Time-Domain Capability Curves

• Time-domain capability curves are quantitative envelopes that constrain system responses over time by specifying limits for overshoot, settling time, and steady-state error.
• They utilize mathematical formulations including polynomial relaxations, sums-of-squares decompositions, and semidefinite programming to certify performance.
• Applications span engineered control systems, power grid regulation, and time-domain astronomical surveys, as exemplified by ULTRASAT and grid code implementations.

Time-domain capability curves are quantitative specifications or envelopes that constrain or characterize the response of a dynamic system—such as an engineered control system or an astronomical time-domain survey—directly in the time domain. They formalize requirements such as maximum allowable overshoot, rise time, settling time, steady-state error, or instrument sensitivity versus observational cadence. These curves are essential in multiple fields, including control theory and time-domain astrophysics, for certifying system performance and enabling objective comparison of system capability across timescales.

1. Definition and Significance of Time-Domain Capability Curves

Time-domain capability curves impose upper and lower envelopes on a system’s output signals, constraining them throughout or on specific subintervals of the time axis. In control system engineering, such requirements may comprise limiting the peak overshoot $M_{\rm allow}$ above a steady-state value $y_{ss}$, bounding the settling time $T_s$ (the earliest $t$ at which $|y(t)-y_{ss}|\le\varepsilon$ holds), rise time constraints, and steady-state error tolerances $|y_{ss}-1| \le \text{tol}$ (Aangenent et al., 2011). In power-grid applications, grid codes mandate converter or generator step-responses to reside above or within piecewise-linear time-domain envelopes after perturbations (e.g., frequency or voltage steps) (Häberle et al., 2023).

In astronomical instrumentation, a capability curve might, for example, show the 5σ limiting magnitude for a survey as a function of sampling cadence (exposure time), thus quantifying the faintest transient that could be detected for phenomena with a given timescale (Shvartzvald et al., 2023). These curves contextualize survey depth, accessible volume, or detection rate as a function of timescale, providing a rigorous basis for inter-comparison between facilities.

2. Mathematical Formulation in Linear Control

Closed-loop linear control systems subject to time-domain constraints can be formulated as feasibility problems over a set of univariate or multivariate polynomial inequalities derived from the system response. For a strictly proper SISO plant $P(s)=b(s)/a(s)$ and a controller $C(s)=d(s)/c(s)$, after nominal closed-loop pole placement, the Youla–Kučera parameter $q(s)$ parametrizes all stabilizing controllers with the desired poles. The step response can be expressed, via partial fractions, in terms of real and (when present) complex-pair pole coefficients, which are affine functions of $q(s)$ (Aangenent et al., 2011).

The system output $y_{ss}$0 is then cast in the surrogate coordinate $y_{ss}$1, so $y_{ss}$2 becomes a univariate or multivariate polynomial. Time-domain bounds, such as $y_{ss}$3 or piecewise time-varying envelopes, are rendered as constraints on these polynomials over $y_{ss}$4 or $y_{ss}$5. Two main relaxations are provided to manage complex poles:

• Exponential-envelope relaxation: Bounds the trigonometric components by their maximal modulus envelopes.
• Multivariate-polynomial relaxation: Uses a change of variables to lift the polynomial into three variables constrained on a semialgebraic set.

These polynomial constraints encode the entire set of permissible system responses with explicit time-domain requirements.

3. SOS and LMI-Based Enforcement

Time-domain constraints in polynomial form can be enforced by representing nonnegativity through sums-of-squares (SOS) decompositions. For a univariate polynomial constraint $y_{ss}$6 on $y_{ss}$7, one seeks $y_{ss}$8 with $y_{ss}$9, $T_s$0 sums of squares (Aangenent et al., 2011). In the multivariate setting, the Positivstellensatz is invoked on each patch of the semialgebraic set: $T_s$1, for SOS multipliers $T_s$2.

The practical solution deploys semidefinite programming (SDP) to check these positivity conditions, as each SOS requirement is encoded as a linear matrix inequality (LMI) over a Gram matrix. Increasing the maximum SOS degree yields a converging hierarchy of SDP relaxations.

The full system, consisting of affine equations relating Youla coefficients to $T_s$3 and the LMI/SOS nonnegativity constraints, is solved as a feasibility SDP. Optimization objectives—such as minimizing steady-state error, maximizing convergence speed, or bounding overshoot—can be incorporated via convex-quadratic or linear cost functions in the SDP (Aangenent et al., 2011).

4. Practical Implementation: Example and Controller Synthesis

A prototypical example involves $T_s$4 and targeting complex-conjugate closed-loop poles. The step response is partial-fraction expanded, the coefficients are written as affine functions of the Youla parameters, and exponential-bound relaxation is used. Envelopes $T_s$5 and $T_s$6 enforce a 1% band around unity. Univariate SOS constraints ensure the step response remains within the band at all $T_s$7.

Solving the resulting SDP for a quadratic cost (combining steady-state and mode suppression terms) yields an explicit $T_s$8, and thus a controller $T_s$9 with significantly improved time-domain performance (zero steady-state error and reduced settling time), as verified by numerical solution in YALMIP + SeDuMi (Aangenent et al., 2011). For other use cases, the multivariate-relaxation hierarchy provides certified global optima under more complex pole configurations.

5. Time-Domain Capability Curves in Power Grids: Piecewise-Linear Envelopes

In power system ancillary service provision, grid codes often specify minimum step-response requirements via time-domain capability curves that are piecewise-linear in form. Given breakpoints $t$0, the required response is imposed as $t$1 for all $t$2 (Häberle et al., 2023). Device- and grid-level requirements—such as maximum ramp rate and activation time—define both the slopes and positions of these segments.

To enable practical realization in converter-based systems, these piecewise-linear curves are mapped to rational transfer functions $t$3 by Laplace transformation of each segment, followed by Padé approximation of time delays. The aggregate target is $t$4, where each term is rational in $t$5. This transfer function is then implemented in the control loop via PI-based matching, tuning controller constants to match the closed-loop response to $t$6.

Performance assessments compare this direct capability-curve-matching approach to classical droop and virtual-inertia schemes, demonstrating that only the former can precisely reproduce the kinks and ramps demanded by grid codes without violating device ramp or saturation limits. Typical performance metrics tracked include rise time, settling time, overshoot, and steady-state error (Häberle et al., 2023).

Method	Meets Grid-Code?	Remarks
TF-matching (min)	Yes	Precisely tracks capability curve
VI+droop (fast)	No	Overshoot, exceeds ramp limit
VI+droop (slow)	No	Activation too slow

6. Astronomical Survey Capability Curves: Example from ULTRASAT

In time-domain astronomy, capability curves quantify the relationship between observational cadence (or integration time) and the limiting sensitivity for detecting transients. For ULTRASAT, the 5σ magnitude limit after exposure time $t$7 is modeled as

$t$8

in the NUV, valid in the background-dominated regime. For instance, 60 s yields $t$9, 600 s yields $|y(t)-y_{ss}|\le\varepsilon$0, and 86400 s (1 day) yields $|y(t)-y_{ss}|\le\varepsilon$1 (Shvartzvald et al., 2023). The accessible survey volume scales as $|y(t)-y_{ss}|\le\varepsilon$2, and the full suite of curves—such as accessible volume $|y(t)-y_{ss}|\le\varepsilon$3 versus cadence—can be constructed from these analytic forms.

Survey grasp, defined as $|y(t)-y_{ss}|\le\varepsilon$4 (where $|y(t)-y_{ss}|\le\varepsilon$5 is instantaneous solid angle and $|y(t)-y_{ss}|\le\varepsilon$6 the limiting flux), permits direct comparison with other facilities. For ULTRASAT, $|y(t)-y_{ss}|\le\varepsilon$7 is normalized to 1, compared to GALEX at $|y(t)-y_{ss}|\le\varepsilon$8 and LSST at $|y(t)-y_{ss}|\le\varepsilon$9, over their respective wavelength regimes.

Time Scale	$|y_{ss}-1| \le \text{tol}$0	$|y_{ss}-1| \le \text{tol}$1 (AB, 5σ)
1 min	60 s	~21.0
10 min	600 s	~22.3
1 day	86,400 s	~25.0
1 month	$|y_{ss}-1| \le \text{tol}$2 s	~26.8

These capability curves enable rigorous quantification of survey performance as a function of timescale and facilitate science trade studies for transient detection sensitivity (Shvartzvald et al., 2023).

7. Design Considerations and Operational Guidelines

For control systems, the granularity in specifying time-domain capability curves (e.g., choice of polynomial degree in $|y_{ss}-1| \le \text{tol}$3, selection of pole locations) allows for design flexibility versus computational tractability. The Padé order in transfer-function realization (typically $|y_{ss}-1| \le \text{tol}$4) represents a balance between fidelity and implementability in converter control (Häberle et al., 2023). In time-domain surveys, factors such as background noise, detector noise, field-of-view, and crowding/confusion at faint magnitudes constrain achievable performance across timescales (Shvartzvald et al., 2023).

Extensions to the capability curve formalism include incorporating deadbands, time-varying activation, or envelope clipping, and adapting requirements in real-time based on grid or survey conditions. Assumptions such as small-signal operation, time-scale separation, and separate P–Q control branches are typically necessary for current implementations.

Time-domain capability curves thus constitute a unifying specification modality for quantifying, certifying, and comparing dynamic system performance—both in engineered regulation systems and in time-domain observational astronomy—across a range of timescales and operational regimes.