Page 783 - Mechanical Engineers' Handbook (Volume 2)

774 Control System Design Using State-Space Methods

is then defined:
An augmented state x a

x
x z x (70)
a
z u
Using Parseval’s theorem, the index of performance, Eq. (61), is transformed to the time
domain to yield a standard LQR problem formulation with constant weighting matrices:

T
T
T
J (xR x 2xR u uR u) dt (71)
1aa
a
a
12a
2a
0
The optimal-control law is obtained by solving the corresponding algebraic matrix Riccati
equation:
u (Kx Kz Kz ) (72)
xx
u u
The states z and z , are dynamically related to x and u, respectively. Hence, the optimal-
u
x
control law has dynamic compensators in addition to linear instantaneous state-variable
feedback (Fig. 8). If the state x is not completely measurable, state estimation is required as
described in Section 5.
The utility of this design method is that it establishes a clear link between features of
the weighting matrices R ( j ) and R ( j ) and the resulting controllers. Gupta has shown
27
1
2
that the compensator poles and zeros are the same as poles and zeros of the transfer functions
P (s) and P (s). For example, if R ( j ) is singular at 0, integral control results. If R (j )
1
1
2
1
is singular at any other frequency , the controller has a notch filter at that frequency. This
1
would be desirable if the controlled system has a known resonant frequency at and we
1
wish to minimize the excitation of the resonance by disturbances. Finally, if R ( j ) is chosen
2
to increase with frequency , the optimal-control law would have reduced control action at
high frequencies. As indicated previously, this is a desirable feature if the controller is to
have good noise suppression and robustness to model errors.
4.4 Robust Servomechanism Control
Davison and Ferguson 21 have proposed a controller structure and formulated a design pro-
cedure for the robust control of servomechanisms. Robustness is defined here to imply as-
ymptotic stability of the closed-loop system and asymptotic tracking of the desired trajectory
for all initial conditions of the controller used and for all variations in the system model
parameters that do not cause the controlled system to become unstable. The system equations
are the time-invariant versions of Eqs. (45) and (46). The disturbance inputs w(t) are modeled
by time-invariant versions of Eqs. (47) and (48) with no provision for either state-dependent
disturbances [B (t) 0 D (t)] or impulsive inputs [ (t) 0]. The desired trajectory y (t)

r

is described by time-invariant versions of Eqs. (50) and (51).
Under certain specified conditions, 21 the robust servomechanism problem is assured of
a solution. The resulting controller structure consists of a servocompensator and stabilizing
compensator (Fig. 9), and the robust control input is given by
u K K
(73)

where and
are the outputs of the servocompensator and stabilizing compensator, respec-
tively, and K and K are constant-gain matrices. The servocompensator is a dynamic con-

troller with the trajectory error as input and its form and parameters are determined from
the state-space models of the system and the disturbance and trajectory inputs. The servo-