Application of numerical analysis to root locus design of feedback control systems

The root locus techniques find a variety of uses in feedback control system design, stability analysis and root determination of high-order algebraic equations This review paper presents the different schemes reported hitherto for Computer-Aided computation of root loci. Amongst I lie different methods available, the Newton-Raphson technique and the algorithm of Pan and Chao are elaborated in detail.

This paper presents a comprehensive analysis of the need for the Padé approximation for continuous-time models with delays, focusing on its critical role in addressing the control challenges posed by time delays. Time delays, often referred to as dead times, transport delays or time lags, are inherent in a wide range of industrial and engineering processes. These delays introduce phase shifts that degrade control performance by reducing control bandwidth and threatening the stability of closed-loop systems. Accurate modelling and compensation of these delays is essential to maintain system stability and ensure effective control. This paper highlights the difficulties that arise when using advanced control techniques such as root locus (RL), linear quadratic regulator (LQR) and H-infinity (H_∞) control in systems with delays. Representing delays in exponential form leads to an infinite number of state problems, complicating the design and analysis of controllers in such systems. To address these challenges, the Padé approximation is proposed as an effective method for approximating time delays with rational polynomials of appropriate order. This approach allows for more accurate simulation, system analysis and controller design, thereby mitigating the problems caused by delays. The paper also provides a detailed comparative analysis between the Padé approximation and Taylor polynomials, demonstrating the superiority of the former in achieving accurate delay modelling and control performance. The results show that the use of Padé approximation not only improves the accuracy of system models, but also improves the robustness and stability of control strategies such as RL, LQR, and H_∞. These results highlight the importance of the Padé approximation as a valuable tool in the design of delay-affected control systems, offering significant advantages for both theoretical and practical applications.

Control systems are pervasive in our lives. Our homes have environmental controls. The appliances we use, such as the washing machine, microwave, etc. carry embedded controllers in them. We fly in airplanes and drive automobiles that extensively use control systems. The industrial plants that produce consumer goods run on process control systems. The recent drive toward automation has increased our reliance on control systems technology. This book discusses control systems design from a model-based perspective for dynamic system models of single-input single-output type. The emphasis in this book is on understanding and applying the techniques that enable the design of effective control systems in multiple engineering disciplines. The book covers both time-domain and the frequency-domain design methods, as well as controller design for both continuous-time and discrete-time systems. MATLAB© and its Control Systems Toolbox are extensively used for design. Technical topics discussed in the book include: • Mathematical models of physical systems • Analysis of transfer function and state variable models • Control systems design objectives • Control system design with root locus • Control system design in the state-space • Control system design of sampled-data systems • Compensator design with frequency response methods

Control systems are pervasive in our lives. Our homes have environmental controls. The appliances we use, such as the washing machine, microwave, etc. carry embedded controllers in them. We fly in airplanes and drive automobiles that extensively use control systems. The industrial plants that produce consumer goods run on process control systems. The recent drive toward automation has increased our reliance on control systems technology. This book discusses control systems design from a model-based perspective for dynamic system models of single-input single-output type. The emphasis in this book is on understanding and applying the techniques that enable the design of effective control systems in multiple engineering disciplines. The book covers both time-domain and the frequency-domain design methods, as well as controller design for both continuous-time and discrete-time systems. MATLAB© and its Control Systems Toolbox are extensively used for design. Technical topics discussed in the book include: • Mathematical models of physical systems • Analysis of transfer function and state variable models • Control systems design objectives • Control system design with root locus • Control system design in the state-space • Control system design of sampled-data systems • Compensator design with frequency response methods

Approaches to the analysis and design of feedback control systems in which measurement quantization is present are discussed. The author regards a quantized measurement of a real number simply as a partial observation of that number and not as a straightforward approximation with associated error. As a consequence, the results have an information theoretic component that is absent from earlier research, the majority of which deals with measurement quantization by means of signal-plus-noise models and least-squares theory. It is possible to compute an upper bound on the asymptotic accuracy (measured as a mutual information) with which the current state of a linear system can be specified given a long record of quantized measurements. In deriving this upper bound, it is shown how input to the system can be used as information-gathering tools, and an observer-like construction for systems with quantized outputs is obtained. The results have some interesting system theoretic ramifications, and they form the foundation for a theory of observability for state-space systems with piecewise constant output. This theory differs from the usual one in several significant ways, and the utility of its attendant formalism is demonstrated by some simple examples. >

Feedback control of functional neuromuscular stimulation has the potential to improve daily function for individuals with spinal cord injuries (SCIs) by enhancing seated stability. Our fully implanted networked neuroprosthesis (NNP) can provide real-time feedback signals for controlling the trunk through accelerometers embedded in modules distributed throughout the trunk. Typically, inertial sensors are aligned with the relevant body segment. However, NNP implanted modules are placed according to surgical constraints and their precise locations and orientations are generally unknown. We have developed a method for calibrating multiple randomly oriented accelerometers and fusing their signals into a measure of trunk orientation. Six accelerometers were externally attached in random orientations to the trunks of six individuals with SCI. Calibration with an optical motion capture system resulted in RMSE below 5° and correlation coefficients above 0.97. Calibration with a handheld goniometer resulted in RMSE of 7° and correlation coefficients above 0.93. Our method can obtain trunk orientation from a network of sensors without a priori knowledge of their relationships to the body anatomical axes. The results of this study will be invaluable in the design of feedback control systems for stabilizing the trunk of individuals with SCI in combination with the NNP implanted technology.

The model of NCS (Networked Control Systems) with state observer is presented when the network-induced random time-varying delays are bounded and the upper bound of which is longer than one sampling period. Thus we established a discrete-time jump system model with Markov delay. Based on this, this paper gave also initial feasible solution of interior-point algorithm to design stabilizing controllers by the optimization method. This paper generalizes previous results on delay-less state observer systems. Theoretical analysis and simulation on a cart and inverted pendulum with random time delay show that feedback control based state observer can make the system stable.

Examines the design of feedback control systems for linear, time-varying systems. The design procedure is developed using a frequency domain description of a linear, time-varying system and the solution of the H/sub /spl infin// optimal control problem. In particular, design guidelines are derived for several basic performance and robustness objectives. These guidelines reflect the time dependence of the frequency response of a linear, time-varying system while retaining the frequency domain intuition of the time-invariant case. This procedure is analogous to the loop shaping design procedure of McFarlane and Clover (1989) for linear, time-invariant systems.

Optimal causal control of a wave energy converter in a random sea

The use of abstract chemical reaction networks (CRNs) as a modelling and design framework for the implementation of computing and control circuits using enzyme-free, entropy driven DNA strand displacement (DSD) reactions is starting to garner widespread attention in the area of synthetic biology. Previous work in this area has demonstrated the theoretical plausibility of using this approach to design biomolecular feedback control systems based on classical proportional-integral (PI) controllers, which may be constructed from CRNs implementing gain, summation and integrator operators. Here, we propose an alternative design approach that utilises the abstract chemical reactions involved in cellular signalling cycles to implement a biomolecular controller - termed a signalling-cycle (SC) controller. We compare the performance of the PI and SC controllers in closed-loop with a nonlinear second-order chemical process. Our results show that the SC controller outperforms the PI controller in terms of both performance and robustness, and also requires fewer abstract chemical reactions to implement, highlighting its potential usefulness in the construction of biomolecular control circuits.

ABSTRACTIn this study, a command tracking error square control scheme is first proposed for analysis and design of feedback control systems. One of the tracking errors is low‐pass filtered and used in the feedback loop for gain adaptation; the other is used in the forward loop for command tracking control. The overall systems are nonlinear feedback systems, and can be reconfigured to an automatic gain control (AGC) loop with command tracking error input. The stability and robustness of the controlled systems are verified by time response, frequency response, and large parameter variation testing with a simple illustrating example and are finally applied to a complicated electro‐hydraulic velocity servo system with large load disturbance.

Summary This paper contains a summary description of the way in which interactive C.A.D. of feedback control systems has been introduced into a number of technological degree courses at the University of Reading. It describes the basic hardware/software available, the ways in which the students are taught and the practical work they are expected to perform.

An analytical framework is described for the design of feedback control systems to meet specified performance criteria in the presence of structured and unstructured uncertainty. Attention is focused upon the linear time invariant, single-input, single-output problem for the purposes of exposition. The framework provides for control of the degree of the stabilizing compensator or controller.

The networked environment presents many new challenges for the design of feedback control systems. A specific problem is that networked control systems (NCSs) are subject to the performance-degrading effects of time-varying, random loop delays. Model predictive controllers are a well-known method for delay compensation, but often exhibit poor performance under these types of loop delay. This paper investigates the use of play-back buffers to remove the uncertainty in the delay. First we study the value of removing all delay uncertainty with comparisons to unbuffered PID control using a bounded- interval delay distribution, and then we explore optimal playback buffer design for bounded-interval and heavy-tailed delay distributions.

A systematic control design methodology is introduced for multi-input/multi-output stable open-loop plants with multiple saturations. The idea is to introduce a supervisor loop so that when the references and/or disturbances are sufficiently small, the control system operates linearly as designed. For signals large enough to cause saturations, the control law is modified in such a way to ensure stability and to preserve, to the extent possible, the behavior of the linear control design. Key benefits of this methodology are: (1) the modified compensator never produces saturating control signals: (2) integrators and/or slow dynamics in the compensator never windup: (3) the directional properties of the controls are maintained; and (4) the closed-loop system has certain guaranteed stability properties. The advantages of the proposed design methodology are illustrated by the simulation of an academic example and the simulation of the multivariable longitudinal control of a modified model of the F-8 aircraft. >