Controller design for stabilizing bilinear systems with state-based switching

Some nonlinear systems can be approximated by switching bilinear systems. In this paper, we proposed a method to design state-based stabilizing controller for switching bilinear systems. Based on the similarity between switching bilinear systems and switching linear systems, corresponding switching linear systems are obtained for switching bilinear systems by applying state-based feedback control laws. Instead, we consider asymptotically stabilizing the corresponding switching linear system through solving a number of relaxed LMI conditions. Stabilizing controllers for switching bilinear systems can be derived based on the results of the corresponding switching linear systems. The stability of the controller is proved step by step through the decreasing of the multiple Lyapunov functions along the state trajectory. The effectiveness of the method is demonstrated by both a theoretical example and an example of urban traffic network with traffic signals.     

Stabilisation des systèmes bilinéaires homogènes

We deal with the problem of stabilizing the bilinear homogeneous systems. We give sufficient conditions for a bilinear system to be stabilizable with homogeneous feedback of degree zero. The explicit design of the feedback is given.     

Bilinear Control and Application to Flexible a.c. Transmission Systems

The purpose of this paper is an integrated overview of bilinear systems (BLS) research which has evolved over the past few decades, and a new result on control of flexible a.c. transmission systems (FACTS) is presented. BLS may be derived in many cases from principles of physics, chemistry, biology, socioeconomics, and engineering. In other cases, BLS are more accurate approximations to nonlinear systems than are traditional linear systems, as shown for example by the added bilinear terms (in state and control) for the Taylor series.While an appropriately designed linear control system may be optimum relative to some quadratic performance index without added constraints, bilinear or parametric control can be designed to improve more global performance and indeed to increase the region of attainable states. Such controllability and stabilization of BLS and of a series line-capacitor controlled FACTS is presented.     

Feedback stabilization of stochastic bilinear systems and of some nonlinear stochastic systems

The purpose of this note, is to derive sufficient conditions for the existence of stabilizing feedback laws for control stochastic bilinear systems and to apply these results to the stabilization of a class of nonlinear stochastic differential systems. The method used in this paper rely on the stochastic Lyapunov machinery     

Stability of the residual free bubble method for bilinear finite elements on rectangular grids

We consider the nature of the stabilizing term arising in theresidual free bubble approach for piecewise bilinear functionson rectangular grids. We show, that on the subspace of piecewiselinear functions the stabilizing term is identical to that inthe streamline diffusion approach. However, on the space ofpiecewise bilinear functions there is a case in which the stabilizingterm is weaker compared to the term used in the streamline diffusionmethod. In the particular case when the direction of the convectionis parallel to a diagonal of the quadrilateral, control is lostover the mixed derivatives in the convection-dominated limit.     

Global Stabilization of a Class of Bilinear Systems in $${\mathbb{R}^3}$$

In this paper, we consider a class of bilinear systems in dimension three which can be an extension of another one in \({\mathbbm{R}^{2}}\). We prove that there exists some homogeneous feedback of degree zero stabilizing the considered class if and only if these feedbacks are constants.     

Design of hybrid regrouping PSO–GA based sub-optimal networked control system with random packet losses

In this paper, a new approach has been presented to design sub-optimal state feedback regulators over networked control systems with random packet losses. The optimal regulator gains, producing guaranteed stability are designed with the nominal discrete time model of a plant using Lyapunov technique which produces a few set of bilinear matrix inequalities (BMIs). In order to reduce the computational complexity of the BMIs, a genetic algorithm (GA) based approach coupled with the standard interior point methods for LMIs has been adopted. A regrouping particle swarm optimization based method is then employed to optimally choose the weighting matrices for the state feedback regulator design that gets passed through the GA based stability checking criteria i.e. the BMIs. This hybrid optimization methodology put forward in this paper not only reduces the computational difficulty of the feasibility checking condition for optimum stabilizing gain selection but also minimizes other time domain performance criteria like expected value of the set-point tracking error with optimum weight selection based LQR design for the nominal system.     

Nonlinear and bilinear models for chemical reactor control

The dynamic behavior of a continuously stirred tank reactor (CSTR) with an exothermic reversible reaction is studied. The balance equations of the reaction lead to a set of highly nonlinear differential equations. For system analysis and control synthesis the dynamic equation are rewritten as state space model. From this nonlinear model a bilinear model is derived. Then, two optimization problems are solved: The time optimal problem for the nonlinear model and the quadratic problem for the bilinear model. In case of the finite time bilinear-quadratic problem a modified Riccati approximation algorithm for a stabilizing feedback controller is presented.     

基于模糊双线性模型的非线性系统的跟踪控制

研究一种基于T-S模糊双线性系统的跟踪控制器设计及稳定性分析.使用分布并行补偿法(PDC)设计了模糊控制器,得到模糊双线性系统跟踪控制渐近稳定的充分条件,仿真结果验证了该方法改进了闭环系统的性能.     

Mixed strategies in maximin testing of robust stabilization performance

We consider algorithms for testing robust stabilization performance for bilinear systems of special form, discrete-continuous systems of control of a dynamic object. We present an example of synthesis of a mixed testing strategy in the space module-orbital station rendezvous problem.     

Stabilization of irregular systems

We consider the stabilization problem for the zero equilibrium of bilinear and affine systems in canonical form. We obtain necessary and sufficient conditions for the stabilizability of second-order bilinear and affine systems in canonical form and generalize these conditions to the case of simultaneous stabilization of a family of bilinear systems and to nth-order bilinear systems.     

Generalization of the Jurdjevic-Quinn theorem and stabilization of bilinear control systems with periodic coefficients: II

We prove sufficient conditions for the uniform global asymptotic stabilization of the origin for bilinear time-varying systems that can be reduced by a Lyapunov transformation to bilinear periodic systems with time-invariant drift, in particular, for bilinear periodic systems with time-varying drift. We obtain corollaries for consistent systems.     

A Sufficient Condition for Stabilization of Bilinear Systems in a Plane

The article considers stabilization of bilinear systems in a plane. A discontinuous-feedback algorithm is proposed. Sufficient conditions for stabilization of bilinear systems are derived. The proposed algorithm stabilizes a system which is not stabilizable in the class of constant controls.     

On the Optimal Stabilisation for a set of Programmed Motions of the Bilinear Control System

During recent years there has been considerable interest in using bilinear systems [1, 2] as mathematical models to represent the dynamic behavior of a wide class of engineering, biological and economic systems. Moreover, there are some methods [3] which may approximate nonlinear control systems by bilinear systems. For the first time Zubov has proposed a method of stabilization control synthesis for a set of programmed motions in linear systems [4]. In papers [5, 6] this method has been developed and used to solve the same problem for bilinear systems. In the present paper the following problems are considered. First, synthesis of nonlinear control as feedback under which the bilinear control system has a given set of programmed and asymptotic stable motions. Because this control is not unique, the second problem concerns optimal stabilization. In this paper a method for the design of nonlinear optimal control is suggested. This control is constructed in the form of a convergent series. The theorem on the sufficient conditions to solve this problem is represented.     

APPROXIMATE CONTROLLABILITY OF INFINITE DIMENSIONAL BILINEAR SYSTEMS

1.IntroductionRecentlymanyresultshavebeenobtainedfordistributedbilinearsystemsl3'7'8].In[3]thecontrollabilityofabilinearsystem%=Aw(t) p(t)Bw(t)wasstudied,whereAistheindnitesimalgeneratorofaCosemigroupofboundedlinearoperatorsT(t)onaBanachspaceX.B:X-- 5isaboundedlinearoperator,andpELI([0,T];R)isacontrol.TheconditionsforelementsofEtobeaccessiblefromagiveninitialstatecoweregiven.Itisclearthattheystudiedthebilinearsysteminaspecialcase.In[8]theystudiedthesysteminamorespecialcasebecausetheyass…     

Optimization of Bilinear Systems Using a Higher-Order Variational Method

This paper derives some optimization results for bilinear systems using a higher-order method by characterizing them over matrix Lie groups. In the derivation of the results, first a bilinear system is transformed to a left-invariant system on matrix Lie groups. Then, the product of exponential representation is used to express this system in canonical form. Next, the conditions for optimality are obtained by the principles of variational calculus. It is demonstrated that closed-form analytical solutions exist for classes of bilinear systems whose Lie algebra are nilpotent.     

Global feedback stabilization of multi-input bilinear systems

This paper mainly investigates the problem of stabilization of homogeneous bilinear systems with multiple inputs. Explicit state feedback laws are given to stabilize the bilinear systems. Meanwhile, an estimate of the convergence speed is obtained under the given feedback laws. Besides, sufficient conditions, which are easy to be verified, are presented for the stabilization of the bilinear systems.     

Bilineare Systeme zur Bestimmung optimaler Werbestrategien

Zusammenfassung In jüngster Zeit wurde den bilinearen Systemen von seiten der Systemtheorie einige Aufmerksamkeit gewidmet. In der Tat findet man zahlreiche Beispiele, nicht nur technisch-physikalische Systeme, sondern auch biologische, ökologische oder sozio-ökonomische Prozesse, die sich durch bilineare Zustandsdifferentialgleichungen vorteilhaft modellieren lassen. Daß dies auch für reale ökonomische Prozesse gilt, soll im folgenden demonstriert werden. Dazu wird in Anlehnung an eine empirische Untersuchung vonVidale/Wolfe [1957] der Einfluß der Werbung auf den Absatz einer Firma als Kontrollproblem mit bilinearer Struktur formuliert und die Bedingungen einer optimalen Werbestrategie untersucht. Die Arbeit soll einen Anreiz geben zum Aufspüren weiterer bilinearer Prozesse in der Ökonomie.

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Summary This paper uses the tool of bilinear systems for solving a marketing problem. It is demonstrated how recent conceptual advances in the class of bilinear systems can be applied for economic processes. This approach is in particular favourable for modelling economic processes with variable structure.We use as an example for such an economic process the study ofVidale/Wolfe [1957] about the impact of advertisement on sales. Using the results of this study a bilinear control problem is formulated. This enables us to come up with the conditions of an optimal advertisement strategy. The purpose of this study is to motivate further investigations for economic problems which can be favourably formulated in a bilinear systems framework.

     

Parametric model order reduction of thermal models using the bilinear interpolatory rational Krylov algorithm

The Bilinear Interpolatory Rational Krylov Algorithm (BIRKA; P. Benner and T. Breiten, Interpolation-based H₂-model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl. 33 (2012), pp. 859–885. doi:10.1137/110836742) is a recently developed method for Model Order Reduction (MOR) of bilinear systems. Here, it is used and further developed for a certain class of parametric systems. As BIRKA does not preserve stability, two different approaches generating stable reduced models are presented. In addition, the convergence for a modified version of BIRKA for large systems is analysed and a method for detecting divergence possibly resulting from this modification is proposed. The behaviour of the algorithm is analysed using a finite element model for the thermal analysis of an electrical motor. The reduction of two different motor models, incorporating seven and thirteen different physical parameters, is performed.