Stability Analysis of Fractional Order Systems Described in the Lur'e Structure

Lur'e systems are feedback interconnection of a linear time-invariant subsystem in the forward path and a memoryless nonlinear one in the feedback path, which have many physical representatives. In this paper, some classical theorems about the L2 input-output stability of integer order Lur'e systems are discussed, and the conditions under which these theorems can be applied in fractional order Lur'e systems with an order between 0 and 1 are investigated. Then, application of circle criterion is compared between Lur'e systems of integer and fractional order using their corresponding Nyquist plots. Furthermore, applying Zames-Falb and generalized Zames-Falb theorems, some classes of stable fractional order Lur'e systems are introduced. Finally, in order to generalize the off-axis circle criterion to fractional order systems, another method is presented to prove one of the theorems used in its overall proof

Uncertain Multi-Agent Systems with Distributed Constrained Optimization Missions and Event-Triggered Communications: Application to Resource Allocation

This paper deals with solving distributed optimization problems with equality constraints by a class of uncertain nonlinear heterogeneous dynamic multi-agent systems. It is assumed that each agent with an uncertain dynamic model has limited information about the main problem and limited access to the information of the state variables of the other agents. A distributed algorithm that guarantees cooperative solving of the constrained optimization problem by the agents is proposed. Via this algorithm, the agents do not need to continuously broadcast their data. It is shown that the proposed algorithm can be useful in solving resource allocation problems

Robust Output Regulation: Optimization-Based Synthesis and Event-Triggered Implementation

We investigate the problem of practical output regulation: Design a controller that brings the system output in the vicinity of a desired target value while keeping the other variables bounded. We consider uncertain systems that are possibly nonlinear and the uncertainty of the linear part is modeled element-wise through a parametric family of matrix boxes. An optimization-based design procedures is proposed that delivers a continuous-time control and estimates the maximal regulation error. We also analyze an event-triggered emulation of this controller, which can be implemented on a digital platform, along with an explicit estimates of the regulation error

Robust Output Regulation: Optimization-Based Synthesis and Event-Triggered Implementation

We investigate the problem of practical output regulation, i.e., to design a controller that brings the system output in the vicinity of a desired target value while keeping the other variables bounded. We consider uncertain systems that are possibly nonlinear and the uncertainty of their linear parts is modeled element-wise through a parametric family of matrix boxes. An optimization-based design procedure is proposed that delivers a continuous-time control and estimates the maximal regulation error. We also analyze an event-triggered emulation of this controller, which can be implemented on a digital platform, along with an explicit estimates of the regulation error

Guest Editorial Introduction to the Special Section on Nonlinear Fractional-Order Circuits and Systems: Advanced Analysis and Effective Implementation

Nowadays, fractional order differential operators, as a generalization for classical differential operators, have established their key-role in modeling, analysis, and implementation of specific circuits and systems in which one typically faces nonlinear behaviors. It enforces to exploit analysis and implementation methods covering simultaneously "fractionality" and "nonlinearity" aspects. This special section, entitled "Nonlinear Fractional Order Circuits and Systems: Advanced Analysis and Effective Implementation," aims at introducing some of these methods

On the Selection of Tuning Methodology of FOPID Controllers for the Control of Higher Order Processes

In this paper, a comparative study is done on the time and frequency domain tuning strategies for fractional order (FO) PID controllers to handle higher order processes. A new fractional order template for reduced parameter modeling of stable minimum/non-minimum phase higher order processes is introduced and its advantage in frequency domain tuning of FOPID controllers is also presented. The time domain optimal tuning of FOPID controllers have also been carried out to handle these higher order processes by performing optimization with various integral performance indices. The paper highlights on the practical control system implementation issues like flexibility of online autotuning, reduced control signal and actuator size, capability of measurement noise filtration, load disturbance suppression, robustness against parameter uncertainties etc. in light of the above tuning methodologies.Comment: 27 pages, 10 figure

Time-Fractional KdV Equation: Formulation and Solution using Variational Methods

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure

A Novel Fractional Order Fuzzy PID Controller and Its Optimal Time Domain Tuning Based on Integral Performance Indices

A novel fractional order (FO) fuzzy Proportional-Integral-Derivative (PID) controller has been proposed in this paper which works on the closed loop error and its fractional derivative as the input and has a fractional integrator in its output. The fractional order differ-integrations in the proposed fuzzy logic controller (FLC) are kept as design variables along with the input-output scaling factors (SF) and are optimized with Genetic Algorithm (GA) while minimizing several integral error indices along with the control signal as the objective function. Simulations studies are carried out to control a delayed nonlinear process and an open loop unstable process with time delay. The closed loop performances and controller efforts in each case are compared with conventional PID, fuzzy PID and PI{\lambda}D{\mu} controller subjected to different integral performance indices. Simulation results show that the proposed fractional order fuzzy PID controller outperforms the others in most cases.Comment: 30 pages, 20 figure

Control of a novel chaotic fractional order system using a state feedback technique

We consider a new fractional order chaotic system displaying an interesting behavior. A necessary condition for the system to remain chaotic is derived. It is found that chaos exists in the system with order less than three. Using the Routh-Hurwitz and the Matignon stability criteria, we analyze the novel chaotic fractional order system and propose a control methodology that is better than the nonlinear counterparts available in the literature, in the sense of simplicity of implementation and analysis. A scalar control input that excites only one of the states is proposed, and sufficient conditions for the controller gain to stabilize the unstable equilibrium points derived. Numerical simulations confirm the theoretical analysis. © 2013 Elsevier Ltd. All rights reserved