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Relay control of systems with parameter uncertainties and disturbances
Automatica, Vol. 5, pp. 755-762. Pergamon Press, 1969. Printed in Great Britain.
Relay Control of Systems with Parameter Uncertainties and Disturbances* Commande ~t relais de syst6mes avec incertitude des param&res et perturbations Relais-Regelungssysteme mit Parameterunsicherheiten und -St6rungen Pe3ie~noe ynpaB.qen/4e C/4CTeMaMrI C neonpe~e~enHocTbrO napaMeTpoB /4 C 1-IOMeXaM/4 D. P. L I N D O R F F t
The design of model tracking systems using relay control is presented. The freedom allowed in choosing the measured state variables results in favorable perfbrmance characteristics both with respect to instrumentation noise and response to disturbances. Summary--This paper presents an approach, using a semidefinite Liapunov function, for designing a model-tracking system with a relay controller. The effect of parameter uncertainties and disturbance inputs is considered. It is shown that, by this method, the state variables need not be in phase variable form, as has been formerly required. This results in a reduction of instrumentation noise entering the system, and a more reasonable solution to the problem of disturbance rejection. Application of the method is made to the control of an unstable system. Specifically, experimental data on control of a cart-supported inverted pendulum is presented.
such as permitting a bound on the system response due to imperfect switching to be determined [1]. In addition, with the phase variable form of the state equations the design approach does not ordinarily place restrictions on the rate of change of parameter values. Nevertheless, two criticisms of the method are that the instrumented phase variables in higher-order systems are generally contaminated with noise, and the disturbance response is not accommodated in the design procedure in a realistic way. By the use of a semidefinite Liapunov function, the rigid requirement on the form of the state variables can be relaxed. The purpose of this paper is to develop this concept, with application to a model-tracking system using a relay controller. Although the theory developed in this paper is applicable to a wider class of problems, the specifc plant to be controlled is pictured in Fig. 1, and consists of an inverted pendulum mounted on a cart whose position is controlled by a D.C. motor. Although design techniques have been proposed for relay control systems, it is believed that this paper extends previous work by generalizing the application of model-tracking systems in the presence of parameter variations and disturbances. To guarantee stability in response to initial conditions the SD design approach necessarily requires some information about the system trajectories, as contrasted with the PD design. One approach to solving this problem, by drawing upon the notion of an invariant set, has been proposed [5]. However, the convergence problem does not lie within the scope of this paper.
1. INTRODUCTION IN THIS paper the use of a semidefinite (SD) quadratic form .is examined further in relation to the relay control problem where the plant is subject to parameter uncertainty [1, 2]. Specifically, consideration is given to the model tracking problem with disturbance inputs, for the case in which the state variables are not defined as phase variables. In previous works, a positive definite (PD) Liapunov function has been used in the design of systems subject to parameter variations [3, 4]. In these works it was found necessary to assume the phase-variable form in writing the state equations for the plant, this being an unavoidable restriction if the effects of parameter uncertainties are to be considered with the PD design approach. However, the use of the PD form offers certain advantages, * Received 27 February 1969; revised 19 April 1969. The original version of this paper was presented at the IFAC Congress which was held in Warsaw, Poland during June 1969. It was recommended for publication in revised form by associate editor P. Parks. t Professor of Electrical Engineering at the University of Connecticut, Storrs, Connecticut, USA. 755
756
D . P . LINDORI t: 2. THE DESIGN TECHNIQUE
The system is diagrammatically represented in Fig. 2. The plant is defined according to the linear
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Model of I ~ I
Compensation
nominal plont Model
FIG. 2. Block diagram of compensated model and plant with relay control.
It is assumed that elements ol'A and 6 can be slox~h varying within known bounds. By permitting m to include feedback signals of the model, il i~ possible to cause an unstable plant to track a stable model using relay control. By choosing the model in this form, it will be seen that the nominal plant can be made to follow the model with zero error. The design problem consists o1' determining values for L and k so that the plant will track the model with zero error when A = 0 , 6---0, and with a bounded error when the elements of A and 6 are within the stated bounds. Hence, it may be said that the plant is to be controlled with limited force so that, in the presence of bounded parameter variations, the error vector will remain bounded. The effect of disturbance d is treated similarly. The design approach is to define a positive semidefinite quadratic form V=½?Z(e).
(7)
Now if the control law can be found which causes the time derivative
differential equation k=Ax+bu+d
(1) ¢=?,)
(8)
where x is an n-dimensional state vector with ~ = dx/dt, u is a scalar forcing function, and d is a
disturbance vector. The elements of A and b can be slowly varying within known bounds. It is assumed that (1) represents a completely controllable system. A need not be a stability matrix. The relay is assumed to be ideal, and hence is characterized by the sign function u=~Lsgny(e)
(2)
where e = s - x , s being the response of the model. L and ? are to be determined. # equals + 1 or - I depending upon the specific problem involved. ?(e), which represents a linear function of the errorstate variables, is defined by ?(e)=k'e
(3)
where k ' = [ k l k 2 . . , k,] denotes the transpose of k. The model is represented by the equation s = As + ~rn(r, s)
(4)
with A and ~ defined as nominal values of A and b. m is a signal which produces the desired response, s, of the model, r is the input to the model. The deviations in A and b are denoted by
A=A-A,
(5)
g=b-p.
(6)
to be negative semidefinite so that V>0, V < 0 except on the hyperplane (switching plane) defined by 7 = 0, then if at some time e lies on the switching plane, its motion will be confined to the switching plane thereafter. The conditions must then be derived which assure stable, bounded, motion on the switching plane. Since stability of motion off the switching plane is not assured by use of a semidefinite V function, it is pertinent to question why V cannot be made a positive definite function. If V were taken to be positive definite as has been previously assumed [1-3], then in the presence of parameter uncertainties 12 could generally be assured of being negative definite only if the vector b in (1) were to contain but one non-zero element as would be the case if (1) were in phase-variable form. In this paper such a restriction on b is not allowed. In the following discussion, it is to be assumed that V is semidefinite as given by (7), and that the error vector is initially on the switching plane. Although no quantitative statement is to be made as to the stability of motion off the switching plane, it is reasonable to assume that the system will be stable for small perturbations about the switching plane. A quantitative treatment of the convergence problem for large deviations about the switching plane has been proposed [5]. This problem lies outside the scope of this paper, however. In formulating the control law, it is necessary to obtain a differential equation for the error vector.
FIG. 1. Photograph of experimental apparatus cart and inverted pendulum.
showing
Relay control of systems with parameter uncertainties and disturbances Thus, using (1), (4), and (5), it follows that +=Ae-As+[Im-bu-d,
(9)
or in scalar form d i = . f i ( e ) + g i ( s , d, m ) - b i u
, i = 1. . . .
, n.
(10)
To insure that chatter motion will take place, i.e. that motion will be constrained to lie on the switching plane, it is sufficient, as mentioned above, to require that V> 0, l?< 0, when ), ¢ 0. This condition will be satisfied if, for 7 ¢ 0, sgn~= - sgn 7.
(11)
Equation (11) can be satisfied if the magnitude of u is large enough to control the sign of ~. Consider the explicit form for y and ~. Thus, upon expanding (3), y=klej+k2ez+
...
+k,e,,
(12)
7=kldl
. . . +k.~..
(13)
whereupon +k2d2-]-
From (10) and (13) it is seen that ~ can be expressed in the form = ~b(e, s,
d, m)- ~ kibiu.
(14)
i=l
Now if u is made to satisfy the following two conditions, L = l u ] > ~b(e,s, d, m) i~_1-kibi
sgn(i~1
kibiu)=sgn7,
(l 5)
(16)
then it follows that (11) will be satisfied. If (15) and (16) are satisfied and if at some time v(e)=0, then the motion thereafter will remain on the switching plane. It is now necessary to establish the conditions for such motion to be stable. Whereas (9) characterizes motion in Euclidian n space (E"), motion on the switching plane can be described in E "-1 space. It follows based on the conditions v(e)=0 and ~(e)=0, that (9) can be reduced dimensionally so that, if represents a vector composed of n - 1 components of e, then (9) can be expressed as a linear differential equation of the form ~.=hl(k, A,~)+h2(k, b, As, d, q5 m).
(17)
757
It should be noted that motion on the switching plane will not generally call for u=0. Since sgn(0) can take on any value between + 1 and - l, it can be said that, for the idealized relay, u = L s g n 7 will in fact satisfy the requirement for motion on the switching plane. For a discussion of the chatter problem, see SCHAEFER[6]. Since s and m are signals generated by the model and d is bounded, it is found that h2 is a bounded forcing function whose magnitude depends on parameter deviations, A and 6, and d. Furthermore, as shown in the Appendix, h2 = 0 if A = [0], 6 = 0 , and d = 0 . If elements o f k are chosen so that (17) is asymptotically stable with h 2 = 0 , then e, as well as e, will be bounded. Finally, with a knowledge of the bounds on e, s, and m, and d, the value of L which is required to satisfy (15) can be determined. This completes the formal design procedure. The details of the method will be elaborated upon in a specific design application. It is noted that, in contrast to previous work [3, 4], by this design procedure the plant will in general track the model with zero error only if the parameters of the model and the plant are equal, and the disturbance is zero. This is in contrast to the PD design in which, with ideal switching, the plant tracks the model with zero error in spite of the presence of parameter deviations and disturbances, provided these remain within certain bounds. The fact that in the PD design the disturbances does not cause the error vector to leave the origin may be contrary to good engineering design, as in the case of the control of a flexible structure subject to disturbing influences. For example, it may not be practical to attempt to control a structure so as to nullify the effects of high frequency disturbances, since the result can be destructive to the structure. Hence it is seen that the SD design offers a more practical result, both with respect to instrumentation problems (noise) and design for disturbance rejection. It is interesting to observe that, in the SD design, once on the switching plane, the response to disturbance remains on the switching plane, and only through imperfect switching or initial conditions, does motion off the switching plane become a problem. In effect, a relay controller has been designed to cause the plant to have a desired nominal response characteristic, i.e. the model response. The effect of parameter uncertainty has been related to the response of a linear differential equation (17). 3. DESIGN APPLICATION The design technique represented in the preceding section will now be applied to a model-tracking system using a relay controller. The plant consists of a cart supporting an inverted pendulum. As
758
D . P . LINDORFF [-~BIL: 1.
may be seen in Fig. 1, the cart, which is lnounted on a track, is connected through a pulley and gearing to a D.C. motor. The objective is to cause the plant to track a linearized model of the nominal plant. This particular plant is of interest because the state variables are most naturally defined other than as phase variables. Thus, referring to Fig. 3,
~;i~__ 1 if =cons'l" R io r-~x'~'/'~n _
U=,o
- -
- ' ' c°;¢. "
•
t dependence
at I
800
none
a l2
3
lqOIqe
a3t
800
I,'1
a32
13
I/1
b~
15
none
d3
15
I/l
According to the design procedure, the nominal values in Table 1 should be assigned to the corresponding terms in (4) pertaining to the model. Thus, (4) becomes
L//
%) ~-
Nominal
+ -v
d. c motor "~l ~(~1 1S1 ~- ~12S2 -{-ill Ftl
FIG. 3. Diagrammatic representation of experimental system.
,~2 = $ 3 $ 3 = ~ 3 1 S 1 -I-9(32S2-~'fl3171
if velocity of the cart is to be controlled, then the state variables logically become
(t9)
where each e~j and fli has the nominal value of the corresponding a u and b i in Table 1. It is noted that the plant is open-loop unstable. In this problem linear feedback was used so that the signal m(t) stabilized the model and produced a desired response (s) to a reference input (r). From (10), (18), (19), the error equations become
al = v m/see x 2 = 0 rad. x3 = 0 rad./sec and
~ = a t te 1+ a l a e 2 + y t - b ~ u u = e. volts. 02-=e 3
Relative to (1) and based on the linearizing assumption (sin0~0, c o s 0 ~ l ) , the equations of the plant can be written in the form [7]
03 = a31 el + a 3 : 2 + 93 -- b3tt
(20)
where "~1 = a l 1Xl -~"a~ 2x2 --~ blu
Qi= - - A i l s 1 - A i 2 s 2 - } - f l l m , i = 1, 3
"~'2 = X3 2 3 = a s IXl + a s z X 2 + b 3 u .
(18)
We have assumed here that d = 0 , since the experimental investigation did not include the effect of disturbances. In order to study the effect of parameter variations, the distance from the pivot point to the center of gravity of the pendulum was made adjustable by repositioning m I on the rod. The nominal values of the parameters, as well as their dependences on l, are shown in Table 1.
and A i j : a u - - o ~ i ] , .]= 1, 2.
Equation (7) in turn becomes
v=½(¢) =½(el +k2e2 +k3e3) 2
(21)
wherein it is arbitrarily assumed that k~ = 1 and
Relay control of systems with parameter uncertainties and disturbances k2>0, k3>0. It follows, upon substituting (20) into the expression for p, that
= d1+k2e2 q-kad3 =attet -[-a12e2 +if1 -blu+k2e3 + ka(a3 l el + a32e2 + g3 -- b3u), whereupon (I 5) becomes L = [u] _> tPt(e) + (fit + k3fl3)m V2(s) bt +k3b3 -
(23)
+kaaa2)e2 +k2e3
and ~Id2
=
(A11 + k3A3 l)Sl + (A12 + kaA32)s2 •
Equation (16) in turn becomes sgn (bl + kaba)u = sgny.
(24)
The equations (23) and (24) form the basis for a control law. However, before discussing these equations further, it will be necessary to obtain the reduced equation (17) which describes motion on the switching plane, assuming that (23) and (24) will be satisfied. Using the equations y(e)=O, ~(e)=O which are valid on the switching plane, it is found that (17) becomes
(l + ~3rt)dl=Ialt +~3(rtr2-a31)]et +ol--~393 +[a12 + ~(r~--a32)]e2 e2 = -rlet - r2e2
(25a)
(25b)
where 1
Yt
k3 '
k2
1"2 k3
and
gi = -Airs I --Ai2s2
+ flim ,
i= 1, 3.
It is required at this point that k2 and k 3 be chosen so that (25) will be stable. If (25) is rewritten in the homogeneous form dt ----~bltet +q~12e2 d2 = t~2te 1 d- ~22e2,
(26)
then the characteristic equation is seen to be
;tz-(¢tt + ¢22)~+(~1¢22-¢2t~2)=0.
k2>
(27)
0.
(28)
Returning to (23) and (24), it is required that conditions be established which guarantee 12<0 with 1' ~ 0. On the assumption that (23) is satisfied, it follows that u will control the sign of ~. For the nominal values of bl, b 3 in Table l, and the requirement k 3 > I, it is seen that (24) is satisfied if sgn u - sgn ~.
with qq(al i + k3a31)el +(a12
Stabifity requires that q511+ t~22 < O, q~l 1(~22 > t~21 ~bl2, which can be shown to be equivalent to requiring for k I = 1 that k 3 > 1,
(22)
759
(29)
In order that (29) can be used for the full range of variations of bt and b3, it is necessary that k 3 be chosen so that (bl+k3b3) will always be of one sign. In this case, k 3 can always be made large enough to meet this requirement. Attention is now directed to (23). In order to determine a sufficiently large value of L, it is necessary to know at least the bounds on q~(e), q~2(s) and m. It is informative first to assume that the plant parameters are at their nominal values. In this case it can be seen that qJ2(s)=0, q~l(e)=0. Hence (23) reduces to
z=l.l_>lm I .
(30)
The statement that q~a(e)=0 in the absence of parameter deviations follows from the fact that the forcing function in (25) then reduces to zero. Thus, for the nominal plant, if e = 0 initially, then the tracking error is zero if (29) and (30) are satisfied. Since the magnitude of m depends upon the size of the input (r), as well as the response time of the model, which is reflected in re(t), the sizing of L requires a knowledge of the specifications placed on the system's performance. At this point in the design, the effect of parameter deviations should be considered. More specifically, the value of L in (23) must be made large enough to override the terms Wt(e ) and qJ2(s). In section 4 results of simulation studies are presented which bear on this aspect of the problem. It is important to note that it is possible to use simulation methods advantageously in the design since the response of the plant in tracking the model is insensitive to L, provided L is large enough. Thus, by making L excessively large in the simulation, the behavior of e(t) can be determined. This information can in turn be used in finding the minimum acceptable value of L. 4. RESULTS BY SIMULATION AND EXPERIMENT The design which has been discussed in the preceding section was completed with the aid of analog-computer simulation and tests on the physical system, in response to a square-wave input to the model.
760
D . P . LINDORFF
The results of the simulation studies are presented in Figs. 4, 5, and 6. In these data the value of L was made excessively large, so as to guarantee that (15) would be satisfied. In Fig. 4 the responses of the state variables of the model and the
l
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a
a
J
a
a
J
A
J
i
l
l
FIG. 5. Traces of simulated responses of the model and the plant with parameter deviation.
FIG. 4. Traces of simulated responses of the model and the plant without parameter deviation.
,
~
I
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~
~ - ~ = - = - - . , ,~
.
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-lle,
nominal plant are shown. The tracking error is seen to be essentially zero as predicted. In Figs. 5 and 6 the value of l in the plant was increased by 40 per cent of its nominal value, and the test signal used in Fig. 4 was again applied. The data in Fig. 5 shows the responses of the same state variables as in Fig. 4, whereas the data in Fig. 6 shows the responses of the error state variables (el, e2, e3). It is interesting to note that the main effect of this particular parameter variation is to alter the initial response of the plant state variables. The spikes which appear in the error responses were used to establish bounds on the error terms appearing in (23).
Fro. 6. Traces of simulated error responses with parameter deviation.
Relay control of systems with parameter uncertainties and disturbances In Fig. 7 response of the physical system in terms of xl (velocity of cart) and x2 (angle of pendulum) is compared with corresponding terms, s I and s 2 of the model, using a value of L determined from the simulation. Since parameter values of the plant could not be determined exactly, some error was to be expected. The high frequency components present in the reponse of the plant are attributed to imperfect switching and coulomb friction.
761
REFERENCES [1] T. M. TAYLOR: Determination of a realistic error bound for a class of imperfect non-linear controllers. Preprints of Tech. Papers, 9th JACC, prepared by IEEE, New York, N.Y. 522-538 (1968). [2] D. P. LINDORFF: Relay Control of Multioutput Systems with Parameter Uncertainties. 4th IFAC Congress, Warsaw, Poland (June 1969). [3] R. V. MONOPOLI:Control of linear plants with zeros and slowly varying parameters. 1EEE Trans. AC-12, 80-83 (1967). [4] D. P. LINDORFF: Control of non-linear multivariable systems. 1EEE Trans. AC-12, N 11, 506-515 (1967). [5] D. P. LINOORFF: Use of semi-definite quadratic forms in designing relay control systems. Proc. Third Princeton Conf. on Information Sciences and Systems, March (1969). [6] J. F. SCHAEFER." Chatter motion of bang-bang controllers. Proc. Second Annual Princeton Conf. on Information Science and Systems 228-231 (1968). [7] R. H. CANNON,JR. : Dynamics of Physical Systems, Chap. 22. McGraw-Hill, New York (1965).
APPENDIX
Conditions for which the bounded forcing function, h 2 iS zero To show that h 2 = 0 in (17) if A = 0 , 8 = 0 , and d = 0 , it is noted that, subject to these conditions, (9) becomes 6=Ae+b(m-u),
(A.1)
or in scalar form
FIG. 7. Responses of model compared with responses of experimental system. 5. CONCLUSIONS The design of a relay controller has been formulated in which a linear plant subject to parameter uncertainties and disturbances is caused to track a model with bounded error. The method can be applied to multi-output systems, or more generally to systems whose state variables are not necessarily available as phase variables. The method is demonstrated by application to a specific design problem involving the stabilization of a cart-supported inverted pendulum. F r o m the theoretical standpoint, further study should be made of the tracking error which is incurred due to the use of an imperfect relay.In addition, because a semidefinite Liapunov function is used in the design, bounds on the motion which result if the error states are not initially on the switching plane are difficult to find. This problem merits further consideration.
~i= ~ aijej+bi(m-u),
search Grant NGL 07-002-002 of the National Aeronautics and Space Administration.
n. (A.2)
It will be assumed that b contains at least two nonzero elements, and for convenience that the equations are ordered so that one of these non-zero elements is designated as b,. It is then possible to write u=lFb,L j=,~ a,jei+b,m-~,].
(A.3)
Using (A.3), u can be eliminated from the first n - 1 equations in (A.2), so that
b "
]
ei= ~ aijej+bi m - ~ t F Z anjej+bnm-en , j=l b.Li=l i=1 .....
(A.4)
n-l,
or in simplified form ei:
Acknowledgement--This work was supported under Re-
i = 1. . . . .
j=l
j=
anj
en,
i=l .... , n-1. (A.5)
762
D . P . LINDORFF
Since u is tacitly assumed to be controlled so that m o t i o n is restricted to the switching plane, it is possible, using the e q u a t i o n s y ( e ) = 0 , ~,(e)=0, to eliminate e, a n d 0, from (A.5). The result can then be put i n n o r m a l form, whence it follows that h2 = 0 subject to the stated assumptions.
R~sum~-Cet article pr6sente une m6thode employant une fonction de Liapunov semi-d6finie, permettant le calcul d'un syst6me d'asservissement / t u n module avec r6gulateur gt relais. I1 consid~m les effets d'incertitude des param6tres et d'entrdes perturbatrices. I1 est montr6 qu'avec cette m6thode, les variables d'6tat ne doivent passe trouver sous la forme de phase variable ainsi que cela 6tait n6cessaire ant6rieuremerit. Ceci r6sulte en une reduction du bruit de l'appareillage entrant dans le syst~me et en une solution plus raisonable du probl6me de rejet des perturbations. Une application de cette mgthode est fare ~tla cornmande d'un syst6me instable. Plus6xactement, rarticlepr6sente desr6sultats experirnentaux r61atifs a la commande d'un pendule inverse soutenu par un chariot.
Zusammenfassung--Behandelt wird eine N/iherung unter Benutzung einer semidefiniten Ljapunow-Funktion und zwar ftir den Entwurf eines Modellfiihrungssystems mit einem Relais-Regler. Die Wirkung yon Parameterunsicherheiten und Eingangsst6rungenwird betrachtet. Gezeigt wird, dal3 bei dieser Methode die Zustandsvariablen nicht wie frtiher erforderlich in der Form von Phasenvariablen vorliegen mtissen. Dies bewirkt eine Reduktion des in das System eintretenden Rauschens seitens der MeBger/iteausrtistung und eine verniJnftigere L6sung des Problems der Stt~rtmgsaufnahme. Angewandt wird die Methode auf die Regelung eines unstabilen Systems. Experimentelle Daten der Regelung eines yon einem kleinen Wagen gestfitzten umgekehrten Pendels sind angegeben. P e 3 m M e - - D T a CTaTb~l npe~naraex MeTO)I, Hcno~b3ylOLt~Hfl qaCTaqHO o n p e ~ e ~ e u H y ~ o qbyHKaH}o J/~lnyHOBa 14 HO3BO~flrOtR~l~t p a c c q e T MoJie~bHOCJ~e,a,qtltei~ CHCTeMbI C peJIe~HblM p e r y r t a T O p O M . O H a pacCMaTpHBaeT BJlllitmte HeonpezleJIeHHOCTH n a p a M e T p o B i4 BXO}JHbIX tJOMeX, gIoKa3bmaeq-ca qTo c 0-TAM MeTOJIOM n e p e M e n H b l e COCTO~HHIt He ,~tOSIXHBI HaxoJI,IITbC~I B Brl/le rtepeMeHHO~ ~ a 3 b l KaK 3TO 6blJIO Heo6xo,r/brdvto paHbUle. 9 T O rlpHBOIJ,I'IT K coKpameHH~O IHyMa a r m a p a T y p b i BBO}II/MOFO B CHCTeMy it K 6 0 a e e rio/Ixojlameb~y p e m e m u o n p o 6 : I e M b I 60pr~6bl c nor~texaMH. , ~ a H o n p n M e H e H n e DTOEO Me r o ~ a K y n p a B s l e H m o r l e y c x o ~ UlfBOfl CHCTeMOfl. T o q H e e , CTaThlt //aeT 3KcneprtMCHTaYlbHble pe3y.qbTaTbI y n p a B ~ e H r l a o6paTrfblM MaflTHHKOM n OR/lep~nBarogi~iMCn KapeRtKOfL
Relay Control of Systems with Parameter Uncertainties and Disturbances* Commande ~t relais de syst6mes avec incertitude des param&res et perturbations Relais-Regelungssysteme mit Parameterunsicherheiten und -St6rungen Pe3ie~noe ynpaB.qen/4e C/4CTeMaMrI C neonpe~e~enHocTbrO napaMeTpoB /4 C 1-IOMeXaM/4 D. P. L I N D O R F F t
The design of model tracking systems using relay control is presented. The freedom allowed in choosing the measured state variables results in favorable perfbrmance characteristics both with respect to instrumentation noise and response to disturbances. Summary--This paper presents an approach, using a semidefinite Liapunov function, for designing a model-tracking system with a relay controller. The effect of parameter uncertainties and disturbance inputs is considered. It is shown that, by this method, the state variables need not be in phase variable form, as has been formerly required. This results in a reduction of instrumentation noise entering the system, and a more reasonable solution to the problem of disturbance rejection. Application of the method is made to the control of an unstable system. Specifically, experimental data on control of a cart-supported inverted pendulum is presented.
such as permitting a bound on the system response due to imperfect switching to be determined [1]. In addition, with the phase variable form of the state equations the design approach does not ordinarily place restrictions on the rate of change of parameter values. Nevertheless, two criticisms of the method are that the instrumented phase variables in higher-order systems are generally contaminated with noise, and the disturbance response is not accommodated in the design procedure in a realistic way. By the use of a semidefinite Liapunov function, the rigid requirement on the form of the state variables can be relaxed. The purpose of this paper is to develop this concept, with application to a model-tracking system using a relay controller. Although the theory developed in this paper is applicable to a wider class of problems, the specifc plant to be controlled is pictured in Fig. 1, and consists of an inverted pendulum mounted on a cart whose position is controlled by a D.C. motor. Although design techniques have been proposed for relay control systems, it is believed that this paper extends previous work by generalizing the application of model-tracking systems in the presence of parameter variations and disturbances. To guarantee stability in response to initial conditions the SD design approach necessarily requires some information about the system trajectories, as contrasted with the PD design. One approach to solving this problem, by drawing upon the notion of an invariant set, has been proposed [5]. However, the convergence problem does not lie within the scope of this paper.
1. INTRODUCTION IN THIS paper the use of a semidefinite (SD) quadratic form .is examined further in relation to the relay control problem where the plant is subject to parameter uncertainty [1, 2]. Specifically, consideration is given to the model tracking problem with disturbance inputs, for the case in which the state variables are not defined as phase variables. In previous works, a positive definite (PD) Liapunov function has been used in the design of systems subject to parameter variations [3, 4]. In these works it was found necessary to assume the phase-variable form in writing the state equations for the plant, this being an unavoidable restriction if the effects of parameter uncertainties are to be considered with the PD design approach. However, the use of the PD form offers certain advantages, * Received 27 February 1969; revised 19 April 1969. The original version of this paper was presented at the IFAC Congress which was held in Warsaw, Poland during June 1969. It was recommended for publication in revised form by associate editor P. Parks. t Professor of Electrical Engineering at the University of Connecticut, Storrs, Connecticut, USA. 755
756
D . P . LINDORI t: 2. THE DESIGN TECHNIQUE
The system is diagrammatically represented in Fig. 2. The plant is defined according to the linear
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Model of I ~ I
Compensation
nominal plont Model
FIG. 2. Block diagram of compensated model and plant with relay control.
It is assumed that elements ol'A and 6 can be slox~h varying within known bounds. By permitting m to include feedback signals of the model, il i~ possible to cause an unstable plant to track a stable model using relay control. By choosing the model in this form, it will be seen that the nominal plant can be made to follow the model with zero error. The design problem consists o1' determining values for L and k so that the plant will track the model with zero error when A = 0 , 6---0, and with a bounded error when the elements of A and 6 are within the stated bounds. Hence, it may be said that the plant is to be controlled with limited force so that, in the presence of bounded parameter variations, the error vector will remain bounded. The effect of disturbance d is treated similarly. The design approach is to define a positive semidefinite quadratic form V=½?Z(e).
(7)
Now if the control law can be found which causes the time derivative
differential equation k=Ax+bu+d
(1) ¢=?,)
(8)
where x is an n-dimensional state vector with ~ = dx/dt, u is a scalar forcing function, and d is a
disturbance vector. The elements of A and b can be slowly varying within known bounds. It is assumed that (1) represents a completely controllable system. A need not be a stability matrix. The relay is assumed to be ideal, and hence is characterized by the sign function u=~Lsgny(e)
(2)
where e = s - x , s being the response of the model. L and ? are to be determined. # equals + 1 or - I depending upon the specific problem involved. ?(e), which represents a linear function of the errorstate variables, is defined by ?(e)=k'e
(3)
where k ' = [ k l k 2 . . , k,] denotes the transpose of k. The model is represented by the equation s = As + ~rn(r, s)
(4)
with A and ~ defined as nominal values of A and b. m is a signal which produces the desired response, s, of the model, r is the input to the model. The deviations in A and b are denoted by
A=A-A,
(5)
g=b-p.
(6)
to be negative semidefinite so that V>0, V < 0 except on the hyperplane (switching plane) defined by 7 = 0, then if at some time e lies on the switching plane, its motion will be confined to the switching plane thereafter. The conditions must then be derived which assure stable, bounded, motion on the switching plane. Since stability of motion off the switching plane is not assured by use of a semidefinite V function, it is pertinent to question why V cannot be made a positive definite function. If V were taken to be positive definite as has been previously assumed [1-3], then in the presence of parameter uncertainties 12 could generally be assured of being negative definite only if the vector b in (1) were to contain but one non-zero element as would be the case if (1) were in phase-variable form. In this paper such a restriction on b is not allowed. In the following discussion, it is to be assumed that V is semidefinite as given by (7), and that the error vector is initially on the switching plane. Although no quantitative statement is to be made as to the stability of motion off the switching plane, it is reasonable to assume that the system will be stable for small perturbations about the switching plane. A quantitative treatment of the convergence problem for large deviations about the switching plane has been proposed [5]. This problem lies outside the scope of this paper, however. In formulating the control law, it is necessary to obtain a differential equation for the error vector.
FIG. 1. Photograph of experimental apparatus cart and inverted pendulum.
showing
Relay control of systems with parameter uncertainties and disturbances Thus, using (1), (4), and (5), it follows that +=Ae-As+[Im-bu-d,
(9)
or in scalar form d i = . f i ( e ) + g i ( s , d, m ) - b i u
, i = 1. . . .
, n.
(10)
To insure that chatter motion will take place, i.e. that motion will be constrained to lie on the switching plane, it is sufficient, as mentioned above, to require that V> 0, l?< 0, when ), ¢ 0. This condition will be satisfied if, for 7 ¢ 0, sgn~= - sgn 7.
(11)
Equation (11) can be satisfied if the magnitude of u is large enough to control the sign of ~. Consider the explicit form for y and ~. Thus, upon expanding (3), y=klej+k2ez+
...
+k,e,,
(12)
7=kldl
. . . +k.~..
(13)
whereupon +k2d2-]-
From (10) and (13) it is seen that ~ can be expressed in the form = ~b(e, s,
d, m)- ~ kibiu.
(14)
i=l
Now if u is made to satisfy the following two conditions, L = l u ] > ~b(e,s, d, m) i~_1-kibi
sgn(i~1
kibiu)=sgn7,
(l 5)
(16)
then it follows that (11) will be satisfied. If (15) and (16) are satisfied and if at some time v(e)=0, then the motion thereafter will remain on the switching plane. It is now necessary to establish the conditions for such motion to be stable. Whereas (9) characterizes motion in Euclidian n space (E"), motion on the switching plane can be described in E "-1 space. It follows based on the conditions v(e)=0 and ~(e)=0, that (9) can be reduced dimensionally so that, if represents a vector composed of n - 1 components of e, then (9) can be expressed as a linear differential equation of the form ~.=hl(k, A,~)+h2(k, b, As, d, q5 m).
(17)
757
It should be noted that motion on the switching plane will not generally call for u=0. Since sgn(0) can take on any value between + 1 and - l, it can be said that, for the idealized relay, u = L s g n 7 will in fact satisfy the requirement for motion on the switching plane. For a discussion of the chatter problem, see SCHAEFER[6]. Since s and m are signals generated by the model and d is bounded, it is found that h2 is a bounded forcing function whose magnitude depends on parameter deviations, A and 6, and d. Furthermore, as shown in the Appendix, h2 = 0 if A = [0], 6 = 0 , and d = 0 . If elements o f k are chosen so that (17) is asymptotically stable with h 2 = 0 , then e, as well as e, will be bounded. Finally, with a knowledge of the bounds on e, s, and m, and d, the value of L which is required to satisfy (15) can be determined. This completes the formal design procedure. The details of the method will be elaborated upon in a specific design application. It is noted that, in contrast to previous work [3, 4], by this design procedure the plant will in general track the model with zero error only if the parameters of the model and the plant are equal, and the disturbance is zero. This is in contrast to the PD design in which, with ideal switching, the plant tracks the model with zero error in spite of the presence of parameter deviations and disturbances, provided these remain within certain bounds. The fact that in the PD design the disturbances does not cause the error vector to leave the origin may be contrary to good engineering design, as in the case of the control of a flexible structure subject to disturbing influences. For example, it may not be practical to attempt to control a structure so as to nullify the effects of high frequency disturbances, since the result can be destructive to the structure. Hence it is seen that the SD design offers a more practical result, both with respect to instrumentation problems (noise) and design for disturbance rejection. It is interesting to observe that, in the SD design, once on the switching plane, the response to disturbance remains on the switching plane, and only through imperfect switching or initial conditions, does motion off the switching plane become a problem. In effect, a relay controller has been designed to cause the plant to have a desired nominal response characteristic, i.e. the model response. The effect of parameter uncertainty has been related to the response of a linear differential equation (17). 3. DESIGN APPLICATION The design technique represented in the preceding section will now be applied to a model-tracking system using a relay controller. The plant consists of a cart supporting an inverted pendulum. As
758
D . P . LINDORFF [-~BIL: 1.
may be seen in Fig. 1, the cart, which is lnounted on a track, is connected through a pulley and gearing to a D.C. motor. The objective is to cause the plant to track a linearized model of the nominal plant. This particular plant is of interest because the state variables are most naturally defined other than as phase variables. Thus, referring to Fig. 3,
~;i~__ 1 if =cons'l" R io r-~x'~'/'~n _
U=,o
- -
- ' ' c°;¢. "
•
t dependence
at I
800
none
a l2
3
lqOIqe
a3t
800
I,'1
a32
13
I/1
b~
15
none
d3
15
I/l
According to the design procedure, the nominal values in Table 1 should be assigned to the corresponding terms in (4) pertaining to the model. Thus, (4) becomes
L//
%) ~-
Nominal
+ -v
d. c motor "~l ~(~1 1S1 ~- ~12S2 -{-ill Ftl
FIG. 3. Diagrammatic representation of experimental system.
,~2 = $ 3 $ 3 = ~ 3 1 S 1 -I-9(32S2-~'fl3171
if velocity of the cart is to be controlled, then the state variables logically become
(t9)
where each e~j and fli has the nominal value of the corresponding a u and b i in Table 1. It is noted that the plant is open-loop unstable. In this problem linear feedback was used so that the signal m(t) stabilized the model and produced a desired response (s) to a reference input (r). From (10), (18), (19), the error equations become
al = v m/see x 2 = 0 rad. x3 = 0 rad./sec and
~ = a t te 1+ a l a e 2 + y t - b ~ u u = e. volts. 02-=e 3
Relative to (1) and based on the linearizing assumption (sin0~0, c o s 0 ~ l ) , the equations of the plant can be written in the form [7]
03 = a31 el + a 3 : 2 + 93 -- b3tt
(20)
where "~1 = a l 1Xl -~"a~ 2x2 --~ blu
Qi= - - A i l s 1 - A i 2 s 2 - } - f l l m , i = 1, 3
"~'2 = X3 2 3 = a s IXl + a s z X 2 + b 3 u .
(18)
We have assumed here that d = 0 , since the experimental investigation did not include the effect of disturbances. In order to study the effect of parameter variations, the distance from the pivot point to the center of gravity of the pendulum was made adjustable by repositioning m I on the rod. The nominal values of the parameters, as well as their dependences on l, are shown in Table 1.
and A i j : a u - - o ~ i ] , .]= 1, 2.
Equation (7) in turn becomes
v=½(¢) =½(el +k2e2 +k3e3) 2
(21)
wherein it is arbitrarily assumed that k~ = 1 and
Relay control of systems with parameter uncertainties and disturbances k2>0, k3>0. It follows, upon substituting (20) into the expression for p, that
= d1+k2e2 q-kad3 =attet -[-a12e2 +if1 -blu+k2e3 + ka(a3 l el + a32e2 + g3 -- b3u), whereupon (I 5) becomes L = [u] _> tPt(e) + (fit + k3fl3)m V2(s) bt +k3b3 -
(23)
+kaaa2)e2 +k2e3
and ~Id2
=
(A11 + k3A3 l)Sl + (A12 + kaA32)s2 •
Equation (16) in turn becomes sgn (bl + kaba)u = sgny.
(24)
The equations (23) and (24) form the basis for a control law. However, before discussing these equations further, it will be necessary to obtain the reduced equation (17) which describes motion on the switching plane, assuming that (23) and (24) will be satisfied. Using the equations y(e)=O, ~(e)=O which are valid on the switching plane, it is found that (17) becomes
(l + ~3rt)dl=Ialt +~3(rtr2-a31)]et +ol--~393 +[a12 + ~(r~--a32)]e2 e2 = -rlet - r2e2
(25a)
(25b)
where 1
Yt
k3 '
k2
1"2 k3
and
gi = -Airs I --Ai2s2
+ flim ,
i= 1, 3.
It is required at this point that k2 and k 3 be chosen so that (25) will be stable. If (25) is rewritten in the homogeneous form dt ----~bltet +q~12e2 d2 = t~2te 1 d- ~22e2,
(26)
then the characteristic equation is seen to be
;tz-(¢tt + ¢22)~+(~1¢22-¢2t~2)=0.
k2>
(27)
0.
(28)
Returning to (23) and (24), it is required that conditions be established which guarantee 12<0 with 1' ~ 0. On the assumption that (23) is satisfied, it follows that u will control the sign of ~. For the nominal values of bl, b 3 in Table l, and the requirement k 3 > I, it is seen that (24) is satisfied if sgn u - sgn ~.
with qq(al i + k3a31)el +(a12
Stabifity requires that q511+ t~22 < O, q~l 1(~22 > t~21 ~bl2, which can be shown to be equivalent to requiring for k I = 1 that k 3 > 1,
(22)
759
(29)
In order that (29) can be used for the full range of variations of bt and b3, it is necessary that k 3 be chosen so that (bl+k3b3) will always be of one sign. In this case, k 3 can always be made large enough to meet this requirement. Attention is now directed to (23). In order to determine a sufficiently large value of L, it is necessary to know at least the bounds on q~(e), q~2(s) and m. It is informative first to assume that the plant parameters are at their nominal values. In this case it can be seen that qJ2(s)=0, q~l(e)=0. Hence (23) reduces to
z=l.l_>lm I .
(30)
The statement that q~a(e)=0 in the absence of parameter deviations follows from the fact that the forcing function in (25) then reduces to zero. Thus, for the nominal plant, if e = 0 initially, then the tracking error is zero if (29) and (30) are satisfied. Since the magnitude of m depends upon the size of the input (r), as well as the response time of the model, which is reflected in re(t), the sizing of L requires a knowledge of the specifications placed on the system's performance. At this point in the design, the effect of parameter deviations should be considered. More specifically, the value of L in (23) must be made large enough to override the terms Wt(e ) and qJ2(s). In section 4 results of simulation studies are presented which bear on this aspect of the problem. It is important to note that it is possible to use simulation methods advantageously in the design since the response of the plant in tracking the model is insensitive to L, provided L is large enough. Thus, by making L excessively large in the simulation, the behavior of e(t) can be determined. This information can in turn be used in finding the minimum acceptable value of L. 4. RESULTS BY SIMULATION AND EXPERIMENT The design which has been discussed in the preceding section was completed with the aid of analog-computer simulation and tests on the physical system, in response to a square-wave input to the model.
760
D . P . LINDORFF
The results of the simulation studies are presented in Figs. 4, 5, and 6. In these data the value of L was made excessively large, so as to guarantee that (15) would be satisfied. In Fig. 4 the responses of the state variables of the model and the
l
J
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~
a
a
J
a
a
J
A
J
i
l
l
FIG. 5. Traces of simulated responses of the model and the plant with parameter deviation.
FIG. 4. Traces of simulated responses of the model and the plant without parameter deviation.
,
~
I
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~
~ - ~ = - = - - . , ,~
.
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-lle,
nominal plant are shown. The tracking error is seen to be essentially zero as predicted. In Figs. 5 and 6 the value of l in the plant was increased by 40 per cent of its nominal value, and the test signal used in Fig. 4 was again applied. The data in Fig. 5 shows the responses of the same state variables as in Fig. 4, whereas the data in Fig. 6 shows the responses of the error state variables (el, e2, e3). It is interesting to note that the main effect of this particular parameter variation is to alter the initial response of the plant state variables. The spikes which appear in the error responses were used to establish bounds on the error terms appearing in (23).
Fro. 6. Traces of simulated error responses with parameter deviation.
Relay control of systems with parameter uncertainties and disturbances In Fig. 7 response of the physical system in terms of xl (velocity of cart) and x2 (angle of pendulum) is compared with corresponding terms, s I and s 2 of the model, using a value of L determined from the simulation. Since parameter values of the plant could not be determined exactly, some error was to be expected. The high frequency components present in the reponse of the plant are attributed to imperfect switching and coulomb friction.
761
REFERENCES [1] T. M. TAYLOR: Determination of a realistic error bound for a class of imperfect non-linear controllers. Preprints of Tech. Papers, 9th JACC, prepared by IEEE, New York, N.Y. 522-538 (1968). [2] D. P. LINDORFF: Relay Control of Multioutput Systems with Parameter Uncertainties. 4th IFAC Congress, Warsaw, Poland (June 1969). [3] R. V. MONOPOLI:Control of linear plants with zeros and slowly varying parameters. 1EEE Trans. AC-12, 80-83 (1967). [4] D. P. LINDORFF: Control of non-linear multivariable systems. 1EEE Trans. AC-12, N 11, 506-515 (1967). [5] D. P. LINOORFF: Use of semi-definite quadratic forms in designing relay control systems. Proc. Third Princeton Conf. on Information Sciences and Systems, March (1969). [6] J. F. SCHAEFER." Chatter motion of bang-bang controllers. Proc. Second Annual Princeton Conf. on Information Science and Systems 228-231 (1968). [7] R. H. CANNON,JR. : Dynamics of Physical Systems, Chap. 22. McGraw-Hill, New York (1965).
APPENDIX
Conditions for which the bounded forcing function, h 2 iS zero To show that h 2 = 0 in (17) if A = 0 , 8 = 0 , and d = 0 , it is noted that, subject to these conditions, (9) becomes 6=Ae+b(m-u),
(A.1)
or in scalar form
FIG. 7. Responses of model compared with responses of experimental system. 5. CONCLUSIONS The design of a relay controller has been formulated in which a linear plant subject to parameter uncertainties and disturbances is caused to track a model with bounded error. The method can be applied to multi-output systems, or more generally to systems whose state variables are not necessarily available as phase variables. The method is demonstrated by application to a specific design problem involving the stabilization of a cart-supported inverted pendulum. F r o m the theoretical standpoint, further study should be made of the tracking error which is incurred due to the use of an imperfect relay.In addition, because a semidefinite Liapunov function is used in the design, bounds on the motion which result if the error states are not initially on the switching plane are difficult to find. This problem merits further consideration.
~i= ~ aijej+bi(m-u),
search Grant NGL 07-002-002 of the National Aeronautics and Space Administration.
n. (A.2)
It will be assumed that b contains at least two nonzero elements, and for convenience that the equations are ordered so that one of these non-zero elements is designated as b,. It is then possible to write u=lFb,L j=,~ a,jei+b,m-~,].
(A.3)
Using (A.3), u can be eliminated from the first n - 1 equations in (A.2), so that
b "
]
ei= ~ aijej+bi m - ~ t F Z anjej+bnm-en , j=l b.Li=l i=1 .....
(A.4)
n-l,
or in simplified form ei:
Acknowledgement--This work was supported under Re-
i = 1. . . . .
j=l
j=
anj
en,
i=l .... , n-1. (A.5)
762
D . P . LINDORFF
Since u is tacitly assumed to be controlled so that m o t i o n is restricted to the switching plane, it is possible, using the e q u a t i o n s y ( e ) = 0 , ~,(e)=0, to eliminate e, a n d 0, from (A.5). The result can then be put i n n o r m a l form, whence it follows that h2 = 0 subject to the stated assumptions.
R~sum~-Cet article pr6sente une m6thode employant une fonction de Liapunov semi-d6finie, permettant le calcul d'un syst6me d'asservissement / t u n module avec r6gulateur gt relais. I1 consid~m les effets d'incertitude des param6tres et d'entrdes perturbatrices. I1 est montr6 qu'avec cette m6thode, les variables d'6tat ne doivent passe trouver sous la forme de phase variable ainsi que cela 6tait n6cessaire ant6rieuremerit. Ceci r6sulte en une reduction du bruit de l'appareillage entrant dans le syst~me et en une solution plus raisonable du probl6me de rejet des perturbations. Une application de cette mgthode est fare ~tla cornmande d'un syst6me instable. Plus6xactement, rarticlepr6sente desr6sultats experirnentaux r61atifs a la commande d'un pendule inverse soutenu par un chariot.
Zusammenfassung--Behandelt wird eine N/iherung unter Benutzung einer semidefiniten Ljapunow-Funktion und zwar ftir den Entwurf eines Modellfiihrungssystems mit einem Relais-Regler. Die Wirkung yon Parameterunsicherheiten und Eingangsst6rungenwird betrachtet. Gezeigt wird, dal3 bei dieser Methode die Zustandsvariablen nicht wie frtiher erforderlich in der Form von Phasenvariablen vorliegen mtissen. Dies bewirkt eine Reduktion des in das System eintretenden Rauschens seitens der MeBger/iteausrtistung und eine verniJnftigere L6sung des Problems der Stt~rtmgsaufnahme. Angewandt wird die Methode auf die Regelung eines unstabilen Systems. Experimentelle Daten der Regelung eines yon einem kleinen Wagen gestfitzten umgekehrten Pendels sind angegeben. P e 3 m M e - - D T a CTaTb~l npe~naraex MeTO)I, Hcno~b3ylOLt~Hfl qaCTaqHO o n p e ~ e ~ e u H y ~ o qbyHKaH}o J/~lnyHOBa 14 HO3BO~flrOtR~l~t p a c c q e T MoJie~bHOCJ~e,a,qtltei~ CHCTeMbI C peJIe~HblM p e r y r t a T O p O M . O H a pacCMaTpHBaeT BJlllitmte HeonpezleJIeHHOCTH n a p a M e T p o B i4 BXO}JHbIX tJOMeX, gIoKa3bmaeq-ca qTo c 0-TAM MeTOJIOM n e p e M e n H b l e COCTO~HHIt He ,~tOSIXHBI HaxoJI,IITbC~I B Brl/le rtepeMeHHO~ ~ a 3 b l KaK 3TO 6blJIO Heo6xo,r/brdvto paHbUle. 9 T O rlpHBOIJ,I'IT K coKpameHH~O IHyMa a r m a p a T y p b i BBO}II/MOFO B CHCTeMy it K 6 0 a e e rio/Ixojlameb~y p e m e m u o n p o 6 : I e M b I 60pr~6bl c nor~texaMH. , ~ a H o n p n M e H e H n e DTOEO Me r o ~ a K y n p a B s l e H m o r l e y c x o ~ UlfBOfl CHCTeMOfl. T o q H e e , CTaThlt //aeT 3KcneprtMCHTaYlbHble pe3y.qbTaTbI y n p a B ~ e H r l a o6paTrfblM MaflTHHKOM n OR/lep~nBarogi~iMCn KapeRtKOfL