Control Theo
227
Part A | 10.1
10. Control Theory and Applications
Nikolaos I. Xiros, Pak-Cheung Edgar An
In this chapter, a presentation of control theory
and engineering as applied to ocean engineering
is given. The chapter starts with the fundamentals
of systems science and theory, i. e., descriptions
of systems in state space and for linear, timeinvariant ones in the frequency domain using
the Laplace transform as well as with ordinary
differential equations with respect to time. Then
stability, controllability, and observability with an
emphasis to linear, time-invariant systems are
presented. Bode plots for sinusoidal steady-state
analysis as well as the root locus technique for proportional gain feedback design are presented. For
single-input, single-output systems, PID control is
introduced as both a pole placement problem as
well as in the framework of conventional Ziegler–
Nichols methods. Pole placement design with the
addition of Luenberger observers is presented for
linear, time-invariant systems with any time of
inputs and outputs. A brief presentation of digital
controller implementations is given. Applications
from ocean engineering include control of autonomous underwater vehicles and autopilots for
surface vessels.
10.1 System Theory ..................................... 227
10.1.1 Definitions and Fundamentals ... 227
10.1.2 The Laplace Transform ............... 229
10.1.3 Linear Time-Invariant Systems.... 230
10.1.4 Multivariable Systems
and State Space ........................ 233
10.1.5 Nonlinear Systems
and Linearization ...................... 235
10.2 Analysis of LTI Systems ......................... 237
10.2.1 Block Diagrams ......................... 237
10.2.2 Stability.................................... 238
10.2.3 Controllability and Observability . 240
10.2.4 Sinusoidal Steady-State
Response and Bode Plots ........... 241
10.2.5 Analysis
of Second-Order Systems ........... 244
10.3 SISO System Controls ............................ 247
10.3.1 Performance Criteria .................. 247
10.3.2 ON/OFF Control .......................... 248
10.3.3 PID Control................................ 249
10.3.4 Ziegler–Nichols’ Methods
for PID Controller Tuning ............ 254
10.3.5 Digital Controller
Implementation ........................ 256
10.3.6 The Root Locus Technique .......... 259
10.4 Pole Placement of LTI Systems .............. 261
10.4.1 Input–Output Decoupling .......... 261
10.4.2 Full-State Feedback ................... 264
10.4.3 Pole Placement Design............... 264
10.5 Course-Keeping Autopilots ................... 267
10.5.1 The Vessel in the Control Loop .... 267
10.5.2 Surface Vessel
State-Space Model .................... 269
10.5.3 PID Autopilots ........................... 271
10.5.4 State Observers and Use
in Autopilots ............................. 274
References................................................... 275
10.1 System Theory
10.1.1 Definitions and Fundamentals
Signal
A signal is any function that contains time t as an independent variable. Typical examples of signals are the
position and velocity of a moving body, for example,
a ship. The symbols used for signals are commonly
lower case letters of the Greek or Latin alphabet. In classical control theory signals are only functions of time.
However, there has been recently a growing interest for
distributed signals, that is, signals that contain more than
one independent variables. For example, a beam supported at one end is described by signals that depend on
both time and the distance x from the support point.
System
A confined space that accepts input signals (or excitations) and generates output signals (or responses) is
227
Part A | 10.1
10. Control Theory and Applications
Nikolaos I. Xiros, Pak-Cheung Edgar An
In this chapter, a presentation of control theory
and engineering as applied to ocean engineering
is given. The chapter starts with the fundamentals
of systems science and theory, i. e., descriptions
of systems in state space and for linear, timeinvariant ones in the frequency domain using
the Laplace transform as well as with ordinary
differential equations with respect to time. Then
stability, controllability, and observability with an
emphasis to linear, time-invariant systems are
presented. Bode plots for sinusoidal steady-state
analysis as well as the root locus technique for proportional gain feedback design are presented. For
single-input, single-output systems, PID control is
introduced as both a pole placement problem as
well as in the framework of conventional Ziegler–
Nichols methods. Pole placement design with the
addition of Luenberger observers is presented for
linear, time-invariant systems with any time of
inputs and outputs. A brief presentation of digital
controller implementations is given. Applications
from ocean engineering include control of autonomous underwater vehicles and autopilots for
surface vessels.
10.1 System Theory ..................................... 227
10.1.1 Definitions and Fundamentals ... 227
10.1.2 The Laplace Transform ............... 229
10.1.3 Linear Time-Invariant Systems.... 230
10.1.4 Multivariable Systems
and State Space ........................ 233
10.1.5 Nonlinear Systems
and Linearization ...................... 235
10.2 Analysis of LTI Systems ......................... 237
10.2.1 Block Diagrams ......................... 237
10.2.2 Stability.................................... 238
10.2.3 Controllability and Observability . 240
10.2.4 Sinusoidal Steady-State
Response and Bode Plots ........... 241
10.2.5 Analysis
of Second-Order Systems ........... 244
10.3 SISO System Controls ............................ 247
10.3.1 Performance Criteria .................. 247
10.3.2 ON/OFF Control .......................... 248
10.3.3 PID Control................................ 249
10.3.4 Ziegler–Nichols’ Methods
for PID Controller Tuning ............ 254
10.3.5 Digital Controller
Implementation ........................ 256
10.3.6 The Root Locus Technique .......... 259
10.4 Pole Placement of LTI Systems .............. 261
10.4.1 Input–Output Decoupling .......... 261
10.4.2 Full-State Feedback ................... 264
10.4.3 Pole Placement Design............... 264
10.5 Course-Keeping Autopilots ................... 267
10.5.1 The Vessel in the Control Loop .... 267
10.5.2 Surface Vessel
State-Space Model .................... 269
10.5.3 PID Autopilots ........................... 271
10.5.4 State Observers and Use
in Autopilots ............................. 274
References................................................... 275
10.1 System Theory
10.1.1 Definitions and Fundamentals
Signal
A signal is any function that contains time t as an independent variable. Typical examples of signals are the
position and velocity of a moving body, for example,
a ship. The symbols used for signals are commonly
lower case letters of the Greek or Latin alphabet. In classical control theory signals are only functions of time.
However, there has been recently a growing interest for
distributed signals, that is, signals that contain more than
one independent variables. For example, a beam supported at one end is described by signals that depend on
both time and the distance x from the support point.
System
A confined space that accepts input signals (or excitations) and generates output signals (or responses) is