optimal controller and filter realizations using finite-precision, floating-point arithmetic

12
King’s College London Centre for Mechatronics and Manufacturing Systems Department of Mechanical Engineering Optimal Finite-precision Controller and Filter Realizations using Floating-point Arithmetic James .F. Whidborne Department of Mechanical Engineering King’s College London Strand, London WC2R 2LS, UK email: [email protected] Da-Wei Gu Department of Engineering University of Leicester Leicester LE1 7RH, U.K. email: [email protected] September 26, 2001 EM2001/07

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King’s College London

Centre for Mechatronics and Manufacturing Systems

Department of Mechanical Engineering

Optimal Finite-precision Controller and FilterRealizations using Floating-point Arithmetic

James .F. WhidborneDepartment of Mechanical Engineering

King’s College LondonStrand, London WC2R 2LS, UK

email: [email protected]

Da-Wei GuDepartment of Engineering

University of LeicesterLeicester LE1 7RH, U.K.

email: [email protected]

September 26, 2001 EM2001/07

Optimal Finite-precision Controller and Filter Implementationsusing Floating-point Arithmetic

J.F. Whidborne and D.-W. Gu

Abstract

In this report, eigenvalue sensitivity measures are proposed that are suitable for assessing thefragility of digital controllers and filters implemented using floating point arithmetic. Floating pointarithmetic parameter uncertainty is shown to be multiplicative. Based on first order eigenvalue sensi-tivity analysis, an upper bound on the eigenvalue perturbations is derived. Consequently, open-loopand closed-loop eigenvalue sensitivity measures are proposed. These measures are dependent uponthe filter/controller realization. Problems of obtaining the optimal realization with respect to boththe open-loop and the closed-loop eigenvalue sensitivity measures are posed. The problem for theopen-loop case is completely solved. The problem for the closed-loop case is solved using nonlinearprogramming. The problems are illustrated with a numerical example.

1 Introduction

The reducing cost and increasing speed of computer hardware means that there is an increasing tendencyfor digital controller implementations to be implemented using machines which utilize floating-pointarithmetic. Although the effects on the control system due to the finite precision resulting from the finiteword-length have been extensively studied for fixed-point implementations, see Istepanian & Whidborne(2001) for a review, there has been little work looking explicitly at the finite-precision effects for floating-point digital controller implementations. Some exceptions include Rink & Chong (1979), Molchanov &Bauer (1995), Faris, Pare, Packard, Ali & How (1998), Williamson (2001). There has been more workin the signal processing area, for example Rao (1996).

It is known that some controller/filter realizations are very sensitive to small errors in the parameters, andthese small errors can even lead to instability. These parameter errors may result from the finite precisionof the computing device. Such controller realizations can be described as fragile (Keel & Bhattacharryya1997). However, a linear system has an infinite number of equivalent realizations. If a digital linearsystem is implemented in the state space form,

������������ ��������, then

������������� �������� ����� ��������

is an equivalent realization for any non singular matrix�

. The effect of the finite precision is actuallydependent upon the realization. Thus, in order to ensure a non-fragile implementation, it is of interest toknow the realization, or matrix

�, which minimizes the effect on the system of the finite precision.

In the next section, floating-point arithmetic is discussed and shown to result in multiplicative perturba-tions on the filter/controller parameters. In Section 3, an upper bound on the eigenvalue perturbations isobtained. In Section 4, a measure of the relative stability based on this upper bound is proposed for digitalfilter implementations, and the problem of minimising this measure for state space realizations is solved.In Section 5, a similar measure for closed-loop controller implementations is proposed. A necessarycondition for minimising the measure is obtained. In the subsequent section, non-linear programming isshown to be effective for solving the problem.

1

Notation�����denotes the floor function, that is, the largest integer less than or equal to

��� ������ ����� ����

denotes the Hadamard product of

and�

��denotes the transpose of a matrix

��

denotes the complex conjugate transpose of a matrix����� ���� denotes the column stacking operator of a matrix� �! �#" %$ � ��& '�

denotes the Frobenius norm of a matrix

�)( &denotes, for a matrix

+*-,, the unique symmetric matrix satisfying

�)( &.*-,and

�)( & �)( &��

2 Floating-point Representation

/ � � 0�0�0 �2143 � � 0�0�0 �5146sign 7 8:9 ;exponent < 7 8:9 ;mantissa =

Figure 1: Floating-point number representation

Numbers in a digital computer are represented by a finite number of bits – the word-length, >@?BA�C . Ina floating-point arithmetic, the word consists of three parts:

i) one bit, / , for the sign of the number,

ii) >:D#?EA C bits for the mantissa, =F?HG , and

iii) >�IJ?HA C bits for the exponent, <K?HA .

Therefore, > � >:D � >:I �ML. The number is typically stored as shown in Figure 1, and with this

representation, the value�

is interpreted as� �ON =QPSR I (1)

where the mantissa is usually normalized so that = ?UT�VXWZY L � . Now, since >[I and >:D are finite, ( > istypically 16, 32 or 64 bits), the set of numbers that is represented by a particular floating-point schemeis not dense on the real line. Thus the set of possible floating-point numbers, \ , is given by

\^] �`_�� �aL �cbKde, VXW � 146f hg � �� R ji C �)kml PnR I ] / ?po , Y Lrq Y �s ?po , Y Lrq Ys<K?EAut I $ I4vxwpy o ,Zq (2)

where < z{?pA ,z<|?}A , < z|~ z< represent the lower and upper limits of the exponent, and

z< � < z � R 143 ��L .Note that unlike fixed-point representation, underflow can occur in floating-point arithmetic.

In the remainder of this report, it is assumed that no underflow or overflow occurs, that is >�I unlimited,so <a?HA . Define the floating-point quantization operator, ��]�G���\ , as� � � � ] � _��5�2� � � � R i I 1 6 ��)k�� R i 1 6 I C �)k�� � � ��, VXWr��Y for

�}���,, Y for� ��, (3)

2

where < � ����� � & � � � � ��L .The quantization error, � , is defined as�@] � � � � � � � � � V (4)

It can be shown easily that the quantization error is bounded by�@~ � � � R ji 1 6 C �)k V (5)

Thus, when a number is implemented in finite-precision floating-point arithmetic, it may be perturbed to� � � � � � �cL ��� � Y � � � ~ ����� V (6)

where� ��� � R ji 146 C �)k . Thus the perturbation is multiplicative, unlike the perturbation resulting using

finite-precision fixed-point arithmetic, which is additive.

3 Eigenvalue sensitivity

In general, the perturbations on the controller parameters resulting from finite-precision implementationwill be very small. Thus, perturbations on the closed-loop system eigenvalues can be approximated byconsidering the first-order term of a Taylor expansion, i.e., the eigenvalue sensitivities to changes in thecontroller parameters. A number of different eigenvalue sensitivity indices have bee proposed for fixed-point digital controller and filter implementations (Mantey 1968, Gevers & Li 1993, Li 1998, Istepanian,Li, Wu & Chu 1998, Whidborne, Istepanian & Wu 2001, Wu, Chen, Li, Istepanian & Chu 2001).

Assume that a controller/filter realization� � ����� �� � is implemented with floating-point arithmetic

with finite precision, that is the actual realization will be � � � � . Then, from (6), each element of�

will beperturbed to

� �cL ��� �,� � � ~ �����.� R ji 1m6 C �)k , and the realization vector will be perturbed to

� � � ���where

�K���! �.

Proposition 1 Let � � � � ?�� be a differentiable function of� ? G���� . Assume that

�is perturbed to ��

where �� � � �cL ���! �. Then, to a first-order Taylor series approximation� � � �� � � � � � � ��� � ��� ���a� � � � ����! �"� &��� �#�� (7)

where� �: � ~ � ���

for all $ , represents the second and higher order terms of the Taylor series and� � � �

is the gradient vector, i.e.,� � � � ] �&% � � � �% � �('�)+*)-,/.10 , (8)

evaluated at�

.

Proof: Taking a first-order Taylor series approximation:

� � �� � � � � � ��� �f hg � 2 % �% � -3 , � �� � � ��� �"� &��� � (9)

Now, from (6), �� � � �cL ��� �, so

� � �� � � � � � ��� ���f hg � � �� � � � 4�! � �"� &��� � V (10)

3

Hence � � � �� � � � � � � ��� � �f hg � � � � � � ��� � ��� � � � �� �"� &��� � �� (11)

~ � ��� � �f hg � � � �� � � ��� � � ���� �"� &��� �#�� (12)

which, by the Cauchy-Schwartz inequality, gives� � � �� � � � � � � � ~ � ��� ��� � � � ����� � ���� �"� &��� �#�� V (13)

If � � 0 � is the system pole/eigenvalue,�

is the infinite-precision parameter vector and �� is the finite-precision parameter vector, then Proposition 1 can be used to measure the relative system stabilitywhen subject to finite-precision implementation using floating-point arithmetic. Based on Proposition1, tractable eigenvalue sensitivity indices can be formulated which are appropriate for finite-precisionfloating-point digital controller and filter implementation.

4 Optimal Digital Filter Realizations

Consider the problem of implementing a digital filter, � ��� � � � * ����� � * �� ���� * � � * , where * ?G ��� � and has no repeated eigenvalues,

� * ?MG ����� , � * ?`G � � � and� * ?MG � ��� . In this report,�� * Y � * Y � * Y � * � is also called a realization of � ��� � . The realizations of � ��� � are not unique, if�� * Y ��* Y ��* Y � * � is a realization of � ��� � , then so is

� �� ��� * � Y � �����* Y ��* � Y � * � for any non-singularsimilarity transformation

� ?pG ��� � . The system poles are simply the eigenvalues of * . The problem

under consideration is to find the similarity transformation such that the realization is has a minimaleigenvalue sensitivity when implemented using finite word-length floating-point arithmetic.

Based on Proposition 1, the following tractable eigenvalue sensitivity index, � , is proposed

� � � * � & �f� g � � � � � (14)

where � � is a non-negative real scalar weighting and

� � � ���� %�� �% * ���� & (15)

where o � ]�$ �ML Y�V�V�V�Y�� q represents the set of unique eigenvalues of * . The measure � is dependent

upon the filter realization, that is, given * � � ���� * � ,

� � ��� ] � �� � �� * � �� & �f� g � � � � � � ��� (16)

where, (Gevers & Li 1993, Li 1998),

� � � ��� ��������� �� � � � �� � ��� ��� ��! � � ��� � ! �"� (17)

and where� � and

! � are the right and left eigenvectors respectively for the # th eigenvalue of * .

4

Problem 1 Given an initial realization��� * , ��* , � * ,

� * � , calculate

� � ��� � ��� ���� ���������� i � k��g � � ��� (18)

and calculate a subsequent similarity transformation� � ��� such that � � ��� � � � � � ��� � .

Theorem 1 The solution to Problem 1 is given by

� � ��� � �f� g � � � � � & �f

� g � � � (19)

and

� � ��� � � ��� � � � �)( &�� (20)

where� � � �

is the matrix of right eigenvectors of� * , � ��� ��� � � � � Y�V�V�V�Y � � � is a diagonal matrix

of the weights and�

is an arbitrary orthogonal matrix.

Proof: From Lemma 6.2 and Theorem 6.1 of Gevers and Li (Gevers & Li 1993, pp137-138), it followsthat � � *#L with equality for all # if

* is normal. From Horn & Johnson (1985, p101),� * � & * �f� g � � � � � & (21)

with equality if * is normal. Clearly, if

* is normal, � is minimal and (19) holds. Theorem 6.2 ofGevers & Li (1993, p141) gives (20).

Remark 1 The requirement for minimal eigenvalue sensitivity for FWL fixed-point arithmetic is alsothat the transition matrix

* is in the normal form (Gevers & Li 1993, p139).

5 Optimal Digital Controller Realizations

"!$#&%

' !(#*),+-%

./.

0 1 0

.22

2 2

3 4

plant

controller

Figure 2: Feedback control system

Consider the linear discrete-time feedback control system shown in Figure 2. Let the plant be 5 ��� � , andlet the controller be

� ��� Y � where

is the parameterization of the controller.

5

Let���� Y ��� Y ��� Y ,�� be a state space description of the strictly proper plant 5 ��� ��� ��� ������� ����� ������

, � ?UG D � D ,� � ?MG D � � and

� � ?`G � � D . Let���� Y ��� Y ��� Y ��� � be a state space description of����� ��� � � ������ � � �� � � � � �

, where � ?EG ��� � ,

� � ?HG ����� , � � ?SG � � � and� � ?HG � ��� .

The transition matrix of the closed loop system isz�� ���� ��� � � ��� ��� � �� � ��� �� Y� � ,, , � � � ,, � �

��� ���� � �� � � ,, � � Y�� �� � � & � z��� � Y (22)

where ] � � � � �� � � Y (23)

In the sequel, it is assumed thatz

has no repeated eigenvalues.

Let the realization�� � Y ��� Y � � Y � � � of

����� �be represented by � � � � �

��� �� Y (24)

then any realization is given by Q� � ,, � �� � � � ���� �� � ,, � Y (25)� � ��� � � Y (26)

where� ?HG ��� � is non-singular.

Let� � � � � �� �cL ��� �� � R ��� � and

! � � � ! �� �cL � ! �� � R ��� � be the right and left eigenvectors respectivelyfor the # th eigenvalue of

zpartitioned such that

� � �cL � Y ! � �cL � ? � D and� � � R � Y ! � � R � ?�� � , i.e., the

partitions correspond to the partitions of

defined by (23). Then, it can be shown (Li 1998, Whidborne,Istepanian & Wu 2001) that2 % � �% � 3 � ��� � � R � ! � � � R � Y (27)2 % � �% ��� 3 � � ��� � � �cL � ! � � � R � Y (28)2 %�� �% ��� 3 � ��� � � R � ! � � �cL � ��� Y (29)2 %�� �% � � 3 � � ��� � � �cL � ! � � �cL � ��� Y (30)

where o � � ] # ��L Y�V�V�V�Y�� � = q represents the set of unique eigenvalues ofz.

Based on Proposition 1, the following tractable eigenvalue sensitivity index, � , is proposed

� �� � ] � � � & � C Df� g � � � � � (31)

6

where � � is a non-negative real scalar weighting and

� � � ���� %�� �% � ���� & � ���� %�� �% � � ���� & � ���� % � �% � � ���� & � ���� %�� �% � � ���� & V (32)

The measure � is dependent upon the controller realization. Given an initial realization�� * ,

��* ,� * ,� * � , then it can be easily shown that� � & ����� � 5 �� � 5 �� ��� ��� � 5 �� � � � �� ��� ��� � 5 � �� � � ��� ��� ��� � � �� � Y (33)

where 5 � � � �and, from (27) – (30), that

� � � ��� � � c�� � R � 5 �� � � � R ��� ��� � ! c�� � R � 5 ! � � R ������� � ��� � ! c�� � R � 5 ! � � R ������ � ��� � � c�� � R � 5 �� � � � R ��� ��� � � � Y (34)

where � � � ��� � � c�� �cL ��� �� ��� � � �cL ��� Y (35)� � � ��� � ! c�� �cL � ��� � �� ! � �cL ��� V (36)

Rearranging gives

� � 5 � � � ��� � 5 �� � 5 �� ��� ��� � 5 �� � � � �� ��� ��� � 5 � �� � � ��� ��� ��� � � �� ���P�� � C Df� g � ��� � 5 �� �� � ��� � 5 �� ��� ��� � 5 �� ���� ��� � 5 �� �� ������� (37)

where ���.� � �)( &� � � � R � � c�� � R � (38) ��.� � �)( &� ! � � R � ! c�� � R � (39)�� E� ! � R � � ��� � � � � � � Y�V�V�V�Y � � C D � � C D � ! c� � R � Y (40)��B� � � R � � ��� � � � � � � Y�V�V�V�Y � � C D � � C D � � c� � R � Y (41)

are all Hermitian, and

�J� � C Df� g � � � � � V (42)

Problem 2 Given an initial realization�� �

,� �

,� �

,� � �

, calculate

� � ��� � � � �� �� � � �� g ����� � � 5 � (43)

where 5 � � � �, and calculate a subsequent similarity transformation

� � ��� such that � � ��� �� � � � ��� � �� ��� � .7

Theorem 2 A necessary condition for the solution of Problem 2 is given by

� � C Df� g � ��� � 5 �� � � � ��� � 5 � ��� ��� � 5 �� ���� ��� � 5 �� � ������� � � 5 �� �� � � �� � � �� � ��� � 5 �� � 5 �� ��� ��� � 5 �� � � � �� ��� ��� � 5 � �� � � ��� ��� ��� � � �� ���

P � � C Df� g � ��� � 5 �� �� �� � � �� ��� 5 �� d � � C Df� g � ��� � 5 �� ��� � ��� � 5 � ��� ��� � 5 �� ��� ��� � 5 �� �� ������� � � 5 �� � � � � �� �� � ��� � 5 �� � 5 �� ��� ��� � 5 �� � � � �� ��� ��� � 5 � �� � � ��� ��� ��� � � �� ���

P � � C Df� g � ��� � 5 �� �� �� � �� � l 5 �� V (44)

Proof: Differentiating � with respect to 5 gives

% �% 5 � � � C Df� g � ��� � 5 �� �� � ��� � 5 � ��� ��� � 5 �� ��� ��� � 5 �� � �������P � � 5 �� �� � 5 �� � 5 �� 5 �� � 5 �� � � � �� 5 �� � � �� � � �� � ��� � 5 �� � 5 �� ��� ��� � 5 �� � � � �� ��� ��� � 5 � �� � � ��� ��� ��� � � �� � �

P � � C Df� g � ��� � 5 �� ��� �� � � � ��� � 5 � � 5 �� � � 5 �� � � � � 5 �� � � 5 �� � V (45)

Any solution 5 of (43) must necessarily satisfy% �% 5 ��, Y (46)

so (44) is obtained.�

Remark 2 There does not appear to be an analytic solution to (44), hence nonlinear programming isused to find a solution to Problem 2.

To solve the problem using nonlinear programming, a search is required over � P � real, positive def-inite symmetric matrices. This can be accomplished by utilizing a Cholesky factorization given by thefollowing theorem (Golub & Van Loan 1989, p 141).

Theorem 3 (Cholesky Factorization) For 5 ?-G ��� � , 5 � 5 � , 5 � ,, there exists a unique lower

triangular ��?SG ��� � with positive diagonal entries such that 5 � ��� � .

Thus a search can be made over the set � ��� Y ��� � ] ��� ?HG i � C �)k � ( & Y ��� ?EG � C .

8

From the optimal 5 � ��� , a corresponding optimal transformation matrix� � ��� where 5 � ��� � � � ��� � �� ���

can be constructed as (Li, Anderson, Gevers & Perkins 1992)� � ��� � 5 �)( &� ��� � (47)

for any orthogonal matrix�

. This provides an extra degree of freedom which could be utilized to find,for example, sparse realizations.

6 Example

The following numerical example is taken from Gevers & Li (1993, pp236-237). The discrete timesystem to be controlled is given by

� � ����� V�� L W�� � WZV� L � � � V��W�R�W ��, V������RL V ,�,�, , , ,, L V ,�,�, , ,, , L V ,�,�, ,

���� Y (48)

� � ��%L , , ,[� � Y (49)���a���, V L�L�L � , V ,�, � � , V L�,���� , V ,�, L � � P L�, �� V (50)

A pole-placement controller is designed to place the closed-loop poles at, V� � ��� N�, V , � W�����Y , V����� � N�, V , L ��W���Y (51)

and a state observer is designed with poles located at, V�� L W�R N�, V� � � � ��Y , V � W�R�R N�, VXR � W�����V (52)

The initial realization of the feedback controller����� �

is given by (to 4 decimal places) � � � � � � � � � � � � �

(53)� ����RZV����� � � WZV������ � RZVXW , ��� ��, V� L ���, VXR � ��� ��, V , R�� � ��, V������� ��, V ,�,������, V ��� ��� , V� � � L ��, V � R���� ��, V ,�, ��R��, V ,�� � , ��, V ,�, � R , V� L � , ��, V ,�, L�,

���� Y (54)

� � ��L V , ��� � , V� � � W , V � , R�� , V , ����� � � P L�,�� Y (55)� � � , V L�� L�� ��, VXR � � L , V , W ,�, , V , � L � � Y (56)� � ��, V (57)

The weights are set to � � �cL � � ��� ��� �cL � � � � � where � ���.� ����� o � � � q .The initial realization has an open-loop pole sensitivity, � � L VXW�� � �pP L�, � . From Theorem 1, theoptimal open-loop pole sensitivity � � ��� � �ZV L ��� � , which can be achieved with the realization (to 4decimal places):

� � ����, V� L ��� ��, V L ����R ��, V ,�� � W ��, V L R���W, V L � � � , V� , W�R ��, VXR�R���� , V , L � L, V , W ,�� , V L ��W , , VXW � L W ��, VXR � L �, VXR , ��� , V , ��W � , VXR�R L�� , VXW�� , W

���� Y (58)

� � � , V��W ,�� , V ,�, � � RZV ,�, R , , VXR���� L � � P L�, � Y (59)��� � , V L�L�,�,�, V , R�R�R �}, V , L ��R � , V , L � � � V (60)

9

The closed-loop pole sensitivity for the initial realization is � � � V��� , � P L�, &x& and for the open-loopoptimal realization, it is � � �ZV � L W�� P L�, & � . Using the MATLAB routine balreal.m, a balancedrealization was obtained (to 4 decimal places):

� � ����, V L�L�L � , VXW�� ,�� ��, V L ��W�� ��, V , W � L��, VXW�� ,�� , V��[R L � , V L ����� , V , � W ,��, V L ��W�� ��, V L ����� , V������ � ��, V L R�� �, V , W � L , V , � W , , V L R�� � , V�� L�� �

���� Y (61)

� � �� R , � V L�� L � � � � VXW�� , � � RZV , ��R�� � � V L�L � � � � Y (62)� � �� R , � V L�� L � � � VXW�� , � � RZV , ��R�� � V L�L � � � V (63)

The closed-loop pole sensitivity for the balanced realization is � ��L VXR�W�� � P L�, ��� .The MATLAB routine fminsearch.m was used with the search procedure proposed in the previoussection to solve Problem 2. An optimal closed-loop pole sensitivity value of � � ��� � � V ��� � � P L�,�� wasobtained with a realization (to 4 decimal places):

�� � ������, V��� �[, ��, V��W���� ��, V ,�, ��� ��, V ,�, L �RZV L�L � � L V����W L ��, V , R�R�R , V ,�, W ��aL V� L ��� , V� L�L � , V��� � � ��, V , � , �L V����R � � RZV��W���� , V��R�R � , V��� ,��

���� Y (64)

� � � L � � VXW , ��� �aL � � V � � � R W � V � ��� , � R�WZV� ����� � � Y (65)� � ��%L W��ZV�� � , � �aL�L �ZV���� , � R���V ��� W , , V����� � � V (66)

7 Discussion and Conclusions

In previous works, the eigenvalue sensitivity approach to obtaining optimal digital filter and controllerrealizations so as to account for the finite precision inherent in digital computing devices has been thor-oughly investigated. However, there has been an assumption that the parameter uncertainty is additive.This assumption is perfectly valid for filter and controller implementations that use fixed-point arith-metic, however, it is shown in this report that for floating-point arithmetic, the parameter uncertainty ismultiplicative. It is becoming increasingly common to use floating-point arithmetic for digital filters andcontrollers. Thus, in this report, the work of Gevers & Li (1993) is extended to obtain optimal floating-point digital filter realizations; and the work of Whidborne, Istepanian & Wu (2001) is extended to obtainoptimal floating-point digital controller realizations.

The methods are demonstrated on a numerical example. It is shown that the optimal open-loop andthe balanced realization result in high closed-loop pole sensitivities. The closed-loop pole sensitivity ofthe balanced controller realization can reduced by three orders of magnitude by means of the proposedmethod for closed-loop controller realizations.

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