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Plant Uncertainty Contribution to QFT Tracking Control Design
Copyright © IFAC Robust Control Design Milan, Italy, 2003
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PLANT UNCERTAINTY CONTRIBUTION TO QFT TRACKING CONTROL DESIGN GiI-Martinez, M. I and Garcia-Sanz, M. z I
System Engineering and Automation Group. Electrical Engineering Department. University ofLa Rioja. Luis de Ulloa. 20. 26004 Logrono. SPAIN Tf:+34 941299496. E-mail: [email protected] 2Automatic Control and Computer Science Department. Public University ofNavarra. Campus Arrosadia. sin. 31006 Pamplona. SPAIN Tf:+ 34948 16938. E-mail: [email protected]
Abstract: Tracking feedback control trade-offs increase dangerously with arbitrarily large uncertainties in the process model. This is shown by the bounds in the QFT domain. In this context, the paper develops a theory on the contribution of the plant uncertainty to the tracking QFT bounds through a serious study of tracking bound formulas. From this, a methodology for uncertainty fragmentation is developed. It hopes to loosen the bound toughness, or the control design trade-offs. Better feedback benefits can be achieved while reducing the high-frequency amplification of noises and disturbances at the plant input at the same time. Different QFT controllers in a controller-scheduler structure should drive different uncertainty boundaries. Copyright © 2003 IFAC Keywords: Quantitative Feedback Theory, Robust Control, Uncertain Dynamical Systems.
I.
[9]. This paper extends the study to any frequency, and is based on the bound fonnulas developed in [5].
INTRODUCTION
Feedback control necessarily implies trade-off designs, even for m.p. SISO plants. (a) A high enough open-loop gain lL=GPI is required at low-medium frequencies for robust command tracking and disturbance rejection. (b) ILl should decrease as fast as possible with frequency to avoid large noise amplifications mainly at the actuator inputs [I]. (c) ILl decrement rate is constrained to ensure robust stability. The controller design trade-offs increase with the uncertainty of the plant and the aggressiveness of the perfonnance. A quantitative fonnulation of the problem as QFT (Quantitative Feedback Theory [2]) is used to define a solvable robust feedback control problem.
Appropriate values for robust tracking and stability specifications within the QFT domain are discussed in Section 2. Section 3 describes the tracking bound fonnulation. The fonnal inequalities are analyzed in Section 4, and the contribution of uncertainty to QFT tracking bounds is inferred. Then, Section 5 summarizes the main rules to accomplish an uncertainty division aimed at improving feedback perfonnance. Section 6 provides an example. Finally, Section 7 includes the main conclusions. 2.
TRACKING SPECIFICATIONS IN QFT
Robust specifications in command tracking are usually the most restrictive ones according to the perfonnance limitations in the presence of arbitrarily large uncertainties. Thus, disturbance or noise rejection problems are ignored in this paper for simplicity reasons. The control structure of Fig.1 needs two-degrees of freedom [10]: a prefilter F and a controller G.
QFT takes the plant uncertainty and closed loop specifications and translates them into certain bounds on the nominal open-loop transfer function [3,4], Bound fonnulas in [5] were developed for automatic bound computation [6]. Based on these bound fonnulas, general feedback objectives were classified in different bound typologies in [7]. The bound typology depends on the nature of the requirement, the toughness of the specification and the size of the uncertainty. Assuming a fixed plant uncertainty, an achievable feedback control can be guaranteed by a proper selection of the specification tolerances [8]. However, arbitrarily large uncertainties will imply a poor-tracking perfonnance. A possible solution is to reduce the uncertainty. Then, a higher perfonnancelower cost controller will be responsible for each partition. A comprehensive study of the contribution of uncertainty to the bounds is required to maximize the benefits of uncertainty division. The uncertainty at the low frequency and its QFT bounds were approached in a graphical way in
Fig 1.
In the presence of certain open-loop uncertainty {P}={PLB}, the controller G aims to reduce the closedloop gain uncertainty {In} according to upper 8rSIIP and lower 8ritif models within the frequencies of interest:
517
· =I L(jm) :5 OT (m), Q T ={m:5 mT} IT(jm)1 1+ L(jm) OTsup (m)
0T (m)
°Tinf(m) ,
(1)
L(jm) =G(jm) P(jm)
Provided that the task of the prefilter F is to place {171} inbetween these models: (2)
The amount of feedback required depends on the tracking specification value t5r=t5rsuplt5rin/ and on the plant gain uncertainty Pmax1pm;n. As it was concluded in [7], QFT represents this requirement by upper bounds or typology n bounds (see Fig. 2b) when Pmax1pmin>t5r. This guarantees a minimum open-loop gain L at any phase for oEOJr. Tracking upper bounds become the dominant ones within the low-middle frequency range, and their aggressiveness increases when t5r is smaller [8]. In contrast, the cost of feedback [1] and stability requirements within the high m range imply the release of t5r. Then, Pmax1pminOJr and according to [7], outer bounds or typology 0 bounds represent it (see Fig.2a). These tracking outer bounds will become negligible compared to stability outer bounds. (a) OIR' bolRl
,.
(b) ",per bolftl
find the contribution of uncertainty to tracking bounds based on this mathematical formulation. The procedure is as follows. A set of discrete frequencies is chosen: {cq, i=l, ...,n}={QT tlQs} from (1) and (3). The border of the cq plant template {P(jcq)} is approximated by a finite m-point set of plants [11]. Each r=I, ... ,m plant can be expressed in its polar form as P,{j(4)=p,{cq)·~6ri.(J)I)=pLB, and likewise the polar form of the controller is G(jcq)=g(cq)'~;=gL4J (the phase controller dependence on the frequency has been suppressed considering that the 41 range is the same for any [4-bounds). The tracking model in (1) takes a discrete value at the frequency cq, t>rt.cq)=t5rsuplt5rint<.cq), being Or> 1 since (},sup> (},in/' Then, two plants in the artemplate are selected: PtFPdLBd, Pe=PeLBe. For a discrete controller phase f/JE l/);[-21t, 0], Inequality (1) is as follows:
I w; = P;PJ(I-
g 2 + 2pePd'
01
r' 8i I
PePd
1·{p,OO'C!H8'l-
)//
)~y ·'80
Fig. 2. QFT bound typologies
Stability requirements are also defined; these requirements imply certain robust phase and gain margins. For example:
MF
(3)
~ 180o(I-.3..cos-,(.Q211, MG ~ 2010g(1 +_11 1r
Os
(4)
OS
~«I
4.
implies too conservative stability margins, whereas I means highly underdamped systems. A ~> I implies outer bounds (see Fig.2a) [7] and, as long as ~ gets closer to I (for appropriate stability margins), these outer bounds become dominant at high frequencies [8]. The outer bound encloses a forbidden area around the stability point.
Os»
3.
Z00,\;+8,)1 (6)
1-
~; COS(q} + Oe)Y - (1- ;1 i(Pi - ~r)
Only real and positive solutions g/ and/or g;z in (6) impose constraints on g at the phase 41, and they configure the ar bound for a fixed pair of plants and the specification requirement (},. Then, the less favorable curve representing the tracking requirement at cq is computed by repeating the procedure for the whole set of pairs of plants in {P(j cq)}. Finally, this G-bound is translated into terms of the nominal loop transmission Lo=Po·G=loL'IIo for a nominal plant Po=poLBo in {Plo The conclusions for G-bounds are also applicable to Lo-bounds. Prior to the final loopshaping, the La-bound intersection amongst the bounds standing for the different robust feedback problems (stability, tracking commands, disturbance rejection,...) must be obtained. A bound plotted with a solid line implies that G(jm) (or Lo(jm) must lie above or on it to meet the particular specification, while a bound plotted with a dashed line means that G(j m) (or Lo(j OJ) must lie below or on it.
:V
/T(jm)1 = I L(jm) :5 0s(m),Qs = {ms E [O,oo]} 1+ L(jm)
01
The unknown parameter g (required magnitude of the controller) can be computed by solving I(J)I=O in (5). Two solutions are obtained: guarantee
(Pe cos(q} + Od) -
·1'.J6O
(5)
.(PeCOS(4J+Bd)- Pd COS(4J+Be)lg+(p; _ PJl ~ 0
g,.,
(c) lower bolRl
;1 I
PLANT UNCERTAINTY AND QFT BOUNDS
As discussed before, tracking specifications yield dominant upper bounds at low frequencies and non-dominant (as regards the stability requirements) outer bounds at high frequencies [7]. The upper bounds become more stringent when it rises and the outer bound when it encloses a bigger area (its upper portion rises, the lower one slows down, and both segments widen in the phase range). A smaller tracking tolerance t5r=(},supl (},inf implies severer bounds according to previous definitions (for a formal study see
TRACKING BOUND FORMULAS
An efficient numerical algorithm was proposed in [5,6] for the automatic computation of bounds. This paper tries to
518
[8]). The purpose of this paper is to examine the contribution of the uncertainty to tracking bound severity. The research is based on the formal procedure of QFT bound computation presented in Section 3, and its foundations are detailed in [12]. The main conclusions are following presented.
compute the bounds in Fig. 3, and thus omIttmg the quantitative height of the bound, the following qualitatively analysis can be made. The appearance of the bounds B I and B2 is typical of low frequencies, where feedback benefits require a strong sensitivity reduction and, thus that PmdPmin»~' The aspect of the bound B 3 is common at middle frequencies, where in spite of the possibility of PmdPmin being significant, the specification ~ begins to increase (high frequency gain cut off), and then, it is easy to meet ~l"?pmdPmin"?~. The appearance of the bounds B 4 and B 5 reminds of outer tracking bounds at high frequencies where PmdPmin«~, and these bounds are nondominant compared to outer robust stability bounds. 4Or------.-------,-----,-------,
30
'.&.o=----~.270=------:'.,00c:------'_90,.------l ~phase(")
Fig. 3. Tracking bounds on the controller for different sizes of the template magnitude uncertainty and no template phase uncertainty
Let's suppose a cq-template with certain phase uncertainty and certain magnitude uncertainty. To simplify the study, let's consider first the cq-template with only certain magnitude uncertainty. Fig. 3 depicts different G-bounds at cq for different sizes of magnitude uncertainty and a constant tracking specification tolerance M cq)= 1.1. For simplicity reasons, let's assume that the cq template is placed on the D phase angle, i.e. {PUcq)}={P(cq)}L{;{cq) with (;{cq)=D. In Fig. 3, the bound BIUcq) stands for {P(cq)}E {D.I,...,I}; B l for {P}E {O.I,...,O.2}; B3 for {P}E {O.I, ,O.1l5}; B 4 for {P}E {D.l,...,O.105}; and B 5 for {P}E {O.I, ,O.lOI}. According to the main results in [7], BU.3 are upper bounds since PmdPmin"?~, being Pmax=max{p(cq)} and Pmin=min{p(cq). However, the outer bounds B o stand for PmdPmin<~' Note that a logarithmic uncertainty converts linear differences Pmax-Pmin into ratios Lm(pmdPmin), where 'LmO' symbolizes '20/ogI00'. Note also that Fig. 3 depicts G-bounds that should be subsequently referred to the nominal plant Po to perform loop shaping onto Lo-bounds.
.10
·~=-----;;.270~----::.,00=-----=_90:-----...J.
00_' phase (") Fig. 4. Tracking bounds on the controller for different sizes of the template phase uncertainty and no template magnitude uncertainty.
Secondly, let's consider now a artemplate only with phase uncertainty. Fig. 4 depicts different bounds for {PU cq)} =p(cq)L {(;{ cq)} and a specification value Mcq)=1.1. The B 6 bound stands for {O}E {-n/6, ...,O} rad; and B 7 for {B}E {-n/I2,...,O} rad; both for the fixed gain p=O.1. And the B 8 bound stands for p= 1 and {B} E {n/6,...,D} rad. Fig. 4 shows first that the templates with phase uncertainty only give outer bounds, since PmdPmin<~, where PmdPmin=1 and 8,. is always greater than 1. Second, Fig. 4 illustrates how an increment in the phase uncertainty produces severer bounds (compare B 6 with B 7). Third, the severest bounds due to phase uncertainty are those with the minimum modulus P (compare B 6 with B 8 ). And finally, if these bounds are added to Fig. 3, the bound portion due to the phase uncertainty would become significant at the valleys of the bound portion due to the magnitude uncertainty.
Fig. 3 illustrates that when PmdPmin( cq) decreases (but the upper bound also goes down (see the sequence Bj, B 2, B3). Using the tracking bound formula (6) and for the case of upper bounds, it is possible to infer [12] that if PmdPmin"?~2"?8r. the upper bound has three peaks (see B I and B 2), whereas it only has one peak (see B 3) if ~l"?PmdPmin"?~' This proves the relaxation of the bounds when the magnitude uncertainty decreases. If PmdPmin keeps on decreasing beyond ~, that is, if ~2"?8i>-Pmax/Pmin, the upper bound becomes an outer bound, placed lower on the dB axe (see B 4 and B 5 ). The outer bound will keep on contracting (less enclosed area) if Pmax/Pmin decreases more with respect to ~ (compare the sequence B 4 and B 5)'
PmdPmin"?~,)
Ignoring the quantitative values {P} and
~
These results of separate studies of phase and magnitude uncertainties will be now combined. Let's consider a ar template {PUcq)}={pLB} with p(cq)E {O.l,...,l} and (;{cq)E {-n/6, ...,O}, and a tracking specification value M cq)= 1.1. Discretizing the {P} domain at three logspace points and {B} at seven points, the intersection arbound on G, called Br, is depicted in Fig. 5 with the darkest line. The main bound contributions of certain pairs of plants are also highlighted. See the simple bound Ba due to the plant pair {PmaxL.Bmin> PminL.Bmax}; the bound Bb due to the plant pair
used to
519
uncertainty can be a difficult task. Some insights were provided in [9].
{PmaxLOmax, PminLOmin}; and the bound Bc due to {PminLOmax,PminLOmin}' Bounds B T, Ba and Bb show that the size of the absolute magnitude uncertainty, !lp=PmdPmin> and also its relative values, {P}E {Pmin>' ..,Pmax}, are responsible for the points of maximum height in the bound. Besides, these maximum points extend along a phase range equal to phase uncertainty, /1(FOmax-Omin' Adding now Bc, the B T bound height at its valley points are due to the absolute phase uncertainty, i.e. /10. The phase placement of the B T bound on [-21t, 0] depends on the relative phase values {o}. However, once computed the bound on L o (add the nominal phase to the bound on G), it can be proven that the maximum points of the Ltrbound always occur at 21t::!:!10 , -n:::!:.!10, and -O+~O rad, whereas the minimum ones occur at -31t12±!10, -1tI2±!10 rad. Exceptionally, if 8/>pmdPmi';>~' Lo-bound maximums are only reached at -1t::!:!10 rad, and the bound slows down to both edges -21t and 0 rad (see curve Bj in Fig. 5). A formal demonstration can be found in [12].
• Few divisions must be performed to obtain as few designs as possible and avoid future unexpected instabilities in on-line switching. • The divisions must be somehow identifiable through certain auxiliary measurable variables, as the scheduler will switch particular controllers on-line. • The robust stability bound of each controller must be computed considering the full uncertainty instead of the reduced uncertainty. This prevents from unexpected on-line failures of the controller-scheduler spoiling performance while preserving the robust stability. 6.
EXAMPLE
Let's consider now the incompressible fluid system in Fig. 6. A servo-valve controls the inflow q;{t) (control variable) demanded by a centrifugal pump. The outflow qa(t) may vary according to step commands within a moderately large range. A linear parameter resistance R plus certain uncertainty can express the non-linear behavior of the load valve. Fluid storage in the tank is represented by a capacitance C. The linear uncertain model of the outflow control through the inflow around certain equilibrium point is: P(s) = Qo(s) Qi(S)
=
1 RC s+l
zos+l'
(7)
where the uncertain time constant is r-=[1, 10] sec. Control Valve
qlt)_~
\'
\~ "?.o~------l--::_270=-----=--':_1.. =-----7....:-----.JL-~
............ n h(t)
Fig. 5. Tracking bounds on the controller due to templates with both, phase and magnitude uncertainties.
L-*---~=t~...,.l==".qD(I)
These results can be easily extended to real templates that do not build rectangular shapes, as shown in the example in Section 6. A qualitatively conclusion of the study is that a smaller uncertainty in the phase and magnitude of the plant reduces the severity of the upper bound (bound height at each phase) that stands for a fixed specification tracking value M at). This proves the expected benefits of the uncertainty division, proposed as a solution to performance limitations due to excessive plant uncertainty and/or ambitious specifications. 5.
Load Valve .0
Capacitance
Resistance
C
R
Fig. 6. Incompressible fluid system
The outflow feedback control requirements are: (i) Robust tracking specifications in (1) and particularized for frequencies 0Jr~7 rad/s as follows:
(2)
OTsup(S) = 0.66(s+ 30)/(i +4s+ 19.75)
(8)
0Tinf(S) = 8400/(s + 3)(s + 4)(s + 10)
(9)
For discrete frequency values .Qr={ atT}={O.l, 0.7, 7} rad/s, the specification models in (8) and (9) yield: ~~su/~in.F{1.0012, 1.0607,3.1287} .
RULES FOR UNCERTAINTY DIVISION
• Divide the magnitude uncertainty of the artemplate to cut off the tracking bound height close to -21t, -1t, or -0 rad. And/or divide the phase uncertainty to reduce the bound height around -31t12 or -1tI2 rad. The option adopted will depend on the type (origin poles) of the system, the characteristics of the plant, or the expected benefits at low or middle frequencies.
(ii) Certain robust stability with minimum margins of
MJ
amplifications of noise or negligible high-frequency disturbances in the control input. In this way, the expected control performance demanded by (i) and (ii) will not be
• The translation of the template phase-magnitude uncertainty into the parameter (i.e. gain, pole, and zero)
520
specifications and, at the same time, minimize the control effort at all the frequencies (in particular (iii) at lLl..?-100).
spoilt by the saturation of the actuator. (b)~O.7rad1.
j+0
10 I
I t
~inalpt"'l
CD
5
I __ 1
i 1i
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- -,- - ~ OO~-
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--:---:---:--~--
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-10
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(d)~100radls
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.... _r)
, , '0 , ,0 - -:- - -, - -: - -i - ,
6Or-~"':";---.----,
-20
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(b)~7rad1s
-20
(c)
0Msk:ln2
I I
...,
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I
-10
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...,
8(j1)
·10 -ijh.,,(?:
E
i
-20
~
-60
plant_M --g..,' '---2~70--'~OO---:':90---'O
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Fig. 7. Plant templates for the fuIl parameter uncertainty -.-=[1, 10], and for its divisions 'l'[=[I, 4.5] and 'l'.F[4.5, 10].
·270
·180
_r)
-90
0
Fig. 8. Bounds and loop shaping designs for plant uncertainty subsets: (a) Full -.-=[1, 10], (b) Division 'l'r[I, 4.5], (c) Division
Considering the parameter uncertainty F[I, 10] and the discrete frequencies ll.F{0.1, 0.7, 7, lOO} rad/s, Fig. 7 depicts the plant iLl-templates for the whole uncertainty and its divisions, marked' , and '0'. For the whole set, note that the uncertainty in the low frequency template {P(jO.I)} is mainly phase uncertainty. The template {P(j0.7)} shows significant phase and magnitude uncertainties. And the templates {P(j7)} and {P(jIOO)} mainly display magnitude uncertainty. Thus, the pole parameter uncertainty in (7) is translated into different magnitude and phase uncertainties at the diverse frequencies. The division of the 'r parameter has a different impact on the templates and finally on the bounds, thus illustrating the insights into the contribution of uncertainty to frequency bounds covered in Section 4. The low frequency template is the reference used to perform the fragmentation, since this frequency requires the toughest tracking performance. Then, taking 'r[=[I, 4.5] and 'rZ= [4.5, 10], the phase uncertainty of the low frequency template {P(jO.I)} is more or less divided into two halves (see Fig. 7a). The magnitude and phase uncertainties of the rest of the templates also decrease (see Figs. 7b,c,d).
'l'.F[4.5, 10].
Only the tracking specification 5,(0.lrad/s)=1.0012 (~ is dismissed) is significant for the bounds B(jO.!) in Fig.8. These bounds look like bounds in Fig. 4 (only the upper part depicted here), since the 0.1 rad/s template mainly exhibits phase uncertainty (no gain uncertainty). As shown in Section 4, the larger the phase uncertainty, the higher the bound; compare B(jO.I) in Fig. 8(a) to Figs. 8(b)(c). If there had been a significant magnitude uncertainty, the largest benefits in terms of phase uncertainty division would have been at -rr/2 rad, that is, at the phase of interest after adding the integrator for zero tracking error. The relaxation of the bound B(jO.I) allows for the slowing down of the curve L,jj oJj. Thus, the control effort decreases at all the frequencies, while preserving the same dynamic accuracy (see Fig. 9). The tracking specification 5,(0.7rad/s)=1.0607 is also the dominant one to configure the bounds B(j0.7). Both phase and magnitude uncertainties are significant in the template {P(j0.7)}. Thus, B(j0.7) looks like to the bounds shown in Fig. 5. The reduction of phase and magnitude uncertainties improves the aggressiveness of the bound (compare Fig. 8(a) to Figs. 8(b)(c».
Fig. 8 displays the bound arrangement for the explicit specifications (i) and (ii): Fig 8(a) before the uncertainty fragmentation, and Fig. 8(b) and 8(c) for the divisions. Fig. 8 also includes the nominal plant P,jjoJj and nominal openloop transfer function L,jj w)=G(j oJjPoCi oJj for the three designs with nominal parameters TorlO, 'rod[=4.5, Todz=IO. According to Fig. 8(a), it is impossible to loop-shape a controller to meet the explicit bounds on robust stability and tracking specifications and cut off the implicit feedback cost in (iii) at the same time. There are two possible solutions. If the uncertainty remains, the tracking tolerances 8r={1.0012, 1.0607, 3.l287} at ll.F{0.1, 0.7, 7} rad/s must be relaxed, and then the tracking performance deteriorates. The other solution is reducing the uncertainty while preserving 5r values. Then, different controllers will be responsible of its operating area of uncertainty. Fig. 8(b) and 8(c) show the relevant designs. The new bound arrangements allow to fulfil the tracking and stability
Both
tracking 5,(7rad/s)=3.1287 and stability specifications contribute to the bound B(j7). The contribution of ~7)=1.3 is always an outer bound. However, 5,(7)=3.1287 produces upper bounds for the full uncertainty and its division I, and outer bounds for division 2. The reasons were extensively discussed in Section 4. The least favorable intersection between tracking and stability bounds is taken. The expected result is reached, that is the aggressiveness of the bound is relaxed. This causes a quick decrease in the high frequency open-loop gain !LoCi co) 1 for 0»7 in division I and for 0»0.7 in division 2 (see Fig. 8). ~7rad/s)=1.3
The only specification at high frequency, ~lOOrad/s)=1.3, configures the bound B(j 100). Note that this bound is computed for the whole uncertainty in the three designs.
521
This aims to preserve the robust stability in spite of any future switching failure in the controllers.
7.
Tracking feedback control trade-offs increase in the presence of large uncertainties. In the context of QFT, this is shown in the bounds. This paper has studied the bound formulation of tracking bounds in order to infer the contribution of uncertainty to the QFT bound aggressiveness. This has led to certain rules for uncertainty division in order to preserve or improve the tracking accuracy and reduce at the same time the control effort within the whole frequency range. The benefits have been illustrated in an example of tracking outflow commands in the outlet of a tank supplied by a centrifugal pump-servo valve system. Further studies would build a controllerscheduler [13] to switch the QFT controllers designed for each division. The final goal was to design accurate and reliable trackers in the presence of arbitrarily large uncertainties.
The loop-shapings LrFGPo performed on the bound arrangements in Figs. 8 give the following controllers: Gr(S)
0.16
(10)
429.5
1 171.4 (S) = 61.98(-S-+ 11 /(s(_s_+ III 0.70 67.36
GdJ(s)
Gd2
= 98.95(-S-+ 11/(S(_S_+ll')
=33.13(-S-+ 1 /(S(-S-+lll
(11)
0.31
(12)
The prefilter F to place all 1L/(1+L) I in-between 8rsup and 8rin/ following (2), (8), (9) is:
1 = (((~+111~ +11
F(s)
(13)
la)
CONCLUSIONS
le) 1
---/-~---,
ACKNOWLEDGEMENTS The authors gratefully appreciate the support given by the Spanish 'Comisi6n Interministerial de Ciencia y Tecnologia (CICYT)' under grant DPI'2000-0785.
I 0.5
....
1
1.5
2
2.5
0.5
....
1
1.5
2
2.5
....
1
1.5
2
REFERENCES
2.5
[I] LM. Horowitz, and M. Sidi, "Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances," 1nl. J. Control, voI. 16, no.2, pp. 287-309, 1972. [2] LM. Horowitz, Synthesis ofFeedback Systems. Academic Press: New York,1963. [3] C.H. Houpis, and SJ. Rassmussen, Quantitative Feedback Theory; Fundamentals and Applications. Marcel Dekker: NY, USA, 1999. [4] O. Yaniv, Quantitative Feedback Design of Linear and Non-linear Control Systems. KIuver Academic Publishers: Massachusetts, USA, 1999. [5] Y. Chait, and O. Yaniv, "Multi-input/single-output computer-aided control design using the Quantitative Feedback Theory," Int. J. Robust & Non-linear Control, vol.3, pp. 47-54, 1993. [6] C. Borghesani, Y. Chait, and O. Yaniv. Quantitative Feedback Theory Toolbox User's Guide. The Math Works Inc., USA; 1994. [7] M. Gil-Martinez, and M. Garcia-Sanz, "Simultaneous meeting of robust control specifications in QFf," Proc. 5th International Symposium on QFT and Robust Frequency Domain Methods, Pamplona, Spain, Aug. 2001, pp.I93-202. Also accepted for the Special Issue on Frequency Domain Methods of the Int. J. Robust Control, expected publication date: 2003. [8] M. Gil-Martinez, and M. Garcia-Sanz, "Robust specification influence on feedback control strategies." in Proc. 15th IFAC World Congress on Automatic Control, Barcelona, Spain, 21-26 July 2002, 6 pages. [9] M. Gil-Martinez, and M. Garcia-Sanz, "Uncertainty fragmentation to reduce static gain in QFf controllers," in Proc. UKACC International Conference on Control 2000, University of Cambridge, UK, Sep.20oo, pp.151(6 pages). [10] LM. Horowitz . "Survey of quantitative feedback theory (QFf)". Int. J. Control, 53 (2),255-291,1991. [I I] J.M. Rodrigues, Y. Chait, and C.V. Hollot. "An efficient algorithm for computing QFf bounds". ASME J. ofDyn. Syst.• Meas.• Control, 119 (3), pp.548-552; 1997. [12] M. Gil-Martinez. Sintesis de controladores robustos mediante el analisis de la compatibilidad de especijicaciones e incertidumbre. PhD Thesis, Public University of Navarra, Spain, 2001. [13] J.R. Wilson, and J.S. Shamma. "Research on gain scheduling". Automatica. 36, pp.1401-1425, 2000.
Fig. 9. Plots (a), (c) and (e): outflow qo tracking performance of G r , Gd/ and Gdb respectively; Plots (b), (d) and (e): control inflow qi demanded by G r , Gd/ and Gdb respectively. The simulations are for the nominal plants ror=lO, ~/=4.5, ~rlO.
Fig. 9 shows the behavior in time of the nominal plants 1"or=1O, 1"Od/=4.5, 1"odF1O. Note the similar performance of qo(t) as a response to a unit command in Figs. 9(a),(c),(e). Figs. 9(b),(d),(f) depict the inflow q,{t) demand. The whole uncertainty system exhibits an inflow qi about hundred times larger than the equilibrium inflow. Centrifugal pumps in fluid systems are often chosen to meet the demand at the equilibrium operating points plus a certain security coefficient, but never to supply a demand one hundred times the nominal one. Then, the dynamic inflow qi depicted in Fig. 9(b) would not be reachable with a physical system, and therefore, the theoretically good performance of outflow qo in Fig. 9(a) would not be possible. In contrast, after performing the uncertainty division, the dynamic inflow is just about ten times larger than the equilibrium inflow, and this demand can be easily satisfied with the centrifugal pump-control valve system. Then, the situations depicted in Figs. 9(c) to 9(f) are quite realistic. The improvements are due to the reduction of the uncertainty in each control design.
522
ELSEVIER
IFAC PUBLICATIONS www elsevier comllocate/ifac
PLANT UNCERTAINTY CONTRIBUTION TO QFT TRACKING CONTROL DESIGN GiI-Martinez, M. I and Garcia-Sanz, M. z I
System Engineering and Automation Group. Electrical Engineering Department. University ofLa Rioja. Luis de Ulloa. 20. 26004 Logrono. SPAIN Tf:+34 941299496. E-mail: [email protected] 2Automatic Control and Computer Science Department. Public University ofNavarra. Campus Arrosadia. sin. 31006 Pamplona. SPAIN Tf:+ 34948 16938. E-mail: [email protected]
Abstract: Tracking feedback control trade-offs increase dangerously with arbitrarily large uncertainties in the process model. This is shown by the bounds in the QFT domain. In this context, the paper develops a theory on the contribution of the plant uncertainty to the tracking QFT bounds through a serious study of tracking bound formulas. From this, a methodology for uncertainty fragmentation is developed. It hopes to loosen the bound toughness, or the control design trade-offs. Better feedback benefits can be achieved while reducing the high-frequency amplification of noises and disturbances at the plant input at the same time. Different QFT controllers in a controller-scheduler structure should drive different uncertainty boundaries. Copyright © 2003 IFAC Keywords: Quantitative Feedback Theory, Robust Control, Uncertain Dynamical Systems.
I.
[9]. This paper extends the study to any frequency, and is based on the bound fonnulas developed in [5].
INTRODUCTION
Feedback control necessarily implies trade-off designs, even for m.p. SISO plants. (a) A high enough open-loop gain lL=GPI is required at low-medium frequencies for robust command tracking and disturbance rejection. (b) ILl should decrease as fast as possible with frequency to avoid large noise amplifications mainly at the actuator inputs [I]. (c) ILl decrement rate is constrained to ensure robust stability. The controller design trade-offs increase with the uncertainty of the plant and the aggressiveness of the perfonnance. A quantitative fonnulation of the problem as QFT (Quantitative Feedback Theory [2]) is used to define a solvable robust feedback control problem.
Appropriate values for robust tracking and stability specifications within the QFT domain are discussed in Section 2. Section 3 describes the tracking bound fonnulation. The fonnal inequalities are analyzed in Section 4, and the contribution of uncertainty to QFT tracking bounds is inferred. Then, Section 5 summarizes the main rules to accomplish an uncertainty division aimed at improving feedback perfonnance. Section 6 provides an example. Finally, Section 7 includes the main conclusions. 2.
TRACKING SPECIFICATIONS IN QFT
Robust specifications in command tracking are usually the most restrictive ones according to the perfonnance limitations in the presence of arbitrarily large uncertainties. Thus, disturbance or noise rejection problems are ignored in this paper for simplicity reasons. The control structure of Fig.1 needs two-degrees of freedom [10]: a prefilter F and a controller G.
QFT takes the plant uncertainty and closed loop specifications and translates them into certain bounds on the nominal open-loop transfer function [3,4], Bound fonnulas in [5] were developed for automatic bound computation [6]. Based on these bound fonnulas, general feedback objectives were classified in different bound typologies in [7]. The bound typology depends on the nature of the requirement, the toughness of the specification and the size of the uncertainty. Assuming a fixed plant uncertainty, an achievable feedback control can be guaranteed by a proper selection of the specification tolerances [8]. However, arbitrarily large uncertainties will imply a poor-tracking perfonnance. A possible solution is to reduce the uncertainty. Then, a higher perfonnancelower cost controller will be responsible for each partition. A comprehensive study of the contribution of uncertainty to the bounds is required to maximize the benefits of uncertainty division. The uncertainty at the low frequency and its QFT bounds were approached in a graphical way in
Fig 1.
In the presence of certain open-loop uncertainty {P}={PLB}, the controller G aims to reduce the closedloop gain uncertainty {In} according to upper 8rSIIP and lower 8ritif models within the frequencies of interest:
517
· =I L(jm) :5 OT (m), Q T ={m:5 mT} IT(jm)1 1+ L(jm) OTsup (m)
0T (m)
°Tinf(m) ,
(1)
L(jm) =G(jm) P(jm)
Provided that the task of the prefilter F is to place {171} inbetween these models: (2)
The amount of feedback required depends on the tracking specification value t5r=t5rsuplt5rin/ and on the plant gain uncertainty Pmax1pm;n. As it was concluded in [7], QFT represents this requirement by upper bounds or typology n bounds (see Fig. 2b) when Pmax1pmin>t5r. This guarantees a minimum open-loop gain L at any phase for oEOJr. Tracking upper bounds become the dominant ones within the low-middle frequency range, and their aggressiveness increases when t5r is smaller [8]. In contrast, the cost of feedback [1] and stability requirements within the high m range imply the release of t5r. Then, Pmax1pmin
,.
(b) ",per bolftl
find the contribution of uncertainty to tracking bounds based on this mathematical formulation. The procedure is as follows. A set of discrete frequencies is chosen: {cq, i=l, ...,n}={QT tlQs} from (1) and (3). The border of the cq plant template {P(jcq)} is approximated by a finite m-point set of plants [11]. Each r=I, ... ,m plant can be expressed in its polar form as P,{j(4)=p,{cq)·~6ri.(J)I)=pLB, and likewise the polar form of the controller is G(jcq)=g(cq)'~;=gL4J (the phase controller dependence on the frequency has been suppressed considering that the 41 range is the same for any [4-bounds). The tracking model in (1) takes a discrete value at the frequency cq, t>rt.cq)=t5rsuplt5rint<.cq), being Or> 1 since (},sup> (},in/' Then, two plants in the artemplate are selected: PtFPdLBd, Pe=PeLBe. For a discrete controller phase f/JE l/);[-21t, 0], Inequality (1) is as follows:
I w; = P;PJ(I-
g 2 + 2pePd'
01
r' 8i I
PePd
1·{p,OO'C!H8'l-
)//
)~y ·'80
Fig. 2. QFT bound typologies
Stability requirements are also defined; these requirements imply certain robust phase and gain margins. For example:
MF
(3)
~ 180o(I-.3..cos-,(.Q211, MG ~ 2010g(1 +_11 1r
Os
(4)
OS
~«I
4.
implies too conservative stability margins, whereas I means highly underdamped systems. A ~> I implies outer bounds (see Fig.2a) [7] and, as long as ~ gets closer to I (for appropriate stability margins), these outer bounds become dominant at high frequencies [8]. The outer bound encloses a forbidden area around the stability point.
Os»
3.
Z00,\;+8,)1 (6)
1-
~; COS(q} + Oe)Y - (1- ;1 i(Pi - ~r)
Only real and positive solutions g/ and/or g;z in (6) impose constraints on g at the phase 41, and they configure the ar bound for a fixed pair of plants and the specification requirement (},. Then, the less favorable curve representing the tracking requirement at cq is computed by repeating the procedure for the whole set of pairs of plants in {P(j cq)}. Finally, this G-bound is translated into terms of the nominal loop transmission Lo=Po·G=loL'IIo for a nominal plant Po=poLBo in {Plo The conclusions for G-bounds are also applicable to Lo-bounds. Prior to the final loopshaping, the La-bound intersection amongst the bounds standing for the different robust feedback problems (stability, tracking commands, disturbance rejection,...) must be obtained. A bound plotted with a solid line implies that G(jm) (or Lo(jm) must lie above or on it to meet the particular specification, while a bound plotted with a dashed line means that G(j m) (or Lo(j OJ) must lie below or on it.
:V
/T(jm)1 = I L(jm) :5 0s(m),Qs = {ms E [O,oo]} 1+ L(jm)
01
The unknown parameter g (required magnitude of the controller) can be computed by solving I(J)I=O in (5). Two solutions are obtained: guarantee
(Pe cos(q} + Od) -
·1'.J6O
(5)
.(PeCOS(4J+Bd)- Pd COS(4J+Be)lg+(p; _ PJl ~ 0
g,.,
(c) lower bolRl
;1 I
PLANT UNCERTAINTY AND QFT BOUNDS
As discussed before, tracking specifications yield dominant upper bounds at low frequencies and non-dominant (as regards the stability requirements) outer bounds at high frequencies [7]. The upper bounds become more stringent when it rises and the outer bound when it encloses a bigger area (its upper portion rises, the lower one slows down, and both segments widen in the phase range). A smaller tracking tolerance t5r=(},supl (},inf implies severer bounds according to previous definitions (for a formal study see
TRACKING BOUND FORMULAS
An efficient numerical algorithm was proposed in [5,6] for the automatic computation of bounds. This paper tries to
518
[8]). The purpose of this paper is to examine the contribution of the uncertainty to tracking bound severity. The research is based on the formal procedure of QFT bound computation presented in Section 3, and its foundations are detailed in [12]. The main conclusions are following presented.
compute the bounds in Fig. 3, and thus omIttmg the quantitative height of the bound, the following qualitatively analysis can be made. The appearance of the bounds B I and B2 is typical of low frequencies, where feedback benefits require a strong sensitivity reduction and, thus that PmdPmin»~' The aspect of the bound B 3 is common at middle frequencies, where in spite of the possibility of PmdPmin being significant, the specification ~ begins to increase (high frequency gain cut off), and then, it is easy to meet ~l"?pmdPmin"?~. The appearance of the bounds B 4 and B 5 reminds of outer tracking bounds at high frequencies where PmdPmin«~, and these bounds are nondominant compared to outer robust stability bounds. 4Or------.-------,-----,-------,
30
'.&.o=----~.270=------:'.,00c:------'_90,.------l ~phase(")
Fig. 3. Tracking bounds on the controller for different sizes of the template magnitude uncertainty and no template phase uncertainty
Let's suppose a cq-template with certain phase uncertainty and certain magnitude uncertainty. To simplify the study, let's consider first the cq-template with only certain magnitude uncertainty. Fig. 3 depicts different G-bounds at cq for different sizes of magnitude uncertainty and a constant tracking specification tolerance M cq)= 1.1. For simplicity reasons, let's assume that the cq template is placed on the D phase angle, i.e. {PUcq)}={P(cq)}L{;{cq) with (;{cq)=D. In Fig. 3, the bound BIUcq) stands for {P(cq)}E {D.I,...,I}; B l for {P}E {O.I,...,O.2}; B3 for {P}E {O.I, ,O.1l5}; B 4 for {P}E {D.l,...,O.105}; and B 5 for {P}E {O.I, ,O.lOI}. According to the main results in [7], BU.3 are upper bounds since PmdPmin"?~, being Pmax=max{p(cq)} and Pmin=min{p(cq). However, the outer bounds B o stand for PmdPmin<~' Note that a logarithmic uncertainty converts linear differences Pmax-Pmin into ratios Lm(pmdPmin), where 'LmO' symbolizes '20/ogI00'. Note also that Fig. 3 depicts G-bounds that should be subsequently referred to the nominal plant Po to perform loop shaping onto Lo-bounds.
.10
·~=-----;;.270~----::.,00=-----=_90:-----...J.
00_' phase (") Fig. 4. Tracking bounds on the controller for different sizes of the template phase uncertainty and no template magnitude uncertainty.
Secondly, let's consider now a artemplate only with phase uncertainty. Fig. 4 depicts different bounds for {PU cq)} =p(cq)L {(;{ cq)} and a specification value Mcq)=1.1. The B 6 bound stands for {O}E {-n/6, ...,O} rad; and B 7 for {B}E {-n/I2,...,O} rad; both for the fixed gain p=O.1. And the B 8 bound stands for p= 1 and {B} E {n/6,...,D} rad. Fig. 4 shows first that the templates with phase uncertainty only give outer bounds, since PmdPmin<~, where PmdPmin=1 and 8,. is always greater than 1. Second, Fig. 4 illustrates how an increment in the phase uncertainty produces severer bounds (compare B 6 with B 7). Third, the severest bounds due to phase uncertainty are those with the minimum modulus P (compare B 6 with B 8 ). And finally, if these bounds are added to Fig. 3, the bound portion due to the phase uncertainty would become significant at the valleys of the bound portion due to the magnitude uncertainty.
Fig. 3 illustrates that when PmdPmin( cq) decreases (but the upper bound also goes down (see the sequence Bj, B 2, B3). Using the tracking bound formula (6) and for the case of upper bounds, it is possible to infer [12] that if PmdPmin"?~2"?8r. the upper bound has three peaks (see B I and B 2), whereas it only has one peak (see B 3) if ~l"?PmdPmin"?~' This proves the relaxation of the bounds when the magnitude uncertainty decreases. If PmdPmin keeps on decreasing beyond ~, that is, if ~2"?8i>-Pmax/Pmin, the upper bound becomes an outer bound, placed lower on the dB axe (see B 4 and B 5 ). The outer bound will keep on contracting (less enclosed area) if Pmax/Pmin decreases more with respect to ~ (compare the sequence B 4 and B 5)'
PmdPmin"?~,)
Ignoring the quantitative values {P} and
~
These results of separate studies of phase and magnitude uncertainties will be now combined. Let's consider a ar template {PUcq)}={pLB} with p(cq)E {O.l,...,l} and (;{cq)E {-n/6, ...,O}, and a tracking specification value M cq)= 1.1. Discretizing the {P} domain at three logspace points and {B} at seven points, the intersection arbound on G, called Br, is depicted in Fig. 5 with the darkest line. The main bound contributions of certain pairs of plants are also highlighted. See the simple bound Ba due to the plant pair {PmaxL.Bmin> PminL.Bmax}; the bound Bb due to the plant pair
used to
519
uncertainty can be a difficult task. Some insights were provided in [9].
{PmaxLOmax, PminLOmin}; and the bound Bc due to {PminLOmax,PminLOmin}' Bounds B T, Ba and Bb show that the size of the absolute magnitude uncertainty, !lp=PmdPmin> and also its relative values, {P}E {Pmin>' ..,Pmax}, are responsible for the points of maximum height in the bound. Besides, these maximum points extend along a phase range equal to phase uncertainty, /1(FOmax-Omin' Adding now Bc, the B T bound height at its valley points are due to the absolute phase uncertainty, i.e. /10. The phase placement of the B T bound on [-21t, 0] depends on the relative phase values {o}. However, once computed the bound on L o (add the nominal phase to the bound on G), it can be proven that the maximum points of the Ltrbound always occur at 21t::!:!10 , -n:::!:.!10, and -O+~O rad, whereas the minimum ones occur at -31t12±!10, -1tI2±!10 rad. Exceptionally, if 8/>pmdPmi';>~' Lo-bound maximums are only reached at -1t::!:!10 rad, and the bound slows down to both edges -21t and 0 rad (see curve Bj in Fig. 5). A formal demonstration can be found in [12].
• Few divisions must be performed to obtain as few designs as possible and avoid future unexpected instabilities in on-line switching. • The divisions must be somehow identifiable through certain auxiliary measurable variables, as the scheduler will switch particular controllers on-line. • The robust stability bound of each controller must be computed considering the full uncertainty instead of the reduced uncertainty. This prevents from unexpected on-line failures of the controller-scheduler spoiling performance while preserving the robust stability. 6.
EXAMPLE
Let's consider now the incompressible fluid system in Fig. 6. A servo-valve controls the inflow q;{t) (control variable) demanded by a centrifugal pump. The outflow qa(t) may vary according to step commands within a moderately large range. A linear parameter resistance R plus certain uncertainty can express the non-linear behavior of the load valve. Fluid storage in the tank is represented by a capacitance C. The linear uncertain model of the outflow control through the inflow around certain equilibrium point is: P(s) = Qo(s) Qi(S)
=
1 RC s+l
zos+l'
(7)
where the uncertain time constant is r-=[1, 10] sec. Control Valve
qlt)_~
\'
\~ "?.o~------l--::_270=-----=--':_1.. =-----7....:-----.JL-~
............ n h(t)
Fig. 5. Tracking bounds on the controller due to templates with both, phase and magnitude uncertainties.
L-*---~=t~...,.l==".qD(I)
These results can be easily extended to real templates that do not build rectangular shapes, as shown in the example in Section 6. A qualitatively conclusion of the study is that a smaller uncertainty in the phase and magnitude of the plant reduces the severity of the upper bound (bound height at each phase) that stands for a fixed specification tracking value M at). This proves the expected benefits of the uncertainty division, proposed as a solution to performance limitations due to excessive plant uncertainty and/or ambitious specifications. 5.
Load Valve .0
Capacitance
Resistance
C
R
Fig. 6. Incompressible fluid system
The outflow feedback control requirements are: (i) Robust tracking specifications in (1) and particularized for frequencies 0Jr~7 rad/s as follows:
(2)
OTsup(S) = 0.66(s+ 30)/(i +4s+ 19.75)
(8)
0Tinf(S) = 8400/(s + 3)(s + 4)(s + 10)
(9)
For discrete frequency values .Qr={ atT}={O.l, 0.7, 7} rad/s, the specification models in (8) and (9) yield: ~~su/~in.F{1.0012, 1.0607,3.1287} .
RULES FOR UNCERTAINTY DIVISION
• Divide the magnitude uncertainty of the artemplate to cut off the tracking bound height close to -21t, -1t, or -0 rad. And/or divide the phase uncertainty to reduce the bound height around -31t12 or -1tI2 rad. The option adopted will depend on the type (origin poles) of the system, the characteristics of the plant, or the expected benefits at low or middle frequencies.
(ii) Certain robust stability with minimum margins of
MJ
amplifications of noise or negligible high-frequency disturbances in the control input. In this way, the expected control performance demanded by (i) and (ii) will not be
• The translation of the template phase-magnitude uncertainty into the parameter (i.e. gain, pole, and zero)
520
specifications and, at the same time, minimize the control effort at all the frequencies (in particular (iii) at lLl..?-100).
spoilt by the saturation of the actuator. (b)~O.7rad1.
j+0
10 I
I t
~inalpt"'l
CD
5
I __ 1
i 1i
0
- -,- - ~ OO~-
-6
--:---:---:--~--
:J
E
,_ _
-60
I I
I I
.O~OO
I
I
I
I
..... -all
I
-20 _.(")
I
..., ...,
;;;-25
i"
I
- -1- -
0
I:
-I~-
I I
I
..,--,--
...,
~-all
e
1-35 ...,
·110
I
_ _ '_ _ _
I I
-100
_ _I _ _
I
J __
..., ....
-60 -65
" "
.... _·r)
-60
-eo
-eo
-110
~
I
...,
-} - ~,- - ~,--
,
,
,
-eo
-70
I
-
iD
"
i
--,--,-, , ' , --,--,--
_1_ _
-100
'0
I
- -~ - -,-, I
-10
-60
(d)~100radls
I
~
I
I
-70
.... _r)
, , '0 , ,0 - -:- - -, - -: - -i - ,
6Or-~"':";---.----,
-20
-10
(b)~7rad1s
-20
(c)
0Msk:ln2
I I
...,
I
Nominalplanl
I I
I
-10
0 •
ONsiOn 1
...,
8(j1)
·10 -ijh.,,(?:
E
i
-20
~
-60
plant_M --g..,' '---2~70--'~OO---:':90---'O
_.r)
Fig. 7. Plant templates for the fuIl parameter uncertainty -.-=[1, 10], and for its divisions 'l'[=[I, 4.5] and 'l'.F[4.5, 10].
·270
·180
_r)
-90
0
Fig. 8. Bounds and loop shaping designs for plant uncertainty subsets: (a) Full -.-=[1, 10], (b) Division 'l'r[I, 4.5], (c) Division
Considering the parameter uncertainty F[I, 10] and the discrete frequencies ll.F{0.1, 0.7, 7, lOO} rad/s, Fig. 7 depicts the plant iLl-templates for the whole uncertainty and its divisions, marked' , and '0'. For the whole set, note that the uncertainty in the low frequency template {P(jO.I)} is mainly phase uncertainty. The template {P(j0.7)} shows significant phase and magnitude uncertainties. And the templates {P(j7)} and {P(jIOO)} mainly display magnitude uncertainty. Thus, the pole parameter uncertainty in (7) is translated into different magnitude and phase uncertainties at the diverse frequencies. The division of the 'r parameter has a different impact on the templates and finally on the bounds, thus illustrating the insights into the contribution of uncertainty to frequency bounds covered in Section 4. The low frequency template is the reference used to perform the fragmentation, since this frequency requires the toughest tracking performance. Then, taking 'r[=[I, 4.5] and 'rZ= [4.5, 10], the phase uncertainty of the low frequency template {P(jO.I)} is more or less divided into two halves (see Fig. 7a). The magnitude and phase uncertainties of the rest of the templates also decrease (see Figs. 7b,c,d).
'l'.F[4.5, 10].
Only the tracking specification 5,(0.lrad/s)=1.0012 (~ is dismissed) is significant for the bounds B(jO.!) in Fig.8. These bounds look like bounds in Fig. 4 (only the upper part depicted here), since the 0.1 rad/s template mainly exhibits phase uncertainty (no gain uncertainty). As shown in Section 4, the larger the phase uncertainty, the higher the bound; compare B(jO.I) in Fig. 8(a) to Figs. 8(b)(c). If there had been a significant magnitude uncertainty, the largest benefits in terms of phase uncertainty division would have been at -rr/2 rad, that is, at the phase of interest after adding the integrator for zero tracking error. The relaxation of the bound B(jO.I) allows for the slowing down of the curve L,jj oJj. Thus, the control effort decreases at all the frequencies, while preserving the same dynamic accuracy (see Fig. 9). The tracking specification 5,(0.7rad/s)=1.0607 is also the dominant one to configure the bounds B(j0.7). Both phase and magnitude uncertainties are significant in the template {P(j0.7)}. Thus, B(j0.7) looks like to the bounds shown in Fig. 5. The reduction of phase and magnitude uncertainties improves the aggressiveness of the bound (compare Fig. 8(a) to Figs. 8(b)(c».
Fig. 8 displays the bound arrangement for the explicit specifications (i) and (ii): Fig 8(a) before the uncertainty fragmentation, and Fig. 8(b) and 8(c) for the divisions. Fig. 8 also includes the nominal plant P,jjoJj and nominal openloop transfer function L,jj w)=G(j oJjPoCi oJj for the three designs with nominal parameters TorlO, 'rod[=4.5, Todz=IO. According to Fig. 8(a), it is impossible to loop-shape a controller to meet the explicit bounds on robust stability and tracking specifications and cut off the implicit feedback cost in (iii) at the same time. There are two possible solutions. If the uncertainty remains, the tracking tolerances 8r={1.0012, 1.0607, 3.l287} at ll.F{0.1, 0.7, 7} rad/s must be relaxed, and then the tracking performance deteriorates. The other solution is reducing the uncertainty while preserving 5r values. Then, different controllers will be responsible of its operating area of uncertainty. Fig. 8(b) and 8(c) show the relevant designs. The new bound arrangements allow to fulfil the tracking and stability
Both
tracking 5,(7rad/s)=3.1287 and stability specifications contribute to the bound B(j7). The contribution of ~7)=1.3 is always an outer bound. However, 5,(7)=3.1287 produces upper bounds for the full uncertainty and its division I, and outer bounds for division 2. The reasons were extensively discussed in Section 4. The least favorable intersection between tracking and stability bounds is taken. The expected result is reached, that is the aggressiveness of the bound is relaxed. This causes a quick decrease in the high frequency open-loop gain !LoCi co) 1 for 0»7 in division I and for 0»0.7 in division 2 (see Fig. 8). ~7rad/s)=1.3
The only specification at high frequency, ~lOOrad/s)=1.3, configures the bound B(j 100). Note that this bound is computed for the whole uncertainty in the three designs.
521
This aims to preserve the robust stability in spite of any future switching failure in the controllers.
7.
Tracking feedback control trade-offs increase in the presence of large uncertainties. In the context of QFT, this is shown in the bounds. This paper has studied the bound formulation of tracking bounds in order to infer the contribution of uncertainty to the QFT bound aggressiveness. This has led to certain rules for uncertainty division in order to preserve or improve the tracking accuracy and reduce at the same time the control effort within the whole frequency range. The benefits have been illustrated in an example of tracking outflow commands in the outlet of a tank supplied by a centrifugal pump-servo valve system. Further studies would build a controllerscheduler [13] to switch the QFT controllers designed for each division. The final goal was to design accurate and reliable trackers in the presence of arbitrarily large uncertainties.
The loop-shapings LrFGPo performed on the bound arrangements in Figs. 8 give the following controllers: Gr(S)
0.16
(10)
429.5
1 171.4 (S) = 61.98(-S-+ 11 /(s(_s_+ III 0.70 67.36
GdJ(s)
Gd2
= 98.95(-S-+ 11/(S(_S_+ll')
=33.13(-S-+ 1 /(S(-S-+lll
(11)
0.31
(12)
The prefilter F to place all 1L/(1+L) I in-between 8rsup and 8rin/ following (2), (8), (9) is:
1 = (((~+111~ +11
F(s)
(13)
la)
CONCLUSIONS
le) 1
---/-~---,
ACKNOWLEDGEMENTS The authors gratefully appreciate the support given by the Spanish 'Comisi6n Interministerial de Ciencia y Tecnologia (CICYT)' under grant DPI'2000-0785.
I 0.5
....
1
1.5
2
2.5
0.5
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1
1.5
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2.5
....
1
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REFERENCES
2.5
[I] LM. Horowitz, and M. Sidi, "Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances," 1nl. J. Control, voI. 16, no.2, pp. 287-309, 1972. [2] LM. Horowitz, Synthesis ofFeedback Systems. Academic Press: New York,1963. [3] C.H. Houpis, and SJ. Rassmussen, Quantitative Feedback Theory; Fundamentals and Applications. Marcel Dekker: NY, USA, 1999. [4] O. Yaniv, Quantitative Feedback Design of Linear and Non-linear Control Systems. KIuver Academic Publishers: Massachusetts, USA, 1999. [5] Y. Chait, and O. Yaniv, "Multi-input/single-output computer-aided control design using the Quantitative Feedback Theory," Int. J. Robust & Non-linear Control, vol.3, pp. 47-54, 1993. [6] C. Borghesani, Y. Chait, and O. Yaniv. Quantitative Feedback Theory Toolbox User's Guide. The Math Works Inc., USA; 1994. [7] M. Gil-Martinez, and M. Garcia-Sanz, "Simultaneous meeting of robust control specifications in QFf," Proc. 5th International Symposium on QFT and Robust Frequency Domain Methods, Pamplona, Spain, Aug. 2001, pp.I93-202. Also accepted for the Special Issue on Frequency Domain Methods of the Int. J. Robust Control, expected publication date: 2003. [8] M. Gil-Martinez, and M. Garcia-Sanz, "Robust specification influence on feedback control strategies." in Proc. 15th IFAC World Congress on Automatic Control, Barcelona, Spain, 21-26 July 2002, 6 pages. [9] M. Gil-Martinez, and M. Garcia-Sanz, "Uncertainty fragmentation to reduce static gain in QFf controllers," in Proc. UKACC International Conference on Control 2000, University of Cambridge, UK, Sep.20oo, pp.151(6 pages). [10] LM. Horowitz . "Survey of quantitative feedback theory (QFf)". Int. J. Control, 53 (2),255-291,1991. [I I] J.M. Rodrigues, Y. Chait, and C.V. Hollot. "An efficient algorithm for computing QFf bounds". ASME J. ofDyn. Syst.• Meas.• Control, 119 (3), pp.548-552; 1997. [12] M. Gil-Martinez. Sintesis de controladores robustos mediante el analisis de la compatibilidad de especijicaciones e incertidumbre. PhD Thesis, Public University of Navarra, Spain, 2001. [13] J.R. Wilson, and J.S. Shamma. "Research on gain scheduling". Automatica. 36, pp.1401-1425, 2000.
Fig. 9. Plots (a), (c) and (e): outflow qo tracking performance of G r , Gd/ and Gdb respectively; Plots (b), (d) and (e): control inflow qi demanded by G r , Gd/ and Gdb respectively. The simulations are for the nominal plants ror=lO, ~/=4.5, ~rlO.
Fig. 9 shows the behavior in time of the nominal plants 1"or=1O, 1"Od/=4.5, 1"odF1O. Note the similar performance of qo(t) as a response to a unit command in Figs. 9(a),(c),(e). Figs. 9(b),(d),(f) depict the inflow q,{t) demand. The whole uncertainty system exhibits an inflow qi about hundred times larger than the equilibrium inflow. Centrifugal pumps in fluid systems are often chosen to meet the demand at the equilibrium operating points plus a certain security coefficient, but never to supply a demand one hundred times the nominal one. Then, the dynamic inflow qi depicted in Fig. 9(b) would not be reachable with a physical system, and therefore, the theoretically good performance of outflow qo in Fig. 9(a) would not be possible. In contrast, after performing the uncertainty division, the dynamic inflow is just about ten times larger than the equilibrium inflow, and this demand can be easily satisfied with the centrifugal pump-control valve system. Then, the situations depicted in Figs. 9(c) to 9(f) are quite realistic. The improvements are due to the reduction of the uncertainty in each control design.
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