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Control Theory operators

20 operators in the control_theory category of the live registry. Each is a named formula you can compose inside a state contract or call directly through POST /api/zeq/compute. KO42 is always on; add up to three more per call (total ≤ 4), per the 7-step protocol.

OperatorDescriptionEquation
CT1Transfer function representing a linear time-invariant system as the ratio of output to input Laplace transforms.G(s) = \frac{Y(s)}{U(s)}
CT10Peak overshoot percentage for an underdamped second-order system expressed as a function of the damping ratio.M_p = e^{-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}}
CT11Rise time approximation for a second-order system inversely proportional to the natural frequency, measuring response speed.t_r \approx \frac{1.8}{\omega_n}
CT12Settling time approximation for a second-order system inversely proportional to the product of damping ratio and natural frequency.t_s \approx \frac{4}{\zeta\omega_n}
CT13State-space representation expressing a dynamic system as first-order matrix differential equations with input, output, and state vectors.\dot{x} = Ax + Bu, \quad y = Cx + Du
CT14Controllability test checking whether the rank of the controllability matrix equals the system order, ensuring all states are reachable.\text{rank}[B, AB, ..., A^{n-1}B] = n
CT15Observability test checking whether all internal states can be inferred from output measurements via the observability matrix rank.\text{rank}[C; CA; ...; CA^{n-1}] = n
CT16Linear-quadratic regulator cost functional balancing state deviation and control effort through weighted integral minimization.J = \int_0^\infty (x^T Q x + u^T R u) dt
CT17Optimal LQR gain matrix computed from the solution of the algebraic Riccati equation, minimizing the quadratic cost functional.K = R^{-1}B^T P
CT18Algebraic Riccati equation whose solution yields the optimal gain for linear-quadratic control and Kalman filtering problems.A^T P + PA - PBR^{-1}B^T P + Q = 0
CT19Nyquist stability criterion determining closed-loop stability by counting encirclements of the critical point in the open-loop frequency response.N_p \circlearrowleft(-1,0) = Z - P
CT2Closed-loop transfer function for negative feedback systems, showing how feedback reshapes the open-loop plant response.H_{cl}(s) = \frac{G(s)}{1 + G(s)H(s)}
CT20Laplace transform converting time-domain differential equations into algebraic equations in the complex frequency domain for system analysis.L\{f(t)\} = \int_0^\infty f(t)e^{-st}dt
CT3Steady-state error computed via the final value theorem, quantifying the persistent tracking error for a given input type.e_{ss} = \lim_{s\to 0} \frac{sR(s)}{1+G(s)H(s)}
CT4Position error constant determining steady-state error for step inputs from the DC gain of the open-loop transfer function.K_p = \lim_{s\to 0} G(s)H(s)
CT5Velocity error constant determining steady-state error for ramp inputs from the open-loop transfer function behavior near the origin.K_v = \lim_{s\to 0} sG(s)H(s)
CT6Acceleration error constant determining steady-state error for parabolic inputs, requiring at least a type-2 system for zero error.K_a = \lim_{s\to 0} s^2G(s)H(s)
CT7PID controller combining proportional, integral, and derivative actions to minimize error in a feedback control loop.u(t) = K_p e(t) + K_i\int e(t)dt + K_d\frac{de(t)}{dt}
CT8Phase margin measuring the additional phase lag at gain crossover that would bring the system to the edge of instability.PM = 180° + \angle G(j\omega_{gc})
CT9Gain margin measuring how much the open-loop gain can increase at the phase crossover frequency before the system becomes unstable.GM = \frac{1}{|G(j\omega_{pc})|}

Compute with one of these​

curl -sS -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["CT1"],"inputs":{}}'

The response carries the bare physics value, its unit and uncertainty, the generated master equation, and a signed envelope you can verify on any node.

See also​