Viable immersion and invariance control for a class of nonlinear systems and its application to aero-engines

This paper presents a novel approach to stabilize a class of nonlinear systems with state constraints. The motivation behind this study is the need to develop a stabilizing state feedback controller that does not require the knowledge of Lyapunov function and can regulate the states to the equilibrium while meeting the constraints. By using an integration of two relatively new tools: immersion and invariance (I&I) theory and viability theory, a sufficient condition for stability and stabilizability of a general nonlinear affine system with state constraints is derived; Then, the related results are exploited to stabilize a class of nonlinear system in feedback form and with state constraints represented by inequalities and the viable I&I stabilizing state feedback controller is obtained constructively. Further, an application to a nonlinear aero-engine model with the temperature constraint is given to illustrate the applicability and the effectiveness of the proposed method. Finally, a comparative simulation is presented, highlighting the advantages of the viable I&I controller.     

A hybrid partial feedback linearization and deadbeat control scheme for a nonlinear gantry crane

This paper presents the design of a hybrid partial feedback linearization and deadbeat control scheme for a nonlinear gantry crane with friction to control its position and sway angle. The partial feedback linearization is used to linearize the nonlinear model and to stabilize its internal dynamics. In many crane applications, it's necessary to accelerate the system response. As a result, this will cause oscillation in the position as well as the sway angle. So, the deadbeat controller is added to get the desirable accelerated response without any oscillation or adverse effects on the internal dynamics stability. By using Lyapunov stability method, the proposed scheme is proved to be globally stable, with converging tracking errors to the desired performance. The simulation results are accomplished to evaluate the effectiveness of the proposed scheme and to demonstrate its reliability to control crane systems with comparative results.     

Network-based control of nonlinear large-scale systems composed of identical subsystems

This paper deals with a control of coupled nonlinear identical systems that admit full exact feedback input-output linearization. The subsystems are linearized using this nonlinear transformation. In the next step, an auxiliary low-dimensional system is derived whose stability implies stability of the original large-scale system. The control law is designed so that the control loops are only local, no information exchange between subsystems is required. Unknown time delay in the feedback are allowed. Two cases are studied: equal time delay for all subsystems or different delay in all subsystems. Results are illustrated by two examples.     

On direct adaptive control design for nonlinear discrete-time uncertain systems

In this paper, we develop a direct adaptive control framework for adaptive stabilization of the MIMO nonlinear uncertain systems, which can be represented as discrete-time normal form with input-to-state zero dynamics. The framework is Lyapunov-based and guarantees partial stability of the closed-loop systems, such that the adaptation of the feedback gains can stabilize the closed-loop system without the knowledge of the system parameters. In addition, our results show that the adaptive feedback laws can be characterized by Kronecker calculus. Two numerical examples are given to demonstrate the efficacy of the proposed framework.     

Stable model predictive control for a nonlinear system

In this paper, a stable model predictive control approach is proposed for constrained highly nonlinear systems. The technique is a modification of the multistep Newton-type control strategy, which was introduced by Li and Biegler. The proposed control technique is applied on a constrained highly nonlinear aerodynamic test bed, the twin rotor MIMO system (TRMS) to show the efficacy of the control technique. Since the accuracy of the plant model is vital in MPC techniques, the nonlinear state space equations of the system are derived considering all possible effective components. The nonlinear model is adaptively linearized during the prediction horizon. The linearized models of the system are employed to form a linear quadratic objective function subject to a set of inequality constraints due to the system input/output limits. The stability of the control system is guaranteed using the terminal equality constraints technique. The satisfactory performance of the proposed control algorithm on the TRMS validates the effectiveness and the reliability of the approach.     

Advanced controller synthesis for fuzzy parameter varying systems

A novel nonlinear time-varying model termed as the fuzzy parameter varying (FPV) system is proposed in this research, which inherits both advantages of the conventional T-S fuzzy system in dealing with nonlinear plants and strengths of the linear parameter varying (LPV) system in handling time-varying features. It is, therefore, an attractive mathematical model to efficiently approximate a nonlinear time-varying plant or to serve as a type of time-varying controller. Using the full block S-procedure, sufficient stability conditions have been derived in the form of linear matrix inequalities (LMIs) to test quadratic stability of the open-loop FPV system. Moreover, sufficient conditions have been derived on synthesizing both state feedback and dynamical output feedback fuzzy gain-scheduling controllers that can stabilize the FPV system. An inverted pendulum with a variable length pole is utilized to demonstrate advantages of the FPV system compared to the conventional T-S fuzzy system in representing a practical time-varying nonlinear plant and to validate the controller synthesis conditions.     

Performance-oriented parallel distributed compensation

The main idea of the original parallel distributed compensation (PDC) method is to partition the dynamics of a nonlinear system into a number of linear subsystems, design a number of state feedback gains for each linear subsystem, and finally generate the overall state feedback gain by fuzzy blending of such gains. A new modification to the original PDC method is proposed here, so that, besides the stability issue, the closed-loop performance of the system can be considered at the design stage. For this purpose, the state feedback gains are not considered constant through the linearized subsystems, rather, based on some prescribed performance criteria, several feedback gains are associated to every subsystem, and the final gain for every subsystem is obtained by fuzzy blending of such gains. The advantage is that, for example, a faster response can be obtained, for a given bound on the control input. Asymptotic stability of the closed loop system is also guaranteed by using the Lyapunov method. To illustrate the effectiveness of the new method, control of a flexible joint robot (FJR) is investigated and superiority of the designed controller over other existing methods is demonstrated.     

Control of a directly excited structural dynamic model of an F-15 tail section

We used two simple control laws based on linear velocity and cubic velocity feedback to suppress the high-amplitude vibrations of a structural dynamic model of the twin-tail assembly of an F-15 fighter when subjected to primary resonance excitations. We developed the nonlinear differential equations of motion and obtained an approximate solution using the method of multiple scales. Then, we conducted bifurcation analyses for the open- and closed-loop responses of the system and investigated theoretically the performance of the control strategies. The theoretical findings indicate that the control laws lead to effective vibration suppression and bifurcation control. Furthermore, we conducted experiments to verify the theoretical analysis. We built a digital control system that consists of a SIMULINK modeling software and a dSPACE controller installed in a personal computer. Actuators made of piezoelectric ceramic material were used. The results show that both laws are effective at suppressing the vibrations. To compare the performance of both techniques, we calculated the power requirements for a simple system.     

Nonlinear curve fitting-based fast robust MPC algorithm for nonlinear system

In the existing efficient robust model predictive control (ERMPC) algorithms (see e.g. [14,31,32]), through offline optimization and online lookup table calculation, a fixed state feedback control law or a linear interpolated control law is applied to a system when the system state lies between two adjacent polyhedrons, which undoubtedly will result in conservativeness of the controller. Faced with this issue, an improved ERMPC algorithm is proposed in this paper, which considers the nonlinearity between the state feedback control laws with respect to polyhedrons and the norm distance from system state to origin, and can provide continuously variable state feedback control law varying with the state. First, a set of polyhedral parameters and their corresponding state feedback control law sequences are obtained offline by solving a set of LMIs optimization problems. Next, for each state feedback control law sequence, a nonlinear fitting function is established offline between the state feedback control law and its serial number. Then a simplified lookup table is constructed offline to save memory space and shorten online computation time of the controller. According to the simplified lookup table and information of the norm distance from system state to origin, we online establish the coordinate of current state in the nonlinear fitting curve for getting current feedback control law, which changes continuously with the state. The proposed ERMPC algorithm is successfully applied to an actual fast-responding linear one stage inverted pendulum (LOSIP) system to verify its effectiveness.     

基于微分包含的约束非线性系统显示预测控制

提出了一种非线性预测控制的新方法.首先基于线性微分包含理论,利用泰勒级数对系统进行线性化,通过对偏导数取最大和最小的方法构造多面体描述的线性时变系统包裹原非线性系统,然后对于多面体描述的线性不确定系统,采用多参数规划的方法建立显示模型预测控制系统.对该方法进行了仿真计算,仿真结果表明采用这种方法可以更好的描述非线性系统...     

Tracking control of nonlinear automobile idle-speed time-delay system via differential geometry approach

This paper is concerned with the problem of global asymptotical tracking of single-input single-output (SISO) nonlinear time-delay control systems. Based on the input-output feedback linearization technique and Lyapunov method for nonlinear state feedback synthesis, a robust globally asymptotical output tracking controller design methodology for a broad class of nonlinear time-delay control systems is developed. The underlying theoretical approaches are the differential geometry approach and the composite Lyapunov approach. One utilizes the parameterized co-ordinate transformation to transform the original nonlinear system into singularly perturbed model and the composite Lyapunov approach is then applied for output tracking. For the view of practical application, the proposed control methodology has been successfully applied to the famous nonlinear automobile idle-speed control system.     

Stabilization for a class of strict-feedback nonlinear systems via the PWM control law

This paper studies the PWM control problem of a class of nonlinear systems. During a modulation period, the PWM control signal maintains a pulse waveform with tunable width and fixed magnitude. The PWM control only possesses finite states, and has relatively limited control capability. This causes the degradation of system performance, and even the instability when implementing into a nonlinear system. We will introduce a novel method to design both the state feedback stabilizer and the output feedback stabilizer for strict-feedback nonlinear systems via the PWM control. The system performance is analyzed in a novel framework and the stability criteria is derived to ensure the system convergence. At last, two examples are considered to illustrate the effectiveness of our proposed method.     

Robust adaptive tracking with an additional plant identifier for a class of nonlinear systems

In this paper, we develop two new model reference adaptive control (MRAC) schemes for a class of nonlinear dynamic systems that is robust with respect to an uncertain state (output) dependent nonlinear perturbations and/or external disturbances with unknown bounds. The design is based on a controller parametrization with an adaptive integral action. Two types of adaptive controllers are considered—the state feedback controller with a plant parameter identifier, and the output feedback controller with a linear observer.     

Incremental passivity-based output regulation for switched nonlinear systems via average dwell-time method

This paper investigates the output regulation problem for a class of switched nonlinear systems with at least a feedback incrementally passive subsystem via average dwell time method. First, the output regulation problem for switched nonlinear system via full information feedback is solved. The stabilizing controllers consist of the state feedback controllers and linear output feedback controllers. In some particular cases, it is unnecessary to verify that all the solutions of the switched nonlinear system converge to the bounded steady-state solution, while we only have to verify the regulated outputs converge to zero directly. Second, a dynamic error-feedback stabilizer for each subsystem and a switched internal model whose subsystems all are incrementally passive are designed to solve the output regulation problem for the switched nonlinear system under a composite switching signal with average dwell times. The stabilizer and the internal model are interconnected in a more simple way and allowed to switch asynchronously. Finally, two examples are provided to show the effectiveness of the obtained results.     

Stabilization of differently structured hybrid neutral stochastic systems by delay feedback control under highly nonlinear condition

This paper mainly studies the stabilization of differently structured highly nonlinear hybrid neutral stochastic systems by delay feedback control. Based on the existing works, our new neutral type stochastic system has completely different highly nonlinear structures in switching subspaces, which is more general and applicable. When such a system is given unstable, we focus on studying the asymptotic and exponential stability criteria by designing a feedback control with a time delay for the underlying system. A simulating example is shown to illustrate the feasibility of these results.     

Limit cycle analysis of nonlinear sampled-data systems by gain-phase margin approach

This work analyzes the limit cycle phenomena of nonlinear sampled-data systems by applying the methods of gain-phase margin testing, the M-locus and the parameter plane. First, a sampled-data control system with nonlinear elements is linearized by the classical method of describing functions. The stability of the equivalent linearized system is then analyzed using the stability equations and the parameter plane method, with adjustable parameters. After the gain-phase margin tester has been added to the forward open-loop system, exactly how the gain-phase margin and the characteristics of the limit cycle are related can be elicited by determining the intersections of the M-locus and the constant gain and phase boundaries. A concise method is presented to solve this problem. The minimum gain-phase margin of the nonlinear sampled-data system at which a limit cycle can occur is investigated. This work indicates that the procedure can be easily extended to analyze the limit cycles of a sampled-data system from a continuous-data system cases considered in the literature. Finally, a sampled-data system with multiple nonlinearities is illustrated to verify the validity of the procedure.     

Stabilizing unstable periodic orbits in large stability domains with dynamic time-delayed feedback control

Dynamic time-delayed feedback control (DDFC) is applied to stabilize the unstable periodic orbits (UPOs) of chaotic systems in large stability domains. The stability domain is defined as certain areas of the parameter space of the feedback strength, in which the UPOs are stabilized. The control effect of the DDFC with a second-order controller system is investigated by considering two control objectives: to broaden the stability domain of the controlled UPOs and to minimize the modulus of the largest Floquet multiplier, which leads to a multi-objective optimization problem (MOP). The MOP is solved with the genetic algorithm. Case studies indicate that the control effect of the DDFC is significantly better than that of original time-delayed feedback control. The DDFC can stabilize the UPOs in a large stability domain and with a small modulus of the largest Floquet multiplier when the adjustable parameters of the controller system are properly designed.     

Sufficient bit rate conditions to stabilize an uncertain scalar nonlinear system based on event triggering

This paper considers to stabilize an uncertain scalar continuous-time nonlinear system with bounded network delay and process noise, which transmits all feedback signals through a digital communication network. In order to save the bandwidth of the feedback network, stability is expected to be maintained at as low as possible feedback bit rate. Based on event triggering, this paper proposes a model-based event-triggered sampling strategy to guarantee the desired input-to-state stability of the concerned system. Due to the bounded network delay, the receiving time instant of a feedback packet cannot be precisely controlled by the sensor, i.e., the receiving time instant is not always equal to its sampling time instant. Their gap, i.e., the network delay, determines how much information can be carried through the receiving time instant and makes great impact on the system’s stability. Sufficient bit rate conditions to stabilize that system are derived. The conditions are determined by the parameter of Lipschitz condition, the upper bound of the network delay and the system uncertainty. Compared with the periodic sampling strategies, a lower bit rate is required by the proposed event-triggered strategy. Simulations are done to verify the effectiveness of the achieved stabilizing bit rate conditions.