Lagrange Stability of a Class of Nonlinear Discrete-time Systems

This article is concerned with robust stability of uncertain discrete-time linear systems. The matrix defining the linear system (system matrix) is assumed to depend affinely on a set of time-invariant unknown parameters lying on a known polytope. Robust stability is investigated by checking whether a certain integer power κ of the uncertain system matrix has spectral norm less than one. This peculiar stability test is shown to be equivalent to the positivity analysis of a homogeneous symmetric matrix polynomial with precisely known coefficients and degree indexed by κ. A unique feature is that no extra variables need to be added to the problems being solved. Numerical experiments reveal that the value of κ needed to test robust stability is mostly independent of the system dimension but grows sharply as the eigenvalues of the uncertain system approach the unit circle. By identifying the proposed stability test with a particular choice of a parameter-dependent Lyapunov function, extra variables can be introduced, yielding linear matrix inequalities optimisation problems of improved convergence.

In this paper, we consider robust stability and stabilization of uncertain Takagi-Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability of a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems and it thus leads to a conservative fuzzy controller design as well. We first consider rather relaxed robust stability conditions of uncertain fuzzy systems. To this end, we introduce an auxiliary system to the original fuzzy time-delay system to obtain generalized delay-dependent stability conditions. Such an auxiliary system has some arbitrary matrices that generalize not only the system representation but also delay-dependent stability conditions. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time-delay, and are given in terms of linear matrix inequalities (LMIs). Then, we compare our delay-dependent stability conditions with other conditions in the literature, and show that our conditions guarantee the stability of a wider class of systems than others. Next, we consider the robust stabilization problem with memoryless and delayed state feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give three examples that illustrate our results.

In this paper, a new robust control method and stability analysis method using the higher order derivatives (HODs) of universal learning networks (ULNs) is discussed. The application of these two methods to a Photovoltaic (PV) supplied, separately-excited DC motor with a constant load torque is also studied. The simulation results of the proposed robust control and stability analysis method showed that the robustness of the DC motor system is improved and its stability can be analyzed easily.

Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays

Because the inter-sample behavior can be considered, the lifting technique now becomes the unique method for design and analysis of sampled-data systems. However, when we consider the robust stability problem by using the small gain theorem, the state transition operator is zero, and the current lifting algorithms can not be used directly. The lifting computation formulas for the case of the robust stability analysis are derived in detail. Further study shows that when the lifting technique is used for robust stability analysis, its result is conservative. A new discretization method for robust stability analysis is proposed in the paper, and it is not conservative when the condition of the small gain theorem is used. Two examples are given here to illustrate the above ideas.

This paper studies the fuzzy H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> filter design problem for signal estimation of nonlinear discrete-time systems with multiple time delays and unknown bounded disturbances. First, the Takagi-Sugeno (T-S) fuzzy model is used to represent the state-space model of nonlinear discrete-time systems with time delays. Next, we design a stable fuzzy H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> filter based on the T-S fuzzy model, which guarantees asymptotic stability and a prescribed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> index for the filtering error system, irrespective of the time delays and uncertain disturbances. A sufficient condition for the existence of such a filter is established by using the linear matrix inequality (LMI) approach. The proposed LMI problem can be efficiently solved with global convergence guarantee using convex optimization techniques such as the interior point algorithm. Simulation examples are provided to illustrate the design procedure of the present method.

This paper demonstrates a real-time hybrid substructuring (RTHS) shake table test to evaluate the seismic performance of a base isolated building. Since RTHS involves a feedback loop in the test implementation, the frequency dependent magnitude and inherent time delay of the actuator dynamics can introduce inaccuracy and instability. The paper presents a robust stability and performance analysis method for the RTHS test. The robust stability method involves casting the actuator dynamics as a multiplicative uncertainty and applying the small gain theorem to derive the sufficient conditions for robust stability and performance. The attractive feature of this robust stability and performance analysis method is that it accommodates linearized modeled or measured frequency response functions for both the physical substructure and actuator dynamics. Significant experimental research has been conducted on base isolators and dampers toward developing high fidelity numerical models. Shake table testing, where the building superstructure is tested while the isolation layer is numerically modeled, can allow for a range of isolation strategies to be examined for a single shake table experiment. Further, recent concerns in base isolation for long period, long duration earthquakes necessitate adding damping at the isolation layer, which can allow higher frequency energy to be transmitted into the superstructure and can result in damage to structural and nonstructural components that can be difficult to numerically model and accurately predict. As such, physical testing of the superstructure while numerically modeling the isolation layer may be desired. The RTHS approach has been previously proposed for base isolated buildings, however, to date it has not been conducted on a base isolated structure isolated at the ground level and where the isolation layer itself is numerically simulated. This configuration provides multiple challenges in the RTHS stability associated with higher physical substructure frequencies and a low numerical to physical mass ratio. This paper demonstrates a base isolated RTHS test and the robust stability and performance analysis necessary to ensure the stability and accuracy. The tests consist of a scaled idealized 4-story superstructure building model placed directly onto a shake table and the isolation layer simulated in MATLAB/Simulink using a dSpace real-time controller.

This paper focuses on the robust stability and the memory feedback stabilization problems for a class of uncertain switched nonlinear systems with multiple time-varying delays. Especially, the considered time delays depend on the subsystem number. Based on a novel common Lyapunov functional, the aggregation techniques, and the Borne and Gentina criterion, new sufficient robust stability and stabilization conditions under arbitrary switching are established. Compared with existing results, the proposed criteria are explicit, simple to use, and obtained without finding a common Lyapunov function for all subsystems through linear matrix inequalities, considered very difficult in this situation. Moreover, compared with the memoryless one, the developed controller guarantees the robust stability of the corresponding closed-loop system with more performance by minimizing the effect of the delays in the system dynamics. Finally, two numerical simulation examples are shown to prove the practical utility and the effectiveness of the proposed theories.

An improved model-free adaptive control for nonlinear systems: An LMI approach

Robust analysis and control for a class of uncertain nonlinear discrete-time systems

In this paper, we consider generalized delay-dependent stability conditions of Takagi-Sugeno fuzzy systems with discrete and distributed time-delays. For fuzzy systems with discrete time-delays, both delay-independent stability conditions and delay-dependent stability conditions have already been obtained. However, no stability condition for fuzzy systems with distributed time-delays has appeared in the literature. This is also true in case of the robust stability for uncertain fuzzy systems with distributed time-delays. With the aid of the stability theorems of linear systems with distributed time-delays, we employ a similar Lyapunov functional for the stability of fuzzy time-delay systems. Then we introduce many free matrices to such a Lyapunov functional, in order to obtain delay-dependent stability conditions. These techniques lead to generalized and less conservative stability conditions. In fact, delay-dependent stability conditions thus obtained are given in linear matrix inequalities(LMIs) and guarantee a wide stability region. We also consider the robust stability of fuzzy time-delay systems with uncertain parameters. Applying the same techniques made on the stability conditions, we obtain delay-dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed time-delays. Moreover, we consider the state feedback stabilization. Based on stability and robust stability conditions, we obtain conditions for the state feedback controller to stabilize the fuzzy time-delay systems. Finally, we take some examples to illustrate our results.

This paper modifies the prototype model-free adaptive control (PMFAC) approach in which system output fluctuation and overshoot (SOFO) is easy to occur. Targeting a class of nonlinear discrete-time single-input single-output systems, historical input/output sensor data are utilized to online construct equivalent explicit data model for control design, making the controller model-free. Then, a novel compensation input term including a compensation gain and a signum function parameterized by incremental output is proposed to soften system inputs, which can in turn restrain the SOFO. Numerical simulations and a speed control experiment for a direct current motor verify that the modified approach can restrain the SOFO successfully and reduce response time compared with the PMFAC.

Observers Synthesis Method for a Class of Nonlinear Discrete-Time Systems with Extension to Observer-Based Control

We propose a congestion control algorithm and design a state feedback controller for regulating the congestion of available bit rate (ABR) traffic in asynchronous transfer mode (ATM) networks. ABR service is specified as the rate-based feedback mechanism for congestion control and provides a control signal to throttled sources from a bottleneck node in order to adjust the throttled source rates based on available capacity. Our main purposes are as follows: (1) to design a state feedback controller; (2) to place the closed-loop poles in any desired regions based on a pole placement constraint; (3) to use the LMI (linear matrix inequalities) approach for finding the optimal state feedback gain K. The simulation results show that, under pole placement and LMI approaches, our state feedback congestion controller for ABR traffic in ATM networks can possibly guarantee the system stability, provide no buffer overflow, determine the optimal source traffic rates and place the closed-loop poles in any desired regions as well.

This paper presents a dynamic output feedback controller with determined order for the stabilization of a class of fractional-order system with nonlinear uncertain parameters with fractional order . Using stability theories of fractional-order systems and linear matrix inequalities (LMIs), some sufficient conditions in the LMI form are deduced to guarantee the robustness and asymptotic stabilization of the system. Designing a dynamic robust controller, along with all its useful features, leads to more unknown parameters in comparison with a static controller and makes controller design procedure more difficult due to more complex constraints that must be solved. In this paper, using proper lemmas and theorems, LMI techniques, and suitable solvers and parsers the difficulty of designing such controllers has been overcome. Simulation results of three different numerical examples illustrate that the proposed sufficient theoretical results are applicable and effective for tackling robust stabilization problems.