An improved version of the singularity-induced bifurcation theorem

It has been shown recently that there is a new type of codimension one bifurcation, called the singularity-induced bifurcation (SIB), arising in parameter dependent differential-algebraic equations (DAEs) of the form x˙=f and 0=g, and which occurs generically when an equilibrium path of the DAE crosses the singular surface defined by g=0 and det g_y=0. The SIB refers to a stability change of the DAE owing to some eigenvalue of a related linearization diverging to infinity when the Jacobian g_y is singular. In this article an improved version (Theorem 1.1) of the SIB theorem with its simple proof is given, based on a decomposition theorem (Theorem 2.1) of parameter dependent polynomials     

The Hopf bifurcation theorem for quasilinear differential-algebraic equations

We prove a generalization of the Hopf bifurcation theorem for quasilinear differential equations (DAEs), i.e. equations of the form A(μ, χ)χ = G(μ, χ) where the matrix A(χ, μ) has constant but not full rank and hence the system cannot be made into an explicit ODE. The paper includes an appendix by J. Ernsthausen addressing the numerical calculation of the Hopf points in the DAE setting.     

On the cobham-semenov theorem

We prove that under reasonable conditions, the Cobham-Semenov theorem remains true for two numeration systems, one of which is nonstandard.     

On the generalized Cayley-Hamilton theorem

The Cayley-Hamilton theorem is extended to the case where twoAandB n times nmatrices are involved. Results similar to the regular case are presented. The results are useful for problems that concern the analysis as well as the synthesis of singular systems, for the definition of a function of two matricesf(A, B)and in various other problems in linear algebra.     

On the Klimushchev–Krasovskii theorem

The Klimushchev–Krasovskii theorem describes a sufficient condition for the asymptotic stability of singularly perturbed systems. Motivated by some subtle examples, this note presents a new necessary and sufficient condition by imposing an additional assumption on the statement of the Klimushchev–Krasovskii theorem. This note also extends the main inertia theorem to singularly perturbed systems.     

On the Shyr–Yu theorem

An alternative proof of Shyr–Yu theorem is given. Some generalizations are also considered using fractional root decompositions and fractional exponents of words.     

On theorem prover-based testing

HOL-TestGen is a specification and test case generation environment extending the interactive theorem prover Isabelle/HOL. As such, Testgen allows for an integrated workflow supporting interactive theorem proving, test case generation, and test data generation. The HOL-TestGen method is two-staged: first, the original formula is partitioned into test cases by transformation into a normal form called test theorem. Second, the test cases are analyzed for ground instances (the test data) satisfying the constraints of the test cases. Particular emphasis is put on the control of explicit test-hypotheses which can be proven over concrete programs. Due to the generality of the underlying framework, our system can be used for black-box unit, sequence, reactive sequence and white-box test scenarios. Although based on particularly clean theoretical foundations, the system can be applied for substantial case-studies.     

On a large-gain theorem

The aim of this paper is to give a general quantitative requirement which the loop gain must satisfy in order to stabilize a given unstable (possibly nonlinear and time-varying) plant, namely that the gain must exceed one.     

On the Lyapunov theorem for singular systems

In this paper, we revisit the Lyapunov theory for singular systems. There are basically two well-known generalized Lyapunov equations used to characterize stability for singular systems. We start with the Lyapunov theorem of the work by Lewis. We show that the Lyapunov equation of that theorem can lead to incorrect conclusion about stability. Some cases where that equation can be used are clarified. We also show that an attempt to correct that theorem with a generalized Lyapunov equation similar to the original one leads naturally to the generalized equation of Takaba et al.     

On the scale-space theorem of Chen and Yan

Chen and Yan presented (ibid., vol.11, p.694-700, 1989) a theorem concerning zero crossings of boundary curvature under morphological openings. In this correspondence, we show by means of a counterexample a flaw in this theorem and suggest how the theorem may be modified to make it correct     

On the irreducibility condition in the structural controllability theorem

It is known that the structural system (A,B) is structurally controllable if and only if the corresponding matrix [A B] is generically full rank and irreducible. In this paper it is shown that the irreducibility condition alone implies that every nonzero mode of (A,B) is generically controllable. This result provides an easy proof to the structural controllability theorem stated above. In addition, it is shown that the basic structure of the Jordan canonical form of (A,B) remains unaffected, in the generic sense, under the variation of the free parameters of (A,B).     

On the fundamental theorem of linear state variable feedback

The important theorem due to Rosenbrock, concerning the limits of linear state variable feedback in modifying the dynamics of constant linear multivariable systems, is obtained from state space considerations. This approach illuminates the results which were originally obtained by polynomial system matrix methods.     

On the role of unification in mechanical theorem proving

Summary A generalization of the Skolem-Herbrand-Gödel Theorem is stated and proved, clarifying the role of unification in resolution. It is shown that any of the pre-resolution theorem-proving procedures of the Davis and Putnam variety can be upgraded to incorporate unification.This work was supported in part by the National Science Foundation and the Advanced Research Projects Agency of the U. S. Government.     

On a theorem of Megan and Zabczyk

It will be proved that for any linear infinite-dimensional control system [xdot](t) = Ax(t) + Bu(t) and for any p ? [1, ∞], the implication that ‘complete stabilizabitity ? the T-controllability operator for L ^p -controls is surjective’ holds true provided A generates a strongly continuous group of bounded linear operators. This extends a theorem by Megan and Zabczyk in several directions. In particular, not necessarily separable Banach spaces are allowed, and also the class of controls which are sufficient to ensure exact controllability is restricted.     

On Herglotz theorem in partially ordered vector spaces

Herglotz theorem is generalized for a monotone σ-complete partially ordered vector space.     

Comments on "On structural controllability theorem"

In the above mentioned paper, Mayeda gave an elegant proof to the structural controllability theorem. This note suggests further simplification of that proof using graph-theoretic interpretation of some determinants.     

On the use of bifurcation analysis for robust controller tuning for nonlinear systems

This paper presents a new approach to robust controller tuning for nonlinear systems. The main idea is to apply numerical bifurcation analysis to the closed-loop process, using the controller tuning parameters, the set points, and parameters describing model uncertainty (parametric as well as unmodeled dynamics) as bifurcation parameters. By analyzing the Hopf bifurcation and saddle-node bifurcation loci, bounds on the controller parameters are identified. These bounds depend upon the type as well as the degree of mismatch that exists between the plant and the model used for controller design.The method is illustrated by tuning a state feedback linearizing controller for an unstable reactor as well as by comparing a proportional-integral (PI) controller and a state feedback linearizing controller applied to a continuously operated fermenter. The feedback linearizing controller can result in better performance than the PI controller if a very accurate model of the process is known and if the operating conditions vary over a significant range. However, stability properties of systems controlled by feedback linearizing controllers can degrade significantly as the mismatch between the plant and the model increases. This is illustrated in the fermenter example by showing that bounds on the tuning parameter of the feedback linearizing controller are significantly tighter than the ones for the PI controller.