Controller Design
Controller design, the topic of this book, is only a part of the broader task of
control engineering. In this chapter we rst give a brief overview of control
engineering, with the goal of describing the context of controller design. We then
give a general discussion of the goals of controller design, and nally an outline
of this book.
1.1
Overview of Control Engineering
The goal of control engineering is to improve, or in some cases enable, the perfor-
mance of a system by the addition of sensors, control processors, and actuators. The
sensors measure or sense various signals in the system and operator commands the
control processors process the sensed signals and drive the actuators, which a ect
the behavior of the system. A schematic diagram of a general control system is
shown in gure 1.1.
This general diagram can represent a wide variety of control systems. The sys-
tem to be controlled might be an aircraft, a large electric power generation and
distribution system, an industrial process, a head positioner for a computer disk
drive, a data network, or an economic system. The signals might be transmitted
via analog or digitally encoded electrical signals, mechanical linkages, or pneumatic
or hydraulic lines. Similarly the control processor or processors could be mechanical,
pneumatic, hydraulic, analog electrical, general-purpose or custom digital comput-
ers.Because the sensor signals can a ect the system to be controlled (via the con-
trol processor and the actuators), the control system shown in gure 1.1 is called
1
2
CHAPTER 1 CONTROL ENGINEERING AND CONTROLLER DESIGN
other signals that aect system
(disturbances)
. . . .
actuators
sensors
s
..
System to
s
..
..
be controlled
..
s
s
actuator
signals
sensed
signals
.
.
.
.
.
Control
.
.
processor(s)
operator display,
.
command signals
warning indicators
.
.
.
.
(operator inputs)
A schematic diagram of a general control system.
Figure
1.1
a feedback or closed-loop control system, which refers to the signal \loop" that cir-
culates clockwise in this gure. In contrast, a control system that has no sensors,
and therefore generates the actuator signals from the command signals alone, is
sometimes called an open-loop control system. Similarly, a control system that has
no actuators, and produces only operator display signals by processing the sensor
signals, is sometimes called a monitoring system.
In industrial settings, it is often the case that the sensor, actuator, and processor
signals are boolean, i.e. assume only two values. Boolean sensors include mechan-
ical and thermal limit switches, proximity switches, thermostats, and pushbutton
switches for operator commands. Actuators that are often con gured as boolean
devices include heaters, motors, pumps, valves, solenoids, alarms, and indicator
lamps. Boolean control processors, referred to as logic controllers, include indus-
trial relay systems, general-purpose microprocessors, and commercial programmable
logic controllers.
In this book, we consider control systems in which the sensor, actuator, and
processor signals assume real values, or at least digital representations of real values.
Many control systems include both types of signals: the real-valued signals that we
will consider, and boolean signals, such as fault or limit alarms and manual override
switches, that we will not consider.
1.1 OVERVIEW OF CONTROL ENGINEERING
3
In control systems that use digital computers as control processors, the signals
are sampled at regular intervals, which may di er for di erent signals. In some cases
these intervals are short enough that the sampled signals are good approximations
of the continuous signals, but in many cases the e ects of this sampling must be
considered in the design of the control system. In this book, we consider control
systems in which all signals are continuous functions of time.
In the next few subsections we brie y describe some of the important tasks that
make up control engineering.
1.1.1
System Design and Control Configuration
Control conguration is the selection and placement of the actuators and sensors on
the system to be controlled, and is an aspect of system design that is very important
to the control engineer. Ideally, a control engineer should be involved in the design of
the system itself, even before the control con guration. Usually, however, this is not
the case: the control engineer is provided with an already designed system and starts
with the control con guration. Many aircraft, for example, are designed to operate
without a control system the control system is intended to improve the performance
(indeed, such control systems are sometimes called stability augmentation systems,
emphasizing the secondary role of the control system).
Actuator Selection and Placement
The control engineer must decide the type and placement of the actuators. In
an industrial process system, for example, the engineer must decide where to put
actuators such as pumps, heaters, and valves. The speci c actuator hardware (or
at least, its relevant characteristics) must also be chosen. Relevant characteristics
include cost, power limit or authority, speed of response, and accuracy of response.
One such choice might be between a crude, powerful pump that is slow to respond,
and a more accurate but less powerful pump that is faster to respond.
Sensor Selection and Placement
The control engineer must also decide which signals in the system will be measured
or sensed, and with what sensor hardware. In an industrial process, for example,
the control engineer might decide which temperatures, ow rates, pressures, and
concentrations to sense. For a mechanical system, it may be possible to choose
where a sensor should be placed, e.g., where an accelerometer is to be positioned on
an aircraft, or where a strain gauge is placed along a beam. The control engineer
may decide the particular type or relevant characteristics of the sensors to be used,
including the type of transducer, and the signal conditioning and data acquisition
hardware. For example, to measure the angle of a shaft, sensor choices include
a potentiometer, a rotary variable di erential transformer, or an 8-bit or 12-bit
4
CHAPTER 1 CONTROL ENGINEERING AND CONTROLLER DESIGN
absolute or di erential shaft encoder. In many cases, sensors are smaller than
actuators, so a change of sensor hardware is a less dramatic revision of the system
design than a change of actuator hardware.
There is not yet a well-developed theory of actuator and sensor selection and
placement, possibly because it is di cult to precisely formulate the problems, and
possibly because the problems are so dependent on available technology. Engineers
use experience, simulation, and trial and error to guide actuator and sensor selection
and placement.
1.1.2
Modeling
The engineer develops mathematical models of
the system to be controlled,
noises or disturbances that may act on the system,
the commands the operator may issue,
desirable or required qualities of the nal system.
These models might be deterministic (e.g., ordinary di erential equations (ODE's),
partial di erential equations (PDE's), or transfer functions), or stochastic or prob-
abilistic (e.g., power spectral densities).
Models are developed in several ways. Physical modeling consists of applying
various laws of physics (e.g., Newton's equations, energy conservation, or ow bal-
ance) to derive ODE or PDE models. Empirical modeling or identication consists
of developing models from observed or collected data. The a priori assumptions used
in empirical modeling can vary from weak to strong: in a \black box" approach,
only a few basic assumptions are made, for example, linearity and time-invariance
of the system, whereas in a physical model identi cation approach, a physical model
structure is assumed, and the observed or collected data is used to determine good
values for these parameters. Mathematical models of a system are often built up
from models of subsystems, which may have been developed using di erent types
of modeling.
Often, several models are developed, varying in complexity and delity. A simple
model might capture some of the basic features and characteristics of the system,
noises, or commands a simple model can simplify the design, simulation, or anal-
ysis of the control system, at the risk of inaccuracy. A complex model could be
very detailed and describe the system accurately, but a complex model can greatly
complicate the design, simulation, or analysis of the system.
1.1 OVERVIEW OF CONTROL ENGINEERING
5
1.1.3
Controller Design
Controller design is the topic of this book. The controller or control law describes
the algorithm or signal processing used by the control processor to generate the
actuator signals from the sensor and command signals it receives.
Controllers vary widely in complexity and e ectiveness. Simple controllers in-
clude the proportional (P), the proportional plus derivative (PD), the proportional
plus integral (PI), and the proportional plus integral plus derivative (PID) con-
trollers, which are widely and e ectively used in many industries. More sophisti-
cated controllers include the linear quadratic regulator (LQR), the estimated-state-
feedback controller, and the linear quadratic Gaussian (LQG) controller. These
sophisticated controllers were rst used in state-of-the-art aerospace systems, but
are only recently being introduced in signi cant numbers.
Controllers are designed by many methods. Simple P or PI controllers have only
a few parameters to specify, and these parameters might be adjusted empirically,
while the control system is operating, using \tuning rules". A controller design
method developed in the 1930's through the 1950's, often called classical controller
design, is based on the 1930's work on the design of vacuum tube feedback am-
pli ers. With these heuristic (but very often successful) techniques, the designer
attempts to synthesize a compensation network or controller with which the closed-
loop system performs well (the terms \synthesize", \compensation", and \network"
were borrowed from ampli er circuit design).
In the 1960's through the present time, state-space or \modern" controller de-
sign methods have been developed. These methods are based on the fact that the
solutions to some optimal control problems can be expressed in the form of a feed-
back law or controller, and the development of e cient computer methods to solve
these optimal control problems.
Over the same time period, researchers and control engineers have developed
methods of controller design that are based on extensive computing, for example,
numerical optimization. This book is about one such method.
1.1.4
Controller Implementation
The signal processing algorithm speci ed by the controller is implemented on the
control processor. Commercially available control processors are generally restricted
to logic control and speci c types of control laws such as PID. Custom control pro-
cessors built from general-purpose microprocessors or analog circuitry can imple-
ment a very wide variety of control laws. General-purpose digital signal processing
(DSP) chips are often used in control processors that implement complex control
laws. Special-purpose chips designed speci cally for control processors are also now
available.
6
CHAPTER 1 CONTROL ENGINEERING AND CONTROLLER DESIGN
1.1.5
Control System Testing, Validation, and Tuning
Control system testing may involve:
extensive computer simulations with a complex, detailed mathematical model,
real-time simulation of the system with the actual control processor operating
(\hardware in the loop"),
real-time simulation of the control processor, connected to the actual system
to be controlled,
eld tests of the control system.
Often the controller is modi ed after installation to optimize the actual perfor-
mance, a process known as tuning.
1.2
Goals of Controller Design
A well designed control system will have desirable performance. Moreover, a well
designed control system will be tolerant of imperfections in the model or changes
that occur in the system. This important quality of a control system is called
robustness.
1.2.1
Performance Specifications
Performance specications describe how the closed-loop system should perform.
Examples of performance speci cations are:
Good regulation against disturbances. The disturbances or noises that act on
the system should have little e ect on some critical variables in the system.
For example, an aircraft may be required to maintain a constant bearing
despite wind gusts, or the variations in the demand on a power generation and
distribution system must not cause excessive variation in the line frequency.
The ability of a control system to attenuate the e ects of disturbances on
some system variables is called regulation.
Desirable responses to commands. Some variables in the system should re-
spond in particular ways to command inputs. For example, a change in the
commanded bearing in an aircraft control system should result in a change in
the aircraft bearing that is su ciently fast and smooth, yet does not exces-
sively overshoot or oscillate.
Critical signals are not too big. Critical signals always include the actuator
signals, and may include other signals in the system. In an industrial process
1.2 GOALS OF CONTROLLER DESIGN
7
control system, for example, an actuator signal that goes to a pump must
remain within the limits of the pump, and a critical pressure in the system
must remain below a safe limit.
Many of these speci cations involve the notion that a signal (or its e ect) is small
this is the subject of chapters 4 and 5.
1.2.2
Robustness Specifications
Robustness specications limit the change in performance of the closed-loop system
that can be caused by changes in the system to be controlled or di erences between
the system to be controlled and its model. Such perturbations of the system to be
controlled include:
The characteristics of the system to be controlled may change, perhaps due
to component drift, aging, or temperature coe cients. For example, the e -
ciency of a pump used in an industrial process control system may decrease,
over its life time, to 70% of its original value.
The system to be controlled may have been inaccurately modeled or identi ed,
possibly intentionally. For example, certain structural modes or nonlinearities
may be ignored in an aircraft dynamics model.
Gross failures, such as a sensor or actuator failure, may occur.
Robustness speci cations can take several forms, for example:
Low dierential sensitivities. The derivative of some closed-loop quantity,
with respect to some system parameter, is small. For example, the response
time of an aircraft bearing to a change in commanded bearing should not be
very sensitive to aerodynamic pressure.
Guaranteed margins. The control system must have the ability to meet some
performance speci cation despite some speci c set of perturbations. For ex-
ample, we may require that the industrial process control system mentioned
above continue to have good regulation of product ow rate despite any de-
crease in pump e ectiveness down to 70%.
1.2.3
Control Law Specifications
In addition to the goals and speci cations described above, there may be constraints
on the control law itself. These control law specications are often related to the
implementation of the controller. Examples include:
The controller has a speci c form, e.g., PID.
8
CHAPTER 1 CONTROL ENGINEERING AND CONTROLLER DESIGN
The controller is linear and time-invariant (LTI).
In a control system with many sensors and actuators, we may require that
each actuator signal depend on only one sensor signal. Such a controller is
called decentralized, and can be implemented using many noncommunicating
control processors.
The controller must be implemented using a particular control processor. This
speci cation limits the complexity of the controller.
1.2.4
The Controller Design Problem
Once the system to be controlled has been designed and modeled, and the designer
has identi ed a set of design goals (consisting of performance goals, robustness re-
quirements, and control law constraints), we can pose the controller design problem:
The controller design problem: Given a model of the system to be
controlled (including its sensors and actuators) and a set of design goals,
nd a suitable controller, or determine that none exists.
Controller design, like all engineering design, involves tradeo s by suitable, we
mean a satisfactory compromise among the design goals. Some of the tradeo s in
controller design are intuitively obvious: e.g., in mechanical systems, it takes larger
actuator signals (forces, torques) to have faster responses to command signals. Many
other tradeo s are not so obvious.
In our description of the controller design problem, we have emphasized the
determination of whether or not there is any controller that provides a suitable
tradeo among the goals. This aspect of the controller design problem can be as
important in control engineering as nding or synthesizing an appropriate controller
when one exists. If it can be determined that no controller can achieve a suitable
tradeo , the designer must:
relax the design goals, or
redesign the system to be controlled, for example by adding or relocating
sensors or actuators.
In practice, existing controller design methods are often successful at nding a
suitable controller, when one exists. These methods depend upon talent, experience,
and a bit of luck on the part of the control engineer. If the control engineer is suc-
cessful and nds a suitable controller, then of course the controller design problem
has been solved. However, if the control engineer fails to design a suitable con-
troller, then he or she cannot be sure that there is no suitable controller, although
the control engineer might suspect this. Another design approach or method (or
indeed, control engineer) could nd a suitable controller.
1.3 CONTROL ENGINEERING AND TECHNOLOGY
9
1.3
Control Engineering and Technology
1.3.1
Some Advances in Technology
Control engineering is driven by available technology, and the pace of the relevant
technology advances is now rapid. In this section we mention a few of the advances
in technology that currently have, or will have, an impact on control engineering.
More speci c details can be found in the Notes and References at the end of this
chapter.
Integrated and Intelligent Sensors
Over the past decade the technology of integrated sensors has been developed. Inte-
grated sensors are built using the techniques of microfabrication originally developed
for integrated circuits they often include the signal conditioning and interface cir-
cuitry on the same chip, in which case they are called intelligent sensors. This
signal conditioning might include, for example, temperature compensation. In-
tegrated sensors promise greater reliability and linearity than many conventional
sensors, and because they are typically cheaper and smaller than conventional sen-
sors, it will be possible to incorporate many more sensors in the design of control
systems than is currently done.
Another example of a new sensor technology is the Global Positioning System
(GPS). GPS position and velocity sensors will soon be available for use in control
systems.
Actuator Technology
Signi cant improvements in actuator technology have been made. For example,
direct-drive brushless DC motors are more linear and have higher bandwidths than
the motors with brushes and gears (and stiction and backlash) that they will replace.
As another example, the trend in aircraft design is towards many actuators, such
as canards and vectored thrust propulsion systems.
Digital Control