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     

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.     

求解不等圆Packing问题的一个启发式算法

求解具有NP难度的圆形packing问题具有很高的理论与实用价值.现提出一个启发式方法,求解了货运中常遇到的矩形区域内的不等圆packing问题.此算法首先将待布局圆按半径大小降序排列,然后用占角动作来逐个放置.通过试探性地放入一个或多个待布局圆,给出了占角动作的度以及更全局的有限枚举策略来评价占角动作的优度.在放置每一个圆时,以贪心的方式选取当前具有最大优度的占角动作来放置.最后用测试算例验证了算法的高效性.     

An algorithm for locating all zeros of a real polynomial

We present a globally convergent algorithm for calculating all zeros of a polynomialp _n ,p _n (z) = ∑ _{v = 0} ⁿ a _v z ^v, with real coefficients. Splittingp _n (exp(it)) into its real and imaginary part we can decide via Euclidean division of Chebyshev expansions and Sturm sequence argumentations whetherp _n has some zeros on the unit circle and how many zeros lie on the boundary and in the interior of it. Hence, by a bisection strategy we get the moduli of all zeros to a prescribed accuracy, and additionally we find the arguments as real zeros of a low degree polynomial. In this way we generate starting approximations for all zeros which in a final step are refined by an iterative process of higher order of convergence (e.g. Newton's or Bairstow's method).     

General optimal attenuation of harmonic disturbance with unknown frequencies

We present algorithms for optimal harmonic disturbance attenuation in standard discrete-time control structure, based on a parametrisation of (marginally) stabilising controllers. The Frobenius norm and the spectral norm of the closed-loop transfer matrix at the disturbance frequencies are minimised. If there is only one frequency of the disturbance, the controller has an observer–based form, which we obtain by solving a static output feedback (SOF) stabilisation control problem. Although the SOF stabilisation problem is hard, the generical case of nonsquare matrix G ₂₂ is solved by linear algebra methods. Numerical simulation results are presented. As a corollary, we transform the control problem with unit circle invariant zeros into a ?_∞ control problem without such zeros. The elimination of the unit circle invariant zeros is based on the fact that matrix Y(zI???A?+?BF)^?1 is stable, where (Y,?F) with Y?≥?0 is a solution of a discrete-time algebraic Riccati system.     

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.     

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.     

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.     

Using feedthrough for near one-step-ahead control of inverselyunstable plants

The one-step-ahead (OSA) control law is modified by the use of feedthrough to accommodate unstable plant zeros. An nth-order discrete-time system is guaranteed to track the desired input in one timestep. This is accomplished by subtracting the denominator dynamics and algebraically canceling the numerator dynamics. A problem occurs with OSA control when the plant has unstable zeros, that is, zeros that are outside the z-plane unit circle. In this case, an unstable zero is canceled and the resulting controller will be unstable. The authors address this problem. The approach taken is specific to OSA control     

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.     

Limiting zero distribution of sampled systems

In this paper, we consider the sampling of a continuous time system with possibly a non-rational transfer function. Limiting zero distribution of the sampled system represented by an FIR model is derived. It is shown that the zeros of the FIR sampled system converge evenly to the unit circle or to the unit disk plus unstable roots of the reciprocal polynomials depending on assumptions.     

Precise Dereverberation Using Multichannel Linear Prediction

In this paper, we discuss the numerical problems posed by the previously reported LInear-predictive Multi-input Equalization (LIME) algorithm when dealing with dereverberation of long room transfer functions (RTF). The LIME algorithm consists of two steps. First, a speech residual is calculated using multichannel linear prediction. The residual is free from the room reverberation effect but it is also excessively whitened because the average speech characteristics have been removed. In the second step, LIME estimates such average speech characteristics to compensate for the excessive whitening. When multiple microphones are used, the speech characteristics are common to all microphones whereas the room reverberation differs for each microphone. LIME estimates the average speech characteristics as the characteristics that are common to all the microphones. Therefore, LIME relies on the hypothesis that there are no zeros common to all channels. However, it is known that RTFs have a large number of zeros close to the unit circle on the z-plane. Consequently, the zeros of the RTFs are distributed in the same regions of the z-plane and, if an insufficient number of microphones are used, the channels would present numerically overlapping zeros. In such a case, the dereverberation algorithm would perform poorly. We discuss the influence of overlapping zeros on the dereverberation performance of LIME. Spatial information can be used to deal with the problem of overlapping zeros. By increasing the number of microphones, the number of overlapping zeros decreases and the dereverberation performance is improved. We also examine the use of cepstral mean normalization for post-processing to reduce the remaining distortions caused by the overlapping zeros     

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.     

On the stability properties of polynomials with perturbed coefficients

Given a polynomialP_{c}(S) = S^{n} + t_{1}S^{n-1} + ... t_{n} = 0which is Hurwitz orP_{d}(Z) = Z^{n} + t_{1}Z^{n-1} + ... t_{n} = 0which has zeros only within or on the unit circle, it is of interest to know how much the coefficients tjcan be perturbed while preserving the stability properties. In this note, a method is presented for obtaining the largest hypersphere centered att^{T} = [t_{1} ... t_{n}]containing only polynomials which are stable.     

On continuity properties of the two-block ℓ1 optimal design

This paper investigates continuity properties of the two-block ℓ₁ optimization resulting from the optimal design of BIBO stability robustness for discrete time systems in the presence of coprime factor perturbations. In general, the two-block ℓ₁ optimal design might lack continuity properties, since it might have no finite dimensional solution. However, this paper shows that for a suitable given truncated order, the two-block ℓ₁ suboptimal design is continuous as a map from the plant to the suboptimal closed loop solution, if the plant has no zeros on the unit circle and has a unique suboptimal solution. Furthermore, if the set of plants is compactness, we show that there exists a uniform bound on the degree of suboptimal solution for all plants in the set such that their deviation of suboptimal value from the optimal one have the same bound, and the two-block ℓ₁ suboptimal design is uniformly continuous. In particular, in the case in which the plant has nonunique solution, a procedure is also developed on how to choose a solution that preserves continuity. The results enable the stability of robust ℓ₁ adaptive scheme to hold in the presence of coprime factor perturbations.     

VC-dimension of exterior visibility

In this paper, we study the Vapnik-Chervonenkis (VC)-dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera. In the past, it has been shown that the VC-dimension of planar visibility systems is bounded by 23 if the cameras are allowed to be anywhere inside a polygon without holes. Here, we consider the case of exterior visibility, where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary. We present results for the cases of cameras lying on a circle containing a polygon (VC-dimension=2) or lying outside the convex hull of a polygon (VC-dimension=5). The main result of this paper concerns the 3D case: We prove that the VC-dimension is unbounded if the cameras lie on a sphere containing the polyhedron, hence the term exterior visibility.     

Fundamental limitation on achievable decentralized performance

A commonly accepted fact is that the diagonal structure of the decentralized controller poses fundamental limitations on the achievable performance, but few quantitative results are available for measuring these limitations. This paper provides a lower bound on the achievable quality of disturbance rejection using a decentralized controller for stable discrete time linear systems with time delays, which do not contain any finite zeros on or outside the unit circle. The proposed result is useful for assessing when full multivariable controllers can provide significantly improved performance, as compared to decentralized controllers. The results are also extended to the case, where the individual subcontrollers are restricted to be PID controllers.     

A new approach to the solution of the l1 controlproblem: the scaled-Q method

We explore an approach for solving multiple input-multiple output (MIMO) l₁ optimal control problems. This approach, which we refer to as the scaled-Q approach, is introduced to alleviate many of the difficulties facing the numerical solution of optimal l₁ control problems. In particular, the computations of multivariable zeros and their directions are no longer required. The scaled-Q method also avoids the pole-zero cancellation difficulties that existing methods based on zero-interpolation face when attempting to recover the optimal controller from an optimal closed-loop map. Because the scaled-Q approach is based on solving a regularized auxiliary problem for which the solution is always guaranteed to exist, it can be used no matter where the system zeros are (including the stability boundary). Upper and lower bounds that converge to the optimal cost are provided, and all solutions are based on finite dimensional linear programming for which efficient software exists