Constructive Stabilization of Quadratic-Input Nonlinear Systems with Bounded Controls

Abstract

Abstract. In this paper, the stabilization of quadratic-input nonlinear systems with bounded controls is considered. According to the type of quadratic-input forms, two cases, namely, positive definite and positive semi-definite, are considered. For the case of positive definiteness, a universal formula for bounded stabilizers is given via a known Lyapunov control function. For the case of positive semidefiniteness, a constructive parametrization of bounded stabilizers is proposed under the assumption that there exists a known Lyapunov control function with respect to a smaller control set than the admissible control set.

Global CLF Stabilization of Nonlinear Systems: A Geometric Point of View

“…However, in view that control ω(x) is not admissible (it is singular at N b ), we consider admissible feedback stabilizers of the general form u(x) = ρ(x) ω(x), where ρ(x) is a rescaling function, that comprehends many of the control formulae proposed in the literature. Then, we propose a family of admissible suboptimal feedback controls u ε (x) given by (24)- (25). Finally, we introduce a regular approximation method to compact convex sets U in order to redesign the control formula u ε (x).…”

“…In fact, in Suárez et al (2002), for the cvs U = B m r (p) defined in (9) (for 1 < p < ∞), the designed control formula is of the form (24). In this case, ρ ε (x) is given by (25) (1) with controls taking values in U ] satisfying the scp, and a(x) and b(x) are regular. Then, for all ε > 0, ρ ε (x) is everywhere continuous and satisfies Hypothesis H. Furthermore, ∀x ∈ R n \N b , if ε → 0 then ρ ε (x) → 1.…”

“…Assume that U ∈ K(R m ) with 0 ∈ intU , V (x) is a clf [for system (1) with controls taking values in U ] satisfying the scp. If a(x), b(x) and h(ξ) are smooth, and q ∈ 2 Z + , then ∀ε > 0, u ε (x) defined by (24)- (25), with ω(x) given by (29), is a family of feedback controls, smooth on R n \(N a ∪ ∂N b ∪ {0}) and continuous on R n , and taking values in U , which render system (1) gas.…”

Optimal Control for Discrete‐time Descriptor Noncausal Systems

“…Theorem 4.1 tells us that when the objective function is linear the optimal controls are bang‐bang for the optimal control problem subject to a linear descriptor noncausal system. The input of the noncausal system is linear in Section 4, while the quadratic input form can be found in a number of practical cases such as power system models studied in [34] and it also attracted the attentions of some researchers at a theoretical level [35]. These results concerning quadratic input form inspired me to investigate an optimal control problem for a descriptor noncausal system with quadratic input variables as the following: where r j ∈ R n , j =0,1,2,…, L are known coefficient vectors, and u ( j )∈ U ad =[−1,1], j =0,1,2,…, L −1.…”