NASA Technical Memorandum 101606

An LQR Controller Design Approach for a

Large Gap Magnetic Suspension System

(LGMSS)

Nelson J. Groom and

Philip R. Schaffner

(NASA-TM-IOI_06) AN L_ p CUNT#OLLCR DESIGN

APPnQACH fOR A LARbE GAP MAGNETIC SUSPENSION

SYSTEM (LGMSS) (NASA) 65 o "CSCL l]n

G3/31

N90-23151

July 1990

National Aeronautics and

Space Administration

Langley Research CenterHampton, VA 23665

https://ntrs.nasa.gov/search.jsp?R=19900019435 2020-07-30T03:35:55+00:00Z

SUMMARY

Two control approaches for a Large Gap Magnetic Suspension System (LGMSS) are investigated andnumerical results are presented. The approaches are based on Linear Quadratic Regulator (LQR) control

theory and include a nonzero set point regulator with constant disturbance input and an integral feedback

regulator. The LGMSS provides five degree-of-freedom control of a cylindrical suspended element whichis composed of permanent magnet material. The magnetic actuators are air core electromagnets mounted

in a planar array.

INTRODUCTION

This paper describes two control approaches for a Large Gap Magnetic Suspension System (LGMSS).

The approaches are based on LQR control theory and include a nonzero set point regulator with constantdisturbance input and an integral feedback regulator. The LGMSS is a conceptual design for a ground-

based experiment which could be used to investigate the technology issues associated with magnetic

suspension at large gaps, accurate suspended element control at large gaps, and accurate position sensing

at large gaps. This technology would be applicable to future efforts which could range from magnetic

suspension of wind tunnel models to advanced spacecraft experiment isolation and pointing systems. The

LGMSS provides five degree-of-freedom control. The suspended element is a cylinder which is composed

of permanent magnet material and the magnetic actuators are air core electromagnets mounted in a

planar array. This configuration is described and an analytical model developed in reference 1. Values of

magnetic fields generated by the electromagnets at the location of the suspended core are presented andthe method of calculation discussed in Appendix A which was written by Colin P. Britcher, Department

of Mechanical Engineering and Mechanics, Old Dominion University, Norfolk, Virginia. All numericalresults which are presented were obtained by using MATRIXx with SYSTEM_BUILD. 1 The MATRIXx

implementation of the system is presented in Appendix B. Numerical results were obtained for positioning

accuracy, with fixed gains, over large angular displacements about the yaw axis.

LGMSS ANALYTICAL MODEL

This section presents an analytical model of the LGMSS. For a detailed development of the model see

reference 1. The suspended element, or core, is a cylinder which is composed of permanent magnetmaterial. The core is suspended over a planar array of five electromagnets mounted in a circular

configuration. Figure 1 is a schematic representation of this system which shows the coordinate systemsand initial alignment. A set of orthogonal x, y, z, body fixed axes defines the motion of the core with

respect to inertial space. The core coordinate system is initially aligned with an orthogonal x, y, z systemfixed in inertial space. A set of orthogonal Xb, Yb, Zb axes, also fixed in inertial space, define the location

of the electromagnet array with respect to the x, y, z system. The Xb and Yb axes are parallel to the x and

y axes respectively and the z b and z axes are aligned. Thc centers of the two axis-systems are separatedby the distance h. The angular acceleration of the core, in core coordinates, can be written as (see ref. 1)

{_} = (1lie) (Vol([_I][Tm]{B} ) + {:rd} ) (1)

where Ic is the core moment of inertia about the axes of symmetry (y and z), Vol is the volume of the

core, M is the magnetization of the core, [113/] is the skew symmetric cross product matrix, [Tm] is the

vector transformation matrix from inertial to core coordinates, {B} is the flux density produced by the

1Use of names of products in this report does not constitute an official endorsement of such products, either expressed orimplied, by the National Aeronautics and Space Administration.

electromagnets,and{T,t } represents external disturbance torques. A bar over a variable indicates that it

is referenced to core coordinates. The translational acceleration of the core, in core coordinates, can bewritten as

where mc is the mass of the core, [OB] is a matrix of the gradients of B, and {-Pd} represents externaldisturbance forces in core coordinates. Under the assumptions of reference 1, {B} can be written as

{B} = (t/Imax)[f(,1{I} (3)

where/max is the maximum coil current, [KB] is a 3 x 5 matrix whose elements represent the values of{B} produced by a corresponding coil driven by the maximum current and the coil currents are

{z}T= Lh I2 z3 I4 I5J (4)

The gradients can be put in the same form by arranging the elements of [OB] as a column vector. Thisresults in

{OB}= (X/Imp)[I_] {I} (5)

where {OB} is a nine element column vector containing the gradients of {B}, and [KoB ] is a 9 x 5 matrix

whose elements represent the values of {OB} produced by a corresponding coil driven by the maximum

current. The simplified notation from reference 1, where Bij represents OBi/Oj, will be used for gradients.Each element of {OB}, for example Bxx, can bc written in the form

Bx.r = (1/Imax)[Kx_.J {I} (6)

where [K_:xJ is a 1 x 5 matrix containing values of B:_:r produced by a corresponding coil driven by themaximum current. A block diagram of the system is shown in figure 2. This model is nonlinear becauseof the coordinate transformations and is of the form

where x is given by

and the input u is given by

2 = f(x, u) (7)

x_r= [fi,, fi: _] _)__ 0,,_0: • y zJ

J'= L5 _r2_-3I,_ZsJ

The states ¢Yu and 0: in (8) are the pitch and yaw angles of the core respectively.

(8)

(9)

Analyses andsimulations of the model were performed using MATRIX A with SYSTEM_BUILD. A description of the

MATRIXx/SYSTEM_BUILD implementation is presented in Appendix B. LGMSS model parameters arepresented in Table A1.

CONTROLLER DESIGN APPROACH

The control approach which was investigated was to linearize thc system about a nominal operatingpoint xo, Uo and then use the resulting linear model to calculate feedback gains to stabilize and control

the core about this point. The equations for computing uo for a given xo are presented in the appendix

of reference 1. The feedback gains for the linearized system were calculated using LQR theory. In thisapproach the system is put in state variable form

_' = Ax + Bu (10)

and a set of full state feedback gains are calculated that minimizes the quadratic cost function

oc 7'J = f (x Qx + uT Ru)dt (11)

Jo

where the weighting matrices, Q and R, are symmetric and positive-semidcfinitc and positive-definite,respectively. The resulting control law is

u = - KI_x (12)

where

K],, = R-1B'r p

and P is a symmetric positive-definite matrix satisfying the matrix Riccati equation (see ref. 2)

(13)

PA + ATP+ Q - I"BR-IBrrP = 0 (14)

The operation of the LGMSS requires that the control system provide variable set point control and

also that it compensate for a constant disturbance (the weight of the core). Two feedback controller

configurations which meet these requirements wcre investigated and arc described below.

Nonzero set point regulator with constant disturbance.- With a constant disturbance, 1'o, the systcmunder consideration becomes

2 = Ax + B u + 1o (15)

The controlled variables arc assumed to be

z = D:c 06)

For the LGMSS, z is defined as

d'= e: y zj (17)

For a set point of Zo, a constant input Uo must be found to hold the states at Xo so that

zo = Dxo (18)

From (15) above, Xo and uo must be related by

0 = Axo + Buo + Io (19)

Next, define the shifted input, shifted state, and shifted controlled variable, respectively, as

' (20)"it _--- It -- 1to

' (21)_l; = £ -- X o

' (22)2 = 2 -- Zo

Solving equations (20)-(22) for u, x, and z, substituting the results into (15) and (16) and using (18) and

(19), it can be shown that the shifted variables satisfy the equations

:i:p = Aft + Bu t + _ (23)

andI

z = Dx' (24)

Assume the shifted regulator problem has a steady-state solution in the form of the control law

!u = -Fx' (25)

that minimizes the quadratic cost function

_0 _ r 1J = (x' 1Qx' + u' TRu')dt (26)

3

whereF is a matrix of constant feedback gains. From (25): and (20)-(22)

u = -Fx + uo + Fxo

which has the form

(27)

where

74 =A-BF

For a stable system A is nonsingular and Xo becomes

Zo then becomes

and solving for ulo results in

xo = (-A)-lBu_o + (-A)-IVo

zo = D(-74)-l Bu_o + D(-7t)-Ivo

' [D(_74)-lB]-lzo [D(-74)-lB]-lD(-74)-lVoU 0 -_

A block diagram of this controller is shown in figure 3.

Integral feedback regulator.- Another controller configuration which was investigated is one which uses

integral feedback. Integral feedback is often used in systems subjected to constant disturbances in order

to minimize steady-state errors (ref. 2). To develop the integral feedback controller equations, assume the

system of (15) with the controlled variables defined by (16). To the system of (15) add the integral statesq, defined by

_) = z' (35)

where z I is defined by (22). A performance function can now bc defined as

f0c¢( -j = zTQz + qT'Qq + uTRu)dt

Assume that the time-invariant control law

u = -Fix - F2q (37)

and from (35)

can be found which stabilizes the augmented system described by (15), (16), and (35), and which minimizes

(36). Since the presence of the constant disturbance vector in (15) has no effect on the asymptotic stabilityof the system

lim q = O (38)t---* cx)

lim z I = O (39)t---* oo

4

(36)

(31)

(32)

(33)

(34)

' (28)U -_ -Fx -I- u o

is to be determincd so that in steady-state the controlled variable z assumeswhere the constant vector u o

the given value zo. Substituting (28) into (15) gives

5c = (A - BF)z + Bu_o + Vo (29)

Since the closed-loop system is assumed stable, as t _ c_ the states approach xo which satisfies

o = 74xo+ Bu'o+ Vo (30)

This means that the control system with the control law (37) has the property that the error in the

controlled variables, due to a constant input disturbance, eventually goes to zero. A block diagram of the

integral feedback regulator is shown in figure 4.

Initial conditions.- The initial conditions on current for the nonzero set point regulator with a given

xo and zo can be determined from equation (34). The initial conditions on the integrator outputs, for

the integral feedback regulator, can be determined from (15) and (37). In equilibrium, assuming a stable

system, (15) reduces to (19). Substituting (37) into (19) results in

0 = _'txo - BF2qo ÷ Vo (40)

where 74 = A - BF1. Solving for xo results in

xo = 74-1BF2qo - 74-1Vo (41)

and zo becomesZo = D74-1BF2qo - D_4-1Vo (42)

Finally, qo becomes

qo= D74-1BF2 zo+ D74-1BF2

.Discrete model.- The equations presented above are for a continuous system and assume a perfect

sensor with zero delay. In order to investigate the effects of a delay introduced by a finite sensor processing

time, a worst case sampling time of 20 samples per second was assumed. The continuous system equations

were discretized and discrete system gains were calculated using this sample time. The sensor delay time

was modeled as a zero order hold (ZOH) on the output of the nonlinear open-loop system for simulation

purposes. A block diagram showing the placement of the ZOH for the integral feedback regulator is shown

in figure 5.

ANALYSIS AND SIMULATION RESULTS AND DISCUSSION

This section presents the analysis and simulation results for the nonzero setpoint and integral feedback

regulator which were obtained by using the MATRIXx implementation described in Appendix B. The

results are for positioning accuracy of the the two approaches, with fixed gains, over large angular

displacements about the yaw axis. The approach was to define, first of all, the range over which a

set of fixed gains was stable for each regulator type. This was accomplished by obtaining linearized

models of the open-loop system at intervals of two degrees yaw. Feedback gains were computed for a

given reference angle and these gains were then used with the linearized models to obtain closed-loop

eigenvalues over a given range of yaw angles approaching the limits of stability. The full nonlinear modelwas then simulated using feedback gains calculated at a given yaw angle. Inputs to the system consisted of

position commands in one degree increments. For more detail on the design model used, see Appendix B.

Continuous System

Nonzero set point regulator.- A plot of the eigenvalues of the nonzero set point regulator with feedback

gains calculated for the model linearized around zero yaw angle, as the core is rotated through increments

of two degrees, is shown in figure 6. The low frequency eigenvalues are shown on an expanded scale

in figure 7. The point at which the eigenvalues cross over the imaginary axis is at a yaw angle of

approximately 42 degrees. A plot of the eigenvalues, with the same feedback gains, as the model is

rotated from zero degrees to -42 degrees is shown in figure 8. The low frequency eigenvalues are shown

on an expanded scale in figure 9. As expected, because of symmetry, these eigenvalue plots are identical

with the previous plots. Further calculations also verified that this pattern repeats every 36 degrees. That

is, fixed gains calculated at zero, 36, 72, ect. are stable over a range of plus and minus 42 degrees around

the angle for which the gains are calculated. The system performance was evaluated by commanding yawposition over a range of 18 degrees in increments of one degree. Time between increments was 20 seconds.

Figure 10 is a plot of the input command and figure 11 is a plot of the actual yaw angle. Figure 12 is a

plot of the difference between command and actual angle. Figures 13 through 16 show the response of thepitch, x, F, and z axes. As can be seen, the errors build up as the yaw angle increases. This is the main

disadvantage of the nonzero set point controller for this particular application. The input which minimizes

the position error for a given command and constant input disturbance is given by equation (34) and is a

function of A which is in turn a function of A. As evident from the block diagram of the analytical model

in figure 2, A is a function of core position. The system response to a command over a negative 18-degree

range was essentially the same and is not shown. The system response about other nominal gain points,at 36-degree intervals in yaw, was also essentially the same.

Integral feedback regulator.- A plot of the eigenvalues for the integral feedback regulator with fixed

gains calculated for the model linearized around zero yaw angle is shown in figure 17. The low frequencyeigenvalues are shown on an expanded scale in figure 18. The symmetry properties for the nonzero set

point regulator discussed above were found to apply to the integral feedback regulator also. The responseof the system with the integral feedback controller with fixed gains calculated at zero degrees, to the

command input shown in figure 19, is shown in figure 20. The yaw error is shown in figure 21 and the

response of the pitch, x, y, and z axes are shown in figures 22 through 25. As can be seen from these

figures, and as predicted by equation (39), after an initial transient the errors all go to zero. As was

the case with the nonzero set point regulator discussed above, system response about other nominal gainpoints (at 36-degree intervals in yaw) was essentially the same.

Discrete System

Integral feedback regulator.- After evaluating the performance of the nonzero set point regulator witha continuous system model, a decision was made to drop this approach from further consideration and

concentrate on the integral feedback regulator for the discrete system evaluation. A plot of the eigenvaluesof the discrete integral feedback regulator with feedback gains calculated for the model linearized around

zero yaw angle, as the core is rotated through increments of two degrees, is shown in figure 26. The

eigenvalues shown have been mapped from the z-plane into the s-plane. The low frequency eigenvalues

are shown on an expanded scale in figure 27. As can be seen from the figures, the range of stable operationis reduced from 42 degrees for the continuous system to approximately 36 degrees for the discrete system.

The discrete system was symmetric about points 36 degrees apart in yaw angle as was the continuous

system. The response of the discrete integral feedback controller with fixed gains, calculated for the model

linearized around zero degrees, to the command input shown in figure 28 is shown in figure 29. The yaw

error is shown in figure 30 and the response of the pitch, x, y, and z axes are shown in figures 31 through

34. The transients in the response of the discrete system were of higher magnitude than the continuoussystem but not by a significant amount. Other than magnitudes, the responses were similar with theerrors returning to zero after initial transients.

CONCLUDING REMARKS

Two control approaches for a Large Gap Magnetic Suspension System (LGMSS) have been investigated

and numerical results presented. The control approaches investigated were a nonzero set point regulator

with constant disturbance input and an integral feedback regulator. The LGMSS provides five degreesof freedom control of a suspended element which is a cylinder composed of permanent magnet material.

The magnetic actuators are air core electromagnets mounted in a planar array. Results were obtained

for both continuous system and discrete system models. The continuous system model assumed a perfect

position sensor with no delays and the discrete system model assumed a worst case delay introduced

by the sensor of fifty milliseconds. All analyses and simulations were performed using MATRIX x with

SystemBUILD. The numerical results are for positioning accuracy, with fixed gains, over large angular

6

displacements about the yaw axis. These results indicate that for the LGMSS investigated, the integral

feedback regulator provides the best performance. Also, for control of yaw over a range of 360 degrees,

scheduled gains will be required. Based on the numerical results, control about a nominal operating point

is possible over a range of plus or minus 18 degrees. Although thc system is stable over a greater range,

transient errors become larger as the yaw angle is increased and plus or minus 18 degrees appears to

be a reasonable limit. Gains can be scheduled over a smaller interval based on particular requirements.

Control of pitch over large angles was not addressed in this investigation and should be the subject of afuture effort.

The results presented in this paper, for the analytical model utilizcd, indicate that delays introduced bythe position sensor, for the worst case value investigated, can be adequately compensated. Future efforts

should include investigations of the effects on performance of: 1) different output feedback approaches,

2) changes in (B) and [0B] with core displacement, 3) power supply and coil dynamics, and 4) sensorhardware characteristics.

APPENDIXA

CALCULATIONOFELECTROMAGNETFIELDSANDGRADIENTS

By

ColinP.BritcherDepartmentof MechanicalEngineeringandMechanics

OldDominionUniversityNorfolk,Virginia

Magneticfieldsexternalto air-coredelectromagnetsof simplegeometrycanbccalculatedaccuratelybytheuseoflineconductormodels.A computerprogramFORCE(ref.5) representssuchelectromagnetsbyanassemblyof straightlineconductorelements.Fieldsandfieldgradientsproducedby eachelementat anypoint,not coincidentwith theelement,maythenbecalculatedexactlyusingtheBiot Savartlaw.Fieldsandfieldgradientcomponentsarethensimplysummedoverall conductorelements.Thisapproachis illustratedin figureA1.

Thisprocedureisaccurateif the number of conductor elements is large enough to properly represent thethree-dimensionality of the electromagnet yet small enough to avoid excessive accumulation of numerical

(rounding) errors. Accuracy generally improves with distance from the electromagnet as the effects ofthe discontinuous representation of current are reduced. Recalculation with greater or lesser numbers of

elements is a straightforward and effective check on the reliability of the solution.

Calculations using FORCE were made for the electromagnet configuration used in the earlier stagesof the design study reported in reference 6. The parameters for this system are summarized in table A1

and a schematic representation is given in figure A2. Calculation results are presented in table A2. It

should be noted that detail adjustments to the configuration and to the design suspension height weremade later in the design study mentioned above but do not substantially affect the results shown here orthe conclusions of this report.

TABLE A1.- LGMSS MODEL PARAMETERS

Core diameter, m ................................. 0.1016Core length, m .....

Core mass (mc),kg.. :. : : : : : : : : : : : : : : : : : : : : : : : • : • • . 0.3048

Core inertia (It), kg-m 2 ..... 22.76................................. .6

Core volume (Vol), m 3 .............................. 2.5(10)_3

Core magnetization (M_:), A/m ......................... 9.5493(10)5Core suspension height (h), m ............................ 0.9144

Electromagnet outer radius, m ............................. 0.386

Electromagnet inner radius, m ............................. 0.173

Electromagnet height, m ............................... 0.493

Maximum electromagnet current (/max), A ....................... 559.5

TABLEA2. - ELECTROMAGNETFIELDSANDGRADIENTS

ElectromagnetB\- By Bz

-.0237

-.0073

+.0192+.0192

-.0073

(Tesla)0

-.0225

-.0139

+.0139

+.0225

+.0218

+.0218+.0218

+.0218

+.0218

13X X t3 X Y B X Z B y y

(Tesla/mcter)-.0101

+.0296

+.0051+.0051

+.0296

0

-.0129

+.0209-.0209

+.0129

+.0544 +.0338+.0168 -.0059

-.0440 +.0186

-.0440 +.0186

+.0168 -.0059

BYz [ Bzz

0 -.0237

+.0518 -.0237

+.0320 -.0237

-.0320 -.0237

-.0518 -.0237

Fields correspond to maximum rated current for each clectromagnct (559.5 Amp.).

9

Y

I2

Z

_ __44mFields calculated_y

at model _-_ 1.1"

centroid

O. 7m

14

15

0.493m

0. 772m O.D.

Figure A1 .-Geometry used for magnetic field calculations.

Line element radius

adjusted to match

field on axis

Calculations in this

paper use 3 elements

radially, 4 axially and

32 circumferentially,

total 384

Figure A2.-Representation of electromagnets by line conductor elements.

APPENDIX B

MATRIXx IMPLEMENTATION

The process of implementing the LGMSS plant model and control approaches, using MATRIXx and

SYSTEM_BUILD, involves the construction of block diagram representations of the system (figure B1)and the controllers (figures B2, B3, and B4). Since version 6.0 of MATRIXx/SYSTEM _BUILD was used

to construct the models, it was necessary to break the implementation into nested super blocks containinga maximum of 6 blocks each (for more detailed information on MATRIXx/SYSTEM_BUILD see refs. 3and 4). Figure B5 shows the actual super blocks of the implementation. Each of the blocks in figure B1can be identified as a block within some level of the nested super blocks shown in figure B5. For thesake of brevity and clarity only the unnested versions of the SYSTEM_BUILD models will be shown anddiscussed from this point on.

There is a correspondence between blocks in the block diagram of the system (figure 2) and blocks,or groups of blocks, in the SYSTEM-BUILD implementation of figure B1. For instance, the leftmost

block in figure 2 is modeled by the super block labeled FIELDS in figure B1. The expression (1/Ima.x) isrepresented by the gain block labeled I MAX in which each of the 5 outputs is calculated by multiplying thecorresponding input by a constant gain factor of (1/Imax). The K matrix multiplication is implementedby the State Space block labeled XYZ which multiplies thc vector of 5 currents by a 5 x 12 matrix to

calculate the vector of 12 outputs corresponding to the 3 field components (Bz, By, B-) and the 9 gradients(Bxz, Bxy, Bxz, B_z, Bu_, Bu: , Bzx, Bzu, Bz:).

One block found in figures B1 and B5 has no corresponding clement in the system block diagram offigure 2. This block, labeled A GRAV outputs a vector [0 0 9] which is passed through a transformationfrom inertial to body coordinates (block TM) and summed with the linear accelerations to model theeffects of gravity on the core.

In order to set up for calculation of the LQR gains, the SYSTEM-BUILD ANALYZE option is invokedon the super block OPEN (fig. B6). This super block contains a super block labeled PLANT, which is

the highest level super block in the plant model of figure B5 (and represents the whole of fig. B1). Superblock OPEN also has a step input block labeled I OFF which sets the initial currents about which the

nonlinear system will be linearized and a gain block labeled CONTIN. The CONTIN block has unity gainand acts as a place holder for a zero order hold super block used in discrete versions of the system as willbe discussed below. The yaw angle (Oz) for linearization is established by setting initial conditions on thecorresponding integrator prior to analysis of the model.

The ANALYZE operation returns from the SYSTEM_BUILD module to MATRIX x where theLINearize function is invoked to calculate the linearized state space representation of the nonlinear system

represented by PLANT. LIN returns an S matrix and NS, the number of states in S (10 for this model).The state space matrix S can be partitioned into

S (B1)

where A and B are the matrices in the state variable representation of the system in equation 10. Thematrix A will be of dimension NS x NS, B will be dimensioned NS x NI, C will be NO x NS and

D will be NO x NI; where NI is the number of inputs (5 currents) and NO is the number of outputs(( X, _ Z) positions and velocities, Y and Z rotation angles and rotational velocities). The MATRIXxfunction SPLIT(S, NS) accomplishes this partitioning and can return A and B only or all four matrices.

For the problem being considered only A and B are of interest since C turns out to be an identity matrix

12

andD is a zero matrix. The output equation

y = Cx + Du

reduces to

y ----X

(B2)

(B3)

since the outputs are the position and velocity variables corresponding to the states, and outputs are notdirectly related to inputs. For the model linearized around zero, the A matrix becomes

A

5x

[0]

5x5

162.17 0 0 0 0

0 162.17 0 0 0

-9.8066 0 0 0 0

0 0 0 0 0

-0.0147 -0.0049 0 0 0

5×5

[o]

(B4)

and the B matrix becomes

B

-0.155 -0.155 -0.155 -0.155 -0.155

0 -0.16 -0.098 0.098 0.16

-0.0019 0.0055 0.001 0.001 0.0055

0 -0.0024 0.0039 -0.0039 0.0024

0.102 0.0031 -0.0082 0.0082 0.0031

5x5

[o]

This model has four nonzero eigenvalues: -12.7346, -12.7346, 12.7346, and 12.7346.

(B5)

A SYSTEM_BUILD implementation of the linearized model was constructed for use in testing and isshown in figure B7. The full nonlinear model was employed for all simulations shown in this document.

The nonzero set point regulator gains can then be calculated using the linearized model representedby A and B, and the Q and R matrices from the quadratic cost function of equation 26. The Q and Rmatrices used for gain calculation were

Q= l(lO) J m6)and

where Q is a 10 x 10 diagonal matrix and R is a 5 x 5 identity matrix. The MATRIXx functionREGULATOR calculates the LQR gain matrix F and the corresponding eigenvalues, EV. As a baseline,the closed-loop eigenvalues for the system with the model linearized around zero and gains calculatedusing (B6) and (B7) are presented in figures B8 and B9. The nonzero set point matrix HCI can then bccalculated. HCI is given by

HCI = [D(-_t)-IB] -1 (B8)

where D is defined in equation 16. For this implementation the only nonzero elements in D are thosecorresponding to X, Y, Z, _u, and 0z. The value of each nonzero element is one. The SYSTEM_BUILD

13

implementation of the nonzero set point controller is shown in figure B2 and corresponds to the

conventional block diagram of figure 3. Currents as well as position and rotation variables are brought tothe output. Inputs and outputs are scaled to appear in inches and degrees while internal quantities aremeters and radians.

The integral feedback continuous LQR gains were calculated similarly by augmenting the linear state

space model, represented by A and B, with 5 new states corresponding to the feedback integrators:

A I _-

t10xl0 J 5 5

[A] f ×' [olI

i )i 5x5

f ×51 [01[0] [I]

(B9)

B !

10x5

[S]

[01

(B10)

The MATRIXx function REGULATOR was again used to calculate LQR gains. The closed-loopeigenvalues for the integral feedback system with the model linearized around zero and gains calculated

using (B6), extended to a 15 x 15 diagonal matrix, and (B7) are presented in figures B10 and Bll.

Figure B3 shows the SYSTEM_BUILD model of the continuous integral feedback controller of figure 4.The delay due to the sensor was modeled by using the MATRIX x DISCRETIZE function which takes as

parameters a state space matrix, S, its order, NS, and a delay TAU which was specified as 0.05 seconds,

and returns a discrete state space matrix SD. A state space matrix for the augmented system, S !,was constructed using A _ and B _ above as well as dummy C _ and D !. The discretized version of this

matrix, SD, is then SPLIT and the resulting AD and BD system matrices used with MATRIXx function

DREGULATOR, similar to REGULATOR discussed above, to calculated LQR feedback gains for thediscrete version of the system. The closed-loop eigenvalues for the discrete system with the model

linearized around zero and gains calculated using (B6), extended to a 15 x 15 diagonal matrix, and

(B7) are presented in figures B12 and B13. The eigenvalues shown have been mapped from the z-plane

into the s-plane. Figure B14 shows the SYSTEM_BUILD model of the discrete integral feedback controllerof figure 5. The discrete super block ZOH is the sensor model. It replaces the CONTIN block and acts

as a Zero Order Hold. ZOH is the only discrete super block used in this otherwise continuous model.

14

II d

i \

Q.0

!

0

°_

0Z

I

°_

I.i_

OR1G!NAL PAGE IS

OF POOR QUALITY15

II

5

IMETRICl15STATE I._

10

/

STATESPACE

5

"'"1PLANT I

{X} =,c

°

/

_ry

"1o

IENGLISII

I 5I

I

{Xoot)

Figure B2.-Nonzero set point regulator.

5

{zo}--_

10

IMETRIClII5 i 5 /I INTEGI I L_t I KRDIS! ]

K L_J STATEI S,-' 1--77]sP'°EI'

_1 I_1 co..E.,s lr :l/Vx/

II IOFF Ii {x}.Jv,_l 10

Figure B3.-Integral feedback regulator.

-4

I,-¢

5

(Zo}--_.

10

STATE _ ="I_-T'_ _._ IKR DIsllK STATE I /5,.

5

I IPLANT I I

l lOaF !1

10

/

lot' l'IyIIzVx

Vy

{X} =< Vz

OzX

10

IENGLISl I

i _ f_lGAIN I r

Figure B4.-Integral feedback regulator with zero order hold.

r=j-r I

!tilting M TM B (Cont,nuous/

[d,t,ng B Ma_T (Continuous)

(Cont,nuous)

Ed,bng MATOPS (Continuous)

E_itinc; PLANT (Conhnuous)

_ditmg TRANSF (Conbnuous)

d,t,nc} INT 2 (Continuous)

. Figure B5.-Nested super block model of nonlinear open-loop system.

[I OFF L6_

F---Step

Io:5I:01

PLANT 12

1:5 o:1ol

IcONTINI=___l2 GAIN

t1:10 0:10

Editing'OPEN (Continuous)

Figure B6.-Open-loop system used for linearization.

[B

STATE

SPACE

11:5o:1ol1_ +

10,,_ +

[A GRAV U

Step

I0:10 I:0

I INT 10 !14 1

NS:IO

I1:10 0:10 t

L [AsTATE

SPACE

t0:10I:1OI

10,_/ v

Figure B7.-Linearized open-loop system.

i,.i

•oJez punoje pez!Jeeu!llepow eql qI!M wels/_s eql jo senleAue6!e dool-pasolo-'sE] eJn6!_-I

ZNVd 7V3N

05 L-- 00_-0 Og- O01 - 0C3_- 00£-: T r I I ? I , _ , 1 i I I _ ! l 1 I I

............................................ i .............. • .............

.................. i ............. i........... :........... :............ ! .............

.................... !............................. i............. . ............... :..............

900 -

900"-

1700-

EO0"

#00"

900"

900"

LO"

m

E

C_

z>;u-<

"13

I--n_<n

)-r_<Z

<

.01

.008 ............................. : ....... :...................... : .............

.006 .....................

.004 ..... .............. i ............ ,............ : ...................... !........

.002 ............ " ........................... !........... ! ........... ........... i ..............

0 v _._

-.002 ................................................................ '.......................................

--,004

-.006 ............ ................. ' ............................ ........

-.008 .......... ,.............. : ........... :.......... _.................................

-.01 , r ' i , i : , : i .......-1.6 -1.4 -1.2 -1 -.8 -.6 -.4 -.2 0

REAL PART

Figure B9.- Low frequency eigenvalues of figure B8.

t,_

I--rv,<13_

>-rY<Z

<

.6

.4

.2

0 x ×;

--.6 _ _ L _ ' ................-350 -500 -250 -200 --150 --100 -50

REAL PART

0

Figure BlO.-Closed-loop eigenvalues for the integral feedback

system with the model linearized around zero.

.6

i--n,,<n

>-rY<z

<

i : : i

.4 ........... :............. :................ ×.... i........... _.......... !..........................

0 ×

-.2 ........ _...................................... :................................... i ...........

--.4, ................ x ..... ........

-.6 ;li_il I

.6 -1.4 -1.2

: i L i i ; : ' _ ,

-1 -.8 -.6 -.4 0

REAL PART

Figure B11 .- Low frequency eigenvalues of figure BIO.

.6

i--n,-<Q.

>.-fY<Z

(.9<

.4

.2

0

--.2

--.4

--.6

............. i ........... i --X

× × × X ;;

Z

i I I I ' J ; ' _ L I

20 -- 1O0 -80 -60

REAL PART

--40 --20E

0

Figure B12.- Closed-loop eigenvalues for the discrete integral feedback system withthe model linearized around zero.

I--

0.

<z

o<

.4 ............"............i...................X....................i.............i...........................

0 X_ X

i

--.4 ..............................................× ................:...........:...............

-1.6 --1.4 -1.2 -1 --.8 -.6 --.4 --.2

REAL PART

0

Figure B13.-Low frequency eigenvalues of figure B12.

t_o0

Editing: Comp (Continuous)

Editing: Open (Continuous)

Editing: Z CTRL (Continuous)

Editing: ZOH (3,screte)

Editing: ENG 10 (Continuous)

Figure B14.-Super Block implementation of intergral feedback regulatorwith zero order hold.

REFERENCES

1. Groom, Nelson J.: Analytical Model Of A Five Degree Of Freedom Magnetic Suspension AndPositioning System. NASA TM-100671, March 1989.

2. Kwakernaak, Huibert; and Sivan, Raphael: Linear Optimal Control Systems. John Wiley & Sons,

Inc., 1972.

3. MATRIX X User's Guide, Version 6.0. Integrated Systems Inc., 1986.

4. SYSTEM_BUILD User's Guide, Version 6.0. Integrated Systems Inc., 1986.

5. Britcher, Colin P.: Some Aspects Of Wind Tunnel Magnetic Suspension Systems With Special

Application At Large Scales. NASA CR-172154, September 1983.

6. Boom, R. W.; Abdelsalam, M. K.; Eyssa, Y. M.; and McIntosh, G. E.: Repulsive Force Support

System Feasibility Study. NASA CR-178400, October 1987.

29

_-,Z,Zb

Xb 72 Yb

Figure 1 .-Initial coordinate system alignment for large gap magneticsuspension system.

12---_

13----_

15"''"

t

By _ 'IBZ'Bx/ax,

1 K B t----aBz/ax

Zmax _ _ax,oy (E_(E _>F--Bey/By (Vol) 13] Tm]-I,I---aBz/aYI-..--aBx/az_---aBy/az,

___ _---aBz/az

[Tm] -I Transformation matrix for vector

in body axes to inertial coordinates.

[TE] = Transformation matrix for body

ex=q ,,ex:-o

_'_'l ez_I _ [ez

ler-_ F! '-' '-

" Vx . X =

-I Vy y ,-['Elm] ' Vz z

rates to Euler rates

f.OI-4

Figure 2.-Block diagram of analytical model of large gap magneticsuspension system.

tO

Vo

Open Loop System (Nonlinear)

:I[D(__)-IBj-1D(__)-I__.f(x,u,t)I x_

Linear System: _)_- Asx+B_u

X--EA-BFJz- Dsx

3U

FX

J J[o(-x)-,.l-' '_ _o

Figure 3.-Nonzero set point regulator.

u0

Open Loop System (Nonlinear)

"1 k=f(x,u,t) x

8U

+

(

+

).+

°•t zz.

z--Dsx

z'z-z 0

_I=Z '

x o

Zo

Figure 4.-Integral feedback regulator.

¢,0

Uo

-4-k = f(x,u,t)

X _IzoHl ,

+

5u (

+

F21_ q _sl_

Z

!

Xo

Zo

Figure 5.-Integral feedback regulator with zero order hold (ZOH).

I.-n'-<

>-n,-<Z

o<

10

8

6

4

2

-4

-6

-8

-10

' X

X

-x

x

x

...................... X-

x

x

X

ix

x

, x..... ..... _ ..................... T .....................................

x

X

-x-

x

X

............ '_.................... :............. _ .......................... x-

xx

-350 -500 -250 -200 -150 --100 --50

REAL PART

0

Figure 6.- Eigenvalues of nonzero setpoint regulator with feedback gainscalculated at zero yaw angle as core is rotated from zero to42 degrees yaw in increments of two degrees.

"9 oJn6!j _o sonleAuo6!o ,_ouonbaJj MO-I-'L aJn6!=l

J.aVd 7V3_

_'-- 9'- 8- L-- _L- "1;,'L -- 9L-- _"

xx

xx

x

x

x

X

x

..... : _5_B ..... _ 1. _ X _A ..... _X X X X

x

X

x

• X

............. ....... : X

XX

XX

X

"m

_lT"m

0

[7'

.

m

6")

Z)>

"U

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817 ol oJez LUOJ] peleloJ s! eJoo se elSUe Me/_ OJaZ le pa:lelnOleO

su!e6 _toeqpaej ql!M JOleln6E_J lu!od 183 oJezuou _o sanleAUaS!3 -'8 eJn6!-i

o

/_Vd qV3_

OCj- O0 L-- Og L-- OOT-- OcjT - 00£- Oc_£-OL--

...........!..................................:...........!...............i.............x

x

X '

X .......... '...................... : ......................... , .....................

X

X

X

L,,.i., il,,,. X "1 X X X X _-_

X '

X i

X

X

x ............. f .......... :........................ :.............. i .............. ..............X , ,

x

x- ............. ,........................................ _ ............ !.............. " .............

X

_

9

g

OL

m

if)z

-<

-o

.8

t--

O.

>-

<zo<

.6

.4

.2

-.4

-.6

-.8

xx

xx

i x

x

x

............... .............. : ........ X ................... , ............

×

' X

X X X"'"' ....... 'XXX .... '

X

.............. - ............... !............ i ...... X

X

X

XX

, XX

X

.8 --1.6 -1.4 -1.2 -1 -.8 -.6

REAL PART

-.4.

Figure 9.-Low frequency eigenvalues of figure 8.

18

16

4

I i i i ; : I J i i q : I I i i I I

1O0 1 50 200 250

TIME (SECONDS)

300 350 400

t,o

Figure lO.-Input to nonzero set point regulator with feedback gainscalculated at zero yaw angle.

018

16

14

12

03ILlbJn_ 100wr_

_-_ 8

<

6

4

2

00 5O I00 150 200 250 300 550

TIME (SECONDS)

400

Figure 11 .-Yaw angle response to command input of figure 10.

($33_030) WOWW3 MVA

0

0

O

OOF3

O

rlZO

O (PO hi

W

or-

OO

O

O

O_

I

I--

OT"

--I

°_

O

tm

o_

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I

ai

Ii,--

Ii

,tl

d_

(/3I,Ib.Jn,"0Ld£3

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.00018

.00016

.00014

.00012

.0001

.OOO08

.00006

.00004

.00002

0

-.000020 50 4OO

Figure 13.-Pitch angle response to command input of figure 10.

03I,IT(JZ

X

.0012

.001

.0008

.0006

.0004

.0002

0

_ L ,

0 50 100 150 200 250 500 550

TIME (SECONDS)

400

Figure 14.-X axis translational response to command input offigure 10.

O3

la,

0

-.0O002

O0W

"I"

0

z

>-

-.00004

-.00006

-.00008

-.0001

-.00012

-.000140 50 100 150 200 250 300

TIME (SECONDS)

350 400

Figure 15.-Y axis translational response to command input offigure 10.

O3WIOZ

N

1.2

.8

.6

.4

.2

00 50 100 150 200 250 300 550 400

TIME (SECONDS)

Figure 16.-Z axis translational response to command input of figure 10.

¢,q

F-rY<n

>-

<Z

0<

10

8

6

4

2

-4

-6

-8

-10-550

Y_

-500 -250 -200 -150 -100 -50 0

t.......................................... X _ .........

X:

X i............................ , ............. J ........ X!

XI

xiI

................ 1............ : ......... :............ ; .................... X!F

x!

- " - : ........... : .......... : ......... X-! .............

' , hJl

............' ...........!............i .......... !........... ............ <..........×-!........x!

X_

, , ' , X

. Xi'n

/

REAL PART

_0

Figure 17.-Eigenvalues of integral feedback regulator with feedback gainscalculated at zero yaw angle as core is rotated from zero to 43

degrees in increments of two degrees.

1.2

I..-

r/

p-It"<Z

0<

.9

.6

.3

0

-.6

-.9

' xx

X:x

XX

×

X '

×

....................... × ....... i .........................

×

............. : ......... :............. : ............... i..... X x ___:..............

: i x x x Xxx:,_,x.._

, _

............ :......... ....................... N ........ :................. ,............

x

x

x

-X ....................................

xX

xX:

X X

.6 -1.4 -1.2 --1 --.8 --.6

REAL PART

Figure 18.-Low frequency eigenvalues of figure 17.

oo

18

16

14

C_Ww 12n_owo

10

Z<

8

©

6

<

4

2

0

0 50 100 150 200 250 500 550 400

TIME (SECONDS)

Figure 19.-Input to integral feedback regulator with feedback gains calculatedat zero yaw angle.

L

I I I I I I I I k I I I I I I

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($33W030) MVA

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o o o 0 oo o o o oo o o o oo o o o oo o o o o

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r-

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X

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[0 _'} LDrO 0 (N0 0

i

c

i

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o

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w

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C

X

NI

L.-

U_

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I-n-

Q_

>-n-<z(D<

15

12

9

6

3

--6

--9

--12

y............................................................... ; __X ......................

. X i

i• ' x !

............................. 4............. '-.............. :........................ x--i ...............X X '

' X

, : X............ _.................... . ......................... . ..............................

X

×

............... ,........................... !............ ;................ x i

, ×

, , × X

............................. !......... .......... ; .......... :............. i ...........X !

I

i×

_.0 - 100 -80 -60 -40 -20 0

REAL PART

2O

¢,i

Figure 26.-Eigenvalues of discrete integral feedback regulator withfeedback gains calculated at zero yaw angle as core isrotated from zero to 36 degrees yaw in increments of twodegrees.

F-

CL

>-0C

Zo

o

1.2

.9

.6

.3

-,6

-.9

x..... •......................................................... X ..................

X

x

X

.... i ..................... .× ........ !..............

r ,a_

X X )< X :X X .... "----' v, - -_,.....

X

: X

X

..................... : ................................. : ..... X ..................

; X

-3,5 -3 -2.5 -2 -1.5 -1 -.5 0

REAL PART

Figure 27.-Low frequency eigenvalues of figure 26.

•elSue Me_ OJeZ le pelelnoleOsu!e6 )loeqpeej 41!MJOlleln6eJ )loeqpeej leJ6elu! eleJ::)s!p ol lndul-'Sg 8JnS!-I

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18

16

14

12

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4

2

00 50 1O0 150 200 25-C) 300 550 4.00

TIME (SECONDS)

Figure 29.-Yaw angle response to command input of figure 28.

o

(533W030) _IOUU3 MY1

0o

o

0

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0

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.000024

.000021

.O0O018

.000015

.000012

.000009

.000006

.000003

0

-.0000030 50 1O0 1,50 350

TIME

200 250 300

(SECONDS)

400

Figure 32.-X axis translational response to command input of figure 28.

i.d

i

1

00

0i.o_-)

0o

0u_

(/1r_Z0

0 (D0 wc_

w

,r---

0o

o

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63

Report Documentation PageN==_,_A===-,=um¢=L_¢I

1 Report NO. 2. Government Accession No.

NASA TM-101606

4 Rile anti SubtiBe

An LQR Controller Design Approach for a large Gap Magnetic SuspensionSystem (LGMSS)

7 Author(s)

Nelson J. Groom and Philip R. Schaffner

9 Performing Organization Name and Address

NASA Langley Research CenterHampton, VA 23665-5225

12 Sponsonng Agency Name and Address

National Aeronautics and Space AdministrationWashington, DC 20546-0001

3. Fleootenrs Catalog No.

5. Report Date

July 19906. Performing Organization Code

8. Perforrning Organization Report No.

15. Supplementary Notes

10. Work Unit No.

505-66-91-02

11. Cont_ac_ or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsonng Agency Code

16. Ab$_ac_

Two conUol approaches for a Large Gap Magnetic Suspension System (LGMSS) are investigated and numerical

results are presented. The approaches are based on Linear Quadratic Regulator (LQR) control theory and include a

nonzero se_oint regulator with constant disturbance input and an integral feedback regulator. The LGMSS which is

considered provides five degree of freedom control. The suspended element is a cylinder which is composed of

3ermanent magnet material and the magnetic actuators are air core electromagnets mounted in a planar array.

I7 Key Words (Suggested by Au_or(s))

Magnetic Suspension; Magnetic Levitation;

Pointing System; Magnetic Suspension Model;

Magnetic Suspension Control System

19. Security Classlf. (of I_is report)

Unclassified

18. Distribution Statement

Unclassified - Unlimited

Subject Category 31

20. Security Classif. (of his page) 21. No. of pages

Unclassified 64

NASA FORM 1626 OCT 86

22. Price

A04

For sale by the National TechnicaJ Information Sen/me, Springfield, VA 22161-2171