Hello friends in this video we are going to study that how we can find out the solution of the time invariant state equations so let’s start with our topic now we know that the state equation for the time invariant system it is given by the formula X dot T equals to AXT plus B U T okay so we have here supporter time-invariant system the state equation is given by we have X dot T equals 2 XT plus bu T okay now this state equation it is classified into two groups first is homogeneous state equation and second is non homogeneous state equation and this classification is based upon whether the input is present or absent okay so we have two types of state equations so it is classified into two groups first is homogeneous state equation now in homogeneous state equation this input UT it is not present so you piss absent here or we can say that UB is equals to zero whereas in the non-homogeneous estate equation UT is present so we can say that you d is not equals to zero in the non homogenous state equation so now we are going to study that how we can find out the solution of the two types of the state equations the homogeneous state equation and the non-homogeneous state equation considering the two cases that whether UB is present or it is absent so first we will study the solution of the homogeneous state equation okay so we have our state equation as X dot T equals 2 XT plus bu T and for homogenous case u T is equals to 0 now if u T is equals then this term become 0 because u B is 0 so X dot T is equal to AXT now taking Laplace transform of this equation we will get blackness transform of X dot T is X is into X s and the Laplace transform of XT is X s so our Laplace transform will become is X s equals to a X s now taking this X s to this side because we have both the terms in X s so we will have s X s minus a X s equals to X of 0 now keep one thing in mind here we have taken the Laplace transform and we have not assumed that the initial conditions are 0 so we will have to consider the initial conditions also here this is X 0 when you take Laplace transform assuming initial conditions to be 0 then this X 0 part is neglected but here we are taking the Laplace transform and we have considered the initial conditions also so we have put an additional term here plus X 0 so this X s minus ax s will be equals to X 0 then we can take X s as common and we are left with case into I will be the identity matrix because a is a matrix and we have to keep both the sides as our matrix to do the subtraction so here we will have s I minus a and this is x0 now let’s bring this si minus a to this right-hand side so we will have X s equals to s minus a inverse X 0 okay now this si minus a inverse is Phi is Phi S is the state transition matrix so we can have here X s is equals to 5 years X 0 where Phi s is si minus a inverse and it is known as the state transition matrix also s T now if we take the Laplace transform of this Phi s then it is inverse Laplace transform then it is 5 T equals 2 inverse Laplace transform of Phi s ok or we can say that it is the Laplace transform of a side minus a inverse so this is our state transition matrix also it is known as the resultant matrix so whenever you are asked that let’s calculate the state transition matrix so it is the formula for the state transition matrix Phi s is equals to X I minus a inverse or if you are asked that Phi P then you have to take the inverse Laplace transform of si minus a inverse now there are some properties of this state transition let’s study because if we take the inverse Laplace transform this state transition matrix Phi T it comes out to be e raised to the power a T okay this is another formula for the state transition matrix so there are some properties of this state transition matrix let us study those properties the first property of the state transition matrix is that phi phi 0 it means we have the value of the state transition matrix phi t is e raised to the power 80 so we are putting this T equals to zero so here also we will put zero so its value will come out to be one so state transition matrix at time P equals to 0 it is one second property is we know that fight is e raised to the power eighty now if we take here it is to the power minus eighty and then calculate its inverse then it will be equivalent to this because what is 5 minus T we have erased to the power minus 80 instead of T we are writing minus T and then we are calculating its inverse so it is similar to this so if Phi T will be equals to if we are taking the inverse of Phi T then it will be equals to Phi of minus D inverse or we can say that Phi of inverse T is equals to Phi of minus T okay third property of transition matrix is Phi T 1 plus T 2 that is instead of T we are writing T 1 plus T 2 so it will be e raised to the power a t1 plus t2 or it will become e raised to the power a t1 plus 82 or we can say that e raised to the power 8 e 1 e raised to the power 8 e 2 or it will be Phi T 1 plus T 2 will be equals to Phi at t1 and Phi at T 2 because e raised to the power a t1 is what fight even and earase to the power 8 e 2 is 5 e 2 so we have this expression now forth property is that Phi T to the power n it means that we have a raise to the power a t and its power is n so it will become e raise to the power n eight T or we can say that it is Phi n T that is instead of T we are writing n T so it will become e raised to the power n a T and it is similar to Phi T raised to the power n so this is the fourth property of the state transition matrix now next we have fifth property Phi T 2 minus T 1 and Phi T 1 minus T naught it is equal to Phi T 2 minus T naught or it is equal to Phi this how we can prove this we have five t2 minus t1 it means a raise to the power a t2 minus t1 then fight even minus t-not means a raise to the power 8 T one minus T naught so if we expand this we will have a raise to the power 8 e to e raised to the power minus 81 earase to the part a t1 and erased to the power a T naught so this raised to the power minus 81 and a th to the power 8 even they will be canceled out and we are getting a raise to the power 8 e 2 and e raised to the power minus 8 e naught so this will become T 2 minus T naught or we can say that it is Phi T 2 minus T naught so we have proved this that Phi T 2 minus T 1 or fight even minus T do when T naught when we multiplied it is equal to 5 T 2 minus T naught similarly it is also equals to Phi T 1 minus T naught and 5 T 2 minus T 1 ok so these are the five properties of the state transition matrix so we have studied the solution of the time invariant homogeneous equation now next we will study the second type of equation state equations non-homogeneous type now in a non-homogeneous state equation UT is not equal to 0 this is the homogenous non-homogeneous case ud is not equals to zero so we have our state equation as X dot T equals to ax plus bu u is present here now taking the Laplace transform of this equation Laplace transform of X naught is s XS equals 2 we have a X so it is a XS plus X of 0 plus B us now take this excess bring this to the left hand side so we will have s XS minus a XS taking this excess come we will have si minus a equals two now taking the inverse Laplace transform because we have to find out the solution so we will take it in solution in the time domain so we will take taking inverse Laplace transform we will get it Moustakas transform of exes is XT then we have the if we bring this si minus a to the right-hand side it will become si minus a inverse and it’s inverse Laplace transform will be the inverse Laplace transform of si minus a inverse then we have X of 0 plus this si minus a inverse will be also multiplied with this B so we have your inverse Laplace transform of si minus a inverse and B and us it is the inverse Laplace transform of hold keep now applying this we have XT equals to Laplace inverse Laplace transform of all the terms plus inverse Laplace transform of this so we are breaking it into two parts first is this part and second is this path this is our fourth Spartan this is second one first we will calculate this inverse Laplace transform now inverse Laplace transform of si minus a inverse this is what our state transition matrix Phi is and inverse Laplace transform of Phi s is what Phi T and X 0 is as it is now we know that this Phi T it is what it is to the power a T and X of 0 so we have obtained the inverse Laplace transform of this first part now second part we have si minus a inverse B mu X so it will be 0 to t we have this this is also our again the state transition matrix and then it is multiplied with B us so if we write it in the Laplace transformation form then it will be 0 to te raised to the power 8 t minus ax bu x DX so this is the inverse Laplace transform of first part this is inverse Laplace transform of second part so a complete solution will become this is XT then inverse Laplace transform of first part is e raised to the power ATX of 0 plus inverse Laplace transform of second part is 0 to te e raised to the power a team – X be UX DX so this equation XT equation it is the solution of the non-homogenous state equation non-homogeneous when we are considering our input beauty so we can say that this equation is the solution of the non-homogenous state equation now if you can see that this equation is having two parts okay first part is this and second part is this so this first part it is representing the solution of the homogenous state equation and the second part it is representing the solution of the non homogenous state equation now in homogeneous state equation we take our input as 0 so in first part you can see that no input term is involved and in the second part we have our input so it is the solution of the non-homogenous state equation so we can say that this equation is having two parts first part is e raise to the power ATX of 0 and this represents the solution of homogeneous state equation and second part is 0 to t this is the solution of non-homogeneous state equation so in this video we have studied that how we can find out the solution of the time invariant state equations the time invariant state equations were of two types homogeneous and not homogeneous and we have studied the solution for both these type of state equations so I hope this topic is clear to you thank you

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you are teaching wrong. phi(s) is not state transition matrix its called Resolvent Matrix. phi(t) is the transition matrix which is inverse laplace of resolvent matrix.