+0.9934
=7394 +0,8465'
The closed loop slep response with in the forward path is found usingstepand shown in Fig. 8.21.Italso meets the specifications, a slow compensator mode excited and the settling time of this system is considerably longerthan the state command structure. The ad"antage of this approach is that the compensator is first order while the estimator approach (a) required a order compensation.
been kept at 0.05. the overshoot would have been about which is still
significantly This illustrates the sensitivity of the system to the pole placement when using the output error command structure.
(e) For design. we note that sample rate of6 kHz is over loo times faster than the desired closed loop natural frequency 12oo rad/sec 30 Hz). Therefore, expedient design method is to use the s-plane and convert the result to the discrete when through. Furthermore, since the resonance is significantly faster than the desired bandwidth 2oo rad/sec can ignore the resonance for the first cut the design and simply find a compensation for a system. frequency ideas. we know thatalead compensation with a ratio of 25 between the zero and will yield a maximum increase in phase of about61)' (Fig. 2.17). We also know that
(Section2.4.4)will translate to a damping ratio of about0.6which will the overshoot specification, For a plant, the phase is 180 everywhere: therefore, desiredPM be obtained if we place the lead compensation so that the maximum phase lead is al the desired crossover point 2oo rad/sec This is accomplishedb\
placing the zero a factor of 5 below 2oo (at 40 rad/sec) and the pole a factor of 5 200(al 1000 thus producing
Integral control is useful in eliminating the steady-state errors due to constant disturbances or reference input commands. Furthermore. most actual control are nonlinear and the input gain
r
and the state matrix vary with time and/or the set The linear analysis which is the subject of this book pertains to perturbations about a set point of the nonlinear plant and the control is a perturbation from nominal value, The of integral control eliminates need to catalog nominal values or to reset the control. Rather, the integral term can be thought of as constantly calculating the value of the control required at the8.5 Integral Control and Disturbance Estimation
322 Chapter8 Design State-Space Methods
• Example8.16
Plant
r.H.L
8.5 Integral Control and Disturbance Estimation 325
[KI K] requires the augmented system matrices (Eq. 8.84): therefore. there will beIl
+
Ipoles to be selected for this portion of the design.If implemented as in Fig. 8.23, the addition of the extra pole foethe integrator state element will typically lead to a deteriorated command input response com-pared to that obtained without integral control. While it is possible to iterate on theIl
+
Iselected control poles until a satisfactory response is obtained, it is also possible to retain the reference response obtained from a non-integral-control design by placing a zero in the controller as shown in Fig. 8.24 so that it cancels the extra pole from the integrator. Note that the feedforward tenn inFig. 8.22 has been replaced by which introduces a zero at = I Using the zerocancel the closed-loop pole that was added for the integral element the excitation of that pole by command inputs. Note that this does not cancel the integral action, it merely eliminates the excitation of the extra root by command inputs. A change in the disturbance, wiU also excite the integral dynamics and the steady-state errors due to either constant disturbances or constant command inputs are eliminated. As always. the integrator output changes until its input.
which is constructed to be the system error. is zero. The configuration of Fig.
8.24 can be changed to replace the feedforward ofrto additional feed forward of eand modified feedback of
Detennine the integral control structure and gains for the satellite control problem using full state feedback. Place the control poles at = 0.8±jO.25.0.9 and use a sample
15 Infact, the system of Fig. 8.22 a zero at 1 - so the selection of zero location corrresponds to a selection of
integrator pole cancellation
Figure8.24
Block diagram for integral control With fUll-state feedback and an added zero the model of the plant with an integrator. thus adding an error integral outpUl
the existing plant state outpu!. This augmented model is then used as calculate the feedback control gains for the augmented state. More speCifically.
to the standard system
x(k
+
=+
rll(k)+
= Hx(kJ.
we augment the state with the integral of the error.e= - r.The integral is simply a summation of aU past values ofe(k)(Eq. 3.15). which in the difference equation
-'/(k
+
I)=-'I(k)+
e(k) = -'/(k)+
Hx(k) - r(k). (8.83)arriving at plant model
]
= ]+ [ ]
lI(k)- [ ] r(k). (8.841The control law. following Eq. (8.75), is
lI(k) = -[K , K] [ -',lk) ]x(k)
+
KNr(k).With this revised definition ofthe system. the design techniques already developed can be used directly for the control law design. Following Figs. 8.15(a) 8. 16(a), it would be implemented as shown in Fig. 8.22 for the full-state feedback case and as shown in Fig. 8.23 for the case where an estimator is used to proVIde The integral is replacing the feedforward tenn.N".and has the additional role of eliminating errors due to
The estimator is based on the unaugmented model and is used to reconstruct the unaugmented state. It will be 11th-order. where11 is the order of the original.
system and requires the placement of 11 poles. On the other hand, the deslgn ot Figure8.23
Block diagram for Integral control with state estimation
Design Csing State-Space
.
ulS 1115
Tim
--- - -- - ---1
Integral Control Disturbance Estimation 327
before 2 sec is identical to the in Example 8.12 with no integral control: the integral action successfully eliminates the steady-state error.
o
---·---1
-I
_ _
-I
0
(bl Figure 8.26
Response of satellite attitude to a unit step input att= 0and a step disturbance att= 2sec with integral control, Example8.16 (a) As In Fig.8.22, ib) with an
zero as In Fig.8.24
Time
(a) This case is the same controller as used in Example 8.12. The only difference is that is step in the disturbance 2sec. Therefore. 151m must be used in orderto multiple inputs. the step inrat (= 0 and the disturbance step in starting at result is in Fig. 8.25. We that the system responds identically to 8.17 the first 2 sec. then there is a error in the output. I' afler the transients from the disturbance die out. output crror can shown the final \alue theorem to
KI=0.5. thus the final value of I 1.5 instead of the commanded value of I.
(b) steady-state error resulting from a disturbance is a classic motivation for the additll'll of integral control. The system model from Example 8.1 is augmented I"
Eq. and used with ackerto the augmented feedback gain [KI KI= 13.7-1 by asking for control roots at =0.8 jO.25.0.9. from 8.12 = [I O]T:therefore. the system design is complete and can implemented accordingtoFig.8.22. of 151m produces the response in
that the result has obtained in thaI there no longer a
in the output. note that it has come with a price: the before disturbance step been degraded. More control used. and the initial increased from the original toabout ofthe additional root at = (c) The implementation is structured shown in Fig. which a zero at = 0.9.
Allother parameters the as Note that the in Fig. I
period ofT
=
0.1 Compute the time responses for a unit inr atr=
0 and disturbance of 5 att= 2sec for (al no integral control.(b)integral control 1IlFig. and(c)integral control as in Fig. with the added zero = +0.9.
Solution.
Figure 8.25 Response of satellite eXample to a unit reference Input t= 0 and step disturbance att= Zsec with no integral control action, Example8.16
-326 Chapter 8 Deslgn Using Me[hods
Chapter Design
(8.90) 18.89) [ x(k
+ I)] [
1I'lk
+
I) = 0= [ H
X,I(k+ I) (8.85)
18.86) where = For purposes of disturbance estimation. we augment the model the disturbance model. so (8.85) and r8.86) become
]
= ]] +
lI(k). (8.87)= [H 0 ] [
] ,
(8.88)+
I) ] ]+
=+
r,lI(k)= H [ x{k) ]
. x"rk)
In the particular case where the disturbance is a constant. these equations reduce to
which can be written
and the discrete model given by
A sinusoiLlal disturbance would have the model
R 329
=
or. in general. we could that the disturbance obeys
= u'=H.ix
r
Jl ] [ X(k)]+ [r ]
0 lI(k).o ] [
x(k) ] ll'(k) .. AlI the ideas of state estimation in Section 8.2 still apply. and any of the estlmatron methods can be used to reconstruct the state consistina of x alldX provided the is The computation of the required
exactly given in Section 8.2. the only change being that the system model IS the augmented one given above by and H,. from Fig. 8.27.
however. that control gmnmatrix. K. is/101obtained using the augmented model. Rather. It tS uSll1g the and
r
associated with the unauomented F G, In falct. the augmented system described by will be uncontrollable. We no Il1ftuence the value of11'by means of the control16 \ irrual plantGUrul.
a fr0m to = l.
Plant