16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 14 Last time: w(t, r)=w(t-T) ()=」v(t-r)x( Let y(r=w(r)x(t-t)dr For the differential system characterized by its equations of state, specialization to invariance means that the system matrices A, B, C are constants. =Cx For AB. constant. y(1)=Cx(D) x()=Φ(t-1)x()+」c(t-)B(ldr The transition matrix can be expressed analytically in this ca ap(t, r)= A(t, r), where (r,r) This is a matrix form of first order constant coefficient differential equation. The solution is the matrix exponential +A(t-)+A(t-r)2+…+4(1-r) Useful for computing a(t) for small enough t-T Page 1 of 6
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 6 Lecture 14 Last time: wt wt (, ) ( ) τ ⇒ −τ 0 () ( ) ( ) ' Let: () ( ) ( ) t yt wt x d t d d yt w xt d τ τ τ τ τ τ τ τ τ τ −∞ ∞ = − ⎧ = − ⎨ − = ′ ⎩ = − ′ ′′ ∫ ∫ For the differential system characterized by its equations of state, specialization to invariance means that the system matrices ABC , , are constants. x Ax Bu y Cx = + = For ABC , , constant: 0 0 0 () () () ( ) ( ) ( ) ( ) t t y t Cx t x t t t x t t Bu d τ τ τ = =Φ − + Φ − ∫ The transition matrix can be expressed analytically in this case. ( , ) ( , ), where ( , ) d t At I dt Φ =Φ Φ = τ τ ττ This is a matrix form of first order, constant coefficient differential equation. The solution is the matrix exponential. ( ) () 2 2 (, ) 1 1 ( ) ( ) ... ( ) ... 2 ! A t At k k t e e I At A t A t k τ τ τ ττ τ − − Φ = =+ − + − + + − + Useful for computing Φ( )t for small enough t −τ
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
16.322 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 ∞ = ∫
16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Stationary statistics Invariant output statistics implies more than stationary inputs and invariant systems; it also implies that the system has been in operation long enough under the present conditions to have exhausted all transients Input-Output Relations for Correlation and Spectral Density Functions Derive autocorrelation of output in terms of autocorrelation of input y()=」v(r)x(-r)dr y=w(t,)xdt, x w(r)dr, R(r)=yoy(t+r) jdr w(r dt w([)x()x(+T-T2) jdr w(r dt w(t2)R(r+T, -T2) ∫drn(z)dzw2)R2(-z2 R2(r)=x()y(t+r) x(o]w(r)x(+T-T,)dr, ∫v(r)R2(a T-tdT Page 3 of 6
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 6 Stationary statistics Invariant output statistics implies more than stationary inputs and invariant systems; it also implies that the system has been in operation long enough under the present conditions to have exhausted all transients. Input-Output Relations for Correlation and Spectral Density Functions Derive autocorrelation of output in terms of autocorrelation of input 1 11 0 1 1 0 1 1 0 () ( ) ( ) ( ) ( ) yt w xt d y w xd xw d τ τ τ τ τ τ τ ∞ ∞ ∞ = − = = ∫ ∫ ∫ 11 2 2 1 2 0 0 11 2 2 12 0 0 ( ) () ( ) ( ) ( )( )( ) () () ( ) yy xx R yt yt d w d w xt xt dw dw R τ τ τ τ τ τ τ ττ τ τ τ τ ττ τ ∞ ∞ ∞ ∞ = + = − +− = +− ∫ ∫ ∫ ∫ 2 11 2 2 12 0 0 () () ( ) xx y dw dw R τ τ τ τ ττ ∞ ∞ = − ∫ ∫ 1 11 0 1 11 0 ( ) () ( ) () ( ) ( ) () ( ) xy xx R xt yt x t w xt d wR d τ τ τ ττ τ τ ττ τ ∞ ∞ = + = +− = − ∫ ∫
16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Transform to get power density spectrum of output. 下=x(dr F(0)x y(o)=」R(r)edr dr dr, w( dr w(t2)R(r+, -T2)e /er j drR(r+T, -T2 )e et+T-a]dr, w(t) ot dr w(t2)e era In first integral only, let Ir=T+T,-T2 Idr'=dz S,(o)=dr'Rm(r)e"er j dr w(r, Jeer dr, w(r2 )e" jera =Sx()F(-)F() F(o)'S( The power spectral density thus does not depend upon phase properties The cross-spectral density function can be derived similarly, to obtain y(o)=F(o)S(o) Mean squared output in time and frequency domain =R, (0)=dr, w(r dt w(t2)R(c -T2) s(ode F(OF(oS(o)de Generally speaking, with linear invariant systems it is easier to work in the transform domain than the time domain -so we shall commonly use the last expression to calculate the mean squared output of a system. However, control engineers are more accustomed to working with Laplace transforms than with Fourier transforms. By making the change of variables s= jo we can cast this expression in that form Page 4 of 6
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 6 Transform to get power density spectrum of output. 1 12 1 2 0 11 2 2 12 0 0 ( ) 12 11 2 2 0 0 first integral ( ) (0) ( ) () ) () ( ) ( ) () () In first integral o j yy yy j xx j jj xx y xw d F x S R ed d dw dw R e dR e d w e d w e ωτ ωτ ω τ τ τ ωτ ωτ τ τ ω ττ τ τ τ τ τ ττ τ τ ττ τ τ τ τ τ ∞ ∞ − −∞ ∞∞ ∞ − −∞ ∞ ∞∞ − +− − −∞ = = = = ( +− = +− ∫ ∫ ∫∫ ∫ ∫ ∫∫ 1 2 1 2 11 2 2 2 = + nly, let () () () ( ) ()( )() () () j j j yy xx xx xx d d S dR e dw e dw e SFF F S ωτ ωτ ωτ τ ττ τ τ τ ω τ τ ττ ττ ω ωω ω ω ∞ ∞∞ − ′ − −∞ −∞ −∞ ⎧ ′ − ⎨ ⎩ ′ = = ′ ′ = − = ∫ ∫∫ The power spectral density thus does not depend upon phase properties. The cross-spectral density function can be derived similarly, to obtain: () () () xy xx S FS ω = ω ω Mean squared output in time and frequency domain 2 11 2 2 12 0 0 (0) ( ) ( ) ( ) 1 ( ) 2 1 ()( ) () 2 yy xx yy xx y R dw dw R S d FF S d τ τ τ τ ττ ω ω π ω ω ωω π ∞ ∞ ∞ −∞ ∞ −∞ == − = = − ∫ ∫ ∫ ∫ Generally speaking, with linear invariant systems it is easier to work in the transform domain than the time domain – so we shall commonly use the last expression to calculate the mean squared output of a system. However, control engineers are more accustomed to working with Laplace transforms than with Fourier transforms. By making the change of variables s j = ω we can cast this expression in that form
16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde 2jT F(SF(s)Si(s)ds We know that S(o) is even. If it is a rational function of @, and we will work exclusively with rational spectra, it is then a rational function of a. So only even powers of o appear in S(o) and thus Sa= which we may call S(s)is derived from S(o) by replacing o by -s F'(s)is the ordinary transfer function of the system -the Laplace transform of its weighting function. Because w(0)=0, I
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 6 2 1 2 1 () ( ) () 2 j xx j xx s s s ds y FF S j j jj F s F s S s ds j π π ∞ − ∞ ∞ −∞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ = − ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ = − ′′ ′ ∫ ∫ We know that ( ) xx S ω is even. If it is a rational function of ω , and we will work exclusively with rational spectra, it is then a rational function of 2 ω . So only even powers of ω appear in ( ) xx S ω and thus xx s S j ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ which we may call ( ) xx S s is derived from ( ) xx S ω by replacing 2 ω by 2 −s . F s ′( )is the ordinary transfer function of the system – the Laplace transform of its weighting function. Because wt t ( ) 0, 0 = ∫ F s ds j F s = π ∑ If F s( ) has a pole of order m at z a = , ( ) 1 1 1 Res( ) ( ) ( ) 1 ! m m m s a d a s a Fs m ds − − = ⎧ ⎫ = − ⎨ ⎬ ⎡ ⎤ ⎣ ⎦ − ⎩ ⎭ F s( ) has a pole of order m at s a = if m is the smallest integer for which
16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde is finite If F(s) is rational and has a Ist order pole at a F(s) D(s) hen Res(a)m=l=lim[(s-a)F(S)I (a-ba-c Page 6 of 6
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 6 lim ( ) ( )m s a s a Fs → ⎡ ⎤ − ⎣ ⎦ is finite. If F s( ) is rational and has a 1st order pole at a , ( ) ( ) ( ) ( ) ( )( )... N s F s D s N s s as b = = − − then Res( ) lim ( ) ( ) 1 [ ] ( ) ( )( )... m s a a s aFs N a a ba c = → = − = − −