《现代控制理论》课程教学资源课程教学资源——教学课件_Chap2_2.3 State Transition Matrix

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