Gregory Hagen

SPILLOVER STABILIZATION IN FINITE-DIMENSIONAL CONTROL AND OBSERVER DESIGN FOR DISSIPATIVE EVOLUTION EQUATIONS ∗

We consider the problem of global stabilization of a semilinear dissipative evolution equation by finite-dimensional control with finite-dimensional outputs. Coupling between the system modes occurs directly through the nonlinearity and also through the control influence functions. Similar modal coupling occurs in the infinite-dimensional error dynamics through the nonlinearity and measurements. For both the control and observer designs, rather than decompose the original system into Fourier modes, we consider Lyapunov functions based on the infinite-dimensional dynamics of the state and error systems, respectively. The inner product terms of the Lyapunov derivative are decomposed into Fourier modes. Upper bounds on the terms representing control and observation spillover are obtained. Linear quadratic regulator (LQR) designs are used to stabilize the state and error systems with these upper bounds. Relations between system and LQR design parameters are given to ensure global stability of the state and error dynamics with robustness with respect to control and observation spillover, respectively. It is shown that the control and observer designs can be combined to yield a globally stabilizing compensator. The control and observer designs are numerically demonstrated on the problem of controlling stall in a model of axial compressors.

Stability Analysis for a String of Coupled Stable Subsystems With Negative Imaginary Frequency Response

For a string of (possibly arbitrarily many) coupled stable subsystems that are equipped with a dynamic property of negative imaginary frequency response, we characterize stability of the string by a dc gain condition that can be expressed as a continued fraction with verifiable convergence properties. Through analysis of the convergence of the continued fraction, we establish stability results for the string with various coupling gains and patterns. The derived results are applied to locally decentralized control of large vehicle platoons, possibly with heterogenous neighboring coupling.

Distributed control design for parabolic evolution equations: application to compressor stall control

We present a theoretical study of distributed control design for semilinear parabolic nonlocal evolution equations with an application to axial compressor stall control using air injection. By taking advantage of the spatial invariance of the equations, a linear controller is constructed (following Bamieh et al.) via linear quadratic control of each Fourier mode. We derive sufficient conditions for the linear controller to stabilize the full nonlinear system. Concepts such as controller decentralization, finite-dimensional implementation, inverse-optimality, and beneficial nonlinearities are discussed. In the second part of this paper, these developments are applied to a model of axial compressor control with air injection. The unactuated model is derived following the work of Moore and Greitzer and the model coupled with air injection actuation follows the works of Behnken et al. and Weigl et al. A numerical study of the control designs is pursued and the comparison of controller performance is discussed. The techniques presented here are expected to be useful for distributed control design of a much broader class of nonlinear reaction-diffusion systems.

Absolute stability via boundary control of a semilinear parabolic PDE

We consider the problem of achieving global absolute stability of an unstable equilibrium solution of a semilinear dissipative parabolic partial differential equation (PDE) through boundary control. The state space of the system is extended in order to write the action of the boundary control as an unbounded operator in an abstract evolution equation. Absolute stability via boundary control is accomplished by analyzing a control Lyapunov function based on the infinite-dimensional dynamics and applying a finite-dimensional linear quadratic regulator (LQR) controller. Sufficient conditions for absolute stability of the infinite-dimensional system are established by the feasibility of two finite-dimensional linear matrix inequalities (LMIs). Numerical results are presented for a Dirichlet boundary controlled system, however the analysis in this work applies to Nuemann and Robin type boundary controllers as well.

Symmetry and Symmetry-Breaking for a Wave Equation with Feedback ∗

This paper is concerned with model-independent approaches for the analysis of inception and sup- pression of oscillations in certain feedback interconnections arising in aerospace and industrial ap- plications. One of the subsystems in the interconnection is known and assumed here to be the wave equation on the circle. The dynamic model of the other subsystem is uncertain, and the approach assumes only its structure, namely its symmetry properties. We show that only the structure (skew- symmetry) of the feedback can be used to explain the instability, and manipulation of the structure (mistuning) can be used to suppress the instability.

Passive Control of Limit Cycle Oscillations in a Thermoacoustic System Using Asymmetry

In this paper we investigate oscillations of a dynamical system containing passive dynamics driven by a positive feedback and how spatial characteristics (i.e., symmetry) affect the amplitude and stability of its nominal limit cycling response. The physical motivation of this problem is thermoacoustic dynamics in a gas turbine combustor. The spatial domain is periodic (passive annular acoustics) which are driven by heat released from a combustion process, and with sufficient driving through this nonlinear feedback a limit cycle is produced which is exhibited by a traveling acoustic wave around this annulus. We show that this response can be controlled passively by spatial perturbation in the symmetry of acoustic parameters. We find the critical parameter values that affect this oscillation, study the bifurcation properties, and subsequently use harmonic balance and temporal averaging to characterize periodic solutions and their stability. In all of these cases, we carry a parameter associated with the spatial symmetry of the acoustics and investigate how this symmetry affects the system response. The contribution of this paper is a unique analysis of a particular physical phenomena, as well as illustrating the equivalence of different nonlinear analysis tools for this analysis. DOI: 10.1115/1.2745399

Uncertainty propagation in a reduced order thermo-acoustic model

We present a thermo-acoustic model on a cylindrical, or annular, domain capable of modeling instabilities of tangential acoustic modes. The model accounts for non-uniform density, damping, rotational flow, and heat-release coupling. It is shown that deliberately introducing spatial variations in some quantities have a similar effect to adding damping to the system. The effects of these symmetry-breaking concepts are evaluated on the model through linear analysis and the net amount of additional damping is computed. We show how various symmetry-breaking concepts are robust with respect to the uncertainty in the model parameters and we examine propagation of uncertainty with respect to a recently defined measure of uncertainty.

A linear model for control of thermoacoustic instabilities on annular domain

We present a distributed linear model of thermoacoustic instability in form of a set of coupled PDEs including an acoustic model based on potential Euler formulation, a fully distributed fuel transport model based on advection equation, and a fuel-sensitive heat release model based on assumption of fixed flame location. The damping in the distributed model is provided on the acoustic boundaries using local acoustic impedance models. The model is suitable for analysis and control of multiple acoustic modes in annular combustors with bluff body stabilized flames and for optimization of fuel control architecture. We also derive a low order model for control using Galerkin projection of the potential Euler equations on finite number of acoustic basis functions and analytically solving the linearized fuel advection equation. The resulting frequency domain model has a form of coupled system involving undamped oscillators representing acoustic modes, distributed delays representing effect of acoustic perturbation on the fuel transport and combustion, and positive real transfer functions representing acoustic impedances of the boundaries. A simple control algorithm to suppress pressure oscillations is derived using the reduced order model.

Absolute stability of a heterogeneous semilinear dissipative parabolic PDE

We analyze absolute stability of the equilibrium solution of a semilinear dissipative parabolic PDE with a spatially varying nonlinearity that satisfies a given sector condition. Stability is shown based on Lyapunov analysis of the infinite-dimensional dynamics. The pertinent linear operators are expressed in terms of their infinite-dimensional matrix representations, some of which have a Toeplitz structure due to the spatial heterogeneity of the nonlinearity. The time derivative of the Lyapunov function is expressed as a sum of finite-dimensional expressions. The analysis is described in terms of finite-dimensional linear matrix inequalities (LMI). Sufficient conditions, in terms of a finite set of finite-dimensional LMI, are given to establish absolute stability. Numerical simulations are presented for a system with Dirichlet boundary conditions with spatially varying saturation nonlinearities.

Control spillover in dissipative evolution equations

We consider the problem of global stabilization of a semilinear dissipative parabolic PDE by finite-dimensional control. The infinite-dimensional system is decomposed into a reduced order finite-dimensional system and a residual system. Coupling between the systems occurs through the nonlinear function and also through the actual controller; a phenomenon known as control spillover. Rather than decompose the original PDE into Fourier modes, we decompose the derivative of a control Lyapunov function. Upper bounds on the terms representing control spillover are obtained. An LQR design is used to stabilize the system with these upper bounds. Relations between system and LQR control design parameters are given to ensure global stability and robustness with respect to control spillover.