System Characteristic Polynomial Research Articles

State feedback control for stabilization of PMSM ‐based servo‐drive with parametric uncertainty using interval analysis

For a class of multivariable uncertain dynamic systems, the parametric uncertainties are belonging to a closed interval with lower and upper boundaries a priori known. Thereby, these systems can be analyzed based on interval structures and interval matrices. In this article, a fully interval analysis–based method is developed and applied to the state feedback control of permanent magnet synchronous motor (PMSM) in order to design a stabilized servo-drive. Indeed, the parametric uncertainties are investigated in state space representation, which represents a simplified and effective way to analyze the robust stability for the interval system. Firstly, a robust state feedback control technique, dedicated to an uncertain system is introduced. For that, the robust controllability test is performed for the interval system by using the linear independency condition of column interval vectors. It is proven that there is a direct correlation between the controllability test and the existence of a robust modal P-regulator for the correction of the uncertain system. It is also shown that it invariably relies on each input's controllability indices and thus their effect on the uncertain system's state variables. In order to ensure stability in a closed loop, the modal P-regulator is designed with possibility of incorporation of an integral action. The modified modal PI regulator has the ability to reject disturbance and guarantee zero-steady-state error for step inputs. In fact, the stability is achieved by placing all coefficients of the system characteristic polynomial within assigned intervals based on Kharitonov's Theorem. The technique provides a matrix gain with interval coefficients for the stabilizing regulator. Finally, the developed approach is applied to position control of linearized model of the PMSM-based servo-drive, presenting parametrical uncertainties. To demonstrate the efficiency of the proposed method, a numerical and graphical comparison of conventional LQR and pole placement, state feedback controllers for the PMSM servo-drive with the robust interval controller is provided. In order to verify the feasibility of the whole proposed technique, calculations and simulations are performed by using Matlab/Intlab toolbox. Real-time simulation is also investigated using Lab-View Compact-RIO.

A novel two-polynomial criteria for higher-order systems stability boundaries detection and control

There are many methods for identifying the stability of complex dynamic systems. Routh and Hurwitz’s criterion is one of the earliest and commonly used analytical tools analysing the stability of dynamic systems. However, it requires tedious and lengthy derivations of all components of the Routh array to solve the stability problem. Therefore, it is not a simple method to define analytically, stability boundaries for the coefficients of the system characteristic equation.The proposed brand-new criterion is an effective alternaztive technique in identifying stabilityhigher-order linear time-invariant dynamic system that binds the coefficients of the system characteristic polynomial at the stability boundaries by means of an additional single constantk. It defines the necessary and sufficient conditions for the absolute stability of higher-order dynamic systems. It also allows the analysing of the system’s precise marginal stabilityor marginal instability condition when the roots are relocated on imaginary jω-axis of s-plane. The criterion proposed bythe authors, in contrast to Routh criteria, simplifies the identification of maximum and minimum stability limits for any coefficient of the higher-order characteristic equation significantly. The derived in the paperstability boundary formulas for the polynomial coefficients are successfully used for the proportional integral derivative (PID) controller with single or multiple gains selections in closed-loop control system.

Synthesis of Discrete Modal SISO Controllers by Engineering Performance Indices

The linear discrete minimum-phase plant with one control input, one measured output and with an unmeasured external disturbance is considered. The problem is stated of the design of the discrete output controller that provides the preset engineering performance indices: control error, settling time, stability margins radius. Analytical formulas are obtained that allow to relate the engineering performance indices with the desired closed system characteristic polynomial roots in the modal control problem when choosing real non-negative roots. At the same time, the specified control error and the settling time lead to the constraints on the maximum absolute value of the desired roots, while the values of the desired roots, providing the given stability margins radius, may not satisfy these constraints, so the specified values of the performance indices should be revised. Examples are given showing the effectiveness of the approach proposed.

On the order reduction of MIMO large-scale systems using block-roots of matrix polynomials

The present paper deals with the problem of approximating linear time invariant MIMO large-scale systems with reduced order system via the help of the so called block-moment matching method based on the dominance exist between solvents of the system characteristic matrix polynomial, where the Block-roots are reconstructed using a new proposed procedure. The validation and study of accurate approximation is done by a specified performance index called pulse energy criterion. The necessary condition for correctness and applicability of the proposed method is the block-controllablity or block-observability. Finally, for the demonstration of the proposed method efficiency a numerical example is illustrated.

Optimal supplementary frequency controller design using the wind farm frequency model and controller parameters stability region

In most of the existing studies, the frequency response in the variable speed wind turbines (VSWTs) is simply realized by changing the torque set-point via appropriate inputs such as frequency deviations signal. However, effective dynamics and systematic process design have not been comprehensively discussed yet. Accordingly, this paper proposes a proportional-derivative frequency controller and investigates its performance in a wind farm consisting of several VSWTs. A band-pass filter is deployed before the proposed controller to avoid responding to either steady state frequency deviations or high rate of change of frequency. To design the controller, the frequency model of the wind farm is first characterized. The proposed controller is then designed based on the obtained open loop system. The stability region associated with the controller parameters is analytically determined by decomposing the closed-loop system's characteristic polynomial into the odd and even parts. The performance of the proposed controller is evaluated through extensive simulations in MATLAB/Simulink environment in a power system comprising a high penetration of VSWTs equipped with the proposed controller. Finally, based on the obtained feasible area and appropriate objective function, the optimal values associated with the controller parameters are determined using the genetic algorithm (GA).

A THEORETICAL-EXPERIMENTAL APPROACH FOR ELASTO-DAMPING PARAMETERS ESTIMATION OF CONE INERTIAL CRUSHER MOUNTING

The present paper deals with estimation of the elasto-damping parameters of a cone inertial crusher mounting. The numerical values of these parameters are crucial for accurate reproduction of the machine vibrational behavior and dynamical model adequacy. Due to the significant difficulties arising during the purely theoretical determination of the stiffness and damping parameters of the rubber vibroisolators it is well-suited to use a theoretical-experimental approach. The developed approach is based on the theoretical determination of the mounting stiffness parameters as a function of two experimentally measured natural frequencies of the mechanical system. The crusher is represented as a six degrees of freedom system with two planes of symmetry. By using the system characteristic polynomial, the theoretical derivation of mathematical relationships for the mechanical system natural frequencies as a function of stiffness, inertial and geometrical parameters is performed. A good agreement is shown when comparing the experimental and the theoretical results for the system kinematical characteristics.

Interval Analysis Approach to Prototype the Robust Control of the Laboratory Overhead Crane

The paper describes the software-hardware equipment and control-measurement solutions elaborated to prototype the laboratory scaled overhead crane control system. The novelty approach to crane dynamic system modelling and fuzzy robust control scheme design is presented. The iterative procedure for designing a fuzzy scheduling control scheme is developed based on the interval analysis of discrete-time closed-loop system characteristic polynomial coefficients in the presence of rope length and mass of a payload variation to select the minimum set of operating points corresponding to the midpoints of membership functions at which the linear controllers are determined through desired poles assignment. The experimental results obtained on the laboratory stand are presented.

Set‐membership parity space approach for fault detection in linear uncertain dynamic systems

SummaryIn this paper, a set‐membership parity space approach for linear uncertain dynamic systems is proposed. First, a set of parity relations derived from the parity space approach is obtained by means of a transformation derived from the system characteristic polynomial. As a result of this transformation, parity relations can be expressed in regressor form. On the one hand, this facilitates the parameter estimation of those relations using a zonotopic set‐membership algorithm. On the other hand, fault detection is then based on checking, at every sample time, the non‐existence of a parameter value in the parameter uncertainty set such that the model is consistent with all the system measurements. The proposed approach is applied to two examples: a first illustrative case study based on a two‐tank system and a more realistic case study based on the wind turbine fault detection and isolation benchmark in order to evaluate its effectiveness. Copyright © 2014 John Wiley & Sons, Ltd.

Interval arithmetic-based fuzzy discrete-time crane control scheme design

Abstract In many manufacturing segments, container terminals and shipping yards the automation of material handling systems is an important element of enhancing productivity, safety and efficiency. The fast, precise and safe transfer of goods in crane operations requires a control application solving the problems, including non-collision trajectory planning and limitation of payload oscillations. The paper presents the interval arithmetic-based method of designing a discrete-time closed-loop anti-sway crane control system based on the fuzzy interpolation of linear controller parameters. The interval analysis of a closed-loop control system characteristic polynomial coefficients deviation from their nominal values is proposed to define a minimum number of fuzzy sets on the scheduling variables universe of discourse and to determine the distribution of triangular-shaped membership functions parameters, which satisfy the acceptable range of performances deterioration in the presence of the system’s parameters variation. The effectiveness of this method was proved in experiments conducted using the PAC system on the laboratory scaled overhead crane.

Evolutionary optimization of interval mathematics-based design of a TSK fuzzy controller for anti-sway crane control

Abstract A hybrid method combining an evolutionary search strategy, interval mathematics and pole assignment-based closed-loop control synthesis is proposed to design a robust TSK fuzzy controller. The design objective is to minimize the number of linear controllers associated with rule conclusions and tune the triangular-shaped membership function parameters of a fuzzy controller to satisfy stability and desired dynamic performances in the presence of system parameter variation. The robust performance objective function is derived based on an interval Diophantine equation. Thus, the objective of a fuzzy logic-based control scheme is to place all the closed-loop control system characteristic polynomial coefficients within desired intervals. The reproduction process in the proposed Evolutionary Algorithm (EA) is based on the arithmetical crossover, uniform and non-uniform mutation along with gene deletion/insertion mutation ensuring a diversity of genomes sizes, as well as a diversity in the parameter space of membership functions. The proposed algorithm was implemented to design a fuzzy logic-based anti-sway crane control system taking into consideration the rope length and the mass of a payload variation. The results of experiments conducted using the EA for different conditions assumed for system parameter intervals and desired closed-loop system performances are compared with results achieved using the iterative procedure which is also described in the paper.

Linear dynamical systems over finite rings

The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem of determining whether the system is a fixed point system can be answered by computing and factoring the system's characteristic polynomial and minimal polynomial. It has become clear recently that the study of finite linear dynamical systems must be extended to embrace finite rings. The difficulty of dealing with an arbitrary finite commutative ring is that it lacks of unique factorization. In this paper, an efficient algorithm is provided for analyzing the cycle structure of a linear dynamical system over a finite commutative ring. In particular, for a given commutative ring R such that | R | = q , where q is a positive integer, the algorithm determines whether a given linear system over R n is a fixed point system or not in time O ( n 3 log ( n log ( q ) ) ) .

Critical stability constraints for discrete-time linear systems

Critical stability constraints are a small set of conditions that are enough to maintain the stability of a system when some parameters are perturbed from a nominal stable setting. The paper uses a recently introduced efficient integer-preserving (IP) form of the Bistritz test to derive critical constraints for stability of discrete-time linear systems. The new procedure produces polynomial (rather than rational) constraints of particularly low degree whose variates are the free parameters (or the literal coefficients) of the system's characteristic polynomial. Comparison with the modified Jury test, also an efficient IP stability test, shows that the constraints are obtained with less computation and, more contributing to simplicity, the constraints appear as polynomials with degrees lower by a factor of two.

Accurate Computation of Traffic Workload Distributions

In traffic modeling, the computation of workload distributions plays a prominent role, since they influence the quality of service parameters. This paper deals with different ways to determine the workload distributions of a GI/GI/1 service system accurately. We discuss two different techniques, the polynomial factorization approach and the Wiener–Hopf factorization. These methods factorize the server system's characteristic polynomial to compute the workload. Starting with classical algorithms to obtain initial approximations, we enhance them using verification techniques to guarantee the correctness of the results.

Using interval arithmetic for robust state feedback design

A problem of a modal P-regulator synthesis for a linear multivariable dynamical system with uncertain (interval) parameters in state space is considered. The designed regulator has to place all coefficients of the system characteristic polynomial within assigned intervals. We have developed the approach proposed earlier in Dugarova and Smagina (Avtomat. i Telemech. 11 (1990) 176) and proved a direct correlation between interval system controllability and existence of robust modal P-regulator.

A New Approach to the Modal Regulator Synthesis for Interval Plant with Scalar Input.

We consider the problem of a closed-loop interval dynamical system stabilisation by a modal regulator which locates all coefficients of the open-loop system characteristic polynomial within assigned intervals. We present an original method for the modal regulator synthesis. This method differs from the analytic method [6] by the smaller number of operations with polynomial matrices. Also the method does not use the system canonical transformation as it is realized in [3], [11].

Decoupling and coefficient assignment for (A, B, C, D) time-delay systems

Using V as the time-delay operator, a linear invariant time-delay system is expressed by a quadruple (A(δ)), B(δ) C(V) D(V)) with the matrices over a polynomial ring in V. This paper studies the decoupling and coefficient assignment of such a system. For a class of time-delay systems non-controllable over the polynomial ring in V but controllable over the field of rational functions in V, we present sufficient conditions under which the system characteristic polynomial may be determined arbitrarily by feasible control laws. Two examples are investigated to illustrate the theory.

Recovering the poles from third-order cumulants of system output

The problem of identifying the poles of single-input/single-output (SISO) linear stochastic systems from the higher-order statistics of noisy observations is considered. It is assumed that the system is driven by an independent and identically distributed non-Gaussian process with nonzero third-order cumulant function at zero lag. There is no other restriction of the probability distribution of the driving noise. The system is assumed to be of known order, causal, and exponentially stable, but is not required to be minimum phase. The system output is observed in additive, possibly non-Gaussian, noise. It is shown that if there are no pole-zero cancellations in the transfer function of the given system, then it is necessary and sufficient for a block Hankel matrix to have rank equal to the system order where the matrix is constructed from a partial set of third-order cumulants of the noisy output sequence. This fundamental result then leads to a linear solution to the problem of estimating the coefficients of the system characteristic polynomial from which the system poles can be found via root-finding.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Iterative method for obtaining the argument of the dominant complex pole-pair

A simple and efficient method is presented for obtaining the argument of the dominant complex pole-pair of a discrete-time system. This new method can be directly applied to either the system characteristic polynomial or the state transition matrix, whichever is available. Moreover, it can easily be extended to extract the values of the arguments of all complex pole-pairs of the system under consideration.

State space representation of interconnected power systems for dynamic interaction studies

In this paper a linearized model of an interconnected power system in state space form is presented in order to analyse the dynamic interactions of its components across the interconnection network. The model, modularly structured, proves flexible and includes detailed descriptions of generators with their control devices, static and dynamic loads. The mathematical representation of the system is formulated such that the effect of interconnection on system characteristic polynomial and system dynamic stability can be systematically investigated. Additionally, a suitable index is defined to measure the degree of dynamic interaction which may occur between the machines of the interconnected power system. A numerical example is included using a four-generator ten-bus system to illustrate the capability of the developed model and the usefulness of the proposed technique.

Comrade matrix and systems excited by colored noise

Stable linear systems described by state variable equations are considered where the input process is stationary and, in general, nonwhite. We show that there exists a natural coordinate basis for the system, the main properties for which imply that the state covariance matrix is a multiple of the identity, the system matrix itself is a comrade matrix, and the orthonormal polynomial set used to define the comrade matrix is obtained from the input spectrum and the system characteristic polynomial.