Digital Adaptive Control System Design For A Particular Class of Hydraulic Systems
Digital Adaptive Control System Design For A Particular Class of Hydraulic Systems
Digital Adaptive Control System Design For A Particular Class of Hydraulic Systems
Tamara Nestorović
Faculty of Mechanical Engineering, University of Niš,
Beogradska 14, 18000 Niš, Yugoslavia, e-mail: tamara@masfak.masfak.ni.ac.yu
Abstract. This paper is devoted to design of adaptive control laws for a particular
class of hydraulic systems. Considered hydraulic systems belong to a class of level-
systems, which were so far mostly considered with the aim of level control, i.e. keeping
the level of a liquid in a reservoir at desired constant value. This control aim could be
achieved by linear feedback gain control system. Mentioned hydraulic level-systems are
now considered from a different point of view, which demands nonlinear adaptive
control system introducing. Namely, if the control aim is to make the level of a liquid in
a reservoir change according to a specified function, then the parameters of the system
under control are variable due to variable liquid level, which demands adaptive
control. The case when the system parameters are unknown is also considered.
Regarding adaptive control design, different approaches were applied. The paper
considers gradient approach (MIT rule for parameter adaptation), reference model
adaptive systems based on hyperstability theory and self-tuning regulators. Design
procedures were considered in discrete form which enables their implementation as a
part of a digital system controled by computer.
For the purpose of adaptive control design, differential equation models of considered
systems were converted into appropriate discrete-time models and afterwards
represented in polinomial form, which is the basis for adaptive control design
procedure. As a result discrete control laws, which provide accomplishing of the
control aim, were obtained.
The verification of designed control laws was accomplished by the numerical
experiment method, i.e. by digital computer simulation, and obtained simulation results
were presented in the work. Simulations were performed for different types of reference
input and compared. On the basis of simulation results appropriate conclusions were
made. For the purpose of the simulation completion, original programs, which
represent software implementation of designed control laws, were written.
1. INTRODUCTION
Adaptive control represents a special type of control with nonlinear feedback in
which the states of the process could be divided into two categories: parameters which
change slowly and state variables which change faster. Many processes are characterized
by the fact that their parameters are variable or unknown, which demands adaptive
systems for their control. Adaptive control systems are nonlinear even if the system under
control represents a linear system or a system which could be described by a linearized
mathematical model. The structure of adaptive systems and the mechanisms used for the
adaptation of unknown or variable parameters impose the nonlinearity of the control
system.
The paper considers several types of adaptive control systems (gradient approach -
MIT rule for parameter adaptation, reference model adaptive systems based on
hyperstability theory and self-tuning regulators) and their application to certain hydraulic
systems. Known hydraulic level-systems are now considered from a different point of
view. Standard hydraulic level control systems provided keeping the level of a liquid in a
reservoir at desired constant value. That could be achieved by linear feedback gain
control system. If the aim of control is to make the level of a liquid in a reservoir change
according to a specified function, then the adaptive control system is needed, because the
parameters of the system under control are variable due to variable liquid level. The case
when the system parameters are unknown is also considered.
control law:
D1
u(kT) = −f(kT)y(kT) + g(kT)ω(kT) (5)
A
Rv
where the controller parameters f and g
h D2 are determined through the following
adaptation mechanism (MIT rule) obtained
Fig. 1. Hydraulic level-system described using gradient approach [1], [9], [10]:
by the single input single output
linear model of the first order
∂e(kT ) q −1
g (kT + T ) = g (kT ) − γe(kT ) = g (kT ) − γ ω(kT )e(kT ) (6)
∂g 1 + a m q −1
∂e(kT ) q −1
f (kT + T ) = f (kT ) − γe(kT ) = f (kT ) + γ y (kT )e(kT ) (7)
∂f 1 + am q −1
The control aim is that the plant output changes according to the output of the
reference model ym.
y! m = −2 ym + 2ω , (9)
Adjustable
Adaptation mechanism
controller
Reference model
q −1 g(k)
ω(k) −1 ym(k) π −γ Σ q
−1
π
bm Σ q 1 + am q −1
+ − + u(k)
Σ
am −
e(k) −1 f(k) −
γ q −1
Σ π Σ q π
1 + am q −1
Plant +
u(k) −1 y(k)
b Σ q
+ −
a
-1
0
-1.5
0 10 20 30 40 50 0 10 20 30 40 50
Time (min) Time (min)
Fig. 3. Simulations of the output of reference model adaptive system based on MIT rule
for hydraulic level-system of the first order (system output − dashed line; output
of the reference model − solid line). a) Reference input: rectangular wave with
height 2, width 10 and period 20 min. Adaptation gain: γ=0.005. b) Reference
input: sine wave, period 2π min, adaptation gain: γ=0.02.
Simulation results in Figure 3 show that quite good tracking of the reference model
output is accomplished after the first cycle, approximately. Simulations performed during
the investigation also showed that designed adaptive system could also be used with step
reference input, i.e. for regulation. Depending on the adaptation gain, different step
responses were obtained and they varied from extremely oscillating to very slow responses.
Adaptive system with adaptation gain γ = 0.001 resulted in acceptable step response.
polynomial form.
A(q−1)y(k+d) = B(q−1)u(k) or A(q−1)y(k) = q−dB(q−1)u(k), y(0) ≠ 0, (10)
− nB m
Bm (q −1 ) = b0m + b1m q −1 + ... + bnmB q . (15)
m
Γ (q −1 ) y m (k + d ) + R(q −1 ) y (k ) y m (k + 1) + γ1 y m (k ) − r0 y (k )
u (k ) = = . (17)
b0 b0
The plant parameter vector is estimated at each sampling interval and it is:
2.5 1.8 d)
c)
1.6
2 1.4 ∧
r0
1.2
1.5
1
0.8
1
0.6
0.5 0.4 ∧
b0
0.2
0 0
0 10 20 30 40 50 0 1 2 3 4 5 6
Time (min) Time (min)
Fig. 4. a) Reference input is unit step function with delay 1. b) Reference input is sine
wave with period 2π. c) Reference input is rectangular wave, period 20, height 2,
width 10. For a), b) and c): output of the reference model − solid line, system
output − dashed line. d) Estimated parameters.
Digital Adaptive Control System Design for a Particular Class of Hydraulic Systems 1429
According to conditions [9], [10] regarding the order of the polynomials in the control
1430 T. NESTOROVIĆ
law, which should be satisfied in order to determine unique and causal control law, after
solving Diophantine equation and determining the controller parameters, the following
control law was obtained in the case of known plant parameters:
15 + 5q −1 356.4208 − 267.411q −1
u (k ) = −1
ω(k ) − y (k ) (25)
1 + 0.9473q 1 + 0.9473q −1
In the case of unknown plant parameters, they are estimated at each sampling interval.
The plant model can be written in the form: (1 + a1q−1 + a2q−2) y(k) = (b0 + b1q−1)u(k−1).
It follows that:
θT = [a1 a2 b0 b1 ] , (27)
φT (k ) = [− y (k − 1) − y (k − 2) u (k − 1) u (k − 2)] . (28)
The controller is designed on the basis of estimated plant parameters at each sampling
interval by solving Diophantine equation and using the procedure described in [9]. The
control law is calculated and applied at each sampling interval in the form:
T (q −1 ) Sˆ (q −1 )
u (k ) = ω(k ) − y (k ) . (30)
Rˆ (q −1 ) Rˆ (q −1 )
Presented design algorithm was implemented through the software program designed
in [9]. The program was used to obtain simulation results for the system output, input and
estimated parameters. Simulation results for rectangular reference input are shown in
Figure 6.
Digital Adaptive Control System Design for a Particular Class of Hydraulic Systems 1431
1.2 1
a) a^ 2
1 0
b)
0.8 -1
a^ 1
0.6 -2
0 5 10 15 20 25 30 35 40
0.4 x 10-3
2.4
^
0.2 b0 c)
2.3 ^
0 b1
-0.2 2.2
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time (min) Time (min)
Fig. 6. Simulation results for rectangular wave reference input, period 2π, height 2,
width 10. a) Output of the reference model − dashed line, system output − solid
line. b) and c) Estimated plant parameters.
REFERENCES
1. Åström K., Wittenmark B., Adaptive Control, Addison-Wesley Publishing Company, New York, 1989.
2. Åström K., Wittenmark B., Computer Controlled Systems, Prentice-Hall, Inc., Englewood Cliffs, 1984.
3. Åström K., Wittenmark B., On Self-tuning Regulators, Automatica, Vol. 9, pp. 185-199, 1973.
4. Debeljković D., Dinamika objekata i procesa − matematički modeli objekata i procesa u sistemima
automatskog upravljanja, Mašinski fakultet, Beograd, 1989.
5. Friedland B., Advanced Control System Design, Prentice Hall, New Jersey, 1996.
6. Kučera V., Diophantine Equations in Control − A survey, Automatica, Vol. 29, No. 6, pp. 1361-1375,
1993.
7. Merrit H. E., Hydraulic Control Systems, John Wiley&Sons, New York, 1967.
8. Nestorović T., Nikolić V., Digital Tracking System Design for the Hydraulic Positioning System of the
Fourth Order, Theoretical and Applied Mechanics, Jugoslovensko društvo za mehaniku, (to be issued).
9. Nestorović T., Application of Digital Control Laws to some groups of Hydraulic Systems and
Possibilities for their Microprocessor Implementation, master’s thesis, Faculty of Mechanical
Engineering, University of Niš, April 2000.
10. Paraskevopulos P. N., Digital Control Systems, Prentice Hall, New Jersey, 1996.
11. Ротач В. Я. Теория автоматического управления тепло-энергетическими процессами,
Энергоатомиздат, Москва, 1985.
12. Vaccaro R. J., Digital Control − A State-Space Approach, McGraw-Hill Inc., Singapore, 1995.
13. Walters R. B., Hydraulic and Electro-Hydraulic Control Systems, Elsevier Applied Science, Elsevier
Science Publishers Ltd., London and New York, 1991.
sa drugačije tačke gledišta, što zahteva uvođenje nelinearnog adaptivnog upravljačkog sistema.
Naime, ukoliko je cilj upravljanja da se nivo tečnosti u rezervoaru menja po unapred zadatoj
funkciji, onda su parametri objekta upravljanja promenljivi zbog promenljivog nivoa tečnosti u
rezevoaru, zbog čega je potrebno uvesti adaptivno upravljanje. Ovde se razmatra i slučaj kada su
parametri objekta upravljanja nepoznati.
Pri projektovanju adaptivnog upravljanja korišćeni su različiti pristupi. U ovom radu raz-
motren je gradijentni pristup (MIT pravilo), adaptivni sistemi sa referentnim modelom zasnovani
na teoriji hiperstabilnosti i samopodešavajući regulatori. Postupci projektovanja razmatrani su u
diskretnom obliku, što omogućava njihovu primenu u sastavu digitalnog sistema kojim upravlja
računar.
U cilju projektovanja adaptivnog upravljanja, modeli razmatranih sistema predstavljeni dife-
rencijalnim jednačinama, prevedeni su u odgovarajuće diskretne modele, a zatim su predstavljeni u
polinomnom obliku koji predstavlja osnovu za projektovanje adaptivnog upravljanja. Kao rezultat
projektovanja dobijeni su diskretni upravljački zakoni kojima se ostvaruje cilj upravljanja.
Verifikacija projektovanih upravljačkih zakona ostvarena je metodom numeričkog eksperi-
menta, odnosno simulacijom na digitalnom računaru, a dobijeni rezultati prikazani su u radu.
Simulacije su izvršene za različite tipove refrentnih signala i izvršeno je njihovo upoređivanje. Na
osnovu rezultata simulacije izvedeni su odgovarajući zaključci. U cilju izvršenja simulacija napisa-
ni su i određeni originalni programi koji predstavljaju softversku implementaciju projektovanih
upravljačkih zakona.