Variance-error quantification for identified poles and zeros

This paper deals with quantification of noise induced errors in identified discrete-time models of causal linear time-invariant systems, where the model error is described by the asymptotic (in data length) variance of the estimated poles and zeros. The main conclusion is that there is a fundamental difference in the accuracy of the estimates depending on whether the zeros and poles lie inside or outside the unit circle. As the model order goes to infinity, the asymptotic variance approaches a finite limit for estimates of zeros and poles having magnitude larger than one, but for zeros and poles strictly inside the unit circle the asymptotic variance grows exponentially with the model order. We analyze how the variance of poles and zeros is affected by model order, model structure and input excitation. We treat general black-box model structures including ARMAX and Box–Jenkins models.     

Simultaneous Stabilization and Constant Reference Tracking of LTI,MIMO Systems

It is shown that any finite number of plants that belong to certain classes of multi‐input multi‐output systems with no zeros in the region of instability can be simultaneously stabilized using linear, time‐invariant integral‐action controllers. These plants may be stable or unstable and their poles are not restricted; they may also have any number of zeros in the stable region of the complex plane. The classes of systems under consideration include plants with blocking or transmission zeros at infinity. The common controller achieves asymptotic tracking of step‐input references with zero steady‐state error and has a low order transfer‐function. Systematic synthesis methods are presented, and a parametrization of all simultaneously stabilizing controllers with integral‐action is also provided.     

Decoupling controller design for linear multivariable plants

Decoupling controller design for linear time-invariant square multiple-input multiple-output (MIMO) plants under the unity-feedback configuration is discussed. For plants with no coincidences of unstable poles and zeros, a simplified necessary and sufficient condition for closed-loop stability is given. The simplified condition leads to a simple parameterization of the set of all achievable decoupled input/output (I/O) maps and an algorithm which allows the design of decoupling controllers to achieve preassigned closed-loop poles     

On structural invariants and the root-loci of linear multivariable systems

The geometric nature of the infinite zeros of the root-loci of linear multivariate systems is investigated using the canonical form derived by Morse (1973). It is shown that an invertible system S(A, B, C) has integer-order infinite zeros in the generic case equal to the controllability indices of a pair (A + KC, B), that suitable choice of proportional output feedback guarantees the absence of other than integer-order zeros and that the orders and asymptotic directions of the infinite zeros are independent of constant state feedback and output injection.     

Robust controller design based on reduced order plants

Two dual controller design methods are proposed for linear, time-invariant, multi-input multi-output systems, where designs based on a reduced order plant robustly stabilizer higher order plants with additional poles or zeros in the stable region. The additional poles (or zeros) are considered as multiplicative perturbations of the reduced plant. The methods are tailored towards closed-loop stability and performance and they yield estimates for the stability robustness and performance of the final design. They can be considered as formalizations of two classical heuristic model reduction techniques. One method neglects a plant-pole sufficiently far to the left of dominant poles and the other cancels a sufficiently small stable plant-zero with a pole at the origin.     

On backward shift algorithm for estimating poles of systems

In this paper, we present an algorithm for estimating poles of linear time-invariant systems by using the backward shift operator. We prove that poles of rational functions, including zeros and multiplicities, are solutions to an algebraic equation which can be obtained by taking backward shift operator to the shifted Cauchy kernels in the unit disc case. The algorithm is accordingly developed for frequency-domain identification. We also prove the robustness of this algorithm. Some illustrative examples are presented to show the efficiency in systems with distinguished and multiple poles.     

Periodic Compensation of Continuous-Time Plants

This note presents a periodic compensator which achieves robust stability for single-input-single-output (SISO), linear time invariant (LTI) plants having both right-half plane (RHP) poles and zeros, a job LTI controllers fail to do. In addition, for strictly proper plants this controller achieves model matching ensuring at the same time that the periodic oscillations present in the plant output are insignificant in magnitude. The design steps are straightforward and linear algebraic in nature     

Controllable and Uncontrollable Poles and Zeros of nD Systems

 We use the behavioural approach to define and characterize controllable and uncontrollable poles and zeros of multidimensional (nD) linear systems. We show a strong relationship between controllable poles and zeros and properties of the transfer function matrix, and we give characterizations of uncontrollable poles and zeros, in particular demonstrating that these have an input decoupling property. Date received: December 13, 1999. Date revised: October 9, 2000.     

Optimal root loci of flexible space structures

It is shown that simple generic properties can be proved for the optimal root loci of flexible space structures (FSS). It is proved that the angles and rates at which closed-loop poles depart from the open-loop poles can be found by inspection for any FSS with sufficiently widely space natural frequencies. Similar results also apply to the angles at which certain loci approach the transmission zeros of any FSS with compatible (physically colocated and coaxial) sensors and actuators, as a consequence of the special properties of the zeros of such systems. Finally, determining the number and orders of the Butterworth configurations of such as FSS is much more straightforward than for a general state-space system: it amounts only to checking whether structural displacements or their rates are the measured outputs. These results depend on the fundamental second-order nature of such systems, and five considerably more insight than is possible a priori for general linear multivariable systems     

Identification of Wiener–Hammerstein models: Two algorithms based on the best split of a linear model applied to the SYSID'09 benchmark problem

This paper describes the identification of Wiener–Hammerstein models and two recently suggested algorithms are applied to the SYSID'09 benchmark data. The most difficult step in the identification process of such block-oriented models is to generate good initial values for the linear dynamic blocks so that local minima are avoided. Both of the considered algorithms obtain good initial estimates by using the best linear approximation (BLA) which can easily be estimated from data. Given the BLA, the two algorithms differ in the way the dynamics are separated into two linear parts. The first algorithm simply considers all possible splits of the dynamics. Each of the splits is used to initialize one Wiener–Hammerstein model using linear least-squares and the best performing model is selected. In the second algorithm, both linear blocks are initialized with the entire BLA model using basis function expansions of the poles and zeros of the BLA. This gives over-parameterized linear blocks and their order is decreased in a model reduction step. Both algorithms are explained and their properties are discussed. They both give good, comparable models on the benchmark data.     

Properties and calculation of transmission zeros of linear multivariable systems

A new definition of transmission zeros for a linear, multivariable, time-invariant system is made which is shown to be equivalent to previous definitions. Based on this new definition of transmission zeros, new properties of transmission zeros of a system are then obtained; in particular, it is shown that a system with an unequal number of inputs and outputs almost always has no transmission zeros and that a system with an equal number of inputs and outputs almost always has either n−1 or n transmission zeros, where n is the order of the system; transmission zeros of cascade systems are then studied, and it is shown how the transmission zeros of a system relate to the poles of a closed loop system subject to high gain output feedback. An application of transmission zeros to the servomechanism problem is also included. A fast, efficient, numerically stable algorithm is then obtained which enables the transmission zeros of high order multivariable systems to be readily obtained. Some numerical examples for a 9th order system are given to illustrate the algorithm.     

Arbitrarily fast and robust tracking by feedback

The output of a singe-input-single-output linear feedback system with more than one pole in excess over the zeros in the loop transmission cannot track arbitrarily fast its input (by the root locus). In this work we extend the linear feedback so that some of the open loop poles may depend on the open loop gain; we call this new class quasi-linear feedback systems. We then derive time domain, pole-zero, and frequency domain conditions which ensure arbitrarily fast and robust tracking by quasi-linear feedback, for an arbitrary number of poles in excess over the zeros. We prove that in a particular case these conditions are equivalent, and that the boundedness in frequency of the closed loop transfer function is no longer necessary for achieving arbitrarily fast tracking. The robustness is to external disturbances and initial conditions, and the open loop has to be minimum phase. Some examples are presented which illustrate these results. They also show that this good performance can be obtained with a reduced control effort, and that quasi-linear feedback can alleviate the limitation on performance of non-minimum phase open loops.     

Limitations on maximal tracking accuracy

This paper studies optimal tracking performance issues pertaining to finite-dimensional, linear, time-invariant feedback control systems. The problem under consideration amounts to determining the minimal tracking error between the output and reference signals of a feedback system, attainable by all possible stabilizing compensators. An integral square error criterion is used as a measure for the tracking error, and explicit expressions are derived for this minimal tracking error with respect to step reference signals. It is shown that plant nonminimum phase zeros have a negative effect on a feedback system's ability to reduce the tracking error, and that in a multivariable system this effect results in a way depending on not only the zero locations, but also the zero directions. It is also shown that if unity feedback structure is used for tracking purposes, plant nonminimum phase zeros and unstable poles can together play a particularly detrimental role in the achievable tracking performance, especially when the zeros and poles are nearby and their directions are closely aligned. On the other hand, if a two parameter controller structure is used, the achievable tracking performance depends only on the plant nonminimum phase zeros     

Pole and zero assignment by state vector feedback

A procedure is described which enables specified poles and zeros of a scalar transfer function of a controllable, observable, linear system to be obtained by using state vector feedback to two inputs. The number of zeros is equal to the number of zeros in the transfer function, before feedback is applied, from one input or the other, whichever is the greater. Those zeros which can be changed, and those which cannot, are identified. The former can be made equal to, or arbitrarily close to, any assigned values, and the poles can be assigned arbitrarily.     

Unbiased estimation of a sinusoid in colored noise via adapted notch filters

Standard notch models for the estimation of a sinusoid in noise are characterized by two poles on the unit circle and two zeros aligned with the poles. These models provide biased estimates of the unknown frequency, the bias increasing with the zero-pole distance. On the other hand, keeping the zeros far from the poles has the beneficial effect of making the identification algorithm less sensitive to the adopted initialization. In this paper we remove the pole-zero alignment constraint. The extra degree of freedom thus obtained is used to alleviate the traditional trade-off between unbiasedness of the estimates and robustness against poor initializations. The proposed technique applies to the case of both white and colored additive noise.     

Control of linear plants with zeros and slowly varying parameters

A reduction of order technique is developed which is useful in controller design for single-input, single-output, linear plants which have transfer functions with zeros and parameters which vary slowly compared to the response time of the plant. The controller design is based on a Liapunov-like method and employs a model reference. The reduction of order technique allows the control signal to be generated from the plant output and only its firstn-m-1derivatives if the plant transfer function has annth order denominator and anmth order numerator and if the transfer function hasmknown fixed poles.     

具有随机延时的网络控制系统的均方可镇定性分析

针对网络系统的可镇定性问题,研究整数步随机延时离散时间线性系统的均方可镇定性.利用Youla参数化与内外分解方法,结合均方小增益定理得到系统输出反馈均方可镇定的充分必要条件.该条件明确给出系统可镇定性与被控对象特性(不稳定极点、非最小相位零点、相对阶)和信道特性(频域信噪比函数)的关系,其中频域信噪比函数在被控对象不稳定极点的取值对可镇定影响甚大.利用仿真算例量化被控对象的非最小相位零点及相对阶对可镇定性的影响,验证可镇定性条件的正确性.     

具有最优动态性能的鲁棒镇定控制器设计

针对ＳＩＳＯ线性离散系统,利用线性规划方法设计具有指定最优动态性能的鲁棒稳定控制器。当线性离散模型的零,极点已知时,将最优动态性能指标在指定输入信号下直接转化为线性规划问题。从而解出最优响应输出序列。最优动态性能指标与鲁棒稳定性的统一使该控制器的设计方法具备了工业应用条件。仿真实例验证了结果的正确性。     

A new approach for nonlinear process identification using orthonormal bases and ordinal splines

A new gray-box method for nonlinear process identification is presented. Industrial deployment for model predictive control (MPC) is the primary focus of this development. For flexibility, the identification accommodates Hammerstein, Wiener and the more general N–L–N block-oriented structures. Instruments comprised of linear and nonlinear combinations of inputs and outputs are also accommodated. Unique to this approach is the utilization of two sets of bases. One is constructed using an estimate of the process poles and the other is constructed using a predefined set of special cubic splines. An intriguing aspect of this formulation is that nonlinear dynamics are implicitly accommodated. In addition, problems associated with identifying the linear portion of the model in conventional block oriented formulations are removed. Because of the bases formulation, it is possible to solve the identification problem for many supported structures by convex optimization and hence avoid the inherent problems of iterative solutions. To insure open-loop unbiased estimates, any structures using output nonlinearities do require an iterative solution. Two test cases from the open literature are presented as are results from plant step-test data on a problematic air separation unit.