Stability and robust stability for systems with a time-varying delay

To concern the stability and robust stability criteria for systems with time-varying delays, this note uses not only the time-varying-delayed state x(t-h(t)) but also the delay-upper-bounded state to exploit all possible information for the relationship among a current state x(t), an exactly delayed state x(t-h(t)), a marginally delayed state , and the derivative of the state , when constructing Lyapunov-Krasovskii functionals and some appropriate integral inequalities, originally suggested by Park (1999. A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Transactions on Automatic Control, 44(4), 876-877). Two fundamental criteria are provided for the cases where no bound of delay derivative is assumed and where an upper bound of delay derivative is assumed. Examples show the resulting criteria outperform all existing ones in the literature.     

Invariance properties in the root sensitivity of time-delay systems with double imaginary roots

If is an eigenvalue of a time-delay system for the delay τ₀ then is also an eigenvalue for the delays τ_k?τ₀+k2π/ω, for any k∈Z. We investigate the sensitivity, periodicity and invariance properties of the root for the case that is a double eigenvalue for some τ_k. It turns out that under natural conditions (the condition that the root exhibits the completely regular splitting property if the delay is perturbed), the presence of a double imaginary root for some delay τ₀ implies that is a simple root for the other delays τ_k, k≠0. Moreover, we show how to characterize the root locus around . The entire local root locus picture can be completely determined from the square root splitting of the double root. We separate the general picture into two cases depending on the sign of a single scalar constant; the imaginary part of the first coefficient in the square root expansion of the double eigenvalue.     

Design of switching sequences for controllability realization of switched linear systems

In this paper, we study the problem of designing switching sequences for controllability of switched linear systems. Each controllable state set of designed switching sequences coincides with the controllable subspace. Both aperiodic and periodic switching sequences are considered. For the aperiodic case, a new approach is proposed to construct switching sequences, and the number of switchings involved in each designed switching sequence is shown to be upper bounded by d(d-d₁+1). Here d is the dimension of the controllable subspace, , where (A_i,B_i) are subsystems. For the periodic case, we show that the controllable subspace can be realized within d switching periods.     

Some new results for system decoupling and pole assignment problems

In a related article, we derived a canonical decomposition of the right invertible system {C,A,B} and applied this canonical decomposition to study the Smith form of the matrix pencil and findout the finite zeros and infinite zeros of P(s), the range of the ranks of P(s) for s∈C, and the controllability of the right invertible system.In this paper, we will apply this canonical decomposition of the right invertible system {C,A,B} to deduce the triangular decouple upon to row permutation, provide some new results of the row-by-row decoupling, and associated pole assignment problems.     

Injectivity analysis using interval analysis: Application to structural identifiability

This paper presents a new numerical algorithm based on interval analysis able to prove that a differentiable function is injective. This algorithm also performs a partition of the domain A in subsets A_i where, for all , the cardinality of is constant. In the context of parameter estimation, we show how this algorithm provides an efficient and numerical method to study the structural identifiability of parametric models.     

Stability of switching infinite-dimensional systems

In this note, we generalize the results from Narendra and Balakrishnan (IEEE Trans. Automatic Control 39 (1994) 2469) to the infinite-dimensional system theoretic setting. The paper gives results on the stability of a switching system of the form , i∈{1,2}, when the infinitesimal generators A₁ and A₂ commute. In addition, the existence of a common quadratic Lyapunov function is demonstrated.     

A combinatorial algorithm for max csp

We consider the problem max csp over multi-valued domains with variables ranging over sets of size s_i?s and constraints involving k_j?k variables. We study two algorithms with approximation ratios A and B, respectively, so we obtain a solution with approximation ratio max(A,B).The first algorithm is based on the linear programming algorithm of Serna, Trevisan, and Xhafa [Proc. 15th Annual Symp. on Theoret. Aspects of Comput. Sci., 1998, pp. 488-498] and gives ratio A which is bounded below by s^1−k. For k=2, our bound in terms of the individual set sizes is the minimum over all constraints involving two variables of , where s₁ and s₂ are the set sizes for the two variables.We then give a simple combinatorial algorithm which has approximation ratio B, with B>A/e. The bound is greater than s^1−k/e in general, and greater than s^1−k(1−(s−1)/2(k−1)) for s?k−1, thus close to the s^1−k linear programming bound for large k. For k=2, the bound is if s=2, 1/2(s−1) if s?3, and in general greater than the minimum of 1/4s₁+1/4s₂ over constraints with set sizes s₁ and s₂, thus within a factor of two of the linear programming bound.For the case of k=2 and s=2 we prove an integrality gap of . This shows that our analysis is tight for any method that uses the linear programming upper bound.     

Explicit convergence rates for MRAC-type systems

It is well known since many years ago that for the linear time-varying system , with A Hurwitz, and B(t) bounded and globally Lipschitz, it is necessary and sufficient for global exponential stability, that B(t) satisfy the so-called persistency of excitation condition. In this note, we revisit this question and provide explicit bounds for the convergence rate and on the overshoot of the transient behaviour of the solutions e(t), θ(t) as functions of the richness of B(·). We believe that the result that we present is useful since knowing convergence rates aids in the construction of converse Lyapunov functions. Moreover, the type of systems that we study here appear for instance in model reference adaptive control (MRAC).