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.     

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.     

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.     

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.     

On robust stability of linear time invariant fractional-order systems with real parametric uncertainties

In this paper, the robust bounded-input bounded-output stability of a large class of linear time invariant fractional order families of systems with real parametric uncertainties is analyzed. The transfer functions contain polynomials in fractional powers of the Laplace variable s, possibly in combination with exponentials of fractional powers of s. Using the concept of the value set and a generalization of the zero exclusion condition theorem, a theorem to check the robust bounded-input bounded-output stability of these families of systems is presented. An upper cutoff frequency for drawing the value sets is provided as well. Finally, two numerical examples are given to illustrate results obtained by the lemma and theorems presented in the paper.     

An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators

This paper introduces an integer order approximation method for numerical implementation of fractional order derivative/integrator operators in control systems. The proposed method is based on fitting the stability boundary locus (SBL) of fractional order derivative/integrator operators and SBL of integer order transfer functions. SBL defines a boundary in the parametric design plane of controller, which separates stable and unstable regions of a feedback control system and SBL analysis is mainly employed to graphically indicate the choice of controller parameters which result in stable operation of the feedback systems. This study reveals that the SBL curves of fractional order operators can be matched with integer order models in a limited frequency range. SBL fitting method provides straightforward solutions to obtain an integer order model approximation of fractional order operators and systems according to matching points from SBL of fractional order systems in desired frequency ranges. Thus, the proposed method can effectively deal with stability preservation problems of approximate models. Illustrative examples are given to show performance of the proposed method and results are compared with the well-known approximation methods developed for fractional order systems. The integer-order approximate modeling of fractional order PID controllers is also illustrated for control applications.     

IMC-PID-fractional-order-filter controllers design for integer order systems

One of the reasons of the great success of standard PID controllers is the presence of simple tuning rules, of the automatic tuning feature and of tables that simplify significantly their design. For the fractional order case, some tuning rules have been proposed in the literature. However, they are not general because they are valid only for some model cases. In this paper, a new approach is investigated. The fractional property is not especially imposed by the controller structure but by the closed loop reference model. The resulting controller is fractional but it has a very interesting structure for its implementation. Indeed, the controller can be decomposed into two transfer functions: an integer transfer function which is generally an integer PID controller and a simple fractional filter.     

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

It is shown that Theorem 3 in the article mentioned in the title is not true and some modification is suggested to eliminate the mistake.     

Identifiability of fractional order systems using input output frequency contents

In this paper, issues related to the identifiability of a fractional order system having its input and output frequency contents are discussed. The effects of the commensurate order α in the identifiability of the model structure and model parameters are analytically studied. It is shown that both identifiabilities (model structure and model parameters) are reduced remarkably for smaller values of α. This phenomenon is observed even though the input signals are rich enough and system belongs to the model set. Our understanding is that the problem arises since differences among different members of the model set fall beyond the practically recognizable precision range. The issue is more problematic when α is smaller and measurements are noisy.     

ROBUST STABILITY ANALYSIS FOR RAILWAY VEHICLE SYSTEMS

The lateral stability for railway vehicle dynamic system with uncertain parameters and nonlinear uncertain force vector is studied by using the Lyapunov stability theory. A robust stability condition for the considered system is derived, and the obtained stability bounds are not necessarily symmetric with respect to the origin in the parameter space. The lateral stability analysis for a railway bogie model is analyzed by using the proposed approach. The symmetric and asymmetric results are both given and the influence of the adjustable parameter β on the stability bounds is also discussed. With the help of the proposed method, the robust stability analysis can provide a reference for the design of the railway vehicle systems.     

Fractional order model reduction approach based on retention of the dominant dynamics: application in IMC based tuning of FOPI and FOPID controllers

Fractional order PI and PID controllers are the most common fractional order controllers used in practice. In this paper, a simple analytical method is proposed for tuning the parameters of these controllers. The proposed method is useful in designing fractional order PI and PID controllers for control of complicated fractional order systems. To achieve the goal, at first a reduction technique is presented for approximating complicated fractional order models. Then, based on the obtained reduced models some analytical rules are suggested to determine the parameters of fractional order PI and PID controllers. Finally, numerical results are given to show the efficiency of the proposed tuning algorithm.     

Impulsive stabilization of fractional differential systems

This paper investigates the impulsive stabilization problem of fractional differential systems (FDSs in short). Both the global exponential stability and ultimate boundedness criteria are established using Lyapunov functions, algebraic inequality techniques and boundedness of Mittag-Leffler functions. It is shown that unstable and unbounded FDSs can be stable and bounded respectively under impulsive control. Examples and simulations are also provided to demonstrate the effectiveness of the derived theoretical results.     

An approach to design controllers for MIMO fractional-order plants based on parameter optimization algorithm

The parameter optimization method for multivariable systems is extended to the controller design problems for multiple input multiple output (MIMO) square fractional-order plants. The algorithm can be applied to search for the optimal parameters of integer-order controllers for fractional-order plants with or without time delays. Two examples are given to present the controller design procedures for MIMO fractional-order systems. Simulation studies show that the integer-order controllers designed are robust to plant gain variations.     

Comment on “Further improvement on delay-range-dependent robust absolute stability for Lur׳e uncertain systems with interval time-varying delays” by P. Liu [ISA Trans. 58(2015)58–66]

In a recent paper (ISA Transactions (2015)58: 58–66), an integral inequality has been proposed to reduce the conservativeness of stability conditions for Lur׳e uncertain systems. We point out that there exist some errors in Theorems 1–8, and the correct theorems are presented. Finally, the allowable maximum admissible upper bound (MAUB) listed in Table 1, Table 2, Table 3, Table 4, Table 5 have been recalculated by using the correct results.     

Stabilization of an inverted pendulum-cart system by fractional PI-state feedback

This paper deals with pole placement PI-state feedback controller design to control an integer order system. The fractional aspect of the control law is introduced by a dynamic state feedback as u(t)=_Kpx(t)+_KIIα(x(t))$u\left(t\right)=K_{p}x\left(t\right)+K_I{\mathfrak{I}}_{\alpha }\left(x\right(t\left)\right)$. The closed loop characteristic polynomial is thus fractional for which the roots are complex to calculate. The proposed method allows us to decompose this polynomial into a first order fractional polynomial and an integer order polynomial of order n−1$n-1$ (n being the order of the integer system). This new stabilization control algorithm is applied for an inverted pendulum-cart test-bed, and the effectiveness and robustness of the proposed control are examined by experiments.     

The application of fractional order control for an air-based contactless actuation system

Industry pushes towards ever faster and more accurate production of thin substrates. Contactless positioning offers advantages, especially in terms of risk of breakage and contamination. A system is considered designed for contactless positioning by floating a silicon wafer on a thin film of air. This paper focuses on the design of a control system, including actuators, sensors and control method, suitable for this purpose. Two cascaded control loops, with decoupled SISO controllers, are implemented for this moving mass controlled on a mass-spring system, which can be modelled as a fourth order system. The SISO controllers are first designed with classic loopshaping tools, which are then modified using fractional control. Two arguments based on examples in this system are given for the application of fractional control. Firstly, to increase the bandwidth of a regular mass-spring system, and secondly to control a plant which behaves fundamentally fractional, such as the moving mass in this cascaded fourth order system. By merely the application of fractionality, the bandwidths are extended by 14.6 % and 62 %, for the inner and outer loop respectively. A closed-loop positioning bandwidth of the wafer of 60 Hz is achieved, resulting in a positioning error of 104 nm (2σ value), which is limited by sensor noise and pressure disturbances. This paper shows how the extension of classic loopshaping tools with fractional control can directly improve the performance, without adding to the complicatedness of the control system. Moreover it demonstrates a working concept of a novel type of contactless actuator.     

Lyapunov theory based robust control of complicated nonlinear mechanical systems with uncertainty

This paper presents a robust control approach for nonlinear uncertain crane systems with a three DOF framework. We deal with an overhead crane in which a trolley located on the top is moved to x- and y-axes independently. We first approximate the nonlinear system model through feedback linearization transformation to simply construct a PD control and then design a robust control system for compensating control deviation feasibly occurring due to modeling error or system perturbation in practice. An adaptive control rule is analytically derived by using Lyapunov stability theory given bounds of system perturbation. We accomplish numerical simulation for evaluating the proposed methodology and demonstrate its superiority by comparing with the traditional control strategy. This paper was recommended for publication in revised form by Associate Editor Dong Hwan Kim Hyun Cheol Cho received a B.S. from the Pukyong National University in 1997, a M.S. from the Dong-A University, Korea in 1999, and a Ph.D. from University of Nevada-Reno, USA in 2006. He is currently a post-doc researcher in the Dept. of Electrical Engineering, Dong-A University. His research interests are in the areas of control systems, neural networks, stochastic process, and signal processing.