Stability Analysis of Interconnected Systems With “Mixed” Negative-Imaginary and Small-Gain Properties

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

In two recent papers, Brayton and Tong developed some significant results which are the basis of a constructive approach in the stability analysis of dynamical systems. Although their algorithm is powerful, it taxes the capabilities of most modern computers when the dimension of a system is high or even when the dimension is moderate in size. In this paper we remove these difficulties to a certain extent by generalizing the work of Brayton and Tong to interconnected dynamical systems. In our approach, the analysis of complex dynamical systems (which may be of high dimension) is accomplished in terms of the qualitative properties of the free subsystems and the interconnections of such systems.

SummaryThe H∞ filtering problem for interconnected systems is addressed in this paper. In order to facilitate the stability analysis of interconnected systems under the conditions of failed interconnections, a strategy related to switched systems is introduced. Assume that the interconnections among these subsystems disconnect based on the mode‐dependent average dwell time (MDADT), a switching signal is implemented to denote the failed interconnections cases. In the aid of the MDADT method, a filter is designed such that the fault detection (FD) system is asymptotically stable and satisfies the weighted H∞ performance. Finally, a numerical simulation is provided to show the effectiveness of the proposed filter.

Stability analysis of interconnected systems by means of aggregatre subsystem models

This paper is concerned with the stability analysis of interconnected systems composed of locally and globally exponentially stable subsystems. Sufficient conditions for not necessarily asymptotic and exponentially asymptotic stability are derived. The problem of estimating the stability region of the composite system is also considered.

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.

Sufficient conditions for the Lyapunov stability of several classes of (a) continuous-time composite systems described by ordinary differential equations, (b) discrete-time composite systems described by difference equations, (c) sampled-data composite systems, and (d) composite systems described by functional differential equations are established. In all cases the objective is the same: to analyze the stability of large-scale composite systems in terms of their lower order subsystems and in terms of their interconnecting structure. In order to demonstrate the usefulness of the present approach, several specific examples are considered.

It is well known that noise is inevitable in gene regulatory networks due to the low-copy numbers of molecules and local environmental fluctuations. The prediction of noise effects is a key issue in ensuring reliable transmission of information. Interlinked positive and negative feedback loops are essential signal transduction motifs in biological networks. Positive feedback loops are generally believed to induce a switch-like behavior, whereas negative feedback loops are thought to suppress noise effects. Here, by using the signal sensitivity (susceptibility) and noise amplification to quantify noise propagation, we analyze an abstract model of the Myc/E2F/MiR-17-92 network that is composed of a coupling between the E2F/Myc positive feedback loop and the E2F/Myc/miR-17-92 negative feedback loop. The role of the feedback loop on noise effects is found to depend on the dynamic properties of the system. When the system is in monostability or bistability with high protein concentrations, noise is consistently suppressed. However, the negative feedback loop reduces this suppression ability (or improves the noise propagation) and enhances signal sensitivity. In the case of excitability, bistability, or monostability, noise is enhanced at low protein concentrations. The negative feedback loop reduces this noise enhancement as well as the signal sensitivity. In all cases, the positive feedback loop acts contrary to the negative feedback loop. We also found that increasing the time scale of the protein module or decreasing the noise autocorrelation time can enhance noise suppression; however, the systems sensitivity remains unchanged. Taken together, our results suggest that the negative/positive feedback mechanisms in coupled feedback loop dynamically buffer noise effects rather than only suppressing or amplifying the noise.

Feedback modules, which appear ubiquitously in biological regulations, are often subject to disturbances from the input, leading to fluctuations in the output. Thus, the question becomes how a feedback system can produce a faithful response with a noisy input. We employed multiple time scale analysis, Fluctuation Dissipation Theorem, linear stability, and numerical simulations to investigate a module with one positive feedback loop driven by an external stimulus, and we obtained a critical quantity in noise attenuation, termed as “signed activation time”. We then studied the signed activation time for a system of two positive feedback loops, a system of one positive feedback loop and one negative feedback loop, and six other existing biological models consisting of multiple components along with positive and negative feedback loops. An inverse relationship is found between the noise amplification rate and the signed activation time, defined as the difference between the deactivation and activation time scales of the noise-free system, normalized by the frequency of noises presented in the input. Thus, the combination of fast activation and slow deactivation provides the best noise attenuation, and it can be attained in a single positive feedback loop system. An additional positive feedback loop often leads to a marked decrease in activation time, decrease or slight increase of deactivation time and allows larger kinetic rate variations for slow deactivation and fast activation. On the other hand, a negative feedback loop may increase the activation and deactivation times. The negative relationship between the noise amplification rate and the signed activation time also holds for the six other biological models with multiple components and feedback loops. This principle may be applicable to other feedback systems.

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.

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.

Lightly damped structures with colocated (position) sensors and (force) actuators can typically be modeled by a (possibly infinite) sum of second order transfer functions. Since the relative degree of these systems is more than unity, they are not directly amenable to standard positive real analysis and since these systems are highly resonant, application of the small gain theorem would be usually very conservative. However, a condition for the internal stability of a positive feedback interconnection of two such systems has been available in the theory of positive position control. This result has important implications on stability robustness due to spill-over dynamics. In this paper, we propose a new stability analysis result which generalizes the positive position control stability result to the feedback interconnection of systems satisfying a certain modified positive-real type condition. Broadly speaking, the result states that a necessary and sufficient condition for the internal stability of the feedback interconnection of such systems is that the DC loop gain is less than unity.

Positive Feedback Regulates Switching of Phosphate Transporters in S. cerevisiae

This paper is concerned with the analysis and synthesis of interconnected systems constructed from heterogeneous positive subsystems and a nonnegative interconnection matrix. We first show that admissibility, to be defined in this paper, is an essential requirement in constructing such interconnected systems. Then, we clarify that the interconnected system is admissible and stable if and only if a Metzler matrix, which is built from the coefficient matrices of positive subsystems and the nonnegative interconnection matrix, is Hurwitz stable. By means of this key result, we further provide several results that characterize the admissibility and stability of the interconnected system in terms of the Frobenius eigenvalue of the interconnection matrix and the weighted L1- induced norm of the positive subsystems again to be defined in this paper. 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., its state converges to a unique strictly positive vector (that is known in advance up to a strictly positive constant multiplicative factor) for any nonnegative and nonzero initial state. As an important consequence of this property, we show that the output of the interconnected system converges to a scalar multiple of the right eigenvector of a nonnegative matrix associated with its Frobenius eigenvalue, where the nonnegative matrix is nothing but the interconnection matrix scaled by the steady-stage gains of the positive subsystems. This result is then naturally and effectively applied to formation control of multiagent systems with positive dynamics. This result can be seen as a generalization of a well-known consensus algorithm that has been basically applied to interconnected systems constructed from integrators.

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