16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde If the input and output are related by an nth order linear differential equation, once can also relate input to output by a set of n linear first order differential equations u(t Sy stem X(1)=A(1)x(1)+B(1)(t) y(1)=C(1)x(1) The solution form is y(1)=C(1)x(1) x(=(46)x)+B()(rdr where a(t, r) satisfie da(t, r)=4(o(l, r), o(t, r)=I note that any system which can be cast in this form is not only mathematically realizable but practically realizable as well. Must add a gain times u to y to get as many zeroes as poles For comparison with the weighting function description, take u and y to b scalars, and take t,=-o0. For stable systems, the transition from -oo to any finite time is zero Specialize the state space model to single-input, single-output (SISO) and d(t,-∞)=0 y()=C(1)x() x(0=ao(l,r)b(r)u(r)dr y(=∫c(,n( r)u(r)dr w(t, r)=c(o p (t, r)b(r) which we recognize by comparison with the earlier expression for y(t) Page 7 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 7 of 8 If the input and output are related by an nth order linear differential equation, once can also relate input to output by a set of n linear first order differential equations. () () () () () () () () x t At xt Bt ut yt Ctxt = + = & The solution form is: 0 0 0 () () () () (, ) ( ) (, ) ( ) ( ) t t yt Ctxt x t tt xt t B u d τ τ ττ = =Φ + Φ∫ where Φ(, ) t τ satisfies: ( , ) ( ) ( , ), ( , ) d t At t I dt Φ =Φ Φ = τ τ ττ Note that any system which can be cast in this form is not only mathematically realizable but practically realizable as well. Must add a gain times u to y to get as many zeroes as poles. For comparison with the weighting function description, take u and y to be scalars, and take 0t = −∞ . For stable systems, the transition from −∞ to any finite time is zero. Specialize the state space model to single-input, single-output (SISO) and 0t → −∞ : (, ) 0 () () () () (, ) ( ) ( ) () () (, ) ( ) ( ) (, ) () (, ) ( ) T t t T T t yt Ct xt xt t b u d yt Ct t b u d wt Ct t b τ τ ττ τ τ ττ τ ττ −∞ −∞ Φ −∞ = = = Φ = Φ = Φ ∫ ∫ which we recognize by comparison with the earlier expression for y t( )