A Template Generation Algorithm For Non-rational Transfer Functions ...

Proceedings of the 36th Conference on Decision & Control San Diego, Califomia USA December 1997

TM13

3:lO

A Template Generation Algorithm for Non-rational Transfer Functions in QFT Designs Gautam Sardar* and P. S. V. Nataraj *

Abstract In several applications of Quantitative feedback theory approach to robust controller synthesis, templates of uncertain non-rational transfer functions are required to be numerically generated. An algorithm is proposed in this paper for generating templates of such transfer functions. The main features of the algorithm are: (i) it is applicable to transfer functions expressible in terms of most standard FORTRAN functions, (ii) nonlinear correlated parametric dependencies are permitted, (iii) it yields templates that are guaranteed to include the actual ones, and (iv) it is simple to implement using any interval arithmetic compiler. Examples are given to demonstrate the capabilities of the proposed algorithm.

1

Thus, there is considerable motivation for developing template generation algorithms for non-rational transfer functions which can provide (safe) including templates of the actual ones. Additionally, as the QFT techniques have the capability t o deal rigorously with non-rational transfer functions, it is desirable that these algorithms take full advantage of this capability by achieving their task without the need for any rational transfer function approximations. In this paper, we propose an algorithm to generate including templates of non-rational transfer functions. Our algorithm is developed in an interval mathematics framework, because of the inherent ability of this branch of computational mathematics to deal with parametric uncertainty in the form of interval numbers. The main features of our algorithm can be summarized as follows :

1. Generality: The expressions for a given transfer function can include most of the standard FORTRAN functions (see Remark 3.1 for an example list); further, nonlinear correlated parametric dependencies are permitted.

Introduction

Horowitz’s Quantitative feedback theory (QFT) [4]is a well-established body of robust feedback synthesis techniques, capable of dealing with a large class of linear and nonlinear plants. In several applications of QFT, templates of non-rational transfer functions are required to be numerically generated. Examples of such applications are found particularly in the area of chemical process control where transportation lags are almost always present. Further, while using the so-called Linear Time-Invariant Equivalent (LTIE) approach [5] for nonlinear plants, the LTLE transfer function set corresponding to the given set of plants and command inputs often turns out t o have a non-rational functional form, for e.g., in chemical reactor and nuclear reactor problems with special command input trajectories. In these problems, the recourse currently available to the QFT designer is to adopt a grid-based method for generating the templates. A difficulty with all grid-based methods is that they are potentially risky, as some critical points of the templates could be missed due to the nature of the grid process. *Systems and Control Engineering, Department of Electrical Engineering, Indian Institute of Technology, Bombay - 400 076, INDIA. Email: gautamOee.iitb.ernet.in, natarajBee.iitb.ernet.in

0-7803-3970-8197 $10.000 1997 IEEE

2. Guaranteed reliability: The generated template is always guaranteed to include all the actual template points. This aspect leads to QFT designs that are safe in practice. 3. No rational function approximation of the given non-rational transfer function needs to be performed.

4. The algorithm is conceptually simple, and is quite straightforward to implement using interval arithmetic compilers such as PASCALXSC [GI.

2

Mathematical Preliminaries

A real interval is a closed and bounded set of real numbers X = [X,5?] = {z E E:X 5 z 5 5 ? } where, are the lower and upper end Points of the - and interval The set of real intervals is denoted by 1%.

x.

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for the last operation o $ Y,", + Y,%

Definition 2.1. W e can treat intervals X and Y as numbers and define the elementary arithmetic operations {+, -, .,/} as follows. A

[X+Y,Z+F]

X.Y

A

[min{XY,XF,Yx,XY},

. x . -1

x

=

Y

1

[=,-I,

1 1

Y E

Y'

We define the absolute value of a complex interval as = and its argument as arg{X} = arctan(

1x1

X+Y

Remark 2.3. W e use rectangular intervals with sides parallel to the coordinate axes. A complex interval could also be defined as a disk in the complex plane described by its radius and midpoint. This gives rise to circular interval arithmetic, see [I].

foroey

If the real number z is in the interval X, we write We call two intervals equal if their corresponding end points are equal. The intersection of two intervals X and Y is empty, X n Y = 8, if either X > or > Else, the intersection of X _ and - Y is again an interval X n Y A [max(x,Y),min(X,Y ) ] . We also define the hull of two intervals as: X U Y [min(X, m a x ( Y , n ] , and set inclusion: X Y if and only if 5 X and 5 y. We further define the w d t h of an interval 4 [X,;iT ] as W ( X )= X - ~ 1 and the absolute value of an interval X as I X max(l z E X.

x.

x),

x

x

Remark 2.1. It is a very important fact that the elementary arithmetic operations are inclusion m o n e tonic. That is, X C X ' , Y C Y' =+ X o Y X' o Y ' , o E /}, provided the operations are well-defined as per Definition 2.1. {+,-,e,

In addition t o the elementary arithmetic operations, there are further common, mostly unary, operations on intervals.

Definition 2.2. Let r(z) be a continuous unary operation o n 92. Then, r(X) [minZExr ( X ) ,max,EX r ( X ) ] defines a unary operation o n I%. Remark 2.2. A list of such unary operations is given in Remark 3.1. The unary interval operations as defined in Definition 2.2 are inclusion monotonic. We next introduce complex intervals, i.e., intervals in the complex plane. Let X r e , X i m E I%. Then, the set X = X,., j X i , is called a complex interval, where j denotes the imaginary unit. The set of complex intervals is denoted by IC . A complex interval may also be written as an ordered pair (X,,, X i m ) of real intervals.

+

Definition 2.3. Let X , Y E IC . Then, the element a w arithmetic operations {+, -, ., /} o n complex in-

tervals are defined as follows. X + Y = Xre + Ke + j ( X i m + X m ) x - y I Xre - Ke + j ( X i m - Xm)

x.y

&

X/Y

A

+j

,

The equality and set inclusion relations are valid if and only if they are valid for both the real and imaginary parts of their operands. The operators for the intersection and hull of two complex intervals may also be defined by reduction to the corresponding operators for the real and imaginary parts. Unary op5rations on IC can also be similarly defined. The key property of inclusion monotonicity holds for the elementary arithmetic operations and unary operations on IC . An interval vector is a vector whose elements are intervals. The set of all n-dimensional real interval vectors is denoted by I%". We use the notation X & ( X I , .. , ,X,) for X E I%". The relations €, are= defined, component (I, wise. The union and hull of two interval vectors are also defined component wise. The intersection of two interval vectors has been defined as, X n Y = { z : zi E X C , & E Y,, V i } . The property of inclusion monotonicity holds for alementary arithmetic operations and unary operations on interval vectors.

2.1

Interval Evaluation and Range Enclosures

Consider a continuous function f. An expression f ( z ) belonging t o f is a calculating procedure that will determine a value of the function f for every argument IC in the domain. We assume that all occurring expressions are composed of finitely many operations and operands for which the corresponding interval operations are well-defined according t o Definitions 2.1 and 2.2. We define an interval evaluation of a function lows.

f

as fol-

Definition 2.4. Let a n expression be given for ,.he

function f. Substitute the corresponding interval X for z in the defining expression for f, and evaluate f us-

XreKe - XimKm + j ( X r c 1 L - Ximyre) (X,,Y,, - X,.,Y,,) XreY,., + XimYzm

y,z,+y,2,

de).

y,2e+yzL

ing interval arithmetic. If all operands are within the domain of definition of the operations given in Definitions 2.1 and 2.2, then this kind of evaluation is culled as an interval arithmetic evaluation, or simply interval evaluation off, and is denoted b y F ( X ) .

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Other functions may be included in this list, provided they are continuous o n each closed interval on which they are defined.

Remark 2.4. A function may have several interval extensions, since it may be defined b y several equivalent arithmetic expressions. Mathematically equivalent expressions do not always give rise to equivalent interval evaluations.

We next define the template of an uncertain transfer function a t a given w .

Remark 2.5. The above definition of interval evaluation off can be extended similarly for continuous functions f ( x ) of the vector variable x E Xn that take on values in C.

Definition 3.1. Given w , define Q(s = j w , A ) = { g ( s = j w , A 1 , . . . ,An) , A i E Ai,i = 1,. . . , n } . Then, the set g ( j w , A ) is a region in the extended logarithmic complex plane (Nichols chart), denoted as B , the template of g ( j w , A I , . . . ,An) at the given w .

From the fact that elementary interval operations and unary functions on interval vectors are inclusion m o n e tonic, two key properties of interval evaluations can readily be derived [l]: Theorem 2.1. (inclusion property ): Let f : Xn -+ C be a continuous function of x E Rn, and let f (x)be an expression for f . Assume that the interval evaluation F ( X ) o f f is well-defined for the interval X E I S n corresponding to x. It then follows that f ( X ) 2 F ( X ) , where 7 ( X ) denote the range o f f over X .

As templates are set representations in the Nichols chart, we shall use the terms 'inclusion' and 'tighter' in the usual setrtheoretic sense. More precisely, we shall say that template A includes a template B, if B C A. Further, let A , B be including templates of a template C. Then, we shall say that A is a tighter including template of C than B , if C C_ A C B.

Remark 2.6. The set equality in Theorem 2.1 holds only in rare cases. I n most cases, a n interval evaluation overestimates the range, i.e., yields an outer estimation of the range.

In the following, we explore the use of interval arithmetic in order t o generate a template that tightly includes the actual one in the Nichols chart.

Theorem 2.2. (inclusion monotonicity): Let f : Rn -P C be a continuous function of x = {XI,.. . ,xn}, and let f ( x ) be a n expression for f . Assume that the interval evaluations F ( X ) , F ( Y ) of f are welldefined for X , Y E ISn. It then follows that F ( X ) G F ( Z ) , for all X , C_ 2, Y,,i = 1,. . . ,n.

3.1

c

3 Initial Developments The class of transfer functions addressed in our work is described below.

Assumption 3.1. Consider a system represented b y the transfer function g ( s , AI,. . . ,A,) with X { A I , . . . ,A,} denoting a real vector of the fundamental system parameters and s denoting the Laplace varioble. The parameters A i are allowed t o v a y independently over given real intervals A;, so that we have an interval parameter vector A = { A I , . . . ,An}. Our stipulation o n the function g is that its interval evaluation G(s = j w , A l , . . . ,An) is well-defined over the given A and at the given frequency w E 3.

=

Remark 3.1. Assumption 3.1 means that g may be constructed from the elementary arithmetic operations and unary functions. The unary functions that can be used include absolute value, square, square root, exponential function, power function, logarithmic functions, trigonometric and inverse trigonometric functions, and hyperbolic and inverse hyperbolic functions.

Form of Transfer Function Expression

First, we can have a straightforward procedure t o generate a t a given w an including template of Q .

Theorem 3.1. Let g ( s , A 1 , . . . ,A,) be a transAt the fer function satisfying Assumption 3.1. given w , find the interval evaluation ( X , Y ) = G ( j w ,A I , . . . ,An) . Translate the complex interval ( X ,Y ) obtained into angle (degrees)- magnitude (absolute) interval ( A ,M ) . Note that ( A ,M ) is an interval vector with dimension of two. An including template of 6 is then given b y Gnaive = ( A , M ) . That is, 6 S 6"""""= ( A , M ) .

Proof: Follows readily from the incluswn property of the interval evaluation of functions given by Theorem 2.1. An empirical fact proves useful in obtaining an including template of that is (usually much) tighter than paive

Empirical Fact 3.1. Let g ( s , AI,. . . ,A,) be a transfer function satisfying Assumption 3.1. Rewrite the expression for g in a f o r m wherein each of the quantities AI,. . . , A n occurs as few times as possible. Call this form as a reduced-occurrence form. A t the given w , find the interval evaluation ( X , Y ) = G ( j w , A I , . . . ,An) using the reducedoccurrence form. Translate the complex interval ( X ,Y )

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found into angle-magnitude interval (A,M ) . An including template of B is then given b y 9'" = ( A ,M ) . Moreover, 6 c 9'" c Bnaiue . Proof: By Theorem 3.1, we have 6 E g"" . That gnaiVeis based on the empirical fact given in g'' [3, P. 361.

elements of X so that the first m of these are the multiply-occurring ones. 3. Choose a subdivision factor N , and accordingly subdivide the corresponding intervals AI,. . . ,Am wiag the uniform subdivision scheme.

4- To begin with, the list L is empty.

Remark 3.2. Centered f o r m s [7] of g can be wed with circular complex interval arithmetic to get tighter including templates of 6 , for suficiently 'small' w(A). However, when w(A) is not 'small', the extra eflort to use them is generally not warmnted /2].

... ,An) where w is the given frequency at which the template is to be found. For the IC-th element of subdiviswn, k = 1,. . . ,Nm,find the magnitude (absolute)Mk and angle (degrees) Ak intervals from FOR ji = , N , DO

5. Set s = j w in g(s,X1,

1 , e . e

IG(jw, Al,jl,* * * ,Amj,, Am+1, Ak = ad.G(jw,Ai,j,,*..,An,j,,,,Am+l,**. ENDFOR and append the items (Ak,Mk) to c . Mk =

3.2

Combining scheme

with

subdivision

We next introduce the uniform subdivision scheme.

Definition 3.2. Let N be a positive integer. A uniform subdivision of the interval vector A = ( A I , . . . ,An) is defined as Ai,j = [lli ( j- l ) w ( A J / N , L + jw(AJ/NI, j = 1 , e . e ,N . We have A, = Aid. Further,

+-

,An)[

,An)}

6. Take the hulls of the magnitude intervals Mk over the elements of subdivision, to get a magnitude interval Tmag.Do likewise for angle intervals Ak to get Tang:

U:=,

N

A = Ujs=l(A1,ji,... > A n J m ) . Remark 3.3. In the subdivision method, we have a tool to compute tighter including templates of g than those got b y merely using special forms of g such as the reduced-occurrence form or centered form. To get tighter inclusions, we combine these forms with the unif o r m subdivision scheme, by appropriately dividing the domain of arguments A I , . . . ,A,, and then taking the unwn of interval evaluations over the elements of subdivision. Details of this idea are found in [7]. Remark 3.4. The uniform subdivision needs to be performed only for those A, that have multiple occurrences in the expression for g , see [7].

4

* * *

Generate the template Gush = (Tang > Tmag 1. 7.

e

0

Choose a subdivision factor r , and uniformly subdivide the angle interval Tanginto angular subdivisions T A , ~. .,. ,TA,,.. Find all the angular subdivisions in which each item (Ak,M k ) of c lies. Using this information, next find the minimum and mwimum magnitudes and T M for , ~ the i-th angular subdivision, i = 1,2,. . . , r . This can be done as follows.

rM,i

(a) Set wTa = w ( T : n g ) . f b ) FOR i = 1 to r DO (This is initialization for the nexi loop, which finds maximum and minimum magnitude f o r each i = 1,2,..., r . )

Main Results

We now give our template generation algorithm.

Algorithm. Template Generation Algorithm

Consider a transfer function g(s,X1,. . . ,An) satisfying Assumption 3.1. Through algebraic manipulations, recast the expression for g so as to minimize the number of occurrences of each parameter Xi. Let m be the number of parameters A i which occur multiply in the reduced-occurrenceform found above. Renumber (with no loss of generality) the

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~

0

0

Form T(i) = 1,.. . , r .

-

( T ~ , i , [ z M , ~ , T ~ , i l),i

Take the union of all the T ( i ) , to get the template g a n g = U't = l T ( i ) .

Lemma 4.1. Let g ( s , A I , . , . ,A,) be a transfer function satisfying Assumption 3.1. Suppose that the interval vector A is uniformly subdivided. Then, at the given w , we have (in the notation of Definition 3.2) N

8. Execute step 7 similarly for the magnitude interto get itemsT(i) -1- ( k , , , T ; i > i ] , T ~ , j , ) , i = r + l , , . . ,r+q, w h e r e q i s t h e n u m b e r o f s u b divisions chosen for Tmag. This yields another %=r+1 ~ ( i ) . template gmag A Ur+q ValTmag,

9. Lastly, the final template is gmag

Galg

=

gang n

.

Remark 4.1. The tightest including set of (that is obtainable f r o m results of the algorithm) is given b y U:=:(&,&) (see step 5). This is proved in Theorem 4.1 below. However, the number of anglemagnitude intervals that this set comprises of is Nm. This gives rise to a template representation that is mually quite unwieldy for the purpose of Q F T bound computations.

U

g ( j w , ~ )c

~ ( j w , ~ l , j i , ,An,j,,) * * *

c

j*=1

N *--lG(j~,~

,. ,An,jn)

1 , j I

* *

A natural extension of Lemma 4.1 is Lemma 4.2. Let g ( s , XI,. . . ,A,) be a transfer function satisfying Assumption 3.1. Further, let m be the number of parameters A i which occur multiply in the f o r m for g . Renumber the elements of X such that the first m of these are the multiply occurring ones. Next, uniformly subdivide the corresponding intervals A I , ., . ,Am. Then, at the given w , we have N

K ~ Q , AG)U ~ ( j o , ~ l , j 1 , ,Am,j.,,,Am+l,**. * * *

,An>

ji=l

Remark 4.2. The simplest including template of g is given b y gush- represented b y a single anglemagnitude interval. However, this template is also the least tight amongst all the including templates generated in the algorithm (cf. Theorem 4.1 ). Remark 4.3. Both g a n g and Qmag are tighter including templates of 0 than Gush (cf. Theorem 4.1). These templates comprise of r and q number of anglemagnitude intervals.

N

G

U G(jw,A1,jI ,

*

,Amdm Am+l, 1

*

,A n )

j,=1

Theorem 4.1. The relations between the actual template and the various templates generated b y the above Template Generation Algorithm are as follows "

B C U ( A k , M k )E

Galg

E Bush

k=l

Remark 4.4. B than either

Galg Bang

is a tighter including template of or Gma9 (cf. Theorem 4.1).

Remark 4.5. To our knowledge, there are no general guidelines available in the literature regarding the choice of values for N , r , q . Based o n experience, we found that the choice N E [lo, 1001, r E [lo, 201, q = r gives reasonably tight including templates for Q F T designs. Remark 4.6. I t is emphasized that even with coarser refinements, the templates g a n g , g m a g , and Galg always include the actual template B . This means that although the resulting Q F T controller design may turn out to be conservative, the controller is nevertheless guaranteed to achieve the given stability and performance specs for the actual plant set. This is in contrast with grid-based methods, where there is always a potential threat to the controller's effectiveness o n the actual plant set, due to some critical points missing in the generated template. The basis of the algorithm is provided by the below Theorem. Before that, we require two Lemmas.

Proofs of Lemma 4.1, Lemma 4.2 and Theorem 4.1 are omitted due t o space restriction. They will appear in a forthcoming paper t o be submitted t o a journal.

Remark 4.7. The subdivision factor N can be chosen differently for different parameters A,, . . . ,A,. Remark 4.8. I n the preceding, we assumed exact interval arithmetic to compute the resulting templates. O n a n actual computer, we w e floating-point or machine interval arithmetic. For implementation of our algorithm in machine interval arithmetic, we use the rules given in Definitions 2.1 to 2.3, using directed rounding to get proper lower and upper bounds. W e refer the reader to 131 for details of machine interval arithmetic.

5

Computation results

For the purpose of computations, we have implemented our Algorithm in Pascal-XSC (Pascal extensions for

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Scientific Computations) [6]. We show the template points computed using MATLAB as dots in all the below Figures.

25

1

Example 1: The first example deals with a rational transfer function having nonlinear correlated parametric dependency. We consider

The template of (1) is t o be found a t w = 2. We chose N = 10,40 for IC, a, respectively, with r, q = 10. The final template GaLg of the Algorithm is plotted in Figure 1. We see from these Figure that indeed G GaLg E Gush . Further, the proposed algorithm gives a satisfactorily tight including template.

Example 2: This example is chosen to demonstrate the capability of the algorithm to deal with the class of non-rational transfer functions. The transfer function is

-60

-40

-20 0 20 Phase (degrees)

40

60

Figure 1: Example 1 : G a b

-24 -_.

22-

C A

n m

2-

3 .-

1.8-

-* a,

c

Fl6E 1 .4 12-

T E [0.01,0.02], a E [ l , 21, b E [0.4,0.6]

1

The template of (2) is t o be generated a t w = 2. We chose N = 10,15,20 for b, T , a, respectively, with r,q = 15. The algorithm yields gaLgplotted as solid lines in Figure 2. From the Figure, it cm be that the algorithm gives a reasonably tight including template.

6

Figure 2: Example 2 :

BaLg

[3] Hammer, R., Hocks, M., Kulisch U., Ratz, D., Numerical Toolbox for Verified Computing I, Springer-Verlag, Berlin Hidelberg, 1993.

Conclusions

[4] Horowitz, I.M., “Survey of Quantitative Feedback Theory (QFT),” International Journal of Control, Vol 53, 1991, pp 255-291.

A fairly general algorithm has been proposed for computation of frequency response templates for nonrational transfer functions having nonlinear correlated parametric dependency. A key aspect of the generated templates is that they axe always including templates of the actual ones. The development of the algorithm has been carried out in an interval mathematics framework. The capabilities of the algorithm are demonstrated by use of two examples. F’rom the result plots of these examples the templates generated are fairly accurate so as t o exhibit limited over design in the Q F T procedure.

[5] Horowitz, I.M., “Synthesis of feedback systems with nonlinear time-varying uncertain plants to satisfy quantitative feedback specifications,’’ Proc. IEEE, VOI 64, 1976, pp 123-130. [6] Klatte, R., Kulisch, U., Neaga, M., Ratz, D., Ullrich, Ch., PASCAL-XSC Language Reference with Examples, Springer-Verlag, Berlin Hidelberg, 199’1. [7] Moore, R.E., Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.

References [l] Alefeld, G. and Herzberger, J., Introduction to Interval Computations, Academic Press, 1983. [2] E. Hansen Global optimization using Interval Analysis, Marcel Dekker, 1992.

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