Page 778 - Mechanical Engineers' Handbook (Volume 2)

4 Extensions of the Linear Quadratic Regulator Problem 769

J [xR x (u Mw) R (u Mw) ˙uR ˙u] dt (43)
T
T
T
1
w
2
w
3
0
where R , R 2 are symmetric positive-semidefinite matrices and R 3 is a symmetric
1
positive-definite matrix. When the system state is augmented to include the vector u M w
w
and the input derivative ˙u is defined to be the new input, the problem reduces to the standard
infinite-time LQR problem. The optimal-control law is a proportional-plus-integral state feed-
back law:
u(t) Kx(t) K x dt (44)
l
where the constant gain matrices K, K are known linear functions of matrices satisfying
l
algebraic Riccati equations. In cases where the complete state x is not measurable, the state
would be estimated from the measured outputs using the methods described in Section 5.
The closed-loop system is asymptotically stable provided that certain controllability and
observability conditions on the matrices A, B, R , and R are satisfied. The gain and phase
1 2
margin results noted earlier for the standard infinite-time LQR problem are valid here as
well.
19
Johnson considers a more general class of disturbances and state-space equations:
˙ x(t) A(t)x(t) B(t)u(t) B (t)w(t) (45)
w
y(t) C(t)x(t) D(t)u(t) D (t)w(t) (46)
w
where w(t) is a vector of disturbance inputs. The disturbance inputs are assumed to be
described by linear time-varying differential equations that constitute a state-space model for
the disturbances:
˙ z (t) A (t)z (t) B (t)x(t) (t) (47)

w(t) C (t)z (t) D (t)x(t) (48)

where z (t) represents the state of the disturbance and (t) is a vector of Dirac delta impulses

occurring at unknown times. The terms including x(t) in the preceding equations enable
cases of state-dependent disturbances to be considered within this framework. The coefficient
matrices are determined experimentally by examination of the records of the disturbances.
This type of description of disturbances constitutes a waveform mode description and is
applicable to a broad class of disturbances of practical interest that are not described well
either by deterministic process models or by stochastic process models. Examples of such
disturbances are piecewise linear, piecewise polynomial, or piecewise periodic signals.
The waveform mode description of disturbances is combined with the system state-
space equations to provide a rather complete description of the system to be controlled and
the inputs affecting its behavior. The exact design of the controller depends on the specific
objectives used to govern the design. If one of the objectives of the control system design
is to counteract as completely as possible the effects of the disturbance inputs, then the
control input is considered to be composed of two parts:
u(t) u (t) u (t) (49)
d
m
where u (t) is the component used to counteract disturbance effects either completely or
d
partially and u (t) is the component used to accomplish other objectives such as closed-loop
m
pole placement. Alternatively, the objective of control system design may be the minimi-
zation of a quadratic index of performance. In either of these cases, the control law would