scholarly journals Sum of Squares Approach for Nonlinear H∞ Control

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Ai-ping Pang ◽  
Zhen He ◽  
Ming-han Zhao ◽  
Guang-xiong Wang ◽  
Qin-mu Wu ◽  
...  
Keyword(s):  
Control Problem ◽  
State Space ◽  
Nonlinear Problem ◽  
Design Process ◽  
Synthesis Method ◽  
Sum Of Squares ◽  
Space Region ◽  
Sufficient Condition ◽  
Set Containment ◽  
Reasonable Result

A proper Hamilton-Jacobi-Isaacs (HJI) inequality must be solved in a nonlinear H∞ control problem. The sum of squares (SOS) method can now be used to solve an analytically unsolvable nonlinear problem. A HJI inequality suitable for SOS approach is derived in the paper. The SOS algorithm for solving the HJI inequality is also provided. Conservativeness of the SOS method is then discussed in the paper. The conservativeness of the SOS approach is caused by the method itself, because it is really a synthesis method over the entire state space. To reduce the conservativeness, a local H∞ design on a restricted state-space region is proposed. But the SOS approach for the local H∞ design also suffers from the conservativeness problem, because the S-procedure for solving the set-containment constraint provides only a sufficient condition. The above-mentioned sources of conservativeness are peculiar for the SOS approaches. So a proper approach must be carefully selected in the design process to get a reasonable result. A design example is also given in the paper.

Guaranteed Avoidance Control and Holding Control

1982 ◽  
Vol 104 (2) ◽  
pp. 166-172 ◽  
Author(s):  
W. E. Schmitendorf ◽  
B. R. Barmish ◽  
B. S. Elenbogen
Keyword(s):  
Control Problem ◽  
State Space ◽  
The State ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Avoidance Control ◽  
Avoidance Problem

This paper considers the problem of steering the state of a system, in the presence of disturbances, so that it avoids a specified subset of the state space. This subset is called the avoidance set and the problem is called the avoidance control problem. An avoidance control is a control which guarantees that the system does not enter the avoidance set regardless of the disturbance. A necessary condition and a sufficient condition for the existence of an avoidance control are given when the disturbance is subject to magnitude constraints. Closely related to the avoidance problem is the holding problem which is concerned with guaranteeing that the state of the system remains within a specified set. We also reinterpret our conditions for the existence of an avoidance control within the context of the holding problem.

scholarly journals A discrete Stochastic Korovkin theorem

1991 ◽  
Vol 14 (4) ◽  
pp. 679-682
Author(s):  
George A. Anastassiou
Keyword(s):  
Stochastic Processes ◽  
State Space ◽  
Sufficient Condition ◽  
Korovkin Theorem ◽  
Index Set ◽  
Real State

In this article we give a sufficient condition for the pointwise−−in the first mean Korovkin property onB0(P), the space of stochastic processes with real state space and countable index setΓand bounded first moments.

The Dimensional Synthesis of Spatial Cable-Driven Parallel Mechanisms

2013 ◽  
Vol 5 (4) ◽  
Author(s):  
K. Azizian ◽  
P. Cardou
Keyword(s):  
Synthesis Method ◽  
Sufficient Condition ◽  
Nonlinear Program ◽  
Parallel Mechanisms ◽  
Dimensional Synthesis ◽  
End Effector ◽  
A Value ◽  
Null Value ◽  
Objective Value

This paper presents a method for the dimensional synthesis of fully constrained spatial cable-driven parallel mechanisms (CDPMs), namely, the problem of finding a geometry whose wrench-closure workspace (WCW) contains a prescribed workspace. The proposed method is an extension to spatial CDPMs of a synthesis method previously published by the authors for planar CDPMs. The WCW of CDPMs is the set of poses for which any wrench can be produced at the end-effector by non-negative cable tensions. A sufficient condition is introduced in order to verify whether a given six-dimensional box, i.e., a box covering point-positions and orientations, is fully inside the WCW of a given spatial CDPM. Then, a nonlinear program is formulated, whose optima represent CDPMs that can reach any point in a set of boxes prescribed by the designer. The objective value of this nonlinear program indicates how well the WCW of the resulting CDPM covers the prescribed box, a null value indicating that none of the WCW is covered and a value greater or equal to one indicating that the full prescribed workspace is covered.

scholarly journals Dynamical modeling and optimal control of landfills

2016 ◽  
Vol 26 (05) ◽  
pp. 901-929 ◽  
Author(s):  
Alain Rapaport ◽  
Terence Bayen ◽  
Matthieu Sebbah ◽  
Andres Donoso-Bravo ◽  
Alfredo Torrico
Keyword(s):  
Optimal Control ◽  
Simple Model ◽  
Control Problem ◽  
State Space ◽  
Optimal Strategy ◽  
Dynamical Modeling ◽  
Minimal Time ◽  
Time Control ◽  
Switching Curve ◽  
Singular Arc

We propose a simple model of landfill and study a minimal time control problem where the re-circulation leachate is the manipulated variable. We propose a scheme to construct the optimal strategy by dividing the state space into three subsets [Formula: see text], [Formula: see text] and the complementary. On [Formula: see text] and [Formula: see text], the optimal control is constant until reaching target, while it can exhibit a singular arc outside these two subsets. Moreover, the singular arc could have a barrier. In this case, we prove the existence of a switching curve that passes through a point of prior saturation under the assumption that the set [Formula: see text] intersects the singular arc. Numerical computations allow then to determine the switching curve and depict the optimal synthesis.

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