On the stability of a convex set of matrices

Dzhafarov, Vakıf; Büyükköroğlu, Taner

doi:10.1016/j.laa.2005.10.044

Cited by 8 publications

(6 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, the supremum of the 2 norm of the set of vectors x satisfying (A(ρ) − λ i I) x = 0 is infinity. Since the constraint x 2 ≤ 1 is present in (16), the solution of problem (16) is 1, contradicting the hypothesis. Proof It follows straightforwardly from Theorem 2 and its proof.…”

Section: Checking Determinist D-stabilitymentioning

confidence: 91%

“…The constraints in (18c) and (18e) are simply the "probabilistic version" of the determinist constraints in (16), and they are used to describe the support of the probability measures Pr ρ,x,λ . The constraint (18d) includes the information in (3) on the (generalized) moments of the probability measures Pr ρ , i.e., P ρ ∈ P ρ .…”

Section: Checking Probabilistic D-stabilitymentioning

confidence: 99%

“…A branch and bound algorithm is then proposed in [12] to solve larger scale problems. Interval and polytopic matrices are considered in [13,14,15,16,17,18,19,20]. The works in [13,14] derive intervals where the real eigenvalues of interval matrices are guaranteed to belong to, and a vertex result is presented in [15] to reduce the computational load in evaluating quadratic stability of interval matrices.…”

mentioning

confidence: 99%

“…Results in [15] can be also used in the case of multiaffine interval matrix uncertainty. Bernstein expansion is used in [16] to check robust nonsingularity of a polytope of real matrices, and sufficient LMI conditions coming from the Lyapunov theory are derived in [17,18,19,20] for checking robust Hurwitz and Schur stability of matrices with polytopic uncertainty. In [21,22,23,24], less conservative LMI conditions to check robust D-stability of uncertain polynomial matrices are derived.…”

mentioning

confidence: 99%

“…Nevertheless, some heuristics can be used to verify, from the solution of problem (43), if the Hamiltonian H has some eigenvalues on the imaginary axis. In fact, when Lasserre's hierarchy is used to relax (deterministic) polynomial optimization problems (like ( 16)), the first order moments m α of the SDP relaxed problem (43) provides, in practice, a good approximation (ρ, x, λ) of the global minimizer (ρ * , x * , λ * ) of the original optimization problem (16). By looking at the first order moments m α associated to the uncertain parameters ρ, we obtain For this values of the uncertainty ρ, we obtain G cl ∞ = 1.013.…”

mentioning

confidence: 99%

See 4 more Smart Citations

A Unified Framework for Deterministic and Probabilistic $\mathscr {D}$-Stability Analysis of Uncertain Polynomial Matrices

Piga¹,

Benavoli²

2017

IEEE Trans. Automat. Contr.

View full text Add to dashboard Cite

ForewordThis report is an extended version of the paper A unified framework for deterministic and probabilistic D-stability analysis of uncertain polynomial matrices submitted by the authors to the IEEE Transactions on Automatic Control. AbstractIn control theory, we are often interested in robust D-stability analysis, which aims at verifying if all the eigenvalues of an uncertain matrix lie in a given region D of the complex plane. Although many algorithms have been developed to provide conditions for an uncertain matrix to be robustly D-stable, the problem of computing the probability of an uncertain matrix to be D-stable is still unexplored. The goal of this paper is to fill this gap by generalizing algorithms for robust D-stability analysis in two directions. First, the only constraint on the stability region D that we impose is that its complement is a semialgebraic set described by polynomial constraints. This comprises main important cases in robust control theory. Second, the D-stability analysis problem is formulated in a probabilistic framework, by assuming that only few probabilistic information is available on the uncertain parameters, such as support and some moments. We will show how to efficiently compute the minimum probability that the matrix is D-stable by using convex relaxations based on the theory of moments. We will also show that standard robust D-stability is a particular case of the more general probabilistic D-stability problem. Application to robustness and probabilistic analysis of dynamical systems is discussed.In control theory, we are often interested in robust D-stability analysis, which aims at verifying if all the eigenvalues of an uncertain matrix lie in a given region D of the complex plane. Although many algorithms have been developed to provide conditions for an uncertain matrix to be robustly D-stable, the problem of computing the probability of an uncertain matrix to be D-stable is still unexplored. The goal of this paper is to fill this gap by generalizing algorithms for robust D-stability analysis in two directions. First, the only constraint on the stability region D that we impose is that its complement is a semialgebraic set described by polynomial constraints. This comprises main important cases in robust control theory. Second, the D-stability analysis problem is formulated in a probabilistic framework, by assuming that only few probabilistic information 1 arXiv:1604.02031v3 [math.OC] 17 Jun 2018 is available on the uncertain parameters, such as support and some moments. We will show how to efficiently compute the minimum probability that the matrix is D-stable by using convex relaxations based on the theory of moments. We will also show that standard robust D-stability is a particular case of the more general probabilistic D-stability problem. Application to robustness and probabilistic analysis of dynamical systems is discussed.To this end, we develop a unified framework for deterministic (robust) and probabilistic D-stability analysis. A semi-infinite linear progr...

show abstract

Section: Checking Determinist D-stabilitymentioning

confidence: 91%

Section: Checking Probabilistic D-stabilitymentioning

confidence: 99%