Control Systems

Temperature up the side, time along the bottom, running left to right. The two numbers at the left-hand end are the top and bottom of the temperature scale. It is not always the same scale, because it is chosen to fit whatever the run did. So compare shapes between runs rather than heights.

The charts answer your pointer. Rest it anywhere on one and a crosshair appears, reading out every line at that moment. The small play button in a chart's top corner walks that crosshair across the whole run, so you can watch the numbers unfold in order. Some charts also carry a round handle you can take hold of and drag, and the slider it belongs to follows your hand.

The solid line is the water itself. The dashed line is what the thermometer says, and the two are not the same thing. A probe sits inside a lump of metal that has to warm up before it can report anything, so the reading trails the truth by a moment. That delay is called lag, and Step 9 is about how much trouble it causes. Three more things are drawn on there. The dashed horizontal line is the target. The pale strip either side of it is a band of close enough. The shaded area along the bottom is how hard the heater is working.

Because water is slow enough to watch. A drone runs this same loop about five hundred times a second and you cannot see any of it. Warm water does it over a minute, which means you can watch a mistake being made, spot why, and change one number.

Nothing in the course is water-specific. Step 11 swaps the tank for a weight on a rail and asks for a position instead of a temperature, using the same three sums. A motor holds a speed, a valve holds a pressure, a quadcopter holds an angle, a cruise control holds 70 miles an hour. All of them are this lesson with different units.

Most electric heating elements have one setting, which is fully on. Per cent power is usually built by switching quickly. On for four tenths of every short period and off for the other six tenths gives four tenths of the heat. The water is far too slow to notice the flicker. Other machines do it with a valve that can be part open, or a dimmer that passes part of the mains.

Either way, the number the controller hands over is a fraction of everything the machine has. That matters later: 100 per cent is a wall. When the sums ask for more, there is nothing there.

Yes, and now you can predict the pattern. The fridge cools until the sensor says it is cold enough, then stops. It sits there warming slowly through the walls and the door seal. It starts again once it has warmed past the far edge of the dead band. The hum is one visit round that loop.

You can hear the trade being made. A freezer full of food is happy to roam a degree or two, so the dead band is wide and the compressor runs a few times an hour. A fish tank heater has a narrow band and clicks far more often, because the fish care more than the peas do.

It is being sloppy on purpose, and it is the right thing to do. Without it, the tiniest wobble in the reading around the target flips the heater, so the switch chatters, and chatter is the one thing that reliably breaks the hardware.

The word for the same trick elsewhere is hysteresis: the point at which a thing turns on is deliberately different from the point at which it turns off. A light switch has it mechanically, which is why it snaps rather than hovering.

In a mechanical relay, two metal contacts slam together and pull apart. Each parting draws a tiny spark that takes a little metal with it, so contacts are rated for a number of operations, often somewhere around a hundred thousand. Divide that by the switching you can hear and you can work out how long the part will last.

A compressor is worse. Starting a motor draws several times its running current, and the motor is hottest during that start. Repeating those starts quickly is called short cycling. It wears a fridge out faster than simply running it. This is why real controllers often add a minimum on-time and a minimum off-time on top of the dead band. They refuse to switch again for a few minutes, whatever the temperature says.

Per cent of power per degree of error, in this lab. That is worth writing down for any gain you meet, because it tells you what the number means. It also stops you carrying a value from one machine to another where it is nonsense.

A motor controller's gain might be volts per millimetre. A drone's might be motor units per degree of tilt. The same tuning number that is calm on one machine can be violent on the next, and the units are the first hint of why.

That is all of it. Double the error and the controller asks for double the power, and no more is promised: nothing about arriving quickly, nothing about arriving accurately.

It is worth checking what fails the test. On and off, from Step 2, is not proportional to anything: at one degree off and at thirty degrees off it does exactly the same thing. That is the whole of what Step 2 was throwing away.

Up to a point, and then it stops being about speed. A higher gain asks for more power at the same distance from the target, so the early part of the run gets steeper. But past a certain gain the run is not slow, it is unstable. The controller reacts hard to a reading that is behind the truth, so the correction lands after the moment it was needed. It then feeds the swing instead of taking it out.

That boundary is not a property of the gain on its own. It depends on how quickly the plant answers and how late the measurement is, which is why Step 9 can wreck a tuning without touching a gain.

That really is what it is. An integral is the total of a quantity that keeps arriving, added up over time. If you are two degrees short for ten seconds, the total gains twenty degree-seconds. A degree-second is a unit like a kilowatt-hour: an amount multiplied by how long it lasted.

The version in a maths class asks what the total would be if you added it up in infinitely small slices, and that question needs machinery. A controller has no need of it. It has a clock ticking every tenth of a second, so it adds a slice every tenth of a second. That is both easier and what actually runs in the machine.

Because of what the total does when the error is small. The proportional term is happy at a small error: small error, small power, and if that power happens to balance the leak, it will sit there for ever. The total is never happy. As long as any error remains, the total keeps growing, so the power keeps rising.

The only place the total stops changing is at zero error. Everywhere else it is still moving, and moving in the direction that reduces the error. So the resting place of a controller with an integral term is the target itself, which is what you wanted all along.

The error turns negative, so the additions turn into subtractions and the total shrinks. It can go through zero and out the other side, which is how an integral term learns to hold a target that needs cooling rather than heating.

That shrinking is slow, though. The total was built up over the whole climb and it comes down at whatever rate the current error allows. A stored number that takes an age to unwind is the fault in Step 8, and it is the price of having a memory at all.

For a controller, yes, and that is all any of them compute. The reading changed by this much, in this much time, so it is moving at that speed. Speed is the rate a position changes at. A derivative is the same question asked about any number at all.

The maths-class version asks what the answer becomes as the gap between the two readings shrinks towards nothing. A controller cannot ask that, because it only has the readings its clock gave it. Every derivative in every real machine is a subtraction and a division, which is why the tick rate is part of the design and not a detail.

Because a sensor is a physical object in an electrically messy world. A probe's wires pick up stray voltage from the mains and from the motor next door. A bubble drifts past the tip. The circuit that turns a voltage into a number has a last digit that has to make its mind up, and sometimes it changes its mind. Nothing is broken. This is what measurement is like.

The result is that each reading is the truth plus a little rubbish, and the rubbish is different every time. Averaging several readings helps, at the cost of making them later, which by Step 9 you will recognise as a real cost rather than a free win.

Because the term that reacts to speed also reacts to nonsense, and nonsense is fast. Practitioners often reach for a controller with only the first two terms, get it good enough, and stop, rather than take on a term that needs filtering to be safe.

When Kd is used, it is usually softened. The reading is smoothed before the subtraction, or the derivative is taken of the measurement rather than of the error, so that changing the target does not cause a jolt. Those are standard tricks, and both exist because of what you are about to do to the heater in this lab.

There is no right answer, and expecting one is a common way to get stuck. Which term dominates is a fact about the moment, not about the tuning: large error favours P, a long-standing error favours I, fast movement favours D.

What is worth checking is whether a term is doing nothing at all across an entire run. A D term that never grows is not helping. An I term that grows without limit is Step 8. A P term that is tiny throughout usually means the gain is too small to be worth having.

The three terms are arithmetic and arithmetic has no idea what a heater is. If the water is above the target, the error is negative, so P is negative. If the water is climbing, D is negative. Those negative numbers mean ask for less, and they are subtracted from what the other terms wanted.

Only after the three are added does the answer meet the machine. A total below zero becomes zero power, because this plant has no cooler, and the bar labels that as clipped. A machine that can push both ways, like the motor in Step 11, uses the negative half of the range instead of throwing it away.

No, and it is not always the best. It is the one worth learning first for three reasons. It needs no model of the machine. It is three numbers a person can tune by hand. It is also what sits inside most of the working equipment in the world.

Above it sit controllers that carry a model of the plant and use it to plan ahead, which is what model predictive control means. Others adjust their own gains while running. All of them still have to answer the questions this course is about. How late is my information. What happens when the machine saturates. What does the disturbance do that nobody told me about.

There are recipes, and they are starting points rather than answers. The best known starts with Ki and Kd turned off. Raise Kp until the loop swings past the target repeatedly without settling, like the swing in Step 2. Note both that gain and how long one swing takes. Those two measurements feed a small table that suggests all three gains. It is named after Ziegler and Nichols, who published it in the 1940s.

Every method of this kind assumes the machine will let you push it into oscillation, which rules out plenty of real plants. It also aims at a compromise nobody asked for. Practitioners use one to get into the right area and then do exactly what this lab does: change one gain, run it, look at the numbers.

Starting from cold is a big, honest, one-off error that every term can see coming a mile off. A draught arrives when the controller has settled and everything looks finished. The error it makes is small, and it does not go away, which is the case that finds out whether your Ki is really doing its job.

Real complaints are nearly always of the second kind. Nobody rings up about the first two minutes after switch-on. They ring up because the room goes cold when the wind is in the east.

It is there because a score made only of temperature can be won by a tuning that destroys the hardware. Add up how far the power moved, tick by tick, across the whole run. That is a measure of how hard the actuator was worked: a valve stem sliding, a motor reversing, a relay clicking.

Real scoring functions in industry include terms like this, alongside energy used and how much the output moved. Deciding what goes into the score is the real design decision, and it is made long before anybody touches a gain.

Anywhere a machine can be asked for something it cannot do. An oven with the door propped open, told to reach 200 degrees. A car's cruise control climbing a hill too steep for the engine it has. A tank being filled by a pump that cannot keep up with the leak.

The famous cases are in aircraft. An autopilot integrating a demand it cannot meet, then meeting it suddenly and reacting to a stored number from a minute ago, has been a factor in real incidents. It is one of the reasons every serious control library has a limit on the stored total. It is also why the word windup is in the manual rather than in a lecture.

That is the other standard fix, and it is often the better one. If the output is already saturated and the error would push it further into the wall, stop adding to the total for that tick, because the addition cannot buy anything. It is usually called conditional integration.

The lab uses the blunter version, a hard limit on the total, because it is easier to see: one number with a ceiling. Both fixes attack the same thing, which is a memory that keeps growing while the growth can no longer change what the machine does.

Big enough that the integral can still ask for full power on its own, and no bigger. That way an honest run, one where the target is reachable, never notices the limit at all: the total does what it always did. The limit only comes into play when the total has grown past anything it could usefully spend.

Written properly, the limit is worked out from the output range and Ki rather than picked by feel, which means it stays correct when somebody retunes Ki later. A limit chosen by feel and forgotten is how a fixed fault comes back three years on.

From several places at once, and they add up. The sensor itself has to change temperature before it can report one. The wires and the converter take a little time. The software may average several readings before believing them. The controller only decides on its tick, so on average it sits on fresh news for half a tick before using it.

Then the plant has its own delay. Heat has to travel from the element to where the probe is. In a long pipe there is a true transport delay. What happens at one end cannot be seen at the other for some number of seconds, however good your sensor is. That kind is the worst kind, and nothing in the controller can shorten it.

No, and the difference is the point of the step. A slow tank is honest: it responds gradually and the controller can watch it respond gradually and act accordingly. Slow is not hard to control.

Lag is dishonest information. The tank may be responding right now while the measurement still shows what it was doing seconds ago, so the controller acts on a world that has moved on. Slow plus truthful is easy. Fast plus late is hard.

First, lower the gains. If your news is old, hard reactions are guesses, so the honest response is to correct more gently and accept a slower loop. That is not defeat, it is arithmetic.

Beyond that there are two routes. Act on causes rather than errors, so that some of your response does not depend on the measurement at all, which is Step 10. Or carry a model of the plant, run it forward, and control the model's guess at where the plant is now rather than the stale reading. That second route is the beginning of the subject called state estimation, and its most famous tool is the Kalman filter.

Somebody measured it once, and the number went into the machine. Open the window, wait for things to settle, write down how much extra power was needed to hold the target. That is the feed forward number. It is a fact about that building rather than about control theory.

Bigger systems store a table or a formula instead of one number: this much extra power per degree of outside cold, this much for that pump running. Filling in that table is unglamorous work and it is where most of the benefit lives.

Probably several times. A shower where you turn the hot tap first because you know the cold pipe is long. An oven preheated before the food goes in, since the recipe knows the food will cool it. A car with a map, easing the throttle on before the hill rather than after the speed drops.

The pattern is always the same. You know something about what is coming, so you act on the knowledge rather than waiting for the consequence. And in every case you keep an eye on the result anyway, because the knowledge might be out of date.

Easily. Too large a number and it overcorrects, so a window that used to cause a dip now causes a spike. The feedback part then has to clean up a mistake that was volunteered. Too early and it acts on a disturbance that has not started yet.

The worst case is a feed forward signal that is wrong in an unnoticed way, because it never gets checked. The feedback part will quietly compensate and everything looks acceptable, until the day the feedback part has nothing spare to compensate with. This is why the two are usually commissioned in order: feedback made good on its own first, then feed forward added and measured for what it actually improved.

Momentum is how much motion a thing has: its mass multiplied by its speed. A heavy trolley moving slowly and a light one moving quickly can be equally hard to stop. To change it, you have to push, and the longer you push the more it changes.

The reason this matters for control is that a moving mass carries energy that has to go somewhere before it can be still. Either something absorbs it, like friction or a controller pushing backwards, or it keeps sloshing between moving one way and moving the other. That sloshing is exactly what the lab shows with Kd at zero.

It has been left out so that the brake is the only thing to look at. Position loops do use one, for the same reason the tank did: friction, or gravity on a lifted arm, leaves a small steady error that only a memory can remove.

But an integral on a mass has to be handled carefully. The stored total keeps pushing after arrival, and a mass responds to a push by moving. So a careless Ki makes an arm creep past its target and hunt back and forth around it. Position loops usually run a fairly small Ki. They nearly always limit the stored total, for the reason you found in Step 8.

Every joint of every robot arm. The read head of a hard disc, which has to move a fraction of a millimetre and stop dead, thousands of times a second. A camera gimbal on a drone. A telescope mount. A lift, where the passengers are part of the mass and the overshoot has to be zero.

They differ in what fights them. An arm has gravity pulling differently at every angle, so the same gains behave differently at each pose. A lift has a mass that changes every trip. A disc head is so light and so fast that the wobble of the arm holding it becomes part of the problem. All of them are the lab in front of you plus one honest complication.

Zoom in. Every smooth curve looks straight if you look at a short enough piece of it, the same way the Earth looks flat from your garden. The slope at a moment is the slope of that tiny straight piece. A ruler laid along it is called a tangent line, and the lab draws one for you.

The controller's version from Step 5 is the honest approximation: take two nearby readings, subtract, divide by the time between them. The shorter the gap, the closer that answer sits to the tangent's slope.

Because physics rarely tells you where a thing will be. It tells you how fast the thing is changing right now, given where it is right now. Heat flow sets how fast temperature changes. Force sets how fast speed changes. Each law is a recipe for a slope.

An equation built out of a quantity and its slopes is called a differential equation, which is Step 13. The simulator behind every lab on this page has been solving one all along: work out the slope, take a small step along it, repeat.

From the balance you found in Step 1. Heat flows in at a rate set by the power. Heat leaks out at a rate set by how much hotter the water is than the room. The difference between the two is spare heat, and spare heat is what raises the temperature. Big difference, steep slope.

That is why the curve flattens as it climbs. The hotter the water gets, the faster it leaks, so the spare shrinks, so the slope shrinks. At the resting level the two rates are equal, the spare is zero, and the slope is zero. Not because heat stopped moving, but because the flows cancel.

Check units before doing algebra. C has units of joules per degree. dT/dt has degrees per second, so their product is joules per second, which is watts. Both terms on the right must also be watts. If one side is metres and the other is seconds, the equation cannot describe the same quantity.

Then check directions. If the water is hotter than the room, T - T_room is positive and the minus sign makes leakage cool the tank. If the heater command rises, the positive K u term makes the slope rise. Those tests catch many sign mistakes before a simulation is run.

It is a model built from a physical law plus assumptions. We assume every part of the water has one temperature, heat loss is proportional to the temperature difference, and the heater responds at once. A large tank with poor mixing violates the first assumption. Radiation and boiling violate the second. A valve or heater with its own delay violates the third.

Engineering does not require a model to contain everything. It requires the model to be accurate enough for the decision being made, and it requires the omitted effects to be named and tested.

With the heater fixed, the tank approaches its resting temperature at a speed set by C/H. That ratio is the time constant, usually written tau. After one tau, a first-order system has completed about 63 per cent of the journey to its new resting value. After about four tau it is within roughly 2 per cent.

A larger heat capacity C makes the tank slower. A larger heat-loss coefficient H makes it respond faster but also demands more heater power to hold the same temperature above the room.

The order tells us how much independent state is needed to predict the future. A first-order thermal model needs one starting value, temperature. A second-order mechanical model needs two, usually position and velocity. In general, an nth-order ODE can be rewritten as n coupled first-order equations.

This is why order matters to a controller. Each independent state may need to be measured or estimated, and each can carry an error that affects the future.

No. Gain changes the forces, but the plant may already store and exchange energy. A spring stores energy when stretched. A mass stores kinetic energy while moving. Low damping lets energy move between those stores, so the system rings even with no controller.

Before retuning, ask whether the model has the right inertia, stiffness, friction and delay. A gain change can hide a bad model without explaining the machine.

Underdamped means the response crosses the final value and usually rings. Critically damped is the boundary that returns without crossing and is fastest among the non-oscillating cases. Overdamped means two slower decays dominate, so it returns without crossing but takes longer.

Critical damping is not automatically best. A camera mount may value no overshoot. A racing actuator may accept some overshoot for speed. The requirement decides.

The Nyquist rule says a sinusoid needs a sample rate greater than twice its frequency to avoid an exact alias. Control engineers usually sample much faster than that, often ten or more times the closed-loop bandwidth, because two points per cycle describe a wave badly and leave little phase room.

The correct rate also includes computation time, sensor filtering, actuator updates and timing jitter. A nominal 100 Hz loop is not a 100 Hz loop if some ticks arrive 30 ms late.

Not necessarily. Forward Euler can make a stable equation grow when dt is too large. That is a failure of the numerical method, not evidence that the real plant explodes. Shrinking dt or using a more stable integration method can separate the two.

The reverse warning also matters. A simulator can look stable because its solver smooths or clips a difficult effect. Always vary the timestep and compare against measurements.

Integrals become sums and derivatives become differences, as Steps 4 and 5 already used. Filters and transfer functions become recurrence equations that remember past inputs and outputs. Their coefficients depend on the chosen sample time.

Methods such as zero-order hold and the bilinear transform preserve different features of the continuous design. The important rule is to state the conversion and sample time. Copying coefficients between loop rates changes the controller.

The complete rule is L{dx/dt} = sX(s) - x(0). The starting value appears as a separate term. Transfer functions usually assume zero initial conditions so the ratio describes only the input-to-output path.

That assumption is useful, not invisible. To predict a real run from a warm start, include the initial condition or use a state-space model that carries it directly.

No. It changes the description, like switching from a street map to coordinates. The time plot is good for seeing when events happen. The s-plane is good for seeing decay, oscillation and algebraic structure.

An inverse Laplace transform returns the time signal. If two forms were transformed correctly, they describe the same behaviour and should agree when simulated.

It hides internal state and normally describes one operating point of a linear time-invariant system. Saturation, friction that changes direction, switching logic and time-varying loads do not fit one exact transfer function.

Engineers still linearise nonlinear systems near an operating point because local transfer functions are useful. They then test the design outside that local region in simulation and hardware.

For a standard second-order model, the pole angle corresponds to damping ratio. Poles close to the imaginary axis decay slowly and ring. Moving the pair further left increases decay. Moving them higher increases oscillation frequency.

That shortcut applies to a dominant pair. Extra poles, zeros, delays and saturation can change the measured response, which is why prediction is followed by simulation and testing.

It guarantees asymptotic stability only for the stated linear model. It does not guarantee acceptable overshoot, actuator effort, noise sensitivity, constraint handling or safety under model error.

A stable aircraft that exceeds an actuator limit or loses too much altitude is still an unacceptable design. Stability is a necessary check, not the entire requirement.

Discrete-time systems use the z-plane. A mode is stable when its pole lies inside the unit circle. Sampling maps continuous poles into discrete poles roughly by z = e^{s dt}.

The left half of the s-plane maps inside the circle. The sample time affects where the pole lands and how accurately the digital model represents fast continuous dynamics.

A root-locus plot traces where closed-loop poles move as one loop gain changes. It makes the trade between speed, damping and instability visible before a gain is chosen. Compensator poles and zeros can reshape the path.

State-space pole placement chooses feedback gains to put controllable closed-loop poles at selected locations. Both methods need sensible target locations, actuator checks and uncertainty tests. Moving a model pole is not the same as proving the hardware can produce that response.

A linear time-invariant system changes a sine wave's size and timing but not its frequency. That makes each frequency an easy test signal. Complicated signals can be split into sines, passed through the system one frequency at a time, and added back together.

This superposition rule depends on linearity. Saturation and switching create new harmonics, so their spectra reveal that the system is no longer behaving linearly.

A steady-state Fourier response assumes the relevant transients settle. If the loop is unstable, that assumption fails. Frequency-domain stability tests use the loop transfer function and conditions such as Nyquist or margins, not a casual spectrum of one finite run.

A finite FFT can still show growing energy, but it does not by itself prove closed-loop stability.

Frequency Domain builds the discrete Fourier transform by hand and then scales it to spectra and spectrograms. Filters connects frequency response to practical low-pass, high-pass and band-pass designs.

This course uses those ideas for loop behaviour: disturbance rejection, noise amplification, delay and stability margins.

A pure delay changes no magnitude, but its phase lag grows with frequency. At angular frequency omega, a delay tau contributes -omega tau radians. A loop that looks safe without the sensor, computation and actuator delays can cross -180 degrees once they are included.

This is the frequency-domain version of Step 9. The controller acts on old information, and at some frequency “old” becomes close to pushing in the wrong direction.

No. Bandwidth describes how quickly the closed loop can follow and reject changes. Sample rate says how often the digital controller runs. The sample rate must be well above the bandwidth so sensing and computation do not consume the phase margin.

Raising sample rate does not make the physical actuator faster. It only reduces one source of delay and gives the digital model more points.

Then one margin number may hide the dangerous crossing. Resonances, flexible structures and filters can make several crossovers. Inspect all of them and use a Nyquist plot or direct closed-loop analysis when the simple reading is ambiguous.

Margins also describe one linearised loop at a time. Check operating points, payloads, temperatures and saturation separately.

A Nyquist plot draws the complex loop response as frequency changes. Its encirclements of the point -1, combined with the open-loop unstable poles, determine closed-loop stability. It handles cases where one crossover and one margin are not enough.

The full rule needs careful contour and sign conventions. For this course, keep the engineering reason: Bode margins are readable summaries, while Nyquist preserves the complex path needed for a complete linear stability test.

A matrix is a compact table of connections. Each row says how one state changes; each column says which state or input contributes. Linear Algebra builds this picture from vectors, transformations, eigenvectors and systems of equations.

The eigenvalues of A are the state-space version of poles. State feedback changes the effective A matrix and can move controllable modes.

Not for this trolley. A sequence of position measurements reveals how position changes, which carries velocity information. An observer combines the model, input commands and measurement error to estimate states that are not measured directly.

Noise and delay limit the estimate, so observability in the algebraic sense does not promise a useful estimate at every bandwidth.

A Kalman filter is an observer that propagates a state estimate and uncertainty through a linear model, then corrects them with measurements and stated noise models. It is covered in context in Autonomous Systems.

A controller and estimator are designed separately under specific linear assumptions, then combined. Their model, timing and noise assumptions still need joint testing.

State feedback uses u = -Kx plus any reference handling. Pole placement chooses K to put controllable modes at desired poles. LQR instead chooses K to minimise an accumulated quadratic cost that weighs state error against control effort.

The LQR weights are design choices, not measured physical constants. Scale states and units carefully, inspect the resulting poles and actuator demand, then test the controller outside the linear model.

A PDE needs rules at the edges. A fixed-temperature end is a Dirichlet boundary. A specified heat flow, including an insulated zero-flow edge, is a Neumann boundary. A convective edge can relate flow to the difference from ambient temperature.

Changing only the boundary conditions can change the whole solution. They are part of the model, not a detail added after the equation.

No. A lumped ODE is often enough when internal differences settle much faster than the control problem. The tank worked because we assumed mixing kept one temperature. Use a distributed model when spatial shape affects sensors, actuators, stress, safety or performance.

Flexible robots, wings, long cables, batteries and buildings often need distributed models somewhere in their design, even if the final embedded controller uses a reduced model.

Finite-difference or finite-element models may have thousands of states. Model-order reduction keeps the modes that matter in the controller's frequency range and discards weak or fast modes carefully.

Discarded flexible modes can still be excited by a controller. Check the full model and hardware for spillover, where control energy enters a neglected mode.

Write the requirements first: variables to hold, disturbances, limits, timing, acceptable errors and hazards. Measure or identify the plant. Start with the simplest controller that can meet the requirements, then test nominal performance, parameter variation, noise, delay, saturation and faults.

Complexity is justified when it buys a measured capability. It also adds software, tuning, computation, monitoring and verification work.

System identification fits a dynamical model from input-output data. It includes classical linear methods, nonlinear grey-box models and learned neural models. The label describes the engineering job, not one algorithm family.

Excite the system safely, keep separate validation runs, check residuals and uncertainty, and repeat across operating conditions. A model that predicts the data used to fit it has not yet passed a useful test.

Cascaded control nests a fast inner loop inside a slower outer loop, such as motor current, then speed, then position. The inner loop must be measurably faster so the outer loop can treat it as a simpler actuator.

Adaptive control changes parameters while operating under stated excitation and stability conditions. Uncertainty-bounded control designs against a stated family of models. Nonlinear methods use the real nonlinear structure instead of one linear operating point. Gain scheduling uses several local designs and rules for moving between them.

Each method answers a different failure of a simpler model. None removes the need to bound uncertainty, rate and amplitude limits, computation, faults and the region in which its guarantees apply.

Simulating Physical Systems covers model fitting, scenario matrices, fault injection, domain randomisation, SIL and HIL. Optimisation supplies the language behind LQR and MPC. Autonomous Systems adds estimation, planning, learned policies, runtime safeguards and safety evidence.

For timing on real hardware, continue to Real-Time Systems. For physical motion, use Robotics and Drones.

The same tank as every other lab. Kp multiplies the error now. Ki multiplies the running total of every error so far. Kd multiplies how fast the error is changing. The three are added, and the sum is clipped to something between nothing and everything the heater has.

The sensor lag slider decides how far behind the truth the reading is. The limit button decides whether the running total has a ceiling. The knock button subtracts a lump of temperature part way through, which is a disturbance with no warning attached.

Three things, and none of them is the controller. What you measure, what you drive, and the numbers that describe how the second affects the first. Swap degrees for revolutions per minute, and heater power for throttle and the three sums are untouched.

What does change is the tuning, because the gains carry units. That is why nobody can hand you a Kp and why every course, including this one, has to hand you a method instead.

Plenty, and it is worth knowing what, so you are not surprised by a real machine. The heater here can change power instantly, while a real valve takes time to move and cannot go faster than it goes. The plant is linear: twice the power gives twice the effect, which is untrue of nearly everything real, including a heater losing heat by radiation. The clock never stutters, and in a real machine a tick can arrive late.

Also, there is exactly one loop. Real plants have several, interfering with each other: heating one tank changes the temperature of the tank next door, so tuning one loop detunes another. That is where this subject stops being a single slider and starts being engineering.

After one practical run, describe the result using the advanced path. Name the state, input and disturbance. Say whether the dominant model is first or second order. Predict what added delay does to phase margin, and list the effects this tank model leaves out. This is how an experiment becomes model evidence.