A package of computer subprograms for control system analysis

Design and analysis of high-order control systems requires a great deal of laborious calculations. Stability analyses are usually conducted with standard techniques such as root locus or frequency response. The calculations involved in these analyses are generally too complex to do by hand. Graphical techniques and approximations may be used, but often introduce unacceptably large errors.

In this paper, an optimal controller is investigate d based on the Linear Quadratic Control method. The controller simulation is execut ed based on a small-scale autonomous model helicopter, Yamaha R-50. Genetic algorithms are employed to generate weighting matrices by optimizing the performance index of the control system. As a result, this method produces optimal weighting matrices which further improve the Linear Quadratic Controller. A full state feedback approach is used in the control design process. A Kalman observer is integrated into the controller to predi ct full state variables, since only a limited number of state variables are measured. Finally, th e performance of the controller is evaluated in the time domain with and without disturbances when the model helicopter is in hovering flight and forward flight. I. Introduction utonomous helicopters offer special advantages for major operations in both civilian and military sect ors, particularly in cases that are considered either in accessible or dangerous for human beings, due to th eir vertical take-off and landing (VTOL) capabilities. The contr ol system of autonomous helicopters plays a key rol e in its ability to carry out the assigned task effectively. This has lead to a growing interest in the design and analysis of the control system of autonomous helicopters. In 1990s, classic control theory was widely used for autonom ous helicopter control systems, which is mainly a succe ssive loop closure approach. The control gains are selected separately using tools such as Root locus, Bode and Nyquist plots. The controller parameters are usual ly tuned based on the controller performance measured either in ti me domain or in frequency domain. This design procedure becomes increasingly difficult as more and more loo ps are added, specifically, when the dynamics invol ve multiple inputs, multiple outputs, or multiple feedback loop s with complex performance criteria. To control an autonomous helicopter modeled as a complex Multiple Input Multiple Output (MIMO) system, the performance criteria can not be easily met by the approach based on classis cont rol theory. Several researchers have developed a number of diff erent ways for the control system design of autonom ous helicopters in the literature. Shim et al 1 presented three different control methods using li near robust, fuzzy logic, and nonlinear tracking controls, for an autonomous helicopter in hover and low speed flight. Johnson a nd Kannan 2

Looking back at the history of control engineering, one finds that technology and ideas combine themselves until they reach a successful result, over the timeline of several decades (Bernstein, 2002). Simple dynamical compensators (such as PID, Phase-Lead, Phase-Lag, etc) have done in the past a remarkably good and efficient job in real-life control applications. Theoretical insight in the closed-loop behavior is provided by powerful and well-developed theories – which are considered as basic knowledge for every control engineer – such as the Root Locus and the Frequency Response techniques. In the past, these theories have been extensively used as analysis tools – e.g. the stability analysis of a given control system based on closed loop poles (Root Locus RL) or Nyquist criterion (Frequency Response FR). However, analysis implies that a controller is already available, irrespective of its design method. Nowadays, thanks to the computational and graphical power of modern computers, many of these theories can be implemented as interactive graphical design tools. In this way, control engineering moves away from being an abstract and mathematical-oriented discipline and it evolves gradually towards a mature engineering discipline. An extensive reflection upon the role of information science in control engineering has been given in (Dormido, 2004). This new way of interactive control education provides practical insights into control systems fundamentals (Wittenmark et al, 1998; Dormido et al, 2002). Such combinations of interactive environment and animation bring visualization to a new level and aid learning and active participation by control engineering students (Kheir et al, 1996; Johansson et al, 1998). Recently, a Root Locus RL toolbox has been introduced in Matlab – in this chapter, a Frequency Response toolbox (FRtool) will be presented. Although some other computer aided design (CAD) tools based on frequency response have been developed in the past (Balakrihsnan & Boyd, 1994; Satoh et al, 1994; Piguet & Gillet, 1997; Pouliz & Pouliezos, 1997; Piguet et al, 1999), the one presented in this contribution is highly interactive, graphical, easy-to-use and posing an elegant simplicity (especially for non-experts, as limited control engineering insight is required). The Matlab® controller design tool based on the root locus method (rltool) cannot handle systems with time-delay without approximating the dead5

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

Design of Continuous and Discrete Optimal Control Systems with Prescribed Transient Response

It is generally accepted that active control systems provide better structural performance when compared to their passive counterparts. On the other hand, the design of active control systems based on linear control theory is highly dependent on the structural properties. For this reason, their performance is expected to be affected more severely by variations in structural properties compared to those of passive systems. These variations can occur due to nonlinear structural behavior, or even before that due to uncertainties in the estimation of these properties and in numerical modeling. The present work is an investigation of the dependency of various control systems used for supplemental energy dissipation to the changes in structural properties. For this purpose, the performance degradations of most common control systems are studied for structures entering their nonlinear response range. The considered control systems include active control using linear quadratic regulator algorithm and passive control using viscous fluid dampers and yielding devices. These control systems are optimized separately for linear and nonlinear structures by minimizing performance indices based on inter-story drift and absolute acceleration response. It is shown that among these control systems, viscous dampers are affected the least by the alteration of structural properties, which may further support their utilization contrary to the more costly active control systems. It is also shown that the amount of performance degradation depends on the selected structural performance index, and more importantly, the severity of the earthquake excitation.

In this paper, the vibration behavior of atomic force microscopy (AFM) micro-cantilever (MC) with a piezoelectric layer in the non-contact mode considering the axial load in the air medium was studied. In the dynamic modes of AFM nano-robot, parameters such as amplitude, frequency and motion phase are used as feedback factors in the control system in order to extract sample surface information and the nanoparticle voltage. Consequently, to extract these parameters, the vibration analysis of the MCs of this nano-robot is essential. Additionally, because of the wide-spread applications of oblique MCs, applying the axial and transverse forces into the model is absolutely necessary. In order to simulate the AFM four-layered piezoelectric MC with the axial displacement, Timoshenko beam theory as well as modified couple stress theory has been used. Considering the axial displacement in the modeling not only leads to more degrees of freedom, but also has increased the amount of calculations. To extract the equations of the system, energy method has been used, and then, the FEM was applied to discrete the equations. To verify the accuracy of the results, the frequency and time responses of FEM have been compared with the results of the 3D mode simulation obtained by COMSOL software and experimental results. The comparison of results indicates that modifying the modeling by considering the axial displacement has increased the accuracy of the frequency and time responses. The error percentage of the first frequency response is 0.42% and the error percentage of the time response is 0.0053%. Furthermore, to examine the effect of MC geometry on its vibrational behavior, frequency and time responses of three common geometries of AFM MCs have been extracted. The results illustrate the improvement of modeling in all three geometries. Then, the sample surface topography of MCs at the first and second excitation frequencies in contact with rectangular roughnesses demonstrated the decline of time delay at the second excitation frequency. The results of the topography simulation in the air environment show that the roughness’s trend is better at the second mode than at the first mode. Finally, the effect of MC angle on the topography of the rectangular MC surfaces in contact with the rectangular roughness in 2D and 3D modes was examined in the presence and the absence of axial load. The results show that in the presence of the axial displacement in modeling, by increasing of MC angle, the topography quality increases drastically.

Macro fiber composite (MFC) has been widely investigated in the active vibration control. Yet, the actuation performance of MFC is a function of temperature. For the active vibration control system using MFC based on the linear piezoelectric equation, if the actuation parameters of MFC are variable, it is easy to cause insufficient control force or control spillover, which degrades the performance of control system. Therefore, an adaptive gain adjustment approach for the active vibration control system is developed, which ensures that the active control system can maintain the expected performance of vibration suppression under the time-varying ambient temperatures. The variation of actuation performance of MFC with temperature is measured by the experiment. The polynomial fitting method is utilized to obtain the functional relationship between the regulator parameters and ambient temperature. The performance of the proposed adaptive gain regulator is verified by the experiment in the temperature environment chamber. In the experiment, a scaled vertical tail model attaching the MFC actuator is as an example, and the fractional order positive position feedback control system is as a representative of the control system without temperature adaptability. When the temperature is −60°C, compared with the control system without regulator, the reduction ratio of response RMS of the control system with adaptive gain regulator is increased by 17.26%.The results indicate that the proposed adaptive gain regulator can avoid the adverse influence of the time.varying ambient temperature on the performance of the control system.

Early control textbooks in Sweden

A novel approach is provided for the analysis of multi-loop control systems by a digital computer introducing operational arrays, in which all the instructions are given through the data input. Changing parameters for optimisation purposes or the introducing of additional networks becomes a matter of changing the operational array in the data, leaving the main programme untouched. Although the method is general, only the frequency response of a control system is considered, but at the same time the way has been left open to expand the method as experience reveals the need for further facilities, i.e. root locus, time response, etc. The programme has been written in Algol, but the basic flow diagram can be easily converted into any other language. Whatever the language is, the programme provides the necessary software design facilities translated into control engineering terms in a readily usable computational form.

The computer program LCAP2 (Linear Controls Analysis Program) provides the analyst with the capability to numerically perform classical linear control analysis techniques such as transfer function manipulation, transfer function evaluation, frequency response, root locus, time response and sampled-data transforms. It is able to deal with continuous and sampled-data systems, including multiloop multirate digital systems, using <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s, z</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</tex> transforms. This program has been used extensively for years at the Aerospace Corporation in the analysis and design of satellite and launch vehicle control systems.

Continuous rolling of plate and strip is a typical complex, multi-variable and multi-objective system, which has shown the characteristics such as complication, large-scale and close coupling increasingly. Therefore, it has become more and more difficult to design complex industrial control systems. This paper describes a multi-agent control system for a strip rolling process. In order to realize the distributed and intelligent control of the system, after analyzing the characteristics and structure of this control system, the multi-agent theory was applied and then a control strategy was designed based on multi-agent interaction. Finally, the partition method, the coordination mechanism, and the model structure of the multi-Agent control system for the strip rolling process were analyzed. This paper concludes with a new control structure and system for strip rolling process, which helps to build up the generally integrated model, to eliminate the coupling among control systems, and to realize more intelligent control.

The root locus method determines all of the roots of the differential equation of a control system by a graphical plot which readily permits synthesis for desired transient response or frequency response. The base points for this plot On the complex plane are the zeros and poles of the open loop transfer function, which are readily available. The locus of roots is a plot of the values of s which make this transfer function equal to - 1 as loop gain is increased from zero to infinity. The plot can be established in approximate form by inspection and the significant parts of the locus calculated accurately and quickly by use of a simple device. For multiple loop systems, one solves the innermost loop first, which then permits the next loop to be solved by another root locus plot. The resultant plot gives a complete picture of the system, which is particularly valuable for unusual systems or those which have wide variations in parameters.

Many practical problems in the field of engineering become so complex that they may be effectively solved only with the aid of a computer. An effective solution depends on the use of an efficient algorithm. Plotting root locus diagrams is such a problem. This thesis presents such an algorithm. Root locus design of feedback control systems is a very powerful tool. Stability of systems under the influence of variables can be easily determined from the root locus diagram. For even moderately complex systems of the type found in practical applications, determination of the locus is extremely difficult if accuracy is required. The difficulty lies in the classical method of graphically determining the location of points on the locus by trial and error. Such a method cannot be efficiently applied to a computer program. The text presents an original algorithm for plotting the root locus of a general system. The algorithm is derived using the combined methods of complex variable algebra and numerical analysis. For each abscissa desired a polynomial is generated. The real roots of this polynomial are the ordinate values for points on the root locus. Root finding methods from numerical analysis enable the solution of the problem to be one of convergent iteration rather than trial and error. Among the material presented is a computer program for solution of the general problem, an example of a completely analytic solution, and a table of solutions for more simple systems. The program inputs are the coefficients of the open loop transfer function and the range and increments of the real axis which are to be swept. The output lists the real and imaginary components of all solution points at each increment of the sweep. Also listed are the magnitude and angle components of the solution point and the value of system gain for which this is a solution. For less complex problems, the method can be applied analytically. This may result in an explicit relation between the real and imaginary components of all solution points or even in a single expression which can be analyzed using the methods of analytic geometry. As with any advance in the theory of problem solving, the ideas presented in the thesis are best applied in conjunction with previous solution methods. Specifically, an idea of the approximate location of the root locus can be obtained using sketching rules which are well known. The method presented

In previous chapters on Bode plot design, we met a class of control design problems which were based on manipulating the open-loop transfer function characteristics to achieve some desired closed-loop performance. For example, we adjusted the open-loop phase to improve the closed-loop system stability margins. We then have to form the closed-loop transfer function and check whether the open-loop control design meets the design specification. Hence there is a need to establish a relationship between the open-loop characteristics, GOL(jω), and the closed-loop characteristics, GCL(jω), in our control design problems. In many cases we have complicated models or sets of frequency response data and the problems of manipulating unwieldy transfer functions or frequency response data files are not easy. However, control design packages like MATLAB make the calculation of response information as simple as typing $$gcl = gol/\left( {1 + gol} \right);$$ where gol and gcl are the open and closed-loop transfer functions respectively. However, traditionally the problem of gaining closed-loop information easily was solved by using a Nichols chart. The Nichols chart is still part of control design packages, since it gives valuable information by graphically relating open-loop and closed-loop information on one plot. We find that Nichols charts are less commonly used than the Bode plot techniques we have been dealing with, but particular industries, such as the aerospace industry, often use Nichols charts to a greater degree. We present here an introduction to the Nichols chart and give several examples of simple design.