A measure of worst-case H∞ performance and of largest acceptable uncertainty

The structured singular value (SSV or μ) is known to be an effective tool for assessing robust performance of linear time-invariant models subject to structured uncertainty. Yet all single μ analysis provides is a bound β on the uncertainty under which stability as well as H_∞ performance level of κ/β are guaranteed, where κ is preselectable. In this paper, we introduce a related quantity ν which provides answers for the following questions: (i) given β, determine the smallest with the property that, for any uncertainty bounded by β, an H∞ performance level of is guaranteed; (ii) conversely, given , determined the largest β with the property that, again, for any uncertainty bounded by β, an H_∞ performance level of is guaranteed. Properties of this quantity are established and approaches to its computation are investigated. Both unstructured uncertainty and structured uncertainty are considered.     

An LMI approach to robust H-infinity control for uncertain singular time-delay systems

The problem of robust H-infinity control for a class of uncertain singular time-delay systems is studied in this paper. A new approach is proposed to describe the relationship between slow and fast subsystems of singular time- delay systems, based on which, a sufficient condition is presented for a singular time-delay system to be regular, impulse free and stable with an H-infinity performance. The robust H-infinity control problem is solved and an explicit expression of the desired state-feedback control law is also given. The obtained results are formulated in terms of strict linear matrix inequalities （LMIs） involving no decomposition of system matrices. A numerical example is given to show the effectiveness of the proposed method.     

An LMI approach to robust H-infinity control for uncertain singular time-delay systems

The problem of robust H-infinity control for a class of uncertain singular time-delay systems is studied in this paper. A new approach is proposed to describe the relationship between slow and fast subsystems of singular time- delay systems, based on which, a sufficient condition is presented for a singular time-delay system to be regular, impulse free and stable with an H-infinity performance. The robust H-infinity control problem is solved and an explicit expression of the desired state-feedback control law is also given. The obtained results are formulated in terms of strict linear matrix inequalities (LMIs) involving no decomposition of system matrices. A numerical example is given to show the effectiveness of the proposed method.     

Singular controls in a given stepwise control problem

The Goursat-Darboux stepwise problem of systems control, i.e., the multi-stage problem of control of systems with distributed parameters, is considered. Second-order formulas for the increment in the quality functional are constructed. In investigating these formulas on needle variations in the control, a number of necessary Pontryagin maximum principle-type optimality conditions is obtained and a singular case is studied.     

New results for near-optimal control of linear multiparameter singularly perturbed systems

In this paper, we consider the linear quadratic optimal control problem for multiparameter singularly perturbed systems in which N lower-level fast subsystems are interconnected through a higher-level slow subsystem. Different from the existing methods, a new method is developed to design a near-optimal controller which does not depend on the unknown small parameters. It is shown that the resulting controller in fact achieves an O(||μ||²) approximation to the optimal cost of the original optimal control problem.     

Decentralized control for two time-scale discrete-time systems

Output feedback design of discrete-time decentralized systems with slow and fast modes is considered. Conditions for the complete separation of slow and fast subsystems are given. The slow and fast subsystem outputs, which are obtained by applying the slow and fast subcontrollers to the corresponding subsystems, will be shown to approximate those of the original system. Also, the composite control, when being applied to the original system, will place the eigenvalues sufficiently close to the desired locations.     

Modelling of non-minimum phase effects in discrete-time norm optimal iterative learning control

The subject of this article is the modelling of the influence of non-minimum phase discrete-time system dynamics on the performance of norm optimal iterative learning control (NOILC) algorithms with the intent of explaining the observed phenomenon and predicting its primary characteristics. It is established that performance in the presence of one or more non-minimum phase plant zeros typically has two phases. These consist of an initial fast monotonic reduction of the L ₂ error norm (mean square error) followed by a very slow asymptotic convergence. Although the norm of the tracking error does eventually converge to zero, the practical implications over a finite number of trials is apparent convergence to a non-zero error. The source of this slow convergence is identified using the singular value distribution of the system's all pass component. A predictive model of the onset of slow convergence behaviour is developed as a set of linear constraints and shown to be valid when the iteration time interval is sufficiently long. The results provide a good prediction of the magnitude of error norm where slow convergence begins. Formulae for this norm and associated error time series are obtained for single-input single-output systems with several non-minimum phase zeros outside the unit circle using Lagrangian techniques. Numerical simulations are given to confirm the validity of the analysis.     

Singularity perturbed large-scale distributed-parameter control systems: an application to nuclear reactor control

A class of distributed-parameter control systems involving interacting fast and slow subsystems is considered and treated by an extension of the singular perturbation method well known in the field of lumped-parameter large scale systems theory. In particular, the method is applied to a power nuclear reactor with one fast, one slow and one delayed neutron group, where, due to that the velocities of fast neutrons are extremely high, a very small parameter multiplies the partial time derivative of the fast neutron flux. The method is useful for the treatment of practical nuclear reactor control problems which require the Riccati equation as an intermediate step.     

含正弦扰动奇异摄动时滞系统的最优减振控制

研究奇异摄动时滞系统在正弦扰动下的最优减振控制问题．基于奇异摄动的快慢分解理论,将原最优控制问题转化为无时滞快子问题和受扰线性时滞慢子问题,通过摄动法和前馈补偿技术求解时滞慢子系统的最优控制问题,得到了系统的前馈反馈组合控制(FFCC)律及其存在唯一性条件．FFCC律由线性解析项和共态向量无穷级数和表示的时滞补偿项组成,其中线性解析项可通过求解Riccati方程和Sylvester方程得到,时滞补偿项通过递推求解共态向量方程得到,仿真算例表明了方法的有效性．     

Decomposition of problems of optimal control and estimation for discrete systems with fast and slow variables

Consideration was given to the linear-quadratic problem of optimal control for the discrete linear system with fast and slow variables under incomplete information about system state. Decomposition of the discrete matrix Riccati equations was carried out. The proposed decomposition algorithm relies on a geometrical approach using the properties of the invariant manifolds of slow and fast motions of the nonlinear multirate discrete systems as basis. The splitting transformation was constructed in the form of asymptotic decomposition in the degrees of a small parameter.     

不确定离散广义系统的鲁棒严格无源控制

利用线性矩阵不等式方法讨论了不确定离散线性广义系统的严格无源性和严格无源控制问题。通过引入松驰变量来描述广义系统的快变子系统和慢变子系统之间的代数关系,给出一个新的保证离散广义系统正则、因果、稳定且严格无源的充分条件,该条件表示为严格线性矩阵不等式的形式,不涉及系统状态矩阵的分解问题。然后利用这一条件,给出了状态反馈鲁棒严格无源控制器的设计方法。仿真实例说明了该方法的有效性。     

Low dimensional disturbance observer‐based control for nonlinear parabolic PDE systems with spatio‐temporal disturbances

In this paper, a design problem of low dimensional disturbance observer‐based control (DOBC) is considered for a class of nonlinear parabolic partial differential equation (PDE) systems with the spatio‐temporal disturbance modeled by an infinite dimensional exosystem of parabolic PDE. Motivated by the fact that the dominant structure of the parabolic PDE is usually characterized by a finite number of degrees of freedom, the modal decomposition method is initially applied to both the PDE system and the PDE exosystem to derive a low dimensional slow system and a low dimensional slow exosystem, which accurately capture the dominant dynamics of the PDE system and the PDE exosystem, respectively. Then, the definition of input‐to‐state stability for the PDE system with the spatio‐temporal disturbance is given to formulate the design objective. Subsequently, based on the derived slow system and slow exosystem, a low dimensional disturbance observer (DO) is constructed to estimate the state of the slow exosystem, and then a low dimensional DOBC is given to compensate the effect of the slow exosystem in order to reject approximately the spatio‐temporal disturbance. Then, a design method of low dimensional DOBC is developed in terms of linear matrix inequality to guarantee that not only the closed‐loop slow system is exponentially stable in the presence of the slow exosystem but also the closed‐loop PDE system is input‐to‐state stable in the presence of the spatio‐temporal disturbance. Finally, simulation results on the control of temperature profile for catalytic rod demonstrate the effectiveness of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Deterministic and stochastic control of discrete-time bilinear systems

Optimal control of a class of time invariant single-input, discrete bilinear systems is investigated in this paper. Both deterministic and stochastic problems are considered.

In the deterministic problem, for the initial state in a certain set ∑₀, the solution is the same as the solution to the associated linear system. The optimal path may be a regular path or a singular path.

The stochastic control problem is considered with perfect state observation, and additive and multiplicative noise in the state equation. It is demonstrated that the presence of noise simplifies the analysis compared to that in the determinstic case.     

An MPC approach to the design of two-layer hierarchical control systems

A methodology for the design of two-layer hierarchical control systems is presented. The high layer corresponds to a system with slow dynamics, whose control inputs must be provided by subsystems with faster dynamics placed at the low layer. Model Predictive Control laws are synthesized for both layers and overall convergence properties are established. The use of different control configurations is also considered by allowing the switching on/off of the subsystems at the low layer. A simulation example is reported to witness the potentialities of the proposed solution.     

Time-scale separation and robust controller design for uncertain nonlinear singularly perturbed systems under perfect state measurements

We study time-scale separation and robust controller design for a class of singularly perturbed nonlinear systems under perfect state measurements. The system dynamics are taken to be jointly linear in the fast state variables, control and disturbance inputs, but nonlinear in the slow state variables. Since global timescale separation may not always be possible for nonlinear singularly perturbed systems, we restrict our attention here to some closed subset of the state space, on which a timescale separation holds for sufficiently small values of the singular perturbation parameter. We construct a slow controller and a composite controller based on the solutions of particular slow and fast games obtained using time-scale separation. For the class of systems for which the slow controller can be selected to be robust with respect to small regular structural perturbations on the slow subsystem, we show under some growth conditions that the composite controller can achieve any desired level of performance that is larger than the maximum of the performance levels for the slow and fast subsystems,. A slow controller, however, is not generally as robust as the composite controller; but, still under some conditions which are delineated in the paper, the fast dynamics can be totally ignored. The paper also presents a numerical example to illustrate the theoretical results.     

Synchronization of singular switched complex networks via impulsive control with all nonsynchronized subnetworks

This paper studies impulsive synchronization of a class of nonlinear singular switched complex networks (SSCNs). With the aid of discretized Lyapunov function method and the matrix generalized inverse technique, an impulsive controller is designed, which works on both the slow and the fast state variables to synchronize SSCNs at impulsive instant. Based on the designed controller, it is shown that the synchronization of SSCNs is achieved in two cases: all subnetworks that are not self‐synchronized and all subnetworks that are self‐synchronized. Moreover, an example is given to validate the effectiveness of the proposed controller.     

On the model-based networked control for singularly perturbed systems

In this paper, the control of a two-time-scale plant, where the sensor is connected to a linear controller/actuator via a network is addressed. The slow and fast systems of singularly perturbed systems are used to produce an estimate of the plant state behavior between transmission times, by which one can reduce the usage of the network. The approximate solutions of the whole systems are derived and it is shown that the whole systems via the network control are generally asymptotically stable as long as their slow and fast systems are both stable. These results are also extended to the case of network delay.     

分数布朗运动干扰下广义随机系统滑模控制

本文考虑分数布朗运动干扰下时滞广义随机系统的基于观测器的滑模控制.首先设计了不受分数布朗运动干扰的状态观测器,然后给出了基于观测器的积分型滑模面函数的定义.为了研究观测器系统有限时间随机有界性,构造了带有二重积分的新型Lyapunov函数来处理分数布朗运动.利用奇异值分解原理,解决了观测器增益矩阵设计问题.利用线性矩阵不等式和Gronwall不等式得到了系统随机有界性的充分条件.同时给出了滑模面函数的有限时间可达性分析.最后的数值仿真验证了所提方法的有效性.     

Stable indirect adaptive interval type-2 fuzzy sliding-based control and synchronization of two different chaotic systems

An indirect approach to adaptive interval type-2 fuzzy sliding mode control is proposed for the stable synchronization of two different chaotic nonlinear systems with different initial conditions under the presence of uncertainties involving process noises and external disturbances. The indirect model-based approach to adaptation is promoted here as a more suitable strategy for the fast changes that occurs in chaotic systems. In other words, the usual direct adaptive strategies may be too slow to respond to the inherently fast changing dynamics of chaotic systems. Using Lyapunov analysis, the sliding mode approach illustrates the asymptotic convergence of synchronization error to zero as well as good robustness against external disturbances. The interval type-2 structure aims to remedy the undesirable chattering phenomenon that is common in most conventional sliding mode control applications. It also provides a more effective equivalent model in the indirect approach, which leads to improved handling of the chaotic variations and uncertainties. Two numerical pairs of chaotic systems, i.e. the Lorenz and Chen’s systems and the Rössler system and modified Chua’s circuit are considered. In particular, in comparison with its type-1 fuzzy counterpart, the control effort is reduced by an average of 26.25% and 17.4% for the synchronization of the two corresponding systems, respectively. Furthermore, the integral of squared error is also improved by an average of 27.2% and 25.33%. This is while convergence time is reduced to less than 0.5 s and 1.5 s.     

A general analysis and control framework for process systems with inventory recycling

In this work, we generalize our previous results concerning the impact of material recycling and energy recovery on plant dynamics and control. We define a generic class of integrated process systems, in which an extensive quantity that obeys conservation laws is recovered from the process output and recycled to the process feed; the operation of the system is assumed to be subject to time‐varying, measurable disturbances. We establish, in this general case, that integration is conducive to the emergence of a two‐time‐scale dynamic behavior and derive reduced‐order models for the dynamics in each time scale. Subsequently, we postulate a hierarchical control framework that exploits these dynamics results in the design of coordinated fast and slow feedback/feedforward controllers and formulate a stability result for the closed‐loop system. We demonstrate these concepts on a case study concerning an energy‐integrated process. Copyright © 2013 John Wiley & Sons, Ltd.