NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

a total closed-loop expression, provided that the plant input, up, is de ned according
to the following interconnection constraint:
up = ; @Vc(qcqp)
: (3.13)
@qp
Thus, the second and most important factabout EL systems is that the closed-loop
system is itself an EL system, and the closed-loop EL parameters (TclVclFcl) are
simply the sum of the parameters of the plant and the control. In other words,
we can shape the energy and dissipation of the closed loop system as desired by
choosing the dynamics of the control in the correct manner: it must comply with
the aforementioned interconnection constraint. In [10] this idea is further re ned
by showing that the injected dissipation function in the control need not necessarily
be a function of the derivative of the plant variables. Rather, a dynamic system in
the control is su cient, under certain passivity conditions on the original plant, to
guarantee closed loop asymptotic stability. In addition, Ortega et.al. [10] add the
constraint thatthere is limited actuator power available, or in other words
up umax: (3.14)
Following the design outlined in [15], a suitable Rayleigh dissipation function Fc was
chosen:
Fc = 1
2ab _
qc
(3.15)
where a and b are parameters that scale the injected damping. Also, the potential
energy function of the control was constructed so as to ensure the stability of the zero
equilibrium point. A su cient condition is that Vcl = Vp + Vc have a global minimum
at zero. Since the potential energy of the plant is a quadratic function of the states,
we chose Vc as
Vc(q3qc) = 1
Z Z qc+bq3
q3
k1 tanh(s)ds + k2 tanh(s)ds: (3.16)
b 0
0
so that it was the sum of two strictly increasing integrals that are minimized at the
zero state. Here tanh(s) was chosen as a saturation function to model the actuator
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