This paper presents a relaxed scheme for controller synthesis of continuous-time systems in the Takagi-Sugeno form, based on non-quadratic Lyapunov functions and a non-PDC control law. The relaxations here provided allow state and input dependence of the membership functions' derivatives, as well as independence on initial conditions when input constraints are needed. Moreover, the controller synthesis is attainable via linear matrix inequalities, which are efficiently solved by commercially available software.
The control problem consists of stabilizing a control system while minimizing the norm of its transfer function. Several solutions to this problem are available. For systems in form, an optimal regulator can be obtained by solving two algebraic Riccati equations. For systems described by , either Wiener-Hopf optimization or projection results can be applied. The optimal regulator is then obtained using operations with proper stable rational matrices: inner-outer factorizations and stable projections. The aim of this paper is to compare the two approaches. It is well understood that the inner-outer factorization is equivalent to solving an algebraic Riccati equation. However, why are the stable projections not needed in the state-space approach? The difference between the two approaches derives from a different construction of doubly coprime, proper stable matrix fractions used to represent the plant. The transfer-function approach takes any doubly coprime fractions, while the state-space approach parameterizes such representations and those selected then obviate the need for stable projections.
We study in this paper Algebraic Riccati Equations associated with single-input single-output linear time-invariant systems bounded in H∞-norm. Our study is focused in the characterization of families of Algebraic Riccati Equations in terms of strictly positive real (of zero relative degree) substitutions applied to the associated H∞-norm bounded system, each substitution characterizing then a particular member of the family. We also consider here Algebraic Riccati Equations associated with systems characterized by both an H∞
-norm constraint and an upper bound on their corresponding McMillan degree.