Feedback Gain Vector Research Articles

Synchronization of A Class of Uncertain Chaotic Systems with Lipschitz Nonlinearities Using State‐Feedback Control Design: A Matrix Inequality Approach

AbstractThis paper proposes a new state‐feedback stabilization control technique for a class of uncertain chaotic systems with Lipschitz nonlinearity conditions. Based on Lyapunov stabilization theory and the linear matrix inequality (LMI) scheme, a new sufficient condition formulated in the form of LMIs is created for the chaos synchronization of chaotic systems with parametric uncertainties and external disturbances on the slave system. Using Barbalat's lemma, the suggested approach guarantees that the slave system synchronizes to the master system at an asymptotical convergence rate. Meanwhile, a criterion to find the proper feedback gain vector F is also provided. A new continuous‐bounded nonlinear function is introduced to cope with the disturbances and uncertainties and obtain a desired control performance, i.e. small steady‐state error and fast settling time. Several criteria are derived to guarantee the asymptotic and robust stability of the uncertain master–slave systems. Furthermore, the proposed controller is independent of the order of the system's model. Numerical simulation results are displayed with an expected satisfactory performance compared to the available methods.

TLBO optimized sliding mode controller for multi-area multi-source nonlinear interconnected AGC system

Abstract This paper represents design of output feedback sliding mode controller (SMC) for multi area multi-source interconnected power system. After designing output feedback SMC, teaching and learning based optimization (TLBO) technique is utilized to optimize feedback gain and switching vector of the controller. The superiority of the proposed approach is shown by comparing the result with output feedback tuned SMC with differential evolution and particle swarm optimization and state feedback SMC tuned with genetic algorithm for a two area thermal interconnected power system. Further, the proposed approach is extended to multi-area multi-source non linear automatic generation control (AGC) system with/without HVDC link. First area consists up thermal, hydro and gas; second area consists up thermal, hydro and nuclear as generating unit. Additionally, the superiority of proposed approach is shown by sensitivity analysis, which is carried out with wide changes in system parameters.

Consensus seeking in a network of discrete-time linear agents with communication noises

This paper studies the mean square consensus of discrete-time linear time-invariant multi-agent systems with communication noises. A distributed consensus protocol, which is composed of the agent's own state feedback and the relative states between the agent and its neighbours, is proposed. A time-varying consensus gain a[k] is applied to attenuate the effect of noises which inherits in the inaccurate measurement of relative states with neighbours. A polynomial, namely ‘parameter polynomial’, is constructed. And its coefficients are the parameters in the feedback gain vector of the proposed protocol. It turns out that the parameter polynomial plays an important role in guaranteeing the consensus of linear multi-agent systems. By the proposed protocol, necessary and sufficient conditions for mean square consensus are presented under different topology conditions: (1) if the communication topology graph has a spanning tree and every node in the graph has at least one parent node, then the mean square consensus can be achieved if and only if ∑∞k = 0a[k] = ∞, ∑∞k = 0a2[k] < ∞ and all roots of the parameter polynomial are in the unit circle; (2) if the communication topology graph has a spanning tree and there exits one node without any parent node (the leader–follower case), then the mean square consensus can be achieved if and only if ∑∞k = 0a[k] = ∞, limk → ∞a[k] = 0 and all roots of the parameter polynomial are in the unit circle; (3) if the communication topology graph does not have a spanning tree, then the mean square consensus can never be achieved. Finally, one simulation example on the multiple aircrafts system is provided to validate the theoretical analysis.

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.

On the Pole-Placement and the Observer for Linear Time-Varying Systems

A simple design method of the Luenberger observer for linear time-varying systems is proposed in this paper. The paper first propose the simple calculation method to derive the pole placement feedback gain vector for linear time-varying systems. For this purpose, it is shown that the pole placement controller can be derived simply by finding some particular "output signal" such that the relative degree from the input to this output is equal to the order of the system. Using this fact, the feedback gain vector can be calculated directly from plant parameters without transforming the system into any standard form. Then, this method is applied to the design of the observer, i.e., because of the duality of linear time-varying system, the state observer can be derived by un-stabilization of the state error equation.

Partial Pole Placement and Controller Norm Optimization over Polynomial Stability Region

AbstractAn arbitrary subset (n – m) of the n closed loop eigenvalues of an nth order continuous time single input linear time invariant (LTI) 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 an ellipsoidal constraint on the characteristic polynomial coefficients through a convex inner approximation for polynomial stability regions. The closed loop locations for these m eigenvalues are chosen through an explicit minimization of the feedback gain vector norm leading to an efficiently solvable semidefinite program. The required controller effort is thus minimized leading to less expensive actuators.

Robust High Bandwidth Discrete-Time Predictive Current Control with Predictive Internal Model—A Unified Approach for Voltage-Source PWM Converters

This paper presents a robust high bandwidth discrete-time predictive current control scheme for voltage-source pulsewidth-modulated (VS-PWM) converters. First, to achieve high bandwidth current control characteristics, a digital predictive current controller with delay compensation is adopted. The compensation method utilizes a current observer with an adaptive internal model for system uncertainties. The predictive nature of both the current observer and the internal model forces the delays elements to be equivalently placed outside the closed loop system. Second, to ensure perfect tracking of the output current in the presence of uncertainties and providing means for attenuating low- and high- frequency system disturbances, the frequency modes of the disturbances to be eliminated should be included in the stable closed loop system. Toward this, an adaptive internal model for the estimated uncertainty dynamics is proposed. To cope with the high bandwidth property of the lump of uncertainties in VS-PWM converter applications, the disturbance slowly varying assumption is relaxed in the proposed controller. The relaxation is achieved by adopting a curbing sliding-mode-based feedback gain vector within the internal model observation system. Comparative evaluation tests were carried out on a grid-connected VS-PWM converter and a direct drive permanent magnet synchronous motor (DD-PMSM) drive system to demonstrate the validity and effectiveness of the proposed control scheme at different operating conditions.

Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal

Abstract In this paper, a drive-response synchronization method with linear output error feedback is presented for synchronizing a class of fractional-order chaotic systems via a scalar transmitted signal. Based on stability theory of fractional-order systems and linear system theory, a necessary and sufficient condition for the existence of the feedback gain vector such that global synchronization between the fractional-order drive system and response system can be achieved and its design method are given. This synchronization approach that is simple, global and theoretically rigorous enables synchronization of fractional-order chaotic systems be achieved in a systematic way and does not require the computation of the conditional Lyapunov exponents. An example is used to illustrate the effectiveness of the proposed synchronization method.

Modifying modal characteristics of sound fields by state feedback control.

State feedback control of a sound field is investigated for improving the irregular distribution of the natural frequencies at low frequencies. A state-space description of a sound field is derived from an inhomogeneous wave equation using the finite element method. The feedback control system is realized as a feedback filter between sensor microphones and control sources using a state estimator for a linear dynamic system. Pole allocation is employed for calculating the state feedback gain vector, such that the roots of the closed-loop system have the desired modal distribution. Computer simulations are performed to demonstrate the control achieved by distributing the lowest undamped natural frequencies in a uniform manner. As a result, the transfer functions with the state feedback control are modified so as to have peaks at identical intervals for the frequency range of interest. A control experiment in an enclosure is carried out with the same objective of control as modeled in the simulations. The experiment verifies that the desired modal distribution can be achieved by introducing the proposed state feedback control into a sound field.

102 新型アクティブサスペンションの構造と制御系の最適化に関する研究

In this paper, we deal with solving optimal design problems of a new type active suspension and its control system based on a modified genetic algorithm. The problems aim to pursue the improvement of vibration isolation performance to the limit. We consider simultaneously to optimize a state feedback gain vector of the control system and structural parameters of the suspension, i.e., stiffness of a spring and damping coefficient of a dash pod. As for active suspension of a 1/4 passenger car model, we do to optimize a feedback control gain vector and structural parameters simultaneously. Then, we show to be able to design an active suspension by which a vehicle body hardly receives the influence of road disturbance.

Design of feedback gain vector of two-state basic PWM multifeedback regulators for large-signal stability

A method for determining boundaries on the feedback gains in a switching regulator with multiple feedback, in order to ensure stability for large changes in the load values, is developed. Using an original approach for time-domain analysis, a large-signal, nonlinear, discrete-time model of the closed-loop converter is obtained; the model's equations are expressed in disturbances of the state variables due to disturbance in the load. The first step toward a global stability design is to find the constraint that ensures local stability around the operating point: the nonlinear terms (products of disturbances) are neglected in the model, the z-transform is applied to the linearized equations, and the condition that the eigenvalues are situated inside the unit circle is imposed. The nonlinear model, including the nonlinearity of the pulsewidth modulation (PWM), due to possible saturation, is then considered. A new constraint in terms of feedback gains is found in order to ensure that one and only one real equilibrium point (i.e., the quiescent operating point) exists. Putting together the constraints, boundaries on the feedback gains are found. The method is developed for converters containing two reactive elements, operating in continuous conduction mode. An example of a boost converter with inductor current and output voltage feedback is presented in detail. The large-signal stability of the design is checked by using the state-plane portrait and time-domain simulation.

Optimality-Preserving Interval Parameters in SI Regulator Systems

Linear quadratic regulators are in certain sense known to possess considerable amount of robustness margins. This paper is concerned with optimality robustness problems for single-input quadratic regulator systems with parameter uncertainties. The uncertainties are assumed to take on value in prescribed intervals. We show that Kharitonov-type extreme point results hold for two kinds of parameters:the coefficients of the open-loop characteristic equation and the elements of the designed optimal feedback gain vector. In both cases, optimality at a few extreme parameter values ensures that property at whole range of parameter values. These are derived from known results in the inverse problems of optimal regulators which for single input case feature geometric interpretations in particular. Numerical examples are provided to illustrate these extremal results.

Optimal simultaneous stabilization of linear single-input systems via linear state feedback control

Simultaneous stabilization of a collection of linear time-invariant systems is considered. The aim is to develop a practical method for the design of a single feedback controller such that all the systems involved are stabilized with good transient responses. Hurwitz's necessary and sufficient conditions are used as the required set of constraints for simultaneous stability. For transient behaviour, we introduce the usual quadratic objective function as performance index for each system. Their sum is the objective function for the collection of systems and the problem is to minimize it by choosing a feedback gain vector subject to boundedness and the Hurwitz stability constraints. A computational technique is proposed for solving this problem. The results obtained give good transient behaviour. For illustration, two numerical examples are solved.

On the design of an optimal automatic generation controller for interconnected power systems

Abstract An optimal controller combining the optimal pole placement technique and the dominant energy mode concept for the automatic generation control of power systems is presented in this paper. Firstly, based on the physical reasoning about the power plant to be controlled, the weighting matrix is roughly selected and the optimal feedback gain vector is found. Then the weighting matrix and thus the feedback gain are sequentially modified to let the dominant energy eigenvalues of the closed-loop system be placed in the desired region. A technique for determining the dominant energy eigenvalues of MIMO systems is proposed. Improper eigenvalue placement can be avoided, and very good control performance can be obtained. The proposed method is applied to design an automatic generation controller for an interconnected thermal power system. The simulation results indicate that very good dynamic responses of the area frequency deviations and tie-line power deviation are achieved.

A homotopy approach for stabilizing single-input systems with control structure constraints

A linear state feedback gain vector used in the control of a single input dynamical system may be constrained because of the way feedback is realized. Some examples of feedback realizations which impose constraints on the gain vector are: static output feedback, constant gain feedback for several operating points of a system, and two-controller feedback. We consider a general class of problems of stabilization of single input dynamical systems with such structural constraints and give a numerical method to solve them. Each of these problems is cast into a problem of solving a system of equalities and inequalities. In this formulation, the coefficients of the quadratic and linear factors of the closed-loop characteristic polynomial are the variables. To solve the system of equalities and inequalities, a continuous realization of the gradient projection method and a barrier method are used under the homotopy framework. Our method is illustrated with an example for each class of control structure constraint.

Simultaneous stabilization of linear single-input systems by linear state feedback control

A necessary and sufficient condition for the existence of a linear state feedback controller which will simultaneously stabilize a finite collection of single-input linear systems is presented. This condition can be checked numerically through the solution of a non-linear programming problem presented here and can be used to determine whether or not a particular simultaneous stabilization problem has a solution. If the stabilization problem has a solution, the method provides a means of constructing a suitable controller. The proposed method is very versatile. It is a simple matter to limit the size of elements of the feedback gain vector and thus ensure that the control action is not excessive. In addition, for simultaneous stabilization problems which have a range of possible solutions, the proposed method makes it possible to select one which yields good performance.

Optimal design method of an active vibration controller.

This paper concerns the optimal design method of an active vibration controller with the optimal regulator theory. This theory is often used to control vibration because it can determine feedback gain vector systematically. However, the problem in the case of determining feedback gain in the design of a control system is that there is no definite method to determine design parameters. Therefore, this paper shows the new design method, by which the required dimensional design parameters can be determined directly from the optimal design chart. First it is found that the similarity law shows the relationship between the dimensional and nondimensional weighting matrix and feedback gain. From this, the response of the nondimensional state variables will be obtained by using the nondimensional weighting matrix and feedback gain. After this, the optimal design chart can be obtained from this response. It is demonstrated theoretically and experimentally that this method is effective for the design of an active vibration controller.

Sub-optimal design of a closed-loop least-upper-bound fuel control

A sub-optimal closed loop controller is proposed for the regulation of a dynamical system, such as a rocket, which uses a strictly bounded non-throttleable control. The system as well as measurements on its states are subject to random disturbances. With the assumption of a fixed constraint on a statistic of the terminal error, the controller attempts to minimize the total fuel consumption. The Control law is prestructured in the form u= i ω sgn k'x̂, where i ω is an indicator function which is unity during control action and zero elsewhere, and where k is a time-varying feedback gain vector. A solution to the problem consists of the computation of i ω and k as a function of a priori known statistics of the measurements, and the minimum cost is simply the integrated value of the value of the i ω indicator functions over the fixed time history of the vehicle's motion.