16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde The solution is y(1)=Cx(1) x(=e(-x()+e(-rBu(o)dr For→∞ x(1)=「e4=)B(r)dr =「e"B(t-r)dr and for a single input, single output (SISO)system, ()=ce If x(t)=e for all past time F(x(t Since w(r)=0 for t<0 for a realizable system, we see that the steady state inusoidal response function, F(o), for a system is the Fourier transform of the weighting function- where the coefficient unity must be used F(o)=w(r)e - dr and w(r) for a stable system is Fourier transformable Then (1)= F(oe/do You can compute the response to any input at all, including transient responses having defined F(o) for all frequencies The static sensitivity of the system is the zero frequency gain, F(O), which is just the integral of the weighting function F(0)=w(r)dr Page 2 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 6 The solution is 0 0 ( ) ( ) 0 () () () ( ) () t At t A t t y t Cx t x t e x t e Bu t d τ τ − − = = + ∫ For 0t → ∞ : ( ) 0 () ( ) ( ) t A t A x t e Bu d e Bu t d τ τ τ τ τ τ − −∞ ∞ ′ = = − ′ ′ ∫ ∫ and for a single input, single output (SISO) system, ( ) T At wt c e b = If ( ) j t x t e ω = for all past time ( ) 0 0 () ( ) ( ) ( ) () j t j jt yt w e d we de F xt ω τ ωτ ω τ τ τ τ ω ∞ − ∞ − = ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ = ∫ ∫ Since w() 0 τ = for τ < 0 for a realizable system, we see that the steady state sinusoidal response function, F( ) ω , for a system is the Fourier transform of the weighting function – where the coefficient unity must be used. ( ) () j F we d ωτ ω τ τ ∞ − −∞ = ∫ and w( ) τ for a stable system is Fourier transformable. Then 1 () ( ) 2 j t wt F e d ω ω ω π ∞ −∞ = ∫ You can compute the response to any input at all, including transient responses, having defined F( ) ω for all frequencies. The static sensitivity of the system is the zero frequency gain, F(0), which is just the integral of the weighting function. 0 F wd (0) ( ) τ τ ∞ = ∫