Stable Inversion Approach Research Articles

Application of stable inversion to flexible manipulators modeled by the absolute nodal coordinate formulation

AbstractCompared to conventional robots, flexible manipulators offer many advantages, such as faster end‐effector velocities and less energy consumption. However, their flexible structure can lead to undesired oscillations. Therefore, the applied control strategy should account for these elasticities. A feedforward controller based on an inverse model of the system is an efficient way to improve the performance. However, unstable internal dynamics arise for many common flexible robots and stable inversion must be applied. In this contribution, an approximation of the original stable inversion approach is proposed. The approximation simplifies the problem setup, since the internal dynamics do not need to be derived explicitly for the definition of the boundary conditions. From a practical point of view, this makes the method applicable to more complex systems with many unactuated degrees of freedom. Flexible manipulators modeled by the absolute nodal coordinate formulation (ANCF) are considered as an application example.

Numerically stable inversion approach to construct Kohn-Sham potentials for given electron densities within a Gaussian basis set framework.

We present a Kohn-Sham (KS) inversion approach to construct KS exchange-correlation potentials corresponding to given electron densities. This method is based on an iterative procedure using linear response to update potentials. All involved quantities, i.e., orbitals, potentials, and response functions, are represented by Gaussian basis functions. In contrast to previous KS inversion methods relying on Gaussian basis sets, the method presented here is numerically stable even for standard basis sets from basis set libraries due to a preprocessing of the auxiliary basis used to represent an exchange-correlation charge density that generates the exchange-correlation potential. The new KS inversion method is applied to reference densities of various atoms and molecules obtained by full configuration interaction or CCSD(T) (coupled cluster singles doubles perturbative triples). The considered examples encompass cases known to be difficult, such as stretched hydrogen or lithium hydride molecules or the beryllium isoelectronic series. For the stretched hydrogen molecule, potentials of benchmark quality are obtained by employing large basis sets. For the carbon monoxide molecule, we show that the correlation potential from the random phase approximation (RPA) is in excellent qualitative and quantitative agreement with the correlation potential from the KS inversion of a CCSD(T) reference density. This indicates that RPA correlation potentials, in contrast to those from semi-local density-functionals, resemble the exact correlation potential. Besides providing exchange-correlation potentials for benchmark purposes, the proposed KS inversion method may be used in density-partition-based quantum embedding and in subsystem density-functional methods because it combines numerical stability with computational efficiency.

Real-Time Implementation of Fractional-Order PID Controller for Magnetic Levitation Plant With Time Delay

The design of a Fractional Order PID controller (FOPID), to stabilize a second-order unstable magnetic levitation plant (MLP) with time delay, is proposed in this paper. The proposed two stage approach involves estimating the lower and upper bounds of the controller gains using an extension of the Hermite-Biehler theorem. Further, the optimal gains within the search space, determined in the first stage, are obtained by formulating the controller design task as a constrained optimization problem seeking to minimize the root mean square error ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RMSE</i> ) of reference tracking, in terms of the ball position measured by the IR Sensor. The optimization problem is solved using a recently proposed agent based evolutionary algorithm, namely, Multi-Agent based Symbiotic Organisms Search (MASOS). It has been confirmed that the theorem is effective in determining the stable ranges of the controller gains, while the MASOS algorithm is able to provide the optimal gain values. The obtained results have been verified by simulation in MATLAB® Simulink®. The findings from the simulation results have been further validated by real-time experiments on the hardware setup of MLP. The comparative analysis with 1-Degree of Freedom (DOF) FOPID, fractional-order Integral-Proportional-Derivative (FOIPD), Linear Quadratic Regulator (LQR), Active Disturbance Rejection Controller (ADRC), Radial Basis Function Sliding Mode Controller (RBFSMC) and Stable Inversion approach using both offline numerical simulation and real-time experiment corroborates the superiority of the FO-IPD controller.

Output Tracking Control for Nonminimum Phase Flexible Air-Breathing Hypersonic Vehicle Models

AbstractBased on the nonlinear stable inversion approach, an exact output tracking control law is designed for flexible air-breathing hypersonic vehicles (FAHVs). This problem is challenging because of the inherent couplings between the propulsion system, the airframe dynamics, and the presence of strong flexibility effects. Because of the complex nonlinear dynamics of the vehicles, a control-oriented model, which can retain the dominant features of the higher-fidelity model, is adopted for the control design. With respect to velocity and altitude as the regulated outputs, a partially linearized model of FAHVs is obtained through input/output linearization. Then the internal dynamics of FAHVs model are constructed and the stability of the zero dynamics are discussed. Based on the noncausal stable inversion approach, the ideal internal dynamics of FAHVs are computed. A state tracking model is derived from the output tracking problem, and an optimal controller is designed. The stability of the closed-loop s...

Active Transient Vibration Control Using a Novel Repetitive Algorithm

In this paper, active control of periodic transient vibration based on repetitive control (RC) algorithm is studied. According to the stability condition of an active vibration control (AVC) system, a novel non-causal stable inversion approach combined with an optimal criterion is used to design the RC controller for an AVC system with a non-minimum phase secondary path. Computer simulations have been carried out to validate the effectiveness of the proposed algorithm. The plant model used in the computer simulations is obtained from a practical AVC system in our laboratory. Simulation results show that the proposed scheme can significantly reduce periodic transient vibration and the convergence rate is acceptable for a non-minimum phase plant.

A functional iterative approach to the tracking control of nonminimum phase switched power converters

In this article, we introduce a methodology that, under appropriate assumptions, provides exact output voltage tracking of smooth periodic references in a class of nonminimum phase, single-input, basic DC-to-DC nonlinear switched power converters by means of a stable inversion approach. Firstly, a \({\fancyscript{C}^1}\) periodic (thus bounded) reference for the nonminimum phase variable that solves the corresponding stable inversion problem, rather than being numerically obtained, is approximated using a Banach’s fixed-point theorem-based iterative procedure that produces an L ∞ norm convergent sequence of \({\fancyscript{C}^1}\) periodic functions that are analytically computable in the closed form. Any element of the sequence may then be used as a reference profile for the nonminimum phase variable in an indirect, state feedback control action that includes exact tracking and stabilizing components as well. In turn, the tracking dynamics of the system yields a sequence of periodic and asymptotically stable output signals that converges in the L ∞ norm to the original output voltage profile. Furthermore, the explicit dependence of the approximate indirect references on the system parameters may provide robustness to piecewise constant disturbances by means of online dynamic compensation. The technique is exemplified on a buck-boost converter and validated through simulation results.

Precision preview-based stable-inversion for nonlinear nonminimum-phase systems: The VTOL example

This article quantifies the importance of the future desired trajectory in determining the exact-output-tracking input for nonlinear, nonminimum-phase systems by using system inversion techniques. It is intuitive that the effect of the desired output's distant-future values, on the output-tracking input at the current time instant, should be small. Therefore, at a current time instant ( t c ) , preview information of the desired output in a finite-time window [ t c , t c + T p ] should be sufficient to compute the output-tracking input with an arbitrarily small prescribed error, if the preview time T p is sufficiently large. The contribution of this article is the quantification of the needed preview time T p by using the benchmark VTOL aircraft model as an example. Additionally, simulation results are presented to evaluate the efficacy of the finite-preview-based stable-inversion approach.

A finite energy property of stable inversion to nonminimum phase nonlinear systems

Stable inversion is a completely new approach to the output tracking control of nonminimum phase nonlinear systems. It not only offers exact reproduction of a given reference output trajectory but also guarantees stability of all external and internal signals. After a brief review of the stable inversion problem, the paper establishes a finite energy property of stable inverses. That is, out of infinitely many possible inverse solutions, the one provided by the stable inversion process is the only one that has finite energy, or, the /spl Lscr//sub 2/(-/spl infin/, +/spl infin/)-norm. The effectiveness of the stable inversion approach in output tracking as compared with the well-known classic inversion and nonlinear regulation approaches is demonstrated by working out an example of a nonlinear nonminimum phase system.