares, prediction-error, subspace — pick the tool that fits the structure.
• Validate. On fresh data. Check residuals, compare simulated vs measured output.
• Iterate. Increase order, try different structures, re-excite if residuals are coloured.

§ 02 Essential Terminology

The shared vocabulary of identification. Skim now, return when you hit it in the wild.

§ 03 Non-parametric Methods

No model structure assumed. You get a curve or a frequency table — not coefficients.

Non-parametric methods estimate the plant's response directly — its impulse response, step response, or frequency response function — without committing to a transfer-function order. They are excellent for discovery, fast sanity checks, and feeding downstream parametric fits with a good initial guess.

Step response

Apply a sudden change in input, record the output. The shape tells you almost everything you need to pick a first-order or second-order starting model:

• Monotonic, no overshoot → first-order or over-damped second-order.
• Damped oscillation → under-damped second-order or higher.
• Long flat, then rise → dead-time / pure delay present.
• Inverse response (dips the wrong way first) → non-minimum-phase zero.

Impulse response

Theoretically elegant — the impulse response is the plant (for LTI). Practically limited because true impulses are hard to apply and SNR is poor. Usually obtained indirectly from correlation analysis rather than from an actual impulse.

Frequency response (Bode from data)

Inject a sinusoid of frequency $\omega_k$, wait for steady state, measure amplitude ratio and phase shift. Repeat for many frequencies. The result is $G(j\omega_k)$ sampled across the band of interest — a Bode plot built empirically. Modern variants use chirp or multisine inputs and FFT-based estimation, often called the Empirical Transfer Function Estimate (ETFE):

ETFE is unbiased but variance-noisy; smoothing with a spectral window (Hamming, Parzen) yields the spectral analysis estimate.

Correlation analysis

If you feed the plant a zero-mean input, the cross-correlation between input and output divided by the input auto-correlation gives the impulse response:

This is the engine behind PRBS-based identification: PRBS has nearly white auto-correlation, so the cross-correlation with the output essentially is the impulse response estimate.

§ 04 The First-Order Model

The workhorse. Two parameters, one exponential, enormous explanatory power.

A surprising fraction of industrial loops are well-approximated by a first-order transfer function with optional dead time:

Three scalars, three physical meanings: gain $K$ (how much the output eventually moves), time constant $\tau$ (how fast), and dead time $L$ (how long before you see anything).

Reading parameters off a step response

For a step of size $U$ applied at $t=0$, the response is $y(t) = KU(1 - e^{-t/\tau})$ for $t \geq L$. Three classic graphical rules:

• Gain. $K = \Delta y_{\infty} / \Delta u$. Read final value minus initial value, divide by input step size.
• Time constant — the 63.2% rule. At $t = \tau$ the output has reached $1 - e^{-1} \approx 0.632$ of its final value. Draw horizontal at 0.632·Δy∞, drop a vertical.
• Time constant — the two-point method. Use $t_{28.3\%}$ and $t_{63.2\%}$: then $\tau = 1.5\,(t_{63.2} - t_{28.3})$ and dead time $L = t_{63.2} - \tau$. More noise-robust than a single-point read.

When first-order is not enough

If the step response has a clear overshoot, an inflection point, or a visible delay followed by an S-shape, the plant needs more than one pole. The S-shape in particular usually signals either (a) dead time, or (b) multiple real poles whose combined behaviour mimics a sluggish second-order. The tangent method (Ziegler-Nichols) handles (b) by fitting an equivalent FOPDT to the steepest-slope tangent line.

§ 05 The Second-Order Model

Two poles, and suddenly the world oscillates.

The canonical underdamped second-order form:

Three knobs: DC gain $K$, natural frequency $\omega_n$ (how fast), and damping ratio $\zeta$ (how oscillatory). The ratio $\zeta$ tells the whole story:

• $\zeta = 0$ — undamped, pure sinusoid forever.
• $0 < \zeta < 1$ — underdamped, decaying oscillation.
• $\zeta = 1$ — critically damped, fastest approach without overshoot.
• $\zeta > 1$ — overdamped, two real poles, slow monotonic rise.

Reading parameters from the step response

For an underdamped step response, the following three relations usually give you everything:

So: measure overshoot $M_p$ and peak time $t_p$ from the plot. Invert the first relation for $\zeta$, then the second for $\omega_n$. Gain $K$ is again just $\Delta y_\infty / \Delta u$.

The logarithmic decrement method

More robust when noise contaminates a single peak reading, use two successive peaks (amplitudes $a_1$ and $a_2$ above steady state):

Higher-order echoes of second-order thinking

Real plants often have many poles. A useful rule: if two poles are much closer to the $j\omega$ axis than all the others, the slow pair dominates — you can treat the plant as second-order for controller design, and push the faster dynamics into an unmodeled residue. This is the foundation of dominant-pole design.

§ 06 Parametric Methods

Commit to a model structure. Crank the data. Get numbers out.

Parametric identification assumes a model form with a finite parameter vector $\theta$ and finds the $\hat\theta$ that minimises a criterion — usually the one-step-ahead prediction error variance. The difference between ARX, ARMAX, OE, and BJ is where you allow the noise to enter the model.

How ARX actually gets solved

Stack the regression $y(t) = \varphi(t)^{\!\top}\theta + e(t)$ where $\varphi(t) = [-y(t-1), \dots, -y(t-n_a),\, u(t-n_k), \dots, u(t-n_k-n_b+1)]^{\!\top}$. Minimising $\sum \varepsilon(t)^{2}$ yields the normal equations:

This is ordinary linear regression — the same mechanics as fitting a polynomial. The beauty of ARX is that a one-line QR solve delivers the parameters. The price is the (usually unrealistic) assumption that the driving noise $e(t)$ is white.

The prediction-error method (PEM)

A general framework: given a model $\mathcal{M}(\theta)$ that produces a one-step prediction $\hat y(t|t-1;\theta)$, choose $\theta$ to minimise

with $\ell(\cdot)$ typically the squared norm. ARX, ARMAX, OE, and BJ all sit inside this framework — they differ only in how the predictor is parameterised. In MATLAB, tfest, armax, oe, bj, and ssest all invoke variants of PEM.

Model-order selection

You want the smallest model that explains the data. Three tools:

• AIC: $\text{AIC} = N\log V_N + 2 d$ where $d$ is the number of free parameters. Penalises complexity lightly.
• BIC (MDL): $\text{BIC} = N\log V_N + d\log N$. Penalises complexity more — preferred when $N$ is large.
• Hankel singular values (subspace methods): plot them; an obvious gap indicates the right order.

§ 07 Input Design & Excitation

A good experiment matters more than a clever algorithm.

No fitting routine can recover dynamics you never excited. The question is not just "what input?" but "what spectrum?" — you need energy in every frequency band you care about modelling.

Amplitude selection — the Goldilocks problem

• Too small: output swamped by noise, SNR collapses, estimates become garbage.
• Too large: plant leaves its linear region — saturation, friction break-away, back-EMF nonlinearity in motors.
• Goldilocks: largest amplitude that still produces linear-looking output. Start at 5% of normal operating span and sweep up while watching for output asymmetry (response to +step ≠ − response to −step).

Duration and record length

Rules of thumb: collect data for at least 10–20 dominant time constants, or at least one full period of your lowest-frequency PRBS, whichever is longer. For statistical confidence, $N \gtrsim 20 \cdot d$ samples where $d$ is the number of parameters.

Sampling rate

Pick $f_s$ so that $f_s / f_{\text{bandwidth}} \in [10, 30]$. Slower and you miss dynamics; faster and you waste data and amplify quantisation noise. Always precede the ADC with an analog anti-alias filter at $f_s/2$.

§ 08 Validation & Model Choice

Never validate on the data you fit. Never.

Fitting is the easy part — anything will fit if you throw enough parameters at it. Validation is what separates a model you can stake a controller on from a pretty curve.

Three validation pillars

• Fresh-data simulation (FIT %). Simulate the model on input data it has never seen. Compare to measured output. FIT above 80% usually means the model is usable.
• Residual whiteness. Compute $R_{\varepsilon\varepsilon}(\tau)$; it should look like a delta at τ=0 and noise everywhere else. Colour in the residuals means the model is missing dynamics.
• Residual-input cross-correlation. $R_{\varepsilon u}(\tau)$ should be zero for $\tau \geq 0$. Non-zero values mean the input is still predicting something the model isn't capturing.

Simulation vs prediction — a subtle distinction

A one-step-ahead prediction uses past measured outputs as inputs. It almost always looks great, because yesterday's output is a superb predictor of today's output. Simulation uses only past inputs and the model's own past predictions — it's the honest test. Always report FIT on simulation, not prediction.

§ 09 Common Pitfalls

The traps every practitioner falls into at least once.

• Forgetting to detrend. A 5 V steady-state bias on a 10 V step test will wreck your least-squares fit. Always subtract operating-point means.
• Insufficient excitation. Identifying a 6-parameter model from a single step is mathematically possible but statistically reckless. Use PRBS.
• Closed-loop bias. With the controller running, noise affects both $y$ and $u$, so standard least-squares is biased. Use OE, BJ, or instrumental-variable methods — or break the loop if you can.
• Sampling too fast. At $f_s \gg$ bandwidth, model poles cluster near $z=1$, numerical conditioning degrades, and noise variance dominates the regression.
• Sampling too slow. You alias dynamics you cared about into frequencies you didn't want. No amount of clever fitting fixes this.
• Over-modelling. Adding states to chase residuals that are really measurement noise. Watch AIC/BIC; watch whether added parameters change sign between datasets.
• Under-modelling. A first-order fit to a system with oscillatory behaviour. Residuals will scream.
• Ignoring dead time. Even 2–3 samples of transport delay, if unmodelled, will ruin a good estimator. Always check cross-correlation for a peak offset from zero.
• Operating-point trap. A nonlinear plant fit linearly at one setpoint may behave completely differently at another. Identify across the envelope, not at a single op-point.

§ 10 Rules of Thumb

Ten commandments, refined by everyone who ever burned a day on bad data.