- Email: [email protected]
Characteristic Phenomena and Model Problems in Nonlinear Control
Copyright © 1996 IFAC 13th Triennial World Congress, San
Franci~co,
4c-045
USA
CHARACTERISTIC PHENOMENA AND MODEL PROBLEMS IN NONLINEAR CONTROL Roger Brockett *,1
* Harvard University, Cambridge, Massachusetts
Abstract. In this expository paper we discuss some of the main ideas in nonlinear centrol that have emerged over the past several decades. Through the use of model prcblems) we attempt to bring out certain characteristic phenomena that distinguish the operation of strongly nonlinear systems from linear onf'..s. Topicc; discussed include controllability as related to Lie brackets) linearization via nonlinear feedback) exact and approximate inversion of nonlinear systems and feedback stabilization. A novel aspect of the paper is the use of the idea of pattern generation as a unifying theme in the control of nonlinear systems. We show that with the help of pattern generation it is possible to associate an inverse system with nonlinear control systems and that the ~.ystematic use of this idea gives considerable insight into control phenomena. Keywords. Inverse SystenLS) Lie Brackets) Nonlinear Control) Pattern Generation) Stabilization
I. INTRODUCTION
Common control problems ranging from speed control of internal combustion engines to the regulation of blood pressure in mannnals are profoundly nonlinear. In many cases of interest OIle can linearize about some steady !\tate) often a periodically varying steady state) and get some insight into the dynamics, but such analyses seldom explain why the system wa..<; df'A<;igned to use nonlinear effects in the first place. For example) it has long been recognized (Brown, 1911) that some provision for pattern generation is an important part of the neural circuitry used to generate and control various animal movements such as walking) breathing, blood circula1 This work wa.s supported in part. by the Na.tiona.1 Science Foundat.ion undel· Engineering Research Center Program, NSF D CDR8803012. by the US Army Research Office under gra.nt DAAL0386-K-0171{Center for Intelllgent Control Systems), a.nd by the Office of Naval Research under Grant NOOO14-1887
3020
tion, peristalsis, etc. yet linear theory provides no insight about this mode of control. Parametric amplification and switched capacitor networks, provide examples from electrical engineering in which pattern generation plays an important role. One application of particular significance today is the use of orche;;trated periodic switching to transform direct current at one voltage to direct current at a different voltage (S"verns and Bloom, 1987). In the domain of mechanics) vibratory motors (Brockett, 1989), provide a class of highly nonlinear examples. Nonlinear situations which require pattern generation differ in a fundamental way from those associated. with the ordinary linear regulation problem. In this paper we concentrate on these diff,,·rences. Recently interest has been focused. on this area because of robotic applications involving wheeled vehicle and object manipulationj and because of spacecraf; control problems involving nonholonomie effects. SpEcific problems of interest include the control of unicycles) parallel parking) control
of autonomous vehicles, and the control of tractor-trailer systems. Space precludes us from going into details but we have provided some references (Leonard and Krishnapr""ad, 1995) and (Murray and Sastry, 1993).
their average value to be zero hut allowing some periodic pattern SUdl as u(t) = ams 21rt and v(t.) = asin(27Tt) then x does not remain at the origin but movE'S in the direction x(l) '" :: (
2. A BRIEF HISTORY There now exists a rather extensive theory of nonlinear systems based on tools developed by differential geometers. [n many cases this theory makes statements about "large signal" behavior but it is even more successful in providing local analysis. Problems that have been treated include questions of controllability, inputoutput representation, feedback stabilizatioll, exact linearization by feedback, as well as problems in optimal control. The theory is, for the most part, developed by doing a careful analysis of the system in the neighborhood of a trajectory and then, if possible, extending the analysis to the whole space. This stands in contrast with some of the most important uses of nonlinear control in that applications often depend on large signal behavior.
i: 10
Drawing on the work ofCaratheodory (1910) and Chow (1938), it was observed in (Hermann, 1963) that some known results in differential geometry had implications for certain nonlinear controllability questions. This different.ial geometric point of view began to receive more attention around 1970 with the appearance of work by engineers and mathematicians devoted to various nOIllinear problems involving modeling, controllability observability) decomposition of systems, etc. An important aspect of this work involved the development of methods by (Lobry, 1972), (Sussman and Jurdjevic, 1972) and (Krener, 1974) for treating problems with a drift term, i.e. to treat systems of the form x(t) = f(x(t))
+ Lg,(x(t))u,(t)
o which are not covered by the Caratheodory-Chow type of result. To capture the familiar linear system controllability criterion in this framework requires one to take into account structural properties of the Lie algebra as in citeBroc:72 and (Hirschorn, 1973). These early results already suggest a role for temporal patterns in designing controls for nonlinear systems. Consider x(t) = 91(X(t))u(t)
+ 92(X(t))v(t)
; x(O) = 0
If we let u(t) = vii) = 0 then after one unit of time x(t) still equals 0 but if we alter"Ul and U2) still constraining
3021
i: 10
92 (0))
One can get a good intuition for the proof by repIa(.. . ing the trigonometric function3 by the simpler functions) u(t) = a·sgn(sin27Tt) and vet) = a·sgn(cos21rt) and doing a careful (tedious) ealclllat.ion) w;;ing repeatedly the fact that to second order in I. the solution x = f(x) is x(t) = x(O)
+ f(x(O))t + ~
DJ I J(x(O))(' 2 Dx x(O)
It is worth repeating that ever, though the average value of Ut and U2 is zero) the pattern of variation about 0 gives rise to a definite displaA:~ement. The coefficient of a 2 /41r in the above approximltion of x(l} is called the Lie bracket and is written
{gr, 92] 2.1 Controllability and Lie Brackets
9, (0) -
=
Dy, aYl Dx ,(', - &;;92
This result can be recast to av·)id the specific role played by trigonometric functions. Notice that if the integral over [0, 1] of both u and v is zero, then the loc-us of points traced out for 0 .::; t :S 1
(J t
((m(t),n(t)) =
o
J t
ulu)du,
v(a)da)
0
generates a closed curve in lR2 and this closed curve defines some area. We consid€r this area to have a sign) positive if the curve in rn, n-space is traversed in the countcrclockwise direction. With this understanding we can assert that L'l. = x( 1) - x(C') is approximately (g" 92] evaluated at x = x(O) times the integral of the area encompassed by the graph ot (rn,1t). When m and n are periodic the curve in lR.2 is traced out repeatedly. If it is circular, the average of the time derivative of x is the product of the square of the amplitude times the angular frequency, but more ,~enerally it is the rate at which (1n, n) sweeps out area. Many examples of similar "area rules') appear in math€,matics and physics. (See (Brockett, 1989))
fO rm
n
)?-:
X
Y
Figure 1. Illustrating the area rule for the three dimensional system x = u; if = v;.i == xv - yu.
2.2 The Impossibility Theorems
We may characterize a second stage in the recent history of nonlinear control as being one in which certain impossibility result.'i were formulated and proven. We illustrate with two examples. The first relates to feedback linearization. It is obvious that a second order b'Ystem such as X + f(x, x) = " can be made linear by feedback. Replace u by 1£ + f(x, x), to get x = u. It may be tempting to think that by means of feedback and change of variables any controllable system can be converted to linear form. However this is not the case. [n fact, the very factors that make nonlinear <...'Ontrollability different from linear controllability stands in the way of such a linearization, The system i: = u, iJ = v) and i = xv - yu, about which we will have more to say later) provides an example of a controllable system that cannot be Iinearized by feedback (Brockett, 1978). The ~ecoIld impossibilit.y re~ult ru;serts that there exist controllable nonlinear systems having the property that no continuous time-invariant control law (linear or nonlinear) wiJ] make a given equilibrium point asymptotically stable. Such rE-suIts served to establish limits on the extent to which intuition abollt linear systems can be transferred to the nonlinear domain (Brockett, 1983).
2.:1 Patterns, Inversion and Slabilizalion
In a more recent, third stage, a number of very interesting new insights have been developed which have lead to new ways to think about the difficulties raised above as well a.c; other questions. The works of (Coron, 1992) and (Liu and Sussmann, 1991),while ostensibly addressing quite different questions, illustrates the effectivene~ of periodic patterns in solving the stabilization problem (Coron) and general tracking prohlems (Llli and Sussmann). In particular, Liu and Sussmann show that under an appropriate controllability hypothesis, one can force the solution trajectory of ",
where.., the (signed) area of the oscillation swept out per unit time determines how mw~h movement is generated in the directions spanned by Ihe Lie brackets. These results show that the character of the tracking problem for controllable systems of the given form is radically different from that of linear systems in that it is possible to find controls such that the x-trajectory approximates arbitrary smooth paths in the state space by using high frequency aspeets of the inputs to generate suitable movement in the directions that are not in the range space of the gi(X), !~act tracking, is impos.5ible unless the number of controls equals, or exceeds, the number of outputs, just as in the linear case. However, the possibility of arbitrarily g.)od approximate tracking gives rise to interesting effecl;s. [n particular it useful to consider dynamical systems that are approximate inverse.5 to such systems.
3. PHE:-IOMENA 3,1 Rectification
By rectification we mean Ollf of the various processes whereby some zero-mean input signals are transformed into some nonzero-mean signals. The word is often associated with the conversion of alternating current to direet current via diodes but thf~ idea we mean to sugge~t here is more accurately cha.racterized by the behavior of the system x(t) = U[t)
y(t) = vlt) i(t) = x(t)v(t) -- y(t)u(t) Because we will be using this example several times we give a block diagram illustrating its input-output form. u
v
x(t) = L9,(X(t))Ui(t) to stay close to a desired path by manipulating the {Ui} in such a way as to have suitable average values while further constraining them by specifying the average values of the rates at which the pairwise components of the integrals of the Ui sweep out areas. In less detailed terms, the basic action required for following a path is that of generating an oscillation about a moving (:enter, the motion of the eenter corrf'sponds to low frequency motion
3022
Figure 2: The nonholonomic i:ltegrator. Suppose that we let tt and v te sinusoids; 'U(t) = - sin t and v(t) = cos(t + 1». For the sake of simplicity, let x(O) = 1, y(O) = sin 1>, z(O) = (I so that x(t) = cost y(t) = sin(/. + 1»
,
z(/) =
J
sin(t)sin(t+,p) + costCA>S(t+,p)dt
o Integrating this last expression we see that
z(l) = tcos,p These explicit calculations exemplify the general claims made above in that z a.cquirf'B a definite sense of direction depending on the phase relationship between 'U and v, even though they are both signals whose average value is zero.
This model is perhaps the most elementary one that captures the essence of the use of patterns in pumps, reciprocating engines, DC-toDe power conversion, etc. Its basic features remain unaltered under a wide variety of modifications. We will discu..").~ a "low pass" version in section 3.3.
puts than they have inputs. By its very nature) the composition of a multiplexer and a dcmultiplexer must be a good approximation of the identity, as long as the input signals are confined to a finite frequency range such as 0 :S; W :S; wc. In this senSt~, then) we may think of the nonholonomic integrator as a demultiplexer because it takes two
Der:t=
Multiplexer
Ple,~
Figure 3. Multiplexing/Demdtiplexing as a many-toone and one-to- many transfolmation.
3.3 Approximate Inversion 3.2 The Multiplexin!J/Dernultiplexing Point of View Many conununication systems send separate signals over the same channel using a process called multiplexing. The exact mechani1::lIn used varies from application to application but one conceptual scheme that serves to illustrate is based on dividing the signal space into frequency ranges and allocating different frequency bands for different signals. Of course this only works if the signals that are to he transmitted can be confined to a subset of the frequency axis. In the ca.~ of ordinary telephon~ communication one may consider the input signals {1tl) U2) ... , un} to lie in the frequency range -3000 :S; f ~ 3000 Hz and then multiplex by multiplying u, by cos omegai,t; IWi -Wj I ~ 6000. The signal on the channel is then of the form x(t) =
L"
tti C'OSWi t
I
In order to recover the original n signals from x onc then needs to de-multiplex. In view of the fact that for T larger than several periods we have
J T
T1
COS Wit
cos Wjtdt
:=::::::
;J
6··
o
we see that we can recover a good approximation of 'Ut from x using ui(l) '"
~
,+T
J,
We now illustrate with an example a technique which makes it possible to construct low frequency approximate inverses for large classes of nonlinear systems even though they have more outputs than inputs. Consider a low-pasf~ version of the systerr: considered above.
;;(1.) = -x(t) y(t) = -y(t) i(t) = -z(t)
+ x(t)v(t)
Multiplexers are systems with more inputs than they have outputs, dernultiplexers are systems with more out-
3023
- y(t)u(t)
suppose we have low frequency inputs a, b) c and we want to find a mapping : (a, b, c) ~ (u, v) such that (u, v) causes the system to track tho~se functions in the sense that the vector (x - a, y - b z - c) is small. Wc will show that this is p=iblc and illustrate the use of such approximate inVerSf'B. The plan for constructing the approximate inverse is to think of tt and v as being the sum of two parts. One part, tj, part that uses the lower frequency portion of the powel spectrum will be approximately tt = a - a, v = b - b. The second part will o{',cupy a frequency band well above that needed for a and b and will be shaped by a combination of amplitude and phase modulation in accordance with the form of c - (ab - ba). We may think of this last expression as being c "corrected" by subtn.cting off the influence of the area defined by the desired path of x and y. Of t."Qurse we can add high f·equency terms to 'U and diffe:~ent types of modulation. Because the dynamics associated with x and y wiJI naturally suppress higher frequency effects more than lower frequency effects, it is useful to think in terms of the ratio of the modu1ation freqUE~ncy and the reciprocal of the time constant associated with the x - y dynamics.
v in different ways using
(c'Oswit)x(t)dt
+ u(t) + v(t)
Modulation becomes much less effective if the frequency falls below Wo = liT. Thinking in tenns of the types of applications mentioned in the introduction, it is clear that one can achieve more movement in the z direction if the frequency is increased along with the magnitude of the modulation term. (Fa...;;ter walking is achieved by lengthing the stride and increasing the steps per unit time.) Specifically, consider the parametrizatioIl of given by ,,(t)
'U
and v
= "o(t) + msin(wt + 1>/2)
vet) = vo(t) + msin(wt -1>/2) Our choice for the amplitude m and frequency W depend on two scaling constants, IJ. and Wo. The first is defined by the scaling required to normalize z so that the magnitude of the desired z response is approximately the same as that of the x -y response. The second can be thought of as the reciprocal of the time constant of the x - y system. In terms of these constants, and d = c - (ab - ba.) make the choices m=,fd;.,. W= Wo (l,d/wo)2
,
Figure 5. Illustrating the ability of the nonholonomic integrator to track a square wave with z while u and 'U track a sine wave and 0, respectively.
3.4 Feedback Linearization
VI -
Take
u
V
bl-~--------~~-
Figure 4. Block diagram of the approximate inverse.
If the frequency content of a and b lie in a range below Wo, then we may approximate the response of x and y by the response of x(t) = -x(t) + "o{t) and yet) = -yet) + vo{t). The response of z on the other hand, is approximately
,
J , J
z{t) =
In figure 5 we show the result..:; of a simulation orresponding to the choices I' = 1, Wo = 30, a = sin .3t, b = 0) c = 3sgn(sin .2t)
The role of feedback in making systems perform better is well known. In some cases "perform better') means perform fLC',cording to a linear law and so it is natural to ask when there exists a feedback eontrol law that renders a given nonlinear system linear. Without going into detail here, we recall that the nonhoLonomic integrator cannot be lillcarized using feedback. However, it is clear that the approximate inverse just defined does line-..arize the low frequency behavior of the system. Moreover, because the cutoff frequency Wo which appears in the construction of the approximate inverse is arbitrary, one can choose the domain of validity of this approximation t.o include frequencies that are as high as one likes. Thus even though the nOlllinearity is so profound that no amount of feedback can remove it without destroying controllability, if we allow ourselves the use of temporal pattern~, the problems can be circumvented. This idea can be formulated mathematically using the idea of an approximate inverse.
m(c)sin(wO' +1>(c))do-m(c)sin(wt - 1>(c))dt
o
+
3.5 Feedback Stabilization
m{c)sin(wO' -1>(c))dO'm,(<:)sin(wt+ 1>{<:))dt
o Of course these integrals cannot be evaluated exactly for arbitrary m and W but from the definition, w > Wo and in a range where m is small (',ompared with w, one can get a good approximation by treating m and w as if they are constant. With this approximation
z(l) '" m 2 /w = c
3024
In the case of linear systems oontrollability implies the existence of a time- invariant linear feedback c..ont.rol law making the closed~loop s:lstem asymptotically sta~ hie. As noted above, in this regard nonlinear systems behave rather differently, [n ,:Coron, 1992) it is shown that time-varying control law~; can be effective in stabilizing where time-invariant ones fail and a general theorem on the stabilizability of driftless nonHnear systems
is establU:!hed . An intuitive explaination can be gained through the following line of reasoning. It is well known and easily verified that if unity negative feedback is applied aroWld an integrator, Le. if we have 8. vector system x(/) = ,,(t) and replace u by -x, then the res1llting system x(t) = -x(t) has sol1ltions which decay to zero. Wc may paraphrase this as "negative feedback stabilizes an integrator". This idea is "robust" in many senses. The subject of frequency domain stability criteria is devoted to showing that certain types of nonlinearities do not destabilize linear systems if the nOlllinearities are not too large. The llonholonomic integrator, when precompensated by an approximate inverse, is an approximat.ion to the identity operal.or. When the system is preceded by an integration in each t-flannel we h 6~ an overaU system that is an approximaten-dimensional iut.egrator . If we apply negative feedback we should get an 3.l::Iymptotically stable system. This suggests the followi ng general idea. Given a controllable system to be sta.bilized, find an approximate, inverse, precede it with a.ll integrator, and apply unity nega.tive feedback. If we apply this thinking to the nonholonomic integrator the c1osed- loop equations are a.
" Figure 6. Simulation of the stabilized nonholonomic integrator. Here x(O) = 3, y(O) = -2, z(O) = 4.
4. R.EFERENCES
Avis H. Cohen, Serge Rossignol, Sten Grillner (1988) . Neura.l Control 0/ Rhythmic Movements in Vertebrates. Wiley. Ne w York. Brockett, R.. (1972). System theory on group manifolds and coset spaces. SIAM Journal Contr. pp. 265284, Brockett, R. W . (1978). Feedback invariant. for nonlinear systems. Proceedings of the /978 !FAC Congn:ss.
3025
Broc.kett, R. W. ( 1981). Conrol theory and singular riemannian geometry. [n New Directions in Applied Mathematics (P. HiI ton and G. Young, Eds.). pp. 11-27. Springer- Verhg. New York. Brockett, R. W. (1983). Asymptotic Stability and Feedback Stabilization. Burkhauser. Boston. Brockett, R. W. (1989). On th,,,ectification of vibratory motion. Sen.~ors and Actv.ators 20, 91- 96. Brown, Craham (1911). Intri nsic factors in the aet of progression of the mauuual. Proc. Royal Soc. L01Ir don,. Coron, J .-M. ( 1992). Global "'ymptotic stabilization for controllable linear systems without drift. Systems and Signals 5 , 295-312. Hermann, R. (1963). On the a<.-cessibility problem in cont rol theory. In : Nonlinear Differential Equations and Nonlinear Mechanics (J .P. LaS.lIe and S. Lefschetz, Eds.). Academic Pr",". New York. Hirschorn, R. (1973). Topological semigroups, sets of generations and controllability. Duke Math Journal pp. 937-947. Krener, A. (H174). Generaliza.tion of chow's theorem and t.he bang-bang theorem t:.l nonlinear control problems. SIAM Journal Con':T. pp. 43- 52. Leonard, :-I. and P. S. Kri shnaprasad (1995). Motion control of drift-free, left-invariant systems on lie groups. IEEE Trans . on Automatic Control pp. 1539-1554 . Liu, Wenshang and H. J . S,,;smann (1991) . Limits of highly oscillatory controls a.nd the approximation of general paths by admi..,ible t rajectories. Proceedings of the 90th IEEE CVC pp. 437-442. Lobry, C. ( 1972). Quelques aspects qualitaifs de la theorie'5 de la cotrunA.nde. L 'Ur!.ivcrsite Scientifique et Medicale. de Grcnoble. Murray, R. and S. Sastr.y (1993). Nouholollomic motion planning - steering using sinusoids. IEEE Transactionson Automatic Contml pp. 700-716. Pearsoll, K. G. (1985). Internetlron~ and locomotion. In: The MOtOT System in Neurobiology (S. Wise E. Evarts and D. Bousfield, Eds.). Elsevier Biomedjeal Press. Amsterdam. ~everns, R. and E. Bloom (1987). OCto DC Converters. Van Nostrand. New York Shapere, A. and F. Wilczek (I ~89) . Geometric phases in physias. World Sci.entific. Sussm,,", H. and V . .Iurdjevic (1972). Controllability of nonlinear systems. Journal oJ Differential Equahow. pp. 9fr-116. von Euler, Kurt (1985). P,.ttern generation during breathing. In: The Moto:- System in NeuTobiology (S. Wise E. Evarts and D . Bousfield, Eds.). ElseviE:'r Biomedical Press. Amsterdam.
Franci~co,
4c-045
USA
CHARACTERISTIC PHENOMENA AND MODEL PROBLEMS IN NONLINEAR CONTROL Roger Brockett *,1
* Harvard University, Cambridge, Massachusetts
Abstract. In this expository paper we discuss some of the main ideas in nonlinear centrol that have emerged over the past several decades. Through the use of model prcblems) we attempt to bring out certain characteristic phenomena that distinguish the operation of strongly nonlinear systems from linear onf'..s. Topicc; discussed include controllability as related to Lie brackets) linearization via nonlinear feedback) exact and approximate inversion of nonlinear systems and feedback stabilization. A novel aspect of the paper is the use of the idea of pattern generation as a unifying theme in the control of nonlinear systems. We show that with the help of pattern generation it is possible to associate an inverse system with nonlinear control systems and that the ~.ystematic use of this idea gives considerable insight into control phenomena. Keywords. Inverse SystenLS) Lie Brackets) Nonlinear Control) Pattern Generation) Stabilization
I. INTRODUCTION
Common control problems ranging from speed control of internal combustion engines to the regulation of blood pressure in mannnals are profoundly nonlinear. In many cases of interest OIle can linearize about some steady !\tate) often a periodically varying steady state) and get some insight into the dynamics, but such analyses seldom explain why the system wa..<; df'A<;igned to use nonlinear effects in the first place. For example) it has long been recognized (Brown, 1911) that some provision for pattern generation is an important part of the neural circuitry used to generate and control various animal movements such as walking) breathing, blood circula1 This work wa.s supported in part. by the Na.tiona.1 Science Foundat.ion undel· Engineering Research Center Program, NSF D CDR8803012. by the US Army Research Office under gra.nt DAAL0386-K-0171{Center for Intelllgent Control Systems), a.nd by the Office of Naval Research under Grant NOOO14-1887
3020
tion, peristalsis, etc. yet linear theory provides no insight about this mode of control. Parametric amplification and switched capacitor networks, provide examples from electrical engineering in which pattern generation plays an important role. One application of particular significance today is the use of orche;;trated periodic switching to transform direct current at one voltage to direct current at a different voltage (S"verns and Bloom, 1987). In the domain of mechanics) vibratory motors (Brockett, 1989), provide a class of highly nonlinear examples. Nonlinear situations which require pattern generation differ in a fundamental way from those associated. with the ordinary linear regulation problem. In this paper we concentrate on these diff,,·rences. Recently interest has been focused. on this area because of robotic applications involving wheeled vehicle and object manipulationj and because of spacecraf; control problems involving nonholonomie effects. SpEcific problems of interest include the control of unicycles) parallel parking) control
of autonomous vehicles, and the control of tractor-trailer systems. Space precludes us from going into details but we have provided some references (Leonard and Krishnapr""ad, 1995) and (Murray and Sastry, 1993).
their average value to be zero hut allowing some periodic pattern SUdl as u(t) = ams 21rt and v(t.) = asin(27Tt) then x does not remain at the origin but movE'S in the direction x(l) '" :: (
2. A BRIEF HISTORY There now exists a rather extensive theory of nonlinear systems based on tools developed by differential geometers. [n many cases this theory makes statements about "large signal" behavior but it is even more successful in providing local analysis. Problems that have been treated include questions of controllability, inputoutput representation, feedback stabilizatioll, exact linearization by feedback, as well as problems in optimal control. The theory is, for the most part, developed by doing a careful analysis of the system in the neighborhood of a trajectory and then, if possible, extending the analysis to the whole space. This stands in contrast with some of the most important uses of nonlinear control in that applications often depend on large signal behavior.
i: 10
Drawing on the work ofCaratheodory (1910) and Chow (1938), it was observed in (Hermann, 1963) that some known results in differential geometry had implications for certain nonlinear controllability questions. This different.ial geometric point of view began to receive more attention around 1970 with the appearance of work by engineers and mathematicians devoted to various nOIllinear problems involving modeling, controllability observability) decomposition of systems, etc. An important aspect of this work involved the development of methods by (Lobry, 1972), (Sussman and Jurdjevic, 1972) and (Krener, 1974) for treating problems with a drift term, i.e. to treat systems of the form x(t) = f(x(t))
+ Lg,(x(t))u,(t)
o which are not covered by the Caratheodory-Chow type of result. To capture the familiar linear system controllability criterion in this framework requires one to take into account structural properties of the Lie algebra as in citeBroc:72 and (Hirschorn, 1973). These early results already suggest a role for temporal patterns in designing controls for nonlinear systems. Consider x(t) = 91(X(t))u(t)
+ 92(X(t))v(t)
; x(O) = 0
If we let u(t) = vii) = 0 then after one unit of time x(t) still equals 0 but if we alter"Ul and U2) still constraining
3021
i: 10
92 (0))
One can get a good intuition for the proof by repIa(.. . ing the trigonometric function3 by the simpler functions) u(t) = a·sgn(sin27Tt) and vet) = a·sgn(cos21rt) and doing a careful (tedious) ealclllat.ion) w;;ing repeatedly the fact that to second order in I. the solution x = f(x) is x(t) = x(O)
+ f(x(O))t + ~
DJ I J(x(O))(' 2 Dx x(O)
It is worth repeating that ever, though the average value of Ut and U2 is zero) the pattern of variation about 0 gives rise to a definite displaA:~ement. The coefficient of a 2 /41r in the above approximltion of x(l} is called the Lie bracket and is written
{gr, 92] 2.1 Controllability and Lie Brackets
9, (0) -
=
Dy, aYl Dx ,(', - &;;92
This result can be recast to av·)id the specific role played by trigonometric functions. Notice that if the integral over [0, 1] of both u and v is zero, then the loc-us of points traced out for 0 .::; t :S 1
(J t
((m(t),n(t)) =
o
J t
ulu)du,
v(a)da)
0
generates a closed curve in lR2 and this closed curve defines some area. We consid€r this area to have a sign) positive if the curve in rn, n-space is traversed in the countcrclockwise direction. With this understanding we can assert that L'l. = x( 1) - x(C') is approximately (g" 92] evaluated at x = x(O) times the integral of the area encompassed by the graph ot (rn,1t). When m and n are periodic the curve in lR.2 is traced out repeatedly. If it is circular, the average of the time derivative of x is the product of the square of the amplitude times the angular frequency, but more ,~enerally it is the rate at which (1n, n) sweeps out area. Many examples of similar "area rules') appear in math€,matics and physics. (See (Brockett, 1989))
fO rm
n
)?-:
X
Y
Figure 1. Illustrating the area rule for the three dimensional system x = u; if = v;.i == xv - yu.
2.2 The Impossibility Theorems
We may characterize a second stage in the recent history of nonlinear control as being one in which certain impossibility result.'i were formulated and proven. We illustrate with two examples. The first relates to feedback linearization. It is obvious that a second order b'Ystem such as X + f(x, x) = " can be made linear by feedback. Replace u by 1£ + f(x, x), to get x = u. It may be tempting to think that by means of feedback and change of variables any controllable system can be converted to linear form. However this is not the case. [n fact, the very factors that make nonlinear <...'Ontrollability different from linear controllability stands in the way of such a linearization, The system i: = u, iJ = v) and i = xv - yu, about which we will have more to say later) provides an example of a controllable system that cannot be Iinearized by feedback (Brockett, 1978). The ~ecoIld impossibilit.y re~ult ru;serts that there exist controllable nonlinear systems having the property that no continuous time-invariant control law (linear or nonlinear) wiJ] make a given equilibrium point asymptotically stable. Such rE-suIts served to establish limits on the extent to which intuition abollt linear systems can be transferred to the nonlinear domain (Brockett, 1983).
2.:1 Patterns, Inversion and Slabilizalion
In a more recent, third stage, a number of very interesting new insights have been developed which have lead to new ways to think about the difficulties raised above as well a.c; other questions. The works of (Coron, 1992) and (Liu and Sussmann, 1991),while ostensibly addressing quite different questions, illustrates the effectivene~ of periodic patterns in solving the stabilization problem (Coron) and general tracking prohlems (Llli and Sussmann). In particular, Liu and Sussmann show that under an appropriate controllability hypothesis, one can force the solution trajectory of ",
where.., the (signed) area of the oscillation swept out per unit time determines how mw~h movement is generated in the directions spanned by Ihe Lie brackets. These results show that the character of the tracking problem for controllable systems of the given form is radically different from that of linear systems in that it is possible to find controls such that the x-trajectory approximates arbitrary smooth paths in the state space by using high frequency aspeets of the inputs to generate suitable movement in the directions that are not in the range space of the gi(X), !~act tracking, is impos.5ible unless the number of controls equals, or exceeds, the number of outputs, just as in the linear case. However, the possibility of arbitrarily g.)od approximate tracking gives rise to interesting effecl;s. [n particular it useful to consider dynamical systems that are approximate inverse.5 to such systems.
3. PHE:-IOMENA 3,1 Rectification
By rectification we mean Ollf of the various processes whereby some zero-mean input signals are transformed into some nonzero-mean signals. The word is often associated with the conversion of alternating current to direet current via diodes but thf~ idea we mean to sugge~t here is more accurately cha.racterized by the behavior of the system x(t) = U[t)
y(t) = vlt) i(t) = x(t)v(t) -- y(t)u(t) Because we will be using this example several times we give a block diagram illustrating its input-output form. u
v
x(t) = L9,(X(t))Ui(t) to stay close to a desired path by manipulating the {Ui} in such a way as to have suitable average values while further constraining them by specifying the average values of the rates at which the pairwise components of the integrals of the Ui sweep out areas. In less detailed terms, the basic action required for following a path is that of generating an oscillation about a moving (:enter, the motion of the eenter corrf'sponds to low frequency motion
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Figure 2: The nonholonomic i:ltegrator. Suppose that we let tt and v te sinusoids; 'U(t) = - sin t and v(t) = cos(t + 1». For the sake of simplicity, let x(O) = 1, y(O) = sin 1>, z(O) = (I so that x(t) = cost y(t) = sin(/. + 1»
,
z(/) =
J
sin(t)sin(t+,p) + costCA>S(t+,p)dt
o Integrating this last expression we see that
z(l) = tcos,p These explicit calculations exemplify the general claims made above in that z a.cquirf'B a definite sense of direction depending on the phase relationship between 'U and v, even though they are both signals whose average value is zero.
This model is perhaps the most elementary one that captures the essence of the use of patterns in pumps, reciprocating engines, DC-toDe power conversion, etc. Its basic features remain unaltered under a wide variety of modifications. We will discu..").~ a "low pass" version in section 3.3.
puts than they have inputs. By its very nature) the composition of a multiplexer and a dcmultiplexer must be a good approximation of the identity, as long as the input signals are confined to a finite frequency range such as 0 :S; W :S; wc. In this senSt~, then) we may think of the nonholonomic integrator as a demultiplexer because it takes two
Der:t=
Multiplexer
Ple,~
Figure 3. Multiplexing/Demdtiplexing as a many-toone and one-to- many transfolmation.
3.3 Approximate Inversion 3.2 The Multiplexin!J/Dernultiplexing Point of View Many conununication systems send separate signals over the same channel using a process called multiplexing. The exact mechani1::lIn used varies from application to application but one conceptual scheme that serves to illustrate is based on dividing the signal space into frequency ranges and allocating different frequency bands for different signals. Of course this only works if the signals that are to he transmitted can be confined to a subset of the frequency axis. In the ca.~ of ordinary telephon~ communication one may consider the input signals {1tl) U2) ... , un} to lie in the frequency range -3000 :S; f ~ 3000 Hz and then multiplex by multiplying u, by cos omegai,t; IWi -Wj I ~ 6000. The signal on the channel is then of the form x(t) =
L"
tti C'OSWi t
I
In order to recover the original n signals from x onc then needs to de-multiplex. In view of the fact that for T larger than several periods we have
J T
T1
COS Wit
cos Wjtdt
:=::::::
;J
6··
o
we see that we can recover a good approximation of 'Ut from x using ui(l) '"
~
,+T
J,
We now illustrate with an example a technique which makes it possible to construct low frequency approximate inverses for large classes of nonlinear systems even though they have more outputs than inputs. Consider a low-pasf~ version of the systerr: considered above.
;;(1.) = -x(t) y(t) = -y(t) i(t) = -z(t)
+ x(t)v(t)
Multiplexers are systems with more inputs than they have outputs, dernultiplexers are systems with more out-
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- y(t)u(t)
suppose we have low frequency inputs a, b) c and we want to find a mapping : (a, b, c) ~ (u, v) such that (u, v) causes the system to track tho~se functions in the sense that the vector (x - a, y - b z - c) is small. Wc will show that this is p=iblc and illustrate the use of such approximate inVerSf'B. The plan for constructing the approximate inverse is to think of tt and v as being the sum of two parts. One part, tj, part that uses the lower frequency portion of the powel spectrum will be approximately tt = a - a, v = b - b. The second part will o{',cupy a frequency band well above that needed for a and b and will be shaped by a combination of amplitude and phase modulation in accordance with the form of c - (ab - ba). We may think of this last expression as being c "corrected" by subtn.cting off the influence of the area defined by the desired path of x and y. Of t."Qurse we can add high f·equency terms to 'U and diffe:~ent types of modulation. Because the dynamics associated with x and y wiJI naturally suppress higher frequency effects more than lower frequency effects, it is useful to think in terms of the ratio of the modu1ation freqUE~ncy and the reciprocal of the time constant associated with the x - y dynamics.
v in different ways using
(c'Oswit)x(t)dt
+ u(t) + v(t)
Modulation becomes much less effective if the frequency falls below Wo = liT. Thinking in tenns of the types of applications mentioned in the introduction, it is clear that one can achieve more movement in the z direction if the frequency is increased along with the magnitude of the modulation term. (Fa...;;ter walking is achieved by lengthing the stride and increasing the steps per unit time.) Specifically, consider the parametrizatioIl of given by ,,(t)
'U
and v
= "o(t) + msin(wt + 1>/2)
vet) = vo(t) + msin(wt -1>/2) Our choice for the amplitude m and frequency W depend on two scaling constants, IJ. and Wo. The first is defined by the scaling required to normalize z so that the magnitude of the desired z response is approximately the same as that of the x -y response. The second can be thought of as the reciprocal of the time constant of the x - y system. In terms of these constants, and d = c - (ab - ba.) make the choices m=,fd;.,. W= Wo (l,d/wo)2
,
Figure 5. Illustrating the ability of the nonholonomic integrator to track a square wave with z while u and 'U track a sine wave and 0, respectively.
3.4 Feedback Linearization
VI -
Take
u
V
bl-~--------~~-
Figure 4. Block diagram of the approximate inverse.
If the frequency content of a and b lie in a range below Wo, then we may approximate the response of x and y by the response of x(t) = -x(t) + "o{t) and yet) = -yet) + vo{t). The response of z on the other hand, is approximately
,
J , J
z{t) =
In figure 5 we show the result..:; of a simulation orresponding to the choices I' = 1, Wo = 30, a = sin .3t, b = 0) c = 3sgn(sin .2t)
The role of feedback in making systems perform better is well known. In some cases "perform better') means perform fLC',cording to a linear law and so it is natural to ask when there exists a feedback eontrol law that renders a given nonlinear system linear. Without going into detail here, we recall that the nonhoLonomic integrator cannot be lillcarized using feedback. However, it is clear that the approximate inverse just defined does line-..arize the low frequency behavior of the system. Moreover, because the cutoff frequency Wo which appears in the construction of the approximate inverse is arbitrary, one can choose the domain of validity of this approximation t.o include frequencies that are as high as one likes. Thus even though the nOlllinearity is so profound that no amount of feedback can remove it without destroying controllability, if we allow ourselves the use of temporal pattern~, the problems can be circumvented. This idea can be formulated mathematically using the idea of an approximate inverse.
m(c)sin(wO' +1>(c))do-m(c)sin(wt - 1>(c))dt
o
+
3.5 Feedback Stabilization
m{c)sin(wO' -1>(c))dO'm,(<:)sin(wt+ 1>{<:))dt
o Of course these integrals cannot be evaluated exactly for arbitrary m and W but from the definition, w > Wo and in a range where m is small (',ompared with w, one can get a good approximation by treating m and w as if they are constant. With this approximation
z(l) '" m 2 /w = c
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In the case of linear systems oontrollability implies the existence of a time- invariant linear feedback c..ont.rol law making the closed~loop s:lstem asymptotically sta~ hie. As noted above, in this regard nonlinear systems behave rather differently, [n ,:Coron, 1992) it is shown that time-varying control law~; can be effective in stabilizing where time-invariant ones fail and a general theorem on the stabilizability of driftless nonHnear systems
is establU:!hed . An intuitive explaination can be gained through the following line of reasoning. It is well known and easily verified that if unity negative feedback is applied aroWld an integrator, Le. if we have 8. vector system x(/) = ,,(t) and replace u by -x, then the res1llting system x(t) = -x(t) has sol1ltions which decay to zero. Wc may paraphrase this as "negative feedback stabilizes an integrator". This idea is "robust" in many senses. The subject of frequency domain stability criteria is devoted to showing that certain types of nonlinearities do not destabilize linear systems if the nOlllinearities are not too large. The llonholonomic integrator, when precompensated by an approximate inverse, is an approximat.ion to the identity operal.or. When the system is preceded by an integration in each t-flannel we h 6~ an overaU system that is an approximaten-dimensional iut.egrator . If we apply negative feedback we should get an 3.l::Iymptotically stable system. This suggests the followi ng general idea. Given a controllable system to be sta.bilized, find an approximate, inverse, precede it with a.ll integrator, and apply unity nega.tive feedback. If we apply this thinking to the nonholonomic integrator the c1osed- loop equations are a.
" Figure 6. Simulation of the stabilized nonholonomic integrator. Here x(O) = 3, y(O) = -2, z(O) = 4.
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Broc.kett, R. W. ( 1981). Conrol theory and singular riemannian geometry. [n New Directions in Applied Mathematics (P. HiI ton and G. Young, Eds.). pp. 11-27. Springer- Verhg. New York. Brockett, R. W. (1983). Asymptotic Stability and Feedback Stabilization. Burkhauser. Boston. Brockett, R. W. (1989). On th,,,ectification of vibratory motion. Sen.~ors and Actv.ators 20, 91- 96. Brown, Craham (1911). Intri nsic factors in the aet of progression of the mauuual. Proc. Royal Soc. L01Ir don,. Coron, J .-M. ( 1992). Global "'ymptotic stabilization for controllable linear systems without drift. Systems and Signals 5 , 295-312. Hermann, R. (1963). On the a<.-cessibility problem in cont rol theory. In : Nonlinear Differential Equations and Nonlinear Mechanics (J .P. LaS.lIe and S. Lefschetz, Eds.). Academic Pr",". New York. Hirschorn, R. (1973). Topological semigroups, sets of generations and controllability. Duke Math Journal pp. 937-947. Krener, A. (H174). Generaliza.tion of chow's theorem and t.he bang-bang theorem t:.l nonlinear control problems. SIAM Journal Con':T. pp. 43- 52. Leonard, :-I. and P. S. Kri shnaprasad (1995). Motion control of drift-free, left-invariant systems on lie groups. IEEE Trans . on Automatic Control pp. 1539-1554 . Liu, Wenshang and H. J . S,,;smann (1991) . Limits of highly oscillatory controls a.nd the approximation of general paths by admi..,ible t rajectories. Proceedings of the 90th IEEE CVC pp. 437-442. Lobry, C. ( 1972). Quelques aspects qualitaifs de la theorie'5 de la cotrunA.nde. L 'Ur!.ivcrsite Scientifique et Medicale. de Grcnoble. Murray, R. and S. Sastr.y (1993). Nouholollomic motion planning - steering using sinusoids. IEEE Transactionson Automatic Contml pp. 700-716. Pearsoll, K. G. (1985). Internetlron~ and locomotion. In: The MOtOT System in Neurobiology (S. Wise E. Evarts and D. Bousfield, Eds.). Elsevier Biomedjeal Press. Amsterdam. ~everns, R. and E. Bloom (1987). OCto DC Converters. Van Nostrand. New York Shapere, A. and F. Wilczek (I ~89) . Geometric phases in physias. World Sci.entific. Sussm,,", H. and V . .Iurdjevic (1972). Controllability of nonlinear systems. Journal oJ Differential Equahow. pp. 9fr-116. von Euler, Kurt (1985). P,.ttern generation during breathing. In: The Moto:- System in NeuTobiology (S. Wise E. Evarts and D . Bousfield, Eds.). ElseviE:'r Biomedical Press. Amsterdam.