Designing MPC controllers by reverse-engineering existing LTI controllers

This technical report presents a method for designing a constrained output-feedback model predictive controller (MPC) that behaves in the same way as an existing baseline stabilising linear time invariant output-feedback controller when constraints are inactive. The baseline controller is cast into an observer-compensator form and an inverse-optimal cost function is used as the basis of the MPC controller. The available degrees of design freedom are explored, and some guidelines provided for the selection of an appropriate observer-compensator realisation that will best allow exploitation of the constraint-handling and redundancy management capabilities of MPC. Consideration is given to output setpoint tracking, and the method is demonstrated with three different multivariable plants of varying complexity.

💡 Research Summary

The paper presents a systematic procedure for turning an existing stabilising linear‑time‑invariant (LTI) output‑feedback controller into a constrained model predictive controller (MPC) that reproduces the exact closed‑loop behaviour of the original controller whenever constraints are inactive. The motivation is clear: many industrial plants already run a well‑tuned LTI controller, and the engineer would like to add the powerful constraint‑handling capabilities of MPC without redesigning the whole control law.

The core idea is to rewrite the given LTI controller (K_0(z)) in an observer‑compensator (observer‑based) form. Two discrete‑time observer structures are discussed: a filter‑form observer, which uses the current measurement to update the state estimate, and a predictor‑form observer, which relies only on the previous measurement, thereby freeing the computation time needed for the MPC optimisation. The paper shows that the predictor form is preferable for digital implementation because the optimisation can start as soon as the new sample arrives, but it requires the controller to be strictly proper (no direct feed‑through). If the original controller has a non‑zero (D_K) term, the authors propose loop‑shifting or insertion of a low‑pass filter to make the controller strictly proper before the observer‑based transformation.

Once the observer‑compensator realisation is obtained, the state‑feedback gain (K_c) and observer gain (K_f) are extracted. These gains are then used to construct an “inverse‑optimal” quadratic cost function for the MPC. By choosing weighting matrices (Q) and (R) such that the infinite‑horizon LQR solution coincides with (K_c), the unconstrained MPC reproduces exactly the dynamics of the original controller. The authors reference classic results (Kalman 1964; Kreindler & Jameson 1972) that any static gain can be reproduced by a quadratic cost with appropriate cross‑terms, and they note more recent LMI‑based techniques that allow the weighting to vary over the prediction horizon (Di‑Cairano & Bemporad 2009).

A crucial degree of freedom in the observer‑based realisation is the matrix (T\in\mathbb{R}^{n_K\times n}) that links the plant state to the controller state. (T) must satisfy a non‑symmetric Riccati equation (-T(A+BD_KC)-TBC_KT+BK_C+AKT=0). The paper provides two design pathways: (1) compute a minimal‑order (T) directly from the given controller realisation, and (2) deliberately shape (T) to achieve desired observer dynamics (e.g., faster convergence, reduced sensitivity). The authors discuss how the choice of (T) influences the placement of observer poles, the level of cross‑coupling between channels in MIMO systems, and the numerical conditioning of the resulting MPC problem.

Reference‑tracking is handled by reverse‑engineering a feed‑forward reference model that, when combined with the observer‑based state feedback, yields the same static gain as the original controller. This allows the MPC to track set‑points without altering the unconstrained behaviour. The paper also warns about unwanted cross‑coupling that can arise when the observer‑compensator is not carefully designed; it proposes using structured (T) matrices or additional decoupling filters to mitigate this issue.

Three case studies illustrate the methodology:

1. Spacecraft attitude control with redundant actuators – The baseline controller uses torque pairs for fault tolerance. The observer‑compensator form enables the MPC to respect torque limits and to reallocate control effort automatically when an actuator fails, while still matching the original closed‑loop response.

2. Inverted pendulum on a cart – An unstable plant with output constraints (e.g., actuator saturation). The reverse‑engineered MPC respects the saturation limits, yet reproduces the fast, aggressive response of the original LTI controller when the pendulum is near the upright equilibrium.

3. Large airliner MIMO control – A highly cross‑coupled aerodynamic model. By selecting an appropriate (T) that reduces inter‑channel coupling, the MPC achieves the same performance as the baseline multivariable controller while handling actuator rate limits and flight‑envelope constraints.

The authors conclude that the presented reverse‑engineering approach provides a practical “plug‑and‑play” pathway for upgrading legacy LTI controllers to constrained MPC. It preserves all the design effort already invested in the original controller, supplies clear guidelines for observer‑compensator realisation, and offers systematic tools for handling reference tracking, cross‑coupling, and constraint activation. This makes the method especially attractive for industrial practitioners who need to introduce constraint handling without discarding proven control designs.