Arman Dabiri

Optimal fractional state feedback control for linear fractional periodic time-delayed systems

This paper proposes a new strategy to design an optimal fractional state feedback control for linear fractional periodic time-delayed (FPTD) systems. Although there exist different techniques to design the state feedback control for linear ordinary periodic time-delayed (OPTD) systems such as discretizing their monodromy matrix, there is no systematic method to design state feedback control for FPTD systems. Moreover, linear OPTD systems have the monodromy operator defined explicitly in a Banach space, and can be discretized in arbitrary basis functions. However, linear FPTD systems do not have any monodromy operator because of the nonlocal properties of fractional operators. In the proposed method, a monodromy matrix is defined for the steady state solution. Then, the efficiency of the proposed control technique is shown by implementing the method to a double inverted pendulum with fractional dampers subject to a periodic retarded follower force.

The spectral parameter estimation method for parameter identification of linear fractional order systems

This paper presents a new method to identify unknown parameters of linear fractional order systems by discretizing it at the Gauss-Lobatto-Chebyshev collocation points. The proposed spectral parametric estimation method benefits from the spectral method exponential convergence feature and results in more accuracy in the estimated parameters and less computational time. The advantages of using the spectral parameter estimation method are shown in two examples. In the first example, a batch of five isothermal creep experimental data for an epoxy is used to estimate a simple solid viscoelastic model with fractional order. In the second example, the state matrix and fractional orders of a linear system with non-commensurate fractional order are estimated by using its input-output data disturbed by unknown white Gaussian measurement noise.

Transition Curve Analysis of Linear Fractional Periodic Time-Delayed Systems Via Explicit Harmonic Balance Method

This paper presents a technique to obtain the transition curves of fractional periodic time-delayed (FPTD) systems based on a proposed explicit harmonic balance (EHB) method. This method gives the analytical Hill matrix of FPTD systems explicitly with a symbolic computation-free algorithm. Furthermore, all linear operations on Fourier basis vectors including fractional order derivative operators and time-delayed operators for a linear FPTD system are obtained. This technique is illustrated with parametrically excited simple and double pendulum systems, with both time-delayed states and fractional damping.

Optimal Periodic-Gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time-Delayed Systems

This paper develops the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control for a class of linear fractional-order periodic time-delayed systems. Although there exist techniques for the state feedback control of linear periodic time-delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for a steady-state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a closed form for FPTD systems, and hence, the short-memory principle along with the fractional Chebyshev collocation method is used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers, in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response.

Stability and Control of Fractional Periodic Time-Delayed Systems

In this chapter, two new methods are proposed to study the stability of linear fractional periodic time-delayed (FPTD) systems. First, the explicit harmonic balance (EHB) method is proposed to find necessary and sufficient conditions for fold, flip, and secondary Hopf transition curves in linear FPTD systems, from which the stability boundaries are obtained as a subset. Transition curves of the fractional damped delayed Mathieu equation are obtained by using the EHB method. Next, an approximated monodromy operator in a Banach space is defined for FPTD systems, which gives the linear map between two solutions. The fractional Chebyshev collocation (FCC) method is proposed to approximate this monodromy operator. The FCC method is outlined and illustrated with three practical problems including obtaining the parametric stability charts of the fractional Hayes equation and the fractional second-order system with delay, and designing an optimal linear periodic gain fractional delayed state feedback control for the damped delayed Mathieu equation.

Chaos Analysis and Control in Fractional Order Systems Using Fractional Chebyshev Collocation Method

In this paper, fractional Chebyshev collocation method is proposed to study Lyapunov exponents (LEs) and chaos in a fractional order system with nonlinearities. For this purpose, the solution of the fractional order system is discretized by N-degree Gauss-Lobatto-Chebyshev (GLC) polynomials where N is an integer number. Then, the discrete orthogonality relationship for the Chebyshev polynomials is used to obtain the fractional Chebyshev differentiation matrix. The differentiation matrix is then used to convert the nonlinear fractional differential equations to a system of nonlinear algebraic equations with the collocation points as the unknowns. The dominant LE (other than the zero LE) that corresponds to the time dimension is then computed by measuring the exponential rate of the trajectory deviations initiated slightly off the attractor point. The proposed technique is implemented to a damped driven pendulum with fractional order damping and the convergence of the dominant LE is studied versus the number of Chebyshev collocation points. The LE analysis is also verified by studying the system time and frequency responses for different values of the bifurcation parameter. Furthermore, the LE obtained by the proposed method for the analogous integer order system is compared with those obtained by the Jacobian technique and Gruwald-Letnikov approximation. Finally a fractional state feedback controller is designed to control the chaotic system to a desired equilibrium or periodic trajectory such that the error dynamics are time invariant or time periodic, respectively. The numerical example studied is the damped driven pendulum with fractional dampers.Copyright

Isothermal Creep Behavior of 3M™ Scotch-Weld™ EC-2216 B\A in Single Lap Joints

This paper studies the isothermal creep behavior of the 3M™ Scotch-Weld™ EC-2216 B\A Gray (EC-2216) epoxy in single lap joints subjected to a uniaxial tensile load. The time-independent mechanical properties of EC-2216 such as the Young’s modulus, shear modulus, and bulk modulus have been reported in the literature, but the results are not consistent. However, to the best knowledge of the authors its shear stress-strain constitutive equation has not been obtained yet. Thus, in this paper, we propose the shear stress-strain constitutive equation of the EC-2216 in single lap joints subjected to uniaxial tensile tests at room temperature (24±3°C). First, a viscoelastic model is estimated in short-term experiments within the viscoelastic regime. Second, the creep deformation of the EC-2216 is modeled by different conventional imperial models. Third, different conventional creep rate models are used to estimate the steady state creep rate of the third stage, which mainly indicates the longevity of the material. Moreover, the failure accumulate strain is obtained based on the maximum strain criterion, for different shear stress levels. Finally, the proposed models are compared to the current published data.Copyright

Computed Torque Control of the Stewart platform with uncertainty for lower extremity robotic rehabilitation

Parallel manipulators play a key role in robotic rehabilitation. In reality, such systems operate under uncertainty due to the changes in the characteristics of the patients and lack of knowledge about the physical and geometrical properties of the system. In this paper, we present a robust control scheme to control a six-degree-of-freedom Stewart platform. In this application, it is aimed to follow a desired pure rotational motion required in the robotic rehabilitation of the foot for patients with diabetic neuropathy. It is assumed that uncertainty exists in the mass of the foot of the patients (the proposed approach can also be used when disturbance exists). To perform this, the method of polynomial chaos expansion (PCE) is extended and integrated with the computed torque control law (CTCL) to control the system. In PCE scheme, uncertainty is introduced to the system by compactly projecting each stochastic response output and random input onto the space of appropriate independent orthogonal polynomial basis functions. CTCL uses a feedback linearization technique which provides the necessary force/torque to enforce the system to follow a prescribed trajectory. This papers presents a successful implementation of the PCE-base CTCL on a Stewart platform. Finally, a comparison between the efficiency and accuracy of the Monte Carlo and PCE is conducted.

An optimal Stewart platform for lower extremity robotic rehabilitation

In this paper, two algorithms are performed to find an optimum design of a six-degree-of-freedom Stewart platform to provide a desired pure rotational motion required in the robotic rehabilitation of the foot for patients with neuropathy. To accomplish this, first, we present the kinematic and the dynamic analysis of the Stewart platform. The dynamic equations are derived by using a customized Lagrange method. Then, physically meaningful objective variables are defined such as the size of the platform, the length of the six links, the maximum stroke of the six linear actuators, the maximum actuator force, and the reachable workspace. This is followed by using two optimization methods (Genetic Algorithm and Monte-Carlo method) to study the aforementioned objective variables, resulting in the optimal solution for the desired orientation motions. Then, the detailed investigation of the effect of changes in these objective variables on the variation of the platform design variables is studied. Finally, in a numerical example, the advantages and disadvantages of using the Genetic Algorithm and the Monte-Carlo method to find the optimal design variables for a custom cost function with weighted objective variables are revealed.

Adaptive neural-fuzzy inference system to control dynamical systems with fractional order dampers

In this paper, an adaptive neural fuzzy inference system (ANFIS)-based control technique is proposed to stabilize dynamical systems with fractional order dampers. For this purpose, a linear quadratic regulator (LQR) is first designed for the analogous linearized integer order systems where the fractional damper is replaced by the combination of an integer spring and an integer damper. Next, the ANFIS-based controller is trained based on the responses of the closed-loop LQR-controlled system under different scenarios such as several initial conditions and/or inputs. Since the number of fuzzy rules increases exponentially by increasing the number of inputs, a fusion function proposed in the literature is used to reduce the number of inputs in the ANFIS-based controller. Hence the number of fuzzy rules is reduced as well. The result of this training is a trained ANFIS-LQR controller that can be used for stabilizing the fractional-order models with fractional order dampers. As an illustrative example, the proposed technique is employed to stabilize an under-actuated double inverted pendulum on the cart with fractional order dampers.