Control Systems Cheat Sheet
The core ideas of Control Systems distilled into a single, scannable reference — perfect for review or quick lookup.
Quick Reference
Feedback Control (Closed-Loop Control)
A control strategy in which the output of the system is measured and fed back to be compared with the desired reference input. The difference (error) is used by the controller to adjust the input to the plant, continuously correcting for disturbances and model uncertainties.
Transfer Function
A mathematical representation of the relationship between the input and output of a linear time-invariant (LTI) system in the Laplace domain. It is expressed as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming zero initial conditions.
PID Controller
The most widely used industrial controller, combining three control actions: Proportional (reacts to the current error), Integral (reacts to the accumulated past error), and Derivative (reacts to the rate of change of error). Tuning the three gains Kp, Ki, and Kd shapes system response.
Stability
A system property indicating that its output remains bounded for bounded inputs (BIBO stability) or that it returns to equilibrium after a disturbance (asymptotic stability). For LTI systems, stability requires that all poles of the closed-loop transfer function lie in the left half of the s-plane.
Bode Plot
A pair of frequency-domain graphs consisting of a magnitude plot (in decibels) and a phase plot (in degrees) versus logarithmic frequency. Bode plots are used to analyze and design controllers by visualizing gain margin, phase margin, bandwidth, and steady-state error characteristics.
Root Locus
A graphical method that plots the trajectories of the closed-loop poles in the complex s-plane as a system parameter (typically the loop gain K) varies from zero to infinity. It reveals how pole locations, and therefore system stability and transient response, change with gain.
State-Space Representation
A mathematical model of a physical system expressed as a set of first-order coupled differential equations using state variables. Represented in matrix form as x-dot = Ax + Bu (state equation) and y = Cx + Du (output equation), it generalizes to multi-input multi-output (MIMO) systems.
Nyquist Stability Criterion
A frequency-domain method that determines the stability of a closed-loop system by examining the open-loop frequency response. By plotting the Nyquist contour of the loop transfer function and counting encirclements of the critical point (-1, 0), one can determine the number of unstable closed-loop poles.
Controllability and Observability
Two fundamental properties of state-space systems. Controllability means every state can be driven to any desired value using the available inputs. Observability means every state can be determined from the outputs and inputs. Both are prerequisites for effective controller and observer design.
Steady-State Error
The difference between the desired output and the actual output of a control system as time approaches infinity. The system type number (number of integrators in the open-loop transfer function) determines the steady-state error for different classes of reference inputs such as steps, ramps, and parabolas.
Key Terms at a Glance
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