Stability, absolute

Global stability of the trivial solution of a non-linear system of ordinary differential equations (or equations of other type), uniform for all systems of a certain class. The term "absolute stability" assumes given a class of systems and an indication of the sense in which stability and uniformity are to be understood. Besides ordinary differential equations one also considers finite-difference equations, integral equations, ordinary differential equations with delay argument, and partial differential equations.

Consider the system described by the differential equation

(1)

and by a certain set of pairs of functions . Here are constant complex matrices of dimensions and , respectively; and are vectors of complex-valued functions of order and , respectively, where is locally summable and is absolutely continuous. In applications , , , are usually real, equation (1) describes the linear part of a system, while the set is determined by the properties of the non-linear blocks of the system. In simple cases there is one non-linear block, which is described by an equation

(2)

( and are scalar functions and is a -dimensional matrix; , , are real). In this case is the set of all pairs for which (2) holds.

Numerous studies of particular non-linear systems have led to the understanding that in the first place one should take in consideration a certain quadratic relation between and . For example, suppose that about the function in (2) it is known only that for all and ,

In this case is the set of all and for which almost-everywhere , where , or, otherwise,

(3)

Below, and is a Hermitian form on . In the general case one considers the class of all pairs of functions satisfying almost-everywhere the local constraint

(4)

as well as the class of pairs of functions satisfying the integral constraint

(5)

(the numbers depend on ). A variety of practically important non-linear blocks ( "air vents" , hysteresis non-linearity, impulse modulators of different types) satisfy a constraint (5), with a suitably chosen form .

Below it is assumed that equation (1) is controllable (cf. [1]), i.e. that the rank of the -dimensional matrix

equals , and also that the following condition of minimal stability is fulfilled: There exists an -dimensional matrix such that is a Hurwitz matrix (i.e. is stable) and

where is the form in (4) or (5). Let , be arbitrary matrices of orders and , respectively, , and form the "output" of the system (1):

(6)

One distinguishes between the real case, when all quantities in (1), (6) and the coefficients of are real, and the complex case, when they are generally complex. The set of all real satisfying (4) (or (5)) is denoted below by (respectively ). Put

The system (1) is called absolutely stable with respect to the output (6) in the class if there exist constants such that (1), (6) and imply that is finite and satisfies the estimate

(7)

Quadratic criteria for absolute stability. For the absolute stability of the system (1) with respect to output (6) in the class (in the real case in the class ) it is necessary and sufficient that

(8)

for all complex , , , and real connected by the relations

(9)

If (8), (9) hold, then one can take in (7) , where the numbers , do not depend on in (5). If and (4) is satisfied as well as (8) (for ), then one has global exponential stability:

(10)

for all , .

Suppose that for all (where is the -dimensional unit matrix). For the absolute stability of the system (1) with respect to the output in the class it is necessary and sufficient that for any , , and any complex , the inequality

(11)

holds. For the class an analogous assertion is true only relative to sufficiency. Necessary and sufficient conditions for the absolute stability in the class are known only for special forms , and an effectively verifiable condition only for (cf. [3], [7]).

From relation (9) it follows that

the element of the matrix is called the frequency characteristic from input to output . Criteria establishing certain properties of the system expressible by the frequency characteristics are called frequency stability criteria. The merit of frequency criteria lies in their usefulness in practical applications and in their invariance under a transformation (, ) of the system (1).

In the real case with , for the class defined by the relation (3) condition (11) reduces to the form

(12)

where is the frequency characteristic from input to output . The frequency criterion (12) (circle criterion) means that the frequency characteristic , , is non-intersecting with the circle with centre at the point and passing through the points , . The condition of minimal stability in this case means asymptotic stability of the linear system (1) with , for some . The criterion (12) is the natural extension of the Mikhailov–Nyquist criterion for non-linear systems (cf. Mikhailov criterion; Nyquist criterion).

Historically, the first frequency criterion of absolute stability for non-linear systems was Popov's criterion for and the class of stationary non-linearities , where (cf. [2]). It has the form:

(13)

The condition of minimal stability in this case is equivalent to requiring the matrix in (1) to be a Hurwitz matrix. This criterion can be simply verified in a geometrical manner.

There exists a definite connection between the frequency criteria (6), (12), (13), and others and the existence of a global Lyapunov function. The frequency criteria of absolute stability usually cover all criteria which can be obtained by means of a Lyapunov function in certain multi-parameter classes of functions. For example, the criterion (12) is a necessary and sufficient condition for the existence of a function

( is an -dimensional matrix, where is the sign for Hermitian conjugation) such that its derivative along the trajectories of the systems (1), (2) with an arbitrary non-linearity (2) (for which ) satisfies the condition

Similarly, Popov's frequency condition (13) includes all criteria which can be established using Lyapunov functions of the form

Many other frequency criteria for absolute stability are known for different classes of non-linearities (cf. [3]–[6]). In particular, they cover many important cases in applications, such as non-unique equilibrium positions (cf. [1]). Frequency criteria of absolute stability allow one to distinguish classes of non-linear systems of a general form for which the fact of global stability is rather simple to establish. E.g., the system (1) with , , , (i.e. an arbitrary system of order at most 3 with a single non-linearity), is globally asymptotically stable if and if any linear system with , , is asymptotically stable. For systems of order 4 (or higher) an analogous assertion is incorrect. Moreover, for , there exists a system (1) and a non-linearity , , such that the matrix of any linearized system with , , is a Hurwitz matrix, while the non-linear system has a periodic solution.

After replacing the condition of minimal stability by the analogous condition of minimal instability, the inequalities (8), (11), (12), (13) become criteria of absolute instability (with a corresponding meaning for the last term). For example, consider the real case with , let the matrix of coefficients of the system (1) with , (i.e. the matrix ) for some , , have eigenvalues in the half-plane , and let the frequency condition (12) be satisfied. Then the system (1), (2) with function satisfying the condition (as well as the system (1), (3)) possesses a solution for which

where the constants , are the same for all systems of the class considered. The corresponding vectors fill a cone , where is a matrix having negative eigenvalues.

Similarly, for the condition (13) is a frequency criterion for the absolute instability of the system (1) with in the class of stationary non-linearities , where , if in (1) the matrix has an eigenvalue in the half-plane .

In the theory of absolute stability there are similar frequency criteria for dissipation, convergence, existence of periodic motions (self-oscillations and forced regimes), and others (cf. e.g. [3], [5] and the references in [1], [3], [5]; see also [8]–[10]).

References

[1]	A.Kh. Gelig, "Stability of non-linear systems with non-unique equilibrium positions" , Moscow (1978) (In Russian)
[2]	M.A. Aizerman, F.R. Gantmakher, "Absolute stability of non-linear control sytems" , Moscow (1963) (In Russian)
[3]	V.A. Yakubovich, , Methods for studing non-linear systems of automatic control , Moscow (1975) pp. 74–180 (In Russian)
[4]	V.M. Popov, "Hyperstability of control systems" , Springer (1973) (Translated from Russian)
[5]	A.A. Voronov, "Stability, controllability, observability" , Moscow (1979) (In Russian)
[6]	V. Rezvan, "Absolute stability of automatic systems with delay" , Moscow (1983) (In Rumanian)
[7]	E.S. Pyatnitskii, "New research on the absolute stability of automatic control systems" Automat. Remote Control : 6 (1968) pp. 885–881 Avtomatika i Telemekhanika : 6 (1968) pp. 5–36
[8]	D.D. Ŝiljak, "Nonlinear systems. Parameter analysis and design" , Wiley (1969)
[9]	K.S. Narendra, I.H. Taylor, "Frequency domain criteria for absolute stability" , Acad. Press (1973)
[10]	J.L. Willems, "Stability theory of dynamical systems" , Nelson (1970)

References

[a1]	J.C. Willems, "Least squares stationary optimal control and the algebraic Riccati equation" IEEE Trans. Aut. Control , AC-16 (1971) pp. 621–634
[a2]	J. La Salle, S. Lefschetz, "Stability by Lyapunov's direct method with applications" , Acad. Press (1961)