Solved Problems in Dynamical Systems and Control
2: Faculty of Engineering, Institute of Engineering, Polytechnic of Porto, Porto, Portugal
3: IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal
This book presents a collection of exercises on dynamical systems, modelling and control. Each topic covered includes a summary of the theoretical background, problems with solutions, and further exercises. Topics covered include: block diagram algebra and system transfer functions; mathematical models; analysis of continuous systems in the time domain; root locus analysis; frequency domain analysis; PID controller synthesis; state space analysis of continuous systems; controller synthesis by pole placement; discrete time systems and the z transform; analysis of nonlinear systems with the describing function method; analysis of nonlinear systems with the phase plane method; and fractional order systems and controllers. Based on triedandtested problems and solutions that the authors use in teaching over 500 students each year, this book is essential reading for advanced students with courses in modelling and control in engineering, applied mathematics, biomathematics and physics.
Inspec keywords: statespace methods; continuous systems; threeterm control; transfer functions; pole assignment; nonlinear control systems; mathematical analysis; algebra; discrete time systems; describing functions
Other keywords: mathematical models; PID controller synthesis; block diagram algebra; nonlinear systems; dynamical systems; pole placement; rootlocus analysis; fractional order systems; system transfer functions; continuous systems; describing function; state space analysis; frequency domain analysis; transform; time domain; discretetime systems; phase plane method
Subjects: Discrete control systems; General and management topics; Algebra; Mathematical analysis; Control system analysis and synthesis methods; Nonlinear control systems
 Book DOI: 10.1049/PBCE107E
 Chapter DOI: 10.1049/PBCE107E
 ISBN: 9781785611742
 eISBN: 9781785611759
 Page count: 350
 Format: PDF

Front Matter
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1 Block diagram algebra and system transfer functions
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We introduce the Laplace transform as a method of converting differential equations in time into algebraic equations in a complex variable. Afterward, we present the concepts of transfer function and block diagram as a means to represent linear timeinvariant (LTI) dynamical systems.

2 Mathematical models
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A mathematical model is a description of a system by means of mathematical concepts and language. The mathematical model can be used to predict the system behavior, to explain the effect of individual components, and to decide about the changes needed to achieve the system specifications.

3 Analysis of continuous systems in the time domain
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In this chapter, the time response of a control system to typical test input signals, namely unitimpulse, unit step, and unit ramp functions, is an important design criterion. In fact, given the system response to these test inputs, we can infer about the system behavior in response to more general real signals.

4 Rootlocus analysis
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Author introduced the rootlocus method as an important tool for the analysis of closedloop feedback systems. In fact, the relative stability and the transient performance are directly related to the location of the closedloop roots of the characteristic equation in the s plane. The main properties of the rootlocus are presented, which are also used as practical sketching rules for quickly obtaining the rootlocus chart by hand.

5 Frequency domain analysis
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Frequency response means system steadystate response to a sinusoidal input. The chapter presents three alternatives for representing graphically the frequency response of a dynamical system, namely Bode, Nyquist and Nichols plots. Afterward, closedloop stability and conditional stability criteria were addressed.

6 PID controller synthesis
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A proportional, integral and derivative (PID) controller is a simple yet versatile feedback compensator that is widely used in industrial control systems. This chapter presents the effect of each PID component on the closedloop dynamics of a feedbackcontrolled system. Afterward, we address different PID tuning methods.

7 State space analysis of continuous systems
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In this chapter, modern control theory represents the system dynamics as a set of coupled firstorder differential equations in a set of internal variables, known as state variables, together with a set of algebraic equations that combine the state into physical output variables.The statespace representation of LTI systems surpasses several limitations of the classical methods that are mostly based on inputoutput descriptions. Moreover, the increase in the number of inputs, or outputs, does not affect the complexity of the statespace representations.

8 Controller synthesis by pole placement
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The pole placement synthesis technique allows placing all closedloop poles at desired locations, so that the system closedloop specifications can be met. Thus, the main advantage of pole placement over other classical synthesis techniques is that we can force both the dominant and the nondominant poles to lie at arbitrary locations.

9 Discretetime systems and Z transform
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This chapter introduces the main theory and tools necessary to deal with computercontrolled systems, namely the Ltransform, discretetime models, controllability and observability conditions, and stability criteria. Most concepts presented for continuoustime systems can be adapted to the discretetime case.

10 Analysis of nonlinear systems with the describing function method
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The describing function (DF) is one method for the analysis of nonlinear systems. The main idea is to study the ratio between a sinusoidal input applied to the system and the fundamental harmonic component of the output. The DF allows the extension of the Nyquist stability criterion to nonlinear systems for detection of limit cycles, namely the prediction of limit cycle amplitude and frequency.

11 Analysis of nonlinear systems with the phase plane method
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The system response can be represented graphically by the locus of x(t) versus x(t), that is, parametrized in t. The pair {x(t), x(t)} corresponds to the coordinates of a point in the socalled phase plane (PP). As time varies in the interval t ∈ [0, ∞[, this point describes a PP trajectory. A family of PP trajectories is called a phase portrait. By means of the PP technique, we can analyze the time response of linear and nonlinear secondorder systems to general input functions.

12 Fractional order systems and controllers
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Derivatives and integrals can be extended to orders which are not integer. These can be used in differential equations to describe the dynamics of a system, or of a controller, in a more supple manner than with integer derivatives and integrals only.

AppendixA
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Presents a collection of tables covering the following aspects: Laplace transforms; Bode diagrams; transfer functions; Nichols plots; ztransforms; nonlinearities; GrünwaldLetnikoff definitions; RiemannLiouville definitions; and Caputo definition.

Back Matter
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