A Conformal Mapping Based Fractional Order Approach for Sub-optimal Tuning of PID Controllers with Guaranteed Dominant Pole Placement

A novel conformal mapping based Fractional Order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PI{\lambda}D{\mu}) controller have been approximated in this paper vis-\`a-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PI{\lambda}D{\mu} controller pushes the open loop zeros of the equivalent PID controller towards greater damping regions which gives a trajectory of the controller zeros and dominant closed loop poles. This trajectory is termed as "M-curve". This phenomena is used to design a two-stage tuning algorithm which reduces the existing PID controller's effort in a significant manner compared to that with a single stage LQR based pole placement method at a desired closed loop damping and frequency.Comment: 23 pages, 7 figures, in press; Communications in Nonlinear Science and Numerical Simulations, 201

Lagrangian Approach to Dispersionless KdV Hierarchy

We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether's theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and pplications) at http://www.emis.de/journals/SIGMA

Dynamical systems theory for nonlinear evolution equations

We observe that the fully nonlinear evolution equations of Rosenau and Hymann, often abbreviated as $K(n,\,m)$ equations, can be reduced to Hamiltonian form only on a zero-energy hypersurface belonging to some potential function associated with the equations. We treat the resulting Hamiltonian equations by the dynamical systems theory and present a phase-space analysis of their stable points. The results of our study demonstrate that the equations can, in general, support both compacton and soliton solutions. For the $K(2,\,2)$ and $K(3,\,3)$ cases one type of solutions can be obtained from the other by continuously varying a parameter of the equations. This is not true for the $K(3,\,2)$ equation for which the parameter can take only negative values. The $K(2,\,3)$ equation does not have any stable point and, in the language of mechanics, represents a particle moving with constant acceleration.Comment: 5 pages, 4 figure

Optimum Weight Selection Based LQR Formulation for the Design of Fractional Order PI{\lambda}D{\mu} Controllers to Handle a Class of Fractional Order Systems

A weighted summation of Integral of Time Multiplied Absolute Error (ITAE) and Integral of Squared Controller Output (ISCO) minimization based time domain optimal tuning of fractional-order (FO) PID or PI{\lambda}D{\mu} controller is proposed in this paper with a Linear Quadratic Regulator (LQR) based technique that minimizes the change in trajectories of the state variables and the control signal. A class of fractional order systems having single non-integer order element which show highly sluggish and oscillatory open loop responses have been tuned with an LQR based FOPID controller. The proposed controller design methodology is compared with the existing time domain optimal tuning techniques with respect to change in the trajectory of state variables, tracking performance for change in set-point, magnitude of control signal and also the capability of load disturbance suppression. A real coded genetic algorithm (GA) has been used for the optimal choice of weighting matrices while designing the quadratic regulator by minimizing the time domain integral performance index. Credible simulation studies have been presented to justify the proposition.Comment: 6 pages, 5 figure

Estimation, Analysis and Smoothing of Self-Similar Network Induced Delays in Feedback Control of Nuclear Reactors

This paper analyzes a nuclear reactor power signal that suffers from network induced random delays in the shared data network while being fed-back to the Reactor Regulating System (RRS). A detailed study is carried out to investigate the self similarity of random delay dynamics due to the network traffic in shared medium. The fractionality or selfsimilarity in the network induced delay that corrupts the measured power signal coming from Self Powered Neutron Detectors (SPND) is estimated and analyzed. As any fractional order randomness is intrinsically different from conventional Gaussian kind of randomness, these delay dynamics need to be handled efficiently, before reaching the controller within the RRS. An attempt has been made to minimize the effect of the randomness in the reactor power transient data with few classes of smoothing filters. The performance measure of the smoothers with fractional order noise consideration is also investigated into.Comment: 6 pages, 6 figure

Fractional Order Modeling of a PHWR Under Step-Back Condition and Control of Its Global Power with a Robust PI{\lambda}D{\mu} Controller

Bulk reduction of reactor power within a small finite time interval under abnormal conditions is referred to as step-back. In this paper, a 500MWe Canadian Deuterium Uranium (CANDU) type Pressurized Heavy Water Reactor (PHWR) is modeled using few variants of Least Square Estimator (LSE) from practical test data under a control rod drop scenario in order to design a control system to achieve a dead-beat response during a stepped reduction of its global power. A new fractional order (FO) model reduction technique is attempted which increases the parametric robustness of the control loop due to lesser modeling error and ensures iso-damped closed loop response with a PI{\lambda}D{\mu} or FOPID controller. Such a controller can, therefore, be used to achieve active step-back under varying load conditions for which the system dynamics change significantly. For closed loop active control of the reduced FO reactor models, the PI{\lambda}D{\mu} controller is shown to perform better than the classical integer order PID controllers and present operating Reactor Regulating System (RRS) due to its robustness against shift in system parameters.Comment: 10 pages, 11 figure