Proportional Control

08/16/17

“How can we have a control method proportional to the error?”

Closed loop control systems respond to external stimuli with the use of an error compared to a setpoint. So how can we use this information to make an easy to use control system? Well, what if we were to base our adjustment to be proportional to the error, such that the higher the error the higher the restoring force? Well, engineers have implemented exactly this in a form known as proportional control and are used in applications such as self-driving vehicles and valve systems.

Routh–Hurwitz stability criterion

08/15/17

“How can we analytically estimate the stability of a transfer function?”

We know that transfer functions have a certain level of stability afforded to them. However, sometimes the functions can be quite complicated. So how can we analytically analyze their stability? Well, let’s think about it. First, let’s take our polynomial. Then, let’s take the coefficients for all of our functions. Then, let’s make a graph with the number of rows equal to the highest exponent and number of columns equal to half rounded up of that. Afterward, let’s place the coefficients on the first two columns such that the greatest coefficient goes on the top left, the next one goes beneath, the next one goes to the right of the first one, the next one goes below, and so on. Now let’s fill in the rest of the table by taking the difference between the coefficient in the first column two rows up multiplied by the coefficient up and right one, and then let’s subtract it from the value directly up two and right one minus the one at the first column one row up while dividing everything by the number directly up one row and put the result in place of the column . Let’s now repeat this pattern until we have the entire table filled out. If any coefficients in the first column are of a different sign, then the system will be unstable. This mathematical tool is known as the Routh–Hurwitz stability criterion and is used in designing all sorts of control systems.

Control System Stability

08/14/17

“How can we measure the stability of a control system?”

Control systems are necessary for the function of society. However, if our system proves to be unstable, then it can cause serious harm to its operation. So how can we measure the control system stability? Well if we take the Laplace transform of the transfer function and observe that there are poles in the right-hand plane, then the exponential part of the function will grow to infinity over time, thereby causing a system malfunction. Control system stability is used to analyze a diverse range of fields ranging from aerospace controls to robotics and even building energy management.

Open loop control

08/13/17

“How can we make a simple control system?”

Control systems can be very complicated in nature due to their reliance on feedback systems. However, if we want, we can make our systems much more flexible if we take away such a mechanism. This is known as an open loop control. An example of an open loop control is a movement mechanism that pushes an object towards a destination regardless of what is in the way. If we were to model this on a control diagram, then the input would go straight to the output and never come back (hence the name open loop)

Linear and Time Invariant Systems

08/13/17

“What is the most ideal form for controls systems?”

There is a motley of types of control systems out there. So before we begin any sort of analysis, let’s start with the most simple form, known as Linear and Time Invariant Systems. LTI systems have three properties.

• Homogeneity If an input signal is scaled by a constant then the output will be scaled by the same constant

• Superposition If two unique inputs are summed together, then the sum of their outputs will be produced.

• Time Invariance The system will perform the same way no matter what the time is.

Unfortunately, Most controls systems are not LTI systems, but they are still important to study due to their easy to solve structure.