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

$$ \tag{1 } \dot{x} ( t) = Ax ( t) + B \xi ( t) $$

and by a certain set $ \mathfrak M $ of pairs of functions $ \{ x ( \cdot ), \xi ( \cdot ) \} $. Here $ A, B $ are constant complex matrices of dimensions $ N \times N $ and $ N \times n $, respectively; $ x ( t) $ and $ \xi ( t) $ are vectors of complex-valued functions of order $ N $ and $ n $, respectively, where $ \xi ( t) $ is locally summable and $ x ( t) $ is absolutely continuous. In applications $ A $, $ B $, $ x ( t) $, $ \xi ( t) $ are usually real, equation (1) describes the linear part of a system, while the set $ \mathfrak M $ 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

$$ \tag{2 } \xi ( t) = \phi [ \sigma ( t), t] \ \textrm{ where } \ \sigma ( t) = Cx ( t) $$

( $ \sigma ( t) $ and $ \xi ( t) $ are scalar functions and $ C $ is a $ ( 1 \times N) $- dimensional matrix; $ \sigma ( t) $, $ \xi ( t) $, $ C $ are real). In this case $ \mathfrak M $ is the set of all pairs $ \{ x ( t), \xi ( t) \} $ 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 $ \xi ( t) $ and $ x ( t) $. For example, suppose that about the function $ \phi ( \sigma , t) $ in (2) it is known only that for all $ t \geq 0 $ and $ \sigma $,

$$ \mu _ {1} \leq { \frac{\phi ( \sigma , t) } \sigma } \leq \mu _ {2} . $$

In this case $ \mathfrak M = \mathfrak M [ \mu _ {1} , \mu _ {2} ] $ is the set of all $ x ( t) $ and $ \xi ( t) $ for which almost-everywhere $ \mu _ {1} \leq \xi ( t)/ \sigma ( t) \leq \mu _ {2} $, where $ \sigma ( t) = Cx ( t) $, or, otherwise,

$$ \tag{3 } [ \mu _ {2} \sigma ( t) - \xi ( t)] [ \xi ( t) - \mu _ {1} \sigma ( t)] \geq 0. $$

Below, $ n \geq 1 $ and $ F ( x, \xi ) $ is a Hermitian form on $ \mathbf C ^ {N} \times \mathbf C ^ {n} $. In the general case one considers the class $ \mathfrak M _ {F,L} $ of all pairs $ \{ x ( t), \xi ( t) \} $ of functions satisfying almost-everywhere the local constraint

$$ \tag{4 } F [ x ( t), \xi ( t)] \geq 0, $$

as well as the class $ \mathfrak M _ {F,I} ( \gamma ) $ of pairs of functions $ x ( t), \xi ( t) $ satisfying the integral constraint

$$ \tag{5 } \exists T _ {k} \rightarrow \infty :\ \int\limits _ { 0 } ^ { {T _ k} } F [ x ( t), \xi ( t)] dt \geq - \gamma $$

(the numbers $ T _ {k} $ depend on $ x ( \cdot ), \xi ( \cdot ) $). 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 $ F ( x, \xi ) $.

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

$$ ( B, AB \dots A ^ {N - 1 } B ) $$

equals $ N $, and also that the following condition of minimal stability is fulfilled: There exists an $ ( n \times N ) $- dimensional matrix $ R $ such that $ A + BR $ is a Hurwitz matrix (i.e. is stable) and

$$ F ( x, Rx) \geq 0 \ \textrm{ for } \textrm{ any } x, $$

where $ F $ is the form in (4) or (5). Let $ D $, $ E $ be arbitrary matrices of orders $ m \times N $ and $ m \times n $, respectively, $ \| D \| + \| E \| \neq 0 $, and form the "output" of the system (1):

$$ \tag{6 } \eta ( t) = Dx ( t) + E \xi ( t). $$

One distinguishes between the real case, when all quantities in (1), (6) and the coefficients of $ F ( x, \xi ) $ are real, and the complex case, when they are generally complex. The set of all real $ x ( \cdot ), \xi ( \cdot ) $ satisfying (4) (or (5)) is denoted below by $ \mathfrak M _ {F,L} ^ \partial $( respectively $ \mathfrak M _ {F,I} ^ \partial ( \gamma ) $). Put

$$ \| \eta ( \cdot ) \| ^ {2} = \ \int\limits _ { 0 } ^ \infty | \eta ( t) | ^ {2} dt. $$

The system (1) is called absolutely stable with respect to the output (6) in the class $ \mathfrak M $ if there exist constants $ C _ {1} , C _ {2} \geq 0 $ such that (1), (6) and $ [ x ( \cdot ), \xi ( \cdot )] \in \mathfrak M $ imply that $ \| \eta ( \cdot ) \| $ is finite and satisfies the estimate

$$ \tag{7 } \| \eta ( \cdot ) \| ^ {2} \leq \ C _ {1} | x ( 0) | ^ {2} + C _ {2} . $$

Quadratic criteria for absolute stability. For the absolute stability of the system (1) with respect to output (6) in the class $ \mathfrak M _ {F,I} ( \gamma ) $( in the real case in the class $ \mathfrak M _ {F,I} ^ \partial ( \gamma ) $) it is necessary and sufficient that

$$ \tag{8 } \exists \delta > 0 :\ F ( \widetilde{x} , \widetilde \xi ) \leq - \delta | \widetilde \eta | ^ {2} $$

for all complex $ \widetilde{x} $, $ \widetilde \xi $, $ \widetilde \eta $, and real $ \omega $ connected by the relations

$$ \tag{9 } i \omega \widetilde{x} = A \widetilde{x} + B \widetilde \xi ,\ \ \widetilde \eta = D \widetilde{x} + E \widetilde \xi . $$

If (8), (9) hold, then one can take in (7) $ C _ {2} = C _ {2} ^ \prime \gamma $, where the numbers $ C _ {1} $, $ C _ {2} ^ \prime $ do not depend on $ \gamma $ in (5). If $ \eta ( t) = x ( t) $ and (4) is satisfied as well as (8) (for $ \widetilde \eta = \widetilde{x} $), then one has global exponential stability:

$$ \tag{10 } \exists C , \epsilon > 0 :\ | x ( t) | \leq Ce ^ {- \epsilon ( t - t _ {0} ) } | x ( t _ {0} ) | $$

for all $ x ( \cdot ) $, $ t \geq t _ {0} $.

Suppose that $ \mathop{\rm det} ( A - i \omega I ) \neq 0 $ for all $ \omega $( where $ I $ is the $ ( N \times N) $- dimensional unit matrix). For the absolute stability of the system (1) with respect to the output $ \eta ( t) = [ x ( t), \xi ( t)] $ in the class $ \mathfrak M _ {F,L} $ it is necessary and sufficient that for any $ \omega $, $ - \infty \leq \omega \leq + \infty $, and any complex $ \widetilde \xi \neq 0 $, the inequality

$$ \tag{11 } F [ ( A - i \omega I ) ^ {-} 1 B \widetilde \xi , \widetilde \xi ] < 0 $$

holds. For the class $ \mathfrak M _ {F,L} ^ \partial $ an analogous assertion is true only relative to sufficiency. Necessary and sufficient conditions for the absolute stability in the class $ \mathfrak M _ {F,L} ^ \partial $ are known only for special forms $ F $, and an effectively verifiable condition only for $ N = 2 $( cf. [3], [7]).

From relation (9) it follows that

$$ \widetilde \eta = \ W ^ {( \eta ) } ( i \omega ) \widetilde \xi \ \ \textrm{ where } \ W ^ {( \eta ) } ( i \omega ) = E + D ( i \omega I - A ) ^ {-} 1 B; $$

the element $ W _ {jk} ^ {( \eta ) } $ of the matrix $ W ^ {( \eta ) } ( i \omega ) $ is called the frequency characteristic from input $ \xi _ {k} $ to output $ \eta _ {j} $. 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 $ x ^ \prime = Sx $( $ S = \textrm{ const } $, $ \mathop{\rm det} S \neq 0 $) of the system (1).

In the real case with $ n = 1 $, for the class $ \mathfrak M [ \mu _ {1} , \mu _ {2} ] $ defined by the relation (3) condition (11) reduces to the form

$$ \tag{12 } \mathop{\rm Re} \{ [ \mu _ {2} \overline{ {W ( i \omega ) }}\; - 1] [ 1 - \mu _ {1} W ( i \omega )] \} > 0, $$

where $ W ( i \omega ) = C ( A - i \omega I) ^ {-} 1 B $ is the frequency characteristic from input $ \xi ( t) $ to output $ [- \sigma ( t)] $. The frequency criterion (12) (circle criterion) means that the frequency characteristic $ W ( i \omega ) $, $ - \infty \leq \omega \leq + \infty $, is non-intersecting with the circle with centre at the point $ (- \mu _ {1} ^ {-} 1 - \mu _ {2} ^ {-} 1 )/2 $ and passing through the points $ (- \mu _ {1} ^ {-} 1 ) $, $ (- \mu _ {2} ^ {-} 1 ) $. The condition of minimal stability in this case means asymptotic stability of the linear system (1) with $ \xi = \mu \sigma $, $ \sigma = Cx $ for some $ \mu \in [ \mu _ {1} , \mu _ {2} ] $. 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 $ n = 1 $ and the class $ M $ of stationary non-linearities $ \xi ( t) = \phi [ \sigma ( t)] $, where $ 0 \leq \sigma \phi ( \sigma ) \leq \mu _ {0} \sigma ^ {2} $( cf. [2]). It has the form:

$$ \tag{13 } \exists \theta :\ \mu _ {0} ^ {-} 1 + \mathop{\rm Re} W ( i \omega ) + \theta \mathop{\rm Re} [ i \omega W ( i \omega )] > 0, $$

$$ 0 \leq \omega \leq \infty . $$

The condition of minimal stability in this case is equivalent to requiring the matrix $ A $ 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

$$ V ( x) = x ^ {*} Hx $$

( $ H = H ^ {*} = \textrm{ const } $ is an $ ( N \times N) $- 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 $ \mu _ {1} \leq \phi ( \sigma , t)/ \sigma \leq \mu _ {2} $) satisfies the condition

$$ \frac{dV ( x) }{dt } < 0 \ \textrm{ for } x \neq 0. $$

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

$$ V ( x) = \ x ^ {*} Hx + \theta \int\limits _ { 0 } ^ \sigma \phi ( \sigma ) d \sigma . $$

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 $ \xi = \phi ( \sigma ) $, $ \sigma = Cx $, $ N \leq 3 $, $ n = 1 $( i.e. an arbitrary system of order at most 3 with a single non-linearity), is globally asymptotically stable if $ \mu _ {1} \leq \phi ^ \prime ( \sigma ) \leq \mu _ {2} $ and if any linear system with $ \xi = \mu \sigma $, $ \mu _ {1} \leq \mu \leq \mu _ {2} $, is asymptotically stable. For systems of order 4 (or higher) an analogous assertion is incorrect. Moreover, for $ N \geq 4 $, $ n = 1 $ there exists a system (1) and a non-linearity $ \xi = \phi ( \sigma ) $, $ \mu _ {1} \leq \phi ^ \prime ( \sigma ) \leq \mu _ {2} $, such that the matrix of any linearized system with $ \xi = \mu \sigma $, $ \mu _ {1} \leq \mu \leq \mu _ {2} $, 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 $ n = 1 $, let the matrix of coefficients of the system (1) with $ \xi = \mu \sigma $, $ \sigma = Cx $( i.e. the matrix $ A + B \mu C $) for some $ \mu $, $ \mu _ {1} \leq \mu \leq \mu _ {2} $, have $ k \geq 1 $ eigenvalues in the half-plane $ \mathop{\rm Re} \lambda > 0 $, and let the frequency condition (12) be satisfied. Then the system (1), (2) with function $ \phi ( \sigma , t) $ satisfying the condition $ \mu _ {1} \leq \phi ( \sigma , t)/ \sigma \leq \mu _ {2} $( as well as the system (1), (3)) possesses a solution $ x ( t) $ for which

$$ | x ( t) | \geq Ce ^ {\epsilon t } | x ( 0) | \ \ \textrm{ for } t \geq 0, $$

where the constants $ C > 0 $, $ \epsilon > 0 $ are the same for all systems of the class considered. The corresponding vectors $ x ( 0) $ fill a cone $ x ( 0) ^ {*} Hx ( 0) < 0 $, where $ H = H ^ {*} $ is a matrix having $ k $ negative eigenvalues.

Similarly, for $ n = 1 $ the condition (13) is a frequency criterion for the absolute instability of the system (1) with $ \sigma = Cx $ in the class of stationary non-linearities $ \xi ( t) = \phi [ \sigma ( t)] $, where $ 0 \leq \sigma \phi ( \sigma ) \leq \mu _ {0} \sigma ^ {2} $, if in (1) the matrix $ A $ has an eigenvalue in the half-plane $ \mathop{\rm Re} \lambda > 0 $.

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]).

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