Obtaining stabilizing stationary controls via finite horizon cost

Costa, Eduardo F.; Val, J.B.R. do

doi:10.1109/acc.2006.1657394

Cited by 2 publications

(6 citation statements)

References 15 publications

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“…Indeed, in the companion paper [8] we show that if f and c are Lipschitz in their arguments and if the control is nonsingular in the sense that there exists ξ such that c t (z, u) ≥ ξd(u), then we can set u ≡ 0 in those concepts with no loss of generality.…”

Section: Remarkmentioning

confidence: 98%

“…[5] and [9]. Here, due to the rather general setup we deal with, we need to consider the following closed loop version [8], [6] and [7].…”

Section: The Controlled System and A Sufficient Condition For Rmentioning

confidence: 99%

“…As regards to the design of r, N -stabilizing controls, we extend some of the results in the companion paper [8] to obtain a sufficient condition for r, N -stability, which relies on the existence of a bound for the cost functional and on a detectability concept. As one would expect, the condition reflects the fact that the smaller is the bound r on the trajectory, the larger is the horizon N demanded by the control to steer the trajectory within the target.…”

Section: Introductionmentioning

confidence: 98%

“…The assumption holds true under basic conditions; for illustration, we mention: uniformly bounded f , or continuous f plus compact Z, or observability plus bounded cost functionals (see [8]). …”

Section: Introductionmentioning

confidence: 99%

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A finite-time stability concept and conditions for finite-time and exponential stability of controlled nonlinear systems

Costa

Val

2006

2006 American Control Conference

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This paper introduces a finite-time stability concept for nonlinear systems and the corresponding notion of stabilizability of controls. The concept generalizes previous finite-time stability as it requires that the state trajectory satisfies a given bound at some time instant in a certain interval, whereas previous notions considers that the system stays within that bound over the entire interval. Here, the bound on the trajectory may represent a region arbitrarily close to the origin, and thus it is in tune with situations where contractive trajectories are required. This feature allows us to relate the proposed concept with the usual exponential stability concept, and simultaneously, with the previous finitetime concepts, thus clarifying the relations among them. As regards to controlled systems, we derive a sufficient condition for stabilizability, with the interpretation that the size of the time interval demanded by the control to drive the trajectory into a specified region is in inverse proportion with the size of the region. Moreover, we present a simple moving horizon implementation for a stabilizing (in the new sense) control that provides an exponentially stable controlled system. For stationary controls we can connect stabilizability in the sense here with the classical exponential sense. We show that the control is exponentially stabilizing whenever it is finite-time stabilizing and stationary. An illustrative example is included.

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Section: Remarkmentioning

confidence: 98%

“…[5] and [9]. Here, due to the rather general setup we deal with, we need to consider the following closed loop version [8], [6] and [7].…”

Section: The Controlled System and A Sufficient Condition For Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A finite-time stability concept and conditions for finite-time and exponential stability of controlled nonlinear systems

Costa

Val

2006

2006 American Control Conference

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“…We also study how those bounds can be extended to the conditional second moments of the system. In the literature, the bounds for the cost C T are connected to basic conditions in different contexts such as conditions for existence of stabilizing receding horizon controls [9,10] or uniform cost convergence [11,12], among others. Note that even in the context of MJLS with complete state observation and general Markov chain there is no available result parallelling the ones obtained here.…”

Section: Introductionmentioning

confidence: 99%

Quadratic costs and second moments of jump linear systems with general Markov chain

Costa

Vargas

Val

2011

Math. Control Signals Syst.

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This paper presents an analytic, systematic approach to handle quadratic functionals associated with Markov jump linear systems with general jumping state. The Markov chain is finite state, but otherwise general, possibly reducible and periodic. We study how the second moment dynamics are affected by the additive noise and the asymptotic behaviour, either oscillatory or invariant, of the Markov chain. The paper comprises a series of evaluations that lead to a tight two-sided bound for quadratic cost functionals. A tight two-sided bound for the norm of the second moment of the system is also obtained. These bounds allow us to show that the long-run average cost is well defined for system that are stable in the mean square sense, in spite of the periodic behaviour of the chain and taking into consideration that it may not be unique, as it may depend on the initial distribution. We also address the important question of approximation of the long-run average cost via adherence of finite horizon costs.

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Obtaining stabilizing stationary controls via finite horizon cost

Cited by 2 publications

References 15 publications

A finite-time stability concept and conditions for finite-time and exponential stability of controlled nonlinear systems

A finite-time stability concept and conditions for finite-time and exponential stability of controlled nonlinear systems

Quadratic costs and second moments of jump linear systems with general Markov chain

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