NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

3.3 Passivity-Based Control
Certain physical systems can be pro tably studied by examining the prop-
erties of their potential energy functions, and by noticing the natural damping and
dissipation present. Passivity-Based Control (PBC) algorithms have yielded simple,
yet robust control laws to a variety of such systems by shaping the energy and dis-
sipation functions. The exible beam system is an example of an under-actuated
Euler-Lagrange system, a system that is completely modeled by the EL equations.
The stability of the equilibrium states of EL systems is solely dependent on their
potential energy functions, and if enough damping is present these equilibria will be
asymptotically stable. Passivity-based control designs seek to exploit these facts to
robustly stabilize complicated nonlinear systems. In particular, the potential energy
function of the closed-loop system is shaped by the controller, and suitable damping
is injected into the system via the control law to achieve the desired stability and
performance objectives. There are several fundamental properties of EL systems that
make this energy-shaping and damping injection practical. First, EL systems are
completely described by their EL parameters: TpVpFpM where the EL equation is
written as follows:
d
dt (@Lp(qp qp) _
) ;
@qp _
@Lp(qp qp) _
@qp
= Mup ; @Fp( qp) _
: (3.11)
@ _
In this equation Lp(qp qp) _ = Tp(qp qp) _ ; Vp(qp qp), _ where Tp is the plant's kinetic
energy, Vp is the plant's potential energy, Fp( qp) _ is the Rayleigh dissipation function
for the plant, and M is a vector of ones and zeros that indicates which of the states
are directly actuated by the plant inputs, up. We can now construct a controller as
another EL system as follows:
d
dt (@Tc(qc qc) _
) ;
@qc _
@Tc(qc qc) _
+
@qc
@Vc(qcqp)
+
@qc
@Fp( qp) _
=0: (3.12)
@qp _
Here, the potential energy function of the control block, Vc, is a function of the plant
states qp and the control state qc, but its not a function of the generalized control
velocity qc. _ This equation can be combined with the EL equation for the plant into
19
qp