Optimal Control
481
where Qt is an n n real, symmetric, positive semidenite weighting matrix for t 2
t0 ; tf and mt is the prespecied desired path of the state vector xt. The vector
et xt mt is the error, which we want to minimize. This may be accomplishe
Digital Control
Figure 12.2
517
Block diagram of a discrete-time system.
computer. In this case the signals uk and yk are number sequences (usually 0 and
1). These types of systems, as we shall see, are described by difference equations.
The term sampled-
Optimal Control
505
1 T
lt r22 tf ; t CT tf Sr12 tf ; t
C tf SCtf f 1 t
CT tf Smtf f 2 t
11:4-19
Finally
ut Ktxt qt
where
Kt R1 tBT tPt
and
qt R1 tBT tlt
Figures 11.5 and 11.6 give an overview of the optimal servomechanism problem. Comparing these gures
Optimal Control
493
As a practical example of an optimal regulator, consider a ground antenna
having a xed orientation. Assume that the antenna undergoes a disturbance, e.g.,
due to a sudden strong wind. As a result, the antenna will be forced to a new po
Digital Control
Figure 12.11
529
Hold circuit block diagrams: (a) hold circuit; (b) idealized hold circuit.
hold circuit shown in Figure 12.11b. Here it is assumed that the signal ut goes
through an ideal sampler T , whose output u t is given by the relat
Digital Control
541
In the case where the system (12.4-1) is stable in accordance with Denition
12.4.1, the point xk0 T is called the equilibrium point. In the case where the system
(12.4-1) is asymptotically stable, the equilibrium point is the origin 0.
Digital Control
_
!
577
d2 L
dt2 J
where is the position (angle of rotation), ! is the rate of the angle of rotation, L
is the externally applied torque, and J is the moment of inertia. If x1 and
_
x2 !, then the state-space description is
!
!
!
!
_
0 1 x
Digital Control
565
specications: gain margin Kg ! 25 dB, phase margin p ! 708, and velocity error
constant Kv 1 sec1 . The sampling period T is chosen to be 0.1 sec.
Solution
^
Let Gs Gh sGp s. Then
&
'
&
'
Ts
1
1
1
^ z Z Gs Z 1 e
^
G
1z Z 2
ss 2
s
s s
Digital Control
553
The remarks above are the basic motives for the implementation of indirect
design techniques for discrete-time controllers mentioned in Sec. 12.6. Indirect techniques take advantage of the knowledge and the experience one has for conti
System Identification
589
Example 13.2.2
Consider a discrete-time system described by the following second-order difference
equation:
yk 2 !2 yk buk
The input uk and the output yk, for N 5, are presented in Table 13.2. Estimate the parameters ! and b.
Sol