NASA/TM-2002-212128

Multirate Flutter Suppression System Design for

the Benchmark Active Controls Technology

Wing

Part I." Theory and Design Procedure

Gregory S. Mason and Martin C. Berg

University of Washington, Seattle, Washington

Vivek Mukhopadhyay

Langley Research Center, Hampton, Virginia

m

December 2002

https://ntrs.nasa.gov/search.jsp?R=20030011273 2018-05-15T15:38:32+00:00Z
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NASA/TM-2002-212128

Multirate Flutter Suppression System Design for

the Benchmark Active Controls Technology

Wing

Part L" Theory and Design Procedure

Gregory S. Mason and Martin C. Berg

University of Washington, Seattle, Washington

Vivek Mukhopadhyay

Langley Research Center, Hampton, Virginia

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

December 2002
Available from:

NASA Center for AeroSpace Information (CASI)

7121 Standard Drive

Hanover, MD 21076-1320

(301) 621-0390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 2216 I-2171

(703) 605-6000
iii

CONTENTS

I. Introduction ................................................................................................................................................. I

2. A Methodology for Designing Multirate Compensators ............................................................................ 3

2.1. Overview ........................................................................................................................................ 3

2.2. Definitions, Assumptions and Notation ......................................................................................... 3

2.3. Modeling a Multirate System ......................................................................................................... 4

2.3. I. The GMCLS .................................................................................................................... 4

2.3.1.1. Sampling Schedule/or a GMCLS .................................................................... 4

2.3.1.2. Digital Processor Gains ..................................................................................... 5

2.3.1.3. State Space Formulation of the GMCLS .......................................................... 6

2.3.1.4. Factored Form of the GMCLS .......................................................................... 6

2.3.1.5. Implementation .................................................................................................. 6

2.3.2. The Equivalent Time-Invariant System (ETIS) .............................................................. 7

2.3.2.1. Implementation .................................................................................................. 7

2.4. Synthesizing a Multirate Compensator .......................................................................................... 7

2.4.1. Compensator Structure and Sampling Schedule Selection .............................................. 8

2.4.1.1. Successive Loop Closures Structure ................................................................. 8

2.4. !.2. Coupled Successive Loop Closures Structure .................................................. 9

2.4.2. Optimizing the Digital Processor Gains .......................................................................... 9

2.4.3. Implementation .............................................................................................................. 10

2.5. Analyzing a Multirate System ...................................................................................................... 10

2.5. I. Gain and Phase Margins ................................................................................................ 11

2.5.2. Singular Values .............................................................................................................. 11

2.5.2.I. Unstructured Singular Value Analysis ............................................................ 12

2.5.2.2. Structured Singular Value Analysis ................................................................ 12

2.5.2.3. Implementation ................................................................................................ 13

2.5.3. Maximum RMS Gain .................................................................................................... 13

2.5.3.1. Implcmemation ................................................................................................ 14

2.5.4. Steady-State Covariance ................................................................................................ 14

2.5.5. Time Domain Simulations ............................................................................................. 14

2.6. Summary ...................................................................................................................................... 14

3. Application of the Multirate Design Methodology to the Design of a Flutter Suppression

System for the BACT wing .................................................................................................................... 17

3.1. Introduction .................................................................................................................................. I7

3.2. The Model Wing and Its Open-Loop Dynamics .......................................................................... 17

3.2.1. Model Wing Description ............................................................................................... 17

3.2.2. Open-Loop Dynamics .................................................................................................... 18

3.3. Design Goals and Constraints ...................................................................................................... 20
iv

3.4. FlutterSuppression SystemDesign............................................................................................. 21

3.4.1. SelectingtheCostFunctionWeights............................................................................. 21

3.4.2. SelectingtheCompensator StuctureandSampleRate................................................. 22

3.4.2.1. Single-Rate(SR)............................................................................................. 22

3.4.2.2. MultirateSuccessiveLoopClosures(MRSLC).............................................. 23

3.4.2.3. MultiratewithMultiplexedInputs(MRMI).................................................... 25

3.4.2.4. Single-RateFaultTolerant(SRFT)................................................................ 25

3.4.3. DesigningaStabilizingCompensator ........................................................................... 26

3.4.4. Optimizing the Digital Processor Gains ........................................................................ 27

3.4.5. Design Iteration Based on Performance and Robustness Analysis ............................... 27

3.5. Design Results ............................................................................................................................. 28

3.5.1. Cost Function Value ...................................................................................................... 28

3.5.2. The Gust Pulse Response ............................................................................................... 30

3.5.3. RMS Gain for Control Surface Deflection and Deflection Rate ................................... 32

3.5.4. Gain and Phase Margins at the Compensator Output .................................................... 33

3.5.5. Robustness at the Compensator Input ............................................................................ 34

3.6. Conclusions .................................................................................................................................. 35

4. Conclusions and Recommendations ......................................................................................................... 37

4.1. Conclusions .................................................................................................................................. 37

4.2. Recommendations for Future Research ....................................................................................... 37

References ...................................................................................................................................................... 39

Appendix A. Design Results ......................................................................................................................... 43

A.1. Single-Rate 2nd Order ................................................................................................................. 43

A.2. Multirate Successive Loop Closures ............................................................................................ 43

A.3. Multirate Multiplexed Input ......................................................................................................... 44

A.4. Single-Rate Fault Tolerant ........................................................................................................... 45

Appendix B.

B.1.

B.2.

B.3.

B.4.

B.5.

M-Files Used to Define the Flutter Suppression System Synthesis Problem ......................... 47

PAPAabcd .................................................................................................................................... 47

FSScomp ...................................................................................................................................... 48

mropt_srORmrmi ......................................................................................................................... 48

mropt mrs!c ................................................................................................................................. 48

mropt srft ..................................................................................................................................... 48
1. INTRODUCTION

To study the effectiveness of various control system design methodologies, the NASA Langley Research

Center initiated the Benchmark Active Controls Project. In this project, the various methodologies will be

applied to design a flutter suppression systems for the Benchmark Active Controls Technology (BACT)Wing

(also called the PAPA wing). Eventually, the designs will be implemented in hardware and tested on the BACT

wing in a wind tunnel.

This report describes a project at the University of Washington to design a multirate flutter suppression

system for the BACT wing. The objective of the project was two fold. First, to develop a methodology for

designing robust multirate compensators, and second, to demonstrate the methodology by applying it to the

design of a multirate flutter suppression system for the BACT wing.

The contributions of this project are

1) Development of an algorithm for synthesizing robust low order multirate control laws. The algorithm

is capable of synthesizing a single compensator which stabilizes both the nominal plant and multiple

plant perturbations.

2) Development of a multirate design methodology, and supporting software, for modeling, analyzing

and synthesizing multirate compensators.

3) Design of a multirate flutter suppression system for NASA's BACT wing which satisfies the specified

design criteria
2

Blank
2. A METHODOLOGY FOR DESIGNING

MULTIRATE COMPENSATORS

2.1. OVERVIEW

Our design methodology defines the general approach a designer would take, and provides the specific

tools needed, to solve a multixate control problem. The general approach dictated by the methodology is to

model a multirate system as an equivalent single-rate system, to synthesize the compensator using parameter

optimization, and to analyze the resulting closed-loop system by applying modified single-rate techniques to a

single-rate equivalent model of the multirate system. A schematic of our multirate design methodology is

shown in Fig. 2.1. In the following paragraphs we first in_oduce the terminology and notation unique to

multirate systems and then discuss each aspect of the design methodology along with the applicable design and

analysis tools.

2.2. DEFINITIONS, ASSUMPTIONS AND NOTATION

A multirate sampled-data system consists of a continuous plant in feedback with a multirate compensator.

A block diagram of such a system is shown in Fig. 2.2 where the signals Ys and Yc are continuous output

vectors, u is the continuous control input vector, w is the continuous process noise, and v is the discrete sensor

noise. The primary components of the multirate system are the continuous plant, the sampling hardware (e.g.

A/D converters), a digital processor (e.g., a computer), and the signal holding hardware (e.g., zero-order-hold

D/A converters). The samplers, digital processor and holds will be referred to together as the "multirate

compensator". We will assume that the plant is linear time-invariant, and that the multi.rate compensator

conforms to the Generalized Multixate Control Law Structure discussed in Section 2.3.1.

SucoessNe J

Loop Closures

(2.4.1.1)

Multiple Plant I

Conditions

I

(2.4.2) I

I

[ Low Order Multirate

Robustness Improvement ] Compensator Synthesis

Quadratic Cost ] [ Al_.4"_m

Function _

(2.4.2) 1 [

Stabilizing Compensator

F:.

i+f Generalized Multirate [ I

Compensator Structure I

(2.3.1)

Complex Sampling I

] Schedules I

t r 1-

L__ .2) J , |

-- -- 2_2a. + I

Function Descnption

LI (2_2) (2.3.2) [

l -- -- -Modeling-

+ ¢ + --+--

[ structured [Gaman0Phasel MaxRMSGa_t[SteadyStatoJJl_meDomaJnll Singular Values [ Margins I (2.5.5) | Covadance J Simulation

<25_) L (2.5.1) J (2.5.4) [ (2.5.5) I

Robustness Analysis Performance Analysis _Analysis. !

Figure 2.1. A multirate design methodology. Section numbers indicated in parentheses
4

i _rnZero.Orcler.Hold(_ ) I

[

+

Continuous Plant Ys(t)

_pler(s) I

v (re, n) I

_ rn,n+ l ) = Az{n)_(m,n ) + Bz(n)y(m,n)

_m,n) = Cz(n)7(m,. ) + Dz_n)y_m,.)

Digital Processor

-- MultirateCompensator--

Figure 2.2. Multirate sampled-data system

As we will later see, multirate systems which satisfy our assumptions are periodically time-varying. To

emphasize their periodic nature we will use a double index notation for the independent variable of a sampled or

discrete signal. For example, given a continuous signal y (t), y(m,n) represents y(t) sampled at the time

t=(mN+n)T; where the integer N is the period of repetition; T is the sampling period; m =0, 1 .... ; and

n--0, 1.... N-1.

The design methodology presented in the following sections provides tools to model the closed loop system

in Fig. 2.2, to compute optimum values ofAz, Bz, C z and D z, and to analyze the performance of the closed-loop

system.

2.3. MODELING A MULTIRATE SYSTEM

Two useful modeling tools are the Generalized Multi.rate Control Law Structure (GMCLS) and the

Equivalent Time-Invariant System (ETIS).

2.3.1. The GMCLS

The GMCLS is a control law structure which describes a multirate compensator of arbitrary dynamic

order, with an independent sampling rate for every compensator input, and independent update rates for every

processor state and compensator output. A multirate compensator with the GMCLS is shown in Fig. 2.2. In

this figure each element of the continuous plant output Ys is sampled at an independent rate. The sampled value

ofys, y, is combined with the current processor state vector, _-, using the state space structure shown in the

figure. Each element of the processor state vector, _-, is updated at an independent rate. The continuous output

from the compensator, represented by the vector u, is formed by holding the output from the digital processor,

_, with a zero-order-hold. Each element of the vector _ can be held at an independent rate to form u.

Conceptually, one can divide the multirate compensator into two parts, the "sampling schedule" and the

digital processor gains. This is the approach used in the GMCLS. The "sampling schedule" is a description of

when each compensator input is sampled and when each compensator output and processor state is updated,

while the digital processor gains determine the dynamics of the digital processor.

2.3.1.1. Sampling Schedule for a GMCLS

In general, the sampling and updating of the elements of Ys, z-, and _ in Fig. 2.2 can occur at any time.

However, to conform to the GMCLS, we require that these sample and update activities occur only at integer

multiples of some fixed time, called the shortest time period (STP). The actual value of the STP is arbitrary, but

it is often a function of the hardware and software used to implement the control law. We also require that the

sampling and updating activities of the sensors, states and outputs repeat themselves after some fixed period of
Ysensor ?= _?= =?_ ]?: :9_ _? Sampler

= sample/ulxlate activity

N=15

Figure 2.3 Example Sampling Schedule

Ysensor#1 I ? (_ ? I ? ? ? Sampler

Ysensor#2 ? I f F ? I 1 I Sampler

0 T 2T 3T 4T 5"r 6TTT

? = sarnl_e/update activity

N=4

Figure 2.4 Aperiodic Sampling Schedule

time. (This requirement disallows, for example, a system whose sampling period is a function of the time

require to execute the control software which might vary with control inputs values.) The period of repetition of

the sampling schedule is called the basic time period (BTP). Finally, we define

BTP

the integer N = _ and the value T = STP (2.1)

In our double index notation, the first index (m) in, for example, y(m,n) indicates the integer number of BTP's

which have elapsed when the sample/update occurred and the second index (n) indicates the integer number of

STP's which have elapsed within the current BTP when the sample/update occurred.

We can represent the sampling schedule for the multirate compensator graphically, as shown in Fig. 2.3.

The figure shows a time line for each sampler, processor state, and zero-order-hold. The time line is divided

into one STP increments. On the left side of the time line is a description of the signal or state being sampled or

updated. On the right side is a description of the particular activity represented by the time line, e.g., state

update, sampler, or zero-order-hold. Circles on each time line indicate when a sample or update activity

associated with that particular signal or state takes place. Usually the sampling schedule is shown for only one

BTP since the sampling schedule repeats itself every BTP.

In most applications, the sampling/updating activities for a given sensor, output or state will be periodic

within the BTP, as is shown in Fig. 2.3. However, the sampling/updating activities do not have to be periodic

within the BTP. The only requirement is that the sampling/updating activities have some period of repetition

(the BTP) and that they occur at integer multiples of the STP. Once the STP and BTP have been selected, the

designer can arbitrarily specify sampling/updating activities at any multiple of the STP within one BTP. An

example of a multirate sampling schedule in which the sampling/updating activities are not periodic within the

BTP is shown in Fig. 2.4. A sampling policy like this might be used to multiplex multiple inputs through a

single analog to digital converter.

2.3.1.2. Digital Processor Gains

The processor gains are the values of the matrices A z, Bz, Cz, and Dz in Fig. 2.2. Like the sampling

schedule, they can be periodically time-varying with a period of repetition of one BTP. Generally, these

matrices are free design parameters which can be adjusted by the designer to improve the performance of the

multirate compensator. The synthesis algorithm discussed in Section 2.4 can be used to calculate optimum

values for these gains.
2.3.1.3. State Space Formulation of the GMCLS

A compensator with the GMCLS can be modeled as a periodically time-varying discrete-time system. The

state space form of the GMCLS is given by

z(m,n+l) = Ag(n)z(m,n) + Bg(n)y(m,n) (2.2a)

u(m,n) = Cg(n)z(m,n) + Dg(n)z(m,n) (2.2b)

where

z(m,n) = [_-(m,n) T y(m,n) T _'(m,n)T] T (2.3)

and _(m,n) is used to model the sample and hold activity from _(m,n) to u(m,n). The form of Ag, Bg, Cg and Dg

is given in [Berg, Mason & Yang 1991] and [Mason & Berg 1992] which are included as Attachments 1 and 2.

We should emphasize that Eqn. (2.2) is used to model the complete sampling/updating activities and

dynamics of a multirate compensator. It would not be used in the actual implementation of the compensator.

When implemented, the sample and hold activities of the inputs and outputs would be performed by appropriate

hardware. The only dynamics to be calculated are those associated with the processor state vector _-.

2.3.1.4. Factored Form of the GMCLS

Equation (2.2) is a convenient form to model the general multirate compensator. The difficulty with

Eqn. (2.2) is that it ties up the digital processor matrices, Az(n ), Bz(n), Cz(n), and Dz(n ), in the model matrices

Ag(n), Bg(n), Cg(n), and Dg(n). The matrices Az(n ), Bz(n ), Cz(n), and Dz(n), which describe the dynamics of

the digital processor, are the unknown design parameters which we will later optimize. We can separate the

processor dynamics matrices from the model mauices as follows.

Define the composite compensator matrix:

and factor P(n) as follows

p(n)=I Dg(n) Cg(n) 1

Bg(n) Ag(n)

P(n)= S1(n)Pz(n)S2(n)+ S3(n)

(2.4)

(2.5)

[ Dz(n) Cz(n) ]

where Pz(n)= (2.6)

Bz(n) az(n)

and S1, $2 and $3 are the switching matrices defined by the sampling schedule for the compensator. Their

exact form is given in [Mason 1992] and [Mason & Berg 1992]

It is important to note the difference between P(n) and Pz(n) in Eqn. (2.5). P(n) is a periodically timevarying matrix defined by Eqn. (2.4). It includes all the information about the processor gains and the

sampling/update schedule. Pz(n) contains only the gains for the processor dynamics and is independent of the

sampling schedule.

2.3.1.5. Implementation

The Generalized Multirate Control Law Structure (GMCLS) provides a framework for dealing with

multiple sample/update rates, time delays, and periodically time-varying gains in a digital control system. It

gives the designer freedom to either select the "sampling schedule" that best solves the problem, or if necessary,

to use the "sampling schedule" dictated by existing hardware and software, with out having to worry about the

bookkeeping involved with multiple rates and time delays.
In practice,theGMCLSis implementedin softwareandisrarelyuseddirectlybythedesigner.The

designerneedonlysupplythesamplingscheduleandvaluesforthedigitalprocessorgainsto providea

completecompensator description.Thisdescriptioncanthenbetransformeddirectlyintoa single-rate

periodicallytime-varyingsystemusingtheGMCLS.

TheGMCLSisusedextensivelybythesynthesisalgorithmdescribedinSection2.4,andbythemodeling

andanalysissoftwarereferredtoinSection2.5.Documentation forthissoftwareisprovidedin Ref.43.

2.3.2. TheEquivalent Time.Iavariant System (ETIS)

A multirate compensator with the periodically time-varying structure discussed in Section 2.3.1.3 can be

further transformed into a single-rate Equivalent Time-lnvariant System (ETIS) with the form shown below

x(m+ 1,0) = A E x(m,O) + B EUE(m,O) (2.7a)

y_m,O) = CE X(m,O) + DEuE(m,O) (2.7b)

where

F ys(m,0)7 7

yE(m,O,=[ ys(m'l) [and: uE(m,O)=[ u(m'l) ] (2.8)

l..ys(m oN-1)_1 l..u(m '_V-1)_

We use the subscript E to denote vectors and matrices strictly associated with the ETIS. See [Meyer & Bun-us

1975] or [Mason 1992] for a definition of AE, BE, CE and D E.

A key feature of an ETIS is that a multirate, or periodically time-varying system will be stable if and only if

its ETIS is stable [Kono 1971]. Also notice that the ETIS input/output vectors are composite vectors

containing the input/output values of the multirate (or periodically-time varying) system at N sampling times.

Consequently, an ETIS is always MIMO even ff the original system is SISO. If the multirate system has p

inputs, q outputs and a sampling period of one STP then the ETIS is a single-rate linear time-invariant system

with Np inputs, Nq outputs and a sampling period of one BTP.

2.3.2.1. Implementation

The ETIS is fundamental to the analysis of multi.rate systems. It allows one to evaluate the performance

and stability of complex systems comprised of multirate, periodically time-varying and/or single-rate

components using only techniques developed for linear time-invariant single-rate systems. For example, to

evaluate the stability of the system in Fig. 2.2, we would first transform the multirate compensator into its ETIS

with a given value for N. Then we would discretize the plant at the STP of the compensator using a zero-orderhold and transform the resulting single-rate system into an ETIS using the BTP of the compensator. Next, the

plant and compensator ETIS's could be combined in feedback just as if they were traditional single-rate

systems. Finally, we could determine the stability of the original multirate sampled-data system from the

eigenvalues of its closed-loop ETIS.

Documentation for software capable of transforming multirate and single-rate systems into their ETIS's is

provided in Ref. 43 (NASA TM 2002-212129)

2.4. SYNTHESIZING A MULTIRATE COMPENSATOR

When designing a multirate compensator for the system in Fig. 2.2 there are three components one must

consider: the compensator structure (this includes the dynamical order of the digital processor), the sampling

schedule, and the values for the digital processor gains. In our design methodology the compensator structure

and sampling schedule are selected by the designer based on the open-loop plant dynamics, the hardware

constraints, if any, and the desired dosed-loop performance. Values for the digital processor gains are
calculatedbyoursynthesisalgorithmsoasto provideoptimumclosed-loopperformanceforthechosen

compensator structureandsamplingschedule. In thefollowingparagraphswediscusscompensator structure

andsamplingscheduleselection, andprovideabriefdescriptionof oursynthesisalgorithm.A complete

discussionofthealgorithmisprovidedinRef.43(NASATM2002-212129).

2.4.1. Compensator Structure and Sampling Schedule Selection

The choice of compensator structure and sampling schedule is problem dependent. It depends on the

hardware constraints, the open-loop plant dynamics, and the design objectives. Two often used multirate

compensator structures are worthy of mention, however. They are successive loop closure and coupled

successive loop closures. (Also see [Berg 1986] for a discussion of successive loop closures.)

2.4.1.1. Successive Loop Closures Structure

The simplest multirate compensator structure is successive loop closures (SLC). This structure consists of

multiple decoupled single-rate control loops, each loop operating at a unique sample/update rate. The state

space representation of a SLC structure with two loops is

Xfast(m+l)_-Iafast O llxfast(m)l+Fbfast O ]_yfast(ra)l (2.9a) Xslow(n + 1)J - aslowJLXslow(n)J L 0 bslow.jLYslow(n)J

Ufast(m)_=[CfoaSt O llxfast(m)l+[dfoaSt O IIYfast(m) 1 Uslow( n )J Cslow.JLXslow( n )J dslow_]LYslow( n )J

(2.9b)

where y represents the sampled input from the sensor and u is the output to the zero-order-hold. The subscripts

fast and slow denote inputs, outputs and states which are sampled/updated at a fast or slow rate, respectively.

SLC is best applied to control problems where the closed-loop dynamics are comprised of some fast and

some slow dynamics with the bandwidths of the two separated by at least a factor of four. In this type of

problem, the "fast" loop(s) of the SLC compensator, operating at a fast sampling/update rate, would be used to

control the high bandwidth dynamics, while the "slow" loop(s), operating at a slower sampling/update rate,

would be used to control the low bandwidth dynamics. Problems such as these usually fall into one of two

categories.

In the first, the open-loop system exhibits both fast and slow dynamics. The multirate compensator is used

to improve the performance of this system without drastically changing the fast or slow bandwidths. An

example of this type of problem is an aircraft yaw damper/modal suppression system. The aircraft is open-loop

stable and has some fast dynamics associated with the flexibility of the airframe and some slower dynamics

associated with the yawing motion of the entire aircraft. A multirate compensator for such a system might

consist of a high bandwidth loop to damp the airframe vibrations and a low bandwidth loop to improve yaw

damping.

In the second type, the open-loop dynamics of the plant are arbitrary, but in feedback with the compensator

the closed-loop system exhibits the characteristic fast and slow dynamics. These systems usually have a

decoupled structure where sets of open-loop modes are strongly controllable and observable with a particular set

of inputs and outputs and weakly controllable and observable with the remaining inputs and outputs. An

example of this type of system is the two link robot arm (TLA) used in [Berg, Amit & Powell 1988], and in

[Yang 1988]. All four of the open-loop poles of the TLA are at the origin of the "s" plane. The plant has two

inputs and two outputs. Only two of the modes can be controlled with any one input. Similarly, only two of

these modes can be observed with any one output. In the multirate design, one input/output pair is used to place
twooftheclosed-looppolesatahighfrequencyandtheotherinput/output pairisusedtoplacetheothertwo

closed-loop poles at a low frequency.

Sample rate selection for the individual control loops of a SLC design follows the same guide fines used in

single-rate sample rate compensator design: the sample rate for each SLC loop should be 5 to 20 times faster

than the closed-loop bandwidth desired for that loop. See [Franklin Powell & Workman 1990] for a discussion

of sample rate selection for single-rate systems.

2.4.1.2. Coupled Successive Loop Closures Structure

The coupled SLC structure is the same as the traditional SLC structure except the designer can include

cross feed terms which couple the fast and slow inputs and outputs of the design. In the state space formulation,

cross coupling is represented by non-zero off diagonal terms in the compensator gain matrices. An example of

a compensator structure with cross feed from the slow sampled sensor to the fast sampled/updated control loop

is given in Eqn. (2.10).

Xslow(n + 1)J [ 0 aslowJLXslow(n)J 0 bslowJ[Yslow(n)J

"*

Uslow(n)Jk o Cslow _,_w<,> 0 dslowj[yslow(n) j (2.10b)

This structure is best applied to systems which have coupling between their fast and slow closed-loop dynamics.

See [Yang 1988] for a discussion of cross feed in the TLA problem.

2.4.2. Optimizing the Digital Processor Gains

Having chosen an appropriate compensator structure and sampling schedule, the designer can use our

synthesis algorithm to calculate optimum values for the digital processor gains Az, Bz, Cz and D z sucla that the

closed-loop system in Fig. 2.2 minimizes a quadratic cost function.

The primary design parameter for the synthesis algorithm is the quadratic cost function. By selecting an

appropriate cost function, the designer can influence the performance of the resulting closed-loop system. The

cost function minimized by our synthesis algorithm has the form

r 01

]= t--_** lira EI[yT(t) [" cu (t)][MT Q:JL M]Fyc(t)]I u(,)jj (2.11)

where .7 is the cost associated with the closed-loop system shown in Fig. 2.2. The vector Yc is the continuous

criterion output and u is the continuous control input. QI, Q2 and M are the cost function weighting matrices

and are free design parameters.

The cost function in Eqn. (2.11) has the same form in a continuous time LQR design. Thus the cost

associated with the optimized multirate compensator and that of an LQR design can be compared directly. The

designer can also use this fact to help select appropriate values for Q1,122 and M.

To improve the robustness of the compensator, the synthesis algorithm can optimize the digital processor

gains for multiple plant conditions simultaneously. The resulting compensator will stabilize the each plant

condition and provide overall optimum performance. This is accomplished by minimizing the new cost

function of Eqn. (2.12) which is the sum of the costs associated with each plant condition.

J=E_=_fimEI[YTci(t) Np u?(t)] [Qi.T MilrYci(t)ll Ri JL ui(n JJ (2.12)

i=I i=l t'->"° [
10

Here Ji is the cost associated with the ith plant perturbation and there are Np plant perturbations.

Optimum values of Az, Bz, Cz, and Dz, occur when

3J 3J 3J o3J OJ

ff-_-_ =0, _z =0, -_z =0, and _-D-_-z=0 or equivalently when -_zz=0 (2.13)

Our algorithm uses a gradient type numerical search and a closed form expression for the gradients in

Eqn. (2.13) to determine values of the digital processor gains such that the conditions in Eqn. (2.13) are

satisfied. Refer to [Mason & Berg 1992] in Attachment 1 for a closed form expression for the gradients in

Eqn. 2.13. The synthesis software uses an iterative process to determine optimum values for the digital

processor gains and the user must provide the software with an initial guess for Az, B z, Cz, and Dz. The initial

guess must stabilize every plant condition considered in Eqn. (2.12).

2.4.3. Implementation

In practice, the steps for designing a compensator with our methodology are

1) Construct a continuous LQ regulator for each plant condition which achieves the desired performance

for that condition.

2) Based on the desired closed-loop dynamics and the constraints imposed by the system hardware,

choose an appropriate compensator structure and sampling schedule.

3) Using the chosen sampling schedule and compensator structure, design a compensator which

stabilizes all plant perturbations. When the desired compensator structure is one of the two structures

discussed in the previous section, the designer can use successive loop closures to find a stabilizing

value for the digital processor gains. In successive loop closures, the plant is stabilized by closing one

loop at a time, from one set of inputs to one set of outputs. To obtain a multirate compensator, each

loop is closed using a different sampling/update rate. When, due either to a complex sampling

schedule, or the complexities of the control problem, successive loop closures cannot be used to find a

stabilizing value for the digital processor gains, use Yang's algorithm (see [Yang 1988]). This may

seem counterproductive at ftrst, since one of the reasons for developing our algorithm was the

computational inefficiencies of Yang's algorithm. However, our experience has shown that, in

general, Yang's algorithm converges to a stabilizing compensator fairly rapidly. It is the computation

time associated with optimization of this stabilizing solution that tends to be excessive.

4) Calculate optimum values for the digital processor gains using the synthesis algorithm of

Section 2.4.2. The cost function weighting matrices for the optimization are the same as those used to

design the LQ regulators in Step 1. The starting point for the optimization is the stabilizing

compensator designed in Step 3.

See Attachment 4 for the complete documentation of the software that implements the synthesis algorithm.

2.5. ANALYZING A MUL IR TE SYSTEM

Multirate system analysis is difficult because the periodic nature of a multirate system implies that a

traditional transfer function does not exist. Thus, common analysis tools such as frequency response or Nyquist

diagrams are not directly applicable to multirate systems. Our solution is to transform the multirate system into

a linear time-invariant single-rate system, the ETIS, and then apply established single-rate analysis techniques

using the Z-Transform of the ETIS. (Note: we write the Z-Transform of an ETIS where N=BTP/STP as

GE (zN).) The following paragraphs discuss five useful tools for analyzing the performance and stability of a

multirate system based on its ETIS.
11

Uncertainty A ]___'X_

Linear

Plant

Multirate

Compensator

F@_re 2.5. Plant/compensator configuration

tETIS ETIS Uncertainty v_ of Plant and Compensator

GE (ZN)

Figure 2.6. Plant/compensator with uncertainty

2.5.1. Gain and Phase Margins

In Section 2.3.2 we noted that a multirate system will be stable if and only if its ETIS is stable. Therefore,

we can determine whether the multirate system is stable by applying the Nyquist criterion to its ETIS. Since all

but trivial ETIS's are MIMO, we must use the multiloop Nyquist stability criterion. The multivariable Nyquist

is a plot of the eigenvalues of the ETIS loop transfer function as the discrete variable z traverses the unit circle

[MacFarlane 1970] [Maciejowske 1990].

When the multirate system is SISO we can obtain traditional gain and phase margins from the multiloop

Nyquist plot. Let GE(Z N) be the ETIS loop transfer function and let A be some constant gain and phase

uncertainty at the plant input. If

_(z) = ke/0 where k e/0 is a scalar (2.14)

then AE(zN) = lke jO (2.15)

where/is and Nx N identity matrix

Now the new loop transfer function with the gain and phase uncertainty of Eqn. (2.15) can be written as

HE (zN)loop = GE (z'N) k eie (2.16)

The multiloop Nyquist plot of HE(zN)loop is just the multiloop Nyquist plot of GE (zN) scaled by the gain k and

rotated by the phase shift 8 - the same as in traditional SISO Nyquist plots. Gain and phase margins for the

multirate system can therefore be obtained from the multiloop Nyquist plot of GE (r N) by determining the

values of k and e which destabilize the ETIS. (See [Thompson 1986] for an alternate derivation using Kranc

operators.)

When the multirate system is MIMO, the gain and phase margins calculated by this procedure apply

simultaneously to all inputs and outputs, and are consequently not realistic measures of robustness. To obtain

realistic measures of robustness for a MIMO multirate system, a norm based approach such as singular value

analysis is required.

2.5.2. Singular Values

Singular values are useful for measuring the robustness of MIMO multirate systems. The key step in

multirate singular value analysis is transforming the multirate system in Fig. 2.5 into an ETIS system which has

the output feedback form shown in Fig. 2.6. Since the multirate system will be stable if and only if its ETIS ,is

stable, the closedqoop system in Fig. 2.5 will be stable for a given value of A provided the closed-loop system

in Fig. 2.6 is stable for a corresponding value of AE. Thus we can use single-rate techniques to evaluate the

robustness of the ETIS system and relate those results directly to the associated multirate system.
12

2.5.2.1. Unstructured Singular Value Analysis

A bound on the smallest value of_(AE) for which AE destabilizes the system shown in Fig. 2.6 can be

calculated using unstructured singular value analysis. This system will be stable for all AE such that

_(AE (zN)) < 1 for all z on the unit circle (2.17)

?s(Ge(zN))

(see [Maciejowski 1989]). This result, however, is only a measure of the size of the smallest destabilizing AE

and is generally not a measure of the size of the smallest destabilizing uncertainty A. Because the input/output

vectors of an ETIS are composite vectors, containing the input/output values of the multi.rate system at N sample

times, AE can be a complex function of the values of A at N sample times. (The relation between LiE and A is

given by Eqn. 2.7.) The size of the smallest destabilizing A E found using unslructured singular value analysis is

only a conservative estimate of the size of the smallest destabilizing A. This estimate accounts for not only the

fictitious perturbations normally associated with unstructured singular values, but also for time-varying and

non-causal perturbations.

Consider the simple case where A is a constant. From Fig. 2.5 we have that

For an ETIS with N=2

w=Av (2.18)

or lw(m'O)l=[ _'' A'21f, v(m'O)l (2.19)

we=aeVe Lw(m,1)j LA2, A22Jtv(m,1)J

A destabilizing AE determined by singular value analysis might, for example, include block diagonal elements

in A E which are unequal, e.g. A11 ¢ A22- This corresponds to a time-varying perturbation because the gain

between w and v varies with time. Another such AE could include non-zero upper block diagonal elements in

AE, e.g. A12,0. This corresponds to a non-causal perturbation because a future input, v(m,1), can affect the

current output w(m,O).

We can eliminate this conservativeness by restricting the allowable perturbations in AE. This leads directly

to structured singular value analysis.

2.5.2.2. Structured Singular Value Analysis

In order for the ETIS uncertainty A E to represent the actual uncertainty A, its structure must obey

Eqn. (2.7). Finding the size of the smallest destabilizing AE subject to Eqn. (2.7) requires the solution of a

structured singular value problem. For the system in Fig. 2.6 we define the slructured singular value, St, as

0 if det(l - GE(zN)AE(zN)) _ 0 for all A _ ABD Ia(GE(zN)) ( (2.20)

l/ suchthat det(t- 6E(zN)/,E(zN)) = O/ otherwise

J

where ABD is the form of the pem_ssible block diagonal perturbations z_ and the structure of AE must satisfy

Eqn. (2.7). The size of the smallest destabilizing perturbation _(Amin) satisfies

1 = supp(GE(zN)) where z N = e j¢ (2.21)

_(Arm) ¢

For a discussion of# and ABD see [Doyle 1982].
13

Unfortunately, evena simplystructureddynamicuncertaintyA(z)transformsto anETISuncertainty,

AE (zN), with a complex structure. For example, if N=2 then the ETIS of A (z) is

AE(Z 2) 1 I A(z) + A(-z) z-I(A(z)- A(-z))" (2.22)

= _Lz(A(z)- A(-z)) A(z) + _(-z)

In order to find the value of_(Amm) using Eqn. (2.21), one must solve Eqn. (2.20) with AE constrained to have

the structure in Eqn. (2.7). Currently there is no general technique for solving this problem. When, however,

the uncertainty, A, is a constant, as is the case for many problems, the ETIS uncertainty, AE, is also a constant

with a repeated block diagonal form.

AE = diag(A, A..... A) with N blocks. (2.23)

There are several good methods for estimating_(Amin) when AE has this block diagonal structure. One simple

method for estimating/.t when A is strictly diagonal is derived in [Safonov 1982]. It is

IZ(GN(zN)) < inf(-_(abs(DGN(zN)D-l)) = 2p(GN(zN)) (2.24)

D

where abs(A) is a matrix such that [abs(A)] ij = IA/jl; Aij is the itj th element of A; and Xp is the Perron-Frobenius

eigenvalue.

2.5.4.3 Implementation

The procedure for performing singular value analysis via the ETIS is as follows

1) Transform the problem into the form shown in Fig. 2.5

2) Discretize the plant at the STP of the compensator and compute the ETIS of the plant using the N of

the compensator

3) Combine the ETIS of the plant and compensator to obtain the closed-loop system shown in Fig. 2.6

4) Use any applicable single-rate singular-value based analysis tool to compute the size of the smallest

destabilizing uncertainty AE.

5) Interpret the results in the light of the fact that the computed results are for an ETIS uncertainty AE

whereas the actual plant uncertainty is A. AE is a function of A as given by Eqn. 2.7 and so the results

might be conservative unless structured singular value analysis is used.

2.5.3. Maximum RMS Gain

The maximum RMS gain of a SISO single-rate system is the maximum gain on that system's Bode plot.

As already noted, a traditional Bode plot cannot be generated for a multirate system. However, the maximum

RMS gain of a SISO multirate system can be computed; it is the H** norm of the ETIS transfer function. This

value, shown in Eqn. (2.25), plays the same role as the maximum Bode plot gain of a single-rate system.

RMS(y(m, n)) RMS(yE( m, 0))

pd_u(P)_o RMS(u(m,n)) = sup = IIGE(ZN)II** (2.25) RMS(uE),0 RMS(uE(m, 0))

Actually, Eqn. (2.25) can be used to calculate the RMS gain of SISO or MIMO systems. It simply states that

the maximum RMS gain of a transfer function G E is equivalent to the H,_ norm of GE. See also the related

work of [Sivashankar & Khargonekar 1991].

Unlike linear time invariant single-rate systems, the discrete input signal resulting in the maximum

multirate RMS gain does not necessarily have the simple form sin(o9 Tin). Instead it is comprised of the sum of
14

sinusoidsof severaldistinctfrequencies.Detailsoncomputingthesignalof maximumRMSgainfor a

multiratesystemaregivenin[-Mason&Berg1992]

2.5.3.] Implementation

One simple method for determining the H._ norm is to plot the maximum singular value of GE as z

traverses the unit circle. H._(GE) is then the peak value on that plot.

It is important to remember that Eqn. (2.25) is a measure of the discrete RMS gain between the discrete

inputs and outputs of interest. Often the designer is interested in calculating the maximum RMS gain between a

continuous input and output of a sampled-data system. A good estimate of the RMS gain in this case can be

found by sampling the continuous input and output of interest at a fast rate. The result is a multirate system -

the input and output of interest are sampled/updated at a fast rate while the other inputs and outputs are sampled

at the rate appropriate for connection to the multirate compensator. (This is also useful for determining the

inter-sample behavior of a sampled-data system.) The maximum RMS gain can then be calculated using the

ETIS of this new system.

2.5.4. Steady-State Covarianee

A common measure of performance is the steady-state covariance of select outputs in response to a

disturbance input. In a multirate system the "steady-state" covariance values are periodically time-varying.

Fortunately, the periodic "steady-state" covariance values at each sample/update time are straightforward to

calculate using the ETIS.

It is easy to show that

E{y(m,O)y(m,O)T }

E{YEyT} =1 E{y(m'I)y(m'O)T}

[E{y(m, N - 1)y(m, 0) T}

E{y(m,O)y(m,1) T} ... E{y(m,O)y(m,N-1) T} ]

E{y(m, 1)y(m, 1)T} E{y(m, 1)y(m, N- 1)T} I (2.26)

• • •

E{y(m, N- 1)y(m, 1)T} .-- E{y(m, N- 1)y(m, N- 1)T}

The diagonal block elements of Eqn. (2.26) contain the steady-state covariance values at each sample/update

time of the corresponding muir.irate system. Therefore, the steady-state covariance values can be found by

calculating the ETIS of the multirate system and computing the steady-state covariance values of the ETIS using

the discrete Lyapunov equation. Refer to [Kwakernaak & Sivan 1972]. Algorithms for calculating discrete

covariance values are widely available (e.g., in Maflab and in Matrixx).

2.5.5. Time Domain Simulations

Time domain "simulations are straightforward to compute using the ETIS and Eqn. (2.7). As noted in

Section 2.5.3, inter-sample behavior can be obtained by sampling the continuous inputs and outputs at an

arbitrarily fast rate. Documentation for the M-File mrsim, which generates a time domain simulation of a

multirate sampled-data system using the ETIS is provided in Ref. 43 (NASA TM 2002-212129).

2.6. SUMMARY

The tools presented in this section form the foundation of our multirate design methodology, and provide a

unified approach to multirate modeling, synthesis and analysis. Using these tools one can model a complex
15

multiratecompensator, determinetheoptimumvaluesofthatcompensator's processor gains,andanalyzeits

robustnessandperformance. In manycasesthemultiratesystemsmodelingandanalysiscanbeperformed

usingcommerciallyavailablesoftwarein conjunctionwith',heETIS. Forthosetoolsspecificto multirate

systems, includingmultiratecompensator synthesis, documentation forcustomsoftwarehasbeenprovidedin

Ref.43(NASATM2002-212129).
16

Blank
17

3. APPLICATION OF THE MULTIRATE DESIGN METHODOLOGY TO THE

DESIGN OF A FLUTTER SUPPRESSION SYSTEM FOR THE BACT WING

3.1. INTRODUCTION

To demonstrate some of the advantages of multirate control and the capabilities of our design methodology,

we designed several flutter suppression systems for NASA's BACT wing using the methodology in Section 2.

A summary of our designs is presented in the following paragraphs. In Section 3.2 we describe the model wing

and its open-loop characteristics. In Section 3.3 we discuss our design goals and conslTaints. In Section 3.4 we

discuss our design approach and the details of the design process. In Section 3.5 we present our flutter

suppression system design results. Finally, we end the chapter with some concluding remarks in Section 3.6.

3.2. THE MODEL WING AND ITS OPEN-LOOP DYNAMICS

3.2.1. Model Wing Description

The BACT wing was developed by NASA Langley for the Benchmark Models Program. It consists of a

rigid airfoil mounted on a flexible base. The base, called the Pitch and Plunge Apparatus (PAPA), provides the

two degrees of freedom needed to model classical wing flutter. Our designs used the single control surface (CS)

located on the trailing edge of the airfoil and two accelerometers, one near the trailing edge (TE) of the airfoil

and one near the leading edge (LE). A diagram of the BACT wing is shown in Fig. 3.1. A detailed description

of the BACT wing can be found in [Durham, Keller, Bennett & Wieseman 1991] and [Bennett, Eckstrom,

Rivera, Dansberry, Farmer & Durham 1991].

The flutter suppression system was designed using a 16th order linear state model of the BACT wing

developed by NASA Langley's Structural Dynamics Division. This model consists of 4 rigid body states

corresponding to the pitch and plunge modes, 6 unsteady aerodynamic states, a second order actuator model, a

second order Dryden filter, and two first order anti-aliasing filters. A block diagram of the mathematical model

is shown in Fig. 3.2 on the following page.

We were provided with 24 different mathematical models of the wing. These models describe the motion

of the wing in freou at 24 different operating points. The operating points include dynamic pressures above and

below the critical flutter pressure at three different mach numbers. See Table 3.1 on the following page for a

summary of the operating points.

Pitch and Plunge Apparatus

/

NACA 0012 Airfoil

Control Surface (CS)

Trailing Edge (TE)

Accelerometer

32 in

Leading Ed_e (LE) 16 in _--X_ AcceleromeTer

Figure 3.1. BACT wing
18

Disturbance Input

ro oooF,-, 2nd order I

CS Comma__ Actuator IJ BACTWing

| 2nd order _-_ 10 th order

L

____ Anti-aJiasing j _ To A/D

Filter 1st order _" Convener

LE Accel J AntJ-aliasing _ To A/D

:]Filter 1st order _"Converter

Figure 3.2. Block diagram of BACT wing

Table 3.1. Operating points for BACT wing. All operating points assume Freon medium

Mach 0.50 75

Mach 0.70 75

Mach 0.78 75

Dynamic Pressure (psf)

(Nominally unstable operating points are in gray)

•_ _ :,_:_i_ _,_!__i_,_ _,,:I_¢,_:_ _ '_?;,,,'!:_:_,_,_,_::_:_: _ : _: :

100 125 136

100 125 141 _::_!

3.2.2. Open-Loop Dynamics

The response of the open-loop BACT wing model at each operating point is characterized by two dominant

modes - the pitch and plunge modes. The poles associated with pitch and plunge at roach 0.5 and 75 psf are

indicated on Figs 3.3-3.4. As the dynamic pressure increases, one pair of these dominant poles moves towards

the right half plane and eventually crosses the imaginary axis at the flutter stability boundary. Figures 3.5-3.6

show the migration of these dominant modes as dynamic pressure increases. The locations of the open-loop

poles not shown in the figures remain relatively constant.

The dominant pitch and plunge modes arc observable at all operating points with either the TIE or the LE

accelerometer outputs and are controllable at all operating points using the CS command input. The zeros of the

CS command to TE accelerometer and the CS command to LE accelerometer transfer functions are shown in

Figs. 3.3-3.4 for an operating poin t of roach 0.5 and 75 psf. As dynamic pressure increases, the non-minimum

phase zeros associated with the TIE accelerometer migrate into the left half plane. The minimum phase zeros

that are associated with the LE accelerometer and located near the dominant poles migrate into the right half

plane. See Figures 3.5-3.6.

At low dynamic pressures the transfer functions from CS command input to both the TIE and LE

accelerometer outputs are non-minimum phase. Non-minimum phase systems are typically more difficult to

control than minimum phase systems. An alternative output is one which measures the difference between the

two accelerometers. This new output is essentially pitch acceleration. The CS command to pitch acceleration

transfer function is minimum phase for all operating points. Figure 3.7 shows the locations of the zeros near the

pitch and plunge modes as dynamic pressure increases. It turns out that the BACT wing is fairly easy to control

using this new output. The problem is that the pitch acceleration output is artificially created and assumes

perfect measurement of TE and LE accelerations. In reality there is some uncertainty in the TE and LE

acceleration measurements that must be accounted for in any design. Therefore we did not use the pitch

acceleration output directly in our designs. We did, however, use the pitch acceleration output to determine an

initial stabilizing compensator for the synthesis algorithm. This is discussed further in Section 3.4.3
19

3oo

20o

0

100 Actuator Pole Actuator Pole

o ........................................................................... _ ....... _ ..............._" - Z,.,

y--

-100 Pitch & Plunge Pole_

0

-200

X open-loop poles

o open-loop zeros

3°_-4_0 -3_0 -3bo -2_,0 -260 -1_0 -tbo -_;0 50

Figure 3.3. Pole/Zero map for open-loop BACT wing at roach 0.50, 75 psf for CS command to TE Accel.

300

200

lo0 Actuator Pole

/

0 x

-100

-200 x open-loop poles

o open-loop zeros

-300

-400 -350 -300

Actuator Pole

_-u .... #!

II

Pitch & Plunge Poles_

"1

o

-25o -2bo -1_o -lbo -_o o so

Figure 3.4. Pole/Zero map for open-loop BACT wing at mach 0.50, 75 psf for CS command to LE Accel.

40

35

30

25

20

15

10

-4

Mach 0.50

_ __ m Mach 0.70

....... Mach 0.78

Arrows .point in direction of

increasing dynamic pressure

.... t-"--- -_c-^-;_x ;_x _.x:._°°-r - "_" "_--_t-_x

x open-loop poles

o open-loop zeros

Figure 3.5. Migration of open-loop poles and zeros for CS command to TE Accel.
2O

4O

-- Mach 0.50

- -- -- Mach 0.70

....... Mach 0.78

Arrows point in direction of

increasing dynamic pressure 30

25

2O

15

10

-4

x=_j....._ ...... ;,."-.... _-^-h/_-_-x -_-xYx open-loop poles

o open-loop zeros

x

0 1

Figure 3.6. Migration of open-loop poles and zeros for CS command to LE Accel.

40

Mach 0.50

35i -- -- -- Mactl 0.70

....... Mach 0.78

Arrows point in direction of

30 increasing dynamic pressure

25

-- -_":-- _- m'___-_-.-_.._. _,o

20 _ ..... _ x .... ;_-x-_,_- _- x _'-_:-"

15

x open-loop poles

o open-loop zeros

10

-4 3 -_ -_ 0 _ 2

Figure 3.7. Migration of open-loop pole and zeros for CS command to TE-LE Accel.

3.3. DESIGN GOALS AND CONSTRAINTS

The goal of the design was to synthesize a multirate flutter suppression system which stabilizes the BACT

wing at all 24 operating points. In addition to stability, NASA Langley specified the following constraints.

Control Activity Constraint: For unity RMS white noise input disturbance (1 in/sec RMS), the steadystate covariance of the CS deflection must not exceed 0.0625 deg 2 (0.25 deg RMS), and the CS

deflection rate must not exceed 65 deg2/sec 2 (8.0 deg/sec RMS).

Sampling Rate Restrictions: The minimum sampling period is 0.005 seconds. For multirate sampling all

sampling periods must be multiples of 0.005 sec.

Computational Delay: All compensators must be designed with a minimum 0.005 second computational

delay.
21

Robustness Constraints: The gain and phase margins at the compensator inputs and output must be _+6db

and +45 ° . At the compensator input, which has two sensors, we use the generalized gain and phase

margins based on the singular value. The specified gain and phase margins correspond to a minimum

value of 0.75 for the maximum singular value of a multiplicative uncertainty at the compensator inputs

(see [Mukhopdhyay & Newsom 1984]).

3.4. FLUTTER SUPPRESSION SYSTEM DESIGN

We used the methodology discussed in Section 2 to design the flutter suppression system. The specific

steps for this design were:

1) Select an LQR cost function such that the BACT wing in feedback with the LQ regulator satisfies the

criterion specified by NASA

2) Choose an appropriate multirate compensator structure and sampling schedule based on this LQR

design

3) Find a set of processor gains so that the compensator stabilizes the BACT wing

4) Synthesize a multirate compensator which minimizes the LQR cost function of step (1) at a few select

operating points using the algorithm discussed in Section 2.4

5) Check the performance and robustness of the closed-loop system

6) Iterate on items (1)-(5) as required

We elaborate on the details of each step in the following paragraphs.

3.4.1. Selecting the Cost Function Weights

The multirate synthesis algorithm finds optimum values of the compensator's digital processor gains by

minimizing a quadratic cost function with respect to those gains. This optimization can be performed for

multiple plant conditions simultaneously. We used the multiple plant capabilities of the algorithm to help

ensure that the compensator stabilizes the wing at all 24 operating points. Instead of using all 24 operating

points for the optimization we used six representative ones. The six include the operating points at the extremes

of mach number and dynamic pressure, and two operating points midway between the extremes. These

operating points are listed in Table 3.2 on the following page. For the fault tolerant design discussed in

Section 3.4.2.4 we included four additional operating points at roach 0.50. These operating points are grayed in

Table 3.2.

For each operating point we selected a unique set of weights for the synthesis algorithm's cost function.

The weights were based on a continuous LQR design which weighted the pitch and plunge modes, and the CS

command input of the BACT wing. The cost function has the form

x t)+ u t)} Ol)

where x = {x 1 x2 x3 x4 }T and the xi are the four states associated with the pitch and plunge mode in a

modalized version of the BACT wing model. States Xl and x2 correspond to the complex conjugate poles which

migrate to the left as dynamic pressure increases, see Fig. 3.5. States x 3 and x4 correspond to the complex

conjugate poles which migrate to the right as dynamic pressure increases, see Fig. 3.5. The latter set of poles

cause instability in the BACT wing at high dynamic pressures. The variable u is the CS command signal.

For each operating point, the weights, Q 1 and Q2, were chosen so that the closed-loop damping of the pitch

and plunge modes was greater than 0.07, and the RMS control constraints specified by NASA were satisfied.

For comparison, the damping in the open-loop BACT wing at the stable dynamic pressure of 75 psf is
22

approximately0.025•Theweightsfor eachoperatingpointwerescaledtoobtainaunityLQRcostfor a6

inch/seeRMSwhitenoisedisturbanceinput.

Table3.2.Costfunctionweights. Grayedoperating points used only for fault tolerant design.

Operating Point State Weight (Q 1) Control Weight (Q 2)

Mach 0.50 75

_. _ ..

Mach 0.50 225

Mach 0.70 125 psf

Mach 0.70 175 psf

Mach 0.78 75 psf

Mach 0.78 225 psf

9sf diagI1.2x10-2 1.2x10 -2 12 12] 610

_sf diag[9.6xl0 -4 9.6x10 "4 9.6x10 -2 9.6x10 -2] 4800

diag[1.3xl0 -2 1.3x10 2 6.4 6.4] 3900

diag[1.9xl0 -3 1.9x10 3 0.56 0.56] 5600

diag[8.8xl0 -2 8.8x10 -2 44 44] 8800

diag[3.3xl0 -4 3.3x10 "4 1.6x10 -2 1.6x10 -2] 26000

3.4.2. Selecting the Compensator Structure and Sample Rate

Traditionally, the design of a multirate compensator structure begins with a successive loop closures

structure and then incorporates cross feed between the loops as necessary. As discussed in Section 2.4.1,

multirate successive loop closures is best applied to problems in which the closed-loop system dynamics can be

separated into some fast dynamics and some slow dynamics. The BACT wing however does not exhibit those

closed-loop characteristics. Closed-loop bode plots, from control input to accelerometer outputs of the BACT

wing in feedback with a LQ Regulator, are shown in Fig. 3.8. The LQ Regulator was designed using the cost

function weights for the mach 0.50 75 psf operating point specified in Table 3.2. Therefore the bode plot _s

representative of the closed-loop dynamics we are trying to achieve with the flutter suppression system. Notice

that the closed-loop dynamics have only one peak - that associated with the pitch and plunge modes - and do not

exhibit the fast and slow dynamics traditionally associated with successive loop closures. Consequently, a

traditional multirate successive loop closure structure is not directly applicable to this problem.

Instead of basing our multirate compensator structure on the closed-loop dynamics of the system, we

selected compensator structures which used different sampling schedules to reduce either the number of

computations or the hardware required to implement the compensator. We designed four compensators: a

single-rate (SR); a multirate successive loop closures type (MRSLC); a multirate with multiplexed inputs

(MRMI); and a single-rate fault tolerant (SRFT). All of these compensators are second order except the fault

tolerant design which is fourth order.

3.4.2.1. Single-Rate (SR)

The single-rate compensator was designed for comparison with the other compensators. A block diagram

of this compensator is shown in Fig. 3.9. The sample/update rate for this compensator is 50 Hz. This rate is

approximately 10 times the frequency of the dominant pitch and plunge modes. The compensator includes a

0.02 second computational delay, which satisfies NASA's computational delay requirement. This was achieved

by constraining the compensator's direct feedthrough term to be zero.

The state space structure of the compensator is

'l(m," + 1)_ __[ 0 1 l#'l(m,n);+FO bl l#TE Accel(m,n) l Z2(m,n+l)j a2j[z2(m,n)j L_ b2j[LEAccel(m,n) j (3.2a)
23

80

70

60

50

40

30,

20

10

0

lOO

CS Cmd Input to TE Acce! Output

i-" CS Cmd Input to LE Accel Output

i 1 1 i i i i i i _ j _ _ i i J i J ii 1 i 1 i i i i

101 10 2 10 3 104

Rad/Sec

Figure 3.8. Bode plot of closed-loop BACT wing with LQ regulator at mach 0.5 75 psf

,e_ _0.0.0_2_ TE Accel

2 nd Order 0.02 sec

CS Cmd Digital Processor

0.02 sec I 0.02 Sec Upclate _ LE Accel sex: I

TEAcee, '9 ,9 ? ,9 ,gSampler

LEAe, eel ? Q, ,9 ,9 ,gSarnl:der

CScmd9 ,9 9 q ,gZOH

Zl, Z2 9 9 9 9 9Stata Updates

S'T'P=0.02 sec _ _ BTP=0.02 sec

N=t

= sarnple/up_ate activity

Figure 3.9. Block diagram and corresponding sampling schedule for the SR compensator

CS Cmd(m,n)=[c 1 c2 [z2(m,n)J

where z-1 and 52 are the digital processor states; TE Accel and LE Accel are the acceleration inputs from the

A/D converters; and CS Cmd is the command output to the zero-order-hold, a i, bi, and q are the free gains

(matrix elements) which were optimized. The other gains were constrained to the values shown. The structure

in Eqn. (3.2) is a minimum realization of the second order compensator. See [Berg, Mason & Yang 1991] for a

discussion of minimum realizations. The sampling schedule for Eqn. (3.2) is shown in Fig. 3.9.

3.4.2.2. Multirate Successive Loop Closures (MRSLC)

The MRSLC compensator was designed to reduce the total number of multiplications per unit time

performed by the compensator's digital processor. The compensator is comprised of two first order loops. Both

Joops have two inputs, TE and LE acceleration, and one output, CS command. One of the ]tops is

sampled/updated at 50 Hz, the same as the single-rate design, and the other is sampled/updated four times

slower at 12.5 Hz. Just as in the single-rate design, the direct feedthrough terms were constrained to be zero,

resulting in a 0.02 second computational delay.
24

CS Cmd

0.08 sec

1st Order

Digital Processor

0.02 Sec Update

1st Order

Digital Processor

0.08 See Update

TEAccel _ _ ? ? ?Sampler

LEAocel _ _ ? _ ?Sampler

_e--0_0"02s_'_--TEAccel CScmds ? I i I ?ZOH

•_..,__-LEAocel CS cmdf ? 9 Q ? Q ZOH

Zl 9 9 9 9 9StateUp date

STP=O.02 sec

I 008 sec _ _ BTP=o.08 sec

N=4

9 = sample/update actMly

Figure 3.10. Block diagram and corresponding sampling schedule for the MRSLC compensator

o

-ft.

c

O

iv50

40

30

20

lO

• • • O • •

• , , , I i i i i i

12345 67 8910

Ratio of Fast to Slow Sample Rates

Figure 3.11. Computational savings with the MRSLC design

The net result of this two loop configuration is a compensator structure just like the single-rate design

except that the digital processor needs to update one of the digital processor states only every fourth

sample/update period. A block diagram of this compensator along with a diagram of its sampling schedule is

-shown in Fig. 3.10. Note that this diagram only illustrates the structure of the compensator - it is not a

schematic of how the compensator would be implemented. When actually implemented, this compensator will

use the same number of D/A and A/D converters as the SR compensator, but will require 37% fewer real-time

multiplications per unit time.

The choice of sample/update rates for the slow loop was arbitrary within the constraints of the GMCLS.

Our goal was simply to reduce the number of multiplications required by the compensator without significantly

degrading its performance. The 12.5 Hz sample/update rate was chosen because it is a good compromise

between the total number of multiplications saved by utilizing this multirate structure and the ratio of the fast to

slow sampling rates. Figure 3.11 shows the percent reduction in the number of multiplication by using the

MRSLC design over the SR design. There is a decreasing return in computational savings as the ratio of the

fast to slow sampling rate increases. In the limit, the compensator degenerates to a first order compensator with

a reduction in multiplications of 50%. Based on Fig. 3.11 we chose a sampling rate ratio of 4.

The state space structure of the compensator which was used for the optimization is

,l(m,n+l)l=[aof o]_=i<,.,.>l+[1 _

asJ[Z2(m,n)J L1 Accel(m, n)J
25

CS Cmd',_E_ _-

O.02s_

_p._____ TE Accel

2 nd Order PTV 0.04 sec

Digital Processor

0.02 Sec Update LE Accel

I 0.04 sec

J 0.02 sec delay

TEAccel _ f _ i _Sampler

LEAccel I _ I _ I Samp let

cs_,-nd? 5' ? ? ?ZOH

•_,"2? f' ? ? ?Stat_Up=t,,,

S_=O.02,_c_ _ 8rP=o._ _e_

N=2

?= sample/update actMty

Figure 3.12. Block diagram and corresponding sampling schedule for the MRMI compensator

_CSCmdf(m,.)l=[_ 0 l[Zl(m,.) l (3.3b) [CS Cmds(m,n) j _ CsJt_2(m,n)J

cs Cmd(m,n) = CS Cmdf(m,n) + CS Cmds(m,n) (3.3c)

where Z'l and z-2 are the digital processor states; TE Accel and LE Accel are the acceleration inputs from the

A/D converters; and CS Cmd is the command output to the zero-order-hold, ai, bi, and ci are the free gains

which were optimized. The other gains were constrained to the values shown. The structure in Eqn. (3.3)

corresponds to the successive loop closures structure of Fig. 3.10. The intermediate outputs CS Cmpf and

CS Cmps were added to ensure that Equ. (3.3) corresponded to Fig. 3.10.

3.4.2.3. Multirate with Multiplexed Inputs (MRMI)

The multirate compensator with multiplex inputs was designed to reduce the number of A/D converters

required to implement the SR design, In this design, the compensator state and output updates occur at 50 Hz.

The outputs of the TE and LE accelerometers are sampled at 25 Hz with a 0.02 second delay between the

sampling of the TE accelerometer output and the LE accelerometer output. Thus, the MRMI requires only one

A/D converter to sample both accelerometer outputs because it can be multiplexed between the two signals. In

addition, the digital processor gains for the MRMI compensator are periodically time-varying. One set of gains

is used when the TE accelerometer output is sampled and another set is used when the LE accelerometer output

is sampled. Just as in the single-rate design, the direct feedthrough terms were constrained to be zero, resulting

in a 0.02 second computational delay. This compensator requires the same number of multiplications per unit

time as the SR design but it uses only one D/A converter. Figure 3.12 shows a block diagram of the MR_MI

compensator.

The state space structure of the MRMI compensator is

z'2(rn'n+l)J ,l(m,n+l)l=[ al(n) 0 a2(n)JLz2(m,n)J 1 l_'_.l(m,n)l+IO L1 b2(n)JLLEAccel(m'n)J bl(n)]_TEAccel(m,n) l(3.4a)

,fZl(m,n)]

CS Cmd(m,n)=[cl(n) c2(n)J_2(ra,n) l (3.4b)

where z-1 and z2 are the digital processor states; TE Accel and LE Accel are the acceleration inputs from the

A/D converters; and CS Cmd is the command output to the zero-order-hold, ai(n), bi(n), and ci(n ) are the free

gains which were optimized. These gains are functions of n because they are periodically time-varying, e.g.

ai(n) = ai(n+2) The other gains were constrained to the values Shown. The sampling schedule for Eqn. (3.4) is

shown in Fig. 3.12.
26

CSCmd_:_--

o._5s_

4 th Order 1._6Too.oo5sec TE Accel

Digital Processor

0.005 Sec Update I 0.005 sec LE Accel

TEAccel ? ? ? ? ?Samper

LEAch( ? ? ? ? ?Sampler

CScmd ? ? ? ? ?ZOH

Zl,Z2, Z3, Z4 ? _ ? ? _ State Updates

STP==O.005 seo--_ _ BTP--0.005 sac

N=I

?= sampte/ul:x;late activity

Figure 3.13. Block diagram and corresponding sampling schedule for the SRFT compensator

3.4.2.4. Single-Rate Fault Tolerant (SRFT)

The single-rate fault tolerant compensator was designed to highlight the multiple plant capability of our

synthesis algorithm. This compensator is fourth order with a sample/update rate of 200 Hz and a 0.005 second

computational delay. A block diagram of the compensator and its corresponding sampling schedule are shown

in Fig. 3.13. The state space representations of the SRFT compensator is similar to the 2nd order single-rate

compensator with the exception that the digital processor is fourth order.

The SRFT compensator is fault tolerant in the sense that it stabilizes all the plant conditions even with one

of the accelerometers disconnected. To achieve fault tolerance for all 24 plant conditions, we optimized the

compensator for 22 simultaneous plant conditions - as opposed to just six for the preceding designs. These

include the six operating points used in the previous designs evaluated at three cases each: 1) both TE and LE

sensors active; 2) only the TE sensor active; and 3) only the LE sensor active. In addition to those 18, we

added four more operating points at roach 0.50 evaluated for the case where only the LE sensor is active. These

operating points are grayed in Table 3.2.

3.4.3. Designing a Stabilizing Compensator

We used the synthesis algorithm presented in Section 2.4 to optimize the gains of the four compensators

discussed in Section 3.4.2. The algorithm requires an initial guess for the compensator's digital processor gains

for which the closed-loop system, the BACT wing and compensator, is stable. The difficulty in finding these

gains is that the closed-loop system must be stable at all operating points used in the optimization.

To get a stabilizing guess for the wing at all operating points we used a boot-strapping technique. First we

found values of the processor gains which stabilized the BACT wing at one operating point. Then we optimized

the gains for the wing at that one operating point using large values for the plant disturbance noise and sensor

noise intensities. The large value of noise intensities introduced uncertainty into the plant. Consequently, the

resulting compensator was more robust than a compensator optimized for a plant with no noise. This new set of

processor gains always stabilized the wing at the original operating point plus at least one other operating point.

We then used the new processor gains as the initial guess to the problem with the wing at two (or more)

operating points. The procedure was continued until the compensator stabilized the plant at all the operating

points and the problem could be solved using realistic noise intensities.

Before beginning the bootstrapping procedure we needed to find a set of processor gains which stabilized

the closed-loop system for at least one operating point. This was straightforward for the SR and MRSLC

compensators. We designed a first order single-rate compensator with pitch acceleration input, CS command

output and a sampling rate of 50 Hz. Recall from Section 3.2.2 that pitch acceleration is essentially the

difference in the TE and LE accelerations. The pole location and gain value of this compensator were found

using root locus. The initial stabilizing guess for the SR design consisted of this f'trst order compensator in

parallel with an arbitrary first order compensator that had an input/output gain of zero. For the MRSLC system,
27

weusedthefirstordercompensator asaninitial guess for the fast loop of the successive loop closures structure,

and an arbitrary first order compensator, with an input/output gain of zero, for the slow loop.

An initial guess for the MRMI processor gains was more difficult to fred than for the SR and MRSLC

compensators. Due to its complex sampling schedule we could not design an initial guess by traditional

methods. Instead, we designed a compensator with the multiplexed structure but with very small gains. Then

we used the bootstrapping technique, beginning with the BACT wing operating at a low dynamic pressure

where it is open-loop stable. Since the compensator gains were very small, they did not destabilize the wing

and could be used as an initial guess. The bootstrapping process for this compensator took several iterations,

verses one or two for the other compensators, because we began with such a poor initial guess.

To obtain an initial guess for the SRFT processor gains we began by designing two 2 nd order

compensators. One stabilized the plant when the LE sensor was disconnected, the other stabilized the plant

when the TE was disconnected. We then combined these two compensators into a single 4 th order design and

adjusted their gains until the new fourth order compensator stabilized the plant when both sensors were active or

when only one or the other was active. Finally this design was used in the bootstrapping procedure discussed

earlier to obtain a single fourth order compensator which stabilized the wing at all operating points.

3.4.4. Optimizing the Digital Processor Gains

We optimized the digital processor gains of the three compensators with the algorithm discussed in

Section 2.4. The optimization used the following parameters:

Plant Conditions: Six simultaneous operating points for the second order designs; 22

simultaneous operating points for the fourth order design. See Table 3.2 and

Section 3.4.2.4

Cost Function Weights: The second order designs used the cost function weights listed in Table 3.2.

The fourth order design used the weights in Table 3.2 for cases where both

the TE and LE sensors were active, and one-tenth those values for cases

where either sensor was inactive

Process Noise PSD value: 36 in2/sec 2 _this is the intensity of the white noise input to the Dryden filter

and was specified by NASA

Sensor Noise PSD value: 0 rad2/sec 4 for initial designs, 240 rad2/sec 4 for final designs. This is

discrete sensor noise for the TE and LE acceleration measurements

Initial Stabilizing Gains: Obtained using root locus and boot strapping, see Section 3.4.3

Compensator Structure: See equations (3.2)-(3.4)

Sampling Schedule: See Figures 3.9, 3.10, 3.12 and 3.13.

Gain Constraints: In all designs the direct feed through terms were constrained to be zero.

Additional gain constraints for each compensator are specified in

Section 3.4.2.

The M-Files which define the above input parameters for the synthesis software presented in Ref. 43 are

documented in Appendix B

3.4.5. Design Iteration Based on Performance and Robustness Analysis

After synthesizing the multirate compensators we evaluated their performance and robustness using the

methods discussed in Section 2.5. One of the robustness measures was the maximum singular value of the

minimum destabilizing multiplicative uncertainty at the compensator inputs (a structured singular value). When

we synthesized the compensators using a sensor noise covariance intensity of zero, the size of the destabilizing
28

gainwasunacceptably small- lessthan0.20fortheBACTwingatsomeoperatingpoints.NASAhadspecified

avalueof0.75.Toimprovetherobustness atthecompensator inputweincreasedthesensornoiseintensityto

240rad2/sec4 andre-optimizedtheprocessor gains.ThisprocedurewasmotivatedbytheLoopTransfer

RecovertechniqueforLQGsystemsdescribedin[Doyle&Stein1981].Theresultsofincreasingthesensor

noisearediscussedinthefollowingSection.

3.5. DESIGN R ESULTS

We designed four compensators using the approach discussed in the previous sections. For review, the four

are the:

1) Single-Rate 2 nd Order (SR)

2) Multirate 2 nd Order Successive Loop Closures (MRSLC)

3) Multirate 2 nd Order Multiplexed Input (MRMI)

4) Single-Rate Fault Tolerant (SRFT)

The structure of each of these compensators was discussed in Section 3.4.2. Optimum values for the digital

processor gains are given in Appendix A.

We looked at five performance and robustness measures:

1) Cost function value

2) Gust pulse response

3) Maximum RMS gain from disturbance to the control surface deflection and deflection rate

4) Gain and phase margins at the compensator output

5) The maximum singular value of the minimum destabilizing multiplicative uncertainty at the

compensator input

Results are presented for three operating points, roach 0.50 132 psf, roach 0.70 146 psi and roach 0.78

151 psf. Each of these operating points is 5 psf above the critical flutter dynamic pressure for the corresponding

mach number, and so the BACT wing is nominally unstable at each of these operating points. It is important to

note that none of these operating points were used for the compensator optimization. Therefore the

compensators were not tuned to these particular operating points. In general, the performance and robustness of

the compensators at these three operating points is indicative of their performance at the remaining 21 operating

points.

3.5.1. Cost Function Value

One measure of the overall steady-state performance of a compensator is the value of the cost function in

Eqn. (3.1) at the optimum value of the digital processor gains. (A value for the cost function is returned by our

synthesis algorithm at the completion of the optimization.) For our 2nd order designs, a "perfect" compensator

would have a cost function value of 6, assuming no sensor noise. The "perfect" fault tolerant design would

have a cost of 7.6 since it optimizes a different cost function. By "perfect" compensators we mean continuous

LQR designs with gain scheduling, i.e., they use a different set of feedback gains at every operating point. We

expect the costs associated with our compensators to be higher since they used discrete sampling, did not use

gain scheduling, and had fictitious sensor noise.

It is more realistic to compare the cost of our compensators to that of a discrete LQG design with fictitious

sensor noise and gain scheduling. This comparison eliminates some of the differences due to sampling and

fictitious sensor noise. The cost associated with the discrete LQG compensator is the lowest cost we can expect
29

foragivensamplingrateandsensornoiselevel.Table3.3summarizes thevaluesof thecostfunctionforthe

discreteLQGandforourfourdesigns.Thecostsassociatedwithoursecondordercompensators aealmost

twicethatof thediscreteLQGdesign.Thisis notsurprisingsincethediscreteLQGis significantlymore

complex- it isa16th order compensator with gain scheduling.

3.5.2. The Gust Pulse Response

The gust pulse response provides an indication of the transient response of the closed-loop system due to a

disturbance input. The gust pulse response was found by simulating the response of the BACT wing in

feedback with the flutter suppression system to a disturbance input pulse with an amplitude of 10 in/sec and a

duration of 0.004 seconds. This simulation was performed using the M-file mrsim described in Attachment 4.

Figures 3.14-3.16 show the response of the BACT wing at mach 0.70 and 146 psf to the specified

disturbance gust pulse. Also shown is the response of the wing with a continuous LQ regulator. The cost

function weights for this LQ regulator design satisfy the same design criterion as was used to optimize the

compensator's gains. (See Section 3.4.1.) We provided response plots for only one operating point. The gust

pulse responses at other operating points are similar to those provided in Figs. 3.14-3.16.

For comparison we also provided a gust pulse response plot for the 2 nd order compensators synthesized

without fictitious sensor noise. Recall that fictitious noise was added to the sensors in order to improve the

robustness at the compensator input. Figure 3.17 shows the pitch response of the BACT wing at mach 0.70 and

146 psf due to a gust pulse disturbance. The primary effect of adding sensor noise is to decrease the damping

of the pitch and plunge modes. The reduction in damping is more prevalent in the pitch response than in the

plunge response.

The gust pulse response plots are shown below.

4

-2

-4

-6

xlO'*

Continuous LQR

Single-Rate

............... MuSirate Successive Loop Closures

.......... Mutt/rate w/MU_l:Xexed inputs

........... Single-Rate Fault Tolerant

Time (see)

Figure 3.14. Plunge gust pulse response at roach 0.70 146 psf
3O

xlO _

6 .

2

O..

-2

-6

0

ti !,'

'.1

i;

:!

o:s _'

.1: I_ r. .

i_,_ r,;, _i; .;\ :-, , .

.i t:' '_ v

Continuous LQR

Single-Rate

............... Multirate Successive Loop Closures

.......... Multimtewl MultiplexedInputs

........... Sin_e-Rate FaultTolerant

Time (sec)

Figure 3.15. Pitch gust pulse response at math 0.70 146 psf

8 ixlO'_

IAA 4

_ ",_. "_ _'. r •

2 i" J _ ,." "_ . \ " _, /"_,

_. V _ _ V v: _: _. .. ,..

Continuous LQR

-6 F I / Single-Rate

I _/ ............... MultJrate Successive Loop Closures

-o j_ [I .......... Mulldrate w/Multiplexed Inputs

J V ........... Single-Rate Fault Tolerant

-10,_ , , , , '

0 0.5 1 1.5 2 2.5 3

Time (sec)

Figure 3.16. Control surface deflection gust pulse response at roach 0.70 146 psf

XI0"4

6

t/ ,"

t_

_0

-2 "

Continuous LQR

-4 ........ Single-Rate

............... Muffirate Successive Loop Closures

.......... Mulfirate w/Multiplexed Inputs

o_ 0 o.'5 Y t._ 2' _

_me (see)

Figure 3.17. Pitch gust pulse responses at Mach 0.70 146 psf for compensator designed without fictitious sensor noise
3]

r_

_a

[-

o0

.C: ,--1

cJo

o

.7. '_ _ _ o,

c'q

,--q OC _0 o_ _

_1 Irl

¢-q ¢,q

.d, e_

r_

0
32

xlO _

7.5

7

6.5

© 6

if)

0

5.5

0 5

4.5

/

/

/

/

/

I ..0

.°"

". ...... .. .. .."

/ . .

/ "'.....°'°

I

I

I

I

l

l

I

0.01 0.02

Single-Rate

..... Multimte Successive Loop Closures

......... Multirate w/Multiplexed Input

o "rime of Sample/Update Ac'dvity

\

\

\

\

\

\

\

%

\

%

o. 3 0.04 0.05 0.06 0.07

Seconds

0.08

Figure 3.18. Steady-state covariance propagation at mach 0.70 146 psf wRh 1 in/sec RMS white noise disturbance

0.001 sec

BACTW_ng

CS Deflect

0.001 sec

CS Deflect. Rat_

0.001 sec

TEAccel

CS Cm0 _._

LE Accel

2ndOrder

Digital Processor

0.02 Se¢ Update I - 0.02 sec

I

_'I TEAccel_ I I I ... _Sampler

i LEAccel_ I I I ... _Sampler

cscmn9 . : : ..._ZOH

_L zl (_ t I I "'" -+--I'_StateUpclate _2 _) I I I ... _StateUpdate

Disturbance ? 9 ,9 9, ...._?r__.___Sampler

cs_a_o. ? ? ? ? ..._?._?.._?zo.

T

STP=-O.O01 S_ .--_ _

_= sarnp_e/upclate activity

Figure 3.19. Block diagram of discrete system for calculating RMS gain and corresponding sampling schedule

3.5.3. RMS Gain for Control Surface Deflection and Deflection Rate

One of NASA's specifications was a limit on the steady-state covariance of the control surface deflection

and deflection rate for a 1 in/sec RMS white noise disturbance. Our closed-loop system consists of a continuous

plant and a discrete compensator. Therefore these steady-state covariances are periodically time-varying. In

Fig. 3.18 we show the steady-state covariance propagation for the BACT wing in feedback with the three

compensators at an operating point of mach 0.70 and 146 psf for a unity RMS white noise disturbance.

We calculated the values of the steady-state covariance at the sample/update times using the method

described in Section 2.5.4. Between the sample/update times of the compensator, the covariances were

propagated using the dynamics of the open-loop continuous BACT wing. The steady-state covariances are only

shown for one BTP of the compensator - they repeat themselves during every BTP of the compensator.

One meaningful interpretation of NASA's specification would be to look at the peak steady-state

covariance value taken from this covariance plot. This value, though, is an upper limit on the closed-loop gain

for a white noise disturbance and is not an accurate indicator of the control activity level. A better measure of
33

controlactivitywouldbethemaximumRMSgaincalculatedusing Eqn. 2.25. This is an exact measure of the

maximum RMS gain for any non-decaying input signal.

In order to apply Eqn. (2.25), which is for a discrete system, to our mixed continuous/discrete system we

created a new discrete multirate system in which the continuous inputs and outputs of interest are sampled very

fast (see Section 2.5.3). We chose a sampling rate for the CS deflection and deflection rate of 1000 Hz. This is

more than twenty times the control surface actuator rolloff frequency. A block diagram of this new discretetime system, with the single-rate compensator of Eqn. (3.2), is shown in Fig. 3.19 along with its sampling

schedule. This new system is now multirate even though the compensator is single-rate. The ETIS for this

system has a sampleupdate rate of 1000 Hz and an N of 20.

We used this new ETIS system to calculate the maximum RMS gain of the original system between the

disturbance and the CS deflection and between the disturbance and the CS deflection rate. The maximum RMS

gains for the BACT wing at three operating points are summarized in Table 3.3. See also the related work of

[Sivashankar & Khargonekar 1991].

3.5.4. Gain and Phase Margins at the Compensator Output

Gain and phase margins were calculated at the compensator output using the ETIS and a multiloop Nyquist

diagram. The ETIS of the plant and compensator were computed independently and then combined in series to

form an ETIS loop transfer function. Gain and phase margins were subsequently measured directly off the

multiloop Nyquist plot of this function. These are traditional gain and phase margins, and assume that the gain

and phase do not vary simultaneously. The details of this technique are given in Section 2.5.1, [Mason 1992],

and [Mason & Berg 1992]

The gain and phase margins for the BACT wing at three operating points are presented in Table 3.3. These

values are typical of the margins at all 24 operating points, although the margins tend to be better at lower

dynamic pressures and slightly worse at higher dynamic pressures. A representative Nyquist diagram is shown

in Fig. 3.20. This particular Nyquist plot has two encirclements of the -1 point because the open-loop plant has

two unstable poles.

-30

,r/ [ s. ,/f----..

-20 .._ 10

i

/

\

\

\

\

"4

L

/

/

/

2

\

#

I

-2

/

' h

Figure 3.20. Multiloop Nyquist for BACT wing at roach 0.70 146 psf with MRMI compensator
34

BACT Wing

Compensator

TE Accel

1

LEAccet J I

Figure 3.21. Uncertainty Model

with Nblocks

Figure 3.22 ETIS uncertainty output feedback model

3.5.5. Robustness at the Compensator Input

The uncertainty at the compensator input was assumed to be a multiplicative perturbation of the form

shown in Fig. 3.21, where k 1 and k2 are complex gains. We transformed this system into the output feedback

form traditionally used in robustness analysis using simple block diagram algebra. However, when the

compensator is multirate we must use the ETIS of the plant, compensator and uncertainty. A block diagram of

this closed-loop ETIS for the multirate flutter suppression system is shown in Fig. 3.22. GE is the loop transfer

function consisting of the compensator and plant ETIS transfer functions connected in series.

Now, given the system in the form shown in Fig. 3.22, we can calculate an exact value for the size of the

smallest destabilizing perturbation [Doyle 1982]. First rewrite AE in Fig. 3.22 as

AE= I 1 kI + 12k2 (3.5)

where I1 = diag{ 1 0 1 0 ... 1 0} with 2N diagonal elements, and where 12 has a similar form. Then it can be

shown that

_r(Arnm)= s maxp I]+ for0__.¢<xand0<0_<2rt; (3.6)

o

where _(Amin) represents the maximum magnitude of the smallest destabilizing k 1 or k2; p is the spectral

radius: and HE (ZN) = (I - GE (zN))-IGE (zN).

We are guaranteed that the system in Fig. 3.21 will remain stable as long as

_[', 0] (3" 0 k2 < O'(Amm ) (3.7)

We are also guaranteed that when Eqn. (3.7) is violated, there exist values of kl and k2 that destabilize the

system in Fig. 3.21.

Equation (3.6) is straightforward to solve with a two dimensional search in $ and (9. The results are given

in Table 3.3. For comparison, the corresponding results for the design without the fictitious sensor noise are

also given in Table 3.3 Notice that the addition of the fictitious noise increases the maximum singular value of

the smallest destabilizinguncertainty by as much as 60%.

Even with the fictitious sensor noise, the robustness at the compensator inputs does not meet NASA's

specification for a maximum singular value of 0.75. We could have improved the robustness at the
35

compensator output further by increasing the fictitious sensor noise level, but we chose not to do so because this

simultaneously reduces the gain and phase margins at the compensator output.

3.6. CONCLUSIONS

The performance and robustness of the three 2nd order compensators are nearly identical. All three

stabilize the BACT wing at all 24 plant conditions and, with the exception of the robustness at the compensator

input, satisfy all of NASA's specifications. From this perspective there is little reason to use the multirate

designs over the single-rate design.

The real advantage of the multirate designs is that they allow the engineer to trade design simplicity for

reductions in real-time computations or a reduction in hardware. The successive loop closures design trades a

reduction in the number of computations for a more complex digital processor program. The multiplexed design

trades one A/D converter for multiplexing hardware and a more complex digital processor program. Depending

on the costs of the hardware, such trades might be very advantageous.

The 4th order fault tolerant design, on the other hand, does not satisfy NASA robustness specifications.

The compensator does, however, meet the robustness specifications to which it was designed. It stabilizes the

BACT wing at all 24 operating point even if one of the accelerometers fails. This type of robustness - to a very

specific perturbation - would be difficult to achieve using more common robustness improvement techniques

such as Loop Transfer Recovery, but was straightforward to achieve using the multiple plant condition

capability of our synthesis algorithm.
36

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37

4. CONCLUSIONS AND RECOMMENDATIONS

4.1. CONCLUSIONS

The principle advantage of multirate control is that it gives the designer freedom to choose a sampling

schedule which best utilizes the available hardware and software. In the flutter suppression system design, for

example, we developed multirate controllers that provide performance comparable to a single-rate design, yet

require either fewer real time multiplications per unit time to implement or require fewer A/D converters.

The disadvantage of multirate control is that this additional flexibility substantially increases the

complexity of design and analysis over the single-rate case. Undoubtedly, the lack of good design and analysis

tools has discouraged many from applying multirate control even when the situation may be ideal for a multirate

design.

In this report we addressed the difficulty of multirate design and analysis by presenting a multirate design

methodology. The methodology specifies a design approach and provides specific tools necessary to apply the

approach to a practical problem. The tools are for modeling a multirate system, for synthesizing a multi.rate

compensator which is robust to plant perturbations, and for analyzing the performance and robustness of a

multirate system. The resulting methodology is powerful and straightforward to apply.

To demonstrate the methodology we applied it to design several multirate compensators for NASA's

BACT wing. Those compensators satisfy the specified design specifications and illustrate some of the benefits

of mulfirate control.

4.2. RECOMMENDATIONS FOR FUTURE RESEARCFi

1) Our synthesis algorithm currently requires a stabilizing initial guess for the digital processor gains.

Obtaining a stabilizing initial guess for those gains can be difficult, especially when the multiple plant

conditions capability of the algorithm is used, because the initial guess must stabilize all plant

conditions simultaneously. Eliminating this requirement would substantially improve the algorithm's

versatility.

2) The singular value analysis of multirate systems leads directly to a structured singular value problem

with repeated blocks. Calculating an exact solution to this problem is difficult for all but the simple

tow parameter case. This is an area which needs further research.
38

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39

REFERENCES

1. Amit, N., and Powell J.D., "Optimal Control of Multirate Systems," Proc. AIAA Guid. Contr. Conf.,

Albuquerque, NM, 1981.

2. Apostolakis, I.S. and Jordan, D., "Multirate system Performance Evaluation Using Singular Value

Analysis," 1990 ACC, pp. 1502-1507.

3. Bennett, R.M., Eckstrom, C.V., Rivera, J.A.Jr., Dansberry, B.E., Farmer, M.G., and Durham, M.H.,

"The Benchmark Aeroelastic Models Program - Description and Highlights of Initial Results," NASA

Tech. Memo. 104180, Dec. 1991

4. Berg, M.C., "Design of Multirate Digital Control Systems," Ph.D. Thesis, Stanford Univ., Stanford,

CA, 1986.

5. Berg, M.C., Amit, N. and Powell, J.D., "Multirate Digital Control System Design," IEEE Trans. Auto.

Contr., Vol AC-33, Dec 1988, pp.1139-I 150.

6. Berg, M.C., and Yang, G.S., "A New Algorithm for Multirate Digital Control Law Synthesis," Proc.

IEEE Conf. Decision Contr., Dec 1988, Austin, TX, pp. 1685-1690.

7. Berg, M.C. and Mason, G.S., "Multirate Sampled-Data Yaw-Damper and Modal Suppression System

Design," Final Report for NASA Langley Research Grant NAG-1-1055, Feb. 1991.

8. Berg, M.C., Mason, G.S. and Yang, G.S., "A New Multirate Sampled-Data Control Law Structure and

Synthesis Algorithm," AIAA Journal of Guidance, Control and Dynamics, Vol. 15, No. 5, 1992,

pp. 1183-1191.

9. Boyd, S., and Doyle, J., "Comparison of peak and RMS gains for discrete-time systems," Systems &

Control Letters, vol 9, 1987, pp. 1-6.

10. Boykin, W.H. and Frazier, B.D., "Analysis of Multiloop, Multirate Sampled-Data Systems," A/AA

Jour. Guid Control & Dyn., Vol. 13, May 1975.

I 1. Dansberry, B.E., "Dynamic characteristics of a Benchmark Models Program Supercritical Wing,"

AIAA Paper No. 92-2368, April, 1992.

12. Dahleh, M.A., Voulgaris, P.G., and Valavani, L.S., "Optimal and Robust Controllers for Periodic and

Multirate Systems," IEEE Trans. Auto. Control, Vol. 37, No 1., January 1992, pp. 90-99.

i3. Doyle, J'., "Analysis of Feedback systems with Structured Uncertainties," lEE Proc., Vol 129, Pt. D,

No. 6, Nov. 1982, pp. 243-250.

14. Doyle, J.C., and Stein, G., "Multivariable Feedback Design: Concepts for a Classical/Modern

Synthesis," IEEE Trans. Auto. Control, Vol AC-26, Mo 1, Feb 1981, pp 4-16.

15. Durham, M.H., Keller, D.F., Bennett, R.M., Wieseman, C.D., "A Status Report on a Model for

Benchmark Active Controls Testing," AIAA Paper No. 91-101 i, April 1991.

16. Franklin, G.F., Powell, J. David, and Workman, M.L., Digital Control of Dynamic Systems, AddisonWesley Pub. Co. 1990.

17. Glasson, D.P., "A New Technique for Multirate Digital Control Design and Sample Rate Selection,"

AIAA Jour. Guid. Control & Dyn., Vol. 5, July-Aug. 1982.
4O

18. Glasson,D.P.,"Development andApplicationofMultirateDigitalControl,"IEEE Control Systems

Magazine, Vol. 3, No. 4, 1983, pp. 2-8.

19. Kalman, R.E. and Bertram, J., "A Unified Approach to the Theory of Sampling Systems," Journal of

the Franklin Institute, Vol 267, May 1957.

20. Kono, M., "Eigenvalue assignment in Linear Periodic Discrete-Time Systems," Int. J. Control, Vol.

32, No. 1, 1980, pp. 149-158.

21. Kranc, G.M. "Compensation of an Error Sampled System by a Multirate Controller," Part II, AIEE

Transactions, Vol. 76, July 1957.

22. Kranc, G.M., "Input-Output Analysis of Multirate Feedback Systems," IRE Trans. Auto. Control, Vol.

AC-3, Nov. 1957, pp. 21-28.

23. Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems. New York, Wiley, 1972

24. Ly, U.L., "A Design Algorithm for Robust Low-Order Controllers." Ph.D. Thesis, Stanford Univ.,

Stanford, CA, 1982.

25. MacFarlane, A.G.J., "Return-Difference and Return-Ratio Matrices and Their Use in Analysis and

Design of Multivariable Feedback Control Systems," lEE Proc., Vol. 117, No. 10, Oct. 1970,

pp.2037-2049.

26. Maciejowski, J.M., Multivariable Feedback Design, Addison-Wesley Publishing Company, Reading

Mass. 1990.

27. Mason, G.S., "Multirate Compensator Synthesis and Analysis", Ph.D. Dissertation, Univ. Washington.

1993

28. Mason, G.S. and Berg, M.C., "Reduced Order Multirate Compensator Synthesis," AIAA Jour. Guid.

Contr. and Dynamics, Vol 15, No. 3 May-June 1992, pp. 700-706.

29. Mason, G.S. and Berg, M.C., "Robustness Analysis of a Multirate Flutter Suppression System," A/AA

Journal of Guidance, Control and Dynamics, Vol. 16, No. 5, 1993, pp. 992-926.

30. Mason, G.S. and Berg, M.C., 'Wlultirate Flutter Suppression System for a Model Wing," accepted for

publication in the A/AA Journal of Guidance, Control and Dynamics.

31. Meyer, R.A., and Burrus, C.S., "A Unified Analysis of Multirate and Periodically Time-Varying

Digital Filters," IEEE Trans. Circuits in Systems, Vol. CAS-22, No. 3, March 1975, pp. 162-168.

32. Mukhopadhyay, V., "Digital Robust Control Law Synthesis Using Constrained Optimization," A/AA

Jour. Guid., Contr. and Dynamics, Vol. 12, March-April 1989, pp. 175-181.

33. Mukhopadhyay, V., and Newsom, J.R., "A Multiloop System Statility Margin Study Using Matrix

Singular Values," A/AA Jour. Guid., Contr. and Dynamics, Vol. 7, Sept-Oct 1984, pp. 582-587.

34. Mukhopadhyay, V., Newsom, J.R., and Abel, I., "A Method for Obtaining Reduced-Order Control

Laws for High-Order Systems Using Optimization," NASA Tech. Paper 1876, August 1981.

35. Narigon, M.L., "Optimal Sample Policies for Multirate Digital Control," Ph.D. Thesis, Stanford Univ.,

Stanford, CA, 1991.

36. Packard, A., "What's New with MU: Structured Uncertainty in Multivariable Control," Ph.D.

Dissertation, Univ. Cal. Berkley, 1988.
37.

38.

39.

40.

41.

42.

43.

41

Safonov. M.,andDoyle,J.C.,"MinimizingConservativeness ofRobusmess SingularValues,"

Multivariable Control: New Concepts and Tools, 1984, pp. 197-207

Safonov, M. G., "Stability Margins of Diagonally Perturbed Multivariable Feedback Systems," lEE

Proc., Vol. 129, Pt. D, No. 6, Nov. 1982, pp. 251-256.

Sivashankar, N. and Khargonekar, P. P., "Induced Norms for Sampled-Data Systems", Automatica, Vol.

28, No. 6, 1992, pp. 1267-1272.

Thompson, P. M., "Gain and Phase margins of Multirate sampled-data Systems," International Journal

of Automatic Control, Vol. 4, No. 3, 1986, pp. 833-846.

Yang, G. S., "A generalized Synthesis Method for Multirate Feedback Control Systems," Ph. D.

Dissertation, University of Washington, Seattle, Washington, 1988.

Zhu, G. and Skelton, R. E., "Robust Properties of Periodic Discrete and Multirate Systems," IEEE Trans.

Auto. Control, Vol. 36, No. 5, May 1992, pp. 610-615.

Mason, G. S., Berg, M. C., and Mukhopadhyay, V., "Multirate Flutter Suppression System Design for the

Benchmark Active Controls Technology Wing: Part H : Methodology Application Software Toolbox,

NASA TM 2002-212129, December 2002.
42

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43

APPENDIX A. DESIGN RESULTS

Following are the state space matrices for the optimized flutter suppression system digital processors

discussed in Section 3.0.

A.1. SINGLE-RATE 2/w ORDER

STP=BTP=0.02 sec; N=I. See Section 3.4.2.1 for a description of the sampling schedule.

_2(m+ I, 0)] - L--o.61542

CS Cmd(m, 0) = 10-5 [2.8302

I. 3562.JL_2(m, o) ] Ll

..,F&(m,o)l

-13.6ZUL_2(m,o)J

-0.872581[TE A cel(m, 0)l

-0.94601J[LE Accel(m, 0).J

A.2. MULTIRATE SUCCESSIVE LOOP CLOSURES

STP=0.02 sec; BTP=0.08 sec; N=4. See Section 3.4.2.2 for a description of the sampling schedule.

Update during first STP of the BTP:

[ _f(m,l) I [0.75673 0 IFze(m,O)l F -I + -4 LZ,(m+ 1,o).J= o -0.47672JLz,(m, 0)] [-10

Zf(m,O)

CS Cmd(m, 0) =10-'412. 35354 -2.5338][_s(m,0)]

Update during second STP of the BTP:

0.37644][TE Accel(m,0)1

0.53661JLLE Accel(m, 0)J

_f .FTE Accel(m, 1)] (m, 2) = 0. 75673%f (m, 1) + [-1 0.37644][LE Accel(m, 1)

_f(m,1)

CS Cmd(m,l)=10_[-2.5338 2.35354][_s(m,O)]

Update during third STP of the BTP:

.['TE Accel(m, 2)1

_f(m, 3) = 0.75673_f(m, 2) + [-1 0.37644][LE Accel(m, 2)J

_f(m,2) CS Cmd(m,2)= 10--4[-2.5338 2.35354][ ]

L_s(m,O)J
44

Update during fourth STP of the BTP:

_f(m+l,0) = 0.75673Zf(m,3)+[-1 0. ]FTEAccel(m'3)l

37644"LLE Accel(m, 3)J

CS Cmd(m,3)= 10-4[-2.5338 2.35354][ zf(m'3)]

LW_(m,O)J

We assumed that #-s is updated during the first STP of the BTP, but it it can be updated during any STP of

the BTP.

A.3. MULI/RATE MULTIPLEXED

STP=0.02 sec; BTP=0.04 sec; N=2. See Section 3.4.2.3 for a description of the sampling schedule

Update during first STP of the BTP: Only the TE Accelerometer is sampled. The LE Accel value is held

from the previous STP.

["'"'I r 0 ,

z2(m,1)J=L--O.14712 0.88072JL_2(m,o) j L-0.75421

cs Cmd(m, 0) = 10-518.6277 -8.7583][ _1(m, 0)]

LZ2(m,o)j

186.761[ TZA_cel(.',,0)l

136.42JLLE Accel(m - 1,1)J

Update during second STP of the BTP: Only the LE Accelerometer is sampled.The TE Accel value is held

from the previous STP.

o , if-_(,-,,,l)l I'-2.5371

Cmd(m,1)= 10--413.7645 --4.6910]r_'(2'l))l CS

1__2 ' .l

-191.091[TE Accel(m,0) 1

- 189.04J[ LE Accel(m, 1) ]
45

A.4. SINGLE-RATE FAULT TOLERANT

STP=0.005 sec; BTP=0.005 sec; N=I. See Section 3.4.2.4 for a description of the sampling schedule.

zl(m+l,0)l ]

z2(m+l,0)/ [

_3(rn+l,0)/= t

z4(m+X,O).] [

CS Cmd(m,0) = [1 0

0

0

0

-0.48177

1 0

0 1

0 0

2.4151 -4. 3750

4.2073

+ 10-5 / -0" 06264

-2.1575

[-3.1600

F l(r ,0 1

^,1_2 (m,0)/

o uj/_3(m,O)/

L_'4(m, O)J

o [|_2(_,o)1

/

3.4415JL_4(,.,,,0)_1

6. 4437]

-1.1393 tLLE Accel(m, 0)J

-2.3865.J
46

Blank
47

APPENDIX B. M-FILES USED TO DEFINE THE FLUTTER

SUPPRESSION SYSTEM SYNTHESIS PROBLEM

n°l.

Format:

Description:

PAPA ABCD

[am, bm, cm, dm, vm]=PAPAabcd (fname, rolioff, form)

Creates state space matrices defining the PAPA wing at operating point specified in fname such

that

x = amx+bmu

y = crux + dmu

where y =

"plunge

pitch

plunge rate

pitch rate

TE accelerometer

LE accelerometer

command to actuator

CS control surface

CS control surface rate

CS control surface accel

mode 1

mode 2

mode 3

mode 4

and{ CS c°mmand } u = Dryden filter input

/nputs:

Ou_u_:

fname text variable containing the name of the operating point of interest, e.g. 'freon_m5_q75'.

fname must have the same name as the file which contains the data

rolloff frequency in rad/sec of first order anti-aliasing roll-off at the sensors. The filter has the

form

rolloff

Yfiltered - s+ rolloff Yunfiltered

form indicates the desired form

if form = 0 : am, bm, cm, dm is unchanged from original data

1 : am, bin, an, tim is block diagonal

2 : am, bm, era, tim is block diagonal with scaled states and outputs

am, bin, cm, dm state space description of the plant

vm transformation matrix used to obtain modal form
48

B.2.

Format:

Description:

Inputs :

Outputs :

B_3.

Format:

Description:

Inputs:

Outputs:

FSSCOMP

[cmp, sz, su, sy, stp, stppbtp]=FSScomp(ctype)

Generates the digital processor gain matrices and sampling schedule description for the four

compensator described in Section 3.

ctype specifies the desired compensator

if ctype = 'sr' then FSScmp returns a description of the 2nd order Single-Rate design

'mrslc' then FSScmp returns a description of the Multirate Successive Loop

Closure design

'mrmi' then FSScmp returns a description of the Multirate w/Multiplexed Input

design

'srft then FSScmp returns a description of the Single-Rate Fault Tolerant

design

crop, sz, su, sy, stp, stppbtp a description of the compensator used by the synthesis algorithm.

See Ref. 43 (NASATM 2002-212129).

MROPT_SR OR MRMI

mropt_srORmrmi

Defines the input data for the 2nd order single-rate compensator or mulfirate compensator with

multiplexed inputs. The user needs to comment and uncomment three lines to switch between

the SR and the MRMI design. These are indicated in the text of the script.

none

Outputs to global variables used by optimization routine and defined in Section 3.3 of

Ref. 43 (NASATM 2002-212129).

B°4.

Format:

Description:

Inputs :

Outputs:

MROPT_MRSLC

mropt_mrsic

Defines the input data for the multirate compensator with successive loop closure form.

none

Outputs to global variables used by optimization routine and defined in Section 3.3 of

Ref. 43 (NASATM 2002-212129).

B.5.

Format:

Description:

Inputs:

Outputs:

MROFr_sRFr

mropt_srft

Defines the input data for the single-rate fault tolerant compensator.

none

Outputs to global variables used by optimization routine and defined in Section 3.3 of

Ref. 43 (NASATM 2002-212129).

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12-2002 Technical Memorandum

4. TITLE AND SUBTITLE

Multirate Flutter Suppression System Design for the Benchmark Active

Controls Technology Wing

Part 1." Theory and Design Procedure

6. AUTHOR(S)

Mason, Gregory S.; Berg, Martin C.; and Mukhopadhyay, V.

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

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National Aeronautics and Space Administration

Washington, DC 20546-0001

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706-17-51-03

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14. ABSTRACT

To study the effectiveness of various control system design methodologies, the NASA Langley Research Center initiated the

Benchmark Active Controls Project. In this project, the various methodologies were applied to design a flutter suppression

system for the Benchmark Active Controls Technology (BACT) Wing. This report describes a project at the University of

Washington to design a multirate suppression system for the BACT wing. The objective of the project was two fold. First, to

develop a methodology for designing robust multirate compensators, and second, to demonstrate the methodology by applying

it to the design ofa multirate flutter suppression system for the BACT wing.

15. SUBJECT TERMS

Cybernetics; Digital Control Systems; Multirate sampling; Flutter suppression; Control law design and optimization; Metlab

toolbox

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