Stabilization problems of nonlinear systems using feedback laws with Wiener processes

2009 ◽  
Author(s):  
Yuki Nishimura ◽  
Kazuki Takehara ◽  
Yuh Yamashita ◽  
Kanya Tanaka ◽  
Yuji Wakasa
Keyword(s):  
Nonlinear Systems ◽  
Wiener Processes ◽  
Feedback Laws

Inverse Optimal Stabilization of Dynamical Systems by Wiener Processes

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
K. D. Do
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Covariance Matrix ◽  
Optimal Stabilization ◽  
Isaacs Equation ◽  
Wiener Processes

Abstract This paper formulates and solves new problems of inverse optimal stabilization and inverse optimal stabilization with gain assignment for nonlinear systems by Wiener processes. First, a theorem is developed to design inverse optimal stabilizers (i.e., covariance matrix multiplied by variance of Wiener processes), where it does not require to solve a Hamilton–Jacobi–Belman equation. Second, another theorem is developed to design inverse optimal stabilizers with gain assignment for nonlinear systems perturbed by both nonvanishing deterministic and stochastic (Wiener processes) disturbances without having to solve a Hamilton–Jacobi–Isaacs equation.

scholarly journals Robust stabilization of nonlinear systems: The LMI approach

2000 ◽  
Vol 6 (5) ◽  
pp. 461-493 ◽  
Author(s):  
D. D. Šiljak ◽  
D. M. Stipanovic
Keyword(s):  
Nonlinear Systems ◽  
Robust Stabilization ◽  
Linear Matrix ◽  
Feedback Gain ◽  
Control Laws ◽  
New Approach ◽  
Quadratic Constraint ◽  
Matching Conditions ◽  
Feedback Laws ◽  
New Formulation

This paper presents a new approach to robust quadratic stabilization of nonlinear systems within the framework of Linear Matrix Inequalities (LMI). The systems are composed of a linear constant part perturbed by an additive nonlinearity which depends discontinuously on both time and state. The only information about the nonlinearity is that it satisfies a quadratic constraint. Our major objective is to show how linear constant feedback laws can be formulated to stabilize this type of systems and, at the same time, maximize the bounds on the nonlinearity which the system can tolerate without going unstable.We shall broaden the new setting to include design of decentralized control laws for robust stabilization of interconnected systems. Again, the LMI methods will be used to maximize the class of uncertain interconnections which leave the overall system connectively stable. It is useful to learn that the proposed LMI formulation “recognizes” the matching conditions by returning a feedback gain matrix for any prescribed bound on the interconnection terms. More importantly, the new formulation provides a suitable setting for robust stabilization of nonlinear systems where the nonlinear perturbations satisfy the generalized matching conditions.

scholarly journals Fundamental Limitations of the Decay of Generalized Energy in Controlled (Discrete–Time) Nonlinear Systems Subject to State and Input Constraints

2019 ◽  
Vol 29 (4) ◽  
pp. 629-639
Author(s):  
István Selek ◽  
Enso Ikonen
Keyword(s):  
Nonlinear Systems ◽  
Discrete Time ◽  
Input Constraints ◽  
Control Performance ◽  
Controlled System ◽  
Loop Control ◽  
Fundamental Limitations ◽  
Hard Constraints ◽  
Input Variables ◽  
Feedback Laws

Abstract This paper is devoted to the analysis of fundamental limitations regarding closed-loop control performance of discrete-time nonlinear systems subject to hard constraints (which are nonlinear in state and manipulated input variables). The control performance for the problem of interest is quantified by the decline (decay) of the generalized energy of the controlled system. The paper develops (upper and lower) barriers bounding the decay of the system’s generalized energy, which can be achieved over a set of asymptotically stabilizing feedback laws. The corresponding problem is treated without the loss of generality, resulting in a theoretical framework that provides a solid basis for practical implementations. To enhance understanding, the main results are illustrated in a simple example.

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