Category: Mathematics

Pulse-width Modulation

Pulse-width Modulation

Pulse-width Modulation

09/18/17

“How can we use a digital signal to control power appliances?”

Using sinusoidal analog signals for control applications has drawbacks. Specifically, the constantly changing signal can cause the resistors on a circuit to heat up and induce damage. However, how can we use our engineering mindset to fix this problem? Well, what if we were to replace this analog system with a discrete one operating at a duty cycle? That way we can imitate the perpetually switching signal while avoiding the issues that come along with it. This type of signaling is known as pulse-width modulation and is one of the fundamental ideas of modern control theory

 

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Quadratic programming

Quadratic programming

Quadratic programming

09/10/17

“How can we optimize a nonlinear phenomenon using math?”

 

Linear programming is great for optimizing first order models. However, most real world systems are actually nonlinear in nature and thus require something further than linear programming. So how can we devise a method new, more optimal method? Well, let’s think about it. First, let’s boil everything down into matrices. Then, let’s introduce their constraints. The equation should now be in the form f(x) = 0.5*x^T*B*x – x^T*b subject to A1*x = c and A2*x = d, where x is the set of all independent variables, B and b are any quadratic objective function on these variables, and A1/c and A2/d are the inequality and equality constraints. Once we have the system set up, we can enter it into a computational package and achieve our results. This method is known as quadratic programming and is frequently used to solve problems fields ranging from energy analysis to finance

Deadband

Deadband

Deadband

09/05/17

“Do some control systems have a zone with no feedback?”
Ideal controls systems are available to take in all possible frequencies. Some controls systems have a zone where the input frequency will return nothing. This region is known as the deadband and can be used to prevent unwanted side-effects.

Phase margin

Phase margin

Phase margin

08/28/17

“How can we measure the difference between a control signal and a half phase shift?”

 

When working with electronic amplifiers, the phase of an input signal might be shifted, which might introduce instability. And if this phase shift is greater than 180 degrees, then the system will be unstable. To standardize all measurements, electronics researchers have introduced the concept of a phase margin, or how far off from a 180-degree phase shift this new phase is. The phase margin can be calculated with the simple equation P_margin = |180-phase|.

Gain Margin

Gain Margin

Gain Margin

08/27/17

“What is the margin of stability for a gain Bode Plot?”

 

One of the most useful features of a Bode Plot is the ability to find the stability of a system. One way to do that is to find the frequency at which the phase shift becomes 180 degrees, get the amplitude of the gain at the point, and then make a gain margin extending out to both sides equal to the magnitude of 1/|Amplitude value|, such that anything within that range will be stable.

Undamped oscillations

Undamped oscillations

Undamped oscillations

08/26/17

“What are steady oscillations called?”

 

Many physical systems exhibit oscillating behavior. However, the natures of these oscillations can be different from one another. And in the most ideal oscillations, the amplitude is constant and unchanging. These oscillations are known as undamped oscillations and are rarely found outside equations and simulations.

Control Time Shifts

Control Time Shifts

Control Time Shifts

08/22/17

“Can we have built-in time delays into control systems?”

 

When working with control systems, sometimes we don’t want all actions to occur instantaneously. For example, we might want to have an elevator door wait to close a few seconds after everyone has entered. This can be modeled as a time shift within the system. A time shift for a function in the time domain can be represented by f(t) = x(t-tau) where tau is the time constant and in the Laplace domain by the equation f(s) = e^(-tau*s) *X(s).