NASA/TM 1998 208744

Predictive Feedback and Feedforward

Control for Systems with Unknown

Disturbances

Jet-Nan Juang and Kenneth W. Eure

Langley Research Center, Hampton, Virginia

December 1998

https://ntrs.nasa.gov/search.jsp?R=19990008647 2018-05-11T16:55:22+00:00Z
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NASA/TM 1998 208744

Predictive Feedback and Feedforward

Control for Systems with Unknown

Disturbances

Jet-Nan Juang and Kenneth W. Eure

Langley Research Center, Hampton, Virginia

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

December 1998
Available from:

NASA Center for AeroSpace Information (CASI)

7121 Standard Drive

Hanover, MD 21076 1320

(301) 6214)390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 22161 2171

(703) 605 6000
Predictive Feedback and Feedforward Control for

Systems with Unknown Disturbances

Jer-Nan Juang *and Kenneth W. Eure

NASA Langley Research Center

Hampton, VA 23681

1 Abstract

Predictive feedback control has been successfully used in the regulation of plate vibrations

when no reference signal is available for feedforward control. However, if a reference signal

is available it may be used to enhance regulation by incorporating a feedforward path in

the feedback controller. Such a controller is known as a hybrid controller. This paper

presents the theory and implementation of the hybrid controller for general linear systems,

in particular for structural vibration induced by acoustic noise. The generalized predictive

control is extended to include a feedforward path in the multi-input multi-output case and

implemented on a single-input single-output test plant to achieve plate vibration regulation.

There are cases in acoustic-induced vibration where the disturbance signal is not available

to be used by the hybrid controller, but a disturbance model is available. In this case

the disturbance model may be used in the feedback controller to enhance performance. In

practice, however, neither the disturbance signal nor the disturbance model is available. This

paper presents the theory of identifying and incorporating the noise model into the feedback

*Principal Scientist, Structural Dynamics Branch.

*Aerospace Technologist, Structural Acoustics Branch, and also Graduate Student, Department of Electrical Engineering, VPI£:SU
controller. Implementations are performed on a test plant and regulation improvements over

the case where no noise model is used are demonstrated.

2 Introduction

Recent advancements in microprocessor technology have made it possible to successfully apply predictive feedback control theory 1-11 to regulate acoustically induced vibrations without

a reference signal or disturbance model. 12 However, if information about the disturbance is

known, this information may be used in the controller design to improve the ability of the

controller to regulate the plant. 13 The internal noise model has been shown to enhance the

performance of the Linear Quadratic Regulator (LQR) 14 if the disturbance states can be

estimated. Predictive control may likewise be extended to take advantage of disturbance

information. The disturbance information may be in the form of a noise model or the input

disturbance signal may be available. Given the present and past values of the disturbance, a

predictor may be used to estimate future disturbance values when there is coherence in the

disturbance signal.

Given the present and past values of the disturbance signal, a finite-difference equation

may be determined to model the correlation in the disturbance signal. With this finitedifference model, future disturbance values may be estimated and incorporated into the

hybrid controller to enhance performance. In the case of single frequency disturbances,

plant regulation may be dramatically improved. As the disturbance bandwidth increases,

the regulation enhancement obtained by the disturbance predictions diminishes.

If the disturbance signal is not measurable or available for feedforward, one important

question arises if the disturbance signal can ever be recovered or estimated from the control

input and output signals. An innovative approach has been developed 15 using the concept
of a deadbeatobserverto identify a systemwith unknown periodic disturbances. Another

approachhasalso beenintroducedla using the conceptof multi-step output prediction for

identification of the systemand periodic disturbances.

The objective of this paper is to presenthybrid predictive control methodsincluding

feedbackand feedforwardthat can be used for systemswith unknown disturbances. This

paper beginswith addressingthe problemsof systemidentification and disturbanceestimation. With unknown periodic disturbances,a new approachis developedto characterizethe

relationship betweenthe control signal and the output measurementsignal. The approach

beginswith the finite-differenceequationin the z-domain. It is shown that the transfer function from the control input to the measurement output can be fully recovered without need

of knowing the disturbances. The disturbance-induced output is shown to be embedded in

the observer Markov parameters for the finite-difference model describing the correlation between the control signal and the output signal. Furthermore, the disturbance-induced output

can be computed and predicted. Several hybrid predictive control techniques are presented

including feedback plus feedforward and feedback with embedded feedforward. These techniques are verified and compared using a simple test for reducing structural vibration induced

by acoustic noise.

3 System Identification and Disturbance Estimation

A generic block diagram of the closed-loop control system is shown in Fig. 1. There are

two fundamental steps involved in the closed-loop system. The first step is to identify a

mathematical model. The second step is to use the identified model to design a controller.

In this paper, the finited difference model is used for the process of system identfication as

well as predictive control designs. The controller may contain the disturbance information in
the form of a feedforward transfer function or the disturbance information may be embedded

in the feedback control parameters. If feedforward is used, the controller needs to have access

to the disturbance signal. If the disturbance information is embedded in the feedback control

parameters, it will be proven that the controller needs no measurement of the disturbance

signal.

The input/output relationship of a linear system is commonly described by a finite

difference model. 18 Given a system with rc control inputs, rd disturbance inputs, and m

outputs, the finite difference model (FDM) for the rc × 1 input u(k), rd × 1 input d(k) and

the m x 1 output y(k) at time k is

y(k) -- OZly(]£ -- 1) + c_2y(k - 2) +... + ozpy(k - p)

@ _0lt(]{) @ flllt(]{ -- 1) + fl2u(k - 2) +... + flpu(k - p)

@ 7od(k) @ 71d(k - 1) @ 72d(k - 2) @... @ "ypd(k - p) (1)

This simply means that the current output can be predicted by the past input and output

time histories. The finite difference model is also often referred to as the ARX model where

AR refers to the AutoRegressive part and X refers to the eXogeneous part. The coefficient

matrices, ai ofm x m, fli ofm x rc, and 7,i ofm x rd for i -- 0, 1,... ,p are commonly referred

to as the observer Markov parameters (OMP) 19-22 or ARX parameters, because they are

closely related to the deadbeat observer.

The equivalent version of Eq. (1) in the z-domain is

- + (2)

where y(z), u(z), and d(z) are the z-transform of the data sequences y(k), u(k), and d(k),

4
respectively,and

_(_) : tm - _1_-1 - _2_-2 ..... _p:-p (3)

/3(z) = /30+/31z-1 +/32z-2 +... +/3pz-p (4)

_(_) : _0+ _1_-1 + _-_ +... + _-_ (5)

where/ira is an m x m identity matirx.

3.1 Single Disturbance Input

For simplicity, let us assume that there is only one disturbance, i.e., rd -- 1, applied to the

system at a certain location. Furthermore, the disturbance signal d(k) for k - 1, 2,...,g

is known to be periodic. For a periodic signal (i.e., correlated signal), there exists a finitedifference model such that

d(k) = ,ld(k - 1)+ ,_d(k - 2)+ ...+ ,,_d(k - _) (6)

where Tt d is the model order which is twice the number of frequencies in the disturbance

signal and _li (i - 1, 2,..., rid) are the constant scalars weighting the past disturbance signal.

The maximum disturbance frequency should be less than the Nyquest frequency which is

one half of the sampling frequency. Equation (6) in the z-domain becomes

,(_)d(_) =0 (7)

where

_l(z) = 1 - _llZ -1 - _12z-2 ..... _lndz -nd

Premultiplying Eq. (2) by _l(Z) and noting Eq. (7) yields

(8)

,(_)_(_)y(_)= ,(_)/3(_)_(_) (9)
or equivalently

where

(10)

,j(_)_(_)_ tm-- _1_-1-- OL2Z - -_ ..... OLpZ --- P nd (11)

= ,j(_)Z(_)= _o+ _-_ + 3_-_ +... + 3_-_-"_ (12)

Note that the order of 6z and/3 is p + na whereas the order of c_ and/3 is p. Equation (10)

indicates that, for a periodic disturbance, there exists a finite-difference model mapping

exactly from the control input to the measurement output without the need of knowing the

disturbance signal. The question arises if this is true for the general case with multiple

disturbance inputs.

3.2 Multiple Disturbance Inputs

Let us now look at the case where ra > 1, i.e., multiple disturbance inputs. First, reformulate

Eq. (2) to become

_(_)y(_)-/3(_)_(_) + _(_) (13)

where

e(z)-7(z)d(z) (14)

Here e(z) is an m × 1 vector, 7(z) is an m × rd matrix, and d(z) is a rn × 1 vector. When d(z)

is the z-transform of a periodic signal (i.e., correlated signal) d(k), there exists an equation

identical to Eq. (7) except that the quantity _l(z) in this case is a rd × rd matrix. As a result,

there should exist a finite-difference equation such that

c(_)_(_)= C(_)7(_)d(_)= C'(_)_j(_)d(_)= 0 (15)
where g(z) is an m × m matrix and g'(z) is an m × rd matrix. Equation (15) implies that

C(_)7(_)d(_)- C'(_)_j(_)d(_)- 0 (16)

or, if d(z) 7/O,

[c(_) c'(_)] __j(_) =0 (17)

Civen7(_) of _ × _dand _j(_)of _d× _., bothC(_)of _ × _ and C'(_)of _ × _. can be

found by solving the null space of the (m + rd) × r d matrix

__](z)_) I (18)

Equation (17) shows that there always exists a matrix g(z) such that Eq. (15) is satisfied.

Let g(z) be defined as

g(z) -- I- Sl Z-1 - _2 Z-2 ..... _p,Z -p' (19)

Each quantity g,i (i - 1, 2,... ,p_) is an m x m matrix. The integer p_ is the order of the

finite-difference model. Assume that the order of the disturbance signal is nd (twice the

number of disturbance frequencies). This implies that the determinant of Eq. (19) can have

the maximum number, nd, of roots. As a result, the relationship between the integers p_ and

nd should be

p_rn >_ nd -- twice the number of disturbance frequencies (20)

Premultiplying Eq. (2) by g(z) and noting Eq. (15) yield

c(_)_(_)y(_)- c(_)_(_)_(_) (21)

or equivalently

_(_)y(_)- _(_)_(_) (22)

7
where

_(_) : c(_)_(_): fm- _1_-1 - _2_-2 ..... _p_-p-p' (23)

/3(z) : £(z)/3(z) :/30 +/31z -1 +/32z -2 +'" +/3pZ -p-p (24)

The order of c_ and/3 is p+p' whereas the order of c_ and/3 is p. Equation (22) indicates that,

for any periodic disturbance, there exists a finite-difference model mapping exactly from the

control input to the measurement output without need of knowing the disturbance signal.

3.3 System Identification

If the matrix g(z)is nonsingular, Eqs. (2) and (22) produce the same system transfer function

from the control input to the measurement output, i.e.,

_(_) = 6(_)-_/3(_)= _(_)-1/3(_) (25)

It is known that the inverse transform of the transfer function G(z) forms a sequence of

system Markov parameters. Equation (25) thus implies that both o_(z)-l/3(z) and o_(z)-l/3(z)

produce the same sequence of system Markov parameters. It is quite easy to compute the

coefficient matrices c_1,..., o_v+v, for c_ and /30,/31,...,/3v+v' for /3 from input and output

sequences, u(k) and y(k) for k - 1, 2,... ,g (see Refs. 18-22). The coefficient matrices

o_1,..., o_v+v, and/30,/31,. •.,/3v+v' do not represent the actual observer Markov parameters,

because they include the information of the periodic disturbance applied to the system.

Nevertheless, they should produce the system Markov parameters for the map from the

control input to the measurement output because of Eq. (25). The system Markov parameters

can be computed recursively as shown in Refs. 18-22. From the computed system Markov

parameters, the actual observer Markov parameters a_,..., ap for a in Eq. (3) and/30,...,/3p

8
for/_ in Eq. (4) can beeasilycomputed. The sameresultsasshownin this sectionhavealso

beenprovenusing the conceptof the deadbeatobserver15or multi-step output prediction.16

3.4 Disturbance Identification

Equation (13) can then be used to estimate the characteristics of the disturbance inputs, i.e,

- - (26)

Using the signal e(k) that represents the inverse z-transform of e(z), the C(z) shown in

Eq. (19) can be determined to form the following finite-difference equation

e(k) -- Cae(k - 1) + C2e(k - 2) +... + _p,e(k -p')

for the disturbance-induced output error. The state-space representation of Eq. (27) is

The eigenvalues of the matrix

I 0 ... 0 0

0 0 ... I 0

_(k - p' + 1)

I 0 ... 0 0

0 0 ... I 0

(27)

(28)

(29)

provide the frequency information of the periodic disturbance applied to the system.

For the case where the disturbance input signal is white and random, the signal may be

considered as having an infinite number of frequencies, i.e., p' ---+o_. For random disturbance

inputs, the computed a and/_ satisfying Eq. (22) are still valid, as long as the integer p' is

chosen sufficiently large. The system Markov parameters computed from a and/_ should

approach the true values.
4 Generalized Predictive Control (GPC)

It is the goal of the system identification technique to determine the observer Markov parameters (0MP) based on input and output data• The 0MP may be estimated using batch least

squares, recursive least squares, or any other appropriate system identification technique

(Ref. [11]). If the 0MP of the plant is known, the future plant outputs may be predicted

using a recursive relationship as shown in Ref. [11], i.e.,

_p+q(]_) = _catp+q(]_) @ ]_atp(]_ -- ]3) @ A_p(]_ -- ]3) @ _dp(]_ - ]3) @ _ddp+q(]_) (30)

Here the vectors _]p+q(]_) and _/p(]_ -- p) are defined as

_p+q(_)

y(k)

y(k + 1)

y(k +p+q- 1)

yp(k-p)

y(k - p)

y(k-p+ 1)

y(k- 1)

and atp+q(k) and dp+q(k) are similar to _p+q(k) with y replaced by at and d, respectively.

The vectors atp(k -p) mad dp(k -p) are similar to yp(k -p) with y replaced by at mad d,

respectively• The matrices T_ and A are

%

_0

/_1) /_0

/_p+q--1) f_(p+q--2) t-'c0 "'" /30

A

OZp O£(p_ 1) " " " 0£1

OL(p1) OL;I) 1 ... OL_ 1)

OL_ p+q-1) CZp___ (p+q--1)l " " " OL_ p+q-1)

10
The matrix Td is similar to Tc with/3 replaced by 7, and/3 and 7) are similar to .4 with a

replaced by/3 and 7, respectively. In Eq. (30), p is the integer related to the system order,

q is an intermediate time step, and the prediction horizon is hp - p + q - 1. The quantity

yp+q(k) is the vector containing the predicted future plant outputs, whereas up+q(k) is the

vector containing the future control inputs yet to be determined, and dp+q(k) is the vector

containing the future disturbance inputs yet to be predicted. Also yp(k -p) is the vector

containing the past plant outputs, up(k -p) is the vector containing the past control inputs,

and dp(k -p) is the vector containing the past disturbance inputs.

The GPC algorithm is based on system output predictions over a finite horizon known

as the prediction horizon. In order to predict the future plant outputs, some assumptions

need to be made about the future control inputs and the future disturbance inputs. In

determining the future control inputs, it is assumed that control is applied only over a finite

horizon known as the control horizon. Beyond the control horizon, the control input is

assumed to be zero. In the GPC algorithm, the control horizon is always equal to or less

than the prediction horizon. In addition to the horizons, a control penalty is introduced

to limit the control effort and stabilize the closed loop system. The cost function to be

minimized in the GPC algorithm is.

J(k) = yy+q(k)yp+ (k)+

p+q- 1

- Z {[y(k+ + j)] + + + j)]} (31)

j 0

In Eq.(31), y(k + j) is the plant output vector, u(k + j) is the control input vector, and

is the control penalty scalar. For simplicity, both prediction horizon and control horizon are

set from 0 to p + q - 1 and the control penalty _ is assumed to be a positive scalar rather

than a matrix. Minimizing Eq.(31) with respect to up+q(k) and solving for up+q(k) will give

11
the control sequenceto be applied to the plant. The first r values of the control sequence

are applied to the r control inputs, the remainder is discarded, and a new control sequence

is calculated the next time step. In order to carry out the above process, the future plant

outputs y(k), y(k + 1),..., y(k + p + q - 1) must be predicted. In the regulation problem,

the desired plant output is zero. Inserting Eq.(30) into Eq.(31), minimizing with respect to

up+q(k) and taking the first r rows results in

_(k) - the first_ rowsof [-(:rc_:rc+ _I)-l:rc_] ×

[_p(k - p) + Ayp(k - p)+ _d_(k - p) + %d_+q(k)] (32)

where I is an identity matrix. When _ - 0, the closed-loop system will be unstable for nonminimum systems because the matrix Tc is rank deficient. The quantity _ must be carefully

tuned to make the system stable.

The formulation given in Eq.(32) assumes that the control horizon is equal to the

prediction horizon. Nevertheless, the control horizon may be chosen to be less than the

prediction horizon resulting in a more stable and sluggish regulator. This is achieved by

reducing the matrix Tc in Eq. (32) to become

_C z

: : '..

(33)

The control sequence determined by using Eq. (33) in Eq. (32) is for a shorter control horizon,

i.e. hc < p + q - 1. Beyond the control horizon the control input is assumed to be zero.

The formulation given in Eq. (32) differs from that given in Ref. [2-3] in two ways.

First, the current control computed by Eq. (32) will be applied at the next time step rather

12
than at the present time step as required in Ref. [2-3]. This is important in implementation

because the present formulation allows time to perform computations. Second, the controller

coefficients are calculated using an intuitively recursive relationship, 11 rather than solving

the Diophantine equation for future predictions? -3 By adjusting the control horizon, the

prediction horizon, and the control cost, Eq. (32) may be tuned to yield the best results for

a given regulation problem.

Equation (32) includes a feedforward path. If the disturbance signal is measurable,

then it is placed in the dp(k -p) vector. This vector contains the last p disturbance measurements. If there is correlation in the disturbance measurements such as the periodic

signal, then this correlation may be used to estimate future disturbance values and fill the

dp+q(k) vector. If there is no correlation in the disturbance signal, then the last term in

Eq. (32) is dropped. Given that correlation does exist, a finite difference model may be used

together with past disturbance measurements to predict future disturbance values. The

finite-difference model is shown in the following

d(k) + _lld(k - 1) + _12d(k - 2) +... + _lndd(k - rid) -- 0 (34)

In Eq. (34), nd is the disturbance order and _]j(j - 1, 2,...,rid) are nd× nd coefficient

matrices. In order to include the last term in Eq. (32), we need the future disturbance signal

that can be estimated by recursively solving Eq. (34) for future values of the disturbance

signal based on past disturbance measurements. As a result, the jth prediction of the

disturbance signal may be calculated by

d(k + j) = j J)d(k- 1)+ j J)d(k- 2)+... + - (35)

The index j will go from 0 to the prediction horizon p+ q- 1. The future disturbance signals

will then be estimated and the last term of Eq. (32) may be used to enhance performance.

13
In practice, the disturbancesignal is generallynot available. The conventionaldesign

strategy is to increasethe closed-loopsystem damping via control feedbackusing the input/output transfer function. The unknown disturbancesaretreated asinput uncertainties.

In mostof cases,they areconsideredaswhite, Gaussian,andrandomnoises.The closed-loop

feedbackthus designedmay be stable but may not satisfy the performancerequirements.

However,it has beenproven in Eq. (22) that the unknown periodic disturbancesare erabeddedin the observerMarkov parameters(OMP) which can beidentified. With the OMP

identified, a new approachis introduced to designa feedbacklaw that includesanembedded

forward to enhancethe performance.The multi-output prediction formulation for Eq. (22)

is identical to Eq. (30) with last two terms dropped. All the terms related to the unknown

disturbancesareembeddedin the coefficientmatricesA and/3 with the expenseof increasing the number of OMP to include the order of the disturbances. The matrix Tc remains

unchanged because it represents the control input/output map. Therefore, if it is not possible to measure the disturbance signal, the control law with embedded forward, Eq. (32),

becomes

-- the first _ rows of [--(TcTTc +/_/)-ITcT ] X

[s p+p,(k - p- p') + myp+ ,(k - p- p')] (36)

where p' is the order of the unknown periodic disturbances.

5 Experimental Results

The GPC algorithm with feedforward and a noise predictor was implemented on a Texas

Instrument C-30 chip. The plant to be regulated is shown in Fig. 2. The box shown in Fig. 2

is made of plexiglass with an aluminum plate on top. The disturbance enters the plant

14
through the loud speakerlocated at the bottom of the plexiglassbox. The control input is

appliedto the plant through the piezomountedonthe bottom centerof the aluminum plate.

The plant output to be regulatedis the accelerometersignaltaken at the top centerof the

aluminum plate.

A block diagram of the plant and control system is shownin Fig. 3. In Fig. 3 the

disturbanced is band limited to 1KHz and enters the plant at point 2. The plant output is

the accelerometer signal taken at point 3. The control input is applied to the plant at point

1 of Fig. 3. Low pass filters and amplifiers were used where appropriate. The controller has

a sample rate of 2.5 KHz. It includes the filtered and amplified accelerometer signal y, and

the disturbance measurement d, and the control signal u.

Since the implementation in this study is not adaptive, we must first estimate the

observer Markov parameters (OMP or ARX) of the plant in order to design a controller.

This is accomplished by applying two independent white noise signals (band limited to

1Kz), to point 2 of Fig. 2 and at u. With both these random inputs applied, the system

output y is measured. The two input data vectors and the one output vector are then used to

approximate a finite-difference model (FDM) of the system. It is important to note that the

FDM represents both the plexiglass box and the filters and amplifiers used in the controller

loop. The form of the model is that of Eq. (1) which in turns yields Eq. (30). The coefficient

matrices in Eq. (30) were then used to find the GPC controller coefficients shown in Eq. (32).

Figure 4 shows the autospectrum of the accelerometer signal y before control (gray

line), and after control (black line). The feedback law is based on a 12th order identified

model with the weighting _ - 0.001. The black solid line in Fig. 4 is the spectrum of

the accelerometer signal when both feedforward and feedback are applied, this is known as

15
hybrid control. The dotted line in Fig. 4 wasobtained with feedbackonly. The disturbance

applied to the plant was band limited white noiseto 1000Hz and thus no finite-difference

model can beusedto predict future disturbances. Thereforethe last term in Eq. (32) was

dropped.

Figure 5 showsthe advantagegained by including the last term of Eq. (32) when

possible.In this case,both bandlimited white noiseto 1000Hz and a sinewavedisturbance

of 800 Hz are applied to the plant through the speaker. In Fig. 5, the noisemodel was

formed basedon a 2nd order FDM obtained by performing a systemidentification on the

disturbancesignal. The resulting FDM and the past disturbancemeasurementswerethen

usedto predict the future disturbancevaluesand incorporate the last term of Eq. (32). As

seenin Fig. 5, this greatly attenuatedthe accelerometersignalat the disturbancefrequency.

However,the performanceof the controller suffersat the upper end of the spectrumdue to

the large amount of control energyapplied at 800Hz.

For the caseof unknown periodic disturbances,a large increasein regulation may

be obtained by performing the systemidentification in the presenceof the disturbance. By

doingthis, a disturbancemodelis implicitly incorporatedinto the identified observerMarkov

parameters.

Figure 6 illustrates the disturbance model obtained by performing the systemidentification with the disturbance on. The figure is a pole/zero plot of the transfer function

which describesthe dynamics betweenthe control input and accelerometeroutput. Both

plots representthe samesystem (samesystem Markov parameters), however,the bottom

plot containsthe disturbancemodel. In this case,the disturbancewasa 200Hz sine wave.

The plant has a mode at 300 Hz and wasmodeledas a 10th order system. Note the pole

16
zerocancellationin the bottom plot. This plot has the samesystemMarkov parametersas

the top plot, but hasdifferentobserverMarkov parameters.

Figure 7 showsthe ability of a feedbackcontroller with an embeddednoisemodelto

cancela sinewavedisturbance. The feedbacklaw is basedon a 20th order identified model

with the weighting A- 0.001. The disturbance in Fig. 7 is a 300 Hz tone. Sincethis is a

resonantfrequencyof the plant, regulation may beperformedwith little control effort. The

advantageof using the embeddednoisemodelmay beseenby comparingthe dotted line to

the solid black line. Figure 7 illustrates the ability of feedbackcontrol to greatly attenuate

a periodic disturbance without the needof a referencesourceasin feedforwardcontrol.

Multiple sine wavesmay also bemodeledin the observerMarkov parameters(OMP),

returned by the systemidentification. Figure.8 illustrates the performanceof a broadband

controller which wasdesignedusing the OMP containing the disturbanceinformation of a

50th order model with the weighting A- 0.001. The disturbance signalentering the plant

of Fig. 8 wasthe sum of a 300 Hz sinewave,a 1100Hz sine wave,and band limited white

noiseto 1000Hz. When the systemidentification wasperformed with the disturbanceon,

the resulting OPM containedthe information of the sinewavedisturbances.As can beseen

by comparingthe solid line to the dotted line in Fig. 8, the embeddednoisemodel greatly

improvedregulation at the frequenciesof both sine wavedisturbances.

The periodic disturbancesin both Fig. 7 and 8 were at resonantfrequenciesof the

plant. If the disturbance is at an off resonantfrequency,regulation may still be performed

if the control actuator has sufficientauthority. The disturbancein Fig. 9 wasa 800Hz tone

plus band limited white noiseto 1000Hz. As can be seenin Fig. 9, control of the broad

band resonantresponseis obtained alongwith a reduction of the off resonance800Hz tone.

17
To comparefeedbackplus embeddeddisturbance model with feedbackplus feedforward,the samenoiseand disturbancecharacteristicsfor Fig. 5 areused.Figure 11illustrates

the regulation performanceof GPC with a 14th order system. The model wasdetermined

basedon input and output data taken in the presenceof white noise band limited to 1000

Hz plus an 800 Hz tone. As can be seenfrom Fig. 11, the controller did not completely

cancelthe 800 Hz tone due to the low systemorder. This results from the fact that there

arenot enoughOMP coefficientsto model both the systemand disturbancecharacteristics.

In practice,whenthe disturbancepropagatesthrough the plant to the accelerometer,disturbanceinformation will becontaminated by the measurementnoise.As a result, an increase

in systemorderis required. Figure 12comparesthe performanceof a 20th order controller

to that of a 30th order controller. All other parametersarethe sameasin Fig. 11. There is

an increasein regulation with an increasein systemorder.

Figure 13showsbroadbandresultsusingthe GPC controller. Here bandlimited white

noisewasappliedasthe disturbance.The blackline is the autospectrumof the accelerometer

signalwhenthe hybrid controllerofEq. (4) wasusedwithout the last term. It isof interestto

comparethe result obtained usingfeedbackonly, (dark gray line) with that of feedbackwith

an embeddednoisemodel, (dotted line). In both casesno feedforwardwasused. However,

the systemidentification wasperformedin thepresenceof the disturbancefor the dotted line.

For the dark gray line, the systemidentification wasdonewith the disturbanceoff. By having

the disturbanceon whilegathering input and output data, the resultingsystemidentification

will incorporate someinformation about how the disturbancepropagatesthrough the plant

to the accelerometer.The controller designwill usethis information to increaseregulation

ascan be seenin when comparingthe dark gray line to the dotted line in Fig. 13.

18
6 Concluding Remarks

It is well known that feedforward control does indeed enhance feedback control. Designing

feedback control based on the transfer function alone will not perform as well as hybrid

control for any linear systems. This paper has shown that feedforward control can be implemented explicitly or implicitly. Explicit implementation requires identification of a disturbance model. A new approach is presented to show that the characteristics of unknown periodic disturbances are predictable and its induced vibration is controllable. In other words,

the propagation of the disturbance through the system is observable and controllable. Given

a sufficiently large system order, a system model can be identified including an embedded

disturbance model. The states of the identified model are observable but not controllable.

When the identified model is reduced to the minimum order, the reduced model provides the

exact transfer function describing the control input to system output. This means that the

system Markov parameters (i.e., pulse response function) are preserved. The disturbanceinduced output can be controlled using either predictive feedback alone or feedback plus

feedforward. In fact, predictive feedback alone can perform as well as the combination of

feedback and feedforward as evidenced by the experimental results. The key reason is that

the predictive feedback controller is capable of implicitly identifying and incorporating the

disturbance model into the control loop. The disadvantage is that the identified model is

larger than the true system model to accommodate unknown disturbances. Note that any

random disturbance can be treated as periodic disturbances with a sufficiently large number

of frequencies. As a summary, the main contribution of this paper is the development of

practical control methods for linear systems with unknown periodic disturbances. These

methods have been theoretically proven and experimentally verified.

19
7 References

1. Clarke, D.W., "Self-tuning Control of Nonminimum-phase Systems," Automatica, Vol.

20, No. 5, pp. 501-517, 1984.

2. Clarke, D.W., Mohtadi, C., and Tufts, P.S., "Generalized Predictive Control - Part I.

The Basic Algorithm," Automatica, Vol. 23, No. 2, pp. 137-148, 1987.

3. Clarke, D.W., Mohtadi, C., and Tufts, P.S., "Generalized Predictive Control - Part II.

Extensions and Interpretations," Automatica, Vol. 23, No. 2, pp. 149-160, 1987.

4. Dong,Y., "Extended Horizon Predictive Control," Ph.D Thesis, Case Western Reserve

University, 1992.

5. Mosca, E., Optimal, Predictive, and adaptive Control, Prentice-Hall, 1995.

6. Wang, W., Henriksen, H., "Direct Adaptive Generalized Predictive Control," Proceedings of the American Control Conference, Chicago, U.S.A., pp. 2402-2406, 1992.

7. Wang, W., "A direct adaptive generalized predictive control algorithm for MIMO systems," International Journal of Control, Vol. 60, No. 6, pp. 1371-1381, 1994.

8. Bialasiewicz, J.T., Horta, L.G., and Phan, M., "Identified Predictive Control," Proceedings of the American Control Conference, Baltimore, Maryland, June 1994.

9. Phan, M. Q., and Juang, J.-N., "Predictive Controllers for Feedback Stabilization,"

Journal of Guidance, Control, and Dynamics, Vol. 21, No. 5, Sept.-Oct., 1998, pp.

747-753.

2O
10. Goodzeit, N. and Phan, M. Q., "System and Periodic Disturbance Identification for

Feedforward-Feedback Control of Flexible Spacecraft," Proceedings of the 35th Aerospace

Sciences Meeting and Ezhibit, Reno, Nevada, January 1997.

11. Juang, J.-N., and Phan, M. Q., "Deadbeat Predictive Controllers,"Jour'nal of Chinese

(Taiwanese) Society of Mechanical Engineers, Vol. 19, No. 1, Jan.-Feb. 1998, pp.

25-37.

12. Eure, K. W., and Juang, J.-N., "Broadband Noise Control Using Predictive Techniques," NASA Technical Memorandum 110320, January 1997.

13. Ellite, S.J., Sutton, T.J., Rafaely, B., and Johnson, M., "Design of Feedback Controllers

using a Feedforward Approach," Active 95, Newport Beach, CA, USA, 1995.

14. Dorato, P., Abdallah, C., and Cerone, V., Linear-Quadratic Control, An Introduction,

Prentice-Hall, 1995.

15. Goodzeit, N. E., and Phan, M. Q., "Exact System Identification in the Presence of

Completely Unknown Periodic Disturbances," Department of Mechanical and Aerospace

Engineering Technical Report No. 2096, Princeton University, Jan. 1997, New Jersey

08544.

16. Phan, M. Q., Goodzeit, N. F., and Juang, J.-N., "Identification of System and Periodic

Disturbance," 1997 ASME Design Engineering Technical Conferences, DETC97/VIB4256, Proceedings of DECTC '97, September 14-17, 1997, Sacramento, California.

17. Favier, G., Dubois, D., "A review of k-Step-ahead Predictors," Automatica, Vol. 26,.

No. 1, pp.75-84, 1990.

21
18. Juang,J.-N., Applied System Identification, Prentice-Hall, 1994.

19. Chen, C.-W., Huang, J.-K., Phan, M. Q. and Juang, J.-N., "Integrated System Identification and Modal State Estimation for Control of Large Flexible Space Structures,"

Journal of Guidance, Control, and Dynamics, Vol. 15, No. 1, pp. 88-95, Jan.-Feb.

1992.

20. Phan, M. Q., Juang, J.-N., and Hyland, D. C., "On Neural Networks in Identification

and Control of Dynamic Systems," Wave Motion, Intelligent Structures, and Nonlinear

Mechanics edited by S. Guran and D. Inman, World Scientific, Massachusetts, USA,

1995, pp. 194-225.

21. Juang, J.-N., Phan, M. Q., Horta, L.G., and Longman, R.W., "Identification of Observer/Kalman Filter Markov Parameters: Theory and Experiments," Journal of Guidance, Control, and Dynamics, Vol. 16, No. 2, March-April 1993, pp. 320-329.

22. Phan, M. Q., Horta, L.G., Juang, J.-N., and Longman, R.W., "Improvement of Observer/Kalman Filter Identification (OKID) by Residual Whitening," Journal of Vibrations and Acoustics, Vol. 117, April 1995, pp. 232-239.

22
Disturbance(d)

Input (u) q Plant

Output (y)

v

_____ System DisturbanceIdentification Estimation &

__ (Feedback/Feedforward) Predictive Control __

Figure 1: Generic block diagram of a closed-loop system

k /

Accelerometer(Error)

Piezo (Control)

Aluminum Plate

8 in..by 8 in.

0.05 m. thick

II

Speaker (Disturbance)

I I 6 in. from plate

Figure 2: Plant to be Regulated

23
3

1 KHz 2.5 KHz

dI

Y

BOX

2

T

1 KHz

_ DISTURBANCE

0-1000 Hz

TX C30 CHIP 2.5 KHz

CONTROLLER _

u

1.) Control signal goes to PZT

acuator

2.) Disturbance enters the plant from the

speaker..

3.) Error is picked up by accelerometer

LPF - low pass filter

HIGH P. AMP - high power amplific

Sample rate is 2.5 KHz

Figure 3: Block Diagram of Control System

24
i ! :i

-80

0 200 400 600 800 1000 1200 1400 1600

Frequency, Hz

Figure 4: Autospectrum of plant output without control (dotted line), feedback only (dot

dash line), and with hybrid control (solid line), 12th order system, 2.5 KHz sample rate.

25
-10 il

-30 ! i

i

-40

-50

-60

-70

-80

0 200 400 600 800 1000 1200 1400 1600

Figure 5: Autospectrum of plant output without control (dotted line) and with hybrid control

plus noise predictor (solid line), 12th order system, 2.5 KHz sample rate.

26
1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

I

-1

t"

I"

r

.w

,-o

0

_,o

i i i

X

X

X

°, _

! ! !

-0.5 0 0.5

Z plane

.2

br

J

0

1.5

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

i

(;,

_' X

0

0

X

_ w

• M

"* X

Q :o

;• X /

"% t"

• %. .,

•*•x ............. Q. ....

I I I I I

-1 -0.5 0 0.5 1

Z plane

0

Figure 6: Pole Zero plots obtained with disturbance off (top plot) and disturbance on (bottom

plot), 10th order system model, 1 KHz sampling rate, 200 Hz sine wave disturbance.

27
dB

60,

5O

4O

3O

20

10

oi

-10

-20

-30

-40

0

I

i ........................ i :i

.I

5O 1O0 150 200 250 300 350 400

Frequency, Hz

Figure 7: Autospectrum of plant output without control (dotted line), feedback only (dashed

line), and feedback with embedded noise model (solid line), 20th order system, 1 KHz sample

rate.

28
10

20

30

dB

10 _ _ , , :, _

......................................................................

-401 " :_:: . :: ;" : ": i - , '.': _'_ _ :: :

60

70

-80 i i , , :i i i

0 200 400 600 800 1000 1200 1400 1600

Frequency, Hz

Figure 8: Autospectrum of plant output without control (dotted line), feedback (dashed

line), and feedback with embedded noise model (solid line), 50th order system, 2.5 KHz

sample rate.

29
dB

-10

-20

-3O

-40

-50

-60

-7O

0

m m m m n

• r

|

• ). .'..

,_ " .

: *, : ) ; :-, :

200 400 600 800 1000 1200 1400 1600

Frequency, Hz

Figure 9: Autospectrum of plant output without control (dotted line) and feedback with

embedded noise model (solid line), 50th order system, 2.5 KHz sample rate.

30
50,

4O

3O

2O

10

dB

0

-10 ....i

-20

-30

0

I

i.

...... : .2_- , • .......

i,"

i

I I I I I I I

200 400 600 800 1000 1200 1400

Frequency, Hz

600

Figure 10: Autospectrum of plant output without control (dotted line), feedback with embedded noise model (dot-dot dash line), and hybrid controller (solid line), 50th order system

for all cases, 2.5 KHz sample rate.

31
-10

-2O

-30

-40

-50

-60

-70

-80 i i i i i i i

0 200 400 600 800 1000 1200 1400 1600

Figure 11' Autospectrum of plant output without control (dotted line), and feedback with

embedded noise model (solid line), 14th order system, 2.5 KHz sample rate.

32
-7O

0 200 400 600 800 1000 1200 1400 1600

Figure 12: Autospectrum of plant output without control (dotted line), and hybrid control

with embedded noise model, 20th order system (thin solid line), 30th order system (thick

solid line), 2.5 KHz sample rate.

33
dB

i i i i i i i

-20 : ............................. :

-25 .: = ....

t

.; rL : i •

-30 ...... ! II : ............. "; .........................

:. ,, / . • ; I

L ;_ _-: .,'zi .r-. . - .,..Jl J

, . If" _ J • • • LJ "

; " " .@ Dr', 4 ] . "

-40]- ,. I :fL: ,, I .,, i_I,, ...,,_. _-,, .,,, 4

• L _ , h .,_ , '. " i '.J : • ,

/,_,,:,.]" ;-::_,, ,, I.I "u_ ,l_t," 1,..,<.._ •

-501- i,¢ -:.-_. i;...," : ,i I I/'l,I '1; " 'trhl I.:;.%,;j_ . tr"J

-55 " r *_: : _.

-65

-70

0 500 1000 1500 2000 2500 3000 3500 4000

Frequency, Hz

Figure 13: Autospectrum of plant output without control (dotted line), feedback (dot-dot

dash line), feedback with embdded noise model (dash-dash dot line), and hybrid control

(solid line), 38th order system, sample rate is 10 KHz.

34
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1. AGENCY USE ONLY (Leave blank ]2. REPORT DATE 3. REPORTTYPE AND DATES COVERED

I

December 1998 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Predictive Feedback and Feedforward Control for Systems with Unknown

Disturbances WU 632-20-21-11

6. AUTHOR(S)

Jer-Nan Juang and Kenneth W. Eure

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

8. PERFORMING ORGANIZATION

REPORT NUMBER

L-17788

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA/TM-1998-208744

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Unclassified-Unlimited

Subject Category 39 Distribution: Standard

Availability: NASA CASI (301) 621-0390

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13. ABSTRACT (Maximum 200 words)

Predictive feedback control has been successfully used in the regulation of plate vibrations when no reference signal is available for feedforward control. However, if a reference signal is available it may be used to enhance regulation by incorporating a feedforward path in the feedback controller. Such a controller is known as a hybrid

controller. This paper presents the theory and implementation of the hybrid controller for general linear systems, in

particular for structural vibration induced by acoustic noise. The generalized predictive control is extended to

include a feedforward path in the multi-input multi-output case and implemented on a single-input single-output

test plant to achieve plate vibration regulation. There are cases in acoustic-induce vibration where the disturbance

signal is not available to be used by the hybrid controller, but a disturbance model is available. In this case the disturbance model may be used in the feedback controller to enhance performance. In practice, however, neither the

disturbance signal nor the disturbance model is available. This paper presents the theory of identifying and incorporating the noise model into the feedback controller. Implementations are performed on a test plant and regulation

improvements over the case where no noise model is used are demonstrated.

14. SUBJECT TERMS

Predictive Control, Real-time Adaptive Control, Feedback and Feedforward Control,

Real-time System Identification

15. NUMBER OF PAGES

39

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A03

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