Semiglobal stabilization of multi-input linear systems with saturated linear state feedback

For multi-input linear systems with eigenvalues in the closed left-half comlex plane, we address the problem of stabilization by a linear state feedback subject to saturation. Using an eigenvalue-generalized eigenvector assignment technique, we prove that such systems can be semiglobally stabilized with a saturated linear state feedback. Based on this result, we propose an algorithm to calculate an -parametrized family of state feedback gain matrices that semiglobally stabilize the system. Two examples are used to illustrate the results.     

Robust global stabilization of linear systems with input saturation via gain scheduling

The problem of robust global stabilization of linear systems subject to input saturation and input‐additive uncertainties is revisited in this paper. By taking advantages of the recently developed parametric Lyapunov equation‐based low gain feedback design method and an existing dynamic gain scheduling technique, a new gain scheduling controller is proposed to solve the problem. In comparison with the existing ?₂‐type gain scheduling controller, which requires the online solution of a state‐dependent nonlinear optimization problem and a state‐dependent ?₂ algebraic Riccati equation (ARE), all the parameters in the proposed controller are determined a priori. In the absence of the input‐additive uncertainties, the proposed controller also partially recovers Teel's ?_∞‐type scheduling approach by solving the problem of global stabilization of linear systems with actuator saturation. The ?_∞‐type scheduling approach achieves robustness not only with non‐input‐additive uncertainties but also requires the closed‐form solution to an ?_∞ ARE. Thus, the proposed scheduling method also addresses the implementation issues of the ?_∞‐type scheduling approach in the absence of non‐input‐additive uncertainties. Copyright © 2009 John Wiley & Sons, Ltd.     

Semiglobal synchronization of multiple generic linear agents with input saturation

This paper investigates the semiglobal synchronization problem for a group of agents with input saturation under directed interaction topology, where each agent is modeled as a generic linear system rather than the single‐integrator or double‐integrator dynamics. The main result is the construction of a feedback coupling gain that achieves semiglobal synchronization if all the agents have identically saturated linear dynamics, which can be of any order. It is shown that the coupling gain obtained via parametric Lyapunov equations can semiglobally synchronize any directed network provided that the interaction topology has a directed spanning tree. Furthermore, due to the use of parametric Lyapunov equations, a convergence rate is analytically obtained. Finally, a simulation example is provided to demonstrate the effectiveness and advantages of our theoretical findings. Copyright © 2013 John Wiley & Sons, Ltd.     

Semiglobal stabilization of linear systems using constrained controls: a parametric optimization approach

A bounded feedback control for asymptotic stabilization of linear systems is derived. The designed control law increases the feedback gain as the controlled trajectory converges towards the origin. A sequence of invariant sets of decreasing size, associated with a (quadratic) Lyapunov function, are defined and related to each of them, the corresponding possible highest gain is chosen, while maintaining the input bounded. Gains as functions of the position are designed by explicitly solving a c-parameterized programming problem. The proposed method allows global asymptotic stabilization of open-loop stable systems, with inputs subject to magnitude bounds and globally bounded rates. In the general case of linear systems that are asymptotic null controllable with bounded input, the semiglobal stabilization is also addressed taking into account the problem of semiglobal rate-limited actuators. The method is illustrated with the global stabilization of an inertial navigator, and the stabilization of a nonlinear model of a crane with hanging load. Copyright © 1999 John Wiley & Sons, Ltd.     

Stabilization of linear systems with input delay and saturation—A parametric Lyapunov equation approach

This paper studies the problem of stabilizing a linear system with delayed and saturating feedback. It is known that the eigenstructure assignment‐based low‐gain feedback law (globally) stabilizes a linear system in the presence of arbitrarily large delay in its input, and semi‐globally stabilizes it when the input is also subject to saturation, as long as all its open‐loop poles are located in the closed left‐half plane. Based on a recently developed parametric Lyapunov equation‐based low‐gain feedback design method, this paper presents alternative, but simpler and more elegant, feedback laws that solve these problems. The advantages of this new approach include its simplicity, the capability of giving explicit conditions to guarantee the stability of the closed‐loop system, and the ease in scheduling the low‐gain parameter on line to achieve global stabilization in the presence of actuator saturation. Copyright © 2009 John Wiley & Sons, Ltd.     

Semiglobal stabilization of linearly uncontrollable and unobservable nonlinear systems via sampled‐data control

This article investigates the problem of using sampled‐data state/output feedback to semiglobally stabilize a class of uncertain nonlinear systems whose linearization around the origin is neither controllable nor observable. For any arbitrarily large bound of initial states, by employing homogeneous domination approach and a homogeneous version of Gronwall‐Bellman inequality, a sampled‐data state feedback controller with appropriate sampling period and scaling gain is constructed to semiglobally stabilize the system. In the case when not all states are available, a reduced‐order sampled‐data observer is constructed to provide estimates for the control law, which can guarantee semiglobal stability of the closed‐loop system with carefully selected sampling period and scaling gain.     

一类非线性系统的输出反馈半局鲁棒镇定

从半局镇定角度研究了一类非线性系统的输出反馈镇定问题,这类系统具有较强的非线性增长特征.利用高增益观测器,把非分离设计原则应用于输出反馈半局镇定控制器设计.在不需系统具有一个一致完全可观状态反馈控制器情况下,给出了输出反馈控制器的设计方法.对任意给定紧子集,所设计的反馈控制器使闭环系统是渐近稳定的且吸引域包含指定的紧子集.最后的仿真实例说明了设计方法的有效性.     

Event‐based global stabilization of linear systems via a saturated linear controller

This paper investigates the problem of event‐based linear control of systems subject to input saturation. First, for discrete‐time systems with neutrally stable or double‐integrator dynamics, novel event‐triggered control algorithms with non‐quadratic event‐triggering conditions are proposed to achieve global stabilization. Compared with the quadratic event‐triggering conditions, the non‐quadratic ones can further reduce unnecessary control updates for the input‐saturated systems. Furthermore, for continuous‐time systems with neutrally stable or double‐integrator dynamics, because an inherent lower bound of the inter‐event time does not exist for systems subject to input saturation, novel event‐triggered control algorithms with an appropriately selected minimum inter‐event time are proposed to achieve global stabilization. Finally, numerical examples are provided to illustrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

A semi-global low-and-high gain design technique for linear systems with input saturation—stabilization and disturbance rejection

In this paper we propose a linear low-and-high gain design technique to solve the semi-global stabilization problem for a class of linear systems subject to input saturation. The central idea behind this new technique is to increase the utilization of the available control capability of the system. The power of this new semi-global design technique is shown by solving the problem of semi-global stabilization with input-additive disturbance rejection which is formulated as a semi-global practical stabilization problem.     

Semi-global stabilization of discrete-time linear systems with position and rate-limited actuators

It is shown via explicit construction of feedback laws that, if a discrete-time linear system is asymptotically null controllable with bounded controls, then, when subject to both actuator position and rate saturation, it is semi-globally stabilizable by linear state feedback. If, in addition, the system is also detectable, then it is semi-globally stabilizable via linear output feedback.     

A parametric Lyapunov equation approach to low gain feedback design for discrete-time systems

Low gain feedback, a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero, has found several applications in constrained control systems, robust control and nonlinear control. In the continuous-time setting, there are currently three ways of constructing low gain feedback laws: the eigenstructure assignment approach, the parametric ARE based approach and the parametric Lyapunov equation based approach. The eigenstructure assignment approach leads to feedback gains explicitly parameterized in the low gain parameter. The parametric ARE based approach results in a Lyapunov function along with the feedback gain, but requires the solution of an ARE for each value of the parameter. The parametric Lyapunov equation based approach possesses the advantages of the first two approaches and results in both an explicitly parameterized feedback gains and a Lyapunov function. The first two approaches have been extended to discrete-time setting. This paper develops the parametric Lyapunov equation based approach to low gain feedback design for discrete-time systems.     

Robust global stabilization of spacecraft rendezvous system via gain scheduling

The problem of robust global stabilization of a spacecraft circular orbit rendezvous system with input saturation and inputadditive uncertainties is studied in this paper. The relative models with saturation nonlinearity are established based on Clohessey-Wiltshire equation. Considering the advantages of the recently developed parametric Lyapunov equation-based low gain feedback design method and an existing high gain scheduling technique, a new robust gain scheduling controller is proposed to solve the robust global stabilization problem. To apply the proposed gain scheduling approaches, only a scalar nonlinear equation is required to be solved. Different from the controller design, simulations have been carried out directly on the nonlinear model of the spacecraft rendezvous operation instead of a linearized one. The effectiveness of the proposed approach is shown.     

Global stabilization of systems containing a double integrator using a saturated linear controller

In this paper, we present a Lyapunov function for systems containing a double integrator and with controller saturation. This Lyapunov function is composed of a positive-semidefinite quadratic term and an integral term. The main result provides a sufficient condition that guarantees a system with a double integrator can be globally stabilized by a saturating linear controller. For a triple-integrator system the saturated linear controller does not satisfy the sufficient condition, which agrees with the known result. Copyright © 1999 John Wiley & Sons, Ltd.     

Semi-global stabilization of linear systems subject to output saturation

It is established that a SISO linear stabilizable and detectable system subject to output saturation can be semi-globally stabilized by linear output feedback if all its invariant zeros are in the closed left-half plane, no matter where the open loop poles are. This result complements a recent result that such systems can always be globally stabilized by discontinuous nonlinear feedback laws, and can be viewed as dual to a well-known result: a linear stabilizable and detectable system subject to input saturation can be semi-globally stabilized by linear output feedback if all its poles are in the open left-half plane, no matter where the invariant zeros are.     

A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation

This paper is concerned with semi-global stabilization of discrete-time linear periodic systems subject to actuator saturation. Provided that the open loop characteristic multipliers are within the closed unit circle, a low gain feedback design approach is proposed to solve the problem by state feedback. Our approach is based on the solution to a parametric discrete-time periodic Lyapunov equation. The proposed approaches not only generalize the corresponding results for time-invariant systems to periodic systems, but also reveal some important intrinsic properties of this class of periodic matrix equations. A numerical example is worked out to illustrate the effectiveness of the proposed approaches.     

Robust semi-global stabilization of linear systems with imperfect actuators

This paper deals with the problem of semi-globally stabilizing a linear system in the presence of input additive uncertainties and disturbance using actuators that have imperfect input output characteristics. The input output characteristics of the actuators used include those of dead zone and saturation and are not precisely known. Two design approaches, the low-and-high-gain approach and the low-gain-based variable structure control approach, are taken in solving the problem.     

An improvement to the low gain design for discrete‐time linear systems in the presence of actuator saturation nonlinearity

A low‐gain design for linear discrete‐time systems subject to input saturation was recently developed to solve both semi‐global stabilization and semi‐global output regulation problems. This paper proposes an improvement to the low‐gain design and determines controllers with the new design that achieve semi‐global output regulation. The improvement is reflected in better utilization of available control capacity and consequently better closed‐loop performance. Copyright © 2000 John Wiley & Sons, Ltd.     

Stabilisation of time-varying linear systems via Lyapunov differential equations

This article studies stabilisation problem for time-varying linear systems via state feedback. Two types of controllers are designed by utilising solutions to Lyapunov differential equations. The first type of feedback controllers involves the unique positive-definite solution to a parametric Lyapunov differential equation, which can be solved when either the state transition matrix of the open-loop system is exactly known, or the future information of the system matrices are accessible in advance. Different from the first class of controllers which may be difficult to implement in practice, the second type of controllers can be easily implemented by solving a state-dependent Lyapunov differential equation with a given positive-definite initial condition. In both cases, explicit conditions are obtained to guarantee the exponentially asymptotic stability of the associated closed-loop systems. Numerical examples show the effectiveness of the proposed approaches.     

线性多变量系统的控制Lyapunov函数构造

提出线性多变量系统控制Lyapunov函数(CLF)构造的一般方法. 先证明可以通过解一类Lyapunov方程, 得到线性系统二次型的CLF. 接着证明了对于线性系统, 这种方法可以提供所有二次型的CLF. 最后证明了若线性系统存在CLF, 那么必存在二次型的CLF. 由此完全解决了线性系统的CLF构造问题.     

Robust output feedback stabilization via a small gain theorem

In this paper, we give sufficient conditions for designing robust globally stabilizing controllers for a class of uncertain systems, consisting of ‘nominal’ nonlinear minimum phase systems perturbed by uncertainties which may affect the equilibrium point of the nominal system (‘biased’ systems). The constructive proof combines a systematic step-by-step procedure, based on H_∞ arguments, with a small gain theorem, recently proved for nonliner systems. At each step, one finds two Lyapunov functions, one for a state-feedback problem and the other one for an output injection problem. Combining these two functions, one derives at each step a Lyapunov function candidate for solving an ouptut feedback stabilization problem. This approach allows one to put into a unified framework many existing results on robust output feedback stabilization. © 1998 John Wiley & Sons, Ltd.