We describe several families of controller design problems that can be solved
rapidly and exactly using standard methods.
12.1
Linear Quadratic Regulator
The linear quadratic regulator (LQR) from optimal control theory can be used to
solve a family of regulator design problems in which the state is accessible and regu-
lation and actuator e ort are each measured by mean-square deviation. A stochastic
formulation of the LQR problem is convenient for us a more usual formulation is
as an optimal control problem (see the Notes and References at the end of this
chapter). The system is described by
_x = Ax + Bu + w
where w is a zero-mean white noise, i.e., w has power spectral density matrix
Sw(!) = I for all !. The state x is available to the controller, so y = x in our
framework.
The LQR cost function is the sum of the steady-state mean-square weighted
state x, and the steady-state mean-square weighted actuator signal u:
Jlqr = lim ;
t
x(t)TQx(t) + u(t)TRu(t)
E
!1
where Q and R are positive semide nite weight matrices the rst term penalizes
deviations of x from zero, and the second term represents the cost of using the
actuator signal. We can express this cost in our framework by forming the regulated
output signal
z = R12u
Q12x
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