Sliding mode control for a nonlinear phase-field system

In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state-feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.Comment: Key words: phase field system, nonlinear boundary value problems, phase transition, sliding mode control, state-feedback control la

Global existence for a singular phase field system related to a sliding mode control problem

In the present contribution we consider a singular phase field system located in a smooth and bounded three-dimensional domain. The entropy balance equation is perturbed by a logarithmic nonlinearity and by the presence of an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. The second equation of the system accounts for the phase dynamics, and it is deduced from a balance law for the microscopic forces that are responsible for the phase transition process. The resulting system is highly nonlinear; the main difficulties lie in the contemporary presence of two nonlinearities, one of which under time derivative, in the entropy balance equation. Consequently, we are able to prove only the existence of solutions. To this aim, we will introduce a backward finite differences scheme and argue on this by proving uniform estimates and passing to the limit on the time step.Comment: Key words: Phase field system; maximal monotone nonlinearities; nonlocal terms; initial and boundary value problem; existence of solution

Nonlinear Control Approaches of Several Power Converters

The exploitation and utilization of renewable energy have become the important measures taken by countries all over the world to solve the contradiction among energy shortage, economic development and ecological environment. As an important bridge and energy conversion channel between renewable energy and electric energy, power converter is a basic form of transformation and control of electric energy, which plays a vital role in the exploitation and utilization of renewable energy. In the past few decades, due to the significance of both theory and practical applications, the study of power converter has become one of the hotspots of research in the field of power electronics and automation. The study of power converter control strategies is an important study research of power converter. The control strategy as the core of the control system directly determines the dynamic and static performance of the power converter. The power converter system is a typical nonlinear system. However, most of power converter control strategies are designed based on linearization control methods, which makes the control system sensitive to system parameter variations, slow dynamic response speed and poor steady-state performance, etc. Thus, in order to further improve the dynamic and static performances of power converters, the investigation of using nonlinear control methods for power converters is a challenging and meaningful work. Based on modern control theory utilizing nonlinear control approaches, this dissertation investigates the nonlinear control strategies design for several typical power converters, and the main contributions are as follows: (1) The important role of power converters in renewable energy power generation systems is introduced. Then the internal and overseas research situations of the control strategies design for the several typical power converters are classified and summarized, where the theoretical significance and the practical application backgrounds are given, and the study structures and contents of this dissertation are presented. (2) Two system models are built for the DC-DC Buck converters, respectively, i.e., the nominal system model and uncertain system model. Based on the nominal system without considering parametric uncertainties, the single-loop adaptive control strategy is built by adaptive and back-stepping control approach, and the double-loop adaptive control strategy is set up by adaptive and sliding mode control approach. Based on the uncertain system model, the single-loop disturbance observer based control strategy is developed using designed disturbance observer and back-stepping control technique, and the double-loop disturbance observer based control strategy is synthesized using designed disturbance observer and sliding mode control method. (3) The control strategy of voltage regulation and current tracking for three phase two-level grid-connected power rectifiers is presented. By using power-invariant Park’s transformation, an averaged mathematical model of power converters is obtained in synchronous reference frame. Then a novel control strategy using adaptive control and technique is proposed to regulate the dc-link output voltage as well as track a desired current reference. More specifically, an efficient adaptive controller is established in the external loop for regulating dc-link output voltage in the presence of external disturbances. A set of controllers are designed in the internal loop to force the input currents track their desired values. (4) A novel robust control strategy is proposed for three-level neutral-point-clamped power rectifiers. The proposed control scheme consists of three control loops, i.e., instantaneous power tracking control loop, voltage regulation loop and voltage balancing loop. First, in the power tracking control loop, a set of adaptive sliding mode controllers are established to drive the active and reactive power tracking their desired values via radial basis function neural network technology. In the voltage regulation loop, an efficient but simple adaptive controller is designed to regulate dc-link output voltage where the load is considered as an external disturbance. Moreover, a composite controller is developed in the voltage balancing loop to ensure imbalance voltages between two dc-link capacitors close to zero, in which a reduced-order observer is used to estimate sinusoidal disturbance improving the converter performance. (5) Based on the second order sliding mode control technique, a novel control strategy is proposed for three-phase power rectifiers under unbalanced grid conditions to achieve cooperative control between power and current. A consolidated control objective which can be flexibly adjust among the degree of oscillation in active and reactive powers and balance of three-phase current is obtained in the stationary frame. Based on the dynamic of the converter and control objective, a control scheme in a cascaded framework is presented, in which an adaptive observer is applied to estimate the positiveand negative-sequence of grid voltage without complex filtering process. In the current tracking loop, the super–twisting algorithm current controller coupled with super-twisting differentiator is implemented to force the currents to their references, featuring a fast dynamic and an improved robustness. Also, in the voltage regulation loop, an effective composite controller is developed for regulation of the output voltage, where a supertwisting observer is used to estimate load disturbance. (6) The problem of regulation output voltage of three-phase two level filtered voltage source inverters is presented using disturbance observer-based integral sliding mode control approach. First, the dynamics of the inverter are reformulated to facilitate the use of the proposed control strategy, which consider the parametric uncertainties of filter. A disturbance observer is designed to estimate the parametric uncertainties and external disturbances. Then, an integral sliding mode surface is established considering the voltage tracking error, its integral and the estimations of the parametric uncertainties and external disturbances. A sliding mode controller is proposed such that the systems are robustness to the admissible uncertainties and disturbances and satisfy the reaching condition. The stability of the closed-loop system is proved based on the Lyapunov theory

Hidden dynamics in models of discontinuity and switching

AbstractSharp switches in behaviour, like impacts, stick–slip motion, or electrical relays, can be modelled by differential equations with discontinuities. A discontinuity approximates fine details of a switching process that lie beyond a bulk empirical model. The theory of piecewise-smooth dynamics describes what happens assuming we can solve the system of equations across its discontinuity. What this typically neglects is that effects which are vanishingly small outside the discontinuity can have an arbitrarily large effect at the discontinuity itself. Here we show that such behaviour can be incorporated within the standard theory through nonlinear terms, and these introduce multiple sliding modes. We show that the nonlinear terms persist in more precise models, for example when the discontinuity is smoothed out. The nonlinear sliding can be eliminated, however, if the model contains an irremovable level of unknown error, which provides a criterion for systems to obey the standard Filippov laws for sliding dynamics at a discontinuity

Output tracking via sliding modes in causal systems with time delay modeled by higher order pade approximations

Output tracking in a SISO causal uncertain nonlinear system with an output subject to a time delay is considered using sliding mode control. A higher order Pade approximation to a delay function with a known time delay is used to construct a model of a transformed system without a time delayed output and is employed in a feedback sliding mode control. This model functions as a predictor of the plant states and the plant output, but is of nonminimum phase due to the application of the Pade approximation. The method of the stable system center is used to stabilize the internal dynamics of this plant model, and a control is developed using a sliding surface to allow the plant to track an arbitrary reference profile given by an exogenous system with a known characteristic equation. Simulations of the system are performed for the plant model using a first, second and third order Pade approximations and a controller in plant feedback mode. Numerical examples for Pade approximations of increasing order are considered and compare

Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems

Switches in real systems take many forms, such as impacts, electronic relays, mitosis, and the implementation of decisions or control strategies. To understand what is lost, and what can be retained, when we model a switch as an instantaneous event, requires a consideration of so-called hidden terms. These are asymptotically vanishing outside the switch, but can be encoded in the form of nonlinear switching terms. A general expression for the switch can be developed in the form of a series of sigmoid functions. We review the key steps in extending the Filippov's method of sliding modes to such systems. We show how even slight nonlinear effects can hugely alter the behaviour of an electronic control circuit, and lead to `hidden' attractors inside the switching surface.Comment: 12 page