Short, precise definitions.

The Laplace-domain ratio $G(s)=Y(s)/U(s)$ of a system's output to its input, assuming zero initial conditions. The poles (roots of the denominator) determine stability; the zeros shape the transient response. Nearly every classical-control tool — Bode plots, root locus, Nyquist criteria — operates on $G(s)$.

Measured at the frequency where $|G(j\omega )|=1$ (the gain crossover), the phase margin is ${\phi }_m=180^{\circ }+\angle G(j{\omega }_c)$. A rule of thumb: aim for 45°–60°. Below 30° the system rings; negative means it's already unstable.

Measured at the frequency where the open-loop phase is $-180^{\circ }$, it's the reciprocal of $|G(j{\omega }_{180})|$ in dB. Paired with phase margin, it gives a quick read on how close a controller is to the edge.

The standard classical-control chart. Reads phase margin, gain margin, bandwidth, and roll-off rate at a glance. Pairs naturally with PID tuning and loop shaping.

For a closed-loop system to be stable, the Nyquist plot of $G(s)$ must encircle the $-1$ point a number of times equal to the number of right-half-plane open-loop poles. Stronger than Bode margins because it works for unstable open-loop systems.

The workhorse controller: $u(t)=K_{p}e(t)+K_i{\int }_0^{t}e(\tau )d\tau +K_d\dot{e}(t)$. Proportional for response, integral to kill steady-state error, derivative for damping. Still runs more of the world's control loops than anything else.

A system described by its internal state $x$ rather than input/output relations. Makes multi-input multi-output systems tractable and is the foundation for LQR, Kalman filtering, and MPC.

Minimizes $J={\int }_0^{\infty }(x^{\top }Qx+u^{\top }Ru)dt$ for $\dot{x}=Ax+Bu$. The solution is a linear state feedback $u=-Kx$, with $K$ computed from the algebraic Riccati equation. Guaranteed infinite gain margin and $\ge 60^{\circ }$ phase margin — a free lunch as far as classical-control robustness goes.

The core idea behind model predictive control. At each timestep you optimize a finite-horizon trajectory, execute only the first control, and resolve the full problem at the next step with updated state. Trades a harder per-step computation for the ability to handle constraints and disturbances online.

Formulated as: ${\min }_{u_{0:N-1}}{\sum }_{k=0}^{N-1}\ell (x_k,u_k)+V_f(x_N)$ subject to $x_{k+1}=f(x_k,u_k)$ and constraints on $x,u$. Solved with quadratic programming (linear MPC) or nonlinear solvers. Dominates in HVAC, process control, autonomous vehicles, and increasingly legged robotics.

A function $V(x)>0$ (except at equilibrium) whose time derivative along trajectories is $\dot{V}(x)\le 0$. If such a function exists, the system is stable; if $\dot{V}<0$, it's asymptotically stable. Finding a valid Lyapunov function is the hard part — increasingly, SOS programming and neural networks are used to search for one.

Given noisy measurements and a linear model, maintains a belief $\mathcal{N}(\hat{x},P)$ over the state that is provably optimal in the MMSE sense. Two steps: predict ($\hat x^-, P^-$) and update (fold in the new measurement via the Kalman gain $K$). Extensions (EKF, UKF, particle filter) handle nonlinearity.

$J_{ij}=\partial f_i/\partial x_j$. In robotics, the manipulator Jacobian maps joint velocities to end-effector velocities. In control, the Jacobian linearizes a nonlinear system around an operating point.

For $\dot{x}=f(x,u)$ at equilibrium $(x^{\ast },u^{\ast })$, the linearization is $\dot{\delta x}\approx A\delta x+B\delta u$ with $A=\partial f/\partial x$ and $B=\partial f/\partial u$. Lets you apply every tool from linear control — local to the operating point only.

For a linear system $\dot{x}=Ax+Bu$, check whether the controllability matrix $[B,AB,A^{2}B,\dots ,A^{n-1}B]$ has full row rank. A practical warning flag: if controllability only holds near the edge of numerical rank, your real system probably has modes you can barely influence.

The dual of controllability: the observability matrix $[C;CA;C{A}^2;\dots ;C{A}^{n-1}]$ must have full column rank. Failure means some state directions are invisible to your sensors — a state estimator can't recover them no matter how clever the filter.