Feedback Control Fundamentals

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1Introduction

This lecture explains negative-feedback control, in which a measured output is compared with a reference and the resulting error determines the input. For a linear time-invariant single-input single-output system, it derives the closed-loop transfer function and examines poles, stability, and steady-state error.

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2Components

A plantPlant is the object whose output $y (t)$ is changed by an input $u (t)$ . A referenceReference $r (t)$ is the desired output, and the errorError $e (t) = r (t) - y (t)$ is the difference between the reference and measured output. Consequently, $r$ and $y$ have the same physical dimension, and so does $e$ .

A controllerController generates the input from the error. Let the plant transfer function be $G (s)$ and the controller be $C (s)$ . A transfer function is the ratio of the Laplace transform of output to that of input for an LTI system under zero initial conditions.

A disturbanceDisturbance is an input that acts on the plant but is not directly chosen by the controller. Responses to disturbances and measurement noise are important, but we first derive the relation from reference to output.

3Closed-Loop Transfer Function

Under unity negative feedback,

E (s) = R (s) - Y (s), U (s) = C (s) E (s), Y (s) = G (s) U (s) .

Eliminating the intermediate variables gives

Y (s) = G (s) C (s) {R (s) - Y (s)},

and hence

[PARSE ERROR: Undefined("Command(\"boxed\")")] T (s) = \frac{Y (s)}{R (s)} = \frac{G (s) C (s)}{1 + G (s) C (s)} .

$L (s) = G (s) C (s)$ is the loop transfer functionLoop transfer function. This expression assumes that the block interconnection is well defined and that $1 + L (s)$ is not identically zero.

4Poles and Stability

A polePole is a value of $s$ at which a rational transfer function in reduced form becomes unbounded. BIBO stabilityBounded-input bounded-output stability means that every bounded input produces a bounded output.

For a finite-dimensional, causal, continuous-time LTI system with a strictly proper rational transfer function, all poles in the open left half-plane imply BIBO stability. A right-half-plane pole produces a divergent component, while an imaginary-axis pole generally prevents BIBO stability. An unstable pole-zero cancellation can hide internal state, so reduction of an input-output transfer function alone does not guarantee internal stability.

Negative feedback does not guarantee stability merely by its sign. The locations of the roots of the closed-loop characteristic equation $1 + L (s) = 0$ in the complex plane must be examined.

5Proportional Control of a First-Order Plant

Let

G (s) = \frac{K}{τ s + 1}, τ > 0,

and use the proportional controller $C (s) = k_{p}$ . Assume that $K k_{p}$ is a dimensionless loop gain. Then

T (s) = \frac{K k_{p}}{τ s + 1 + K k_{p}},

whose closed-loop pole is

s = - \frac{1 + K k_{p}}{τ} .

The system is therefore stable when $1 + K k_{p} > 0$ . Within the range $K k_{p} > 0$ , increasing the gain accelerates the response of this ideal model.

For a unit-step reference, if the closed loop is stable and the conditions of the final-value theorem hold, then

e_{s s} = \frac{1}{1 + K k_{p}} .

The steady-state error is generally nonzero for finite proportional gain. Increasing gain also affects sensitivity to modeling error, noise, and saturation, so response speed alone cannot determine the gain.

6Key Points

Negative feedback supplies the error $e = r - y$ to the controller.
Under unity feedback, the closed-loop transfer function is $G C / (1 + G C)$ .
Stability is determined from the closed-loop poles, not from the label “negative feedback.”
Proportional control involves trade-offs among response speed, steady-state error, noise, and saturation.

7Next Lecture

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