Tracking control design for fractional order systems: A passivity-based port-Hamiltonian framework

This article focuses on the design of tracking control for chaotic fractional order systems subjected to perturbations in a port-Hamiltonian framework. The fractional order systems of general form are modeled into port-controlled Hamiltonian form. Then, the extended results on the dissipativity, energy balance, and passivity of the fractional order systems are proved and presented in this paper. The port-controlled Hamiltonian form of the fractional order systems are proved to be asymptotically stable via energy balancing concept. Furthermore, a tracking controller is designed for the fractional order port-controlled Hamiltonian form by utilizing the matching conditions of the port-Hamiltonian systems. Stability of the system is established and analyzed explicitly for the closed-loop system with the help of direct Lyapunov method. Finally, an application example is solved with simulation results and discussions to prove the effectiveness of the propounded control design approach.

An innovative modulating functions method for pseudo-state estimation of fractional order systems

In this paper, the objective is to estimate the pseudo-state of fractional order systems defined by the Caputo fractional derivative from discrete noisy output measurement. For this purpose, an innovative modulating functions method is proposed, which can provide non-asymptotic estimation within finite-time and is robust against corrupting noises. First, the proposed method is directly applied to the Brunovsky's observable canonical form of the considered system. Then, the initial value of the pseudo-state is exactly expressed by an algebraic integral formula, based on which the pseudo-state is estimated. Second, the properties and construction of the required modulating functions are studied. Furthermore, error analysis is provided in discrete noise cases, which is useful for improving the estimation accuracy. In order to show the advantages of the proposed method, two numerical examples are given, where both rational order and irrational order dynamical systems are considered. After selecting the design parameters using the provided noise error bound, the pseudo-states of considered systems are estimated. The fractional order Luenberger-like observer and the fractional order H_∞-like observer are also applied. Better than the applied fractional order observers, the proposed method can guarantee the convergence speed and robustness at the same time.

Lyapunov stability criteria in terms of class K functions for Riemann-Liouville nabla fractional order systems

This paper focuses on the problem of stability analysis for Riemann-Liouville nabla fractional order systems. On one hand, a useful comparison principle is built and then a rigorous proof is constructed for the well-known Lyapunov stability criterion in terms of class K functions. On the other hand, the constraint of the Lyapunov function is refined using a positive constant γ₄ or a sequence h(k), resulting two practical theorems. Finally, three illustrative examples are given to show the applicability of the proposed method.

Design of the modified fractional central difference Kalman filters under stochastic colored noises

For state estimation of discrete nonlinear fractional stochastic systems, this study presents two innovative modified fractional central difference Kalman filters. We consider a complicated scenario where the process noise or measurement noise in the system become colored noise. Firstly, the nonlinear function is linearized by utilizing the Stirling polynomial interpolation formula. Thus there is no need to calculate the Jacobi matrix for both algorithms, which means very few application limitations. Then, based on the augmented-state method, we develop an augmented state fractional central difference Kalman filter under the scenario of colored process noise. Afterwards, a state estimation algorithm for handling stochastic systems containing colored measurement noise is put forward by using the measurement expansion method. Finally, to perform the superiority of the developed algorithms, several simulations are carried out. As well, the algorithms derived in this paper are contrasted with the original fractional central difference Kalman filter and three other algorithms. Notably, a simulation with engineering significance for the state-of-charge estimation for lithium-ion batteries is also introduced, Aside from the commonly used numerical simulation. The results verify the superiority of the developed algorithms in sense of estimation accuracy and real-time performance.

Positivity problem for the one dimensional heat transfer process

In the paper the positivity problem of the model of an one dimensional heat transfer process is addressed. Such a problem has not been considered yet. The considered thermal process is described by the fractional order state equation, derived from parabolic heat equation with homogenous Neumann boundary conditions and distributed control and observation. The internal and external positivity of the model depend on heater and sensor location as well as the size of the model. It is proved that the external positivity of the considered system can be achieved without internal positivity. Conditions of the internal and external positivity are proposed and proved. Theoretical considerations are supported by experiments. Experiments were done using the real system containing typical industrial components. The proposed results can be applied in real temperature measurements, for example in thermal cameras.

Stability and robust stabilization of uncertain switched fractional order systems

In this paper, the stability and robust stabilization of switched fractional order systems are concerned. Firstly, two stability theorems for switched fractional order systems with order 0<α<1 and 1<α<2 under the arbitrary switching law are given. Secondly, the relationship between the stability of switched integer order systems and that of switched fractional order systems is obtained. Finally, the robust stabilization of uncertain switched fractional order systems under the common switching law is further discussed. The state feedback control gains are obtained under both the sensor and actuator faults in terms of linear matrix inequalities. A practical electrical circuit example and four numerical simulation examples are presented to show the effectiveness of our results.

Identification of fractional-order transfer functions using exponentially modulated signals with arbitrary excitation waveforms

This paper proposes a new identification method based on an exponential modulation scheme for the determination of the coefficients and exponents of a fractional-order transfer function. The proposed approach has a broader scope of application compared to a previous method based on step response data, in that it allows for the use of arbitrary input signals. Moreover, it dispenses with the need for repeated simulations during the search for the best fractional exponents, which significantly reduces the computational workload involved in the identification process. Two examples involving measurement noise at the observed system output are presented to illustrate the effectiveness of the proposed method when compared to a conventional output-error optimization approach based on the polytope algorithm. In both examples, the proposed method is found to provide a better trade-off between computational workload and accuracy of the parameter estimates for different realizations of the noise.

A numerical approximation method for fractional order systems with new distributions of zeros and poles

A multiple-zero-pole (MZP) method is proposed for general SISO fractional order dynamic systems in this paper. Based on amplitude-frequency curve, a new rational approach to fractional differentiator is designed. There are three advantages of MZP method. 1) A more generalized form of approximation system is proposed by design the distribution of zeros and poles in a new way. 2) The same fractional differentiator can be approximated in many different forms. 3) The robustness of the approximation system is enhanced by using integer order integrators to construct fractional differentiator. The feasibility of the method is assessed in the illustrative examples, and the simulations prove the effectiveness.

Adaptive output feedback control for fractional-order multi-agent systems

This study investigates the leader-following consensus problem in fractional-order multi-agent systems (MAS). First, a distributed observer is presented to estimate the state variables by using only the output information of MAS. Second, to overcome the difficulty of selecting an appropriate Lyapunov function for fractional order MAS, we propose a new state transformation technique to reconstruct the observer dynamic model. Then a new distributed frequency-dependent Lyapunov function is introduced in accordance with this model. Based on this new Lyapunov function, an adaptive vector backstepping controller design procedure is presented. Lastly, a series of virtual signals is generated to compensate for input nonlinearities including dead zones, quantization, and saturation by the fractional order auxiliary systems. Simulation results confirm the validity of the theoretical analysis.

Estimation and disturbance rejection performance for fractional order fuzzy systems

This paper gives attention to the issues of output tracking and disturbance rejection performance for a class of fractional order Takagi-Sugeno fuzzy systems in the presence of time-varying delay and unknown external disturbances. More specifically, a new configuration of a fractional order modified repetitive controller that incorporates an improved equivalent-input-disturbance estimator and gain fluctuations in its design is proposed to perform disturbance rejection for the addressed system. By introducing a continuous frequency distributed equivalent model and using the Lyapunov-Krasovskii stability theory, a new set of sufficient conditions ensuring robust asymptotic stability of the resulting closed-loop system is obtained in the framework of linear matrix inequalities. Finally, a numerical example is presented to validate the developed theoretical results, where it is shown that the obtained conditions could force the considered system output to exactly track the given any kind of reference signal by compensating the unknown external disturbance.

Fixed pole based modeling and simulation schemes for fractional order systems

This paper mainly investigates the numerical implementation issue of fractional order systems. First, a pattern of fixed pole schemes are developed to approximate fractional integrator/differentiator, whose common is that the poles keep constant for different α. Then, two solutions are proposed to improve the approximation performance around α=0. Afterwards, the simulation schemes are introduced for two kinds of fractional order systems. In those schemes, the configuration problem of nonzero initial value is considered. Finally, a fair and solid comparison to the classical approximation methods is presented, demonstrating the effectiveness and efficiency of the elaborated algorithms.

Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order α: The 0<α<1 case

This paper presents three different necessary and sufficient conditions for the admissibility and robust stabilization of singular fractional order systems (FOS) with the fractional order α:0<α<1 case. Two results are obtained in terms of strict linear matrix inequalities (LMIs) without equality constraint. The system uncertainties considered are norm bounded instead of interval uncertainties. The equivalence between quadratic admissibility and general quadric stability for FOS are derived. A condition is not only strict LMI condition without quality constraint but also avoid a singularity trouble caused by the superfluous solved variable. When α=1 and E=I, the three results reduce to the conditions of stability and robust stabilization of normal integer order systems. Numerical examples are given to verify the effectiveness of the criteria. With the approaches proposed in this technical note, we can analyze and design singular fractional order systems with similar way to the normal integer order systems.

Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems

In this paper, controllability and observability matrices for pseudo upper or lower triangular multi-order fractional systems are derived. It is demonstrated that these systems are controllable and observable if and only if their controllability and observability matrices are full rank. In other words, the rank of these matrices should be equal to the inner dimension of their corresponding state space realizations. To reduce the computational complexities, these matrices are converted to simplified matrices with smaller dimensions. Numerical examples are provided to show the usefulness of the mentioned matrices for controllability and observability analysis of this case of multi-order fractional systems. These examples clarify that the duality concept is not necessarily true for these special systems.

Fractional order uncertainty estimator based hierarchical sliding mode design for a class of fractional order non-holonomic chained system

This paper proposes a novel fractional order sliding mode control approach to address the issues of stabilization as well as tracking of an N-dimensional extended chained form of fractional order non-holonomic system. Firstly, the hierarchical fractional order terminal sliding manifolds are selected to procure the desired objectives in finite time. Then, a sliding mode control law is formulated which provides robustness against various system uncertainties or external disturbances. In addition, a novel fractional order uncertainty estimator is deduced mathematically to estimate and mitigate the effects of uncertainties, which also excludes the requirement of their upper bounds. Due to the omission of discontinuous control action, the proposed algorithm ensures a chatter-free control input. Moreover, the finite time stability of the closed loop system has been proved analytically through well known Mittag-Leffler and Fractional Lyapunov theorems. Finally, the proposed methodology is validated with MATLAB simulations on two examples including an application of fractional order non-holonomic wheeled mobile robot and its performances are also compared with the existing control approach.

An efficient algorithm for low-order direct discrete-time implementation of fractional order transfer functions

Fractional order systems become increasingly popular due to their versatility in modelling and control applications across various disciplines. However, the bottleneck in deploying these tools in practice is related to their implementation on real-life systems. Numerical approximations are employed but their complexity no longer match the attractive simplicity of the original fractional order systems. This paper proposes a low-order, computationally stable and efficient method for direct approximation of general order (fractional order) systems in the form of discrete-time rational transfer functions, e.g. processes, controllers. A fair comparison to other direct discretization methods is presented, demonstrating its added value with respect to the state of art.

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 fast controller design via nonlinear fractional differential equations

A new method for linear system controller design is proposed whereby the closed-loop system achieves both robustness and fast response. The robustness performance considered here means the damping ratio of closed-loop system can keep its desired value under system parameter perturbation, while the fast response, represented by rise time of system output, can be improved by tuning the controller parameter. We exploit techniques from both the nonlinear systems control and the fractional order systems control to derive a novel nonlinear fractional order controller. For theoretical analysis of the closed-loop system performance, two comparison theorems are developed for a class of fractional differential equations. Moreover, the rise time of the closed-loop system can be estimated, which facilitates our controller design to satisfy the fast response performance and maintain the robustness. Finally, numerical examples are given to illustrate the effectiveness of our methods.

Generalized characteristic ratios assignment for commensurate fractional order systems with one zero

In this paper, a new method for determination of the desired characteristic equation and zero location of commensurate fractional order systems is presented. The concept of the characteristic ratio is extended for zero-including commensurate fractional order systems. The generalized version of characteristic ratios is defined such that the time-scaling property of characteristic ratios is also preserved. The monotonicity of the magnitude frequency response is employed to assign the generalized characteristic ratios for commensurate fractional order transfer functions with one zero. A simple pattern for characteristic ratios is proposed to reach a non-overshooting step response. Then, the proposed pattern is revisited to reach a low overshoot (say for example 2%) step response. Finally, zero-including controllers such as fractional order PI or lag (lead) controllers are designed using generalized characteristic ratios assignment method. Numerical simulations are provided to show the efficiency of the so designed controllers.

An innovative fixed-pole numerical approximation for fractional order systems

A novel numerical approximation scheme is proposed for fractional order systems by the concept of identification. An identical equation is derived firstly, from which one can obtain the exact state space model of fractional order systems. It reveals the nature of the approximation problem, and then provides an effective scheme to obtain the desired model. This research project also focuses on solving a knotty but crucial issue, i.e., the initial value problem of fractional order systems. The results generated by the study prove that it can reduce to the Caputo case by selecting some specific initial values. A careful simulation study is reported to illustrate the effectiveness of the proposed scheme. To exhibit the superiority clearly, the results are compared with that of the published fixed-pole finite model method.