Control, from first principles.
A guided course in modeling, feedback, design, and validation. Read a chapter, predict what will happen, then test it in the lab.
Learn to reason about a control loop.
Begin with a physical system and a feedback question. Represent its changing state with equations, use transfer functions to study input and output, and check every design against stability, limits, and model error.
The path moves from feedback and modeling to response and classical design, then PID, state-space design, robustness, and digital validation.
Before you begin
Comfort with algebra, basic calculus, simple physical laws, and basic matrices helps. The first chapters introduce the control vocabulary and state representation.
By the end
Build and linearize a model; interpret time and frequency responses; choose a controller; explain stability, estimation, robustness, and sampling limits.
How to study
Read the ideas in order. Before each lab, predict the effect of one change; compare the result with your prediction and explain the difference.
Introduction to control
Start with the feedback loop: what it measures, changes, and cannot overcome.
Mathematical and physical modeling
Build equations from physical systems, introduce states, and connect time-domain models to transfer functions.
From hardware to equations
A model keeps the mechanisms that matter for your question and leaves the rest out.
States and the cart-pole model
A state records enough information to predict the next motion once the input is known.
A useful model near one operating point
Linearization is a local approximation, not a claim that a nonlinear machine has become linear.
Tank balance and local linear models
A nonlinear gravity-drained tank beside its local linear prediction. Explore operating points, bounded pump flow and PI disturbance rejection.
RLC dynamics and electrical resonance
An ideal series RLC circuit. Connect electrical damping and resonance to the mechanical system you already know.
What LTI really promises
Linear and time-invariant are two separate tests. Passing both lets us reuse one response to predict many others.
A controller cannot read tomorrow's measurement
Causality says the present output can depend on present and past inputs, but not on an input that has not happened.
Build a response from tiny impulses
The impulse response is an LTI system's reusable response pattern; convolution adds shifted copies of it.
Why use the Laplace transform?
It turns differential equations into algebra while preserving the story of transients and stability.
Transfer functions, poles and zeros
Poles describe the system's natural modes; zeros reshape how inputs excite them.
Follow the signal, then reduce the blocks
Predict series, parallel and feedback outputs before switching topology. Then change only H from 1 to 2. Which error does feedback reduce?
Time response and stability
Read transients and steady error, then test what stability claims actually mean.
Read a step response
A single target change reveals speed, overshoot, settling and residual error.
Stability and steady-state error
A response can be bounded yet inaccurate, or accurate in theory but unreachable with real actuator limits.
Bounded input, bounded output: what is being promised?
BIBO stability is a statement about every bounded input and the observed output, not about one attractive plot.
Find stability without solving every root
Keep a=2 and b=3. Compare c=4, 6 and 8. Why does positivity of every coefficient fail to guarantee stability?
Count integrators, then check the error
Compare type 0/1/2 at K=2 with a step, then a ramp. Why can a loop track a fixed target but lag behind a moving one?
Root locus and frequency design
Design with moving poles, Bode plots, and loop margins.
Root locus: follow the poles
Root locus shows where closed-loop poles move as a gain changes.
Lead and lag compensation
Shape the loop across frequency instead of increasing one gain everywhere.
Why analyze frequency and phase?
A complex system can be probed one sine wave at a time.
Bode plots: gain and phase
Two aligned plots show how a loop responds across slow and fast inputs.
Why −180° matters: Nyquist and margins
Negative feedback can become reinforcing feedback when the returning signal is inverted and strong enough.
PID and practical implementation
Tune proportional, integral, and derivative action against real actuator and thermal limits.
PID, saturation and windup
P reacts to error now, I remembers error, D anticipates measured motion.
Thermostats, PI and heating limits
A heater makes three ideas tangible: a system stores energy, holding a target requires continued input, and an actuator cannot do everything a controller asks.
State-space control and estimation
Check controllability and observability before feedback, observers, and optimal control.
Can you move it? Can you see it?
Controllability concerns the actuator; observability concerns the sensors. Both matter before state feedback and estimation.
State feedback by pole placement
A position-sensed mass–spring–damper with designed state feedback and a Luenberger observer. Compare true and estimated states, noise and actuator limits.
Luenberger observers and separation
A position-sensed mass–spring–damper with designed state feedback and a Luenberger observer. Compare true and estimated states, noise and actuator limits.
LQR: decide what a good balance costs
LQR calculates a state-feedback gain from a linear model and a stated tradeoff between state error and control effort.
Kalman filter: infer hidden states
A sensor cannot always reveal every state directly.
LQG: control what you estimate
Combine LQR state feedback with a Kalman state estimate.
Sensitivity and robustness
Understand tradeoffs when disturbances, noise, and model error enter the loop.
Tracking, disturbance and sensitivity
A target, a load and sensor noise do not enter the feedback loop at the same place.
Will your controller survive a different motor?
Nominal performance describes one model. Robustness asks what remains true across a stated set of plausible models.
Give tracking its own path
Keep feedback gain fixed. Change reference feedforward and compare target tracking with the separate disturbance response.
Digital control and model validation
Move from continuous models to sampled control and test whether the model predicts new trials.
A controller has a clock
A digital controller observes and updates at intervals while the physical plant keeps moving.
From the left half-plane to the unit circle
Keep the same continuous mode and change the sampling period. Does a smaller discrete pole mean faster motion in seconds?
When samples tell a different story
Sample a 7 Hz sine at 10 Hz. Predict the apparent frequency, then increase the sampling rate.
A model must survive another test
Estimating a plant from one response is the beginning of a model, not the end of validation.
This independently written course adapts the broad topic progression in Ogata’s Modern Control Engineering (5th ed.) and MIT's Analysis and Design of Feedback Control Systems syllabus. Publisher contents ↗ · MIT OCW ↗