Quadratic Performance Function Research Articles

Krein-space based robust [formula omitted] fault estimation for two-dimensional uncertain linear discrete time-varying systems

In this study, the robust H∞ fault estimation problem for two-dimensional linear time-varying systems with norm-bounded unknown input, measurement noise, and time-varying process uncertainty is investigated. By introducing an equivalent auxiliary system and a new certain indefinite quadratic form performance function, the system uncertainty can be appropriately considered into the new performance function and the fault estimator design is converted to the minimization problem of a quadratic form. Based on the partially equivalence property between the deterministic quadratic form problem and the Krein space estimation theory, the two-dimensional H∞ fault estimation problem can be solved via signal deconvolution in Krein space. Through employing projection operation and Riccati-like difference equation in two dimensions, both the recursive form fault estimator and the explicit condition for existence of the estimator are derived. One Darboux equation example is provided to illustrate the effectiveness of the proposed fault estimator.

Switching-iterative learning control method for discrete-time switching system

In this paper, we propose a new monotonically convergent switching iterative learning control for a class of linear discrete time switched system. It is assumed that the considered switched systems are operated during a finite time interval repetitively, and then the iterative learning control scheme can be introduced. After the switched system is transformed into a 2D repetitive system, sufficient conditions in terms of linear matrix inequalities (LMIs) are derived by using a Lyapunov functional approach and a quadratic performance function. It is shown that if certain LMIs are met, the tracking error $$l_2$$ norm converges monotonically to zero between (subsystem/iteration), while the switching learning gains could be determined directly by solving the LMIs.The integrated design of this SILC scheme is transformed into a robust monotonic stabilizability problem (RMS) of an uncertain switched system. A numerical simulation example is established shown the effectiveness of the proposed method .

Output‐feedback model predictive control for stochastic systems with multiplicative and additive uncertainty

SummaryThis paper studies the output‐feedback model predictive control (MPC) design problem for linear systems with multiplicative and additive random uncertainty. We first present an off‐line optimization algorithm to optimize feedback gains of the observer and the dual‐mode control policy. After that, by defining a cuboid tube whose center and boundary are both time‐varying variables, we develop a set sequence with increased freedom to contain stochastic system trajectories. A quadratic performance function with analytic upper and lower bounds is minimized such that it decreases exponentially to a finite range under the expectation. The resulting MPC algorithms are proved to guarantee practically stochastic input‐to‐state stability. A numerical example of the wind turbine model illustrates the properties of the MPC algorithms.

Leader-following guaranteed-performance consensus design for singular multi-agent systems with Lipschitz nonlinear dynamics

Leader-following guaranteed-performance consensus problems for singular multi-agent systems with Lipschitz nonlinear dynamics are investigated. To obtain leader-following guaranteed-performance consensus, the consensus protocol and quadratic performance function based on state errors are proposed. Then, leader-following consensus problems of a nonlinear singular multi-agent system are transformed into asymptotic stabilization problems of a reduced-order subsystem by a new state decomposition method, and the impacts of the Lipschitz nonlinear dynamics are eliminated by using the structure property of the transformation matrix and the Lipschitz condition. Moreover, a guaranteed-performance consensualization criterion based on the Riccati inequality is given and the guaranteed-performance cost with a new structure is determined. Finally, a numerical simulation is shown to verify the validity of theoretical results.

Guaranteed Cost Anti-windup Stabilization of Discrete Delayed Cellular Neural Networks

This paper studies the problem of guaranteed cost anti-windup stabilization of discrete delayed cellular neural networks. Saturation degree function is initially presented and the convex hull theory is applied to handle the saturated terms of discrete delayed cellular neural networks. Accordingly, after choosing a common quadratic performance function, the paper designs a guaranteed cost stabilization controller in the absence of input saturation on the basis of Lyapunov–Krasovskii theorem and linear matrix inequality formulation. Then a static state feedback anti-windup compensation is derived, which guarantee a guaranteed cost and the estimation of the asymptotic stability region for the closed-loop system. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed design technique.

Exponentially stable guaranteed cost control for continuous and discrete-time Takagi–Sugeno fuzzy systems

This paper investigates exponentially stable guaranteed cost control (GCC) for a class of nonlinear systems which is represented by Takagi–Sugeno (T–S) fuzzy systems. State feedback controllers of parallel distributed compensation (PDC) structure are designed by the means of GCC for continuous and discrete-time T–S fuzzy systems respectively. GCC methods in this paper adopt quadratic performance functions, which take effects of control efforts, regulation errors and convergence rates into consideration simultaneously, to provide desirable performance and fast response for closed-loop systems. Sufficient design conditions that guarantee exponential stabilities of resulting closed-loop systems with predefined convergence rates are presented. By setting different convergence rates, response speed of the closed-loop nonlinear system can be adjusted. The proposed design procedures are eventually converted into linear matrix inequalities (LMIs) problems of minimizing upper bounds of the guaranteed cost functions. Finally, a well-known nonlinear benchmark control example and a truck–trailer example demonstrate the effectiveness and feasibility of all the proposed methods.

Monotonically convergent hybrid ILC for uncertain discrete-time switched systems with state delay

This paper is mainly devoted to a monotonically convergent iterative learning control (ILC) design for a class of uncertain discrete-time switched systems with state delay (UDTSDSs). By taking advantage of output error and state information, a hybrid ILC law for a class of UDTSDSs is proposed. After the ILC process is transformed into a 2D system, sufficient conditions in terms of linear matrix inequalities (LMIs) are derived by using a multiple Lyapunov–Krasovskii-like functional approach and a quadratic performance function. It is shown that if certain LMIs are met, the tracking error 2-norm converges monotonically to zero along the iteration direction, while the learning gains could be determined directly by solving the LMIs. The simulation results are provided to illustrate the theoretical analysis.

A Method for Improving Computational Accuracy of Reliability Based on the Theory of the Minimum Cut Set

According to the nonlinearity and the multiple extreme value points of performance function,it is dealt with piecewise and a new computing method of failure probability based on the minimum cut set is put forwards. Multiple quadratic performance functions are used to replace the original complicated performance function piecewise,and the combination of failure domains of these quadratic performance functions is as close as to the failure domain of original performance function. The reliability indexes of quadratic performance functions are calculated by second order reliability method. All the minimum cut sets to describe the original failure domain are obtained by using the failure domains of quadratic performance functions according to the concept of the minimum cut set. All the minimum cut sets of equivalent failure domains were obtained to express the original failure domain by substituting quadratic failure domains with equivalent failure domains. Reliability index calculated by first order reliability method is redefined so that it could be suitable to the case that the origin of coordinates is in failure domain. And then,the combination of failure probabilities of multiple equivalent failure domains could be computed and the approximate probability of the original failure domain could be obtained. The example shows that the method is feasible and the calculation accuracy of failure probability could be improved greatly.

Vibration Control of a Flexible Beam Using Model Predictive Control

Model predictive control is applied to suppress the vibration of a flexible link with piezoelectric actuators and strain gage transducer. The state-space dynamic model of the system was derived by using finite element method and experimental modal test. On the basis of the model, model predictive controller is designed taking into account the uncertain disturbance and measurement noise. The discrete prediction model is derived from the state-space equation of the system, and the future output is obtained from the model. The uncertain external disturbance and measurement noise are white noise signal, the Kalman filter estimator is designed to estimate the state variables of the system. A standard quadratic programming optimization problem is formed where the performance index function minimizes a quadratic performance function that trades off controller performance and control effort. The constraints are the control input voltage and its change rate. Finally, the optimization problem is solved to obtain the optimal control output. A MIMO control system is built using dSPACE DS1103 platform, and experimental tests are performed. The performances of the controller are verified experimentally. The results of experiment show the effectiveness of the controller.

Vibration Control of a Flexible Beam Using Model Predictive Control

Model predictive control is applied to suppress the vibration of a flexible link with piezoelectric actuators and strain gage transducer. The state-space dynamic model of the system was derived by using finite element method and experimental modal test. On the basis of the model, model predictive controller is designed taking into account the uncertain disturbance and measurement noise. The discrete prediction model is derived from the state-space equation of the system, and the future output is obtained from the model. The uncertain external disturbance and measurement noise are white noise signal, the Kalman filter estimator is designed to estimate the state variables of the system. A standard quadratic programming optimization problem is formed where the performance index function minimizes a quadratic performance function that trades off controller performance and control effort. The constraints are the control input voltage and its change rate. Finally, the optimization problem is solved to obtain the optimal control output. A MIMO control system is built using dSPACE DS1103 platform, and experimental tests are performed. The performances of the controller are verified experimentally. The results of experiment show the effectiveness of the controller.

熱延ダウンコイラへの電気油圧サーボシステムの適用

Many kinds of electro-hydraulic servo systems are applied to rolling mills such as a hot strip mill, a cold strip mill and a seamless tube mill because of quick response, high power and high positioning accuracy. However when applying an electro-hydraulic servo system to a hot strip downcoiler, it was found that the controlled object has low natural frequency and is lightly damped so that stable control is very difficult. Optimal control theory has been recognized as a powerful tool for system synthesis in such a case. This paper first shows that the existing design method of optimal control is effective for designing an electro-hydraulic servo controller, in which the proper weighting matrices in the quadratic performance function can be determined directly from a physical model. A new design method is then proposed by which the state feedback gains can be obtained directly based on the characteristics of the controlled object. The effectiveness of the controller is demonstrated experimentally.

Guaranteed Cost Piecewise Fuzzy Controller Design

AbstractA common approach for stability and performance control design in Takagi- Sugeno fuzzy models is to solve an LMI problem which proves global stability and an upper bound of a given quadratic performance function for any shape of membership functions. This paper improves efficiency of the controllers in Takagi-Sugeno fuzzy models, by having into account the shape of the membership functions in a particular region of interest. A multiple-gain piecewise PDC controller is designed for several particular regions ensuring global stability as well as increased performance as the state approaches the origin.

Design of non‐fragile guaranteed‐cost repetitive‐control system based on two‐dimensional model

AbstractThis paper concerns a new method of repetitive control based on two‐dimensional (2D) system theory. First, a 2D model is presented that enables the independent adjustment of control, which happens within a repetition period, and learning, which happens between periods. Next, the problem of designing a repetitive‐control law is formulated as a state‐feedback design problem for the 2D model. An existence condition and a method of designing a robust repetitive‐control law for a plant containing time‐invariant structured uncertainties are established by combining 2D system theory with linear matrix inequalities. Then, based on those results, a non‐fragile guaranteed‐cost repetitive‐control law is derived. The controller gain to be designed is assumed to have additive gain variations. It guarantees that the value of a quadratic performance function is less than a specified upper bound for all admissible uncertainties. The main feature of this approach is that it enables the control action and the learning process to be adjusted independently by the direct tuning of the weighting matrices in the quadratic cost function. Finally, a numerical example demonstrates the validity of this approach.Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

An output-feedback fuzzy approach to guaranteed cost control of vehicle lateral motion

This paper introduces a fuzzy guaranteed cost control approach for automated steering of a highway vehicle. The Takagi–Sugeno (T–S) fuzzy model is utilized to depict the dynamics of nonlinear time-varying lateral system. Based on the fuzzy model, an observer-based fuzzy controller is developed so that without knowing road’s curvature the vehicle can track center of the present lane on a curved highway section. Integrating H ∞ control and optimal control strategies, the fuzzy controller and observer are formulated by solving a minimization problem, which is to minimize a given quadratic performance function. Sufficient conditions to ensure minimum upper bound of the performance function are derived in terms of linear matrix inequalities (LMIs). Therefore, the designing work can be efficiently completed by applying the convex optimization techniques. Verified by computer simulation, the proposed method can perform well in driving safety and ride comfort.

Finite-Dimensional Constrained Fuzzy Control for a Class of Nonlinear Distributed Process Systems

This correspondence studies the problem of finite-dimensional constrained fuzzy control for a class of systems described by nonlinear parabolic partial differential equations (PDEs). Initially, Galerkin's method is applied to the PDE system to derive a nonlinear ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, a systematic modeling procedure is given to construct exactly a Takagi-Sugeno (T-S) fuzzy model for the finite-dimensional ODE system under state constraints. Then, based on the T-S fuzzy model, a sufficient condition for the existence of a stabilizing fuzzy controller is derived, which guarantees that the state constraints are satisfied and provides an upper bound on the quadratic performance function for the finite-dimensional slow system. The resulting fuzzy controllers can also guarantee the exponential stability of the closed-loop PDE system. Moreover, a local optimization algorithm based on the linear matrix inequalities is proposed to compute the feedback gain matrices of a suboptimal fuzzy controller in the sense of minimizing the quadratic performance bound. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.

A Profit Index for Assessing the Benefits of Process Control

In industrial processes, the potential benefit that arises from improved control is assessed before the installation of a new control strategy. Estimating the economic benefit motivates the investment in a new control system and minimizes the risk of the new technology not living up to its promises. In this paper, a method for a priori estimation of the benefit from improved process control is presented. The method builds on conventional estimation techniques and evaluates the performance function associated with the controlled variable by assuming a reduction in variance. Three frequently used performance functions, namely, a quadratic performance function, a linear performance function with constraints, and the so-called clifftent performance function, are investigated. The profit index as a function of the variance is computed, and the implications of a reduction in variance for each performance function are assessed. A further contribution of this article is the analytical derivation of the optimal op...

Decentralized robust guaranteed cost control for a class of interconnected singular large-scale systems with time-delay and parameter uncertainty

Abstract The decentralized robust guaranteed cost control problem is studied for a class of interconnected singular large-scale systems with time-delay and norm-bounded time-invariant parameter uncertainty under a given quadratic cost performance function. The problem that is addressed in this study is to design a decentralized robust guaranteed cost state feedback controller such that the closed-loop system is not only regular, impulse-free and stable, but also guarantees an adequate level of performance for all admissible uncertainties. A sufficient condition for the existence of the decentralized robust guaranteed cost state feedback controllers is proposed in terms of a linear matrix inequality (LMI) via LMI approach. When this condition is feasible, the desired state feedback decentralized robust guaranteed cost controller gain matrices can be obtained. Finally, an illustrative example is provided to demonstrate the effectiveness of the proposed approach.

Energy minimization control for a discrete chaotic system

A general framework algorithm is proposed for energy minimization control for a discrete chaotic system. A quadratic performance function is first given and the chaotic system is decomposed into a linear and a nonlinear parts. Then, the twolevel algorithm is presented to solve the nonlinear optimal control problem: The first level predicates the nonlinear part of the chaos system; the second level solves a nonlinear quadratic optimization control problem by dynamic programming. The solution is fed back into the first level. The first level re-estimates the nonlinear part according to the solution from the second level. The information has been exchanged between the two levels by this means such that the optimal control law is obtained eventually. This method not only can make the control of chaos system be realized but also makes the energy consumed minimal during the whole control process.Simulations show the effectiveness of this algorithms.

Stability of steepest descent with momentum for quadratic functions

This paper analyzes the effect of momentum on steepest descent training for quadratic performance functions. We demonstrate that there always exists a momentum coefficient that will stabilize the steepest descent algorithm, regardless of the value of the learning rate. We also demonstrate how the value of the momentum coefficient changes the convergence properties of the algorithm.

Convergence Properties of Two-Stage Stochastic Programming

This paper considers a procedure of two-stage stochastic programming in which the performance function to be optimized is replaced by its empirical mean. This procedure converts a stochastic optimization problem into a deterministic one for which many methods are available. Another strength of the method is that there is essentially no requirement on the distribution of the random variables involved. Exponential convergence for the probability of deviation of the empirical optimum from the true optimum is established using large deviation techniques. Explicit bounds on the convergence rates are obtained for the case of quadratic performance functions. Finally, numerical results are presented for the famous news vendor problem, which lends experimental evidence supporting exponential convergence.