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.     

An iterative method for the finite-time bilinear-quadratic control problem

For bilinear control systems with quadratic cost, the so-called bilinear-quadratic problems, a feedback controller for the finite-time case is designed. An iteration procedure in close proximity to the Riccati approach is presented, and the proof of convergence is outlined. The potential of the new method is discussed, and the design procedure is illustrated for two examples.     

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.     

Feedback stabilization of non-homogeneous bilinear systems with a finite time delay

This paper investigates the feedback stabilization of non-homogeneous delayed bilinear systems in Hilbert state space. More precisely, under an observability-like assumption, we prove the exponential and strong stability of the solution by using bounded feedback control. Partial stabilization is discussed as well. The proofs of the main results are based on the decomposition method. The decay estimates of the stabilized solutions are obtained. Finally, some examples are presented.     

Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations

Stability results are given for a class of feedback systems arising from the regulation of time-varying discrete-time systems using optimal infinite-horizon and moving-horizon feedback laws. The class is characterized by joint constraints on the state and the control, a general nonlinear cost function and nonlinear equations of motion possessing two special properties. It is shown that weak conditions on the cost function and the constraints are sufficient to guarantee uniform asymptotic stability of both the optimal infinite-horizon and moving-horizon feedback systems. The infinite-horizon cost associated with the moving-horizon feedback law approaches the optimal infinite-horizon cost as the moving horizon is extended.     

A Bilinear Control Problem Solved by the Fractional Step Scheme

We consider a control problem for which the cost functional consists of locally Lipschitz functions and is governed by a bilinear multivalued differential equation. We study this problem using the fractional step scheme. Two particular examples are solved using the proposed method.     

Quantum Optimal Control Problems with a Sparsity Cost Functional

In this article, the investigation of a class of quantum optimal control problems with L¹ sparsity cost functionals is presented. The focus is on quantum systems modeled by Schrödinger-type equations with a bilinear control structure as it appears in many applications in nuclear magnetic resonance spectroscopy, quantum imaging, quantum computing, and in chemical and photochemical processes. In these problems, the choice of L¹ control spaces promotes sparse optimal control functions that are conveniently produced by laboratory pulse shapers. The characterization of L¹ quantum optimal controls and an efficient numerical semi-smooth Newton solution procedure are discussed.     

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.     

Stochastic stability of positive Markov jump linear systems with fixed dwell time

This paper studies the stochastic stability of positive Markov jump linear systems with a fixed dwell time. By constructing an auxiliary system that originated from the initial system with state jumps, sufficient and necessary conditions of stochastic stability for positive Markov jump linear systems are obtained with both exactly known and partially known transition rates. The main idea in the latter case is applying a convex combination to convert bilinear programming into linear programming problems. On this basis, multiple piecewise linear co-positive Lyapunov functions are provided to achieve less conservative results. Then state feedback controller is designed to stabilize the positive Markov jump linear systems by solving linear programming problems. Numerical examples are presented to illustrate the viability of our conclusions.     

随机线性系统保成本控制的线性矩阵不等式方法

考虑具有二次成本函数的随机线性系统,研究了状态反馈控制的保证成本控制问题.依据线性矩阵不等式得到了保证成本控制器存在的充分条件,最后得到了随机线性闭环系统保证成本最小的最优保证成本控制律的表达式.     

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.     

Comments on the feedback stabilization for bilinear control systems

Necessary and sufficient condition are given for state feedback stabilization of distributed control bilinear systems. The results are based on the theory of linear and nonlinear contraction semigroups.     

广义双线性系统的输出变结构控制

首次利用广义Lyapunov函数方法针对不确定广义双线性系统的输出变结构控制问题进行研究.首先,选取设计了带有滑动模动态补偿器的切换函数,保证了系统在准切换流形上的渐近稳定性.其次,在不确定参数和扰动范数有界的条件下,设计了变结构控制器,使得在控制下闭环系统在有限时间内实现滑模运动.最后,通过数值算例说明了设计方法的合理性和有效性.     

Controlling hyperchaos in hyperchaotic Lorenz system using feedback controllers

In this paper, the control strategies for hyperchaotic Lorenz system is investigated. The ordinary, dislocated, enhancing and speed feedback controls are used to suppress hyperchaos to unstable equilibrium. The Routh-Hurwitz criterion is used to derive the conditions of stability of the controlled hyperchaotic systems. It is found that the coefficients of enhancing feedback control and dislocated feedback control may be smaller than those of ordinary feedback control, so, the complexity and cost of the system control are reduced. Numerical simulations are given to illustrate the effectiveness of the proposed controllers.     

Asymptotical Stability in Probability for Stochastic Bilinear Systems with Markovian Switching

This paper deals with asymptotical stability in probability in the large for stochastic bilinear systems. Some new criteria for asymptotical stability of such systems have been established in the inequality of mathematic expectation. A sufficient condition for bilinear stochastic jump systems to be asymptotically stable in probability in the large in Markovian switching laws is derived in a couple of Riccati-like inequalities by introducing a nonlinear state feedback controller. An illustrative example shows the effectiveness of the method.     

Robust guaranteed cost control for uncertain linear differential systems of neutral type

In this paper, the robust guaranteed cost control problem for a class of uncertain linear differential systems of neutral type with a given quadratic cost functions is investigated. The uncertainty is assumed to be norm-bounded and time-varying nonlinear. The problem is to design a state feedback control laws such that the closed-loop system is robustly stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainty and time delay. A criterion for the existence of such controllers is derived based on the matrix inequality approach combined with the Lyapunov method. A parameterized characterization of the robust guaranteed cost controllers is given in terms of the feasible solutions to the certain matrix inequalities. A numerical example is given to illustrate the proposed method.     

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.     

Stability of bilinear time-delay systems

In this paper, the stability of the differential bilinear time-delaysystems is first studied. We consider time-varying bilineartime-delay systems with output feedback. The input or controlu(t)is not only a signal but also an input with output feedback.The analysis is given by using norm-transformation methods.     

Global stabilization of switched control systems with time delay

In this paper, the stabilization problem of switched control systems with time delay is investigated for both linear and nonlinear cases. First, a new global stabilizability concept with respect to state feedback and switching law is given. Then, based on multiple Lyapunov functions and delay inequalities, the state feedback controller and the switching law are devised to make sure that the resulting closed-loop switched control systems with time delay are globally asymptotically stable and exponentially stable.     

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.