Mohamed A. Mabrok

Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures With Free Body Motion

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented.

Sampling-Based Learning Control for Quantum Systems With Uncertainties

Robust control design for quantum systems has been recognized as a key task in the development of practical quantum technology. In this paper, we present a systematic numerical methodology of sampling-based learning control (SLC) for control design of quantum systems with uncertainties. The SLC method includes two steps of training and testing. In the training step, an augmented system is constructed using artificial samples generated by sampling uncertainty parameters according to a given distribution. A gradient flow-based learning algorithm is developed to find the control for the augmented system. In the process of testing, a number of additional samples are tested to evaluate the control performance, where these samples are obtained through sampling the uncertainty parameters according to a possible distribution. The SLC method is applied to three significant examples of quantum robust control, including state preparation in a three-level quantum system, robust entanglement generation in a two-qubit superconducting circuit, and quantum entanglement control in a two-atom system interacting with a quantized field in a cavity. Numerical results demonstrate the effectiveness of the SLC approach even when uncertainties are quite large, and show its potential for robust control design of quantum systems.

Spectral Conditions for Negative Imaginary Systems With Applications to Nanopositioning

The negative imaginary (NI) property is exhibited by many systems such as flexible structures with force actuators and position sensors and can be used to prove the robust stability of flexible structure control systems. In this paper, we derive methods to check for the NI and strict negative imaginary (SNI) properties in both the single-input single-output as well as multi-input multi-output cases. The proposed methods are based on spectral conditions on a corresponding Hamiltonian matrix obtained for a given system transfer function matrix. Under certain conditions, a given transfer function matrix satisfies the NI property if and only if the corresponding Hamiltonian matrix has no pure imaginary eigenvalues with odd multiplicity. It is also shown that a given transfer function matrix satisfies the SNI property if and only if the corresponding Hamiltonian matrix has no eigenvalues on the imaginary axis, except at the origin. The results of this paper are applied to check the NI property in two nanopositioning applications.

A new stability result for the feedback interconnection of negative imaginary systems with a pole at the origin

This paper is concerned with stability conditions for the positive feedback interconnection of negative imaginary systems. A generalization of the negative imaginary lemma is derived, which remains true even if the transfer function has poles on the imaginary axis including the origin. A sufficient condition for the internal stability of a feedback interconnection for NI systems including a pole at the origin is given and an illustrative example is presented to support the result.

Enforcing negative imaginary dynamics on mathematical system models

Flexible structures with collocated force actuators and position sensors lead to negative imaginary dynamics. However, in some cases, the mathematical models obtained for these systems, for example, using system identification methods may not yield a negative imaginary system. This paper provides two methods for enforcing negative imaginary dynamics on such mathematical models, given that it is known that the underlying dynamics ought to belong to this system class. The first method is based on a study of the spectral properties of Hamiltonian matrices. A test for checking the negativity of the imaginary part of a corresponding transfer function matrix is first developed. If an associated Hamiltonian matrix has pure imaginary axis eigenvalues, the mathematical model loses the negative imaginary property in some frequency bands. In such cases, a first-order perturbation method is proposed for iteratively collapsing the frequency bands whose negative imaginary property is violated and finally displacing the eigenvalues of the Hamiltonian matrix away from the imaginary axis, thus restoring the negative imaginary dynamics. In the second method, direct spectral properties of the imaginary part of a transfer function are used to identify the frequency bands where the negative imaginary properties are violated. A pointwise-in-frequency scheme is then proposed to restore the negative imaginary system properties in the mathematical model.

A generalized negative imaginary lemma and Riccati-based static state-feedback negative imaginary synthesis

Abstract In this paper, we present a generalized negative imaginary lemma based on a generalized negative imaginary system definition. Then, an algebraic Riccati equation method is given to determine if a system is negative imaginary. Also, a state feedback control procedure is presented that stabilizes an uncertain system and leads to the satisfaction of the negative imaginary property. The controller synthesis procedure is based on the proposed negative imaginary lemma. Using this procedure, the closed-loop system can be guaranteed to be robustly stable against any strict negative imaginary uncertainty, such as in the case of unmodeled spill-over dynamics in a lightly damped flexible structure. A numerical example is presented to illustrate the usefulness of the results.

Spectral Conditions for the Negative Imaginary Property of Transfer Function Matrices

Abstract This paper derives some necessary and sufficient conditions for linear time invariant systems to have the negative imaginary property in both the single-input-single-output as well as the multi-input-multi-output cases. The conditions for a system to be negative imaginary are described in terms of spectral conditions obtained for a given transfer function matrix.

Enforcing a system model to be negative imaginary via perturbation of Hamiltonian matrices

Flexible structure dynamics with collocated force actuators and position sensors lead to negative imaginary (NI) systems. However, in some cases, the models obtained for these systems may not satisfy the NI property. This paper provides a new method for enforcing such models to be NI. The results are based on a study of the spectral properties of related Hamiltonian matrices. A test for the negativity of the imaginary part of a corresponding transfer function matrix is first performed by checking for the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues, the system is not NI. In such cases, a first-order perturbation is presented for the precise characterization of frequency bands where violations of the NI property occur. This characterization is then used for the design of an iterative perturbation scheme for state matrices aimed at displacing the imaginary eigenvalues of the Hamiltonian matrix away from the imaginary axis.

Robust entanglement control between two atoms in a cavity using sampling-based learning control

In this paper, a sampling-based learning control (SLC) algorithm is used to find a robust control law that can steer a quantum system with uncertainties into a maximally entangled state. The quantum system under consideration consists of two two-level atoms interacting with a quantized electromagnetic field. In the sampling-based learning control method, an artificial system is constructed based on the quantum system with uncertainties and an optimal control law is learned for the artificial system. Some additional samples which are generated by sampling the uncertainty parameters are used to test the performance of the optimal control law. Numerical results demonstrate the effectiveness of the SLC method in finding a robust control law for entanglement generation between two atoms in a cavity in the presence of a quantized field.

Entanglement Generation in Uncertain Quantum Systems Using Sampling-Based Learning Control

Abstract In this paper, we develop a control algorithm to generate entanglement in a quantum system with uncertainties. The system under consideration is an uncertain system of two two-level atoms interacting with each other through a dipole-dipole interaction. The sampling-based learning control (SLC) strategy is employed to find a control law. An SLC strategy contains two steps of training and evaluation. In the training step, we obtain several samples to construct an augmented system by sampling the uncertainties according to a possible distribution of the uncertainty parameters and learn an optimal control law by maximizing the performance index. In the evaluation step, we apply the obtained control law from the training step to additional samples through randomly sampling the uncertainties. Numerical results are presented showing the success of the SLC method in control design for generating entanglement.