AME-558

Topic 1: Review of Classical Control Theory

Introduction

This is a brief review of prerequisite material. Hopefully you know most everything regarding this topic. Still it is important that you do the homework to refresh your memory and identify minor deficiencies.

Reading: Brogan's Chapter 2 is a very condensed overview of classical control theory using Laplace and Z-transform methods. We shall cover Z-transforms and discrete-time systems in Topic 2, so you can skip those parts for now. Other than that you should read the chapter.

If you need a more detailed review of the material in Chapter 2 of Brogan, it is covered in Ogata, Chapters 3, 4, 5, 6, 7, and 8. All this material is supposed to be prerequisite for this course, but we will review it quickly in this Topic. Test #1 will be confined to these prerequisite materials.

Objectives: You should be able to solve all of the problems given at the end of this lesson with confidence. You are then nominally prepared to take this course!

Linear Systems Modeled by Transfer Functions

A system is linear if two simultaneous inputs produce an output equal to the sum of the outputs produced when the same inputs are applied separately. This is the principle of superposition . One consequence is that the amplitude of the output must be proportional to that of the input. Real systems are never truly linear, because they cannot respond proportionately to inputs of arbitrary size.

The most important tool in classical control theory is the modeling of linear systems in the Laplace transform domain:

[Block Diagram 1]

$R(s)$ is the Laplace transform of the regulation signal $r(t)$ : $R (s) = L {r (t)}$

$C(s)$ is the Laplace transform of the controlled signal $c(t)$ : $C (s) = L {c (t)}$

$G(s)$ is the transfer function, defined by: $G (s) = \frac{C (s)}{R (s)}$

The superposition principle follows:

$L \{r_{1} (t) + r_{2} (t)\} \times G (s) = [R_{1} (s) + R_{2} (s)] \times G (s) = C_{1} (s) + C_{2} (s) = L \{c_{1} (t) + c_{2} (t)\}$

Feedback Control Systems

[Block Diagram 2]

We need the following definitions:

Feedforward Transfer Function: $\frac{C (s)}{E (s)} = K G (s)$

Feedback Transfer Function: $\frac{F (s)}{C (s)} = H (s)$

Open-Loop Transfer Function: $\frac{F (s)}{E (s)} = K G (s) H (s)$

Closed-Loop Transfer Function: $\frac{C (s)}{R (s)} = \frac{K G (s)}{1 + K G (s) H (s)}$

The first two definitions should be obvious from the block diagram. The open-loop transfer function is the transfer function between the input and the feedback signal, if the connection of the feedback were severed, or opened.

You should derive: the closed-loop transfer function.

Poles and Zeros

Write the transfer function F(s) as a ratio of polynomials:

$F (s) = \frac{f_{n} (s)}{f_{d} (s)}$

Then the values of $s$ (roots) that satisfy the equation $f_{n} (s) = 0$ are called the zeros of the transfer function, and the roots of the equation $f_{d} (s) = 0$ are called the poles of the tranfer function.

Applying this to the feedback system shown previously, the open-loop transfer function is:

$K G (s) H (s) = \frac{K g_{n} (s) h_{n} (s)}{g_{d} (s) h_{d} (s)}$

Thus the zeroes and poles of the open-loop transfer function are respectively the unions of the zeros and of the poles of the feedforward and the feedback transfer functions.

For the closed-loop transfer function, we have:

$\frac{C (s)}{R (s)} = \frac{K G (s)}{1 + K G (s) H (s)} = \frac{K g_{n} (s) h_{d} (s)}{g_{d} (s) h_{d} (s) + K g_{n} (s) h_{n} (s)}$

From this we see that the zeroes are equal to the union of the zeroes of the feedforward transfer function and the poles of the feedback transfer function. The poles of the closed-loop transfer function, however, depend on the value of the gain $K$ . For the case $K = 0$ , the poles are the same as the open-loop poles. For the case $K = \infty$ , the poles are the same as the open-loop zeros. The solutions of:

$g_{d} (s) h_{d} (s) + K g_{n} (s) h_{n} (s) = 0$ ,

or equivalently,

$1 + K \frac{g_{n} (s) h_{n} (s)}{g_{d} (s) h_{d} (s)} = 0$ .

for all positive values of $K$ form the root locus of the closed-loop transfer function. The root locus will consist of curves starting at the open-loop poles ( $K = 0$ , roots of $g_{d} (s) h_{d} (s) = 0$ ) and ending at the open-loop zeros ( $K = \infty$ , roots of $g_{n} (s) h_{n} (s) = 0$ ).

Root Locus Properties

The root locus equation is

$1 + K F (s) = 1 + K \frac{Q (s)}{P (s)} = 0$

The open-loop poles are the roots of P(s) = 0. The open-loop zeros are the roots of Q(s) = 0.

For example, suppose that the closed-loop characteristic function of a system of interest is

$s^{3} + (K + 2) s^{2} + K s + 4 = 0$

or equivalently

$(s^{3} + 2 s^{2} + 4) + K (s^{2} + s) = 0$

The roots of this equation are the closed-loop eigenvalues, and thus of considerable interest. The corresponding root locus equation is

$1 + K \frac{s^{2} + s}{s^{3} + 2 s^{2} + 4} = 1 + K \frac{s (s + 1)}{(s + 2.5943) (s^{2} - 0.5943 s + 1.5418)} = 0$

Here is a list of root locus properties.

Symmetry: The root locus is symmetric about the real axis because any complex roots occur in complex conjugate pairs.
Real-axis segments: Sections of the real axis belong to the root locus if the number to the right of the section of real poles and zeros of the open-loop transfer function F(s) is odd.
Branches: The root locus contains $N = max (N_{z}, N_{p})$ branches, where $N_{z}$ is the number of zeros $\{z_{1}, z_{2}, ...\}$ and $N_{z}$ is the number of poles $\{p_{1}, p_{2}, ...\}$ . Usually there are more poles than zeros.
Starting and ending points: Each branch starts (K = 0) at a pole of F(s) and ends (K = ∞) at a zero of F(s). If there are more poles than zeros then $N_{p} - N_{z}$ branches to go infinity along straight asymptotes.
Asymptotes: The aforementioned straight-line asymptotes intersect at a common real point known as the asymptote centroid $\overline{s}$ determined by $\overline{s} = \frac{\sum_{j = 1}^{N_{p}} p_{j} - \sum_{i = 1}^{N_{z}} z_{i}}{N_{p} - N_{z}}$ . The angles of the asymptotes are $θ_{M} = \frac{2 M + 1}{N_{p} - N_{z}}$ for $M = 0, 1, ..., | N_{p} - N_{z} | - 1$ .
Breakpoints: Breakpoints are intersections of root locus branches. Breakpoints may occur where $\frac{d K (s)}{d s} = 0$ , where $K (s) = \frac{- P (s)}{Q (s)}$ .
Imaginary-axis intercepts: Values of K corresponding to crossings of the imaginary axis (and associated roots) can be found using the to-be-reviewed Routh-Hurwitz stability criterion.
Angle at a complex pole or zero: The angle of departure of a branch from a pole $p_{k}$ is $θ_{d} = 180^{°} + \sum_{i} ∠ (p_{k} - z_{i}) - \sum_{j \neq k} ∠ (p_{k} - p_{j})$ . The angle of arrival of a branch to a zero $z_{k}$ is $θ_{a} = 180^{°} + \sum_{j} ∠ (z_{k} - p_{j}) - \sum_{i \neq k} ∠ (z_{k} - z_{i})$ .