Control of unstable mechanical system Control of pendulum]

I:-1T. J. CONTROL,

1976,

VOL.

23,

No.5,

673-692

Control of unstable mechanical system Control of pendulum] SHOZO MORI,t HIROYOSHI NISHlHARAt and KATSUHISA FURUTAt

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This paper relates to the design of a control system for a mechanical system which contains an unstable mode and to an experiment for demonstrating that the control theory may be applied to practical real systems. In this paper t.he object treated is the control of the pendulum-cart system, which has been studied by many control theorists and engineers as an inverted pendulum problem, and a composite control system is designed not only to stabilize the inverted pendulum but to swing the pendulum from the natural pendent position up to r.he inverted posit.ion, which is actually made. The experimental results are presented.

Introduction In the development of the control theory, its applicability to practical problems has been often demonstrated by synthesizing control systems for practical systems with simple mechanisms. A typical example of such a system to be controlled is an inverted pendulum, particularly as an example of an unstable system. Many control systems for stabilizing the inverted pendulum have been presented. Schaefer and Cannon (1966) designed a control system to stabilize a bending beam in the inverted position using bang-bang type control. Koenigsberg and Frederick's (1970) controller for the same system was based on the output feedback. Further, Strugeon and Loscutoff (1972) designed a control system to stabilize a double inverted pendulum using an observer-regulator type dynamic stabilizer. These examples until now have used linear models and stimulate the further application of modern control theory to more complex systems. This paper shows how to control a pendulum system, considering the overall characteristics including its non-linear property and so the model considered is non-linear and more complex but may represent the real system more closely. The objective of the control described in this paper is not only to stabilize the inverted pendulum carried by a cart in its 'inverted position in the centre of the cart-moving area, but also to transfer the pendulum from its natural stable equilibrium point, i.e. its pendent position, into the zone in which the pendulum can be stabilized about its inverted position, i.e. its inherently unstable equilibrium point. The authors developed an idea for synthesizing such a control system, and experimentally manufactured the real control system. The control system consists of the following three controllers: (1) a feedforward controller that is to 'swing up' the pendulum from its pending I,

Received 28 July 1975. t This research is in part supported by a grant from the Ministry of Education, 1975.

t Department of Control Engineering, Tokyo Institute of Technology, Tokyo, Japan 152. CON.

5F

8. Mori et al.

674

position to its inverted position; (2) an observer-regulator type dynamic stabilizer; and (:3) a switching mechanism which. switches the above two modes of controls. System description and problem statement The subject system consists of (I) a cart moving along a line on a monomil of limited length, (2) a pendulum hinged to the cart so as to rotate in the plane containing that line, and (:3) a cart-driving means which contains a d.e. motor, It pulley-belt transmission system and a d.c. power amplifier. This mechanical system is schematically shown in Fig. 1 and also shown in Fig. 2.

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2.

CART

POTENTIOMETER

IDLEH PULLEY

u

MONOHAIL

Figure 1.

D.C. MOTOH

Experimental cart-pendulum system.

Under the assumptions that the pendulum is a rigid body and that the driving force is proportional to the input voltage to the amplifier and is directly applied to the cart without any delay, a four-dimensional vector x whose components are the position of the cart 1', the angle of the pendulum e, and their velocities l' and 0, i.e. x = (r,

e, 1', 0)'

(1 )t

can be considered as the state of this system, and the input 'voltage 1t to the d.c. power amplifier can be considered as the system input. The origin of the cart position r is the centre of the range where it can move and the origin of the pendulum angle e is the upright position. Assuming that the friction of the cart is proportional only to the velocity l' and the friction generating the pivot axis is proportional to the angular

t ( . ) denotes matrix.

dkit and ( .. ) d2ldt2.

()' designates transpose of a vector or. a

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Control

(jiG

0/ 'Unstable mechanical system

Figure 2. Cart-pendulum experimental system.

velocity () of the pendulum, the following non-linear differential equations are obtained : (M 0 + M df + Mil cos ()(j = -

Ff + Jl! Il()2

~in () + On }. (2)

Mil cos

()f +

(J

+ 1ll P)B =

- C() + M Ilg S1l1 ()

which describe the dynamics of the system. The definitions and the values of the parameters 1'1-1 0, P, G, .MI , l, J, C and g in eqns. (2) are listed in Table I with other parameters of the system. Equations (2) are rewritten as the ordinary differential equation x=f(x, u)

(3)

describing the system dynamics, where

f l =x a f2=x. fa = .p(X2)(fl. a2 sin x 2 cos x 2 + fl. aaX3 + 0. 3, cos x 2x, + (La5 sin

X 2X.

2

+ bau)

f. = .p(X2)(fl . 2 sin x 2 + fI..,1 cos x 2xa + a•• x. + (L.5 sin x 2 cos X2X.2

+b. cos

.p(x2 ) = (l

X 21t)

+ f3 sin 2 X 2)-1

In eqn. (3'), f i denotes the ith component of f(x, 1t) and the parameters "t» b, and f3 are listed in Table 2. The system considered is subjected to the restrictions (4)

where Irl =ro corresponds to both ends of the monorail and 'Uo denotes the maximum possible input value to the amplifier without any sat.urntion, those limit values also being listed in Table I. 5F2

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676

S. Mori et al. Cart.t

Mass Friction constant Gain constant Region of cart movement Maximum input voltage

Mo F G ±ro ±uo

Pendulum

Mass Length between the axis and the centre of gravity Moment of inertia about the centre of gravity Friction constant Acceleration of gravity

M,

J C (J

0'48 3·83 8·41 ±0·5 ±0-7

Kg Kg/s N/V m V

0·16

Kg

0·25

m

0·0043 0·00218 9'8

Kg-rn" Kg.m 2/s m/s 2

t Thc parameters of the cart cxcept "o are equivalent parameters observed from the input terminal of the power amplifier connected to the cart driving motor', in consideration of thc effects of the moment of inertia of the motor rotor and the pulley', the motor armature resistance, the gain of the amplifier, etc. Table I. Parameter, of cart-pendulum system. The parameters listed in Table 1 can be identified by experiments such as the step response test of the cart and the small amplitude natural oscillation of the pendulum. The identified parameters are presented in Table l. The objective of the control in this paper is to drive the pendulum from the pendent position to the upright position and to keep the pendulum in that position. The problem of synthesizing such a control. system can be divided into the following problems. The first problem is how to keep the pendulum in its upright position and the cart in its central position, that is, how to regulate the system at its origin x = 0 so that a stable zone may be created around the origin which is inherently unstable. In short, the problem is to design a stabilizer which may stabilize the inherently unstable origin of the system. As the controller for this purpose, a linear feedback controller com bined with a state observer is em ployed, based on the partial state observation by the use of two potentiometers connected to the motor rotor and the pendulum axis. Accordingly, the output equation y = h(x)

(5)

is added to the system eqn. (3), where y is a two-dimensional vector whose elements represent the output voltage of the potentiometers. The second problem is to drive the system state from the natural stable state xN=(r, -71',0,0)', (6) to the stable zone which is generated in the neighbourhood of the origin by the above stabilizer. In order to solve this problem it is not essentially necessary to pursue the 'optimality', since the problem is to find out the possible input {u(t): 0:;:; t:;:; T} driving the state x,y to the origin state x = o. Some kind of optimality is, however, necessary in order to avoid redundancy and unreality. For this purpose a programme controller is employed. Naturally, for this stage of control the non-linear model described by eqn. (3) must be used. The third problem is how to switch those two kinds or two modes of controllers mentioned above.

Control 01 unstable mechanical system

677

3. Structure of stabilizer As mentioned in the Introduction, many types of controllers have been proposed as stabilizers which keep the pendulum carried by the cart in its upright position. The authors designed a linear feedback controller combined with a state observer, similar to that by Sturgeon and Loscutoff (1972) who proposed an observer-controller designed by the pole allocation method. The authors' method of designing depends more on the performance index of the system. In designing the stabilizer, the linearized model about the origin x=Ax+ Bu

(7)

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is used, where

A=~I = iJx .-0

0

0

1

0

0

0

0

1

0 0 B- iJfl

0

a 32

a 33

a..

0

a 42

a 43

a 44

-

- iJ7t .-0-

(7')

b3 b4

The a;/s and b/s in eqn. (7') are parameters listed in Table 2. The characteristic roots of the matrix A are shown in Fig. 3. The origin of the system is a saddle-point type of unstable equilibrium point. It can also be easily seen that the linear system defined by eqn. (7) is completely controllable.

-10

-5 ! )(

.----R:flf+=' > TI

T~

T3

T~

T"

Figure 11. A candidate for swinging-up control.

Control

0/ unstable

mechanical sysfem

trying to solve the adjoint equation satisfying the boundary conditions, introduce the following bang-bang control sequence: (0~t