NCS - 01a - Why Nonlinear Control

Assalamu alaikum. I am Muhammad Abid and the title of this course is Non-Linear Control Systems. Before we talk about non-linear control systems, there is one basic question. That is, why do we need to study non-linear control? All of you have taken at least one course in linear control systems and you are quite expert in designing controllers for linear systems. So now we have a question that why do we need to study this new course which is a little bit more, as far as difficulty level is concerned, it is a little bit more difficult. So why do we need to study this course? So the first important point is that non-linear control gives you an improved control. What do we mean by improved control? So you remember that linearized models become less accurate for wider operating points. Actually, almost all real physical processes, they have non-linear mathematical models. And what do you do is that you linearize the non-linear model at an operating point. A general non-linear model is given by this equation, x dot is equal to f of x, where f of x is some non-linear function of x. So you remember the approach, the Taylor series expansion of a non-linear function that can be about some operating point that is given by this expression. That is the non-linear function evaluated at the equilibrium point or operating point plus the derivative of this function with respect to its argument x and this also evaluated at the equilibrium point multiplied by x minus x0 divided by 1 factorial plus the second derivative and likewise higher order terms. So in linear control, what do you do is that you ignore the higher order terms and the reason is that if this x minus x0 is smaller, that is, you are working near the operating point, near the equilibrium point, then this difference will be smaller. Its square power will be further smaller and its cubic power and higher order powers will be even smaller and smaller and you can just ignore these terms. Therefore, this non-linear function is approximately equal to this relation, that is, we have ignored the higher order terms. And furthermore, by noting that we can bring this term to the left-hand side and therefore fx minus fx0 that is equal to this derivative of this function with respect to x multiplied by x minus x0. We have simply rearranged this equation to write it in this particular form and if you define x minus x0 to be equal to delta x and delta x dot is equal to fx minus fx0 and then delta x dot is approximately equal to this relation. What is dimension of f? Same dimension has this state vector, that is, generally we denote the dimension of states by n. So this is n by 1 vector and what is derivative of a vector function with respect to a vector? You get a matrix. So this thing is a matrix and therefore this expression is very similar to the one which you see for linear systems, that is, x dot is equal to some constant matrix A multiplied by x. So this is how you linearize the non-linear systems and then utilize the linear control theory for the design of controller for those systems. However, this theory, linear control theory becomes less applicable if we move away from the operating point. For example, here I think it is visible to you. Here we have a non-linear graph f of x. What we do is that we approximate this non-linear curve by a straight line. So you see that as we move at the operating point, this black curve that is approximately equal to this blue curve, that is, the linear approximation of non-linear function is approximately the same as the non-linear function. However, if we move away from this operating point, this black curve becomes significantly different from the blue curve, that is, the linear approximation of non-linear models that becomes less applicable. So one reason for using non-linear control is this one, that is, linear control theory becomes less applicable if we have wider operating range. Another reason for using this non-linear control is that there are some nonlinearities which are called hard nonlinearities. And we cannot obtain linear approximation for these hard nonlinearities. So what are hard nonlinearities? Hard nonlinearities, for example, this saturation, saturation nonlinearity. What is saturation nonlinearity? That is depicted over here. That is, up to certain point, x and the output of that nonlinearity is the same. These are linearly related. However, beyond that thing, if you increase the input, output does not increase. That is the saturation. You are well familiar with this kind of nonlinearity. And this nonlinearity is very common in physical processes. Every process has certain saturation limit. And that is always there in physical processes. Another kind of very common nonlinearity is dead zone nonlinearity. What is dead zone nonlinearity? Dead zone nonlinearity means that for certain range of inputs, there is no output. That is very common. For example, in motors, what you expect is that for the mathematical model that generally you derive is that as you apply small voltages, there is small induced torque or small velocity. If you increase the applied voltage, there is a proportional increase in the torque. However, what happens in real motors is that up to certain range of input voltages, there is no output velocity. There is no induced torque. There is dead zone. And after that dead zone, the motor starts behaving like a linear element. And again, this kind of nonlinearity is very common in physical processes, in pumps, in motors, in all other real processes. Other examples are backlash and hysteresis nonlinearity. What is backlash nonlinearity? What is backlash? In gears, for example, ideally what happens is that if you rotate input pulley, there is proportional rotation in the output pulley. However, what happens in reality is that there is gap between the teeth of two pulleys. So as input pulley starts moving, there is no displacement in the output pulley for a certain range. And after that, there is a proportional increase in the displacement. And if you reverse the direction of displacement of input pulley, so there is a backlash. And similar phenomena is there in electrical circuits, which is called hysteresis. So these are nonlinearities. And such nonlinearities cannot be linearly approximated. For example, if we want to linearly approximate this kind of nonlinearity at, for example, this operating point, what will happen? Its linear approximation is zero at this operating point. Therefore, such kind of nonlinearities, these are called hard nonlinearities. And we cannot find linear approximation to these nonlinearities. And therefore, linear control theory will not be applicable for these systems. Another advantage of using nonlinear control is that some of the nonlinear techniques can easily handle uncertainties. And sometimes nonlinearities are intentionally added to physical processes to handle uncertainties. What are uncertainties? Before I talk about that, some examples of nonlinearities, on-off control. On-off control, you may have seen it, is very common in many processes. So in iron, you want to maintain temperature at a particular level. And there is an on-off controller. If the temperature rises above or reaches at the desired level, then that thing is switched off. And if it drops below a certain level, then you turn on the heater. So you see that that kind of controller can easily handle uncertainties. If there is some error in the sensor which is measuring the temperature, that can be easily handled. Or even if there is some uncertainty in the heating element, so that can be easily handled by these kind of nonlinearities. So sometimes nonlinearities are intentionally added to handle uncertainties. Another reason to use nonlinear control is that it is less expensive. Why less expensive? If there are sensors and actuators which are more closer to linear sensors and actuators, however, those will be more expensive. Those will be more expensive. And therefore, we prefer not to use these expensive sensors and actuators. Rather, we can use nonlinear sensors and actuators and use nonlinear control theory. That will be less expensive. There was previously one limitation of nonlinear control techniques. This limitation was that you know that linear controllers, when designed, those can be easily implemented by, for example, using operational amplifiers. You can easily implement a transfer function by using operational amplifiers. However, nonlinear equations cannot be implemented by operational amplifiers. So previously this was a limitation. But now we have modern computers. We have digital processors which can implement any kind of nonlinear equation, nonlinear differential equation with theoretically any complexity. So therefore, now this limitation is no more there.