Riccati Equation


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Riccati equation

[ri′käd·ē i‚kwā·zhən]
(mathematics)
A first-order differential equation having the form y ′ = A0(x) + A1(x) y + A2(x) y 2; every second-order linear differential equation can be transformed into an equation of this form.
A matrix equation of the form dP (t)/ dt + P (t) F (t) + F T (t) P (t) -P (t) G (t) R -1(t) G T (t) P (t) + Q (t) = 0, which frequently arises in control and estimation theory.

Riccati Equation

 

a first-order ordinary differential equation of the form

where a, b, and a are constants. This equation was first investigated by J. Riccati in 1724; certain special cases were studied earlier. In work done in 1724 and 1725, D. Bernoulli established that equation (*) can be integrated in terms of elementary functions when α = – 2 or α = – 4k/(2k – 1), where k is an integer. J. Liouville proved in 1841 that, for other values of α, the equation (*) cannot be solved by quadrature, that is, by applying a finite number of algebraic operations, transformations of variables, and indefinite integrations to elementary functions; a general solution can be expressed in terms of cylindrical functions.

The differential equation

where P(x), Q(x), and R(x) are continuous functions, is called the generalized Riccati equation. When P(x) = 0, the generalized Riccati equation is a linear differential equation; when R(x) = 0, it is the Bernoulli equation. In these two cases, the equation is integrable in closed form. Other cases of the integrability of the generalized Riccati equation have also been studied.

REFERENCE

Kamke, E. Spravochnik po obyknovennym differentsial’nym uravneniiam, 4th ed. Moscow, 1971. (Translated from German.)
References in periodicals archive ?
Therefore we can conclude here that there is neat linkage between Schrodinger equation, Maxwell equation, Riccati equation via biquaternion expression [22, 23, 24].
Nekoo, "Nonlinear suboptimal control of fully coupled non-affine six-DOF autonomous underwater vehicle using the state-dependent Riccati equation," Ocean Engineering, vol.
Authors presented rational solutions of (1) with the aid of the generalized Riccati equation mapping method [42].
where P is the solution of the famous Riccati equation.
Petersen, "A Riccati equation approach to the design of stabilizing controllers and observers for a class of uncertain linear systems," IEEE Transactions on Automatic Control, vol.
For optimal chemotherapy, the state dependent Riccati equation (SDRE) based optimal control technique is applied to the nonlinear model.
We seek a solution of this nonlinear Riccati equation in the form
They cover a general theory of doubling algorithms, discrete-time and continuous-time algebraic Riccati equations, the M-matrix algebraic Riccati equation, other algebraic Riccati equations, and nonlinear matrix equations X + BX-1A=Q.
where: K and P are the gain matrix of the optimal control and solution matrix of the Riccati equation, respectively.
where S is the solution of the Riccati equation and the matrices A and B are, respectively, the coefficient matrix of the state variable X based on Equations (1-7) and the coefficient matrix of control forces U*.Matrices Q, R, and N can be obtained from Equation (21).
In this work, we shall use the following solutions of Riccati equation: