Robust stability and stabilization of fractional order linear systems with positive real uncertainty

This paper investigates the robust stability and stabilization of fractional order linear systems with positive real uncertainty. Firstly, sufficient conditions for the asymptotical stability of such uncertain fractional order systems are presented. Secondly, the existence conditions and design methods of the state feedback controller, static output feedback controller and observer-based controller for asymptotically stabilizing such uncertain fractional order systems are derived. The results are obtained in terms of linear matrix inequalities. Finally, some numerical examples are given to validate the proposed theoretical results.     

A numerical investigation for robust stability of fractional-order uncertain systems

This study presents numerical methods for robust stability analysis of closed loop control systems with parameter uncertainty. Methods are based on scan sampling of interval characteristic polynomials from the hypercube of parameter space. Exposed-edge polynomial sampling is used to reduce the computational complexity of robust stability analysis. Computer experiments are used for demonstration of the proposed robust stability test procedures.     

Probabilistic robust stabilization of fractional order systems with interval uncertainty

This study investigates effects of fractional order perturbation on the robust stability of linear time invariant systems with interval uncertainty. For this propose, a probabilistic stability analysis method based on characteristic root region accommodation in the first Riemann sheet is developed for interval systems. Stability probability distribution is calculated with respect to value of fractional order. Thus, we can figure out the fractional order interval, which makes the system robust stable. Moreover, the dependence of robust stability on the fractional order perturbation is analyzed by calculating the order sensitivity of characteristic polynomials. This probabilistic approach is also used to develop a robust stabilization algorithm based on parametric perturbation strategy. We present numerical examples demonstrating utilization of stability probability distribution in robust stabilization problems of interval uncertain systems.     

Practical stability analysis of fractional-order impulsive control systems

In this paper we obtain sufficient conditions for practical stability of a nonlinear system of differential equations of fractional order subject to impulse effects. Our results provide a design method of impulsive control law which practically stabilizes the impulse free fractional-order system.     

Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations

With power mapping (conformal mapping), stability analyses of fractional order linear time invariant (LTI) systems are carried out by consideration of the root locus of expanded degree integer order polynomials in the principal Riemann sheet. However, it is essential to show the left half plane (LHP) stability analysis of fractional order characteristic polynomials in the s plane in order to close the gap emerging in stability analyses of fractional order and integer order systems. In this study, after briefly discussing the relation between the characteristic root orientations and the system stability, the author presents a methodology to establish principal characteristic polynomials to perform the LHP stability analysis of fractional order systems. The principal characteristic polynomials are formed by factorizing principal characteristic roots. Then, the LHP stability analysis of fractional order systems can be carried out by using the root equivalency of fractional order principal characteristic polynomials. Illustrative examples are presented to explain how to find equivalent roots of fractional order principal characteristic polynomials in order to carry out the LHP stability analyses of fractional order nominal and interval systems.     

Robust stability criterion for fractional-order systems with interval uncertain coefficients and a time-delay

This study investigates the robust stability of fractional-order systems with interval coefficients and a time-delay. By the Minkowski Sum, the vertices of value set with respect to the characteristic function of the investigated fractional-order system are offered, avoiding the calculations of the redundant vertices. Meanwhile, a function depending on the obtained vertices is defined to represent the position relationship between the origin and the value set. Based on the zero exclusion principle, we propose sufficient and necessary conditions to determine the robust stability of fractional-order systems with interval uncertain coefficients and a time-delay. Finally, illustrative examples are offered to verify the effectiveness of the proposed robust stability criterion.     

Author’s reply: A comment on the “Robust stability analysis of fractional order interval polynomials”, by Nusret Tan et al.

This author’s reply addresses the comment given in the note mentioned in the title. Theorem 3 given in Tan et al. (2009) [1] uses zero exclusion principle for the stability analysis of Fractional Order Interval Polynomial (FOIP). We show that the constant degree assumption is exist in the definition of zero exclusion principle. Although it has not been clearly stated in Tan et al. (2009) [1] that FOIP of Eq. (1) is a constant degree polynomial, this condition is implicit in the zero exclusion principle. Therefore, Theorem 3 is true under the constant degree assumption which is a requirement for zero exclusion principle.     

The design of NMSS fractional-order predictive functional controller for unstable systems with time delay

Open-loop unstable systems are more difficult to control than stable processes. In presence of time delay and uncertainty, the complexity of problem increases. In this paper, non-minimal state space predictive functional control (NMSS-PFC) infrastructure has been generalized for control of unstable systems with time delay. At first, NMSS system representation has been extended for a so-called coprime-factorized equivalent model of the unstable processes. Then, the proposed NMSS-fractional-order PFC (NMSS-FOPFC) has been formulated via a fractional order cost function in terms of the output tracking error vector. In the developed formulation and via the NMSS structure, the constraints on inputs of the system could be easily formulated and employment of fractional order cost function led to improved performance even in case of disturbances, perturbations or uncertainties. Closed loop robust stability analysis was also performed based on small gain theorem and verified via simulations. Simulation examples show that the proposed NMSS-FOPFC results in improved nominal and perturbed responses compared to conventional methods. Comparison has been carried out by considering integral of absolute error (IAE) and integral of squared error (ISE) as well as step response transient and steady state properties and the control effort.     

Robust stability analysis of fractional order interval polynomials

The paper deals with the robust stability analysis of a Fractional Order Interval Polynomial (FOIP) family. Some new results are presented for testing the Bounded Input Bounded Output (BIBO) stability of dynamical control systems whose characteristic polynomials are fractional order polynomials with interval uncertainty structure. It is shown that the Kharitonov theorem is not applicable for this type of polynomial. A procedure is given for computation of the value set of FOIP. Based on the value set, an algorithm is presented for testing the stability of FOIP. The results presented in the paper are useful for the analysis and design of Fractional Order Interval Control Systems (FOICS). Examples are given to show how the proposed method can be used to assess the effects of parametric variations on the stability in feedback loops with fractional order interval transfer functions.     

Adaptive sliding mode control for finite-time stability of quad-rotor UAVs with parametric uncertainties

Adaptive control methods are developed for stability and tracking control of flight systems in the presence of parametric uncertainties. This paper offers a design technique of adaptive sliding mode control (ASMC) for finite-time stabilization of unmanned aerial vehicle (UAV) systems with parametric uncertainties. Applying the Lyapunov stability concept and finite-time convergence idea, the recommended control method guarantees that the states of the quad-rotor UAV are converged to the origin with a finite-time convergence rate. Furthermore, an adaptive-tuning scheme is advised to guesstimate the unknown parameters of the quad-rotor UAV at any moment. Finally, simulation results are presented to exhibit the helpfulness of the offered technique compared to the previous methods.     

Finite time stability and controller design for nonlinear impulsive sampled-data systems with applications

This paper investigates the finite time stability (FTS) for nonlinear impulsive sampled-data systems. By constructing an appropriated Lyapunov function and employing average impulsive interval (AII) method, some FTS criteria for the nonlinear impulsive sampled-data systems are derived in terms of linear matrix inequalities (LMIs), which can be easily verified via the LMI toolbox. The hybrid controller including sampled-data controller and impulsive controller is designed via the established LMIs. Moreover, the impulse effect considered in this paper including stabilizing impulse and destabilizing impulse. Our developed results are less conservative than the recent work in the literature. Finally, two numerical examples are provided to show the applications of the proposed criteria.     

Diagnosis of delay-deadline failures in real time discrete event models

In this paper a method for fault detection and diagnosis (FDD) of real time systems has been developed. A modeling framework termed as real time discrete event system (RTDES) model is presented and a mechanism for FDD of the same has been developed. The use of RTDES framework for FDD is an extension of the works reported in the discrete event system (DES) literature, which are based on finite state machines (FSM). FDD of RTDES models are suited for real time systems because of their capability of representing timing faults leading to failures in terms of erroneous delays and deadlines, which FSM-based ones cannot address. The concept of measurement restriction of variables is introduced for RTDES and the consequent equivalence of states and indistinguishability of transitions have been characterized. Faults are modeled in terms of an unmeasurable condition variable in the state map. Diagnosability is defined and the procedure of constructing a diagnoser is provided. A checkable property of the diagnoser is shown to be a necessary and sufficient condition for diagnosability. The methodology is illustrated with an example of a hydraulic cylinder.     

Extension of the root-locus method to a certain class of fractional-order systems

In this paper, the well-known root-locus method is developed for the special subset of linear time-invariant systems commonly known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variable s. Such systems are defined on a Riemann surface because of their multi-valued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root-locus method such as asymptotes, roots condition on the real axis and breakaway points are extended to the fractional case. It is also shown that the proposed method can assess the closed-loop stability of fractional-order systems in the presence of a varying gain in the loop. Moreover, the effect of perturbation on the root loci is discussed. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.     

A novel fractional order PID navigation guidance law by finite time stability approach

In this paper, a novel fractional order proportional–integral–differential navigation guidance law utilizing finite time stability approach is presented in order to achieve robust performance for intercepting incoming targets. The proposed guidance law is designed following three-loop guidance and control scheme, considering the interceptor’s nonlinear 6 degrees-of-freedom model. In the outer loop, normal acceleration commands are generated by the proposed guidance law. In the intermediate loop, these commands are converted into equivalent body rate commands, which are tracked by dynamic inversion based autopilot in the inner loop. A fractional order circle criterion is developed for the finite time stability analysis of this proposed guidance law, whose stability conditions give an analytical bound for the flight up time in which stability can be insured. Extensive 6 degrees-of-freedom simulations and a variety of comparison studies against maneuvering targets are implemented to demonstrate the effectiveness of the proposed guidance law. The simulation results show that the proposed guidance law has better performance when comparing with the proportional navigation and proportional–integral–differential navigation guidance laws.     

CRONE control of continuous linear time periodic systems: application to a testing bench

Frequency methods only are used here for the study and control of continuous linear time periodic systems. Using time varying frequency responses defined by L. A. Zadeh in the 1950s, the second generation CRONE control is extended to the control of linear time periodic systems. This control strategy ensures, for the closed-loop system, a near stationary behavior, performances set by the designer, and robustness of performances to gain variations of the plant. An application of the proposed control strategy to a testing bench shows its efficiency.     

Robust stabilizing first-order controllers for a class of time delay systems

In this paper, stabilizing regions of a first-order controller for an all poles system with time delay are computed via parametric methods. First, the admissible ranges of one of the controller’s parameters are obtained. Then, for a fixed value of this parameter, stabilizing regions in the remaining two parameters are determined using the D-decomposition method. Phase and gain margin specifications are then included in the design. Finally, robust stabilizing first-order controllers are determined for uncertain plants with an interval type uncertainty in the coefficients. Examples are given to illustrate the effectiveness of the proposed method.     

A new fractional-order sliding mode controller via a nonlinear disturbance observer for a class of dynamical systems with mismatched disturbances

This paper investigates the stabilization and disturbance rejection for a class of fractional-order nonlinear dynamical systems with mismatched disturbances. To fulfill this purpose a new fractional-order sliding mode control (FOSMC) based on a nonlinear disturbance observer is proposed. In order to design the suitable fractional-order sliding mode controller, a proper switching surface is introduced. Afterward, by using the sliding mode theory and Lyapunov stability theory, a robust fractional-order control law via a nonlinear disturbance observer is proposed to assure the existence of the sliding motion in finite time. The proposed fractional-order sliding mode controller exposes better control performance, ensures fast and robust stability of the closed-loop system, eliminates the disturbances and diminishes the chattering problem. Finally, the effectiveness of the proposed fractional-order controller is depicted via numerical simulation results of practical example and is compared with some other controllers.     

Improved delay-range-dependent stability analysis of a time-delay system with norm bounded uncertainty

This paper presents improved robust delay-range-dependent stability analysis of an uncertain linear time-delay system following two different existing approaches – (i) non-delay partitioning (NDP) and (ii) delay partitioning (DP). The derived criterion (for both the approaches) proposes judicious use of integral inequality to approximate the uncertain limits of integration arising out of the time-derivative of Lyapunov–Krasovskii (LK) functionals to obtain less conservative results. Further, the present work compares both the approaches in terms of relative merits as well as highlights tradeoff for achieving higher delay bound and (or) reducing number of decision variables without losing conservatism in delay bound results. The analysis and discussion presented in the paper are validated by considering relevant numerical examples.     

Asymptotic stability and stabilizability of nonlinear systems with delay

This paper is concerned with asymptotic stability and stabilizability of a class of nonlinear dynamical systems with fixed delay in state variable. New sufficient conditions are established in terms of the system parameters such as the eigenvalues of the linear operator, delay parameter, and bounds on the nonlinear parts. Finally, examples are given to testify the effectiveness of the proposed theory.