Robust Computer Control of an

Inverted Pendulum
G.A. Medrano-Cerda

M odem control theory is now well developed to provide a

genuine design technique for synthesizing practical con-
trol systems. Moreover, the advent of fast and powerful comput-
nism (dc motor, power amplifier, and pulley-belt transmission)
and an aluminum rod (0.9 m long) which is free to rotate about a
pivot attached to the cart. An aluminum disk (with its own shaft) is
ers, as well as reliable software packages, has considerably attached at the top of the pendulum and, together with the end
reduced the computational burden for designing and implement- buffers, allows automatic start-up of the balancing controller. The
ing sophisticated controllers. end buffers also prevent damaging the pendulum when the balanc-
This article illustrates the design of a robust digital controller ing controller is switched off. The robustness of the control system
using a blend of state space and frequency response methods. can be experimentally tested by varying or removing the mass of
The specific application is that of balancing an inverted pendu- the disk. Collisions between the pendulum and the buffers intro-
lum on a moving cart while controlling the cart position. The in- duce torque disturbancesinto the system. Disturbance torques also
verted pendulum stabilization problem has been the subject of arise when the pendulum is subjected to sudden ‘pushes’ of short
previous research [2], [3], [9], [ l l ] , [13], [16], [18]. However, time duration. The rail inclination and disturbance torques can be
the robustness properties of such controllers have not been dis- incorporated in the system’s model as external disturbances which
cussed in sufficient detail. the controller must eliminate. The rail inclination may be assumed
This work investigates issues regarding robust stability (stabil- to be a small but unknown constant, while disturbance torques will
ity margins) and robust performance (disturbance attenuation). in general contain high frequency components. The system in-
Furthermore, the effectiveness of a simple nonlinear compensa- cludes three sensors for measuring the pendulum angle, the cart
tion scheme for counteracting Coulomb friction is investigated. position and the cart velocity.
The controller design is based on an approximate discrete-time Based on Lagrange’s formulation, a set of nonlinear differen-
linear model of the pendulum-cart system. Design calculations tial equations describing the pendulum-cart system dynamics
and analysis are carried out using Matlab/toolboxes and M-files can be obtained. The derivation follows along the developments
developed by the author. Experimental results are also discussed. in [13] and is therefore omitted. The corresponding nonlinear
The design outline is as follows: model including the motor drive unit is given by
1) The model is augmented with an ‘integrator’ to counteract
effects of constant rail inclinations. (M, + mpli- mJ cos(@- a18+ m,I sin(0 - a)6*
2 ) Design state feedback, using linear quadratic regulator the-
ory to achieve time response and relative stability specifications.
+(M, + m,)g sina + bi + F, sign(r) = au (1)
3) Design reduced order observer to reconstruct the missing
state variables, while preserving stability margins achieved by
state feedback. (J, + m,12)6- m,l cos(8 - a)r- m,gl sin8 + c6 = z.
4) Assess disturbance rejection properties of the closed-loop (2)
system for different observer dynamics. Parameter definitions and values are given in Table 1.
5) Investigate the effects of known neglected dynamics, e.g., To design the balancing controller a linearized model about
signal conditioning circuitry, with respect to stability margins the upright pendulum position is used. For sufficiently small an-
and performance.
gles 0 and a,sin0 3 0, sina 5 a , cos(0 - a ) G 1 and sin(8 - a)E
6) Design additional nonlinear compensation to reduce Cou-
lomb friction effects. (0-a). In addition, for small angular velocities 6, the term
@mJ sin(0 - a ) may be neglected; hence (1)-(2) yield the state
System Description and Modeling space model
The pendulum-cart system is depicted in Fig. 1 and consists of
a cart moving along an inclined rail (0.5 m long), a drive mecha- x(t) = Ax(t) +B,u(t) +B,a(t) + B , T ( ~+BF,Fs
) sign(i-(t))
(34
Mathematical expressions throughout this article are not italic
by author preference.

58 IEEE Control Systems

 quantization errors. An analog filter at the output of the D/A con-
verter may be used for smoothing the control signal and analog
filters prior to the A/D converter need to be incorporated to re-
duce aliasing effects. In our system, both the A/Dand D/A are 12
bit bipolar (*lo V) converters. The overall A/Dconversion time
is about 35 microseconds for each signal (including multiplexor
and zero order hold). The D/A has a settling time smaller than
one microsecond and contains a zero order hold at its output.
Each sensor signal has been suitably scaled within the range
+1OV. The nominal scale factor and corresponding sensor gain
are taken into account in the computer control software. First or-
der anti-aliasing filters are used for each measured signal. At the
output of the D/A converter a first order filter is also used to
smooth the control signal. Suitable cut-off frequencies will be
specified in a later section. Taking these filters into account, an
extended state space representation is obtained:

x'(t) = A'x'(t) +B:u(t) +B:a(t) +B",(t) +BE5Fssign(i(t))

(54

Fig. I . Inverted pendulum-cart system. y"(t) = C'x'(t) (5b)

' ,
wherex=[r 8 i 61, y = [ r 0 - a i], x(t) = CxXe(t), (5c)

0 where X' = [v,, x' V),,~I',vin is the input filter state, vOUt is the

'1 state of the anti-aliasing filters

'
A=l 0 0 0
B,=
0 2.805 -5.2563 0 ' 3.7545
10 18548 -7.7283 01 55200
Ba=[ 0 ],BT=l

-12.6157 05678 and

-185480 3.7540

"' = [-0:678l'
-03862
c=o
[
1 0 0 0
1 0 0
0 0 1 0
[
D, = :l].

(3c)

The model (3a) shows how the disturbances interact with the sys-
tem dynamics. Eq. (3b) also shows that the angle sensor reading
is directly affected by the rail inclination (this sensor is mounted
on the cart and calibrated to read zero when the rail is horizontal).
Since BF, = -B, I a, this model also suggests that when Fs is
known, a simple control law of the form

(4)

eliminates the Coulomb friction term. Generally, Fs is not accu-

rately known and it may also vary along the length of the rail, hence,
in practice, (4) will not completely cancel out the last term in (3a)
and limit cycle oscillations can occur in the closed loop system.
- -w lL r" C

For implementing a digital control system, a suitable com-

puter interface is required. This interface consists of signal con-
ditioning and data conversion circuitry: signal scaling and
filtering, A/D and D/A converters, multiplexors and samplehold
devices. Signal scaling is desirable for reducing the effects of

June 1999 59
 ~ ~~

ai" is a scalar, aoutis a 3x3 diagonal matrix, I3 and 14 denote 3x3 Table 1. Parameter definitions and values.
and 4x4 identity matrixes, respectively. For this model a com-
MC 2.3900 Kg Cart mass including motor intertia and
pensation scheme similar to (4) will not cancel the Coulomb fric-
tion term, even when Fs is known. pulley-belt drive

mp 0.9400 Kg Pendulum mass

Model Discretization
JP 0.0727 Kg-m2 Pendulum moment of inertia about
Digital controllers can be derived either by discretization of a
given continuous-time controller or they could be directly devel- center of gravitv
oped using a discrete-time system model. In either case a suitable 1 0.5358 m Distance between pivot and pendulum
sampling time interval has to be selected. In this article we follow center of gravity
the second approach. For the pendulum-cart system designs with
sampling time intervals ts of 5 ms and I O ms have been imple- 9.8100 m/s2 Acceleration of gravity
mented. We only present the best performance results forts = 5 9.7220 N N Gain of motor drive system
ms (200 Hz sampling frequency). To simplify the disturbance (neglecting motor inductance and
analysis the models (3) and ( 5 )are discretized under the assump- amplifier dynamics)
tion that the disturbances remain constant throughout the sam-
pling time interval. This is reasonable for the rail inclination but b 13.6111 Kg/s Cart's viscous friction coefficient
not necessarily true for disturbance torques. The discrete-time (including motor's bemf)
model associated with (3) takes the form C = 0 Kg-m2/s Pendulum's viscous friction coefficient

x(k+l) =@x(k)+Tuu(k)+raa(k)+r,~(k)+N(r(k);k) z 1f.fN-m Disturbance toraue

(64 U V Input voltage

y(k) = Cx(k) + D,a(k), r m Cart position along the rail

(6b)
e rad Pendulum angle with respect to the
where N(i(k);k) is a function of the cart velocityifo), 0 E vertical
w , , (k+l)t,)
a rad Rail inclination
N(i(k);k) = 6:' F, F sign (r(o + kt,))do.
eA(t,-ub '

(6c)
Fs N Coulomb friction coefficient

If Fs is known and the cart's velocity is not reversed during the

sampling time interval it can be shown that the control law In further developments the model (6) is used to design the
balancing controller. The model (8) represents the system more
u(k) = u,(k) + L s i g n (i(k)) accurately and is used in the final assessment of the closed loop
9.722 (7) system properties as well as to assist in the selection of suitable
filter cut-off frequencies.
will cancel out the term N(r(k); k). Note however that one of the
controller objectives is to achieve r = 0, hence as i(o)+ 0 cart Controller Design
velocity reversals are likely to occur more often and the compen-
sation (7) will become less effective (also motor stiction would
State Feedback and Integrator
have a larger impact on the performance of the control system).
To counteract the effects of constant rail inclinations a dis-
The corresponding discrete-time model including the filter
crete-time integrator is appended to the model (cD,T;).The inte-
dynamics is:
grator takes the form

w(k + 1) = w(k) + r(k). (9)

Here w denotes the integrator state and r is the cart's position.

y'(k) = C'xe(k) (8b) The augmented system (W,r,")is controllable, and stabilizing
state feedback controllers can be designed using discrete-time
linear-quadratic regulator theory: minimize with respect to U the
x(k) = C"x'(k) (8c) criterion

J(u) = C[x"(k)'Qx"(k) + Ru*(k)],

k=O

For this model a compensation scheme similar to (7) will be less where x"(k)' = [x(k)' w(k)], R is a positive scalar and Q is a pos-
effective in cancelling the term N'(i(k);k) even when Fs is itive semi-definite matrix. The optimal state feedback takes the
known and no cart velocity reversals occur. form [ l ]

60 IEEE Control Systems

 Cart Position: State Feedback Simulation Pendulum Angle: State Feedback Simulation
0.2 . . . . - . . 0.04

0.15 0.03

0.1 0.02

rad
m 0.05 0.01

0 0

-0.05 -0.01

4.’ -0.02
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
Seconds Seconds

Cart Velocitv: State Feedback Simulation Pendulum Anaular Velocitv: State Feedback Simulation
0.4 0.6

0.3 0.5

0.4
0.2
0.3
d.
S 0.1
radls 0.2
0
0.1

4.1 0

-0.2 -0.1

Seconds Seconds

Control Signal: State Feedback Simulation

7

Volts 3

-1
0 1 2 3 4 5 6 7 8 9 10
Seconds

Fig. 3. State feedback simulation.

June 1999 61
 u(k) = -[F, F,]xa(k)
, ,
[F, ,P(Da,
F,]=(R+r," ,PrUa)-'rua (10)

where Pis the unique positive definite solution of the discrete al- we obtain from the block diagram in Fig. 2
gebraic Riccati equation
I , I ,
P = aapaa- aaprua(R+ r; Pr;)-'r; paa+ Q.
U(Z)= -11 + K(z)G,,(z)l-'K(z>[G,,(z)a(z)+ GxT(z)7(z)1.
The selection of Q and R affect both the performance and the
(14)
relative stability of the closed loop system [ 11. For simplicity Q is Disturbance transmission characteristics are therefore quanti-
chosen to be a diagonal matrix. The following specifications are fied via magnitude Bode diagrams of the entries in the correspond-
used to obtain a suitable feedback matrix. The transfer from the ing transfer function matrixes (13)-(14). The relative stability of
equilibrium state ~'(0)'= [0.17 0 0 0 01 (i.e., pendulum bal- the closed loop system can also be evaluated in terms of the
anced but cart is at one end of the rail) to the zero equilibrium Nyquist plot of the return difference at the plant input,
state should satisfy: (l+K(z)Gxu(z)),or equivalently using the Nyquist diagram of the
loop gain at the plant input, K(z)Gxu(z).Note that here we have
Irl I 0 . 2 m , 101 I O.OSrad, li-(I 0 5 m / s , used the return difference at the plant input, however the same re-
I
101I Irad/s, Iu 5 6volt. sults can be obtained using the return difference at the plant output
since (l+K(z)Gxu(z)) = det(I+G,,(z)K(z)). The final state feed-
back that has been chosen is a compromise between speed of re-
Occasionally the limits on the control signal U may be exceeded sponse and good relative stabilityproperties. The actual values are
during short time intervals. In this formulation the controller will
bring the pendulum-cart system to the zero equilibrium state. Q = diag([10 2000 2 25 0.001]), R = 01,
Note however that by replacing r(k) by (r(k)-ref) in (9)-(10) the
controller would drive the system to the equilibrium state [F, F,]=[-35.8128 174.7939 - 353298 321409 - 0.0885].
[ref 0 0 0 O r (here ref denotes a constant setpoint specifying
the desired cart position). (15)
The relative stability properties of the closed loop system and These values for Q and R have been obtained after extensive
disturbance transmission characteristics can be quantified via trials. Simulation results for this feedback are shown in Fig. 3. In
frequency response diagrams. Combining (9)-( 10) and taking particular the large overshoot in the cart's position indicates that
Z-transforms we have the transfer of the cart from one extreme of the rail to the opposite
end would violate the rail's physical limits (k0.25 m). In such a
U(Z)= -K(z)x(z) (11) case the cart position setpoint can be smoothed by a suitable unity
gain low pass digital filter. Alternatively,the overshoot could also
be reduced by adjusting the state weights (for example, by mod-
K(z) = F, + F,[1 0 0 O](z -l)-'. (12) estly increasing the cart position weight Ql1 and decreasing the in-
tegrator weight Q55). We point out that increasing Ql1 produces
Defining larger angles and velocities during the transient while decreasing

2, I

-16 -14 -12 -10 -8 -6 4 -2 0 2

Real
(b)

Fig. 4. Loop gain Nyquist plots: ( a ) Overall Nyquist (not scaled), (b) Nyquist in the neighbourhood of the critical point (-1,O).

62 IEEE Control Systems

Qss reduces the closed-loop system response to disturbances. Fig. X(k) = L,y(k) + L,v(k), (17)
4(a) shows a typical Nyquist diagram for the open loop gain
K(z)Gxu(z).Since the open loop system has one eigenvalue out- where K(k) denotes the estimate of x(k), E is a scalar with magni-
side the unit circle, the requirement for closed loop stability is that tude less than one and E, H, M, L1, L2 satisfy
the critical point (-1,O) should be enclosed once in an
anticlockwise direction. Fig. 4(b) shows the Nyquist plo TO- ET - MC = 0 (18)
controller (15) in the neighborhood of the critical point
line). The nominal design has approximately a phase
67.5 degrees and tolerates substantial gain fluctuation H = TT, (19)
8.9). We point out that continuous-time optimal regulators
have-in theory-impressive relative stability properties, how-
ever discrete-time optimal regulators do not necessarily exhibit -1
similar properties 111. Disturbance tansmission plots for the state [L, L*l=[;] .
feedback controller (15) are shown in Figs. 5 and 6 (dashed lines).
(20)
Only the effects on cart position and pendulum angle are included
in this article. Fig. 5 shows excellent attenuation properties with The matrix is a design for which the inverse in (20)
respect to rail inclinations, however, these plots assume that the exists. Replacing x(k) in by ?(k) and taking Z-transforms,
state are measured. Torque disturbancetrans- the observer based controller (9)-(lo), (16)-( 17) takes the form
mission plots are shown in Fig. 6. For frequencies up to 0.01 Hz
disturbance torques are sufficiently attenuated.
w(z) = [ 1 0 O](z - l)-' y(z)
Robust Reduced Order Observer %(k)= L,(zI - E)-'Hu(z) + [L, + L,(zI - E)-'M]y(z)
It is well known that observer based controllers may yield dis- U(Z) = -F,~(z)- F,w(z).
appointing stability margins [ 13, [6], [ 121, [ 141, hence some cau-
tion is required when selecting the observer dynamics and design
Substitution of the first two equations in the third expression
parameters. For the pendulum-cart system a reduced order ob-
server is to be designed so as to preserve the relative stability gives' after Some
properties achieved by the state feedback gains (15).In this sense
we refer to this observer as a robust observer. The design is to be u(z) = -K(z)y(z) (21)
carried out using the nominal model (@,T,,C). The pair(@,C)be-
ing observable is sufficient to guarantee the existence of a re-
K(z) = [ 1 + F,L,(A - E)-' HI-'
duced order observer with arbitrarily chosen dynamics. For the
inverted pendulum-cart system the observer order is one and has X(F,(L, + L,(zI-E) ' M ) + F,[1 0 O](Z-l)-'].
(22)
the general form
Defining G,,(z) = CA(z)ru, the loop gain at the plant input is
v(k+l) = Ev(k)+Hu(k)+My(k) (16) given by

-20 -

Db
\

120- /

140- , /

0.001 0.01 0.001 0.01 0.1 1 10 100

Frequency (Hz)

Controller
Filters

Fig. 5. Magnitude Bode diagrams: Rail inclination amplitude 0.25 rad.

June 1999 63
 State Feedback

Fig. 6. Magnitude Bode diagrams: Torque disturbance amplitude 1Nm.

K(z)G,,(z) = [ I + F,L,(zI - E).' H1-l

x[(FX+ F,[1 0 0 O](Z-I)~')A(Z)T,,-F,L,(ZI-E)~'H].
(23)
In the above derivation we have used the observer equations
(18)-(20) in the equivalent form
then from the block diagram in Fig. 2 we have
(zl - E)-' MC(z1- @)-I = T(z1- - (ZI - E)-' T
@)-I

L,C + L,T = I

and the identity

u(z) = -11 +K(z)G,,(z)l-'K(z)[G,,(z)a(z) +Gy,(z)T(z)l.
[l 0 O]C=[l 0 0 01. (25)
The algorithm for designing the reduced order observer is
For H = 0, (23) is identical to the loop gain for the state feed- summarized below.
back controller; hence the relative stability properties achieved Let T = [tl t2 t3 t4 and M = [ml m2 ms], for a given E and H,
by (9)-( 10) are fully recovered, i.e., full loop transfer recovery equations (18)-(19) can be rewritten using Kronecker tensor
(LTR) is accomplished whenever H = 0. Note that unlike the products [ 171 as
standard LTR methods in [7] and [ 151, the recovery using H does
not require squaring up the plant by adding dummy inputs. Re-
duced order observers with H = 0 and arbitrary selection of the
eigenvalues of E can be designed for observable systems if
rank(CT,) = rank(rJ and the triple (@,T,,C)has no invariant ze- wherep = [ t , t, t, t, m, m2 m3Y,
ros [5], [lo]. For our system these conditions are satisfied.
It is interesting to note that when H is not zero, (23) indicates
how the relative stability may be degraded by inclusion of the ob-
server in the feedback loop. In some cases an approximate LTR
at the plant input can be achieved by selecting H to be 'small,' but
I, 8 @' - E 8 I, -I, 8 C'
'1x3 1' (27)

there are cases when this might fail. Examples can be found where IE and ID denote identity matrixes of sizes consistent with
within the class of unstable systems that are not strongly stabil- E and @, respectively.
izable [19], [8], [14], i.e., systems that can only be stabilized by To guarantee the existence of the inverse in (20) we only need
unstable controllers. to set t/l different from zero, for simplicity we take t4 = 1. Also,
The disturbance transmission properties for the observer since the observer is to estimate the pendulum angular velocity,
based controller are evaluated from the corresponding closed ml can be set to zero, thus making the observer independent of
loop transfer function matrices. Let the cart's position. These additional constraints can be written as

64 IEEE Control Systems

Fig. 7. Experimental results: Transient response.

E = 0.9, T = [0 - 20.0350 - 13964 1.01,

M = [0 - 1.9350 - 0.13961

where
(28)
r' O 0 I 01

Eqs. (26)-(29) yield Disturbance transmission plots for the observer based control-
ler are shown in Figs. 5 and 6 (dash-dot lines). The effects of rail
inclinations are shown in Fig. 5. These plots indicate that the
controller is quite effective in eliminating the effects of con-
stant rail inclinations. Fig. 6 shows disturbance torque trans-
mission plots. There is a small degradation in the low frequency
region and for frequencies above 2 Hz there is a modest im-
The matrix S is now square and depends only on the chosen provement.
observer dynamics. For a given E and setting H = 0, solving
(30) yields a unique solution whenever S'' exists. The above Filter Cut-off Frequency Selection
scheme is easily implemented as a Matlab M-file. In our imple- Since the smoothing and anti-aliasing filters are not taken into
mentation the observer eigenvalue was chosen as E = 0.9 and account in the design of the controller, the relative stability of the
represents a compromise between the speed of response of the closed loop system will degrade as the cut-off frequencies de-
observer and disturbance transmission characteristics of the crease. An upper bound for the cut-off frequencies is usually
closed loop system. In general, as E+O, the closed loop system taken as one-half of the sampling frequency (in our case 100 Hz).
becomes more sensitive to disturbances. The actual values for To select suitable frequencies we assess the loop gain and distur-
the observer are bance transmission properties of the observer based controller

June 1999 65
applied to the model with the filters. The controller (22) is now and Y is a constant to be selected experimentally. As pointed out
acting on the model (8). Defining previously, (35) will not completely eliminate the terms arising
from Coulomb friction, but as the experimental results will show
Ae(z) = (zl-W-', G:,(z) =CxAe(z)r:, G:,(z) =CxX(z)r:, substantial reductions in the amplitude of the limit cycles can be
obtained by suitably selecting the gain Y.
G:(z) = cxx(z)r:,
G;JZ) = ceAe(z)r;, G;,(z) = cex(z)r:,
G:,(z) = CeAe(z)rz, Q'(z) = G:,(z)[l +K(z)G;,(z)]-'K(z), Experimental Results
The observer based controller was implemented in a 386
the disturbance tansmission properties of the closed loop system DW20 MHz personal computer in the C language. The overall
are given by program includes code to capture experimental data and store
this information in the hard disk. The program is compiled to op-
timize execution speed but not memory requirements. All calcu-
lations are carried out in floating point arithmetic using a math
co-processor. In all experiments the observer and integrator ini-
tial conditions were set to zero. The transient response for an ini-
tial condition x(0) = [0.171 0.02 0 O r is displayed in Fig. 7,
while Fig. 8 shows the cart position after 300 seconds. These fig-
ures indicate that the design specifications given earlier are satis-
fied. The gain Y for the nonlinear compensation scheme (35)
was experimentally determined and set to 0.8. This value reduces
K(z)G;"(z). (34) the amplitude of the cart's position limit cycle by a factor of 3.
Additional experiments were carried out varying the sampling
The Nyquist plot of the loop gain (34) is shown in Fig. 4(b) (solid time interval from 3 ms to 13 ms, as well as removing the alumi-
line) for ain = 259.4707 rads (41.3 Hz) and aout = 227.2727 13 num disk at the top of the pendulum (0.227 kg). In all cases the
rads (36.17 Hz). The phase margin for this case is about 30 de- system performance was stable, confirming the robustness prop-
grees and the upper gain margin is reduced to 4. The magnitude erties of the control system.
Bode plots for rail inclination and torque disturbances are dis-
played in Figs. 5 and 6 (solid lines).
Conclusions
A robust computer control system for balancing a single in-
Coulomb Friction Compensation
verted pendulum on a moving cart was successfully designed us-
It is well known that Coulomb friction (including stiction)
ing a blend of state space and frequency domain methods. In
can produce limit cycle oscillations in a control system. To re-
particular, the controller design illustrates how the available
duce the amplitude of these limit cycles the state feedback gains
freedom inherent in reduced order observer theory can be used to
could be increased at the expense of reducing the relative stabil-
avoid degrading relative stability of observer based controllers
ity of the closed loop system. Alternatively,suitable dither signals
and whenever possible to fully recover state feedback robustness
could be used to reduce the amplitude of these oscillations [4]. In
at the plant inputs (the case H = 0). The analysis of c l o s e d loop
this section we consider a simple scheme based on velocity-sign
disturbance transmission properties provides useful benchmarks
compensation using the signal at the output of the corresponding
for the stabilization problem of the now classical pendulum-cart
anti-aliasing filter, i.e.,
system. Experimental results show that a simple velocity-sign
compensation with a fixed gain Y can be quite effective in reduc-
u(k) = + Ysign([O 0 lly'(k)), u,(z) = -K(z)y'(z),
ing limit cycle oscillations arising from nonlinear friction. Cur-
(35)
rent research work involves techniques for on-line estimation of
the gain and the use of neural networks to counteract Coulomb
Cart Position Limit Cycle: Experimental Results friction effects. Future work will focus on modeling the effects of
0.04 I * - Coulomb friction as an external disturbance and the design of a

0.02
0.03
0.01
t
1 f
1
/
/
' \
\
\
control system which is insensitive to this type of disturbance.
Double and triple inverted pendulum-cart systems are also being
developed.

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66 IEEE Control Systems

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