H,
Control
of
an Experimental
Inverted Pendulum' with Dry Friction
Gert-Wim van der Linden and Paul
E
Lambrechts
n important problem in the application of linear control
Ath
eory on mechanical systems is the occurrence of dry fric- tion. Several authors have studied this problem (e&, [7], [18]). Here we consider the incorporation of dry friction in a standard plant setting, and the application of Hm optimal control design to find a controller that is insensitive to dry friction. The approach will be applied to an inverted pendulum setup, to show that an important improvement of behavior can be obtained.First, it will be shown that the application of an H, optimal controller does not result in a satisfactory behavior if dry friction is not taken into account and cart position reference error mini- mization (tracking) is demanded. Next, we will consider only the effect of dry friction, leading to a controller that is insensitive to this effect, but has no reference tracking property. Finally a trade-off between the tracking objective and the insensitivity to dry friction is performed, resulting in a controller with good behavior in both respects. This behavior has been validated by means of implementation on a laboratory setup.
The inverted pendulum problem has been considered a suit- able test-case for controller design methods by many authors, like Furuta et al. [SI, Wang [ 191, and Meier zu Farwig and Unbehau- wen [ 131. It is a popular problem because the setup is simple, yet has interesting features like instability, geometric nonlinearity, and dry friction.
Also, H, optimal control, as initiated by Zames [20], is well established in literature (see for instance [2], [3], [lo]). This has led to several successful applications, as reported by Morari and Doyle [ 141 and Skogestad et al. [ 161.
We will start by giving some preliminaries on Hm theory. Then the experimental setup is described in the next section, and a linear model of the setup is introduced. The dry friction phenome- non is covered separately, resulting in a suitable description, to be used at the control design stage. Then results of the implemen- tation of the final controller on the experimental setup are pre- sented and some conclusions are given in the last section.
Preliminaries
Standard Plant ConfigurationFig. 1 shows the general framework that we will use in this article. Any control problem within a linear setting may be written in this form.
Presented at the First IEEE Conference on Control Applica- tions, Dayton, OH, September 13-16, 1992. The authors are with the Mechanical Engineering Systems und Control Group, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands.
YA
Fig. 1. Standard plant ( P ) with controller ( K ) and uncertainty
(A).
The standard plant P incorporates the plant and provides an
interconnection structure, defining the way in which the uncer-
tainty block A and the controller K act on the system.
The inputs to the standard plant are: the output of the uncer- tainty block UA; the disturbances acting on the standard plant w,
such as reference signals and noise; and, finally, the controller- The outputs of the standard plant are: the input to the uncer- tainty block Y A ; the control objectives
z ,
such as tracking error and control effort; and the measurable signals YK.The closed loop transfer function from w to
z
will be denoted as T&). The control problem can then be stated as the minimi- zation over all possible controllers K ( s ) of TW&) in some given norm, for the worst case A. This uncertainty block A is allowedto be any arbitrary element of a given set, representing the difference between the linear model and reality.
OUtpUt UK.
H, Control Design Method
In order to find a solution to the control problem represented by the standard plant, the H, control design method is used. The goal is to find the controller which minimizes the transfer func- tion TWXs) with respect to the infinity norm, denoted by 11.1L and defined as
IIH(s)ll, := sup omax(H(s))
where
omax
denotes largest singular value and C+ denotes the closed right half of the complex plane. For a more elaborate introduction to H, control, see [2,10]; an in depth discussion can be found in [ 171.To be able to optimize certain design goals (usually perform- ance), under some constraints, such as robustness and noise sensitivity reduction, the standard plant (in fact the weight func- tions reflecting the goals which are to be optimized) may depend on a free parameter y E
R.
By defining y such that the demand with respect to one or more design goals increases when y increases, we may define the
H, control design problem as finding the largest y such that there exists a controller K ( s ) for which
An important consequence of (2) for the sequel of this article is that the inifinity norm of the uncertainty must be bounded. Otherwise, no solution can be found.
If no uncertainty block is present, a direct computational solution is available, based on the solution of two algebraic Ricatti equations. This solution has been implemented in the software package MATLAB386 [ 1 I ] using a small adaption of the algorithm given by Glover and Doyle 161 (see also [3]).
Experimental Setup
Description
The inverted pendulum setup (see Fig. 2). can be split into four parts, which will be considered consecutively: mechanical part, sensors with accompaning electronics, digital control sys-
ings have a considerable amount of friction (the drivetrain also contributes to this) and will have to be taken into account at the control design stage.
The drivetrain consists of a toothed belt, actuated by an electrical servomotor. The belt is attached to the cart, and com- bined with a toothed wheel on the motor it provides slip-free traction. Motor, drivetrain, and cart have an equivalent mass of 2 kg.
Sensors with Electronics. The measurement signals are ob-
tained from three sensors:
1 . A magneto-restriction sensor alongside the guiding bars for measuring the linear displacement of the cart.
2. A rotary variable differential transformer (RVDT) at the rotation point of the pendulum, measuring the angle.
3. A tacho generator on the servomotor, measuring the angular velocity of the motor, which is approximately linear dependent on the cart velocity.
All sensors are sufficiently accurate (less than 1% deviation)
and have a high bandwidth (well exceeding 100 Hz).
Digital Corztrol System. The digital control system is a DSP-
based system of dSPACE GmbH [4], with an IBM-AT compat- ible host (see Fig. 2). Accompaning software provides discretization, scaling (see, for instance, [SI) and assembly source code generation almost automatically.
The use of fast A/D and D/A converters in combination with a Texas Instruments TMS320C25 DSP provided adequate com- putational speed under all circumstances.
Actuator with Power Amplifei: The actuator is a 400-W DC electromotor connected to a voltage driven power amplifier. Under normal circumstances, the motor with amplifier has a linear behavior, with a high bandwidth (well above the eigenfre- quencies of the mechanical part).
x x n
Fig. 2. Inverted pendulum setup.
tem, and the actuator with power amplifier.
Mechanical Part. The mechanical part of the setup consists
of a cart, a pendulum, and a drive-train. The hollow steel pendu- lum has a weight of 0.6 kg and a length of 57cm from the center of rotation to the tip. The pendulum can rotate in the vertical plane by means of low friction rollerbearings. The angular deviation is limited to +15" from the vertical.The aluminum cart can move along steel guiding bars, also using rollerbearings. These bear-
Mathematical Model
Since the dynamics of the sensors and the actuator are rela- tively fast, we will combine a dynamical model of the mechanical part with static models of all other components. A nonlinear rigid body model of cart and pendulum has been derived by means of Kane's method [ 9 ] , with the software package Autolev [ 151. This model can simply be linearized manually. In combination with the static models, this results in the following fourth-order state space model:
G:{
The state
5
consists, respectively, of pendulum anglea,
angular velocitya,
cart positions, and velocity .U. The control input u equals the controller output (a voltage on the amplifier) and the disturbance input v equals the disturbance force on the cart (mainly friction forces). The output y consists of all measurable variables:a,
x, andX.
Note that G,(s) is the transfer function from the control input to the measurable outputs, and G,js) from the disturbance input to the measurable outputs. This linear mathematical model will be the central item in the standard plants as considered in control design phase.Dry Friction
To be able to incorporate dry friction in the standard plant setting, a characterization of this phenomenon is necessary. Since we will be using H, control design, this characterization must be linear, possibly accompanied by a bounded perturbation (in Hm sense) on that linear description to capture the modeling error as uncertainty.
First, an attempt is made to preserve the feedback nature of dry friction, and construct an appropriate model which relates the state and inputs of the system to the dry friction force. However, it will be shown that this is impossible and an altemative ap- proach will be proposed.
Characterization of Dry Friction
Generally speaking, a dry friction force occurs at the plane of contact between to surfaces. The force acts opposite to the direction of motion, and is highly nonlinearly dependent on the velocity. Two cases must be distinguished:
Case I : the relative velocity
x
of the bodies is zero (sticfion).If a force is applied on a body at rest, it will not move until the applied force exceeds some limit (stiction or backlash force). This can be modeled as a dead-zone effect.
Case 2: the velocity is not equal to zero (Cou/omhfi.ic.tinri). When a body (with dry friction) is moving at a certain velocity, a more or less constant Coulomb friction force Fc will act on it, having the opposite direction a2 the direction of motion. Usually the magnitude of this force F is dependent on the velocity, especially at low speeds. Of course it is also dependent on the surface structure and lubrication. To simplify the dry friction model, the magnitude of
fe
Coulomb friction force will be chosen constant (so Fc, = -Fsign(:\-)).Futher refinements on the friction model can be made, but are not of concern here. For more information on modeling, simula- tion, and compensation of dry friction, see Gotzmann and Meyer [7] and Walrath [ 181.
Uncertainty Model of Dry Friction
The uncertainty model of dry friction should address both Case 1 and 2. For both cases, the friction force F is dependent on the state of the plant
6
and in case of stiction also dependent on the applied control input u. This results in Fig. 3. In case 1 (zero velocity), the dead-zone effect can be modeled quite simply: the magnitude of the friction force will never exceed the applied force, so the dead-zone can be written as an uncertain gain between zero and one. times the transfer function from(6,
u ) to the applied force.In case 2 (non-zero velocity) however, we have the problem that around
S
= 0, the Coulomb friction function has a disconti-I I I
1
Fig. 3. Model of plant with friction as uncertainty.
nuity (because of the effect of stiction). This implies that if we model the force F due to Coulomb friction with an uncertain gain:
we find for the extremes of the operating conditions:
( 7 )
When modeling the uncertain gain k as a nominal gain ko with a
perterbation term @ k , the latter is unbounded:
Hence this model does not fit the H, control design theory. Simplified Dry Friction Model
As the uncertainty model is not suitable to incorporate the dry
friction effect in a standard plant setting, an alternative solution must be found. The usual approach to the reduction of dry friction influence is the construction a disturbance observer. In that case an extra state is added to the controller to be able to estimate the dry friction force (hence the intemal model of dry friction is an integrator). The result then resembles the servocompensator approach of Davison [ I ] for a step-like disturbance input signal, which uses extra dynamics to describe this signal.
To be able to incorporate the dry friction phenomenon in the standard plant setting, but without the necessity of adding dy- namics, a similar approach is proposed here:
Model the dry friction force as an external disturbance force. Hence, add a (possibly weighted) disturbance input to the standard plant.
Thus, only where the dry friction force acts on the plant is emphasized. As with the disturbance observer approach, the knowledge of the feedback nature of dry friction is lost.
The implications of this approach on the control design of an inverted pendulum setup are covered next.
Fig. 4 . Standaizlplaiir.
Control Design
In order to arrive at a satisfying controller, a few steps have to be made. Each step will consist of the construction of a standard plant and the calculation of the accompaning Hm opti- mal controller. The complete standard plant is given in Fig. 4. Inputs to the standard plant are: n : measurement noise; d,, plant output disturbance (position reference); d,: input disturbance (friction force); and u , the control input. Outputs are: ?: the weighted positon error; the weighted control effort; and finally
y c the outputs of the plant available to the controller K . The weight functions W.. follow directly from these definitions. Note that not all inputs will be used in the first two design steps.
The design will be based on a fixed linear model of the setup since, apart from the dry friction, no uncertain effects have to be modeled to obtain sufficient accuracy.
The control of the inverted pendulum setup will have two goals: stabilization of the pendulum, and positioning of the cart. Also, the control energy must be limited, and measurement errors should not be amplified too much. Furthermore, the controller must be insensitive to the effect of dry friction.
The obtained controller will be tested on a nonlinear MA- TRIXX SystemBuild [ 121 model, incorporating the nonlinear equations of motion of cart, pendulum and drive train with dry friction.
Tracking Controller
To show that it is necessary to take dry friction into account. first acontroller will be designed to meet the tracking objective only. This results in a standard plant as depicted in Fig. 4, but without the input
d;. For the objectives and the matching weight functions. see Table I.
To obtain good reference following properties, the transfer function from d,, toy, (the cart position) must be small, especially at low frequencies. This may be achieved by minimizing the
Table I
Tracking Objective and Weight Functions
'
~ _ - _ _
r
7-------
-SO Reference signal weight WV ( S .
Y
) = ___k-
- ~ L _ - - -y-'
s+
~ 1Control ener limitation
w,,
= 5Noise sensitivit limitation
'
w,,,
= lo-'/I==
Position error weight ~,
Wpc = [O 1 01pp
transfer function from do to
e.
Therefore, Wr s is chosen to be a first order weighting function, with the cross-over frequency parameterized by y. This frequency can be found by solving50
(10) which has the solution
This implies that (given W,, = [0 1 01) the closed loop system has a minimal bandwidth of SOX
y
rad/s. The optimal solution will find the maximal bandwidth, given the other weight func- tions.The static position error can be found by evaluating the closed loop transfer function from d, to Z at w = 0. resulting in an error of less than 1/50 = 2%.
Also note that only the posirion error is weighted. Since
stability is guaranteed a priori if an Hm optimal solution exists, the pendulum angle need not be weighted. In fact, if the pendu- lum angle is weighted, it will reduce the reference following properties. since pendulum and cart are inherently coupled.
To conform to the Glover and Doyle demands [6] for existence of a solution, an extemal force-input must be added, to prevent a transmission zero at s = 0. A small weight (W,f= will be put on the cart disturbance force d;. such that this will not influence the result.
The standard plant P ( s , y) (neglecting the cart disturbance
T ; ( s ) := ( 1
+
K(s)Gu(s))-'K(.s)G,s(s)T ! ) := ( I
+
C1,(s)K(s))-lGu(s)K(s) the resulting closed loop transfer function isTheH, iteration yields an optimal yof 0.0334, so the bandwidth must be at least 50x
y
= 1.5 rad/s, such that in the time domain a rise time of at most 2 s is expected. This is confirmed by the0.15 0.1
-
0.05 0 -0.05-i----
-1---
-
! / - -I
:/i
/ I tracking-
_ - -
disturbanceI
.---combination!
-
1 time[SI
6 - 3 ar 0 0, s o -3Fig. 5 . Simulated step responses.
0 2 0 1
-
E x o I 0 1 0 2 0 5 0 25-
a, ar a 0 F I 0 25 0 0 5 1 1 5 2 2 5 3 time[SI
Fig. 6 . Force disturbance responses
step responses (the solid lines in Fig. 5). However, as we can see in the response to a unit force disturbance on the linear system (Fig. 6), the tracking controller allows a large excursion of the cart and pendulum, to compensate the applied force. This might indicate a sensitivity to dry friction.
0 1 0.05
-
E
O
-0.05 -0 1-
a, 0 a, 5 2.5 0 -2 5 -5 1 0.5 0 -0.5 0 0.5 1 1.5 2 2.5 3 time[SI
time Is] 0 0 5 1 1 5 2 2.5 3 time [SIFig. 7. Nonlinear simulations.
Indeed, a simulation of the controller applied on the nonlinear model shows unsatisfactory behavior; see Fig. 7. There is a large
limit cycle, due to the combination of dry friction and instability of the plant. Therefore the approach, as sketched in the previous section, will be applied to improve this behavior.
Disturbance Attenuating Controller
To find out how much dry friction force influence reduction can be obtained, the tracking objective will be temporarily re- moved and attention is put on the disturbance force only, so do is
Table I1
~ Disturbance Attenuation Objectives and Weight Functions Cart force weight
not used in Fig. 4. The weight functions can be found in Table 11. The standard plant P(s, y) now appeares as:
The resulting closed loop transfer function is
H, optimization then results in an optimal y of 4.1 I . Note that
this y is the maximum weight on the cart disturbance force. under the restriction that IIT,.As, y)lL I 1. The nonlinear response to a nonzero initial angle (fig. 7 ) shows a quick recovery of the angular deviation, without excessive control effort. Striking is the immediate reaction of the controller on a change in direction of movement of the cart, directly coun- teracting the dry friction force. There is a small drift of the cart left, which results in a very slow limit cycle, with minimal excursions of the pendulum angle. As expected, the force responses (Fig. 6) show a very small excursion of cart and pendulum.
Combined Controller
To find a proper trade-off between the tracking objective and insensitivity to dry friction, a combination of the previous two standard plants is made, resulting in Fig. 4 with all inputs used. The matching weights for the combination of performance and disturbance attenuation are presented in Table 111. An acceptable trade-off was obtainable with a disturbance force weight of 1 . The standard plant for the combined controller then is:
y---
-
-
,
I4
so
I
1
Table 111Combined Objectives and Weight Functions
- ~ _ _ _
Reference signal weight Cart force we4ht
Control energy limitation
1
W , , = S- i
-1
Noise ,ensitivity reduction!
W,,, = I O ' 1Povtion error weight
1
W,,, = [O I 01 1I
wfx
( S *Y
= ___y-' \ + 1
-]pLPL----
I
~ _ _
resulting in the closed loop transfer function:
I , I 1 1 0 0 5 1 1 5 2 2 5 3 3 5 time [SI 0 0 1 , 0 01 0 5 1 1 5 2 2 5 3 3 5 time Is]
Fig. 8. Limit c.ycle coritrnlled setup
It is remarkable that in this configuration an optimal y of 0.0329 is obtained: this implies that the bandwidth of the closed loop system is only slightly smaller than in the case of the tracking controller (y = 0.0334). This is confinned in the step reponses (Fig. 5 ) : they can hardly be distighuised from the tracking
controller.
Fig. 6 shows a 30 times reduction of the cart excursion, compared to the tracking controller. As a result, the simulated limit cycle is drastically reduced (see Fig. 7). Note the 1 0 0 ~ magnification. This controller will therefore be applied to the experimental setup in the next section.
Implementation of the Controller
The controller as obtained in the previous section is a fourth order, linear, proper continuous time controller in state-space description which was implemented by means of the digital control system.
First the resulting limit cycle of the closed loop system is shown in Fig. 8; while a simulation of the tracking controller showed a limit cycle of about 180 mm, the implemented com- bined controller has reduced this to about 5 mm, which is a
tremendous improvement.
Interesting may be the fact that traditional methods of reduc- ing the influence of dry friction effects, being dither and veloc- ity-sign c o m p e n s a t i o n , o n l y resulted in a f a c t o r o f 2 improvement. This is a valuable gain, but not of the order of magnitude which was reached by a redesign of the controller.
When the measured signals of Fig. 8 are compared to the simulation results of the combined controller in Fig. 7, the limit cycle shows some differences. Both the amplitude and shape of the responses are different. This is probably due to unmodeled effects, such as measurement offset, vibration of the toothed belt and some friction in the pendulum bearings.
Next, the step response to a step on the position reference signal is shown in Fig. 9. Within 1 .S s the cart has reached the reference position.
Influence of Dry Friction Can Be Reduced
It is shown that the influence of dry friction on a controlled mechanical system can be largely reduced, within a linear setting. The standard plant approach has been very useful for this, in that it allows adding or removing objectives, simply by adding ortime
[SI
Fig. 9. Step response controlled setupremoving weight functions. By altering weight functions one can find a proper trade-off between various objectives.
There appears to be a strong connection between cart distur- bance force influence reduction, and the magnitude of the limit cycle. By adding a weighted disturbance input to the standard plant, an attempt has been made to incorporate the effect of dry friction. Based on this extended standard plant, an H, optimal controller was calculated, which resulted in a satisfying imple- mentation on an experimental inverted pendulum setup.
Since the standard plant setting is not limited to H- optimiza- tion only, the approach to dry friction as presented here can also be applied in combination with other optimization techniques available for this framework, such as H? optimization or p-syn- thesis.
References
[ I ] E.J. Davison and A. Goldenberg, “Robust control of a general servo- mechanism problem: The servo compensator,” Autoniutic~z. vol. 1 1 . pp. 4 6 1 4 7 1 , 1975.
[2] J.C. Doyle , B.A. Francis. and A.R. Tannenbaum. F e ~ d h ~ ~ k Control Theory. Maxwell MacMillan. 1992.
[3] J.C. Doyle, K. Glover. P.P. Khargonekar. B.A. Franci5, “State-\pace solutions to standard H2/H- control problems.” /E€/? fi-uns Auto. Control. vol. AC-34, no. 8, pp. 831-847. 1989.
141 dSPACE, DSP-CITpm Hurdfiur-r murruul. dSPACE digital signal proc- essing and control engineering GmbH. An der Schtinen Aussicht 2. D-4790 Paderbom. West-Germany. 1989.
[SI K. Furuta. T. Ochai, and N. Ono. “Attitude control of a triple inverted pendulum,” I n t . J . Control, vol. 39, pp. 135 1-1365. 1984.
[6] K . Glover and J.C. Doyle, “State-space formulae for all stabilizing controllers that satisfy an H- norm bound and relations to risk sensitivity”. Sysr. Conrr-nlLerts., vol. I I . pp. 167-172. 1988.
[7] W.H. Giitzmann and H. Meyer, “Zur digitalen Simulation von Ueber- gaengen zwischen Gleit- und Haftreibung,” A z r t o n i o t r . s i c ~ ~ . u r i , ~ . s l ~ c ~ f n / k u t , vol. 37, no. IO, pp. 399400, 1989.
[8] H. Hanselmann, “Implementation of digital controllers - A surveq.” Automarica, vol. 23, pp. 7-32. 1987.
191 T.R. Kane, D.A. Levinson, Dynamicx T h e o y and Applications. New York: McGraw-Hill, 1985.
[ 101 J.M. Maciejowski, Mulrii,ariah/e Feedback DesrKn. U.K.: Addison Wesley, 1989.
[ 111 MATLAB,MATLABrefirent.emanuul.The MathWorksInc.,MA, 1987.
[ 121 MATRIXx. User Guide MATR1X.v 1‘6.0. Integrated Systems Inc., CA, 1986.
[ 131 H. Meier zu Farwig and H. Unbehauen. “Discrete computer control of a triple inverted pendulum,” Optimal Control Applic. Meth., vol. I I , pp.
157-171. 1990.
[ 141 M. Morari and J.C. Doyle, “A unifying framework for control system design under uncertainty and its implications for chemical process control,” in P/-oc.. CPC / I / . Assilomar CA, 1986. pp. 5-5 I .
[ 151 D.B. Schaechter. D.A. Levinson. T.R. Kane, Autolei. Uiser’s Manual, Online Dynamics. Inc., Sunnyvale, CA, 1991.
[ 16) S. Skogestad, M. Morari. and J.C. Doyle, “Robust control of ill-condi- tioned plants: High purity distillation,” / € E € Tr-ans. Auto. Control, vol. AC-33, pp. 1092-1 105, 1988.
[ 171 A.A. Stoorvogel. The H- Control Problem: A State Spac.e Approach. Englewood Cliffs. NJ: Prentice Hall. 1992.
1181 C.D. Walrath, “Adaptive bearing friction compensation based on recent knowledge of dynamic friction,” Automotica. vol. 20, pp. 717-727, 1984. [ 191 J . Wang, “Optimally robust output feedback control for SISO nonlinear systems.” in Proc. IEEE Cor$ Deci.rion und C(n?lr0/., vol. , no. , pp. 2 3 7 6 2 3 8 1. 1989.
l20] G. Zames. “Feedback and optimal sensitivity: Model reference transfor- mations. multiplicative \eminorms, and approxiinate inverses,’’ /E€€ Trans.
Auto. Contwl, vol. AC-26, no. 2, pp, 301-320. 1981.
Gert-Wim van der Linden is a Ph.D. student at the
Delft University of Technology, The Netherlands, in the Mechanical Engineering Systems and Control Group, where he received the M.S. degree in me- chanical engineering in 1991. His current research interest is the implication of task-based robotics on controller design methodologies. Specific areas of interest include nonlinear control theory, passivity based control methods, and DSP-based digital con- trol systems.
Paul E Lambrechts graduated in 1987 in the Me-
chanical Engineering Systems and Control Group at the Delft University of Technology. The Nether- lands. His M.S. theGs on the control of the tape transportation system of a digital video recorder, performed at the Philip Research Laboratories in Eindhoven, was awarded that year‘s prize for best graduation work by the Mechanical Engineering And Marine Technology Department ofthe Dutch Royal Institute of Engineers (KIVI). He is currently a Ph.D. student at the Delft Mechanical Engineering Systems and Control Group, temporarily working at the Dutch Defense Research Institute PML- TNO. His main research interest is the application of robust control theory like H- and p design methods on mechanical servosystems using state-of- the-an digital control hardware.
