# bilinear time-delay systems

## Top PDF bilinear time-delay systems: ### Stabilization Criteria for Bilinear Systems with Time varying Delay

Abstract In this paper we consider the problem of stabilizing a bilinear system with time- varying delay via linear state feedback control. Based on the Lyapunov method, a delay-dependent criterion for determining the stabilization of system is obtained in terms of linear matrix inequalities (LMIs) and used to express the relationships between the terms in the Leibniz-Newton formula, which can be easily solved by efficient convex optimization algorithms. From the numerical examples, the obtained results have some significant improvements over the recent literatures. ### VARIOUS FUZZY MODELING TECHNIQUES AND FUZZY CONTROLLER DESIGN FOR NONLINEAR SYSTEM

649 | P a g e analysis and controller design methods for nonlinear time-delay systems. A typical approach for the analysis and synthesis of nonlinear system with time delay is the local linearization approach. First a linearization model on the nominal operating point is gotten and then a linear feedback control is designed for this linear model. Fuzzy logic control is another approach to obtain nonlinear control systems, especially in the presence of incomplete knowledge of the plant or even of the precise control action appropriate to a given situation. First the nonlinear plant is represented by a dynamic fuzzy model. In this type of fuzzy model, local dynamics in different state- space regions are represented by linear models. The overall model of the system is achieved by fuzzy “blending” of these fuzzy models. The idea is that for each local linear model, a linear feedback control is designed. The resulting overall controller, which is nonlinear in general, is again a fuzzy blending of each individual linear controller. In this paper, we extend the above procedure to the nonlinear systems with time delay. But it is impossible to approximate some equations by a linear model then we have to go for bilinear model. As stated, a bilinear system is different from a linear one and is considered a type of nonlinear system. In recent years, bilinear systems and controls have been widely applied to a wide variety of fields, for example, bioengineering, biochemistry, nuclear engineering and socio-economics. A bilinear system exists between nonlinear and linear systems, and its dynamic is simpler than that of nonlinear systems. In fact, “the bilinear systems provide much better approximation of the original nonlinear systems than the linear systems,” and this is why we are investigating the T–S fuzzy bilinear system. It is well known that fuzzy control has been attracting increasing attention in the study and application of the stabilization of nonlinear systems , . ### Dimensional reduction of nonlinear time delay systems

B(Ψ(s), Φ(θ)), one can see that (Ψ(s), Φ(θ))B = B(Ψ(s), Φ(θ)). From this, it readily fol- lows that B = B, if and only if the inner product matrix (Ψ(s), Φ(θ)) = I, is the iden- tity matrix. The elements of the inner product matrix ( Ψ (s), Φ (θ)) are the bilinear pair- ing in (2.6c), namely, (Ψ(s), Φ(θ)) = (ψ j (s), φ κ (θ)), j,κ = 1, 2. Oftentimes, the value of ### On stability analysis of discrete time uncertain switched nonlinear time delay systems

Switched systems are an important class of hybrid systems. Generally speaking, a switched system is composed of a family of subsystems described by diﬀerential or diﬀerence equations and a switching rule orchestrating the switching between the subsystems that have attracted much attention in control theory and practice during recent decades. Switched systems can be eﬃciently used to model many practical systems which are in- herently multi-model in the sense that several dynamical systems are required to describe their behavior. For example, many physical processes exhibit switched and hybrid nature. Switched systems have strong engineering background in various areas and are often used as a uniﬁed modeling tool for a great number of real-world systems, such as power elec- tronics, chemical processes, mechanical systems, automotive industry, aircraft and air traﬃc control and many other ﬁelds [–]. ### Lyapunov-Krasovskii Approach for Delay-Dependent Stability Analysis of Nonlinear Time-Delay Systems

It is well known that there are several challenges in extension of Lyapunov-Krasovskii functionals for the stability analysis of nonlinear system with time-delay. Further complication arises when delay-dependent sta- bility is considered. There are a few works that have investigated delay-independent stability and delay- dependent stability, however, with more conservative criteria [11-14]. ### H∞ Finite Time Control for Switched Linear Systems with Time Varying Delay

A switched system is a special kind of hybrid system, which is composed of a family of subsystems and a switching sequence orchestrating the switching between the subsystems. Recently, switched systems have re- ceived a great deal of attention, and commonly been found in automotive engine control systems, network control, process control, traffic control, etc. Many im- portant progress and remarkable results have been made on basic problems concerning stability and design of switched systems [1-10]. For recent progress, readers can refer to survey papers [11-13] and the references therein. Many Lyapunov function techniques are effective tools dealing with switched systems [14-17]. Average dwell time and dwell time (DT) approaches were employed to study the stability and stabilization of time-dependent switched systems [18-20]. ### Hopf bifurcation control for a class of delay differential systems with discrete-time delayed feedback controller

This paper is concerned with asymptotical stabilization for a class of delay differential equations, which undergo Hopf bifurcation at equilibrium as delay increasing. Two types of controllers, continuous-time and discrete-time delay feedback controllers, are presented. Although discrete-time control problems have been discussed by several authors, to the best of our knowledge, so few controllers relate to both delay and sampling period, and the method of Hopf bifurcation has not been seen. Here, we first give a range of control parameter which ensures the asymptotical stability of equilibrium for the continuous- time controlled system. And then, for the discrete-time controller we also obtain an efficient control interval provided that the sampling period is sufficiently small. Meanwhile, we try our best to estimate a well bound on sampling period and get a more complete conclusion. Finally, the theoretical results are applied to a physiological system to illustrate the effectiveness of the two controllers. ### Delay Dependent Anti windup Synthesis for Time varying Delay Systems with Saturating Actuators

This paper concentrates on anti-windup methodologies to stabilize time-delay systems in the presence of control constraints: in this context we can just cite [2, 14, 13]. The anti-windup compensation is known to be a efficient technique to cope with undesirable effects (on performance and stability) produced by actuator saturation in control loops. The basic idea underlining anti-windup designs is that when the control saturates, it is temporarily modified using an anti-windup compensator in order to recover, as much as possible, the performance expected on the basis of the unsaturated system. Motivated by this, we consider L 2 -gain analysis and anti-windup ### Delayed Decomposition Approach to Delay Dependent Stability for Time Varying Delay Descriptor Systems

Time-delays are often encountered in various dynamic systems, such as manufacturing systems, economic systems, biological systems, networked control systems, and so on. The time-delay is frequently a source of instability and performance deterioration. Therefore, stability analysis and controller synthesis for time-delay system have been one of the most challenging issues [1-31]. The main aim of these studies is to achieve a maximum admissible upper bound (MAUB) such that the system under consideration is globally asymptotically stable for any time delay less than the MAUB. On the other hand, the descriptor system is referred to as a singular system, a generalized state-space system or a semi-state system. It is commonly encountered ### Representations of solutions to linear and bilinear difference equations and systems of bilinear difference equations

There has been a recent custom that the bilinear diﬀerence equation is called Riccati dif- ference equation. During the last two decades several colleagues, some of them who used the terminology, asked me if the attribution is correct, bearing in mind that it is known that there were not so many investigations on diﬀerence equations during the life of Ric- cati and that nobody has seen a paper written by him on the topic. The frequent question and recent studies of solvability of diﬀerence equations motivated me to conduct a con- siderable, but, of course, not thorough, literature investigation, to try to give a possible answer for the “open problem” in history of diﬀerence equations. ### Regional controllability of distributed bilinear systems*

Abstract. In this paper, we consider the problem of optimal regional controllability of a distributed bilinear system evolving on Ω. The question is to obtain a control with minimum energy that drives such a system from an initial state to a final state close to a desired one in finite time, only on a subregion ω of Ω. Our purpose is to prove that a regional optimal control exists and characterized in both bounded and unbounded cases. The obtained results are successfully illustrated by simulations. ### New delay dependent observer based control for uncertain stochastic time delay systems

This article considers state estimation for a class of Itô-type stochastic systems subject to time delay and parameter uncertainties. The system states are unmeasured. The stochas- tic system involves parameter uncertainties and time delay, and they are dependent on the state. The objective is to design a robust observer such that the dynamics of the estimation error is guaranteed to be asymptotically stable in the mean square. Attention is focused on the design of the gain matrix and the state-feedback controller. This paper derived an ob- server design method of uncertain stochastic time-delay systems by constructing a proper Lyapunov–Krasovskii functional and by making use of the free weighting matrix method. In , the authors obtained some theorems, but conclusions are independent of time delay. Consequently, this will largely restrict the applying area of the conclusions. The in- novation of this paper is that the delay-dependent suﬃcient condition for the existence of such a state observer for any admissible uncertainties is given. We present a new method to estimate the stochastic systems. Based on the new criterion, a delay-dependent con- dition for the existence of state observer is derived in terms of a linear matrix inequality (LMI), therefore it is in the sense of being conservative reduced. A numerical example is exploited to show the validity of the results obtained. ### Load Frequency Control of Three Area Power System using Fuzzy Logic Controller

Understanding of the frequency dynamics of power systems is essential for several issues including frequency control design and estimation of the system inertia through measurements. Consider a power system with several conventional synchronous generators. If this power system is subjected to a disturbance such as a sudden outage of a generator, then dynamical changes in the system start instantaneously. These dynamics are mainly caused by the instantaneous power imbalance between the instantaneous generation and consumption of electric power. Consequently, the remaining synchronous generators are subjected to acceleration and deceleration effects. Due to the strong connection between the mechanical and electrical frequency, the changes in the rotor speed results in changes in the electrical frequency. Eventually, the power balance and the frequency are restored in the system has sufficient capacity to compensate the lost generation. This control action is called primary Automatic Load Frequency Control (ALFC). In this direction, the load-frequency control (LFC) is one of important control problems in concerning the integration of wind power turbine in a multi-area power system [2, 8, 18, 20, 21].The increasing need for electrical energy in the twenty-first century, as well as limited fossil fuel reserves, very high transportation and fuel cost and the increasing concerns with environmental issues for the reduction of carbon dioxide (CO2) and other greenhouse gasses, causes fast development in the area of renewable energy sources (RESs). One of the adaptive and nonlinear intelligent control techniques that can be effectively applicable in the frequency control design is reinforcement ### Robust l2-l∞ Filtering for Discrete-Time Delay Systems

The problem of robust 𝑙 2 -𝑙 ∞ filtering for discrete-time system with interval time-varying delay and uncertainty is investigated, where the time delay and uncertainty considered are varying in a given interval and norm-bounded, respectively. The filtering problem based on the 𝑙 2 -𝑙 ∞ performance is to design a filter such that the filtering error system is asymptotically stable with minimizing the peak value of the estimation error for all possible bounded energy disturbances. Firstly, sufficient 𝑙 2 -𝑙 ∞ performance analysis condition is established in terms of linear matrix inequalities (LMIs) for discrete-time delay systems by utilizing reciprocally convex approach. Then a less conservative result is obtained by introducing some variables to decouple the Lyapunov matrices and the filtering error system matrices. Moreover, the robust 𝑙 2 -𝑙 ∞ filter is designed for systems with time-varying delay and uncertainty. Finally, a numerical example is given to demonstrate the effectiveness of the filter design method. ### Nonlinearity Tests For Bilinear Time Series Data

results for bilinear models do not give an outright decision of rejecting or accepting the null hypothesis of linearity since the percentages of acceptance is not less than 5%. This means that Keenan’s test may still classify nonlinear data set as linear data set. On the other hand, as the bilinear coefficients get larger, the F-test performs better as the percentages of rejection of null hypothesis become smaller than 5%. Several authors such as Chan and Tong  and Tsay  raised the same issue, saying not one nonlinearity test is enough to detect nonlinearity in a data set. Nonetheless, it is expected that the nonlinearity test will suggest whether a data set is linear or otherwise. If Keenan’s or F-test does suggest that the data is nonlinear, we expect that a bilinear model will improve the modelling of the data set. Otherwise, bilinear model may still be modelled before further inspection is carried out. ### Stability of a Class of Neutral Time Delay Systems with a Robust Control

This paper deal with the stability problem of neutral time- delay systems. Based on the Lyapunov-Krasovskii functional theory, new theorems are proposed for a type of neutral delay systems with robust time-delay control. New delay-dependent stability conditions are developed for the system without time- delay control in first time and with time-delay control in second time. Linear matrix inequality approaches are used to solve the stability problem in these cases. Numerical examples illustrate that the proposed methods are effective and lead to less conservative results. ### Delay-partitioning approach to stability criteria for T-S fuzzy systems with time-varying delay

Received 6 January 2015 ; Revised 12 March 2015 ; Accepted 18 April 2015 Abstract. This paper is concerned with the stability criteria of T-S fuzzy systems with time-varying delay by delay-partitioning approach. Based on Finsler’s lemma, LMI approach and an appropriate augmented LKF established in the framework of state vector augmentation, some tighter bounding inequalities such as Seuret- Wirtinger’s integral inequality and Peng-Park’s integral inequality are employed to deal with (time-varying) delay-dependent integral items. Therefore, less conserva- tive delay-dependent stability criteria are obtained in terms of LMIs, which can be solved efficiently with the Matlab LMI toolbox. Finally, one numerical example is provided to show that the proposed conditions are less conservative than existing ones. ### Constrained Feedback Stabilization for Bilinear Parabolic Systems

In this paper, we shall study the stabilization and the robustness of a constrained feedback control for bilinear parabolic systems defined on a Hilbert state space. Then, we shall show that stabiliz- ing such a system reduces stabilization only in its projection on a suitable subspace. For this pur- pose, a new constrained stabilizing feedback control that allows a polynomial decay estimate of the stabilized state is given. Also, the robustness of the considered control is discussed. An illu- strating example and simulations are presented. ### Proportional integral plus (PIP) control of time delay systems

the Engineering and Physical Sciences Research Council 9 Chotai, A. and Young, P. C. Pole-placement design for time grant number GR/J10136. delay systems using a generalised, discrete-time Smith pre- dictor. In IEE Conference Publication 285 of Control 88, 1988, pp. 218–223. ### Active fault-tolerant control for switched systems with time delay

select different controllers according to different component failures, and therefore better performance of the closed-loop system can be expected. If an AFTC is designed properly, it will be able to deal with unforeseen faults and maintain the system stability and acceptable level of performance in the presence of fault. Some preliminary results on AFTC can be found in - and references therein. Compared with the fruitful FTC results for various dynamic systems, relatively few efforts were made to investigate FTC issue for switched systems.  and  considered the passive FTC issue for discrete-time switched systems. In , passive FTC for switched nonlinear systems in lower triangular was studied. Switched system belongs to hybrid system, which consists of several subsystems and a switching signal that specifies which subsystem will be activated along the system trajectory at each instant of time. Many real-world process and systems can be modeled as switched systems, including chemical processes, computer controlled systems, switched circuits, and so on. During the past three decades, fruitful theoretic results have been reported for switched systems, for examples - and references therein. On the other hand, time delays are the inherent features of many physical process and the big sources of instability and poor performances. Meanwhile, switched systems with time delay have strong engineering background, such as in network control systems  and power systems . More recently, many theoretical studies were conducted for switched systems with time delays -.