Journal of Dynamic Systems,

Measurement and

Control

On the Inverse of a Special Class of Bilinear Systems1

C.S. Hsu2 and R.R. Mohler3

In this paper, the design of inverse system for a special class of bilinear control processes is examined. It is shown that if the rank of the input matrix B is unity, then the inverse system of a bilinear process is a completely controllable linear time-invariant system. A simplified model of immune response is used as an example to illustrate the design procedure and its biomedical significance. Result of computer simulation is also presented.

Introduction During recent years, much progress has been made in the

study of bilinear systems (BLS), including controllability, observability, optimal control, stabilisation, observer design and so on among many others. However, the inverse system design of BLS, which is of practical importance, goes relatively unnoticed. To the authors' knowledge, the most significant contribution is due to Professor Hirschorn of Queen's University [1,2].

Investigation of a "system" employing a mathematical model normally grants that the input (control) and output (measurement) are known, at least in the probabilistic sense. This is the essence of the wellknown "black-box approach." Formally speaking, the object is to construct a model that will accurately represent the underlying system by way of, for instance, estimating the parameters, realizing the system structure, etc. However, very often the situation is not so. The communication system is an illustrative example. The main concern is to retrieve the message (input) based on the output (the received signal which may be corrupted by noises).

To this, it is remarkable that the question of in-vertibility—when the output of a control system uniquely determines the input—is of practical as well as theoretical interest. The construction of an 'inverse model' which is able to predict the unknown input of a model from measured output data is particularly important in biological modeling [3]. Fish [4] made a model of the movement of uranium in the body, in which uranium was picked by the body from the surroundings and appeared in the urine. The output could be

This research was supported by National Science Foundation under Grant ENG 74-15530.

Department of Electrical Engineering, Washington State University, Pullman, Wash. 99164.

Department of Electrical and Computer Engineering, Oregon State University, Corvallis, Ore. 97330.

Contributed by the Dynamic Systems and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Dynamic Systems and Control Division, April 7, 1980.

measured, but the history of the input was unknown. In immunology, the measurable output data usually is the an­tibody concentration in the serum, the threshold (input or parametric control) which determines the cell stimulations and differentiations is unknown [5,6].

Theory There is a considerable amount of literature dealing with

the construction of an inverse system for linear system. The main result is summarized as follows without proof.

Theorem 1. Consider the SISO completely controllable linear system,

x{t) =Ax(t) + bu(t);x(G) = x0,x(t)eR"

y(t)=cTx(t) (3.1)

If the relative order a of (3.1) is finite, i.e. a<a>, then there is a unique inverse system which is also completely con­trollable and is defined by: x{t) =[A-bcTAa /cTAa-lb)\x(t)

+ (l/cMa-'b)bM(f),x(0) = x0

y(t) = - (cTAa/cTAa-[b)x{t) + (l/cTAa~[b)u(t)(3.2)

Letii(t) =y{a) (t), theny(t)=u(t). Here the relative order of a linear system is referred to as

the difference in the degree of numerator and denominator polynomials of its transfer function G(s) = cT(Is—A) ~'b. Theorem 1 shows the design of the inverse system of given linear system is available if certain conditions are met. The extension of constructing the inverse system for a bilinear system is a much more difficult task. Hirschorn's results which generalize the Theorem 1 are:

Theorem 2 [1]. Consider SISO BLS,

x(t) =Ax(t) +«(OBx(0,x(0) = x0ei?n

y(t)=cTx(t) (3.3) If the BLS (3.3) is invertible, then its relative order is a<e». Here the relative order of a BLS is referred to as the least positive integer k such that cfad/T^B ^ 0 or a = » if crafiftfl = OforallA:>0. If «<oo and cTadA

a~lBx0 ^ 0, then the BLS is invertible with inverse system defined by

x(0=fc(x(0)+«(Ob(x(0) ,x(0) = x06R"

y(t)=d(x(t))+u(t)e(x{t)) (3.4) where

n(.x{t))=Ax(t)~(crAax(t)/cTAa-lBx(t))Bx(t)

b{x(t))=(l/cTAa~,Bx(t))Bx(t) d{x(t)) = - (cTAax{t)/cTAa-lBx(t))

e(x(t)) = l/cTAa-lBx(t) (3.5)

Journal of Dynamic Systems, Measurement, and Control JUNE 1981, Vol. 102/103

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If u(t)x = yia) (0,theny(t) = u(t). Proof of this theorem is omitted here since it is quite

lengthy. Readers who are interested in the proof are referred to the original paper [1]. There are a few consequences of Theorem 2 which are of particular interest. First it can be proved that when u(t) =j>(a) ( 0 , the state x(t)=\(t), the state of the original BLS (3.3). Thus the inverse system acts as state observer for the BLS, a result of which itself is of practical importance. Secondly, the inverse system (3.4) is in general a highly nonlinear system. The vector fields a ( x ( 0 ) and b ( x ( 0 ) may not be complete, that is, there is a possibility that the solution to (3.4) may go without bound at a certain finite time. Thirdly, the above theorem presents a sufficient condition for inverting vector BLS (3.3) in case where a<oo, but this condition is far from being necessary. Fourthly, Hirschorn raises the question whether or not an invertible BLS has a bilinear inverse system. This is indeed only a specific question of the more subtle one, that is, what is the connection (if any) of BLS and its inverse as far as their structural aspects are concerned. For instance, the inverse system is linear and controllable (see Theorem 1). Can this connection be carried over to bilinear systems? While aforementioned problems are interesting of their own right, the last question is treated here.

The inverse system (3.4) is obviously a linear-analytic system which includes BLS as a special case. In order to have the inverse system of simpler structure, it is reasonable to impose conditions on the matrices A and B of the original BLS (3.3). It is shown here that if the rank of B is unity, then the inverse system (3.4) is much simplified.

First consider the following fact in matric theory.

Lemma. Any n-dimensional square matrix of rank one can be uniquely (within a scalar factor) expressed as a product of a column and a row «-vector.

Theorem 3. Consider the SISO BLS (3.3) with relative order a if the system is invertible and rank (B) = 1, then the inverse system of (3.3) is a linear time-invariant system with nonlinear output defined by (3.4) with

x x ( t ) , x 2 ( t ) 10

«• t

x 3 ( t ) x 10 16

Fig. 1 Simulation of BLS (4.1) and its inverse system, the dots denote the estimated values.

a(x(0)

b(x

rf(x(0)

o(-i(t\ \

- ( " •

(0) =

C

lcMc

cTAa~

1

cTA«x

,JxtO

v (0

TAa-ilmTx(t)

1 c M a - ' I m 7 x ( 0

(3.6a)

(3.6b)

(3.6c)

(3.6d)

where B = lm T

Theorem 3 is obtained by applying the above lemma into Theorem 2. The assumption that rank (B) = 1 seems very restrictive, but BLS with this property stands out as special BLS of particular interest both in theory and in practice. For instance, discrete BLS with rank (B) = 1 has been extensively studied as far as the controllability and optimal control are concerned [7, 8]. Many natural bilinear systems do satisfy this rank asumption as can be found in the next section and elsewhere [9].

It is observed from this theorem that the inverse system (3.6a) and (3.6b) is exactly the same as that for the constant linear system (see Theorem 1). The following stronger result can be established.

Theorem 4. Consider the BLS (3.3) as described in Theorem 3, then the inverse system of (3.3) is a linear time invariant

Data: x1 (0) = 4 x Kr,x2 (0) = 0,x3(0) = 0,« = 0.1, « ' = 3 . 6 x 1 0 6 , a " = 3 . 6 X 1 0 7 , T 1 = 1 0 5 , T 2 = 5 0 .

Unit of each state variable is number of cells, unit of time is hour.

system which is completely controllable if the original system is completely controllable.

The proof of this theorem is based on the following two remarks, and is somewhat straightforward:

"'I) '1 .

= Rank (1,/H A"~l\) Remark 1. Rank (1,/4I, A" where,4 = A- (lcTAa)/cTAa-

Remark 2. If the BLS (3.3) with rank (B) = 1 is completely controllable, then Rank (\,A\, v4"-1I) = n.

In other words, if BLS (3.3) with rank (B) = 1 is com­pletely controllable, then by the above remarks Rank (1./41, A"~li) = n. The inverse system of (3.3), which is a constant linear system by Theorem 3, is thus completely controllable as a consequence of the wellknown rank test.

Numerical Example. A simplified immune model is adopted here to illustrate the inverse design of a BLS [10]. Consider the following model of immune response,

x,(f) = a ( 2 u ( f ) - l ) x , ( f ) x,(f),x,(0) = x10

j f 2 ( 0 = « ( l - « ( 0 ) * i ( 0 *2(O,*2(0) = 0 T2

(4.1)

104/Vol. 102, JUNE 1981 Transactions of the ASME

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input u ( t ) h

TB—rr To" - t ^ t

Fig. 2 Input function u(t) reproduced from the output of the original system (4.1). Data used is the same specified in Fig. 1.

i 3 (0 = a'xx{t) + cx"x2(t),xi(P) = 0,te[0J]

where x{(t), x2(t) and x3(t), respectively, denote the population density of large lymphocytes, plasma cells and antibody. The input u(t) is the fraction of cells that remain large lymphocytes (i.e. 0 <«(/) <1), 1 -u(t) is the fraction that differentiate into plasma cells. Definitions of rate constants a.,T\, r2, a', and a" are obvious from the'context.

Model (4.1) can be written in the form of (3.3), withx(0 = (*,(/), x 2 ( / ) , x 3 ( 0 ) \ x n Cxr,o,0,0)and

A =

0 0

1 — 0 7"2

" 0

B =

2& 0 0

-a 0 0

0 0 0

c=

In practice, the input u(t) is not known, and the antibody concentration x3(t) is the only measurable state. For this particular example, rank (B) = 1, the relative order a = 2 if 2a' = a ", and therefore, Theorem 3 can be used to estimate X\(t),x2(t) andu(t).

Since the relative order a = 2, it follows that if u{t) = x3

<2)(0> then y(t) = u(t). The input to the inverse system involves second derivative, a first order approximation is used for the numerical simulation. Figures 1 and 2 show the simulation result using a set of immunological data. It is seen that the inverse system favorably estimates the inaccessible state and input variables. However, on Fig. 2 there is a discrepancy between the reproduced input and the actual one for times larger than 80 hours. This seems to be attributed from the discontinuity of the input at / = 80 hours, and the integration algorithm used in simulation incurs significant numerical errors.

References 1 Hirschorn, R. M., "Invertibility of Control Systems on Lie Groups,"

SIAMJ. Control & Optimization, Vol. 15, No. 6, 1977, pp. 1034-1049. 2 Hirschorn, R. M., "Invertibility of Nonlinear Control Systems," SIAM

J. Control & Optimization, Vol. 17, No. 2, 1979, pp. 289-297. 3 Osburn, J. O., "Biological Modeling: A Program to Calculate the Input

from Observations on the Output," Proc. Iowa Acad. Sci., Vol. 80, No. 2, 1973, pp. 87-90.

4 Fish, B. R., "Applications of an Analog Computer to Analysis of Distribution and Excretion Data," Health Physics, Vol. 1, 1958, pp. 276-281.

5 Waltman, P. and Butz, E., "A Threshold Model of Antigen-Antibody Dynamics," J. Theor. Biol., Vol. 65, 1977, pp. 499-512.

6 DeLisi, C , "Some Mathematical Problems in the Initiation and Regulation of the Immune Response," Math Biosciences, Vol. 35, 1977, pp. 1-26.

7 Goka, T., Tarn, T. J., and Zaborszky, J., "On the Controllability of a Class of Discrete Bilinear Systems," Automatica, Vol. 9, 1973, pp. 615-622.

8 Evans, M. E. and Murphy, D.N.P., "Controllability of Discrete Time Inhomogeneous Bilinear Systems," Automatica, Vol. 14, 1978, pp. 147-151.

9 Mohler, R. R., Bilinear Control Processes, Academic Press, New York, 1973.

10 Hsu, C. S., Bilinear Control Processes with Application to Immunology, Ph.D. thesis, Oregon State University, 1978.

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