Model Predictive Control (MPC)

Why MPC matters

Classical control tunes a fixed law — a PID gain triple, a lead-lag network — and hopes the plant stays inside the region where that law behaves. The moment the interesting engineering shows up, that assumption breaks: a valve saturates at 100 % open, a motor current hits its thermal limit, a drone's tilt cannot exceed 30° before it stalls its own lift, a distillation column must not flood. Those are constraints, and a fixed feedback law has no principled way to respect them. It clamps, it winds up, it fights itself when several loops share one actuator.

MPC turns control into repeated forecasting. It carries an explicit model, looks a fixed number of steps into the future, and asks an optimizer the concrete question a good operator asks: given where I am now and everything I am not allowed to violate, what is the best sequence of moves? Because the answer is recomputed every sample from the newest measurement, MPC is still feedback — but feedback that plans. The economic payoff is real: refiners run units closer to their true limits because the controller knows exactly where those limits are, which is worth millions of dollars per year on a single large unit.

• Constraints are first-class. Actuator limits, slew rates, and output bounds live inside the optimization, not in bolted-on logic.
• Multivariable by construction. One controller coordinates many coupled inputs and outputs; no loop-pairing or decoupling network needed.
• Look-ahead. Known future setpoint changes and measured disturbances are used before they hit, not just reacted to.
• Economic operation. Running at the true constraint edge, not a conservative margin, is where the money is.
• Systematic tuning. Weights on error and effort have clear physical meaning, unlike hand-tuned gain triples.

How MPC works, step by step

Every sample instant k, the controller runs the same four-stage cycle:

1. Measure / estimate the state. Read the sensors, and use an observer or Kalman filter to reconstruct the full state vector x(k) and estimate current disturbances. This is the "where am I now" that anchors the forecast.
2. Predict. Push the plant model forward N steps for a candidate input sequence, producing the predicted output trajectory. The model is the crystal ball; its fidelity sets the ceiling on performance.
3. Optimize. Solve for the input sequence that minimizes the cost J while satisfying every constraint. For a linear model with quadratic cost this is a convex quadratic program with a unique global optimum.
4. Apply the first move, then discard the rest. Send only u(k) — the first element of the optimal sequence — to the actuators. Wait one sample, take a fresh measurement, and return to stage 1. The horizon has receded by one step.

Throwing away everything but the first move looks wasteful, but it is exactly what makes MPC robust: the discarded plan was based on a prediction, and the next measurement is truth. Re-solving from the new state folds every disturbance and every model error back into the loop.

The cost function and the QP

The heart of MPC is the optimization solved at each step. In the standard linear-quadratic form the plant is the discrete state-space model

x(k+1) = A x(k) + B u(k),   y(k) = C x(k)

and the controller minimizes, over the input sequence, the finite-horizon cost:

J = Σ_i=1..N ‖ y(k+i) − r(k+i) ‖²_Q + Σ_i=0..M−1 ‖ Δu(k+i) ‖²_R

subject to u_min ≤ u ≤ u_max, Δu_min ≤ Δu ≤ Δu_max, and y_min ≤ y ≤ y_max. Every symbol:

Symbol	Meaning	Units / notes
x(k)	State vector at sample k	dimension n (engineering units)
u(k)	Control input (manipulated variable)	dimension m (e.g. valve %, volts)
Δu	Input move, u(k) − u(k−1)	per-sample change; slew rate × Ts
y(k)	Controlled output	dimension p (measured variable)
r(k+i)	Reference / setpoint trajectory	can be a known future profile
N	Prediction horizon	samples; ~10–30, covers settling
M	Control horizon (≤ N)	free moves; ~2–5, rest held constant
Q ⪰ 0	Output-error weight	p×p; larger = tighter tracking
R ≻ 0	Move-suppression weight	m×m; larger = gentler, slower action
A, B, C	Plant model matrices	from first principles or system ID
Ts	Sample time	seconds; loop period

Because J is quadratic and every constraint is linear, this is a convex quadratic program (QP) in the stacked move vector. Its solution is unique; active-set or interior-point solvers return it in microseconds for small problems and single-digit milliseconds for hundreds of variables. Two tuning ratios do most of the work: the Q/R balance trades tracking aggressiveness against control smoothness, and the horizon length trades foresight against compute. As a sanity check, if you delete all the constraints and let N→∞, the MPC law collapses exactly onto the infinite-horizon LQR — MPC is constrained LQR, re-solved online.

Worked example: a temperature loop with a rate limit

Take a heating process modelled as a first-order plant with a 30 s time constant, sampled at T_s = 1 s, and a heater whose power can move no faster than 5 % per second (Δu limit) and never exceed 100 %. Suppose an operator commands a 40 °C step in setpoint.

• PID would slam the heater toward full power, hit the 100 % ceiling, and its integrator would wind up; the temperature overshoots while the integrator unwinds. Anti-windup helps but is reactive.
• MPC predicts, at t = 0, that driving the heater hard now will overshoot in ~40 s. It plans a ramp that respects the 5 %/s slew and the 100 % ceiling, easing off before the target so the temperature arrives without overshoot. It never violated a limit because the limits were constraints in the QP, not afterthoughts.

Set N = 30 (one full time constant of foresight), M = 3 free moves, Q = 1 on the temperature error and R = 0.1 on the heater move. Increase R and the approach gets gentler and slower; increase Q and it gets crisper but pushes harder against the slew limit. Those two numbers, plus the horizon, are the whole tuning story — and each has a physical meaning, unlike a hand-tuned K_p, K_i, K_d triple.

MPC vs PID vs LQR — a spec comparison

Property	PID	LQR	MPC
Plant model needed	No	Yes (state-space)	Yes (state-space or step response)
Look-ahead / prediction	None	Infinite (implicit)	Finite horizon N, explicit
Hard constraints	No (clamp + anti-windup)	No	Yes — built into the QP
Multivariable / coupled	Poorly (loop pairing)	Natively	Natively
Compute per step	A few multiply-adds	One matrix-vector product	Solve a QP
Typical loop rate	kHz–MHz	kHz	Hz (process) to kHz (embedded)
Tuning knobs	Kp, Ki, Kd	Q, R weights	Q, R, N, M, constraints

Common misconceptions and failure modes

• "Every step is optimal, so it must be stable." False. A finite-horizon optimum is not the infinite-horizon optimum; a short horizon or bad weights can be unstable. Stability is engineered in with a terminal cost (the LQR value function) plus a terminal set, or a long-enough horizon.
• "MPC removes the need for a good model." The opposite — MPC's ceiling is set by model fidelity. Bad predictions steer bad moves. Integrating-disturbance models and an observer are what give it PID-like zero offset despite mismatch.
• "Constraints are always satisfiable." A disturbance can drive the plant into a state where no feasible input exists. Naïve MPC then throws an infeasible QP and stalls. Real controllers use soft constraints (slack variables with a heavy penalty) so output limits can be relaxed gracefully while input limits stay hard.
• "It's too slow for fast systems." Historically true, now obsolete. Explicit MPC precomputes the piecewise-affine solution offline for a lookup table, and modern embedded QP solvers run linear MPC at kilohertz rates on automotive and robotics ECUs.
• "Longer horizon is always better." Beyond the settling time, extra horizon mostly adds compute and can amplify model error at the far end. Match N to the dominant time constant, not to a bigger number.

Variants worth knowing

• Linear MPC / DMC / GPC. The workhorses. Dynamic Matrix Control (step-response models) and Generalized Predictive Control are the classic process-industry formulations.
• Nonlinear MPC (NMPC). A nonlinear model turns the QP into a nonlinear program; real-time iteration and SQP schemes make it tractable for robots and vehicles.
• Explicit MPC. Solve the parametric QP offline; the controller is a lookup of affine feedback laws — no online solver, ideal for tiny fast systems.
• Robust / tube MPC. Plans a nominal trajectory plus a bounded "tube" so guarantees hold under disturbance and model mismatch.
• Economic MPC. The cost is a direct economic objective (profit, energy) rather than tracking error — the plant is driven to the economically optimal operating point.