Variable-Gain Controllers for Nonlinear Systems Using the T–S Fuzzy Model

This correspondence proposes two novel control schemes with variable state-feedback gain to stabilize a Takagi-Sugeno (T-S) fuzzy system. The T-S fuzzy model is expressed as a linear plant with nonlinear disturbance terms in both schemes. In controller I, the T-S fuzzy model is expressed as a linear plant around a nominal plant arbitrarily selected from the set of linear subsystems that the T-S fuzzy model consists of. The variable gain then becomes a function of a gain parameter that is computed to neutralize the effect of disturbance term, which is, in essence, the deviation of the actual system dynamics from the nominal plant as the system traverses a specific trajectory. This controller is shown to stabilize the T-S fuzzy model. In controller II, individual linear subsystems are locally stabilized. Fuzzy blending of individual control actions is shown to make the T-S fuzzy system Lyapunov stable. Although applicability of both control schemes depends on the norm bound of unmatched state disturbance, this constraint is relaxed further in controller II. The efficacy of controllers I and II has been tested on two nonlinear systems     

Variable-gain controllers for nonlinear systems using the T-S fuzzy model.

This correspondence proposes two novel control schemes with variable state-feedback gain to stabilize a Takagi-Sugeno (T-S) fuzzy system. The T-S fuzzy model is expressed as a linear plant with nonlinear disturbance terms in both schemes. In controller I, the T-S fuzzy model is expressed as a linear plant around a nominal plant arbitrarily selected from the set of linear subsystems that the T-S fuzzy model consists of. The variable gain then becomes a function of a gain parameter that is computed to neutralize the effect of disturbance term, which is, in essence, the deviation of the actual system dynamics from the nominal plant as the system traverses a specific trajectory. This controller is shown to stabilize the T-S fuzzy model. In controller II, individual linear subsystems are locally stabilized. Fuzzy blending of individual control actions is shown to make the T-S fuzzy system Lyapunov stable. Although applicability of both control schemes depends on the norm bound of unmatched state disturbance, this constraint is relaxed further in controller II. The efficacy of controllers I and II has been tested on two nonlinear systems.     

基于向量Lyapunov函数方法的模糊控制系统稳定性分析和设计

采用模糊动态模型对连续时间非线性系统进行模糊控制,对闭环模糊系统的稳定性进行分析,并给出系统化的控制器设计程序,在一系列局部模型通过模糊隶属函数连接得到的连续的全局模型中,全面考虑其它关联子系统对标称线性系统的摄动,并利用向量Lyapunov函数的概念和方法,得到了闭环模糊系统稳定的充分条件;仿真例子验证了该设计方法的正确性。     

Robust convergence of discrete‐time delayed switched nonlinear systems and its applications to cascade systems

Systems in real world are always subject to perturbations. This paper addresses the convergence properties of a class of nominal systems with perturbations. It is assumed that the nominal system is a switched nonlinear exponentially stable one with time‐varying delays and that the perturbation exponentially or asymptotically converges to zero. It is revealed that the trajectories of the perturbed system behave as the perturbation, ie, trajectories exponentially or asymptotically converge to zero, depending on the property of perturbation. Applying these results to cascade switched nonlinear systems, it is shown that such systems are exponentially stable if and only if all the subsystems, obtained by removing the coupling terms, are exponentially stable. A similar conclusion is presented for systems being asymptotically stable. Finally, a numerical example illustrates the proposed theoretical results.     

基于模糊动态模型的多变量系统模糊控制

采用模糊动态模型对多变量复杂非线性系统进行模糊控制.首先针对局部线性动态模型设计状态反馈控制器,然后利用模糊推理确定整个系统的控制;在一系列局部模型通过模糊隶属函数连接得到的连续的全局模型中,全面考虑其它关联子系统对标称线性系统的摄动,并利用大系统分散控制关联稳定性的概念和方法,得到了闭环模糊系统稳定的充分条件.仿真例子验证了该设计方法的正确性.     

Real-Time Implementation of Decoupled Controllers for Multirotor Aircrafts

In this paper, a practical controller design method is presented to control six degrees of freedom multirotor aircrafts–quadrotors. The quadrotor system is divided into four subsystems; that is, the longitudinal, lateral, yaw, and height subsystems. Then, a linear and decoupled nominal model is obtained for each subsystem, whereas coupling and nonlinear dynamics, parametric perturbations, and external disturbances are considered as uncertainties. For each subsystem, a decoupled robust controller is then proposed. Although there exist coupling between each channel, the output tracking errors of the four subsystems are proven to ultimately converge into a-priori set neighborhood of the origin. Finally, real-time implementation results of the decoupled controller are given to demonstrate its viability and applicability.     

Open loop dynamic trajectory generator for a fuzzy gain scheduler

This paper presents a dynamic trajectory generator for a nonlinear system to be controlled by a fuzzy gain scheduler composed of a set of local linear Takagi–Sugeno (TS) fuzzy controllers. The local control laws are designed for the error system including the desired state and the corresponding desired control input. The task of the open loop dynamic trajectory generator is the generation of a sequence of control inputs along a predefined dynamic trajectory of the nominal nonlinear system. While the desired state is normally given, the corresponding desired control input may not always be computable in an explicit or unique way. With the proposed method the desired control input is approximated by an inverse fuzzy model of the nominal system. The model is built on the basis of a combination of c-elliptotype and Gustafson–Kessel clustering and a subsequent identification of local linear and affine TS models. In a next one-step ahead optimization loop the approximated control input is corrected by an analytical forward model of the nominal system.     

A decentralized approach to integrated flight control synthesis

A new approach to decoupled control of large-scale non-linear systems is applied to a dynamic flight control. Control synthesis is performed in two steps. First the nominal, programmed control is synthesized using the complete model of flight dynamics. This nominal control should realize the nominal trajectory under ideal conditions with no perturbations. In the second step the tracking of the nominal trajectory is realized. The system is viewed as a set of decoupled subsystems and for each subsystem local control is synthesized. Then, the stability of the overall system is analyzed and the global control is introduced to compensate coupling among some of the subsystems. A particular choice of subsystems in a case of flight control and of local and global control synthesis is proposed. A simulation of flight control with the proposed control law is also presented.     

Linear dynamically varying LQ control of nonlinear systems overcompact sets

Linear-quadratic controllers for tracking natural and composite trajectories of nonlinear dynamical systems evoluting over compact sets are developed. Typically, such systems exhibit “complicated dynamics”, i.e., have nontrivial recurrence. The controllers, which use small perturbations of the nominal dynamics as input actuators, are based on modeling the tracking error as a linear dynamically varying (LDV) system. Necessary and sufficient conditions for the existence of such a controller are linked to the existence of a bounded solution to a functional algebraic Riccati equation (FARE). It is shown that, despite the lack of continuity of the asymptotic trajectory relative to initial conditions, the cost to stabilize about the trajectory, as given by the solution to the FARE, is continuous. An ergodic theory method for solving the FARE is presented. Furthermore, it is shown that wrapping the LDV controller around the nonlinear system secures a stable tracking dynamics. Finally, an example of controlling the Henon map to follow an aperiodic orbit is presented     

ROBUST GAIN‐SCHEDULED CONTROL OF A VERTICAL TAKEOFF AIRCRAFT WITH ACTUATOR SATURATION VIA THE LMI METHOD

This paper presents a robust gain‐scheduled approach for the control of a vertical/short takeoff. and landing (V/STOL) aircraft. The nonlinear aircraft dynamics exhibit non‐minimum phase characteristics arising from the parasitic coupling effect between the aircraft's lateral force and rolling moment. The undesired coupling effect also causes modelling uncertainy of the aircraft dynamics. The nonlinear aircraft dynamics are considered to be composed of a nominal linear parameter varying (LPV) system and a linear system with a norm bounded uncertainy matrix multiplied by the parasitic uncertain non‐minimum phase coupling parameter. The nominal LPV system is considered to be affinely dependent on a measurable varying parameter. The ranges of the varying parameter and its variation as well as its parasitic induced uncertain matrix are addressed by introducing the parameter‐dependent invariant ellipsoid interpretation for dealing with the issue of affinely quadratic stabilization. In this paper, the relations among the magnitude of actuator saturation, the maximum achievable relative stability, and the sustainable coupling uncertainty are investigated for the considered robust gain‐scheduled design.     

Decentralized model reference adaptive control without restrictionon subsystem relative degrees

When the direct model reference adaptive control (MRAC) scheme with first-order local estimators is employed to design totally decentralized controllers, the stability result can only be applied to a system with all of its nominal subsystem relative degrees less than or equal to two. In this paper, this restriction is relaxed and it is achieved by employing the parameter projection together with static normalization. To implement the local controllers, no a priori knowledge of the subsystem unmodeled dynamics and no information exchange between subsystems are required. Global stability is established for the closed-loop system and small in the mean tracking error is ensured. With this analysis, the class of interactions and subsystem unmodeled dynamics can be enlarged to include those having infinite memory     

Incremental passivity and output regulation for switched nonlinear systems

This paper studies incremental passivity and global output regulation for switched nonlinear systems, whose subsystems are not required to be incrementally passive. A concept of incremental passivity for switched systems is put forward. First, a switched system is rendered incrementally passive by the design of a state-dependent switching law. Second, the feedback incremental passification is achieved by the design of a state-dependent switching law and a set of state feedback controllers. Finally, we show that once the incremental passivity for switched nonlinear systems is assured, the output regulation problem is solved by the design of global nonlinear regulator controllers comprising two components: the steady-state control and the linear output feedback stabilising controllers, even though the problem for none of subsystems is solvable. Two examples are presented to illustrate the effectiveness of the proposed approach.     

Trajectory tracking of leader–follower formations characterized by constant line-of-sight angles

A group of identical unicycles is controlled by means of local feedback laws that require measurements of the unicycles relative positions and speeds. Vehicle interconnections are considered unilateral and are modeled by means of a directed acyclic graph. Although not needed for the implementation of the controllers, the desired trajectory of each vehicle is derived from the requirement that the formation must rotate with the leader while ensuring that the relative positions and line-of-sight angles between unicycles are time-invariant. Exponential convergence of the actual trajectories to a ball centered on the desired trajectories is obtained by computing the Jacobian of the nonlinear dynamics and using results from contraction theory. Instrumental to this derivation is the subsystem feedback decomposition interpretation of the plant model. In this context, convergence depends on the uniform negative definiteness and strict positive realness of the forward and feedback subsystems. Such properties are obtained provided a set of linear and bilinear matrix inequalities as well as kinematic constraints are satisfied. A numerical example illustrates the convergence property of a leader–follower formation.     

Nonlinear system stabilization via hierarchical switching control

A nonlinear control system design framework predicated on a hierarchical switching controller architecture parameterized over a set of moving system equilibria is developed. Specifically, using equilibria-dependent Lyapunov functions, a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems. The switching nonlinear controller architecture is designed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function over a given switching set induced by the parameterized system equilibria. The proposed framework provides a rigorous alternative to designing gain-scheduled feedback controllers and guarantees local and global closed-loop system stability for general nonlinear systems     

Multilinear model decomposition of MIMO nonlinear systems and its implication for multilinear model-based control

In order to accomplish the multilinear model decomposition of MIMO nonlinear processes with multiple scheduling variables, a systematic division algorithm based on gap metric together with a supporting dichotomy gridding algorithm is proposed by using the gap metric as a measuring tool. For a prescribed distance level, this gap metric based division algorithm effectively decomposes a MIMO nonlinear system into a set of linear subsystems which provide enough model information for multilinear model-based controller design without linear model redundancy. Based on the linear models, a set of linear MPC controllers are designed and combined into a global controller for setpoint tracking control. Two benchmark nonlinear processes are studied to demonstrate the effectiveness of the proposed method.     

Robust stability and stabilization of discrete-time non-linear systems: The LMI approach

The purpose of this paper is to convert the problem of robust stability of a discrete-time system under non-linear perturbation to a constrained convex optimization problem involving linear matrix inequalities (LMI). The nominal system is linear and time-invariant, while the perturbation is an uncertain non-linear time-varying function which satisfies a quadratic constraint. We show how the proposed LMI framework can be used to select a quadratic Lyapunov function which allows for the least restrictive non-linear constraints. When the nominal system is unstable the framework can be used to design a linear state feedback which stabilizes the system with the same maximal results regarding the class of non-linear perturbations. Of particular interest in this context is our ability to use the LMI formulation for stabilization of interconnected systems composed of linear subsystems with uncertain non-linear and time-varying coupling. By assuming stabilizability of the subsystems we can produce local control laws under decentralized information structure constraints dictated by the subsystems. Again, the stabilizing feedback laws produce a closed-loop system that is maximally robust with respect to the size of the uncertain interconnection terms.     

参数不确定机器人分散鲁棒跟踪控制

提出了一种新的参数不确定机器人分散控制器设计方法．首先将关节子系统的动力学模型分解为人工标称模型和非线性时变不确定模型两部分；然后分别设计相应的标称控制器和鲁棒补偿器．标称控制器使得标称闭环系统具有理想的跟踪性能；鲁棒补偿器可以抑制参数不确定和关节间非线性耦合等因素的影响，实现鲁棒跟踪．所设计的控制器只需要局部关节的位置反馈，具有易于实现和可在线调整的优点．仿真结果说明了该方法的有效性．     

永磁同步电机高效非线性模型预测控制

永磁电机控制器要求电机有很强的转速跟踪能力,并且要保证系统参数变化及负荷扰动下系统的鲁棒性. 永磁电机包含很多不确定因素,是强耦合的非线性系统,传统的线性控制器很难对其进行控制. 针对永磁电机的转速控制构造非线性模型预测控制方法. 非线性永磁电机模型通过输入-输出反馈线性化策略解耦成为新的线性系统. 为保证可行解的收敛性,提出一种迭代二次规划方法来处理由输入-输出反馈线性化导致的非线性约束. 仿真结果表明,控制器能有效降低计算负担,具有很好的动态控制性能,能抑制转矩脉动,并保证在参数变化和负荷扰动下控制系统的鲁棒性.     

Robust stabilization of nonlinear cascaded systems

This paper considers the problem of robust stabilization for an uncertain nonlinear system which is a cascaded interconnection of two subsystems. Both of the subsystems are allowed to be nonlinear, multi-variable, and containing uncertain parameters. We present a new approach to designing stabilizing controllers which assure both robust global asymptotic stability and local quadratic stability. Compared with existing results, the assumptions required for such robust stabilizing controllers to exist are significantly simplified.     

Robust stability and controller synthesis of discrete linear switched systems with dwell time

Sufficient conditions are derived for the robust stability of discrete-time, switched, linear systems with dwell time in the presence of polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, is assigned to each of the subsystems. This function is allowed to be time-varying and piecewise linear during the dwell time and it becomes time invariant afterwards. Asymptotic stability conditions are obtained in terms of linear matrix inequalities for the nominal set of subsystems. These conditions are then extended to the case where the subsystems encounter polytopic type parameter uncertainties. The developed method is applied to l ₂-gain analysis where a bounded real lemma is derived, and to H _∞ control and estimation, both for the nominal and the uncertain cases.