What should the system do?
Setpoint or trajectory?
The desired value can be constant or time-varying.
Side 33
Control Theory
A study of how dynamic systems are steered. Control theory measures state, compares it with a target, acts on the error and asks whether the resulting feedback makes the system stable, fast and robust.
state→error→controller→feedback→stability
06control lenses
03PID terms
05stability ideas
33Side
Control begins with a gap between actual and desired state.
A closed loop observes the system, computes error and changes input in response.
What is happening now?
Measure output or state.
Measurement quality and delay shape how well control can work.
How far from target?
Reference − measured output.
Error converts deviation into a signal the controller can act on.
What corrective action?
Map error to input.
Controller design determines speed, overshoot, stability and sensitivity.
How does the system respond?
Dynamics matter.
The controlled process transforms input into changing output over time.
A controller acts on a moving system, not a static equation.
Dynamics determine how quickly and in what pattern a system responds to input.
Minimum variables needed to predict future evolution.
State summarizes the system’s internal condition at a moment.
What can be manipulated?
Control inputs influence the system’s trajectory.
What is observed or regulated?
Outputs may be only part of the full internal state.
How quickly does response unfold?
First-order systems approach new conditions at characteristic rates.
Action takes time to appear.
Delay can destabilize otherwise reasonable feedback.
External influence not chosen by controller.
Good control rejects disturbances without excessive corrective action.
PID control combines present, accumulated and anticipated error.
The proportional–integral–derivative family is widely used because three simple terms capture distinct correction behaviors.
P
Proportional: act on current error.
Larger deviation produces larger corrective action, improving responsiveness but potentially increasing overshoot.
I
Integral: accumulate past error.
Integral action removes persistent offset but can wind up and destabilize response if too aggressive.
D
Derivative: react to rate of change.
Derivative action anticipates motion and can damp oscillation, but is sensitive to noisy measurements.
PID intuitioncontrol = present error + accumulated error + error trend
Fast control is useless if it destabilizes the system.
Stability asks whether disturbances decay, persist or grow over time.
Deviations decay.
The system returns toward equilibrium after small disturbances.
Deviations persist.
Oscillation may continue without growing or shrinking.
Deviations grow.
Feedback amplifies error rather than containing it.
Correction passes the target.
Aggressive control can create oscillation around the setpoint.
How long until response stays close?
Speed must be balanced against overshoot and noise sensitivity.
How much more feedback before instability?
Margins quantify distance from dangerous feedback settings.
State-space models expose internal dynamics.
Instead of describing only input–output behavior, state-space representation tracks how internal variables evolve.
State vector
Collect the variables needed to characterize internal condition.
State equation
Describe how current state and input determine the next state.
Output equation
Map internal state into observable measurements.
Controllability
Ask whether available inputs can move the system through the state space as needed.
Observability
Ask whether internal state can be reconstructed from available measurements.
Real systems are never modeled perfectly.
Robust control asks whether acceptable performance survives uncertainty, parameter drift and unmodeled disturbances.
The plant differs from the equation.
Controllers should not depend on implausibly exact parameter knowledge.
Sensors are imperfect.
Filtering can reduce noise while adding delay, creating another design trade-off.
Actuators have limits.
Real controllers cannot demand infinite force, voltage, flow or speed.
Controller parameters can change.
Adaptive control updates behavior as the plant or environment changes.
Act before error appears.
Known disturbances can be countered directly rather than waiting for feedback to detect their effect.
Performance vs robustness.
Highly tuned controllers can perform brilliantly under nominal conditions yet fail badly when assumptions drift.
Feedback SystemsÅström & Murray · modern control introduction
Modern Control EngineeringKatsuhiko Ogata · classical and state-space control
Control Systems EngineeringNorman Nise · applied foundation
Linear System Theory and DesignChi-Tsong Chen · state-space methods