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ABSTRACT: We consider set-point regulation and L <sub>2</sub> robust stability properties of a class of reset control systems consisting of a minimum-phase relative degree-one linear SISO plant controlled by a novel first-order reset element (FORE). These results rely on necessary and sufficient conditions for exponential and L <sub>2</sub> finite gain stability of a class of planar reset systems consisting of a scalar linear plant controlled by the novel FORE. We show that the L <sub>2</sub> gain of the planar reset system decreases to zero as the pole and/or the gain of the FORE are increased to infinity. A number of stability results, including Lyapunov conditions for L <sub>p</sub> and exponential stability, for a larger class of reset and hybrid systems are presented and used to prove our main results.
IEEE Transactions on Automatic Control 12/2011; · 2.11 Impact Factor
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ABSTRACT: It is well known that controlling the attitude of a rigid body is subject to topological constraints. We illustrate, with examples, the problems that arise when using continuous and (memoryless) discontinuous quaternion-based state-feedback control laws for global attitude stabilization. We propose a quaternion-based hybrid feedback scheme that solves the global attitude tracking problem in three scenarios: full state measurements, only measurements of attitude, and measurements of attitude with angular velocity measurements corrupted by a constant bias. In each case, the hybrid feedback is dynamic and incorporates hysteresis-based switching using a single binary logic variable for each quaternion error state. When only attitude measurements are available or the angular rate is corrupted by a constant bias, the proposed controller is observer-based and incorporates an additional quaternion filter and bias observer. The hysteresis mechanism enables the proposed scheme to simultaneously avoid the “unwinding phenomenon” and sensitivity to arbitrarily small measurement noise that is present in discontinuous feedbacks. These properties are shown using a general framework for hybrid systems, and the results are demonstrated by simulation.
IEEE Transactions on Automatic Control 12/2011; · 2.11 Impact Factor
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ABSTRACT: We consider control systems for which we know two stabilizing output feedback controllers. One is globally asymptotically stabilizing, while the other one is only locally asymptotically stabilizing. We look for a composite output feedback control law that is equal to the local feedback on a neighborhood of the origin and that is globally asymptotically stabilizing. Since we want some robustness with respect to measurement noise, actuator errors, and external disturbances, we need to consider hybrid output feedback controllers. Under an input-output-to-state stability assumption, we exhibit a solution of this uniting problem by means of a dynamic hybrid output feedback controller. Then, we particularize our study to linear control systems with saturation at the input for which we know two stabilizing output feedback controllers. One is a nonlinear globally asymptotically stabilizing controller, while the other one is a high-performance linear-only locally asymptotically stabilizing controller. We specify numerically tractable conditions to solve this uniting problem. Finally, we illustrate our main results by means of numerical examples.
IEEE Transactions on Automatic Control 08/2011; · 2.11 Impact Factor
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ABSTRACT: The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Quaternion-based feedback control laws that are inconsistent (i.e. do not have a unique value for a given attitude) require an additional mechanism that lifts a continuous attitude trajectory to the unit quaternion space. Lifting mechanisms that are memoryless, for example, selecting the quaternion having positive scalar component, have a limited domain where they remain injective and, when used globally, introduce discontinuities into the closed-loop system. We show that such discontinuities can be exploited by an arbitrarily small measurement disturbance to stabilize attitudes far from the desired attitude and destroy "global" attractivity properties.
American Control Conference (ACC), 2011; 08/2011
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ABSTRACT: The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Often, quaternion-based feedbacks require an additional mechanism that lifts a continuous attitude path to the unit quaternion space. When this mechanism is memoryless, it has a limited domain where it remains injective and leads to discontinuities when used globally. To remedy this limitation, we propose a hybrid-dynamic algorithm for lifting a continuous attitude path to the unit quaternion space. We show that this hybrid-dynamic mechanism allows us to directly translate quaternion-based controllers and their asymptotic stability properties (obtained in the unit-quaternion space) to the actual rigid-body-attitude space. We also show that when quaternion-based controllers are not designed to account for the double covering of the rigid-body-attitude space by a unit-quaternion parameterization, they can give rise to the unwinding phenomenon, which we characterize in terms of the projection of asymptotically stable sets.
American Control Conference (ACC), 2011; 08/2011
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ABSTRACT: This technical note develops a unified Lyapunov approach to analysis of self-induced oscillations and stability for systems with piecewise linear elements. For self-induced oscillation within a global or regional attractor, invariant level sets of a piecewise quadratic Lyapunov function are obtained to bound the attractor via linear matrix inequality based optimization. The analysis results for self-induced oscillations are easily adapted to global or regional stability analysis.
IEEE Transactions on Automatic Control 01/2011; · 2.11 Impact Factor
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ABSTRACT: There are many communication imperfections in networked control systems (NCS) such as varying transmission delays, varying sampling/transmission intervals, packet loss, communication constraints and quantization effects. Most of the available literature on NCS focuses on only some of these aspects, while ignoring the others. In this paper we present a general framework that incorporates communication constraints, varying transmission intervals and varying delays. Based on a newly developed NCS model including all these network phenomena, we will provide an explicit construction of a continuum of Lyapunov functions. Based on this continuum of Lyapunov functions we will derive bounds on the maximally allowable transmission interval (MATI) and the maximally allowable delay (MAD) that guarantee stability of the NCS in the presence of communication constraints. The developed theory includes recently improved results for delay-free NCS as a special case. After considering stability, we also study semi-global practical stability (under weaker conditions) and performance of the NCS in terms of Lp gains from disturbance inputs to controlled outputs. The developed results lead to tradeoff curves between MATI, MAD and performance gains that depend on the used protocol. These tradeoff curves provide quantitative information that supports the network designer when selecting appropriate networks and protocols guaranteeing stability and a desirable level of performance, while being robust to specified variations in delays and transmission intervals. The complete design procedure will be illustrated using a benchmark example.
IEEE Transactions on Automatic Control 09/2010; · 2.11 Impact Factor
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ABSTRACT: We apply recent results on robust global asymptotic stabilization of the attitude of a single rigid body to the problem of synchronizing the attitude of a network of rigid bodies using graph-local information. The proposed synchronization scheme relies on a hysteretic hybrid feedback based on the unit quaternion representation of rigid body attitude to achieve a global synchronization result that is robust to measurement noise. While the hysteretic feedback manages a trade-off between robustness to measurement noise and unwinding, the scheme necessitates the communication of a single binary logic variable between neighboring rigid bodies.
American Control Conference (ACC), 2010; 08/2010
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ABSTRACT: A hybrid algorithm inspired by is given that identifies a parameter for a class of nonlinear systems in finite time. We show that this algorithm contains a compact globally asymptotically stable set that is robust to small perturbations. We further give a persistency of excitation condition that ensures convergence of the parameter estimate.
American Control Conference (ACC), 2010; 08/2010
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ABSTRACT: In this paper we address the sampled-data regional and global H<sub>∞</sub> synthesis problem for a class of linear plants subject to input saturation, where the sampling and hold rates are synchronous. Such a sampled-data system is expressed as a jump system, which is further described as a class of hybrid systems. Based on Lyapunov theorems for hybrid systems, a Lyapunov function is constructed and it is proved that the H<sub>∞</sub> problem is equivalent to a purely discrete-time synthesis problem. The proposed synthesis approach is cast as a convex optimization over Linear Matrix Inequalities (LMIs), which leads to an output feedback controller with an internal deadzone loop, achieving stability and desired performance. The effectiveness of the proposed techniques is illustrated by one example consisting in a mechanical system with different sample-and-hold rates.
American Control Conference (ACC), 2010; 08/2010
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ABSTRACT: The problem of robust, global asymptotic stabilization of a rigid body is hampered by major topological obstructions. These obstructions prevent a continuous state feedback from solving the problem and also lead to robustness issues when (non-hybrid) discontinuous feedback is applied. In this paper, we extend a hybrid control strategy proposed in a companion paper for robust, global asymptotic stabilization of rigid body attitude to the case where translation is also considered. Through Lyapunov analysis, we develop quaternion-based hysteretic hybrid control laws in the kinematic and dynamic settings. In the dynamic setting, two control laws are derived: one from an energy-based Lyapunov function and one by backstepping. Robustness to measurement noise is asserted by employing recently developed stability theory for hybrid systems. A comparison between discontinuous and hysteretic feedback under measurement noise is shown in simulation.
Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on; 01/2010
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ABSTRACT: Global asymptotic stabilization of the attitude of a rigid body is hindered by major topological obstructions. In fact, this task is impossible to accomplish with continuous state feedback. Moreover, when the attitude is parametrized with unit quaternions, it becomes impossible to design a globally stabilizing state feedback (even discontinuous) that is robust to measurement noise. In this paper, we present a quaternion-based hysteretic hybrid feedback that robustly globally asymptotically stabilizes the attitude of a rigid body. The hybrid control laws are derived through Lyapunov analysis in kinematic and dynamic settings. In the dynamic setting, we provide two control laws: one derived from an energy-based Lyapunov function and another based on backstepping. Analyzing the change in these Lyapunov functions due to switching of a logic variable yields a straightforward form for state-based hysteresis. A simulation study demonstrates how hysteresis provides robustness to measurement noise and highlights differences between the energy-based and backstepping control laws.
Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on; 01/2010
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ABSTRACT: There are many communication imperfections in networked control systems (NCSs) such as varying sampling/transmission intervals, varying delays, possible packet loss, communication constraints and quantization effects. Most of the available literature on NCSs focuses on only some of these phenomena, while ignoring the others, although recently some papers appeared that consider at least three of these phenomena. In one paper time-varying delays, time-varying transmission intervals and communication constraints are considered, while in an other time-varying transmission intervals, communication constraints and quantization effects are studied. As both approaches are based on the same underlying hybrid modeling framework, it will be shown here that the models can be combined in a unifying hybrid model including the five mentioned network phenomena under some restrictions. On the basis of this model, stability will be analyzed of the closed-loop system in which the controller is obtained using an emulation approach. The analysis provides tradeoffs between the maximally allowable transmission interval (MATI), the maximally allowable delay (MAD) and the quantization parameters, while still guaranteeing closed-loop stability.
Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on; 01/2010
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ABSTRACT: A theorem on nested Matrosov functions is extended to time-varying hybrid systems. It provides sufficient conditions for uniform global asymptotic stability of a compact set. An application to parameter identification with state resets is made and illustrated on an example.
IEEE Transactions on Automatic Control 08/2009; · 2.11 Impact Factor
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ABSTRACT: This paper develops a unified Lyapunov approach to analysis of self-induced oscillations and stability for systems with piecewise linear components. For self-induced oscillation within a global or regional attractor, invariant level sets of a piecewise quadratic Lyapunov function are obtained to bound the attractor via linear matrix inequality based optimization. The analysis results for self-induced oscillations are easily adapted to global or regional stability analysis.
American Control Conference, 2009. ACC '09.; 07/2009
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ABSTRACT: The purpose of this note is to apply recent results on stabilization of networked control systems to obtain an explicit formula for the maximum allowable sampling period (MASP) that guarantees stability of a nonlinear sampled-data system with an emulated controller. Such formulas are of great value to control practitioners.
IEEE Transactions on Automatic Control 04/2009; · 2.11 Impact Factor
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ABSTRACT: A family of hybrid control algorithms is developed that steer a nonholonomic autonomous vehicle to the source of a scalar signal present in the environment. In an idealized setting, we develop a general hybrid control scheme that globally asymptotically stabilizes the vehicle position about the source. Pursuing a practical implementation, a series of perturbations to the family of controllers is introduced, resulting in a semi-global practical stability of the vehicle position about the source. An example of a recently developed conjugate direction-based controller fitting into this family is developed and demonstrated by simulation and experiment.
Decision and Control, 2008. CDC 2008. 47th IEEE Conference on; 01/2009
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ABSTRACT: This paper studies a class of hybrid systems with linear (or linear plus saturated linear) continuous and discrete dynamics, which are determined by a flow map and jump map, and state-triggered jumps. One motivation for considering this class of systems is that they can model control systems with a relay-type hysteresis element. Based on Lyapunov theorems for hybrid systems, a Lyapunov function is constructed that effectively incorporates the feature of the jumps. Global asymptotic stability analysis is presented for the case when the flow map is linear, and local asymptotic stability analysis is presented for the case when the flow map is linear plus saturated linear. The stability conditions are derived as matrix inequalities. A numerical example is presented to illustrate the hybrid modeling process for a system experiencing hysteresis. Simulations confirm the effectiveness of the proposed analysis tools and demonstrate the potential of the Lyapunov function.
Decision and Control, 2008. CDC 2008. 47th IEEE Conference on; 01/2009
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ABSTRACT: This paper describes an algorithm for achieving robust, global asymptotic stabilization in nonlinear control systems by supervising the actions of a family of hybrid controllers. The family is such that the regions over which they operate cover the state space in an appropriate sense. Moreover, their behavior is such that they can be scheduled to move the state of the system toward a desirable region, whether it be an equilibrium point or a compact set. In establishing our main result, we use the concept of "events" for hybrid systems and show that, under mild assumptions, stability of a system without events is preserved when a finite number of events are incorporated. The algorithm is applied to robust, global stabilization problems involving vehicle orientation, position and orientation of a mobile robot, and the inverted configuration of a pendulum.
Decision and Control, 2008. CDC 2008. 47th IEEE Conference on; 01/2009
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ABSTRACT: A Lyapunov-based small-gain theorem is presented for hybrid systems modelled using a recently proposed framework. Lyapunov small-gain theorems for continuous-time and discrete-time systems are special cases of our result. Several examples including networked control systems and reset systems are presented to illustrate our main result. Our results are general and they apply to a range of other situations.
Decision and Control, 2008. CDC 2008. 47th IEEE Conference on; 01/2009
