Closed-loop Eigenvalues Research Articles

Non‐linear controller approach for an autonomous battery‐assisted photovoltaic system feeding an AC load with a non‐linear component

The conventional controller design of the autonomous photovoltaic (PV) energy system feeding a load is usually based on a dual-loop cascade controller structure. The controllers are designed based on the small signal equations of the non-linear system. However, most of the elements comprising the energy system are non-linear in nature such that controllers based on small signal models are not effective in all operability regions. Since the PV-battery-assisted energy system has four inputs and five controlled outputs, it is a non-square mult-input–multi-output system as well. To consider the non-linear behaviour of the system, the concept of feedback linearisation control algorithm is used which provides satisfactory performance, ensures input–output decoupled control on each of the outputs and gives the opportunity to achieve any level of dynamic performance through appropriate locations of the closed-loop eigenvalues and zeros. Different modes of operation of the autonomous, battery-assisted PV system and their controls are investigated and their boundaries of operation are determined. The proposed control algorithm is programmed to operate in all the modes of operation and it is shown that for various load demands, solar irradiation levels and the status of the battery, the control system set forth faithfully follows the reference points. These assertions are validated with computer simulation results.

Eigenvalue assignment in linear descriptor systems using dynamic compensators

An approach for eigenvalue assignment in strongly controllable and observable linear descriptor systems using dynamic compensators is proposed. Parametric expressions for the controller coefficient matrices are given. The approach assigns the full number of distinct finite closed-loop eigenvalues, guarantees the closed-loop regularity and overcomes the defects of some previous works. In addition, using the proposed eigenvalue assignment approach, a sufficient condition for generic eigenvalue assignability using dynamic compensators is proved.

Modal analysis of active flexible multibody systems containing PID controllers with non-collocated sensors and actuators

A method for performing modal analysis of undamped active flexible multibody systems with collocated sensors and actuators in a finite element environment was recently developed by the authors. In this paper, the theory is further expanded to include systems with non-collocated sensors and actuators, damping and steady-state error elimination. The closed-loop eigenvalue problem for active flexible multibody systems with multiple-input multiple-output proportional-integral-derivative (PID) feedback type controllers and multiple degrees of freedom finite element models is solved.

LMI-Based Control for Grid-Connected Converters With LCL Filters Under Uncertain Parameters

This paper presents a linear matrix inequality-based procedure for the design of state feedback current controllers that are robust to uncertainties on grid inductance for grid-connected converters with LCL filters. The state feedback control law includes the filter state variables, the states of resonant controllers, and considers also the delay from digital control implementation. As the main contribution here, the proposed strategy allows to deal with resonant controllers of arbitrary dimension, and provides results complying with IEEE Standard 1547 requirements. The control synthesis condition takes into account the assignment of the closed-loop eigenvalues in a circular region with radius r, located inside the unit circle, and the control design relies only on the choice of the value of the radius in the interval 0 <; r ≤ 1, providing all the state feedback gains in a simple and efficient procedure.

A unified method for optimal arbitrary pole placement

We consider the classic problem of pole placement by state feedback. We offer an eigenstructure assignment algorithm to obtain a novel parametric form for the pole-placing feedback matrix that can deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. This parametric formula is then exploited to introduce an unconstrained nonlinear optimisation algorithm to obtain a feedback matrix that delivers the desired pole placement with optimal robustness and minimum gain. Lastly we compare the performance of our method against several others from the recent literature.

Localization of Eigenvalues in Small Specified Regions of Complex Plane by State Feedback Matrix

This paper is concerned with the problem of designing discrete-time control systems with closed-loop eigenvalues in a prescribed region of stability. First, we obtain a state feedback matrix which assigns all the eigenvalues to zero, and then by elementary similarity operations we find a state feedback which assigns the eigenvalues inside a circle with center and radius. This new algorithm can also be used for the placement of closed-loop eigenvalues in a specified disc in z-plane for discrete-time linear systems. Some illustrative examples are presented to show the advantages of this new technique.

Damping of inter-area oscillations in multimachine systems

A classical lead-lag power system stabiliser is used for demonstration in this paper. Four machines ten bus system and ten machine 39 bus systems are considered. The stabiliser parameters are selected in such a manner to damp the inter area and rotor oscillations. The problem of selecting the stabiliser parameters is converted into a simple optimisation problem with an eigenvalue-based objective function and it is proposed to employ tabu search and simulated annealing for solving optimisation problem. The objective function allows the selection of the stabiliser parameters to optimally place the closed-loop eigenvalues in the left hand side of the complex s-plane. The tuning results of various combinations of machines are compared. Stability is confirmed through eigenvalue analysis and simulation results.

Comparison and Optimal Design of SSSC Controller Based on ICA and PSO for Power System Dynamic Stability Improvement

A new imperialist competitive algorithm (ICA)-based approach is proposed for optimal selection of the static synchronous series compensator (SSSC) damping controller parameters in order to shift the closed loop eigenvalues toward the desired stability. The optimal selection of the parameters for the SSSC controllers is converted to an optimization problem which is solved by recently developed evolutionary ICA method. This optimization algorithm has a strong ability to find the most optimistic results for dynamic stability improvement. Single machine infinite bus (SMIB) system has been considered to examine the operation of proposed controllers. The input power variation of generator is considered as a disturbance. The effectiveness of the proposed controller for damping low frequency oscillations is tested and results compared with particle swarm optimization (PSO). Also, the performance of proposed method is tested in dierent loading conditions. In addition, the potential and superiority of the proposed ICA method over the PSO is demonstrated. Also, performance of ICA in 10 times run is the same as in 1 time run. The simulation results analysis show that the designed ICA based SSSC damping controller has an excellent capability in damping low frequency oscillations and enhance rapidly and greatly the dynamic stability of the power systems, in compare to PSO technique.

Stabilization Of Dynamic Systems By Localization Of Eigenvalues In A Specified Interval

This paper is concerned with the problem of designing linear time-invariant control systems with closed-loop eigenvalues in a prescribed region of stability. First, we obtain a state feedback matrix which assigns all the eigenvalues to zero, and then by elementary similarity operations we find a state feedback which assigns the eigenvalues in the interval shown in figure 1. This new algorithm can also be used for the placement of closed-loop eigenvalues in a specified interval in z-plane and can be employed for large-scale linear time-invariant control systems. Some illustrative examples are presented to show the advantages of this new technique.

A method for controller parameter estimation based on perturbations

Simulation and prediction of eigenfrequencies and mode shapes for active flexible multibody systems is an important task in disciplines such as robotics and aerospace engineering. A challenge is to accurately include both controller effects and flexible body dynamics in a multidisciplinary system model appropriate for modal analysis. A method for performing modal analyses of such systems in a finite element environment was recently developed by the authors. On issue is, however, that for engineers working in a finite element environment, the controller properties are not always explicitly available prior to modal analyses. The authors encountered this problem when working with the design of a particular offshore windmill. The controller for the windmill was delivered in the form of a dynamic link library (dll) from a third party provider, and when performing virtual testing of the windmill design, it was of great importance to use the "real" controller in the form of the provided dll, rather than re-model it in for instance Simulink or EASY5. This paper presents a method for estimating the controller parameters of PID-type controllers when solving the closed-loop eigenvalue problem for active flexible multibody systems in a finite element environment. The method is based on applying incremental changes, perturbations, to relevant system variables while recording reactions from other system variables. In this work, the theory of the method is derived and the method is tested through several numerical examples.

Optimal Automatic Generation Control of Interconnected Power System Considering New Structures of Matrix Q

This article presents optimal automatic generation control regulator designs considering different methods of designing state cost-weighting matrix “Q.” The various aspects of observability and controllability of the system states, which influence the system dynamics directly or indirectly, are incorporated in the design of optimal automatic generation control regulators. For the investigation, a two-area interconnected power system with identical plants consisting of non-reheat turbines is considered. An AC transmission line in parallel a with DC transmission link is taken as area interconnection. The full state feedback control strategy has been applied to get optimal gains of the automatic generation control regulators. The pattern of closed-loop system eigenvalues, optimal gains, and system dynamic response plots in the wake of load disturbance are achieved with the implementation of the different optimal automatic generation control regulators to analyze the system stability and its dynamic performance.

Theoretical and experimental vibration analysis of rotating beams with combined ACLD and Stressed Layer Damping treatment

Active constrained layer damping (ACLD) treatment increases the efficiency of passive constrained layer damping (PCLD) treatment, but in case of circuit failure, only the decreased efficiency of PCLD treatment is available. The efficiency of the ordinary PCLD treatment can be enhanced by adding a stressed poly vinyl chloride (PVC) layers on the base beam instead of using viscoelastic materials. Hamilton Principle in conjunction with finite element method is used to derive the non-linear differential equations of motion for a rotating beam. Using proportional feedback controllers, the complex closed loop eigenvalue problem is developed and solved numerically. The effect of rotational speed of the beam, initial strain and other parameters of the PVC layer is investigated. To prove the effectiveness of the new technique, experimental investigations have also been carried out.

Robust control stability using the error loop

This paper formulates the error loop as a tool for designing robust stability control systems in the presence of structured and unstructured uncertainties. The error loop indicates that uncertainties can be accommodated through the design of the noise estimator, which is the unique feedback channel from plant to control. The real-time model that is embedded in the control unit and the noise estimator constitute a state predictor. The embedded model consists of a controllable dynamics plus a disturbance dynamics fed by the noise estimator. It is shown that causality constraint prevents perfect cancellation of causal uncertainties (unknown disturbance), but makes the control law which is fed by the state predictor to play a role, thus offering a further degree of freedom. Employing asymptotic expansions of the closed-loop transfer functions, simple, explicit design formulae derive from stability inequalities. They relate closed-loop eigenvalues to model parameters and requirements, and define an admissible frequency band for the state predictor bandwidth. This paper restricts formulation to the univariate case. A simple example is provided with simulated and experimental data.

Minimum norm partial quadratic eigenvalue assignment with time delay in vibrating structures using the receptance and the system matrices

The partial quadratic eigenvalue assignment problem (PQEAP) is to compute a pair of feedback matrices such that a small number of unwanted eigenvalues in a structure are reassigned to suitable locations while keeping the remaining large number of eigenvalues and the associated eigenvectors unchanged. The problem arises in active vibration control of structures. For real-life applications, it is not enough just to compute the feedback matrices. They should be computed in such a way that both closed-loop eigenvalue sensitivity and feedback norms are as small as possible. Also, for practical effectiveness, the time-delay between the measurement of the state and implementation of the feedback controller should be considered while solving the PQEAP. These problems are usually solved using only system matrices and do not necessarily take advantage of the receptances which are available by measurements.In this paper, we propose hybrid methods, combining the system matrices and measured receptances, for solutions of the multi-input PQEAP and the minimum-norm PQEAP, both for systems with and without time-delay. These hybrid methods are more efficient than the standard methods which only use the system matrices and not the receptances. These hybrid methods also offer several other computational advantages over the standard methods. Our results generalize the recent work by Ram et al. [Partial pole placement with time delay in structures using the receptance and the system matrices, Linear Algebra and its Applications 434 (2011) 1689–1696]. The results of numerical experiments demonstrate the effectiveness of the proposed methods.

Investigation of the Damping of Electromechanical Oscillations Using Power System Stabilizers (PSS) in Nigerian 330 kV Electrical Network

The study simu lated the behaviour of power system stabilizer (PSS) on automatic voltage regulator (A VR) and excitation system. It also developed an algorithm to investigate the transient and dynamic stability of the power systems. This was with a view to providing informat ion of damping rotor oscillations of synchronous generators. Three types of power systems were investigated: a single-machine-infinite-bus system with and without PSS, two generators connected to a load with various types of excitation controls and a mult imachine power system typified by Nigerian 330-kV electrical network. Tabu-search technique was used to tune the PSS-parameters for a single machine connected to an infinite bus operating at three different loading conditions. The objective function allowed the selection of the stabilizer parameters to optimally place the closed-loop eigenvalues in the left-hand side of a vertical line in the complex s -plane. The effectiveness of this suggested technique was confirmed through eigenvalue analysis. Time-do main simulat ions were also carried out on two generators connected to a load using MATLAB/Simu lin k. Two types of PSS were used for these simulat ions: generic and mult iband. The power systems were subjected to a balanced three-phase fault (most severe fault). The investigation was further extended to Nigerian 330-kV electrical network.

Optimal Automatic Generation Control of Interconnected Power System with Asynchronous Tie-lines under Deregulated Environment

This article presents the design of optimal automatic generation control regulators for an interconnected power system operating in a deregulated environment. An extra-high-voltage AC transmission line in parallel with a high-voltage DC link is considered as an area interconnection between the two areas. The dynamic model of the power system under consideration is developed by incorporating various types of possible market transaction environments. The designed optimal automatic generation control regulators are implemented, and dynamic system responses for various system states are obtained considering load perturbation in one of the areas. The patterns of open-loop and closed-loop eigenvalues are also determined to ensure system stability with and without optimal automatic generation control regulators. From the investigations carried out in the work, it is inferred that the dynamics performance of the system under consideration has an appreciable improvement when a high-voltage DC link is incorporated in parallel with an extra-high-voltage AC line as an interconnection between two areas as compared to that when only an extra-high-voltage AC line is considered as an area interconnection. The frequency deviations are more in the area in which distribution companies violate the contracts. Moreover, optimal automatic generation control regulators are able to demonstrate the matching of demand with generation under a deregulated environment with different market transactions. The power system model under investigation is marginally stable in open-loop operating mode, while it attains good stability margins in closed-loop mode.

Application of matrix perturbation theory in robust control of large-scale systems

The aim of this paper is to develop new results for robust centralized and decentralized control of large-scale systems using matrix perturbation theory. A new robustness measure is proposed which is more appropriate for large-scale systems than the existing measures. The use of this new index allows for an evaluation of system eigenvalue sensitivity to perturbations without calculating the eigenvectors and the condition number of the modal matrix. In addition, a novel robust decentralized controller design method is proposed to assign the closed-loop eigenvalues of the large-scale system in a desired region. This method is based on eigenstructure assignment of isolated subsystems. The assignment of overall closed-loop poles in the desired region is guaranteed provided that certain sufficient conditions for the isolated subsystems are satisfied. Simulation results are given to show the efficiency of the new methods.

Partial Pole Placement with Controller Optimization

An arbitrary subset (n - m) of the (n) closed loop eigenvalues of an n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> order continuous time single input linear time invariant system is to be placed using full state feedback, at pre-specified locations in the complex plane. The remaining m closed loop eigenvalues can be placed anywhere inside a pre-defined region in the complex plane. This region constraint on the unspecified poles is translated into a linear matrix inequality constraint on the feedback gains through a convex inner approximation of the polynomial stability region. The closed loop locations for these m eigenvalues are optimized to obtain a minimum norm feedback gain vector. This reduces the controller effort leading to less expensive actuators required to be installed in the control system. The proposed algorithm is illustrated on a linearized model of a 4-machine, 2-area power system example.

Impacts of Wind Power Variations on Frequency Related Power System Operations

Wind power is seen as the most cost effective way to generate electricity from renewable sources. The wind turbine prime mover, wind, is uncontrollable as compared to the conventional power plant prime mover. Therefore, it becomes very important to carry out investigations on the dynamic behavior of wind power generating systems. In this paper, the dynamic model of 1 MVA unit is extrapolated from 100 kW unit existing in NASA –Lewis Research Centre. The various types of investigations are carried out to study the dynamic performance of various states of the model considering variations in the wind speed. At the outset of the work, state space model of the system is developed. To study the dynamic behavior of the system, optimal controllers are designed using full state feedback control strategy. Following the controller designs, the closed loop system eigenvalues and dynamic response plots are obtained. The Strip Eigenvalue Assignment method is applied to design sub-optimal controllers using feedback of few states which are accessible for their observation and measurement. The comparative study of closed loop eigenvalues and dynamic response plots obtained for various operating conditions shows a comparable system dynamic performance. The optimal controllers are designed for various operating conditions using pole placement technique. The dynamic response plots and closed loop eigenvalues are obtained for various system states considering various operating conditions. The investigations of these reveal that the implementation of optimal controllers offer not only good dynamic performance, also ensure system dynamic stability. KeywordsThe Strip Eigenvalue Assignment method; dynamic response plots; optimal controllers; eigenvalue

CONDITIONING OF STATE FEEDBACK POLE ASSIGNMENT PROBLEMS

In [26,27,35], condition numbers and perturbation bounds were produced for the state feedback pole assignment problem (SFPAP), for the single- and multi-input cases with simple closed-loop eigenvalues. In this paper, we consider the same problem in a different approach with weaker assumptions, producing simpler condition numbers and perturbation results. For the SFPAP, we shall show that the absolute condition number $\kappa \leq c_0 \|B^{\dagger}\| $ $ \left[ \kappa_X + \left(1 + \|F\|^2 \right)^{1/2} \right]$, where the closed-loop system matrix $A+BF = X \Lambda X^{-1}$, the closed-loop spectrum in $\Lambda$ is pre-determined, $\kappa_X \equiv\|X\| \|X^{-1}\|$, the operators $P_c (\cdot) \equiv (A+BF)(\cdot) - (\cdot) \Lambda$ and $\mathcal{N}(\cdot) \equiv (I - BB^{\dagger}) P_c(\cdot)$, and $c_0 \equiv \|I(\cdot) -P_c\left[ \mathcal{N}^{\dagger}(I - BB^{\dagger}) (\cdot) \right]\|$. With $c_B \!\equiv\!\|B\| \|B^{\dagger}\|$ and $c_1 \!\equiv\! (\|B\| \|F\|)^{-1}$, the relative condition number $\kappa_r \!\leq c_0 c_B \left[ c_1 \kappa_X \|\Lambda\| + \right. \left. \left( c_1^2 \|A\|^2 + 1 \right)^{1/2} \right]$. With $B$ well-conditioned and $\Lambda$ well chosen, $\kappa$ and $\kappa_r$ can be small even when $\Lambda$ (not necessary in Jordan form) possesses defective eigenvalues, depending on $c_0$. Consequently, the SFPAP is not intrinsically ill-conditioned. Similar results were obtained in [23], although differentiability was not established for its local perturbation analysis. Simple as well as general multiple closed-loop eigenvalues are treated.