Questions: Model Predictive Control (Advanced)

Question 1

An MPC controller with a 20-step prediction horizon predicts that in step 19, an actuator will saturate. How does MPC respond, and how does this differ from a PID controller's response?

• A. MPC ignores the saturation prediction because it only cares about the current step; PID ignores saturation until it actually occurs
• B. MPC backspreads the constraint: it reduces current control effort to avoid saturation 19 steps ahead, planning an alternative path now; PID continues applying maximum effort until saturation occurs, then windup occurs
• C. Both MPC and PID respond identically — saturation handling is independent of control algorithm
• D. MPC increases control effort to overcome anticipated saturation; PID holds output constant

Answer: (B)

MPC solves an optimization problem over the horizon, respecting all predicted constraints. If the optimizer predicts saturation at step 19, it adjusts the entire control trajectory now to avoid it — a fundamentally different approach from PID. PID is reactive: it applies control proportional to current error and only responds to saturation as it occurs, risking integrator windup. This lookahead-and-backspread capability is why MPC excels for systems with delay or where saturation is frequent. The trade-off is computational cost: the optimization must be solved every time step.

Question 2

Increasing the prediction horizon from 15 to 50 steps in your MPC formulation sometimes decreases closed-loop performance rather than improving it. Why?

• A. Longer horizons introduce future uncertainties that corrupt the control calculation
• B. Numerical ill-conditioning and round-off error accumulate; the QP solver becomes less accurate; worse, the objective function becomes less sensitive to near-term control moves, prioritizing far-future performance
• C. MPC is designed only for short horizons by mathematical principle
• D. The plant dynamics are undefined beyond 15 steps, making longer horizons meaningless

Answer: (B)

The QP Hessian matrix grows with horizon length, and for long horizons or ill-conditioned dynamics, the matrix becomes poorly conditioned — small numerical errors are amplified. More subtly, minimizing cost over 50 steps means the near-term control moves (steps 0–5) have diffuse influence over the cost, while far-future moves dominate — the solver may find a solution that defers control effort, degrading near-term performance. This is why robust MPC often includes a terminal penalty — an additional cost term enforcing stability behavior at the horizon end — which effectively decouples horizon length from stability and allows shorter horizons without sacrificing performance.

Question 3

An MPC formulation uses a finite prediction horizon but claims asymptotic stability. Under what conditions is this guaranteed?

• T. True
• F. False

Answer: True

MPC with a finite horizon can guarantee asymptotic stability if the formulation includes a terminal equality constraint (state at the end of the horizon is in a small neighborhood of the origin) or a terminal cost function that dominates beyond the horizon, combined with conditions on the system (e.g., controllability, observability). These are non-trivial conditions, but under them, finite-horizon MPC stability is as rigorous as LQR with infinite horizon. Most practical MPC implementations omit these conditions for simplicity, accepting only local stability and hoping the horizon is long enough.

Question 4

Nonlinear MPC requires solving a nonlinear program at every time step, which is computationally expensive. Therefore, NMPC is not suitable for real-time control of fast systems.

• T. True
• F. False

Answer: False

While NMPC is computationally more demanding than linear MPC, advances in optimization algorithms (Convex-Concave Procedure, SQP with warm-starting from the previous time step, explicit NMPC precomputation) have made real-time NMPC feasible for systems with time constants in the 10–100 millisecond range. The key is algorithm selection: sequential convex approximation or sensitivity-based SQP warm-starting can solve the optimization in milliseconds on modern hardware. For very fast systems (kilohertz control loops), explicit MPC (precomputed piecewise-affine or polynomial control laws) or linearized MPC may be necessary, but NMPC is increasingly viable.

Question 5

Explain the receding-horizon principle and why MPC stability depends on both the prediction horizon and the terminal cost or terminal constraint in the optimization problem.

Model answer: The receding-horizon principle: at time k, MPC solves an optimization over k:k+N (a window of N future steps), implements the first optimal control move, then shifts the window to k+1:k+N+1 and resolves. This 'rolling' horizon allows adaptation to disturbances and nonlinearities that emerge, unlike open-loop optimization. However, optimizing only over a finite horizon creates a subtle stability issue: at step k+N−1, the optimal control might steer the state toward a large cost at step k+N, then abandon it (since step k+N is outside the horizon). Without a terminal cost or terminal constraint, the optimizer has no incentive to leave the system in a stabilizable state beyond the horizon — the system can diverge after the prediction window ends. The terminal constraint (forcing the state into a small region, typically near the origin) or terminal cost (penalizing the predicted state at the horizon end as if it were the start of an infinite-horizon problem) closes this loophole, ensuring that the policy computed at each time step is stabilizing over the full infinite time.

This is why seemingly stable MPC implementations sometimes fail: when the horizon is not long enough relative to system dynamics, or when the terminal condition is too loose, the finite-horizon optimization can produce a control sequence that appears optimal locally but is globally destabilizing. Modern MPC software computes the required minimum horizon or chooses terminal conditions automatically, but the underlying issue is fundamental to finite-horizon optimization.