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Proceedings of the 2001 IEEEInternationalConference on Control ApplicationsSeptember 5-7,2001 Mexico City, Mexico

Implementation of Servo Systems for ControllingaDouble Purpose Nonlinear Plant: Inverted

Pendulum a nd Cran eArturo Rojas-Moreno FernandoS. MerchAn-Gordillo Leonard0D. Gushiken-Gibu

Universidad Nacional de IngenieriaSecci6n de Postgrado de la Facultad de Ingenieria EMctrica y Electr6nica

AV. Tupac Amaru 210, Lima 33, Per6postgradofieeQuni.edu.pe arojasQuni.edu.pe

http://fiee.uni.edu.pe/Posgrado/index.htrn http://fiee.uni.edu.pe/728681F

+ Control y ’Abstract- This paper develops design proced ures ofProportional-Integral Servo Syste ms ( S S s for s h o r t ) for con-trolling a double purpose nonlinear plant: Inverted Pendu-lum and Crane (IP C). Such a plant can be described bynonlinear differential equations, where the nonlinear termscomplicate th e analytical aspects of modeling and controllerdesign. However, based on th e linearized plant model, wecan configurate discrete-time S S s by combining linear con-trollers with linear observers. E xperim ental results demon-s t ra te that each S S is able to stabilize an IP or a Crane)mounted on a servomotor-dr iven car t . Three S S s configu-rations are developed: A Steady-Sta te Quadrat ic Opt imalController (SSQOC) with a Quadrat ic Opt imal Sta te Ob-server (QOSO), a Pole Placement Controller (P PC ) witha Full-Order State Observer FOSO), a n d a P P C w it h a

Fig. 1. The IP C plant (IP view).

Application of th e Taylor’s expansion theorem for th e o peration point x, i into 4) yields

Minimum-Order Sta te Observer (MOSO).

I. MODELINGHE I P c PLANT

af 2dU

- a) + . . .

( 5 )

ig. 1 depicts the nonlinear IP plant, which may be 2 = f z, ) + I >

described by t he following nonlinear model [ l ]

M2g(sen6) - M2S(cos6) - 16 = 0

- M2(sen6)i2 + M 2 ( C O S 6 e - =l ) If we choose j z , 0 = 0, ) , hen equation ( 5 ) becomes

af o O af o,o)U = A x + B u 6)(2)

3) j , - 0,O) = A xdX d U

F = K , K A U - J22 - B,i

where:where

1M2 = mele +m,-

2 r 0 1 0 0 1M i = m, + m e + m u ;

J1 = Je + J v ;

Table I describes all variables and parameters of theIP C plant. Define the st at e variables x1 = 6, 5 2 = 6,

23 = 2 and 5 4 = i wherex

and 5 3 are measurable vari-ables. The n, the expression of x (with S = sin x andV = cos X I ) is found to be

r 0 1

The corresponding discrete-time state-space description of6) for a sampling time T takes on the form

x k + 1) = Gx k) +Hu k); y k) = Cx k) 9)

where I is the discrete time and y s the output vector. TheCrane plant is the Fig. 1 with the rod in resting (stable)

0-7803-6733-2/01/$10.00 001 IEEE 884

http://fiee.uni.edu.pe/Posgrado/index.htrn

http://fiee.uni.edu.pe/728681F

http://fiee.uni.edu.pe/728681F

http://fiee.uni.edu.pe/Posgrado/index.htrn

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Symbol

I ,1,g

Fig. 2 shows the SS used in thi s pape r, where r k ) is astepwise reference signal, I is the Identity matrix, z is theshift operator, K is the gain matrix of th e st ate feedbackcontroller, KI is the gain of the integral controller, and 5is the e stimated s tat e vector. For the sequel, let us assumetha t % = x . Th e linear model of the IP C plant in theFig. 2 is described by

x k + 1) = Gx k) +Hu k); y k) = Cx k) (12)

Description Value/Relat ion

rod length 0.767 mdistance not usedeart h gravity 9.8 N

m e I sphere mass

I m, I car t mass 0.92 rr

not usedm,, I rod mass 0.063095 krr

position. Following th e outlined procedure , then A and Bmatrices for the Crane plant result

~ r pJ pJ,,

r 0 1 0 0 1

radius of the pulley

equivalent m.0.i

0.0648 mm.0.i. of the pulley m p r 9 l 2

Jm + n (J0 + D )

11. THE PI SERVO SYSTEM SS)

J ,J,,

Fig. 2. The PI servo system (SS for short)

m.0.i. of th e spherem.0.i of the rod m 2 3

The control law u (k ) s found to be

~ k ) -Kx k) +Krv k); K = [Kl K2 . . . K,] 13)

where the variable w k) verifies

w k) = v k - 1) + r ( k ) - y k)

The output equation takes on the augmented form

(14)

Using the fact t hat for k + CO, r k ) = r ( k + l ) = ... thenthe state-space equation of the SS is found t o be

k + 1) = CZ k) + Hw k); w k) = -K k) (16)

111. THE POLE LACEMENT ONTROLLER P P C )

The state feedback gain matrix k [K - K I ] of thePP C can be calculated from [2]

a = [ 0 * . . 0 1 ]iM- (G) (21)

@ e)G n + alGn--l + . . . +a 4 G +a,I 2 2 )

where n = 5 is the order of the SS (note that the order ofthe IP C plant is 4). T he controllability matrix M needsto be of full order, i.e.,

r a n k M = rank [ i G H . . . G - 'f i ] = n (23)

The parameters a 1 . . . , a , can be obtained from

z - PI). . 2 - ,) = z n + a 1 z n - l + . . + a - 1 2 + a ,

where P I . . . , p , are the desired closed-loop poles of theSS. For both SSs, one with the IP plant and the otherwith the Crane plant, the desired poles are chosen to be:/ ~ 1 , 2 = 0.9967 O.O058i, p 3 4 5 = 0.98.

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IV. THE STEADY-STATE U ADRATI C PTIMALCONTROLLER SSQOC)

Assuming that: The SS is completely state controllable(i.e., equation (23) is satisfied), Q is a n x n positive semi-definite real symmetric matrix, and R is a positive con-st an t, then th e minimization of th e following quadr atic cost

function

K O= @(Gbb)

leads to the steady-state Riccati equation

P = Q + G T P G - G T P H [ R+ H T P H ] - ' H T P G (25)

and to the the steady-state feedback controller K

K = [ R+ H T P H ] - ' H T P G (26)

Q may be a diagonal matrix. For the 1P case R is set to100 while diagonal elements of Q are set to 200, 0 , 100,0, 0.01. For the Crane case R is set to 1 while diagonal

elements of Q are set t o 500, 0, 200, 0, 0.1.

V. THE FULL-ORDER TATE OBSERVER FOSO)

Fig. 3 depicts the block diagram of the FOSO , where

x k + 1) = Gx k) + Hu k); y k) = Cx k) (27)

(28)i k + 1) = (G - K,C)ji(k) + H u ( k ) + K e y ( k )

Note that K , is the state observer gain matrix. Define

e k ) = z k )- Z k ) (29)

Subtracting (28) from (27) and using (29), we can obtainthe error equation of the FOSO

e k + 1 ) = (G - K,C)e(k) (30)

The K , matrix needs to be chosen such tha t t he eigenvaluesof the charac teristic equation of the FOSO

det[zI - G + K,C] = 0 (31)

make the error e k ) zero with sufficient speed. Such a ma-trix K, can be computed from a]

K = @ ( G ) N - l[ 0 0 . . . 0 1 1 (32)

@ ( G )= G + Q ~ G ~ - '* . + ( Y, - ~ G+ CW,I (33)

where m = 4 is the order of the linear IP C plant Theobservability matri x needs t o be of full order

r a n k N = r a n k [CT GTCT ... (GT) -lCT] = m (34)

and the parameters 0 1 , . . r can be obtained from

( z - p 1 ) . . . ( . - p r n ) = Z + LY 1 Z - - 1 + . . . + LY m _ 1 z + c r,

where p1 , . . . , p m are th e desired closed-loop poles of t heFOSO. For the I P plant t he desired poles are chosen to be;p1,2 = 0.7165, p3 = 0.9, p4 = 0, while for the Crane plantare: pl ,2 = 0.1353, p3 = 0.9, p 4 = 0.

- -0 0

GabGbb 0 0

=o-' .

- - 1 , -Gab

GabGE-P-2 0 0

- GabG;-'-' - - 1 > 1

Fig. 3. The full-order state observer (FOSO).

VI. THE MINIMUM-ORDER TATE OBSERVER MOSO)

Let us consider that the state vector x k) in (27) canbe partitioned into two parts: A pvector xa k) containingthe measurable states (in our case, states z1 and z3) anda ( m - p)-vector xb k) holding the unmeasurable states.

Therefore, the partitioned state-space description of thelinear IP C plant becomes

xa k + 1) xa k)[Xb k+ 1) ] = [2: 2; [Xb k) ] [2 ](35)

provided tha t th e rank of matrix 0 is m - . For our case,p = 2. Parameters 6 1 , . . , m - p can be obtained from thecharacteristic equation of the MOSO

- Gbb +KOG,bI = 2 - pi). . . Z - m - p ) =

zrn-' + ~z - ' - - I +. . + m - p - l ~ + (38)

where , ~ ~ l , . . . , p ~ - ~re the desired eigenvalues of the

MOSO. For the IP plant, the desired poles are chosen to be:p1,2 = 0.7165, while for the Cr ane plant are: p1,2 = 0.1353.

VII. THE QUADRATIC PTIMAL TATE OBSERVER

The Q O S O employs the configuration of th e FOSO de-

39)

Q O S O )

picted in Fig. 3. From equation (31), we can formulate

d e t [ t I - G + K,C] = d e t [ z I - G~ + c ~ K , T ]

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On using (16), the characteristic equation of th e SS is foundto be

ciet[zI - G + fiK1 (40)

Comparing equations (40) and (39), dual expressicns ofequations (25) and (26) can be obtained replacing GT by

G, H by C T, K by KT, and P by Pe, i.e.,

Pe = Qe+GPeGT-GPeCTIR,+CPeCT]-lCPeGT 41)

K , = [Re+CPeCT]-lGPeCT (42)

To determine on-line Pe (or P given by equation (25)),we can use the recursive form of the Riccati equation

Pe(k + 1) = Q e +GPe(k)GT -

GPe (k)CT [Re +CPe( k)CT]-'CPe (k)GT (43)

Re and Q e ay be chosen to be diagonal matrices. For theIP case, diagonal elements of R e re set to 1 , 10 while forQe are set to 1, 0 , 1000, 0.9, 1000. For the Crane case,diagonal elements of Re are set to 1, 1 while for Q are sett o 1000, 1500, 100, 1500.

VIII. HARDWAREN D SOFTWARE F THE SS

Fig.4 depicts the h ardware configuration of the SS whichincludes a DC servo motor, a H-type PWM (pulse WidthModulation) amplifier, a sensor block with two optical en-coders for sensing signed rod and ca rt positions, a LabPC+Input /Out put interface card, and a Pentium PC. The in-formation from the sensor block is transmitted in form oftwo pulse tr ain s, one of them leading the o the r by 7r/2 rad.

PLANT :INVERTED

CRANE

ANALOGOUTPUTS

DIGITALPORTS

FORINPUTSOUTPUTS

MOTORSENSOR

ROD16 bits

PB SENSORPB7

PC6PC7

Fs SAMPLING FRECUENCYCOUNTER /TEMPORIZER

The designed SSs were simulated using MATLAB fortesting control performance and SS behavior before im-plementati on. All source files for real-time implementa-tion ware writ ten in C-code. Unavoidable inherent nonlin-earities, like servomotor saturation and coulombic friction,ware compensa ted by software.

IX. EXPERIMENTAL ESULTS

All experiment al results were obtained by set tin g th ereference position of the cart to 1.5 m and the samplingfrequency t o 200 Hz (0.05 ms). Rod up and and roddown positions correspond to zero angular positions ofth e I P and of th e Cra ne, respectively. Th e following fig-ures illustrate cart and rod positions for each designed SS.Recall that the acronyms PPC, SSQOC, FOSO, MOSO,and QOSO stand for pole placement controller, steady-sta te qu adrati c optimal controller, full-order sta te observer,minimum-order stat e observer, and quadratic optimal sta teobserver, respectively. Observe th at all the designed SSsare able to stabilize either the IP or th e Crane mounted on

a servomotor-driven cart.

X. CONCLUSIONS

In light of the experimental results, we can assure thatit is possible t o find a linear process model capable of c a ptur ing significant features of th e act ual nonlinear process.Such a process model permit to employ linear controllersand observers.

Despite the restriction imposed by the linearization pro-cedure, we found th at t he designed SSs are able to stabilizeangular positions of the rod up to 20 .

Experimental results demonstrated th at the control per-formance of the configurations IP+ SSQOC+QOSO (fig-ures 9 and 10) and Crane+S SQOC+QO SO (figures 15 and

16) are superior th an th e control performance of th e otherconfigurations.

2

I._

a 0 5 t

I I I

Fig. 5. Cart position m). SS: IP+PPC+FOSO.

Fig. 4. Hardware implementation of the SS.

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Fig. 6. Rod position (rad). SS: IP+PPC+FOSO.

0 4

B 0 3 -E

0 2 4 6 8 10 12 14 16 18 20-0

Time s)

02

-03-

0 4

Fig. 7. Cart position m). S S : IP+PPC+MOSO

REFERENCES[l] A . Rojas-Moreno, Control Avanzado - Diserio y Aplicaciones

en Tiempo Real, Independent Publication, Lima, Perd, 2001,http://fiee.uni.edu.pe/728681F.

21 K. Ogata, Designing Linear Control Systems with MATLAB,Prentice Hall, Englewood Cliffs, New Jersey 07632, 1994.

0 5 I

-0 5k 2 6 10 1 l h 16 l d0Time 5 )

Fig. 8. Rod position (rad). S S : IP+PPC+MOSO.

2 ,

Fig. 9. Cart position m). SS: IP+SSQOC+QOSO.

0 4 5 5

q , , I , I-0 0 2 4 6 8 10 12 14 16 18 20Time 5)

Fig. 10. Rod position (rad). SS: IP+SSQOC+QOSO.

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06-

9 0 4 -

0= 0 2

C

Fig. 11. Cart position m). SS: CranefPPC tFOSO.

4

02

04

-06-

Fig. 12. Rod position (rad). SS: Crane+PPC+FOSO

-

o

2

1 5

1

0C

0 5 -

-00 2 4 6 8 10 12 14 16 18

Time (s)

Fig. 13. Cart position m). SS: Crane+PPC+MOSO.

Fig. 14. Rod position (rad). SS: Crane+PPC+MOSO

-0 L 0 2 4 6 8 Time0 (s) 12 14 16 18

Fig. 15. Cart position m). SS: Crane+SSQOC+QOSO.

06

Fig. 16. Rod position (rad). SS: Crane+SSQOC+QOSO.

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