On optimal nonlinear feedback regulation of linear plants

Salehi, S.V.; Ryan, E. P.

doi:10.1109/tac.1982.1103108

Cited by 40 publications

(21 citation statements)

References 4 publications

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“…We are now ready to state the main result of this paper. Versions of this result have either appeared without proof or have been used implicitly in [3], [4], [11], [18], [19]. …”

Section: It Follows From the Above Two Facts That If Y(a) ∈ ω κ Thementioning

confidence: 99%

Finite-Time Stability of Continuous Autonomous Systems

Bhat

Bernstein²

2000

SIAM J. Control Optim.

4,239

2,080

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Abstract. Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Hölder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.Key words. stability, finite-time stability, non-Lipschitzian dynamics AMS subject classifications. 34D99, 93D99PII. S03630129973213581. Introduction. The object of this paper is to provide a rigorous foundation for the theory of finite-time stability of continuous autonomous systems and motivate a closer examination of finite-time stability as a possible objective in control design.Classical optimal control theory provides several examples of systems that exhibit convergence to the equilibrium in finite time [17]. A well-known example is the double integrator with bang-bang time-optimal feedback control [2]. These examples typically involve dynamics that are discontinuous. Discontinuous dynamics, besides making a rigorous analysis difficult (see [9]), may also lead to chattering [10] or excite high frequency dynamics in applications involving flexible structures. Reference [8] considers finite-time stabilization using time-varying feedback controllers. However, it is well known that the stability analysis of time-varying systems is more complicated than that of autonomous systems. Therefore, with simplicity as well as applications in mind, we focus on continuous autonomous systems.Finite [21] suggests, based on a scalar example, that systems with finite-settling-time dynamics possess better disturbance rejection and robustness properties. However, no precise results exist for multidimensional systems. This paper attempts to fill these gaps.In section 2, we define finite-time stability for equilibria of continuous autonomous systems that have unique solutions in forward time. Continuity and forward uniqueness render the solutions continuous functions of the initial conditions, so that the

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“…We are now ready to state the main result of this paper. Versions of this result have either appeared without proof or have been used implicitly in [3], [4], [11], [18], [19]. …”

Section: It Follows From the Above Two Facts That If Y(a) ∈ ω κ Thementioning

confidence: 99%

Finite-Time Stability of Continuous Autonomous Systems

Bhat

Bernstein²

2000

SIAM J. Control Optim.

4,239

2,080

View full text Add to dashboard Cite

show abstract

“…Next, since V (·) is continuously differentiable and, by (31) and (32), V (0) is a local minimum of V (·), it follows that V (0) = 0, and hence, it follows from (34) and (36) that φ(0) = 0, which implies (10). Finally, it follows from (14), (29), and (36) that…”

Section: Finite-time Stabilization For Affine Dynamical Systems Amentioning

confidence: 91%

“…Finally, we explore connections of our approach with inverse optimal control [24]- [28], wherein we parametrize a family of finite-time stabilizing sublinear controllers that minimize a derived cost functional involving subquadratic terms. Subquadratic performance criteria have been studied in [11], [29], [30] and have been shown to permit a unified treatment of a broad range of design goals.…”

Section: Introductionmentioning

confidence: 99%

Finite-Time Stabilization and Optimal Feedback Control

Haddad

L’Afflitto

2016

IEEE Trans. Automat. Contr.

105

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Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite-time convergence implies nonuniqueness of system solutions in reverse time, such systems possess non-Lipschitzian dynamics. Sufficient conditions for finite-time stability have been developed in the literature using continuous Lyapunov functions. In this paper, we develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finitetime stability and finite-time stabilization. Finite-time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that satisfies a differential inequality involving fractional powers. This Lyapunov function can clearly be seen to be the solution to a partial differential equation that corresponds to a steady-state form of the Hamilton-Jacobi-Bellman equation, and hence, guaranteeing both finite-time stability and optimality.

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“…It has since generated many research activities in the following two decades. When optimal feedback nonlinear regulation problem of linear time invariant system is studied, control laws with finite time stability was proposed in [8]. Later theorem on finite time stability of nonlinear control systems based on Lyapunov stability analysis method was introduced in [9].…”

Section: Introductionmentioning

confidence: 99%

Guidance Law Design Based on Finite Time Convergence

Yang¹,

Chen²,

Liu³

et al. 2015

2015 International Conference on Computer Science and Mechanical Automation (CSMA)

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Traditional guidance laws are designed based on Lyapunov theorem. They will guide the line of sight (LOS) angular rate to converge to zero when time approaches infinity. In this paper, guidance laws with finite time convergence are proposed in terms of finite time stability theory of nonlinear control systems. It is proved that the LOS angular rate will converge to zero before the final guidance ends with the designed guidance laws. Simulation results show that the guidance laws are highly effective.

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On optimal nonlinear feedback regulation of linear plants

Cited by 40 publications

References 4 publications

Finite-Time Stability of Continuous Autonomous Systems

Finite-Time Stability of Continuous Autonomous Systems

Finite-Time Stabilization and Optimal Feedback Control

Guidance Law Design Based on Finite Time Convergence

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