Control Systems

Transcription

1 6.5 Control Systems Last Time: Introdction Motivation Corse Overview Project Math. Descriptions of Systems ~ Review Classification of Systems Linear Systems LTI Systems The notion of state and state variables were introdced Today: Matrix Operations -- Fndamental to Linear lgebra Determinant Matrix Mltiplication Eigenvale Rank Math. Descriptions of Systems ~ Review LTI Systems: State Variable Description Linearization

2 Operations on Matrices The classical control theory is based on Laplace transform and z-transform also called freqency-domain approach The modern control theory is established pon Linear lgebra State-space approach, or time domain approach linear time invariant system can be described as x(t) & x(t) + B(t) y(t) Cx(t) + D(t) Systems properties all characterized with the matrices,b,c,d. Matrices: Sqare and non-sqare by (or ) matrices: by matrices: a b c, d e f n n m matrix: a a L a a a a M M O M a a L a, a a a a m L m n n nm by matrix: n: the nmber of rows; m: the nmber of colmns. n : a colmn vector; m : a row vector. ddition and sbtraction: element by element Mltiplication is not by element. a c e b d f a ij : element at the i-th row, j-th colmn 4

3 Matrix Mltiplication: a bx ax + by c d y cx + dy a b x a + bv ax + by c d v y c + dv cx + dy Prodct of a n matrix a n matrix is a scalar. Prodct of a n matrix and a n matrix is an n n matrix a b c y ax + by + cz z Generally, B B [ ] x x xa xb xc y[ a b c] ya yb yc z za zb zc Yo cannot mltiply any two matrices. They have to be compatible: to get B, the nmber of colmns of mst eqal to the nmber 5 of rows of B, e.g., : k n, B: n m. Let be an 4 matrix, B be an 4 matrix 4 a e , B b f 9 c g d h B [ B B ] [ ] [ ] [ ] B B B [ ] B [ B B ] B B B B B B B B B B B B a+ b+ c+ 4d e+ f + g+ 4h B B 5a+ 6b+ 7c+ 8d 5e+ 6f + 7g+ 8h B B 9a+ b+ c+ d 9e+ f + g+ h 6

4 How abot B? 4 a e , B b f 9 c g d h B [ B B ] B is defined only if the nmber of colmns of B is the Same as the nmber of rows of. Sppose B [ B B B],, then B B B B B + B + B [ ] Bt each B, B, B Not defined at all for this case. is a matrix. 7 How abot B B? [ ] [ ]? B B B B B If and B are compatible, (B defined), then B and B are not defined. Be carefl. Correct partition is important. Compatibility is essential. Example:? 8 4

5 Prodct of block partitioned matrices: Sppose that and B are partitioned compatibly as, B B B B, B Then B B B + B + B B + B + B B Compatibly partitioned means that the partition of the colmns of is the same as the partition of the rows of B 9 Determinant: scalar defined for a sqare matrix det a b ad bc; c d a b c det d e f aei + dhc + gbf gec ahf dbi g h i a b c a b c d e f d e f g h i g h i a b c d e f Exercise: det

6 Determinant of a trianglar matrix: det? 8 9 ll zero below the diagonal, or all zero above the diagonal det is the prodct of the diagonal elements. The determinant can be simplified by making the matrix a diagonal one throgh elementary operation that preserve the determinant. If an entire row or an entire colmn is, the determinant is. Elementary operations that preserve the determinant: det( ) det ) dd one row scaled by a nmber to another row dd row to row ) dd one colmn scaled by a nmber to another colmn dd colmn (-) to colmn dd row (-) to row Row mins row (/4)

7 Determinant of the prodct of matrices: det( B) det det B det( BC) det det B det C How abot det(+b), det(-b)? Determinant of a block trianglar matrix: * * * * * det? * 4 ssme that,,, 4 are all sqare. 9 Examples: det 5, det n example: det det 4 8 det 5 det dd row (-) to row dd row (-) to row dd row (-) to row 4 7

8 Why elementary operations preserve the determinant? Becase elementary operation is eqivalent to mltiplying the matrix with another one whose determinant is. a d g b e h c f i dd colmn scaled by x to colmn a + cx d + fx g + ix Since det, x a b c a b c a+ cx b c det d e f det d e f det d + fx e f g h i g h i x g+ ix h i What abot a b c d e f? x g h i b e h c f i a d g b e h c f i x 5 If two rows (colmns) are switched, the determinant changes the sign; If a whole row (colmn) is scaled by a nmber k, the determinant is scaled by a nmber k. What if the whole matrix is scaled by a nmber k? Exercise: x det x ax bx x + c det 4 sqare matrix is said to be nonsinglar if its determinant is nonzero. 6 8

9 Eigenvale of a sqare matrix: s is an eigenvale of if det[ si ] L I L M M O M L If is n n, det[si-] is a polynomial of order n. n eigenvale is a root of the polynomial. has n eigenvales., s s + det[ si ] det s + s + ( s + )( s + ) Roots are s and s. Exercise: 7 The inverse of a sqare matrix: If det, has a niqe inverse X sch that XXI, denoted as X. a b, d b c d ad bc c a Solving a system of eqations: ax + by e a bx e cx + dy f c d y f a b a bx a b e c d c d y c d f x a b e y c d f X Exercise: What is the inverse of b cosθ sinθ X sinθ? cosθ b 8 9

10 The inverse of a block partitioned matrix: B ssme that and are sqare and nonsinglar, then X for certain X. What is X? 9 Sb-matrix of a matrix a by matrix a by sqare matrix There are? by sb-matrices? by sb-matrices by matrices

11 Rank: The rank of M is the highest dimension of a sqare sb-matrix whose determinant is nonzero. denoted as ρ(m) For example, a 4 matrix sb-matrices sbmatrices sb-matrices Sppose that the nmber of sb-matrices with nonzero det is N() the nmber of sb-matrices with nonzero det is N() the nmber of sb-matrices with nonzero det is N() If N(), then ρ(m) If N() and N(), ρ(m) If N()N() and N(), ρ(m) ρ(m) only if M 4 7 M ?? Yo need to work from the highest order sbmatrices. The procedre stops whenever yo find one nonzero det. Example : 4 sbmatrices: ; ; ; 4 4 det det det 4 ; 4 det 8 ρ() Example : ll the sbmatrices have det. nd there is at least one nonsinglar sbmatrix ρ() It can be very tedios to check all sb-matrices. systematic way to find the rank is to se elementary operation to transform the matrix into a special form,.e.g., block diagonal.

12 Elementary operations that preserve the rank: 4 ) dd one row scaled by a nmber to another row 4 + ll operations that keep the determinant or scale the determinant by nonzero nmber dd row scaled by 4 to row ) dd one colmn scaled by a nmber to another colmn 4 4 (-) 4 + ) Exchange two colmns or two rows 4 7 M dd row scaled by - to row dd row scaled by - to row 4 7 dd row scaled by - to row 4 7 Rank <, bt Rank det 4 4 4

13 Sppose M is n m. rank(m) min{m,n} If M is mltiplied with a nonsinglar matrix (sqare and has non-zero determinant), the rank is preserved. This is why elementary operations preserve the rank. Since there are sally many sb-matrices, a systematic procedre to compte the rank is to se elementary operation to make it in pper or lower trianglar form. nother simple operation that preserves the rank is to reorder the colmns or rows. [ ] X, X [ ] Y, Y? ρ ρ ; ρ( [ ] ) ρ( [ ] ) This operation is eqivalent to mltiply the matrix with a matrix whose determinant is or -. Example: ? Today: Matrix Operations -- Fndamental to Linear lgebra Determinant Matrix Mltiplication Eigenvale Rank Math. Descriptions of Systems ~ Review LTI Systems: State Variable Description Linearization Modeling of Selected Systems Electrical Circits Mechanical Systems Integrator/Differentiator Realization Operational mplifiers 6

14 State Variable Description Start with a general lmped system: x(t) & h(x(t),(t),t) y(t) f (x(t),(t),t) If the system is linear, the above redces to: x(t) & (t)x(t) + B(t)(t) y(t) C(t)x(t) + D(t)(t) If the system is linear and time-invariant, then: x(t) & x(t) + B(t) y(t) Cx(t) + D(t) 7 To find an LTI system's response to a particlar inpt (t), we can se Laplace transform: sxˆ(s) x xˆ(s) + Bû(s) ŷ(s) Cxˆ(s) + Dû(s) Solve the above linear algebraic eqations: xˆ(s) ŷ(s) ( si ) Bû(s) + ( si ) C( si ) B + D û(s) + x [ ] C( si ) x Transfer fnction matrix Ĝ(s) x is the information needed to determine x(t) and y(t) for t>, apart from the inpt (t). x is the state 8 4

15 .4 Linearization There are many reslts on linear systems while nonlinear systems are generally difficlt to analyze What to do with a nonlinear system described by x(t) & h(x(t),(t), t) y(t) f (x(t),(t), t) Linearization. How? Under what conditions? Using Taylor series expansion based on a nominal trajectory, ignoring second order terms and higher Effects are not bad if first order Taylor series expansion is a reasonable approximation over the dration nder consideration 9 Sppose that with x o (t) and o (t), we have o (t) h(xo(t),o(t), t) Sppose that the inpt is pertrbed to o (t) + (t) ssme the soltion is x o (t) + x(t), with x(t) satisfying o (t) + x(t) & h(xo(t) + x(t),o(t) + (t), t) h h h(xo (t),o(t), t) + x x o o h h h h h h.... x x x n p h h h h h h h.. h.. x x x, n x p : :.. : : :.. : hn hn hn h n hn h.... n x x xn p ~ Jacobians 5

16 Then the pertrbed system can be described by h h x(t) & x + x o o ~ linear system The above is valid if the first order Taylor series expansion works ot well within the time dration nder consideration. It may lead to wrong prediction. What to do with the otpt y(t) f(x(t), (t), t)? The otpt eqation can be similarly linearized, bt most often there is no need for linearization nless with otpt feedback There is another approach to deal with nonlinear time-varying systems: Conservative bt reliable Example: model for a pendlm x θ(the angle), x & θ (anglar velocity), x θ The state is x x & θ The model is derived from Newton s law, θ l mg h( x) x torqe force arm g h( x) sin x+ cos x l ml Linearize the system at x, x,, h h h,, x x h g g h h ( cos x sin x ),, cos x x l ml l x ml ml 6

17 h h h h x x h g, x h h h l ml x x θ l Linearized system: x g x + & l ml T θ l m g T Exercise: Linearize the following system at x,. x; g mg sin x + cos x sin( x x ) + l ml ml x4; g 4 sin x + cos x l m l m g sin x sin( x x ) Linear Differential Inclsion (LDI) n LTI system: x(t) & x(t) + B(t) y(t) Cx(t) + D(t) y C B x D In many sitations,,b,c,d are not constant, bt nonlinear time varying, and/or depend on a parameter α, sch as, x(t) & (x,α, t)x(t) + B(x,α, t)(t) y(t) C(x, α, t)x(t) + D(x,α, t)(t) We can find a set Ω sch that (x,α, t) C(x, α, t) B(x,α, t) Ω D(x, α, t) The system satisfies y C B x : D C B Ω D 4 7

18 y C B x : D C B Ω D This is a linear differential inclsion (LDI) n LDI ses a set of linear systems to describe a complicated nonlinear system. In many cases Ω is a polytope: the behavior of an LDI can be characterized by finite many linear systems, e,g., x(t) & ix(t) + Bi(t) y(t) C x(t) + D (t), i i i, L, N Like a polygon, its properties are determined by finite many vertices. 5 Example: model for a pendlm h( x) x g h( x) sinx+ cosx l ml sin x g x + ( x) x+ B( x) x cos x l x ml If the angle is restricted between and π/4, we can write x [ ( x), B( x) ] : x [, π /4] 6 8

19 9 7 Today: Matrix Operations -- Fndamental to Linear lgebra Determinant, mltiplication, eigenvale, rank Math. Descriptions of Systems ~ Review LTI Systems: State Variable Description Linearization Next Time: Modeling of Selected Systems Continos-time systems (.5) Electrical circits, mechanical systems, integrator/differentiator realization, op amps Discrete-Time systems (.6) difference eqations, simple financial systems dvanced Linear lgebra, Chapter 8 Homework Set #:. Compte the eigenvales for. Compte the ranks for. Compte the determinant for (Use elementary operation to simplify the matrices for Pb. and.), 8 4, 4,, 4,