OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 D (p) = a0p n + a 1p n–1 + …+ a n-1p + an (11) where n – the degree of polynomial and it is also the order of the differential equation, for the case represented by the expression (8), (10), n = 8 . Assuming p = jω , we transform the characteristic polynomial into a complex frequency polynomial: D (jω) = a0(jω) n + a 1(jω) n-1 +…+ a n-1(jω) + an . In case of stable systems, the hodograph of the Mikhailov vector has the property of starting from the point U(0) = an, V(0) = 0. As ω increases from zero to infinity, the point M(U,V) moves to the left so that the curve tends to cover the origin, while moving away from it. If we draw the radius vector from the origin to the point M(U,V), it turns out that the radius vector rotates counterclockwise, continuously increasing. The Mikhailov criterion itself is formulated as follows: when the frequency changes from zero to infinity, the Mikhailov hodograph begins on the real axis at point an, sequentially passes counterclockwise n quadrants of the complex plane without passing through zero, and goes to infinity in the nth quadrant, the system is stable. In case of unstable systems, the curves do not cover the origin, while if the hodograph starts from the origin or passes through the origin, the system is on the stability boundary. To assess the stability of the control system by the Mikhailov method, it is necessary to determine the characteristic polynomial of the control system, described by the system of equations (8), (10). As this system is nonlinear, the first thing that is required is to linearize this system of equations in some vicinity of the equilibrium point, which is done below. 2 11 12 13 11 1 3 2 2 12 1 3 13 1 1 2 21 22 23 21 2 3 2 2 (1 )( ) cos( ) ( ) cos( ) 0, [ (1 )( ) sin( ) ] [ v v jT p p p p jT p d x dx dy dz m h h h x c e h t dt dt dt dt y c S h c z Q t S d y dx dy dz m h h h x c e h t dt dt dt dt y c − ω − ω + + + + + χ − ρ + ρµ + ϕ σ α + + + χ ρ + ρµ + ϕ σ + + χ ρµα = + + + + + χ − ρ + ρµ + φ σ α + + 22 2 3 23 2 1 2 31 32 33 31 3 0 3 2 2 32 3 0 3 33 3 1 1 2 3 ( ) sin( ) ] 0, (1 )( ) ( ) ( ) ( ) ( ) 0, 2 ( v p p p jT t t p t p t t p p f f p S h c z Q t S d z dx dy dz m h h h x c e k k h t dt dt dt dt k y c S k k h c z Q t S t h T − ω + χ ρ + ρµ + φ σ + + χ ρµα = + + + + + χ − ρ + ρµ + + ∆ σ α + ∆ + + χ ρ + ρµ + + ∆ σ + + χ ρµα + α − α σ = 2 1 2 1 2 0 3 0 0 3 2 1 3 1 0 3 3 0 0 3 2 0 3 1 0 3 2 ) ( ) ( ) ( ) ( ) ( ) (1 )( ) ( ) (1 )( ) ( ) v v v p p t t p jT p p t t p jT c t t p c jT c t t d Q dQ dz T T T k t S k k k h t dt dt dt dz k t S k k k h t e dt x e V k k k h t V k x e V k k k h t − ω − ω − ω + + + χ ρ + ρµ + + ∆ σ + + χ ρ + ρµ + + ∆ σ + + χ − ρ + ρµ + + ∆ σ α + + χ − ρ + ρµ + + ∆ σ α 1 3 0 3 3 1 0 1 3 3 1 1 2 3 3 1 1 1 2 3 1 ( ) ( ) ( ) ( ) 1 ( ) 2 ( ) 0, 2 v v v jT p c jT p c t t c p c t t c t p p c f f p c jT t p p c f f p c V k e y S kV k k kV h y S k V k k k V h e k Q t S kV t h kV k Qe t S k V t h k V − ω − ω − ω + + χ ρ + ρµ + + ∆ σ + χ ρ + ρµ + + ∆ σ + ∆ + + χ ρµα + α − α σ + ∆ + χ ρµα + α − α σ = (12)
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