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We recapitulate the notion of phase change rate maximization and demonstrate the usefulness of its solution on analyzing the robust instability of a cyclic network of multi-agent systems subject to a homogenous multiplicative perturbation. Subsequently, we apply the phase change rate maximization result to two practical applications. The first is a magnetic levitation system, while the second is a repressilator with time-delay in synthetic biology.

This paper is concerned with a new optimization problem named "phase change rate maximization" for single-input-single-output linear time-invariant systems. The problem relates to two control problems, namely robust instability analysis against stable perturbations and minimum norm strong stabilization. We define an index of the instability margin called "robust instability radius (RIR)" as the smallest $H_\infty$ norm of a stable perturbation that stabilizes a given unstable system. This paper has two main contributions. It is first shown that the problem of finding the exact RIR via the small-gain condition can be transformed into the problem of maximizing the phase change rate at the peak frequency with a phase constraint. Then, we show that the maximum is attained by a first-order all-pass function and derive conditions, under which the RIR can be exactly characterized, in terms of the phase change rate. Two practical applications are provided to illustrate the utility of our results.

In many practical control applications, the performance level of a closed-loop system degrades over time due to the change of plant characteristics. Thus, there is a strong need for redesigning a controller without going through the system modeling process, which is often difficult for closed-loop systems. Reinforcement learning (RL) is one of the promising approaches that enable model-free redesign of optimal controllers for nonlinear dynamical systems based only on the measurement of the closed-loop system. However, the learning process of RL usually requires a considerable number of trial-and-error experiments using the poorly controlled system that may accumulate wear on the plant. To overcome this limitation, we propose a model-free two-step design approach that improves the transient learning performance of RL in an optimal regulator redesign problem for unknown nonlinear systems. Specifically, we first design a linear control law that attains some degree of control performance in a model-free manner, and then, train the nonlinear optimal control law with online RL by using the designed linear control law in parallel. We introduce an offline RL algorithm for the design of the linear control law and theoretically guarantee its convergence to the LQR controller under mild assumptions. Numerical simulations show that the proposed approach improves the transient learning performance and efficiency in hyperparameter tuning of RL.

This paper is concerned with robust instability analysis for linear multi-agent dynamical systems with cyclic structure. This relates to interesting and important periodic oscillation phenomena in biology and neuronal science, since the nonlinear phenomena often occur when the linearized model around an equilibrium point is unstable. We first make a problem setting on the analysis and define the notion of robust instability radius (RIR) as a quantitative measure for maximum allowable stable dynamic perturbation in terms of the H-infinity norm. After showing lower bounds of the RIR, we derive the exact RIR, which is analytic and scalable, for first order time-lag agents. Finally, we make a remark on the potential applicability to some classes of higher order systems.

Oscillatory chemical reactions often serve as a timing clock of cellular processes in living cells. The temporal dynamics of protein concentration levels is thus of great interest in biology. Here we propose a theoretical framework to analyze the frequency, phase and amplitude of oscillatory protein concentrations in gene regulatory networks with negative cyclic feedback. We first formulate the analysis framework of oscillation profiles based on multivariable harmonic balance. With this framework, the frequency, phase and amplitude are obtained analytically in terms of kinetic constants of the reactions despite the nonlinearity of the dynamics. These results are demonstrated with the Pentilator and Hes7 self-repression network, and it is shown that the developed analysis method indeed predicts the profiles of the oscillations. A distinctive feature of the presented result is that the waveform of oscillations is analytically obtained for a broad class of biochemical systems. Thus, it is easy to see how the waveform is determined from the system's parameters and structures. We present general biological insights that are applicable for any gene regulatory networks with negative cyclic feedback., Comment: Appendix A and some references have been added

This paper is concerned with conditions for the existence of oscillations in gene regulatory networks with negative cyclic feedback, where time delays in transcription, translation and translocation process are explicitly considered. The primary goal of this paper is to propose systematic analysis tools that are useful for a broad class of cyclic gene regulatory networks, and to provide novel biological insights. To this end, we adopt a simplified model that is suitable for capturing the essence of a large class of gene regulatory networks. It is first shown that local instability of the unique equilibrium state results in oscillations based on a Poincare-Bendixson type theorem. Then, a graphical existence condition, which is equivalent to the local instability of a unique equilibrium, is derived. Based on the graphical condition, the existence condition is analytically presented in terms of biochemical parameters. This allows us to find the dimensionless parameters that primarily affect the existence of oscillations, and to provide biological insights. The analytic conditions and biological insights are illustrated with two existing biochemical networks, Repressilator and the Hes7 gene regulatory networks.