- Email: [email protected]
The Effect of Uncertainty on Controller Performance
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
Copyright C0 1999 IF~t\C 14th Triennial 'rVorld Congress,
l3c~ilng~
14th World Congress of IFAC
N-7a-12-l
P.R. China
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFOR.l\1ANCE Rohit S. Patwardhan, Sirish L. Shah and Biao Huang
Department of Chemical and ./'vfate7'ials Engineering llni'lJersity of Alberta, Edmonton, Canada, T6G 2G6. email: [email protected]
Abstract: In 1110St model-based control sC}lemes, the desig"n performance is rarely achieved due to the presence of Inudel plant mismatch. In this study the effect of modelling uncertainty on achieved controller performance is investigated. Particular attention is paid to uncertainty in the time delay. With the design performance as the reference point, bounds are derived on the achieved variances for the manipulated variables and controlled outputs, in the presence of model plant mismatch. For the deterministic case, similar bounds are obtained on the 2-norm (sum of errors squared) of the output errors. A simulation example is presented to illustrate the results. Copyright
~)
19991r:4C
Keywords: Uncertainty, Performance, Robustness, Time Delay
1.
I~TRODlJCTIO~
The objective of controller design is to achieve satisfa.ctory performance under feedback. The characteristics of a process decide the choice of one or IIlore design objectives. Conventionally a single measure of the tracking/regulatory performance in either the time domain or the frequency dOJnaill fOTlllS the design objective. 1.1oT8 often than not a process has features which limit the achievable performance of any controller. i\..mongst these limiting features are the presence of time delay~ IlOllminimum phase zeros~ constraints on the inputs and outputs, etc.
At the outset it is llnportant to kno\v the contribution of these featl1res to'\vards the best a.chievable performance. The knowledge of the best achievable pcrforll1allce then serves as a benchmark. This benchnlark is useful in assessing the performance of any controller. 'The contribution of the time delay term towards minimum achievable variance is V\TcII knOV,tll for linear processes, and forms tIle basis of the InilliIllUrIl VariaI1Ce benchmarking in performance assessment (Harris, 1989; Huang et ai., 1997).
In practice many components of the model, including the time delay, are never precisely known. The process tilnc delay may be time varyillg due to changing flOVi.r rates. Furthermore any chemical process is inherently nonlinear, and is approximated by a fixed linear model only in the neighbourhood of the operating point. These and other factors lead to the pre..~ence of uncertainty in modelling. A controller whose design is based on the nominal model of the process is unlikely to deliver the design perfOrnlaIlCe in the presence of model plant IuisIIlatch (1tIP1f). The presence of uncertainty leads to a gap between the designed and the achieved performance. The knowledge of the C'.olltributioIl of UIlceTtainty to\vaTds the possible deterioration of controller performance is vital to performance assessment and diagnosis (Ko~ub, 1996)~
The effect of uncertainty on closed loop stability is "veIl known and is the topic of several books and articles (sec for example - Doylc et al., 1992; J\Jlorari and Zafiriou~ 1989). The objective of this work is to quantify the effect of modelling nncertainty on the achieved performance of a lin-
ear model based controller. For the regulatory
6962
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
14th World Congress of IFAC
The sensitivity function is defined as
S
y
==
(1 +PC)-l
and the complementary sensitivity fUIlction or the closed loop transfer function as
Fig. 1.
The conventional feedback system
case with stochastic inputs, the use of Parseval's theorem is mstrumental in obtaining bounds on the achieved input and output variances, relative to the design ca..r;;e. For the case of deterministic inputs, the induced Hoc norm definition is used to derive bounds on the achieved performance in relation to the design performance, witll the 2-norrIl (sum of squared errors (SSE)) as a measure of performance. A simulation example is used to illustrate the effects of uncertainty in time delay on controller perforlllance, in t.he context of minimum variance benchmark as a measure of performance. The rest of this paper is organized as follows: Section 2 gives the preliminaries required for furtIler analysis. TIlls includes discussion of conventional feedback control, introduction to ~ome common norms for signals and systems and the internal model control framework which wc adopt throughout the report. I I I section 3 \ve derive bounds for the Vlorst case achieved performance for an uncertain systeln.The effect of uncertainty in tirne delay is llighlighted. The results are extended to the case of det.erministic inputs. A simulation example, in section 5, is used to illustrate the lnaill results. Sectioll 6 gives concluding rernarks.
Throughout this paper we restrict ourselves to the class of discrete~ stable systems v.rhich are finite diInensional linear tirHe invariant. (FDLTI). For the sake of brevity, ",re ha"ve omitted the argmnent z in ¥lTitjng the relations between the signals and the transfer functions for discrete systellls. The theory developed here is applicable to SI80 systems.
2. PRELI1/I INAR1ES Figure 1 shows the standard feedback configuration~ rThe measurement error m is assumed to be unknown and most frequently described with a white noise process. The disturbance d is made up of measured and tmmeasured disturbances. For a 8180 process, the closed loop relations bet"veen the inputs and the outputs can be ~vrittell as:
(1
+ PC)y == d + peer -
m,)
Therefore "re have y
== Bd + Tr - Tm
(1)
For a discussion on trade-offs involved in design of the sensitivity function, S, see I\1aciejowski (1989). Some idea of the "size" of a signal and the relevant transfer functions, such as the sensitivity function, are needed in order to quantify performance of a control system. Two norms for capturing the magnitude of a signal are:
2-norm oo-norm
j
IIul12 = {-u(O)2 + u(1)2 + ....} 1/2 lull(X) == suplu(k)1
(2)
k
TIle follu\ving tv.ro systeIll I10TIllS provide v..rays of capturing the mag-'nitude of a linear time invariant system (Chen and Francis, 1995): 21i"
2-nOTm
IIPI!z = (2~ ~
C<)-norm
IIP ~r
I CXJ
J
IP(e
jW
)1 2 dw)1/2
o p(O)2 + P(1)2
-
-sup
IP( e jW)( --
w
+ .... sup
Il u 112#O
(3)
11Yll2 -11-1-1 11.
2
The 2-noIIIl of a signal is iIlvariant with respect to the Z-transforrn. In addition to basic properties that define a norm, the (X)-norm obeys the following property,
(4) A detailed discussion on signal and system nOTIIlS and their properties, for univariate as ,"Tell as lnultivariatc systems, can be found in Boyd and Barrat (1991) . The internal model control (IMC) framework provides a way of rc-parametrizing the conventional controller (IVlorari and Zafiriou~ 1989). It is \vell known that this approach is equi vdlent to the celebrated Youla~I(ucera parameterization of all stabilizing controllers for a stable system~ Figure 2 shows the 11\1[0 franle"vork. The controller Q in the 111:C framevlork is related to the conventional feedback controller C in Figure 1 by
Q
=
0(1
+ pC)-l
(5)
where P is the nominal model of the process. For stable plants a stable Q guarantees internal stability. The nominal sensitivity function and
6963
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
y
14th World Congress of IFAC
In the following discussion 1,-ve rest.rict ourselves to the case where there are no setpoint changes, i.e. the regulatory case, and the disturbance is a realization of a white noise process (see Figure 2). Using Parseval's Theorem ,"ve have
+
Fig. 2. The internal model control stucture the complimentary sensitivity function can be expressed ill ternls of Q as
(6) The main advantage of this parametrization is that the sensitivity and the complementary sen-
In a similar \vay we get the following lower bound.
Var(y) 2: III
sitivity function are linear vvith respect to Q, the I~!{C controller. There is a one-ta-one corre-
spondence between the IMC controller and the cOllventional controller C.
c
~
Therefore the achieved perforluance is bound by 1
3. EFFECT OF UNCERTAINTY IN THE PLANT MODEL
No identification procedure or mechanistic model can be relied upon to deliver all exact Illodel of the actual physical system. This gives rise to modelling uncertainty. Uncertainty can be structured or unstructured. T"vo COlnIllon ,vays of representing unstructured uncertainty are given below, where P is the model and P is the plant. Additive Ullcertaillty: P 1:fultiplicative uncertainy: P
< Var(y) < II(
111 + QLlI,'~ -
Q(1 - FQ)-l
In the remaining discussion V\.re use the 11/IC fraulcvlork for analysis l)urposes.
== f> + ~ (7) == P(l + ~Tn)
+ ~AII;" Var(!/)
Var(y) -
1
-+ Q~
)-] 11 2 00·
(11) The above expression quantifies the bounds on achieved variance, in relation to the design variance, in the presence of modeling uncertainties. The performance bOUlld can be further simplified and expressed in terulS of IIQ~IICXJ for the case \vhere IIQ6.lloo < 1 (Boyd and Barrat, 1991).
(1
+
1 < l/ar(y) < IIQ~llCX))2 - V ar(y) ~ (1 -
1_ _--::::-
1IQ.6. Iloo )2 (12)
Using a similar approach Vle can define an upper bound for the increase ill. input energy (variance) due to uncertainty.
In this section we anal.yze the effect of modelliIlg uncertainty on the performance of a con-
trol system. In particular we derive bounds on the achieved performance in terms of the design performance and the CX)-norm of a certain transfer
function. The achieved sensitivity and the cOlllpleruentary sensitivity functions, in the presence of the additive uncertainty, are given by
S == (1 - QP)(l 7'
== QP(! +
+ QA)-l
Q~)-l
(8)
Thus the achieved sensitivity function is related to the design sensitivity function in eq. 6 by
S ~ 8(1
Once again we have the following tilue domain bounds on input variance due to rvIPJ\.-1
+ Q~)-l
(9)
Thus wc have arrived at the sanle perforulance bounds for both, the input and output variances. The geometry of this region i~ highlighte
6964
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
14th World Congress of IFAC
Worst Case Performance
Let p ~ jIQL\jloo: Robustness ~Ieasure.
Var(y)
The smaller the value of p the more robust is the control system, and conversely) the closer it is to unit-v the closer it is to instability due to IVIP1tL In ter~s of the robustness lneasure the performance bound (eq. 11) can now be re\vritten as:
DC:iign Performance
Var(u)
Fig. 3.
Illustration of the effect of modelling uncertainty on achieved performance
3 1 . If the MP:NI was negligible this region would be very small and the bigger it is the larger the possible degradation of the COlltrol systenl in terms of loss in performance.
Remark 1. The lo\,,~er bound derived in eqns. 11, being less than unity, does not guarantee that performance is \vorse ill presence of nlodel plant rniSluatch. This is an interesting observation since it implies that the achieved performance may actually be better thaIl the designed performance. For example this can ha.ppen when the delay is overestimated (see simulation results). Remark 2. It should be noted that eqns. 11 13 give bounds on the achieved performance that t.hese ean be conservative. The actual (or gain) in performance ,"vould depend on spcctrulll of the disturbances in relation to spectruIil of the uncertainty.
(14)
and and loss the
the
In case the above performance bounds indicate
a significant change in design performance, the next step is to determine the robustness of the existing controller with respect to the estimated uncertainty. Let us assume that the controller design was done v.rith the nominal model as the plant. A controller C i~ said to provide rob'lLst stability if it provides stability for every plant in the (uncertain) set. The small gain theorem (SGT) provides a ,vay of assessing robust stability.
Theorern 1. For the Illultiplicative uIlcertainty model, C provides robust stability iff Ilt~1nli,::o < 1, Vv~hcre T is the nominal complementary sensitivity functioIl Pro~f. For proof see Doyle et al. (1992) .
A simple measure of robustness follows from application of th.c SeT to additive uncertainty case. For robust stability Vi/re have 1 T'he design performance is shown in relation to the design curve for the linear quadratic regulatur (LQR). The LQR performance curve givefl the minimum achievable input-output variances for a linear process for different input weightings.
Var(y) < Var(iJ) -
1 (1 _ p)2
(15)
Therefore the slnaller the value of p( «.. l),the achieved performance \vill be close to the design performance. It is well known that the design perforInance itself is lower for a system with a large robustness margin. On the other hand if the feedback system has poor robustness properties, Le. p ~ 1, the higher will be the ULlcertainty in performance i.e~ we pay the price for being less robust by a possible loss in the design performance. There is a higher risk of instability if there is a further change in the plant dynamics. The above discussion portrays the traditional trade-off between performance aIld robustness in a different light.
UnceTta-inty in the
til1~e
delay
As an illustration of the ideas presented earlier, wc discuss here the special case of uncertainty in the process tilne delay. In practice uIlcertainty in time delay is very common and it is an interesting exercise to investigate the effect of this uncertainty on controller performance. First "ve rederive the results in the previous section in terms on the multiplicative uncertainty modeL The multiplicative uncertainty is particularly well suited for represellting variable delay. The Inultiplicative uncertainty is of the form: P == P(l
+ !:J.~)
The Illultiplicative uncertainty is related to the additive uncert.ainty by ~ == P~rn
(16)
Therefore we have, 1
+ Q6t. == 1 + QP6. rn == 1 + T~7n
The uncertainty in time delay can be characterized via the multiplicative uncertainty in the follo\\ring way~ P:::::::Fz-O,O:::; B S
::::::: P(l
+ z-()
-
e
Ir1 aYo:
(17)
1)
hence we have,
6965
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
~Jn ~m(ej1L')
== z-{) -1 == e-j()w -
14th World Congress ofIFAC
(18) 1 Design
= cos(Bw) - j sin(Bw) - 1 l~m(ejW)' = I cos(8w) - j sin(Bw) - 1}
Achieved Worst Cast:
== y'2 - 2 cos(f)~v)
For the worst case uncertainty in the delay, B == Ornax , r~'Tn.(eJ·W)r is a periodic function of frequency and takes the maximllIll 'values, l~m(ejW)1 == 2, at the frequencies w == en71 _ lJsing this I)Toperty the bounds on the \vorst ~~e performance can be derived 2 as:
Rohustness
Perronnflnce (a)
Fig. 4.
(b)
The perfonnance and robustness properties of the
control system \vith a unit delay mismatch.
e;;::;= S(r
+ d)
(21)
<::?
e = 8(1 + Q~)-l(r + d)
Ilcll~
1
III + Q~ll~ s Irell~ ~ 11(1 + Q~
)-1 2
1100
4. SI1tIULATION R.ESDLTS
Var(y) 1 (1 + 211 TI1)2 S VarCY) S (1 _ 211 T I1)2 1
(19)
Notice that the above expression has only the complenlentary 'sensitivity function appearing in the upper and lo\ver bounds. The uncertainty expression does not appear explicitly in the above inequality, i.e., the bounds are independent of the actual magnitude of the uncertainty in the delay. It should be noted here that these are very conservative bounds.
This is a simulation example from Desborough and Harris (1992), \vith a nOlninal time delay of tVilO samples:
"
y(k)
+
2)
1-0.2z- 1
l-z
(1 r.
z-l
== Tr
(20)
e = r - y == (1 - T)r ::= Sr e == S(l + Q~) -lr <;=} 1 lIe"~ III + Q~Il~ ::; flell~
$
J
1('
1 -r
Q~)
-1
2
//00
Thus we have a worst case bound on the sum of the squared errors (2-norm) using the same transfer function (l+Q~)-l as before. The salne limit can be re-derived. for the case where \ve have deterrrllnistic disturbances. In this derivation norm. 2
Copyright 1999 IFAC
11./1 is used
to denote
11.1100
+ 1< z-2)
T==QP~-~--~ (1 - z - l Kz-2)
+
An additional delay \vas introduced in the "plane'. y(k)
== u(k -
3)
+
1 - O.2z- 1
1 _ z-1
the
a(k)
Thus we have the multiplicative uncertainty of the form -1
Figure 4 shows the effect of the time delay mismatch on performance and stability of the designed controller. Figure 4 (a) ShOWH the achieved performance index in relation to the nominal performance for different controller gains. The performance index is based on the minimum variance bcnchn'larking (Harris, 1989). T'his performance index is based on the follo"ving idea. For K:::::::O.2, "v"'e have
. . (~)
y
1
~ ~ 1-
== (1 l
a(k)
Kz- 2
""
~m::::;:; Z-l
y
-1
A simple integral controller .6.u(k) :::= -Ky(k) was used for control. The nonlillal transfer functions are given by:
Effect of uncertainty in the presence of de terrninistic in,puts In a practical setting a process is subject to different kiIuls of inputs such as sctpoint changes, random ,"valk type disturbances, measureIIlent noise, etc. In this section '\-ve consider the effect of modelling Ullcertainty ill the presence of deterministic inputs. Consider the case V\rhere V\re have sctpoint changes as the only exogenol1s inputs.
== u(k -
1 - O.2z(k) Z - l + O.2z- 2 a
+ O.8z- 1 + O.6z- 2 + 0.44z- 3 + ' .. )a(k)
00-
For
K==O~5,
the closed loop response is
g~ven
by
6966
ISBN: 0 08 043248 4
14th World Congress ofIFAC
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
1
A.(k) _
1 - O.2za(k) - 1 - z-l + O.5z- 2 ::::::- (1 + O~8z-1 + O~3z-2 - O.lz- 3
y
Design SSl7 2.431 Achio:w~ S~E"
+ ... )a(k)
44tP.
..
'
-
-
Design
-_ .• Achiow.'ed
Notice that the first two terrns in each case are identicaL In fact they will be the same irrespective of the controller. Thus the minimum achievable variance for the process with two sanlple delays is given by
(i~v ~ (1
+ O.82)a~ ==
1.64cr~
(a)
2 :::=
••
O"Tn1' (J~
~ 1
The design curve in figure 4 Ca) \vas based on the I)erformance index for the nominal case and the achieved curve corresponds to the unit S8.IIlpIe delay mismatch. The worst case bound \vas derived using eq. 11 . The controller shows a significant degradation in performance due to an increase in the time delay of a sample interval. Figure 4 (b) sho'\vs the robllstness properties of the integral controller for a unit sample time delay mismatch. As the controller gain increases, the robustness margin deteriorates. For K=::::O.2, the COlltrolleI is higllly TO bth.,;;t and t.he achieved perforlllance is close to the design performance (see figure 4 (a)). For K==O.4 the robustness lnargill is small and the achieved performance is significantly different from the design performance. Figure 5 (a) shows the achieved and the design servo response when the tilne delay is underestimated. The achieved performance is worse than the design performance.
Ilerl~ ~ 1. 8117 ~ Uell~
'I( 1 + T.. ~rn )-1'12 lloo !\
~4. 8 4
VVe also considered the case where the delay is overestimated i.e. the delay is assullled to be 2 units when the actual delay is 1 Ul1it. Figure 5 (b) ShOVilS the achieved and design closed loop step responses for the case of time delay overestimation (L\m ;;= zl - 1).
l'ell~
lj'e"""22 ~ 0.6428
I
2:
1
,.
1\(1 + T~rrt)ll~
= 0.3486
It is interesting to note that when the time delay is overestimated the performance improves over the design case whereas ""-'"hen the delay is tUlderestimated the performance deteriorates. Thus this exalnple serves to illustrate the point that achieved perforInallce can improve even in the presence of mismatch and both the upper arId lower limits in eq. 11 are meaningful.
Design •. -. Achico.'w
Cb)
Fig. 5.
Cornparison of achieved and designed closed loop step responses for the case where (a) the delay is underestimated a.nd Cb) the delay is overestimated.
The performance index is then calculated as a ratio of the miniIuUIll achievable variance to the achieved variance.
PI
De:d,gn SS£:' ~431 A,,1tit:\la::l SSE". U
5. CONCLUSIONS In this lvork we have considered the contributions of the modelling uncertainty on the achieved performance of a control systenl. The presence of uncertainty in the modelling step is reflected in uncertain controller performance. Time domain bounds on the achieved perforlIlance are obtained based on the knowledge of the uncertainty. As a special case, the effect of varying time delay is captured using multiplicative uncerta.inty. Silnilar performance hounds are deri'v-ed for the case of deterministic inputs~ A simulation example is presented to illustrate tile effect of uncertainty in time delay on achieved perforIllance. 'York is currently in progress to obtain performance bounds for the multivariate case.
6. REFERENCES Boyd, S. P. and C. H. Barrat (1991). Linear Controller Design: Limits of Performance. Prentice Hall. NY. Desborough, L. and 'T. Harris (1992). Performance assessment measure for univariate feedback control. Can. J. Chem·. Eng. 70, 605-616. Doyle, J. C., B. A. Frallcis and A. R. Tannenbaum (1992). Feedback Control Theory. MacMillan Publishing Co.. NY. Harris~ T. J . (1989). Assessment of closed loop performance. Can. J. Chern. Eng. 67, 856861. Huang, B., S. L. Shah and K. Y. Kwok (1997). Good, bad or optimal~r performance asseSSloent of miIno processes. ~4utomatica 33, 1175-1183. Kozub, D. J. (1996). Controller performance monitoring and diagnosis experiences and challenges. In: Proceedings of CPC- v:. Lake Tahoe, CA. Maciejov.rski, J. M. (1989). A1ultivariate Feedback Design. Addison- Welsley. NY. ~lorari, M~ and E~ Zafiriou (1989). Robust Process Control. Prentice-HalL NY.
6967
Copyright 1999 IFAC
ISBN: 008 0432484
Copyright C0 1999 IF~t\C 14th Triennial 'rVorld Congress,
l3c~ilng~
14th World Congress of IFAC
N-7a-12-l
P.R. China
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFOR.l\1ANCE Rohit S. Patwardhan, Sirish L. Shah and Biao Huang
Department of Chemical and ./'vfate7'ials Engineering llni'lJersity of Alberta, Edmonton, Canada, T6G 2G6. email: [email protected]
Abstract: In 1110St model-based control sC}lemes, the desig"n performance is rarely achieved due to the presence of Inudel plant mismatch. In this study the effect of modelling uncertainty on achieved controller performance is investigated. Particular attention is paid to uncertainty in the time delay. With the design performance as the reference point, bounds are derived on the achieved variances for the manipulated variables and controlled outputs, in the presence of model plant mismatch. For the deterministic case, similar bounds are obtained on the 2-norm (sum of errors squared) of the output errors. A simulation example is presented to illustrate the results. Copyright
~)
19991r:4C
Keywords: Uncertainty, Performance, Robustness, Time Delay
1.
I~TRODlJCTIO~
The objective of controller design is to achieve satisfa.ctory performance under feedback. The characteristics of a process decide the choice of one or IIlore design objectives. Conventionally a single measure of the tracking/regulatory performance in either the time domain or the frequency dOJnaill fOTlllS the design objective. 1.1oT8 often than not a process has features which limit the achievable performance of any controller. i\..mongst these limiting features are the presence of time delay~ IlOllminimum phase zeros~ constraints on the inputs and outputs, etc.
At the outset it is llnportant to kno\v the contribution of these featl1res to'\vards the best a.chievable performance. The knowledge of the best achievable pcrforll1allce then serves as a benchmark. This benchnlark is useful in assessing the performance of any controller. 'The contribution of the time delay term towards minimum achievable variance is V\TcII knOV,tll for linear processes, and forms tIle basis of the InilliIllUrIl VariaI1Ce benchmarking in performance assessment (Harris, 1989; Huang et ai., 1997).
In practice many components of the model, including the time delay, are never precisely known. The process tilnc delay may be time varyillg due to changing flOVi.r rates. Furthermore any chemical process is inherently nonlinear, and is approximated by a fixed linear model only in the neighbourhood of the operating point. These and other factors lead to the pre..~ence of uncertainty in modelling. A controller whose design is based on the nominal model of the process is unlikely to deliver the design perfOrnlaIlCe in the presence of model plant IuisIIlatch (1tIP1f). The presence of uncertainty leads to a gap between the designed and the achieved performance. The knowledge of the C'.olltributioIl of UIlceTtainty to\vaTds the possible deterioration of controller performance is vital to performance assessment and diagnosis (Ko~ub, 1996)~
The effect of uncertainty on closed loop stability is "veIl known and is the topic of several books and articles (sec for example - Doylc et al., 1992; J\Jlorari and Zafiriou~ 1989). The objective of this work is to quantify the effect of modelling nncertainty on the achieved performance of a lin-
ear model based controller. For the regulatory
6962
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
14th World Congress of IFAC
The sensitivity function is defined as
S
y
==
(1 +PC)-l
and the complementary sensitivity fUIlction or the closed loop transfer function as
Fig. 1.
The conventional feedback system
case with stochastic inputs, the use of Parseval's theorem is mstrumental in obtaining bounds on the achieved input and output variances, relative to the design ca..r;;e. For the case of deterministic inputs, the induced Hoc norm definition is used to derive bounds on the achieved performance in relation to the design performance, witll the 2-norrIl (sum of squared errors (SSE)) as a measure of performance. A simulation example is used to illustrate the effects of uncertainty in time delay on controller perforlllance, in t.he context of minimum variance benchmark as a measure of performance. The rest of this paper is organized as follows: Section 2 gives the preliminaries required for furtIler analysis. TIlls includes discussion of conventional feedback control, introduction to ~ome common norms for signals and systems and the internal model control framework which wc adopt throughout the report. I I I section 3 \ve derive bounds for the Vlorst case achieved performance for an uncertain systeln.The effect of uncertainty in tirne delay is llighlighted. The results are extended to the case of det.erministic inputs. A simulation example, in section 5, is used to illustrate the lnaill results. Sectioll 6 gives concluding rernarks.
Throughout this paper we restrict ourselves to the class of discrete~ stable systems v.rhich are finite diInensional linear tirHe invariant. (FDLTI). For the sake of brevity, ",re ha"ve omitted the argmnent z in ¥lTitjng the relations between the signals and the transfer functions for discrete systellls. The theory developed here is applicable to SI80 systems.
2. PRELI1/I INAR1ES Figure 1 shows the standard feedback configuration~ rThe measurement error m is assumed to be unknown and most frequently described with a white noise process. The disturbance d is made up of measured and tmmeasured disturbances. For a 8180 process, the closed loop relations bet"veen the inputs and the outputs can be ~vrittell as:
(1
+ PC)y == d + peer -
m,)
Therefore "re have y
== Bd + Tr - Tm
(1)
For a discussion on trade-offs involved in design of the sensitivity function, S, see I\1aciejowski (1989). Some idea of the "size" of a signal and the relevant transfer functions, such as the sensitivity function, are needed in order to quantify performance of a control system. Two norms for capturing the magnitude of a signal are:
2-norm oo-norm
j
IIul12 = {-u(O)2 + u(1)2 + ....} 1/2 lull(X) == suplu(k)1
(2)
k
TIle follu\ving tv.ro systeIll I10TIllS provide v..rays of capturing the mag-'nitude of a linear time invariant system (Chen and Francis, 1995): 21i"
2-nOTm
IIPI!z = (2~ ~
C<)-norm
IIP ~r
I CXJ
J
IP(e
jW
)1 2 dw)1/2
o p(O)2 + P(1)2
-
-sup
IP( e jW)( --
w
+ .... sup
Il u 112#O
(3)
11Yll2 -11-1-1 11.
2
The 2-noIIIl of a signal is iIlvariant with respect to the Z-transforrn. In addition to basic properties that define a norm, the (X)-norm obeys the following property,
(4) A detailed discussion on signal and system nOTIIlS and their properties, for univariate as ,"Tell as lnultivariatc systems, can be found in Boyd and Barrat (1991) . The internal model control (IMC) framework provides a way of rc-parametrizing the conventional controller (IVlorari and Zafiriou~ 1989). It is \vell known that this approach is equi vdlent to the celebrated Youla~I(ucera parameterization of all stabilizing controllers for a stable system~ Figure 2 shows the 11\1[0 franle"vork. The controller Q in the 111:C framevlork is related to the conventional feedback controller C in Figure 1 by
Q
=
0(1
+ pC)-l
(5)
where P is the nominal model of the process. For stable plants a stable Q guarantees internal stability. The nominal sensitivity function and
6963
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
y
14th World Congress of IFAC
In the following discussion 1,-ve rest.rict ourselves to the case where there are no setpoint changes, i.e. the regulatory case, and the disturbance is a realization of a white noise process (see Figure 2). Using Parseval's Theorem ,"ve have
+
Fig. 2. The internal model control stucture the complimentary sensitivity function can be expressed ill ternls of Q as
(6) The main advantage of this parametrization is that the sensitivity and the complementary sen-
In a similar \vay we get the following lower bound.
Var(y) 2: III
sitivity function are linear vvith respect to Q, the I~!{C controller. There is a one-ta-one corre-
spondence between the IMC controller and the cOllventional controller C.
c
~
Therefore the achieved perforluance is bound by 1
3. EFFECT OF UNCERTAINTY IN THE PLANT MODEL
No identification procedure or mechanistic model can be relied upon to deliver all exact Illodel of the actual physical system. This gives rise to modelling uncertainty. Uncertainty can be structured or unstructured. T"vo COlnIllon ,vays of representing unstructured uncertainty are given below, where P is the model and P is the plant. Additive Ullcertaillty: P 1:fultiplicative uncertainy: P
< Var(y) < II(
111 + QLlI,'~ -
Q(1 - FQ)-l
In the remaining discussion V\.re use the 11/IC fraulcvlork for analysis l)urposes.
== f> + ~ (7) == P(l + ~Tn)
+ ~AII;" Var(!/)
Var(y) -
1
-+ Q~
)-] 11 2 00·
(11) The above expression quantifies the bounds on achieved variance, in relation to the design variance, in the presence of modeling uncertainties. The performance bOUlld can be further simplified and expressed in terulS of IIQ~IICXJ for the case \vhere IIQ6.lloo < 1 (Boyd and Barrat, 1991).
(1
+
1 < l/ar(y) < IIQ~llCX))2 - V ar(y) ~ (1 -
1_ _--::::-
1IQ.6. Iloo )2 (12)
Using a similar approach Vle can define an upper bound for the increase ill. input energy (variance) due to uncertainty.
In this section we anal.yze the effect of modelliIlg uncertainty on the performance of a con-
trol system. In particular we derive bounds on the achieved performance in terms of the design performance and the CX)-norm of a certain transfer
function. The achieved sensitivity and the cOlllpleruentary sensitivity functions, in the presence of the additive uncertainty, are given by
S == (1 - QP)(l 7'
== QP(! +
+ QA)-l
Q~)-l
(8)
Thus the achieved sensitivity function is related to the design sensitivity function in eq. 6 by
S ~ 8(1
Once again we have the following tilue domain bounds on input variance due to rvIPJ\.-1
+ Q~)-l
(9)
Thus wc have arrived at the sanle perforulance bounds for both, the input and output variances. The geometry of this region i~ highlighte
6964
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
14th World Congress of IFAC
Worst Case Performance
Let p ~ jIQL\jloo: Robustness ~Ieasure.
Var(y)
The smaller the value of p the more robust is the control system, and conversely) the closer it is to unit-v the closer it is to instability due to IVIP1tL In ter~s of the robustness lneasure the performance bound (eq. 11) can now be re\vritten as:
DC:iign Performance
Var(u)
Fig. 3.
Illustration of the effect of modelling uncertainty on achieved performance
3 1 . If the MP:NI was negligible this region would be very small and the bigger it is the larger the possible degradation of the COlltrol systenl in terms of loss in performance.
Remark 1. The lo\,,~er bound derived in eqns. 11, being less than unity, does not guarantee that performance is \vorse ill presence of nlodel plant rniSluatch. This is an interesting observation since it implies that the achieved performance may actually be better thaIl the designed performance. For example this can ha.ppen when the delay is overestimated (see simulation results). Remark 2. It should be noted that eqns. 11 13 give bounds on the achieved performance that t.hese ean be conservative. The actual (or gain) in performance ,"vould depend on spcctrulll of the disturbances in relation to spectruIil of the uncertainty.
(14)
and and loss the
the
In case the above performance bounds indicate
a significant change in design performance, the next step is to determine the robustness of the existing controller with respect to the estimated uncertainty. Let us assume that the controller design was done v.rith the nominal model as the plant. A controller C i~ said to provide rob'lLst stability if it provides stability for every plant in the (uncertain) set. The small gain theorem (SGT) provides a ,vay of assessing robust stability.
Theorern 1. For the Illultiplicative uIlcertainty model, C provides robust stability iff Ilt~1nli,::o < 1, Vv~hcre T is the nominal complementary sensitivity functioIl Pro~f. For proof see Doyle et al. (1992) .
A simple measure of robustness follows from application of th.c SeT to additive uncertainty case. For robust stability Vi/re have 1 T'he design performance is shown in relation to the design curve for the linear quadratic regulatur (LQR). The LQR performance curve givefl the minimum achievable input-output variances for a linear process for different input weightings.
Var(y) < Var(iJ) -
1 (1 _ p)2
(15)
Therefore the slnaller the value of p( «.. l),the achieved performance \vill be close to the design performance. It is well known that the design perforInance itself is lower for a system with a large robustness margin. On the other hand if the feedback system has poor robustness properties, Le. p ~ 1, the higher will be the ULlcertainty in performance i.e~ we pay the price for being less robust by a possible loss in the design performance. There is a higher risk of instability if there is a further change in the plant dynamics. The above discussion portrays the traditional trade-off between performance aIld robustness in a different light.
UnceTta-inty in the
til1~e
delay
As an illustration of the ideas presented earlier, wc discuss here the special case of uncertainty in the process tilne delay. In practice uIlcertainty in time delay is very common and it is an interesting exercise to investigate the effect of this uncertainty on controller performance. First "ve rederive the results in the previous section in terms on the multiplicative uncertainty modeL The multiplicative uncertainty is particularly well suited for represellting variable delay. The Inultiplicative uncertainty is of the form: P == P(l
+ !:J.~)
The Illultiplicative uncertainty is related to the additive uncert.ainty by ~ == P~rn
(16)
Therefore we have, 1
+ Q6t. == 1 + QP6. rn == 1 + T~7n
The uncertainty in time delay can be characterized via the multiplicative uncertainty in the follo\\ring way~ P:::::::Fz-O,O:::; B S
::::::: P(l
+ z-()
-
e
Ir1 aYo:
(17)
1)
hence we have,
6965
Copyright 1999 IFAC
ISBN: 0 08 043248 4
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
~Jn ~m(ej1L')
== z-{) -1 == e-j()w -
14th World Congress ofIFAC
(18) 1 Design
= cos(Bw) - j sin(Bw) - 1 l~m(ejW)' = I cos(8w) - j sin(Bw) - 1}
Achieved Worst Cast:
== y'2 - 2 cos(f)~v)
For the worst case uncertainty in the delay, B == Ornax , r~'Tn.(eJ·W)r is a periodic function of frequency and takes the maximllIll 'values, l~m(ejW)1 == 2, at the frequencies w == en71 _ lJsing this I)Toperty the bounds on the \vorst ~~e performance can be derived 2 as:
Rohustness
Perronnflnce (a)
Fig. 4.
(b)
The perfonnance and robustness properties of the
control system \vith a unit delay mismatch.
e;;::;= S(r
+ d)
(21)
<::?
e = 8(1 + Q~)-l(r + d)
Ilcll~
1
III + Q~ll~ s Irell~ ~ 11(1 + Q~
)-1 2
1100
4. SI1tIULATION R.ESDLTS
Var(y) 1 (1 + 211 TI1)2 S VarCY) S (1 _ 211 T I1)2 1
(19)
Notice that the above expression has only the complenlentary 'sensitivity function appearing in the upper and lo\ver bounds. The uncertainty expression does not appear explicitly in the above inequality, i.e., the bounds are independent of the actual magnitude of the uncertainty in the delay. It should be noted here that these are very conservative bounds.
This is a simulation example from Desborough and Harris (1992), \vith a nOlninal time delay of tVilO samples:
"
y(k)
+
2)
1-0.2z- 1
l-z
(1 r.
z-l
== Tr
(20)
e = r - y == (1 - T)r ::= Sr e == S(l + Q~) -lr <;=} 1 lIe"~ III + Q~Il~ ::; flell~
$
J
1('
1 -r
Q~)
-1
2
//00
Thus we have a worst case bound on the sum of the squared errors (2-norm) using the same transfer function (l+Q~)-l as before. The salne limit can be re-derived. for the case where \ve have deterrrllnistic disturbances. In this derivation norm. 2
Copyright 1999 IFAC
11./1 is used
to denote
11.1100
+ 1< z-2)
T==QP~-~--~ (1 - z - l Kz-2)
+
An additional delay \vas introduced in the "plane'. y(k)
== u(k -
3)
+
1 - O.2z- 1
1 _ z-1
the
a(k)
Thus we have the multiplicative uncertainty of the form -1
Figure 4 shows the effect of the time delay mismatch on performance and stability of the designed controller. Figure 4 (a) ShOWH the achieved performance index in relation to the nominal performance for different controller gains. The performance index is based on the minimum variance bcnchn'larking (Harris, 1989). T'his performance index is based on the follo"ving idea. For K:::::::O.2, "v"'e have
. . (~)
y
1
~ ~ 1-
== (1 l
a(k)
Kz- 2
""
~m::::;:; Z-l
y
-1
A simple integral controller .6.u(k) :::= -Ky(k) was used for control. The nonlillal transfer functions are given by:
Effect of uncertainty in the presence of de terrninistic in,puts In a practical setting a process is subject to different kiIuls of inputs such as sctpoint changes, random ,"valk type disturbances, measureIIlent noise, etc. In this section '\-ve consider the effect of modelling Ullcertainty ill the presence of deterministic inputs. Consider the case V\rhere V\re have sctpoint changes as the only exogenol1s inputs.
== u(k -
1 - O.2z(k) Z - l + O.2z- 2 a
+ O.8z- 1 + O.6z- 2 + 0.44z- 3 + ' .. )a(k)
00-
For
K==O~5,
the closed loop response is
g~ven
by
6966
ISBN: 0 08 043248 4
14th World Congress ofIFAC
THE EFFECT OF UNCERTAINTY ON CONTROLLER PERFORMANC...
1
A.(k) _
1 - O.2za(k) - 1 - z-l + O.5z- 2 ::::::- (1 + O~8z-1 + O~3z-2 - O.lz- 3
y
Design SSl7 2.431 Achio:w~ S~E"
+ ... )a(k)
44tP.
..
'
-
-
Design
-_ .• Achiow.'ed
Notice that the first two terrns in each case are identicaL In fact they will be the same irrespective of the controller. Thus the minimum achievable variance for the process with two sanlple delays is given by
(i~v ~ (1
+ O.82)a~ ==
1.64cr~
(a)
2 :::=
••
O"Tn1' (J~
~ 1
The design curve in figure 4 Ca) \vas based on the I)erformance index for the nominal case and the achieved curve corresponds to the unit S8.IIlpIe delay mismatch. The worst case bound \vas derived using eq. 11 . The controller shows a significant degradation in performance due to an increase in the time delay of a sample interval. Figure 4 (b) sho'\vs the robllstness properties of the integral controller for a unit sample time delay mismatch. As the controller gain increases, the robustness margin deteriorates. For K=::::O.2, the COlltrolleI is higllly TO bth.,;;t and t.he achieved perforlllance is close to the design performance (see figure 4 (a)). For K==O.4 the robustness lnargill is small and the achieved performance is significantly different from the design performance. Figure 5 (a) shows the achieved and the design servo response when the tilne delay is underestimated. The achieved performance is worse than the design performance.
Ilerl~ ~ 1. 8117 ~ Uell~
'I( 1 + T.. ~rn )-1'12 lloo !\
~4. 8 4
VVe also considered the case where the delay is overestimated i.e. the delay is assullled to be 2 units when the actual delay is 1 Ul1it. Figure 5 (b) ShOVilS the achieved and design closed loop step responses for the case of time delay overestimation (L\m ;;= zl - 1).
l'ell~
lj'e"""22 ~ 0.6428
I
2:
1
,.
1\(1 + T~rrt)ll~
= 0.3486
It is interesting to note that when the time delay is overestimated the performance improves over the design case whereas ""-'"hen the delay is tUlderestimated the performance deteriorates. Thus this exalnple serves to illustrate the point that achieved perforInallce can improve even in the presence of mismatch and both the upper arId lower limits in eq. 11 are meaningful.
Design •. -. Achico.'w
Cb)
Fig. 5.
Cornparison of achieved and designed closed loop step responses for the case where (a) the delay is underestimated a.nd Cb) the delay is overestimated.
The performance index is then calculated as a ratio of the miniIuUIll achievable variance to the achieved variance.
PI
De:d,gn SS£:' ~431 A,,1tit:\la::l SSE". U
5. CONCLUSIONS In this lvork we have considered the contributions of the modelling uncertainty on the achieved performance of a control systenl. The presence of uncertainty in the modelling step is reflected in uncertain controller performance. Time domain bounds on the achieved perforlIlance are obtained based on the knowledge of the uncertainty. As a special case, the effect of varying time delay is captured using multiplicative uncerta.inty. Silnilar performance hounds are deri'v-ed for the case of deterministic inputs~ A simulation example is presented to illustrate tile effect of uncertainty in time delay on achieved perforIllance. 'York is currently in progress to obtain performance bounds for the multivariate case.
6. REFERENCES Boyd, S. P. and C. H. Barrat (1991). Linear Controller Design: Limits of Performance. Prentice Hall. NY. Desborough, L. and 'T. Harris (1992). Performance assessment measure for univariate feedback control. Can. J. Chem·. Eng. 70, 605-616. Doyle, J. C., B. A. Frallcis and A. R. Tannenbaum (1992). Feedback Control Theory. MacMillan Publishing Co.. NY. Harris~ T. J . (1989). Assessment of closed loop performance. Can. J. Chern. Eng. 67, 856861. Huang, B., S. L. Shah and K. Y. Kwok (1997). Good, bad or optimal~r performance asseSSloent of miIno processes. ~4utomatica 33, 1175-1183. Kozub, D. J. (1996). Controller performance monitoring and diagnosis experiences and challenges. In: Proceedings of CPC- v:. Lake Tahoe, CA. Maciejov.rski, J. M. (1989). A1ultivariate Feedback Design. Addison- Welsley. NY. ~lorari, M~ and E~ Zafiriou (1989). Robust Process Control. Prentice-HalL NY.
6967
Copyright 1999 IFAC
ISBN: 008 0432484