Strong Stabilization of the Acrobot at the Down–Up Equilibrium Point⋆

In this paper, we investigate the strong stabilization of folded configurations (two links fold onto each other) of the Acrobot, a two-link underactuated gymnastic robot, utilizing a single sensor to obtain a single output. We show that this strong stabilization problem is equivalent to that of a fourth-order linear plant having a pair of conjugate imaginary poles and a pair of symmetric real poles around the origin, with its zeros that vary depending on the sensor's location. We derive an analytical expression determining all the sensor's location range ensuring the strong stabilizability of the linear plant. For both cases where the finite zeros symmetric about the origin are located within a certain range on either the real or imaginary axes, we accordingly provide parameterisation schemes for minimal-order strongly stabilizing controllers based on the Liénard–Chipart criterion. We present two numerical examples and simulation results to validate the robustness of this strong stabilization approach.

A strongly stabilization is the control method to make control systems stable using stable controllers. Using this method, it is possible to construct a highly reliable control system because it has robustness. The controller needs a pole at the origin to follow the step reference input without steady-state error. A semi-strongly stabilizing controller is proposed that has a pole at the origin and the others in the open left half plane. In some cases, the controller needs to have poles on the imaginary axis; for example, the control systems follow sinusoidal signals and the self-repairing control systems with faulty sensors using resonance. Therefore, Kimura proposed extended semi-strongly stabilizing controllers with poles on the imaginary axis and other poles in the open left-half plane. Niiyama propose control system with failure detection by adapting the extended semi-strongly stabilizing controller to the self-repairing control system with failure sensor using resonance. However, this method remains an error between the output and the reference input. To eliminate this error the controller needs a pole at the origin and two pairs of poles on the imaginary axis. In this paper, to overcome this problem, we propose an extended semi-strongly stabilizing controller with a pole at the origin and two pairs of poles on the imaginary axis. In addition, we clarify the parameterization of all extended semi-strongly stabilizable plants and of all extended semi-strongly stabilizing controllers for extended semi-strongly stabilizable plants.

We consider a system described by the one-dimensional linear wave equation in a bounded domain with appropriate boundary conditions. To stabilize the system, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the proposed controller is a proper rational function of the complex variable and may contain a single pole at the origin and a pair of complex conjugate poles on the imaginary axis, provided that the residues corresponding to these poles are nonnegative; the rest of the transfer function is required to be a strictly positive real function. We then show that depending on the location of the pole on the imaginary axis, the closed-loop system is asymptotically stable. We also consider the case where the output of the controller is corrupted by a disturbance and show that it may be possible to attenuate the effect of the disturbance at the output if we choose the controller transfer function appropriately. We also present some numerical simulation results which support this argument.

Abstract. Microbial-explicit soil organic carbon (SOC) cycling models are increasingly being recognized for their advantages over linear models in describing SOC dynamics. These models are known to exhibit oscillations, but it is not clear when they yield stable vs. unstable equilibrium points (EPs) – i.e., EPs that exist analytically but are not stable in relation to small perturbations and cannot be reached by transient simulations. The occurrence of such unstable EPs can lead to unexpected model behavior in transient simulations or unrealistic predictions of steady-state soil organic carbon (SOC) stocks. Here, we ask when and why unstable EPs can occur in an archetypal microbial-explicit model (representing SOC, dissolved OC (DOC), microbial biomass, and extracellular enzymes) and some simplified versions of it. Further, if a model formulation allows for physically meaningful but unstable EPs, can we find constraints in the model parameters (i.e., environmental conditions and microbial traits) that ensure stability of the EPs? We use analytical, numerical, and descriptive tools to answer these questions. We found that instability can occur when the resupply of a growth substrate (DOC) is (via a positive feedback loop) dependent on its abundance. We identified a conservative, sufficient condition in terms of model parameters to ensure the stability of EPs. Principally, three distinct strategies can avoid instability: (1) neglecting explicit DOC dynamics, (2) biomass-independent uptake rate, or (3) correlation between parameter values to obey the stability criterion. While the first two approaches simplify some mechanistic processes, the third approach points to the interactive effects of environmental conditions and parameters describing microbial physiology, highlighting the relevance of basic ecological principles for the avoidance of unrealistic (i.e., unstable) simulation outcomes. These insights can help to improve the applicability of microbial-explicit models, aid our understanding of the dynamics of these models, and highlight the relation between mathematical requirements and (in silico) microbial ecology.

A stable multimodel scheme control for the regulation of the transient behavior of a tunnel-diode trigger circuit

This study presents a study of equilibrium points, periodic orbits, stabilities, and manifolds in a rotating plane-symmetric potential field. It has been found that the dynamical behaviour near equilibrium points is completely determined by the structure of the submanifolds and subspaces. The non-degenerate equilibrium points are classified into twelve cases. The necessary and sufficient conditions for linearly stable, non-resonant unstable and resonant equilibrium points are established. Furthermore, the results show that a resonant equilibrium point is a Hopf bifurcation point. In addition, if the rotating speed changes, two non-degenerate equilibria may collide and annihilate each other. The theory developed here is lastly applied to two particular cases, motions around a rotating, homogeneous cube and the asteroid 1620 Geographos. We found that the mutual annihilation of equilibrium points occurs as the rotating speed increases, and then the first surface shedding begins near the intersection point of the −x axis and the surface. The results can be applied to planetary science, including the birth and evolution of the minor bodies in the Solar system, the rotational breakup and surface mass shedding of asteroids, etc.

Multiple μ-stability analysis of complex-valued neural networks with unbounded time-varying delays

Chaotic dynamics and chaos control for the fractional-order geomagnetic field model

Continuity and equilibrium stability

Quantifying the unstable in linearized nonlinear systems

Stability analysis of high-order Hopfield type neural networks with uncertainty

A topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented. The stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points (and/or closed orbits) on the stability boundary. Several necessary and sufficient conditions are derived to determine whether a given equilibrium point (or closed orbit) is on the stability boundary. A method for finding the stability region on the basis of these results is proposed. The method, when feasible, will find the exact stability region, rather than a subset of it as in the Lyapunov theory approach. Several examples are given to illustrate the theoretical prediction. >

In order to understand the complex interactions between different technologies in a communications market, it is of fundamental importance to understand how technologies affect the demand of users and competition between network service providers (NSPs). To this end, we analyze user subscription dynamics and revenue maximization in monopoly and duopoly communications markets. First, by considering a monopoly market with only one NSP, we investigate the impact of technologies on the users' dynamic subscription. It is shown that, for any price charged by the NSP, there exists a unique equilibrium point of the considered user subscription dynamics. We also provide a sufficient condition under which the user subscription dynamics converges to the equilibrium point starting from any initial point. We then derive upper and lower bounds on the optimal price and market share that maximize the NSP's revenue. Next, we turn to the analysis of a duopoly market and show that, for any charged prices, the equilibrium point of the considered user subscription dynamics exists and is unique. As in a monopoly market, we derive a sufficient condition on the technologies of the NSPs that ensures the user subscription dynamics to reach the equilibrium point. Then, we model the NSP competition using a non-cooperative game, in which the two NSPs choose their market shares independently, and provide a sufficient condition that guarantees the existence of at least one pure Nash equilibrium in the market competition game.

In this paper, we obtain new sufficient conditions ensuring existence, uniqueness, and global asymptotic stability (GAS) of the equilibrium point for a general class of delayed neural networks (DNNs) via nonsmooth analysis, which makes full use of the Lipschitz property of functions defining DNNs. Based on this new tool of nonsmooth analysis, we first obtain a couple of general results concerning the existence and uniqueness of the equilibrium point. Then those results are applied to show that existence assumptions on the equilibrium point in some existing sufficient conditions ensuring GAS are actually unnecessary; and some strong assumptions such as the boundedness of activation functions in some other existing sufficient conditions can be actually dropped. Finally, we derive some new sufficient conditions which are easy to check. Comparison with some related existing results is conducted and advantages are illustrated with examples. Throughout our paper, spectral properties of the matrix (A + Atau) play an important role, which is a distinguished feature from previous studies. Here, A and Atau are, respectively, the feedback and the delayed feedback matrix defining the neural network under consideration.

If V is a Lyapunov function of an equation du/dt = u′ = Zu in a Banach space then asymptotic stability of an equilibrium point may be easily proved if it is known that sup(V′) < 0 on sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is proved for a bounded infinitesimal generator Z under a weaker assumption V′ ≤ 0 (which alone implies ordinary stability only) if some observability condition, involving Z and the Frechet derivative of V′, is satisfied. The proof is based on an extension of LaSalle′s invariance principle, which yields convergence in a weak topology and uses a strongly continuous Lyapunov function. The theory is illustrated with an example of an integro‐differential equation of interest in the theory of chemical processes. In this case strong asymptotic stability is deduced from the weak one and explicit sufficient conditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbert space, strong asymptotic stability is proved under the following assumptions Z*+Z is weakly negative definite and Ker Z = {0}. The proof is based on spectral theory.