M. De la Sen

About Robust Stability of Caputo Linear Fractional Dynamic Systems with Time Delays through Fixed Point Theory

This paper investigates the global stability and the global asymptotic stability independent of the sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional order possessing internal point delays. The investigation is performed via fixed point theory in a complete metric space by defining appropriate nonexpansive or contractive self-mappings from initial conditions to points of the state-trajectory solution. The existence of a unique fixed point leading to a globally asymptotically stable equilibrium point is investigated, in particular, under easily testable sufficiency-type stability conditions. The study is performed for both the uncontrolled case and the controlled case under a wide class of state feedback laws.This paper investigates the global stability and the global asymptotic stability independent of the sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional order possessing internal point delays. The investigation is performed via fixed point theory in a complete metric space by defining appropriate nonexpansive or contractive self-mappings from initial conditions to points of the state-trajectory solution. The existence of a unique fixed point leading to a globally asymptotically stable equilibrium point is investigated, in particular, under easily testable sufficiency-type stability conditions. The study is performed for both the uncontrolled case and the controlled case under a wide class of state feedback laws.

Preserving positive realness through discretization

The importance of positive real transfer functions relies on the fact that they are associated with positive linear systems. Those systems possess the property that their input-output product time-integral, which is a measure of the total enerty, is nonnegative. Such a property can be also formulated in the discrete context. It is shown that a discrete positive real transfer function is obtained from a positive real continuous one of relative order zero being strictly stable poles via discretization by a sampler and zero-order hold device provided that the direct input-output transmission gain is sufficiently large. It is also proved that a discrete positive real transfer function may be obtained from a stable continuous one of relative order zero and high direct input-output gain which posses simple complex conjugate critically stable poles even in the case that this one is not positive real. For that purpose, the use of an appropriate phase-lag or phase lead compensating network for the continuous transfer function may be required to ensure positive realness of the discrete transfer function.It is shown that a discrete positive real transfer function is obtained from a positive real continuous one of relative order zero being strictly stable poles via discretization by a sampler and zero-order hold device provided that the direct input-output transmission gain is sufficiently large. Also, a discrete positive real transfer function may be obtained from a stable continuous one of relative order zero and high direct input-output gain which possess simple complex conjugate critically stable poles even in the case that this one is not positive real. For that purpose, the use of an appropriate phase lag or phase lead compensating network for the continuous transfer function may be required to ensure positive realness of the discrete transfer function.

Quadratic stability and stabilization of switched dynamic systems with uncommensurate internal point delays

Abstract This paper deals with the quadratic stability and linear state-feedback and output-feedback stabilization of switched delayed linear dynamic systems with, in general, a finite number of non-commensurate constant internal point delays. The results are obtained based on Lyapunov’s stability analysis via appropriate Krasovsky–Lyapunov’s functionals and the related stability study is performed to obtain both delay independent and delay dependent results. It is proved that the stabilizing switching rule is arbitrary if all the switched subsystems are quadratically stable and that it exists a (in general, non-unique) stabilizing switching law when the system is polytopic, stable at some interior point of the polytope but with non-necessarily stable parameterizations at the vertices defining the subsystems. It is also proved that two subsystems individually parametrized in different polytopic-type sets which are not quadratically stable might be stabilized by a (in general, non-unique) switching law provided that a convexity-type condition is fulfilled at each existing pair of vertices, one corresponding to each subsystem. Some extensions are provided for a typical class of neutral time-delay systems.

Linking Contractive Self-Mappings and Cyclic Meir-Keeler Contractions with Kannan Self-Mappings

Some mutual relations between p-cyclic contractive self-mappings, p-cyclic Kannan self-mappings, and Meir-Keeler p-cyclic contractions are stated. On the other hand, related results about the existence of the best proximity points and existence and uniqueness of fixed points are also formulated.Some mutual relations between p-cyclic contractive self-mappings, p-cyclic Kannan self-mappings, and Meir-Keeler p-cyclic contractions are stated. On the other hand, related results about the existence of the best proximity points and existence and uniqueness of fixed points are also formulated.

On the robust adaptive stabilization of a class of nominally first-order hybrid systems

This paper presents a robust adaptive stabilization scheme for a class of nominally first-order and controllable hybrid time-invariant linear systems involving both continuous and discrete parametrizations and signals. The usual assumptions of inverse stability of the plant and knowledge of the high-frequency gain are not required. The design philosophy relies on the separation of the contributions of continuous and discrete dynamics to the output by generating two associate additive terms to build the control action. The controllability of the nominally estimated plant model is maintained by using a hysteresis switching function. This allows relaxing the assumption of stability of the plant inverse.

The reachability and observability of hybrid multirate sampling linear systems

This paper investigates algebraic sufficient, necessary, and sufficient conditions for sampling time reachability and observability in hybrid multirate sampling data linear systems for the cases of fast input and fast output sampling rates. Some implications in the context of closed-loop pole-placement under both properties under multirate sampling are heuristically discussed.

A note on the stability of linear time-delay systems with impulsive inputs

This brief derives sufficiency-type stability results for time-delay linear systems with constant point delays under impulsive inputs of impulses of state-dependent amplitudes occurring separately through time. It is proved that the amplitudes of the impulses and the time intervals between impulses may be chosen sufficiently large if the delay-free dynamics is sufficiently stable compared to the delayed one. It is also seen that faster input impulses of appropriate amplitudes and signs may achieve stabilization if the delay-free dynamics is not exponentially stable.

On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays

This paper deals with the positivity properties of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate constant point delays. The main idea behind the investigation is that its main body is performed based on the construction of the whole state-space trajectory solution without using as usual equivalence or similarity transformations on the matrix of dynamics in order to split the state-trajectory solution into two parts, one being typically associated with a nilpotent matrix. In that way, the whole state trajectory solution contains impulsive terms associated with the initial conditions and inputs. Some extensions concerning positivity aspects are given for a special canonical form which separates the dynamics associated with the nilpotent matrix obtained from an equivalence transformation on the singular matrix of the dynamic system.

Output feedback sliding mode control of base isolated structures

Abstract The paper addresses the problem of designing stabilizing output feedback controllers for a class of nonlinear seismically excited base isolated structures with unknown parametrical uncertainties. Two types (non-adaptive and adaptive) of sliding mode controllers are presented. In both schemes, only the information on the displacements and velocities of the base and the first floor is used in the controller design. A numerical example is given to illustrate the effectiveness of the proposed strategy to a 10-storey frictional base isolated structure being subject to the El Centro earthquake.

Sliding-Mode Control of Wave Power Generation Plants

The worldwide demand for energy requires alternatives to fossil fuels and nuclear fission, so renewable resources, particularly ocean energy, are called to play a relevant role in a near future. In particular, the oscillating water column (OWC) is one of the most promising devices to harness energy from the sea, as it is the case of the Nereida project plant, located in the Basque coast of Mutriku. This kind of devices consists of a particular type of turbine and a doubly fed induction generator. The turbogenerator module is usually controlled using a traditional proportional-integral (PI)-based vector control scheme, which requires an accurate knowledge of the system parameters and lacks of robustness, limiting, in some cases, the power extraction. To avoid these drawbacks, a novel sliding-mode-control-based vector control scheme for the OWC plant is presented in this paper. This variable-structure control is intrinsically robust under parameter uncertainties, which always appear in real systems, and presents a convenient disturbance rejection. The stability of the proposed controller is analyzed using the Lyapunov theory. The performance of the control scheme presented is proved by comparing it to the traditional PI-based vector control scheme in a series of representative maximum power generation case studies. Both numerical simulations and experimental results show that the proposed solution provides high-performance dynamic characteristics, improving the power extraction in spite of parameter uncertainties and system disturbances.