Scheduled control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies

Attenuation of sinusoidal disturbances with uncertain yet online measurable frequencies is considered. The disturbances are modeled as the outputs of an undisturbed parameter-dependent exogenous system with a skew-symmetric system matrix, obtained in response to nonzero initial conditions. The problem is formulated for a parameter-dependent plant as the synthesis of a parameter-dependent controller in a way to ensure internal stability as well as a desired level of steady-state disturbance attenuation in the face of all admissible parameter variations. The solvability of this problem is first related to the existence of bounded solutions to a matrix differential regulator equation subject to an asymptotic norm constraint. Reformulating this as a parameter-dependent state-feedback like synthesis, based on which suitable solutions to the differential regulator equation can be obtained online, tractable solvability conditions are then provided in the form of parameter-dependent matrix inequalities. Controllers that solve the generalized asymptotic regulation problem are also parameterized in terms of the suitable solutions of the differential regulator equation and some free parameter-dependent matrices that are to be designed off-line to ensure stability. A procedure is then developed to design the free parameters in a way to achieve desirable transient behavior. The use of the developed synthesis procedure is illustrated on a simplified version of the course control problem in ship steering.     

On the properties of the identically singular Lagrange problem

We consider the identically singular Lagrange problem of the calculus of variations. It is investigated how the conjugate points and the Jacobi equation index are related to the solvability conditions of the appropriate matrix Riccati equation and the reducibility of the functional to a perfect square. The criterion of the trajectory variation smallness at a small variation of the functional in the neighborhood of the extremal is obtained.     

基于观测器的不确定广义时滞系统保成本控制

讨论了基于观测器的一类不确定广义时滞系统的保成本控制问题,给出了系统保成本控制器的设计方法.控制器的构造使得闭环系统是鲁棒稳定的.利用线性矩阵不等式以及Lyapunov函数方法,给出了鲁棒保成本控制器存在的充分条件以及相应的可保成本指标,并证明了这个条件等价于一组线性矩阵不等式(LM Is)的可解性问题,同时推广了已知的相关结果.最后以算例验证了设计方法的有效性.     

Solvability and asymptotic behavior of generalized Riccatiequations arising in indefinite stochastic LQ controls

The optimal control problem in a finite time horizon with an indefinite quadratic cost function for a linear system subject to multiplicative noise on both the state and control can be solved via a constrained matrix differential Riccati equation. In this paper, we provide general necessary and sufficient conditions for the solvability of this generalized differential Riccati equation. Furthermore, its asymptotic behavior is investigated along with its connection to the generalized algebraic Riccati equation associated with the linear quadratic control problem in finite time horizon. Examples are presented to illustrate the results established     

H performance of preview control systems

Performance of the preview control is investigated in terms of H^∞-criterion. In the infinite-horizon time setting, the limitation of H^∞ performance in preview control action is characterized with the Hamiltonian matrix. By employing the property of Hamiltonian matrix, the solvability and analytic solution for the preview full-information control problem are clarified based on an auxiliary introduced matrix Riccati equation. The property of resulting control law is illustrated with a numerical example.     

Analytical synthesis of robust controllers of the given accuracy under external perturbations

An approach that allows finding a set of robust statistic feedback controllers that ensure the given control accuracy for the controlled parameters in a multiconnected control system under bounded parametric and external perturbations of various classes is proposed. To describe the set of controllers of the given accuracy, the results of analytically solving the matrix equations based on matrix canonization and parametrization of the Lyapunov equation are used. The solvability conditions of the synthesis problem ensure the robustness and optimal suppression of the perturbations. They are homogeneously represented as linear matrix inequalities obtained based on the Lyapunov functions. A methodological example is given.     

Guaranteed roll-off in a class of high-gain feedback design problems

A number of synthesis problems associated with (almost) disturbance decoupling by state or measurement feedback is considered. Starting from a mathematical definition of the notion of high-frequency roll-off, known results on the solvability of these problems are generalized to the situation in which we require their solvability together with a certain guaranteed roll-off between disturbance and control. The conditions are formulated in terms of the solvability and approximate solvability of certain matrix equations in rational functions.     

Delay‐dependent exponential H ∞ filtering for discrete‐time switched delay systems

In this paper, the problem of delay‐dependent exponential H _∞ filtering for discrete‐time switched delay systems is investigated under average dwell time switching signals. Time delay under consideration is interval time‐varying in the states. By introducing a proper factor to construct a novel Lyapunov‐Krasovskii function and using average dwell time approach, sufficient conditions for the solvability of this problem, dependent on the upper and lower bounds of time‐varying delay, are obtained in terms of linear matrix inequalities. A numerical example is presented to demonstrate the effectiveness of the developed results. Copyright © 2011 John Wiley & Sons, Ltd.     

Robust reliable control of uncertain nonlinear stochastic systems subject to actuator fault

This paper addresses the issue of robust reliable stabilization for a class of uncertain nonlinear stochastic systems with both discrete and distributed time-varying delays and possible occurrence of actuator faults. By constructing a new Lyapunov functional and using linear matrix inequality technique, a new set of sufficient conditions is established for the stochastic stability of the uncertain nonlinear stochastic systems. Then, sufficient conditions are obtained for the solvability of the robust stabilization problem via robust reliable controller. More precisely, the derived control law guarantees the robust stabilization of nonlinear stochastic systems in the presence of known actuator failure matrix and uncertainties. Further, the results are extended to study the stabilization of stochastic systems with unknown actuator failure matrix. Moreover, the obtained criteria are formulated in terms of LMIs and also the reliable controller can be designed in terms of the solutions to certain linear matrix inequalities. Finally, numerical examples with simulation result are presented to demonstrate the validity and less conservatism of the obtained results.     

Assigning Frequencies via Determinantal Equations: New Counterexamples and Invariants

The problem of arbitrary pole-zero assignment by fixed structure compensators has been addressed so far in terms of either necessary or sufficient conditions. The strongest existing condition is based on the rank of the differential of a related map on a degenerate controller and holds true generically when the degrees of freedom of the compensator exceed the number of frequencies to be assigned (mp>n for the output feedback problem). The complete (nongeneric) solvability of the problem is still open, even when complex controllers are considered. A simple necessary solvability condition is that the linear map, appearing as a factor of the main determinantal map, defining the assignment problem, is onto. Here we examine the special problem of assignment of matrix pencil zeros via diagonal perturbations and we present a new necessary and sufficient condition for complex solvability in terms of a new invariant involving the minors of this linear map. Based on this result, we demonstrate that for the important case where the degrees of freedom of the controller are equal to the number of frequencies to be assigned, the surjectivity of this linear map although it is necessary, it is not sufficient for the solvability of the problem     

Quadratic stabilization of a collection of linear systems

The paper considers the problem of simultaneous quadratic (SQ) stablizability of a finite collection linear time-invariant (LTI) system by a static state feedback controller. At first, the solvability conditions for SQ stablization are derived in terms of the solution of certain reduced order matrix inequalities. The existence conditions for the solution of such matrix inequalities are then investigated. Based on that, two new classes of systems are characterized for which a SQ stabilization problem is solvable. These class of systems are (1) partially commutative systems and (2) partially normal systems. These systems are shown to be different from matched uncertain systems and also do not possess any generalised antisymmetric configurations. Both existence and computational algorithm for designing a state feedback controller are given.     

Robust Control of Permanent Magnet Synchronous Motors

In this paper, permanent magnet synchronous motors (PMSMs) are investigated. According to the feature of PMSMs, a novel state equation of PMSMs is obtained by choosing suitable state variables. Based on the state equation, robust controllers are designed via interval matrix and PI control idea. In terms of bilinear matrix inequations, sufficient conditions for the existence of the robust controller are derived. In order to reduce the conservation and the dependence on parameter, the control inputs of PMSMs are divided into two parts, a feedforward control input and a feedback control input, and relevant sufficient conditions for the existence of the controller are obtained. Because of the suitable choice of state variables, the proposed control strategies can cope with the load uncertainty and have robustness for disturbance. Finally, simulations are carried out via Matlab/Simulink soft to verify the effectiveness of the proposed control strategies. The performance of the proposed control strategies are demonstrated by the simulation results.      

Almost disturbance decoupling with internal stability: frequencydomain conditions

The problem is that of determining a dynamic measurement feedback which makes the H^∞-norm of the disturbance-input-to-regulated-output transfer matrix arbitrarily small while achieving internal stability. it is shown that the solvability condition in the frequency domain for this problem is a purely algebraic one and can be formulated in terms of a two-sided matrix matching equation involving polynomial system matrices. This is known to be a zero cancellation condition. A synthesis procedure for the compensator in the frequency domain is also given     

Decentralized stabilization for expanding construction oflarge-scale systems

A decentralized stabilization problem compatible with expanding the construction of large-scale systems is considered. It is assumed that a large-scale system is obtained by adding new subsystems one after another. When a new subsystem is connected to an already interconnected portion, a local controller is designed for the new subsystem so that both the subsystem and the resultant expanded system are stable. The proper-stable-factorization approach is used for calculating the local controllers. Sufficient conditions on subsystems for such stabilization are presented. They are expressed in terms of the solvability of linear matrix equations over the ring of stable rational matrices     

Consensus of a class of second‐order nonlinear heterogeneous multi‐agent systems with uncertainty and communication delay

In this paper, a consensus problem is studied for a group of second‐order nonlinear heterogeneous agents with non‐uniform time delay in communication links and uncertainty in agent dynamics. We design a class of novel decentralized control protocols for the consensus problem whose solvability is converted into stability analysis of an associated closed‐loop system with uncertainty and time delay. Using an explicitly constructed Lyapunov functional, the stability conditions or the solvability conditions of the consensus problem are given in terms of a set of linear matrix inequalities apart from a small number of scalar parameters that appear nonlinearly. Furthermore, the linear matrix inequalities are theoretically verified to be solvable when the communication delay is sufficiently small. The effectiveness of the proposed control protocol is illustrated by numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.     

Reprint of “The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions”

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.     

广义离散系统的状态反馈鲁棒H∞控制

针对一类范数有界参数不确定性的广义离散线性系统,研究了该系统的状态反馈鲁棒H∞控制问题。利用线性矩阵不等式（LMI）的方法,得到了问题可解的条件,并给出了相应的状态反馈控制律。在一定条件下,所得的状态反馈鲁棒H∞控制律使广义离散线性系统对所有容许的不确定性参数,能够保证闭环系统正则、具有因果关系并且渐进稳定,同时其传递函数矩阵能够满足给定的H∞性能指标。正常离散线性系统的相对应结果可作为本文结果的特殊形式。     

Game theory approach to state and input simultaneous estimation for discrete-time LTV systems

This article aims to provide a simple approach realising state observation and input estimation simultaneously for discrete-time LTV systems in H_∞ setting. Through solving a two-player zero sum differential game, appealing results are obtained in two folds. First, necessary and sufficient solvability conditions for state and input simultaneous estimation problem are given in terms of solution to a set of difference Riccati recursion. Second, one estimator is presented with special innovation structure, where innovation information is used to update state observation tuned by gain matrix and to provide input estimation through a projector matrix, where gain matrix and projector matrix are constructed from solution to difference Riccati recursion. At last, simulation results are provided to justify proposed approach.     

Reliable filtering with extended dissipativity for uncertain systems with discrete and distributed delays

This paper deals with the problem of reliable filter with extended dissipativity for uncertain systems with discrete and distributed delays, and sensor-failure model of this system is assumed to be concerned with Markovian behaviour. First, based on a novel Lyapunov–Krasovskii functional, a sufficient condition, which ensures that the filtering error system is stochastically stable and extended dissipative, is obtained. Second, mode-dependent conditions for the solvability to the reliable filter with extended dissipativity problem are given in terms of linear matrix inequalities (LMIs). The desired filter parameters can be derived by using feasible solutions to the presented LMIs. Finally, two numerical examples are given to illustrate the effectiveness of the filter design method.