Riccati difference equation

Nonlinear Difference Equations Riccati Difference Equation The Riccati equation yx-t-iyx + 1 + byx + c = 0 is a nonhnear... 

Example This equation is obtained in distillation problems, among others, in which the number of theoretical plates is required. If the relative volatility is assumed to be constant, the plates are theoretically perfect, and the molal liquid and vapor rates are constant, then a material balance around the nth plate of the enriching section yields a Riccati difference equation. 

S. Bittanti, P. Coaneri, and G.D. Nicolao. The difference periodic Riccati equation for the periodic prediction problem. IEEE Trans. Automat. Contr., 33 706-712, 1988. 

In this chapter, we develop analytical solution methods, which have very close analogs with methods used for linear ODEs. A few nonlinear difference equations can be reduced to linear form (the Riccati analog) and the analogous Euler-Equidimensional finite-difference equation also exists. For linear equations, we again exploit the property of superposition. Thus, our general solutions will be composed of a linear combination of complementary and particular solutions. 

Very few nonlinear equations yield analytical solutions, so graphical or trial-error solution methods are often used. There are a few nonlinear finite difference equations, which can be reduced to linear form by elementary variable transformation. Foremost among these is the famous Riccati equation... 

Chan, S.W., G.C. Goodwin, K.S. Sin. 1984. Convergence properties of the Riccati difference equation in optimal filtering of nonstabilizable systems. IEEE Transactions on Automatic Control AC-29 10-18. 

A nonlinear equation, which arises in both continuous and staged (i.e., finite difference) processes, is Riccati s equation... 

With this in mind, and with the additional approximation which consists in neglecting the overlap between POOs coming from different LMOs, i.e., (S) S, direct RPA Riccati equations of Eq. 12 become ... 

It is then a further approximation to tell that the POOs coming from different LMOs have a negligible overlap, i.e., that = 5y. The Riccati equations become ... 

---