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Quantitative feedback design for tracking error tolerance
Automatica 36 (2000) 319}326
Technical Communique
Quantitative feedback design for tracking error toleranceq Eduard Eitelberg NOY Business, 58 Baines Road, Durban 4001, South Africa Received 22 December 1998; received in "nal form 13 August 1999
Abstract The feedback system tracking error is related rigorously to the sensitivity function S"1/(1#¸). A two design degrees of freedom design procedure is presented that guarantees frequency-domain tracking error tolerances despite uncertainties in the feedback and feed-forward components of the system. This procedure is solidly based on and "ts into the general framework of the Quantitative Feedback Theory of Horowitz (1991, International Journal of Control 53(2), 255}291; 1993, Quantitative Feedback Design Theory (QFT). Boulder: QFT Publications). ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Sensitivity; Tracking error; Tolerances; Quantitative feedback theory
1. Introduction * 2DOF feedback control
The plant input is given by
In general, a SISO feedback system designer has to design two signal paths to the plant input * the feedforward from the reference r and the feedback from the measured output y. There are many ways to implement these two design degrees of freedom (2DOF). Fig. 1 shows a structure that is well suited for a quantitative step-by-step design. Availability of any measured disturbance would add another degree(s) of design freedom. In cases, where lower- and upper-case versions of the same letter are used, the former denotes a timedomain signal and the latter denotes its Laplace transform. The loop transfer function is de"ned as ¸"¸(s)"G(s)P(s)H(s).
(1)
The plant output is given by
C
D
¸ F 1 >" R!N # [DH#> ] 01 1#¸ H 1#¸ P ! ; . 1#¸ 01
(2)
q This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Prof. Peter Dorato. E-mail address: [email protected] (E. Eitelberg)
C
D
¸ ¸/P F R!N!DH!> # ; . ;" 01 1#¸ 01 1#¸ H
(3)
The primary control requirement is to achieve D>!RDP0 (4) within some suitably dexned frequency-domain specixcations * see Section 3 for details. The two main obstacles for satisfying Eq. (4) are unknown disturbances DH, > and ; , and the lack of knowledge about the plant 01 01 transfer function P * they are the additive and multiplicative uncertainties, respectively, in the design and operation of the plant. Reduction of the plant output sensitivity with respect to these two kinds of uncertainty of feedback is called regulating control or regulation. Additional equipment is needed for implementing regulating control * its uncertainty too must be included in a quantitative design procedure. This paper emphasises the central importance of the system sensitivity function S"1/(1#¸) in quantitative control system design. The main novelty is in relating the system tracking error to the sensitivity function. A simple 2DOF procedure is presented that permits derivation of design specixcations for the sensitivity function from given tracking error bounds and the uncertainty of all relevant control system components. There is no over-design when the tracking speci"cations contain the practical requirement of zero nominal error and the uncertainty of F/H is unstructured. A detailed design example is included.
0005-1098/00/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 1 4 9 - 1
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E. Eitelberg / Automatica 36 (2000) 319}326
or expected value. The quantitative feedback theory's (QFT) &nominal' is arbitrary but constant whereas in economics it is arbitrary and changing. d denotes the F relative uncertainty of F/H with respect to its nominal. Horowitz (1963, 1993) de"nes the relative transfer sensitivity as
Fig. 1. A 2DOF SISO feedback structure with the feed-forward controller F, feedback controller G and the sensor H with sensor noise/error n. The transfer function P describes small signal behaviour of the plant around the operating condition Mu , y N. The given plant 01 01 is indicated by heavy lines.
2. The sensitivity function and relative tracking error The sensitivity function is de"ned by 1 S(s)" , 1#¸(s)
(5)
where S determines the transfer of the equivalent output disturbance DH to the controlled output >. With no feedback, ¸"0 and there is no disturbance modi"cation. Only high gain feedback loop (D¸D<1) leads to small sensitivity and to disturbance reduction: The command/reference transfer to system output is de"ned by ¸(s) F(s) ¹(s)" . 1#¸(s) H(s)
(6)
With high gain feedback loop (D¸D<1), ¹PF/H. Therefore, the uncertainties of the feed-forward path F and of the return path H must be small * zero uncertainty is implicitly assumed in practically all of the ¤t' control theoretical literature. There will be significant uncertainty in the plant P and perhaps even in the feedback controller G, leading to the uncertainty of the loop transfer function ¸3M¸ N. The set of all possible i loop transfer functions is denoted by M¸ N and the varii ation is de"ned as * ¸"¸ !¸ , * is used similarly ik i k ik to denote the variation in any other transfer function, such as * ¹"¹ !¹ (where ¹ "¸ /(1#¸ )F/H) or ik i k j j j * P"P !P . ij i j Small uncertainties of complex equipment are often di$cult to specify in highly structured manner and it can be more productive/economical to exploit any structure of the uncertainty during system calibration. Therefore, the uncertainty of F/H is described here by
C D
F(s) F(s) " H(s) H(s)
(1#d (s)) F
(7)
/ The subscript &n' in Eq. (7) denotes &nominal ' in the usual engineering sense * the most important, probable
* ¹(s)/¹ (s) 1 ik k " "S (s). (8) i * ¸(s)/¸ (s) 1#¸ (s) ik k i Eq. (8) is correct only if F/H has no uncertainty * d "0. F This de"nition is identical to that in Eq. (5). Small sensitivity over the entire range of loop uncertainties is necessary not only for disturbance reduction but also for reliable command tracking. Before Horowitz, the transfer sensitivity used was employed to de"ne in"nitesimally small variations as a di!erential relative sensitivity S"(d ln ¹)/(d ln ¸). Although the uncertainty of F and H must be small, they cannot be assumed to be insignixcant. One can observe, that in many real systems, the feedback loop ¸ can easily take care of very large uncertainties of P and G to an extent that the ¹-uncertainty is dominated by the uncertainty of F/H in Eq. (6). For example, an industrystandard PID temperature control loop has in"nite lowfrequency gain * ¸/(1#¸)"1P¹"F/H * but an aged thermocouple and transmitter (H) may be in error by a few K. A slightly more involved real example is shooting with non-co-located gun and tracker on a moving platform: the reference for the gun positioning feedback loops is calculated from the tracker data after complicated coordinate transforms that require platform inertial position measurements * the associated transformation uncertainty (in F) can easily dominate the uncertainty of ¸/(1#¸). Below, the tracking error is related to the relevant individual multiplicative uncertainties of the whole control system. For clarity of presentation, sensor noise and other additive uncertainty in Eq. (2) are ignored for the rest of this paper. The primary control requirement in Eq. (4) can be based on the relative tracking error E . rk With the denotation > "¹ R, k k R!> k "1!¹ E " 3k k R ¸ k "1! 1#¸
CD F H
¸ k ! 1#¸
CD F H
d . (9) F / k / k De"ne the nominal relative error for nominal F/H at d "0: F F ¸ k . (10) E "1! /k 1#¸ H / k Substituting this back into Eq. (9) yields
CD
E "E !(1!E )d . 3k /k /k F
(11)
E. Eitelberg / Automatica 36 (2000) 319}326
The di!erence between two nominal relative errors can be expressed as
A
BC D CD
¸ F ¸ i ! k E !E " /k /i 1#¸ H 1#¸ / i k F ¸ /¸ !1 ¸ k " i k 1#¸ 1#¸ H k / i ¸ i !1 (1!E ). "S /k i ¸ k Hence,
A
A
B
B
¸ i !1 (1!E )#E E "S /k i ¸ /k /i k and substituting this back into Eq. (11) yields
CA
3. The design speci5cations and bounds (13)
B D
B
guarantees that ¸( ju) cannot come too close to the Nyquist point. For example, M"2 (6 dB) is equivalent to a gain margin of 2 (6 dB for gain increase), or 2/3 (!3.5 dB for gain decrease). M"J2 (3 dB) is equivalent to a gain margin of 3.4 (10 dB for gain increase), or 0.59 (!4.6 dB for gain decrease). The corresponding e!ective phase margins are approximately p/6 for M"2 and p/4 for M"J2.
(12)
¸ i !1 !d (1!E )#E . E " S (14) 3k i ¸ F /k /i k In practical designs, mostly, the P-uncertainty dominates the H- and G-uncertainties * ¸ /¸ +P /P * and i k i k small errors are speci"ed * E ;1. In this case, Eq. (14) /k simpli"es to
A
321
P i !1 !d #E . E +S (15) 3k i P F /i k Apart from E , there are two contributions to the /i relative error E * the feed-forward uncertainty 3k d (Dd D(DE D ) and the (nominal) system variability F F 3 .!9 * ¹/¹ "S * ¸/¸ at d "0. ik k i ik k F Eq. (15) is possibly the most important contribution of this paper and, in Section 3, it will be utilised in a new quantitative design procedure. Basically, the sensitivity S has to be designed for achieving suitably speci"ed (worst case, most likely, best possible, or other) system relative error characteristics for given plant and control equipment uncertainty. However, arbitrary speci"cations are not achievable. For example, it can be shown for stable ¸(s) with at least two excess poles over zeros and no poles on the imaginary axis that the integral of the logarithmic sensitivity magnitude (the Bode integral) :=lnDS( ju)D du"0, 0 see Horowitz (1963 or 1993). There is relatively more sensitivity increase in case of open-loop unstable systems that have poles a of multiplicity n in the right half-plane i i * :=lnDS( ju)D du"p + n Re a '0. See Freudenberg 0 i i i and Looze (1985) for a proof, or Horowitz (1993, p. 294) for suggestion of a di!erent proof that is closer to his original derivation. It happens to be very convenient that the same sensitivity function can be used for stability margin speci"cation as well. The general upper bound for the entire sensitivity set, with a given M'1 (M '0 dB), $B 1 DS ( ju)D" (M (16) k D1#¸ ( ju)D k
There are two simple design approaches. 1. Specify the sensitivity DS( ju)D for some (usually low) frequency range, derive the corresponding high gain ¸( ju) and then reduce D¸( ju)D for uPR as fast as possible. If the resulting closed-loop system cannot be stabilised with su$cient margins as de"ned by Eq. (16) (due to right half-plane zeros or dead-time/transportdelay) then relax the speci"cation(s). 2. Determine the maximum possible gain cross-over frequency u from the non-minimum phase-lag proper'# ties of the plant (Horowitz, 1993; Eitelberg, 1999), or by actual testing and then increase D¸( ju)D for uP0 as fast as possible. If the resulting closed-loop system has lower sensitivity than necessary and the plant input is distressed by noise, then reduce the gain cross-over frequency u . '# In real systems, plant sections often share inputs. Designing a high-performance fast loop around one plant section will couple this section's uncertainty to all sections that share the controlled input. This can be unacceptable and a compromise must be found between the requirements of the various sections. Level control of storage and processing vessels in the (chemical) process industry is an example * in fact, perfect level control loop performance would make a storage tank super#uous. Speci"cations for &compromising' level control loops have been derived by Eitelberg (1992). The low-frequency sensitivity speci"cation for the xrst approach can be derived from system output absolute and relative error speci"cations * only the latter is considered in this paper. A possible approach to deriving sensitivity bounds from the relative error equation (15) is to choose ¸ at the most common, important, or probable i operating condition and call it the nominal loop transfer function ¸ (another possibility is mentioned in Eitelberg ) (1999)). The rough outline of this procedure is explained next. d in Eq. (15) is a speci"ed set of complex numbers F around zero. After any design of the nominal loop ¸ "¸ the system variability term S (P /P !1) in ) i ) ) k Eq. (15) is a "xed set of complex numbers (k is not "xed) and it does not depend on the design of the pre-"lter F. In the second (2DOF) design step, the nominal relative
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E. Eitelberg / Automatica 36 (2000) 319}326
error E in Eq. (15), if found to be signi"cant in relation /h to the other two terms, can be forced to vanish by choosing F "H (1#¸ )/¸ (in the relevant low-fre/ / ) ) quency range) * this does not a!ect the other two terms of Eq. (15). Out of this discussion, the following procedure is proposed. The maximum permitted relative tracking error magnitude is denoted by DE D and the maximum un3 .!9 structured feed-forward uncertainty is denoted by Dd D F .!9 * both can be functions of frequency u and are speci"ed up to some maximum frequency u . .!9 Since the nominal [F/H] is the expected case and the / nominal ¸ is the most common, important, or probable ) case, it is sensible to demand zero nominal tracking error * E "0. This leads directly to the nominal sensitivity /) speci"cation for the feedback loop design ("rst degree of freedom):
K
K
1 DS ( ju)D" 4M ) 3% 1#¸ ( ju) ) DE D !Dd D 3 .!9 F .!9 , u(u . " (17) .!9 DP ( ju)/P ( ju)!1D ) k .!9 The feed-forward controller (second degree of freedom) results as F ( ju)"H ( ju)(1#¸~1( ju)), u(u . (18) / / ) .!9 One can argue that this procedure leads to over-design in two ways. The structures of d and S (P /P !1) F ) ) k may be such that DS (P /P !1)!d D ( ) ) k F .!9 DS (P /P !1)D #Dd D . This is true and relevant in ) ) k .!9 F .!9 principle, but in many practical situations d structure is F not su$ciently well known for pro"table exploitation. In fact, its maximum magnitude may not be known either, in which case something like a few standard deviations may have to be used. The other over-design arises when the &most important plant case' cannot be de"ned. Since structured plant uncertainty is not necessarily symmetrical around (an arbitrary) nominal, one can generally "nd a nonzero E so that DE #S (P /P !1)!d D ( /) /) ) ) k F .!9 DS (P /P !1)!d D . However, if the nominal is with) ) k F .!9 in the plant set, the improved error cannot be less than half of the error with zero nominal that corresponds to Eq. (18). At a given frequency, one can avoid this type of over-design by choosing a nominal P so that P~1 is in ) ) the centre of the smallest circle that encloses the set MP~1N * for high loop gain MS (P /P !1)N+(GH)~1 k ) ) k ](M1/P N!1/P ). Unfortunately, this P~1 may not be in ) k ) the ¢re' at other frequencies. From the practical point of view, the author does not know of an application where this slight over-design is a real problem * compared to all other real problems. In high precision control applications it can be quite acceptable to achieve better than speci"ed performance by good system calibration procedures that exploit
hidden or changing subsystem uncertainty structures and sizes. This is not possible in the present case, when zero nominal tracking error is part of the speci"cation and only Dd D is known. F The stability bound M on sensitivity S in Eq. (16) can be transformed uniquely into bounds on any arbitrarily chosen nominal loop transfer function ¸ * this is one of ) the important facts and activities in the Quantitative Feedback Theory of Horowitz (1993). A two degree of freedom relative error speci"cation and design has not yet been considered by the QFT developers in a direct manner. Among the numerous published options, the D¹D-speci"cation comes closest but it alone cannot theoretically guarantee better than 200% relative output error in the frequency domain * see Fig. 2. For various nonquantitative (¹-phase related) reasons, the actual error of the "nal design is usually very much smaller. Both of the above sensitivity bounds are uni"ed for a single quantitative design purpose by choosing ¸ of ) Eq. (17) as the nominal for the entire design. Transformation of the bounds and the subsequent design of the nominal ¸ ( ju) can be carried out on the ) sensitivity chart (also known as the inverse Nichols chart) * see the example in Section 4. At any given frequency, the stability bounds for ¸ ( ju) are found by sliding the ) template of MP( ju)N along (outside) the relevant M curves and recording the path of the handle P ( ju) (the handle ) of the template). Learning this manual procedure is important for understanding the principles and for interpretating the observed performance of implemented (imperfect) designs. It can be automated by a numerical procedure that was developed by Chait and Yaniv (1993). M needs no transformation, but has to be derived 3% from the uncertain plant set. A convenient procedure is obtained by observing in Eq. (17) that
K
K
1 M "[DE D !Dd D ] . 3% 3 .!9 F .!9 1#(!P ( ju))/P ( ju) ) k .*/ (19) Thus, for any given frequency, do the following on the sensitivity chart:
Fig. 2. Relationship between the transfer and error magnitude tolerances on the complex plane of ¹.
E. Eitelberg / Automatica 36 (2000) 319}326
1. Rotate the template MP( ju)N by p around the nominal P ( ju). ) 2. Shift the rotated template so that the handle P ( ju) ) coincides with the Nyquist point (!p, 0 dB). 3. Find the minimum value DSD of DSD that the (ro.*/,$B tated and shifted) template reaches * the handle is at the maximum #R. 4. M "[DE D !Dd D ] #DSD . 3%,$B 3 .!9 F .!9 $B .*/,$B This procedure, too, can be automated * see the example in Section 4. The design of the nominal loop transfer function ¸ ( ju) is carried out so that none of ) the two sets of (performance and stability) bounds are violated * these sets can be combined into a single set before the design step. In the framework of the second approach, only the stability bound (16) has to be satis"ed quantitatively for all P from the set MP N. De"nition of M is not di$cult. k k The choice of the nominal is free. Using the most common, important, or probable P has some merit, but the ) need for the set bound transformation can be obviated by using the worst-case P 3MP N! In designs with an uncon8# k ditional phase margin (see Eitelberg, 1999), P has 8# simultaneously the largest (non-minimum) phase-lag below the phase cross-over frequency and the highest gain above the gain cross-over frequency. Di!erent design situations can lead to di!erent useful de"nitions of the worst case P . A worst-case plant does not 8# necessarily exist in the set MP N. This is the (not unusual) k case in practice when ¸ is conditionally stable or close to it. It is often still possible to avoid the set bound transformation when one acquires the skill of working simultaneously with two worst-case loops * one that is approximately the closest to the stability margins for frequencies below gain cross-over and the other above gain cross-over. Maximising the low-frequency gain of any (nominal or worst-case) ¸ will obviously minimise all low-frequency sensitivities in Eq. (17), as well as the corresponding errors, virtually independent of Arg ¸. It is assumed that the gain uncertainty of P is bounded from below * it cannot become zero. The "rst approach or one similar to it is favoured by some #ight control design circles and the Quantitative Feedback Theory adherents. Something similar to the second approach is generally favoured in the process control and instrumentation circles. Both approaches are strongly linked by the phase and gain relationships of transfer functions * see Bode (1945), Horowitz (1963, 1993) or Eitelberg (1999).
4. A design example Consider an elastic mechanical positioning system (such as a radar antenna or an overhead crane) with the
323
transfer function 10 P(s)" . 1#2d(s/u )#(s/u )2 # #
(20)
Due to load and sti!ness variations the corner frequency can vary within the range u 3[0.04 0.1]. The # damping factor is known only approximately within the range d3[0.1 0.4]. The nominal condition is (chosen as) u "0.08 and d "0.2. Positioning systems generally / / have signi"cant o!sets, due to gravitational force and other reasons. Therefore, it is reasonable to specify at steady state D¸(0)DPR.
(21)
Assume that non-steady-state additive disturbances are insigni"cant and that the overall relative error speci"cation is 2% for u between 0 and 0.1. Let us budget a quarter of this for H and F * meaning Dd D "0.5%. F .!9 The remainder is left for variability: [DE D !Dd D ]"0.015"!36 dB r .!9 F .!9 for u3[0 0.1].
(22)
Use M"6 dB as the stability margin. Often, a constant sensitivity speci"cation is most di$cult to achieve at the highest speci"ed frequency. Alternatively, achievement of the sensitivity speci"cation at the highest frequency may automatically satisfy it for the entire frequency range. Therefore, let us &do the design' at u"0.1 and check at other frequencies afterwards. The sensitivity bound M is 3% found as follows. The plant uncertainty range (the template) at u"0.1 is de"ned in Table 1 and shown in Fig. 3. The rotated and shifted template is shown on the sensitivity chart in Fig. 4. Now, M "!36!16"!52 dB. 3%,$B
(23)
The above calculation results and graphical presentations are easily obtained in Matlab by executing the following few commands: r wc".04:.01:.1; d".1:.1:.4; w".1; r [WM, DM]"meshgrid(wc, d); r P"10./((1!(w./WM). ) 2)#jH2H DMHw./WM); % template r Ppi"angle(P)/pi; PdB"20Hlog10(abs(P)); %Arg(P)&DPD r Ph"10/((1!(w/0.08) )2)#jH2H0.2Hw/0.08); %handle (nom) r Phpi"angle(Ph)/pi;PhdB"20Hlog10(abs(Ph)); r plot(Ppi, PdB,&x', Phpi, PhdB,&o')
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E. Eitelberg / Automatica 36 (2000) 319}326 Table 1 Plant uncertainty range at u"0.1 Arg P in p radians and DPD in dB u # d
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.1
!0.97 5.6
!0.96 10.4
!0.94 14.9
!0.91 19.3
!0.87 24.2
!0.76 29.8
!0.50 34.0
0.2
!0.94 5.4
!0.92 10.2
!0.89 14.4
!0.84 18.5
!0.77 22.5
!0.65 26.0
!0.50 28.0
0.3
!0.91 5.3
!0.88 9.8
!0.84 13.8
!0.78 17.4
!0.70 20.6
!0.61 23.0
!0.50 24.4
0.4
!0.88 5.0
!0.84 9.4
!0.80 13.1
!0.74 16.2
!0.66 18.8
!0.58 20.7
!0.50 21.9
Fig. 3. Plant uncertainty range (the template): &]' at u"0.1 and &#' at u"1. The nominal (the handle) is encircled.
Fig. 4. De"nition !16 dB.
The shifting and rotation of the template is carried out by:
pected around u+1. This approximate analysis helps us to decide at what frequencies to derive the stability bounds for ¸ . ) Clearly, we need plant templates at u51. Fig. 3 shows the template at u"1 (the plant frequency response points are shifted up by 50 dB). The phase uncertainty is very small and will become smaller for u'1. The magnitude uncertainty is essentially due to the &high-frequency' uncertainty in P(hf )+!10u2/u2. In this example, there # is no dead-time or right half-plane zeros, hence, all &higher'-frequency templates are almost identical and lead to the same stability bound for these frequencies as is shown in Fig. 5. In order to satisfy the !52 dB relative error bound, a gain of 52}22.5 dB "29.5 dB is needed. ¸ ( ju)"30P ( ju) with P (s)"10/ (1#5s#156.25s2) )1 ) ) is shown in Fig. 6 as L1. Next, we need about 0.2p lead at u"0.7. From standard Bode plots, one "nds that
r Psr"Ph./P; r Psrpi"angle(Psr)/pi; PsrdB"20Hlog 10(abs(Psr)); r plot(Psrpi!1,PsrdB,*x@) The graphical procedure in Fig. 4 is very necessary for developing an understanding of the relationships between uncertainty and sensitivity and for practical design and tuning skills. A more exact value !16.3 for DSD .*/,$B can also be calculated directly in Matlab by executing: r SmdB"20Hlog10(min(min(abs(1./(1!Psr))))) Eq. (23) means that ¸ ( j0.1) must have a magnitude ) of 52 dB (or greater). From Fig. 3 we see that the set M¸( j0.1)N will have a corresponding magnitude range of 32 to 62 dB. Assuming an approximate average slope of !30 dB/dec, the earliest gain cross-over is to be ex-
of
DSD "D1/(1#(!P (ju))/P ( ju))D " .*/,$B / k .*/
E. Eitelberg / Automatica 36 (2000) 319}326
Fig. 5. Design bounds for ¸ ( ju). )
Fig. 6. Design of ¸ ( ju). )
(1#s/u ) gives 0.2p lead at 0.7u . Hence u "1 # # # and G (s)"30(1#s). The corresponding ¸ ( ju)" 2 )2 G ( ju)P ( ju) is shown in Fig. 6 as L2. 2 ) A strictly proper controller is obtained by adding a second-order lag term. The "rst attempt is made with the damping factor d"0.7 and desired lag of about 0.25p at u"5. From standard Bode plots, one "nds that 1/(1#2ds/u #s2/u2) gives 0.25p lag at about 0.7u . # # # Hence u "5/0.7"7. This is modi"ed slightly to # d"0.8 and u "10, yielding G (s)"30(1#s)/ # 3 (1#1.6 s/10#s2/100). The corresponding ¸ ( ju)" )3 G ( ju)P ( jx) is shown in Fig. 6 as L3. 3 ) Finally, in"nite steady-state loop gain is obtained by adding the integrating term (1#u /s). In order to # have an unconditionally stable design, at least for ¸ , we would not like to have more than about 0.1p ) of additional lag at u"0.2. From standard Bode plots, one "nds that (1#u /s) gives a little less than # 0.1p lag at 4u . Hence u "0.2/4"0.05 and # # G (s)"30(1#s)(1#0.05/s)/(1#1.6s/10#s2/100). The 4 corresponding "nal ¸ ( ju)"G ( ju)P ( ju) is shown in )4 4 ) Fig. 6 as L4.
325
The "nal design ¸ ( ju) appears to be slightly under)4 designed at and above u"5. This was done on purpose * to allow for some additional non-minimum phase-lag from a fast sampling digital controller implementation. An even greater phase reserve should be left when low sampling rate is a digital controller implementation requirement. Furthermore, ¸ ( ju) is slightly under-de)4 signed between u"0.7 and u"5 * reducing phase lead and making the corresponding slope of ¸ steeper in this frequency range would require a higher-order controller transfer function. The tracking transfer function ¹"¸/(1#¸) (with F/H"1) is displayed in Fig. 7 for the speci"ed range of the uncertain d and u . # The set of ¹"¸/(1#¸) at u"0.1 is displayed in Fig. 8 for the speci"ed range of the uncertain d and u . # For the four combinations of the extreme d and u , as # well as for the nominal plant, the entire frequency response is shown for u3[0 0.1]. The maximum distance to ¸ /(1#¸ ) is 0.015"1.5%. Fig. 8 con"rms that the ) ) largest error is (in this case) at the highest frequency of the speci"cation, u"0.1. The maximum deviation from a reference would be 1.7% because ¹ O1. However, ) 1.5% can be achieved by shifting ¹ to 1 with the help of ) an appropriately designed pre-"lter F. At lower frequencies, ¸ /(1#¸ ) is so close to 1 that it should not be ) ) shifted at all. In the presented example, we could simply specify a constant DF/HD"0.998 for all frequencies within u3[0 0.1]. For this to make sense, F and H would have to be implemented so that the accuracy of the above ratio is better than 0.1%! This leads to the practical necessity of (regular) calibrations of the sensor H, the pre-"lter F, or both. The ability of the above designed feedback system to track time-domain references within and beyond the speci"ed frequency range [0, 0.1] is shown in Fig. 9 for the two extreme and two intermediate values of the plant's u * the plant's damping factor uncertainty # has negligible e!ect on the system output. For simplicity, DF/HD"1 in the following simulations. The
Fig. 7. Bode plots of ¹"¸/(1#¸).
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corresponding plant inputs and outputs are shown in Fig. 10. With less than 1% error, the output traces cannot be separated by the naked eye.
5. Conclusion
Fig. 8. Arithmetic plane representation of ¹"¸/(1#¸) at u"0.1. The nominal ¹ "¸ /(1#¸ ) is encircled. ) ) h
A simple 2DOF system sensitivity based design procedure has been developed and demonstrated that guarantees frequency-domain tracking error tolerances despite uncertainties in the feedback and feedforward components of the system. There is no over-design when the tracking speci"cations contain the practical requirement of zero nominal error and the combined return path and feed-forward uncertainty is unstructured. On the one hand, this procedure is solidly based on and "ts into the general framework of the QFT of Horowitz (1991, 1993). On the other hand, the presented procedure competes with one of the earliest 2DOF design procedures of the QFT * the one that is based on the command transfer magnitude tolerances.
References
Fig. 9. Reference tracking relative error Dy(t)!r(t)D/maxDr(t)D in %. r(t)"2 sin(0.06t)#0.7 sin(0.1t)#0.005 sin(t); maxDr(t)D"2.705.
Fig. 10. Plant inputs (top part) and outputs (bottom part) that correspond to Fig. 9.
Bode, H. (1945). Network analysis and feedback amplixer design. New York: van Nostrand. Chait, Y., & Yaniv, O. (1993). Multi-input/single-output computer aided control design using the Quantitative Feedback Theory. International Journal of Robust and Nonlinear Control, 3, 47}54. Eitelberg, E. (1992). Storage size from process control bandwidth speci"cation and the other way around. QFT Symposium, Wright-Patterson Air Force Base, Dayton, Ohio, USA (pp. 264}274). Eitelberg, E. (1999). Load sharing control. Durban: NOYB Press. Freudenberg, J. S., & Looze, D. P. (1985). RHP poles and zeros and design tradeo!s in feedback systems. IEEE Automattic Control, 30, 555}565. Horowitz, I. (1963). Synthesis of feedback systems. New York: Academic Press. Horowitz, I. (1991). Survey of quantitative feedback theory (QFT). International Journal of Control, 53(2), 255}291. Horowitz, I. (1993). Quantitative feedback design theory (QFT). Boulder: QFT Publications.
Technical Communique
Quantitative feedback design for tracking error toleranceq Eduard Eitelberg NOY Business, 58 Baines Road, Durban 4001, South Africa Received 22 December 1998; received in "nal form 13 August 1999
Abstract The feedback system tracking error is related rigorously to the sensitivity function S"1/(1#¸). A two design degrees of freedom design procedure is presented that guarantees frequency-domain tracking error tolerances despite uncertainties in the feedback and feed-forward components of the system. This procedure is solidly based on and "ts into the general framework of the Quantitative Feedback Theory of Horowitz (1991, International Journal of Control 53(2), 255}291; 1993, Quantitative Feedback Design Theory (QFT). Boulder: QFT Publications). ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Sensitivity; Tracking error; Tolerances; Quantitative feedback theory
1. Introduction * 2DOF feedback control
The plant input is given by
In general, a SISO feedback system designer has to design two signal paths to the plant input * the feedforward from the reference r and the feedback from the measured output y. There are many ways to implement these two design degrees of freedom (2DOF). Fig. 1 shows a structure that is well suited for a quantitative step-by-step design. Availability of any measured disturbance would add another degree(s) of design freedom. In cases, where lower- and upper-case versions of the same letter are used, the former denotes a timedomain signal and the latter denotes its Laplace transform. The loop transfer function is de"ned as ¸"¸(s)"G(s)P(s)H(s).
(1)
The plant output is given by
C
D
¸ F 1 >" R!N # [DH#> ] 01 1#¸ H 1#¸ P ! ; . 1#¸ 01
(2)
q This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Prof. Peter Dorato. E-mail address: [email protected] (E. Eitelberg)
C
D
¸ ¸/P F R!N!DH!> # ; . ;" 01 1#¸ 01 1#¸ H
(3)
The primary control requirement is to achieve D>!RDP0 (4) within some suitably dexned frequency-domain specixcations * see Section 3 for details. The two main obstacles for satisfying Eq. (4) are unknown disturbances DH, > and ; , and the lack of knowledge about the plant 01 01 transfer function P * they are the additive and multiplicative uncertainties, respectively, in the design and operation of the plant. Reduction of the plant output sensitivity with respect to these two kinds of uncertainty of feedback is called regulating control or regulation. Additional equipment is needed for implementing regulating control * its uncertainty too must be included in a quantitative design procedure. This paper emphasises the central importance of the system sensitivity function S"1/(1#¸) in quantitative control system design. The main novelty is in relating the system tracking error to the sensitivity function. A simple 2DOF procedure is presented that permits derivation of design specixcations for the sensitivity function from given tracking error bounds and the uncertainty of all relevant control system components. There is no over-design when the tracking speci"cations contain the practical requirement of zero nominal error and the uncertainty of F/H is unstructured. A detailed design example is included.
0005-1098/00/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 1 4 9 - 1
320
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or expected value. The quantitative feedback theory's (QFT) &nominal' is arbitrary but constant whereas in economics it is arbitrary and changing. d denotes the F relative uncertainty of F/H with respect to its nominal. Horowitz (1963, 1993) de"nes the relative transfer sensitivity as
Fig. 1. A 2DOF SISO feedback structure with the feed-forward controller F, feedback controller G and the sensor H with sensor noise/error n. The transfer function P describes small signal behaviour of the plant around the operating condition Mu , y N. The given plant 01 01 is indicated by heavy lines.
2. The sensitivity function and relative tracking error The sensitivity function is de"ned by 1 S(s)" , 1#¸(s)
(5)
where S determines the transfer of the equivalent output disturbance DH to the controlled output >. With no feedback, ¸"0 and there is no disturbance modi"cation. Only high gain feedback loop (D¸D<1) leads to small sensitivity and to disturbance reduction: The command/reference transfer to system output is de"ned by ¸(s) F(s) ¹(s)" . 1#¸(s) H(s)
(6)
With high gain feedback loop (D¸D<1), ¹PF/H. Therefore, the uncertainties of the feed-forward path F and of the return path H must be small * zero uncertainty is implicitly assumed in practically all of the ¤t' control theoretical literature. There will be significant uncertainty in the plant P and perhaps even in the feedback controller G, leading to the uncertainty of the loop transfer function ¸3M¸ N. The set of all possible i loop transfer functions is denoted by M¸ N and the varii ation is de"ned as * ¸"¸ !¸ , * is used similarly ik i k ik to denote the variation in any other transfer function, such as * ¹"¹ !¹ (where ¹ "¸ /(1#¸ )F/H) or ik i k j j j * P"P !P . ij i j Small uncertainties of complex equipment are often di$cult to specify in highly structured manner and it can be more productive/economical to exploit any structure of the uncertainty during system calibration. Therefore, the uncertainty of F/H is described here by
C D
F(s) F(s) " H(s) H(s)
(1#d (s)) F
(7)
/ The subscript &n' in Eq. (7) denotes &nominal ' in the usual engineering sense * the most important, probable
* ¹(s)/¹ (s) 1 ik k " "S (s). (8) i * ¸(s)/¸ (s) 1#¸ (s) ik k i Eq. (8) is correct only if F/H has no uncertainty * d "0. F This de"nition is identical to that in Eq. (5). Small sensitivity over the entire range of loop uncertainties is necessary not only for disturbance reduction but also for reliable command tracking. Before Horowitz, the transfer sensitivity used was employed to de"ne in"nitesimally small variations as a di!erential relative sensitivity S"(d ln ¹)/(d ln ¸). Although the uncertainty of F and H must be small, they cannot be assumed to be insignixcant. One can observe, that in many real systems, the feedback loop ¸ can easily take care of very large uncertainties of P and G to an extent that the ¹-uncertainty is dominated by the uncertainty of F/H in Eq. (6). For example, an industrystandard PID temperature control loop has in"nite lowfrequency gain * ¸/(1#¸)"1P¹"F/H * but an aged thermocouple and transmitter (H) may be in error by a few K. A slightly more involved real example is shooting with non-co-located gun and tracker on a moving platform: the reference for the gun positioning feedback loops is calculated from the tracker data after complicated coordinate transforms that require platform inertial position measurements * the associated transformation uncertainty (in F) can easily dominate the uncertainty of ¸/(1#¸). Below, the tracking error is related to the relevant individual multiplicative uncertainties of the whole control system. For clarity of presentation, sensor noise and other additive uncertainty in Eq. (2) are ignored for the rest of this paper. The primary control requirement in Eq. (4) can be based on the relative tracking error E . rk With the denotation > "¹ R, k k R!> k "1!¹ E " 3k k R ¸ k "1! 1#¸
CD F H
¸ k ! 1#¸
CD F H
d . (9) F / k / k De"ne the nominal relative error for nominal F/H at d "0: F F ¸ k . (10) E "1! /k 1#¸ H / k Substituting this back into Eq. (9) yields
CD
E "E !(1!E )d . 3k /k /k F
(11)
E. Eitelberg / Automatica 36 (2000) 319}326
The di!erence between two nominal relative errors can be expressed as
A
BC D CD
¸ F ¸ i ! k E !E " /k /i 1#¸ H 1#¸ / i k F ¸ /¸ !1 ¸ k " i k 1#¸ 1#¸ H k / i ¸ i !1 (1!E ). "S /k i ¸ k Hence,
A
A
B
B
¸ i !1 (1!E )#E E "S /k i ¸ /k /i k and substituting this back into Eq. (11) yields
CA
3. The design speci5cations and bounds (13)
B D
B
guarantees that ¸( ju) cannot come too close to the Nyquist point. For example, M"2 (6 dB) is equivalent to a gain margin of 2 (6 dB for gain increase), or 2/3 (!3.5 dB for gain decrease). M"J2 (3 dB) is equivalent to a gain margin of 3.4 (10 dB for gain increase), or 0.59 (!4.6 dB for gain decrease). The corresponding e!ective phase margins are approximately p/6 for M"2 and p/4 for M"J2.
(12)
¸ i !1 !d (1!E )#E . E " S (14) 3k i ¸ F /k /i k In practical designs, mostly, the P-uncertainty dominates the H- and G-uncertainties * ¸ /¸ +P /P * and i k i k small errors are speci"ed * E ;1. In this case, Eq. (14) /k simpli"es to
A
321
P i !1 !d #E . E +S (15) 3k i P F /i k Apart from E , there are two contributions to the /i relative error E * the feed-forward uncertainty 3k d (Dd D(DE D ) and the (nominal) system variability F F 3 .!9 * ¹/¹ "S * ¸/¸ at d "0. ik k i ik k F Eq. (15) is possibly the most important contribution of this paper and, in Section 3, it will be utilised in a new quantitative design procedure. Basically, the sensitivity S has to be designed for achieving suitably speci"ed (worst case, most likely, best possible, or other) system relative error characteristics for given plant and control equipment uncertainty. However, arbitrary speci"cations are not achievable. For example, it can be shown for stable ¸(s) with at least two excess poles over zeros and no poles on the imaginary axis that the integral of the logarithmic sensitivity magnitude (the Bode integral) :=lnDS( ju)D du"0, 0 see Horowitz (1963 or 1993). There is relatively more sensitivity increase in case of open-loop unstable systems that have poles a of multiplicity n in the right half-plane i i * :=lnDS( ju)D du"p + n Re a '0. See Freudenberg 0 i i i and Looze (1985) for a proof, or Horowitz (1993, p. 294) for suggestion of a di!erent proof that is closer to his original derivation. It happens to be very convenient that the same sensitivity function can be used for stability margin speci"cation as well. The general upper bound for the entire sensitivity set, with a given M'1 (M '0 dB), $B 1 DS ( ju)D" (M (16) k D1#¸ ( ju)D k
There are two simple design approaches. 1. Specify the sensitivity DS( ju)D for some (usually low) frequency range, derive the corresponding high gain ¸( ju) and then reduce D¸( ju)D for uPR as fast as possible. If the resulting closed-loop system cannot be stabilised with su$cient margins as de"ned by Eq. (16) (due to right half-plane zeros or dead-time/transportdelay) then relax the speci"cation(s). 2. Determine the maximum possible gain cross-over frequency u from the non-minimum phase-lag proper'# ties of the plant (Horowitz, 1993; Eitelberg, 1999), or by actual testing and then increase D¸( ju)D for uP0 as fast as possible. If the resulting closed-loop system has lower sensitivity than necessary and the plant input is distressed by noise, then reduce the gain cross-over frequency u . '# In real systems, plant sections often share inputs. Designing a high-performance fast loop around one plant section will couple this section's uncertainty to all sections that share the controlled input. This can be unacceptable and a compromise must be found between the requirements of the various sections. Level control of storage and processing vessels in the (chemical) process industry is an example * in fact, perfect level control loop performance would make a storage tank super#uous. Speci"cations for &compromising' level control loops have been derived by Eitelberg (1992). The low-frequency sensitivity speci"cation for the xrst approach can be derived from system output absolute and relative error speci"cations * only the latter is considered in this paper. A possible approach to deriving sensitivity bounds from the relative error equation (15) is to choose ¸ at the most common, important, or probable i operating condition and call it the nominal loop transfer function ¸ (another possibility is mentioned in Eitelberg ) (1999)). The rough outline of this procedure is explained next. d in Eq. (15) is a speci"ed set of complex numbers F around zero. After any design of the nominal loop ¸ "¸ the system variability term S (P /P !1) in ) i ) ) k Eq. (15) is a "xed set of complex numbers (k is not "xed) and it does not depend on the design of the pre-"lter F. In the second (2DOF) design step, the nominal relative
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E. Eitelberg / Automatica 36 (2000) 319}326
error E in Eq. (15), if found to be signi"cant in relation /h to the other two terms, can be forced to vanish by choosing F "H (1#¸ )/¸ (in the relevant low-fre/ / ) ) quency range) * this does not a!ect the other two terms of Eq. (15). Out of this discussion, the following procedure is proposed. The maximum permitted relative tracking error magnitude is denoted by DE D and the maximum un3 .!9 structured feed-forward uncertainty is denoted by Dd D F .!9 * both can be functions of frequency u and are speci"ed up to some maximum frequency u . .!9 Since the nominal [F/H] is the expected case and the / nominal ¸ is the most common, important, or probable ) case, it is sensible to demand zero nominal tracking error * E "0. This leads directly to the nominal sensitivity /) speci"cation for the feedback loop design ("rst degree of freedom):
K
K
1 DS ( ju)D" 4M ) 3% 1#¸ ( ju) ) DE D !Dd D 3 .!9 F .!9 , u(u . " (17) .!9 DP ( ju)/P ( ju)!1D ) k .!9 The feed-forward controller (second degree of freedom) results as F ( ju)"H ( ju)(1#¸~1( ju)), u(u . (18) / / ) .!9 One can argue that this procedure leads to over-design in two ways. The structures of d and S (P /P !1) F ) ) k may be such that DS (P /P !1)!d D ( ) ) k F .!9 DS (P /P !1)D #Dd D . This is true and relevant in ) ) k .!9 F .!9 principle, but in many practical situations d structure is F not su$ciently well known for pro"table exploitation. In fact, its maximum magnitude may not be known either, in which case something like a few standard deviations may have to be used. The other over-design arises when the &most important plant case' cannot be de"ned. Since structured plant uncertainty is not necessarily symmetrical around (an arbitrary) nominal, one can generally "nd a nonzero E so that DE #S (P /P !1)!d D ( /) /) ) ) k F .!9 DS (P /P !1)!d D . However, if the nominal is with) ) k F .!9 in the plant set, the improved error cannot be less than half of the error with zero nominal that corresponds to Eq. (18). At a given frequency, one can avoid this type of over-design by choosing a nominal P so that P~1 is in ) ) the centre of the smallest circle that encloses the set MP~1N * for high loop gain MS (P /P !1)N+(GH)~1 k ) ) k ](M1/P N!1/P ). Unfortunately, this P~1 may not be in ) k ) the ¢re' at other frequencies. From the practical point of view, the author does not know of an application where this slight over-design is a real problem * compared to all other real problems. In high precision control applications it can be quite acceptable to achieve better than speci"ed performance by good system calibration procedures that exploit
hidden or changing subsystem uncertainty structures and sizes. This is not possible in the present case, when zero nominal tracking error is part of the speci"cation and only Dd D is known. F The stability bound M on sensitivity S in Eq. (16) can be transformed uniquely into bounds on any arbitrarily chosen nominal loop transfer function ¸ * this is one of ) the important facts and activities in the Quantitative Feedback Theory of Horowitz (1993). A two degree of freedom relative error speci"cation and design has not yet been considered by the QFT developers in a direct manner. Among the numerous published options, the D¹D-speci"cation comes closest but it alone cannot theoretically guarantee better than 200% relative output error in the frequency domain * see Fig. 2. For various nonquantitative (¹-phase related) reasons, the actual error of the "nal design is usually very much smaller. Both of the above sensitivity bounds are uni"ed for a single quantitative design purpose by choosing ¸ of ) Eq. (17) as the nominal for the entire design. Transformation of the bounds and the subsequent design of the nominal ¸ ( ju) can be carried out on the ) sensitivity chart (also known as the inverse Nichols chart) * see the example in Section 4. At any given frequency, the stability bounds for ¸ ( ju) are found by sliding the ) template of MP( ju)N along (outside) the relevant M curves and recording the path of the handle P ( ju) (the handle ) of the template). Learning this manual procedure is important for understanding the principles and for interpretating the observed performance of implemented (imperfect) designs. It can be automated by a numerical procedure that was developed by Chait and Yaniv (1993). M needs no transformation, but has to be derived 3% from the uncertain plant set. A convenient procedure is obtained by observing in Eq. (17) that
K
K
1 M "[DE D !Dd D ] . 3% 3 .!9 F .!9 1#(!P ( ju))/P ( ju) ) k .*/ (19) Thus, for any given frequency, do the following on the sensitivity chart:
Fig. 2. Relationship between the transfer and error magnitude tolerances on the complex plane of ¹.
E. Eitelberg / Automatica 36 (2000) 319}326
1. Rotate the template MP( ju)N by p around the nominal P ( ju). ) 2. Shift the rotated template so that the handle P ( ju) ) coincides with the Nyquist point (!p, 0 dB). 3. Find the minimum value DSD of DSD that the (ro.*/,$B tated and shifted) template reaches * the handle is at the maximum #R. 4. M "[DE D !Dd D ] #DSD . 3%,$B 3 .!9 F .!9 $B .*/,$B This procedure, too, can be automated * see the example in Section 4. The design of the nominal loop transfer function ¸ ( ju) is carried out so that none of ) the two sets of (performance and stability) bounds are violated * these sets can be combined into a single set before the design step. In the framework of the second approach, only the stability bound (16) has to be satis"ed quantitatively for all P from the set MP N. De"nition of M is not di$cult. k k The choice of the nominal is free. Using the most common, important, or probable P has some merit, but the ) need for the set bound transformation can be obviated by using the worst-case P 3MP N! In designs with an uncon8# k ditional phase margin (see Eitelberg, 1999), P has 8# simultaneously the largest (non-minimum) phase-lag below the phase cross-over frequency and the highest gain above the gain cross-over frequency. Di!erent design situations can lead to di!erent useful de"nitions of the worst case P . A worst-case plant does not 8# necessarily exist in the set MP N. This is the (not unusual) k case in practice when ¸ is conditionally stable or close to it. It is often still possible to avoid the set bound transformation when one acquires the skill of working simultaneously with two worst-case loops * one that is approximately the closest to the stability margins for frequencies below gain cross-over and the other above gain cross-over. Maximising the low-frequency gain of any (nominal or worst-case) ¸ will obviously minimise all low-frequency sensitivities in Eq. (17), as well as the corresponding errors, virtually independent of Arg ¸. It is assumed that the gain uncertainty of P is bounded from below * it cannot become zero. The "rst approach or one similar to it is favoured by some #ight control design circles and the Quantitative Feedback Theory adherents. Something similar to the second approach is generally favoured in the process control and instrumentation circles. Both approaches are strongly linked by the phase and gain relationships of transfer functions * see Bode (1945), Horowitz (1963, 1993) or Eitelberg (1999).
4. A design example Consider an elastic mechanical positioning system (such as a radar antenna or an overhead crane) with the
323
transfer function 10 P(s)" . 1#2d(s/u )#(s/u )2 # #
(20)
Due to load and sti!ness variations the corner frequency can vary within the range u 3[0.04 0.1]. The # damping factor is known only approximately within the range d3[0.1 0.4]. The nominal condition is (chosen as) u "0.08 and d "0.2. Positioning systems generally / / have signi"cant o!sets, due to gravitational force and other reasons. Therefore, it is reasonable to specify at steady state D¸(0)DPR.
(21)
Assume that non-steady-state additive disturbances are insigni"cant and that the overall relative error speci"cation is 2% for u between 0 and 0.1. Let us budget a quarter of this for H and F * meaning Dd D "0.5%. F .!9 The remainder is left for variability: [DE D !Dd D ]"0.015"!36 dB r .!9 F .!9 for u3[0 0.1].
(22)
Use M"6 dB as the stability margin. Often, a constant sensitivity speci"cation is most di$cult to achieve at the highest speci"ed frequency. Alternatively, achievement of the sensitivity speci"cation at the highest frequency may automatically satisfy it for the entire frequency range. Therefore, let us &do the design' at u"0.1 and check at other frequencies afterwards. The sensitivity bound M is 3% found as follows. The plant uncertainty range (the template) at u"0.1 is de"ned in Table 1 and shown in Fig. 3. The rotated and shifted template is shown on the sensitivity chart in Fig. 4. Now, M "!36!16"!52 dB. 3%,$B
(23)
The above calculation results and graphical presentations are easily obtained in Matlab by executing the following few commands: r wc".04:.01:.1; d".1:.1:.4; w".1; r [WM, DM]"meshgrid(wc, d); r P"10./((1!(w./WM). ) 2)#jH2H DMHw./WM); % template r Ppi"angle(P)/pi; PdB"20Hlog10(abs(P)); %Arg(P)&DPD r Ph"10/((1!(w/0.08) )2)#jH2H0.2Hw/0.08); %handle (nom) r Phpi"angle(Ph)/pi;PhdB"20Hlog10(abs(Ph)); r plot(Ppi, PdB,&x', Phpi, PhdB,&o')
324
E. Eitelberg / Automatica 36 (2000) 319}326 Table 1 Plant uncertainty range at u"0.1 Arg P in p radians and DPD in dB u # d
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.1
!0.97 5.6
!0.96 10.4
!0.94 14.9
!0.91 19.3
!0.87 24.2
!0.76 29.8
!0.50 34.0
0.2
!0.94 5.4
!0.92 10.2
!0.89 14.4
!0.84 18.5
!0.77 22.5
!0.65 26.0
!0.50 28.0
0.3
!0.91 5.3
!0.88 9.8
!0.84 13.8
!0.78 17.4
!0.70 20.6
!0.61 23.0
!0.50 24.4
0.4
!0.88 5.0
!0.84 9.4
!0.80 13.1
!0.74 16.2
!0.66 18.8
!0.58 20.7
!0.50 21.9
Fig. 3. Plant uncertainty range (the template): &]' at u"0.1 and &#' at u"1. The nominal (the handle) is encircled.
Fig. 4. De"nition !16 dB.
The shifting and rotation of the template is carried out by:
pected around u+1. This approximate analysis helps us to decide at what frequencies to derive the stability bounds for ¸ . ) Clearly, we need plant templates at u51. Fig. 3 shows the template at u"1 (the plant frequency response points are shifted up by 50 dB). The phase uncertainty is very small and will become smaller for u'1. The magnitude uncertainty is essentially due to the &high-frequency' uncertainty in P(hf )+!10u2/u2. In this example, there # is no dead-time or right half-plane zeros, hence, all &higher'-frequency templates are almost identical and lead to the same stability bound for these frequencies as is shown in Fig. 5. In order to satisfy the !52 dB relative error bound, a gain of 52}22.5 dB "29.5 dB is needed. ¸ ( ju)"30P ( ju) with P (s)"10/ (1#5s#156.25s2) )1 ) ) is shown in Fig. 6 as L1. Next, we need about 0.2p lead at u"0.7. From standard Bode plots, one "nds that
r Psr"Ph./P; r Psrpi"angle(Psr)/pi; PsrdB"20Hlog 10(abs(Psr)); r plot(Psrpi!1,PsrdB,*x@) The graphical procedure in Fig. 4 is very necessary for developing an understanding of the relationships between uncertainty and sensitivity and for practical design and tuning skills. A more exact value !16.3 for DSD .*/,$B can also be calculated directly in Matlab by executing: r SmdB"20Hlog10(min(min(abs(1./(1!Psr))))) Eq. (23) means that ¸ ( j0.1) must have a magnitude ) of 52 dB (or greater). From Fig. 3 we see that the set M¸( j0.1)N will have a corresponding magnitude range of 32 to 62 dB. Assuming an approximate average slope of !30 dB/dec, the earliest gain cross-over is to be ex-
of
DSD "D1/(1#(!P (ju))/P ( ju))D " .*/,$B / k .*/
E. Eitelberg / Automatica 36 (2000) 319}326
Fig. 5. Design bounds for ¸ ( ju). )
Fig. 6. Design of ¸ ( ju). )
(1#s/u ) gives 0.2p lead at 0.7u . Hence u "1 # # # and G (s)"30(1#s). The corresponding ¸ ( ju)" 2 )2 G ( ju)P ( ju) is shown in Fig. 6 as L2. 2 ) A strictly proper controller is obtained by adding a second-order lag term. The "rst attempt is made with the damping factor d"0.7 and desired lag of about 0.25p at u"5. From standard Bode plots, one "nds that 1/(1#2ds/u #s2/u2) gives 0.25p lag at about 0.7u . # # # Hence u "5/0.7"7. This is modi"ed slightly to # d"0.8 and u "10, yielding G (s)"30(1#s)/ # 3 (1#1.6 s/10#s2/100). The corresponding ¸ ( ju)" )3 G ( ju)P ( jx) is shown in Fig. 6 as L3. 3 ) Finally, in"nite steady-state loop gain is obtained by adding the integrating term (1#u /s). In order to # have an unconditionally stable design, at least for ¸ , we would not like to have more than about 0.1p ) of additional lag at u"0.2. From standard Bode plots, one "nds that (1#u /s) gives a little less than # 0.1p lag at 4u . Hence u "0.2/4"0.05 and # # G (s)"30(1#s)(1#0.05/s)/(1#1.6s/10#s2/100). The 4 corresponding "nal ¸ ( ju)"G ( ju)P ( ju) is shown in )4 4 ) Fig. 6 as L4.
325
The "nal design ¸ ( ju) appears to be slightly under)4 designed at and above u"5. This was done on purpose * to allow for some additional non-minimum phase-lag from a fast sampling digital controller implementation. An even greater phase reserve should be left when low sampling rate is a digital controller implementation requirement. Furthermore, ¸ ( ju) is slightly under-de)4 signed between u"0.7 and u"5 * reducing phase lead and making the corresponding slope of ¸ steeper in this frequency range would require a higher-order controller transfer function. The tracking transfer function ¹"¸/(1#¸) (with F/H"1) is displayed in Fig. 7 for the speci"ed range of the uncertain d and u . # The set of ¹"¸/(1#¸) at u"0.1 is displayed in Fig. 8 for the speci"ed range of the uncertain d and u . # For the four combinations of the extreme d and u , as # well as for the nominal plant, the entire frequency response is shown for u3[0 0.1]. The maximum distance to ¸ /(1#¸ ) is 0.015"1.5%. Fig. 8 con"rms that the ) ) largest error is (in this case) at the highest frequency of the speci"cation, u"0.1. The maximum deviation from a reference would be 1.7% because ¹ O1. However, ) 1.5% can be achieved by shifting ¹ to 1 with the help of ) an appropriately designed pre-"lter F. At lower frequencies, ¸ /(1#¸ ) is so close to 1 that it should not be ) ) shifted at all. In the presented example, we could simply specify a constant DF/HD"0.998 for all frequencies within u3[0 0.1]. For this to make sense, F and H would have to be implemented so that the accuracy of the above ratio is better than 0.1%! This leads to the practical necessity of (regular) calibrations of the sensor H, the pre-"lter F, or both. The ability of the above designed feedback system to track time-domain references within and beyond the speci"ed frequency range [0, 0.1] is shown in Fig. 9 for the two extreme and two intermediate values of the plant's u * the plant's damping factor uncertainty # has negligible e!ect on the system output. For simplicity, DF/HD"1 in the following simulations. The
Fig. 7. Bode plots of ¹"¸/(1#¸).
326
E. Eitelberg / Automatica 36 (2000) 319}326
corresponding plant inputs and outputs are shown in Fig. 10. With less than 1% error, the output traces cannot be separated by the naked eye.
5. Conclusion
Fig. 8. Arithmetic plane representation of ¹"¸/(1#¸) at u"0.1. The nominal ¹ "¸ /(1#¸ ) is encircled. ) ) h
A simple 2DOF system sensitivity based design procedure has been developed and demonstrated that guarantees frequency-domain tracking error tolerances despite uncertainties in the feedback and feedforward components of the system. There is no over-design when the tracking speci"cations contain the practical requirement of zero nominal error and the combined return path and feed-forward uncertainty is unstructured. On the one hand, this procedure is solidly based on and "ts into the general framework of the QFT of Horowitz (1991, 1993). On the other hand, the presented procedure competes with one of the earliest 2DOF design procedures of the QFT * the one that is based on the command transfer magnitude tolerances.
References
Fig. 9. Reference tracking relative error Dy(t)!r(t)D/maxDr(t)D in %. r(t)"2 sin(0.06t)#0.7 sin(0.1t)#0.005 sin(t); maxDr(t)D"2.705.
Fig. 10. Plant inputs (top part) and outputs (bottom part) that correspond to Fig. 9.
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