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Microcomputer based self-tuning and self-selecting controllers
Automatica, Vol. 16, pp. 1 8 Pergamon Press Ltd. 1980. Printed in Great Britain © International Federation of Automatic Control
0005-1098/80/0101-0001 $02.00/0
Microcomputer Based Self-Tuning and Self-Selecting Controllers* A. H. G L A T T F E L D E R t, F. H U G U E N I N $
and W. S C H A U F E L B E R G E R $
Classical control system design techniques, upgraded to include cases with large parameter and state variations, and self-tuning and self-selecting controllers may be successfully implemented on microprocessors as indicated by a survey of industrial applications. Key Words--Adaptive control; computer control; control engineering; computer applications; direct digital control; industrial control; Lyapunov methods; microprocessors; PID control; self-adjusting systems; stability criteria. Abstract~lassical control system design techniques are extended to include cases with large parameter and state variations. New results on stability of adaptive gain control systems are obtained and a new method to design self-selecting controllers is presented. It is then shown that microcomputers are well suited for hybrid implementation of the resulting control systems. Several examples and a short survey on industrial applications are also given.
1. INTRODUCTION T HE PAPER demonstrates how self-selecting and selftuning control systems can be designed and implemeted with microcomputers. That the classical methods of continuous- and discrete-time control system design can be extended to include cases with large parameter and state variations is shown in the first part. The stability problem is investigated for the selftuning regulators. That the systems resulting from these design methods can easily be realized with the aid of microprocessors is outlined in the second part. Only the most critical part of the self-tuning control system, the adaptive controller, is realized digitally. The hybrid solutions that are obtained in this way allow control of fast processes. The microprocessor is used as a building block in this approach. This is in contrast to a completely digital design as reported in (Astr6m and co-
workers, 1977). The simple structures of the selfselecting controllers are implemented digitally. Hardware and software requirements are formulated. Three examples are presented in the third part. Industrial applications of the methods presented in this paper are surveyed. The methods that are discussed are straightforward and m o d u l a r extensions of classical control system design techniques. This fact results in considerable advantages during the design, implementation, tuning and operation phase of a system. These advantages are confirmed by several industrial applications and by the fact that students are able to implement adaptive controllers on laboratory scale plants within a few hours (Schaufelberger, 1977).
2. SELF-TUNING CONTROLLERS Different techniques for the design of simple self-tuning regulators are presented in this section. The adaptation problem for a system with u n k n o w n or slowly varying gain will be treated extensively. The stability of the resulting control systems will be investigated. The methods can easily be extended to the multiparameter case. 2.1 A
*Received May 25 1978; revised January 18 1979; revised May 29 1979. The original version of this paper was presented at the 5th IFAC/IFIP International Conference on Digital Computer Applications to Process Control which was held in The Hague, Netherlands during June 1977. The published Proceedings of this IFAC Meeting may be ordered from: North Holland Publishing Company, P.O.B. 103, Amsterdam--West Netherlands. This paper was recommended for publication in revised form by associate editor A. Longmuir. ~Escher Wyss Ltd, Research Department, CH-8023 Ziirich, Switzerland. SFachgruppe fiir Automatik der ETH, CH-8092 Ziirich, Switzerland.
stability property of linear, self-tuning regulators
If a linear control system can be designed in such a way, that a constant asymptotically stable system results for a long time period, then a certain kind of stability results. The system is in that case eventually asymptotically stable in the whole in the sense of La Salle (La Salle and Rath, 1963). This fact justifies to a certain extent the separation of the adaptive control problem into an identification and a controller design problem.
A. H. G L A T T F E L D E R , F. HUGUENIN and W. S C H A U F E L B E R G E R The above statement will now be formalized For = A(t)x + B(t)u(t)
(2.1)
with A(t) Ao
B(t) u(t)
bounded, continuous, lim,_~~ A(t) = Ao all eigenvalues in the left half plane bounded, continuous, limt .... B(t) = Bo bounded, continuous
The proof is analogous to the proof in the continuous-time case. 2.2 Gain adaptation by identification The system represented in Fig. l is investigated in this section. The basic control system, consisting of R and G, has to be designed asymptotically stable for p = l/k. Eventual stability results in that case for limt. ~ p(t)= 1/k. A more detailed definition of all parts in Fig. 1 is given in Huguenin and Schaufelberger (1977).
the following holds: The differential equation for e with e = x - z and = Aoz +
C
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
,,
B0u
6=Aoe+(A(t)-Ao)(e+z)+ (B(t)-Bo)u is eventually asymptotically stable in the whole in the sense of La Salle (.La Salle and Rath, 1963), i.e. e(t) is finite for all times and lim e(t)=O. The proof is a straightforward extension of the proof given in Huguenin and Schaufelberger (1977), chapter 3 in combination with theorem 7 from La Salle and Rath (1963). The same Lyapunov-function and the same technique of completing of square terms may be used. The proof is omitted. The corresponding theorem for the discrete time case is the following: For x(k+ 1)=A(k)x(k)+B(k)u(k)
(2.2)
P,
FIG. 1. Adaptive control for a plant with unknown gain. Process: kG(s):O< kmin< k~ kmax, G(s): aymptotically stablc. G(0)4-0 Input signal r 0 nonzero, constant. Controller RI.s): R (0) 4=0. Compensating gain p(t I.
The identification stage with inputs v and y and output q will be investigated first. The stability question for the total system will then be discussed. With the differential equations
y =e~ xk;
u(k)
P process
i~M = Apx M +bey bounded, limk~ ~ A ( k ) = A o all eigenvalues inside the unit circle bounded, limk_~o~ B(k ) = B o bounded
(2.3)
for the process and
with A(k) Ao B(k)
Ap asymptotically stable
= Aex + bpv
M
model
y~ = c x x M
(2.4)
for the model; letting e = x M - x , w = q - l / k one obtains the following equations of the perturbed motion for the identification stage:
the following holds: The difference equation for e = x - z and
= Ape ~, = hy( - yw + c T e).
(2.51
z(k + 1) : Aoz(k ) + Bou(k ) e(k + 1 ) = Aoe(k ) + (A ( k ) - Ao)(e(k) + z(k))
The Lyapunov-function
+ (B(k ) - Bo)u(k ) is eventually asymptotically stable in the whole, i.e. e(k) is finite for all times and lim e(k)=O.
1
2
V=h W +eTPe
(2.6)
A~P+pAp+cpeXe= - Q
(2.7)
with
Microcomputer based self-tuning and self-selecting controllers yields
< - y2w2 - eTQe.
(2.8)
is therefore negative semidefinite. The identification stage is uniformly stable in the whole. From
e~,e=e(t )=c~ exp (Aet )eo
(2.9)
and therefore
I~(t)l<~exp(-flt)
~,fl>o
(2.10)
tation of a gain is shown in Fig. 2. The controller acts on the model and the process is adjusted to the model. This design resulted from stability analysis. The following two conditions guarantee asymptotic stability in the whole for the error e = x M - x between the state vector x M of the model and x of the process: (i) The basic control system, consisting of R and G has to be asymptotically stable. (ii) The control system depicted in Fig. 3 has to be asymptotically stable for all ro and k. Design can be done by root locus technique.
equation (2.5) may be written as
= hy(-yw + e(t)).
(2.11 )
That the condition
z = ~ y2dt = ~
(2.12)
o
is sufficient for asymptotic stability can be proved with v = w2/h for equation (2.11). These results can be summarized as follows: The identification stage represented in Fig. 1 is uniformly stable in the whole. It is uniformly asymptotically stable in the whole when equation (2.12) holds. An investigation of the basic control system shows that equation (2.12) holds for all ro4:0. The total system differential equation augmented by ,~__y2 is used in the proof. It can be shown that no steady-state solution with ~ = 0 and z finite can exist. The limiter in Fig. 1 is important for this proof. With this, it is shown that the system in Fig. 1, if properly designed, reaches its desired steadystate operating conditions. Classical control system design techniques can be used to design the system. The restrictions put on G(s)-are very small, i.e. asymptotic stability and G(0) 4=0. The adaptive controller marked by the dashed line in Fig. i is replaced by a digital circuit with equivalent convergence properties in the implementation. See Astr6m and co-workers (1977); Mendel (1973); Graupe and Fogel (1976); S6derstr6m and co-workers (1978) for descriptions of the well-known gradient and least squares identification methods that can be used to identify q from y and YM recursively. Methods for the discrete design of the entire identification stage can be found in Astr6m and co-workers (1977) and S6derstr6m and co-workers (1978). 2.3 Model control Another simple control system for the adap-
?
FIG. 2. Model Control. Process kG(s); G(s) asymptotically stable, G(0)#0. Model G(s), Controller R(s). Input signal r 0 nonzero, constant.
FIG. 3. Control system used during the design phase. Input signal y o = l i m ~ y(t). Gain u 0 =limt~oo u(t). Process kG(s).
This stability property is a direct consequence of the following theorem: For the differential equation
R=F(x)
(2.13)
with xX= (yV, zX), and representable as
~=f(y)
(2.14)
=g(z) + G(x)y the following holds: The differential equation (2.13) is asymptotically stable in the whole, if it is stable in the sense of Lagrange (La Salle and Lefschetz, 1967) and if
A. H. GLATTFELDER, F. H U G U E N I N and W. SCHAUFELBERGER
~=fly)
(2.15)
and
=g(z) are both asymptotically stable in the whole. The proof may be found in Wehrli and Schaufelberger (1970). In the case at hand, Lagrange stability can easily be proved by using a theorem stated in (Bellman, 1953) for differential equations of the form i~--Ax + B(t)x.
The differential equations (2.15) are linear and present no problem. A simple numerical integration procedure can be used to replace the adaptive controller marked in Fig. 2. It is also easy to design a discrete controller for steady-state operation of the basic control system consisting of the controller R and the process G. This yields eventual stability of the adaptation loop according to Section 2.1. 2.4 Model reference control The adaptive system is presented in Fig. 4. The adaptive loop tunes the gain p of an otherwise classical control system by model-adjustment,
t- . . . . . . . . . . .
-I
L It .
.
.
.
.
.
U
FIG. 4. Model reference adaptive gain control. Process kG(s), Controller R(,s), Model G(s), Compensator C(s), Input signal ro nonzero, constant.
Global asymptotic stability for the adaptive loop can be proved for strictly positive transfer functions G(s) and for transfer functions that can be made strictly positive by a suitable compensator C(s) (Huguenin and Schaufelberger, 1977; Landau, 1969; Popov, 1973; Anderson, 1968). The theory of hyperstability or of passive systems is used. ¢I)(u)=u is normally chosen. This leads to a gain proportional to u 2 in the adaptive loop. A better choice of (I) will lead to a smaller dependence on u. Stability can be proved for slowly varying u(t) if the process is not hyperstable.
The augmented error signal method outlined in Monopoli (1974) may be used if the process is not hyperstable. The adaptive controller in Fig. 4 can be replaced by a simple numerical integration technique or the adaptive loop of the system can be designed by discrete techniques (Landau and B6thoux, 1975; Ionescu and Monopoli, 1977). 2.5 Sinusoidal perturbation ,system The corresponding system is represented in Fig. 5. A perturbation signal ~ is sent to the control system. The corresponding signal component is extracted at y. Filtering yields an estimate a proportional to p,k. a is adjusted to A 0•
FIG. 5. Adaptive gain control with sinusoidal perturbation system. Process kG(s), Controller R(s), Input signal ro: constant.
The describing function can be used for a stability discussion (Eveleigh, 1967). Examples may be found in Schaufelberger (1977) and Glattfelder (1970). 2.6 Multiparameter adjustment The methods presented in this section can easily be extended to the multiparameter case. Stability investigations are much more difficult in these cases. Applications of the methods discussed in this paper may be found in Schaufelberger (1977); Maletinsky and Schaufelberger (1974); Maletinsky ( 1975); Unbehauen and co-workers (1976); Landau and Unbehauen (1974); Glattfelder (1970). 3. SELF-SELECTING CONTROLLERS
Self-selecting control systems are treated in detail in Glattfelder (1974). Only the most important facts are summarized here. The design procedure is explained for the system in Fig. 6. The output signal y has to be controlled and the state variable I has to satisfy certain constraints. A self-selecting proportionalintegral (PI) controller will be designed. The following procedure is used:
Microcomputer based self-tuning and self-selecting controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,] I
FIG. 6. Self-selectingcontrolsystem.ProcessFG(s)withF(s)hyperstableand G(s) stable. (i) A linear PI-controller (ke, Tp, setpoint r) is designed for the stable plant FG(s). FG(s) must therefore be reasonably well controllable by this type of controller. (ii) The constraints are taken into account by designing a pair of controllers (kL, TL) for F(s) with k L ~ k P. This is possible, since F(s) is hyperstable. Setpoint values L T and L,L have to be selected according to the upper and lower limits of the state variable t. (iii) The self-selection is realized by, cf. Fig. 6 w = Min {Max (v, v,L),vT} This guarantees that only the appropriate control loop will be closed. (iv) The system is asymptotically stable if
L~
u1
FrG, 7. Overflowprotection for the integrator in Fig. 6 by a self-selectingcontroller. (y, I) is convenient to study the behaviour of the system. The design technique that has been briefly summarized in this section has been used in numerous examples and in several industrial applications.
4. IMPLEMENTATIONUSING A MICROCOMPUTER The methods that were presented in Sections 2 and 3 can easily be adapted for implementations on microcomputers. The analog solutions may be replaced by equivalent digital ones or by digital solutions with improved properties. Accuracy and drift problems will be reduced when digital solutions are used.
4.1 The microcomputer system A small hybrid computer system is used for development, testing and implementation of the algorithms described above. It consists of an EAI Mini Ac analog computer, an extension containing several adaptation units (multiplierintegrator-multiplier combinations) and an Intellec 80 Microcomputer development system. The development system has a complete process control interface with A/D and D/A converters, a real time clock and an interrupt system.
6
A . H . GLATTFELDER, F. HUGUENIN and W. SCHAUFELBERGER /SOFTWARE ~SUPPORT
INTEGER16bit IARITHMETIC PACKAGE
"~ /REAL TIME ) ~ MULTITASKING // ~ERATING SYSTEM/
1
l
FLOATING POINT24bit ARITHMETIC PACKAGE
sT2.0UT,_____.ES
: store - decout - de n
I- mln [ - max I- hys
1
float
DEBU )
RTC
driver user support
I/0
adcldac text
I/0
user support
fixx fixx2
IdlACROS write readpal convv
fabs
FIG. 8. Software support used to implement the controllers. 4.2 Algorithm development on the microcomputer A library of subroutines is used to program the different control algorithms efficiently. Integer and floating arithmetic operation packages and drivers for the different I/O devices and for the interface are available. A set of some 30 standard macros has been created. With these macros, the adaptive controllers can be coded from the block-diagrams Fig. 1 to Fig. 7 very easily. In this macro code a typical program as used in the examples would need less than 30 lines of code. The macros are expanded by Intels Marco Assembler. The software support system is represented in Fig. 8. If short sampling intervals are needed, interrupt driven programs are used. A real time operating system has also been developed. Up to 15 user tasks can run simultaneously and they can be controlled from the TTY. Special emphasis has been put on the fact that the operating system is well suited for control applications.
resistive load. Integer arithmetic proved to be sufficient for the adaptive controller in this case (Huguenin and Schaufelberger, 1977). A typical result is represented in Fig. 10. The sampling interval is 11 ms and the length of the program is 600 bytes for an I N T E L 8080 Microprocessor. I.O
p
~
P-f(n)
0.5 I000
L
2000 3000 n, r.p.m. FIC~. 10. Model reference adaptive generator control. Resulting gain P as a function of turbine speed n.
5. EXAMPLES 5.1 Self-tuning controllers Two very simple examples for the process represented in Fig. 9 will be gfven. Numerous examples of multiparameter cases may be found in the references of Section 2.6. (i) Modelreference control. The control system works according to Fig. 4 for a generator with FIG. 11. Generator control with gain adaptation by identification. Response to rectangular input r(t). Top: low speed.
bottom: high speed. (1) non adaptive, (2) adaptive.
FiG. 9. Process with variable gain. Turbine T, Generator G.
(ii) State variable filter identification. For the generator of Fig. 9 with complex load an adap-
7
Microcomputer based self-tuning and self-selecting controllers tive control system according to Fig. 1 was realized (Huguenin and Schaufelberger, 1977). Floating arithmetic was used t o implement a recursive filtered least squares algorithm for the estimation of q from y and YM. A typical result is represented in Fig. 11. The sampling interval is 23 ms and the length of the program is 1 K bytes. The programs used in these two examples are interrupt driven. Use of the real time operating system will be made in the next example.
Xz
~Ul
I150%
5.2 Self-selecting controller A process according to Fig. 12 with the three U2
~ "50%
! L1
""
D- . . . . . . .
ul
x2
)
~ ........
.....
U2
-I
FIG. 12. Process for level control. Input signals Ul, •2, 1/3. Output signals x~, x2.
input signals u 1, u 2 and u 3 and the two output signals xl and x 2 was built to test several selfselecting controller structures. Figure 13 shows a typical response to large set point steps on the level controller for x 2 with constraints 111" on ul and IZT on U2 as proposed in Fig. 7 (Thaler, 1978). The length of the program is 8 K bytes. This includes the real time operating system with TTY oriented monitor, the floating point package and the user program of about 1 K. Only a few of the possibilities of the operating system are used in this simple example (Huguenin, 1978). The sampling interval is 60 ms.
6. APPLICATIONS IN INDUSTRY 6.1 Self-tuning controllers Early attempts to use the methods investigated in this paper are summarized in Landau and Unbehauen (1974). These methods have especially been used to control electrical drives (Speth, 1972) and machine tool systems (Stute
FIG. 13. Result of level control experiment for rectangular reference value. Level x2 is controlled by inlet valve uI and outlet valve u2. and G6tz, 1973), which are both examples of systems with varying gain. Today the use for control of turbogenerators (D'Ans and co-workers, 1977) and for heating and ventilation control systems is investigated. 6.2 Self-selecting controllers A first application (Glattfelder and Gross, 1974) was to limit level excursions on a buffer reservoir by means of the outflow valve. The regulator was built from standard electromechnanical control equipment. It has reduced operator interference and fluid losses to virtually nil since its installation four years ago. In another application (Glattfelder and Gross, 1975) the combined pressure and power control of a steam generator was upgraded to cope with large load swings, avoiding replacement of plant subsystems. The controller was implemented with standard electronic control equipment and has been on line for three years. A third example (Glattfelder and Gross, 1977) is the power control of a district heating system with variable configuration and large flow transients absorbed by sequential use of the available thermal storage capacities. 6.3 Combined controllers The methods presented in this paper can be combined to form self-tuning and self-selecting controllers. These controllers may for example be used in machine tool systems (Stute and G6tz, 1973; Pressmann and Williams, 1977) or for control of pH values (Thaler, 1978) or purifi-
A. H . G L A T T F E L D E R , F . H U G U E N I N
cation of waste water (Thaler, 1978; Gujer and Erni, 1978).
7. CONCLUSIONS
Several methods to design self-tuning and selfselecting controllers have been presented and investigated in this paper. New results have been obtained in stability analysis of self-tuning controllers. All methods are straightforward extensions of classical control system design methods. Microcomputers are well suited for the implementation of the corresponding control systems. The resulting hybrid systems are transparent and modular with respect to hardware and software. This fact has led to a considerable interest from industry. The algorithms that are obtained from the theoretical development can easily be implemented on minimum microcomputer systems or on single board computers. Decentralized adaptive control can thus be realized. New hardware designs will soon speed up operation by a factor of 1 ~ 1 0 0 compared to the installation that was used in this paper leading to sampling intervals well below 1 ms (Huguenin, 1979). The main limitation of the methods presented in this paper is that the control problem has to be broken down into isolated subproblems and that these subproblems are solved on a priority basis. Despite this drawback these methods are very attractive for many small to medium size problems. REFERENCES Anderson, B. D. (1968). A simplified viewpoint of hyperstability. IEEE Trans. Aut. Control, AC-13, 292 294. Astr6m, K. J., U. Borrison, L. Ljung and B. Wittenmark (1977). Theory and applications of self-tuning regulators. Automatica, 13, 457M76. Bellman, R. (1953). Stability Theory (~f DifJerential Equations. McGraw-Hilt. D'Ans, G., H. Glavitsch and L. M. Panis (1977). An evaluation of present day excitation system control of turbogenerators. Proc. IFAC Melbourne, 1977, 199 203. Eveleigh, V. (1967). Adaptive Control and Optimization Techniques. McGraw-Hill. Glattfelder, A. H. (1970). Zur Adaptierung von Eingr6ssenregelkreisen mit harmonischen Prtifsignalen. Regelungstechnik, 11,485~493. Glattfelder, A. H. (1974). Regelungssysteme mit Begrenzungen. R. Oldenbourg, Mtinchen. Glattfelder, A. H. and L. Gross (1974). Regelschaltungen zur K ondensatr tickverteilung und zum Ausgleich der Massenverluste in einem W~irmeverbundnetz. BrennstoffWiirrne-Krafi, 26, 472~476. Glattfelder, A. H. and L. Gross (1975). WeitbereichLeistungsregelung eines Trommelkessels. Entwurf, Realisierung und Betriebserfahrungen. Brennstoff-W~irmeKraft, 27, 379 382.
and W. S C H A U F E L B E R G E R
Glattfelder, A. H. and L. Gross (1977). Weitbereichs-Regelung eines W/irmeverbundnetzes. BrennstoffWiirme-KrqlL 29, 27 33. Graupe, D. and E. Fogel (1976). A unified sequential identification structure based on convergence considerations. Automatica, 12, 53-59. Gujer, W. and P. Erni (1978). Nitrification in the activated sludge process. Proc. 9th International Cor~ferenee on Water Pollution Research, Stockholm. Huguenin, F. (1978). Entwicklung yon Regelalgorithmen ffir Mikrorechner. Elektroniker, 11/78, 11 20. Huguenin, F. (1979). Die Implementation von Regelalgorithmen mit Mikrorechnern: Probleme und L6sungsvorschlS_ge. Elektroniker, 1/79, 1 6. Huguenin, F. and W. Schaufelberger (1977). Model reference adaptive gain control using microcomputers. Proe. IFAC The Hague, 497 504. lonescu, T. and R. Monopoli (1977). Discrete model reference adaptive control with an augmented error signal. Automatica, 13, 507 517. Landau, J. D. (1969). A hyperstability criterion for model reference adaptive control systems. IEEE Trans. Aut, Control, AC-14, 552 555. Landau, J. D. and H. Unbehauen (1974). The devclopmcnt of adaptive systems in Germany and France. ASME Tran.s. Journal of Dynamic Systems, Measurement amt Control, 96, 405 413. Landau, J. D. and G. B6thoux (1975). Algorithms lbr discrete time model reference adaptive systems. Proc. IFAC Boston, 1975, ID, Paper 58.4. La Salle, J. P. and R. J. Rath (1963). Eventual stability. Proceedings of the Second Congress of IFAC, Basle. 1963, Part I, 550~ 560. La Salle, J. P. and S. Lefschetz (1967). Die Stabilitiitstheorie l~on Ljapunov. Bibliographisches lnstitut Mannheim. Maletinsky, V. (1975). On-line parameter estimation of continuous processes. Proc. IFAC Boston, 1975, liB, Paper 11.6. Maletinsky, V. and W. Schaufelberger (1974). Suboptimum adaptive control. Proc. lEA(" Ziirich, 1974, Part I, 129 143. Springer, Berlin. Mendel, J. (1973). Discrete Techniques ~?/ Parameter Estimation. M. Dekker, New York. Monopoli, R. V. (1974). Model reference adaptivc control with an augmented error signal. IEEE Trans. Aut. Control, AC-19, 474484. Popov, V. M. (1973). Hyperstability ~?I Control Systems. Springer. Pressmann, R. S. and J. E. Williams (1977). Numerical Control and Computer Aided Manufacturing. Wiley. Schaufelberger, W. (19771. Laboratory experiments in adaptive control. Proc. IFAC Barcelona, 1977, 1 13. S~,derstr6m, T., L. Ljung and I. Gustavsson (1978). A theoretical analysis of recursive identification methods. Automatica. 14, 231 244. Speth, W. (1972). Adaptivregelkreise in der Antriebstechnik. lnterkama 71, R. Oldenbourg, Mianchen, 240 249. Stute, G. and F. G. G6tz, (1973). Anwendung adaptiver Systeme bei spanenden Werkzeugmaschinen. VDI/VDEAussprachetag Freiburg, 1973, 263 278. Thaler, E. (1978). Einsatz von Mikrorechnern ftir die adaptive Regelung von industriellen Prozessen. Faehhericht AIE/ETH 78. Unbehauen, H., B. G6hring and B. Bauer (1974). Parametersehiitzvel?lhhren zur Systemidentil~kation. R. Oldenbourg, Mtinchen. Unbehauen, H., Ch. Schmid and F. B6ttiger, 11976). Comparison and applications of DDC algorithms for a heat exchanger. Automatica, 12, 393 402. Utkin, V. I. (1977). Variable structure systems with sliding modes. IEEE Trans. Aut. Control, AC-22, 212 222. Wehrli, Ch. and W. Schaufelberger (1970). Stabilit~it modelladaptiver Systeme nach der direkten Methode. ZAMP, 21, Fasc. 3, 593 598.
0005-1098/80/0101-0001 $02.00/0
Microcomputer Based Self-Tuning and Self-Selecting Controllers* A. H. G L A T T F E L D E R t, F. H U G U E N I N $
and W. S C H A U F E L B E R G E R $
Classical control system design techniques, upgraded to include cases with large parameter and state variations, and self-tuning and self-selecting controllers may be successfully implemented on microprocessors as indicated by a survey of industrial applications. Key Words--Adaptive control; computer control; control engineering; computer applications; direct digital control; industrial control; Lyapunov methods; microprocessors; PID control; self-adjusting systems; stability criteria. Abstract~lassical control system design techniques are extended to include cases with large parameter and state variations. New results on stability of adaptive gain control systems are obtained and a new method to design self-selecting controllers is presented. It is then shown that microcomputers are well suited for hybrid implementation of the resulting control systems. Several examples and a short survey on industrial applications are also given.
1. INTRODUCTION T HE PAPER demonstrates how self-selecting and selftuning control systems can be designed and implemeted with microcomputers. That the classical methods of continuous- and discrete-time control system design can be extended to include cases with large parameter and state variations is shown in the first part. The stability problem is investigated for the selftuning regulators. That the systems resulting from these design methods can easily be realized with the aid of microprocessors is outlined in the second part. Only the most critical part of the self-tuning control system, the adaptive controller, is realized digitally. The hybrid solutions that are obtained in this way allow control of fast processes. The microprocessor is used as a building block in this approach. This is in contrast to a completely digital design as reported in (Astr6m and co-
workers, 1977). The simple structures of the selfselecting controllers are implemented digitally. Hardware and software requirements are formulated. Three examples are presented in the third part. Industrial applications of the methods presented in this paper are surveyed. The methods that are discussed are straightforward and m o d u l a r extensions of classical control system design techniques. This fact results in considerable advantages during the design, implementation, tuning and operation phase of a system. These advantages are confirmed by several industrial applications and by the fact that students are able to implement adaptive controllers on laboratory scale plants within a few hours (Schaufelberger, 1977).
2. SELF-TUNING CONTROLLERS Different techniques for the design of simple self-tuning regulators are presented in this section. The adaptation problem for a system with u n k n o w n or slowly varying gain will be treated extensively. The stability of the resulting control systems will be investigated. The methods can easily be extended to the multiparameter case. 2.1 A
*Received May 25 1978; revised January 18 1979; revised May 29 1979. The original version of this paper was presented at the 5th IFAC/IFIP International Conference on Digital Computer Applications to Process Control which was held in The Hague, Netherlands during June 1977. The published Proceedings of this IFAC Meeting may be ordered from: North Holland Publishing Company, P.O.B. 103, Amsterdam--West Netherlands. This paper was recommended for publication in revised form by associate editor A. Longmuir. ~Escher Wyss Ltd, Research Department, CH-8023 Ziirich, Switzerland. SFachgruppe fiir Automatik der ETH, CH-8092 Ziirich, Switzerland.
stability property of linear, self-tuning regulators
If a linear control system can be designed in such a way, that a constant asymptotically stable system results for a long time period, then a certain kind of stability results. The system is in that case eventually asymptotically stable in the whole in the sense of La Salle (La Salle and Rath, 1963). This fact justifies to a certain extent the separation of the adaptive control problem into an identification and a controller design problem.
A. H. G L A T T F E L D E R , F. HUGUENIN and W. S C H A U F E L B E R G E R The above statement will now be formalized For = A(t)x + B(t)u(t)
(2.1)
with A(t) Ao
B(t) u(t)
bounded, continuous, lim,_~~ A(t) = Ao all eigenvalues in the left half plane bounded, continuous, limt .... B(t) = Bo bounded, continuous
The proof is analogous to the proof in the continuous-time case. 2.2 Gain adaptation by identification The system represented in Fig. l is investigated in this section. The basic control system, consisting of R and G, has to be designed asymptotically stable for p = l/k. Eventual stability results in that case for limt. ~ p(t)= 1/k. A more detailed definition of all parts in Fig. 1 is given in Huguenin and Schaufelberger (1977).
the following holds: The differential equation for e with e = x - z and = Aoz +
C
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
,,
B0u
6=Aoe+(A(t)-Ao)(e+z)+ (B(t)-Bo)u is eventually asymptotically stable in the whole in the sense of La Salle (.La Salle and Rath, 1963), i.e. e(t) is finite for all times and lim e(t)=O. The proof is a straightforward extension of the proof given in Huguenin and Schaufelberger (1977), chapter 3 in combination with theorem 7 from La Salle and Rath (1963). The same Lyapunov-function and the same technique of completing of square terms may be used. The proof is omitted. The corresponding theorem for the discrete time case is the following: For x(k+ 1)=A(k)x(k)+B(k)u(k)
(2.2)
P,
FIG. 1. Adaptive control for a plant with unknown gain. Process: kG(s):O< kmin< k~ kmax, G(s): aymptotically stablc. G(0)4-0 Input signal r 0 nonzero, constant. Controller RI.s): R (0) 4=0. Compensating gain p(t I.
The identification stage with inputs v and y and output q will be investigated first. The stability question for the total system will then be discussed. With the differential equations
y =e~ xk;
u(k)
P process
i~M = Apx M +bey bounded, limk~ ~ A ( k ) = A o all eigenvalues inside the unit circle bounded, limk_~o~ B(k ) = B o bounded
(2.3)
for the process and
with A(k) Ao B(k)
Ap asymptotically stable
= Aex + bpv
M
model
y~ = c x x M
(2.4)
for the model; letting e = x M - x , w = q - l / k one obtains the following equations of the perturbed motion for the identification stage:
the following holds: The difference equation for e = x - z and
= Ape ~, = hy( - yw + c T e).
(2.51
z(k + 1) : Aoz(k ) + Bou(k ) e(k + 1 ) = Aoe(k ) + (A ( k ) - Ao)(e(k) + z(k))
The Lyapunov-function
+ (B(k ) - Bo)u(k ) is eventually asymptotically stable in the whole, i.e. e(k) is finite for all times and lim e(k)=O.
1
2
V=h W +eTPe
(2.6)
A~P+pAp+cpeXe= - Q
(2.7)
with
Microcomputer based self-tuning and self-selecting controllers yields
< - y2w2 - eTQe.
(2.8)
is therefore negative semidefinite. The identification stage is uniformly stable in the whole. From
e~,e=e(t )=c~ exp (Aet )eo
(2.9)
and therefore
I~(t)l<~exp(-flt)
~,fl>o
(2.10)
tation of a gain is shown in Fig. 2. The controller acts on the model and the process is adjusted to the model. This design resulted from stability analysis. The following two conditions guarantee asymptotic stability in the whole for the error e = x M - x between the state vector x M of the model and x of the process: (i) The basic control system, consisting of R and G has to be asymptotically stable. (ii) The control system depicted in Fig. 3 has to be asymptotically stable for all ro and k. Design can be done by root locus technique.
equation (2.5) may be written as
= hy(-yw + e(t)).
(2.11 )
That the condition
z = ~ y2dt = ~
(2.12)
o
is sufficient for asymptotic stability can be proved with v = w2/h for equation (2.11). These results can be summarized as follows: The identification stage represented in Fig. 1 is uniformly stable in the whole. It is uniformly asymptotically stable in the whole when equation (2.12) holds. An investigation of the basic control system shows that equation (2.12) holds for all ro4:0. The total system differential equation augmented by ,~__y2 is used in the proof. It can be shown that no steady-state solution with ~ = 0 and z finite can exist. The limiter in Fig. 1 is important for this proof. With this, it is shown that the system in Fig. 1, if properly designed, reaches its desired steadystate operating conditions. Classical control system design techniques can be used to design the system. The restrictions put on G(s)-are very small, i.e. asymptotic stability and G(0) 4=0. The adaptive controller marked by the dashed line in Fig. i is replaced by a digital circuit with equivalent convergence properties in the implementation. See Astr6m and co-workers (1977); Mendel (1973); Graupe and Fogel (1976); S6derstr6m and co-workers (1978) for descriptions of the well-known gradient and least squares identification methods that can be used to identify q from y and YM recursively. Methods for the discrete design of the entire identification stage can be found in Astr6m and co-workers (1977) and S6derstr6m and co-workers (1978). 2.3 Model control Another simple control system for the adap-
?
FIG. 2. Model Control. Process kG(s); G(s) asymptotically stable, G(0)#0. Model G(s), Controller R(s). Input signal r 0 nonzero, constant.
FIG. 3. Control system used during the design phase. Input signal y o = l i m ~ y(t). Gain u 0 =limt~oo u(t). Process kG(s).
This stability property is a direct consequence of the following theorem: For the differential equation
R=F(x)
(2.13)
with xX= (yV, zX), and representable as
~=f(y)
(2.14)
=g(z) + G(x)y the following holds: The differential equation (2.13) is asymptotically stable in the whole, if it is stable in the sense of Lagrange (La Salle and Lefschetz, 1967) and if
A. H. GLATTFELDER, F. H U G U E N I N and W. SCHAUFELBERGER
~=fly)
(2.15)
and
=g(z) are both asymptotically stable in the whole. The proof may be found in Wehrli and Schaufelberger (1970). In the case at hand, Lagrange stability can easily be proved by using a theorem stated in (Bellman, 1953) for differential equations of the form i~--Ax + B(t)x.
The differential equations (2.15) are linear and present no problem. A simple numerical integration procedure can be used to replace the adaptive controller marked in Fig. 2. It is also easy to design a discrete controller for steady-state operation of the basic control system consisting of the controller R and the process G. This yields eventual stability of the adaptation loop according to Section 2.1. 2.4 Model reference control The adaptive system is presented in Fig. 4. The adaptive loop tunes the gain p of an otherwise classical control system by model-adjustment,
t- . . . . . . . . . . .
-I
L It .
.
.
.
.
.
U
FIG. 4. Model reference adaptive gain control. Process kG(s), Controller R(,s), Model G(s), Compensator C(s), Input signal ro nonzero, constant.
Global asymptotic stability for the adaptive loop can be proved for strictly positive transfer functions G(s) and for transfer functions that can be made strictly positive by a suitable compensator C(s) (Huguenin and Schaufelberger, 1977; Landau, 1969; Popov, 1973; Anderson, 1968). The theory of hyperstability or of passive systems is used. ¢I)(u)=u is normally chosen. This leads to a gain proportional to u 2 in the adaptive loop. A better choice of (I) will lead to a smaller dependence on u. Stability can be proved for slowly varying u(t) if the process is not hyperstable.
The augmented error signal method outlined in Monopoli (1974) may be used if the process is not hyperstable. The adaptive controller in Fig. 4 can be replaced by a simple numerical integration technique or the adaptive loop of the system can be designed by discrete techniques (Landau and B6thoux, 1975; Ionescu and Monopoli, 1977). 2.5 Sinusoidal perturbation ,system The corresponding system is represented in Fig. 5. A perturbation signal ~ is sent to the control system. The corresponding signal component is extracted at y. Filtering yields an estimate a proportional to p,k. a is adjusted to A 0•
FIG. 5. Adaptive gain control with sinusoidal perturbation system. Process kG(s), Controller R(s), Input signal ro: constant.
The describing function can be used for a stability discussion (Eveleigh, 1967). Examples may be found in Schaufelberger (1977) and Glattfelder (1970). 2.6 Multiparameter adjustment The methods presented in this section can easily be extended to the multiparameter case. Stability investigations are much more difficult in these cases. Applications of the methods discussed in this paper may be found in Schaufelberger (1977); Maletinsky and Schaufelberger (1974); Maletinsky ( 1975); Unbehauen and co-workers (1976); Landau and Unbehauen (1974); Glattfelder (1970). 3. SELF-SELECTING CONTROLLERS
Self-selecting control systems are treated in detail in Glattfelder (1974). Only the most important facts are summarized here. The design procedure is explained for the system in Fig. 6. The output signal y has to be controlled and the state variable I has to satisfy certain constraints. A self-selecting proportionalintegral (PI) controller will be designed. The following procedure is used:
Microcomputer based self-tuning and self-selecting controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,] I
FIG. 6. Self-selectingcontrolsystem.ProcessFG(s)withF(s)hyperstableand G(s) stable. (i) A linear PI-controller (ke, Tp, setpoint r) is designed for the stable plant FG(s). FG(s) must therefore be reasonably well controllable by this type of controller. (ii) The constraints are taken into account by designing a pair of controllers (kL, TL) for F(s) with k L ~ k P. This is possible, since F(s) is hyperstable. Setpoint values L T and L,L have to be selected according to the upper and lower limits of the state variable t. (iii) The self-selection is realized by, cf. Fig. 6 w = Min {Max (v, v,L),vT} This guarantees that only the appropriate control loop will be closed. (iv) The system is asymptotically stable if
L~
u1
FrG, 7. Overflowprotection for the integrator in Fig. 6 by a self-selectingcontroller. (y, I) is convenient to study the behaviour of the system. The design technique that has been briefly summarized in this section has been used in numerous examples and in several industrial applications.
4. IMPLEMENTATIONUSING A MICROCOMPUTER The methods that were presented in Sections 2 and 3 can easily be adapted for implementations on microcomputers. The analog solutions may be replaced by equivalent digital ones or by digital solutions with improved properties. Accuracy and drift problems will be reduced when digital solutions are used.
4.1 The microcomputer system A small hybrid computer system is used for development, testing and implementation of the algorithms described above. It consists of an EAI Mini Ac analog computer, an extension containing several adaptation units (multiplierintegrator-multiplier combinations) and an Intellec 80 Microcomputer development system. The development system has a complete process control interface with A/D and D/A converters, a real time clock and an interrupt system.
6
A . H . GLATTFELDER, F. HUGUENIN and W. SCHAUFELBERGER /SOFTWARE ~SUPPORT
INTEGER16bit IARITHMETIC PACKAGE
"~ /REAL TIME ) ~ MULTITASKING // ~ERATING SYSTEM/
1
l
FLOATING POINT24bit ARITHMETIC PACKAGE
sT2.0UT,_____.ES
: store - decout - de n
I- mln [ - max I- hys
1
float
DEBU )
RTC
driver user support
I/0
adcldac text
I/0
user support
fixx fixx2
IdlACROS write readpal convv
fabs
FIG. 8. Software support used to implement the controllers. 4.2 Algorithm development on the microcomputer A library of subroutines is used to program the different control algorithms efficiently. Integer and floating arithmetic operation packages and drivers for the different I/O devices and for the interface are available. A set of some 30 standard macros has been created. With these macros, the adaptive controllers can be coded from the block-diagrams Fig. 1 to Fig. 7 very easily. In this macro code a typical program as used in the examples would need less than 30 lines of code. The macros are expanded by Intels Marco Assembler. The software support system is represented in Fig. 8. If short sampling intervals are needed, interrupt driven programs are used. A real time operating system has also been developed. Up to 15 user tasks can run simultaneously and they can be controlled from the TTY. Special emphasis has been put on the fact that the operating system is well suited for control applications.
resistive load. Integer arithmetic proved to be sufficient for the adaptive controller in this case (Huguenin and Schaufelberger, 1977). A typical result is represented in Fig. 10. The sampling interval is 11 ms and the length of the program is 600 bytes for an I N T E L 8080 Microprocessor. I.O
p
~
P-f(n)
0.5 I000
L
2000 3000 n, r.p.m. FIC~. 10. Model reference adaptive generator control. Resulting gain P as a function of turbine speed n.
5. EXAMPLES 5.1 Self-tuning controllers Two very simple examples for the process represented in Fig. 9 will be gfven. Numerous examples of multiparameter cases may be found in the references of Section 2.6. (i) Modelreference control. The control system works according to Fig. 4 for a generator with FIG. 11. Generator control with gain adaptation by identification. Response to rectangular input r(t). Top: low speed.
bottom: high speed. (1) non adaptive, (2) adaptive.
FiG. 9. Process with variable gain. Turbine T, Generator G.
(ii) State variable filter identification. For the generator of Fig. 9 with complex load an adap-
7
Microcomputer based self-tuning and self-selecting controllers tive control system according to Fig. 1 was realized (Huguenin and Schaufelberger, 1977). Floating arithmetic was used t o implement a recursive filtered least squares algorithm for the estimation of q from y and YM. A typical result is represented in Fig. 11. The sampling interval is 23 ms and the length of the program is 1 K bytes. The programs used in these two examples are interrupt driven. Use of the real time operating system will be made in the next example.
Xz
~Ul
I150%
5.2 Self-selecting controller A process according to Fig. 12 with the three U2
~ "50%
! L1
""
D- . . . . . . .
ul
x2
)
~ ........
.....
U2
-I
FIG. 12. Process for level control. Input signals Ul, •2, 1/3. Output signals x~, x2.
input signals u 1, u 2 and u 3 and the two output signals xl and x 2 was built to test several selfselecting controller structures. Figure 13 shows a typical response to large set point steps on the level controller for x 2 with constraints 111" on ul and IZT on U2 as proposed in Fig. 7 (Thaler, 1978). The length of the program is 8 K bytes. This includes the real time operating system with TTY oriented monitor, the floating point package and the user program of about 1 K. Only a few of the possibilities of the operating system are used in this simple example (Huguenin, 1978). The sampling interval is 60 ms.
6. APPLICATIONS IN INDUSTRY 6.1 Self-tuning controllers Early attempts to use the methods investigated in this paper are summarized in Landau and Unbehauen (1974). These methods have especially been used to control electrical drives (Speth, 1972) and machine tool systems (Stute
FIG. 13. Result of level control experiment for rectangular reference value. Level x2 is controlled by inlet valve uI and outlet valve u2. and G6tz, 1973), which are both examples of systems with varying gain. Today the use for control of turbogenerators (D'Ans and co-workers, 1977) and for heating and ventilation control systems is investigated. 6.2 Self-selecting controllers A first application (Glattfelder and Gross, 1974) was to limit level excursions on a buffer reservoir by means of the outflow valve. The regulator was built from standard electromechnanical control equipment. It has reduced operator interference and fluid losses to virtually nil since its installation four years ago. In another application (Glattfelder and Gross, 1975) the combined pressure and power control of a steam generator was upgraded to cope with large load swings, avoiding replacement of plant subsystems. The controller was implemented with standard electronic control equipment and has been on line for three years. A third example (Glattfelder and Gross, 1977) is the power control of a district heating system with variable configuration and large flow transients absorbed by sequential use of the available thermal storage capacities. 6.3 Combined controllers The methods presented in this paper can be combined to form self-tuning and self-selecting controllers. These controllers may for example be used in machine tool systems (Stute and G6tz, 1973; Pressmann and Williams, 1977) or for control of pH values (Thaler, 1978) or purifi-
A. H . G L A T T F E L D E R , F . H U G U E N I N
cation of waste water (Thaler, 1978; Gujer and Erni, 1978).
7. CONCLUSIONS
Several methods to design self-tuning and selfselecting controllers have been presented and investigated in this paper. New results have been obtained in stability analysis of self-tuning controllers. All methods are straightforward extensions of classical control system design methods. Microcomputers are well suited for the implementation of the corresponding control systems. The resulting hybrid systems are transparent and modular with respect to hardware and software. This fact has led to a considerable interest from industry. The algorithms that are obtained from the theoretical development can easily be implemented on minimum microcomputer systems or on single board computers. Decentralized adaptive control can thus be realized. New hardware designs will soon speed up operation by a factor of 1 ~ 1 0 0 compared to the installation that was used in this paper leading to sampling intervals well below 1 ms (Huguenin, 1979). The main limitation of the methods presented in this paper is that the control problem has to be broken down into isolated subproblems and that these subproblems are solved on a priority basis. Despite this drawback these methods are very attractive for many small to medium size problems. REFERENCES Anderson, B. D. (1968). A simplified viewpoint of hyperstability. IEEE Trans. Aut. Control, AC-13, 292 294. Astr6m, K. J., U. Borrison, L. Ljung and B. Wittenmark (1977). Theory and applications of self-tuning regulators. Automatica, 13, 457M76. Bellman, R. (1953). Stability Theory (~f DifJerential Equations. McGraw-Hilt. D'Ans, G., H. Glavitsch and L. M. Panis (1977). An evaluation of present day excitation system control of turbogenerators. Proc. IFAC Melbourne, 1977, 199 203. Eveleigh, V. (1967). Adaptive Control and Optimization Techniques. McGraw-Hill. Glattfelder, A. H. (1970). Zur Adaptierung von Eingr6ssenregelkreisen mit harmonischen Prtifsignalen. Regelungstechnik, 11,485~493. Glattfelder, A. H. (1974). Regelungssysteme mit Begrenzungen. R. Oldenbourg, Mtinchen. Glattfelder, A. H. and L. Gross (1974). Regelschaltungen zur K ondensatr tickverteilung und zum Ausgleich der Massenverluste in einem W~irmeverbundnetz. BrennstoffWiirrne-Krafi, 26, 472~476. Glattfelder, A. H. and L. Gross (1975). WeitbereichLeistungsregelung eines Trommelkessels. Entwurf, Realisierung und Betriebserfahrungen. Brennstoff-W~irmeKraft, 27, 379 382.
and W. S C H A U F E L B E R G E R
Glattfelder, A. H. and L. Gross (1977). Weitbereichs-Regelung eines W/irmeverbundnetzes. BrennstoffWiirme-KrqlL 29, 27 33. Graupe, D. and E. Fogel (1976). A unified sequential identification structure based on convergence considerations. Automatica, 12, 53-59. Gujer, W. and P. Erni (1978). Nitrification in the activated sludge process. Proc. 9th International Cor~ferenee on Water Pollution Research, Stockholm. Huguenin, F. (1978). Entwicklung yon Regelalgorithmen ffir Mikrorechner. Elektroniker, 11/78, 11 20. Huguenin, F. (1979). Die Implementation von Regelalgorithmen mit Mikrorechnern: Probleme und L6sungsvorschlS_ge. Elektroniker, 1/79, 1 6. Huguenin, F. and W. Schaufelberger (1977). Model reference adaptive gain control using microcomputers. Proe. IFAC The Hague, 497 504. lonescu, T. and R. Monopoli (1977). Discrete model reference adaptive control with an augmented error signal. Automatica, 13, 507 517. Landau, J. D. (1969). A hyperstability criterion for model reference adaptive control systems. IEEE Trans. Aut, Control, AC-14, 552 555. Landau, J. D. and H. Unbehauen (1974). The devclopmcnt of adaptive systems in Germany and France. ASME Tran.s. Journal of Dynamic Systems, Measurement amt Control, 96, 405 413. Landau, J. D. and G. B6thoux (1975). Algorithms lbr discrete time model reference adaptive systems. Proc. IFAC Boston, 1975, ID, Paper 58.4. La Salle, J. P. and R. J. Rath (1963). Eventual stability. Proceedings of the Second Congress of IFAC, Basle. 1963, Part I, 550~ 560. La Salle, J. P. and S. Lefschetz (1967). Die Stabilitiitstheorie l~on Ljapunov. Bibliographisches lnstitut Mannheim. Maletinsky, V. (1975). On-line parameter estimation of continuous processes. Proc. IFAC Boston, 1975, liB, Paper 11.6. Maletinsky, V. and W. Schaufelberger (1974). Suboptimum adaptive control. Proc. lEA(" Ziirich, 1974, Part I, 129 143. Springer, Berlin. Mendel, J. (1973). Discrete Techniques ~?/ Parameter Estimation. M. Dekker, New York. Monopoli, R. V. (1974). Model reference adaptivc control with an augmented error signal. IEEE Trans. Aut. Control, AC-19, 474484. Popov, V. M. (1973). Hyperstability ~?I Control Systems. Springer. Pressmann, R. S. and J. E. Williams (1977). Numerical Control and Computer Aided Manufacturing. Wiley. Schaufelberger, W. (19771. Laboratory experiments in adaptive control. Proc. IFAC Barcelona, 1977, 1 13. S~,derstr6m, T., L. Ljung and I. Gustavsson (1978). A theoretical analysis of recursive identification methods. Automatica. 14, 231 244. Speth, W. (1972). Adaptivregelkreise in der Antriebstechnik. lnterkama 71, R. Oldenbourg, Mianchen, 240 249. Stute, G. and F. G. G6tz, (1973). Anwendung adaptiver Systeme bei spanenden Werkzeugmaschinen. VDI/VDEAussprachetag Freiburg, 1973, 263 278. Thaler, E. (1978). Einsatz von Mikrorechnern ftir die adaptive Regelung von industriellen Prozessen. Faehhericht AIE/ETH 78. Unbehauen, H., B. G6hring and B. Bauer (1974). Parametersehiitzvel?lhhren zur Systemidentil~kation. R. Oldenbourg, Mtinchen. Unbehauen, H., Ch. Schmid and F. B6ttiger, 11976). Comparison and applications of DDC algorithms for a heat exchanger. Automatica, 12, 393 402. Utkin, V. I. (1977). Variable structure systems with sliding modes. IEEE Trans. Aut. Control, AC-22, 212 222. Wehrli, Ch. and W. Schaufelberger (1970). Stabilit~it modelladaptiver Systeme nach der direkten Methode. ZAMP, 21, Fasc. 3, 593 598.