Topics Preview

This thread will give you an idea what will be covered in the upcoming class. If you want to get prepared, read the chapters that are suggested here.

20 thoughts on “Topics Preview

  1. Up to 1-17: We have jumped around a bit, but there is reason behind the apparent madness. We initially started with an overview and some terminology (Chapter 1.1 and 1.2), then we covered two-point controls (Chapter 1.4). To provide a reasonable example what we can control with a two point controller, I used the waterbath example (Chapter 5), but limited myself to 5.1, i.e., the waterbath differential equation itself.

    Now, we are moving into linear controls, still using the waterbath example, but in the Laplace domain. Chapters 5 and 6 are parallel chapters, one gives the example in the time domain, the other uses Laplace-domain methods with the same example. We covered Chapter 6.1 and are moving into Chapter 6.2.

  2. For 1-19: We will determine the closed-loop equation with P-control (Chapter 6, Eq. 6.14). I want to show

    * That the single pole moves to the left (faster) when $k_p$ increases
    * That $k_p$ has practical limits
    * That P-control cannot eliminate a small steady-state error

    We will use the final value theorem for the steady-state behavior.

  3. For 1-24: Review of the P-controlled waterbath and its steady-state response (Chapter 6, Eqns 6.15 and 6.16).

    * Step response: Inverse Laplace transform (Eqns 6.11, 6.12, 6.13)
    * The basic idea what changes when we use an integral element in the controller (Chapter 6.4)

  4. For 1-26: We will continue to examine the integrator as a means to drive the steady-state error to zero. We will also take an initial, brief look at the dynamic response that becomes more complex as a consequence, but without going in depth (this will be done with a different example at a later time)

    * Figure 6.3 as a generalized example of the waterbath control loop (but see also Quiz 2, which follows the exact same principle)
    * Chapter 6.4, pages 95-96

    Most likely, we’ll have time to start formal block diagram manipulation (Chapter 7).

  5. For 1-31: We discussed driving the steady-state error to zero by means of an integrator element. Analysis of the closed-loop function showed, however, that we now have two poles in the s-plane, i.e., a second- order system. This has a major influence on the dynamic response and I originally wanted to use the spring-mass-damper system as analogy.

    It appears that there are differences in preparation between students in this class. I decided to take a closer look at the spring-mass-damper system and, in analogy, the electrical RLC circuit. We will therefore make a detour to Chapter 2.2 (pages 18, 19, 20), and we will look at the influence of the damper on the locations of the poles in the s-plane and what this means for the step response. This is entirely Linear Systems, and not everything is covered in the book. As before, lecture notes will be made available.

  6. For 2-02: Short recap of the PI controller and how it makes a second-order system. After that, we start with block diagrams (Chapter 7) — with associated homework.

  7. For 2-07: The topic will be linearization (Chapter 8). This is highly relevant for the semester projects, because your physical functions are highly nonlinear. The inverted pendulum/upright robot, for example, has the sine function in its constituent equations. The levitator’s magnetic force is proportional to $I^2/d^2$ where $I$ is the coil current and $d$ is the magnet-object distance.

    We will discuss how we deal with those physical systems that violate our assumption of linearity.

  8. For 2-09: Our next focus will be the dynamic response of a controlled system. With the examples we will for now focus on second-order systems. One example is presented in Chapter 9, an example that represents many electromechanical systems, such as robot arms. Equally interesting will be an underdamped system, for example, a hard disk head or electromagnetic autofocus system.

    Following the outline in Chapter 9, we will examine how feedback control affects the pole locations in the $s$-plane and what influence this has on the response to a step input.

  9. For 2-14: We continue the second-order system. Notably, we will examine the step response, much like the step response of a spring-mass-damper system. We’ll look at the overdamped, critically damped, and underdamped step response and what characterizes these. In addition, we examine quantitative performance criteria, such as rise time, time to first peak etc. See Chapter 9, section 9.4 (pp. 127 — 133).

  10. For 2-16: We continue examining second-order responses and quantitative metrics: Rise time, settling time, time to first peak, overshoot percent. In addition, we will cover time-integrated error metrics. Same book chapter,

  11. For 2-21: Maybe (and maybe not) we will cover one design example.

    In either case, I want to take the positioner example (Ch. 9) and move on to a PI controller. With the integral component, we will get a third pole in the transfer function, and I’ll show you that this third pole can wreak havoc.

    This brings us to the next subject: Stability.

  12. For 2-23: Stability continued. Specifically, the Routh-Hurwitz scheme (at this point it does not look as if we can fully cover time-discrete systems). We will talk about cases where zero-valued coefficients prevent us from computing the $b_{n-1}$ etc. We will also look at more examples how to obtain the stable range of, e.g., controller coefficients.

    If there is still time, I’ll introduce Scilab and Xcos — I think we are ready for it now.

  13. For 2-28: PID controller or not? We’ll be using Scilab to examine damping of an oscillatory system (example continued from last Thursday). This example is not in the book in this form.

    To reduce the tendency to oscillate, we used the D-term to augment the first-order coefficient and thus increase damping. But is this the best solution? Perhaps a compensator in the feedback path can provide better performance?

  14. For 3-03: It looks like we could use a few more examples. Dynamic response is important, so we’ll go over pole placement versus step response once again.

  15. For 3-21: It is now time to look into some computer aid. I will present how we can build a block diagram in Scilab/Xcos and show you how the response of a linear system can be simulated.

  16. Skipping to 4-04 and 4-06: We’ll continue with the root locus method. Presently, we are collecting the sketching rules (and how they are rooted in either the magnitude criterion or the angle criterion. On 4-04, we got to the point where we know which sections of the real axis are root locus.

    Next (4-06), we need to look at the asymptotes, at potential branchoff points, and at departure angles.

  17. For 4-11: I took it somewhat slow last Thursday with so many people absent. So, same again, asymptotes, at potential branchoff points, and at departure angles, which completes root locus analysis.

  18. For 4-18, 4-20 and 4-25: Wrapping up the semester with the last topic. On 4-18, we will take a look at the semester projects (venue TBD).

    On 4-20 and 4-25, we will look at frequency response, Bode diagrams, gain and phase margin, and relative stability. This is in Chapter 11, excluding Sections 11.7 and 11.8.

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