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Handling plant-model mismatch in Koopman Lyapunov-based model predictive control via offset-free control framework
Sang Hwan Son, Abhinav Narasingam, Joseph Sang-Il Kwon
TL;DR
Finite-dimensional approximations to the Koopman operator leave plant-model mismatch in KLMPC. The paper adds disturbance estimation and Lyapunov-constrained control, and simulations show zero steady-state offset where nominal KLMPC has considerable offsets.
Problem
Finite-dimensional EDMD approximations cannot fully represent the Koopman operator, so plant-model mismatch is inherent in KLMPC.
Source: Conclusions · p. 11Method
The framework augments the EDMD lifted model with a disturbance estimate, computes disturbance-adjusted targets, and uses a Lyapunov-constrained optimal control problem.
Source: Conclusions · p. 11Results
The proposed scheme showed zero steady-state offset in simulation, while nominal KLMPC showed considerable set-point offsets in controlled variables c and T.
Source: §3.2 · p. 10Takeaways & Limitations
The simulation supports offset-free KLMPC as a direction for addressing mismatch in Koopman operator-based predictive control.
Source: Conclusions · p. 11The zero-offset analysis assumes set-points are far from process bounds, with process constraints and the first Lyapunov constraint inactive at the optimum.
Source: §2.2 · p. 7
Abstract
from arXiv · showhide
Koopman operator theory enables a global linear representation of a given nonlinear dynamical system by transforming the nonlinear dynamics into a higher dimensional observable function space where the evolution of observable functions is governed by an infinite-dimensional linear operator. For practical application of Koopman operator theory, various data-driven methods have been developed to derive lifted state-space models via approximation to the Koopman operator. Based on approximate models, several Koopman-based model predictive control (KMPC) schemes have been proposed. However, since a finite-dimensional approximation to the infinite-dimensional Koopman operator cannot fully represent a nonlinear dynamical system, plant-model mismatch inherently exists in these KMPC schemes and negatively influences the performance of control systems. In this work, we present offset-free Koopman Lyapunov-based model predictive control (KLMPC) framework that addresses the inherent plant-model mismatch in KMPC schemes using an offset-free control framework based on a disturbance estimator approach and ensures feasibility and stability of the control system by applying Lyapunov constraints to the optimal control problem. The zero steady-state offset condition of the developed framework is mathematically examined. The effectiveness of the developed framework is also demonstrated by comparing the closed-loop results of the proposed offset-free KLMPC and the nominal KLMPC.
1. Preliminaries
The preliminaries construct a finite-dimensional lifted linear predictor from data using EDMD, then formulate KLMPC with Lyapunov constraints to support feasibility and stability.
1.1. Data-driven linear model identification of nonlinear systems
- EDMD provides a finite-dimensional data-driven approximation to the Koopman operator for practical lifted-space modeling.Source: §1.1 · p. 3
- A linear lifted-state predictor is obtained by minimizing over A and B after discarding components associated with future inputs.Source: §1.1 · p. 3
- The output matrix C is fitted as a least-squares projection of y onto the observable-function span, yielding a linear output model.Source: §1.1 · p. 3
1.2. KLMPC design
- KLMPC combines EDMD with Lyapunov-based MPC and formulates its finite-horizon problem as a convex quadratic program with Lyapunov constraints.Source: §1.2 · pp. 3–4
- Lyapunov constraints keep future states within a stability region and ensure the MPC-induced next Lyapunov value is no greater than that from a stabilizing control law.Source: §1.2 · p. 4
- EDMD makes the Lyapunov constraints linear by including the Lyapunov function in the observable library as a lifted state.Source: §1.2 · p. 4
2. Offset-free Koopman Lyapunov-based MPC
This section presents offset-free KLMPC to address plant-model mismatch by augmenting the lifted model with disturbance estimation and updating its target and Lyapunov constraints. It derives a condition for, and establishes, zero steady-state offset.
2.1. Offset-free KLMPC framework
- A disturbance-augmented lifted model and a stable estimator of lifted state and disturbance are used to address plant-model mismatch.The estimator gains Lz and Ld are chosen to make the estimator stable.Source: §2.1 · pp. 4–5
- The target objective corrects the lifted state directly, and the Lyapunov function, constraints, and stabilizing law are updated as the target and disturbance estimates change.The direct lifted-state correction replaces output correction because output correction has insignificant influence on the higher-dimensional lifted-state target.Source: §2.1 · pp. 5–6
- The stabilizing control law achieves perfect set-point tracking at steady state with the specified gains.The proof concludes that the controlled system tracks the set-point perfectly at steady state.Source: §2.1 · p. 6
- The finite-horizon optimization includes updated Lyapunov constraints and yields an input that drives the original system to steady state with zero offset despite plant-model mismatch.The constraints are formulated from the updated Lyapunov function and stabilizing control law.Source: §2.1 · p. 6
2.2. Zero steady-state offset condition
- Zero steady-state offset is guaranteed when estimator gains Lz and Ld satisfy the stated null-space condition.Theorem 5 gives this sufficient condition for the proposed offset-free KLMPC.Source: §2.2 · p. 8
- The offset-free KLMPC system yields zero offset at steady state.This follows from equation (80).Source: §2.2 · p. 8
3. Numerical example
The CSTR simulation compares offset-free and nominal KLMPC after EDMD identifies a lifted model from virtual-plant data. The lifted model retains prediction error, while offset-free KLMPC achieves zero steady-state offset for the controlled outputs despite plant-model mismatch.
3. Numerical example
- The simulation uses a virtual CSTR with unavailable detailed dynamics, tracking outlet concentration c and reactor temperature T through jacket temperature Tc and outlet flow rate F.The measured h value is supplied to the disturbance estimator but is not controlled.Source: §3 · pp. 8–9, §3.1 · p. 9, §3.2 · p. 10
3.1. Data-driven model identification through EDMD
- The lifted model is identified from 500,000 data sets collected across 1,000 virtual-CSTR operation trajectories, each simulated for 500 minutes with one-minute sampling.The observable library includes the state and a Lyapunov function, allowing Lyapunov constraints to be expressed linearly.Source: §3.1 · p. 9
- EDMD prediction errors remain during 15–30 and 30–45 minutes, attributed to finite-dimensional approximation of the Koopman operator.The model predicts the virtual CSTR behavior for most of the validation trajectory.Source: §3.1 · pp. 9–10, Figure 2 · p. 9
- The output NRMSE values are [0.1319, 0.0969, 0.0142], and h is predicted more accurately than c and T.The paper attributes h's lower error to its relatively simple, linear dynamics.Source: §3.1 · p. 10
3.2. Closed-loop simulation results
- The closed-loop controller uses a disturbance-augmented lifted model, estimator, target problem, and optimal control problem with prediction horizon N = 10.Output constraints are 0.81 ≤ c ≤ 0.92, 320 ≤ T ≤ 330, and 0.4 ≤ h ≤ 1.2.Source: §3.2 · p. 10
- Offset-free KLMPC achieves zero steady-state offset in c and T, whereas nominal KLMPC shows considerable set-point offsets under plant-model mismatch.The set point for c changes every 25 minutes, while T is held at 324.5 K.Source: §3.2 · p. 10, Figure 3 · p. 10
- Disturbance estimates take time to settle after set-point changes, but converge to proper values at steady state and support offset-free tracking.The paper notes temporary deviations near 50, 75, and 150 minutes.Source: §3.2 · pp. 10–11, Figure 4 · p. 10
Conclusions
The proposed offset-free KLMPC addresses mismatch inherent in EDMD-based finite-dimensional Koopman models while retaining Lyapunov constraints for feasibility and stability. It updates targets using disturbance estimates and is reported effective against nominal KLMPC in numerical simulations.
Conclusions
- An EDMD-derived lifted model is augmented with a disturbance model; estimates from measured prediction error update lifted-state and input targets before Lyapunov-constrained optimal control.The optimal control problem tracks the set-point while compensating for mismatch and ensuring feasibility and stability.Source: Conclusions · p. 11
- Because the steady-state target changes with the disturbance estimate, the framework develops updated Lyapunov-function and stabilizing-control-law formulations and mathematically analyzes zero steady-state offset.Source: Conclusions · p. 11
- Numerical simulations show the proposed offset-free KLMPC is effective at addressing plant-model mismatch compared with nominal KLMPC.The paper notes that mismatch-related prediction inaccuracy can be exacerbated when states are lifted to the observable space.Source: Conclusions · p. 11
Appendix A. Derivation of (62)
The appendix reformulates the optimization using predicted lifted states, then derives optimality relations through KKT conditions and substitutions.
Appendix A. Derivation of (62)
- The reformulation expresses predicted lifted states explicitly in the objective and inequality constraints, with the input sequence denoted by U.Source: Appendix A · p. 11
- The derivation applies first-order KKT conditions to P̂z∞ and rearranges and substitutes relations to obtain further expressions.The passage identifies v as the dual variable and i as the constraint index.Source: Appendix A · p. 11
- When the Lyapunov-function constraint is active at the optimum, complementary slackness yields the stated condition before subsequent substitutions and the final input expression.Source: Appendix A · pp. 11–12