García, Prett and Morari place regulatory control inside a hierarchy. Measurements enter a control layer; optimisation supplies economically desirable operating conditions; production planning and logistics work on longer timescales. The online control problem is to update manipulated variables so that several performance criteria remain acceptable while feed composition, equipment condition, market priorities and plant dynamics change. Those criteria can encode economics, environmental and safety limits, equipment protection, product specifications and operator preferences. A single scalar set-point is therefore a small special case.

At an economic optimum, several limits commonly meet. A furnace should run hot enough for throughput but not above a metallurgical temperature. A distillation column should meet purity without exhausting steam or flooding a tray. A compressor should approach capacity without crossing surge. Conventional loops can be surrounded by overrides, selectors, split ranges and hand-built exception logic, but the survey warns that such structures become costly to design, explain and maintain when many variables interact.

Model-predictive control offers a common calculation. Use an explicit dynamic model to forecast how proposed future input changes affect outputs. Choose the sequence that best satisfies a stated performance criterion while respecting constraints. Send the first move to the plant. At the next sample, compare prediction with measurement and solve the shifted problem again. The authors connect industrial MPC to Richalet and colleagues’ Model Predictive Heuristic Control, Shell’s Dynamic Matrix Control, and Propoi’s earlier moving-horizon formulation.

This history matters because the architecture grew from operating needs, not from one universal theorem. Industrial plants are often stable and overdamped relative to aerospace examples, yet they have uncertain models, coupled variables and economic incentives to operate near boundaries. MPC made those features legible to the controller and to the engineer tuning it. Its display of predicted futures could also help operators understand why a move was proposed. Transparency here means inspectable predictions and constraints, not a guarantee that every consequence is understood.

The survey moves among state-space, transfer-matrix, impulse-response and step-response descriptions. In a discrete state model, x(k)=Ax(k−1)+Bu(k−1) and y(k)=Cx(k). Eliminating the state gives the transfer matrix P(z)=C(zI−A)⁻¹B. For a stable process, an impulse response can be represented by coefficients that describe each later output response to one input pulse. A step response instead accumulates those coefficients and naturally pairs with input increments Δu(k)=u(k)−u(k−1).

These are not rival definitions of MPC. They reorganise the same convolutional forecast and affect identification, memory and computation. State space stores a compact latent summary, useful when physical states or minimal realisations are available. A transfer matrix exposes input–output algebra. Impulse and step coefficients can be identified from plant tests without asserting that each coefficient is a physical state. The early commercial algorithms favoured finite response arrays because stable chemical processes often settle and because the resulting predictions are linear in future moves.

A truncated response carries a condition: effects beyond the stored horizon must be negligible or otherwise represented. If a slow tail remains, treating it as zero biases the prediction. Integrating processes require care because a step response never settles. A state model can represent such dynamics compactly; a finite stable-response table may not. “Model representation affects computation, not concept” therefore does not mean representation is consequence-free. The chosen form determines what is easy to estimate, which dynamics are discarded and how efficiently an online problem can be solved.

The explicit model also creates an audit boundary. A controller can distinguish its predicted output from the measured output, attribute the difference to a disturbance estimate, and show which proposed move causes which predicted change. That separability is central to the survey’s family resemblance across algorithms. A purely reactive nonlinear mapping from current sensor readings to actuators might work well, but without an identifiable model used for future prediction it is not MPC in the paper’s sense.

At sample k the controller predicts outputs across p future intervals. It chooses m future manipulated-variable changes, often with m no larger than p. A cost rewards closeness to a reference and penalises violent or expensive moves. Constraints restrict inputs, input increments or predicted outputs. Once a solution is found, the plant receives Δu(k), the first planned increment. The remaining entries are conditional intentions, not commands waiting in a queue.

At k+1 a measurement arrives. The model state or disturbance estimate is corrected, the horizon shifts, and the entire optimisation is solved again. This is how feedback enters an apparently open-loop calculation. The feedback law may not be written as one compact formula, especially when active constraints change, but measurement continually changes the optimisation’s initial condition. The sequence protects the first move from foreseeable delayed consequences; repeated replanning protects later moves from stale forecasts.

Why not apply the full sequence? Because disturbances occur, models err and priorities change. A valve may stick, feed quality may shift or an output may move faster than predicted. Executing a plan blindly would convert a feedback controller into an open-loop scheduler. Conversely, why calculate beyond one move? Because present action has memory. A move that looks harmless at k+1 can create an unavoidable violation at k+4. The horizon is an instrument for assigning future consequences to the decision made now.

Prediction horizon and move horizon are therefore distinct. A long prediction horizon sees far enough for slow effects. A shorter move horizon says that after m decisions the manipulated input will be held or follow a specified terminal convention. This reduces variables and can regularise behaviour. The survey records stability results involving sufficiently small m and sufficiently large p under particular single-input, no-delay conditions; it does not promote “longer is always stable” as a law.

The DMC objective has two principal parts. Across the prediction horizon, weighted squared differences compare predicted outputs with desired values. Across the move horizon, another weighted quadratic term penalises Δu. The first asks for tracking; the second discourages unnecessary motion, distributes action across coupled actuators and helps moderate sensitivity to model error. The weights are design choices, not probabilities produced by the plant.

The forecast separates three contributions. Known past moves continue to influence future outputs through the response model. Future candidate moves contribute linearly through shifted response coefficients. A disturbance correction accounts for the difference between the latest measured output and the model’s current prediction. Basic DMC assumes this additive discrepancy persists across the future horizon. That simple rule supplies offset correction, but it also reveals a modelling choice: a constant output disturbance is not the same as a dynamic disturbance model.

Input bounds, rate limits and predicted-output bounds can all be written as linear inequalities in the decision vector. Together with a positive quadratic objective they define a quadratic programme. An actuator diagnosed as stuck can be fixed at its current value by an equality or paired inequalities; the optimiser then asks other inputs to compensate. This is more systematic than allowing an unaware controller to demand an impossible move and clipping it afterwards.

The word “dynamic matrix” refers to the arranged response coefficients that map proposed increments to horizon outputs. It is not an additional physical law. Its columns show how one move echoes through later predictions; its block form shows cross-coupling between manipulated and controlled variables. Ill-conditioning warns that different move combinations produce nearly indistinguishable output effects, making optimisation sensitive. More coefficients do not automatically cure that geometry.

Where DMC commonly uses a step response acting on input increments, MAC can use an impulse response acting on the input itself. The survey also contrasts DMC’s move horizon m with a MAC formulation that sets the number of optimised moves equal to the prediction horizon p. These algebraic rearrangements can describe related input–output behaviour, yet they change the size and conditioning of the online calculation and the knobs available to the engineer.

The latest measured output generally differs from the model’s one-step prediction. If the complete discrepancy is carried forward as a constant additive disturbance, the controller responds rapidly and can remove steady offset, but it may also chase high-frequency measurement noise or modelling artefacts. MAC filters the disturbance estimate. A tuning parameter α decides how quickly the estimated discrepancy changes. Faster correction increases bandwidth; stronger filtering moderates the response and can improve robustness to neglected dynamics.

This filter illustrates a theme that returns in the survey’s Internal Model Control discussion: robustness should have a physical tuning interpretation. Rather than treating a long vector of arbitrary weights as mystical, an engineer can ask how quickly the loop should respond compared with uncertain plant dynamics. That does not solve robust control in general. It makes one trade-off visible and adjustable.

Inferential Control, also included in the survey’s family, emphasises estimating controlled quantities that may not be measured continuously from other available variables and a model. The label does not turn control into epistemology. It addresses instrumentation and state or output reconstruction inside an industrial loop. Across DMC, MAC and related forms, the survey looks for common requirements: explicit modelling, multivariable handling, offset behaviour, constraint treatment, computability and robustness under plant–model mismatch.

When constraints are inactive, the quadratic horizon problem has an explicit linear least-squares solution. Repeating it with the same horizons and weights produces the same mapping from signals to the current move. That mapping can be represented in Internal Model Control form: a model runs in parallel with the plant, the difference between measured and model output acts as a disturbance signal, and a stable controller Q shapes the response.

The paper gathers three useful facts. First, IMC and the familiar feedback arrangement are equivalent under a transformation between Q and the classical controller C, where the necessary inverses exist. Second, with a perfect model, important closed-loop transfer relations are affine in Q. Third, for a stable plant, internal stability can be characterised through the stability of the corresponding Q; in the perfect-model case a stable Q gives a stable closed loop. These facts make controller synthesis and robustness reasoning easier without granting perfection to the model.

The equivalence removes an inflated claim. An unconstrained MPC design cannot outperform every classical feedback controller merely because it calculates a horizon; there is a classical controller that reproduces the same closed-loop behaviour. Its practical merits can still include a natural specification, an engineer-friendly model and convenient tuning. The genuinely distinctive case begins when an inequality becomes active and the feedback mapping changes with the state.

The usual one-degree-of-freedom structure also couples reference tracking and disturbance rejection through the same error signal. A two-degree-of-freedom form uses separate Q₁ and Q₂ paths, allowing set-point response and disturbance response to be shaped differently. This matters because a factory may want gradual product-grade transitions but rapid rejection of feed disturbances. Treating both as the same error can force an unnecessary compromise.

Linear-quadratic control starts from a state model and a quadratic performance criterion. Its infinite-horizon solution produces a time-invariant gain through a Riccati equation. MPC retains a finite moving horizon, solves a possibly large least-squares problem and applies only the first move. With the appropriate augmented state and weights, their laws align. Time invariance arrives differently: LQC lets the terminal horizon recede to infinity, whereas MPC restarts an identical finite problem after each sample.

The computational implications differ. LQC recursively computes matrices of state dimension and can be economical for compact realisations. Response-model MPC may solve a larger online system whose dimension follows outputs, moves and horizons. Yet its prediction and move horizons can be easier for a process engineer to interpret than abstract matrix penalties. The move horizon m says how many independent future adjustments are permitted; increasing its penalty says how costly rapid actuator motion is.

The equivalence does not erase stochastic distinctions. A full linear-quadratic-Gaussian design uses process and measurement covariance assumptions and an estimator. Basic response-model MPC often uses a much simpler constant output-disturbance correction. When stochastic assumptions are credible, LQG can exploit them. When covariances are merely tuning dials, neither name supplies a principled advantage. Both can include the integral-like mechanism needed to reject sustained step disturbances.

Stability claims require conditions. A deadbeat inverse can be stable only when inversion does not expose non-minimum-phase zeros; even then, an aggressive sampled response can ripple badly between samples. Sufficient input penalty can stabilise certain designs. The survey also cites results where a sufficiently small move horizon and sufficiently large prediction horizon stabilise specified single-input systems without delay. Those are design results, not a proof that any constrained multivariable MPC is stable.

A squared two-norm yields the familiar quadratic programme. Its smooth convex geometry supports an explicit unconstrained solution and familiar analysis. It also spreads error: several modest deviations can be preferable to one large actuator move. In a long multivariable horizon, however, the matrices and storage required by 1989 solvers could be substantial. Computational judgement was part of controller design.

An absolute one-norm turns the problem into a linear programme after auxiliary variables represent positive and negative deviations. The arithmetic may be simpler, but the control law is piecewise linear and no single unconstrained formula summarises it. Brosilow’s proposed construction first defines an ideal unconstrained response, then minimises the one-norm distance between the feasible predicted output and that ideal. When constraints are inactive, deviation is zero, so the original linear design remains visible.

An infinity norm minimises the largest weighted deviation. One auxiliary bound sits above every absolute error, again producing a linear programme. This is meaningful when one peak temperature, pressure or quality excursion dominates risk. A two-norm can tolerate a conspicuous peak if it reduces many smaller misses; a one-norm totals absolute deviations but can still trade one peak for dispersed improvement. The infinity norm states that the worst weighted miss is the object of concern.

The survey expresses a preference for constrained algorithms whose inactive-constraint behaviour has a known linear form, citing QDMC and the Brosilow construction. That preference supports analysis and commissioning: an engineer can validate the nominal loop before adding limits. Yet the authors also acknowledge limited evidence for sweeping comparisons among norms. The right metric depends on what loss actually means, not on which optimiser is fashionable.

In a linear model, additive disturbance and model error can be separated conveniently in the output equation. In a nonlinear model their effects can depend on state and input, so that simple addition no longer captures the discrepancy. One architecture runs a nonlinear simulator in parallel with the plant and updates its correction from measured output. The quality of future predictions then depends on both the simulator and the rule used to reconcile it with reality.

The first route is nonlinear optimal control. Classical variational methods become awkward with state inequalities and online computation. One can discretise the future input, simulate the nonlinear model and solve a nonlinear programme. That makes few structural assumptions but may require repeated simulation and numerical gradients. Local minima, initialisation and computation time enter the controller’s safety case.

The second route retains the quadratic DMC calculation around a local linearisation while a nonlinear model supplies a better nominal prediction. This can be practical when the operating region changes but local response remains well behaved. It is not globally nonlinear control: the move sensitivity used by the optimiser is still local. A candidate sequence that travels far from the linearisation can invalidate its own forecast.

The third route seeks inversion of a nonlinear operator: filter the desired response and solve Pu=r for the input. Existence, uniqueness and numerical construction are central. Methods using higher derivatives amplify measurement noise and were judged unsuitable for online work; contraction and Newton-like iterations had promising simulations. The survey’s final assessment is deliberately cautious: the surface had barely been scratched. Nominal stability, constraints and plant–model mismatch remained open.

A long impulse or step response may look richly empirical, but additional coefficients are not themselves robustness. If they estimate the wrong operating regime or contain poorly determined directions, the controller can be confidently fragile. Robustness comes from how model uncertainty interacts with loop bandwidth, plant geometry and constraints. MAC and IMC filters help because they expose a direct speed-versus-uncertainty trade-off; they do not abolish that trade-off.

Ill-conditioned multivariable plants are a central linear problem. If two actuators have nearly the same effect, small model changes can rotate the supposedly best allocation dramatically. Off-diagonal filter terms may be needed, and robust optimisation can become non-convex and non-differentiable. The authors call for physically meaningful links among uncertainty, performance and tuning rather than relying only on nominal case studies.

Constraints make model error more dangerous. A predicted output can satisfy every model inequality while the real output violates a safety boundary. Tightening limits can create margin, but how much depends on uncertainty and disturbance assumptions. The optimisation can also become infeasible: no sequence satisfies every declared bound. A production controller needs a hierarchy of hard and soft constraints, a method for exposing which requirement failed, and a safe recovery action. The 1989 survey identifies infeasibility handling as unresolved rather than pretending that a generic LP or QP always returns an acceptable answer.

The nonlinear agenda is larger still: nominal stability, model uncertainty, state constraints and computational burden interact. These limitations preserve the paper’s central achievement. MPC supplies a disciplined interface among a model, a forecast, an objective and explicit constraints. It makes assumptions inspectable. But repeated optimisation is a control mechanism, not evidence of comprehension. A controller can choose an excellent first move without representing reasons, possessing autonomous ends or having any experience at all.