ABSTRACT
The stability concept of Lyapunov [88] was established and mostly studied for the ISO systems. It has the following main qualitative characteristics:
It concerns the internal dynamical behavior, i.e., the state variation that is the motion, of the system.
It concerns the system behavior in the nominal regime in terms of total coordinates (e.g., I), i.e., in the free regime in terms of deviations (e.g., i, i = I – I N ), Subsection 3.5.1, e . g . , I ( t ) = I N ( t ) , i . e . , i ( t ) = 0 M , ∀ t ∈ T 0 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/eq1410.tif"/>
It treats the influence of nonzero initial conditions on the system dynamical behavior.
It concerns the system dynamical behavior over the unbounded time interval T 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/eq1411.tif"/> . If the system dynamical behavior is satisfactory over T 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/eq1412.tif"/> , then it is satisfactory also on any subinterval of T 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/eq1413.tif"/> .
It allows any permitted upper bound ε, ε ∈ ℜ + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/eq1414.tif"/> , of the instantaneous deviation x(t), or y(t), of a real total systems behavior X(t), or Y(t), from the total nominal behavior X d (t), or Y d (t), respectively.
It demands the existence of the appropriate upper bound δ, δ = δ ( ε ) ∈ ℜ + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/eq1415.tif"/> , of the initial deviation x 0, or y 0, of the total initial system behavior X 0, or Y 0, from the total initial desired behavior X d0, or Y d0.
240Analogously, a positive real number α specifies the arbitrarily requested lower bound of the instantaneous closeness of a real system behavior X(t), or Y(t), to the desired behavior X d (t), or Y d (t), where α = ε − 1 ∈ ℜ + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/eq1416.tif"/> .
A positive real number β = β(α) specifies the lower bound of the initial closeness of X 0, or Y 0, to X d0, or Y d0, which corresponds to α, where β = δ − 1 ∈ ℜ + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/eq1417.tif"/> .
If and only if for arbitrarily chosen permitted upper bound ε, there exists an appropriate upper bound δ of the initial deviation x 0, or y 0, such that the norm ║x 0║, or ║y 0║ less than δ guarantees that the norm ║x(t)║, or ║y(t)║, is less than the chosen permitted upper bound ε at every t ∈ T 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/eq1418.tif"/> , then the desired behavior X d (t), or Y d (t), respectively, is stable (the linear system is limiting stable, in other words the linear system is on the boundary of stability, equivalently: it is critically stable). See Fig. 13.1. If and only if additionally, δ → ∞ as ε → ∞, then the stability is global (in the whole). Equivalently, the zero vector deviation x = 0 n , or y = 0 N , is stable; see Fig. 13.2.
If and only if there exists a Δ-neighborhood of the initial desired behavior X d (0), or Y d (0), such that for every initial condition X 0, or Y 0, from the Δ-neighborhood, the corresponding system dynamical behavior asymptotically approaches the desired behavior X d (t), or Y d (t), as t → ∞, then the desired behavior X d (t), or Y d (t), is attractive; see Fig. 13.3. Equivalently, then, and only then, the zero vector deviation x = 0 n , or y = 0 N , is attractive; see Fig. 13.4. If and only if this 241 242holds for any initial conditions, i.e., for Δ = ∞, then the attraction is global, i.e., in the whole.
If the desired behavior X d (t), or Y d (t), is both stable and (globally) attractive, then it is (globally) asymptotically stable (which is called also asymptotically stable in the whole, the linear system is stable), respectively. Equivalently, then, and only then, the zero vector deviation x = 0 n , or y = 0 N , is (globally) asymptotically stable, respectively.
If and only if additionally the desired behavior X d (t), or Y d (t), is globally stable, then the desired behavior Y d (t) is strictly globally asymptotically stable. Equivalently, then, and only then, the zero vector deviation x = 0 n , or y = 0 N , is strictly globally asymptotically stable.
A stable total desired behavior <bold>X</bold> <sub> <italic>d</italic> </sub>(<italic>t</italic>), or <bold>Y</bold> <sub> <italic>d</italic> </sub>(<italic>t</italic>). https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/fig13_1.tif"/> The zero vector deviation <bold>x</bold> = <bold>0</bold> <sub> <italic>n</italic> </sub>, or <bold>y</bold> = <bold>0</bold> <sub> <italic>N</italic> </sub>, is stable. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/fig13_2.tif"/> The total desired behavior <bold>X</bold> <sub> <italic>d</italic> </sub>(<italic>t</italic>), or <bold>Y</bold> <sub> <italic>d</italic> </sub>(<italic>t</italic>), is attractive. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/fig13_3.tif"/> The zero vector deviation <bold>x</bold> = <bold>0</bold> <sub> <italic>n</italic> </sub>, or <bold>y</bold> = <bold>0</bold> <sub> <italic>N</italic> </sub>, is attractive. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315116556/90c0bd96-ab37-432d-97ec-3597bc845302/content/fig13_4.tif"/>