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Yiming Meng | Applied Mathematics, University of Waterloo
Bifurcation and Robust Control of Instabilities in the Presence of Uncertainties
In real-world applications, nominal mathematical models that are used to describe the state behaviors of dynamical systems are usually less robust to deal with environmental disruptions. Uncertainties, such as imprecision of signals, Gaussian-type white noise, and observation errors, may be injected into the systems and create substantial impacts on stability and safety etc. To better understand and hence robustly eliminate the potential negative impact, this thesis aims to develop novel control methods and bifurcation analysis for general nonlinear systems that are subjected to such types of perturbations.
The research is motivated by an application problem based on the perturbed Moore-Greitzer full model for detection and control of modern jet engine compressor instabilities. This commonly used mathematical model consists of a partial differential equation (PDE), which describes the behavior of disturbances in the inlet region of compression systems, and a twodimensional ordinary differential equation, which describes the coupling of the disturbances within the mean flow. It allows us to consider the two-dimensional subspace and the infinitedimensional subspace driven by the PDE separately under certain conditions. In particular, determined by compressor geometry, as the throttle coefficient decreases, the jet engine compressors with the absence of noise may exhibit potentially three types of Hopf bifurcation within, respectively, each of the two subspaces or both. Compared to the finite-dimensional cases, the Hopf bifurcation in infinite-dimensional systems with stochastic perturbations is not well understood. On the other hand, subjected to the finite-dimensional subspace, control strategies regarding stabilization and performance improvement in the presence of uncertainties need to be developed.
The first main aspect of the research addresses the verification and control synthesis of more complex tasks with ω-regular linear-time properties besides stabilization problems for more general perturbed finite-dimensional nonlinear systems. Rigorous abstraction-based formal methods compute with guarantees a set of initial states from which the trajectories satisfy or a controller exists to realize the given specification, however, at the cost of heavy state-space discretization and potential difficulties of adjusting the speed of the dynamical flows. This thesis proposes discretization-free Lyapunov methods to handle verification and control synthesis for building-block specifications such as safety, stability, reachability, and reach-and-stay specifications. In the presence of non-stochastic and stochastic perturbations, respectively, rigorous analysis is conducted upon the fundamental mathematical guarantees of satisfying the above mentioned specifications using Lyapunov-like functions. A comparison between the proposed Lyapunov method and formal methods is illustrated via numerical simulations for the case with non-stochastic perturbations.
In terms of formal verification and control synthesis for stochastic systems, the current literature focuses on developing sound abstraction techniques for discrete-time stochastic dynamics without extra uncertain signals. However, soundness thus far has only been shown for preserving the satisfaction probability of certain types of temporal-logic specification. We focus on more general discrete-time nonlinear stochastic systems and present constructive finite-state abstractions for verifying or control synthesis of probabilistic satisfaction with respect to general ω-regular linear-time properties. Instead of imposing stability assumptions, we analyze the probabilistic properties from the topological view of metrizable space of probability measures. Such abstractions are both sound and approximately complete. That is, given a concrete discrete-time stochastic system and an arbitrarily small L1-perturbation of this system, there exists a family of finite-state Markov chains whose set of satisfaction probabilities contains that of the original system and meanwhile is contained by that of the slightly perturbed system. A direct consequence is that, given a probabilistic linear-time specification, initializing within the winning/losing region of the abstraction system can guarantee a satisfaction/dissatisfaction for the original system. We make an interesting observation that, unlike the deterministic case, point-mass (Dirac) perturbations cannot fulfill the purpose of robust completeness.
The second aspect of the research addresses the bifurcation analysis in parabolic stochastic partial differential equations (SPDEs). We consider cases with small additive and multiplicative space-time noise, respectively, and conduct a local bifurcation analysis via a multiscale technique. In the presence of small additive noise, we make assumptions that the noise only acts on the stable fast-varying modes. We apply homogenization techniques based on recent advances for systems with one-dimensional critical mode directly to the perturbed Moore-Greitzer full model. We rigorously develop low-dimensional approximations using a multiscale analysis approach near the deterministic Hopf bifurcation point that occurs within the infinite-dimensional subspace. We also show that the reduced-dimension approximation model contains a multiplicative noise.
To better understand the long-term behavior of SPDEs near the deterministic Hopf bifurcation point and demonstrate the stochastic Hopf bifurcations under the impact of small multiplicative noise, we focus on the system with only cubic nonlinearities and use a different approach other than stochastic averaging/homogenization. We propose a simplified equation that has the same linearization as the original equation and prove the error bounds. It can be shown that the stable marginals do have a small impact on determining the stochastic bifurcation points. This approximation scheme does not reduce the stochastic effects from the stable modes to point-mass perturbations, and can be allied with the almost-sure exponential stability of the trivial solution to analyze the stochastic bifurcation diagram as the noise becomes smaller.