Multivariable Gyroscope Control by Individual Channel Design

J. Liceaga*, E. Liceaga** and L. Amézquita*

Abstract—. In this paper the analysis, design and implementation of a linear MIMO controller for the ECP750 gyroscope is addressed. This mechanism can be representedby a 4x2 MIMO model. It is shown that such a model can be decomposed into 2 SISO systems in series with a 2x2 MIMO system. Based on the above, a design composed of two internalSISO control loops and a 2x2 MIMO control system is presented. The two SISO control loops are obtained byapplying classical techniques, while the MIMO controller is designed by applying individual channel design (ICD).

Real time results showing the control system performanceare included.

Index Terms— Control Application, Multivariable Control, Gyroscope Control, Individual Channel Design.

I. INTRODUCTION

he role of gyroscopes in control engineering has been limited to sensing devices. However there are areas of

applications, such as the automotive industry, where the use of a “controlled” gyroscope has many advantages. On theother hand, linear, low order, stable and minimum-phasecontrollers have the advantage of better and easier real timeimplementation, specially for multivariable systems. Astrategy that allows to obtain such controllers, in asystematic and transparent manner, in addition to designspecifications, is Individual Channel Design (ICD) [2]. Thisframework is devoted to square MIMO systems.Unfortunately, the gyroscope control problem posed by theECP750 gyroscope [4], is represented by a non-square 4x2 control model. Below a strategy based on thedecomposition of the non-square process model is proposed in order to apply ICD.

In a previous report the highly non linear nature of theECP 750 gyroscope was discussed [1]. In addition, thelinear models representing the gyroscope in three different operating conditions were estimated. These representationsconsist of 4x2 MIMO linear transfer matrices. In [1] it was also shown that such models can be decomposed into twoSISO systems in series with a 2x2 MIMO system.

The present paper shows the linear control system designin one operating point of the ECP750 laboratory gyroscopebased on the models reported in [1]. In Section II a briefdescription of the model obtained in [1] is introduced. The control specifications are defined in Section III. The design of the internal control loops is presented in Section IV. In sections V and VI the design of the multivariable controlleris described. The internal loops of Section III are factors ofthe diagonal multivariable controller. The robustness characteristics are established in terms of gain and phasemargins. Real time and simulation results are included insection VII. These are followed by some conclusions. Anappendix on ICD ends the paper.

II. GYROSCOPE MODEL AND CONTROL DESIGNSPECIFICATIONS

The schematic representation of the ECP 750 gyroscopeis shown in Figure 1. The system inputs are the supplyvoltages, and , of the two driving motors. Themeasurable variables are the angular positions and velocities of the axes; therefore, the angular positions ofaxis 2, 3 and 4, and the angular velocity of axis 1 areconsidered the process outputs. The linear model of theECP750 gyroscope at the operating point defined by

1V 2V

2 20q , 3 0q and has the structure of the following 4x2 multivariable matrix transfer function (the axis position does not affect the linear representation) [1]:

1 480w rpm

4q

1 11 12

2 12 22 1

3 31 32 2

4 41 42

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

w s g s g s

q s g s g s V s

q s g s g s V s

q s g s g s

(2.1)

In [1] it was shown that the transfer functions 12 ( )g s

and 21( )g s are negligible (these terms correspond to thecoupling between two orthogonal axes). Thus (2.1) can be simplified into:

3 31 32 1

24 41 42

1 11 1

2 22 2

( ) ( ) ( ) ( )( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

q s g s g s V s

V sq s g s g s

w s g s V s

q s g s V s

(2.2)

In the last two equations the angular velocity of axis-1( ) and the position of axis-2 ( ) are measured.On the other hand, and are known control inputs.Thus equation (2.2) can be written as [1]:

1( )w s 2 ( )q s

1V 2V

T

lamezquita@itesm.mx

*Departamento de Ingeniería Mecatrónica Instituto Tecnológico y de Estudios Superiores de Monterrey Campus

Estado de México.jliceaga@itesm.mx

**SEPI-ESIME Ticomán IPN, Ticomán 600, Col. S. J.Ticomán, C.P. 07340, México, D.F., México

eliceaga@ipn.mx

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TB1.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 785

31 32

3 111 22

24 41 42

11 22

( ) ( )( ) ( )( ) ( )

( )( ) ( ) ( )( ) ( )

g s g sq s w sg s g s

q sq s g s g sg s g s

(2.3)

1 11 1( ) ( ) ( )w s g s V s (2.4)

2 22 2( ) ( ) ( )q s g s V s (2.5) That is, the model (2.2) can be represented by two SISOsystems in series with a 2x2 MIMO system as shown inFigure 2. In [1] it was shown that:

11512( 27.9)( )

( 347.6)( 0.99)s

g ss s

,

22 2128.6( 320.4)( )

1.041 590s

g ss s

,

3111

11

-7.949( 347.62)( 0.99)

( 27.9)(s+1.4636)(s+0.49)

( )( )

( )s s

s

g sG s

g s,

2 2

2

0.2816( 226)( -452s+51187)( +s+594)4222

( +5.9s+680)( 1326)(s+321)( 0.324)22

( )( )

( )s s s

s s s

g sG s

g s,

2 2

232

1222

.03349( -452.5s+51187)( +s+ 594.5)

( 321)( 0.046)( 1.3s+ 646)

( )( )

( )s s

s s s

g sG s

g s and

4121

11

1.927( 347.6)( .999)

( 27.9)( 14)( 1.4)

( )( )

( )s s

s s s

g sG s

g s .

III. CONTROL SPECIFICATIONS

Gyroscopes are commonly used as measurement devices in aerospace and automotive engineering. In particular, Level 1 handling qualities of helicopters is an interestingand demanding set of control design specifications [5].These specifications will be considered as follows:

It is required a bandwidth between 2 to 8 rad/s for outputs and is required.3q 4q

Gain and phase margins over 12dB and 50o

respectively are established. Such measures of robustness can be used for MIMO system in the ICDframework.

As the inputs of subsystems (2.3) are the outputs of thetwo SISO systems, and , the bandwidth ofsystems (2.4) and (2.5) should be of the same order as thoserequired for system (2.3). On the other hand the saturationof the input voltages, V

1( )w s 2 ( )q s

1 and V2, must be avoided.

IV. SISO CONTROLLER DESIGN

In order to guarantee that the dynamics of and satisfy the above mentioned requirements, two

internal control loops are included as indicated in Figure 5.By applying classical control theory techniques thefollowing controllers are obtained:

1( )w s

2 ( )q s

2

1 2

0.00148( 493.9 721.9)( )

40.35

s ss

s sc (4.1)

2

2 2

0.00416( 1.99 590)( )

24.13

s ss

s sc (4.2)

The gain and phase margins are dB and more than 75o

for both systems as shown in Figures 3 and 4. Thecorresponding closed loop systems are denoted as:

( ) ( )( )

1 ( ) (i ii

ii ii

c s g sT s

c s g s), where (4.3) 1,2i

V. ICD MULTIVARIABLE CONTROL SYSTEM

ICD can be applied if the internal closed-loop controlsystems (4.3) are considered as factors of the multivariablediagonal controller. That is:

( ) ( ) ( )i ii ik s k s T s , where (5.1) 1,2iThe resulting structure, which is shown in Figure 5, can beanalysed according to the standard 2x2 ICD structure (Figure 6). Then, the gamma function is [2,3]:

12 21

11 22

( ) ( )( )( ) ( )

G s G ss

G s G s(5.2)

and the individual channels are:( ) ( ) ( ) 1 ( ) ( )i i ii jC s k s G s s h s (5.3)

where i,j=1,2 (with and h( ) ( )

i j) ( )1 ( ) (

i iii

i ii

k s G ss

k s G s)

A. Analysis of gamma The Nyquist plot of gamma neither encircles nor passes

near the point (1,0) (Figure 7). Moreover, the Nyquist plotof gamma lies practically on the LHP. However, gammahas three RHPP’s due to G22(s). This indicates the existence of non-minimal phase zeros in the channels. Thus, it is notpossible to design arbitrary high bandwidth stabilizingcontrollers. Fortunately, those zeros are at a very highfrequency –about 226 rad/s- compared with the bandwidthrequirements of 2 to 8 rad/s. In addition, the low gain natureof (s) implies that the MIMO system has low overallcoupling and structural robustness [2].

In order to preserve the dynamical structure of thechannels, the encirclements of (s) and ( ) ( )is h s to thepoint (1,0) should be the same [2,3].

It has been shown that stability and robustness of thecontrol system are related to the characteristics of (s) [2,3].In particular, for the gyroscope model ECP750, it isrequired that the channels should be stable, should be stable and

( )ih s( ) ( )is h s should not encircle the point (1,0).

Note that further restrictions and more general resultsregarding the existence of controllers using ICD, as well asthe results regarding stability and robustness, are presented in [2,3].

VI. ICD CONTROLLER DESIGN

The above requirements are achieved using thefollowing stable and minimum-phase controllers:

2

2

8.84( 4.6 1.3)( )11

101.4

s ss

s sk (6.1)

786

2

2

-1.827 ( 30. 30.6)( )22

196

s ss

s sk (6.2)

Figures 8 and 9 show the phase and gain margins of where k( ) ( )i iik s G s 1(s) and k2(s) are defined by (5.1).

Figures 10a and 10b present the Nyquist plots ofh1(s) ( )s and h2(s) ( )s . These plots do not encircle thepoint (1,0).

The closed loops of the individual channels C1(s) and C2(s) represent the relationships q3/R1 and q4/R2, where R1and R2 are the desired angular positions of Axes 3 and 4.

Figures 11 and 12 show the phase and gain margins ofC1(s) and C2(s). It is relevant to note that the bandwidth ofeach channel complies with the design goals. Thebandwidth of C1(s) is higher than the bandwidth of C2(s).

Finally, it can be noted that the fact of analysing internal SISO closed-loops as part of the diagonal MIMO controllerallows a transparent analysis and design of controllers for the system in the context of ICD. The closed loops Ti(s) are designed in such a way that, together with controllers k11(s)and k22(s), the system bandwidth requirements are achieved.

The design results are summarized in Table I.Table I.

Measure C1 k1G11 h1 C2 k2G22 h2

BW[rad/s] 7.46 7.27 2 1.99

GM [dB] 28.8 28.8 21.8 23 48.3

PM [deg] 50.3 53.4 57.3 56

The channels coupling is expressed by [2,3]:( )( ) 1 ( )

( ) 1 ( ) ( )iji

jj i jj

G sy sh s

R s C s G s(6.3)

as described in Figure 13. Where y1=q3 and y2=q4Figures 14 and 15 show that the coupling between the

channels is lower than -12dB. Separating the channels bandwidths facilitates the

controller design and reduces the coupling of the betweenthe channels [2]. In the present design, the requirementsallowed the use of different bandwidths for each channel.

In general, the perturbation of reference R2 over y1 is given by equation (6.3). Increasing the bandwidth of channel C1 and decreasing the bandwidth of channel C2,which is similar to the bandwidth of h2, will minimize theeffect of R2 over y1.

VII. SIMULATION AND REAL TIME RESPONSES

Figures 16 to 19 present the real time and simulatedresponses of q3, q4, w1 and q2. w1 and q2 are internal variables of the control scheme, and q3 and q4 are the controlled outputs as shown in Figure 5. It should be notedthat the real time response of w1 contains high frequencysensor’s noise.

Moreover, the responses of the control efforts Vi do notsaturate ( -10< Vi <10 ), as shown in Figures 20 and 21.

Figures 22 and 23 show the coupling between thechannels. The response of q3 when R2 is a step of 25o is

presented in Figure 22. In addition, the response of q4 when R1 is a step of 5o is shown in Figure 23.

The differences between digital simulation and the realtime responses are due to dead zones of the actuators (theDC motors), the non-linear dynamics [1] and the precisionof the sensors.

VIII. CONCLUSIONS

A MIMO controller has been designed for the ECP750gyroscope. The design was obtained according to a 4x2 linear model previously reported. It is shown here that thismodel can be decomposed into two SISO systems in series with a 2x2 system. With such a representation, it becamepossible to design a multivariable control system in theframework of Individual Channel Design. This included thedesign of two internal control loops, whose closed-loopdynamics are a part of the multivariable controller.

The controllers were obtained by applying classicaltechniques, and they are low order stable, minimum-phaseand robust controllers.

The control system has been implemented in real time.The real time results where satisfactory and the control goals were achieved.

APPENDIX I: ICD INTRODUCTION

The control structure shown in Figure 6 can also be represented, without loss of information, by the SISOsystems of Figure 13 [2] where

( ) )( )

1 (i ii

ii ii )

K s G sH s

K G s (I.1)

Moreover, the following definition is possible:( ) ( ) ( )(1 ( ) )

1, 2 1, 2i i iiC s K s G s s H

i j i jj (I.2)

where12 21

11 22

( ) ( )( )

( ) ( )G s G s

sG s G s

(I.3)

The equations Ci(s) are defined as “Individual channels”.The closed loop of C1(s) fully represents the input/outputrelation between R1 and y1 including all the internal closedloop interactions.

The study of such a system, by using the presenteddecompositions, allows the designer to use classical SISO frequency analysis methods to design a MIMO controller.

REFERENCES

[1] J. Liceaga and L. Amezquita “Analysis and Modelling of aGyroscope”, IASTED CA2005, pp. 165-170

[2] J. O’Reilly.,and W.E. Leithead, 1991,” Multivariable control byIndividual Channel Design”, Int. J. Control, 54, pp. 1-46

[3] E. Liceaga-Castro, J. Liceaga-Castro, and C.E. Ugalde “Beyond the Existence of Diagonal Controllers: From the Relative Gain Array to the Multivariable Structure Function” Submitted for Publication

[4] T. R. Parks, “Manual For Model 750”, Ed. ECP 1999.[5] J. Liceaga- Castro, C. Verde, J. O’Reilly and W. E. Liethead,

“Helicopter Flight Control Using Individual Channel Design”. IEEProc. Control Theory Appl. Vol. 142, 58-72 1995.

787

Fig. 1. Schematic representation of ECP 750 gyroscope

Fig. 2. Model structure for ECP750 gyroscope.

Fig. 3. Phase and gain margins of c1g11

Fig. 4. Phase and gain margins of c2g22

Fig. 5. Proposed control scheme.

Fig. 6. 2X2 system with diagonal controller.

Fig. 7. Nyquist plot of gamma Function

788

Fig. 8. Phase and gain margins of k1G11

Fig. 9. Phase and gain margins of k2G22

Fig. 10a 10b. Nyquist plots of h1 and h2

Fig. 11. Phase and Gain margins of C1

Fig. 12. Phase and Gain margins of C2

Fig. 13. Decomposed system for output yi

Fig. 14. Bode Diagram of the sensitivity of channel C1 and perturbationfrom R2

Fig. 15. Bode Diagram of the sensitivity of channel C2 and perturbationfrom R1

789

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

0

1

2

3

4

5

6

7

time

q3(degrees)

Real timeDigital simulation

Fig. 16. Real-time and simulated response of q3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

time

q4(degrees)

Digital simulationReal time

Fig. 17. Real-time and simulated response of q4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5462

464

466

468

470

472

474

476

478

480

482

time

W1(RPM)

Digital simulationReal time

Fig. 18. Real-time and simulated response of w1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-5

-4

-3

-2

-1

0

1

time

q2(degrees)

Digital simulationReal time

Fig. 19. Real-time and simulated response of q2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

-8

-6

-4

-2

0

2

4

time

V1(volts)

Real timeDigital simulation

Fig. 20. Real-time and simulated response of V1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-5

-4

-3

-2

-1

0

1

time

V2(votls)

Digital simulationReal time

Fig. 21. Real-time and simulated response of V2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

-4

-3

-2

-1

0

1

2

time

q3(degrees) Real time

Digital simulation

Fig. 22. Real-time and simulated response of q3 from reference R2

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time

q4(degrees)

Digital simulationReal time

Fig. 23. Real-time and simulated response of q4 from reference R1

790