Classical Feedback Control with Nonlinear Multi-Loop Systems

With MATLAB® and Simulink®, Third Edition

Boris J. Lurie, Paul Enright

• Publisher: CRC Press
• Year: 2019
• eBook ISBN: 9781351011839
• Topic: Physical Sciences
• Subtopic: Mechanics

Classical Feedback Control with Nonlinear Multi-Loop Systems describes the design of high-performance feedback control systems, emphasizing the frequency-domain approach widely used in practical engineering. It presents design methods for high-order nonlinear single- and multi-loop controllers with efficient analog and digital implementations. Bode integrals are employed to estimate the available system performance and to determine the ideal frequency responses that maximize the disturbance rejection and feedback bandwidth. Nonlinear dynamic compensators provide global stability and improve transient responses. This book serves as a unique text for an advanced course in control system engineering, and as a valuable reference for practicing engineers competing in today's industrial environment.

1

Feedback and Sensitivity

Chapter 1 introduces the basics of feedback control. The purpose of feedback is to make the output insensitive to plant parameter variations and disturbances. Negative, positive, and large feedback are defined and discussed along with sensitivity and disturbance rejection. The notions of frequency response, the Nyquist diagram, and the Nichols chart are introduced. (The Nyquist stability criterion is presented in Chapter 3.)

Feedback control and block diagram algebra are explained at an elementary level in Appendix 1, which can be used as an introduction to this chapter. Laplace transfer functions are described in Appendix 2.

1.1 Feedback Control System

It is best to begin with an example. Figure 1.1a depicts a servomechanism regulating the elevation of an antenna. Figure 1.1b shows a block diagram for this control system made of cascaded elements, i.e., links. The capital letters stand for the signals’ Laplace transforms and also for the transfer functions of the linear links.

FIGURE 1.1
Single-loop feedback system.

There is one input command U ₁, which is the commanded elevation angle, and just one output U ₂, which is the actual elevation of the antenna, so the system is said to be single-input single-output (SISO). Evidently there is one feedback loop, and so the system is also referred to as single-loop.

The feedback path contains some sort of sensor for the output variable and has the transfer function B. Ideally, the measured output value BU ₂ equals the commanded value U ₁, and the error E = U ₁ – BU ₂, at the output of the summer, is zero. In practice, most of the time the error is nonzero but small.

The error is amplified by the compensator C and applied to the actuator A, in this case a motor regulator (driver) and a motor, respectively. The motor rotates the plant P, the antenna itself, which is the object of the control. The compensator, actuator, and plant make up the forward path with the transfer function CAP.

The return signal, which goes into the summer from the feedback path, is BU ₂ = TE, where the product T = CAPB = BU ₂/E, is called the loop transfer function or the return ratio.

The output of the summer is:

$$E=U_1-ET$$

(1.1)

so that the error can be expressed as

$$E=\frac{U_1}{T+1}=\frac{U_1}{F}$$

(1.2)

where F = T + 1 is the return difference. Its magnitude |F| is the feedback. It is seen that when the feedback is large, the error is small.

If the feedback path was not present, the output U ₂ would simply equal the product CAPU ₁, and the system would be referred to as open-loop.

Example 1.1

A servomechanism for steering a toy car (using wires) is shown in Figure 1.2. The command voltage U ₁ is regulated by a joystick potentiometer. Another identical potentiometer (angle sensor) placed on the shaft of the motor produces voltage U _angle proportional to the shaft rotation angle. The feedback makes the error small, so that the sensor voltage approximates the input voltage, and therefore the motor shaft angle tracks the joystick-commanded angle.

FIGURE 1.2
Joystick control of a steering mechanism.

This arrangement of a motor with an angle sensor is often called servomotor, or simply servo. Similar servos are used for animation purposes in movie production.

The system of regulating aircraft-control surfaces using joysticks and servos was termed “fly by wire” when it was first introduced to replace bulky mechanical gears and cables. The required high reliability was achieved by using four independent parallel analog electrical circuits.

The telecommunication link between the control box and the servo can certainly also be wireless.

Example 1.2

A phase-locked loop (PLL) is shown in Figure 1.3. The plant here is a voltage-controlled oscillator (VCO).

FIGURE 1.3
Phase-locked loop.

The VCO is an ac generator whose frequency is proportional to the voltage applied ...

Table of contents

• Cover
• Half-Title
• Series
• Title
• Copyright
• Contents
• Preface
• To Instructors
• Authors
• 1 Feedback and Sensitivity
• 2 Feedforward, Multi-Loop, and MIMO Systems
• 3 Frequency Response Methods
• 4 Shaping the Loop Frequency Response
• 5 Compensator Design
• 6 Analog Controller Implementation
• 7 Linear Links and System Simulation
• 8 Introduction to Alternative Methods of Controller Design
• 9 Adaptive Systems
• 10 Provision of Global Stability
• 11 Describing Functions
• 12 Process Instability
• 13 Multiwindow Controllers
• 14 Nonlinear Multi-Loop Systems with Uncertainty
• Appendix 1: Feedback Control, Elementary Treatment
• Appendix 2: Frequency Responses
• Appendix 3: Causal Systems, Passive Systems and Positive Real Functions, and Collocated Control
• Appendix 4: Derivation of Bode Integrals
• Appendix 5: Program for Phase Calculation
• Appendix 6: Generic Single-Loop Feedback System
• Appendix 7: Effect of Feedback on Mobility
• Appendix 8: Regulation
• Appendix 9: Balanced Bridge Feedback
• Appendix 10: Phase-Gain Relation for Describing Functions
• Appendix 11: Discussions
• Appendix 12: Design Sequence
• Appendix 13: Examples
• Appendix 14: Bode Step Toolbox
• Appendix 15: Nonlinear Multi-Loop Feedback Control (Patent Application)
• Bibliography
• Notation
• Index