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Bilinear transformation

Bilinear transformation in fact creates a discrete-time system that behaves like a continuous-time system in both frequency and time domain.

A continuous-time causal filter is stable if the poles of its transfer function fall in the left half of the complex s-plane. A discrete-time causal filter is stable if the poles of its transfer function fall inside the unit circle in the complex z-plane. The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane. Thus, filters designed in the continuous-time domain that are stable are converted to filters in the discrete-time domain that preserve that stability.

Likewise, a continuous-time filter is minimum-phase if the zeros of its transfer function fall in the left half of the complex s-plane. A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase.

+ vzoreček

Normalization

Normalization is accomplished by dividing observed data by the same constant factor to reduce its maximum value to unity and remove the physical units.

Poznámky

When the system is stable: Eventually, as time goes to infinity, the function goes to zero.

When the system is unstable: Eventually, as time goes to infinity, the function goes to infinity.

When the system is marginally stable: Eventually, as time goes to infinity, the function doesn’t go to zero nor to infinity.

Step function: In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

What does a controller do? It takes the error (feedback) and turns it into a command (which is then send to the plant. Controller designers design controllers to eliminate the error to zero as the time progresses.

Advantages of FEED FORWARD control: The benefits of feedforward control are significant and can often justify the extra cost, time and effort required to implement the technology.
Energy consumption by the feedforward control system and its driver is typically substantially lower than with other controls.
Stability is enhanced such that the controlled device can be built with lighter weight and springier materials while still being highly accurate and able to operate at high speeds. Other benefits include lower maintenance costs and substantial reduction in hysteresis.

PI controller

Advantages and disadvantages of I term

- Integral term deteriorate dynamic properties of the controller.
It slows down control action.

+ In some cases it is the only possibility how to reach zero steady state error.
It is helpful in disturbance attenuation.

PID controllers = proportional integral derivative