Stability of Linear Fractional Differential Equations with Delays: a coupled Parabolic-Hyperbolic PDEs formulation

Fractional differential equations with delays are ubiquitous in physical systems, a recent example being time-domain impedance boundary conditions in aeroacoustics. This work focuses on the derivation of delay-independent stability conditions by relying on infinite-dimensional realisations of both the delay (transport equation, hyperbolic) and the fractional derivative (diffusive representation, parabolic). The stability of the coupled parabolic-hyperbolic PDE is studied using straightforward energy methods. The main result applies to the vector-valued case. As a numerical illustration, an eigenvalue approach to the stability of fractional delay systems is presented

Backstepping Control and Transformation of Multi-Input Multi-Output Affine Nonlinear Systems into a Strict Feedback Form

This dissertation presents an improved method for controlling multi-input multi-output affine nonlinear systems. A method based on Lie derivatives of the system\u27s outputs is proposed to transform the system into an equivalent strict feedback form. This enables using backstepping control approaches based on Lyapunov stability and integrator backstepping theory to be applied. The geometrical coordinate transformation of multi-input multi-output affine nonlinear systems into strict feedback form has not been detailed in previous publications. In this research, a new approach is presented that extends the transformation process of single-input single-output nonlinear. A general algorithm of the transformation process is formulated. The research will consider square feedback linearizable multi-input multi-output systems where the number of inputs equals to the number of outputs. The preliminary mathematical tools, necessary and sufficient feedback linearizability conditions, as well as a step-by-step transformation process is explained in this research. The approach is applied to the Western Electricity Coordinating Council (WECC) 3-machine nonlinear power system model. Detailed simulation results indicate that the proposed design method is effective in stabilizing the WECC power system when subjected to large disturbances

On the global uniform stability analysis of non-autonomous dynamical systems: A survey

In this survey, we introduce the notion of stability of time varying nonlinear systems. In particular we investigate the notion of global practical exponential stability for non-autonomous systems. The proposed approach for stability analysis is based on the determination of the bounds of perturbations that characterize the asymptotic convergence of the solutions to a closed ball centered at the origin

Stability-preserving model reduction for linear and nonlinear systems arising in analog circuit applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 221-229).Despite the increasing presence of RF and analog components in personal wireless electronics, such as mobile communication devices, the automated design and optimization of such systems is still an extremely challenging task. This is primarily due to the presence of both parasitic elements and highly nonlinear elements, which makes simulation computationally expensive and slow. The ability to generate parameterized reduced order models of analog systems could serve as a first step toward the automatic and accurate characterization of geometrically complex components and subcircuits, eventually enabling their synthesis and optimization. This thesis presents techniques for reduced order modeling of linear and nonlinear systems arising in analog applications. Emphasis is placed on developing techniques capable of preserving important system properties, such as stability, and parameter dependence in the reduced models. The first technique is a projection-based model reduction approach for linear systems aimed at generating stable and passive models from large linear systems described by indefinite, and possibly even mildly unstable, matrices. For such systems, existing techniques are either prohibitively computationally expensive or incapable of guaranteeing stability and passivity. By forcing the reduced model to be described by definite matrices, we are able to derive a pair of stability constraints that are linear in terms of projection matrices.(cont.) These constraints can be used to formulate a semidefinite optimization problem whose solution is an optimal stabilizing projection framework. The second technique is a projection-based model reduction approach for highly nonlinear systems that is based on the trajectory piecewise linear (TPWL) method. Enforcing stability in nonlinear reduced models is an extremely difficult task that is typically ignored in most existing techniques. Our approach utilizes a new nonlinear projection in order to ensure stability in each of the local models used to describe the nonlinear reduced model. The TPWL approach is also extended to handle parameterized models, and a sensitivity-based training system is presented that allows us to efficiently select inputs and parameter values for training. Lastly, we present a system identification approach to model reduction for both linear and nonlinear systems. This approach utilizes given time-domain data, such as input/output samples generated from transient simulation, in order to identify a compact stable model that best fits the given data. Our procedure is based on minimization of a quantity referred to as the 'robust equation error', which, provided the model is incrementally stable, serves as up upper bound for a measure of the accuracy of the identified model termed 'linearized output error'. Minimization of this bound, subject to an incremental stability constraint, can be cast as a semidefinite optimization problem.by Bradley Neil Bond.Ph.D

Feedback Control for Average Output Systems

In this work we propose new methods for the design of economic Nonlinear Model Predictive Control (NMPC) feedback schemes for Average Output Optimal Control Problems (AOCPs). AOCPs are Optimal Control Problems (OCPs) defined on infinite time horizons with averaging performance critera as objective functionals. Such problems arise frequently for continuously operating systems such as for example power plants. Due to the infinite time horizon and the resulting intrinsic nonuniqueness of solutions, the design of appropriate NMPC schemes for AOCPs is challenging. Often, the analysis of the closed-loop behavior of economic NMPC schemes depends on dissipativity conditions on the dynamical system and the associated performance criterion, which sometimes can be hard to check. The methods we develop are based on the observation that periodic solutions exhibit excellent approximation properties for AOCPs, which is exploited by splitting the time horizon and the objective functional of the NMPC subproblems into a transient and a periodic part. For the analysis of the closed-loop behavior of the resulting controller we develop new methods that essentially work by showing that the (appropriately defined) difference of two subsequent NMPC subproblem solutions vanishes asymptotically. Complementary to many other economic NMPC schemes, this approach is not based on dissipativity assumptions on the dynamical system and the associated performance criterion but rather on assumptions on existence of periodic orbits, controllability of the dynamical system, and uniqueness of the NMPC subproblem solutions itself. As a result, we can show that the economic performance of the closed-loop system is equal to the economic performance of the optimal periodic solutions. Furthermore, the approach is extended in two directions. First, we consider the general setting of a parameter dependent dynamical system where the parameter can be subject to change during operation. This parameter change can lead to a change in the optimal periodic behavior, in particular also to a change of the optimal period, which we take into account by including the period as an optimization variable in the NMPC subproblem. Second, we show that the approach can also be applied to systems with time-dependent periodic performance criteria. All the described methods are implemented within the MATLAB NMPC toolkit MLI and are applied to a number of demanding applications. The simulation results confirm that the generated closed-loop trajectories perform economically equally well as the optimal periodic trajectories