Characteristic polynomial assignment for plants with semialgebraic uncertainty: A robust diophantine equation approach

In this paper, we address the problem of robust characteristic polynomial assignment for LTI systems whose parameters are assumed to belong to a semialgebraic uncertainty region. The objective is to design a dynamic fixed‐order controller in order to constrain the coefficients of the closed‐loop characteristic polynomial within prescribed intervals. First, necessary conditions on the plant parameters for the existence of a robust controller are reviewed, and it is shown that such conditions are satisfied if and only if a suitable Sylvester matrix is nonsingular for all possible values of the uncertain plant parameters. The problem of checking such a robust nonsingularity condition is formulated in terms of a nonconvex optimization problem. Then, the set of all feasible robust controllers is sought through the solution to a suitable robust diophantine equation. Convex relaxation techniques based on sum‐of‐square decomposition of positive polynomials are used to efficiently solve the formulated optimization problems by means of semidefinite programming. The presented approach provides a generalization of the results previously proposed in the literature on the problem of assigning the characteristic polynomial in the presence of plant parametric uncertainty. Copyright © 2014 John Wiley & Sons, Ltd.     

一类不确定系统的鲁棒D-稳定保守性研究

针对一类多面体不确定线性连续系统,研究了系统鲁棒D-稳定问题.为降低设计的保守性,引入参数相关Lyapunov函数,给出了系统鲁棒D-稳定的充分条件.通过求解一组线性矩阵不等式,得到了系统鲁棒D-稳定的状态反馈控制器,使得闭环系统鲁棒D-稳定.对某卫星姿态控制系统的仿真结果表明了该方法的有效性.     

不确定时滞系统的鲁棒H∞控制

基于Lyapunov稳定性理论,研究一类同时具有状态非线性不确定和线性不确定时变时滞系统的鲁棒控制器的设计问题。在非线性不确定性满足增益有界条件下,得到该类时变时滞系统依赖于时变延迟导数的最大值的满足鲁棒H∞性能的一个充分条件。通过求解一个线性矩阵不等式-LMI,即可获得鲁棒H∞控制器。给出的两个具体算例说明该方法的有效性。     

多项式理论在导弹稳定性能评估中的应用

针对线性参数不确定性系统的鲁棒稳定性问题,提出一种新的分析方法,并对其在导弹自动驾驶仪鲁棒稳定性评估中的可行性进行了研究。首先给出线性参数不确定性系统在其特征多项式系数不相关情况下的鲁棒稳定性的充要条件,并通过自适应网格划分算法将此条件与鲁棒D-稳定性理论结合,得到基于多项式理论的评估算法。将算法用于导弹自动驾驶仪鲁棒稳定性评估,得到了不同攻角下导弹在全包线范围内的稳定区域。和只能在离散点进行评估的传统评估方法比较,结果表明提出的算法可以在全包线范围内连续进行评估,从而发现一些隐蔽的不稳定区域。     

Stability and Stabilization of Fractional‐Order Systems with Different Derivative Orders: An LMI Approach

Stability and stabilization analysis of fractional‐order linear time‐invariant (FO‐LTI) systems with different derivative orders is studied in this paper. First, by using an appropriate linear matrix function, a single‐order equivalent system for the given different‐order system is introduced by which a new stability condition is obtained that is easier to check in practice than the conditions known up to now. Then the stabilization problem of fractional‐order linear systems with different fractional orders via a dynamic output feedback controller with a predetermined order is investigated, utilizing the proposed stability criterion. The proposed stability and stabilization theorems are applicable to FO‐LTI systems with different fractional orders in one or both of 0 <  α  < 1 and 1 ≤  α  < 2 intervals. Finally, some numerical examples are presented to confirm the obtained analytical results.     

Exponential stability of impulsive systems with application to uncertain sampled-data systems

We establish exponential stability of nonlinear time-varying impulsive systems by employing Lyapunov functions with discontinuity at the impulse times. Our stability conditions have the property that when specialized to linear impulsive systems, the stability tests can be formulated as Linear Matrix Inequalities (LMIs). Then we consider LTI uncertain sampled-data systems in which there are two sources of uncertainty: the values of the process parameters can be unknown while satisfying a polytopic condition and the sampling intervals can be uncertain and variable. We model such systems as linear impulsive systems and we apply our theorem to the analysis and state-feedback stabilization. We find a positive constant which determines an upper bound on the sampling intervals for which the stability of the closed loop is guaranteed. The control design LMIs also provide controller gains that can be used to stabilize the process. We also consider sampled-data systems with constant sampling intervals and provide results that are less conservative than the ones obtained for variable sampling intervals.     

降维观测器状态反馈系统的鲁棒容错控制

该文研究一类不确定参数的状态反馈系统的鲁棒容错控制问题，即设计反馈控制器，使闭环系统对可允的参数摄动具有鲁棒性，同时对传感器失效具有完整性。采用Luenberger降维观测器实现了系统状态估计，解决了系统状态不易直接测得的问题。基于Riccati方程和Lyapunov渐近稳定性理论讨论了线性不确定系统在故障情况下的鲁棒镇定问题，给出了传感器失效情况下，系统仍能保持稳定的充分条件，并在此基础上给出了鲁棒容错控制器的设计方法，以设计实例说明了文中设计方法的有效性。     

Delta算子时滞系统的可靠D--镇定

研究同时含有执行器故障和传感器故障的Delta算子时滞系统的鲁棒可靠D-镇定问题.利用矩阵特征值理论和线性矩阵不等式方法,给出了Delta算子时滞系统D-稳定的充分条件,进而,针对同时含有执行器故障和传感器故障的Delta算子范数有界参数不确定时滞系统,设计鲁棒可靠D-镇定状态反馈控制器,使得对于所有容许的不确定性、执行器故障和传感器故障,Delta算子闭环时滞系统的极点均落在复平面指定圆盘区域内.最后,通过数值算例验证了控制器设计方法的可行性和有效性.     

Decentralized tracking control for linear uncertain interconnected systems with time delays

This paper investigates the robust output tracking problem for a class of large‐scale linear uncertain systems with interactions and time delays. Time delays exist in both states and controls. Using the Riccati equation, a procedure for determining decentralized linear control laws is presented such that the closed‐loop system asymptotically tracks the reference output and rejects any constant but unknown disturbances. The main feature of this approach is that the uncertain systems may contain time delays in both states and controls as well as in interactions between subsystems. A numerical example is included to show the results. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Fractional‐Order Dynamic Output Feedback Sliding Mode Control Design for Robust Stabilization of Uncertain Fractional‐Order Nonlinear Systems

A robust fractional‐order dynamic output feedback sliding mode control (DOF‐SMC) technique is introduced in this paper for uncertain fractional‐order nonlinear systems. The control law consists of two parts: a linear part and a nonlinear part. The former is generated by the fractional‐order dynamics of the controller and the latter is related to the switching control component. The proposed DOF‐SMC ensures the asymptotical stability of the fractional‐order closed‐loop system whilst it is guaranteed that the system states hit the switching manifold in finite time. Finally, numerical simulation results are presented to illustrate the effectiveness of the proposed method.     

一类非等阶分数阶非线性系统的神经网络控制

讨论一类不确定非线性分数阶非等阶（noncommensurate）的系统的控制问题。假设系统含的不确定包括正实不确定（positive real uncertainty）项和非线性函数完全未知,首先利用RBF神经网络近似未知非线性函数,再基于系统的连续频率分布模型将分数阶系统转化为等价的无穷维分布状态变量的整数阶系统,结合间接Lyapunov方法及线性矩阵不等式（LMI）方法,给出了系统鲁棒渐近稳定的充分条件。理论和实例仿真验证了方法的有效性。     

A Lyapunov functional approach to robust stability analysis of continuous-time uncertain linear systems in polytopic domains

In this paper, a sufficient linear matrix inequality (LMI) condition is presented for robust stability analysis of continuous-time linear time-invariant (LTI) systems in polytopic domains. The underlying idea behind the proposed approach is to introduce a family of complex functions which map the closed right-hand side of the complex plane into the inside of the closed unit circle centered at the origin. Then, the mapping properties are used to assure that all the eigenvalues of a system are located in the open left-hand side of the complex plane. Examples show the validity of the proposed condition.     

Robustness of linear uncertain sampled‐data control systems with generalized sampled‐data hold functions

This paper investigates robust stability of linear time‐invariant (LTI) uncertain sampled‐data control systems with generalized sampled‐data hold functions (GSHFs). A new sufficient condition for robust stability of such systems is developed. Unlike that of most of the previous works, it directly uses the data of the continuous‐time plant and therefore it is less conservative. The condition is expressed in terms of the spectral radius of a certain matrix and is shown to be a unimodal function of a free parameter. This property enabled us to use standard one‐dimensional optimization algorithm to perform the proposed test. Copyright © 2006 John Wiley & Sons, Ltd.     

LMI conditions for quadratic and robust D-stability of interval systems

Sufficient conditions for the quadratic D-stability and further robust D-stability of interval systems are presented in this paper. This robust D-stability condition is based on a parameter-dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities （LMIs） defined at a series of partial-vertex-based interval matrices other than the total vertex matrices as in previous results. The results contain the usual quadratic and robust stability of continuous-time and discrete-time interval systems as particular cases. The illustrative example shows that this method is effective and less conservative for checking the quadratic and robust D-stability of interval systems.     

Robust stability analysis of uncertain multiorder fractional systems: Young and Jensen inequalities approach

Robust stability analysis of multiorder fractional linear time‐invariant systems is studied in this paper. In the present study, first, conservative stability boundaries with respect to the eigenvalues of a dynamic matrix for this kind of systems are found by using Young and Jensen inequalities. Then, considering uncertainty on the dynamic matrix, fractional orders, and fractional derivative coefficients, some sufficient conditions are derived for the stability analysis of uncertain multiorder fractional systems. Numerical examples are presented to confirm the obtained analytical results.     

基于线性矩阵不等式的不确定关联系统的分散鲁棒镇定

应用线性矩阵不等式（LMI）方法研究不确定性关联大系统的分散便棒镇定问题。系统中不稳定项具有数值界,可不满足匹配条件。基于不确定项的表达形式,给出了其可分散状态反馈镇定的充分条件,即一组LMIs有解。在此基础上,通过求第一凸优化问题,提出了具有较小反馈增益的分散稳定化状态反馈控制律的设计方法。仿真示例说明了该方法的有效性和优越性。     

Stability of nonlinear feedback systems in a Volterra representation

Presented in this paper is a stability condition for a class of nonlinear feedback systems where the plant dynamics can be represented by a finite series of Volterra kernels. The class of Volterra kernels are limited to p‐linear stable operators and may contain pure delays. The stability condition requires that the linear kernel is non‐zero and that the closed loop characteristic equation associated with the linearized system is stable. Next, a sufficient condition is developed to upper bound the infinity‐norm of an external disturbance signal thereby guaranteeing that the internal and output signals of the closed loop nonlinear system are contained in L_∞. These results are then demonstrated on a design example. A frequency domain controller design procedure is also developed using these results where the trade‐off between performance and stability are considered for this class of nonlinear feedback systems. Copyright © 2000 John Wiley & Sons, Ltd.     

基于Delta算子系统TS模型的鲁棒稳定性研究

针对一类不确定Delta算子系统,建立其TS模糊模型,并讨论了闭环系统的鲁棒稳定性.首先将一类基于Delta算子描述的系统,表示成TS模糊动态不确定系统形式,然后利用Lyapunov函数法,给出了系统渐近稳定的充分条件.该条件是线性矩阵不等式(LMI)形式的,可以很方便地利用Matlab工具箱进行求解.所得的结果将连续和离散系统统一到Delta算子框架中.数值仿真表明了该方法的有效性和可行性.     

Stabilization of Non‐Linear Fractional‐Order Uncertain Systems

In this paper, we present a stabilization method on the non‐linear fractional‐order uncertain systems. Firstly, a sufficient condition for the robust asymptotic stabilization of the non‐linear fractional‐order uncertain system is presented based on direct Lyapunov approach. Secondly, utilising the matrix's singular value decomposition (SVD) method, the systematic robust stabilization design algorithm is then proposed. Finally, two numerical examples are provided to illustrate the efficiency and advantage of the proposed algorithm.