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114
On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems
"... We consider the problem of common linear copositive function existence for positive switched linear systems. In particular, we present a necessary and sufficient condition for the existence of such a function for switched systems with two constituent linear time-invariant (LTI) systems. A number of ..."
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Cited by 55 (4 self)
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We consider the problem of common linear copositive function existence for positive switched linear systems. In particular, we present a necessary and sufficient condition for the existence of such a function for switched systems with two constituent linear time-invariant (LTI) systems. A number of applications of this result are also given.
On the Stability of Positive Linear Switched Systems Under Arbitrary Switching Laws
"... We consider n-dimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this conditi ..."
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Cited by 25 (5 self)
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We consider n-dimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this condition is also sufficient. Recently, Gurvits, Shorten, and Mason showed that this conjecture is true for the case n = 2, but is not true in general. Their results imply that there exists some minimal integer np such that the conjecture is true for all n < np, but is not true for n = np. We show that np = 3.
Stabilization of Continuous-Time Switched Linear Positive Systems
, 2009
"... In this paper we tackle a few problems related to linear positive switched systems. First, we provide a result on state-feedback stabilization of autonomous linear positive switched systems through piecewise linear co-positive Lyapunov functions. This is accompanied by a side result on the existence ..."
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Cited by 25 (5 self)
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In this paper we tackle a few problems related to linear positive switched systems. First, we provide a result on state-feedback stabilization of autonomous linear positive switched systems through piecewise linear co-positive Lyapunov functions. This is accompanied by a side result on the existence of a switching law guaranteeing an upper bound to the optimal L1 cost. Then, the induced L1 guaranteed cost cost is tackled, through constrained piecewise linear co-positive Lyapunov functions. The optimal L1 cost control is finally studied via Hamiltonian function analysis.
Nice reachability for planar bilinear control systems with applications to planar linear switched systems
- 2007, submitted. [Online]. Available: http://www.eng.tau.ac. il/ ∼ michaelm
"... We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a “nice ” control. Specifically, a control that is a concatenation of a bang arc with either (1) a bang-bang control that is periodic after the third switch; or (2) a p ..."
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Cited by 13 (9 self)
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We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a “nice ” control. Specifically, a control that is a concatenation of a bang arc with either (1) a bang-bang control that is periodic after the third switch; or (2) a piecewise constant control with no more than two discontinuities. Under the additional assumption that the bilinear system is positive (or invariant for any proper cone), we show that the reachable set is spanned by a concatenation of a bang arc with either (1) a bang-bang control with no more than two discontinuities; or (2) a piecewise constant control with no more than two discontinuities. In particular, any point in the reachable set can be reached using a piecewise-constant control with no more than three discontinuities. Several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our main result. We demonstrate this using one example.
POLYNOMIAL-TIME COMPUTATION OF THE JOINT SPECTRAL RADIUS FOR SOME SETS OF NONNEGATIVE MATRICES
, 2009
"... We propose two simple upper bounds for the joint spectral radius of sets of nonnegative matrices. These bounds, the joint column radius and the joint row radius, can be computed in polynomial time as solutions of convex optimization problems. We show that these bounds are within a factor 1/n of the ..."
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Cited by 13 (2 self)
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We propose two simple upper bounds for the joint spectral radius of sets of nonnegative matrices. These bounds, the joint column radius and the joint row radius, can be computed in polynomial time as solutions of convex optimization problems. We show that these bounds are within a factor 1/n of the exact value, where n is the size of the matrices. Moreover, for sets of matrices with independent column uncertainties or with independent row uncertainties, the corresponding bounds coincide with the joint spectral radius. In these cases, the joint spectral radius is also given by the largest spectral radius of the matrices in the set. As a by-product of these results, we propose a polynomial-time technique for solving Boolean optimization problems related to the spectral radius. We also describe economics and engineering applications of our results.
Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions. Annals of Applied Probability
, 2012
"... ar ..."
On the stabilization of persistently excited linear systems
- SIAM J. Control Optim
"... We consider control systems of the type ˙x = Ax+α(t)bu, where u ∈ R, (A, b) is a controllable pair and α is an unknown time-varying signal with values in [0, 1] satisfying a persistent excitation condition i.e., ∫ t+T t α(s)ds ≥ µ for every t ≥ 0, with 0 < µ ≤ T independent on t. We prove that s ..."
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Cited by 11 (7 self)
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We consider control systems of the type ˙x = Ax+α(t)bu, where u ∈ R, (A, b) is a controllable pair and α is an unknown time-varying signal with values in [0, 1] satisfying a persistent excitation condition i.e., ∫ t+T t α(s)ds ≥ µ for every t ≥ 0, with 0 < µ ≤ T independent on t. We prove that such a system is stabilizable with a linear feedback depending only on the pair (T, µ) if the eigenvalues of A have non-positive real part. We also show that stabilizability does not hold for arbitrary matrices A. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter µ/T. 1
On the marginal instability of linear switched systems. Systems and Control Letters, to appear
, 2012
"... a b s t r a c t Stability properties for continuous-time linear switched systems are at first determined by the (largest) Lyapunov exponent associated with the system, which is the analogue of the joint spectral radius for the discrete-time case. The purpose of this paper is to provide a characteri ..."
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Cited by 8 (0 self)
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a b s t r a c t Stability properties for continuous-time linear switched systems are at first determined by the (largest) Lyapunov exponent associated with the system, which is the analogue of the joint spectral radius for the discrete-time case. The purpose of this paper is to provide a characterization of marginally unstable systems, i.e., systems for which the Lyapunov exponent is equal to zero and there exists an unbounded trajectory, and to analyze the asymptotic behavior of their trajectories. Our main contribution consists in pointing out a resonance phenomenon associated with marginal instability. In the course of our study, we derive an upper bound of the state at time t, which is polynomial in t and whose degree is computed from the resonance structure of the system. We also derive analogous results for discrete-time linear switched systems.
On the convergence of linear switched systems
- IEEE Trans. Automat. Control
"... This paper investigates sufficient conditions for the convergence to zero of the trajecto-ries of linear switched systems. We provide a collection of results that use weak dwell-time, dwell-time, strong dwell-time, permanent and persistent excitation hypothesis. The ob-tained results are shown to be ..."
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Cited by 7 (0 self)
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This paper investigates sufficient conditions for the convergence to zero of the trajecto-ries of linear switched systems. We provide a collection of results that use weak dwell-time, dwell-time, strong dwell-time, permanent and persistent excitation hypothesis. The ob-tained results are shown to be tight by counterexample. Finally, we apply our result to the three-cell converter.
