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    New Directions in Dynamical Systems, Automatic Control and Singular Perturbations
    New Directions in Dynamical Systems, Automatic Control and Singular Perturbations
    New Directions in Dynamical Systems, Automatic Control and Singular Perturbations
    Ebook186 pages2 hours

    New Directions in Dynamical Systems, Automatic Control and Singular Perturbations

    By John O’Reilly

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    With this short book, Professor O'Reilly brings his considerable engineering experience to bear upon three subjects close to his heart: dynamical systems, automatic control and singular perturbations. New results of a fundamental and unifying nature are presented in all three areas. New directions are thereby established. Due care is taken of historical context and motivations. This highly readable book with its compelling physical narrative is divided into two parts, Part 1 and Part 2, for the reader’s convenience. Aimed primarily at engineers, this unusually affordable book should be read by every postgraduate.

    Part 1 sets out the fundamental conditions that small-signal physically realisable dynamical system models must satisfy. These fundamental conditions are causality and non-singularity. They apply to all small-signal dynamical system models, as for example arise in electrical networks. Another important example is automatic control. Part 1 of this book also re-interprets the classic works of Nyquist and Bode to establish that the uncontrolled system must also not be singular; nor must the controlled system encounter singularity. But Part 1 goes much further. It shows that these fundamental properties, in particular non-singularity, must obtain for all small-signal system models regardless of how many inputs and outputs the system happens to have. So, small-signal automatic control for instance is all of a piece. It is that simple. 

    As for singular perturbations in Part 2, these little fellows simply pop up all over the place, sometimes where you least expect them. New associated low-frequency and high-frequency system transfer-function models are presented with almost insolent ease. Part 2 achieves for the frequency domain what standard singular perturbation theory does for the time domain. Moreover, even the standard nonlinear singularly perturbed system model does not escape scrutiny. Which model to choose? It could be important. Part 2 is an indispensable aid to modellers across the engineering spectrum seeking generic low-frequency and high-frequency models for what they do.

    LanguageEnglish
    PublisherTroubador Publishing Ltd
    Release dateSep 14, 2022
    ISBN9781803139494
    New Directions in Dynamical Systems, Automatic Control and Singular Perturbations
    Author

    John O’Reilly

    John O'Reilly is Emeritus Professor of Engineering at the University of Glasgow, UK. One of the greatest engineers of his generation, Professor O'Reilly achieved international standing in several areas of control theory before moving to power systems and power electronics. He is a past Editor of the International Journal of Control.

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      New Directions in Dynamical Systems, Automatic Control and Singular Perturbations - John O’Reilly

      PART 1

      DYNAMICAL SYSTEMS AND AUTOMATIC CONTROL

      PREFACE TO PART 1

      Part 1 of this book sets out the fundamental conditions that small-signal physically realisable dynamical system models must satisfy. These fundamental conditions are causality and non-singularity. They apply to all small-signal dynamical system models; for example, those that arise in electrical networks.

      Another important example is automatic control. The key developments in twentieth-century automatic control are the works of Nyquist (1932) and Bode (1940) for systems with a single input and a single output. The paper by Nyquist (1932) in particular specifies causality, but otherwise confines its attention to the stability issue for which it is justly famous. Part 1 of this book, with its physical narrative and due care of historical context, reinterprets these classic works to establish that the uncontrolled system must also not be singular, nor must the controlled system encounter singularity.

      However, Part 1 of this book goes much further. It shows that these fundamental properties – in particular, non-singularity – must obtain for all small-signal system models, regardless of how many inputs and outputs the system happens to have. Feedback must preserve the independence of the system outputs. So, small-signal automatic control is all of a piece. It is that simple. Moreover, it naturally leads to the coordinated decentralised control of large systems – for example, power systems, with their many inputs and many outputs – with consummate ease. The original methods of Nyquist (1932) and Bode (1940) then apply directly to all such multi-input multi-output control systems, without adaptation.

      ONE

      PHYSICAL DYNAMICAL SYSTEMS

      SUMMARY

      This chapter sets out the two fundamental properties that a physical dynamical system should possess. These two fundamental properties are, firstly, that the dynamical system should be physically realisable or causal, and secondly, that the outputs of the dynamical system should be independent.

      1.1 INTRODUCTION

      In every theoretical investigation of a real physical system, we are always forced to simplify and idealise, to a greater or smaller extent, the true properties of the system.

      Theory of Oscillators

      A. Andronov, A. A. Vitt and S. E. Khaikin (2011, P. xv)

      In our quest to develop mathematical models for our chosen purpose, we often forget that these very models are but idealisations of the actual system we seek to describe. Such is the sentiment expressed previously in the opening sentence of Andronov, Vitt and Khaikin (2011) for physical dynamical systems in 1937 (in Russian).

      Furthermore, this idealisation of physical dynamical systems must necessarily extend to their properties. It is with these limitations of model description in mind that we proceed to examine two fundamental properties of dynamical system models that, in their way, are most often merely assumed. These two fundamental properties are causality and independence of the outputs of the dynamical system under consideration. Our examination of these two properties has engineering application in its sights, but may also be of interest to the physical and biological sciences.

      While the concept of a dynamical system took shape in Newtonian mechanics, it is Poincaré (1892–1899), in his study of celestial mechanics, whom is widely regarded as the pioneer of modern dynamical systems. Important work on the stability of dynamical systems was also presented in 1899 by Lyapunov (1992). The methods of Poincaré and Lyapunov remain the bedrock of contemporary studies of dynamical systems. Mention should also be made of the classic paper of Van der Pol (1926) on nonlinear oscillators as they arose in triode valves (tubes). Other important developments from both a mathematical and physical standpoint include the works of Birkhoff (1927) and Smale (1967).

      But it is to publications with an engineering bent that our attention is now directed. The book of Andronov, Vitt and Khaikin (2011) on oscillators already referred to contains examples drawn from electrical circuits, pendulums and steam engines. Closer to the present day, the many publications of Nayfeh, among them Nayfeh and Mook (1979), on nonlinear dynamical systems have had an enormous inpact on structural mechanics as they arise in aircraft, ships and engineering structures of all kinds. Likewise, the work of Chua (1969) on nonlinear network theory has exerted a great influence on electrical engineering. For a recent introduction to nonlinear networks, see Muthuswamy and Banerjee (2019). Structural mechanics and nonlinear networks may be at opposite ends of the engineering spectrum, but they often share the common property of possessing a physical dynamical system model.

      Chapter 1 is organised as follows. Section 1.2 defines what is meant by a dynamical system model, in both its nonlinear and linearised form, with an example drawn from robotics. The fundamental physical system properties of causality and independence of system outputs are then presented in Section 1.3; a most useful application to automatic control of these causality and independence properties is provided in Chapter 2. Conclusions are outlined in Section 1.4.

      1.2 DYNAMICAL SYSTEM MODELS

      Dynamical models of physical systems are invariably nonlinear in their dynamical behaviour. They can be represented by the following system of ordinary differential equations, with initial condition x(t0) = x⁰ , given by

      where t ≥ t0 denotes time, x(t) is the system state vector, u(t) is a vector of system inputs and y(t) is the corresponding vector of system outputs.

      It is observed in the dynamical system model (1.1) and (1.2) that there is an equal number m of system inputs and system outputs. Each of these m system outputs is paired with a particular system input. The most appropriate pairing of system input to system output is determined by the physical context.

      For the nonlinear dynamical system model (1.1) and (1.2) to be physically realisable, it is required that the system model be causal; in other words, the system output y(t) can only depend upon past and present system inputs u(τ), τ ≤ t, but not upon future system inputs u(τ), τ > t.

      Also, as noted, the nonlinear dynamical system model (1.1) and (1.2) contains several inputs (forcing functions) and several outputs (responses). It is therefore also reasonable to require that the system outputs can be manipulated independently by suitable choices of system inputs; for example, as in a robotic arm (Zivanovic and Vukobratovic, 2006). There seems little point in considering any system output that depends upon other system outputs. The following example illustrates these points.

      Example 2.1: Consider the two-link planar manipulator robot arm depicted in Figure 1.1. The equations of motion that describe the dynamical response of the manipulator arm to input joint torques take the form of the nonlinear dynamical system model (1.1) and (1.2). The two system inputs are the joint actuator torques, τ1 and τ2, provided by electrical motors and applied to Joint 1 and Joint 2, respectively. The two corresponding system outputs – the motions, so to speak – are the link angular rotations, θ1 and θ2, in each of the two links of the manipulator.

      Naturally, the nonlinear manipulator arm system of Figure 1.1 is physically realisable or causal. There can be no change in the link angular rotations θ1 and θ2 until after a change in the actuator torque inputs τ1 and τ2 has occurred. No output response occurs before an input stimulus is applied.

      Moreover, it is entirely reasonable to wish to manipulate independently the output angular rotations θ1 and θ2 in each of the two links by way of appropriate joint actuator torques τ1 and τ2.

      Figure 1.1 A two-link planar manipulator robot arm

      In this short chapter on physically realisable dynamical systems, these ideas assume a more concrete form if we focus the discussion on the small-signal or linearised system models resulting from the linearisation of (1.1) and (1.2) about a suitable operating point.

      Let xe, ue and ye be the equilibrium values of x, u and y, respectively, of the nonlinear dynamical system (1.1) and (1.2) for a constant input ue; that is,

      Consider small deviations

      about the equilibrium values ( xe , ue , ye ) . By Taylor series expansions, where only first-order terms are retained, these small deviations δx , δu and δy about the equilibrium values ( xe , ue , ye ) are approximately described by the small-signal or linearised dynamical model

      where the partial derivatives ∂f / ∂x, ∂f / ∂u and ∂h / ∂x – evaluated at x = xe , u = ue and y = ye – are constant matrices. Properties that are local to the equilibrium point ( xe , ue , ye ) of the original nonlinear dynamical system (1.1) and (1.2) can be deduced from the linearised dynamical model (1.6) and (1.7). It is therefore to the small-signal model (1.6) and (1.7) that we henceforth confine our attention.

      It is convenient to rewrite the small-signal or linearised dynamical model (1.6) and (1.7) as the linear time-invariant dynamical system model

      which differs from (1.6) and (1.7) only in the obvious change of notation with constant matrices A, B and C.

      It is also the case, as we shall see in Section 1.3, that the linear time-invariant dynamical system model (1.8) and (1.9) exhibits further interesting properties when transformed into the complex domain. Taking one-sided Laplace transforms of (1.8) and (1.9), we have

      in which s = jω denotes the complex variable. Ignoring the initial conditions, it is then a simple matter to rearrange the model (1.10) and (1.11) as the strictly proper transfer-function matrix G(s) from the input u(s) to the output y(s) given by

      where

      Of course, the transfer-function elements of gij(s) of the m-input m-output transfer-function system matrix G(s) of (1.12), as depicted in Figure 1.2, may either come from the differential equation system model (1.10) and (1.11), if known, or experimentally from measured frequency-response data.

      Since, as noted previously, it makes no sense for the system model G(s) of (1.12) to possess either redundant system inputs or redundant system outputs, the system model matrix G(s) is a square m x m matrix.

      Henceforth, we consider in the complex domain lumped m-input m-output linear time-invariant dynamical systems of the form described by the square m x m transfer-function matrix G(s) of Figure 1.2 where s is the complex variable s = jω. It is assumed in Figure 1.2 that

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