Comments on "On blocking zeros in linear multivariable systems"

It is suggested that the name "blocking zeros" in the above technical note is not appropriate and should be changed in order to avoid confusion in the literature.     

Decentralized blocking zeros and the decentralized strongstabilization problem

This paper is concerned with a new system theoretic concept, decentralized blocking zeros, and its applications in the design of decentralized controllers for linear time-invariant finite-dimensional systems. The concept of decentralized blocking zeros is a generalization of its centralized counterpart to multichannel systems under decentralized control. Decentralized blocking zeros are defined as the common blocking zeros of the main diagonal transfer matrices and various complementary transfer matrices of a given plant. As an application of this concept, we consider the decentralized strong stabilization problem (DSSP) where the objective is to stabilize a plant using a stable decentralized controller. It is shown that a parity interlacing property should be satisfied among the real unstable poles and real unstable decentralized blocking zeros of the plant for the DSSP to be solvable. That parity interlacing property is also sufficient for the solution of the DSSP for a large class of plants satisfying a certain connectivity condition. The DSSP is exploited in the solution of a special decentralized simultaneous stabilization problem, called the decentralized concurrent stabilization problem (DCSP). Various applications of the DCSP in the design of controllers for large-scale systems are also discussed     

Comments on `On blocking zeros in linear multivariable systems'[with reply]

Reference is made to a recently published correspondence by P.G. Ferreira in which concern was expressed about the name `blocking zeros' given in a paper by R.V. Patel (see ibid., vol.AC-31, p.239-41, Mar. 1986) to a newly defined class of zeros for multivariable linear systems. The commenter refers to a few remarks about some `well-known' results by Ferreira, and objects to the lack of precision in the wording of two of these remarks. Counterexamples to these `well-known' results are presented. In replying, the original author claims that Example 1 is not a counterexample to his statement, and objects to some lack of precision in Martin's words     

On zeros of pulse transfer functions

The famous Åstrom-Hagander-Sternby theorem (1984) on limiting zeros of the pulse transfer function is extended by determining the accuracy of the asymptotic results for both the discretization and the intrinsic zeros when the sampling interval is small. Closed form formulas are derived that express the degree of the principal term of Taylor expansion of the difference between the true zeros and asymptotic ones as a function of the relative degree of the underlying continuous-time system and the value of the corresponding coefficient itself. Certain known results on asymptotic zeros are shown to be particular cases of the result presented     

On undershoot and nonminimum phase zeros

In this note we derive a simple necessary and sufficient condition for a stable system to exhibit an undershooting step response. Specifically, we show that undershoot occurs if and only if the plant has an odd number of real right-half plane zeros.     

On overshoot and nonminimum phase zeros

It is widely known that real nonminimum phase zeros lead to step response undershoot, and that the size of the undershoot necessarily tends to infinity as the settling time tends to zero. In this note, we show that the presence of two or more real nonminimum phase zeros can lead to step response overshoot in addition to undershoot. A lower bound on the overshoot is derived, and it is shown that the overshoot, like the undershoot, necessarily tends to infinity as the settling time tends to zero. The results are derived for single-input-single-output linear time-invariant continuous-time systems, and apply to both open-loop control and general two degree-of-freedom closed-loop control.     

On the simultaneous calculation of two zeros

Differentiable functions are approximated by parabolas the zeros of which are approximations for two zeros of the function. Using the calculated zeros for the construction of new approximating parabolas, it arises an iteration method with quadratic convergence at single zeros.     

On determining the zeros of state-space systems

The zeros of a linear, time-invariant, state-space system are defined, and a matrix rank test for determining their locus is formally established and illustrated by example.     

On the stabilization of discrete system zeros

The stabilization of discrete system zeros is commonly achieved by allowing the input pattern to deviate from zero-order hold (ZOH). In this paper, we show that this design technique may not be practical at small sampling times when the continuous system has unstable zero(s). Deviation of the input pattern from ZOH may also cause undesirable fluctuation in the inter-sampling-time output. In order to reduce this effect, we present a method which uses linear quadratic optimization for the determination of suitable zero locations.     

On zeros of exponential polynomials and quantum algorithms

We calculate the zeros of an exponential polynomial of some variables by a classical algorithm and quantum algorithms which are based on the method of van Dam and Shparlinski, they treated the case of two variables, and compare with the time complexity of those cases. Further we consider the ratio (classical/quantum) of the exponent in the time complexity. Then we can observe the ratio is virtually 2 when the number of the variables is sufficiently large.     

On the computation of transmission zeros of linear multivariable systems

Corrected and recomputed transmission zeros of a previously published numerical example of a realistic physical system are published for possible use as a test case for new computational algorithms.     

On zeros and responses of linear regulators and linear observers

In the pole assignment of the single input and output regulator, there is a case where the integrated square output becomes infinite and the response of the output has an infinitely large maximal amplitude when all poles of the closed loop system are assigned to negative infinity in the numericals-plane. This paper deeply investigates this problem from the linear structure point of view and applies the obtained results to the pole specification problem of observers.     

On interval polynomials with no zeros in the unit disc

We give a necessary condition for an interval polynomial to have no zeros in the closed unit disc. The condition is expressed in terms of the two first intervals     

On a test of programs for numerical zeros of polynomials

We describe a test method by which a program for polynomial equations can be examined with its specification about the input parameters and the output information. The test is systematically performed under desired conditions, and gives a practical criterion of a judgement on the reliability of the program.     

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.