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Automatic Controls 1

Control systems: feedback, sensitivity and models

Open and closed loop, loop transfer functions with proof, sensitivity and disturbance rejection, the DC motor model, linearisation.

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A control system makes a physical quantity follow a desired value: the temperature of an oven, the speed of a motor, the altitude of a drone, the output voltage of a power supply. The key idea is feedback: continuously measure the result and correct the action according to the error. In this lesson we see why feedback works, how the transfer functions of a loop are written and how a model of a physical system is built.

Open loop and closed loop

In open loop the command is computed in advance: a toaster heats for a fixed time, without knowing whether the bread is ready. It works only if the system model is exact and there are no disturbances.

In closed loop a sensor measures the output y, it is compared with the reference r, and the controller C(s) decides the command u based on the error e = r - y. The plant G(s) is the system to be controlled, and d represents the disturbances acting on the output.

Block diagram of a feedback control loop with a disturbance at the output
The reference r enters the summing junction, where the measured output is subtracted; the error e enters the controller C(s), which produces the command u for the plant G(s). A disturbance d adds to the plant output, and the output y returns to the summing junction through the feedback loop.

The loop transfer functions

Theorem. With the notation of the diagram and L(s) = C(s)G(s) (the loop function),

    \[ Y(s) = \frac{L}{1 + L}\,R(s) + \frac{1}{1 + L}\,D(s) \]

Proof. From the diagram Y = GU + D, U = CE, E = R - Y. Substituting, Y = L(R - Y) + D, that is Y(1 + L) = LR + D. ∎

The two functions have names:

    \[ T(s) = \frac{L}{1 + L}\quad\text{(complementary sensitivity)},\qquad S(s) = \frac{1}{1 + L}\quad\text{(sensitivity)} \]

and S + T = 1. If the loop gain |L| is large, T \approx 1 (the output follows the reference) and S \approx 0 (the disturbance is almost removed).

Why feedback works

Disturbance rejection. A disturbance that in open loop would reach the output in full is reduced |1 + L| times.

Insensitivity to plant variations. The real plant is never identical to the model: components age, the load changes.

Theorem. The relative change of T due to a relative change of G is reduced by the factor S:

    \[ \frac{dT/T}{dG/G} = \frac{1}{1 + L} = S \]

Proof. T = CG/(1 + CG), so \frac{dT}{dG} = \frac{C(1 + CG) - CG\cdot C}{(1 + CG)^2} = \frac{C}{(1 + CG)^2}. Multiplying by G/T = (1 + CG)/C gives 1/(1 + CG). ∎

With a loop gain of 100, a 10% error in the plant becomes a 0.1% error in the closed-loop behaviour. It is the same principle as feedback operational amplifiers, whose gain depends only on two resistors and not on the transistor.

The price. Feedback can make a stable system unstable (corrections that are too strong and too late make the loop oscillate), and it brings sensor noise to the output, with transfer function -T. The next lessons are about handling these two problems.

Models of physical systems

Designing the controller requires a model of the plant. The physical equations are written, usually differential equations, and their Laplace transform is taken with zero initial conditions (Mathematical Methods lessons).

Example: the DC motor. The armature circuit has resistance R and inductance L_a; the motor generates a back EMF K\omega and a torque Ki (the same constant K for both, in SI units); the rotor has inertia J and viscous friction b:

    \[ v = Ri + L_a\frac{di}{dt} + K\omega,\qquad J\frac{d\omega}{dt} = Ki - b\omega \]

Transforming and eliminating I(s):

    \[ G(s) = \frac{\Omega(s)}{V(s)} = \frac{K}{(L_as + R)(Js + b) + K^2} \]

a second-order system. With negligible inductance it becomes first order, \frac{K}{RJs + Rb + K^2}, with a single mechanical time constant.

Linearisation

Many systems are not linear: a pendulum has the equation ml^2\ddot\theta = -mgl\sin\theta + \tau. The methods of this course apply to linear systems, and are used by linearising around an equilibrium point with a first-order Taylor expansion (Analysis 1 lesson): near \theta = 0, \sin\theta \approx \theta, and the equation becomes linear. A controller designed on the linearised model works as long as the system stays close to equilibrium, and feedback itself helps keep it there.

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