Andrzej Banaszuk

An adaptive algorithm for control of combustion instability

We propose an adaptive algorithm for control of combustion instability suitable for reduction of acoustic pressure oscillations in gas turbine engines, and main burners and augmentors of jet engines over a large range of operating conditions, and supply an experimental demonstration of oscillation attenuation, the first for a large industrial-scale gas turbine combustor. The algorithm consists of an Extended Kalman Filter based frequency tracking observer to determine the in-phase component, the quadrature component, and the magnitude of the acoustic mode of interest, and a phase shifting controller actuating fuel-flow, with the controller phase tuned using extremum-seeking. The paper also identifies a closed-loop model with phase-shifting control of combustion instability from experimental data; supplies stability analysis of the adaptive scheme based upon the identified model, and stable extremum-seeking designs used in experiments.

Hearing the clusters of a graph: A distributed algorithm

We propose a novel distributed algorithm to cluster graphs. The algorithm recovers the solution obtained from spectral clustering without the need for expensive eigenvalue/vector computations. We prove that, by propagating waves through the graph, a local fast Fourier transform yields the local component of every eigenvector of the Laplacian matrix, thus providing clustering information. For large graphs, the proposed algorithm is orders of magnitude faster than random walk based approaches. We prove the equivalence of the proposed algorithm to spectral clustering and derive convergence rates. We demonstrate the benefit of using this decentralized clustering algorithm for community detection in social graphs, accelerating distributed estimation in sensor networks and efficient computation of distributed multi-agent search strategies.

Adaptive control of combustion instability using extremum-seeking

We present results of experiment with two distinct extremum-seeking adaptive algorithms for control of combustion instability suitable for reduction of acoustic pressure oscillations in gas turbine over large range of operating conditions. The algorithms consists of a frequency tracking extended Kalman filter to determine the in-phase component, the quadrature component, and the magnitude of the acoustic mode of interest, and a phase shifting controller with the controller phase tuned using an extremum-seeking algorithms. Even though the algorithms were designed specifically for control of combustion instability in gas turbine engines, they are applicable for control of oscillations of systems whose oscillation frequency and optimal control phase shift depends on operating conditions, and which are driven by strong broad-band disturbance. The algorithms have been tested in combustion experiments involving full-scale engine hardware and during simulated fast engine transients.

Linear and nonlinear analysis of controlled combustion processes. II. Nonlinear analysis

The results of analysis using a reduced-order model of combustion instability derived at UTRC and experiments with active control using fuel modulation motivated a study of what performance is achievable using active control. Limitations due to lightly damped or unstable eigenvalues, delay, disturbances, and limited actuator authority and bandwidth are studied. In this part of the paper we focus on linear analysis of a combustion model and study effect of delay in the model and limited actuator bandwidth.

Approximate Feedback Linearization: A Homotopy Operator Approach

In this paper, we present an approach for finding feedback linearizable systems that approximate a given single-input nonlinear system on a given compact region of the state space. First, we show that if the system is close to being involutive, then it is also close to being linearizable. Rather than working directly with the characteristic distribution of the system, we work with characteristic one-forms, i.e., with the one-forms annihilating the characteristic distribution. We show that homotopy operators can be used to decompose a given characteristic one-form into an exact and an antiexact part. The exact part is used to define a change of coordinates to a normal form that looks like a linearizable part plus nonlinear perturbation terms. The nonlinear terms in this normal form depend continuously on the antiexact part, and they vanish whenever the antiexact part does. Thus, the antiexact part of a given characteristic one-form is a measure of nonlinearizability of the system. If the nonlinear terms are small, by neglecting them we obtain a linearizable system approximating the original system. One can design control for the original system by designing it for the approximating linearizable system and applying it to the original one. We apply this approach for design of locally stabilizing feedback laws for nonlinear systems that are close to being linearizable.

Optimal mixing in recirculation zones

Coarse-scale mixing in a recirculation zone is described with a simple vortex model. Time-dependent forcing is employed to change the vortex motion and mixing properties. An optimal mixing problem is defined in which the flux across the recirculation region shall be maximized under the side-constraints of bounded vortex motion and bounded actuation. Concepts of control theory and chaotic advection are used to achieve this goal. In particular, controllability is proven with a transformation into flat coordinates. Thus, a feedforward law for the optimal trajectory and a feedback law for its stabilization are derived. Observability of the vortex motion is indicated by a dynamic observer. Mixing in the optimized flow is studied using Poincare maps. The low-frequency modulations to vortex motion are shown to substantially increase mixing in the average. Generalizations of the mathematical framework for mixing optimization are suggested for a larger class of models and flows.

A Backstepping Controller for a Nonlinear Partial Differential Equation Model of Compression System Instabilities

We prove the existence and uniqueness of solutions in Sobolev spaces for the Moore--Greitzer nonlinear partial differential equation (PDE) model for compression system instabilities with mild conditions on the shape of the compressor characteristic and on the throttle control. To achieve this, the model is reformulated as an evolution equation on a Banach space. Using this new representation, we design a backstepping control of the model. Global stabilization of any axisymmetric equilibrium to the right of the peak of the compressor characteristic is achieved. We also prove that the dynamics can be restricted to the small neighborhood of the point on the left of the peak of the compressor characteristic. Thus, it is possible to restrict the magnitude of stall to arbitrary small values. In addition, finite-dimensional Galerkin projections of the partial differential equation model are studied. It is shown that truncated control laws stabilize truncated models. Numerical simulations of the model with and without control are presented.

The disturbance decoupling problem for implicit linear discrete-time systems

The disturbance decoupling problem for implicit linear discrete-time systems is studied in detail. Necessary and sufficient conditions for the problem to be solved are given. The results are obtained with the aid of the concepts of almost invariant, sliding and coasting subspaces for implicit systems.

Markov Chains, Entropy, and Fundamental Limitations in Nonlinear Stabilization

In this paper, we propose a novel methodology for establishing fundamental limitations in nonlinear stabilization. To aid the analysis, we express the stabilization problem as control of Markov chains. Using Markov chains, we derive the limitations as certain maximum probability bounds or as positive conditional entropy of the certain signals in the feedback loop. The former is related to the infeasibility of the asymptotic stabilization in the presence of quantization and the latter to the Bode integral formula. In either cases, it is shown that uncertainty - associated here with the unstable eigenvalues of the linearization - leads to fundamental limitations.

On nonlinear observers with approximately linear error dynamics

We discuss the design of observers for autonomous nonlinear systems with one output. New necessary and sufficient conditions for existence of a solution to the observer error linearization problem are given. In the case when an exact linearization of the observer error dynamics is impossible, a simple least-squares scheme is proposed that yields an observer whose error dynamics is approximately linear. An algorithm for observer construction is provided. An example of observer design for a mass-spring system with nonlinear spring and dynamic friction forces is given.