Stability and robustness of hybrid systems using discrete-time Lyapunov techniques

Driven by recent telecommunication demand, shared communication channels are more abundant today than ever before. These shared wired/wireless networks for communication are being exploited more and more in control systems to connect sensors, controllers and actuators resulting in so-called networked control systems (NCSs). NCSs replace the more traditional control systems where dedicated point-to-point (wired) connections are being used. The advantages of using NCS technology are inexpensive and easily modifiable communication links which allow control algorithms to be easily implemented in situations where dedicated connections are not possible (either economically or physically). However, the drawback is that the control system is susceptible to undesirable (possibly destabilizing) side-effects such as time-varying transmission intervals, time-varying delays, packet dropouts, quantization and a shared communication medium. These network-induced effects undermine fundamental assumptions on which traditional control theory is built and, therefore, it is essential to develop new techniques and tools that can be used to analyze and/or design control systems which communicate via a shared network. This thesis advances NCS analysis and design methodologies by contributing new theoretical developments, new software tools and new experimental validation results. The first new theoretical development is in the area of analyzing robust stability properties with respect to network-induced effects. A sum of squares (SOS) approach for a class of nonlinear NCSs incorporating bounded time-varying delays, bounded time-varying transmission intervals and a shared communication medium is developed. Mathematical models that describe these nonlinear NCSs are cast into suitable hybrid system formulations. Based on these hybrid system formulations, (families of) Lyapunov functions are constructed using SOS techniques. Amongst other benefits, it is shown that this technique improves the guaranteed robustness margins compared to existing work. The second new theoretical development concerns the design of networkaware decentralized controllers guaranteeing robust stability properties with respect to network-induced effects. We develop one of the first approaches based on semidefinite programming techniques to synthesize stabilizing decentralized observer-based output-feedback controllers for linear plants where the controllers, sensors and actuators are connected via a shared communication network subject to time-varying transmission intervals and delays. To effectively deal with the shared communication medium, a switched observer structure is adopted that switches based on the transmitted measured outputs and a switched controller structure is also adopted that switches based on the transmitted control inputs at each transmission time. By taking a discrete-time switched linear system perspective on modeling these decentralized NCSs, we are able to derive a general model that captures all these networked and decentralized control aspects. We provide linear matrix inequality (LMI)-based synthesis conditions which, if satisfied, provide stabilizing observer-based controllers that are both decentralized and robust to network effects. Regarding new software tools, the first prototype of a toolbox is developed to automate (robust) stability analysis (and controller design) for NCSs. Specifically, it is shown that the toolbox can be employed to efficiently verify if a linear time-invariant (LTI) plant and an LTI controller interconnected with a shared network are robust to certain network imperfections. The main intention of the toolbox is to make the available theory readily accessible to and applicable for the general control community. Additionally, the chosen software structure enables the incorporation of custom models or custom stability/performance analysis conditions in an easy manner, thereby allowing the control community to contribute and to further develop the toolbox. Finally, an experimental case study involving a wirelessly controlled inverted pendulum/cart system is investigated. In particular, the communication network itself is analyzed such that the network-induced effects can be characterized in terms of bounds on the transmission intervals and transmission delays. Based on these bounds, the prototype toolbox is applied to analyze the robustness regions for different performance specifications, which aid in tuning the controller to achieve more closed-loop robustness with respect to the network-induced effects. This leads to a validation of the developed theory in an experimental setting. In addition to the validation of the developed theory, many new insights into the network behavior are obtained and explained, thereby raising new interesting questions for future research on NCSs.

This paper deals with stability analysis of hybrid systems. Such systems are characterized by a combination of continuous dynamics and logic based switching between discrete modes. Lyapunov theory is a well known methodology for the stability analysis of linear and nonlinear systems in control system literature. Construction of Lyapunov functions for hybrid systems is generally a difficult task, but once these functions are defined, stabilization of the system is straight-forward. The sum of squares (SOS) decomposition and semidefinite programming has also provided an efficient methodology for analysis of nonlinear systems. The computational method used in this paper relies on the SOS decomposition of multivariate polynomials. By using SOS, we construct a (some) Lyapunov function(s) for the hybrid system. The reduction techniques provide numerical solution of large-scale instances; otherwise they will be practically unsolvable. The introduced method can be used for hybrid systems with linear or nonlinear vector fields. Some examples are given to demonstrate the capabilities of the proposed approach.

Robust fault detection for uncertain switched systems

Relaxed linear matrix inequalities (LMIs) conditions for fuzzy observer-based controller design are proposed based on a kind of improved T-S fuzzy model structure. The improved structure included the original T-S fuzzy model and enough large bandwidth pre- and post-filters. By this structure fuzzy observer-based controller design can be transformed into LMIs optimization problem. Compared with earlier results, it includes the less number of LMIs that equals the number of fuzzy rules plus one positive definition constraint of Lyapunov function. Therefore, it provides us with less conservative results for fuzzy observer-based controller design. Finally, a numerical example is demonstrated to show the efficiency of proposed method.

This paper addresses stability analysis and stabilization for Takagi-Sugeno (T-S) fuzzy systems with parametric uncertainties via a so-called fuzzy Lyapunov function which is a multiple Lyapunov function. The fuzzy Lyapunov function is defined by fuzzily blending quadratic Lyapunov functions. First, the Takagi-Sugeno (T-S) fuzzy model with parametric uncertainties is used as the model for the uncertain nonlinear system. Based on the fuzzy Lyapunov function approach and a parallel distributed compensation (PDC) scheme, we give stabilization conditions for closed-loop fuzzy systems with parametric uncertainties. Second, all the conditions are formulated in the format of linear matrix inequalities (LMIs) and contain upper bounds of the time derivative of premise membership functions as LMI variables. Finally, the T-S fuzzy model of the chaotic Lorenz system, which has complex nonlinearity, is developed as a test bed. A numerical example of the chaotic Lorenz system is given to illustrate the utility of the fuzzy Lyapunov function approach.

This paper presents results for stability and robustness of hybrid systems consisting of nonlinear subsystems. Present stability results that are extensions of classical Lyapunov theory are restricted to certain kinds of hybrid systems. For example, it is required that all subsystems have the same equilibrium point. In this paper these results are generalized, but more importantly, it is shown how Lyapunov functions for hybrid systems can be solved by linear matrix inequalities (LMI). Furthermore, it is shown how uncertainties around the nominal switch sets can be handled by introducing acceptable switch regions as additional stability conditions. The theory is illustrated by an example.

This paper introduces an $H_\infty$ fuzzy control synthesis method for a nonlinear large-scale system with a reduced number of linear matrix inequalities (LMIs). It is well known that a nonlinear large-scale system can be transformed to a Takagi–Sugeno (T–S) fuzzy system by using “sector nonlinearity” or “local approximation in fuzzy partition spaces” methods. Next, in order to achieve the fuzzy control design for this T–S fuzzy system, we solve the stabilization conditions that are represented by the LMIs. However, if the number of LMIs is large, the control design process may become very complicated. In this study, based on the Lyapunov method and S -procedure, several theorems are proposed for the synthesis of parallel distributed compensation (PDC)-type fuzzy control such that the nonlinear large-scale system achieves $H_\infty$ control performance, and the number of LMIs to be solved is reduced explicitly. As a result, the control design process will become much easier. Furthermore, if the modeling error between the nonlinear system and T–S fuzzy system exists, the robust $H_\infty$ control performance and the number reduction of LMIs are also achieved by the proposed theorem. Several examples are presented in this paper to show the number reduction effect of LMIs and the effectiveness of the proposed controller synthesis.

In this note, we address the stability problem for a class of hybrid systems, which contains continuous and logical dynamics. First, we introduce the so-called hybrid system in detail, and give the algebraic form of the system utilizing the semi-tensor product (STP, i.e., Cheng product) of matrices. Next, a new stability concept for the hybrid system is proposed by extending the definitions of set stability and exponential stability. After that, several stability criteria are presented by employing Lyapunov-like functions and STP. Accordingly, the stability conditions in some special cases are derived. Finally, the theoretical results are illustrated by some numerical examples.

In this paper we study the problem of designing gain-scheduled H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> filters for a class of parameter-varying discrete-time systems. We utilize a parameter-dependent Lyapunov function to establish a new characterization of asymptotic stability with H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> norm bound for the linear parameter-varying (LPV) systems. On the basis of the stability with H8 performance characterization established, we provide sufficient conditions for the solvability of the filter design problem by means of a finite number of linear matrix inequalities (LMIs). Then we show that solving a set of LMIs paves the way for designing a desired gain-scheduled H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> filter. We also use a numerical example to demonstrate the application of the presented filter design method.

We propose a novel restriction and relaxation (RAR) method for robust relay beamforming design in cognitive two-way relay networks with imperfect channel state information (CSI). Due to the uncertainty in CSI, the self-interference cannot be completely canceled. Hence, both channel uncertainties and residual self-interference should be considered. The proposed RAR method aims to minimize the transmit power of relay nodes while guaranteeing the worst-case signal-to-interference plus noise ratio (SINR) of secondary users (SUs) as well as satisfying the interference temperature (IT) constraints. Specifically, in the restriction step of RAR, infinite number of complicated constraints are reformulated into a finite number of linear matrix inequalities (LMIs). In the relaxation step of RAR, the nonconvex robust design problem is transformed into a convex semidefinite program (SDP), which can be efficiently solved by interior-point methods. Simulation results verify the effectiveness and robustness of the proposed method.

A method for designing FDI filters for polytopic LPV models

This paper presents theory and algorithms for the determination of coprime factorizations for parameter-dependent systems. We shall see that the obtention of coprime factors for polytopic and parameter-dependent systems reduces to a generalized H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> filtering problem. The coprime factors are obtained thank to a static observer matrix gain that ensures a certain H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm attenuation over the parameter space. The calculation of such an observer gain is done through a finite number of linear matrix inequalities (LMIs). An example illustrates the theoretical results

Gain-scheduled control for morphing aircraft via switching polytopic linear parameter-varying systems

A constant gain Kalman filter is proposed in this paper concerned with the problem of the complexity and large calculation in nonlinear system estimation. With the introduction of linear parameter varying (LPV) model and tensor product (TP) model transformation method, the nonlinear system is represented by a linear polytopic model. The transformation directly leads to the reduction of the conservativeness for the linear polytopic model gained by the parameter bounds method and avoids solving infinite number of linear matrix inequalities (LMIs). Moreover, a constant gain filter is developed based on the EKF and robust H2 filtering, which greatly reduces the calculation number. Finally, an example is employed to illustrate the effectiveness of the proposed filter.KeywordsNonlinear systemLPVTPRobust H2 filterPolytope

Output Feedback Guaranteed Cost Control for Linear Uncertain System with Input Saturation