There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Tables of integrals

Review of fractional PID controller

Design of fractional order PID controller for automatic regulator voltage system based on multi-objective extremal optimization

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 modelling 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.

On the selection of tuning methodology of FOPID controllers for the control of higher order processes.

PI λ D μ controller design for integer and fractional plants using piecewise orthogonal functions

A fractional state space realization method with block pulse basis

The aim of this paper is to determine the feedforward and state feedback suboptimal time control for a subset of bilinear systems, namely, the control sequence and reaching time. This paper proposes a method that uses Block pulse functions as an orthogonal base. The bilinear system is projected along that base. The mathematical integration is transformed into a product of matrices. An algebraic system of equations is obtained. This system together with specified constraints is treated as an optimization problem. The parameters to determine are the final time, the control sequence, and the states trajectories. The obtained results via the newly proposed method are compared to known analytical solutions.

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Time Optimal Control Laws for Bilinear Systems

In this article, generalised operational matrices of the orthogonal block pulse basis, approximating the Riemann-Liouville formula accurately, are exploited for the modelling and control synthesis of fractional dynamic systems. The main advantage of using this tool is that it allows us to transform the analytical fractional differential calculus into an algebraic one relatively easier to solve. The developments presented have led to the formulation of two optimisation problems. In fact, in the first part, the solution of a least square algorithm could provide a reduced integer transfer function approximating the considered fractional system. While in the second part, the proposed non-linear function to be minimised may be helpful for the state space parameters setting of PIλDμ controllers. Simulations in both cases are exhibited to show the effectiveness of the detailed techniques.

Block pulse-based techniques for modelling and synthesis of non-integer systems

In this paper, a new method is introduced to design static output tracking controllers for a class of non-linear polynomial time-delay systems. The proposed technique is based on the projection of the controlled system and the associated linear reference model that it should follow over a basis of block-pulse functions. The useful properties of these orthogonal functions such as operational matrices jointly used with the Kronecker tensor product may transform the non-linear delay differential equations into linear algebraic equations depending only on parameters of the feedback regulator. The least-squares method is then used for determination of the unknown parameters. Sufficient conditions for the practical stability of the closed-loop system are derived, and a domain of attraction is estimated. The implementation of the proposed method is illustrated on a double inverted pendulums benchmark as well as a two-degree-of- freedom mass-spring-damper system. The simulation results show the effectiven...

Mohamed Karim Bouafoura

Papers

PI λ D μ controller design for integer and fractional plants using piecewise orthogonal functions

A fractional state space realization method with block pulse basis

Time Optimal Control Laws for Bilinear Systems

Block pulse-based techniques for modelling and synthesis of non-integer systems

Static output tracking control for non-linear polynomial time-delay systems via block-pulse functions