534 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
5.13 For the unity feedback system with
k
G(s) = ~s+ ) s+4~s+7)
design a PlOD controller that will give time for peak amplitude of 1.2 s and 15%
over~hoot with zero steady-state error for a unit-step input. Plot the unit-step
response for the basic and PID-compensated systems.
5.14 For the system shown in Fig. P5.14, deterrrune the values of gain kh and
k so that the minor loop operates with a damping ratio of 0.707 and the entire
closed-loop system has 15qo overshoot.
Fig. P5.14 Control system for Exercises 5.14 and 5.15-
5.15 Determine the rate gyro gain kr. for the system shown in Fig. P5.14 so that
the compensated system operates at one-third the settling time compared to the
basic system while continuing to have the same 15% overshoot.
5.16 Given the linear time-invariant system
x(t) : Ax(t) + Bu(t)
find (a) eigenvalues of the matrix A, (b) the state transition matrix Q(t), and (c)
state vector x(t) for the foUowing cases:
(i) A = [ 01
(ii) A=[;. -.l]
B=[:] x(0)-.
B = [:]
. 1 -1 (
(iii) A- O 1 -j B=
00-
x(0) -
x,0,= [:]
LINEAR SYSTEMS, THEORY, AND-DESIGN: A BRIEF REVIEW 535
]+[:]u,t,
y=[l O 0] [x-[:]
Can this system be transformed into phase-variable form? If so, find the transfor-
mation z -. Px so that the transformed system z - Azz + Bzu(t), y = Czz is ir]
phase-variable form.
5.18 Represent the following system in state-space, phase- variable form:
d3X 2d2X 3dx
d~3 + dt2 + dt +5x - u(t)
5.19Giventhe O;-:l[x::l+[{lu,t,
Y=[l O 0][x-."]
Can this system be transformed to dual-phase variable form? If so, find the trans-
formation z -. Px such that the transformed system z = Azz + Bzu(t), y = Czz
is in dual phase-variable form.
5.20 Design a phase-variable, full-state feedback controller for the plant given
by
-1- 0.8)
~
G(s)=~+, ;~((sS~)(s+5)
to yield a 15% overshoot with a settling time of 0.8 s.
5.21 Design a full-state feedback observer for the plant
1
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