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
Disturbance Estimation
An alternate approach to state augmentation is to estimate the disturbance in the estimator and then touse that estimate in the control law so to the error to zero in Fig. 8.27. is called disturbance rejection.
This approach yields that are equivalentto integral control when the disturbance is a constant. After the estimate. converges. the feedback of value as shown in Fig. 8.27 will cancel the actual disturbance. and the will behave in the steady state as if no disturbance were present. Therefore.
system will have no steady-state error. assuming. of course. that the
error was due to a described by the assumed equation used in the estimator. It is important to notice that while a disturbance may. in general.
at any point in the plant equations, the control can apply a signal to cancel it at the control input. To reconcile facts. we introduce the "input disturbance. This is a virtual signal applied at the control input which produce the same stale outpUlat y the actual disturbance does. TheIl.
when the control applies the negative of the virtual disturbance, the effect of real disturbance at the output is cancelled and the error is driven to zero. To obtain an estimate of the virtual disturbance. we build the estimator with the equations of the virtual disturbance included.
Disturbances other than constant biases can be modeled. included in the estimator equations. estimated along with the plant state. and their effect on errors eliminated in steady-state, If we assume the disturbance is a constant.
continuous model is quite simple:
Itshould be clear from the discussion and the example that the implementation of integral control is given by Fig. 8.24 where the zero the integrator closed-loop root.
disturbance rejection
8.5.2
disturbance modeling
Figure 8.27
Block diagram for Input disturbance rejection
330 Chapter 8 Design Lsing
H;=[ I Il ] 1', = and
F = [
ImegralControl DIsturbance 331
K= [to.25
•
input: is no as in fact. j identjcal to
Fig. 8.26(bJ up until the disturbance t= sec. Note further that the disturbance estimate. the actual disturbance value asymptotically. thai in this the steady state error due to the reference input made to be zero the calculation of and
not robust to small parameter changes in the way hy control.
Example8.17shows that disturbance estimation can be used to estimate constant disturbance input and then10use that estimate so to reject the effect of the disturbance on errors. When the disturbance a constant. this approach essentially duplicates the function of integral control. The following example shows how disturbance estimation can be used to estimate the value of the disturbance when it is a sinusoid. The estimate is to cancel the effect of the disturbance, creating disturbance rejection.
For a spinning satellite. a from 011 the as
sinusoid the spin frequency. The attilUde dynamics become order as the are now coupled: however. for spin dynamics may approximated to as mE,ample 8.17.
Determine rejection control gainsfor attitude
witha from of is spinning
15 rpm. Place thc at = 0.8±jO.25and a period ofT= 0.1 as before. Use a predictor estimator and the estimator poles at = ± 0.9± ill.I.
The location of the to can a
frequency ,fthc sinusoidal disturbance is known to be stable in magnitude and Bypicking those estimator at a slow fre4ueney. the
estimate will not much to higher frequency disturbances.
PIOlthe hIstory of the estimate of the disturbanee. and system
output to verify in sleady-state. is no from the Put in a
command ofI at r=5 to that the input will unwanted estimator errors. Examine the of the 8th order and what of represent.
Solution. The feedback for the unaugmented computed as in Example 8./7 be
disturbance acts at the control input so tbere no need for the of the virtual disturbance. It is modeled hy choosing
• Example 8.18 [Cl
-1
Solution. For of finding the control gain. we use the un augmented model III
Example 8.1 and. therefore. find the same value of K=[10.25 For ,,:
designing the estimator. augment the plant according to Eqs. (8.89) andr8.90) find that the desired poles yield = [1.3 StruclUnng the control InFIg.
and applying the inputs specified above.ISlmyields the response shown FIg.
Note the similarity to E,ample 8.16 shown in Figs. 8.26Ia) andIbl. The <hsturbance approach also eliminates the steady-state error. But the early response to the referell"c Detennine the bias rejection control gains for altitude control problelll Place the control poles =0.8±jO.25and a sample period ofT= 0.1 sec. ., predictor estimator and place the estimator poles at = ± 0.9. the responses for a unit step in ,. at r
=
0sec and a step disturbance ot5 at r=
compare the results with the integral control in Example 8.16.
I
Time (sec) 0
input. and must live with whatever value nature deals hence. the system is uncontrollable. Our plan is not to control U'.buttouse the value of in a feedforward control scheme to eliminate its effect on
errors. This basic idea works if is a constant. a sinusoid.orany combinationuf functions that can be generated by a linear model. It works regardless of the actual acts since the design is based on the virtual disturbance The only constraint is that the disturbance state. be observable.
• Example8.17
Figure 8.28 Response of example to a unit reference Input att= 0 and a step disturbance att= 2 sec with bias estimation as In Fig.8.27,Example817
(S.91)
= [I 0]
= 0
I].
and the j
=
= [HH..J.
matrices. of c2d compute the discrete matrices and which.
with poles rhe estimator gain computed as
and Disturbance Estimalion 333
= lJ.OS55
The time response of the described Fig. is found by of ISlm
of complete consi,ts of the as well as of
Solution. As in Example 8.18. the feedbackfor the un augmented isK= r10.25 The output di.sturbance modeled augmenting continuous with the matrices
control system. The of the estimator in this case is to use the contaminated measurement to reconstruct the error-free state for use in the The estimate of the sensor error is ignored by the controller.
• Example 8.19
an attitude on a principal of inertia.
attitude a sinusoidal component as the naturally spins about its principal axis. Typically, itIS 10spin about thc principal axis: therefore. it
to magnitude of the to that of the
the from S.18. but replace Ihe disturbance
sinusoidal measurement error of0.3 at 15 rpm. Again a command of
I atJ 5 Place poles at =O.S:r:jO.25. in 8.18. and a
of T = 0.1 a predictor estimator and place the esrimator poles at
• = ±jO.2.0.95 ± jO.05
Figure8.30
Block diagram for sensor disturbance rejection
•
output
++ +
The previous example had a sinusoidal disturbance torque acting on the input to the system.Itis also possible to have a measurement or sensor error that sinusoidal in nature that one would like to eliminate as a source of error to
(8.871 and (8.881. Use of acker with and and results in
= 6.S07
The time response of the system described by Fig. 8.27 found use where.
state of the complete consists the augmented state as well as the 01 state. order The feedback of can be
= [K I OJ.Figure 8.29 results.
the takes"about sec to the correct a
error in output due to the disturbance until that time. step 5 has estimate quality and therefore the the step was
Without disturbance there would steady error of
superimposed on the output.
The of the closed loop order system
O.9±0.lj. 0.98S±0.156j.
The 6 represent those selected in the control and estimation design. The last1\\
represent the discrete equivalent of pure oscillation 15 rpm. which are unchanged
Figure 8.29
Response of satellite to a sinusoidal Input disturbance with disturbance rejection as in Fig.8.27,Example 8.18
332 Chapter8 Design Using State-Space \lethods
Plant
r
Estimator
8.5 lntegral Control and Disturhancc ESlimauon 335
Virtual
Plant Actual
model
,
using the unaugmented plant described by and
r.
but the feedback signal is the system errore.Byestimating pand feeding that estimate into the plant with the control as shown tn FIg, 8,32(a). the effect ofpis eliminated in this model ande 0 steady The actual situation that is implemented is shown in Fig. 8_32(b) here the reference IS subtracted from the output to form the error,e. and the
lal
(h)
Figure 8.32
Block diagramforsensor disturbance following, (a) The pretend model, and(b)the
implementation model
+
\.'.
disturbance +
A example of the use of the estimator to achieve zero steady error arises when it is desirable to track a reference signal withas little error possible and it is known that the signal follows some persistent pattern. Since it would usually take some control effort to follow such a signal. the system would normally exhibit a following error of sufficient magnitude to produce the required control effort. This following error can be eliminated if the systematic pattern can be modeled and estimated. then used in a feedforward manner produce desired control effort, This is called reference following, The idea is the same with disturbance rejection except that the error is now not the output only but difference between the reference and the output. The idea again is to construct virtual reference. p.at the conlrol input which would produce the system error at the plant output. as shown in Figure 8.32(a), The feedback gainK is designed
augmented state. an 8th system. of ignored, 8.31
in figure that about to converge correct
and that there noticeable error in the output due to measurement error until that 5 has no on estimate quality therefore the response to the is precisely as it was originally designed. Without disturbance rejection. there would beena steady sinusoidal error of about0,2 as controller attempted to follow the sinusoidal
reference following
Figure 8.31
Response of satellite to a sinusoidal sensor disturbance with disturbance rejection as in Fig 830,Example 8.19
--_.- - - - - - +
334 8 DesIgn Usmg \Iethods
336 DCSlgn l'slng \[eth0ds
marked in Fig. that. Once has Converged. there no noticeable
following error.
Many feedback control systems a pure time delay. .. imbedded in some of the loop. A one cycle delay analyzed in Section 7.3.2 by adding a to the system model and using the root locus method to show that the stability of the system was decreased when no change to the compensation made.
We can also analyze the effect of delays with frequency response methods as in Section 7.4 by reducing the phase by This analysis easily shows that. if110
changes are made to the compensation. the phase margin and hence stability will be reduced. Either design method can be used to modify the compensation so the response is improved.
For state-space design. we saw in Section that. for an actuator one state element must be added to the model for each cycle of delay or fraction thereof. In other words. a delay of0< ). Twill require an increase in the order of the system model byI.a deJay ofT < 2T will increase order by 2.
and so on. the placement approach, we can assign any desired pole locations to the system. Therefore. we are able to achieve the same closed-loop poles in a system with as one without delays: there are extra
8.6 Effect of Delays
---..
Figure 8.33
Response of disk driveto sinusoidal reference follOWingas Fig,8.32,
8.20
- - - -
.
!OOO
the at rpm. a that follows on the
th amount. P,ek the I t
prctend that disturbance is same in Example that
= and H" = [ I 0] .
= rpm or rad/sec. The difference here compared to Example
suhtracted from the oulput alld to Its 0
In other words. want to follow r in Fig. . h ' .
. U' F Gwith c2d yields and
r
fur the head To t eiterations show that poles with a natural of 1000 radisec and a 0.6 damping con\'ened to the digital domain. produce t
The resulting feedback K=
But to include the reference in the control u= Kx-p.
is constructed A wbas th'ounct.
,,·plane poles. . had natural frequencies ot. '000I d - ra. ot 0]
ofacker with 1012.9
- • • d h F' found of 151m
Thetime of the descn e y 19.0.. d. 1
II h . te fthe aucornente state.
the of the augmenled state we I e 0
order . I h. '11 du 'e = rand
The of the feedforward to t at pro C . 'h' 'h
. . - h II produce the track
() we an ot an p.t at . .
-traCking reference., tcedton' ard ot -. ' . . p . _- - I '.This the
estimate of the viJ1uai is subtracted from the con.tro!. Therehfore.
h . ·t d state e= 0 t at will as ifpwas canceled. so t at s ea y. . - . . . followingr.
h d' k T· . there a small
computer must ont e ) piC . . ' .
of tracks and rotational center of the disk. thus in the tracks that the read head folluw. transter tunctlon
input command and the is
• Example8.20
(8.92)
(8.93)
(894)
(8.95) x(k
+
I) =+
rll(k)= Hx(k).
to Eq. (8.92). where is one more state element and is the value of that delayed by two cycles. Therefore. for a system given by
The model of a one delay of a quantity is
Effect 339
where is the delayed version of and is an additional state element that is added to the system model. The model for more than one can be obtained by adding more similar equations and state elements. So. for two cycles of delay.
we would add
the system model including a two-cycle delay
[
= [ ]+[ l
tt(k).+
I) 0 I 0 0J
=
[0 0 I] ] .
where is the output delayed by two cycles. Any number of cycles of delay can be achieved easily by scheme.
Ifa sensor had delay that not an integer number of cycles, it would not the sampled value until the next sample instance. Therefore. sensor delays must be an integer number of samples.
Note that. due to the column of zeros, the augmented system matrix.
in Eg. (8.94) will always be singular. a fact that will difficulties when calculating gains for a current estimator usingackerinM\TLAB.
17 will in Chapter thaia will difficulty ith dlqr.
Examine the to a command inpntrof satellite control with one cycle of delay at Place two of Ihe contra] pDles at = ±0.25j. lias the for Examples and 8.12. and the additiDnal pole for the delay at = O. the for
prediction estimator = ± Ihe for 8.5 8.12. the
additional for the delay =o.Compare the results with Example the same slep without a delay.
• Example 8.21 ci
delay
poles in the system with delays. Those extra poles can slow down the response even if they are selected as fast as possible, that is. = O. The response system to command inputs and disturbances is also affected by the location of the delays, that is. whether they are part of the control actuator or the sensor. Figure
shows systems with the in the two locations.
8.6.1 Sensor Delays
For the sensor delay. Fig. an estimator can be usedtoreconstruct thc entire state: therefore. the undelayed state is available for control. The system can be respond to command inputs. ,.. in exactly the way that a system would respond without a sensor delay because the estimator sees the command through the feedforward ofttand not depend on the delayed output to detect it. Therefore. no estimator error is excited and the undelayed state estImate accurate. On the other hand. a disturbance input.w.will not usually be seen the estimator until the output. responds: therefore. the estimator error
delay model increased by the delay.
Figure8.34 System With delays, (a) sensor delay,
(blactuator delay
338 Chapter 8 Design Usmg State-Space I\lethods
!l/5 0
·1
()
An example of a in common use with a delay is the fuel injection control for an automobile engine. Here, the sensed value of the fuel-air ratio in the exhaust is delayed by the piston motion and the time for the exhaust stream to reach the sensor. The primary disturbance is the motion of the throttle by the driver's foot; however. this motion can be sensed and used as feed forward to the estimator. Thus the estimator structure is capable of instantaneous rejection of the throttle disturbance in spite of the significant delay of the exhaust sensor.
(See Fekete, 1995.)
8.6 Effect llf 341