Optimally designed Lyapunov–Krasovskii terminal costs for robust stable–feasible model predictive control of uncertain time-delay nonlinear dynamical systems

Abstract

This article details a new Model Predictive Control algorithm ensuring robust stability and control feasibility for uncertain nonlinear multi-input multi-output dynamical systems considering uncertain time-delay effects. The proposed control algorithm is based on construction of a Lyapunov–Krasovskii functional as terminal cost. Incorporation of this terminal cost into the Model Predictive Control optimization problem and calculation of the associated admissible set result in robust feasibility and robust stability of closed-loop system in presence of uncertain time-delay effects and bounded disturbance signals. The Lyapunov–Krasovskii functional term is constructed with respect to predicted sliding functions over the prediction horizon and considers the effects of dynamical variations over the prediction horizon in generation of control inputs. As dynamical variations are investigated in a sample-to-sample basis, feasible sliding regions are updated at each sample as well. Finally, based on expression of sliding functions as a combination of dynamical variations and input-based terms, required control inputs are calculated in the admissible bound by the optimization algorithm. Construction of control scheme on this basis permits straightforward calculation of robust stability and feasibility conditions for a general class of uncertain nonlinear system in finite prediction horizon whereas in the previous works, often-restrictive conditions were considered for the investigated dynamical systems. Numerical illustrations indicate precision and efficiency of control algorithm and improved stability and convergence rate for multivariable nonlinear dynamical systems considering uncertain time-delay effects. Finally, hardware-in-the-loop implementation indicates applicability of the proposed scheme in real-time control applications particularly in case appropriate compromises between optimality and calculation speed are considered.

Active predictive vibration suppression algorithm for structural stability and tracking control of nonlinear multivariable continuum‐mechanics mobile systems

“…Depending on the nature of dynamical expressions, the process of calculating supremum and infimum may be conducted analytically, semi‐analytically or even offline. Such approaches would significantly reduce the required computational burden 32,33 Remark In conventional nonlinear MPC techniques, calculation of admissible terminal cost often requires employing intensive optimization solvers as simultaneously obtaining admissible terminal cost alongside with other decision variables is often analytically impossible 32,34 .…”

“…Alternatively, suboptimal inputs may be assigned by selecting an input in the admissible bound satisfying stability conditions expressed in Theorem 2. In case the appropriate trade‐off between optimality and intensity of numerical calculations can be maintained, MPC schemes are implementable in real‐time even using inexpensive software 32,33 Remark The robust stability condition Equation ) is a second order algebraic equation as a function of α u .…”