Optimal Cotrol, Guidace ad Estimatio Lecture 2 Overview of SS Approach ad Matrix heory Prof. Radhakat Padhi Dept. of Aerospace Egieerig Idia Istitute of Sciece - Bagalore State Space Represetatio Prof. Radhakat Padhi Dept. of Aerospace Egieerig Idia Istitute of Sciece - Bagalore
State Space Represetatio Iput variable: Maipulative (cotrol) No-maipulative (oise) cotrol oise System Cotroller Output variable: Variables of iterest that ca be either be measured or calculated State variable: Miimum set of parameters which completely summarize the system s status. Y Y Z * 3 State Space Represetatio for Dyamical Systems Noliear System Xɺ = f ( X, U ) Y = h( X, U ) X R, Y R p U R m Liear System Xɺ = AX + BU Y = CX + DU A - System matrix- x B C D - Iput matrix- x m - Output matrix- p x - Feed forward matrix p x m 4
State Variable Selectio ypically, the umber of state variables (i.e. the order of the system) is equal to the umber of idepedet eergy storage elemets. However, there are exceptios! Is there a restrictio o the selectio of the state variables? YES! All state variables should be liearly idepedet ad they must collectively describe the system completely. 5 Liearizatio of Noliear Systems Prof. Radhakat Padhi Dept. of Aerospace Egieerig Idia Istitute of Sciece - Bagalore
Problem statemet Problem: Give a oliear system Xɺ = f X U (, ) Derive a approximate liear system Xɺ = AX + BU ( X U ) about a "Operatig Poit", 0 0 Note: A operatig poit is a poit through which the system trajectory passes. 7 Liearizatio: Geeral Systems System havig cotrol iput Xɺ = f X, U, f, X R, U R Referece poit: X 0, U 0 aylor series expasio: f ( X + X, U + U ) 0 0 f f = f ( X 0, U 0) + X + U + HO X U ( X U ) ( X, U ) 0, 0 0 0 m 8
A Liearizatio f f Xɺ 0 + Xɺ f ( X 0, U 0) + X + U X U Xɺ = A X + B U Re-defie: X X, U U his leads to Xɺ = AX + BU f1 f1 x1 x f = X = ( X0, U0 ) f f x x 1 ( X 0, U0 ) ( X 0, U0 ) ( X 0, U0 ) B m f1 f1 u1 u m f = U = ( X 0, U0 ) f f u u 1 m ( X0, U0 ) 9 Review of Matrix heory Prof. Radhakat Padhi Dept. of Aerospace Egieerig Idia Istitute of Sciece - Bagalore
Defiitios Symmetric matrix Sigular matrix A = A A = 0 Iverse of a matrix: Orthogoal matrix B is iverse of A iff AB = BA = I 1 A = adj A / A AA = A A = I cosθ siθ θ Example: ( θ ) = siθ cosθ Result: Colums of a orthogoal matrix are orthoormal. y y x x 11 Eigevalues ad Eigevectors Matrices also act as liear operators with stretchig ad rotatio operatios. y1 x1 y1 Y = AX = A Q = y 2 x 2 y 2 X P x 1 = x 2 12
Eigevalues ad Eigevectors Matrices also act as liear operators with stretchig ad rotatio operatios. Questio: Ca we fid a directio (vector), alog which the matrix will act oly as a stretchig operator? Aswer: If such a solutio exists, the AX = λ X λi A X = For otrivial solutio, we should have 0 λi A = 0 Utility: Stability ad cotrol, Model reductio, Pricipal compoet aalysis etc. 13 Eigevalues ad Eigevectors: Some useful properties If λ1, λ2,, λ are eigevalues of A the for ay positive iteger m, λ m 1, λ m 2,, λ m m are eigevalues of A If A is a osigular matrix with eigevalues 1 1 1 λ, the 1, 2... 1, λ2,, λ λ λ λ are 1 eigevalues of A For triagular matrix, the eigevalues are the diagoal elemets 14
15 Eigevalues ad Eigevectors: Some useful properties If a matrix is symmetric, its eigevalues are all REAL. Moreover, it has liearly-idepedet eigevectors. A A If has real eigevalues ad real orthogoal eigevectors, the the matrix is symmetric A A ad AA are always positive semi defiite. If A is a positive defiite symmetric matrix, the every pricipal sub-matrix of A is also symmetric ad positive defiite. I particular, the diagoal elemets of A are positive. 16
ermiology Defiitio Properties of Eigevalues Positive defiite A > 0 Positive semidefiite A 0 X X AX > 0 X 0 AX 0 X 0 λ > 0, i λ 0, i i i Negative defiite A < 0 X AX < 0 X 0 λ < 0, i i Negative semidefiite A 0 X AX 0 X 0 λ 0, i i 17 Vector Norms Vector orm is a real valued fuctio with the followig properties: (a) X > 0 ad X = 0 if ad oly if X = 0 (b) α X = α X (c) X + Y X + Y X, Y 18
Vector Norm X = x + x + + x ( l orm) 1 2 3 1 2 1 2 2 2 ( ) 3 3 3 ( ) X = x + x + + x ( l orm) p p p ( 1 2 ) 1/ 2 1 2 2 X = x + x + + x ( l orm) 1/3 1 2 3 ( 1 2 ) 1/ p X = x + x + + x ( l orm) p 1/ X = x + x + + x = max x ( orm) i i p l 19 Matrix/Operator/Iduced Norms Defiitio: A Properties: A X = max = max X 0 X X = 1 (a) A > 0 ad A = 0 oly if A = 0 (b) α A = α A (c) A + B A + B (d) AB A B ( A X ) 20
Matrix/Operator/Iduced Norms 1-Norm A 1 1 j 2-Norm A -Norm A = max aij : Largest of the absolute colum sums 1 i i = 1 ( A) = σ 2 max : Largest Sigular Value = max aij : Largest of the absolute row sums j= 1 21 Sigular Values σ A λ A A, if A is real λ * ( A A), if A is complex * Both A A ad A A are positive semidefiite, ad hece, their eigevalues are always o-egative. For sigular value computatio, oly positive square roots eed to be foud out. 22
Least Square Solutios System: AX = b where A R, X R, b R m m Case 1: ( m = ad A 0) (No. of equatios = No. of variables) Uique solutio: X = A 1 b 23 Least square solutios Case 2: (uder costraied problem) (No. of equatios < No. of variables) I this case, there are ifiitely may solutios. Oe way to get a meaigful solutio is to formulate the followig optimizatio problem: Miimize Solutio ( m < ) J = X, Subject to AX = b 2 his solutio WILL satisfy the equatio 1 + + X = A b, where A = A AA right pseudo iverse AX = b exactly. 24
Least square solutios Case 2: ( m > ) (over costraied problem) (No. of equatios > No. of variables) I this case, there is o solutio. However, oe way to get a meaigful (error miimizig) solutio is to formulate the followig optimizatio problem: Miimize: J = AX b 2 Solutio: 1 + + X = A b, where A = A A A left pseudo iverse his solutio eed ot satisfy the equatio AX = b exactly. 25 Geeralized/Pseudo Iverse Left pseudo iverse: Right pseudo iverse: A A + Properties: ( a) A A A = A ( b) A A A = A + + + + + = = + + ( c) A A = A A + + ( d) A A = A A 1 ( A A) A A ( AA ) 1 + 1 =, if is square ad 0 e A A A A 26
Vector/Matrix Calculus: Defiitios [ ] X t x ( t) x ( t) x ( t) 1 2 [ ɺ ] X ɺ t x ɺ ( t) x ɺ ( t) x ( t) 1 2 X ( τ ) dτ x1 ( τ ) dτ x2 ( τ ) dτ x ( τ ) dτ t t t t 0 0 0 0 27 Vector/Matrix Calculus: Defiitios A t a11 ( t) a1 ( t) = am 1( t) am ( t) t 0 aɺ 11( t) aɺ 1 ( t) Aɺ ( t) aɺ m1( t) aɺ m ( t) t t a11 ( τ ) dτ a1 ( τ ) dτ 0 0 A( τ ) dτ t t am 1( τ ) dτ am ( τ ) dτ 0 0 28
Vector/Matrix Calculus: Some Useful Results d ( A ( t ) + B ( t )) = A ɺ( t ) + B ɺ( t ) dt d ( A ( t ) B ( t )) = A ɺ( t ) B ( t ) + A ( t ) B ɺ( t ) dt d dt da( t) dt 1 1 1 A t = A t A t 29 Vector/Matrix Calculus: Defiitios R ( ) [ 1 ] f ( X ) If f X, the f / X f / x f / x is called the "gradiet" of. m If f X f1 X fm X R, the f1 / x1 f1 / x f f X X fm / x1 fm / x is called the "Jacobia matrix" of f X with respect to X. 30
Vector/Matrix Calculus: Defiitios If f ( X ) R, the 2 2 2 f f f 2 x1 x1 x2 x1 x 2 2 f f 2 f x x x x 2 2 1 2 X 2 2 2 f f f 2 x x x x x 1 2 is called the "Hessia matrix" of f. ( X ) 31 Vector/Matrix Calculus: Derivative Rules X X b X = X b = b X ( AX ) = A ( X AX ) = ( A + A ) X X 1 If A = A, X AX = AX X 2 32
Vector/Matrix Calculus: Derivative Rules f g X X X Corollary : ( f ( X ) g( X )) = g ( X ) + f ( X ) g f ( C g( X )) = C, ( f ( X ) C) = C X X X X f g X X X ( f ( X ) Q g( X )) = Q g ( X ) + Q f ( X ) 33 Vector/Matrix Calculus: Derivative Rules If G( X ) R, X R, U R p m m G1 G2 Gm ( G( X ) U ) = u1 u2 u X X + + + X X where G G G 1 2 m G 1 m m If f ( X ) R, g( X ) R, X R, U R g f [ f ( X ) g( X ) U ] = f + g U X X X 34 m
Vector/Matrix Calculus: Chai Rules If F f ( X ) R, f ( X ) R, X R F f F = X X f 1 1 1 1 m If F f ( X ) R, f ( X ) R, X R F f F = X X f 1 m m 1 If F f ( X ) R, f ( X ) R, X R F F f = X f X p p m m p m 35 haks for the Attetio.!! 36