Feedback Stabilization of Quasi-Integrable Hamiltonian Systems

2003 ◽  
Vol 70 (1) ◽  
pp. 129-136 ◽  
Author(s):  
W. Q. Zhu ◽  
Z. L. Huang
Keyword(s):  
Feedback Control ◽  
Cost Function ◽  
Hamiltonian Systems ◽  
Infinite Horizon ◽  
Integrable Hamiltonian System ◽  
First Integrals ◽  
Feedback Stabilization ◽  
Largest Lyapunov Exponent ◽  
Dynamical Programming ◽  
Integrable Hamiltonian Systems

A procedure for designing a feedback control to asymptotically stabilize with probability one quasi-integrable Hamiltonian system is proposed. First, a set of averaged Ito^ stochastic differential equations for controlled first integrals is derived from given equations of motion of the system by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Second, a dynamical programming equation for infinite horizon performance index with unknown cost function is established based on the stochastic dynamical programming principle. Third, the asymptotic stability with probability one of the optimally controlled system is analyzed by evaluating the largest Lyapunov exponent of the fully averaged Ito^ equations for the first integrals. Finally, the cost function and feedback control law are determined by the requirement of stabilization of the system. An example is worked out in detail to illustrate the application of the proposed procedure and the effect of optimal control on the stability of the system.

A class of C∞-integrable Hamiltonian systems

1989 ◽  
Vol 113 (3-4) ◽  
pp. 293-314 ◽  
Author(s):  
W. M. Oliva ◽  
M. S. A. C. Castilla
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
First Integrals ◽  
Integrable Hamiltonian Systems ◽  
Integral Of Motion ◽  
Toda System ◽  
Nonnegative Coefficients ◽  
Special Case ◽  
Three Degrees Of Freedom

SynopsisWe discuss the C∞ complete integrability of Hamiltonian systems of type q = —grad V(q) = F(q), in which the closure of the cone generated (with nonnegative coefficients) by the vectors F(q), q ϵ ℝn, does not contain a line. The components of the asymptotic velocities are first integrals and the main aim is to prove their smoothness as functions of the initial conditions. The Toda-like system with potential V(q)=ΣNi=1 exp(fi∣ q) is a special case of the considered systems ifthe cone C(f1,…,fN)={ΣNi=1cifi,ci≧0} does notcontain a line. In any number of degrees of freedom, if C(f1,…,fN) has amplitude not too large (ang (fi, fj ≦π/2i,j=1,2,…, N), the first integrals are C∞ functions. In two degrees of freedom, without restriction on the amplitude of the cone, C∞-integrability is proved even in a case in which it is known that there is no other meromorphic integral of motion independent of energy. In three degrees of freedom the C∞-integrability of a deformation of the classic nonperiodic Toda system is proved. Some other examples are also discussed.

Asymptotic Stability With Probability One of Random-Time-Delay-Controlled Quasi-Integrable Hamiltonian Systems

2016 ◽  
Vol 83 (9) ◽  
Author(s):  
R. H. Huan ◽  
W. Q. Zhu ◽  
R. C. Hu ◽  
Z. G. Ying
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Hamiltonian Systems ◽  
Hybrid System ◽  
Networked Control System ◽  
Random Time ◽  
Largest Lyapunov Exponent ◽  
Integrable Hamiltonian Systems ◽  
Markov Jump ◽  
Solution Property

A new procedure for determining the asymptotic stability with probability one of random-time-delay-controlled quasi-integrable Hamiltonian systems is proposed. Such a system is formulated as continuous–discrete hybrid system and the random time delay is modeled as a Markov jump process. A three-step approximation is taken to simplify such hybrid system: (i) the randomly periodic approximate solution property of the system is used to convert the random time delay control into the control without time delay but with delay time as parameter; (ii) a limit theorem is used to transform the hybrid system with Markov jump parameter into one without jump parameter; and (iii) the stochastic averaging method for quasi-integrable Hamiltonian systems is applied to reduce the system into a set of averaged Itô stochastic differential equations. An approximate expression for the largest Lyapunov exponent of the system is derived from the linearized averaged Itô equations and the necessary and sufficient condition for the asymptotic stability with probability one of the system is obtained. The application and effectiveness of the proposed procedure are demonstrated by using an example of stochastically driven two-degrees-of-freedom networked control system (NCS) with random time delay.

COMPLETELY AND PARTIALLY INTEGRABLE HAMILTONIAN SYSTEMS IN THE NONCOMPACT CASE

2004 ◽  
Vol 01 (03) ◽  
pp. 167-183 ◽  
Author(s):  
EMANUELE FIORANI
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Integrable Hamiltonian System ◽  
Time Dependent ◽  
Integrable Hamiltonian Systems ◽  
Invariant Submanifold ◽  
Completely Integrable ◽  
Completely Integrable Hamiltonian System

Action-angle coordinates are shown to exist around an instantly compact invariant submanifold of a time-dependent completely integrable Hamiltonian system. Partially integrable Hamiltonian systems are also considered in the noncompact case; a comparison is made with other possible approaches. Results on symplectically complete foliations contained in the Appendix A can be used to give alternative proofs of some propositions.

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