Systems with constraints

Self-excited vibration often emerges in closed-loop systems due to parameter points out of the stable region in the parameter space. Consequently, based on control theory, various analysis methods have been developed to determine the parameter condition of equilibrium stability and to calculate the amplitude and the frequency of self-excited vibration in high-order closedloop systems, of which frequency method is a very useful tool. Particularly, for the transfer function is invariant in nonsingular linear coordinate transformation for a given dynamic system, it may be obtained in a simple manner either from system equations or experimentally. Usually, the block diagram is a convenient model for the frequency method. Thus, the contents arranged in this chapter are divided into five sections: the first is devoted to establishing mathematical models in frequency domain for closed-loop systems; the second provides a frequency criterion of equilibrium stability for linear closed-loop systems; the third recommends a frequency criterion of equilibrium stability for nonlinear closed-loop systems; the fourth introduces a well-known describing function method used to calculate the amplitude and the frequency of self-excited vibration in closed-loop systems; and the fifth gives a quadratic form optimal algorithm to determine the control parameters, which can prevent the controlled system from self-excited vibration.

The aim of this paper is to discuss the methodology of Fault Detection and Diagnosis (FDD) in dynamic systems. Fault diagnosis is characterized as a control system that controls the ability to adapt system component faults automatically. Fault detection is an implementation of using the error signals, where when error signal is zero or approximately zero. A coupled water tank system was used as a study case model for implementing and testing the proposed methodology. The developed system should generate a set of signals to notify the process operator about the faults that are occurring, enabling changes in control strategy or control parameters. Due to the damage risks involved with sensors, actuators and structural faults of the real plant, the data set of the faults are computationally generated and the results will be collected from numerical simulations of the process model. A Recurrent Neural Networks (RNN) is employed in this paper for modeling the used system. This paper shows how to determine the structure and how to estimate the results using the gradient-based algorithm which it is allowed to obtain a Neural Network with relatively small modeling uncertainly. It describes how to increase the stability of dynamic systems using FDD based on RNN. Faults in the study case are faults in the sensors, faults in the actuators and the structural faults. The faults which are presented in the controlled system are reduced to 98% from the default value.

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

This paper presents a wind-turbine model for energy capture and switching simulation at lull wind conditions. The model includes aerodynamics as well as the control parts. It allows to study turbine behavior dependent on different controller settings and wind-speed characteristics. Based on such simulations, control parameters can be optimized regarding energy capture and turbine profits near cut-in wind speed. This was applied to a standard 1.5-MW wind turbine. Model validation was done using measurement data of different wind turbines. The model is kept as simple as possible considering the modeling of system components. However, all simplifications taken are allowable for energy capture and switching simulation. They guarantee shortest simulation times.

This paper studies the proportional-derivative (PD) control method in the control field, which can be applied in industrial automation, robot control, aerospace, process control, and other fields. This paper discusses the PD control to adjust the motor angle and the fuzzy proportional-integral (PI)-PD control of the brushless direct current (DC) motor. How to apply PD control theory to practice is a part worth exploring. In the traditional control field, proportional-integral-derivative (PID) control parameters cannot adjust the changes of controlled parameters, and the traditional PID control must cooperate with three parameters at the same time to work. Therefore, in this paper, PD control is applied to the adjustment of motor angle, and PI-PD fuzzy control is used to adjust the controller parameters in real time. This method can improve the response speed of the controller, apply the control theory to the actual motor control, and improve the anti-interference ability. This research can improve the application scope of PD control and combine it with a variety of control theories, to achieve a better effect of the control system, optimize the response speed of the motor, and achieve the target speed faster.

SummaryAny concept of optimality on a set is based on an ‘ordering’ of the set. In its most general form such an ‘ordering’ will be termed a ‘preference’ on a set and it will simply be a binary relation whose purpose it is to introduce a hierarchy among the elements of the set. In finite-dimensional multicriteria decision problems one basically deals with a mapping g(.):D → a, where D is some decision set and a = {a∈RN: a=g(d),d∈D } is the set of corresponding criterion values, e.g. D may be a functional constraint set in programming or a set of admissible controls in control theory. The optimality concept introduced here consists of a preference ≲ on a and the definition of preference optimal decisions d*∈D as those for which g(d*)=a* is a least or a minimal element of a with respect to ≲ .

In existence theorems for optimal solutions in control theory, concepts of upper semicontinuity of variable sets in Euclidean spaces have been used. In the present paper, a scale of concepts is introduced which is intermediate between Kuratovski's upper semicontinuity and Cesari's upper semicontinuity. First, the relationships and properties of the concepts are studied. Then, examples of problems of control with distributed and boundary parameters are given showing the relevance of the same concepts.

Bowing is the main element in sound production during a violin performance and one of the most basic and important expressive resources for the musician. In the lowest level, control parameters such as force, velocity, or bow-bridge distance are directly determining the characteristics of the sound. In a higher level, bowing strokes constitute one of the main mechanisms for structuring the performance. There are many different kinds of bowing strokes, and they are commonly classified into on-string, if the attack happens with the bow on the string and off-string, if the bow is bouncing. From a database of violin performances containing multimodal data including sound and gestures, a set of spectral features and instrumental controls is extracted and the database is segmented into intra-note segments (attack, sustain, and release). A characterization of bowing strokes and a comparison between bouncing and on-string strokes in terms of bowing controls and sound at the intra-note segments is presented.

Examples from robotics research on bipedal walkers have shown that complex control programs integrating an explicit multibody model into the control program can be outperformed by very simple reinforced learning schemas applied to a ‘fit’ mechanical structure. This seems to show that in robotic and, in general, controlled system there are relations between the information metrics of the control programs and the system dynamics. The relations shown in this paper could help to understand the relations involved by the phase space ‘footprint’ of dynamical systems and the complexity, in the information theory sense, of their control programs, and possibly to ease, giving a quantitative guidance, the development of new more ‘intelligent’ artefacts. This is relevant to control theory and practice as the control of bipedal walkers have proven to be eased in term of complexity of control programs by the so called ‘passive walker’ approaches which leverage of the intrinsic dynamic characteristics of the walker systems. It is relevant to AI research, too, where embodiment behavior based and emergent controls are widely used in the practice of robotics, but still miss a comprehensive quantitative analysis framework. Bipedal walkers are examples of non linear mechanical dynamical systems. The equation derived here could be regarded as an hint of a general connection between control and information theory, so far not completely investigated.

Three trends concerning applications of Systems Theory in education and e-learning are analysed. The first is the holistic General Systems Theory which studies the structure and the behaviour of systems, and identifying the components and the interconnections within systems. The second is represented by the Dynamical Systems Theory which uses concepts such as attractors, bifurcations or constraints to create a framework for the study of the critical components of the learning process. But being too general, these directions cannot provide concrete mathematical models able to give solutions for the management and improvement of this process. The third direction which tries to solve this problem is developed in this paper, continuing the author's two previous papers. An e-learning innovative model is provided using the tools of the Systems and Control Theory. The inputs, outputs and the states which correspond to the e-learning process are described and the equations of the state-space representation of the model are derived. This model is applied to the foreign languages e-learning. The controllability of the system is analysed by means of a suitable Gramian. A necessary and sufficient condition of controllability is presented and in the case of the controllable systems a formula is given for the optimal control of the problem of minimum energy transfer of states. Two programs are implemented using Control Systems Toolbox of the Matlab software, one which generates the e-learning system and the other which computes the optimal solution of the transfer problem. This solution allows teachers and students to schedule their e-learning tasks in order to acquire a desired level of knowledge.

At beginning, the kinematic model and dynamic model of a differential mobile robot (DMR), and the dynamic model of left‐ and right‐wheel DC motors are combined to be the controlled system. The control inputs of the proposed controlled system are the input voltages for the left‐ and right‐wheel motors. The (indirect) outputs are the two‐dimensional (2D) position and orientation of a DMR. Owing to the under‐actuated characteristic, the direct reference input (i.e. two desired motor currents) using the first sliding surface is designed, so that the 2D position and orientation of the DMR are simultaneously controlled by two motor currents (i.e. the direct output). On the other hand, the second sliding surface is designed as the linear dynamics of tracking error of motor currents. Under completely (or partially) known frictions and uncertainties, the hierarchical sliding‐mode under‐actuated control with suitable conditions is designed, such that two motor currents asymptotically track two desired motor currents, respectively. Then the asymptotic tracking for the 2D position and orientation of a DMR is achieved. The simulations for various trajectories, completely and partially known frictions and uncertainties, and control parameters are presented to evaluate the effectiveness and robustness of the proposed method.

[abstFig src='/00280005/11.jpg' width='300' text='ANMPC controller' ] The parameter-tuning method we discuss is for an Adaptive Nonlinear Model Predictive Controller (ANMPC). The MPC is optimization-based controller and decides control input to realize system output that tracks a reference trajectory through “optimal computation.” The reference trajectory is ideal trajectory of system output to converge on a desired value, i.e. controlled system performance depends on the reference trajectory. As a MPC controller which applies to the nonlinear systems, our group has already proposed an adaptive nonlinear MPC (ANMPC) for a tracking control problem of nonlinear two-link planar manipulators. This ANMPC uses a new reference trajectory having control parameters that must be tuned based on the desired controlled system’s responses and properties. To reduce troublesome parameter tuning, we propose new parameter-tuning method for ANMPC by a quantitative analysis of the relationship between a system’s behavior and ANMPC parameters. Numerically simulating the two-link nonlinear manipulator’s tracking control under various conditions demonstrates that proposed tuning method tunes the ANMPC effectively.

THE SAMPLING PERIOD AS A CONTROL PARAMETER

First- and second-order numerical methods for optimizing controlled dynamical systems with parameters are discussed. In unconstrained-parameter problems, the control parameters are optimized by applying the conjugate gradient method. A more accurate numerical solution in these problems is produced by Newton’s method based on a second-order functional increment formula. Next, a general optimal control problem with state constraints and parameters involved on the righthand sides of the controlled system and in the initial conditions is considered. This complicated problem is reduced to a mathematical programming one, followed by the search for optimal parameter values and control functions by applying a multimethod algorithm. The performance of the proposed technique is demonstrated by solving application problems.

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.