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.     

Properties of the Realization of Inner Functions

 In this paper we investigate fundamental properties of state-space realizations for inner functions. We derive necessary and sufficient conditions for the inner function to have a realization such that the associated C ₀-semigroup is exponentially stable. Furthermore, we give necessary and sufficient conditions on the inner function such that the C ₀-semigroup is a group. Combining these results, we have that the C ₀-semigroup is an exponentially stable C ₀-group if and only if the inner function is the product of a constant of modulus one and a Blaschke product for which the zeros satisfy the Carleson–Newman condition and the zeros lie in a vertical strip bounded away from the imaginary axis. Date received: January 11, 1999. Date revised: May 16, 2002. RID="*" ID="*"This paper was supported by the Volkswagen Stiftung (RiP program at Oberwolfach) and by the Deutsche Forschungsgemeinschaft.     

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.     

Computation of invariant zeros of linear,time-invariant,multivariable systems

In this paper a numerical method is presented for computing the invariant zeros of a controllable linear, time-invariant, multivariable system described by the 4-tuplo (A, B, C, D) or the triple (A, B, C). The method is based on the fact. that a controllable system can be made maximally unobservable by means of state variable feedback, thereby causing the cancellation of the invariant zeros by an equal number of the system poles. The invariant zeros are obtained as the eigenvalues of a matrix of the same dimension as the number of invariant zeros. The method is applicable to both multivariable as well as single-input, single-output systems. Examples are given to illustrate the use of the method.     

Algebraic properties of singular systems subject to decentralized output feedback

This paper investigates algebraic properties of decentralized singular systems (DSSs) under local static output feedback. New concepts of the geometric multiplicity (GM) of the finite decentralized fixed mode (DFM), the decentralized output feedback variable polynomial (DVP), and the finite decentralized output feedback cycle index (DCI) of the DSS are defined. The formulas for determining the GM and the DCI are given in terms of the DFM and the system matrices. It is shown that almost any decentralized output feedback can make the zeros of the DVP (i.e., the closed-loop poles that are variable) distinct and away from any given finite set in the complex plane. It is also shown that the finite DFM of the DSS are those uncontrollable and/or observable finite modes of the closed-loop DSS through any single channel. Finally, the minimal number of the inputs and outputs that guarantee the finite modes of the closed-loop DSS controllable and observable is shown to be the finite DCI of the DSS. An illustrative example is provided.     

On undershoot in SISO systems

We show that the step response of an asymptotically stable strictly proper nonminimum phase continuous time system with m₁ real open right half plane zeros, no conjugate complex zeros and poles, and with real open left half plane zeros and poles satisfying a given “bracketing” condition, exhibits undershoot exactly int m₁+½ times. Both qualitative and quantitative arguments are also introduced to highlight the main issues under discussion. Simple examples are given to illustrate and complement the main result of this paper     

Linear Operator Inequalities for Strongly Stable Weakly Regular Linear Systems

We consider the question of the existence of solutions to certain linear operator inequalities (Lur'e equations) for strongly stable, weakly regular linear systems with generating operators A, B, C, 0. These operator inequalities are related to the spectral factorization of an associated Popov function and to singular optimal control problems with a nonnegative definite quadratic cost functional. We split our problem into two subproblems: the existence of spectral factors of the nonnegative Popov function and the existence of a certain extended output map. Sufficient conditions for the solvability of the first problem are known and for the case that A has compact resolvent and its eigenvectors form a Riesz basis for the state space, we give an explicit solution to the second problem in terms of A, B, C and the spectral factor. The applicability of these results is demonstrated by various heat equation examples satisfying a positive-real condition. If (A, B) is approximately controllable, we obtain an alternative criterion for the existence of an extended output operator which is applicable to retarded systems. The above results have been used to design adaptive observers for positive-real infinite-dimensional systems. Date received: July 25, 1997. Date revised: February 10, 2001.     

Integral constraints and performance limits on complementary sensitivity: discrete-time systems

We study performance limitation issues for multivariable discrete-time feedback systems. The complementary sensitivity function is employed as a performance measure, and Bode and Poisson-type integral inequalities and -type performance limits are derived. The results exhibit frequency-dependent constraints as well as best achievable limits on the complementary sensitivity function, which are shown to be determined by nonminimum phase zeros, unstable poles, and time delays. In particular, the directions of such zeros and poles are seen to play a central role to this effect.     

用零极点动态控制波形图显示的Java小程序实现方法

用Java小程序实现了通过拉普拉斯逆变换和z逆变换中s域或z域的零极点动态控制波形图显示的过程。为了增强人机界面的交互性和稳定性,程序采用了Java语言中的鼠标事件来捕获零极点坐标从而动态控制波形图显示,并且也应用了双缓冲技术来消除擦旧画新过程所引起的画面闪烁。实际应用表明,用户通过此程序可以更加方便直观地理解诸如系统稳定性等一系列问题,达到传统教学系统所没有的效果。     

Properties of modulated and demodulated systems with implications to feedback limitations

It is well known that the poles, zeros and delay of a system play an important role in determining the associated feedback performance limitations. In this paper, we first derive an approximate transfer function for a modulated and demodulated system of a particular form. We next analyse the behaviour of the poles, zeros and delay of this transfer function when the modulation frequency is varied. Some implications of these results are also briefly discussed.     

Effect of the interdependence between subnets on the structural controllability of complex networks

Controllability is a hot issue in complex networks. In this paper, without changing the number and the positions of the controllers, the effect of interdependence (viz., inter-coupling relations) between two directed subnets that are controllable or uncontrollable on the structural controllability of complex networks is investigated. According to the Kalman rank criterion and PBH rank criterion, some sufficient conditions and necessary conditions for the structural controllability or uncontrollability of interdependent networks are presented. The results show that (1) if the uncontrollable and controllable nodes in one subnet are inter-coupled with the controllable and uncontrollable nodes in the other subnet, respectively, and the directions are from controllable nodes to the uncontrollable nodes, then the interdependent network may be controllable; (2) the interdependent network can be obtained structurally controllable if there are simply directed paths covering all the uncontrollable nodes of the two subnets, separately; and (3) the interdependent networks are structurally controllable no matter what kind of inter-couplings between two controllable subnets. All of these results are adaptive to undirected subnets. The final example verifies the effectiveness and correctness of the results provided in this paper.     

Controllability and local accessibility - a normal form approach

Given a system with an uncontrollable linearization at the origin, we study the controllability of the system at equilibria around the origin. If the uncontrollable mode is nonzero, we prove that the system always has other equilibria around the origin. We also prove that these equilibria are linearly controllable provided a coefficient in the normal form is nonzero. Thus, the system is qualitatively changed from being linearly uncontrollable to linearly controllable when the equilibrium point is moved from the origin to a different one. This is called a bifurcation of controllability. As an application of the bifurcation, systems with a positive uncontrollable mode can be stabilized at a nearby equilibrium point. In the last part of this paper, simple sufficient conditions are proved for local accessibility of systems with an uncontrollable mode. Necessary conditions of controllability and local accessibility are also proved for systems with a convergent normal form.     

Double‐layer microstrip ultra‐wideband filtering power divider with high selectivity and large isolation

In this article, a compact double‐layer microstrip ultra‐wideband (UWB) filtering power divider with high selectivity and isolation is proposed. The filtering power divider consists of a multimode resonator at the top layer coupled with a pair of branch lines at the bottom through a slotline in the middle ground. The slotline provides strong coupling between the two layers and equally distributes the power to two branch lines. The resistor loaded about a quarter‐wavelength away from the slotline achieves high isolation within UWB range. The UWB filtering properties with controllable transmission poles and zeros as well as power splitting with enhanced isolation have been analyzed. The adjustable transmission zeros of the filter unit enables the bandwidth control of the filtering power divider. Finally, a UWB filtering power divider operating at 3.1 to 10.6 GHz has been designed, fabricated, and measured. It achieves a compact size of only 26 × 28 mm², high isolation about 20 dB, and good out‐of‐band suppression of 40 dB.     

Achievable performance of multivariable systems with unstable zeros and poles

This paper examines the fundamental limitations imposed by unstable (right half plane; RHP) zeros and poles in multivariable feedback systems. We generalize previously known controller-independent lower bounds on the H     

Pole‐zero analysis and wavelength scaling of carbon nanotube antennas

Carbon nanotube (CN) antennas have applications in the THz electromagnetic spectrum. Nanotubes have a highly dispersive and frequency dependent conductivity model. In this article, we compare the poles and zeros in the input impedance of CN antennas at different lengths. We used model‐based parameter estimation to approximate the input impedance of the antenna with a rational function in the complex frequency domain. Despite dispersive conductivity of CN, the imaginary part of the poles and zeros are respectively the integer multiples and odd multiples of the imaginary part of the first pole and zero. However, the real part of poles is almost constant, while the pattern was not observed for the real part of zeros. We also show that CN dipoles operating between 43 and 53 GHz are well matched if the source impedance is much higher than 50 ohms, and even higher than 12.9 kΩ. The fundamental resonances (f₀) of CN dipoles plotted versus their inverse‐half‐length (1/L) are linearly related, but the intercept of the fitted straight line is non‐zero unlike that for perfect electric conductor (PEC) dipoles. This leads to non‐linear variation in wavelength scaling of CN dipoles. The resonant CN antennas are relatively much shorter than PEC dipoles.     

线性控制系统规范结构的新解释与它的算法

应用绝对能观性的概念可以对线性控制系统的规范结构赋于新的解释。文中指出,规范结构由四部分组成,即绝对能观但不能控;绝对能控但不能观;绝对能观与能控;绝对不能观不能控。应用我们引入的方法,规范结构可以用矩阵的初等变换来实现。     

Controllability for a Class of Parallelly Connected Polynomial Systems

Null controllability for a class of parallelly connected discrete-time polynomial systems is considered. We prove for this class of systems that a necessary and sufficient condition for null controllability of the parallel connection is that all its subsystems are null controllable. Consequently, the controllability test splits into a number of easy-to-check tests for the subsystems. The test for complete controllability is also presented and it is subtly different from the null controllability test. A similar statement is then given for complete controllability of a class of parallelly connected continuous-time polynomial systems. The result is somewhat unexpected when compared with the classical linear systems result. We identify the phenomenon which shows the difference between the linear and nonlinear cases. Date received: January 22, 1997. Date revised: January 14, 1998.     

Analysis of Over- and Underdetermined Nonlinear Differential-Algebraic Systems with Application to Nonlinear Control Problems

 We study over- and underdetermined systems of nonlinear differential-algebraic equations. Such equations arise in many applications in circuit and multibody system simulation, in particular when automatic model generation is used, or in the analysis and solution of control problems in the behavior framework.?We give a general (local) existence and uniqueness theory and apply the results to analyze when nonlinear implicit control problems can be made regular by state or output feedback.?The theoretical analysis also leads immediately to numerical methods for the simulation as well as the construction of regularizing feedbacks. Date received: February 21, 2000. Date revised: November 14, 2000.     

分散固定阻塞零点与分散强镇定

研究多通道系统的分散强镇定问题.对集中控制系统的阻塞零点概念提出一种推广的 定义.在此基础上,对分散控制系统引入了关于通道及系统的分散固定阻塞零点等概念,揭 示出分散系统的阻塞零点与反馈结构的关系.利用这种关系,在比已有结论更弱的假定下,给 出了可分散强镇定的充分必要条件及相应控制器求解的一般步骤.     

Zeros and Poles of Linear Continuous-Time Periodic Systems: Definitions and Properties

The paper deals with definitions of zeros and poles and their features in finite-dimensional linear continuous-time periodic (FDLCP) systems under a harmonic framework. More precisely, system and transfer zeros and poles in the harmonic wave-to-wave sense are defined on what we call the regularized harmonic system operators and the harmonic transfer operators of FDLCP systems by means of regularized determinants; then their composition and properties related to system structures are examined via the Floquet theory and controllability/observability decompositions of FDLCP systems. The study shows that under mild assumptions, the harmonic transfer operators of FDLCP systems are analytic and meromorphic, on which zeros and poles are well-defined. Basic zero/pole relationships are established, which are similar to their linear time-invariant counterparts and in particular explicate some interesting harmonic wave-to-wave behaviors of FDLCP systems. The results are significant in analysis and synthesis of FDLCP systems when the harmonic approach is adopted.