Quadratic Integral Performance Index Research Articles

Asymptotics of the Solution to the Minimization Problem of the Integral Quadratic Performance Index on Trajectories of a Quasi-Linear System

The problem of minimizing an integral quadratic performance index on the trajectories of a quasi-linear dynamic system with a small parameter multiplying the nonlinearities subject to terminal constraints is considered. Asymptotic approximations of the optimal control in this problem in the form of open loop control and feedback are constructed. The computations are reduced to solving the unperturbed linear-quadratic problem, integrating systems of linear differential equations, and finding the roots of nonsingular linear algebraic systems.

Cheap control problem for micro drone quadcopter

The paper describes a broad class of problems with a cheap control. In contrast to existing works in this field, a simple solution of the problem of synthesis of optimal control with an integral quadratic performance index is proposed. As an example, we consider the micro drone quadcopter model.

Lyapunov matrices approach to the parametric optimization of a neutral system with two delays

In the paper a Lyapunov matrices approach to the parametric optimization problem of a neutral system with two delays and with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the neutral system with two delays. The Lyapunov functional is determined by means of the Lyapunov matrix.

Quadratic Optimal Control with Disturbance Attenuation for Uncertain Continuous-Time T-S Fuzzy Systems

ABSTRACTIn this study, an algebraically computational method is proposed to synthesize non-parallel-distributed-compensation (non-PDC) fuzzy controller such that (1) the prescribed disturbance attenuation level for the uncertain continuous-time Takagi-Sugeno (T-S) fuzzy system can be achieved, and (2) a quadratic integral performance index for nominal T-S fuzzy model-based control system can be minimized. We first derive relaxed linear matrix inequality (LMI) conditions by non-quadratic Lyapunov function and non-PDC fuzzy controller to meet prescribed disturbance attenuation level for the uncertain T-S systems. Then by using LMIs and orthogonal function array, the robust quadratic optimal control with disturbance attenuation for uncertain T-S fuzzy system is transformed into constrained-optimization problem represented by algebraic equations and LMI constraints. For static constrained-optimization problem, the HTGA is employed to search the gains non-PDC controllers. Therefore, the robust optimal controller design problem can be greatly simplified with the proposed method. The design example is given to demonstrate the applicability of the proposed approach.

Lyapunov Matrices Approach to the Parametric Optimization of a System with Two Delays

AbstractIn the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with two commensurate delays and a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of the Lyapunov functional for the initial function of time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix.

Lyapunov matrices approach to the parametric optimization of a neutral system

In the paper a Lyapunov matrices approach to the parametric optimization problem of a neutral system with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the neutral system. The Lyapunov functional is determined by means of the Lyapunov matrix.

Lyapunov matrices approach to the parametric optimization of time-delay systems

AbstractIn the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix

Dynamic Triggering Mechanisms for Event-Triggered Control

In this technical note, we present a new class of event triggering mechanisms for event-triggered control systems. This class is characterized by the introduction of an internal dynamic variable, which motivates the proposed name of dynamic event triggering mechanism. The stability of the resulting closed-loop system is proved and the influence of design parameters on the decay rate of the Lyapunov function is discussed. For linear systems, we establish a lower bound on the inter-execution time as a function of the parameters. The influence of these parameters on a quadratic integral performance index is also studied. Some simulation results are provided for illustration of the theoretical claims.

The Yakubovich quadratic criterion, F-stability and multi-agent consensus.

Generalizing numerous results on absolute stability of nonlinear Lurie systems, Vladimir A. Yakubovich established a fundamental abstract criterion of absolute stability under uncertain nonlinearities. His Quadratic Criterion offers elegant conditions for the uniform output L2-stability in a wide class of uncertain nonlinearities, obeying anytime ("local") or integral quadratic constraints; these conditions are based on either quadratic Lyapunov functions whose existence is proved via the KYP lemma, or the method of integral quadratic constraints dating back to the works by V.M. Popov. In the present paper, we extend the Yakubovich quadratic criterion, replacing the output L2-stability by boundedness of some quadratic integral performance index, defined by a quadratic form F and referred to as F -stability. We demonstrate that our F -stability criteria enables one to obtain easily many stability criteria for the “critical” case, where the frequency-domain inequality is non-strict and usual L2-stability of the solution cannot be proved, e.g., we derive the circle and Popov's stability criteria in this degenerate case. All of these criteria are especially effective for scalar nonlinearities, nevertheless, we demonstrate that our approach is also applicable to a class of large scale nonlinear multi-agent networks, and derive the “networked” circle criterion for such networks as a consequence of our results.

Application of the perturbation method for the minimization of an integral quadratic functional on the trajectories of a quasilinear system

The problem of optimization of the transient process in a quasilinear system containing a small parameter multiplying the nonlinearities is considered. This problem is to find a feasible control with the minimal value of the integral quadratic performance index. Asymptotic approximations to the optimal open-loop control and optimal feedback are constructed. Computations are reduced to the solution of the undisturbed linear-quadratic problem, integration of systems of linear differential equations, and solving linear algebraic systems of equations.

Parametric optimization of a neutral system with two delays and PD-controller

In this paper a parametric optimization problem for a linear neutral system with two delays with an integral quadratic performance index is formulated and solved. The method of computing of a performance index value bases on determining of a Lyapunov functional defined on a state space such that its value for an initial state is equal to a performance index value. In the paper a form of a Lyapunov functional is assumed and a method of computing its coefficients is given. An example illustrating the application of discussed theory is presented. It concerns the system with a PD-controller designed to control a plant with two delays both retarded and neutral type. For such system a value of considered performance index is determined.

Parametric optimization of neutral linear system with two delays with P-controller

Parametric optimization of neutral linear system with two delays with P-controller In the paper the problem of parametric optimization of a linear neutral system with two delays accomplished with respect to the integral quadratic performance index is formulated and solved. A method of computing the value of performance index relies on determining such a Lyapunov functional defined on the state space that its value for the initial state is equal to that of performance index. In the paper the form of Lyapunov functional is assumed and a method of computing its coefficients is given. An example illustrating the application of the theory discussed is presented. It concerns a system with the P-controller designed to control a plant with two delays both retarded and neutral type. For such a system the value of the considered performance index is determined.

Robust Quadratic-Optimal Control of TS-Fuzzy-Model-Based Dynamic Systems With Both Elemental Parametric Uncertainties and Norm-Bounded Approximation Error

This paper considers the design problem of the robust quadratic-optimal parallel-distributed-compensation (PDC) controllers for Takagi-Sugeno (TS) fuzzy-model-based control systems with both elemental parametric uncertainties and norm-bounded approximation error. By complementarily fusing the robust stabilizability condition, the orthogonal functions approach (OFA), and the hybrid Taguchi genetic algorithm (HTGA), an integrative method is presented in this paper to design the robust quadratic-optimal PDC controllers such that 1) the uncertain TS-fuzzy-model-based control systems can be robustly stabilized, and 2) a quadratic integral performance index for the nominal TS-fuzzy-model-based control systems can be minimized. In this paper, the robust stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). By using the OFA and the LMI-based robust stabilizability condition, the robust quadratic-optimal PDC control problem for the uncertain TS-fuzzy-model-based dynamic systems is transformed into a static constrained-optimization problem represented by the algebraic equations with constraint of LMI-based robust stabilizability condition, thus greatly simplifying the robust optimal PDC control design problem. Then, for the static constrained-optimization problem, the HTGA is employed to find the robust quadratic-optimal PDC controllers of the uncertain TS-fuzzy-model-based control systems. Two design examples of the robust quadratic-optimal PDC controllers for an uncertain inverted pendulum system and an uncertain nonlinear mass-spring-damper mechanical system are given to demonstrate the applicability of the proposed integrative approach.

Design of Optimal PID Controllers for PWM Feedback Systems With Bilinear Plants

This paper considers the optimal proportional-integral-derivative (PID) controller design for a class of pulsewidth-modulation (PWM) feedback systems with bilinear plants by using the orthogonal-functions approach (OFA). Based on the OFA, an algorithm only involving algebraic computation is derived in this paper for analyzing the PWM feedback systems with bilinear plants, and thus, the optimal PID controller design problem for a class of PWM feedback systems with bilinear plants is transformed into a static-parameters optimization problem represented by algebraic equations; this greatly simplifies the optimal PID control design problem. Then, for the static optimization problem, the hybrid Taguchi-genetic algorithm (HTGA) is employed to find both the optimal parameters of the PID controllers and the optimal PWM patterns for the PWM feedback systems with bilinear plants under the criterion of minimizing an integral quadratic performance index. The proposed new method, which integrates the OFA and the HTGA, is nondifferential, nonintegral, straightforward, and well-adapted to computer implementation. The computational complexity can, therefore, be reduced remarkably. An illustrative example for the electrohydraulic servo PWM feedback systems is given to show that the proposed method is an effective approach.

Design of Optimal Controllers for Takagi–Sugeno Fuzzy-Model-Based Systems

By the use of the elegant operational properties of the orthogonal functions, a direct computational algorithm for solving the Takagi-Sugeno (TS) fuzzy-model-based feedback dynamic equations is first developed in this paper. The basic idea is that the state variables are expressed in terms of the orthogonal functions. The new method simplifies the procedure of solving the TS fuzzy-model-based feedback dynamic equations into the successive solution of a system of recursive formulas taking only two terms of the expansion coefficients. Based on the presented recursive formulas, the developed computational algorithm only involves the straightforward algebraic computation. Then, the developed algorithm is integrated with the hybrid Taguchi-genetic algorithm (HTGA) to design both the quadratic optimal fuzzy parallel-distributed-compensation (PDC) controller and the quadratic-optimal non-PDC controller (quadratic optimal linear-state feedback controller) of the TS fuzzy-model-based control systems under the criterion of minimizing a quadratic integral performance index, where the quadratic integral performance index is also converted into the algebraic form by using the orthogonal-function approach (OFA). The proposed new approach, which integrates the OFA and the HTGA, is nondifferential, nonintegral, straightforward, and well adapted to the computer implementation. The computational complexity can, therefore, be reduced remarkably. Thus, this proposed approach facilitates the design tasks of the quadratic optimal controllers for the TS fuzzy-model-based control systems. A design example of the quadratic optimal controllers for the translational oscillator system with an eccentric rotational proof mass actuator is given to demonstrate the applicability of the proposed approach

Phase Portrait Modeling of a Nonlinear System with a Dynamic Fuzzy Network

Fuzzy logic and neural networks are two important technologies for modeling and control of dynamical systems and have been constrained by the non-dynamical nature of their some popular architectures. There exist problems such as large rule bases (i.e., curse of dimensionality), long training times, the need to determine buffer lengths. This article proposes to overcome these major problems in phase portrait modeling of a nonlinear system with a dynamic fuzzy network (DFN) with unconstrained connectivity and with dynamic fuzzy processing units called “feurons”. Nonlinear physical system properties can be encapsulated by DFN. As an example, DFN has been used as the modeler for some nonlinear physical system such as chaotic, limit cycle, oscillator. The minimization of an integral quadratic performance index subject to dynamic equality constraints is considered for a phase portrait modeling application. For gradient computation adjoint sensitivity method has been used. Its computational complexity is significantly less than direct sensitivity method, but it requires a backward integration capability. We used first and approximate second order gradient-based methods including Broyden–Fletcher–Golfarb–Shanno algorithm to update the parameters of the dynamic fuzzy networks yielding faster rate of convergence

Optimal Robot Speed Trajectory by Minimization of the Actuator Motor Electromechanical Losses

This paper considers the control problem of a robotic manipulator with separately excited dc motor drives as actuators. An innovative method is proposed which achieves robot speed-control requirements, with simultaneous minimization of total electromechanical losses, while the drives follow the desired speed profiles of the robot joints under various loads and random load disturbances. If there is no demand for a specific speed profile, the optimal speed trajectory is determined by minimizing an electromechanical losses criterion. Controllable energy losses, such as armature copper losses, armature iron losses, field copper losses, stray load losses, brush load losses, friction and windage losses, can be expressed proportionally to the squares of the armature and the field (exciting) currents, the angular velocity and the magnetic field flux. The controllable energy loss term is also included in the optimal control integral quadratic performance index, defined for the whole operation period. Thus the appropriate control signals required for following the desired trajectory by simultaneous energy loss minimization for the whole operation interval are achieved. Two case studies of optimal robot control with and without minimization of actuator energy losses are presented and compared, showing the energy savings that can be achieved by the proposed methodology.

Direct Computation of the Performance Index for an Optimally Controlled Active Suspension with Preview Applied to a Half-Car Model

A quadratic integral performance index is defined for a linear active preview–type suspension and a series of matrix expressions derived for its evaluation by means of MATLAB or some similar computer program in the case of a unit step input to the system. The computation, which is both fast and accurate compared to simulation, requires the solution of Lyapunov– and Riccati–type equations. Some examples of numerical computation are given and these show excellent agreement with published results. The conclusion features a useful computer program.

Two-degree-of-freedom design method of LQI servo systems: Use of frequency-dependent weighting matrices for sensitivity reduction

Recently, the present authors proposed a new method to design two-degree-of-freedom robust servo systems for step references and step disturbances. In this method, the tracking characteristics for step references are determined optimally with respect to a quadratic-integral performance index, while the feedback characteristics for step disturbances are determined also optimally with respect to another quadratic-integral performance index. The paper extends these results by introducing frequency-dependent weighting matrices into the latter performance index. By employing appropriate frequency-dependent weighting matrices, some sort of loop-shaping can be achieved for the sensitivity characteristics, and thus more design freedom is obtained. In particular, loop-shaping ability and tuning ability can be separated in some sense. A numerical example is also studied to illustrate the results.

Quadratic guaranteed cost control with robust pole placement in a disk

The paper is concerned with the design of robust state feedback controllers for a class of uncertain systems with norm bounded uncertainty, where uncertainty enters in both the state and input matrices. The controller is designed to place the closed loop poles of the uncertain system in a specified disk and guarantees an upper bound on a quadratic cost function. A procedure is given for the construction of a controller which minimises a bound on an integral quadratic performance index.