# Module 7: Discrete State Space Models Lecture Note 3

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1 Modul 7: Discrt Stat Spac Modls Lctur Not 3 1 Charactristic Equation, ignvalus and ign vctors For a discrt stat spac modl, th charactristic quation is dfind as zi A 0 Th roots of th charactristic quation ar th ignvalus of matrix A If dt(a) 0, i.., A is nonsingular and λ 1, λ 2,, λ n ar th ignvalus of A, thn, 1 1 λ 2,, λ n will b th ignvalus of A Eignvalus of A and A T ar sam whn A is a ral matrix. 3. If A is a ral symmtric matrix thn all its ignvalus ar ral. Th n 1 vctor v i which satisfis th matrix quation λ 1, Av i λ i v i (1) whr λ i, i 1,2,,n dnots th i th ignvalu, is calld th ign vctor of A associatd with th ignvalu λ i. If ignvalus ar distinct, thy can b solvd dirctly from quation (1). Proprtis of ign vctors 1. An ign vctor cannot b a null vctor. 2. If v i is an ign vctor of A thn mv i is also an ign vctor of A whr m is a scalar. 3. If A has n distinct ignvalus, thn th n ign vctors ar linarly indpndnt. Eign vctors of multipl ordr ignvalus Whn th matrix A an ignvalu λ of multiplicity m, a full st of linarly indpndnt may not xist. Th numbr of linarly indpndnt ign vctors is qual to th dgnracy d of λi A. Th dgnracy is dfind as d n r whr n is th dimnsion of A and r is th rank of λi A. Furthrmor, 1 d m I. Kar 1

2 2 Similarity Transformation and Diagonalization Squar matrics A and Ā ar similar if AP PĀ or, Ā P 1 AP and, A PĀP 1 Th non-singular matrix P is calld similarity transformation matrix. It should b notd that ignvalus of a squar matrix A ar not altrd by similarity transformation. Diagonalization: If th systm matrix A of a stat variabl modl is diagonal thn th stat dynamics ar dcoupld from ach othr and solving th stat quations bcom much mor simplr. In gnral, if A has distinct ignvalus, it can b diagonalizd using similarity transformation. Considr a squar matrix A which has distinct ignvalus λ 1, λ 2,... λ n. It is rquird to find a transformation matrix P which will convrt A into a diagonal form λ Λ 0 λ λ n through similarity transformation AP PΛ. If v 1, v 2,..., v n ar th ignvctors of matrix A corrsponding to ignvalus λ 1, λ 2,... λ n, thn w know Av i λ i v i. This givs λ A[v 1 v 2... v n ] [v 1 v 2... v n ] 0 λ λ n Thus P [v 1 v 2... v n ]. Considr th following stat modl. x(k +1) Ax(k)+Bu(k) If P transforms th stat vctor x(k) to z(k) through th rlation thn th modifid stat spac modl bcoms x(k) Pz(k),or, z(k) P 1 x(k) whr P 1 AP Λ. z(k +1) P 1 APz(k)+P 1 Bu(k) 3 Computation of Φ(t) W hav sn that to driv th stat spac modl of a sampld data systm, w nd to know th continuous tim stat transition matrix Φ(t) At. I. Kar 2

3 3.1 Using Invrs Laplac Transform For th systm ẋ(t) Ax(t)+Bu(t), th stat transition matrix At can b computd as, At L 1{ (si A) 1} 3.2 Using Similarity Transformation If Λ is th diagonal rprsntation of th matrix A, thn Λ P 1 AP. Whn a matrix is in diagonal form, computation of stat transition matrix is straight forward: λ 1t Λt 0 λ 2t λnt Givn Λt, w can show that At P Λt P 1 At I +At+ 1 2! A2 t P 1 At P P [I 1 +At+ 1 ] 2! A2 t P I +P 1 APt+ 1 2! P 1 APP 1 APt I +Λt+ 1 2! Λ2 t Λt At P Λt P Using Caly Hamilton Thorm Evry squar matrix A satisfis its own charactristic quation. If th charactristic quation is thn, (λ) λi A λ n +α 1 λ n 1 + +α n 0 (A) A n +α 1 A n 1 + +α n I 0 Application: Evaluation of any function f(λ) and f(a) f(λ) a 0 +a 1 λ+a 2 λ 2 + +a n λ n + ordr f(λ) (λ) f(λ) q(λ) (λ)+g(λ) g(λ) q(λ)+ g(λ) (λ) β 0 +β 1 λ+ +β n 1 λ n 1 ordr n 1 I. Kar 3

4 If A has distinct ignvalus λ 1,,λ n, thn, f(λ i ) g(λ i ), i 1,,n Th solution will giv ris to β 0,β 1,,β n 1, thn f(a) β 0 I +β 1 A+ +β n 1 A n 1 If thr ar multipl roots (multiplicity 2), thn Exampl 1: f(λ i ) g(λ i ) (2) f(λ i ) g(λ i ) λ i λ i (3) If A thn comput th stat transition matrix using Caly Hamilton Thorm. (λ) λi A λ λ λ 3 Thn using (2) and (3), w can writ This implis Solving th abov quations Thn (λ 1)2 (λ 2) 0 λ 1 1 (with multiplicity 2), λ 2 2 Lt f(λ) λt and g(λ) β 0 +β 1 λ+β 2 λ 2 f(λ 1 ) g(λ 1 ) f(λ 1 ) g(λ 1 ) λ 1 λ 1 f(λ 2 ) g(λ 2 ) t β 0 +β 1 +β 2 (λ 1 1) t t β 1 +2β 2 (λ 1 1) 2t β 0 +2β 1 +4β 2 (λ 2 2) β 0 2t 2t t, β 1 3t t +2 t 2 2t, β 2 2t t t t At β 0 I +β 1 A+β 2 A 2 2 t 2t 0 2 t 2 2t 0 t 0 2t t 0 2 2t t I. Kar 4

5 Exampl 2 For th systm ẋ(t) Ax(t)+Bu(t), whr A diffrnt tchniqus. Solution: Eignvalus of matrix A ar 1±j1. Mthod 1 [ s 1 1 At L 1 (si A) 1 L 1 1 s 1 [ ] L 1 1 s 1 1 s 2 2s+2 1 s 1 [ ] s 1 1 L 1 (s 1) 2 +1 (s 1) (s 1) 2 +1 [ t cos t t sin t t sin t t cos t s 1 (s 1) 2 +1 ] [ ] 1 1. comput 1 1 At using 3 Mthod 2 [ ] At P Λt P 1 whr Λt (1+j)t 0 0 (1 j)t. Eign valus ar 1 ± j. Th corrsponding ignvctors ar found by using quation Av i λ i v i as follows: [ ][ ] [ ] 1 1 v1 v1 (1+j) 1 1 Taking v 1 1, w gt v 2 j. So, th ignvctor corrsponding to 1 + j is [ ] 1 corrsponding to 1 j is. Th transformation matrix is givn by j [ ] 1 1 P [v 1 v 2 ] P 1 1 [ ] 1 j j j 2 1 j Now, v 2 v 2 ] 1 At P Λt P 1 1 [ ][ ][ ] 1 1 (1+j)t 0 1 j 2 j j 0 (1 j)t 1 j 1 [ ][ ] (1+j)t (1 j)t 1 j 2 j (1+j)t j (1 j)t 1 j 1 [ (1+j)t + (1 j)t j ( (1+j)t (1 j)t) ] 2 j ( (1+j)t (1 j)t) (1+j)t + (1 j)t 1 [ ] 2 t cos t j(j) t 2sin t 2 t (j)(j)2sin t 2 t cos t [ ] t cos t t sin t t sin t t cos t [ ] 1 and th on j I. Kar 5

6 Mthod 3: Caly Hamilton Thorm Th ignvalus ar λ 1,2 1±j. Solving, λ 1t β 0 +β 1 λ 1 λ 2t β 0 +β 1 λ 2 Hnc, β (1+j)(1+j)t (1 j)(1 j)t β 1 1 ( (1+j)t (1 j)t) 2j At β 0 I +β 1 A [ ] t cos t t sint t sint t cos t W will now show through an xampl how to driv discrt stat quation from a continuous on. Exampl: Considr th following stat modl of a continuous tim systm. ẋ(t) y(t) x 1 (t) [ ] 1 1 x(t)+ 0 2 [ ] 0 u(t) 1 If th systm is undr a sampling procss with priod T, driv th discrt stat modl of th systm. To driv th discrt stat spac modl, lt us first comput th stat transition matrix of th continuous tim systm using Caly Hamilton Thorm. This implis (λ) λi A λ λ 2 (λ 1)(λ 2) 0 λ 1 1, λ 2 2 Lt f(λ) λt t β 0 +β 1 (λ 1 1) 2t β 0 +2β 1 (λ 2 2) Solving th abov quations β 1 2t t β 0 2 t 2t I. Kar 6

7 Thn At β 0 I +β 1 A [ ] t 2t t 0 2t Thus th discrt stat matrix A is givn as [ ] T A Φ(T) 2T T 0 2T Th discrt input matrix B can b computd as T [ ] 0 B Θ(T) Φ(T t ) dt 0 1 T [ ][ ] T. t 2T. 2t T. t T dt. 2t 1 [ ][ ] T T T T [ ] 0.5 2T T T 0.5 Th discrt stat quation is thus dscribd by [ T x((k +1)T) 2T T 0 2T y(kt) [ 1 0 ] x(kt) Whn T 1, th stat quations bcom x(k +1) ] x(kt)+ [ ] x(k) y(k) [ 1 0 ] x(k) [ ] 0.5 2T T T u(kt) 0.5 [ ] 1.48 u(k) 3.19 I. Kar 7

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