The e-parameter method, with the introduction o fan extended Liapunov equation, pro vides an effective tool for a suboptimal design of nonlinear feedback systems. The method applies also to a coupled systems design.
Summary--An approximation method is presented to construct an optimal state regulator for a nonlinear system with quadratic performance index. The nonlinearity is taken to be a perturbation to the system, and a parameter e is introduced to stand for it. By making use of a power-series expansion in e, a sequence of partial differential equations is derived whose solutions form a suboptimal feedback law. Given a polynomial nonlinearity, the partial differential equations are reduced to ordinary differential equations by separation of variables. The zero-order terms yield a well known Riccati equation. Higher-order equations are transformed into conventional type linear equations, owing to a lemma regarding an extended Liapunov equation. It is demonstrated that the l-th order approximation for the feedback law results in the (2โข+ 1)th order approximation to the optimal performance.
The procedure developed has a wide variety of applications. As one of the straightforward applications, the synthesis of a suboptimal control is discussed for a largescale system as composed of several subsystems of lower dimensions.
Three examples attached illustrate several features of the method.