Copyright © IFAC Nonlinear Control Systems, Stuttgart, Germany, 2004
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IFAC PUBLICATIONS www.elsevier.com/locatelifac
GAIN SCHEDULED CONSTRAINED CONTROLLER FOR SISO PLANTS Mikuhis Huba Slovak University of Technology in Bratislava, Fac. Electr. Eng. & Info. Technology Ilkovicova 3, 812 19 Bratislava,
[email protected]
Abstract: The paper considers design and tuning of simple constrained controllers for plants with dominant 2nd order dynamics. They represent hybrid solutions with dynamics ranging from the relay minimum time systems to linear pole assignment ones. There understanding requires just basic knowledge of the 2 nd order trajectories patterns. In contrast to the other known solutions the controllers derived are appropriate also for extremely fast application and easy to tune by a procedure that generalizes the wellknown method by Ziegler and Nichols. Copyright © 2004 IFAC Keywords: Constraints, continuous control, minimum-time control, pole assignment
1. INTRODUCTION
integration. The topic is still under development: there does not exist a generally accepted definition of this phenomenon, which results in many different solutions to the problem (Kothare et al., 1994). Originally they have been appropriate just for controlling stable systems. In the case, when the control signal hits both the upper and the lower control constraints, they led to instability (Ronnback, 1996). Last paper by Hippe (2003) solves also this problem by introducing additional circuit for the reference signal shaping, which should avoid control saturation. His paper is also interesting from the terminology point of view, when he speaks not just about the integrator windup, but also about the plant windup. (This classification is, however, not generally accepted.)
In many applications, the controller design is dominated by the existing constraints imposed on the control signal amplitude, its rate, or on the state variables. So, it is not to wonder that after some decades of the dominancy of the linear theory the research community turned back to this topic. Despite this, the present textbooks lack on design procedures, which would reflect the importance of the control constraints and be simultaneously simple enough to be accepted by broad practice. The impact of tills situation is evident e.g. by inflation of different "optimal" PID tunings appearing at practically each control conference, or in conclusions noted by Astrom and Hagglund (1995): " ...derivative action is frequently switched off for the simple reason that it is difficult to tune properly..." It is easy to show that using the linear controller design, the PD controller cannot be properly tuned in real (constrained) situations!
The model predictive control has originally been proposed to solve complex industrial control problems, whereby it has many time demonstrated its abilities. It is, however, superfluous to use this complex approach in looking for simple solutions to simple problems! Furthermore, long pre-computation times up to hundreds of seconds mentioned by Bemporad et all (2002) exclude its application in controlling fast processes with update requirements (acting disturbances, operating point changes in controlling nonlinear systems). Looking for other alternatives, the minimum time control should be mentioned. For more than two
Bemporad et all (2002) mention that the most popular approaches for designing non-linear controllers for linear system with constraints fall into two categories: anti-windup and model predictive control. The anti-windup solutions originate from the aim to suppress an overfluous integration of an integrating controller. They considered linear controller design and an additional circuit to limit
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decades it dominated design of constrained systems. Its intuitive interpretation (the system must be fully accelerated up to the switching curve and than fully braked by the opposite limit control value) was easy to understand and so it was well accepted by practice. Still it can be found in practically each standard textbook about nonlinear control. So, in order to advance the design of simple constrained controllers, one should ask: "What was wrong with this control" and "which its features should be kept also by more sophisticated controllers"? The answer would surely involve high sensitivity to different parasitic phenomena resulting e.g. into relay chattering, or overshoot occurrence during the transient responses and oscillations in the neighborhood of the demanded states. The other nd point is that in controlling 2 order systems one has expect 2 phases of control: acceleration and braking.
2. PROBLEM SPECIFICATION Let us consider a closed loop system
:t[;J=[-~o _~J;J=ARX
(1)
with origin as the reference state. It is composed of a plant = Ax+ bu (2) and of the linear controller
x
u =r t x ; r t
= [ro
r1 ]
(3)
Its main task is usually to guarantee stable equilibrium at the origin, what corresponds to the closed loop poles with negative real parts. These can, however, specify the close loop dynamics also more precisely.
2.1 ReaL cLosed Loop poLes
As the 3rd basic solution, different realizations of hybrid control schemes could be mentioned, with logic-based supervisors switching between two, or several different controllers. Solutions combining the first known constrained controllers - the relay minimum time ones - with "softer" type of control can be traced out up to the 50-ties (see e.g. Smith, 1958). Continuous scheduling of the controller parameters can yet extend the switching procedure (see e.g. IGendl and Schneider, 1972). Actually, also the piecewise linear (affine) controllers resulting from the constrained model predictive control can be included into this group (Bemporad et all, 2002, EIFarra et al, 2003, etc.).
For a~ > 4ao there exist two real poles of the characteristic polynomial A(s) = s2 + ats + ao
a l2 = ,
-at
± ~a~ -4ao 2ao
(4)
The eigenvector
v = -[all - A]-t b
(5)
traces out a line L: at x = 0; a 1- v . The linear pole assignment controller (Huba, 1998; Huba and Bistak, 1998; Huba et all, 1998) decreases oriented distance p (with sign depending on the position with respect to this line) of the representative point x(t) from L according to
This list of possible solutions is, of course, far from to be complete. Here, new simple solutions will be introduced combining the relay on-off control (characteristic by application of limit control values) and the linear pole assignment control with the dynamics patterns characterized by the chosen poles. The overall control can be characterized as the minimum time one with additional constraints on the rate of state (control signal) changes specified by the closed loop poles. It could also be specified as the control, which decreases distance of the representative point to the next lower invariant set with rate proportional to this distance. The controllers presented guarantee continuity of control over the whole range of operation. Furthennore, they represent a natural construction: besides of the closed loop poles and control constraints they do not require any additional design parameters. Their implementation by means of standard computer technique enables sampling period in range of ms and an easy continuous update.
~=a2P;p=atx
(6)
For points of the line L traced out by (5) it decreases the distance from the origin according to
d:ro =alPo ; Po =Ixlsignx
(7)
All transient can be split into two 1st order ones. In parallel to mechanical systems, the transient towards the line L can be denoted as acceleration, the transient along the line towards the origin as braking. Because the couple of poles can be numbered in two ways, two ordered combination of the closed loop poles [at, a 2 ] yield two different lines L. The linear transients are, however, always dominated by the line with the lower slope associated with the "slower" pole in (7). The controller vector r t does not depend on the order of the poles.
In order to simplify the introductory framework and the design treatment, the procedure is explained by the example of a double integrator control. Although it can be directly expanded to controlling other 2nd order systems and generalized also to higher order ones, within the chapter devoted to the controller tuning a simplified step-response based extensions to higher order SISO systems with maximally two unstable poles is shown.
(8)
(9)
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2.2 Complex poles of the closed loop dynamics
The controller is given as
rt =
For a~ < 4ao ' the closed loop poles are complex
at.
H
=-2±j ao
SI.2
at -4 =
(10)
Let us introduce
=
t.r;;; =two; Wo =.r;;; > 0 ,. (= 2F; = ~ 2wo _a_I-
dxo dr Xo
=
[0 1] -1
- 2(
rIx 1= [x]; y dr
2(Wo ]
(22)
The main difference to the real poles case is given by the fact that the overall transient corresponds just to a single phase of a systematic decrease of the state vector module. The corresponding trajectory has shape of a logarithmic spiral and it does not depend on initial states. In designing controller for the real pole pair the requirement on the distance decrease (67) is applied twice, while in the complex pole case just once to the associated quadratic fonn (21). It is well known that the quadratic criteria lead to underdamped systems.
= ~ -(± j~J =wo[-(± jW] r
-[w5
(11 )
x . 0'
(12)
y
2.3 Constraints in control
= dx = _1 dx d-r
Wo dt
nd
Let us consider a 2 order system (2) with origin as the reference state and the control signal UE (U t ,V 2 ); VI <0
Solution of this equation is
w 2x 2 + [y + ~J2 = C 2e
f
2 arctg y+?x (J}X
(J)
being constrained with U I and U 2 as the limit
(13)
y= dx =_I_dx; W=~I-(2 d-r Wo dt In polar coordinates one gets xt = Cl.«; x2 = y + ~
values, which can be simply expressed as U
x)
(24)
The aim is to bring the system from an initial state [xo,O] to the origin in the minimal time by respecting
- = R sin B ( 14) R 2 = X- t 2 + X- 2 2 ; X- t = R cos B ; x2 whereby {e 1'.,,R = Ce(J) B = arct ~2 = arctg~ (15)
the dynamics of the state and control signal changes specified by the closed loop poles at , a 2 (or
Wo [- (
± jW]) with negative real parts.
(tJX
XI
Putting such limiter (24) into the 20d order loop, its behavior can become useless or unstable. The dynamics specified by the closed loop poles can be guaranteed just over the invariant set of linear control. Two lines B j; j = 1,2 parallel to vector
or x = Ae-(t" cos(mr + a)
y = -(Ae-(t" cos(mr + a)- m4e-(t" sin(mr + a) The values C, A and a
= sat{r t
(16)
are given by the initial
conditions. After substituting (16) into (14) B = -a - OJ! (17) Working just with the modified system (12), the poles are PI.2 =-( ± jW (18)
z .1 r satisfying to r t z = 0 limit the strip-like zone of proportional control Pb . Just the line segment of
Obviously, the solution (15) satisfies equations
can be considered as the reference braking trajectory, and as the target for the 1sl phase of control.
dR (fo fo -=C-e(J) (-w)=-(Ce(J) =-(R d-r w
the lines traced out by v with vertices X
j ;
j
= 1,2
(25)
In defining shape of invariant set of linear control, it is important to identify vertices of Bj , in which the trajectories of the closed loop system have tangent parallel to z. These points can be defined by
(19)
dB
-=-w d-r It means that the module of the state vector decreases with the velocity proportional to its actual value by the coefficient ( and, simultaneously, it rotates with
r' Pd
= U j ; r' :: = r' (A + br' )Pj = 0
(26)
The invariant set of linear control is then limited by B j; j = 1,2 and by the trajectories of the closed loop
the angular velocity ill. In these new variables, the module and argument can be expressed as x2 R = ~-tx x; B = arctg-=-
d= v V
(1) crossing the vertices Pcl
(20)
.
Xl
The first of Eqs.( 19) can also be expressed as
For the double integrator and real closed loop poles
d(i'iL r ~-,xx
dR _ I ----d-r 2~iti d-r
Bj
--~
d-r
J
V J._ ; j=1,2 1 i+_ x- -1 + ( at a aa 2
or
d(i'i) = -2s(i'i)
:
(21)
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t 2
(27)
.
pJ
o
=
r
j
2 a~ +a;2 +a\a 2
a\ a2
U . ; j
a l +a 2
J
= 1,2
according to (6). For this phase, one can again allow saturation according to the slogan by Schneider. In deriving appropriate algorithm one can exploit the non-unique distance definition of a point to curve, e.g. by looking for simple solutions. One solution, derived for the double integrator and based on measuring the distance in direction of the x -axis as p = x - x b ' when dp ap dx ap dX . i - = - - + - - = x - - u = a2 p (33) dt ax dt ax dt Uj
(28)
a 1a 2
For the choice of complex poles
2( U·J B·: x + - i + =0 J OJ ~H2 o UJ'O 4
Pt j = o
r
-11
(2 w~ _ 2(
(29)
U ' . J' = 1 2 J '
leads to the controller equation
(30)
,
I
Wo
X-Xb]
u= [ l-a 2 - , - V j = l-a 2
x-2"
x
3. DYNAMICS DESOMPOSITION u = r t X;
It is well known (see e.g. Hippe, 2003) that in controlling I sI order systems the control saturation does not cause any problem. It just slows down the transient responses, so that one can fully accept slogan: "What you are not able to do today, let it for tomorrow" (Schneider, 1971). This slogan is, however, not acceptable for the braking phase of the 2nd order processes, since it leads to overshoot or instability. One way to deal with this problem is the invariant set based approach (Huba et aI., 1998; Huba, 1999; Huba & Bistak, 1999; Huba et aI., 1999)
;
(j2
Vj
u-;+ a\2 .
x
J .
(V 2 VI J
vj;xe: - , at a l
XE (U 2 , U1 J
(34)
a\ a\
whereby j = (3 + sign(x))/ 2 (35) It is interesting to note that the resulting control quality depends on the order of the closed loop poles!
4. NONLINEAR REFERENCE SIGNAL SHAPING In the case of complex poles the 1sI order invariant sets (lines) do not exist and the dynamics cannot be split into two 1sI order ones. RBC will now be constructed from the limit linear trajectory that
The movement along L is formally described by the 1sI order Eq. (7). The movement toward L is described by another 1sI order dynamics (6). Under constrained control, this, however, holds just for a line segment of L in Pb constrained with the
d (31). The
touches boundaries of Ph in the point P
final linear segment will be combined with a segment of limit braking constructed according to
5
vertices X ~ X (25). The rest of L does not belong to an invariant set of constrained control and so it cannot be considered as the target for the 1SI phase of control (acceleration). Looking for analogies with the 1SI order saturating control, it will be further assumed that during a part of the braking phase the control signal was set to one of the limit values U j . Starting
xt (1') = e- pd +U
-T
f e - A&bdl9; j = 1,2 (36) o Pole assignment dynamics of the acceleration phase can be constructed in two ways. Both are characteristic with a shift of the reference point AT
j
oy
(center of the actual spiral X 00 = [xoo trajectory in the acceleration phase) that is equivalent to nonlinear reference shaping.
from X ~ with u = U j with the time increasing in the inverse direction, one gets corresponding points
xt (1')= e-
The 1sI solution (Huba, 2004a,b) is based on a shift of the proportional band in such a way that the line traced out through the representative point x in a direction defined by a chosen vector w crosses the boundary Bj of the controller with shifted origin at
-T AT
X ~ +U j fe-A&bdl9; j
= 1,2
(31)
o Together with the line segment X
dO,
these points
I
belong to an invariant set with control sequence equivalent to the 1SI order constrained pole assignment control. These points will be denoted as the reference braking curve (RBC). After eliminating T , RBC of the double integrator becomes
XE(~:. ~:)
i
X2-~6) 2U j
+x j 0
a point of RBC (Fig.!) Uj
(37)
In this way, it is possible to derive controller characterized by r-------
(32)
.
X2+(~J x~(!!l,!!.L) w j
=rt(x b - X~)
Xb
Uj
{I-
2(2 +
=-
x~J 1
(38) Wo
X60 = {XllJ5+u j - 20 j {1- 2(2 + X~5 ]}/llJ5 (39)
al al
The aim of the acceleration phase is to decrease the distance p of the representative point to RBC
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.
[{
u=-2(OJox+U J 1-2(
")
x(JJ1j
1-2(-+U;
'JJ
(
40
due to Ziegler & Nichols (1942), or by the relay experiments (Astrom & Hagglund, 1995).
)
The controller tuning requires determination of the plant gain Ks and of an appropriate value of the closed loop values. Both can be based on the approximation of the measured process reaction curve by the hTrrmodel
According to the chosen vector w, this controller will be used just for xE
J
4(2 - 1 (4(2 -1 ( -oo'--;;r-U 1 u --;;r-U 2 ,oo
J (41)
s( S ) -- -K. .s, e -Tds
Close to the origin, linear controller (22) is used. B·' 1
(43)
SXb
Using the requirement of a triple dominant pole, it is possible to derive controller coefficients 0.079 0.461 (44) TO =----.., ; 'i =- - -
KsTJ -
KsTJ
The same controller can be get by closed loop poles
W
x
x
al,2
=-(0.23±jO.16)ITd
(45)
or by the characteristic polynomial coefficients ( = 0.82 ~ Wo = 0.28 I TJ (46) Since the up to now available constrained controllers allowed just choice of real poles, (Huba et aI., 1998; Huba Bistak, 1999; Huba et aI., 1999), they used approximation of (45) by the real part (47) a',2 = -0.2311 Td
Fig.I. Scheduling controller parameters by shifting the boundary of the proportional zone of control The 2nd solution could be constructed in such a way that a spiral trajectory crossing the initial point x with a center of rotation shifted to X 00
6. EXPERIMENTAL VERlFICATION
= [xoo 0]/
would touch RBC in a point x b ' when
u=rt(xb-Xdo)=u j
The achieved results have been tested by driving cart of the inverted pendulum system, but with pendulum removed (Fig.2).
(42)
Strictly speaking, the 1si modification does not fully preserve the required pole assignment dynamics in the transition from full acceleration to braking (due to the interaction with the dynamics of the point
X 60)' The advantage is its simplicity. For the 2
nd
problem, an analytical solution was not found up to now. This is, however, not substantial, since in dealing with time-delayed systems even such solution fully preserving dynamics of the pole assignment control would need additional compensation of increasing velocity in crossing the proportional band for diameters many time exceeding that one corresponding to
Fig.2. Plant controlled The controller parameters have been found by approximating the initial phase of the measured step response (Fig.3) by the transfer function (43) with K s = 5 and Td = 0.07 S and by determining the
pj .
real control signal, all previous computations have to be evaluated for U j = KsU rj ; j = 1,2 , then
closed loop poles according to (46) and according to (47). The corresponding closed loop responses for step-wise constant setpoint w, output y and x=y-w are in Fig.4. It is to see that due to higher gains correspondi ng to complex poles also the steady state error caused by friction decreases.
completed by inverse formula u r = u I K s and finally
7. CONCLUSIONS
Since the controller equations have been derived for Ks = I, what corresponds to u = K.\.u r , u r being
limited to gi ven constraints. It was shown that the linear pole assignment control and the minimum time control can be naturally combined within a new concept of the invariant-sets based control. The previous works dealing with the design specified by real closed loop poles have now been extended for the case of complex ones. Implementation of derived controllers by means of standard computer technique enables to work with sampling period in range of fractions of ms and an continuous update to compensate acting disturbances and parameter changes due to system nonlinearity.
5. CONTROLLER TUNING Huba et al. (1998) show, how it is possible to apply such algorithms also for controlling time delayed and higher order systems and how it is possible to tune such controllers by step response based experiments. Huba (2003) has generalized this approach also for controller tuning by the ultimate sensitivity approach
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2.5
10
X
3
by the Grant KEGA 3/0057/02. Part of the results was achieved during the author's stay with the Department of Control Engineering of the cru Prague (Pror. Sebek) under support of the CEEPUS CZ103 project (Coordin. Prof. Bilek).
x{t)
15
REFERENCES 05
- - - - - -...
Bemporad, A., Morari, M., Dua, V. and Pistikopoulos, E.N. (2002). The explicit linear quadratic regulator for constrained systems. Automatica, 38, 3-20. EI-Farra, N.H., Mhaskar, P. and Christofides, P.D. (2003). Uniting bounded control and MPC for stabilisation of constrained linear systems. Automatica, 40, 101-1 10. Hippe, P. (2003). Windup prevention for unstable systems. Automatica 39, 11, 1849-2019. Huba,M., Bistak,P., Skachova,Z., Zakova,K. (1998). Predictive Antiwindup PI-and PlO-Controllers based on 11 and 12 Models With Dead Time. 6 th IEEE Mediterranean Conference on Control and Systems, Alghero. Huba, M. (1999). Dynamical Classes in the Minimum Time Pole Assignment Control. In: Computing Anticipatory Systems - CASYS '98. Woodbury: American Institute of Physics. Huba, M. - Bistak, P.: Dynamic Classes in the PID Control. In: Proceedings of the 1999 American Control Conference. San Diego: AACC, 1999. Huba, M., Sovisova, D. and I. Oravec (1999). Invariant Sets Based Concept of the Pole Assignment Control. In: European Control Conference ECC'99. VDINDE DUsseldorf. Huba, M.( 2003). Constrained systems design. Vol.] Basic controllers. Vol.2 Basic structures. STU Bratislava. Huba, M. (2004a). Design of Gain Scheduled Pole Assignment for SISO Systems. EMCSR'04 Vienna, 743-748. Huba, M. (2004b). Gain Scheduled Constrained Controller For Siso Plants. Int. Conf. Rediscover'2004, Dubrovnik - Cavtat, Croatia. Kiendl,H. und G.Schneider (1972). Synthese nichtlinearer Regler flir die Regelstrecke constJs2 aufgrund ineinandergeschachtelter abgeschlossener Gebiete beschrankter StellgroBe. at 20, 289-296. Kothare, M. V., Campo, P. J., Morari, M., & Nett, C. N. (1994). A unified framework for the study of anti-windup designs. Automatica , 30, 18691883. Schneider,G. (1971). Eine suboptimale Methode zur Synthese van Abtastsystemen mit beschrankter StellgroBe. Regelungstechnik19, No.8, 322-338. Smith,O.1 .M. (1958). Feedback control svstems. McGraw-Hill, New York. ~ Slotine, J.1. and W.Li (1991). Applied Nonlinear Control. Prentice Hall, Englewood Cliffs, N.1. Ziegler,J.G. and Nichols,N.B. (1942). Optimum settings for automatic controllers. Trans. ASME, 759-768.
t
Fig.3. Approximation of the initial phase of the measured step response by 12 Td -model Posltion
0.8
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~
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Y 0.6 [mJ04~ 0.2'
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i
o
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6
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r
o
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oi •
-
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3 qs]
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u
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o~-r-----'r-----r------'I~rt==t==t=::::3
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,5,:--1
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_ _. L . - _ - L - - - - l J . . - - l . - _ - L _ - - - - l _ - - - - l 4
qs)
FigA. Simple position control - experimental results Although the approach can be interpreted as a variable structure, a gain scheduling and a reference signal shaping method, it does not introduce any new design parameter, so that also the tuning procedure remains simple. It is even simpler to tune than in the linear case, since the closed loop poles can be specified independently from the given control constraints! By the example of a mechanical positioning system it was shown, how the new control can minimize friction influence by applying high controller gains without facing problems due to the control saturation in processing large transients. Since the design uses just the well-known behavior patterns of the linear pole assignment control and of the on-off control, which are well established in the engineering community, it is to assume that the new controllers will be also well accepted in practice. In the presented form, in controlling time delayed system the controller derived can lead to overshoot also in the case of the poles (46) due to increasing circumferential speed and constant width of the proportional band. This problem can easily be compensated by an additional gain scheduling (Huba, 2004b).
8. ACKNOWLEDGMENTS This work has been partially supported by the Slovak Scientific Grant Agency, Grant No. 1/7621/20 and
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