The problemofcross-machine-direction (CD) responsecontrol in paper manu facturing is ideallysuited for the proposed IMC robust-controller-design procedure.
Models relating CD paper sheet properties to actuator adjustments are typically given by P^pl(s) = p(s)P^pl, where p(s) are uncertain actuator dynamics with time-delay (as in Equation 5), and is an interaction matrix given byEquation 7.
I Pm . . . P2 Pi P2 V 0 . . . ... 0 Pm . . . P2 Pi j
ra×re
Uncertain interaction parameters in Pç'p are bounded real numbers given by
Pi ∈ ∙ (§)
Since models P^'^i are typically of very large dimension n, μ-synthesis and model- inverse-based techniques for controller design lead to hugely complicated control al
gorithms. Moreover, models Pq'™ are characterized by high condition numbers, so robust performance can be difficult to achievewith model-inverse-based controllers.
Tight boundson the singularvaluesofpositive-definite models Pq'^d despite interac tion parameter uncertainties enable application of theIMC robust-controller-design procedure. The design procedureleads to desirable diagonal and bandedcontrollers with valuable robust-stability, robust-performance, and failure-tolerance properties.
If the robust-performance-analysis problem involving with uncertain in
teraction parameters were written in terms of M and Δ as in Figure 3, matrix Δ would contain m repeated real blocks of dimension n × n in addition to real blocks corresponding to uncertain scalar dynamic parameters - a problem entirely beyond the capabilities of existing software for calculatingμ [Μ]. Application ofμ-synthesis to design either a diagonal or banded controllerfor this problem is equally hopeless without the results in this work.
Operators can produce thinner papercloser to specificationswhenvariations in CD sheet properties are eliminated. Successful application of the proposed controller- design procedure to CD paper-responsecontrol is therefore an opportunity for sig nificant, beneficial impact on a major industry.
†This 140 million metric tons is a 1982 production Figure from Statistical Year
book 1983/84- In 1985 the United States consumed 84.1 million cords of pulpwood in production of 76.5 million short tons of paper and paperboard [Statistical Ab stract of the United States, 1987]. George S. Witham, Sr., author of a classic 1942 text Modern Pulp and Paper Making estimates “average Adirondack Spruce runs 15 to 20 markets to the acre” [Witham, 1942]. Therefore, a one-percent reduction in fiber usage saves approximately
1,400,000 metric tons paper× 84.1 cords pulpwood 76.5 short tons paper
0.907 short ton 1 metric ton
× , f 3 markets wood
— 1,396,000 cords pulpwood× I --- ---—
∖ 1 cord wood
= 4,188,000 markets pulpwood
— 209,000 acres average Adirondack Spruce
= 327 square miles forest.
region-mapping methods are discussed. Figures in Chapter II illustrate each step in the algorithm. Chapter III presents the IMC controller-design procedure for ro
bust performancebasedonthe regions 7r(ιω). The performance/robustness tradeoff associated with selection of the IMC filter tuning parameter is illustrated. Perfor mance of the Smith predictorresultingfrom the design procedure iscompared with that of a PID controller for different levels of uncertainty. In Chapter ΓV the design and tuning of Smith predictors are addressed in detail. Translation of parameter uncertainty into norm-bounded uncertainty is discussed - a particularly useful for
mula for multiplicative error in the first-order with time-delay model 6 is derived.
A method for selecting the form of performance weight W2(s) is suggested. Use ful tables of controller tuning parameters are presented in tabular form for a wide range of uncertainty levels in the first-order with time-delay model. The Smith predictor controller resulting from the IMC design procedure is compared with the μ-optimal controller inChapter ΓV. In Chapter V the IMC robust-controller-design procedure is applied to the problem of CD response control in paper manufac
turing. Characteristics of the physical process that relate to model structure are discussed. Properties of special matrix forms that model CD response interactions are established. In particular, bounds on the singular values of uncertain CD re
sponse models are developed. The bounds enable design of desirable decentralized and banded controllers for large-dimension CD response control problems. Robust stability, robust performance, and robust failuretolerance of the controllersdespite interaction parameter uncertainties are proven. The importance ofa dimensionless
Carey, E. W., with C. R. Bietry and H. W. Stoll, “Performance Factors Asso ciated with Profile Control of Basis Weight on a Paper Machine,”
Tappi, 58, June 1975, pp. 75-78.
Chen, S., Control System Designfor Multivariable Uncertain Processes, Ph.D.
Thesis (Supervised by C. B. Brosilow), Case Western Reserve Uni
versity, August 1984.
Doyle, J. C., “Analysis of Feedback Systems with Structured Uncertainties,”
Proc. Inst. Elect. Engrs., Pt. D, 129, 1982, pp. 242-250.
Doyle, J. C., “A Review ofμ for Case Studies in Robust Control,” IFAC World Congress, Munich, July 1987.
Doyle, J. C., and A. Packard, “Uncertain Multivariable Systems from a State Space Perspective,” Proceedings of the American Control Confer
ence, Minneapolis, June 10-12, 1987, pp. 2147-2152.
East, D. J., “On the Determination of Plant Variation Bounds for Optimum Loop Synthesis,” International Journal of Control, 35, 1982, pp.
891-908.
Fan, M. K., and A. L. Tits, “Characterization and Efficient Computation of the Structured Singular Value,” IEEE Transactions on Automatic
Control, AC-31, August 1986, pp. 734-743.
Horowitz, I., “Quantitative Feedback Theory,” IEE Proc., D-129, November 1982, pp. 215-226.
Verlag, to be published 1988.
Saeki, M., “A Method of Robust Stability Analysis with Highly Structured Uncertainties,” IEEE Transactions on Automatic Control, AC-31, October 1986, pp. 935-940.
Sideris, A., and R. R. E. de Gaston, “Multivariable Stability Margin Calcu lation with Uncertain Correlated Parameters,” IEEE Transactions on Automatic Control, to be submitted 1987.
Smith, O. J. M., “Closer Control of Loops with Dead Time,” Chemical Engi
neering Progress, 53, 1957, pp. 217-219.
Statistical Abstract of the UnitedStates, U. S. Department of Commerce, 107th ed., 1987,pp. 660-661.
Statistical Yearbook 1983/1984, Department of International Economics and Social Affairs, (United Nations: New York), 34th ed., pp. 686- 687.
Wallace, B. W., “Economic Benefits Offered by Computerized Profile Control of Weight, Moisture, and Caliper,” Tappi, 64, 1981, pp. 79-83.
Witham, G. S., Modern Pulp and Paper Making·. A Practical Treatise, (Rein
hold Publishing Co.: New York), 2nd Ed., 1942.
Figure 1: Standard feedback control structure with nominal process P(s), multiplicative uncertainty W1(s)∆1, controller K (s), disturbances d(s) and out
puts y(s).
Figure 2: Standard feedback control structure in Figure 1 with performance requirement W2(s)∆2 in the loop between jz(∙s) and d(s).
Figure 3: The block structures for μ-synthesis (top) and μ-analysis (bottom) are equivalent to that in Figure 2.
GAIN UNCERTAINTY
Figure 4: Twenty percent gain uncertainty in e-β∕(s +1) for sixty frequencies in therange 0.01 to 10.
DÏLXÏ UNCERTAINTY
10.
Figure 5: Twenty percent time-delay uncertainty in e-i∕(s + 1) for sixty frequencies in the range 0.01 to
IIMM-COHSIλHT UNCERTAINTY
Figure 6: Twenty percent time-constant uncer
tainty in e~t∕(s + 1) for sixty frequencies in the range 0.01 to 10
GAIN AHO DELAY UNCERTAINTY REGIONS
Figure 7: Twenty percent gain and time-delayun
certainty in e~*∕(s+l) for sixty frequencies in the range 0.01 to 10
SλIH AND T I MI - COH S TAHT UNCERTAINTY
Figure 8: Twenty percent gain and time-constant uncertainty in e ∕(s + l) for sixty frequencies in the range 0.01 to 10.
DIlλY AUD T I M B - C O N S T λ M T ÜNCÏRTAIHIÏ
Figure 9: Twenty percent time-delay and time- constant uncertainty in e-'∕(s +1) for sixty frequencies in the range 0.01 to 10.
κ»>as
Figure 10: Twenty percent gain, time-delay, and time-constant uncertainty in e~a∕{s + 1) for sixty fre quencies in the range 0.01 to 10.
Figure 11: Internal Model Control (IMC) parame
terization of standard feedback control system with process p(s), nominal model p(s), controller q(s)t set- point r(s), disturbances d(s) and output y(s).
MODEL PARAMETER UNCERTAINTIES
Daniel L. Laughlin and Manfred Morari
Abstract
Regions on the complex plane are employed in a versionof the Nyquist stability test for control systems when the system model contains parameter uncertainties.
Generally, the uncertainparameters are real and bounded and appear as numerator and denominator coefficients in a rational function of frequency. Through the func
tion, theparameters decribe a setof model points at each frequency. Theproblemof locating such sets is defined for a modeluseful in describing the frequency response of physical processes. A new algorithm for locating region boundaries enclosing the set of models is presented. Boundary points connected by straight line segments are determined by the algorithm. The algorithm offers two principal advantages over previously available methods: 1) The region boundaries are guaranteed to en close the set of models, and 2) the algorithm preserves concave sides on the region enclosing the set.