**Fuzzy Control**

Utilização na Robótica Industrial

(Parte **1** de 7)

Design Methods for Control Systems

Okko H. Bosgra Delft University of Technology Delft, The Netherlands

Huibert Kwakernaak University of Twente Enschede, The Netherlands

Gjerrit Meinsma University of Twente Enschede, The Netherlands

Notes for a course of the Dutch Institute of Systems and Control Winter term 2001–2002 i i

Preface

Part of these notes were developed for a course of the Dutch Network on Systems and Control with the title “Robust control and H1 optimization,” which was taught in the Spring of 1991. These first notes were adapted and much expanded for a course with the title “Design Methods for Control Systems,” first taught in the Spring of 1994. They were thoroughly revised for the Winter 1995–1996 course. For the Winter 1996–1997 course Chapter 4 was extensively revised and expanded, and a number of corrections and small additions were made to the other chapters. In the Winter 1997–1998 edition some material was added to Chapter 4 but otherwise there were minor changes only. The changes in the 1999–2000 version were limited to a number of minor corrections. In the 2000–2001 version an index and an appendix were added and Chapter 4 was revised. A couple of mistakes were corrected for the 2001–2002 issue.

The aim of the course is to present a mature overview of several important design techniques for linear control systems, varying from classical to “post-modern.” The emphasis is on ideas, methodology, results, and strong and weak points, not on proof techniques.

All the numerical examples were prepared using MATLAB. For many examples and exercises the Control Toolbox is needed. For Chapter 6 the Robust Control Toolbox or the -Tools toolbox is indispensable.

iv iv

Contents

1.1. Introduction | 1 |

1.2. Basic feedback theory | 3 |

1.3. Closed-loop stability | 1 |

1.4. Stability robustness | 20 |

1.5. Frequency response design goals | 27 |

1.6. Loop shaping | 34 |

1.7. Limits of performance | 40 |

1.8. Two-degrees-of-freedom feedback systems | 47 |

1.9. Conclusions | 52 |

1.10. Appendix: Proofs | 52 |

1. Introduction to Feedback Control Theory 1

2.1. Introduction | 59 |

2.2. Steady state error behavior | 60 |

2.3. Integral control | 64 |

2.4. Frequency response plots | 69 |

2.5. Classical control system design | 79 |

2.6. Lead, lag, and lag-lead compensation | 81 |

2.7. The root locus approach to parameter selection | 87 |

2.8. The Guillemin-Truxal design procedure | 89 |

2.9. Quantitative feedback theory (QFT) | 92 |

2.10. Concluding remarks | 100 |

2. Classical Control System Design 59

3.1. Introduction | 103 |

3.2. LQ theory | 104 |

3.3. LQG Theory | 115 |

3.4. H2 optimization | 124 |

3.5. Feedback system design by H2 optimization | 127 |

3.6. Examples and applications | 133 |

3.7. Appendix: Proofs | 139 |

3. LQ, LQG and H2 Control System Design 103

4.1. Introduction | 153 |

4.2. Poles and zeros of multivariable systems | 158 |

4.3. Norms of signals and systems | 167 |

4.4. BIBO and internal stability | 174 |

4.5. MIMO structural requirements and design methods | 177 |

4. Multivariable Control System Design 153 v

4.6. Appendix: Proofs and Derivations | 192 |

Contents

5.1. Introduction | 197 |

5.2. Parametric robustness analysis | 198 |

5.3. The basic perturbation model | 208 |

5.4. The small gain theorem | 210 |

5.5. Stability robustness of the basic perturbation model | 213 |

5.6. Stability robustness of feedback systems | 221 |

5.7. Structured singular value robustness analysis | 232 |

5.8. Combined performance and stability robustness | 240 |

5.9. Appendix: Proofs | 247 |

5. Uncertainty Models and Robustness 197

6.1. Introduction | 249 |

6.2. The mixed sensitivity problem | 250 |

6.3. The standard H1 optimal regulation problem | 258 |

6.4. Frequency domain solution of the standard problem | 264 |

6.5. State space solution | 272 |

6.6. Optimal solutions to the H1 problem | 275 |

6.7. Integral control and high-frequency roll-off | 279 |

6.8. -Synthesis | 288 |

6.9. An application of -synthesis | 292 |

6.10. Appendix: Proofs | 304 |

A.1. Basic matrix results | 311 |

A.2. Three matrix lemmas | 312 |

A. Matrices 311 Index 324

1. Introduction to Feedback Control Theory

Overview – Feedback is an essential element of automatic control systems. The primary requirements for feedback control systems are stability, performance and robustness.

The design targets for linear time-invariant feedback systems may be phrased in terms of frequency response design goals and loop shaping. The design targets need to be consistent with the limits of performance imposed by physical realizability.

Extra degrees of freedom in the feedback system configuration introduce more flexibility.

1.1. Introduction

Designing a control system is a creative process involving a number of choices and decisions. These choices depend on the properties of the system that is to be controlled and on the requirements that are to be satisfied by the controlled system. The decisions imply compromises between conflicting requirements. The design of a control system involves the following steps:

1. Characterize the system boundary, that is, specify the scope of the control problem and of the system to be controlled.

2. Establish the type and the placement of actuators in the system, and thus specify the inputs that control the system.

3. Formulate a model for the dynamic behavior of the system, possibly including a description of its uncertainty.

4. Decide on the type and the placement of sensors in the system, and thus specify the variables that are available for feedforward or feedback.

5. Formulate a model for the disturbances and noise signals that affect the system. 6. Specify or choose the class of command signals that are to be followed by certain outputs.

7. Decide upon the functional structure and the character of the controller, also in dependence on its technical implementation.

1. Introduction to Feedback Control Theory

8. Specify the desirable or required properties and qualities of the control system.

In several of these steps it is crucial to derive useful mathematical models of systems, signals and performance requirements. For the success of a control system design the depth of understanding of the dynamical properties of the system and the signals often is more important than the ap riori qualifications of the particular design method.

The models of systems we consider are in general linear and time-invariant. Sometimes they are the result of physical modelling obtained by application of first principles and basic laws. On other occasions they follow from experimental or empirical modelling involving experimentation on a real plant or process, data gathering, and fitting models using methods for system identification.

Some of the steps may need to be performed repeatedly. The reason is that they involve design decisions whose consequences only become clear at later steps. It may then be necessary or useful to revise an earlier decision. Design thus is a process of gaining experience and developing understanding and expertise that leads to a proper balance between conflicting targets and requirements.

The functional specifications for control systems depend on the application. We distinguish different types of control systems:

Regulator systems. The primary function of a regulator system is to keep a designated output within tolerances at a predetermined value despite the effects of load changes and other disturbances.

Servo or positioning systems. In a servo system or positioning control system the system is designed to change the value of an output as commanded by a reference input signal, and in addition is required to act as a regulator system.

Tracking systems. In this case the reference signal is not predetermined but presents itself as a measured or observed signal to be tracked by an output.

Feedback is an essential element of automatic control. This is why § 1.2 presents an elementary survey of a number of basic issues in feedback control theory. These include robustness, linearity and bandwidth improvement, and disturbance reduction.

Stability is a primary requirement for automatic control systems. After recalling in x 1.3 various definitions of stability we review several well known ways of determining stability, including the Nyquist criterion.

In view of the importance of stability we elaborate in x 1.4 on the notion of stability robustness. First we recall several classical and more recent notions of stability margin. More refined results follow by using the Nyquist criterion to establish conditions for robust stability with respect to loop gain perturbations and inverse loop gain perturbations.

For single-input single-output feedback systems realizing the most important design targets may be viewed as a process of loop shaping of a one-degree-of-freedom feedback loop. The targets include

closed-loop stability, disturbance attenuation, stability robustness, within the limitations set by limitations plant capacity, corruption by measurement noise.

1.2. Basic feedback theory

Further design targets, which may require a two-degree-of-freedom configuration, are further targets satisfactory closed-loop response, robustness of the closed-loop response.

Loop shaping and prefilter design are discussed in § 1.5. This section introduces various important closed-loop system functions such as the sensitivity function, the complementary sensitivity function, and the input sensitivity function.

Certain properties of the plant, in particular its pole-zero pattern, impose inherent restrictions on the closed-loop performance. In x 1.7 the limitations that right-half plane poles and zeros imply are reviewed. Ignoring these limitations may well lead to unrealistic design specifications. These results deserve more attention than they generally receive.

112 and 2-degree-of-freedom feedback systems, designed for positioning and tracking, are discussedi nS ection1 .8.

1.2. Basic feedback theory

1.2.1. Introduction

In this section feedback theory is introduced at a low conceptual level1. It is shown how the simple idea of feedback has far-reaching technical implications.

Example 1.2.1 (Cruise control system). Figure 1.1 shows a block diagram of an automobile cruise control system, which is used to maintain the speed of a vehicle automatically at a constant level. The speed v of the car depends on the throttle opening u. The throttle opening is controlled by the cruise controller in such a way that the throttle opening is increased if the difference vr − v between the reference speed vr and the actual speed is positive, and decreased if the difference is negative.

This feedback mechanism is meant to correct automatically any deviations of the actual vehicle speed from the desired cruise speed.

vr vr − v u throttle opening reference speed vcruise controller car

Figure 1.1.: Block diagram of the cruise control system

For later use we set up a simple model of the cruising vehicle that accounts for the major physical effects. By Newton’s law where m is the mass of the car, the derivative Pv of the speed v its acceleration, and Ftotal the total force exerted on the car in forward direction. The total force may be expressed as

Ftotal.t/ D cu.t/ − v2.t/: (1.2) 1This section has been adapted from Section 1.2 of Kwakernaak and Sivan (1991).

1. Introduction to Feedback Control Theory

The first term cu.t/ represents the propulsion force of the engine, and is proportional to the throttle opening u.t/, with proportionality constant c. The throttle opening varies between 0 (shut) and 1 (fully open). The second term v2.t/ is caused by air resistance. The friction force is proportional to the square of the speed of the car, with the friction coefficient. Substitution of Ftotal into Newton’s law results in

If u.t/ D 1, t 0, then the speed has a corresponding steady-state value vmax, which satisfies

0 D c− v2 max. Hence, vmax D p c= . Defining wD v vmax (1.4) as the speed expressed as a fraction of the top speed, the differential equation reduces to where T D m=p c. A typical practical value for T is T D 10 [s].

We linearize the differential equation (1.5). To a constant throttle setting u0 corresponds a steady-state cruise speed w0 such that 0 D u0 − w20.L et u D u0 CQu and w D w0 CQw, with jQwj w0. Substitution into (1.5) while neglecting second-order terms yields

T PQw.t/ DQu.t/ − 2w0 Qw.t/: (1.6) Omitting the circumflexes we thus have the first-order linear differential equation

Pw D− 1 wC 1

with

The time constant strongly depends on the operating conditions. If the cruise speed increases from 25% to 75% of the top speed then decreases from 20 [s] to 6.7 [s].

Exercise 1.2.2 (Acceleration curve). Show that the solution of the scaled differential equation (1.5) for a constant maximal throttle position

w.t/ D tanh. t

Plot the scaled speed w as a function of t for T D 10 [s]. Is this a powerful car?

1.2.2. Feedback configurations

To understandand analyze feedback we first consider the configurationof Fig. 1.2(a). The signal r is an external control input. The “plant” is a given system, whose output is to be controlled. Often the function of this part of the feedback system is to provide power, and its dynamical properties

1.2. Basic feedback theory r e u forward compensator forward compensator return compensator y plant plant (a)

(b) Figure 1.2.: Feedback configurations: (a) General. (b) Unit feedback r e γ plant y

(a) (b) return compensator

Figure 1.3.: (a) Feedback configuration with input-output maps. (b) Equivalent unit feedback configuration are not always favorable. The output y of the plant is fed back via the return compensator and subtracted from the external input r. The difference e is called the error signal and is fed to the plant via the forward compensator.

The system of Fig. 1.2(b), in which the return compensator is a unit gain, is said to have unit feedback.

Example 1.2.3 (Unit feedback system). The cruise control system of Fig. 1.1 is a unit feedback system.

For the purposes of this subsection we reduce the configuration of Fig. 1.2(a) to that of

pensator has the IO map | The control input r, the error signal e and the output signal y usually |

Fig. 1.3(a), where the forward compensator has been absorbed into the plant. The plant is represented as an input-output-mapping system with input-output (IO) map , while the return comall are time signals. Correspondingly, and are IO maps of dynamical systems, mapping time signals to time signals.

The feedback system is represented by the equations y D .e/; e D r − .y/: (1.1) These equations may or may not have a solution e and y for any given control input r. If a solution

1. Introduction to Feedback Control Theory exists, the error signal e satisfies the equation e = r − . .e//,o r e C γ.e/ D r: (1.12)

Here γ D , with denoting map composition, is the IO map of the series connection of the plant followed by the return compensator, and is called the loop IO map. Equation (1.12) reduces the feedback system to a unit feedback system as in Fig. 1.3(b). Note that because γ maps time functions into time functions, (1.12) is a functional equation for the time signal e. We refer to it as the feedback equation.

1.2.3. High-gain feedback

Feedback is most effective if the loop IO map γ has “large gain.” We shall see that one of the important consequences of this is that the map from the external input r to the output y is approximately the inverse −1 of the IO map of the return compensator. Hence, the IO map from the control input r to the control system output y is almost independent of the plant IO map. Suppose that for a given class of external input signals r the feedback equation e C γ.e/ D r (1.13) has a solution e. Suppose also that for this class of signals the “gain” of the map γ is large, that is, kγ.e/k kek; (1.14) with k k some norm on the signal space in which e is defined. This class of signals generally consists of signals that are limited in bandwidth and in amplitude. Then in (1.13) we may neglect the first term on the left, so that

Since by assumption kek k γ.e/k this implies that kek krk: (1.16)

In words: If the gain is large then the error e is small compared with the control input r. Going back to the configuration of Fig. 1.3(a), we see that this implies that .y/ r,o r where −1 is the inverse of the map (assuming that it exists).

Note that it is assumed that the feedback equation has a bounded solution2 e for every bounded r. This is not necessarily always the case. If e is bounded for every bounded r then the closedloop system by definition is BIBO stable3. Hence, the existence of solutions to the feedback equation is equivalent to the (BIBO) stability of the closed-loop system.

Note also that generally the gain may only be expected to be large for a class of error signals, denoted E. The class usually consists of band- and amplitude-limited signals, and depends on the “capacity” of the plant.

2A signal is bounded if its norm is finite. Norms of signals are discussed in § 4.3. See also x 1.3.

3A system is BIBO (bounded-input bounded-output) stable if every bounded input results in a bounded output (see x 1.3).

1.2. Basic feedback theory

Example1.2.4 (Proportionalcontrolof the cruise controlsystem). A simple form of feedback that works reasonably well but not more than that for the cruise control system of Example 1.2.1 is proportional feedback. This means that the throttle opening is controlled according to with the gain g a constant and u0 a nominal throttle setting. Denote w0 as the steady-state cruising speed corresponding to the nominal throttle setting u0, and write w.t/ D w0 CQw.t/ as in Example 1.2.1. Setting Qr.t/ D r.t/ − w0 we have

(Parte **1** de 7)