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Control Theory Cheat Sheet

The core ideas of Control Theory distilled into a single, scannable reference — perfect for review or quick lookup.

PiqCue — piqcue.com/control-theory/cheatsheet

Quick Reference

Feedback Loop

A mechanism in which the output of a system is measured and fed back to the input to reduce the error between the actual output and the desired reference. Negative feedback reduces deviations and promotes stability, while positive feedback amplifies deviations.

Transfer Function

A mathematical representation of the relationship between the input and output of a linear time-invariant (LTI) system in the Laplace domain, expressed as the ratio of the output's Laplace transform to the input's Laplace transform assuming zero initial conditions.

PID Controller

A Proportional-Integral-Derivative controller that calculates an error signal as the difference between a measured process variable and a desired set point, then applies a correction based on proportional, integral, and derivative terms to minimize the error over time.

Stability

A system property indicating that bounded inputs produce bounded outputs (BIBO stability) or that the system's state returns to equilibrium after a perturbation (Lyapunov stability). An unstable system's output grows without bound in response to small disturbances.

State-Space Representation

A mathematical model of a system expressed as a set of first-order differential equations using state variables, an input vector, and an output vector. It is written in matrix form as dx/dt = Ax + Bu and y = Cx + Du.

Controllability

A system property indicating whether it is possible to drive the system from any initial state to any desired final state in finite time using appropriate control inputs. A system is controllable if and only if the controllability matrix has full rank.

Observability

A system property indicating whether the complete internal state of the system can be determined from its outputs over a finite time interval. A system is observable if and only if the observability matrix has full rank.

Bode Plot

A pair of logarithmic graphs (magnitude in decibels and phase in degrees versus frequency) used to analyze the frequency response of a linear system. Bode plots reveal gain margin, phase margin, bandwidth, and resonance behavior.

Root Locus

A graphical method for examining how the roots of the characteristic equation (poles of the closed-loop system) move in the complex plane as a system parameter, typically the loop gain, is varied from zero to infinity.

Kalman Filter

An optimal recursive algorithm that estimates the internal state of a linear dynamical system from a series of noisy measurements. It combines predictions from a mathematical model with sensor data, weighting each by their respective uncertainties.

Key Terms at a Glance

Actuator:A device that converts a control signal into physical action, such as a motor, valve, or heater, to influence the plant.
Bandwidth:The range of frequencies over which a system responds effectively, typically defined as the frequency where the gain drops to -3 dB from its maximum value.
Block Diagram:A graphical representation of a control system showing the interconnection of components using blocks, summing junctions, and signal flow lines.
Characteristic Equation:The polynomial equation obtained by setting the denominator of the closed-loop transfer function to zero. Its roots (the poles) determine system stability.
Closed-Loop System:A control system that uses feedback from the output to adjust the control input, enabling error correction and disturbance rejection.
Compensator:A controller component designed to reshape the system's frequency response or root locus to achieve desired performance, such as lead, lag, or lead-lag compensators.
Controllability:The ability to drive a system from any initial state to any desired state in finite time using admissible control inputs.
Damping Ratio:A dimensionless measure describing how oscillations in a second-order system decay over time. Values less than 1 indicate underdamped behavior.
Disturbance:An unwanted signal or input that affects the system output, which the controller must reject or minimize.
Eigenvalue:A scalar associated with the system matrix A in state-space representation. The eigenvalues of A are the system poles and determine stability and dynamic behavior.
Frequency Response:The steady-state response of a system to sinusoidal inputs of varying frequency, characterized by gain and phase shift as functions of frequency.
Gain:The ratio of output magnitude to input magnitude in a system, often expressed in decibels (dB). In a controller, it determines the strength of the corrective action.
Impulse Response:The output of a system when the input is a Dirac delta function. For LTI systems, the impulse response completely characterizes the system's behavior.
Laplace Transform:An integral transform converting time-domain functions into complex-frequency-domain functions, enabling algebraic manipulation of differential equations.
Lyapunov Stability:A stability framework where a positive-definite energy-like function (Lyapunov function) is shown to decrease along system trajectories, proving convergence to equilibrium.

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Control Theory Cheat Sheet - Quick Reference Guide | PiqCue