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.     

Error Analysis of a Derivative-Free Algorithm for Computing Zeros of Holomorphic Functions

We consider the quadrature method developed by Kravanja and Van Barel (Computing 63(1):69–91, 1999) for computing all the zeros of a holomorphic function that lie inside the unit circle. The algorithm uses only the function values and no (first or higher order) derivatives. Information about the location of the zeros is obtained from certain integrals along the unit circle. In numerical computations these are replaced by their trapezoidal rule approximations. We investigate the resulting quadrature error. Our error analysis shows that the zeros located inside the unit circle do not affect the accuracy of the computed approximations whereas the quadrature error related to the zeros located outside the unit circle tends to zero exponentially as the number of quadrature points tends to infinity.     

The fundamental role of general orthonormal bases in systemidentification

The purpose of this paper is threefold. Firstly, it is to establish that contrary to what might be expected, the accuracy of well-known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure. Secondly, it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros. This is achieved by employing certain new results on generalized Fourier series and the asymptotic properties of Toeplitz-like matrices in such a way that the new variance expressions presented here encompass pre-existing ones as special cases. Via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification. Namely, that orthonormal bases are much more than an implementational option offering improved numerical properties. In fact, they are an intrinsic part of estimation since, as shown here, orthonormal bases quantify the asymptotic variability of the estimates whether or not they are actually employed in calculating them     

A constrained recursive pseudo-linear regression scheme for on-line parameter estimation in adaptive control

In adaptive control of systems with poles close to the unit circle, application of the recursive estimation techniques can lead to excursions of the poles of the identified model outside the unit circle even when the process is open loop stable. These excursions can be of two types. The poles of the deterministic component of the model can drift outside unit circle even when the process has no unstable modes. Alternatively, the poles and/or zeros of the unmeasured disturbance (noise) model can drift outside the unit circle. In either case, the identified model is not suitable for on-line controller adaptation. In this work, a novel constrained recursive formulation is proposed for on-line parameter estimation based on the pseudo-linear regression (PLR) approach. The efficacy of the proposed approach is demonstrated by conducting experimental studies on a benchmark laboratory scale heater-mixer setup. The analysis of the open and closed loop experimental results reveals that the proposed constrained parameter estimation scheme provides a systematic and computationally attractive approach to ensure that the identified model parameters are restricted to the feasible region.     

柔性机械臂动力学奇异性渐近行为的比较研究

通过对柔性机械臂动力学系统的零极点及其模态参量渐近行为的解析分析,讨论了 模态截取和零点截取有限维近似模型动力学奇异性关于阶次的渐近行为,并进行了比较研 究.此外,还就截取模型的高阶项对逆动力学计算力矩病态行为的影响进行了数值比较分析.     

Root locus near isolated pole-zero dipoles and a counterintuitive case of digital control compensation

A convenient characterization is given for root loci associated with open-loop pole-zero dipoles. A particular dipole-locus effect is shown to influence the design of a digital PID controller for an oscillatory plant; the design is a counterintuitive one in which zeros of the compensator are placed outside the unit circle in the neighborhood of the plant poles.     

Pseudo Complex Cepstrum Using Discrete Cosine Transform

Two new algorithms are proposed, which obtain pseudo complex cepstrum using Discrete Cosine Transform (DCT). We call this as the Discrete Cosine Transformed Cepstrum (DCTC). In the first algorithm, we apply the relation between Discrete Fourier Transform (DFT) and DCT. Computing the complex cepstrum using Fourier transform needs the unwrapped phase. The calculation of the unwrapped phase is difficult whenever multiple zeros and poles occur near or on the unit circle. Since DCT is a real function, its phase can only be 0 or π and the phase is unwrapped by representing the negative sign by exp (−jπ) and the positive sign by exp (j0) . The second algorithm obviates the need for DFT and obtains DCTC by representing the DCT sequence itself by magnitude and phase components. Phase is unwrapped in the same way as the first algorithm. We have tested DCTC on a simulated system that has multiple poles and zeros near or on the unit circle. The results show that DCTC matches the theoretical complex cepstrum more closely than the DFT based complex cepstrum. We have explored possible uses for DCTC in obtaining the pitch contour of syllables, words and sentences. It is shown that the spectral envelope obtained from the first few coefficients matches reasonably with the envelope of the signal spectrum under consideration, and thus can be used in applications, where faithful reproduction of the spectral envelope is not critical. We also examine the utility of DCTC as feature set for speaker identification. The identification rate with DCTC as feature vector was higher than that with linear prediction-derived cepstral coefficients.     

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.     

Discrete time convolution control systems

In this paper, we study a general class of linear time invariant discrete time distributed systems. We consider both single-input-single-output (SISO) and multi-input-multi-output (MIMO) systems, and study design procedures. We develop a commutative algebra of transfer function, b(p₀), for a general class of SISO discrete time convolution systems, which covers sampled distributed systems and, of course, lumped systems as a special case. Each element of b(p₀) is formulated as a ratio of two elements in an algebra l₁?(p₀) of causal p₀-stable transfer functions. We demonstrate that l₁?(p₀) indeed a euclidean ring, give necessary and sufficient conditions for coprimeness between elements in l₁?(p₀) and characterize poles and zeros for elements in b(p₀). In contrast to the algebra l₁ the algebra b(p₀) includes both stable and unstable systems; furthermore since p₀<1 this formulation allows us to study the dominant poles inside the unit disc of the complex plane. We study next MIMO systems whose transfer functions are matrices with elements in b(p₀). We establish the matrix fraction representation theory and use it to develop : the dynamic interpretation of poles and transmission zeros, the feedback interconnection of such MIMO systems, and the problem of controller design to achieve stabilization (analogous to arbitrary closed-loop eigenvalue assignment), asymptotic tracking and disturbance rejection ; finally, for the case of stable square plants, we show how to achieve complete decoupling with detailed pole assignment and finite settling time, subject to, of course, the limitations imposed by the plant transmission zeros outside the open unit disc.     

Further results on the optimal rejection of persistent boundeddisturbances

The problem of designing controllers that optimally reject persistent disturbances is studied. The focus is on the case where the plant to be controlled has either zeros or poles on the stability boundary, i.e. the unit circle in the discrete-time case and the extended jω axis in the continuous-time case. For the discrete-time case, the problem of minimizing a cost functional of the form ∥f-rg∥₁, where the transform g˜ of g has some unit circle zeros is studied. A previously published dual problem formulation is extended, and it is shown that an optimal controller need not exist. The construction of a sequence of suboptimal controllers whose performance approaches the unattainable infimum of the cost function is studied. It is shown that two results which hold in the case of H_∞ optimization do not hold in the presented situation. Specifically, the introduction of unit circle zeros can increase the value of the infimum, even when every unit circle zero of g˜ is also a zero of f˜, and a sequence of controllers constructed in an obvious fashion fails to be an optimizing sequence. Similar results are obtained for the continuous-time case     

On the solutions of the rational covariance extension problem corresponding to pseudopolynomials having boundary zeros

In this note, we study the rational covariance extension problem with degree bound when the chosen pseudopolynomial of degree at most n has zeros on the boundary of the unit circle and derive some new theoretical results for this special case. In particular, a necessary and sufficient condition for a solution to be bounded (i.e., has no poles on the unit circle) is established. Our approach is based on convex optimization, similar in spirit to the recent development of a theory of generalized interpolation with a complexity constraint. However, the two treatments do not proceed in the same way and there are important differences between them which we discuss herein. An implication of our results is that bounded solutions can be computed via methods that have been developed for pseudopolynomials which are free of zeros on the boundary, extending the utility of those methods. Numerical examples are provided for illustration.     

Comparison of total least squares and instrumental variable methods for parameter estimation of transfer function models

The estimation of the parameters of a transfer function model is considered. Relationships between the total least squares (TLS) and instrumental variable (IV) approaches are outlined. Both methods are able to compute strongly consistent parameter estimates. TLS can be considered as a variation on the IV method where the IV are functions of the time instant and the estimated model parameters. TLS computes strongly consistent estimates of the true model parameters if the outputs and possibly the inputs are independently disturbed by discrete, stationary white noise with zero mean and equal variance. The IV need not be generated. Hence TLS is much simpler to use but more restrictive (IV allows arbitrary noise models) and computationally not so attractive. Next, simulation results are presented comparing the short sample accuracy properties of both methods. When the outputs and possibly the inputs are disturbed by stationary zero mean while noise, TLS outperforms the ordinary IV methods. The accuracy becomes comparable by extending the IV sufficiently. The superiority of TLS is particularly clear in cases where the zeros of the polynomial operating on the outputs are close to the unit circle or where both the inputs and outputs are noisy.     

Linear time invariant system order reduction using multipoint step response matching

The proposed method of linear time invariant system order reduction is based on multipoint step response matching for both pole and zero evaluation of the reduced order system (ROS). Depending on the number of zeros and poles of the ROS, the number of points is selected on the time axis of the unit step response such that the unknown poles and zeros can be determined by solving a set of non-linear equations using Newton's method. Numerous examples have been solved using the proposed technique and the results are encouraging.     

Design of feedback systems with non-minimum-phase unstable plants†

Feedback systems with right half-plane poles and zeros may have inherently very poor sensitivity properties. In the design procedure presented, the closed-loop poles are-restricted to two possible regions in the complex plane. One region is s≤ ?σ. σ>0. A second is the interior and boundary of a circle in the left half-plane. The design is optimum in the sense of maximizing the gain factor uncertainty, for which the restriction is satisfied. The design procedure is very simple to execute and results in loop transmission poles and zeros which are symmetrical with respect to the boundary of the forbidden region. The closed-loop poles lie entirely on the boundary, over the range of gain uncertainty.     

Order estimation for linear time-invariant systems using frequencydomain identification methods

A two-step coarse-fine order estimation technique is proposed to determine the order of the numerator and the denominator polynomials of rational transfer function models for single-input/single-output (SISO) linear time-invariant systems. The coarse order estimation is based on rank detection by verification of the stochastic significance of the singular values of a linearized problem. The fine order estimation is based on a statistical analysis of the maximum likelihood cost function. The method is tested on measurements of low-(4 zeros, 6 poles) and high- (58 poles, 58 zeros) order systems     

Robust Schur stability of a polytope of polynomials

The main objective of the authors is to provide a necessary and sufficient condition for a polytope of polynomials to have all its zeros inside the unit circle. The criterion obtained serves as a discrete-time counterpart for results in S. Bialas (1985) and F. Fu and B.R. Barmish (1987) for the continuous case. Also, the results are reduced to operations on (n-1)×(n-1) matrices. It is concluded that, by the edge result of A.C. Bartlett et al. (1987), it suffices to check the exposed edges in order to determine whether a polytope of polynomials has all its zeros in a simply connected region D     

Finite model order accuracy in Hammerstein model estimation

Hammerstein models is one of the most commonly used model classes used for identifying nonlinear systems. A static input nonlinearity followed by a linear dynamical part is an adequate way to model many real-life systems. This paper investigates the asymptotic (in terms of sample size) variance of Hammerstein model estimates. The work extends earlier results by Ninness and Gibson (2002) in the following ways. Not only frequency function estimation but estimation of general quantities is considered. The expressions are not restricted to be valid asymptotically in the model order. In addition, the results cover model structures having noise models and allow for data generated under feedback. The increase in variance due to the estimation of the input nonlinearity is characterized. In particular, under open loop operation, white additive noise and the assumption of a separable process, it is shown that the variance increase is exactly a term that was observed in Ninness and Gibson (2002) to result in good agreement with simulations. This term vanishes in the formal asymptotic in model order analysis in Ninness and Gibson (2002).