Explicit constructions of global stabilization and nonlinear H∞ control laws for a class of nonminimum phase nonlinear multivariable systems

This paper investigates a global stabilization problem and a nonlinear H_∞ control problem for a class of nonminimum phase nonlinear multivariable systems. To avoid the complicated recursive design procedure, an asymptotic time‐scale and eigenstructure assignment method is adopted to construct the control laws for the stabilization problem and the nonlinear H_∞ control problem. A sufficient solvability condition is established onthe unstable zero dynamics of the system for global stabilization problem and nonlinear H_∞ control problem, respectively. Moreover, based on the sufficient solvability condition, an upper bound of the achievable L₂‐gain is estimated for the nonlinear H_∞ control problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Path-Following for Nonlinear Systems With Unstable Zero Dynamics

In the path-following problem formulated in this note, it is required that the error between the system output and the desired geometric path eventually be less than any prespecified constant. If in a nonlinear multiple-input-multiple-output (MIMO) system the output derivatives do not enter into its zero dynamics, a condition relating path geometry and stabilizability of the zero dynamics is given under which a solution to this problem exists. The solution is obtained by combining input-to-state stability and hybrid system methodologies     

Reduced-order controllers for the H∞ control problem with unstable invariant zeros

This paper addresses the existence and design methods of reduced-order controllers for the H_∞ control problem with unstable invariant zeros in the state-space realization of the transfer function matrix from the control input to the controlled error or from the exogenous input to the observation output, where the realization is induced from a stabilizable and detectable realization of the generalized plant. This paper presents a new controller degree bound for the H_∞ control problem in terms of the minimal rank of the system matrix pencils of these two transfer function matrices in the unstable region. When the unstable invariant zero exists, this paper shows that reduced-order controllers with orders strictly less than that of the generalized plant exist if the H_∞ control problem is solvable. Moreover, this paper shows that the computational problem of finding the controllers with the new degree bound is convex by providing two linear matrix inequality-based design methods (algorithms) for constructing the reduced-order controllers. The results developed in this paper are valid both for the continuous- and discrete-time H_∞ control problems.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> DOF control of stochastic systems with time-varying delay and <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> disturbance

This paper considers the problem of H _∞ dynamic output feedback (DOF) control for a class of stochastic systems with time-varying delay and L _∞ disturbance. A new delay-dependent sufficient condition for the existence of the DOF controller is derived and the controller design method is given in the form of bilinear matrix inequalities (BMIs). Moreover, a variable step size path-following algorithm is proposed to solve the BMI problem. Numerical examples are given to illustrate the effectiveness of the proposed methods.     

Passivity-based designs for synchronized path-following

We consider a formation control system where individual systems are controlled by a path-following design and the path variables are to be synchronized. We first show a passivity property for the path-following system, and next, combine this with a passivity-based synchronization algorithm developed in Arcak [2007. Passivity as a design tool for group coordination. IEEE Transactions on Automatic Control, in press]. The passivity approach expands the classes of synchronization schemes available to the designer. This generality offers the possibility to optimize controllers to, e.g., improve robustness and performance. Two designs are developed in the proposed passivity framework: the first employs the path error information in the synchronization loop, while the second only uses synchronization errors. A sampled-data design, where the path variables are updated in discrete-time and the path-following controllers are updated in continuous time, is also developed.     

A solvability condition of an extended control problem using Riccati inequalities

In this paper, we consider an H infinity problem using unstable weighting functions. Such weights are used to achieve asymptoticaly disturbance at enuation and/or asymptotically reference tracking, and they cannot be handled by the standard H infinity control theory. For this problem, we present a necessary and sufficient condition to exist a solution using Riccati inequalities. This condition is derived under no unnecessary assumption to enlarge the applicability to various control problems. Moreover, introducing the assumption that the image of the zero vectors of G₁₂ and G₂₁ are unique, we obtain a LMI condition in order to check the solvability numerically. A numerical example is presented to demonstrate how to calculate a controller.     

A highly scalable path-following controller for N-trailers with off-axle hitching

The paper presents a highly scalable nonlinear cascaded-like path-following feedback controller for N-trailer robotic vehicles equipped with arbitrary number of off-axle hitched trailers. In contrast to the other path-following control laws proposed in the literature for N-trailer robots, the presented control approach does not require determination of the shortest distance to a reference path. By introducing the so-called segment-platooning reference paths, and under the sign-homogeneity assumption for hitching offsets, the asymptotic following is guaranteed for both constant- and varying-curvature reference paths using either backward or forward vehicle motion strategy with a guidance point fixed on the last trailer. The paper contains experimental results obtained with a 3-trailer laboratory-scale vehicle.     

Performance limitations in reference tracking and path following for nonlinear systems

We investigate limits of performance in reference tracking and path following and highlight an essential difference between them. For a class of nonlinear systems, we show that in reference tracking, the smallest achievable L₂ norm of the tracking error is equal to the least amount of control energy needed to stabilize the zero dynamics of the error system. We then show that this fundamental performance limitation does not exist when the control objective is to force the output to follow a geometric path without a timing law assigned to it. This is true even when an additional desired speed assignment is required to be satisfied asymptotically or in finite time.     

Nonlinear feedback control and trajectory tracking of vehicle

This paper mainly studies nonlinear feedback control applied to the nonlinear vehicle dynamics with varying velocity. The main objective of this study is the stabilisation of longitudinal, lateral and yaw angular vehicle velocities. To this end, a nonlinear vehicle model is developed which takes both the lateral and longitudinal vehicle dynamics into account. Based on this model, a method to build a nonlinear state feedback control is first designed by which the complexity of system structure can be simplified. The obtained system is then synthesised by the combined Lyapunov–LaSalle method. The simulation results show that the proposed control can improve stability and comfort of vehicle driving. Moreover, this paper presents a lemma which ensures the trajectory tracking and path-following problem for vehicle. It can also be exploited simultaneously to solve both the tracking and path-following control problems of the vehicle ride and driving stability. We also show how the results of the lemma can be applied to solve the path-following problem, in which the vehicle converges and follows a designed path. The effectiveness of the proposed lemma for trajectory tracking is clearly demonstrated by simulation results.     

Intelligent complementary sliding-mode control with dead-zone parameter modification

This paper proposes an intelligent complementary sliding-mode control (ICSMC) system which is composed of a computed controller and a robust controller. The computed controller includes a neural dynamics estimator and the robust compensator is designed to prove a finite L₂-gain property. The neural dynamics estimator uses a recurrent neural fuzzy inference network (RNFIN) to approximate the unknown system term in the sense of the Lyapunov function. In traditional neural network learning process, an over-trained neural network would force the parameters to drift and the system may become unstable eventually. To resolve this problem, a dead-zone parameter modification is proposed for the parameter tuning process to stop when tracking performance index is smaller than performance threshold. To investigate the capabilities of the proposed ICSMC approach, the ICSMC system is applied to a one-link robotic manipulator and a DC motor driver. The simulation and experimental results show that favorable control performance can be achieved in the sense of the L₂-gain robust control approach by the proposed ICSMC scheme.     

离轴式拖车移动机器人的任意路径跟踪控制

离轴式拖车移动机器人属于非完整系统,当车头线速度随时间变化且过零变号时,难以用一个控制器实现系统对期望路径的跟踪.本文研究离轴式拖车移动机器人系统的任意路径跟踪问题.首先由系统和虚拟小车的运动学方程得到误差状态模型,线性化后用坐标变换将其化为标准型,然后基于Lyapunov方法构造出一种跟踪控制律.只要车头的运动线速度有界且不趋于零,其导数有界,则所设计的控制律就可以保证系统跟踪任意的期望路径,且跟踪误差最终一致有界,最终界的大小与期望路径的曲率变化率成比例.当期望路径的曲率变化率为零或趋于零时,所设计的控制律可以保证拖车移动机器人指数收敛到期望路径.仿真结果证实了控制律的有效性.     

A simple method of tuning series cascade controllers for unstable systems

A simple method is proposed to design P/PI controllers for a series cascade control system for unstable first order plus time-delay （FOPTD） systems. In this paper, the controller design for unstable FOPTD systems cascaded in series with stable/unstable FOPTD systems is considered. The proposed method is based on equating the coefficients of corresponding powers of s and s2 in the numerator to α1 and α2 times those of the denominator of the closed-loop transfer function for a servo problem. The open loop system consists of an unstable FOPTD system cascaded in series with a stable/unstable FOPTD system. Only two tuning parameters （α1 and α2） are required for the design of controllers. The closed-loop performances are evaluated for both the servo and regulatory problems and the performances are found to be better than that of the well established synthesis method. The robustness for uncertainty in the model parameters is studied and compared with that of the controllers designed by the synthesis method.     

Output feedback control of a class of nonminimum‐phase nonlinear systems

The problem of global stabilization by output feedback is investigated in this paper for a class of nonminimum‐phase nonlinear systems. The system under consideration has a cascade configuration that consists of a driven system known as the inverse dynamics and a driving system. It is proved that although the zero dynamics may be unstable, there is an output feedback controller, globally stabilizing the nonminimum‐phase system if both driven and driving systems have a lower‐triangular form and satisfy a Lipschitz‐like condition, and the inverse dynamics satisfy a stronger version of input‐to‐state stabilizability condition. A design procedure is provided for the construction of an n‐dimensional dynamic output feedback compensator. Examples and simulations are also given to validate the effectiveness of the proposed output feedback controller. Copyright © 2016 John Wiley & Sons, Ltd.     

Receding horizon time-optimal control for a class of differentially flat systems

A closed-loop, time-optimal path-following control scheme is proposed for a class of constrained differentially flat systems. Within a receding horizon framework, a finite horizon optimisation problem is solved at each sample, using available state feedback and feedforward path information. Irrespective of horizon length, the proposed formulation guarantees exact path-following. Moreover, the requirements under which the proposed algorithm achieves minimum-time path-following are established. Simulations conducted with a rigid X–Y table model confirm the theoretical results.     

Disturbance tolerance and H ∞ control of port-controlled hamiltonian systems in the presence of actuator saturation

This paper investigates the disturbance tolerance and H _∞ control of multi-input Port-Controlled Hamiltonian (PCH) systems in the presence of actuator saturation which may be not openloop stable. A simple condition is derived under which trajectories starting from the origin will remain inside an ellipsoid. The disturbance tolerance ability of the closed-loop system under a given feedback control law is measured by the size of this ellipsoid. Based on the above mentioned condition, the problem of disturbance tolerance can be expressed in the form of the linear matrix inequalities (LMIs) optimization problem with constraints. In addition, an H _∞ control approach is presented to attenuate the disturbances, and disturbance rejection ability in terms of L ₂ gain is also determined by the solution of an LMI optimization problem. Study of an illustrative example with simulations shows the effectiveness of the methods proposed.     

Global almost disturbance decoupling with stability for non minimum-phase single-input single-output nonlinear systems

The so-called problem of almost disturbance decoupling with internal stability (ADDPS) is the following one. Given a system and an (arbitrarily small) number γ > 0, find a feedback law yielding a closed loop system which is stable and in which the gain (in the L₂ sense) between the exogenous input and the regulated output is less than or equal to γ. The complete solution of this problem has been known since a long time in the case of linear systems. In the case of nonlinear systems, the only global results available so far in the literature were about SISO systems having an asymptotically stable zero dynamics. In this paper, a new set of results are presented, dealing with nonlinear SISO systems having a possibly unstable zero dynamics, which include the (general) class of linear SISO systems as a special case.     

Using feedthrough for near one-step-ahead control of inverselyunstable plants

The one-step-ahead (OSA) control law is modified by the use of feedthrough to accommodate unstable plant zeros. An nth-order discrete-time system is guaranteed to track the desired input in one timestep. This is accomplished by subtracting the denominator dynamics and algebraically canceling the numerator dynamics. A problem occurs with OSA control when the plant has unstable zeros, that is, zeros that are outside the z-plane unit circle. In this case, an unstable zero is canceled and the resulting controller will be unstable. The authors address this problem. The approach taken is specific to OSA control     

L2–L∞ fuzzy control for Markov jump systems with neutral time-delays

The L₂–L_∞ fuzzy control problem is considered for nonlinear stochastic Markov jump systems with neutral time-delays. By means of Takagi–Sugeno fuzzy models, the fuzzy controller systems and the overall closed-loop fuzzy dynamics are constructed. A sufficient condition is firstly established on the stochastic stability using stochastic Lyapunov–Krasovskii functional. Then in terms of linear matrix inequalities techniques, the sufficient conditions on the existence of mode-dependent state feedback L₂–L_∞ fuzzy controller are presented and proved respectively for constant and time varying case. Finally, the design problems are formulated as optimization algorithms. Simulation results are exploited to illustrate the effectiveness of the developed techniques.     

Zero dynamics and funnel control of linear differential-algebraic systems

We study the class of linear differential-algebraic m-input m-output systems which have a transfer function with proper inverse. A sufficient condition for the transfer function to have proper inverse is that the system has ??strict and non-positive relative degree??. We present two main results: first, a so-called ??zero dynamics form?? is derived; this form is??within the class of system equivalence??a simple form of the DAE; it is a counterpart to the well-known Byrnes?CIsidori form for ODE systems with strictly proper transfer function. The ??zero dynamics form?? is exploited to characterize structural properties such as asymptotically stable zero dynamics, minimum phase, and high-gain stabilizability. The zero dynamics are characterized by (A, E, B)-invariant subspaces. Secondly, it is shown that the ??funnel controller?? (that is a static non-linear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.     

Robust H∞ control of uncertain linear impulsive stochastic systems

This paper develops robust stability theorems and robust H_∞ control theory for uncertain impulsive stochastic systems. The parametric uncertainties are assumed to be time varying and norm bounded. Impulsive stochastic systems can be divided into three cases, namely, the systems with stable/stabilizable continuous‐time stochastic dynamics and unstable/unstabilizable discrete‐time dynamics, the systems with unstable/unstabilizable continuous dynamics and stable/stabilizable discrete‐time dynamics, and the systems in which both the continuous‐time stochastic dynamics and the discrete‐time dynamics are stable/stabilizable. Sufficient conditions for robust exponential stability and robust stabilization for uncertain impulsive stochastic systems are derived in terms of an average dwell‐time condition. Then, a linear matrix inequality‐based approach to the design of a robust H_∞ controller for each system is presented. Finally, the numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.