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Dynamics and stability state-space of a controlled inverted pendulum
Pergamon
hzl J. .f%;on-LmecrMcchanrc.<, Vol. 31, No. 2. pp. 215.227, 1996 Copyright ST; 1996 Elsrvier Saence Ltd Prmlrd m Great Bntam. All nghts mewed 0020-7462:96 515.00 + 0.00
0020-7462(95)00055-O
DYNAMICS AND STABILITY STATE-SPACE OF A CONTROLLED INVERTED PENDULUM M. G. Henders*
and A. C. Soudack”
* MGH Engineering ’ Department
Services, Box 1599. Cl 83, Medicine Hat, Alberta. Canada TIA 7Y5: of Electrical Engineering, University of British Columbia, Vancouver, British Columbia, Canada V6T 124 (Received
17 Auyust 1994)
Abstract-A study is conducted into the dynamics of an inverted pendulum under the regulation of a linear-feedback control system. Analysis of the resulting dynamic equations indicates the presence of stable regions within the four-dimensional state space of the system, which are confirmed through simulation. System trajectories are examined in this state space, and complex, apparently chaotic behaviour is observed outside the stable regions of the state space. An algorithm is developed to map the boundaries of the stable state-space hyper-volume; this algorithm is broadly applicable to the study of other systems. Results are presented, generated by executing the algorithm on the inverted pendulum dynamics. These comprise a series of three-dimensional plots, outlining the four-dimensional stability volume. The boundaries of this volume are seen to exhibit a jagged, possibly fractal, morphology. Keywords: pendulum
dynamics,
stability
1.
of motion,
space state methods
INTRODUCTION
The “inverted pendulum” (Fig. 1) is commonly used as a tutorial problem in control theory [l, 21, where its inherently unstable, highly non-linear dynamics provide interest for the student. A number of researchers have also studied various aspects of the inverted pendulum, in a broad spectrum of settings [3-61. It is perhaps surprising that little work seems to have been done with the most basic implementation of control of the inverted pendulum, that of a simple linear controller designed through pole-placement. This implementation is considered in the textbooks, but discussion of the system dynamics seems to be limited to the region where linearization is applicable. There is generally no mention of the system behaviour outside this region, except perhaps to suggest that “the pendulum falls over”. In earlier work [7], we examined a limited case of this system using phase-plane analysis, and demonstrated the existence of some interesting global dynamics in the system. In this paper, results are presented from a more comprehensive treatment of this problem, showing that, as far as the system dynamics are concerned, its apparent simplicity is quite deceptive. It is demonstrated that the pole-placement controller creates an infinite number of attractor points in the state space of the system, with each one apparently having a finite basin of attraction. Convoluted, possibly chaotic state trajectories are seen outside these attracting basins. The intent of the controller design is to create an attractor at the origin of the state space; the full extent of its basin of attraction is mapped algorithmically and the results are presented. This origin-bound attractor basin exhibits a jagged, possibly fractal, morphology. This paper comprises six major sections, including the Introduction. In Section 2, the background assumptions of the study are outlined, and an analytical treatment of the system is presented in Section 3. Section 4 discusses simulation results, with discussion of the basin-mapping algorithm and a presentation of its results found in Section 5. Finally, Section 6 concludes with a summary of-and commentary on--the results presented. 215
M. G. Henders
216
and A. C. Soudack
Mass 5 M
Fig. 1. Inverted
pendulum:
2. SYSTEM
general
arrangement
DETAILS
For the purposes of the work presented here, the inverted pendulum system is completely idealized. With reference to Fig. 1, the pendulum is assumed to be capable of unlimited rotation (no limits on 101) and the lateral excursion of the cart (x) is also unlimited. The system is assumed to be dissipationless, and there are no restrictions on the applied control force F. One further important detail is that the pendulum angle, 8, is considered as an absolute, rather than a modulo-360” number. The nature of the applied control force is significantly affected by the treatment of this item. Because much of the work described here is based on simulation, it is necessary to define a “reference parameter set”. In all simulations, the cart mass, M, is 1.0 kg, the pendulum mass, m, is 0.1 kg, the pendulum length, r, is 1.0 m, and gravitational acceleration, g, is 9.80665 m.sP2. The numerical work was performed using a specially developed code running on a personal computer, with a fourth-order Runge-Kutta integrator [8] forming the core of the simulation routines.
3. ANALYTICAL
TREATMENT
The analysis in our earlier work [7] used the system parameters (m, M, r, g, and F) directly. The treatment presented here uses normalized values, with ~1= m/M, y = r/g, and P E F/M, for somewhat better clarity. Using the convention that 4 = dq/dt and 4 = d2qldt2, the open-loop dynamic equations of the inverted pendulum system’ are stated here, without derivation2 in the normalized form: j;_=
prg2sin0
+ P - PgsinQcosO 1 + ,u sin2
e
/j=
g(1 + p)sin 0 - p~r8’ sin H cos Q - E cos 0 r(1 + pusin
e) (1)
3.1. Controller design A linear-feedback controller is designed for this system, using standard pole-placement techniques, and all of the work presented here assumes a single, multiplicity-four eigenvalue at (- 1 + j0). (See [7] for notes regarding behaviour with other pole locations.) For the 8 81’ = 0, resulting in: purposes of this design, the system is linearized at [x i
(*)
1 These equations differ significantly from those presented in [l], but follow 121; we suspect flaws in [I]. ’ Reference [9] contains comprehensive derivations for all of the analysis presented in this paper.
Controlled inverted pendulum state-space
217
This equation has the general form i = [A + BG]z, where Gz = E. Solving for G, the feedback coefficients are obtained as: G = [A
B
C
II] = [y
4~
r[6 + Y + (1 + CL)/?] 4r(l + Y)].
(3)
On insertion of these coefficients, the following equations are obtained for the closed-loop system dynamics: jI_=
p&sin%
+ y(x + 4f) + r[6 + y + (1 + p)/y]% + 4r(l + y)%- ~gsin%cos% 1 + p sin’ 6
(4)
8’= ((1 + p)/r))sin 6 - ~1%’sin 6 cos 6 - {(x + 4i)/g + [6 + y + ((1 + p)/r)] 6 + 4(1 + $8) cos 6 1 + p sin’ 8 (5) Examination of these equations shows that the system state variables are coupled, requiring that a four-dimensional state space be used for full understanding of the inverted pendulum dynamics. 3.2. Singularity analysis The closed-loop equations, (4) and (5), are subjected to a local linearization in the vicinity of the system’s singular points to provide insight into the “in-the-small” dynamics of the system. The singular points of the system, at which all derivatives are zero, must also be determined. To find these points, it suffices to set the i and 8 equations, above, equal to zero, with i and 8 also set equal to zero. The following set of simultaneous non-linear equations is solved for singularity locations x, and 6,: From 2:
yx,+r[6+y+(l
From 8:
C(1 +
+~)/~]%p-~gsin%,cos%,=O
PYYIsin6, - {
(6) (7)
Multiplying equation (6) by cos 6, and equation (7) by r, and eliminating terms, the result is obtained that g[l + ~(1 - cos’ %,)]sin 6, = 0. Since p must be > 0 for any physically reasonable system, this equation admits solutions only at 6, = + nz rad, where n=0,1,2,. . ., the corresponding x solutions occur for xP = f [6 + y + (1 + p)/v]gmc m. These solutions represent an infinite number of singular points, spaced at equal intervals along the 4-space line defined by x = fg[6 + y + (1 + p)/u I%, f = 0, %= 0. Linearization of the system equations is accomplished by means of a Taylor series expansion about a genera1 singularity, followed by elimination of non-linear terms. This singularity is specified as (qp + q,,), where qp is the actual singularity location, and q,, is an infinitesimal deviation from it. The result [using the “condensed” form, equation (3), of the state feedback coefficients] is the following pair of equations, linear in the d-subscripted state deviation variables: d3i,, -= dxd d%, a=
AxI + Bfd + %*[prdi cos 6, + C - pg cos (2%,)] + Bd(2pdp sin 6, + D _-&+ ~(21, %dsin eP cos 6, + id sin2 6,)
(8)
- [(AxI + E&)/r] cos 6, - 0, [(D/r) cos 6, + 2~8, sin 6, cos %,I %, + ,LL(~%~ 4, sin 6, cos 6, + 4, sin2 6,) + %d{[(Ax, + B1, + C%, + o%,)/r] sin 6, - [(C/r) - (1 + p)u)lr]cos 6, - p%f cos (26,)) 0, + p(2%, 4, sin eP cos 6, + 4, sin’ 6,) (9)
Recasting these equations in matrix form as first-order DES, one obtains:
218
M. G. Henders
0 1 + p sin’ 0,
and A. C. Soudack
A
0
-(A/r)
cos 0,
B
0
-(B/r)
cos 8,
c - pg cos (20,)
0
X
0
[(Ax,
C(W) 0
1 + ,~1sin’ OP
which may be further
reduced
-(D/r)
(10) sinfIp (1 + /4lYl cos 0, 1
f C0,)/r] cos Qp
to
using the known locations of the singular points of the system. Using the first-order form found in equation (ll), the characteristic equation for this system may be written in terms of a general singularity eigenvalue 2, resulting in: L”+4[(-l)“(l This equation number, n:
+y)--]I.“+[(-1)“(6+y)-y]L2+4(-l)”%+(-l)”=O. exhibits
two distinct
forms, distinguished
(12) by the value of the singularity
For n even:
,I4 + 4i3 + 6;“’ + 42 + 1 = 0
(13)
For n odd:
J4 - 4(1 + 2,/)A3 - (6 + 2,j)i2 - 41%- 1 = 0.
(14)
When n is even, the characteristic equation matches that used in the process of designing the controller in the first place, and it therefore has a single, multiplicity-four root at (- 1 + j 0), independent of any system parameters. Unfortunately, the symbolic root solution for the odd-n case is too complex to be useful; a numerical solution [S], using y as specified by the reference parameter set, yields a pair of real roots, at A,, = - 0.4360 and I1 = 5.9716, plus a complex conjugate root pair at J2. A3 = - 0.3599 +j 0.5045. The even-n case is characterized by four negative real roots, indicating that these singularities act as nodal attractors in all four dimensions of the state space. Because this case includes the origin, where four-dimensional nodal behaviour is expected by design, it also serves to validate the design process in this respect. The odd-n eigenvalues characterize a four-dimensional hybrid of a saddle point and a focal point attractor. In the vicinity of these singularities, state trajectories are drawn in directly along one dimension, attracted along spiral paths through two more dimensions, and expelled from the neighborhood of the singular point along the fourth dimension of the state space. At this point, one might attempt to obtain a global description of the system dynamics based on the singularity map, as was done in [7]. The complexity of the four-dimensional state space of the system, however, creates profound difficulties in understanding, with sufficient clarity, the structure of the spatial features created by the singularities. We note only that it is interesting that two, and only two, sets of eigenvalues characterize all of the infinite number of singular points found in this system. This is notably different from the case examined in [7], in which each singular point exhibited different eigenvalues; it indicates that, for an infinite number of state points, the system exhibits local behaviour identical to that seen near the state-space origin.
4. SIMULATION
STUDIES
We begin with simulation results from the neighborhood of the state-space origin, which, as the “design attractor” of the system, is of obvious interest. As already noted, state trajectories in this region are expected to behave in ways characteristic of a nodal singularity, in four-space. It is unavoidably difficult to visualize exactly what this means, given the
Controlled
inverted
pendulum
219
state-space
“extra” dimension involved. The characteristic stable and unstable of a two-dimensional phase-plane diagram become four-dimensional ing “sheets” in the state space.
singularity attracting
asymptotes and repell-
4.1. Singularity-local hehaviour Figures 2 and 3 show examples of a typical state trajectory in the vicinity of the origin, and, by extension, in the vicinity of any of the even-n singular points of the system. Each of the figures shows three of the state dimensions, ignoring the fourth and thereby performing a four-space-to-three-space projection. The actual state trajectory is shown as a solid line, while the dashed lines show projections of the trajectory onto the principal state planes of the space. This trajectory results from an initial state vector of [O.O, 0.0,51.3,0.0], which lies just inside the bounds of stability for this system. One may observe that the trajectory does appear to approach the origin directly (without spiraling) once it gets into the immediate vicinity of the origin, which is consistent with the nodal behaviour expected there. The gyrations that it exhibits in getting near the origin, however, suggest that the asymptote sheets of the singularity possess a rather complex structure in this region, and there seems to be little that one may do to gain further insight into this structure. Asymptote-tracing techniques, so useful in two-dimensional phase-plane analysis, break down when confronted with three or more dimensions, and we are left with essentially empirical observations of trajectory behaviour in the state space. In the empirical style, then, Fig. 4 shows a pair of typical trajectories computed for reverse time-flow from the vicinity of the singularity at n = 1, at [x i 0 e] = [ -520.334 0.0 180.0 0.0). The left-hand trace in the figure starts at (x, 0) = (-520.3, 180.0), and the right-hand one begins at (x, 0) = (-520.4, 180.0); the trajectories are followed through (x 0 6) space. It will be noted that the x-value increases very rapidly near the outer ends of these traces, and in actual fact, these solutions exhibit an essentially monotonic increase in the cart velocity, 2, with the cart position, x, changing accordingly. One might hope that this reverse-time solution from the vicinity of an unstable singularity would map the stable asymptotes of the point. (Recall, from the singularity analysis, that
Fig. 2. Origin-bound
state trajectory:
(x i 0) space.
220
M. G. Henders
and A. C. Soudack
Initial Theta = 51.3 Degrees Fig. 3. Origin-bound
state trajectory:
(s H .?) space.
the odd-n point is a combination of a stable focus and a saddle point, exhibiting dimensions.) Again, however, the dimensionality of the state space contrives meaningful interpretation of the results. Similarly, forward-time solutions from also leave the local space quickly and offer no further insight into the singularity These trajectories, however, subsequently exhibit complex and rather bizarre which is discussed in the next sub-section.
three stable to thwart this vicinity behaviour. behaviour,
4.2. Behaviour ut non-singular points Figure 5 illustrates a state trajectory in the (0, 0) space, representative of forward-time solutions from points outside the origin-bound basin of attraction. This includes solutions from the vicinity of the odd-n singular points, and we hereafter group all of these points together under the overall category of “unstable points”. On examination of Fig. 5, it becomes evident that the state trajectory is highly irregular, but somewhat repetitive. This quasi-cyclic behaviour immediately suggests the presence of dynamic chaos [lo]. As an aside, we note that the behaviour seen in Fig. 5 is not the only type observed for unstable initial states. A different trajectory type, which we have chosen to identify as a “whorl”, has been observed under certain conditions. This behaviour, which appears to comprise an infinite expanding/contracting spiral in the (0, 4) phase space, seems to be relatively rare, and is not discussed further here. An example of this behaviour may be seen by performing a forward-time solution from the initial state [O.O, 0.0, 225.0, 0.01 [9]. We now consider more closely the apparent chaos in Fig. 5. Plotting the same trajectory in (8,# space, with 0 taken as a module-360” variable (Fig. 6), appears to confirm its chaotic nature. This interpretation, however, is subject to question for two reasons. Firstly, while Fig. 6 shows an apparently bounded state trajectory, the full four-dimensional trajectory is most emphatically not bounded; Fig. 7 extends Fig. 6 into the x-dimension, demonstrating that the system state expands indefinitely there. Extended solution runs also tend to expand somewhat in the (0,e) space, although this effect may be due to an accumulation of numerical error over the duration of the run. As chaotic trajectories are generally considered to be bounded, this behaviour raises questions about the “chaos” interpretation.
Controlled
inverted
pendulum
state-space
221
2 l
I---I
Odd-n Singularity
,’
Fig. 4. Odd-n
-1500
-1250
singularity:
-1000
-750
Pendulum Fig. 5. Pseudo-chaotic
reverse-time
-500
Angle
trajectory:
Theta
traces
-250
0
250
500
(Degrees)
absolute-sense
0 value.
Secondly, and perhaps more significantly, the use of a modulo interpretation for 0 is in direct conflict with the fundamental assumptions of this study. During design of the feedback controller for the system, the control force E is explicity constructed in the form Ax + B1 + CQ + 04, which is linear in 0, not in (0 mod 360). This feedback function increases indefinitely with increasing 0, rather than periodically resetting, as a modulo function does. Given this, it becomes difficult to defend the validity of the modulo-360chaos interpretation.
M. G. Henders
222
and A. C. Soudack
180
270" Fig. 6. Pseudo-chaotic
trajectory:
module-360
0 value
25fJJ zoo0
1500
loo0
500
90”
‘----._. ___L.__---
Fig. 7. Pseudo-chaotic
trajectory:
module
fJ value. with x extension
At this point, the question arises: “If it isn’t chaos, what is it? How, exactly, should these results be interpreted?” We do not have a definitive answer to this, and would suggest that this problem could benefit from the attention of researchers specializing in chaotic dynamics. It is worth noting that, in addition to their pseudo-cyclical evolution, these trajectories exhibit sensitive dependence on initial conditions, the classic hallmark of chaos [lo]. As
Controlled
inverted
pendulum
state-space
223
a demonstration of this, we have observed that trajectories having an initial state difference in the fourth decimal place quickly diverge into completely different solutions (though their overall structures remain sirnilar). To summarize this section, we compare these four-dimensional results with the essentially two-dimensional results obtained in [7]. In [7], the existence of a global state attractor was demonstrated, which is absent from the four-dimensional system studied here. Instead, this system appears to exhibit an infinite number of local attractors, each possessing a finite basin of attraction, and the surrounding regions of the state space are occupied by one or more “quasi-strange” attractors.
5. STABILITY
STATE-SPACE
MAPPING
is evident, from the results presented thus far, that general insight into the behaviour of this system is not readily obtained. Having obtained some idea of its overall behaviour, however, we wish to determine the bounds of its regions of stability.3 More precisely, we wish to obtain some sort of description of the set of initial states for which the linear controller is able to stabilize the inverted pendulum. In [7], it was possible to obtain an exact theoretical description of this set through phase-plane analysis, but, as has already been indicated, this is not feasible for the present system. Instead, an algorithmic technique is used, which systematically searches the space of initial conditions, looking for those initial state vectors which lead to stability. It
5.1. The STABVOL algorithm A new, computationally efficient algorithm was developed, which maps the stability volume from the inside, beginning from a single, known, stable point. For the inverted pendulum system, this point is conveniently taken at the origin of the state space, which is stable by design. Beginning from the initial point, the algorithm systematically extends its search outwards along all of the state-space dimensions until it encounters unstable states, thus mapping all of the stable states of the system. (Stability is assessed through use of a set of heuristics developed from observation of the simulated system.) In filling this region, the mapping algorithm, dubbed STABVOL (for “STABility VOLume”), is somewhat analogous to a “seed-fill” algorithm, well-known in computer graphics circles [ 111. A comprehensive explanation of the STABVOL algorithm may be found in [9], which contains our definitive work on this algorithm, including complete source code. Executing the STABVOL algorithm against the inverted pendulum dynamics, a large data set is obtained, representing all of the vectors of initial conditions encompassed by the system’s origin-bound basin of attraction. Comparing the size of this data set with the size of the rectangular-grid set required to span the same volume, we found that, for these dynamics, the STABVOL algorithm achieved an advantage ratio of about 50: 1. That is, gridded sampling would have required 50 times more computation than the STABVOL algorithm to obtain the same result. STABVOL itself ran for 76 h on a 40 MHz i860 processor (scalar mode), implying a run time of about five months for gridded sampling! As an aside, we note that preliminary investigations indicated that the stability volume of the inverted pendulum would be bounded along all four state axes. If one or more dimensions of the region had been unbounded, the algorithm would have run indefinitely, trying to map an infinite volume. It is vital to consider this detail if one contemplates executing STABVOL against a different dynamic system. We would also note in passing that the dimensionality of the STABVOL algorithm is readily scaled up or down, and the algorithm may therefore be readily applied to other dynamic systems. This restriction on unbounded sets can, of course, be bypassed by putting artificial limits on any unbounded state dimensions. To accomplish this, all that is required is that the
‘The terms “stable” and “stability” are used rather loosely in this section. In general, these terms are used to indicate conditions or processes for which the linear regulator element of the studied control system is able to successfully bring the plant state to its desired setpoint; that is. the origin of the state space.
224
M. G. Henders
and A. C. Soudack
stability heuristics must return “Unstable” whenever the desired limits are exceeded. Note, however, that such limiting may break the continuity of the stability region, giving unexpected results. Suppose, for example, that it is desired to map a given stability region only between two values, A and B, of state p, because the region appears to extend to infinity along the p-axis. Suppose, furthermore, that the stability region loops back and forth along the p-axis several times, passing between A and B, and beyond, each time. In this case, installing limits at p = A and p = B and initiating the STABVOL algorithm at any of the stable points between A and B would result in the mapping only of that portion (loop) of the stability region in which the initial point lay. One might expect the algorithm to map all stable points having a p-coordinate within the specified limits, but it would be unable to do this because of the artificial discontinuities imposed on the stability region.
5.2. The inverted-pendulum data set The initial state vectors mapped for this system are, naturally, four-dimensional, and the data set is sectioned along the i-axis for presentation. This results in a series of threedimensional data “slices”, which, in turn, are projected into two dimensions and plotted. Each data point in the three-dimensional sets is represented by a three-dimensional set of crossed axes, which extend out to span the volume of space sampled by that particular point. Figures 8-l 1 show samples of the three-dimensional data slices, taken at pi = 0, f 25 and 75, respectively. These figures actually show two representations of the data; the dark-grey areas map the three-dimensional stability volume, while the lighter grey areas plot the projections of this volume against the principal planes of the state space. The figures may give the impression of shaded rendering, but this is a misleading artefact of variations in the density of stable data points across the plots. While these slices cannot convey more than a general sense of the overall four-dimensional structure of the data set, they do provide some significant information. One may note the symmetry of the stability region, which might be expected, given the nature of the studied system. It is also worth noting that the stability region is quite extensive. While large, the stability region is finite, in all dimensions (at the resolution level of this sampling,
Fig. 8. Origin
stability
region: section at li = 0.
Controlled inverted pendulum state-space
big. 9. Origin stability region: section at X, = - 25.
Fig. 10. Origin stabihty region: section at -i-i= 25.
at least). This is apparent in the three plotted dimensions, and has also been observed along the fourth dimension. Slices taken at pi = k 150 show only a sparse scatteration of data points, and the data set ends just beyond this. The overall structure of Ihe attractor basin appears to comprise a four-dimensional extension of the spiraling two-dimensional basin described in [7], but the detailed structure is
226
M. G. Henders
Fig. 11. Origin
stability
and A. C. Soudack
region: section
at 1, = 75.
substantially more complex. The boundaries of the basin are highly irregular, appearing to form tendrils of various sizes in the state space, and it is quite likely that, at higher resolution, additional details would emerge. We believe it probable that the basin boundaries are actually fractal in nature. Such fractal boundaries possess high spatial frequencies, which are unavoidably aliased by the sampling process of the STABVOL algorithm, and detail is thereby lost. There are various ways of controlling this effect, including adaptive sampling [12] and spatial filtering [ll], but these add substantially to the complexity of the sampling process, and we have not yet pursued them. It should be noted that, contrary to appearances, the stability basin comprises a single, connected data set. The data slices give the appearance of containing isolated islands of stability, but this is illusory. It is easy to demonstrate that this must be so, reasoning as follows: given an island of stability, it must be true, by definition of stability, that state trajectories originating within the island lead ultimately to the origin of the state space. Such trajectories are continuous, and there thus exists at least one series of state vectors defining a continuous path from within the island to within the immediate vicinity of the origin. This, however, is a logical contradiction; if the island is isolated, such a path cannot exist, but if the path does not exist, then points on the island cannot be stable. The sectioned data contain the illusion of islands for the same reason that a topographic section taken through a rugged landscape appears to contain islands. The sectioning plane (or hyperplane) cuts off the “bottom” of the topography, and the inherent continuity of the data set is lost. It is necessary to recognize that, in sectioning the data set, a large amount of data are lost, and mental interpolation is required to “fill in” between the slices. To reduce this loss, one might consider working with the entire four-dimensional data set, projecting from four-space into three-space, then performing a polygonization and rendering on the resulting three-dimensional data object. The polygonization operation in particular is likely to be non-trivial, however, and, given that the four-dimensional data set contains approximately 215,000 points, the computational cost of this approach may be excessive. As a final item, it has already been noted that the inverted pendulum system contains an infinite number of singular points identical to the origin attractor. Each of these points must possess its own basin of attraction similar-if not identical-to the origin-bound basin.
Controlled
inverted
pendulum
state-space
221
These basins are not connected; each is separate and (presumably) as finite as the originbound basin. The STABVOL algorithm could be applied to map any of these basins, but certain program modifications would be required, and we have not pursued this.
6. CONCLUSION
This paper has reported results from the study of the dynamics of an inverted pendulum, stabilized by a linear-feedback control system. It was demonstrated that the system possesses an infinite number of attractor points in its state space, each of these being surrounded by a finite basin of attraction. Outside these attracting basin, the system state was seen to evolve along convoluted, possibly chaotic, trajectories. Outlines were given of an algorithm for mapping the origin-bound attracting basin. This algorithm is more efficient than rectangular-gridded sampling, and may be useful in studying other systems. Lastly, results were presented showing the structure of the four-dimensional region of system stability, centered at the origin of the system’s state space. This origin-bound attractor basin was seen to exhibit a jagged, possibly fractal, morphology. Although the stabilized inverted pendulum is commonly treated as an almost trivial tutorial problem in control theory, the results presented here show the existence of surprisingly complex and interesting behaviour in its dynamics.
REFERENCES 1. C. Chen, Linear System Theory and Design. Holt, Rinehart and Winston, New York (1984). Edition. Addison-Wesley, New York (1980). 2. R. C. Dorf, Modern Control Systems-Third 3. Q. Feng and K. Yamafuji, Design and simulation of control systems of an inverted pendulum. Robotica 6, 235-241 (1988). inverted pendulum model for evaluating upright postural stability, 4. H. Hemami and A. Katbab, Constrained Trans. ASME, J. Dynamic Syst.. Measurement. Contr. 104, 343-349 (1982). and K. Furuta, Control of unstable mechanical system: control of pendulum. Int. J. 5. S. Mori, H. Nishihara Control 23, 613-692 (1976). 6. J. W. Watts, Control of an inverted pendulum. In American Societyfor Engineering, Education, 92nd ASEE Annual Conference Proceedings, Vol. 2, Engineering Education, Preparation for Life, 24-28 June (1984). behaviour of an inverted pendulum with linear stabilization. 1. M. G. Henders and A. C. Soudack, “In-the-large” Int. J. Non-Linear Mech. 27, 129-138 (1992). 8. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in C-the Art ofScientific _ Computing. Cambridge University Press, Cambridge (1988). dvnamics of the inverted nendulum under linear-feedback stabilization. Thesis. 9. M. G. Henders. Nonlinear Masters of Applied Science, Department of Electrical’Engineering, University of British Columbia (1991) 10. P. Bergi, Y. Pomeau and C. Vidal, Order Within Chaos. Wiley Interscience, New York (1984). 11. A. S. Glassner (ed.), Graphics Gems. Academic Press, New York (1990). Polygonization of implicit surfaces. Computer Aided Geomet. Design 5, 341-355 (1988). 12. J. Bloomenthal,
hzl J. .f%;on-LmecrMcchanrc.<, Vol. 31, No. 2. pp. 215.227, 1996 Copyright ST; 1996 Elsrvier Saence Ltd Prmlrd m Great Bntam. All nghts mewed 0020-7462:96 515.00 + 0.00
0020-7462(95)00055-O
DYNAMICS AND STABILITY STATE-SPACE OF A CONTROLLED INVERTED PENDULUM M. G. Henders*
and A. C. Soudack”
* MGH Engineering ’ Department
Services, Box 1599. Cl 83, Medicine Hat, Alberta. Canada TIA 7Y5: of Electrical Engineering, University of British Columbia, Vancouver, British Columbia, Canada V6T 124 (Received
17 Auyust 1994)
Abstract-A study is conducted into the dynamics of an inverted pendulum under the regulation of a linear-feedback control system. Analysis of the resulting dynamic equations indicates the presence of stable regions within the four-dimensional state space of the system, which are confirmed through simulation. System trajectories are examined in this state space, and complex, apparently chaotic behaviour is observed outside the stable regions of the state space. An algorithm is developed to map the boundaries of the stable state-space hyper-volume; this algorithm is broadly applicable to the study of other systems. Results are presented, generated by executing the algorithm on the inverted pendulum dynamics. These comprise a series of three-dimensional plots, outlining the four-dimensional stability volume. The boundaries of this volume are seen to exhibit a jagged, possibly fractal, morphology. Keywords: pendulum
dynamics,
stability
1.
of motion,
space state methods
INTRODUCTION
The “inverted pendulum” (Fig. 1) is commonly used as a tutorial problem in control theory [l, 21, where its inherently unstable, highly non-linear dynamics provide interest for the student. A number of researchers have also studied various aspects of the inverted pendulum, in a broad spectrum of settings [3-61. It is perhaps surprising that little work seems to have been done with the most basic implementation of control of the inverted pendulum, that of a simple linear controller designed through pole-placement. This implementation is considered in the textbooks, but discussion of the system dynamics seems to be limited to the region where linearization is applicable. There is generally no mention of the system behaviour outside this region, except perhaps to suggest that “the pendulum falls over”. In earlier work [7], we examined a limited case of this system using phase-plane analysis, and demonstrated the existence of some interesting global dynamics in the system. In this paper, results are presented from a more comprehensive treatment of this problem, showing that, as far as the system dynamics are concerned, its apparent simplicity is quite deceptive. It is demonstrated that the pole-placement controller creates an infinite number of attractor points in the state space of the system, with each one apparently having a finite basin of attraction. Convoluted, possibly chaotic state trajectories are seen outside these attracting basins. The intent of the controller design is to create an attractor at the origin of the state space; the full extent of its basin of attraction is mapped algorithmically and the results are presented. This origin-bound attractor basin exhibits a jagged, possibly fractal, morphology. This paper comprises six major sections, including the Introduction. In Section 2, the background assumptions of the study are outlined, and an analytical treatment of the system is presented in Section 3. Section 4 discusses simulation results, with discussion of the basin-mapping algorithm and a presentation of its results found in Section 5. Finally, Section 6 concludes with a summary of-and commentary on--the results presented. 215
M. G. Henders
216
and A. C. Soudack
Mass 5 M
Fig. 1. Inverted
pendulum:
2. SYSTEM
general
arrangement
DETAILS
For the purposes of the work presented here, the inverted pendulum system is completely idealized. With reference to Fig. 1, the pendulum is assumed to be capable of unlimited rotation (no limits on 101) and the lateral excursion of the cart (x) is also unlimited. The system is assumed to be dissipationless, and there are no restrictions on the applied control force F. One further important detail is that the pendulum angle, 8, is considered as an absolute, rather than a modulo-360” number. The nature of the applied control force is significantly affected by the treatment of this item. Because much of the work described here is based on simulation, it is necessary to define a “reference parameter set”. In all simulations, the cart mass, M, is 1.0 kg, the pendulum mass, m, is 0.1 kg, the pendulum length, r, is 1.0 m, and gravitational acceleration, g, is 9.80665 m.sP2. The numerical work was performed using a specially developed code running on a personal computer, with a fourth-order Runge-Kutta integrator [8] forming the core of the simulation routines.
3. ANALYTICAL
TREATMENT
The analysis in our earlier work [7] used the system parameters (m, M, r, g, and F) directly. The treatment presented here uses normalized values, with ~1= m/M, y = r/g, and P E F/M, for somewhat better clarity. Using the convention that 4 = dq/dt and 4 = d2qldt2, the open-loop dynamic equations of the inverted pendulum system’ are stated here, without derivation2 in the normalized form: j;_=
prg2sin0
+ P - PgsinQcosO 1 + ,u sin2
e
/j=
g(1 + p)sin 0 - p~r8’ sin H cos Q - E cos 0 r(1 + pusin
e) (1)
3.1. Controller design A linear-feedback controller is designed for this system, using standard pole-placement techniques, and all of the work presented here assumes a single, multiplicity-four eigenvalue at (- 1 + j0). (See [7] for notes regarding behaviour with other pole locations.) For the 8 81’ = 0, resulting in: purposes of this design, the system is linearized at [x i
(*)
1 These equations differ significantly from those presented in [l], but follow 121; we suspect flaws in [I]. ’ Reference [9] contains comprehensive derivations for all of the analysis presented in this paper.
Controlled inverted pendulum state-space
217
This equation has the general form i = [A + BG]z, where Gz = E. Solving for G, the feedback coefficients are obtained as: G = [A
B
C
II] = [y
4~
r[6 + Y + (1 + CL)/?] 4r(l + Y)].
(3)
On insertion of these coefficients, the following equations are obtained for the closed-loop system dynamics: jI_=
p&sin%
+ y(x + 4f) + r[6 + y + (1 + p)/y]% + 4r(l + y)%- ~gsin%cos% 1 + p sin’ 6
(4)
8’= ((1 + p)/r))sin 6 - ~1%’sin 6 cos 6 - {(x + 4i)/g + [6 + y + ((1 + p)/r)] 6 + 4(1 + $8) cos 6 1 + p sin’ 8 (5) Examination of these equations shows that the system state variables are coupled, requiring that a four-dimensional state space be used for full understanding of the inverted pendulum dynamics. 3.2. Singularity analysis The closed-loop equations, (4) and (5), are subjected to a local linearization in the vicinity of the system’s singular points to provide insight into the “in-the-small” dynamics of the system. The singular points of the system, at which all derivatives are zero, must also be determined. To find these points, it suffices to set the i and 8 equations, above, equal to zero, with i and 8 also set equal to zero. The following set of simultaneous non-linear equations is solved for singularity locations x, and 6,: From 2:
yx,+r[6+y+(l
From 8:
C(1 +
+~)/~]%p-~gsin%,cos%,=O
PYYIsin6, - {
(6) (7)
Multiplying equation (6) by cos 6, and equation (7) by r, and eliminating terms, the result is obtained that g[l + ~(1 - cos’ %,)]sin 6, = 0. Since p must be > 0 for any physically reasonable system, this equation admits solutions only at 6, = + nz rad, where n=0,1,2,. . ., the corresponding x solutions occur for xP = f [6 + y + (1 + p)/v]gmc m. These solutions represent an infinite number of singular points, spaced at equal intervals along the 4-space line defined by x = fg[6 + y + (1 + p)/u I%, f = 0, %= 0. Linearization of the system equations is accomplished by means of a Taylor series expansion about a genera1 singularity, followed by elimination of non-linear terms. This singularity is specified as (qp + q,,), where qp is the actual singularity location, and q,, is an infinitesimal deviation from it. The result [using the “condensed” form, equation (3), of the state feedback coefficients] is the following pair of equations, linear in the d-subscripted state deviation variables: d3i,, -= dxd d%, a=
AxI + Bfd + %*[prdi cos 6, + C - pg cos (2%,)] + Bd(2pdp sin 6, + D _-&+ ~(21, %dsin eP cos 6, + id sin2 6,)
(8)
- [(AxI + E&)/r] cos 6, - 0, [(D/r) cos 6, + 2~8, sin 6, cos %,I %, + ,LL(~%~ 4, sin 6, cos 6, + 4, sin2 6,) + %d{[(Ax, + B1, + C%, + o%,)/r] sin 6, - [(C/r) - (1 + p)u)lr]cos 6, - p%f cos (26,)) 0, + p(2%, 4, sin eP cos 6, + 4, sin’ 6,) (9)
Recasting these equations in matrix form as first-order DES, one obtains:
218
M. G. Henders
0 1 + p sin’ 0,
and A. C. Soudack
A
0
-(A/r)
cos 0,
B
0
-(B/r)
cos 8,
c - pg cos (20,)
0
X
0
[(Ax,
C(W) 0
1 + ,~1sin’ OP
which may be further
reduced
-(D/r)
(10) sinfIp (1 + /4lYl cos 0, 1
f C0,)/r] cos Qp
to
using the known locations of the singular points of the system. Using the first-order form found in equation (ll), the characteristic equation for this system may be written in terms of a general singularity eigenvalue 2, resulting in: L”+4[(-l)“(l This equation number, n:
+y)--]I.“+[(-1)“(6+y)-y]L2+4(-l)”%+(-l)”=O. exhibits
two distinct
forms, distinguished
(12) by the value of the singularity
For n even:
,I4 + 4i3 + 6;“’ + 42 + 1 = 0
(13)
For n odd:
J4 - 4(1 + 2,/)A3 - (6 + 2,j)i2 - 41%- 1 = 0.
(14)
When n is even, the characteristic equation matches that used in the process of designing the controller in the first place, and it therefore has a single, multiplicity-four root at (- 1 + j 0), independent of any system parameters. Unfortunately, the symbolic root solution for the odd-n case is too complex to be useful; a numerical solution [S], using y as specified by the reference parameter set, yields a pair of real roots, at A,, = - 0.4360 and I1 = 5.9716, plus a complex conjugate root pair at J2. A3 = - 0.3599 +j 0.5045. The even-n case is characterized by four negative real roots, indicating that these singularities act as nodal attractors in all four dimensions of the state space. Because this case includes the origin, where four-dimensional nodal behaviour is expected by design, it also serves to validate the design process in this respect. The odd-n eigenvalues characterize a four-dimensional hybrid of a saddle point and a focal point attractor. In the vicinity of these singularities, state trajectories are drawn in directly along one dimension, attracted along spiral paths through two more dimensions, and expelled from the neighborhood of the singular point along the fourth dimension of the state space. At this point, one might attempt to obtain a global description of the system dynamics based on the singularity map, as was done in [7]. The complexity of the four-dimensional state space of the system, however, creates profound difficulties in understanding, with sufficient clarity, the structure of the spatial features created by the singularities. We note only that it is interesting that two, and only two, sets of eigenvalues characterize all of the infinite number of singular points found in this system. This is notably different from the case examined in [7], in which each singular point exhibited different eigenvalues; it indicates that, for an infinite number of state points, the system exhibits local behaviour identical to that seen near the state-space origin.
4. SIMULATION
STUDIES
We begin with simulation results from the neighborhood of the state-space origin, which, as the “design attractor” of the system, is of obvious interest. As already noted, state trajectories in this region are expected to behave in ways characteristic of a nodal singularity, in four-space. It is unavoidably difficult to visualize exactly what this means, given the
Controlled
inverted
pendulum
219
state-space
“extra” dimension involved. The characteristic stable and unstable of a two-dimensional phase-plane diagram become four-dimensional ing “sheets” in the state space.
singularity attracting
asymptotes and repell-
4.1. Singularity-local hehaviour Figures 2 and 3 show examples of a typical state trajectory in the vicinity of the origin, and, by extension, in the vicinity of any of the even-n singular points of the system. Each of the figures shows three of the state dimensions, ignoring the fourth and thereby performing a four-space-to-three-space projection. The actual state trajectory is shown as a solid line, while the dashed lines show projections of the trajectory onto the principal state planes of the space. This trajectory results from an initial state vector of [O.O, 0.0,51.3,0.0], which lies just inside the bounds of stability for this system. One may observe that the trajectory does appear to approach the origin directly (without spiraling) once it gets into the immediate vicinity of the origin, which is consistent with the nodal behaviour expected there. The gyrations that it exhibits in getting near the origin, however, suggest that the asymptote sheets of the singularity possess a rather complex structure in this region, and there seems to be little that one may do to gain further insight into this structure. Asymptote-tracing techniques, so useful in two-dimensional phase-plane analysis, break down when confronted with three or more dimensions, and we are left with essentially empirical observations of trajectory behaviour in the state space. In the empirical style, then, Fig. 4 shows a pair of typical trajectories computed for reverse time-flow from the vicinity of the singularity at n = 1, at [x i 0 e] = [ -520.334 0.0 180.0 0.0). The left-hand trace in the figure starts at (x, 0) = (-520.3, 180.0), and the right-hand one begins at (x, 0) = (-520.4, 180.0); the trajectories are followed through (x 0 6) space. It will be noted that the x-value increases very rapidly near the outer ends of these traces, and in actual fact, these solutions exhibit an essentially monotonic increase in the cart velocity, 2, with the cart position, x, changing accordingly. One might hope that this reverse-time solution from the vicinity of an unstable singularity would map the stable asymptotes of the point. (Recall, from the singularity analysis, that
Fig. 2. Origin-bound
state trajectory:
(x i 0) space.
220
M. G. Henders
and A. C. Soudack
Initial Theta = 51.3 Degrees Fig. 3. Origin-bound
state trajectory:
(s H .?) space.
the odd-n point is a combination of a stable focus and a saddle point, exhibiting dimensions.) Again, however, the dimensionality of the state space contrives meaningful interpretation of the results. Similarly, forward-time solutions from also leave the local space quickly and offer no further insight into the singularity These trajectories, however, subsequently exhibit complex and rather bizarre which is discussed in the next sub-section.
three stable to thwart this vicinity behaviour. behaviour,
4.2. Behaviour ut non-singular points Figure 5 illustrates a state trajectory in the (0, 0) space, representative of forward-time solutions from points outside the origin-bound basin of attraction. This includes solutions from the vicinity of the odd-n singular points, and we hereafter group all of these points together under the overall category of “unstable points”. On examination of Fig. 5, it becomes evident that the state trajectory is highly irregular, but somewhat repetitive. This quasi-cyclic behaviour immediately suggests the presence of dynamic chaos [lo]. As an aside, we note that the behaviour seen in Fig. 5 is not the only type observed for unstable initial states. A different trajectory type, which we have chosen to identify as a “whorl”, has been observed under certain conditions. This behaviour, which appears to comprise an infinite expanding/contracting spiral in the (0, 4) phase space, seems to be relatively rare, and is not discussed further here. An example of this behaviour may be seen by performing a forward-time solution from the initial state [O.O, 0.0, 225.0, 0.01 [9]. We now consider more closely the apparent chaos in Fig. 5. Plotting the same trajectory in (8,# space, with 0 taken as a module-360” variable (Fig. 6), appears to confirm its chaotic nature. This interpretation, however, is subject to question for two reasons. Firstly, while Fig. 6 shows an apparently bounded state trajectory, the full four-dimensional trajectory is most emphatically not bounded; Fig. 7 extends Fig. 6 into the x-dimension, demonstrating that the system state expands indefinitely there. Extended solution runs also tend to expand somewhat in the (0,e) space, although this effect may be due to an accumulation of numerical error over the duration of the run. As chaotic trajectories are generally considered to be bounded, this behaviour raises questions about the “chaos” interpretation.
Controlled
inverted
pendulum
state-space
221
2 l
I---I
Odd-n Singularity
,’
Fig. 4. Odd-n
-1500
-1250
singularity:
-1000
-750
Pendulum Fig. 5. Pseudo-chaotic
reverse-time
-500
Angle
trajectory:
Theta
traces
-250
0
250
500
(Degrees)
absolute-sense
0 value.
Secondly, and perhaps more significantly, the use of a modulo interpretation for 0 is in direct conflict with the fundamental assumptions of this study. During design of the feedback controller for the system, the control force E is explicity constructed in the form Ax + B1 + CQ + 04, which is linear in 0, not in (0 mod 360). This feedback function increases indefinitely with increasing 0, rather than periodically resetting, as a modulo function does. Given this, it becomes difficult to defend the validity of the modulo-360chaos interpretation.
M. G. Henders
222
and A. C. Soudack
180
270" Fig. 6. Pseudo-chaotic
trajectory:
module-360
0 value
25fJJ zoo0
1500
loo0
500
90”
‘----._. ___L.__---
Fig. 7. Pseudo-chaotic
trajectory:
module
fJ value. with x extension
At this point, the question arises: “If it isn’t chaos, what is it? How, exactly, should these results be interpreted?” We do not have a definitive answer to this, and would suggest that this problem could benefit from the attention of researchers specializing in chaotic dynamics. It is worth noting that, in addition to their pseudo-cyclical evolution, these trajectories exhibit sensitive dependence on initial conditions, the classic hallmark of chaos [lo]. As
Controlled
inverted
pendulum
state-space
223
a demonstration of this, we have observed that trajectories having an initial state difference in the fourth decimal place quickly diverge into completely different solutions (though their overall structures remain sirnilar). To summarize this section, we compare these four-dimensional results with the essentially two-dimensional results obtained in [7]. In [7], the existence of a global state attractor was demonstrated, which is absent from the four-dimensional system studied here. Instead, this system appears to exhibit an infinite number of local attractors, each possessing a finite basin of attraction, and the surrounding regions of the state space are occupied by one or more “quasi-strange” attractors.
5. STABILITY
STATE-SPACE
MAPPING
is evident, from the results presented thus far, that general insight into the behaviour of this system is not readily obtained. Having obtained some idea of its overall behaviour, however, we wish to determine the bounds of its regions of stability.3 More precisely, we wish to obtain some sort of description of the set of initial states for which the linear controller is able to stabilize the inverted pendulum. In [7], it was possible to obtain an exact theoretical description of this set through phase-plane analysis, but, as has already been indicated, this is not feasible for the present system. Instead, an algorithmic technique is used, which systematically searches the space of initial conditions, looking for those initial state vectors which lead to stability. It
5.1. The STABVOL algorithm A new, computationally efficient algorithm was developed, which maps the stability volume from the inside, beginning from a single, known, stable point. For the inverted pendulum system, this point is conveniently taken at the origin of the state space, which is stable by design. Beginning from the initial point, the algorithm systematically extends its search outwards along all of the state-space dimensions until it encounters unstable states, thus mapping all of the stable states of the system. (Stability is assessed through use of a set of heuristics developed from observation of the simulated system.) In filling this region, the mapping algorithm, dubbed STABVOL (for “STABility VOLume”), is somewhat analogous to a “seed-fill” algorithm, well-known in computer graphics circles [ 111. A comprehensive explanation of the STABVOL algorithm may be found in [9], which contains our definitive work on this algorithm, including complete source code. Executing the STABVOL algorithm against the inverted pendulum dynamics, a large data set is obtained, representing all of the vectors of initial conditions encompassed by the system’s origin-bound basin of attraction. Comparing the size of this data set with the size of the rectangular-grid set required to span the same volume, we found that, for these dynamics, the STABVOL algorithm achieved an advantage ratio of about 50: 1. That is, gridded sampling would have required 50 times more computation than the STABVOL algorithm to obtain the same result. STABVOL itself ran for 76 h on a 40 MHz i860 processor (scalar mode), implying a run time of about five months for gridded sampling! As an aside, we note that preliminary investigations indicated that the stability volume of the inverted pendulum would be bounded along all four state axes. If one or more dimensions of the region had been unbounded, the algorithm would have run indefinitely, trying to map an infinite volume. It is vital to consider this detail if one contemplates executing STABVOL against a different dynamic system. We would also note in passing that the dimensionality of the STABVOL algorithm is readily scaled up or down, and the algorithm may therefore be readily applied to other dynamic systems. This restriction on unbounded sets can, of course, be bypassed by putting artificial limits on any unbounded state dimensions. To accomplish this, all that is required is that the
‘The terms “stable” and “stability” are used rather loosely in this section. In general, these terms are used to indicate conditions or processes for which the linear regulator element of the studied control system is able to successfully bring the plant state to its desired setpoint; that is. the origin of the state space.
224
M. G. Henders
and A. C. Soudack
stability heuristics must return “Unstable” whenever the desired limits are exceeded. Note, however, that such limiting may break the continuity of the stability region, giving unexpected results. Suppose, for example, that it is desired to map a given stability region only between two values, A and B, of state p, because the region appears to extend to infinity along the p-axis. Suppose, furthermore, that the stability region loops back and forth along the p-axis several times, passing between A and B, and beyond, each time. In this case, installing limits at p = A and p = B and initiating the STABVOL algorithm at any of the stable points between A and B would result in the mapping only of that portion (loop) of the stability region in which the initial point lay. One might expect the algorithm to map all stable points having a p-coordinate within the specified limits, but it would be unable to do this because of the artificial discontinuities imposed on the stability region.
5.2. The inverted-pendulum data set The initial state vectors mapped for this system are, naturally, four-dimensional, and the data set is sectioned along the i-axis for presentation. This results in a series of threedimensional data “slices”, which, in turn, are projected into two dimensions and plotted. Each data point in the three-dimensional sets is represented by a three-dimensional set of crossed axes, which extend out to span the volume of space sampled by that particular point. Figures 8-l 1 show samples of the three-dimensional data slices, taken at pi = 0, f 25 and 75, respectively. These figures actually show two representations of the data; the dark-grey areas map the three-dimensional stability volume, while the lighter grey areas plot the projections of this volume against the principal planes of the state space. The figures may give the impression of shaded rendering, but this is a misleading artefact of variations in the density of stable data points across the plots. While these slices cannot convey more than a general sense of the overall four-dimensional structure of the data set, they do provide some significant information. One may note the symmetry of the stability region, which might be expected, given the nature of the studied system. It is also worth noting that the stability region is quite extensive. While large, the stability region is finite, in all dimensions (at the resolution level of this sampling,
Fig. 8. Origin
stability
region: section at li = 0.
Controlled inverted pendulum state-space
big. 9. Origin stability region: section at X, = - 25.
Fig. 10. Origin stabihty region: section at -i-i= 25.
at least). This is apparent in the three plotted dimensions, and has also been observed along the fourth dimension. Slices taken at pi = k 150 show only a sparse scatteration of data points, and the data set ends just beyond this. The overall structure of Ihe attractor basin appears to comprise a four-dimensional extension of the spiraling two-dimensional basin described in [7], but the detailed structure is
226
M. G. Henders
Fig. 11. Origin
stability
and A. C. Soudack
region: section
at 1, = 75.
substantially more complex. The boundaries of the basin are highly irregular, appearing to form tendrils of various sizes in the state space, and it is quite likely that, at higher resolution, additional details would emerge. We believe it probable that the basin boundaries are actually fractal in nature. Such fractal boundaries possess high spatial frequencies, which are unavoidably aliased by the sampling process of the STABVOL algorithm, and detail is thereby lost. There are various ways of controlling this effect, including adaptive sampling [12] and spatial filtering [ll], but these add substantially to the complexity of the sampling process, and we have not yet pursued them. It should be noted that, contrary to appearances, the stability basin comprises a single, connected data set. The data slices give the appearance of containing isolated islands of stability, but this is illusory. It is easy to demonstrate that this must be so, reasoning as follows: given an island of stability, it must be true, by definition of stability, that state trajectories originating within the island lead ultimately to the origin of the state space. Such trajectories are continuous, and there thus exists at least one series of state vectors defining a continuous path from within the island to within the immediate vicinity of the origin. This, however, is a logical contradiction; if the island is isolated, such a path cannot exist, but if the path does not exist, then points on the island cannot be stable. The sectioned data contain the illusion of islands for the same reason that a topographic section taken through a rugged landscape appears to contain islands. The sectioning plane (or hyperplane) cuts off the “bottom” of the topography, and the inherent continuity of the data set is lost. It is necessary to recognize that, in sectioning the data set, a large amount of data are lost, and mental interpolation is required to “fill in” between the slices. To reduce this loss, one might consider working with the entire four-dimensional data set, projecting from four-space into three-space, then performing a polygonization and rendering on the resulting three-dimensional data object. The polygonization operation in particular is likely to be non-trivial, however, and, given that the four-dimensional data set contains approximately 215,000 points, the computational cost of this approach may be excessive. As a final item, it has already been noted that the inverted pendulum system contains an infinite number of singular points identical to the origin attractor. Each of these points must possess its own basin of attraction similar-if not identical-to the origin-bound basin.
Controlled
inverted
pendulum
state-space
221
These basins are not connected; each is separate and (presumably) as finite as the originbound basin. The STABVOL algorithm could be applied to map any of these basins, but certain program modifications would be required, and we have not pursued this.
6. CONCLUSION
This paper has reported results from the study of the dynamics of an inverted pendulum, stabilized by a linear-feedback control system. It was demonstrated that the system possesses an infinite number of attractor points in its state space, each of these being surrounded by a finite basin of attraction. Outside these attracting basin, the system state was seen to evolve along convoluted, possibly chaotic, trajectories. Outlines were given of an algorithm for mapping the origin-bound attracting basin. This algorithm is more efficient than rectangular-gridded sampling, and may be useful in studying other systems. Lastly, results were presented showing the structure of the four-dimensional region of system stability, centered at the origin of the system’s state space. This origin-bound attractor basin was seen to exhibit a jagged, possibly fractal, morphology. Although the stabilized inverted pendulum is commonly treated as an almost trivial tutorial problem in control theory, the results presented here show the existence of surprisingly complex and interesting behaviour in its dynamics.
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