Morad Nazari

Spacecraft Attitude Stabilization Using Nonlinear Delayed Multiactuator Control and Inverse Dynamics

The dynamics of a rigid spacecraft with nonlinear delayed multiactuator feedback control are studied in this paper. It is assumed that the time delay occurs in one of the actuators, whereas the other actuator has a negligible time delay. Therefore, a nonlinear feedback controller using both delayed and nondelayed states is sought for the controlled system to have the desired linear delayed closed-loop dynamics using an inverse dynamics approach. The closed-loop stability is shown to be approximated by a second-order linear delay differential equation for the modified Rodriguez parameter attitude coordinates for which the Hsu–Bhatt–Vyshnegradskii stability chart can be used to choose the control gains that result in a stable closed-loop response. An analytical derivation of the boundaries of this chart for the case of no derivative feedback control is shown, whereas a numerical method is used to obtain the stability chart for the general case. Then, to achieve a specified performance, the criteria for a cr...

Response and Stability Analysis of Periodic Delayed Systems With Discontinuous Distributed Delay

In this paper, the analysis of delay differential equations with periodic coefficients and discontinuous distributed delay is carried out through discretization by the Chebyshev spectral continuous time approximation (ChSCTA). These features are introduced in the delayed Mathieu equation with discontinuous distributed delay which is used as an illustrative example. The efficiency of stability analysis is improved by using shifted Chebyshev polynomials for computing the monodromy matrix, as well as the adaptive meshing of the parameter plane. An idea for a method for numerical integration of periodic DDEs with discontinuous distributed delay based on existing MATLAB functions is proposed. [DOI: 10.1115/1.4005925]

Decentralized Consensus Control of a Rigid-Body Spacecraft Formation with Communication Delay

The decentralized consensus control of a formation of rigid-body spacecraft is studied in the framework of geometric mechanics while accounting for a constant communication time delay between spacecraft. The relative position and attitude (relative pose) are represented on the Lie group SE(3) and the communication topology is modeled as a digraph. The consensus problem is converted into a local stabilization problem of the error dynamics associated with the Lie algebra se(3) in the form of linear time-invariant delay differential equations with a single discrete delay in the case of a circular orbit, whereas it is in the form of linear time-periodic delay differential equations in the case of an elliptic orbit, in which the stability may be assessed using infinite-dimensional Floquet theory. The proposed technique is applied to the consensus control of four spacecraft in the vicinity of a Molniya orbit.

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.

Effects of Damaged Boundaries on the Free Vibration of Kirchhoff Plates: Comparison of Perturbation and Spectral Collocation Solutions

In order to compare numerical and analytical results for the free vibration analysis of Kirchhoff plates with both partially and completely damaged boundaries, the Chebyshev collocation and perturbation methods are utilized in this paper, where the damaged boundaries are represented by distributed translational and torsional springs. In the Chebyshev collocation method, the convergence studies are performed to determine the sufficient number of the grid points used. In the perturbation method, the small perturbation parameter is defined in terms of the damage parameter of the plate, and a sequence of recurrent linear boundary value problems is obtained which is further solved by the separation of variables technique. The results of the two methods are in good agreement for small values of the damage parameter as well as with the results in the literature for the undamaged case. The case of mixed damaged boundary conditions is also treated by the Chebyshev collocation method.

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.

Optimal Transfers with Guidance to the Earth-Moon L1 and L3 Libration Points using Invariant Manifolds: A Preliminary Study

Two optimization methods are presented for designing fuel-optimal impulsive transfers from low earth orbit to the Earth-Moon libration points L1 and L3 using the stable manifolds found in the Circular Restricted Three Body Problem. The two methods used are a grid search and a genetic algorithm. All transfers require one maneuver to leave LEO and a second to transfer onto the stable manifold. The trajectories discussed have total changes in velocity between 3.48 km/s and 3.72 km/s. An LQR based guidance scheme is presented which maintains the spacecraft on the stable manifold in the presence of injection errors in burn magnitude and direction. Monte Carlo analyses are conducted to further demonstrate effectiveness of the guidance scheme. The total ∆V costs to use the guidance scheme average 12.34 m/s and 10.88 m/s for L1 and L3 libration point transfers, respectively, when a thrust magnitude Gaussian dispersion with 1σ = 1% error is combined with in-plane and out-of-plane thrust direction Gaussian dispersions with 1σ = 1 degree error.