Construction of control design models from engineering simulation models

Math1 Comput. Modelling, Vol. 14, pp. 413-417, Printed in Great Britain

1990

0895-7177/90 $3.00 + 0.00 Pergamon Press plc

CONSTRUCTION OF CONTROL DESIGN MODELS FROM ENGINEERING SIMULATION MODELS Robert A. Smoak Xavier R. Ollat Tennessee Technological University, Department of Mechanical Engineering, Box 5014, Cookeville, TN 38505 ABSTRACT. This paper addresses the problem of extracting appropriate control design models from engineering simulation models. Engineering simulation models are typically hiqh order and non-linear, and freouentlv stiff. Linearizing these models produces candidates for control design models which usually have numerical properties which make them unusable for control design. A group of methods are discussed which can reduce the order and improve the conditioning of the linear models. The methods are used to reduce models for water level in a drum boiler and a nuclear steam generator. Keywords.

Simulation, control, linearization, model order reduction.

INTRODUCTION commonplace Simulation models have become They have been tested and in engineering. validated in different areas On tt(t~""Z-fS~~""~t! chemical plants, etc.). would appear that they would be the appropriate candidates to use for control design with the modern methods (robust linear quadratic regulator, H-infinity norm).

requirements (see Fig 2).

“2

Poor conditioning together with the numerical realities of controllability and observability this problem difficult. The make more alternative, to redevelop a simpler simulation model of adequate fidelitv for a oarticular expensive. The problem, is ’ prohibitively Modular Modelinq System (MMS) and the Advanced Continuous Simulation Language (ACSL) [Mitchell 81, Babcock 851, the industry standards, can produce a simulation model of a Dower olant in a short period of time but the resulting MMS model is nonlinear and quite stiff. This paper will use the feedwater control for problem as the motivation producing simolified models from full nonlinear models The models of fossil and nuclear plants. will be constructed for use with H-infinity or Loop Transfer Recovery methods.

Fig. 1

The control problem is to control the level of water in the steam drum of a boiler (fossil plant) or steam generator (nuclear plant) to be constant in the presence of disturbances of steam flow and energy input. For the H-infinity method [Francis 87, Safonov 88, Safonov ,261, the control objective is: Given a transfer matrix G(s), find a stabilizing controller K(s) such that the closed-loop transfer function Tylul ,is internally stable and its H-infinity norm is less than or equal to one (see Fig 1).

‘WI I

Odb

For the Loop Transfer Recovery method [Safonov 88, Safonov 81, Alouani 88, Doyle 81, Lehtomaki 871, the control objective is: Find a regulator KR(s) and a Kalman Filter KF(s) such that the singular values of the system meet some performance and robustness

Fig.

2 Singular

Value

Speck

on S and T.

Proc. 7th Inr. Conf. on Mathematical and Computer Modelling

414

II PROBLEMS -Most models are made for simulation purposes programmer The physical principles. from conservation equations for a large writes control volumes (each component number of Afterward, all these "simple" of the plant). models are linked together to generate the state simulation oroaram. Sometimes extra the component models must variables, not-in be added to allow the connection. The resulting hiah model is order (100 to 200 state *nonlinear, stiff and ill variables), This approach is a fasit w;; conditioned. to generate good simulation models. an easy way because libraries of preprogrammed This approach takes components already exist. only 2 to 3 man months of work instead of years to generate a full plant model. These models are appropriate to test new designs they but are (control or component) inappropriate for control design purposes. The complete models are useless for design purposes because they are to big for the results in particular problem. This controllability and observability problems: subsystems Controllability some really affected by the inputs,

are

Observability - some parts not affect the outputs.

model

of

the

il

= rA11

k2

IA21

L

Y

A12-j

X1

A221

X2

+

LJ

_I

= [c1

C2j

X1

p;1" IB2I

+

DIJ

x2 [I set

x2

=

0

4

= [All - A12A22-1A211x1 + [B1 - A22-1 B2]U

not do

Also, there are stiffness and ill conditioning problems. Stiffness - the ratio of the largest eigenvalue to the smallest of the associated linear model is a large number, combination of conditioning the Ill controllability, observability, and stiffness problems. The next problem arises from the nonlinearity of the system. Most of the control design methods are developed for linear time invariant systems. The nonlinear model must be linearized point which implies a around an operating different model for each operating point and also different control parameters. The control design techniques can generate They usually require numerical problems also. the solution of Riccati and Lyapunov equations, factorizations inner-outer and YLese oroblems are dif?F!l?e to solve if the size of the model becomes large. It can become imoossible to calculate the coefficients of the characteristic oolvnomial for some models. H-infinity design requ-ires the factorizations. The LQG/LTR methods require the solution of the Riccati equations. Large models and widely spaced eigenvalues can make the factorization uncontrollable and impossible. Nearly unobservable models make the Riccati equation impossible to solve. III

Using the 'ANALYZ' command control problem. in ACSL and defining a set of state variables, inputs and outputs, he can generate a linear After reoresentina the subsrstem. model producing ’ a linear model, the unnecessary state variables introduced for the simulation need to be removed by setting their derivatives equal to zero and solving the equations.

ORDER REDUCTION METHODS

The simulation models with ACSL generated and MMS can be used to develop a linear model The user at a particular operating point. must define the subsystem suitable for the

Y = [cl

- C2

42-l AllI

X1

+ [D C2 A22-1 B2]U The next step is to determine the slow and In most cases the fast part of the system. fast part can be neglected without introducing The fast part is determined a big error. bv the magnitude of the eiaenvalues of the system and- it is the choice- of the user to This limit can be chosen by fix the limit. comparing impulse or step response, singular values, etc., of the slow part and the initial The fast part is removed by setting model. The new model is its derivative to zero. given by: G(s)

=

Gf(s) =

[A, B, C, Dl = Q(s) + G(s)(s) [Af. Bfs Cf, Dfl

Gs(s) = [AS, B,, D,> &I F(s) = [A,,B,,

C,. CD, + Df - Cf

A&l

The order can be reduced further by using Several algorithms approximation techniques. Most of them are exist and can be used. preprogrammed in the Matlab Robust Control Toolbox [Safonov 881. The balanced model reduction is based on Moore's reduction model "balanced-transformation" procedure [Moore 871. This is based on an ordered Schur decomposition of the product controllability observability and the of grammians to compute orthonormal bases, enabling or non-balanced computation of a balanced The reduced realization of the reduced model. with and minimal stable, model ’ observabilit -c ntrollability grammi and diagiiaf TSafonov 88, Laub 87af.' e#~A

hoc.

7th Int. Conf. on Mathematical and Computer Modelling

the frequency domain L-infinity norm bound of the reduced model bound is by [Glover 86, Enns 841:

error given

A Q-Markov Cover is a realization for any q>l. in Appendix).

415 cost equivalent (see Algorithm

IV RESULTS 2

I IG (jw) - F(jw)I Jm 5

y i=k+l

ui

where the oi are the square roots of eigenvalues of the reachability observability grammians.

the and

Optimal Hankel minimum degree approximation without balancing is based on singular value decompositions [Sanofov 871. Hankel The singular values of the system G(s) are the singular values of the product of the grammian. observability and controllability The Hankel norm minimal degree approximatiom problem is, given any_ positive number p, finda state space model G(s) such that the greatest Hankel singular value of the error system (G(s)--6(s)) is at most p. The state space realization of a minimal degree approximation can be given in terms of Moore balanced state space realization of G(s). Glover [84] proved that every kth order reduced model G(s) had the property:

I IG(jw) - ~(&)I I 5

2

Y

01

i=k+l here, the oi are the the product of the reachability grammians.

singular values of observability and

The relative error model reduction [Sanofov 883 is based on the phase matching relative error methods developed by Green and on the Balance Stochastic Truncation technique of Desai [17]. Given an nth order square and stable system G(s), find the kth order reduced model 6(s) such that the Il(G-l(G-G))l Jm is bounded by: re1ative error

I IG-1 (jw)(G(jw) - F(jw)) I I 2 n -ii i=k+l

1 + oi -1 1 - oi

The Q-Markov cover [Skelton (1989)] is based on the principle of matching the first q derivatives of the impulse response Ce*tB at t=O and the first derivatives of the output correlation Ce*9 BC' at t=o. The transient properties of the reduced model will not exactly match those of the original system but will start off at t=O in the "right direction" up to q derivatives of y(t) and the steady state properties will not exactly match those of the original system but will match the steady state quantity

The procedure above was used to create linear problem models for the feedwater control for a fossil and a nuclear plant. MMS models already existed and had been validated for these plants. The subsystem used in each case to generate the linear model included the boiler or steam generator and part of the feedwater train down to the boiler feedpump (valve, pipes, variables The state feedwater heaters). were the same as those of ACSL program, the speed demand, inputs were boiler feedpump energy perturbation, an steam flow a plant the nuclear perturbation and for feedwater control valve position. The outputs were drum or steam generator level, steam flow, feedwater flow, and for the nuclear The ACSL linearization plant steam pressure. created a 15th order model for the boiler and a 32th order model for the steam generator. Removing the unnecessary states, introduced by the connective module in MMS, dropped the number of states to 12 for the boiler and 30 for the steam generator. The fast part was also removed and the order dropped to 8 for the boiler and 25 for the steam The number of fast states removed generator. was such that the reduced model has the same singular values in the frequency range studied and the same step responses. The reduced models were still considered to big to work with the algorithms for modern The approximation techniques were control. The Relative error model reduction used. method was discarded because it did not give The other techniques results. acceptable were was $e;;;y;;da s~~~e~~Jd,'y~ode:he,i~i~~~ The steam generator was reduced the methods. to a 10th order model with the Hankel and Schur reduction techniques and to a 17th The order model with the Q-Markow cover. following graphs show the singular values of the systems and some step responses. Finally, the user must chose the reduced model the most appropriate for his problem. Fossil Boiler Model

'nb

nu Ro=

1 i=l

,;

yi (t) yi' (t) dt

plus q-l derivatives of the output correlation.

Fig. 3

Froqumlcy - radlanhc Maximum singular value of G(S)

416

Proc. 7th Int. Conf. on Mathematical and Computer Modelling Nuclear Steam Generator Model

Fig.

Fig. 4

3

Minimum singular value of G (s)

Response to a step change in boiler feed pump speed demand

Fig. 6

Maximum singular values

Fig. 7

Singular value

+S&urmodd

0

Fig. 5

so

loo

!mo

-told~

nmo

Response to a 1% steam flow perturbation

Fig.

a

Minimum singular value

Proc. 7th Int. Co& on Mathematical and Computer Modelling

f4 f

a s a

d 1!l

!I Fig. 9

Response to a step change in feedwater pump speed demand

REFERENCES Linear Alouani, A., "Lecture Notes: Multivariable Controll" Department of Electrical Engineering, TTU, Cookeville, TN 1988. Desai, U.8. and Pal, D., "A Transformation Approach to Stochastic Model Reduction," IEEE Trans. on Automat. contr., AC-29, 1984. Doyle, J., and Stein, G., "Multivariable Feedback for a Design: Concepts Classical/Modern Synthesis, "IEEE Trans. on Autonmtic Co&r., AC-269 pp. 4-16, 1981. Enns, D. F., "Model Reduction with Balanced Realizations: An Error Bound and Frequency Weighted Generalization," Proc. IEEE Conf.

on Decision

proc.

IEEE Conf.

On

Safonov, M. G. and Chiang, R. Y "Robust-Control Toolbox", The Math Wo;ks, Inc., 1988. Safonov, M. G., and Chiang, R. Y., "A Schur Method for Balanced Model Reduction," PTOC. American Contr. Conf., June 15-17, Trans. on Automat. 1988; alSO IEEE contr., 1988. Safonov, M. G., Joncheere, E. A., Verma, M and Limebeer, D. J. N., "Synthesis of' Positive Real Multivariable Feedback Systems". Int. J. Control, vol. 45, no. 3, pp. 817-842, 1987. Safonov, M. G., Laub, A. J., and Hartmann. G. "Feedback Properties of Multivariable Systems: The Role and Use of the Return Difference Trans. Matrix," IEEE OR Automat. contr., AC-26, pp. 47-65, 1981. Skelton, R. E., "Dynamic Systems Control", John Wiley and Sons, New York, 1989. "Advanced Simulation Continous Language". Reference Manual, Mitchell and Gauthier Associates, Inc. 1981. "Modular Modeling System", Babcock and Wilcox, 1985. APPENDIX If (A,B,c) is any asymptotical ly stable and controllable system, then AR A LAR, BR i LB, CR i CR, = L i u1*woq.

R A = XL*(LXL*)-1

and

and Control,

Las Vegas, NV, Dec. 12-14, 1984. Francis, B. A., "A Course in H, Control Theory, Springer-Verday, 1987. Glover, K., Optimal "All Hankel NOt-lil Approximation of Linear Multivariable their L--Error Bounds", Svstems, and Int. J. Control.Vol. 39, no. 6, PP. 1115-1193, 1984. Laub, A. J., Heath, M. T., Page, C. C.. and Ward, R. C., "Computation of Balancing Transformations and Other Applications of Simultaneous Diagonalization Algorithms, "IEEE ~rans. Automat. contr., AC-32, pp. 115-122, lG7. Lehtomaki, N.A., Sandell, Jr., and Athans. M "Robustness Results ’ L;iear-Quadratic Gausian Basil Multivariable Control IEEE Designs," Trans. on Automat. contr.. AC-269 PP. 75-92, 1981. Moore, B. C., "Principal Component Analysis System: Controllability, Linear &.ervability, and Model Reduction,' IEEE Trans. on Automat. contr., AC-26, pp. 17-31, 1981. Chiang, R. Y., Limebeer, Safonov, M. G., D. J. N., "Hankel Model Reduction Without Balancing, A Descriptor Approach," Decision

Control.

,

Los Angeles, CA, Dec. 9-11, 1987. Safonov, M. G. and Chiang, R. Y., "Model Reduction for Robust Control: A Schur American Relative-Error Method, proc. contr. copf., June 15-17, 1988.

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is a q-Markov COVER.

f