Economic MPC of nonlinear systems with nonmonotonic Lyapunov functions and its application to HVAC control

Abstract

Summary This paper proposes a Lyapunov‐based economic model predictive control (MPC) scheme for nonlinear systems with nonmonotonic Lyapunov functions. Relaxed Lyapunov‐based constraints are used in the MPC formulation to improve the economic performance. These constraints will enforce a Lyapunov decrease after every few steps. Recursive feasibility and asymptotical convergence to the steady state can be achieved using Lyapunov‐like stability analysis. The proposed economic MPC can be applied to minimize energy consumption in heating ventilation and air conditioning control of commercial buildings. The Lyapunov‐based constraints in the online MPC problem enable the tracking of the desired set‐point temperature. The performance is demonstrated by a virtual building composed of 2 adjacent zones.

Lyapunov‐based EMPC of constrained nonlinear systems with changing economic costs

“…Based on the construction of the local controllers in the set χ, it is obtained from Lemma 1 that there exists at least a parameter μ i ∈(0,D 1,i ] × (0,D 2,i ] or μ j ∈(0,D 1,j ] × (0, D 2,j ] such that the constraints in (14) or (15) are satisfied, respectively. This implies that the optimization problem (14) or ( 15) is iteratively feasible for any x (t k )∈χ.…”

“…Given the sampling interval T s > 0 and prediction step N > 0; let p(t −1 ) be associated with the economic cost at initial time t 0 = 0 and k = 0. (14) to get the optimal solution μ i * (t k ) and go to…”

“…In the double-layer structure, the Lyapunov-based EMPC in [12] is modified such that several (economic) MPC controllers use different prediction horizons to stabilize the closed-loop systems at the best economical equilibrium point [13]. More recently, some Lyapunov-based contractive functions have been imposed on the EMPC optimization problems to guarantee stability of the best economic equilibrium point [3,[14][15][16]. Note that the contractive functions are calculated online according to the EMPC solutions at adjacent sampling times.…”

Relaxations of Conditions of Lyapunov Functions

“…In [18], the authors introduced the Almost LF concept and related the volume of the region where the LF is increased with a region where the trajectories converged, and they improved it in [19]. Nondecreasing LFs were also used for switched or hybrid systems in [20], for linear time-varying systems in [21], for arbitrary discrete time systems in [22] and for model predictive control in [23]. To our best knowledge, there does not exists any work that relaxes both monotonicity and positivity of LFs.…”

Tube‐based robust economic model predictive control with practical and relaxed stability guarantees and its application to smart grid

“…Robustness has been paid attention only by exploiting the inherent robustness of a tracking MPC. In Reference 14, economic performance has been improved using relaxed Lyapunov‐based constraints with application to nonlinear systems. The constraints let Lyapunov function decrease after every few samples.…”