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 non-linear systems with the describing function method; analysis of nonlinear systems with the phase plane method; and fractional order systems and controllers. Based on tried-and-tested 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: state-space methods; continuous systems; three-term 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; root-locus analysis; fractional order systems; system transfer functions; continuous systems; describing function; state space analysis; frequency domain analysis; transform; time domain; discrete-time systems; phase plane method
Subjects: Discrete control systems; General and management topics; Algebra; Mathematical analysis; Control system analysis and synthesis methods; Nonlinear control systems
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 time-invariant (LTI) dynamical systems.
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.
In this chapter, the time response of a control system to typical test input signals, namely unit-impulse, 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.
Author introduced the root-locus method as an important tool for the analysis of closed-loop feedback systems. In fact, the relative stability and the transient performance are directly related to the location of the closed-loop roots of the characteristic equation in the s plane. The main properties of the root-locus are presented, which are also used as practical sketching rules for quickly obtaining the root-locus chart by hand.
Frequency response means system steady-state 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, closed-loop stability and conditional stability criteria were addressed.
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 closed-loop dynamics of a feedback-controlled system. Afterward, we address different PID tuning methods.
In this chapter, modern control theory represents the system dynamics as a set of coupled first-order 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 state-space representation of LTI systems surpasses several limitations of the classical methods that are mostly based on input-output descriptions. Moreover, the increase in the number of inputs, or outputs, does not affect the complexity of the state-space representations.
The pole placement synthesis technique allows placing all closed-loop poles at desired locations, so that the system closed-loop 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 non-dominant poles to lie at arbitrary locations.
This chapter introduces the main theory and tools necessary to deal with computer-controlled systems, namely the L-transform, discrete-time models, controllability and observability conditions, and stability criteria. Most concepts presented for continuous-time systems can be adapted to the discrete-time case.
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.
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 so-called 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 second-order systems to general input functions.
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.
Presents a collection of tables covering the following aspects: Laplace transforms; Bode diagrams; transfer functions; Nichols plots; z-transforms; nonlinearities; Grünwald-Letnikoff definitions; Riemann-Liouville definitions; and Caputo definition.