Design of Embedded Robust Control Systems Using MATLAB® / Simulink®
Robust control theory allows for changes in a system whilst maintaining stability and performance. Applications of this technique are very important for dependable embedded systems, making technologies such as drones and other autonomous systems with sophisticated embedded controllers and systems relatively commonplace. The aim of this book is to present the theoretical and practical aspects of embedded robust control design and implementation with the aid of MATLAB® and SIMULINK®. It covers methods suitable for practical implementations, combining knowledge from control system design and computer engineering to describe the entire design cycle. Three extended case studies are developed in depth: embedded control of a tank physical model; robust control of a miniature helicopter; and robust control of twowheeled robots. These are taken from the area of motion control but the book may be also used by designers in other areas. Some knowledge of Linear Control Theory is assumed and knowledge of C programming is desirable but to make the book accessible to engineers new to the field and to students, the authors avoid complicated mathematical proofs and overwhelming computer architecture technical details. All programs used in the examples and case studies are freely downloadable to help with the assimilation of the book contents.
Inspec keywords: program compilers; control engineering computing; mathematics computing; embedded systems; robust control; programming
Other keywords: automated code generation; complex highorder controllers; programming environment; highperformance closedloop dynamics; processors; embedded robust control system design; MatlabSimulink; error free code
Subjects: Compilers, interpreters and other processors; Control engineering computing; General and management topics; Systems analysis and programming; Mathematics computing; Stability in control theory
 Book DOI: 10.1049/PBCE113E
 Chapter DOI: 10.1049/PBCE113E
 ISBN: 9781785613302
 eISBN: 9781785613319
 Page count: 534
 Format: PDF

Front Matter
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1 Embedded control systems
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In this chapter, we make a concise overview of embedded control systems and discuss some aspects of the corresponding hardware and software which is used in these systems. The embedded control systems are digital systems and their performance is affected by sampling and quantization errors. That is why, we present some basic elements of fixedpoint and floatingpoint computations and describe the rounding errors associated with these computations. In case of fixedpoint arithmetic, the emphasis is put on the scaling problem, which is the most important issue in using such arithmetic. We describe briefly the stages of embedded controller design, controller simulation, and implementation.

2 System modeling
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This chapter is devoted to the mathematical description of the basic elements and processes pertaining to the embedded control systems. The models obtained as a result of this description are important for the design of controllers which have to ensure the necessary performance and robustness of the closedloop system. The main point of the chapter is the derivation of adequate continuoustime and discretetime models of the plant, sensors, and actuators. For this aim, we implement various analytic and numeric tools available in control theory and control engineering practice. These tools include modeling, linearization, and discretization of dynamic plants, system identification, modeling of uncertain systems, and stochastic modeling. We demonstrate the usage of different MATLAB® functions and Simulink® blocks intended to build accurate and reliable models of embedded system components.

3 Performance requirements and design limitations
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In this chapter, we consider briefly some important issues concerning the performance requirements to closedloop linear systems and the fundamental design limitations in achieving the control aims. The performance specifications are formulated in continuoustime due to the clear physical interpretation in this case. First, we present the relatively simple case of singleinputsingleoutput (SISO) systems which are well studied in the classical control theory. The tradeoffs in the design of such systems are shown in some details. Then, we discuss the more complicated case of multipleinputmultipleoutput (MIMO) systems whose performance is investigated by using the singular value plots of certain closedloop transfer function matrices. An important issue in controller design is the closedloop system performance in presence of different uncertainties. At the end of the chapter, we present some elements of the contemporary approach to the robustness analysis of uncertain linear systems based on the small gain theorem and structured singular value (SSV).

4 Controller design
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The controller synthesis is probably the most difficult and time consuming stage of embedded control system design. In this chapter, we present the design and analysis of five different discretetime controllers which may be implemented successfully in embedded systems. To compare the controller properties, they are applied in single precision to steer one and the same system, namely, the cartpendulum system presented in Chapter 2. This system has some peculiarities which lead to difficulties in the implementation of design methods. One of our primary goals is to investigate the behavior of the corresponding closedloop systems in presence of plant uncertainty.

5 Case study 1: embedded control of tank physical model
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This chapter presents development and experimental evaluation of lowcost embedded system for control of liquid level in a model of tank. The plant is a physical laboratory model of water tank produced by Lucas Nülle Company [134]. The liquid level is controlled in wide range by designed and, H_{∞} controllers. The control algorithms are implemented in lowcost control kit Arduino Mega 2560 [135]. Software in MATLAB/Simulink® environment is developed for generation of control code. Some additional simple hardware devices are developed too. These devices provide appropriate voltage level of analogue signals which are exchanging between physical model of tank and control kit. Controllers are designed on the basis of the linear discretetime blackbox model derived from experimental data by one of the identification techniques described in Appendix D. The main advantage of this technique is that we obtain loworder models of plant and noise. The noise model is used to design appropriate Kalman filter that reduces significantly the sensitivity of control signal to the noise, which is very important for correct exploitation of the actuator. Results from simulation of the closedloop system as well as experimental results obtained from realtime implementation of designed controllers are given. They confirm embedded control system performance in the whole working range.

6 Case study 2: robust control of miniature helicopter
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The aim of this case study is to present in detail the μ synthesis of a highorder integral attitude controller of a miniature helicopter and to demonstrate results from the hardwareintheloop simulation of the helicopter control system. The μ controller designed for hovering allows to suppress efficiently strong wind disturbances in the presence of 15 percent input multiplicative uncertainty. A simple position controller is added to ensure tracking of a desired trajectory in the 3D space. The results from hardwareintheloop simulation are close to the results from the doubleprecision simulation of helicopter control system in Simulink®. It is shown that even for large deviations of the helicopter variables from their trim values in hovering the control system has acceptable performance. The software platform developed allows to implement easily different sensors, servoactuators, and control laws and to investigate the closedloop system behavior in the presence of different disturbances, noises, and parameter variations.

7 Case study 3: robust control of twowheeled robot
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This case study presents the design and experimental evaluation of two controllers for vertical stabilization of twowheeled robot. The first one is a conventional linear quadratic Gaussian (LQG) controller with 17thorder Kalman filter used for state estimation. This controller ensures robust stability of the closedloop system and good nominal performance. The second one is a μ controller ensuring both robust stability and robust performance. Due to the lack of accurate analytical robot model, the controllers design is based on models derived by closedloop identification from experimental data. The robot uncertainty is approximated by an input multiplicative uncertainty which leads to a μ controller of order 44, subsequently reduced to 30. The yaw motion is controlled by using a proportionalintegral (PI) controller on the basis of yaw angle estimate obtained by a separate second order Kalman filter. A software in MATLAB®/Simulink® environment is developed for generation of control code which is embedded in the Texas Instruments Digital Signal Controller TMS320F28335. Results from the simulation of the closedloop system as well as experimental results obtained during the realtime implementation of the designed controllers are given. The theoretical investigation and experimental results confirm that the closedloop system achieves robustness in respect to the uncertainties related to the identified robot model.

Appendix A: Elements of matrix analysis
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In this appendix, we summarize some standard results of matrix analysis, used in the book.

Appendix B: Elements of linear system theory
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The article presents without proofs some basic results from the theory of linear control systems. The following areas are covered: linear continuous timeinvariant control system (continuous LTI system); stability; controllability and observability; Lyapunov equations; and poles and zeros.

Appendix C: Stochastic processes
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In this appendix, we present some notation and basic facts pertaining to the theory of stochastic processes. For a systematic and rigorous treatment of this subject, the reader should consult the references given at the end of the appendix.

Appendix D: Identification of linear models
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Our aim is to introduce the reader into the practical aspects of identification process and to describe how to perform identification using the capabilities of MATLAB® and Simulink®. First, we shall briefly present identification of blackbox models based on the most often used in practice model structures and methods for their parameters estimation. Next, we describe shortly the graybox identification approach based on the linear statespace model.

Appendix E: Interfacing IMU with target microcontroller
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Analog devices MEMS IMU ADIS16405 integrates three gyroscopes and three accelerometers orthogonally oriented and also include three magnetometers. A functional diagram of the ADIS16405 IMU is presented.

Appendix F: Measuring angular velocity with hall encoder
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The rotational Hall encoder converts the angular velocity to electrical signal. Hall elements are semiconductor devices which change its conductivity as a function of applied magnetic field. A magnetic disk is coupled to the shaft of an electric motor. Along the disk boundary, there are evenly distributed magnetic pole pairs (N or S). Close to the magnetic disk, there are mounted both Hall elements which generate a logical signal reflecting current polarity (N or S) generating two shifted square pulse waveforms (Figure F.1). Their frequency (2/Timp) is proportional to the angular velocity and phase shift τ depends on the direction of rotation.

References
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Back Matter
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