Control Theory for Kids!

Preface

This is (the outline of) a textbook that I intend to write after attaining control theory enlightenment (which may take several lifetimes). In the meantime, it serves as a means of documenting my own thoughts, questions and knowledge of signals, systems and control theory.

I’ve also designed a book cover! (But that’s not gonna be ready for upload just yet… :D)

Future work will include an adaptation of the exposition, made suitable for real kids (not joking here).

A note to eyes that wander further downwards…

…into the depths of this page: Take heed! These are a Stranger’s thoughts through which you wade, and for the present a dark swamp they make. All ground is but muddy cake, and what paths leading through the fog are mostly fake. Therefore, enter at your own peril, pleasure yourself only in confusion, and take little with you when you leave…

Contents

Introduction

Motivation/Setting the stage
  • What are some things that Man has wanted to do? (Examples involving automatic control.)
  • Some historical examples and developments to be discussed.
  • What difficulties were faced, and what theoretical and practical developments are notable, especially in response to those difficulties?
  • One notable development would be the advent of the computer and its uses in control, and the resulting need for the theory to handle discrete signals.
The big picture at a glance
  • Provide a general overview of the structure of the theoretical framework concerning signals, systems, and control. The rest of the book is simply an elaboration of this preview.

Describing Signals

What is a signal?
  • Signal as a representation of some aspect of reality.
  • A record of things perceived under various circumstances (e.g. time, space, computer memory address??)
Concerning the issue of representing signals
  • Need to find suitable mathematical vocabulary to describe these things, and hence the relevant tools may then be put to use towards the analysis (and synthesis?) of signals.
  • The mathematical description of a signal should contain the correspondence between stuff perceived and circumstance of that which was perceived (i.e. signal is data indexed by (mapped to) circumstance).
Real-valued signals
  • Data as physical quantities. What is a quantity?
  • Why are numbers useful? (Relate intuition about the concept of quantity, with a pedagogically convenient axiomatic formulation of number systems)
  • Why are real numbers useful? (ditto above)
  • Circumstances are usually ‘continuous’ quantities, being often time and space
  • So we have a mapping of vector of dimension m to another vector of dimension n, both over the real field.
  • (What about the complex numbers and other kinds of quantities?)
  • These mappings are known as ‘functions’ (Strictly speaking, what sort of mappings are actually functions?)
  • These functions may be arbitrary mappings between real numbers, but they’re often continuous functions, with some relationship that may be expressed in a succinct manner using the language that the algebra of real numbers provides.
  • Ultimately, they’re all just defined as sequences of arithmetic operations, and sometimes they have patterns, though they don’t always need to.
  • Some of these are finite series, others are infinite, and certain infinite series with some kind of pattern are interesting enough to be given their own mathematical shorthand (eg. trig functions). These infinite series functions are called the transcendental functions, for they ‘transcend algebra’ (how so?)
  • Of course signals in real life don’t look like that. So how?
Interlude: Mapping the real world onto the computer
  • Nature of the computer – discrete time operation, quantised values.
  • Therefore, how should computers be interfaced with the real world, and what that all means (This is where we talk about signal acquisition, sampling and conversion.)
  • Number representations – fixed and floating point.
Real-valued functions on the computer
  • Discretely ‘indexed’ functions
  • (Unsure about format of presentation…)
Other kinds of signals
  • Are there any other interesting signals that are represented in any other interesting ways?

Analysing Signals

What is analysis?
  • What exactly needs to be analysed? Depends on what we want to know/need to do.
  • Might wish to analyse some aspect of a signal, or to analyse several signals at once, making comparisons between them for example.
Statistical analysis
  • What are statistics, what are they good for?
  • Why do we want to do this with signals?
  • Maybe a reason is the presence of some element of randomness/uncertainty in the signal, and the need extract patterns from the uncertainty…
  • What examples involving randomness? Perhaps in noise, and also other kinds of information-carrying signals where the contents cannot reasonably be known exactly in advance?
  • Now describe the various statistical analysis tools in detail (in all cases, explain why we want to use them and why they are useful, and what to use them for):
    • mean, variance
    • energy, power
    • auto and cross-correlation – comparing signals
Decomposition into building blocks
  • Breaking things down into smaller pieces as a method of analysis. A ‘reductionist’ approach. Why?
  • Overview of the various methods, and how they are distinguished (types of building blocks, capabilities and generality of the methods)
  • Introduction to the mathematical framework that underlies this approach – spectral analysis, integral transforms, basis functions, vectors in Hilbert space…all the way to functional analysis (?)
Fourier Analysis
  • Standard introduction to Fourier series/transform (relating it to the general framework described above)
  • Discrete Fourier transform/series (another relevant discussion about computers here)
  • Fast Fourier transform (and here also)
Laplace transform
  • Standard introduction to Laplace transform (relating it to the general framework described above).
  • Discuss the relationship with and generalisation from Fourier analysis
  • Additional topic: significance of one-sided vs bilateral transform.
  • Discrete counterpart to the Laplace transform: The z-transform (another relevant discussion about computers here)
[Outline Note]

For each of the transform methods, the standard discussion should include these topics:

  • How to decompose to the building blocks, mathematical definition
  • How to build a signal from its building blocks, mathematical definition of inverse transform
  • Present and discuss related theorems
  • Present example signals and their transforms, and explain why they look like that intuitively, to confirm the intuitive interpretation of what’s going on from all that math (possible, or naive???)

Describing Systems

What is a system?
  • An abstraction, a relationship between signals (e.g. between physical quantities, or maybe something less obvious).
  • Different kinds of systems, different types of signals (example of a system with an interesting combination of signals, especially including the spatial sort)
  • What does it mean to describe relationships between signals? (The mathematical representation of a signal opens up mathematical tools of expression.)
Representing systems
  • Representing a system means finding a way to describe the relationship between signals. (How about probablistic systems? How will the overall picture accomodate that?)
  • The concept of the ‘input aperture’ – the region of data points from the set of input signals upon which a given point in the output signal depends
  • Elaboration and formalisation of the ‘input aperture’ concept
  • For the special case of an input aperture with a single data point, maybe we should give it a name – ‘minimal aperture’?
  • Significance for temporal signals (i.e. data that is indexed wholly or partially by time) – due to the causality of systems, current output can at most depend on past inputs before the current instant
  • Mathematical definition of a causal system in terms of its ‘temporal aperture’ (there’ll be some mathematical subtleties to settle over here about whether ‘past’ includes ‘current’ value)
  • Still on the topic of temporal systems, talk about systems with and without memory (the latter has ‘minimal aperture’)
  • Those without memory, we can just describe them with a static mapping (wise to bring this up
    here?)
  • What about time-varying systems? and other oddities? like adaptive systems, etc..?]
  • [If the discussion here is too general to take it, it may help to begin with a more familiar
    description involving purely temporal signals] (btw…could we even have ‘multi-temporal’ signals?
    with multiple time indices…this could possibly apply to things such as relativity)
  • (Mathematical) methods of describing relationships between signals
  • If the aperture is minimal, then the relationship is simply a mapping between the domains of the input and output signals
  • Otherwise, the representation of the system behaviour can be categorised as direct or
    recursive
  • Direct method – a mapping between input values in the input aperture and the output for any point
  • Recursive method (not sure exactly how to define this right now)
    • May be more compact, but how powerful is it?
    • Mapping is not explicit, but implied.
  • What is the relationship between the direct and recursive descriptions? [IMPT]
  • Now we can talk about various kinds of systems (i.e. systems with different kinds of signals)
    and mathematical methods available to describe their behaviour
Systems with discretely-indexed signals
  • [Note: Either this section is about a particular kind of system, or it is sufficiently general to be a top level discussion. For now I think it applies to a particular kind: a discretely-indexed system. Regardless, this is probably not the place to talk about discrete-time systems and digital computers!]
  • State machines – the Mealy and Moore versions
  • Turing machines – essentially a state machine with feedback
Causal systems with real-valued, continuously-indexed signals
  • [Note: Hopefully it will be possible to treat spatial and all sorts of other signals too.]
  • Definition of  causality – in terms of the aperture.
  • [**Note: If we want to deal with signals with multiple indices then we may need partial differential equations! I wonder how to deal with these things.]
  • [**Note: For now the discussion below will deal with signals having only a single index. But KIV the rest.]
  • [NOTE: This section should cover both SISO and MIMO systems.]
  • If the aperture is minimal, then the relationship is simply a mapping between reals, which in general can be arbitrary of course, but usually in reality these things probably satisfy certain qualities, such as continuity, and the presence of a pattern that can be described mathematically in some manner.
  • The representation of system behaviour can be categorised as direct or recursive
  • Direct method – Convolution integral
  • Recursive method – Differential equations. Mapping is not explicit, but implied. Needs a starting state to work (i.e. initial conditions). Solve equation to figure out explicit behaviour.
  • Recursive method may be more compact, but how powerful is it?
  • General stuff about differential equations.
  • Types of differential equations.
  • A little discussion of methods of solution.
  • The various significances of the order of a differential equation.
  • Present some examples of DE models relating to real life things, e.g. mechanical things, electric circuits
  • General qualitative behaviour of systems (input and output)
  • Discuss this with respect to the solution of n-th order differential equations
  • State space representation.
  • What is it? Decomposing an n-th order ODE into a system of n 1st order ODEs.
  • Why is this representation useful? Poincare’s idea – why did he do it?
  • Breaks down the recursive representation into uniform simple chunks (really? nonlinear scary things too?)
  • Concept of the state and how this functions as memory.
  • Explain how it is that the seemingly infinite memory required is finally compressed to a few state variables (ie. memory elements?)
  • No. of state variables – i.e. the amount of linearly independent pieces of information
  • The various canonical forms (are there really an infinite number of state variable representations?)
  • Understand how the various canonical forms give us different ways to intuitively understand how a particular system works in terms of how it ‘makes use of’ its state/memory
  • Decomposition of a system into building blocks (really??)
  • Utility of this for synthesis/simulation of systems. This was Lord Kelvin’s idea, and also the use of integrators to make these things practical to make.
  • General qualitative behaviour of systems…this is where we talk about phase portraits, and the complete catalog of behaviours.
  • Fundamental question revisited: What is the relationship between the direct and recursive descriptions?
Linear systems with real-valued signals
  • [Note: actually this goes deeper into the area of causal real-valued systems, but maybe for the sake of formatting it should be at the same section level. But to give a good idea of what we’re covering, it may help to provide a diagram showing the part of the big picture of the classes of systems we’re covering]
  • Note: this should cover the multivariable case…i.e. both SISO and MIMO systems
  • Definition of linearity
  • [Note: there’s an urgency to explain what’s the big deal about linearity, but we also want to go through the discussion below first…? Let’s try explaining it here first and see if everything flows.]
  • What’s the big deal about linearity? General strategy for analysing linear systems.
  • Describing linear systems.
  • Direct method – Impulse response and convolution integral. And relate this to the part about decomposing time domain signals into impulses (from section on analysing signals).
  • Recursive method – Linear differential equations. Mapping is not explicit, but implied. Needs a starting state to work (i.e. initial conditions). Solve equation to figure out explicit behaviour.
  • Recursive method may be more compact, but how powerful is it?
  • [Note: Lots of structural repetition in the content to be expected in the following.]
  • General stuff about differential equations.
  • Types of differential equations.
  • A little discussion of methods of solution.
  • The various significances of the order of a differential equation.
  • State space representation – may re-iterate some stuff, particularly about the state and memory
  • The various canonical forms (are there really an infinite number of state variable representations?)
  • Understand how the various canonical forms give us different ways to intuitively understand how a particular system works in terms of how it ‘makes use of’ its state/memory
  • [Notes: This would be different from the general (and possibly nonlinear) case, so we need to provide another specific discussion here specifically pertaining to linear systems.
  • Also, all the discussions above about the state space representation must apply to both SISO and MIMO systems.]
  • Solution of the state equations (compare this with the direct solution of the n-th order equations)
  • [KIV the Matrix Fraction Descriptions…]
  • [KIV and what about the more general stuff called the Polynomial Matrix Descriptions?]
  • Development of the frequency domain representation
  • First, spot a few patterns in the solutions of the linear ODEs, which should give us some hints leading to the frequency domain approach (probably the same ones picked up by Heaviside, which led to his Operational Calculus).
  • Important to highlight the important pre-requisite of linearity for this to work.
  • It may be useful to discuss the historical evolution of this stuff into modern frequency domain methods.
  • Talk about the frequency domain approach proper, and approach from various perspectives:
  • How transform methods help solve differential equations without convolution integral
  • This transform method is a direct method of description – specifying a mapping between input and output, but in a different domain.
  • How come Laplace transform can both decompose signals, and transform differential equations
  • Talk about the initial and final value theorems
  • Develop an intuitive understanding of the s-domain description of the stuff
  • Relation with Fourier transform and the steady-state behaviour
  • Remember to talk about transfer functions for MIMO systems
  • Clarify the relationship between all the methods; transfer function, differential equations, state space representation, impulse response
  • [DIGITAL COMPUTERS: A big issue is how to treat the computers stuff here.]
  • First we need to understand the use of digital computers and their relation to the task of describing systems.
  • Somehow the discussion relating to digital computers will probably involve the following:
    • How to represent differential equations. ie. as difference equations
    • How to do convolution
    • How to do the frequency domain representation, ie. the z-transform and all the intuitive discussions that similarly appear for the discussion of the Laplace transform

Analysing Systems

What about analysis?
  • What sorts of things are worth analysing?
  • [Note: Maybe we should note that the things we analyse here are not exhaustive, but are rather of general interest. Other more specific things like controllability and observability could of course be considered an analysis of the properties of the system, but they occur in a rather specific context, so that it would be better to talk about these later, in their own more specific sections. Really??]
  • [Note: An alternative to discussing the behaviour of the various systems above, is to talk about the full spectrum of qualitative behaviour of systems down here. That means to discuss stuff like limit cycles and bifurcations and chaos etc here.]
  • [See how things work out]
Frequency domain analysis of linear systems
  • What are we trying to do here?
  • Bode plots. What ELSE?
  • Stability of systems with real-valued signals [NOTE: Following discussion has to cover the multivariable case too!]
  • What’s the intuitive meaning of stability? Why is stability important? In terms of controlling systems, instability means a loss of control.
  • Discussion of various notions of stability
  • More formal definitions of stability:
    • Lyapunov
    • BIBO
    • Asymptotic
  • General discussion: How can we stabilise a system?
Stability of linear systems
  • What makes a linear system unstable? From the purely mathematical perspective, look at the equations and see how the transient term dominates (and that’s how we lose control).
  • So, under what situations for the particular case of linear systems, will will get stability as mathematically defined? (refer to mathematical defns of stability)
    • Hence, the various stability criteria / tests:
    • Routh-Hurwitz
    • Nyquist criterion
    • poles in the s-plane
  • [Have to find a way to make clear the relationships between the various ways of representing the system, and what stability means in terms of all these ways. e.g. what stability means in terms of the eigenvalues of the matrix in the state representation.]
  • Stability as a quantifiable quality: Gain margin, phase margin.
Stability of nonlinear systems
  • [Note: More reading to be done before topics here can be properly organised]
  • Should address the issue of why it is so difficult to determine stability of nonlinear systems.
  • Perhaps its because the solutions are hard to find, and they’re all so different, unlike linear systems, where there’s a natural part and a forced part of the total solution?
    Stability in a linear system depends entirely on the system itself, whereas in nonlinear case, it depends also on how we interact with the system (in what sense, exactly?)
Stability of digitally implemented systems
  • [Note: Not sure how to properly fit this into the discussion. Really depends on the treatment in previous sections.]

Controlling Systems

What is control?
  • [don’t really know how to talk about controlling systems at the most general level, so for now, just
    leave a little bit here]
  • What is control? Getting a system to behave it the way we want it to, whatever that means…
    Indeed, it may mean many things. For a start perhaps, how does one specify what is desired behaviour?
  • The second thing is how to get it to behave? By interacting with it of course! But what does interaction mean?
    Connections between systems could be unidirectional, but more generally it could also be bidirectional (i.e. there is interaction between them). In another sense, this would be called feedback.
  • By the way, only with information about what’s going on can we really check and ensure that the desired behaviour is being achieved. That means, we do not make assumptions about the success of our actions in influencing the system being controlled. In other words, these things are open loop and closed loop / feedback control. Stuff being controlled is the plant, and the other thing is called the controller.
Controlling systems with real-valued signals
  • [Note: Remember to cover SISO and MIMO systems]
  • Perhaps some preliminary discussion of the various practical things we wish to do. And this should fit nicely with the discussion at the beginning of the book about what is control.
  • Over here, we will adopt a rather narrow notion of control: solving the tracking problem.
  • The set up: controller, actuators, sensors and plant [insert diagram]
  • Measuring the performance of the controller.
  • [Note: For the following measures, it is important to relate this closely to the tracking problem. And various scenarios that may occur during practical use.]
  • The step response. Why is that useful?
  • Time domain measures for the step response – various characteristics like the settling time, risetime etc.
  • Error-based performance indices, e.g. ISE, ITAE, etc.
  • Error numbers? [does this belong here or in the linear section?]
  • Other kinds of performance requirements:
    • Sensitivity and robustness
    • Control effort
  • We will treat the control of linear and nonlinear systems separately
Control of linear systems
  • [Note: We need to say something about the use of the summer block somewhere. Don’t take its presence for granted!]
  • Open loop control: plant inversion.
  • But doesn’t work very well. And even if it does, it isn’t likely to be stable.
  • Furthermore, plant inversion is difficult, if its a complicated plant, the inverse is also complicated. Whereas the thing about closed loop is that, simple controllers can actually accomplish the task! [Maybe there’s more to talk about this point here. I think it is rather significant.]
  • Closed loop control
  • Cascade control
  • Various PID controllers, and why these things are effective.
  • Phase lead and lag compensators, what these things are for.
  • Show the overall closed loop transfer function, and the time domain representation, how the DE looks like and the frequency domain behaviour.
  • How has the impulse response been affected?
  • Overall, what does the addition of control and the loop do to the transfer function?  I.e. How does addition of the controller shape the transfer function of the total system?
  • How the gains for the PID affect the overall system in all three representations.
  • How to design the PID gains.
  • Relationship between the step response (and maybe other performance measures), and the system parameters, such as the factors and the PID gains.
  • [Diagram about how to move the poles along lines of constant damping ratio, risetime, etc.]
  • Explain how all this shows that feedback control enables us to accomplish the tracking task. And with an economy of effort.
  • Stability of the control system.
  • Gain and phase margins from Bode plots revisited.
  • Nyquist analysis and the vector summation method for cascading systems.
  • State feedback control
  • In cascade control, the effect on the resulting system is limited.
  • Now we can do state feedback.
  • Technique allows for arbitrary pole placement! (Proof please.)
  • Explanation of why this is possible, and more powerful than the cascade control method.
  • Sure this pole placement thing is powerful, but there’s a tradeoff in terms of control effort etc.
  • Calculating the state feedback gains (and using the various formulas).
  • Controllability, observability and stabilisability concepts.
  • Various schemes of state feedback controllers, used together with PID control methods.
  • A very intuitive and deep elaboration needs to be put here regarding the intricacies of what state feedback does to the behaviour of a system, in the three representations of a system.
  • Need to return to the discussion of the various canonical forms again.
  • Now we can place the poles anywhere, how to decide where to place them?
  • One way (the so-called ‘pole-placement’ method) is to refer to tables with pole positions, optimised for various performance indices (as described in the previous part, on ‘Analysing Systems’).
  • Aanother way is the LQR/LQG thing
  • Another way is from transfer functions (huh???)
  • State estimation. And why you may have to do this.
  • How to build observers.
  • Reduced-order observers.
  • The optimal estimation problem.
  • Kalman-Bucy filters.
  • [lots of stuff to write about for both SISO and MIMO case]
  • Kalman separation principle
  • [how about feedforward control?]
  • [need to also address the issue of the apparent circularity of the feedback arrangement, and various other ways of interpreting the stuff, like imagining that a particular data point in the signal could travel serveral times around the loop, thereby creating ever higher order terms in the infinite series that defines the response…hmm whatever all that means. and Also, Black’s interpretation that we have a lot of gain, which is mostly thrown away via negative feedback. Is that really a good interpretation?]
  • -[how about DIGITAL control systems?]
Control of nonlinear systems
  • [Really don’t know enough to properly organise the topics here. here’s a smattering of them:]
  • Feedback control
  • Nonlinear observers
  • Feedback linearisation
  • Sliding mode

Other general notes

  • It seems that we can talk about feedback from many starting points negative feedback, and the use of very high gain, which would be attrited by the negative feedback…is this a nice way of looking at things?
  • There should be some talk about various impt theorems that define the boundaries of what can and cannot be done e.g. bode and his theorem about arbitrary phase and magnitude response.
  • Some mention of decibels should be made (how to get a good idea of relating decibels to magnitude?)
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