Analysis of Lur'e dominant systems in the frequency domain

Frequency domain analysis of linear time-invariant (LTI) systems in feedback with static nonlinearities is a classical and fruitful topic of nonlinear systems theory. We generalize this approach beyond equilibrium stability analysis with the aim of characterizing feedback systems whose asymptotic behavior is low dimensional. We illustrate the theory with a generalization of the circle criterion for the analysis of multistable and oscillatory Lur'e feedback systems.ERC Grant Agreement Switchlet n. 67064

Robust control design with real parameter uncertainty using absolute stability theory

The purpose of this thesis is to investigate an extension of mu theory for robust control design by considering systems with linear and nonlinear real parameter uncertainties. In the process, explicit connections are made between mixed mu and absolute stability theory. In particular, it is shown that the upper bounds for mixed mu are a generalization of results from absolute stability theory. Both state space and frequency domain criteria are developed for several nonlinearities and stability multipliers using the wealth of literature on absolute stability theory and the concepts of supply rates and storage functions. The state space conditions are expressed in terms of Riccati equations and parameter-dependent Lyapunov functions. For controller synthesis, these stability conditions are used to form an overbound of the H2 performance objective. A geometric interpretation of the equivalent frequency domain criteria in terms of off-axis circles clarifies the important role of the multiplier and shows that both the magnitude and phase of the uncertainty are considered. A numerical algorithm is developed to design robust controllers that minimize the bound on an H2 cost functional and satisfy an analysis test based on the Popov stability multiplier. The controller and multiplier coefficients are optimized simultaneously, which avoids the iteration and curve-fitting procedures required by the D-K procedure of mu synthesis. Several benchmark problems and experiments on the Middeck Active Control Experiment at M.I.T. demonstrate that these controllers achieve good robust performance and guaranteed stability bounds

Applications of frequency domain stability criteria in the design of nonlinear feedback systems

The Popov criterion for absolute stability of nonlinear feedback systems is applied to several example problems. Model transformations such as pole shifting and zero shifting extend the class of systems to which the criterion applies. Extensions of the criterion having simple graphical interpretations yield stronger results for systems with constant monotonic slope-bounded nonlinearities. Additional extensions lacking simple graphical interpretations in the complex plane are also demonstrated by example. Stability throughout a region in parameter space is discussed, and the Kalman conjecture is verified for a new class of systems. The Popov criterion is also used to prove BIBO stability, process stability, and degree of stability. The conservatism of the criterion, i. e., the margin of actual performance beyond guaranteed performance, is discussed in the light of simulation results. An interactive computer program is developed to make the Popov criterion, along with two of its extensions, a convenient tool for the design of stable systems. The user has the options of completely automatic parameter adjustment or intervention at any stage of the procedure --Abstract, page ii

Incorporation of nonlinear load models and identification of the inter-area mode phenomenon in the transient energy function method

Considerable progress has been made in first swing power system transient stability assessment using the transient energy function (TEF) method. However, in most of the work reported in the literature, this technique has been applied to the classical power system model. Some of the recent developments of the TEF method include application to stressed large-scale power systems, incorporation of the effects of the exciter, etc. In the classical model, the loads are modelled as constant impedances. This dissertation aims at removing some of the modeling restrictions of the TEF method by incorporating the effects of nonlinear load models;The basic approach consists of representing the effect of nonlinear load models on the stable equilibrium point (SEP) and the controlling unstable equilibrium point (UEP) solution. An alternate network solution procedure is used to reflect the effect of the nonlinear load models, via current injections at the internal generator nodes. The power corresponding to these injections is then included in the mismatch equation for the SEP and UEP solution. A new expression for the TEF is developed to include the term corresponding to the nonlinear load components;The proposed technique has been tested on a 4-generator system and a 17-generator system. The results of these tests compared well with those obtained by time simulation;The other aspect of this research work deals with the application of the TEF method to stressed large-scale power systems. The theory of modal analysis has been used to identify the inter-area mode phenomenon of stressed systems. The inter-area mode phenomenon can be described as the tendency of a large group of generators (including the small group severely disturbed by the fault initially) to separate from the rest of the system as a result of instability. The post disturbance network of stressed systems are characterized by weak synchronizing forces caused by large transmission impedances, and thus generators away from the fault location may also separate from the system;The proposed technique has been tested on three test systems, the 17-generator Iowa system, the 50-generator Ontario-Hydro system and the 126-generator western USA system. Both stressed and unstressed systems have been considered in testing this technique;The technique gives reasonably correct indication of the inter-area mode phenomenon

Optimized state feedback regulation of 3DOF helicopter system via extremum seeking

In this paper, an optimized state feedback regulation of a 3 degree of freedom (DOF) helicopter is designed via extremum seeking (ES) technique. Multi-parameter ES is applied to optimize the tracking performance via tuning State Vector Feedback with Integration of the Control Error (SVFBICE). Discrete multivariable version of ES is developed to minimize a cost function that measures the performance of the controller. The cost function is a function of the error between the actual and desired axis positions. The controller parameters are updated online as the optimization takes place. This method significantly decreases the time in obtaining optimal controller parameters. Simulations were conducted for the online optimization under both fixed and varying operating conditions. The results demonstrate the usefulness of using ES for preserving the maximum attainable performance