Estimation of Model-Plant Uncertainty and its Role in Robust Design of Predictive Control

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

ESTIMATION OF MODEL-PLANT UNCERTAINTY AND ITS ROLE IN ROBUST DESIGN OF PREDICTIVE CONTROL P. Banerjee and S.L. Shah Departm£nt of Chemical Engineering, University of Alberta, Edmonton, T6G 2G6, Canada

Abstract. This paper blends the ideas of estimation of model-plant uncertainty from process data and the robust design of generalized predictive control (GPC) bascd on the small gain thcorem. Estimation of the model uncertainty is bascd on the classical signal processing techniqucs for SISO linear time invariant systems. For a noise free case it is shown hy simulations that with proper choice of data windows, data overlapping and probing or input excitation, the spectral estimatc of the model plant uncertainty is inexcellcnt agreement with the actual uncertainty. The estimated uncertainty spectrum with the stability bounds are thcn used to provide simple graphical based tuning guidelines for robust dcsign of GPc. A bcttcr estimate of thc uncertainty prevents conservative design and the number of GPC tuning parameters give enough flexibility to shape the stability bound for robust perfomlance. Evaluation of this technique is illustrated via simulation and cxpcrimcntal applications. Key words. Small gain thcorem, model-plant unccrtainty cstimation. DFT and GPc.

thrust of this paper is on the utilization and subsequent use of this information for robust design of controllers.

I. INTRODUCTION The performance and rohustness of a modcl based controller depends on how well a model is able to capture the dynamics of a plant. In a realistic situation a mathematical model cannot totally emulate a physical process, and the problems of stahility and performance in a system mostly manifest from this model-plant uncertainty. Furthermore for a real time control application, simpler models such as linear input-output tmnsfcr functions arc desirable. But the performance and stability limitations are often significant for such linear models due to the constraints on model order and structure.

Model-hased long range predictive controllers (LRPC) (Cutler et aI, 1988; Clarke, 1988; Garcia et ai, 1989) have gained widespread acceptance and success in the chemical process industries. To fully exploit the advantages offered by such controllers, it becomes meaningful to explore their robust design guidelines that can be easily adopted in industry. The ideas of robust design for linear quadratic optimal controllers within the SGT framework was provided by Bitmead et al (1990). Robust design of observer prefiIter for the generalized predictive control (GPC) (Clarke et aI, 1987) using the SGT tool was proposed by Robinson et aI, 1991, for mean-level and dead-beat performance. To the best knowledge of the authors, this is the first complete and systematic application paper where a sound estimate of the model-plant uncertainty and rohust design techniques for GPC are combined and supported by simulation and experimental evaluations.

The norm bounded small gain theorem (SGT) (Desoer et aI, 1975; Morari et aI, 1989; Ooyle et aI, 1992) provides a sufficient condition for stability in the presence of model uncertainty. Ignoring the accurate uncertainty can lead to instability whereas a conservative bound can affect the performance of a system. One of the several approaches proposed for the quantification of such uncertainties (Kosut, 1987; Goodwin et aI, 1989; Goberdhansingh et aI, 1992) is the discrete fourier tmnsform (OFT) of the output error by Kosut, 1987. The OFT approach provides one of the reliable ways of estimating the spectrum of the uncertainty. Kosut's result indicate a good fit of the uncertainty over a certain frequency band for a noise free case. In this paper it is shown that with the classical signal processing smoothing techniques (Jenkins et aI, 1968; Oppenheim et aI, 1975) it is possible to obtain a good estimate of the uncertainty over the entire frequency range of interest. A major

This paper is organized as follows: Section two is concerned with the problem of estimating the model uncertainty using direct signal processing methods. The factors governing the estimation of smooth spectrum are outlined as guidelines for better estimation of the uncertainty. Through simulations it is shown that the OFT methods with proper smoothing techniques give excellent estimate of the uncertainty. The formulation of GPC into an SGT framework for robustness analysis is discussed in section three. Simple graphical based tuning 83

guidelines are provided for stable performance of GPC. Experimental evaluation of the synthesized design results are discussed in section four, followed by concluding remarks in section five.

Where K is the number of data segments and M refers to the number of frequency points per segment i. The energy of the window lag function w(k) is given by: (2.7)

2. SIGNAL PROCESSING APPROACH FOR THE ESTIMATION OF MODEL PLANT MISMATCH Consider the true linear plant: y(t)

= G(q -')u(t)+u(t)

and the windowed data Et'(k) in (2.6) for the i 'h data segment is given by: (2.1)

E,W(k)

u-.

= L

l'i(k)w(k)l'

ie!)

(2.8)

N

" ",0

Where G(q - I) is the plant transfer function and u(t) represents the Gaussian white noise. Let the identified model based on the least square technique (Ljung, 1987) for the process data ZN = {y(t),u(t) , t = l.N} be denoted by G(q-I). The additive mismatch G(q-I) between G(q-I) and G(q-I) is then given by: G(q-') = G(q -')-G(q-')

Often the price paid for the decrease in variance in the estimated spectrum is an increase in the bias of the periodogram because the windowing operation modifies the original data sequence. Jenkins et aI, 1968, have provided the following expressions for the 95% confidence limit (L(w» on the estimated gain:

(2.2)

The spectrum of the estimated uncertainty

2

is given by the following relation (Kosut, 1987): Gu(e -jj = ,.(W)

probability for the dofs 2 & v - 2 and

Where the subscript M in GM(e-Jo ,) in (2.3) represents

- 11 I 3.012 1.357) f 2,._2- ex ,\ . + v + v 2

(2.4)

of G(e-JOJ). The error spectrum ,u(w) in (2.3) is

UB(w)=C u (e-jj(1+L(w»

where

e(t) = G(q-I)U(t) + u(t). Assuming the data set Z~ to

(2.5)

IGuvjjl =(,,(W»)o. ~ •• (w)

Where E(w) is DFT of e(t) and U(w), is complex conjugate ofDFT of u(t). Similarly the expression for uu(w) can be defined. Usually the Fast Fourier Transform (FFf) algorithms (Oppenheim et aI, 1975) are used to enhance the computational speed of DFT. The periodogram thus estimated has significant variance. Smoothing techniques such as the convolution of data Z~ with a window function,

k.tMI ~

IG~(l'-jjl =~Cu(l'_jjI2 + ::~:~r

= 0,1, ... M

(2,12)

(2.13)

The above equation shows that in presence of noise u(t), one really obtains a biased estimate IG!(e-Jjl instead of

IGM(e -JUl)I where the bias is proportional

to (uu(w)/uu(w»O.5 that gets added to IGM(e-J" . This bias term is also affected by the shape of the input spectrum uu(w). For example the effect ofroll of in uu(w) at the higher frequencies would be to amplify the noise and therefore add more bias in the estimated IGM(e-JUl)1 at those frequencies. The noise in y(t) gets

2Et(k)U:W(k) k

(2,11)

Since e(t) is corrupted with noise u(t), the above equation can also be written as:

averaging of periodograms computed for each of the split data segments of equal length etc. are used to reduce the problems of variance and spectral leakages in the spectrum. The expression for the smoothened periodogram by Welch's method is given by (Oppenheim et aI, 1975): ,.(k) =

(2.10)

The spectral estimate based on limited data length deteriorates considerabl y in the presence of noise. To examine the effect of u(1), consider the following magnitude spectrum:

be stationary and zero centered, the estimated •.(w) is given by (Oppenheim et aI, 1975): I . ,.(w) = NE(w)U(w)

is the

The above relation is derived on the basis of data fitting and has been observed to give results within an error of ±O.46%. The upper bound for the estimated uncertainty is then given by:

Where [i(.) is the expected operator and we would like to expect that GM(e-JUl ) be an unbiased estimate

= l.N),

k;u

coherency spectrum. We propose the following empirical relation to calculate f2,v-2:

the total number of frequency points in the estimated spectrum. The true and estimated uncertainty are related as:

estimated based on Z~ = {e(t),u(t),t

(2.9)

Where v is degree of freedom (dof) for the specified window, f2.v-2 is the Fisher's distribution for 95%

(2.3)

•.(w)

(1-I;.(w)J~O~

L(w)= -2(/2.. - 2) -~-( vk,.(w)

GM(e-JUl )

transferred to e(t) and thereby deteriorating the SNR in the signal e(t). For the extreme situation, when

(2.6)

84

G(q-l) = ().

the error signal is just the noise u(1). Therefore the effect of noise is to make the estimated IGM(e-iOO)1 more conservative. Data filtering with a phase removal strategy is one of the ways of alleviating the problems of noise.

2.1. Simulation Example In a realistic situation. the true uncertainty is usually unknown and hence the effectiveness of the proposed scheme to estimate G(q-l) cannot be evaluated. The following example is contrived to examine the effectiveness of the DFf approach to estimate G(e-iw ). Let the plant be a third order process: 1/(s+ I )(3s+ 1)(5s+ 1). whose discrete equivalent for a sample time of Ts=l is: G

,0-' Nonnoliud Fr"IU." C)'

Fig. 2. Estimate of the uncertainty with frequency points equal to the data points.

O.OO77q -'+O.0212q -2+0.0036q -J (q )- 1-1.903Iq-'+1.1514q 2-0.2158q -J -I

10- 1

_

(2.14)

The following first order model with two numerator parameters was identified using the RLS algorithm by providing a square wave type excitation: d( q

-1)= O.0419q -'+O.0719q -2

1-0.8969q

,.o

- - - ~ ... lre.tIow •• 0 . . ." ' " - - - - - .J wi'f4low • • .sq~ DtI.'Iar ------- 6 wiMDW'. "Cl'

Fig. 3. Estimate of uncertainty with averaging. (2.15)

1

The effect of some common widows on the estimation of IG(e-iOO)1 are shown in figures 4 and 5 for 256 frequency points with 50% data overlapping. The figures show that the Hanning and Blackman windows give lesser variance than the Hamming and Bartlett windows respectively.

The Nyquist plot of the plant. model and the uncertainty regions are shown in figure 1.

iii

-D .•

Fig. I. Nyquist plot of the plant, model and the /IIIC1?rtainty regions.

Direct use of the relations (2.3 and 2.5) results in a large variance in IGM(e -iw)l. where the data window 10~~'-:_.,----~-~~.O""'_'------'~" NormaUz.J Fr.qu'lf c y

is rectangular and the number of frequency points (in the range ±wN ) are the same as the number of data points (N=1024) as shown in figure 2. The frequency range [O.7tJ along the abscissa arc normalized by the Nyquist frequency WN • A considerable reduction in

Fig. 4. Uncertainty estimates for Hanning and Hamming windows.

the variance of IGM(e -iOO)1 is achieved when the data is split into segments of equal length and their periodograms averaged as shown in figure 3. The figure shows that as more number of data segments are created. the spectral estimate becomes smoother but at the cost of bias. Figure 3 also shows that a further reduction in the variance is possible when the data segments are overlapped by 50%.

- - - - - a.rtl.tl wirulDW

Normalir;.,d Fr.qu,"cy

Fig. 5. Uncertainty estimates f or Blackman and Bartlell windows .

85

The effect of input excitations for different fundamental frequencies is shown in figure 6. It is seen in figures 2-5 that the spectrum of I(j M(e -j"')1 fail

10

'

0) SNR=9. 74 (Z) SNR=O. 97

10 0

to match the true I (j (e -jW)1 at the lower frequencies. In figure 6 it is shown that a smoother input spectrum (Hanning windowed) due to low frequency input at 0.0041t gives a better estimate at all the frequencies including the notch in the low frequency range.

:i

~

~

10- 1 10- 2 true uncertainly

10-3L-~~~~~--~~~~~

10- 1

10- 2

100

Normalized Frequency

Fig. 7. Uncertainty estimates f or tlte two SNR values. 10°

10- 1

- - - - . 0 .016

TT ,M/ riwI. TT ,ad/ru,..

I H)

- - - 0 .004

IT ,Ad/ tiNe.

( RI

- - - - - - O .OQ.I

:i

INJ

l10-

(1) SNR=9. 74

2

(2) SNR=O.97

true uncertDinly

Fig. 6. Uncertainty estimates f or different inpllts alld tlte effect of willdows.

10-3L-~~~~~--~~~~

10- 2

10-1 Norn,aUzed

100

"" r~qu#'ncy

Fig. 8. Smootht'fll'd IIllcertainty estimates from filtered eft).

The effect of additive noise is illustrated by considering two sets of SNR values at 9.55 and 95 .51 in yet) which reduces to 0.97 and 9.74 respectively in e(t). The spectrums of I(j M(e -jW)1 in figure 7

linear form : T(q - I)~U(t)

= R(q -')W(t)-S(q -')y(t)

(3 . 1)

The above equation is derived on the assumption that all future set points are same as the current set point wet). The polynomial coefficients R(q - '),S(q - I) and T(q - ') are obtained by equating (3.1) with the control law of GPC (Clarke et aI, 1987):

corresponding to the specified SNRs show that cJ>w(w)/cJ>uuCw) significantly contributes to the bias in the estimated uncertainty. The estimated I (j V i'")I based on the filtered sequence of the noisy data

z:

are shown in figure 8. A third order butterworth filter with a phase-Iag removal strategy is employed to filter out the noise. Cut off frequencies of 0.3 and 0.1 arc used for the SNRs 9.74 and 0.97 respectively. Figure 8 illustrates that I (j M (e - j "') I based on pre-fiItered data

(3 .2)

Where G ,[ and A arc the step response matrix, free response vector and the controller weighting respectively. The equivalent linear form of GPC is shown in figure 9.

is in good agreement with the I(j (e -jW)I. The authors have observed that the DFT method performs much better than many of the parametric methods (even for very high order models) in presence of noise and for prefiltered data. Further work in this direction is presently in progress.

l

3. ROBUST TUNING OF GPC LRPC strategies are based on the minimization of a multi step quadratic cost function. All such predictive strategies require a knowledge of the process model. Many LRPC algorithms such as DMC, MOCCA etc. work with actual step response coefficients. This is a non-parsimonious representation of a process. GPC on the other hand uses input-output transfer function model of the process, that not only is a parsimonious representation but allows one to express it in the conventional feedback form . Therefore the stability of GPC can be examined using the SGT criterion and a controller can be designed that is robust to the model-plant uncertainty (Bitmead et aI , 1990; Robinson et aI, 1991). McIntosh et al (1991) and others (Mohtadi. 1988) have analyzed the role of various tuning parameters of GPC by rearranging the predictive control law into the following equivalent general

Fig. 9. Feed hack structllre ofGPC.

The polynomials R (q - I) , S(q -I) and T(q - I) in (3.1) are functions of the model parameters as well as the GPC tuning knobs such as NI' N 2, NU, Cc(q-I) , Aetc. For the additive uncertainty (j (q -I) (2 .2), the designed feedback loop M(q - ') is formed by lumping the elements shown within the dashed boundary in figure 9 as (Banerjee et aI, 1992): M( -I) =

q

S(q -')

~T(q -')+G(q -')S(q " )

33 ( . )

Under the small gain criterion, a sufficient condition for stability for an additive perturbation is: ju IG Vj jMV ,. < 1

86

'VOJE

[D,rel

(3.4)

Alternately for a SISO system an equivalent representation is: IC(e -ijt

<1_'_-'.j 1=1 aT(e-i j+C(e -i)S'(e-ij 1 S(e-'j

\;f

roE 10,1t)

M(e

(3.5)

Similarly

for

the

multiplicative

perturbation

(G"'(q-I) = G(q-I)/G(q- I», the stability condition

becomes: Fig. 10. Effect of NU on stability bound

(3.6)

And

for

the

feedback

(Gf(q-I) = -G(q- I)/G(q- I)G(q - I», condition can be expressed as:

perturbation the stability

1(t.T(:i~-:~~~i~ie-ijff(e-ijl < I

(3.7)

_ Stability criteria (3.5) for the additive uncertainty IG(e-'jl is one of the simplest form that can be used as a graphical tuning procedure for robust design of GPC. The advantage of such a procedure is that the controller necessarily need not conform to the following H ~ condition thus making the design less conservative:

le (e -i"') If' <

I-I-. I M(e -' j

SampMn,l'rI/#,..,al

Fig. I J. Effect of NU on Servo tracking

4. EXPERIMENTAL RESULTS The experimental set up consists of a computer-interfaced, pilot scale heater in which the cold water is heated by a submerged steam coil in an insulated vessel. The plant is interfaced toan IBM-PC running a real time (QNX) operating system that executes the GPC algorithm under an in-house developed software called MUL nCON.

(3 .8)

The effect of some of the GPC tuning parameters on the stability of the system in presence of uncertainty has been examined by Banerjee and Shah, 1992, and it has been shown through simulations that the frequency based robustness measures provided by these tuning parameters conform to the discussions in Clarke, 1989 and Mclntosh et ai, 1991.

The open loop process is suitably excited as a first step in the experiment to identify a model and thereby obtain an estimate of IG(e -ioo ) I through the proposed signal processing techniques. The process response for square type excitation is shown in figure 12. This information is used to select robust tuning parameters, which are then implemented on the plant. A MA TLAB based design tool box has been developed that utilizes experimental data for robust design of the GPC. Different sets of GPC tuning parameters that were tried on the plant are shown in Table-I.

The effect of control horizon NU, that has not been examined earlier (in Banerjee et aI, 1992) are shown in figures 10 and 11 for the system (2.14-2.15). It is known that NU adds to the aggressiveness in the control action. For an open loop stable system, NU= I is generally recommended. Figure 10 shows the graphical representation of (3.5) for NU = {I, 2} and controller settings of NI = 1, N2 = 5, A = 0 and Cc(q-I) = 1 - O.8q- l. The corresponding time domain

simulations are shown in figure 11. The figure shows that for the above settings when NU is increased to 2, the spectrum of IIIM(e -iW)1 comes closer to IG(e -iW)1 thus making the controller more active and confirming that NU adds to the aggressiveness in the control action. For NU=2.IIIM (e -iW)1 just touches the uncertainty, thus giving rise to the expected oscillatory action. This oscillatory response is shown to be stabilized for NU=2 when a second order filter

J .. 1 t

J Fig. 12. Plant input output Data.

Cc(q-I) = (1- O.8q- I)2 is selected which pulls away

Figure 13 shows that for case-I, the stability bound

IIIM(e -iW)1 from IG(e -iOl)1 as shown in the figures 10 and 11. The second order filter however cannot prevent the large initial control move due to NU=2.

IIIM(e -JlO)1 almost intersects the uncertainty bound

thus giving rise to ringing in the controller as shown in figure 14. Ringing is not observed for a step down in the setpoint because the valve reaches its lower

87

CASE N2

NU

A

Cc(q - I)

I

5

I

0

I

2

5

I

0

1- O.8q -'

3

5

2

0

1 -O.8q -'

4

5

1

0.5

I-O.8q -'

Table · 1 (Different caseJ afG Pe t//fling parameters).

saturation limit. thus exhibiting the inherent nonlinearity in the system. The CcCq -l) filter in case-2 adds to the robustness in the system hy increasing the stability margin. For NU=2 (in case-3) the controller becomes more active and hrings the spectrum of 11IM(e -i"')1 closer to the uncertainty, therehy limiting the robustness of this controller settings. The contribution of A towards enhancing the rohustness in the system can be seen for case-4, where the controller action is heavily detuned and the corresponding stahility margin is very wide.

Stahility conditions for GPC within SGT framework are presented and evaluated experimentally on a pilot scale heater system. The fact that NU makes the controller more active is verified through simulation in frequency domain. It is experimentally shown that the GPC tuning parameters such as N 2• CcCq - l) and A that are known to endow rohustness also increases the stability margin hetween IIIM(e -i'")1 and IG(e -iW)I. The experimental results show that this method can be implemented on an open loop stable processes. 6. REFERENCES Bitmead.R.R .. M.Gevers and V.Wertz, (1990). Adaptive Optimal Control-Th,' Thinking Man '.s GPc. Prentice-Hall . Ban erjce.P. and S.L.Shah, (1992). Tuning Guidelines for Robust Generatized Predic(ive Control. Proc. CDC, 3233-3234 Clarke .D. W ..CM ohtadi and P.S. Tuffs, (1987). Generalized Pred ict ive Control-Part 1 and 11. AlllOmatica, 23,137-160. Clarke.D.W .. (1988). Application of Generalized Predictive Control to Indu stri al Processes. IEEE Control Systems Magazine, 8 .49-55. Cutl er,C R. , and R.B .llawkins, (1988), Apptication of Large Predicti ve Controlle r to a lIydrocracker Second Stage Reactor. Proc . ACe. 21\4-291.

",,,

Dcsoc r.CA . and M.Vidyasagar, (1975). Feedba ck Systems: 11I!,lIt-Ollt!,lIt Pro!,erties . Academic Press. -~.~- · · .•••~~ • . 3

Doyk.J .C. B.A.Fr;Ulcis and A.R.Tannenbaum, (1992). Feedback COlltrol Theory. Macmillan Publishing Co .. New York . (iareia,CE .. D.M. l'rctt and M.Morari , (1989). Model Predictive Control: '111l'01), and Pract ice - A Survey. Autamatica, 25, 335-348.

IO~~'"'_.c-~---~~,O'"7_,-----_......J,,"· Nurnr.n l i:. ,d "'r"qu,n ry

Fig. 13. Uncertainty a lld stability hOllnd, far different ' ases.

Gobcrdh;U1singh .E .. L.Wang and W.R.Cluelt. (1992). Robust Frequ ency Dom ai n Identification. Chem. E1Igg. Sci , 47, 191\'1- 1'1'1'1. Goodwin.G.C and M.E.Salgado . (1989). Quantification of Unce rt aint y in Estimation usi ng an Embedding Principle. Proc. ACe. 141 6-1421.

j

knkins.G.M. and D.G. Watts. (1968). Spectral Analysis and its A!,f'lim(i,,1I.llolden-Day. Kosut.R.L. .(ll) I\7). Adaptive Uncertainty Modeling: On-Line Rohust Control Design. Proc . ACe. 245-250.

J

I

J.jun &.L. . (1'11\7). Sysfl'f/J Idc1Itijica tia1l : Theory f or the User. I'renticc -lI all. Englewond Cliffs. NJ. I\1dntosh .A.R .. S.L.Shah and D.G.Fisher, (1991). Anatysis and Tuning of Adaptive GPC C<111. 1. afChem . Engg, 69, 97-110.

Fig. 14. Serva trackillg for differellt cases.

5. CONCLUSIONS Classical signal processing methods with appropriate smoothing tec hniques, such as the choice of windows and its lag, data overlapping and proper input give an excellent estimate of Ic; (e" "")1over the entire frequency range [0, rei . A bias proportional to (w(w)/uu(W))05 is introduced in the estimated IG(e-i"')1 in presence of additive noise u(t). The bias in IGM(e -i"')1 due to additive white noise can be reduced by prefilterin g the data.

Morari .M. and E.Zafiriou. (1989). RO/>lL