An expert system for interaction analysis of multivariable systems

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SlGNAL PROCESSING, VOL. 5 , 41--62 (1991)

AN EXPERT SYSTEM FOR INTERACTION ANALYSIS OF MULTIVARIABLE SYSTEMS

J . LIESLEHTO AND H. N. KOIVO Department of Electrical Engineering, Tampere University of Technology, SF-33101 Tampere, Finland

SUMMARY

In this paper a demonstration expert system for interaction analysis of time-invariant, linear, rnultivariable systems has been developed. The principle is to choose an area that is significant in control design but still limited to be implementable. To our knowledge no such systems have been reported in the literature. The paper first briefly describes the general idea of the developed expert system. Next the software structure of the expert system is given. A short description of the interaction analysis and conclusions finish the paper.

KEY WORDS Expert systems Interaction analysis Multivariable controller design

1. INTRODUCTION

Interaction analysis forms a very important part of control system analysis. Once the system model is known and linearized, the question about control structure arises. Can one manage with scalar controllers or are interactions so severe that a multivariable controller is needed? This is a very relevant and important question, since it will also effect the realization of the control system.

Scalar controllers are straightforward to implement and are found as standard components in digital automation components. Multivariable controllers need more software and thinking. They are not standard modules; they have to be implemented individually with quite a bit of care. Thus interaction analysis plays a very significant role in the first phase of control design.

There are a number of established results on interaction results in the literature starting with Bristol’s’ classical result. MCAVOY’S~ book summarizes quite a few results. More recently, Tung and Edgar3 and Lau et u I . ~ have generalized Bristol’s ideas for the dynamic case. The concept of block relative gain (BRG) has been introduced as an interaction measure by Manousiouthakis et al. BRG generalizes Bristol’s relative gain to block pairing of inputs and outputs which are not necessarily SISO structures. Grosdidier and Morari6 have studied the use of structured singular value (SSV) as an interaction measure. Mijares et a/. ’ developed an interaction measure which is based on the difficulty caused by the interaction terms (the off-diagonal elements) in finding the inverse of the steady state gain matrix. Further, Johnston and Barton’ study interaction analysis from the state-space point of view.

All the aforementioned interaction analysis methods have their strong and weak points. That is why the control system designer has to be able to use several of these methods. The designer has to know the limitations of each method. Knowledge is also required when the numerical

0890-63271 9 1 lo1004 1 -22$05 .OO 0 1991 by John Wiley & Sons, Ltd.

Received September 1989 Revised November 1990

42 J. LIESLEHTO AND H. N. KOIVO

results are interpreted. The control engineer must also know how the results from different methods can be combined. As we can see, a thorough interaction analysis requires not only numerical calculations but also a considerable amount of knowledge and expertise. Expert systems offer an interesting technique to transfer this knowledge to less-than-expert users of the control design software.

In this study a demonstration expert system for interaction analysis of time-invariant, linear, multivariable systems is developed to test these ideas. On the basis of the given mathematical model, either a transfer function or a state space equation, the expert system selects appropriate analysis procedures. Interaction analysis methods included in the expert system are Bristol’s relative gain array, Tung and Edgar’s method, singular value analysis proposed by Lau et al. and the method presented by Johnston and Barton. With these methods the expert system is able to analyse both steady state and dynamic interactions from different types of models. After execution of these procedures, numerical results from them are interpreted. Then interpreted results from different analysis methods are combined. On the basis of this information the expert system infers whether the system under study is highly interactive or not. I f interactions are small enough, the expert system selects input-output pairings for separate SISO control loops. The developed knowledge base for interaction analysis is presented in the Appendix. The first version of the expert system was implemented in Scheme- dialect of LISP on an HP 9000/550 computer. The second version was implemented using Common Lisp on a Symbolics 3670 computer. Numerical methods are coded in Fortran.

2. STRUCTURE OF THE EXPERT SYSTEM

The general structure of the expert system is shown in Figure 1. The working memory contains the facts that describe the system under study and the state

of the design process. The elements of the working memory are natural language sentences. These are implemented using lists of symbols, which is a typical data structure in LISP. A typical element in the working memory is

(TYPE OF THE MODEL IS TRANSFER-FUNCTION-MATRIX)

The knowledge base is rule-based. Every rule has one or more premises and one or more conclusions. Premises and conclusions are natural language sentences. This makes the rule base easy to read. It is possible to use wild cards and variables in rules. This adds flexibility to the knowledge representation. An example of a rule is

(MATRIX G IS NOT A SQUARE MATRIX) (SINGULAR VALUES OF THE SYSTEM ARE CLEARLY DISTINCT AT FREQUENCY 0)

(RULE 35 (IF

(TYPE OF THE MODEL IS TRANSFER-FUNCTION-MATRIX) (SINGULAR VALUE ANALYSIS SHOWS THAT INPUT (> X) AND OUTPUT (> Y) CAN BE USED AS AN 10-PAIR FOR SISO-CONTROL IN STEADY-STATE) (INPUT (<X I AND OUTPUT (<Y) CAN BE USED AS AN 10-PAIR FOR SISO-CONTROL IN STEADY-STATE)

(THEN

A pattern-matching algorithm9 compares rule premises and working memory elements. It finds out the combinations of values for the variables to satisfy all premises. These values are substituted for the variables in conclusions.

EXPERT SYSTEM FOR INTERACTION ANALYSIS 43

A forward-chaining inference engineg fires a rule if all premises match one or more working memory elements. Forward chaining begin by asserting all the rules whose if-clauses are true. It then checks to determine what additional rules might be true given the facts it has already established. This process is repeated until the programme reaches a goal or runs out of new possibilities. Three types of conclusions can be made: a fact is added into the working memory, a fact is removed from the working memory or a LISP function is evaluated.

The inference engine also maintains a history list of deductions. The expert system is able to explain its reasoning process using this list. It is able to tell the user how a certain fact was concluded or why a certain fact was needed in the reasoning process.

The inference engine can also execute numerical interaction analysis methods. All numerical methods are coded in Fortran. This is done for two reasons. Typically Fortran is faster than LISP in numerical calculations. We also wanted to use available packages for numerical calculations and most of them are coded in Fortran.

The large amount of numerical data typically produced by these procedures is filtered through feature extraction programmes. These programmes pick up the most interesting features from numerical data and make necessary additions into the working memory.

3. INTERACTION ANALYSIS METHODS

Consider the linear, time-invariant system of Figure 2 given in a state space form

i= A X + BU

y = ex (2)

where the state of x is an n-vector, the output y is a p-vector and the input u is an rn-vector. The system can also be given in an equivalent transfer function form

Y(s) = G(s)U(S) (3)

t- yp

y = cx --I Figure 2

44 .I. LIESLEHTO AND H. N. KOIVO

where the transfer function G(s) = C(sZ- A ) - ' B . Interaction analysis is used to determine how strongly different controls affect measured outputs.

If interactions are strong, compensators should be used to decrease them or control algorithms independent of interactions should be used. If interactions are weak, scalar controllers may be used in different control loops. In the latter case the designer should choose the appropriate input-output pairing based on the interaction analysis in the literature.

A number of different methods have been presented for interaction analysis. None of these is a panacea - all have strengths and weaknesses. None of them can be universally applied to all systems. To obtain reliable interaction analysis results, it is necessary to use different methods and combine the results.

The constructed expert system contains four different methods for studying interaction analysis. Perhaps the best known and most widely used method is the one suggested by Bristol. It uses only static gains and is clear and simple. An extension of Bristol's method is a method by Tung and Edgar3 which also considers the dynamics of the plant. Both methods are limited to systems with equal numbers of inputs and outputs. Lau et u I . ~ suggest a method based on the singular value decomposition. This can also handle systems which have different numbers of inputs and outputs. All three use a transfer function for representing the system.

The fourth method, of Johnston and Barton,' is a structural interaction analysis method which uses state space representation. In this method only direct gains between inputs and outputs are examined.

A brief review of all the methods is given next.

3. I Bristol's relative gain array

The simplest method is the relative gain array. Only steady state gains &-e measured or computed. The change in the open-loop gain of a particular input-output pair is considered when the other control loops are closed. If the gain of a particular input-output pair remains unchanged, the other control loops do not have an effect on it.

The basic strategy of the static relative gain array is to choose a control loop in which the manipulated variable U, and the controlled variable y , are most sensitive to each other and hence less sensitive to other input-output pairs. Because of interaction effects, one must consider the so-called open loop sensitivity

as well as the closed-loop sensitivity

where u k , k Z , indicates that all controllers except uj are held constant. A measure of relative sensitivity is given by the relative gain array whose elements are defined as

The system model is given in the transfer function form of equation (3). The elements of the relative gain array can be calculated from the open-loop steady state gain matrix as

O I Z J = $ZlrJl ( 5 )

where $lJ and r,, are elements of the matrices G(0) and [G(O)] -' respectively.


Bristol has shown that the proper input-output ( U j - y i ) pair for single-loop control is the one having the largest positive aij-value. It is worthwhile to mention that values much larger than unity result in poor controllability. If the values are negative, the system acts in this interval like a non-minimum phase system when the other loops are on. The values for selected input-output pairs should be close to unity. Since the sum of elements in any column or row is unity, this implies that the other elements are close to zero. The designer must then decide when the premise aij= 1 is satisfied.

To give an idea of the use of relative gain, consider a 2 x 2 system. For this the relative gain matrix array becomes

The interaction index a is computed from the steady state open-loop gains as

(7)

The interaction index can be given the following interpretation. When the output y1 (or y z ) is changed into a new value, this causes an effect of

g1 1(0)g22(0) g11(0)gz2(0) - glz(o)gzl(o)

a =

I a I x 100% I4 + I1 -4

in the steady state change caused by the control u1. The rest goes for the corresponding change caused by the control u2.

The usefulness of multivariable control can be considered doubtful if one control causes more than, say, 80% of the output. This gives the limits 0.800 < a < 1.333 used by the expert system. These default values can be changed by the user.

The relative gain array can be calculated only if the system under study has equal numbers of inputs and outputs. The relative gain analysis is a steady state analysis and does not explicitly include dynamic effects. Other methods have to be used to analyse dynamic effects.

The method is useful to identify a number of situations when interactions are so strong as to require multivariable control. If the analysis does not reveal interactions, further work is needed by using methods to uncover dynamic interactions. If only results from Bristol's relative gain array analysis are available, it cannot be deduced that the system exhibits no strong interactions.

Example 1. Consider an example of a system with a transfer function

4/(s+2) l / (s+ 1) s + 4) 9/(s + 3)

G(s) =

On the basis of the gain matrix of the open-loop system the interaction index is computed to be

= 1 a091 2 x 3 a = 2 x 3 - 1 ~ 0 . 5

The relative gain matrix then becomes

1.091 -0.091 -0.091 1.091

46 J. LIESLEHTO AND H. N. KOlVO

The relative effect of U I (UZ) on output yl (y2) is therefore 92%. According to Bristol's method, no strong interactions are present. The recommended pairings are then ul-yl and UZ-yz.

3.2. Tung and Edgar's method

Tung and Edgar have extended Bristol's method to include dynamic effects. Let the system be described by equations (1) and (2). The system is assumed to be controllable and observable.

Consider a change in set point from zero to y o and let the required control change necessary to bring about this set point change be u = uo. Hence both y o and uo are constant vectors bound together by y o = C( - A ) - ' B u o . With zero initial conditions the output response is, in the frequency domain,

Y(s ) = CX(s) = C(s1- A)-'BuO/s = C(s1- A ) - ' [C( - A ) - ' B ] -lyO/s = G(s)[G(O)] - ' y0 /s (9)

where G(s) is the process transfer function matrix. For the ith element of the output Y(s ) , y;(s), this implies

where $ij(s) and F k j are elements of the transfer function matrix G(s) and its inverse at steady state, G-'(0), respectively.

Now consider a step change in yPonly. The corresponding response of yi is

The kth term in the summation results from the kth controller. The above equation indicates that if y; is to be controlled by controller ur, the term +irrr;/s should be the dominant term.

A dynamic relative gain matrix can be formed from the pervious equation. With each ith row of the matrix formed from the ith component of the previous equation, the following dynamic relative gain matrix can be defined:

where

a i j (s ) = +ij(s)I'j;/s, i = 1,2, ..., m, j = 1'2, . .., m (13)

Input u j and output yi are a proper pairing for single-loop control if a i j ( S ) is the dominant term over the whole frequency range studied. By setting s = 0 we obtain Bristol's relative gain matrix.

The expert system uses the following interpretation to find the dominant input for each output.

As indicated above, each output should be attached to the input giving the dominant term


in the sum ( 1 1 ) . In the frequency domain this means that from each row of the dynamic relative gain matrix the element having a significantly larger gain in the frequency range of interest is sought. When the input-output pairing is made, the input indicated by the column is attached to the output corresponding to the row of this element. The system under consideration has strong interactions if an output does not have a corresponding dominant input or if an input is dominant for more than one output.

The user must pick the frequency range of interest. The expert system determines a default frequency range using the phase and characteristic frequencies of the zeros and poles of the transfer function matrix. The values of the phase and characteristic frequencies indicate the distances of the appropriate zeros and poles from the origin. The amplitude curve of the transfer function element goes through a significant change of direction in the frequency range in question. To cover the frequency ranges of all elements, the default range in the expert system begins from a decade below the lowest phase or characteristic frequency found and ends one decade above the highest.

The user must also consider how big a change in the relative gain makes the element dominant compared with other elements. As a default assumption the expert system reasons the input to be dominant if its relative gain over the whole frequency range of interest is more than 80% of the sum of the total gains effecting a particular output or

nt

k = l Iaij(jo)l > 0.8 C I QikCjW) 1 , ~0 < < ~1 (14)

where w0 is the lower and 01 the upper limit for the frequency range. The user can change this interpretation by changing the default coefficient 0.8.

Because Tung and Edgar’s method is an extension of Bristol’s method, it has similar restrictions to Bristol’s method. The system under study has to have as many inputs as outputs. The effect of delays is ignored because the analysis is based only on gains.

Example 2. Consider again the system given in Example 1. The dynamic relative gain matrix of the system is

2 . 1 8 2 0 / ~ ( ~ + 2) - 0 * 0 9 0 9 / ~ ( ~ + 1)) G(s ) [G(O)] -’ - = - 0 . 3 6 3 6 / ~ ( ~ + 4) 3 * 2 7 2 4 / ~ ( ~ + 3)

The gain diagrams of the elements of the matrix G(s) [G(O)] -’ are displayed in Figure 3 in the range 0.1-40 rad s-I. Figure 3 indicates that the diagonal elements have significantly stronger gains than the other elements.

The method of Tung and Edgar then implies that there are no strong dynamic interactions in the system. The recommended pairings are ul-yl and UZ-yz.

3.3. Interaction analysis based on singular value analysis

Both the static and dynamic relative gain arrays are limited to the case of equal numbers of inputs and outputs. A method based on the singular value decomposition removes this restriction. It also takes into account the effect of delays. The method is based on the statement that the interactions of the system are small if the singular vectors differ only a little from the basis vectors in the frequency domain of interest.

Consider the frequency domain representation of the system

Y (s ) = G (s ) U ( S )


0.1 i.0 r0.o t i . 0

I0

g o .s &.I0

-20

-10

-* 4 0

-60 1.0 10.0 IW.0 0. I

I 0 h

8 0 1 - 1 0

-20

-so - 4 0

-so

-60 0.1 1.0 10.0 100.0

fnrlucney (ma/s) Fig

where Y ( s ) is the p-dimensional output vector, U(s ) is the rn-dimensional control vector and G(s) is the ( p x m)-dimensional transfer function matrix.

The transfer function matrix G(s) can be written in the singular value decomposition form

G(s) = Z(s)A(s)VT(s) (16)

where superscript T denotes transposition and

A(s) = rt) -), q = rankG(s) < min(m, n ) m - y

Here A(s) is a diagonal q x q matrix whose entries are the singular values of G(s) defined as the square roots of the eigenvalues of the matrix G'r(s)G(s) (or C(s)GT(s)). Let Z(s) be the n x n matrix consisting of the eigenvectors of G(s)GT(s) and V(s) the m x rn matrix consisting of the eigenvectors of GT(s)G(s).

Let al(s), u2(s), ..., u,(s) be the distinct singular values of G(s). Then Z(s ) and V(s) can be partitioned as

where zi(s) and ui(s) ( i = 1, ..., q ) are the right- and left-hand-side singular vectors which correspond to the ith singular value. The remaining Z j ( S ) ( j = q + 1, ..., rn) and u,(s) ( j = q + 1, .. ., m) are the decomposition vectors which correspond to the zero singular values.


Now the transfer function G(s) can be written as a sum of dyads

G(s) = 9 a;(s)zi(s)uf(s) I

A singular value decomposition of the matrix G(s) defines an input space spanned by a set of orthonormal basis vectors ul(s), uz(s), ..., uq(s), an output space spanned by a set of orthonormal basis vectors ZI (s), ,zz(s), .. ., zq(s) and a gain space defined by the set of singular values (TI (s), u2(s), . . ., uq(s). A one-to-one correspondence is established between these spaces

The transfer function matrix has the following geometrical interpretation (Figure 4). An input vector in the direction of ui'(s) propagates through the input space, is scaled by the gain u;(s) and reappears in the output direction z;(s).

( u i' (s)-ai (S 1-z; (~1) .

Lau et al.4 have shown that

where ( W;(s),Ek[) = [eFTz;(s)] [u?(s)er] and ek is the unit vector, k = 1, ..., n. Superscripts refer to the vector dimensions. The product ( W;(s), Ekl) may be interpreted geometrically as a measure of the alignment of the singular decomposition vectors z;(s) and ui'(s) to the standard basis vectors in the appropriate space. The equation expresses the gain between the kth output and the Ith input in terms related to the internal structure of the system.

Suppose that for some io and for s = jo

I ( W/"ciW),Ek/) I = 1

This implies that the ioth dyad is aligned closely with the basis dyad defined by the kth output and the Ith input or alternatively defined by the (k,l)th loop. Since the basis vectors vl(s), uz(s), ..., vq(s) and zl(s),zz(s), ..., zq(s) form orthonormal sets, it follows that

I (wjcio),E/c[) 12: 0, j z i I ( WidjW), E s p ) I = 0, p z I, s z k

Thus we can conclude that except when the system is poorly conditioned, the (k,l)th loop interacts minimally with other loops which we may select to control the system. The (k,l)th

jrTfEy-'c. v; 0-q 2,

input input rotation scaling output rotation output

Figure 4


loop is a natural loop of the system and input u, and output y k are a proper pair for single-loop control.

The angle Or,] between the dyad W;,, and the basis dyad is defined by

Bio = c0s-l I ( Wi,, E k / ) I (20)

For the control UI and output yk to form an input-output pair, Bj , should be as small as possible over the frequency range of interest. It is for the user to decide the limiting value of the angle for the interaction measure. Lau et al. propose a value of 15O. This corresponds to 1 ( Wio, E k l ) 1 > 0.965.

For the overall interaction measure Lau et al. suggest

The interaction measure is computed from the weighted averages of the input-output pairs chosen from different dyads. The weights are the singular values of the corresponding dyads.

Remark: The same sort of measure is used by Belletrutti and MacFarlane" in their

As in other frequency domain techniques, the user must pick the frequency range of interest. The expert system chooses this in the way presented in connection with the method of Tung and Edgar.

As mentioned before, the main advantages compared with the previous methods are that this method can handle systems with unequal numbers of inputs and outputs and also time delay systems. It suffers from the following deficiency. If the singular values are too close to each other, the method is very sensitive and not too reliable.

characteristic locus method.

Example 3. The system of Example 1 is again studied. The value of the transfer function matrix at steady state is

G(O)= ( 1) 0.5 3

The singular value decomposition of G(0) is -0.5152 -0.5152 -0.8571 -0.8571

Z(0) =

0 1.6111

-0.4274 -0.9041 '(O) = (- 0.9041 0.4274

The relationship between the singular vectors and the basis vectors is presented in Figure 5 . It is apparent that the angles ( u , , el), (212, e z ) , (zl, el) and (a, ez) are quite large, indicating strong system interactions.


Figure 5

.S 1 ::: 0 .I

0 .o 0.1 1.0 IOJ iooa

B il :: 0.2

oa 0.1 i.o da 1d.0

frrquency (rads) fnquency Figure 6

4 ::: .$ 0.1

p 0.4

*= 0.2

0 .O 0.1 1.0 0.0 100.0

E ' . O

s

5 0.2

0 .o 0.t 1.a 0.0 tm.0

." 0.e

.Ei o .a

0 .0 0

frequency (radls) frequency ( d s )

Figure 7


The dyadic decomposition of the transfer function matrix in steady state is

0-2202 0.4658) + 0.7749 -0.3663 .61 ( -0.4658 0.2202 '(')= 3*4139 (0.3663 0.7749

The dominating elements in the dyads are ( Wl, E22) 1 = 1 ( WZ, El l ) I = 0.7749. This then implies connecting input U I to output y1 and u2 to yz. Since the angle 0 is quite large (cos-I 0.7749 = 39'), a multivariable controller should be used to remove interactions.

A more extensive analysis is shown in Figures6 and 7, where the values of the dyadic elements are displayed over the frequency range 0.1-40 rad s-l. The figures indicate that the analysis performed on zero frequency is also true over the whole frequency range since max I ( WI, E n > I = max I ( W2,Ell) I = 0.9272.

3.4. Structural interaction analysis

In the previous methods the mathematical system model is the transfer function model. The methods can also be applied for the state space models since their transfer function models are unique. The transformation in the opposite way is not unique. The transfer function can be represented by a number of state space models. A state space model may include more information about the interactions than the transfer function matrix.

Let us examine the example of Figure 8. The steady state gain matrix of the system is

G(0) = ('i -:) The relative gain matrix of the system is

(::;; ::;;) This would seem to indicate that no strong interactions exist. The recommended pair selections would be U I - ~ I and UZ-y2. Figure 8 indicates that most of the effect of control u1 on y~ is transmitted into output y2 via the state x2. This means that the system is in fact quite strongly interactive.

The basic idea of the structural interaction analysis is to find the direct gain between an input and output. According to Johnston and Barton,8 this is the actual amount of useful gain which can be obtained directly from the input. Any other gain is due to output-output interactions. In Figure 8 an example of the latter is the coupling from y2 to y t .

The direct gain between each of the inputs and the ith output y , is determined by setting the state variables corresponding to all other outputs in the state space model equal to zero. The state space representation must be in the form where the outputs are directly state variables and not their linear combinations. In Figure 8 the direct gain from U I to yl is 1 and from u2 to y l is -20. Correspondingly the direct gain from U I to y2 is 5 and from 2.42 to y2 is 9.

Figure 8


Consider the state space model

i= A X + BU y = c x

This produces a modified state space model

with vector dimensions x: = (n - r + 1 ) x 1, u: = m x 1 and y: = 1 x 1. The corresponding modified transfer function vector is

g:(s) = C f ( d - A,*)-'B,*

Each element in g:(s) will give the direct dynamic response of the current output yi to each input u, = 1, ..., m.

The matrix of direct gains then becomes

I G*W I = (I g: I , e a . 9 I g: I) The elements of the interaction matrix of direct gains DGM(s) is formed by dividing the corresponding absolute value of the direct gain by the sum of all other absolute values of direct gains related to the same input.

If the element DGM,(s)= 1 over the whole frequency range of interest, then output y , is mostly influenced by ut . This is then also the best input-output pairing. If each output has a dominant input and an input is not dominant for more than one output, interactions in the system can be considered negligible. Johnston and Barton do not indicate how large an interaction index is dominant. The expert system uses a value of 0.8. This implies that 80% of the total gain of a particular input is directed for a certain output. The value can be changed by the user.

Another parameter to be determined by the user is the frequency range. If the user does not specify the range, the expert system performs the analysis in addition to zero frequency also for the frequency range which begins one decade below the lowest frequency and continues one decade above the highest frequency determined by the eigenvalues of the matrix A . This default value can be changed by the user.

The method is not limited to the case of equal numbers of inputs and outputs. The most severe limitation is the form of the state space representation. The outputs must also be state variables. Another limitation is that currently the method has not been generalized for the delay case.

Example 4 . representation

Let us consider the following two example systems. The first has the state space

-3.5 - 1 . 5 0.5 0.5 - 5 . 0 -5 .0

U 1.0 1.0 -4.0 -1-0 0.5 0.5 1 . 5 - 1 . 5 -1.0 - 8 . 5

x= ( y = o o ( O O 1 0 O)

54 J . LIESLEHTO AND H. N . KOIVO

the second

U

-2.60 - 1.05 0.95 0.10 - 1.00 0.10 0.10 0.10 -4.00 -0.10 0.50 0.95 1.05 -2.40 -19 -49

y = o o l o (" O 0)x

Remark. It is worthwhile to observe that both systems have the transfer function of equation

The interaction matrix DGM of direct gains is computed at zero frequency first. By setting (8) in Example 1.

x3 = 0, the transformed state space matrices related to output yl are obtained:

-3.5 - 1 . 5 0.5 -5.0 -0.5 A T = ( 1.0 -1.0 I - O ) , B : = ( 4 - 0 l - O ) , C:=(O 1 0)

0.5 -1.0 1.0 -1.0 -8.5

The direct gains affecting output yI at zero frequency are

I rT(0) I = I C?( -A ? ) - ' B : I = (1 ~ 0 0 5.00)

By setting state variable xz equal to zero, the transformed state space matrices related to y2

become

- 3 . 5 -0 .5 -5.0 -0.5 A:= ( 1.0 -4 .0 y: : ) , B : = ( 2.0 9*0) , C:=(O 1 0)

0.5 1.5 - 1 . 5 -1.0 -8.5

The direct gains affecting output y2 at zero frequency are

1 gT(0) I = 1 C;( - A :)-'B: 1 = ( 0 . 3 3 2.92)

Combining the above, the direct gain matrix of the first system at zero frequency is

Ig:(O)( - 1.00 5-00 G * = (1 g2*(0) I) - ( 0 . 3 3 2.92)

Finally, the direct gain interaction matrix is computed at zero frequency to be

The direct gain matrix of the second example system is computed in an analogous manner to be

(G*(O)l z (' *90 Oa40) 0.48 2.99

and the direct gain interaction matrix is

0.80 0.12 DGM(o)= (0.20 0.88)


The two direct gain matrices at zero frequency indicate the interactions of the example systems are quite different in spite of the fact that the transfer function matrix is the same for both. In the first system both inputs strongly affect output y l . Since y2 has no dominant input, the system exhibits significant interactions.

The second system clearly has smaller interactions. The pairing should be performed so that output yl corresponds to input u1 and y2 to u2. It must, however, be observed that the

::: .Fi

o .a

0 .o

Figure 9

K 4.0

# 0..

B *.* 8 .g 0.6

.B 0.2

0 .o 0.1 1.0 0 .0 mom0

frequency ( d s )

3 1.0

3 0 . 8

1:: .!!

4.2

0 .o 0.1 1.0 10.0 tm.0

fiquency (rads) frequency ( d s )

Figure 10

56 J . LIESLEHTO AND H. N . KOlVO

interaction measure between u1 and y1 at zero frequency is smaller than the limiting value of 0.8 used by the expert system as a default value.

Figures 9 and 10 display graphically the interaction measures of the example systems in the frequency range 0. I-40rad s - ' . The strong interactions in the first example concentrate at lower frequencies in Figure 9. At higher frequencies the effect of input u1 concentrates mainly on y1 and similarly the effect of uz on yz.

Figure 10 indicates that the effect of control u1 is more dominant over the whole frequency range and similarly that of u~ on yz. Again it must be mentioned that the interaction measure over the whole frequency range of interest is smaller than the default value of 0 - 8 of the expert system.

4. FUNCTIONAL DESCRIPTION OF THE EXPERT SYSTEM

A typical interaction analysis session can be divided into four steps: a selection of interaction analysis methods, numerical calculations, an interpretation of the numerical results and a combination of the results from different interaction analysis methods.

First the expert system selects appropriate interaction analysis methods. This selection is based on the mathematical limitations of each method. For example, steady state interactions should be analysed with the singular value decomposition method only when the values of the steady state gain matrix are clearly distinct. The Rule 29 is used to check this:

(RULE 29 (IF (INTERACTION ANALYSIS IS RECOMMENDED)

(TYPE OF THE MODEL IS TRANSFER-FUNCTION-MATRIX) (SINGULAR VALUES OF THE SYSTEM ARE CLEARLY DISTINCT AT FREQUENCY 0))

(THEN (EXECUTE SVA-TF-SS 0.965)))

The next step is the execution of the numerical calculations. The computational methods of the interaction analysis expert system have been programmed with Fortran, unlike the other programmes. There are two reasons for this. Fortran is significantly faster the LISP in the computer used to implement the expert system. The other reason is that already existing programmes have been used and these are coded in Fortran.

A deficiency of the LISP interpreter is that it does not offer ready-made interfaces to programmes coded with other programming languages. Therefore the interfaces between the LISP programmes and the Fortran programmes have been implemented by using the files shown in Figure 11.

LISP programmes write the information to be delivered to the Fortran programme into a file called STOF, from which the Fortran programme read them. After the execution the Fortran programmes write the information to be returned to the LISP programmes into a file called FTOS, from which the LISP programmes read it.

The computational methods of the interaction analysis produce a fair amount of numerical results. The size of the working memory would grow far too much if the numerical data were included in their entirety. The deduction procedure slows down as the working memory grows, because then pattern matching must be performed for more elements. Therefore the numerical results are filtered through special programmes attached to the working memory. These pick the essential features from the numerical data and make only the necessary additions to the working memory. In this way the working memory can be kept to a reasonable size.

Next the results from the numerical calculations are interpreted. This is done by comparing


Figure 1 1

different interaction measures with the limit values known by the expert system. The expert system has a set of default limit values for different interaction measures. Because the knowledge base is a simple text file, these limit values can be easily changed by the user. For example the aforementioned Rule 29 specifies 0.965 to be the limit value of the results from the singular value decomposition method. A typical interpretation rule is Rule 30:

(RULE 30 (IF (STEADY STATE SINGULAR VALUE ANALYSIS INTERACTION INDEX

(> SI) BETWEEN INPUT (> X) AND OUTPUT (> Y) IS LARGER THAN (> LB)) (STEADY STATE SINGULAR VALUE ANALYSIS INTERACTION INDEX OF INPUT (< XI IS THE LARGEST FOR OUTPUT (< Y)))

(THEN (SINGULAR VALUE ANALYSIS SHOWS THAT INPUT (< X) AND OUTPUT (< Y) CAN BE USED AS AN 10-PAIR FOR SISO-CONTROL IN STEADY-STATE)))

The last step is to combine the results from different interaction analysis methods. The expert system does this in a straightforward way by checking that input-output pairings proposed by

(RULE 33 (IF

(THEN

different methods are the same. This is done for example by Rule 33:

(MATRIX G IS A SQUARE MATRIX) (SINGULAR VALUES OF THE SYSTEM ARE CLEARLY DISTINCT AT FREQUENCY 0) (TYPE OF THE MODEL IS TRANSFER-FUNCTION-MATRIX) (BRISTOL TEST SHOWS THAT INPUT (> X AND OUTPUT (> Y) CAN BE

(SINGULAR VALUE ANALYSIS SHOWS THAT INPUT (> X) AND OUTPUT USED AS AN 10-PAIR FOR SISO-CONTROL IN STEADY STATE)

(> Y) CAN BE USED AS AN 10-PAIR FOR SISO-CONTROL IN STEADY-STATE) (INPUT (< X) AND OUTPUT (< Y) CAN BE USED AS AN 10-PAIR FOR SISO-CONTROL IN STEADY-STATE)

58 J . LIESLEHTO AND H. N. KOlVO

Finally the results from the steady state and dynamic interaction analysis are combined:

(RULE 53 (IF INPUT (> X AND OUTPUT (> Y) CAN BE USED AS AN IO-PAIR FOR

SISO-CONTROL IN STEADY STATE) (INPUT (< X) AND OUTPUT (< Y) CAN BE USED AS AN IO-PAIR FOR SI SO-CONTRO L D U RI N G TR AN SI ENT) ) (INPUT (< X) AND OUTPUT (< Y) CAN BE USED AS AN IO-PAIR FOR SISO-CONTROL)))

(THEN

5 . CONCLUSIONS

None of the presented methods is alone sufficient to be used in interaction analysis for linear systems. The expert system must choose the most applicable methods among the ones given for the system to be investigated. The methods can be chosen in two ways. The first is to analyse the system to be investigated using all applicable methods. The second is to set the methods in order of preference and use the best method only for the analysis in the particular situation. The expert system uses mostly the first method because no knowledge of the preference of methods is available. Such a preference list might also be based on different sorts of appreciations rather than exact knowledge. An exception for this is the case when the model is given in the state space form. In this case only the structural interaction analysis is used. The reason for this is that it is the only included method utilizing all the state space information.

Analysis methods typically produce numerical results. The analysis occurs by comparing the obtained numerical results with the comparison values of the interaction indices related to a particular method. There is surprisingly little information in the literature about the interpretation of numerical results. The interaction indices given as numerical values are stated to describe interactions appearing in the system, but exact comparison values and rules for interpretation are not given. This makes it harder to put together the knowledge base.

The exact interpretations and comparison values may also be faulted. Suppose that the comparison value in Bristol’s method used to consider a 2 x 2 system is 0.8. This causes an interpretation that the value 0-79 for the interaction measure reveals strong interactions in the system, while on the other hand the value 0.81 can be a sign of weak interactions. Such problems can be avoided in the expert system by using an inference engine, which can handle uncertain knowledge.

The last phase in the analysis is combining the results collected from different interaction analyses. This can be done in several ways. In this study the collected knowledge joins together the different results by weighting the possible problems caused by interactions. This means that the system under consideration is judged to have strong interactions if even one analysis method results in such a deduction. Input-output pairings are accepted only if all applied methods suggest the same pairings.

APPENDIX: EXAMPLE RUNS

The example studied is the one discussed in Section 3 (equation (8)). In the computer run the user-given information is boldfaced. The expert system deductions are indicated with capital letters. If the deductions have resulted from firing a rule, this is mentioned. Otherwise it is a question of knowledge elements formed by the connection programmes based on input information and numerical computing. Comment lines in italic have been added to help understanding.


The example run describes the interaction analysis of the system. The expert system is started by adding the start element to the working memory.

(assert '((start)))

The user is frrst asked about the system type.

RULE 3 SAYS

Type of the model is transfer function matrix (EXECUTE READ-SYSTEM-MODEL-TYPE)

The knowledge element telling about the model type is added to the working memory.

(TYPE OF THE MODEL IS TRANSFER-FUNCTION-MATRIX)

The model dimensions are asked next.

RULE 4 SAYS

Number of inputs is 2 Number of outputs is 2

(EXECUTE READ-MODEL-DIMENSIONS 'TRANSFER-FUNCTION-MATRIX)

Knowledge elements covering dimensions are added to the working memory.

(DIMENSIONS OF MATRIX G ARE KNOWN) (SYSTEM HAS MORE THAN ONE INPUT) (SYSTEM HAS MORE THAN ONE OUTPUT) (MATRIX G HAS AS MANY ROWS AS COLUMNS)

Rules 5 and 8 make deductions connected with dimensions.

RULE 5 SAYS

RULE 8 SAYS (DIMENSIONS OF THE MODEL ARE KNOWN)

(MATRIX G IS A SQUARE-MATRIX)

Model parameters are asked to be supplied by the user.

RULE 9 SAYS

Element 1 , 1 of the transfer function matrix: List of numerator coefficients is (4) List of denominator coefficients is (1 2) Delay is 0 Element 1 , 2 of the transfer function matrix: List of numerator coefficients is (1) List of denominator coefficients is (1 1) Delay is 0 Element 2 , 1 of the transfer function matrix: List of numeratar coefficients is (2) List of denominator coefficients is (1 4) Delay is 0 Element 2,2 of the transfer function matrix: List of numerator coefficients is (9) List of denominator coefficients is (1 3) Delay is 0

(EXECUTE READ-MODEL 'TRANSFER-FUNCTION-MATRIX)

The facts concerning zeros and poles of the transfer function elements are added to the working memory.

(ZEROS OF THE TRANSFER FUNCTION BETWEEN INPUT 2 AND OUTPUT 2 ARE FOLLOWING 0) (ZEROS OF THE TRANSFER FUNCTION BETWEEN INPUT 1 AND OUTPUT 2 ARE FOLLOWING 0 )

60 J . LIESLEHTO AND H. N. KOIVO

(ZEROS OF THE TRANSFER FUNCTION BETWEEN INPUT 2 AND OUTPUT 1 ARE FOLLOWING 0) (ZEROS OF THE TRANSFER FUNCTION BETWEEN INPUT 1 AND OUTPUT 1 ARE FOLLOWING 0) (POLES OF THE TRANSFER FUNCTION BETWEEN

(POLES OF THE TRANSFER FUNCTION BETWEEN



INPUT 2 AND OUTPUT 2 ARE FOLLOWING ( ( - 3 .0 ) ) )

INPUT 1 AND OUTPUT 2 ARE FOLLOWING ((-4.0)))

INPUT 2 AND OUTPUT 1 ARE FOLLOWING (( - 1 .O) ) )

INPUT 1 AND OUTPUT 1 ARE FOLLOWING (( - 2 .0 ) ) )

The information concerning singular values of the system are added to the working memory and matrix G is acknowledged to be known.

(MINIMUM SINGULAR VALUE RATIO 2 . 1 1898 AT FREQUENCY 0 IS GREATER THAN 1.001) (MINIMUM SINGULAR VALUE RATIO 2.03802 BETWEEN FREQUENCIES * 1 AND 40. IS GREATER THAN 1 .001) (MATRIX G IS KNOWN)

Since matrix G is known, so also is the transfer function matrix model. RULE 10 SAYS (MODEL IS KNOWN)

Since the system has several inputs and outputs, interaction analysis is recommended. RULE 12 SAYS (SYSTEM IS A MlMO SYSTEM) RULE 16 SAYS (INTERACTION-ANALYSIS IS RECOMMENDED)

The singular values of the system are judged to be clearly different in the frequency range of interest. RULE 17 SAYS (SINGULAR VALUES OF THE SYSTEM ARE CLEARLY DISTINCT AT FREQUENCY 0) RULE 19 SAYS (SINGULAR VALUES OF THE SYSTEM ARE CLEARLY DISTINCT BETWEEN FREQUENCIES ’ 1 AND 40.)

System interactions are first considered by using the simplest method, that of Bristol.

RULE 21 SAYS (EXECUTE BRISTOL .8 1 .333)

The knowledge elements derived from the numerical results of Bristol’s test are added to the working memory.

(INPUT 2 IS AVAILABLE) (INPUT 1 IS AVAILABLE) (INPUT 1 HAS THE LARGEST RELATIVE GAIN FOR OUTPUT 1) (BRISTOL RELATIVE GAIN 1 m0909 BETWEEN INPUT 1 AND OUTPUT 1 IS BETWEEN (INPUT 2 HAS THE LARGEST RELATIVE GAIN FOR OUTPUT 2) (BRISTOL RELATIVE GAIN 1 .0909 BETWEEN INPUT 2 AND OUTPUT 2 IS BETWEEN .8 AND 1 * 333)

8 AND 1 .333)

The recommended input-output pairings are deducted on the basis of Bristol’s method.

RULE 22 SAYS (BRISTOL TEST SHOWS THAT INPUT 2 AND OUTPUT 2 CAN BE USED AS AN 10-PAIR FOR SISO-CONTROL IN STEADY-STATE)


RULE 22 SAYS (NOT INPUT 2 IS AVAILABLE) RULE 22 SAYS (BRISTOL TEST SHOWS THAT INPUT 1 AND OUTPUT 1 CAN BE USED AS AN 10-PAIR FOR SISO-CONTROL IN STEADY-STATE) RULE 22 SAYS (NOT INPUT 1 IS AVAILABLE)

System interactions are next considered on the basis of singular value decomposition.

RULE 29 SAYS (EXECUTE SVA-TF-SS 965)

Interconnection programmes add the knowledge elements corresponding to the numerical results obtained to the working memory.

(STEADY-STATE SINGULAR VALUE ANALYSIS INTERACTION INDEX OF INPUT 1 IS THE LARGEST FOR OUTPUT 1) (STEADY-STATE SINGULAR VALUE ANALYSIS INTERACTION INDEX * 774868 BETWEEN INPUT 1 AND OUTPUT 1 IS LESS THAN .965) (STEADY-STATE SINGULAR VALUE ANALYSIS INTERACTION INDEX OF INPUT 2 IS THE LARGEST FOR OUTPUT 2)

BETWEEN INPUT 2 AND OUTPUT 2 IS LESS THAN .965) (TOTAL INTERACTION MEASURE .774868 FROM SINGULAR VALUE ANALYSIS IN

(STEADY-STATE SINGULAR VALUE ANALYSIS INTERACTION INDEX ‘77486

STEADY-STATE IS LESS THAN ‘965 )

On the basis of small interaction indices it is deduced that outputs have no dominating inputs.

RULE 31 SAYS (SINGULAR VALUE ANALYSIS SHOWS THAT NO INPUT-OUTPUT CAN BE FOUND FOR OUTPUT 2 IN STEADY-STATE) RULE 31 SAYS (SINGULAR VALUE ANALYSIS SHOWS THAT NO INPUT-PAIR CAN BE FOUND FOR OUTPUT 1 IN STEADY-STATE)

(NO INPUT-PAIR CAN BE FOUND FOR OUTPUT 1)

(NO INPUT-PAIR CAN BE FOUND FOR OUTPUT 2)

RULE 32 SAYS

RULE 32 SAYS

Since outputs have no dominating inputs, use of SISO controllers is not recommended.

RULE 2 SAYS (DUE TO INTERACTIONS SYSTEM CAN NOT BE CONTROLLED BY SISO-CONTROLLERS)

Since the analysis indicates that there are strong interactions in the system, the reasoning is stopped. This happens by emptying the conjlict set containing the rules under consideration.

RULE 1 SAYS (EXECUTE SET! RULE-STACK NIL)

REFERENCES

1. Bristol, E. H., ‘On a new measure of interaction for multivariable process control’, IEEE Trans. Automatic

2. McAvoy, T. J . , Interaction Analysis, Instrument Society of America, Research Triangle Park, NC, 1983. 3. Tung, L. S. and T. F. Edgar, ‘Analysis of control-output interactions in dynamic systems’, AIChE J . , 27,

4. Lau, H., J. Alvarez and K . F. Jensen, ‘Synthesis of control structures by singular value analysis: dynamic

5 . Manousiouthakis, V., B. Savage and Y. Arkun, ‘Synthesis of decentralized process control structures using the

Contra/, 11, 133-134 (1966).

690-693 (1981).

measures of sensitivity and interaction’, AIChE J. , 31, 427-439 (1985).

concept of Block Relative Gain’, AIChE J. , 32, 991-1003 (1986).

62 J . LIESLEHTO AND H. N. KOIVO

6. Grosdidier, P. and M. Morari, ‘Interaction measures for systems under decentralized control’, Automatica, 22,

7. Mijares, G . , J. D. Cole, N. W. Naugle, H. A. Preisig and C. D. Holland, ‘A new criterion for the pairing of

8. Johnston, R. D. and G. W. Barton, ‘Structural interaction analysis’, Int. J . Control, 41, 1005-1013 (1985) 9. Winston P. H. and B. K. P. Horn, LISP, (2nd edn,) Addison-Wesley, Reading, MA, 1984.

10. MacFarlane, A. G. J. and J. J. Belletruth, ‘The characteristic locus design method’, Autornatica, 9, 575-588

309-319 (1986).

control and manipulated variables’, AIChE J . , 32 1439-1449 (1986).

(1973).

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An expert system for interaction analysis of multivariable systems