The design of linear controllers with symbolic algebra (original) (raw)

Symbolic analysis and design of control systems using Mathematica

International Journal of Control, 2006

Contemporary computer tools can generate a tremendous amount of numerical data so the user might easily lose insight into the phenomenon being investigated. Those who use powerful computer algebra systems must thoroughly understand the assumptions that underlie the software. In this paper, the role and importance of symbolic computation in control engineering and signal processing is exemplified. Real-life application examples are presented in which systems are symbolically solved and simulated with Mathematica. We introduce an original approach to algorithm development, system design and symbolic processing that successfully overcomes some problems encountered in the traditional approach. Benefits of symbolic methods and the role of computer algebra systems are highlighted from the viewpoint of both academia and industry.

Software for Control System Analysis and Design: Symbol Manipulation

Wiley Encyclopedia of Electrical and Electronics Engineering, 1999

The computer revolution has radically changed every area of engineering and control/systems engineering is not an exception. Indeed, computers have become essential tools in modeling, analysis, design and implementation of control systems. In particular, they enable the design engineer to tackle problems of ever increasing complexity . The topic of this article is to highlight the effectiveness of symbolic computation software, or the so called computer algebra, in the analysis and design of control systems.

Symbolic Computation For The Analysis AndSynthesis Of Nonlinear Control Systems

WIT transactions on engineering sciences, 1999

Symbolic computation has proved to be an effective tool for handling large mathematical expressions which are too complex for calculation by hand. Especially in the field of nonlinear control the use of computer algebra programs is unavoidable. Since the memory requirements are over polynomial with respect to the problem size, it is important to develop algorithms that optimally fit the symbolic treatment of these problems. Hence, the developers of algorithms in computer algebra programs must free themselves from the well known numerical solutions and they have to take care of the special features of computer algebra. In this sense the present contribution presents some selected algorithms for the analysis and design of a special class of nonlinear systems, the so called affine input systems.

Special issue on the use of computer algebra systems for computer aided control system design

International Journal of Control, 2006

The importance of the continuing and growing need in the systems and control community for reliable algorithms and robust numerical software for increasingly challenging applications is well known and has already been reported elsewhere (IEEE Control Systems Magazine, Vol. 24, Issue 1). However, we have all had the experience of working on a mathematical project where an increased number of symbolic manipulations was needed. In a simple case, the required computation might have been to compute the Laplace transform or the inverse Laplace transform of a function, or to find the transfer function matrix for a given system topology where parameters are included. In a more demanding situation the required computation might have been to find the parametric family of solutions of a polynomial matrix Diophantine equation resulting from a variety of control problems such as those associated with stabilization, decoupling, model matching, tracking and regulation, or to compute the Smith McMillan form of a rational transfer function matrix in order to obtain a better insight into a number of structural properties of a system. The desire to use a computer to perform long and tedious mathematical computations such as the above led to the establishment of a new area of research whose main objective is the development: (a) of systems (software and hardware) for symbolic mathematical computations, and (b) of efficient symbolic algorithms for the solution of mathematically formulated problems. This new subject area is referred to by a variety of terms such as symbolic computations, computer algebra, algebraic algorithms to name a few. During the last four decades this subject area has accomplished important steps and it is still continuing its evolution process.

Symbolic Computations on Rings of Rational Functions and Applications in Control Engineering

2009

A collection of algorithms implemented in Mathematica 7.0, freely available over the internet, and capable to manipulate rational functions and solve related control problems using polynomial analysis and design methods is presented. The package provides all the necessary functionality and tools in order to use the theory of \(\it \Omega-\) stable functions, and is expected to provide the necessary framework for the development of several other algorithms that solve specific control problems.

A transfer function computational algorithm for linear control systems

IEEE Control Systems Magazine, 1995

he problem of computing the closed T loop transfer function of a linear system represented by its block diagram or by its flow graph is of importance in CAD programs designed for the analysis of linear systems. Mason's rule [ 11 is one of the available techniques to perform such calculations, but it requires searching into the graph for different paths and loops and becomes slow as system complexity increases. In this paper, a method is developed to efficiently compute the closed loop transfer function and intermediate transfer functions of a linear system.

Integrated Design of Symbolic Controllers for Nonlinear Systems

IEEE Transactions on Automatic Control, 2012

Symbolic models of continuous and hybrid systems have been studied for a long time, because they provide a formal approach to solve control problems where software and hardware interact with the physical world. While being powerful, this approach often encounters some limitations in concrete applications, because of the large size of the symbolic models needed to be constructed. Inspired by on-the-fly techniques for verification and control of finite state machines, in this note we propose an algorithm that integrates the construction of the symbolic models with the design of the symbolic controllers. Computational complexity of the proposed algorithm is discussed and an illustrative example is included.

Symbolic computation based control design tool for switched systems

2011

This paper presents a hybrid system controller design tool for finding stabilizing state feedback controllers for a class of switched system. The tool is based symbolic computation which avoids problems that may arise from rounding errors prevalent in numerical computation. Attention is focused on finding the existence of common Lyapunov function (CLFs), as this ensures stability for arbitrary switching sequences between subsystems. The switching system considered here is restricted to single input single output switching systems having second order linear subsystems.

Symbolic computation environment for nonlinear L2 control. application examples

2003

In this work a computer based environment for symbolic calculation applied to nonlinear L2 control is described. A very practical design and analysis methodology for nonlinear H∞ control is presented. Starting from a model of the plant in affine form, and a state feedback controller (or state estimated feedback controller), a design procedure is established using nonlinear L2 control theory. Functions hz(x), kuz(x) and parameter γ are used as design parameters. Due to the fact that exhaustive symbolic computation is necessary in nonlinear L2 control, the Maple program has been used. The application is also based on Matlab and Simulink in order to implement functions from Control Toolboxes and simulation facilities. An additional advantage of the control theory used and the design procedure implemented in this work is that it can be applied to scalar and multivariable nonlinear systems. The design methodology and the software environment are tested by means of the simulations studies...

A Full Symbolic Feedback Control Design

Port-Said Engineering Research Journal, 2021

Although Symbolic analysis suffers from higher complexity, higher resource requirements and longer execution than the numerical analysis, it is proven more accurate and general and it is recommended to use it. It is also proven that the feedback gains are very crucial to stabilize any system and it is important to measure them accurately. This paper introduces the design of a full symbolic feedback control system based on pole placement method. The feedback gains of parameter varying control system are estimated using three alternative algorithms; Direct substitution, Bass-Gura and Ackerman formula, where the gains can be changed according to the parameters which are measured online. Experiments were conducted on the aircraft pitch control as an application to estimate the feedback gain to stabilize the system. The results demonstrate that the symbolic solution reduces the complexity by a significant margin and produces the same results assumed in the compared research.