# Observability analysis of nonlinear systems by linear methods.

Abstract: The paper is devoted to the development of method to analyze the observability of nonlinear dynamic systems. The main idea of this method is as follows: the observability of the initial nonlinear system linear part is checked and if it is unobservable, it is examined whether or not the nonlinear term violates this conclusion.

Key words: nonlinear dynamic systems, observability, linear systems

1. INTRODUCTION

The problem of observability is successfully solved for wide class of dynamic systems--linear, linearly-analytical, polynomial, etc. (Isidori, 1989; Kwakernaak & Sivan, 1972; Sontag, 1979). However this problem is not solved for systems with no differential nonlinearities. In this paper a new approach to solve the problem of observability for such class of systems is suggested.

Consider a class of nonlinear dynamic systems described by the following equations:

x(t+1) = Fx(t) + Gu(t) + C x [phi](Ax(t),u(t)), y(t) = Hx(t) (1)

where x [member of] X [subset] [R.sup.n], u [member of] U, y [member of] Y are the vectors of state, control, and output; F, G, and H are constant matrices; A is a row matrix; C is a column matrix: if C[i] [not equal to] 0, the right hand-side of the equation for the i-th component of the state vector contains the nonlinear term [phi](Ax(t),u(t)), C[i] = 0 otherwise. Denote the system (1) as [SIGMA] = (F,G,H,C,A).

There are many definitions of observability (Sontag, 1979), the following one is used in this paper. The system [SIGMA] is observable if for each states x([t.sub.0]) and x'([t.sub.0]) the control u(t), [t.sub.0] [less than or equal to] t < [infinity], exists such that H(x([t.sub.0]),u(t)) [not equal to] H(x'([t.sub.0]),u(t)) where H(x([t.sub.0]),t(t)) is an output sequence produced by the system in the initial state x([t.sub.0]) under the control u(t).

2. MAIN RELATIONSHIPS

Let T be a matrix realizing a linear homomorphism T: [SIGMA] [right arrow] [[SIGMA].sub.*] = ([F.sub.*],[G.sub.*],[H.sub.*],[C.sub.*],[A.sub.*]) according to

[for all]t [x.sub.*](t) = Tx(t)

where the system [[SIGMA].sub.*] is described by the equations

[x.sub.*](t+1) = [F.sub.*][x.sub.*](t) + [G.sub.*][x.sub.*](t) + [C.sub.*] x [phi]([A.sub.*][x.sub.*](t),u(t)), y(t) = [H.sub.*][x.sub.*](t).

It can be shown that the following relationships hold:

[F.sub.*]T = TF, [G.sub.*] = TG, H = [H.sub.*]T, [C.sub.*] = TC, (2)

A = [A.sub.*]T. (3)

If TC = 0, then [[SIGMA].sub.*] is a linear system.

The model (1) without the nonlinear term [phi](Ax(t),u(t)) will be named the linear part of the system [SIGMA].

3. OBSERVABILITY ANALYSIS

The main idea of the suggested approach is as follows: the observability of the linear part is checked and if it is unobservable, it is examined whether or not the nonlinear term violates this conclusion.

Suppose that

Rank(V) < n (4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is the observability matrix, that is the linear part is unobservable.

Denote by T the matrix of maximal rank made up from all linear independent rows of the matrix V. Consider the canonical decomposition of the unobservable linear part (Fig. 1) (Kwakernaak & Sivan, 1972). The e vector of the observable subsystem [[SIGMA].sub.*1] is formed by the matrix T as follows:

[x.sub.*1] = Tx.

The nonlinear term [phi](Ax(t),u(t)) is transformed by this matrix and is added to the decomposition as an additional connection between subsystems. Consider three cases.

1. An argument of the function [phi]([A.sub.*][x.sub.*](t),u(t)) is formed on the basis of the subsystem [[SIGMA].sub.*1] state vector and is independent of the subsystem [[SIGMA].sub.*2] state vector. In this case the additional connection does not violate the existing connection shown in Fig. 1. Sufficient condition to this is as follows: the row matrix A is a line combination of the matrix T rows, i.e. (3) holds for the certain matrix [A.sub.*]. This is equivalent to the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because rank(T) = rank(V), the last condition can be rewritten as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

2. Condition (5) does not hold. In this case the additional connection does not violate the existing connection if it is a member unobservable subsystem [[SIGMA].sub.*2] only. According to (2), the contribution of the nonlinear term in the subsystem [[SIGMA].sub.*1] is [C.sub.*] = TC. Therefore the condition above holds if

[C.sub.*] = TC = 0,

or

VC = 0. (6)

3. Conditions (5) and (6) do not e that the subsystem [[SIGMA].sub.*2] can be presented as shown in Fig. 2, and the argument of the nonlinear function [phi]([A.sub.*][x.sub.*](t),u(t)) is formed on the basis of the subsystems [[SIGMA].sub.*1] and [[SIGMA].sub.*21] state vectors. In this case, the subsystem [[SIGMA].sub.*22] remains unobservable in the presence of the additional connection.

To obtain such decomposition of the subsystem [[SIGMA].sub.*2], consider the variable z(t) = Ax(t) as a fictitious output of the system [SIGMA] and extend the real output y(t):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The extended observability matrix [V.sub.0] is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Clearly, condition (5) with V = [V.sub.0] holds, therefore if

rank([V.sub.0]) < n, (8)

then the decomposition of the subsystem [[SIGMA].sub.*2] shown in Fig. 2 is possible and the system [SIGMA] is unobservable.

Thus the sufficient condition of the system [SIGMA] unobservability are (4) and (5) or (4) and (6), or (8).

4. GENERALIZATIONS

1. The nonlinear function in (1) contains several terms [A.sub.1]x(t), [A.sub.2]x(t), [A.sub.r]x(t):

[phi]([A.sub.1]x(t),[A.sub.2]x(t), ..., [A.sub.r]x(t),u(t)),

[FIGURE 1 OMITTED]

for example, [phi](x) = sin[x.sub.1] x ln[x.sub.2]. In this case one has to obtain the matrix A in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

and used as shown above.

2. The nonlinear term in (1) is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e. there are r nonlinearities with the corresponding matrices [{[A.sub.i]}.sup.r.sub.1]; C[i,j] [not equal to] 0 if the right hand-side of the i-th component of the system [SIGMA] state vector contains the nonlinear term [[phi].sub.j]([A.sub.j]x(t),u(t)), C[i,j] = 0 otherwise; [[phi].sub.j] is a scalar nonlinear function. In this case one has to check condition (3) with the matrix A from (9). If it holds, then the system [SIGMA] is unobservable.

Assume that condition (5) does not hold for some row matrix [A.sub.j]; in this case one has to check condition (6) with the j-th column of the matrix C. If it holds, then the system [SIGMA] is unobservable.

Assume that condition (5) does not hold with the matrices [A.sub.j1], [A.sub.j2], ..., [A.sub.jk] and condition (6) does not hold with the [j.sub.1], [j.sub.2], ..., [j.sub.k] of the matrix C. Then the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

must be formed and used instead of the matrix A in (7) to obtain the matrix [H.sub.0]. Then the extended observability matrix [V.sub.0] must be obtained and used to check the system [SIGMA] observability.

5. CONCLUSION

It can be shown that the suggested approach works in the continuous-time case where the system is described as

[??](t) = Fx(t) + Gu(t) + C x [phi](Ax(t),u(t)). (10)

On the basis ofthe linear systems duality principle (Kwakernaak & Sivan, 1972), one can obtain similar results for the controllability problem for the systems (1) and (10).

Acknowledgement

The paper is supported by grants RFBR 05-08-18240 and 07-08-00102

6. REFERENCES

Isidori, A. (1989). Nonlinear control systems. London: Springer Verlag.

Kwakernaak H. and Sivan R. (1972). Linear optimal control systems.--New York: A Division of John & Sons, Inc.

Sontag E. (1979). On the Observability of Polynomial Systems. SIAM J. Control and Optimization. Vol.17. P.139-151.