Time-Domain Response, Transient Specifications & Performance Integrals

A transient specification is an experiment definition. It must identify the input, initial state, operating point, final value, tolerance, and observed signal. “Settles in two seconds” is ambiguous until the settling band and requirement to remain inside it are stated; “10% overshoot” is ambiguous if the final change is zero or the response starts from a nonzero bias.

Metrics are measured relative to stated initial and final values. The first entry into a tolerance band is not settling unless the response stays there.

Separate the response components #

For an LTI system, output is the sum of zero-input response from initial state and zero-state response from external inputs. In state space,

$$x(t)=e^{At}x(0)+\int_0^t e^{A(t-\tau)}Bu(\tau)\,d\tau.$$

This distinction matters during restart and mode transfer. A transfer-function step response normally assumes zero initial conditions; a real controller can begin with stored integrator, observer, filter, and actuator states.

Also distinguish the maps being tested:

• reference step $r\to y$ for command following;
• load step $d\to y$ for regulation;
• noise burst $n\to y,u$ for measurement sensitivity;
• initial-condition response for recovery from displaced state;
• actuator command $r,d,n\to u$ for authority, wear, and saturation.

Explicit metric definitions #

For initial output $y_i$ and final output $y_f$, normalize

$$y_N(t)=\frac{y(t)-y_i}{y_f-y_i}.$$

One common convention is:

• rise time: time from $y_N=0.1$ to $y_N=0.9$;
• peak time: time of maximum excursion in the specified direction;
• percentage overshoot: $100\max(0,\max_t y_N(t)-1)$;
• settling time: earliest $T$ such that $|y(t)-y_f|\le\epsilon|y_f-y_i|$ for every $t\ge T$;
• steady-state error: $\lim_{t\to\infty}(r-y)$ when the limit exists.

MathWorks currently uses 10–90% rise time and a default 2% settling threshold for standard stepinfo cases, but permits explicit thresholds and documents special behavior for feedthrough, zero final value, unstable zeros, large overshoot, MIMO channels, and unstable systems. GTK always states the convention rather than treating a tool default as physics.

Second-order estimates and their boundary #

For the canonical underdamped system

$$G(s)=\frac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2},$$

$$M_p=\exp\!\left(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\right),\qquad t_p=\frac{\pi}{\omega_n\sqrt{1-\zeta^2}}.$$

An exponential-envelope estimate for settling is

$$t_s\approx\frac{-\ln(\epsilon)}{\zeta\omega_n},$$

so $\epsilon=0.02$ gives about $3.9/(\zeta\omega_n)$. These relations assume a normalized dominant second-order response with no important zero, delay, additional mode, saturation, or unusual numerator.

Worked calculation #

Let $\zeta=0.5$ and $\omega_n=4\ \mathrm{rad/s}$. Then

$$\omega_d=4\sqrt{0.75}=3.464\ \mathrm{rad/s},$$

$$M_p=e^{-\pi(0.5)/\sqrt{0.75}}=0.163\quad(16.3\%),$$

$$t_p=\pi/3.464=0.907\ \mathrm{s},$$

and the 2% envelope estimate is $t_s\approx3.9/(2)=1.95\ \mathrm{s}$. A computed response should be checked against these predictions. A discrepancy can reveal a noncanonical numerator, extra pole, wrong units, or a different metric definition.

Zeros and additional poles #

A right-half-plane zero can produce undershoot, so rise time measured only in the final direction hides the initial wrong-way motion. A left-half-plane zero near the dominant poles can increase overshoot. A supposedly “fast” extra pole can still matter if it lies near crossover or if its residue in the measured channel is large.

Dominance is therefore not just geometric proximity to the imaginary axis. It depends on excitation, observation, residues, and the performance time scale. Compare the full model with the reduced approximation before using second-order formulas.

Integral performance criteria #

For tracking error $e(t)$ over stated horizon $[0,T]$:

$$\mathrm{IAE}=\int_0^T |e(t)|dt,\qquad \mathrm{ISE}=\int_0^T e^2(t)dt,$$

$$\mathrm{ITAE}=\int_0^T t|e(t)|dt,\qquad \mathrm{ITSE}=\int_0^T te^2(t)dt.$$

Squaring penalizes large errors strongly; absolute error is less dominated by peaks; time weighting penalizes lingering tails. None is a complete design objective. A controller can reduce error integral by using excessive or chattering control. Include effort and rate penalties such as

$$J=\int_0^T \left(q_e e^2+q_u u^2+q_{\dot u}\dot u^2\right)dt,$$

and enforce hard constraints separately. Do not hide unit changes inside weights: $e^2$ and $u^2$ have different physical dimensions.

Nonlinear and stochastic performance #

With saturation, friction, deadband, switching, or quantization, response metrics depend on amplitude and direction. Test multiple operating points and both increasing/decreasing steps. With noise, a single realization gives unstable metrics; report distributions or confidence intervals over representative trials and separate deterministic bias from variance.

Limit cycles may satisfy a mean-error requirement while failing wear, acoustic, or safety limits. Peak state constraints can fail even when the measured output stays within tolerance.

A complete time-domain test record #

Record:

1. plant/controller version and operating point;
2. input waveform, amplitude, direction, and application location;
3. all initial controller and plant states;
4. sample rate, latency, filtering, and measurement units;
5. metric definitions, tolerance bands, and time horizon;
6. output, error, control command, saturation/rate flags, and safety-relevant states;
7. expected model response and the residual from measurement.

Time response is where a controller becomes observable, but only a defined experiment makes the observation reusable.