CHAPTER2 TIME RESPONSE OF THE LTISYSTEMCONTENT> 2.1 Time Response of the LTI HomogeneousSystem> 2.2 Calculation of the Matrix ExponentialFunction> 2.3 State Transition Matrix> 2.4 Time Response of the LTI System
CHAPTER2 TIME RESPONSE OF THE LTI SYSTEM • CONTENT 2.1 Time Response of the LTI Homogeneous System 2.2 Calculation of the Matrix Exponential Function 2.3 State Transition Matrix 2.4 Time Response of the LTI System
2.3 State Transition Matrix2.3.1 Definition of State Transition MatrixDefinition 2.1 The state transition matrix @(t-t) isdefined as a matrix that satisfies the conditions shown as.o(t-t) = AD(t-t)(0) = It≥t ≥0When the initial time t, = , The state transition matrix can bewritten as @(t) .Basedon thedefinition ofthe matrix exponentialfunctionA't?At3eAt = I +At +2!3!
2.3 State Transition Matrix 2.3.1 Definition of State Transition Matrix
2.3.l Definition of State Transition MatrixBased on the definition of the matrix exponential functionA't3A?t?eAt=I+At+2!3!the derivationofitcan be obtained as(At)?(At) ^ddA1k!2!dtdtAk,k-1A't?A?tEA+2!(k -1)!(At)*(At)?AtA[l2!K!1!=AeAtA(to-to) = ISimilarly, we also have
2.3.1 Definition of State Transition Matrix
2.3.1 Definition of State Transition MatrixDefinition 2.1 The state transition matrix Φ(t-t.) isdefinedas a matrix that satisfies the conditions shown as.o(t-t) = AQ(t-t):@(0) = It≥t.≥0When the initial time t, = O , The state transition matrix can bewritten as Φ(t) .we obtain another expression of the state transition matrix foitheLTIsystemas+Φ(t) = eAtΦ(t - t.) = e A(r-to)or
2.3.1 Definition of State Transition Matrix
2.3.1 Definition of State Transition MatrixConsider the response of the LTIhomogeneous systemX(t) = AX(t)with the initial condition X(t.) = X(O)X(t) = e4r X(0)X(t) = e4(-to) (t.)is the solution of the LTIhomogenous systemIf the initial condition of the LTI homogenous system is themoregeneralcase X(t), +Consequently, the solution of the LTI homogenous system canberepresented as.X(t) =@(t - to)X(to)X() = Φ(t)X(O)
2.3.1 Definition of State Transition Matrix
2.3.1 Definition of State Transition MatrixConsider the response of the LTIhomogeneous systemX(t) = AX(t)with the initial condition X(t.) = X(O)X(t) = Φ(t- t)X(to)X (t) = d(t)X(O)Since the state transition matrix satisfies the homogenous stateequation, it represents the free response of the system. In otherwords, it governs the response that is excited by the initialconditions only.+
2.3.1 Definition of State Transition Matrix
2.3.1 Definition of State Transition MatrixConsider the response of the LTIhomogeneous systemX(t) = AX(t)with the initial condition X(t.) = X(O)X(t) = Φ(t-t.)X(t)X(t) = Φ(t)X(0)The transition matrix is dependent only upon the system matrixA, and, therefore, is sometimes referred to as the transitionmatrixofA.+As the name implies, the transition matrix @(t-to)completely defines the transition of the state from the initialtime to to any time t when the inputs arezero
2.3.1 Definition of State Transition Matrix
2.3.2 Properties of the State TransitionMatrixX(t) = Φ(t-to)X(to)X(t) = Φ(t)X(0)It is observed from the last section that @(t) plays a key rolein finding the solution for a given LTI system. This section willpresent some properties of the state transition matrix.1. Φ-1(t) = Φ(-t)Φ(t)Φ(-t) =eAt .e-At = ISinceProof.Φ(-t) =Φ-1(t)Thus
2.3.2 Properties of the State Transition Matrix
2.3.2 Properties of the State TransitionMatrixX(t) = Φ(t -t.)X(to)X(t) = Φ(t)X(0)2. Φ(ti +t,) =(t)Φ(t2)Φ(ti + t,) = e4(+2) = e4h .e4t2 = Φ(t)Φ(t,)Proof.for any to, t, t,3. Φ(tz -t)Φ(t -to) =@(tz -t。)Proof.Φ(t, - t)(t - t.) = e4(2-h) . e4(t-t0) = e4(2-) = d(t, - t.)4. [Φ(t)]* =Φ(kt)ekAt = Φ(kt)AT[D(t)]* = eAt .edt ,Proof
2.3.2 Properties of the State Transition Matrix