16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 6 Lecture 14 Last time: (, ) ( )wt wtτ τ?? 0 () ( ) ( ) ' Let: () ( ) ( ) t yt wt x d t dd yt w xt d τ ττ ττ ττ τ ττ ?∞ ∞ =? =? ? ? ′?= ? ′′′=? ∫ ∫ For the differential system characterized by its equations of state, specialization to invariance means that the system matrices ,,ABC are constants. x Ax Bu yCx =+ =  For ,,ABC constant: 0 00 () () () ( ) ( ) ( ) ( ) t t yt Cxt x tttxt tBudτ ττ = =Φ ? + Φ ? ∫ The transition matrix can be expressed analytically in this case. ( , ) ( , ), where ( , ) d tAt I dt ττ τΦ=Φ Φ= This is a matrix form of first order, constant coefficient differential equation. The solution is the matrix exponential. () () 2 2 (, ) 11 ( ) ( ) ... ( ) ... 2! At At k k te eIAt At At k τ τ τ ττ τ ? ? Φ= =+ ? + ? ++ ? + Useful for computing ()tΦ for small enough t τ? . 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 6 The solution is 0 0 () () 0 () () () ( ) () t At t At t yt Cxt x te xt e Butd τ τ ? ? = =+ ∫ For 0 t →∞: () 0 () ( ) () t At A x teBud eBut d τ τ τ τ τ τ ? ?∞ ∞ ′ = ′′=? ∫ ∫ and for a single input, single output (SISO) system, () TAt wt ce b= If () jt x te ω = for all past time () 0 0 () ( ) () ()() jt jjt yt w e d we de Fxt ωτ ωτω ττ ττ ω ∞ ? ∞ ? = ?? = ?? ?? = ∫ ∫ Since () 0w τ = for 0τ < 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 Fwed ωτ ω ττ ∞ ? ?∞ = ∫ and ()w τ for a stable system is Fourier transformable. Then 1 () ( ) 2 jt 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, (0)F , which is just the integral of the weighting function. 0 (0) ( )Fwdτ τ ∞ = ∫ 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 111 0 11 0 11 0 () ( ) ( ) () () yt w xt d ywxd xw d τ ττ ττ ττ ∞ ∞ ∞ =? = = ∫ ∫ ∫ 11 2 2 1 2 00 11 2 2 12 00 () ()( ) () ()( )( ) () () ( ) yy xx Rytyt dw dw xt xt dw dw R ττ τ τττ τ ττ ττ ττ τττ ∞∞ ∞∞ =+ =?+? =+ ∫∫ ∫∫ 2 11 2 2 12 00 () () ( ) xx ydwdwRτ τττ ττ ∞∞ =? ∫∫ 111 0 111 0 () ()( ) () ( ) ( ) () ( ) xy xx Rxtyt x tw xt d wR d ττ τ ττ τ ττττ ∞ ∞ =+ =+? =? ∫ ∫ 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 00 () 12 11 2 2 00 first integral () (0) () () )()( ) () () () In first integral o j yy yy j xx jjj xx yxwd Fx SRed ddw dw R e dR e d w e d w e ωτ ωτ ω ττ τ ωτ ωτ ττ ωττ τττ ττ τττ ττττ ττ ττ ∞ ∞ ? ?∞ ∞∞ ∞ ? ?∞ ∞ ?+? ? ?∞ = = = =( +? =+? ∫ ∫ ∫∫ ∫ ∫  12 12 11 2 2 2 =+ nly, let () () () ( ) ()( )() () () jjj yy xx xx xx dd SdRedwedwe SFF FS ωτ ωτωτ τττ τ ττ ωττ ττ ττ ωωω ωω ∞∞∞ ′ ?? ?∞ ?∞ ?∞ ′ ? ? ? ′= ? ′′= =? = ∫∫∫ The power spectral density thus does not depend upon phase properties. The cross-spectral density function can be derived similarly, to obtain: () () () xy xx SFSω ωω= Mean squared output in time and frequency domain 2 11 2 2 12 00 (0) ( ) ( ) ( ) 1 () 2 1 ()( ) () 2 yy xx yy xx yR dw dwR Sd FF S d τ τττ ττ ωω π ωωωω 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 sjω= we can cast this expression in that form. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 6 2 1 2 1 () ( ) () 2 j xx j xx sssds yFFS j jjj FsF sS sds 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 Ss is derived from () xx S ω by replacing 2 ω by 2 s? . ()Fs′ is the ordinary transfer function of the system – the Laplace transform of its weighting function. Because () 0, 0wt t= < . We shall drop the primes from now on. 2 22 44 1 () ( ) () 2 in ( ) j xx j xx yFsFsSsd j s Ss s π ω ω ∞ ?∞ =? ?=? ? ? = ? ? ∫ Integrating the output spectrum General method Cauchy Residue Theorem ()( ) 2 residues of ( ) at the poles enclosed in the contour C C Fsds j Fsπ= ∑ ∫> If ()Fs has a pole of order m at za= , () 1 1 1 Res( ) ( ) ( ) 1! m m m s a d asaF mds ? ? = ?? ??=? ?? ?? ? ?? ()Fs has a pole of order m at sa= if m is the smallest integer for which 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 6 ()lim ( ) m sa saFs → ?? ? ?? is finite. If ()Fs is rational and has a 1 st order pole at a , () () () () ( )( )... Ns Fs Ds Ns sasb = = ?? then [ ] 1 Res( ) lim ( ) ( ) () ( )( )... m sa asaFs Na abac = → =? = ??