# Robust Control Design With MATLAB

Design a controller for the plant G described in Robust Controller Design. This plant is a first-order system with an uncertain time constant. The plant also has some uncertain dynamic deviations from first-order behavior beyond about 9 rad/s.

## Robust Control Design with MATLAB

Because of the nominal first-order behavior of the plant, choose a PI control architecture. For a desired closed-loop damping ratio ξ and natural frequency ωn, the design equations for the proportional and integral gains (based on the nominal open-loop time constant of 0.2) are:

The nominal closed-loop bandwidth achieved by C2 is in a region where G has significant model uncertainty. It is therefore expected that the model variations cause significant degradations in the closed-loop performance with that controller. To examine the performance, form the closed-loop systems and plot the step responses of samples of the resulting systems.

Written for graduate students and professionals, Robust Control Design with MATLAB offers improved guidance in robust control design. The book helps readers who want to learn how to deal with robust control design problems without spending a lot of time in researching complex theoretical developments. Topics include basic methods and theory, modeling of uncertain systems, robust design specifications, and loop-shaping design procedures.

By sharing their experiences in industrial cases with minimum recourse to complicated theories and formulae, the authors convey essential ideas and useful insights into robust industrial control systems design using major H-infinity optimization and related methods allowing you quickly to move on with your own challenges.

Robust Control Design with MATLAB is for graduate students and practising engineers who want to learn how to deal with robust control design problems without spending a lot of time in researching complex theoretical developments.

Conventional passive suspensions use a spring and damper between the car body and wheel assembly. The spring-damper characteristics are selected to emphasize one of several conflicting objectives such as passenger comfort, road handling, and suspension deflection. Active suspensions allow the designer to balance these objectives using a feedback-controller hydraulic actuator between the chassis and wheel assembly.

The hydraulic actuator used for active suspension control is connected between the body mass mb and the wheel assembly mass mw. The nominal actuator dynamics are represented by the first-order transfer function 1/(1+s/60) with a maximum displacement of 0.05 m.

Observe that the body acceleration is smallest for the controller emphasizing passenger comfort and largest for the controller emphasizing suspension deflection. The "balanced" design achieves a good compromise between body acceleration and suspension deflection.

Next use μ-synthesis to design a controller that achieves robust performance for the entire family of actuator models. The robust controller is synthesized with the musyn function using the uncertain model qcaric(:,:,2) corresponding to "balanced" performance (β=0.5).

The robust controller Krob has relatively high order compared to the plant. You can use the model reduction functions to find a lower-order controller that achieves the same level of robust performance. Use reduce to generate approximations of various orders.

Next use robgain to compute the robust performance margin for each reduced-order approximation. The performance goals are met when the closed-loop gain is less than γ=1. The robust performance margin measures how much uncertainty can be sustained without degrading performance (exceeding γ=1). A margin of 1 or more indicates that we can sustain 100% of the specified uncertainty.

Alternatively, you can use musyn to directly tune low-order controllers. This is often more effective than a-posteriori reduction of the full-order controller Krob. For example, tune a third-order controller to optimize its robust performance.

You choose the weighting functions W1,W2,W3 to shape the frequency responses for tracking and disturbance rejection, controller effort, and noise reduction and robustness, respectively. For details about how to choose weighting functions, see Mixed-Sensitivity Loop Shaping.

Weighting functions, specified as dynamic system models. Choose the weighting functions W1,W2,W3 to shape the frequency responses for tracking and disturbance rejection, controller effort, and noise reduction and robustness. Typically:

The toolbox automatically tunes both SISO and MIMO controllers for plant models with uncertainty. Controllers can include decentralized, fixed-structure controllers with multiple tunable blocks spanning multiple feedback loops.

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 common-place. 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 two-wheeled 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.

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 fixed-point and floating-point computations and describe the rounding errors associated with these computations. In case of fixed-point 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.

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 closed-loop system. The main point of the chapter is the derivation of adequate continuous-time and discrete-time 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.

In this chapter, we consider briefly some important issues concerning the performance requirements to closed-loop linear systems and the fundamental design limitations in achieving the control aims. The performance specifications are formulated in continuous-time due to the clear physical interpretation in this case. First, we present the relatively simple case of single-input-single-output (SISO) systems which are well studied in the classical control theory. The trade-offs in the design of such systems are shown in some details. Then, we discuss the more complicated case of multiple-input-multiple-output (MIMO) systems whose performance is investigated by using the singular value plots of certain closed-loop transfer function matrices. An important issue in controller design is the closed-loop 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).

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 discrete-time 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 cart-pendulum 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 closed-loop systems in presence of plant uncertainty.

The aim of this case study is to present in detail the μ synthesis of a high-order integral attitude controller of a miniature helicopter and to demonstrate results from the hardware-in-the-loop 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 hardware-in-the-loop simulation are close to the results from the double-precision 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 closed-loop system behavior in the presence of different disturbances, noises, and parameter variations. 041b061a72