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4.6 Optimal Outer-Loop Design 247

Note particularly that the relative magnitudes of x(t) and u(t) in the closed-
loop system can be traded off. Indeed, if R is relatively larger than Q p and Q v ,
the control effort in the PI (4.6.4) is weighted more heavily that the state.
Then the optimal control will attempt to keep u(t) smaller by selecting smaller
control gains; thus the response time will increase. On the other hand, selecting
a smaller R will increase the PD gains and make the error vanish more quickly.
If Q p , Q ν , and R are diagonal, so then are the PD gains K p , K v . The LQ
approach with nondiagonal Q p , Q v , and R affords the possibility of outer
feedback loops that are coupled between the joints, which can sometimes
improve performance. Another important feature of LQ design is the
guaranteed robustness mentioned in the theorem. This can be very useful in
approximate computed-torque design where

(4.6.16)

and and can be simplified versions of M(q) and N(q, ). The performance
of such a controller with an LQ-design outer loop can be expected to surpass
that of a controller designed using arbitrary choices for K p and K v . This robust
aspect of LQ design is explored in the problems.
It is important to note that this LQ design results in minimum closed-loop
energy in terms of e(t), (t), and u(t). However, the actual control input into
the robot arm is

(4.6.17)

Although the energy in τ(t) is not minimized using this approach, we can use
some norm inequalities to write

(4.6.18)

so that keeping small ||u(t)|| might be expected to make smaller. A more
formal statement can be made taking into account the bounds on ||M(q)|| and
given in Table 3.3.1.
Since the energy in is not formally minimized in this approach, it is
considered as a suboptimal approach with respect to the actual arm dynamics,
although with respect to the error system and u(t) it is optimal. An optimal
control approach that weights e(t) τ(t) and in the PI is given in [Johansson
1990].
We have derived an LQ controller using a computed-torque (i.e., feedback
linearization) approach. An alternative approach that yields the same Riccati-
equation-based design is to employ the full nonlinear arm dynamics and find
an approximate (i.e., time-invariant) solution to the Hamilton-Jacobi-Bellman
equation [Luo and Saridis 1985]; [Luo et al. 1986].

Copyright © 2004 by Marcel Dekker, Inc.