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The vector function Φ (t, t0 ) is the transition matrix. Demonstration of the steps for obtaining this solution may be performed using the method of constants variation. 66): t y(t) = C(t) ⋅ Φ (t, t0 ) ⋅ x(t0 ) + ∫ C(t) ⋅ Φ (t, τ ) ⋅ B(τ ) ⋅ u(τ ) ⋅ dτ + D(t) ⋅ u(t). 70) t yzero-state (t) = ∫ C(t) ⋅ Φ (t, τ ) ⋅ B(τ ) ⋅ u(τ ) ⋅ dτ + D(t) ⋅ u(t). 71) t0 The term yzero-input is named zero-input response and represents the response of the system having the input equal to zero but nonzero initial conditions.
Definition. 28), to be BIBO stable are: (i) the degree of the P polynomial is less than or equal to the degree of the Q polynomial, (ii) the poles of the system have strictly negative real part, (iii) the system does not have any characteristic roots with strictly positive real part that cancel with zeros, (iv) every characteristic root cancelling with a zero and having a zero real part is only of first order of multiplicity. Definition. 28) are (i) the degree of the P polynomial is less than or equal to the degree of the Q polynomial, (ii) the characteristic roots of the Q polynomial have strictly negative real part.
It may be noticed that as ε →0 the v(t) function approaches the unit step function and the δ0 (t) function approaches the form of an impulse with an infinitely small 2 Basics of systems theory | 25 δ0 (t) 1/Ɛ 0 Fig. 12. Signal δ0 (t) having the form of a unit impulse with finite duration and amplitude. t Ɛ duration (approaching zero) and an infinite amplitude, but having a finite and equal to unity area. 21) ε →0 and having the representation shown in Fig. 13. δ(t) 0 Fig. 13. The elementary unit impulse δ (t) also named the Dirac impulse.