Assessment and Future Directions of Nonlinear Model Predictive ...

Nonlinear Model Predictive Control: A Passivity-Based Approach 159can therefore not be used as a control Lyapunov function. Another possibilitybased on [6, 14] is to choose ϕ(x(t + T )) = ∫ ∞t+T (q(x(τ)) + hT (x(τ))h(x(τ)))dτ,where u = −h(x) is the stabilizing feedback of the system (1) and x is thecorresponding closed loop system trajectory.5 ExampleIn this section, the passivity-based nonlinear model predictive scheme is appliedto control a quadruple tank system. The dynamical model of the quadruple tanksystems [7] is given byẋ = f(x)+g(x)u,with⎡f(x) = ⎢⎣√ √ ⎤A 2gx11+ a3A 2gx3√ 1√A 2gx22+ a4A 2gx42 √ ⎥− a3A 2gx33⎦ ,√− a4A 2gx44− a1− a2⎡g(x) =⎢⎣γ 1A 10γ02A 20(1−γ 2)A 3(1−γ 1)A 10and x =[x 1 ,x 2 ,x 3 ,x 4 ] T , u =[u 1 ,u 2 ] T . The variables x i , i =1, ..., 4representthe water levels of the tanks, u i ,i=1, 2 the control inputs, A i ,i=1, ..., 4thecross-sections of the tanks, a i ,i=1, ..., 4 the cross-sections of the outlet holes,γ i =0.4, i=1, 2 positive constants, and g = 981 cmsthe gravitational constant.2The parameter values of A i and a i are taken from the laboratory experiment[4] and are given in Table 1. In order to control the quadruple tank system viathe passivity-based nonlinear model predictive control, one has to search for afictitious passive output. By using the storage function S(x) = 1 2 x2 3 + 1 2 x2 4 ,oneobtains for x 3 ,x 4 ≥ 0 [the levels of the tanks cannot be negative].⎤⎥⎦ ,Table 1. Parameter values of the systemA ia ii =150.3 cm 2 0.2 cm 2i =250.3 cm 2 0.2 cm 2i =328.3 cm 2 0.1 cm 2i =428.3 cm 2 0.1 cm 2