(434d) An Offset-Free Control Framework for Koopman Lyapunov-Based Model Predictive Control

Sang Hwan Son^1,2, Abhinav Narasingam ^1,2, and Joseph Sang-Il Kwon^1,2

1) Artie McFerrin Department of Chemical Engineering, Texas A&M University College Station, TX

2) Texas A&M Energy Institute, Texas A&M University, College Station, TX

Koopman and Von Neumann showed that a nonlinear system, when lifted to an infinite-dimensional function space, can be represented by a linear operator, called Koopman operator, which describes the temporal evolution of the lifted states (called observables) [1,2]. Recently, various data-driven schemes have been developed to derive finite-dimensional approximations of the (infinite-dimensional) Koopman operator for nonlinear dynamical systems which has sparked its practical implementation [3-5]. Furthermore, this Koopman formalism enables a linear representation of any nonlinear constraints on control inputs and states within an optimal control problem [6]. Based on these valuable features, Narasingam and Kwon proposed a novel Koopman Lyapunov-based model predictive control (KLMPC) for stabilization of nonlinear systems [7]. The main advantage of KLMPC is that the nonlinearity of systems and Lyapunov constraints can be handled in a linear manner using the extended dynamic mode decomposition (EDMD) proposed in [8]. Additionally, a theoretical analysis of the feasibility and a criterion that guarantees closed-loop stability of the original system in the Lyapunov sense based on the stability of the Koopman system has also been established in [9].

However, a finite-dimensional approximation through data-driven numerical methods (e.g. EDMD) cannot completely represent the infinite-dimensional Koopman operator. Therefore, any data-driven model based on the Koopman formalism always has prediction error due to the inherent model-plant mismatch. Consequently, the closed-loop performance of a controller designed with an approximate Koopman model is considerably dependent on its ability to handle this mismatch. One way to effectively handle this discrepancy is using the offset-free MPC strategy proposed in [10-12]. Therefore, in this study, we present a novel control scheme that integrates the framework of offset-free MPC to KLMPC to address model-plant mismatch and disturbance during process operation. This is especially important because the presence of any disturbance can exacerbate the prediction error when we perform the nonlinear transformation of the states to the Koopman space. To tackle this, first, we introduce an additional model to account for disturbance dynamics and model-plant mismatch and augment it with the Koopman model of system state. Based on this integrated model, we then design a disturbance estimator to estimate the disturbance-augmented state from the measurement, and a target calculator to derive the target state and input which can achieve offset-free reference tracking under the effect of disturbance. Then, the derived target state and input are used within the objective function of the optimal control problem of KLMPC to drive the system state to the target state. Additionally, as the target state and input are updated according to the estimated disturbance value, the Lyapunov function and stabilizing control law used in constraints within the KLMPC formulation are also continuously updated. Therefore, the proposed controller can achieve the offset-free reference tracking under disturbance and model-plant mismatch existence while effectively addressing the nonlinearity of the system and Lyapunov constraints with the Koopman model. Furthermore, the closed-loop stability of the system can be ensured through the stabilizing property of Lyapunov-based control law, thus leveraging the best of offset-free and KLMPC based control schemes. The superiority of the proposed method in handling the model-plant mismatch and the presence of disturbance is successfully demonstrated via numerical examples.

REFERENCES

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