An algorithm for robust pole assignment via polynomial approach

Some points of the paper by N.K. Nichols (see ibid., vol.AC-31, p.643-5, 1986), concerning the robust pole assignment of linear multiinput systems, are clarified. It is stressed that the minimization of the condition number of the closed-loop eigenvector matrix does not necessarily lead to robustness of the pole assignment. It is shown why the computational method, which Nichols claims is robust, is in fact numerically unstable with respect to the determination of the gain matrix. In replying, Nichols presents arguments to support the choice of the conditioning of the closed-loop poles as a measure of robustness and to show that the methods of J Kautsky, N. K. Nichols and P. VanDooren (1985) are stable in the sense that they produce accurate solutions to well-conditioned problems.

Pole assignment is a well-established controller synthesis technique used in control engineering. An important drawback of the pole assignment technique is poor numerical accuracy of the resulting compensators especially for high order systems, when fixed precision arithmetic is used in calculations. Obviously there are several parameters that affect the performance of the pole assignment algorithms. It has been, for example, already shown that increasing the order of the system causes an exponential increase in the numerical errors made in pole assignment. In this paper, the effect of the ratio of real poles to all poles of the open-loop (and closed-loop) system (real pole ratio - RPR) to the performance of the pole assignment algorithms is investigated for the first time.

CHAPTER TEN - ROOTS OF POLYNOMIALS

A new pole-placement algorithm for multi-input controllable systems is presented. The algorithm can be used to obtain real-valued feedback gains for arbitrarily specified real and complex, distinct or multiple eigenvalues. The multi-input controllable time-invariant system is converted to a single-input system. Pole assignment is achieved by making use of certain properties of the equation TF - AT = D , where A is the open-loop system matrix and F represents a matrix having the desired closed-loop eigenvalues.

In this paper, the Riemann analytic continuation formula (RACF) is derived from Euler’s quadratic equation. A nonlinear function and a polynomial function that were required in the derivation were also obtained. The connections between the roots of Euler’s quadratic equation and the Riemann zeta function (RZF) are also presented in this paper. The method of partial summation was applied to the series that was obtained from the transformation of Euler’s quadratic equation (EQE). This led to the derivation of the RACF. A general equation for the generation of the zeros of the analytic continuation formula of the Riemann zeta equation via a polynomial approach was also derived and thus presented in this work. An expression, which was based on a polynomial function and the products of prime numbers, was also obtained. The obtained function thus afforded us an alternative approach to defining the analytic continuation formula of the Riemann Zeta equation (ACFR). With the new representation, the Riemann zeta function was shown to be a type of function. We were able to show that the solutions of the RACF are connected to some algebraic functions, and these algebraic functions were shown to be connected to the polynomial and the nonlinear functions. The tables and graphs of the numerical values of the polynomial and the nonlinear function were computed for a generating parameter, k, and shown to be some types of the solutions of some algebraic functions. In conclusion, the RZF was redefined as the product of a derived function, R(tn,s), and it was shown to be dependent on the obtained polynomial function.

В работе развивается полиномиальный подход к синтезу оптимальных регуляторов пониженного порядка для одноканальных линейных стационарных систем, описываемых дифференциально-алгебраическими уравнениями. Критические корневые диаграммы и корневые многочлены применялись для синтеза оптимальных регуляторов в классических системах автоматического управления, а также в системе, описываемой дифференциально-алгебраическими уравнениями (ДАУ системе), включающей неустойчивый объект управления, заданный неправильной передаточной дробью с числителем шестой и знаменателем четвертой степени, и ПИ3-регулятор (пропорционально-интегральный). Здесь этот же объект стабилизируется ПИ-регулятором, который включает апериодическую составляющую и обеспечивает лучшую степень устойчивости замкнутой системы и возможность упрощения его структуры без существенного снижения эффективности. Схема метода по сравнению с классическими системами управления не меняется; переходная характеристика подтверждает астатизм и безымпульсность ДАУ-системы; некоторые особенности отмечены в заключении. We explore a polynomial approach for design of optimal low-order controllers for SISO LTI DAE control systems. The critical root diagram technique and root polynomials were developed for optimal controllers design of classical control systems; both notions were successfully used for design of the DAE system including an unstable plant defined by an incorrect transfer fraction with a 6th degree numerator and a 4th degree denominator, closed by a PI3-controller. In our paper the same plant is stabilized by a PI-controller, which includes an aperiodic term; it provides better relative stability of the closed loop system and a possibility of simplifying controller’s structure without noticeable decrease of efficiency. The scheme of the method is the same as in classical control systems; it involves forming a root polynomial and then a system of polynomial equations; the solution or lack of it allows establishing whether pole location of a closed-loop system is realized in accordance with the critical root diagram; optimal and sub optimal values of control parameters in the case are found using the same equations. The transient response confirms an a statism and impulse free ness of the closed loop DAE system. Thus, the polynomial approach turns out to be a useful alternative to state-space design methods in DAE systems. Some features and problems of DAE control system polynomial design are mentioned in the conclusion.

A pole assignment algorithm was recently developed by considering the sensitivity of closed loop eigenvalues to each gain element. The theory is extended here to incorporate the Linear Quadratic Regulator formulation so that the closed loop system satisfies optimality as well as eigenvalue constraints. The approach is to compute the sensitivity of the Hamiltonian matrix to changes in the state weighting matrix, in contrast to the sensitivity of the closed loop system in the pole placement problem. Examples from a six degree-of-freedom lumped-mass model and an Euler-Bernoulli pinned beam system are investigated to demonstrate the procedure. It is shown that the resulting closed loop system behavior sufficiently approximates the desired response.

A method is presented to place a number of system eigenvalues by means of output feedback, and to use any remaining design freedom to minimize a specified performance functional. The solution for this combination of optimal control and pole assignment is shown to be similar to that for the complete state feedback case and uses a novel pole assignment algorithm.

This paper considers a new algorithm for discrete time adaptive pole assignment which incorporates the internal model principle and hence allows for rejection of deterministically modelled disturbances and tracking of simple reference signals. A complete global stability analysis is given.

The combination of the automatic control theory courses, simulation verification and practical implementation of the designed controller algorithms in real-time conditions is very important for training of the control engineers.This contribution present structure and usage of Self-tuning Controllers Simulink Library (STCSL) for real time control.The STCSL was created for design, simulation verification and especially realtime implementation of single input -single output (SISO) digital self-tuning controllers.The proposed adaptive controllers, which are included in the library, can be divided into three groups (PID controllers, controllers based on the polynomial approach and the controllers derived on the other approaches (minimum variance etc.).This Library is very successfully used in Adaptive Control Course in education practice for design and verification of self-tuning control systems in simulation and real-time conditions.It is suitable also for design and verification of industrial digital controllers.The highly nonlinear laboratory model, the DR300 Speed Control with Variable Load, has been chosen as example for real-time control.

Considering that the link states of some practical networks are not measurable, it is impossible to control the links system directly to achieve structural balance. In order to handle this problem, the state observer design and the structural balance tracking control for the links system with complex dynamical networks are investigated. The dynamical model of the links system is molded as the Riccati matrix difference equation, which contains a coupling term associated with external stimulations. For the discrete-time model, the pole assignment algorithm is adopted to calculate the gain matrix for the observer of the links system. It could guarantee that the estimation error of observer for links system is globally asymptotically stable. Further, combining the effective estimation information of observer and external stimulations, an asymptotically tracking controller is proposed, which could achieve the asymptotic tracking of the reference structure balance network to the links system. Finally, the validity of the proposed method is verified by the simulation example.

Comparison of two polynomial approaches in performance analysis for periodic piecewise polynomial systems

The problem of pole assignment in linear, multivariable systems using an unconstrained-rank feedback matrix is considered. The effect of output feedback of unspecified rank on the characteristic polynomial of a multivariable system is first studied. The results are then used to derive a recursive algorithm for pole assignment without any restrictions on the rank of the output feedback matrix used. The method is based on the pseudoinverse concept for obtaining least-squares solutions of sets of linear equations, and is computationally efficient.

An efficient computational algorithm for pole assignment of linear n-th order m-input systems is proposed. The algorithm requires a small multiple of n3 operations and approximately n2 + (n + m)2 working precision words space. A Fortran program is developed and tested for high-order and ill-conditioned problems.

The presence of non-conforming components instead of, or in addition to, the usual assembly errors results in N- or W-shaped hazard rate (HR) functions rather than the usual bathtub (i.e. U-shaped) ones. Although there have been numerous models for bathtub-shaped HR functions, N- and W-shaped HR functions are usually modelled using mixtures of two or more distributions. While this approach does sometimes lead to tidy interpretation, there can be a degree of overparameterization, with consequent problems in stability and fitting. For this reason, the present paper revisits the natural approach of modelling N- and W-shaped HR functions using polynomial functions of degree three or four. Although the non-negativity of the hazard rate function becomes non-trivial, this ensures a minimal number of parameters. The polynomial approach also allows the use of a parametric model without imposing a particular shape of hazard rate function on the data, which usually requires a non-parametric approach. The possible hazard rate shapes obtainable are characterized, and detailed formulae for local minima and maxima of the functions provided. The performance of the models is compared to that of several generalizations of the Weibull distribution, with promising results.