Some applications of fractional calculus on control problems in robotics and system stability

In recent years, there have been extensive research activities related to applications of fractional calculus (FC), [5] in nonlinear dynamics, mechatronics as well as control theory. In this paper, they are presented recently obtained results which are related to applications of fractional calculus in mechanics - specially stability and control issues. Some of these results [1-4] are presented at the Fifth symposium of fractional differentiation and its applications FDA2012, was held at the Hohai University, Nanjing, China in the period of May 14-May 17, 2012. Also, fractional order dynamic systems and controllers have been increasing in interest in many areas of science and engineering in the last few years.In that way, our objective of using fractional calculus is to apply the fractional order controller to enhance the system control performance as well as it has better disturbance rejection ratios and less sensitivity to plant parameter variations. First, they are introduced and obtained the new algorithms of fractional order PID control based on genetic algorithms in the position control of a 3 DOF’s robotic system driven by DC motors. Then, the main task is to find out the optimal settings for a fractional PI D controller in order to fulfill the proposed design specifications for the closed-loop system. In addition, this method allows the optimal design of all major parameters of a fractional PID controller and then enhances the flexibility and capability of the PID controller. Last,in simulations, they are compared step responses of these two optimal controllers where it will be shown that fractional order PID controller improves transient response as well as provides more robustness in than conventional PID. Second, we propose sufficient conditions for finite time stability for the (non)homogeneous fractional order systems with time delay. Specially, the problem of finite time stability with respect to some of the variables (partial stability) is considered. Namely, along with the formulation of the problem of stability to all variables, Lyapunov also formulated a more general problem on the stability to a given part of variables (but not all variables) determining the state of a system,[6].The problem of the stability of motion with respect to some of the variables also known as partial stability arises naturally in applications. So, in this presentation, it will be proposed finite time partial stability test procedure of perturbed (non) linear (non)autonomous time varying delay fractional order systems. Time-delay is assumed to be varying with time but its upper bound is assumed to be known over given time interval. New stability criteria for this class of fractional order systems will be derived using “classical” Bellman-Gronwall inequality,as well as another another suitable inequality, [7]. Last,a numerical example is provided to illustrate the application of the proposed stability procedure. Third, some attention is devoted to the problem of stability of linear discrete-time fractional order systems is addressed, [8]. It was shown that some stability criteria for discrete time-delay systems could be applied with small changes to discrete fractional order state-space systems. Accordingly, simple conditions for the stability and robust stability of a particular class of linear discrete time-delay systems are derived. These results are modified and used for checking the stability of discrete-time fractional order systems. The systems under consideration involve time delays in the state and parameter uncertainties. The parameter uncertainties are assumed to be time-varying and norm bounded

Some applications of fractional calculus on control problems in robotics and system stability

In recent years, there have been extensive research activities related to applications of fractional calculus (FC), [5] in nonlinear dynamics, mechatronics as well as control theory. In this paper, they are presented recently obtained results which are related to applications of fractional calculus in mechanics - specially stability and control issues. Some of these results [1-4] are presented at the Fifth symposium of fractional differentiation and its applications FDA2012, was held at the Hohai University, Nanjing, China in the period of May 14-May 17, 2012. Also, fractional order dynamic systems and controllers have been increasing in interest in many areas of science and engineering in the last few years.In that way, our objective of using fractional calculus is to apply the fractional order controller to enhance the system control performance as well as it has better disturbance rejection ratios and less sensitivity to plant parameter variations. First, they are introduced and obtained the new algorithms of fractional order PID control based on genetic algorithms in the position control of a 3 DOF’s robotic system driven by DC motors. Then, the main task is to find out the optimal settings for a fractional PI D controller in order to fulfill the proposed design specifications for the closed-loop system. In addition, this method allows the optimal design of all major parameters of a fractional PID controller and then enhances the flexibility and capability of the PID controller. Last,in simulations, they are compared step responses of these two optimal controllers where it will be shown that fractional order PID controller improves transient response as well as provides more robustness in than conventional PID. Second, we propose sufficient conditions for finite time stability for the (non)homogeneous fractional order systems with time delay. Specially, the problem of finite time stability with respect to some of the variables (partial stability) is considered. Namely, along with the formulation of the problem of stability to all variables, Lyapunov also formulated a more general problem on the stability to a given part of variables (but not all variables) determining the state of a system,[6].The problem of the stability of motion with respect to some of the variables also known as partial stability arises naturally in applications. So, in this presentation, it will be proposed finite time partial stability test procedure of perturbed (non) linear (non)autonomous time varying delay fractional order systems. Time-delay is assumed to be varying with time but its upper bound is assumed to be known over given time interval. New stability criteria for this class of fractional order systems will be derived using “classical” Bellman-Gronwall inequality,as well as another another suitable inequality, [7]. Last,a numerical example is provided to illustrate the application of the proposed stability procedure. Third, some attention is devoted to the problem of stability of linear discrete-time fractional order systems is addressed, [8]. It was shown that some stability criteria for discrete time-delay systems could be applied with small changes to discrete fractional order state-space systems. Accordingly, simple conditions for the stability and robust stability of a particular class of linear discrete time-delay systems are derived. These results are modified and used for checking the stability of discrete-time fractional order systems. The systems under consideration involve time delays in the state and parameter uncertainties. The parameter uncertainties are assumed to be time-varying and norm bounded

Stabilising Model Predictive Control for Discrete-time Fractional-order Systems

In this paper we propose a model predictive control scheme for constrained fractional-order discrete-time systems. We prove that all constraints are satisfied at all time instants and we prescribe conditions for the origin to be an asymptotically stable equilibrium point of the controlled system. We employ a finite-dimensional approximation of the original infinite-dimensional dynamics for which the approximation error can become arbitrarily small. We use the approximate dynamics to design a tube-based model predictive controller which steers the system state to a neighbourhood of the origin of controlled size. We finally derive stability conditions for the MPC-controlled system which are computationally tractable and account for the infinite dimensional nature of the fractional-order system and the state and input constraints. The proposed control methodology guarantees asymptotic stability of the discrete-time fractional order system, satisfaction of the prescribed constraints and recursive feasibility

Parameter-Dependent Lyapunov Functions for Linear Systems With Constant Uncertainties

Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties

Stabilization of Linear Systems with Structured Perturbations

The problem of stabilization of linear systems with bounded structured uncertainties are considered in this paper. Two notions of stability, denoted quadratic stability (Q-stability) and μ-stability, are considered, and corresponding notions of stabilizability and detectability are defined. In both cases, the output feedback stabilization problem is reduced via a separation argument to two simpler problems: full information (FI) and full control (FC). The set of all stabilizing controllers can be parametrized as a linear fractional transformation (LFT) on a free stable parameter. For Q-stability, stabilizability and detectability can in turn be characterized by Linear Matrix Inequalities (LMIs), and the FI and FC Q-stabilization problems can be solved using the corresponding LMIs. In the standard one-dimensional case the results in this paper reduce to well-known results on controller parametrization using state-space methods, although the development here relies more heavily on elegant LFT machinery and avoids the need for coprime factorizations