[MUSIC] A special class of these functions fp and hp, which is a very powerful class for analysis on the signs of control systems is when these functions are linear. In that case, we can say a lot about the solutions to the system from a given initial condition, and understand how the state will change over time according to the objects that define these functions. When the functions, Fp an hp satisfy, The following, Equalities, For some matrices of a perfect dimensions Ap, Bp, Mp, and Np. Then, The physical component, Is said to be, Linear. And time invariant. Linearity is a property that will allow us to characterize the effect of different inputs from different initial states, when these inputs are somewhat added in a linear fashion to the system. In a simple terms, if you know the solution for an input for a given initial estate, and then you know the solution for another input for another linear estate. If you're linearly combined this information, then you can determine what will be the solution for the system when that information is applied in the linearly combined way. Invariance corresponds to the independence on the initial time of the system. These are notions that are very natural in control theory courses. We will not get into the detail, but to us one of the key things is that when we have this linearity type of a structure, we can actually solve analytically for the solutions to the system. So more importantly given an initial state, And an input, The solution. Is unique. And specifically given by, The following equation. First notice that when this Ap is a scalar, then this term here is just exponential of that constant Ap times t. When this is a matrix, which is the case when c is a vector, then this is the exponential. Of a matrix. And there are techniques again from systems and signals literature that will allow us to compute that. When there is no input into the system, then the change of the state will only be governed by the initial state and the information Ap. That's because when the input is equal to 0, this interval will go to 0. When the input is now 0, the effect of the input is given by the center which comes from a convolution between a signal and the system's input response, which is again a topic also related to dynamical systems and control. But for us, what will this say, especially when the system has this scalar state is that, we'll just be able to solve analytically for the function of time. And interestingly, we might be able to say, where does Z converge as t gets larger and larger? This is an expression that holds for every t in the domain of the input. If the input is defined from 0 to plus infinity for t, then this will correspond to the entire space of the time access. Well, naturally to define the output of the system all we need to do is to multiply this expression by Mp. And then add to that the effect of the input to the system. And this will also be defined for all t in the domain of the input. Considering different cases of this matrices will be useful, but we would do that in the next video, where we're going to see a scalar example of where can analytically solve for these expressions. But when there is no input, it is clear to see that if this Ap generates a growing exponential like one is a scalar and is a positive sign, this differential will grow with time. Therefore, the state will actually diverge, while when that scalar has a negative sign, then this exponential will converge to 0. [MUSIC]
