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
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-
lt r22 tf ; t CT tf Sr12 tf ; t
C tf SCtf f 1 t
CT tf Smtf f 2 t
ut Ktxt qt
Kt R1 tBT tPt
qt R1 tBT tlt
Figures 11.5 and 11.6 give an overview of the optimal servomechanism problem. Comparing these gures
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
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
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.
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
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.
Let Gs Gh sGp s. Then
^ z Z Gs Z 1 e
1z Z 2
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
Consider a discrete-time system described by the following second-order difference
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.