in this lecture I will be discussing linear and time invariant systems or LTI LTI is a title given to a class of systems that respond in a certain way when subjected to an arbitrary input all LTI systems have the following defining properties these are homogeneity superposition which is also referred to as additivity and time and variance you might not be too familiar with any or all of these words but don't let that discourage you because the concept of LTI is actually very simple I'll provide a mathematical description of each and Show an example on how we use the property we'll start with two criteria that describe linearity homogeneity and superposition let's say you have an LTI system represented here in block diagram form and you subject it to an arbitrary input X of T and it produces output y of T homogeneity states that if you scale the input by factor a then the output would be scaled by the same factor for example if you doubled the input then the output would also double here graphically we can see that for this system if we subjected it to a step input of one the output happens to be a sinusoid with amplitude one and likewise if we subed it to a step input of two the output would also double to amplitude two now let's say that the input X1 produces output y1 and input X2 produces output Y2 then superposition or additivity states that if you add the two inputs together X1 plus X2 then the output would also be the superposition of the two separate outputs y1 + Y2 sometimes homogeneity and superposition are combined to state that if the input is scaled and summed then the output will also be scaled and summed by the same amount if this system meets these two criteria this is called a linear system time and variance refers to the system behaving the same regardless of when in time and action takes place for example if you send a slinky down the stairs the toy will behave the same regardless of whether it's 5:00 or 6:00 this can be written mathematically by saying that an input of X of T minus a produces an output of Y of T minus a or in other words two similar inputs translated in time will produce the same output also Al translated in time sometimes time invariance is referred to by a more generic term called translation invariance which also covers translation in space as well so these three restrictions are all that needs to be met in order to have an LTI system however these restrictions on the system are very severe so severe in fact that practically no real world system fully meets them there's always some aspect of nonlinearity or time variance in the real world so that begs the question why are LTI systems important I think Richard feeman stated it best when he said linear systems are important because we can solve them that is to say we have an entire arsenal of mathematical tools that are designed for and capable of solving complex linear systems whereas we can only solve very simple and contrived non-ti systems but equally important to being able to solve them is that a wide range of physical systems can be approximated very accurately by an LTI model and this is very powerful in deed for example LTI systems can be characterized by its response to an Impulse function I'll go over this fully in the lecture on transfer functions but now I'll explain it briefly just so you can appreciate the power of LTI systems if you impart an Impulse onto a system such as hitting a mass that is sitting on a table with a hammer which would send the mass moving along with an instantaneous velocity and then you observe the response we would say this is the impulse response of the system and if there was friction between the mass and the table then the resp response might look something like this because of the principle of time invariance we know that if we hit the mass with a hammer at time zero and then we hit it again at time one we can expect the same response at both times also because of the principle of homogeneity we know that if we hit the mass twice as hard then we would expect a response that was twice as large and now finally because of the principle of superp position the full response of the system would just be the summation of both signals in this case it's the thick yellow line so you can see that this method Works easily with one or two impulses but what if we have an input into the system that wasn't just made up of one or two discrete impulses but was a continuous ramp and further if we broke that ramp into a series of sequential impulses then with an LTI system we can say that the output is the summation of the responses to each of those individual impulses now for the sake of this lecture I drew these impulses very discreetly in practice each of these impulses would be infinitely close to each other and the height would be infinitesimally small so when you added it up you would have the time response of the system and when you do this summation in the time domain this is called convolution which can be a very difficult integration however if we first trans transform the impulse response and the input signal to the S domain using the Lelo transform then convolution becomes multiplication now this is the basis behind transfer functions and why in the S domain they are only applicable to LTI systems so how can we take advantage of this knowledge as a controls engineer you'll be responsible for Designing the controller that's probably obvious but what you might also be responsible for is creating the requirement ments to which a different group will build the sensors and actuators in the plant or you might be responsible for choosing commercially made control system elements or even if you're lucky actually get to design and build your own sensors and actuators in each of these cases it is almost always beneficial to design components close enough to being LTI that the non-ti behavior can be ignored or to put it another way to design Hardware that can be modeled as an LTI system so that the standard mathematical tools such as transfer functions can be used to design and tune the controller with as little error as possible let me leave you with a common example of a nonlinear spring if you plotted a real Spring's stretched length versus its restoring force it would most likely have a shape similar to this when you stretched it to far eventually the material will give and start to deform elastically if you compress it too far the coils will start bunching up and physically impact each other somewhere in between in what could be considered the operating region of the spring there will be a nice nearly linear relationship between force and distance if you were selecting a spring for your project you might specify the region of linearity that is the range of forces that the spring will likely see and how linear that region needs to be so hopefully you have a better understanding of what an LTI system is and why they're so important to designing a control control system future lectures will show how to linearize a nonlinear system and how to recognize and deal with time and variance once again thanks for following along with this lecture if you have any questions or comments please leave them below and I'll try to get to them and also feel free to leave suggestions for future lessons