Browsing by Subject "Stability Analysis"
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Item Open Access Analysis and control of periodic gaits in legged robots(Bilkent University, 2017-11) Hamzaçebi, HasanThe analysis, identi cation and control of legged locomotion have been an interest for various researchers towards building legged robots that move like the animals do in nature. The extensive studies on understanding legged locomotion led to some mathematical models, such as the Spring-Loaded Inverted Pendulum (SLIP) template (and its various derivatives), that can be used to identify, analyze and control legged locomotor systems. Despite their seemingly simple nature, as being a simple point mass attached to a massless spring from dynamics perspective, the SLIP model constitutes a restricted three-body problem formulation, whose non-integrability has been proven long before. Thus, researchers came up with approximate analytical solutions or they used some other different techniques such as partial feedback linearization for the sake of obtaining analytical Poincar e return maps that govern the motion of the desired legged locomotor system. In the first part of this thesis, we consider a SLIP-based legged locomotion model, which we call as Multi-Actuated Dissipative SLIP (MD-SLIP) that extends the simple SLIP model with two additional actuators. The first one is a linear actuator attached serially to the leg spring to ensure direct control on the compression and decompression of the leg spring. The second actuator is a rotatory one that is attached to hip, which provides ability to inject some torque inputs to the system dynamics, which is mainly inspired by biological legged locomotor systems. Following the analysis of MD-SLIP model, we utilize a partial feedback linearization strategy by which we can cancel some nonlinear dynamics of the legged locomotion model and obtain exact analytical solutions without needing any approximation. Having exact analytical solutions is crucial to investigate stability characteristics of the MD-SLIP model during its hopping gait behavior. We illustrate and compare the applicability of our solutions with open-loop and closedloop hopping performances on various rough terrain simulations. Finally, we show how the MD-SLIP model can be anchored to bipedal legged locomotion models, where we assign two independent MD-SLIP models to each leg and investigate the system performance under their simultaneous but independent control. The proposed bipedal legged locomotion model is called as Multi-Actuated Dissipative Bipedal SLIP (MDB-SLIP) model. The key idea here is that we can still utilize the partial feedback linearization concept that we applied for the original MD-SLIP model and ensure exact analytical solutions for the MDB-SLIP model as well. We also provide detailed investigations for open-loop and closed-loop walking gait performance of the MDB-SLIP model on different noisy terrain profiles.Item Open Access Analysis of two types of cyclic biological system models with time delays(Bilkent University, 2011) Ahsen, Mehmet ErenIn this thesis, we perform the stability analysis of two types of cyclic biological processes involving time delays. We analyze the genetic regulatory network having nonlinearities with negative Schwarzian derivatives. Using preliminary results on Schwarzian derivatives, we present necessary conditions implying the global stability and existence of periodic solutions regarding the genetic regulatory network. We also analyze homogenous genetic regulatory network and prove some stability conditions which only depend on the parameters of the nonlinearity function. In the thesis, we also perform a local stability analysis of a dynamical model of erythropoiesis which is another type of cyclic system involving time delay. We prove that the system has a unique fixed point which is locally stable if the time delay is less than a certain critical value, which is analytically computed from the parameters of the model. By the help of simulations, existence of periodic solutions are shown for delays greater than this critical value.Item Open Access Identification, stability analysis and control of linear time periodic systems via harmonic transfer functions(Bilkent University, 2017-08) Hıdır, Elvan KuzucuMany important systems encountered in nature such as wind turbines, helicopter rotors, power networks or nonlinear systems which are linearized around periodic orbit can be modeled as linear time periodic (LTP) systems. Such systems have been analyzed and discussed from analytical viewpoint extensively in the literature. However, only a few method are available in the literature for the identi cation of LTP systems which utilize input/output measurements. Especially, due to obtaining analytical solutions for LTP systems are quite challenging, utilization of experimental data to identify, analyze and stabilize such systems may be preferable. To achieve this aim, the utilization of harmonic transfer functions (HTFs) of LTP systems can be quite helpful. In the rst part of this thesis, we aim to obtain harmonic transfer functions (HTFs) of LTP systems via data-driven approach by using only input and output data of the system. In this respect, we rst present the identi cation procedure of HTFs by using single cosine input signal with a speci c frequency. However, because of the fact that this method requires multiple experiments in order to cover desired frequency range, we propose a formula for the sum of cosine input signal including di erent frequencies which their output components do not coincide. Then, we present the prediction performance of the estimated HTFs by using single cosine and sum of cosine input signals according to analytical solution of HTFs. In the second part of the thesis, our goal is to utilize harmonic transfer functions in order to analyze and design controllers which stabilize and enhance the performance of LTP systems. In this regard, we implement well known Nyquist stability criterion which is based on eigenloci of HTFs. As an illustrative example, we consider the well-known (unstable) damped Mathieu equation and design P, PD and PID controllers by using obtained Nyquist diagram. Finally, for the unknown LTP systems whose state space model may not be available, we seek to design a novel methodology, where we can obtain Nyquist plots of unknown LTP systems via input-output data analysis using the concept of HTFs. Then, we design PD controllers for the unknown LTP system by using Nyquist diagram in order to enhance the performance and increase the robustness. We illustrate the performance results of these controllers in time domain simulations.Item Open Access PID controller design for first order unstable time delay systems(Bilkent University, 2009) Arslan, Gül EzgiIn this thesis, problem of designing P, PI and PD-like controllers for switched first order unstable systems with time delay is studied. For each type of controller, the problem is solved in two steps. First, the set of stabilizing controllers for the class of plants considered is determined using different approaches. Then, an appropriate controller inside this set is chosen such that the feedback systems satisfies a desired property, which is for example gain and phase margin maximization or the dwell time minimization. In the first part, we focus on PI controllers and tune the PI controller parameters in order to maximize the gain and phase margins. The observations in this part show that a P controller is adequate to maximize gain and phase margins. Then, we move on to the problem of tuning P, PI and PD-like (first order stable) controller parameters such that the switched feedback system is stabilized and the dwell time (minimum required time between consequent switchings to ensure stability) is minimized. For this purpose, a dwell-time based stability condition of [39] is used for the class of switched time delay systems. We show that a proportional controller can be found with this method, but a PI controller is not feasible. Finally, we focus on the design of PD-like controllers for switched first order unstable systems with time delays. The proposed method finds the values of PD-like (first order stable) controller parameters which minimize an upper bound of the dwell time. The conservatism analysis of this method is done by time domain simulations. The results show that the calculated upper bound for the dwell time is close to the lower bound of the dwell time observed by simulations. In addition, we compare the obtained PD-like controller results with some alternative PD and first order controller design techniques proposed in the literature.Item Open Access Stability analysis and controller design for the heat equation with time delayed feedback(Bilkent University, 2010) Çalışkan, Sina YamaçIn this thesis, the stability analysis for the system defined by the heat equation with time delayed feedback is performed. In the first part of the thesis, stability conditions in terms of LMI conditions which are obtained from the analysis in time domain, are explained. Necessary and sufficient conditions for stability are obtained using a frequency domain analysis. In the second part of the thesis, robust stability conditions are obtained for the system with parametric uncertainty. In the third part, an 퐻∞ controller design procedure is given for this type of plants described by the heat equation with time delayed feedback. Finally, the results are illustrated with simulations.Item Open Access Stability and dwell time analysis of switched time delay systems(Bilkent University, 2007) Tapkan, Osman SiraceddinIn this thesis we deal with stability analysis of switched feedback system with time delays. We assume that, at any given time for each “candidate” system a controller is designed and a fixed feedback system is obtained until the next switching instant. We investigate the conservativeness of an LMI-based stability test for the time delay systems. This test is used for the dwell time analysis. After obtaining the limitations of this test, we find the exact bounds of allowable parameters appearing in the LMI-based test, in order to optimize the dwell time. For this purpose we consider simple first order systems and higher order systems separately. We also consider the LQR-based switched feedback controllers with time delays and investigate the effects of weighting matrices Q and R on the dwell time.