Input-output stability properties of networked control systems

Results on input-output L/sub p/ stability of networked control systems (NCS) are presented for a large class of network scheduling protocols. It is shown that static protocols and a recently considered dynamical protocol called try-once-discard belong to this class. Our results provide a unifying framework for generating new scheduling protocols that preserve L/sub p/ stability properties of the system if a design parameter is chosen sufficiently small. The most general version of our results can be used to treat NCS with data packet dropouts. The model of NCS and, in particular, of the scheduling protocol that we use appears to be novel and we believe that it will be useful in further study of these systems. The proof technique we use is based on the small gain theorem and it lends itself to an easy interpretation. We prove that our results are guaranteed to be better than existing results in the literature and we illustrate this via an example of a batch reactor.     

Interconnected linear multivariable systems with delays: System properties and input-output stability

We consider an arbitrary interconnection of linear, time-invariant, distributed, multivariable subsystems; each subsystem is represented either by a strictly proper rational transfer function or by a finite sum of pure delay terms. Such models occur in transportation systems, in chemical processes, in production systems, in communication systems, and in models of boilers in nuclear power plants. Allowing inputs to be applied to any interconnection point, we show that, except for a special case, the input-output transfer function of such a system is in a class extensively stuided in the literature. Next we obtain necessary and sufficient conditions for input-output stability. An example illustrates the application of the test.     

基于Volterra级数的离散MIMO非线性闭环系统的稳定性

基于非线性传递函数矩阵的Volterra级数表示,利用多维Z变换,讨论了一类MIMO离散非线性系统的闭环稳定性,给出了直接利用开环稳定性来判别系统闭环稳定性的新方法,并用实例仿真来验证其有效性。     

基于广义频率响应函数的非线性控制系统的闭环稳定性研究

本文基于广义频率响应函数，讨论了一类非线性控制系统的闭环性问题，给出了输入输出频域稳定性判据条件，最后，利用仿真例子对结论进行了验证。     

Uniform approximation and the circle criterion

The input-output maps G of causal time-invariant systems are considered, and a complete characterization of those maps that can be uniformly approximated by the maps of certain simple structures is given. The criterion is that G must satisfy certain continuity and approximately-finite-memory conditions. It is proved that the conditions are satisfied by the input-output maps of control systems of a familiar type containing a sector nonlinearity for which the circle condition for stability is met. In particular, this shows that such feedback systems, with inputs drawn from a certain large set of bounded functions, possess arbitrarily good finite Volterra-series (or radial-basis-function) approximations     

Multipurpose deadbeat controllers for multivariable systems

A simple and direct deadbeat control design algorithm for discrete-time multivariable systems is introduced. Our brief was to synthesize a realizable controller to achieve the following objectives: (1) input-output decoupling, (2) deadbeat tracking of any prespecified class of changeable deterministic reference signals and (3) changeable deterministic disturbances rejection where the changeable reference signals and disturbances can be different at each channel. Since the internal stability requirement is satisfied, our design algorithm can easily handle both unstable and non-minimum phase systems. Some constraints on the transfer function are also derived to make sure that the derived controller is realizable. Since all solutions can be described in parametric form, some of the performance criteria can be combined.     

Direct computation of infimum in discrete-time H∞-optimization using measurement feedback

A direct and non-iterative method for the computation of the infimum for a class of discrete-time H_∞ optimal control problem is considered in this paper. The problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems. The method is applicable to systems where (i) the transfer function from the disturbance input to the measurement output is free of unit circle invariant zeros and left invertible, and (ii) the transfer function from the control input to the controlled output of the given system is free of the unit circle invariant zeros and right invertible.     

Distributed system transfer functions of exponential order

σo is a real number. We construct a transfer function algebra of fractions, viz. F?(σo), for modelling possibly unstable distributed systems such that (i) [fcirc] in F?(σo) is holomorphic in Re s ≧ σo, (i.e. is σo-stable), iff [fcirc] is σo-exponentially stable, and (ii) we allow delay in the direct input-output transmission of the system. This algebra is (a) a restriction of the algebra B?(σo) developed by Callier and Desoer (1978, 1980 a), (b) an extension of the algebra of proper rational functions such that the exponential order properties of the latter transfer functions of lumped systems are maintained. The algebra F? (σo) can be used for modelling and feedback system design. It is shown that standard semigroup systems are better modelled by a transfer function in F?(σo) rather than B?(σo).     

非均匀采样数据系统的新型模型描述方法

提升技术是处理非均匀采样数据（Non-uniformly sampled-data,NUSD）系统的标准工具.然而,提升状态空间模型存在因果约束问题,相应的提升传递函数模型结构复杂,且参数数目过多.因此,它们不便于非均匀采样数据系统的辨识与控制.通过引入时变后移算子,本文提出了一种输入输出表达的新型模型描述方法.该模型能够克服提升系统模型的缺点,使得传统单率系统的辨识和控制方法能够推广到非均匀采样数据系统中.仿真结果表明了新模型的优越性和有效性.     

Optimal control of systems with a unitary semigroup and with colocated control and observation

We solve the quadratic optimal control problem on an infinite time interval for a class of linear systems whose state space is a Hilbert space and whose operator semigroup is unitary. The difficulty is that the systems in this class, having unbounded control and observation operators, may be ill-posed. We show that there is a surprisingly simple solution to the problem (the optimal feedback turns out to be output feedback). Our approach is to use a change of variables which transforms the system into a one which, according to recent research, is known to be conservative. We show that, under a mild assumption, the transfer function of this conservative system is inner, and then it follows that the optimal control of this conservative system is trivial. We give an example with the wave equation on an n-dimensional domain, with Neumann control and Dirichlet observation of the velocity.     

Equivalence relations between learnability,output-dissipativity and strict positive realness

This paper was originally motivated by aiming at explicating why a simple iterative learning control scheme for complicated robot dynamics with strong non-linearities works well in acquiring any given desired motion over a finite or infinite time duration or any periodic motion. To gain a physical insight into the problem, a class of linear dynamical systems with specified input and output of the same dimension is treated by defining two properties: output-dissipativity and learnability. It is then shown that the former implies the latter and furthermore, for a class of linear systems with single input and single output, they are equivalent to each other and each of them is also equivalent to strict positive realness of input-output transfer function. For a class of MIMO (multiple inputs and multiple outputs) systems, it is possible to prove that each of these properties is equivalent to strict positive realness of the input-output transfer function matrix if it is strictly proper or otherwise its direct term from input to output satisfies an extra condtion.     

Analysis Of Linear Iterative Learning Control Schemes Using Repetitive Process Theory

The purposes of this paper are (i) to critically review existing results on the use of the systems theory for repetitive processes in the analysis of a wide class of linear iterative control laws, and (ii) to present some new results on controller design using this general approach. This paper first presents results on the stability and convergence properties of a general class of iterative learning control schemes using, in the main, theory first developed for the subclass of so‐called differential and discrete linear repetitive processes. A general learning law that uses information from the current and a finite number of previous trials is considered and the results are interpreted in terms of basic systems theoretic concepts such as the relative degree and minimum phase characteristics. It is also shown that a number of other approaches reported in the literature are, in fact, special cases of the results obtained in the repetitive process setting. In the second part of the paper, new results on controller design are given based on 2D transfer function matrices together with new results on the robustness of norm optimal iterative learning control schemes.     

Semi-cancellable fractions in system theory

In this paper, we propose to define the transfer function of a system in a new way, namely as a semi-cancellable fraction (SCF), i.e. a fraction in which unstable pole-zero cancellations are not allowed. Within this new framework, it is possible to be rigorous and coherent in dealing with the stabilizability of a system from an input-output viewpoint, not focusing on the stabilizability of infinite dimensional input-output systems     

Energy methods for stability of bilinear systems with oscillatory inputs

A large body of recent literature has been devoted to the topic of motions of mechanical systems forced by oscillatory inputs (see, for example, References 7, 12, 15, 17 and 29). A common feature in most of this work (with Reference 7 being the exception) is that the results apply principally to kinematic problems, and the equations of motion do not involve drift terms. The main object of study in this paper is the class of bilinear control systems with oscillatory (periodic) inputs. We shall show that included in this class are models of mechanical system dynamics which are obtained through a process of reduction and linearization about operating points. In particular, we study the stability of such systems under high-frequency, periodic forcing. The averaged potential is defined for linearizations of the systems on the (reduced) configuration space, and it is shown that stable motions of the forced system are associated with minimum values of this quantity. We use both classical averaging theory as well as a novel geometric argument to provide parallel but independent assessments of the use of the averaged potential in carrying out a stability analysis. A salient feature of the geometric approach is that we are able to justify the use of an energy-like quantity to determine Lyapunov stability in conservative mechanical systems. This makes contact with a growing body of literature on the use of energy methods for stability and control design (e.g. References 7, 8, 18, 24, 26 and 27). We briefly describe the connection with previous work on the averaged potential.     

An input-output view of robustness in adaptive control

The stability and robustness properties of adaptive control systems are examined using input-output stability theory, i.e. passivity and small-gain theory. A generic adaptive error system is developed based on the concept of a tuned system—an ideal converged (nonadaptive) closed-loop system. Using this error system with passivity theory gives conditions for global stability where only boundedness (in norm) is required on the external inputs, e.g. disturbance, reference and initial conditions. Small gain theory is used to develop local stability results where the magnitudes of the external inputs are restricted. In the global case, a particular system operator (not the plant) is required to be strictly-passive, a condition which is unlikely to hold in actual use due to unmodeled dynamics. The local results, however, are not so restricted and allow for unmodeled dynamics. In this latter case an estimate of the stability margin is given under a persistent excitation condition.     

Min-max model predictive control for constrained nonlinear systems via multiple LPV embeddings

A min-max model predictive control strategy is proposed for a class of constrained nonlinear system whose trajectories can be embedded within those of a bank of linear parameter varying (LPV) models. The embedding LPV models can yield much better approximation of the nonlinear system dynamics than a single LTV model. For each LPV model, a parameter-dependent Lyapunov function is introduced to obtain poly-quadratically stable control law and to guarantee the feasibility and stability of the original nonlinear system. This approach can greatly reduce computational burden in traditional nonlinear predictive control strategy. Finally a simulation example illustrating the strategy is presented. Supported by the National Natural Science Foundation of China (Grant Nos. 60774015, 60825302, 60674018), the National High-Tech Research & Development Program of China (Grant No. 2007AA041403), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060248001), and partly by Shanghai Natural Science Foundation (Grant No. 07JC14016)     

Control of discrete time nonlinear systems with a time-varying structure

In this paper, we present a control methodology for a class of discrete time nonlinear systems that depend on a possibly exogenous scheduling variable. This class of systems consists of an interpolation of nonlinear dynamic equations in strict feedback form, and it may represent systems with a time-varying nonlinear structure. Moreover, this class of systems is able to represent some cases of gain scheduling control, Takagi-Sugeno fuzzy systems, as well as input-output realizations of nonlinear systems which are approximated via localized linearizations. We present two control theorems, one using what we call a “global” approach (akin to traditional backstepping), and a “local” approach, our main result, where backstepping is again used but the control law is an interpolation of local control terms. An aircraft wing rock regulation problem with varying angle of attack is used to illustrate and compare the two approaches.