Closed-loop stability analysis of discrete-time negative imaginary systems

A new definition of discrete-time negative imaginary (NI) systems is provided. This definition characterizes the dissipative property of a zero-order hold sampled continuous-time NI system. Under some assumptions, asymptotic stability can be guaranteed for the closed-loop interconnection of an NI system and an output strictly negative imaginary system, with one of them having a one step advance. In the case of linear systems, we also provide necessary and sufficient frequency-domain and LMI conditions under which the definition is satisfied. Also provided is a simple DC gain condition for the stability results in the linear case.

In this letter, an analytical framework is proposed to examine stability of two stable, linear time invariant systems interconnected in positive feedback where the systems have "mixed" properties of negative-imaginary and small-gain. Using the notion of dissipativity, the interconnection of systems is guaranteed to be finite-gain stable under the condition that the dc loop gain is contractive. This work builds on Griggs, and exploits a new set of frequency dependent triplets that was introduced in above reference to "mix" two unconditional stability statements, i.e., small-gain and passivity. Unlike the above reference the present work explores the important question of how a conditional stability statement as needed when two negative-imaginary systems are connected in a feedback loop can be "mixed" with an unconditional stability statement as needed when two contractive systems are connected in a feedback loop. The usefulness of the proposed analytical framework is demonstrated via a numerical example.

A new lemma is proposed to determine the negative imaginary (NI) properties of time-delay systems of retarded and neutral types. The lemma is employed for stability analysis in the context of positive feedback interconnection of infinite-mode NI systems and time-delay systems. A cantilever beam with a collocated piezoelectric sensor and actuator pair is studied as an infinite-mode NI system. A novel time-delay controller is proposed to be used within positive position feedback (PPF) and parametrized based on H∞-norm optimization targeting the vibration suppression of the flexible beam. The proposed lemma demonstrates that the designed controller is NI. We also present a corollary to ensure the stability of the closed-loop beam system with the proposed controller using Lyapunov and dissipation based stability approaches. The theoretical results are further verified by simulations. Moreover, the efficacy of the controller is evaluated against a state-of-the-art controller for vibration suppression. Simulation results show that the proposed controller achieves better performance in suppressing vibrations without significant change in robustness.

Modulated–demodulated control: Q control of an AFM microcantilever

This paper is concerned with the analysis of large-scale interconnected systems constructed from positive subsystems and a nonnegative interconnection matrix. We first show that the interconnected system is admissible and stable if and only if a Metzler matrix built from the coefficient matrices of the positive subsystems and the interconnection matrix is Hurwitz stable. By means of this key lemma, we further provide several results that characterize the admissibility and stability of interconnected systems in terms of the weighted L1-induced norm of each positive subsystem and the Frobenius eigenvalue of the interconnection matrix. Moreover, in the case where every subsystem is SISO, we provide explicit conditions under which the interconnected system has the property of persistence, i.e., the state of the interconnected system converges to a unique strictly positive vector (up to a strictly positive constant multiplicative factor) irrespective of nonnegative and nonzero initial states. We illustrate the effectiveness of the persistence results via formation control of multi-agent systems.

We study the stability analysis and control synthesis of uncertain discrete-time two-dimensional (2D) systems. The mathematical model of the discrete-time 2D system is established upon the well-known Roesser model, and the uncertainty phenomenon, which appears typically in practical environments, is modeled by a convex bounded (polytope type) uncertain domain. The stability analysis and control synthesis of uncertain discrete-time 2D systems are then developed by applying the Lyapunov stability theory. In the processes of stability analysis and control synthesis, the obtained stability/stabilzaition conditions become less conservative by applying some novel relaxed techniques. Moreover, the obtained results are formulated in the form of linear matrix inequalities, which can be easily solved via standard numerical software. Finally, numerical examples are given to demonstrate the effectiveness of the obtained results.

This paper presents the application of an integral resonant controller (IRC) to the problem of frequency locking an optical cavity. Such frequency locking is necessary to ensure maximum constructive interference inside the cavity. This is achieved by altering the resonant frequency of the optical cavity to match the frequency of the input laser. The optical cavity is characterized as a negative imaginary system in the desired frequency range and a strict negative imaginary IRC is designed for locking such a cavity. We show in simulation that the internal stability of the positive feedback interconnection of the plant and the IRC is guaranteed as long as a certain DC gain condition is satisfied, as mentioned in previous theoretical results. Experimental frequency response data for the plant, comprising of a three-mirror ring cavity and a piezoelectric actuator is used. This data is used to obtain a model for the plant using methods of system identification which is then used to design an IRC. A unit step response is obtained to check for the performance of the controller in closed-loop.

Stability analysis of discrete-time systems with arbitrary delay kernels based on kernel-related summation inequality and model transformation

This paper is concerned with stability criterion of discrete-time descriptor systems in terms of LMIs. Though there have been proposed several LMIs for stability of discrete-time descriptor systems, their use is seemingly limited in comparison with LMIs for continuous-time descriptor systems. A remedy of those existing results is that they are not able to be linearized with respect to the coefficient matrices of the descriptor equation. This makes it difficult to apply them to synthesis problems, while LMIs for analysis of state-space systems or those for continuous-time descriptor systems enjoy change-of-variable methods to obtain LMIs for synthesis. In this paper, a new LMI is presented for admissibility of discrete-time descriptor systems that has such a useful feature of affineness of coefficient matrices in the inequality. After showing the main result with the new LMI, we apply it to several problems on discrete-time descriptor systems, namely, state-feedback synthesis, stability analysis of δ-operator systems and D-stability problems generalized to descriptor systems, and a generalization of quadratic stabilization. Some of these are illustrated via numerical examples.

This paper mainly focuses on the attitude control problem of rigid spacecraft, and conducts the design and research of control methods based on the theory of nonlinear negative imaginary systems. A feedback control law is designed for the attitude control model of the rigid spacecraft, ensuring that the closed-loop system satisfies the nonlinear negative imaginary property. Subsequently, according to the stability theory of nonlinear negative imaginary interconnected systems, a strictly negative imaginary controller is designed to achieve the asymptotic stability of the attitude of the rigid spacecraft during its operation. The numerical simulation results verify the effectiveness of the proposed control synthesis method for the rigid spacecraft system.

A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis

Robustness in the face of uncertainties is an integral part of designing a real-time control system. Based on negative imaginary (NI) systems theory, we design robust and adaptive control systems for accurate trajectory tracking of a quadcopter aerial vehicle. Considering the challenging dynamics of unmanned aerial vehicles, we employ knowledge-based fuzzy inference systems (FIS) to facilitate automatic tuning in our SNI controllers, leading to the development of adaptive SNI control systems. Unlike fixed-gain controllers that have no ability to adapt to the variations in environmental conditions or changes in the dynamics of the plant, our adaptive SNI controllers are able to perform self-tuning to constantly update their parameters. The concept of adaptive autopilots will enhance the ability of the closed-loop control systems to accommodate large uncertainties. To demonstrate their efficacy, we design and implement our adaptive SNI controllers in the three-position control loops of the AR.Drone quadcopter after conducting extensive computer simulations. We also perform a rigorous comparative study with respect to the performance of fixed-gain SNI controllers, fixed-gain NI systems, in addition to model-predictive-control systems, and proportional integral derivative (PID) control systems as our benchmarks. To complete the study, we conduct a stability analysis based on Kharitonov's Theorem.

In this brief note, a probability theory approach is proposed to investigate the stability analysis issue for a class of networked sampled-data systems with packet dropouts. The continuous-time linear system under consideration is first transformed into a discrete-time system, where the system matrix of the discrete-time system is characterized by stochastic characteristic. Then, by the matrix exponential computation, an equivalent discrete-time stochastic system is established. Based on this, a controller is designed by the law of total expectation, which guarantees the stochastic stability of the closed-loop stochastic system. Finally, a numerical example and an example using power grid with IEEE 4-bus line are used to verify the effectiveness and applicability of the proposed design algorithm.

Stability Analysis of Takagi-Sugeno Fuzzy Systems With Linear State-Space Submodels

Practical stability analysis and synthesis for nonlinear singular systems .- 3 Strongly absolute stability analysis for Lur'e singular .- 4 Input-to-state stability analysis and design for Lur'e singular systems .- 5 Observer design for nonlinear singular systems .- 6 Absolute stability of Lur'e singularly perturbed systems .- 7 Multi-objective control for T-S fuzzy singularly perturbed systems. 4 Input-to-state stability analysis and design for Lur'e singular systems .- 5 Observer design for nonlinear singular systems .- 6 Absolute stability of Lur'e singularly perturbed systems .- 7 Multi-objective control for T-S fuzzy singularly perturbed systems.