scholarly journals Indefinite LQ Control for Discrete-Time Stochastic Systems via Semidefinite Programming

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Shaowei Zhou ◽  
Weihai Zhang
Keyword(s):  
Semidefinite Programming ◽  
Riccati Equation ◽  
Discrete Time ◽  
Time Horizon ◽  
Stochastic Systems ◽  
Positive Semidefinite ◽  
Algebraic Riccati Equation ◽  
Infinite Time ◽  
Lq Problem ◽  
Lq Control

This paper is concerned with a discrete-time indefinite stochastic LQ problem in an infinite-time horizon. A generalized stochastic algebraic Riccati equation (GSARE) that involves the Moore-Penrose inverse of a matrix and a positive semidefinite constraint is introduced. We mainly use a semidefinite-programming- (SDP-) based approach to study corresponding problems. Several relations among SDP complementary duality, the GSARE, and the optimality of LQ problem are established.

scholarly journals Discrete-Time Indefinite Stochastic LQ Control via SDP and LMI Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Shaowei Zhou ◽  
Weihai Zhang
Keyword(s):  
Discrete Time ◽  
Time Horizon ◽  
Algebraic Riccati Equation ◽  
Mean Square ◽  
Infinite Time ◽  
Lq Problem ◽  
The Cost ◽  
And Control ◽  
Indefinite Stochastic Lq Control ◽  
Lq Control

This paper studies a discrete-time stochastic LQ problem over an infinite time horizon with state-and control-dependent noises, whereas the weighting matrices in the cost function are allowed to be indefinite. We mainly use semidefinite programming (SDP) and its duality to treat corresponding problems. Several relations among stability, SDP complementary duality, the existence of the solution to stochastic algebraic Riccati equation (SARE), and the optimality of LQ problem are established. We can test mean square stabilizability and solve SARE via SDP by LMIs method.

scholarly journals Minimum System Sensitivity Study of Linear Discrete Time Systems for Fault Detection

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Xiaobo Li ◽  
Hugh H. T. Liu
Keyword(s):  
Fault Detection ◽  
Riccati Equation ◽  
Discrete Time ◽  
Time Horizon ◽  
Linear Time ◽  
Sensitivity Study ◽  
Algebraic Riccati Equation ◽  
System Sensitivity ◽  
Discrete Time Systems ◽  
Time Systems

Fault detection is a critical step in the fault diagnosis of modern complex systems. An important notion in fault detection is the smallest gain of system sensitivity, denoted asℋ−index, which measures the worst fault sensitivity. This paper is concerned with characterizingℋ−index for linear discrete time systems. First, a necessary and sufficient condition on the lower bound ofℋ−index in finite time horizon for linear discrete time-varying systems is developed. It is characterized in terms of the existence of solution to a backward difference Riccati equation with an inequality constraint. The result is further extended to systems with unknown initial condition based on a modifiedℋ−index. In addition, for linear time-invariant systems in infinite time horizon, based on the definition of theℋ−index in frequency domain, a condition in terms of algebraic Riccati equation is developed. In comparison with the well-known bounded real lemma, it is found thatℋ−index is not completely dual toℋ∞norm. Finally, several numerical examples are given to illustrate the main results.

scholarly journals Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Li Wang
Keyword(s):  
Riccati Equation ◽  
Positive Semidefinite ◽  
Algebraic Riccati Equation ◽  
Positive Semidefinite Matrix ◽  
Matrix Inequalities ◽  
Numerical Examples ◽  
Upper Solution ◽  
Solution Bound ◽  
Semidefinite Matrix ◽  
Two Parameter

The continuous coupled algebraic Riccati equation (CCARE) has wide applications in control theory and linear systems. In this paper, by a constructed positive semidefinite matrix, matrix inequalities, and matrix eigenvalue inequalities, we propose a new two-parameter-type upper solution bound of the CCARE. Next, we present an iterative algorithm for finding the tighter upper solution bound of CCARE, prove its boundedness, and analyse its monotonicity and convergence. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.

Discrete Time-Coupled State-Dependent Riccati Equation Control of Nonlinear Mechatronic Systems

2020 ◽  
Vol 142 (8) ◽  
Author(s):  
Xin Wang
Keyword(s):  
Riccati Equation ◽  
Discrete Time ◽  
Disturbance Attenuation ◽  
Infinite Time ◽  
Mechatronic Systems ◽  
Feedback Controllers ◽  
Control Laws ◽  
State Dependent ◽  
Time Optimal ◽  
Nonlinear Plant

Abstract A discrete-time-coupled state-dependent Riccati equation (CSDRE) control strategy is structured in this paper for synthesizing state feedback controllers satisfying the combined nonlinear quadratic regulator (NLQR) and H∞ robust control performance objectives. Under smoothness assumptions, the nonlinear plant dynamics can be formulated into state-dependent coefficient form through direct parameterization. By solving a pair of coupled state-dependent Riccati equations, the optimal stabilizing solutions can achieve inherent stability, nonlinear quadratic optimality, and H∞ disturbance attenuation performance. The established two-player Nash's game theory is utilized for developing both of the finite and infinite time optimal control laws. Furuta swing-up pendulum, a representative nonholonomic underactuated nonlinear system, is stabilized in real-time for validating the robustness and potential of proposed approach in mechatronics applications.

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