Basics of a Control System

Open loop control system.

Suppose the controller action is independent of the output, then the system is said to be an open loop control system. Example air cooler, fan, tube light and traffic lights. In general, any system which does not consist any sensor.

Closed loop control system.

A system in which the controller action completely depends on the output is called as closed loop control system. Example any system which consists of sensors.

Feedback network.

Feedback network is the property of closed loop control system which brings the output to the input and compares it with the reference input so that appropriate control action is formed in order to make the error zero. Whenever this system input is equal to the system output, the system is said to be stable. The feedback network may act as a transducer by converting energy from one form to another form. The best feedback network is a unit negative feedback network. The feedback network consists of R, L, C components. The maximum value of the feedback ratio is one.

Transfer function.

The mathematical equivalent model of the system is called as transfer function. The order of the transfer function represents how many in number of storage elements are present in the system. Usually, L and C components comprise the storage elements of a system. Therefore, the order of a transfer function is nothing but the number of L and C components of the system. Also note that if in case two components are connected in parallel, it must be considered as one unit, while finding the order of the network.

Another definition: the transfer function of a linear time invariant system is also defined as the ratio of Laplace transform of the output to the Laplace transform of the input, with all the initial conditions equals to 0. This condition is required in order to maintain the linearity of the system.

 Linear time invariant.

The linear time invariant system is comprised of RLC components as the system gives rise to linear transfer characteristic and the values of the RLC complements do not vary with time. While analysing the transfer function of the system, initial condition must be zero in order to get the linear transfer characteristic.

Another definition of transfer function: the transfer function of a system is defined as the Laplace transform of the impulse response, with all the initial conditions taken as zero. 

Impulse response is nothing but finding the output of the system, taking impulse signal as input. Another names for impulse response are system response, natural response, free forced response. If in case the input to the system is either a step signal, ramp signal or a parabolic signal, then the response is called as forced response.

Basics

The standard form of the system is represented in the form of Open loop transfer function. This representation is called as time constant form.

The type of the system is nothing but the number of poles at origin. The order of the system is nothing but a number of poles in the S plane.

Characteristic equation.

If you equate the denominator of the transfer function to 0, it gives the characteristic equation of the system. The characteristic equation is used to analyse the system behaviour. The roots of the characteristic equation are nothing but the poles.

Pole

It is the negative inverse of the systems time constant, at which the magnitude of the transfer function becomes infinitely. That is when you substitute the pole, the value of the transfer function becomes in finite

Zero

it is the net negative of inverse of time constant at which the magnitude of the transfer function becomes equals to 0.

Time constant.

It is used to analyse the system behaviour. Whenever the time constant of the system is very large, then the system has a slow response, it takes large time to reach the steady-state. In general, any system takes five times the time constant in order to reach the steady-state. Also, time constant is the negative inverse of the real part of the dominant pole.

Insignificant pole

when you consider the S plane, whichever pole lies towards the left side of the real axis is said to be the insignificant pole. The insignificant pole will have the smallest time constant, and the dominant pole will have the largest time constant. Note that: the insignificant pole time constant must be five times smaller than the dominant pole time constant. Generally, the insignificant poles are neglected, and it does not make any much difference to the system response. First, the insignificant pole should be converted into the time constant form and then it should be neglected.

Poles are considered in order to analyse whether the system is stable or not, to find the system response, to find the system time constant. The system time constants are used in order to draw the response of the system.

If you notice the system response of any system, then it contains only the poles terms in its exponent. Therefore, the system stability depends only on the poles.

While finding the system response, system stability, system time constant consider only the poles, but not zeros because the system response consists of only poles response terms and no zero response terms are presented the response.

Note

Adding poles.

By selecting proper RLC values. The components can be connected either in series or parallel.

Movement of poles.

Changing the values of RLC values. If the pole moves towards the origin, then we have to check RLC values.

Addition of poles or zeros.

It is nothing but connecting RLC components either in series or parallel. Connecting the RLC components in series is nothing but connecting them in the forward path.

Absolutely stable system.

The system is stable for all the values of the system parameter K from 0 to infinity.

Conditionally stable system.

The system is stable only for certain regions of system parameter.

In a complex conjugate pole, the real part of the pole always gives the system time constant and the imaginary part of the pole gives the frequency of oscillation.

Whenever the poles lying on the imaginary axis are non repetitive, the system response is of constant amplitude, the system is said to be marginally stable. The frequency of oscillation are also called as undamaged oscillation or natural frequency of oscillation.

Whenever the poles are complex conjugate in the left-hand side of the S plane, then the system response is an exponential decay. Frequency of oscillation are called as damped oscillation.

Time constant is defined only for a stable system. Whenever the poles lie on the right side of the S plane. The system is said to be unstable.

System stability

Find the transfer function, get the system response, draw the system response and then talk about stability.

Linear momentum and its conservation

Introduction 

Sometimes interaction between two bodies is so fast that it is difficult to measure the forces acting between them and the time of interaction. So in order to predict the speed and direction of motion after interaction the concept of the linear momentum and its conservation is used.

What do you mean by linear momentum?

We all know that position, acceleration and velocity are physical quantities used to describe motion of an object. Now, I will be introducing another physical quantity that will relate body’s state of motion with the forces acting on the body. We know that, one body has the capability to exert force on another body on its way, this capability is measured by linear momentum.

 What is an impulse?

 If you see someone playing billiards, the amount of linear momentum received by the ball depends on the knock it receives from the cue. You can also notice that the linear momentum of the ball varies after interacting with another ball or with the edge of the table. So, the physical quantity which measures this variation of linear momentum is called an impulse.

Conservation of linear momentum 

There might be a situation where all the balls in the table will be moving at the same time. The sum of the linear momenta of each ball constitutes the linear momentum of the system. As you know, linear momentum is a vector. So the linear momentum of each ball must be taken as vector, while adding. Suppose, there exists a situation when few balls collide, some balls can accelerate while other balls can decelerate, then what happens to the linear momentum of the system? Try to find the mass and speed of the balls before collision and try to detect the speed of any one of the ball after interaction.

Conclusion about linear momentum and impulse

 I have defined linear momentum and impulse. So when the linear momentum of a body is increased then it is receiving positive impulse, whereas when the linear momentum of the body is decreased, then it is receiving negative impulse. Principle of conservation of linear momentum: that total linear momentum of system remains constant when the system does not receive any external impulse.

Elastic collision

There might be some energy lost into the environment when two bodies collide. There might be some heat loss or energy wasted which can be neglected. The collision is considered elastic if the total kinetic energy remains conserved. In this case the linear momentum and kinetic energy are used to determine the velocity of each particle after the collision. Not all the collisions are elastic in nature. In certain cases the variation of energy are so small that it is very hard to detect, such cases are also considered as elastic collisions.

 Completely inelastic collision 

the loss of energy is maximum and cannot be neglected the collision is set to be inelastic. For example a bullet shot on a block of wood.

 A real collision 

 it is needed elastic nor completely inelastic. There is a coefficient K, which is called as coefficient of restitution. This measures the degree of elasticity. It can take values from 0 to 1. There is another coefficient called as impact coefficient which also takes values from 0 to 1. Change of these two parameters explains any type of collision occurring between two bodies.

 Conclusion 

there are two cases in which it is possible to determine how a particle will move after collision. Frontal elastic collision in which both kinetic energy and linear momentum are conserved. Completely inelastic collision where both the particles remained stuck to each other even after collision. This case results maximum loss of energy. Real collision is determined by the coefficient of restitution. This will oscillate in between frontal elastic collision and completely inelastic collision

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