6.5 Control Systems Lst Time: trix Opertions Fundmentl to Liner Algebr Determinnt trix ultipliction Eigenvlue nk th. Descriptions of Systems ~ eview LTI Systems: Stte Vrible Description Lineriztion Tody: odeling of Selected Systems Continuoustime systems (.5) Electricl circuits echnicl systems Integrtor/Differentitor reliztion Opertionl mplifiers DiscreteTime systems (.6) Derive sttespce equtions difference equtions Two simple finncil systems Liner Algebr Chpter Liner spces over field Liner dependence
.5 odeling of Selected Systems We will briefly go over the following systems Electricl Circuits echnicl Systems Integrtor/Differentitor eliztion Opertionl Amplifiers For ny of the bove system we derive stte spce description: x(t) & Ax(t) Bu(t) y(t) Cx(t) Du(t) Different engineering systems re unified into the sme frmework to be ddressed by system nd control theory. Electricl Circuits Stte vribles? u(t) C i of L nd v of C v How to describe the evolution of the stte vribles? di L vl u v dt dv v C ic i dt In mtrix form: x& Output eqution: i L x& Ax Bu y Cx Du y di v u Stte Eqution: Two firstorder dt L L differentil equtions in terms dv v i of stte vribles nd input dt C C di i L L u dv dt v dt C C x i y v [ ] u v
Steps to obtin stte nd output equtions: Step : Pick {i L v C } s stte vribles Step : dil L vl f(il vc u) Liner functions dt dvc By using KVL nd KCL C ic f(il vc u) dt Step : dil (/L)f (i Cv Lu) dt dvc (/C)f (i Cv Lu) dt Step : Put the bove in mtrix form Step 5: Do the sme thing for y in terms of stte vribles nd input nd put in mtrix form 5 Exmple Stte vribles? i i nd v Stte nd output equtions? di L vl dt u di i v i dt L di L di vl i dt v i dt L dv dv v C i C i i i i dt dt C C di dt L di dv dt dt C x& L C L i L i u L v x i L L v u(t) C y L L v v L u i i y [ ] i i v x& Ax Bu y Cx Du 6
echnicl Systems y(t) u(t) Elements: Spring dshpot nd mss Spring: y(t) position positive direction Dshpot: y(t) y'(t) f D Dy' opposite direction ~ D: Dmping coefficient f S Ky opposite direction ~ Hooke s lw K: Stiffness y(t) f N (t) ss: Newton s lw of motion LTI elements LTI systems & y f N ~ Net force Liner differentil equtions with constnt coefficients 7 Ky Dy' K D y(t) u(t) y(t) u(t) x& x x& & y& dx K dx dt dt How to describe the system? Free body digrm: & y u Ky Dy& Number of stte vribles? Which ones? x D x stte vribles: x y x x ' u Ky Dy& u Kx Dx u D K && y y& y u ~ Input/Output description y [ ] u x x x 8
Steps to obtin stte nd output equtions: Step : Determine ALL junctions nd lbel the displcement of ech one Step : Drw free body digrm for ech rigid body to obtin the net force on it Step : Apply Newton's lw of motion to ech rigid body Step : Select the displcement nd velocity s stte vribles nd write the stte nd output equtions in mtrix form For rottionl systems: τ Jα τ: Torque Tngentil force rm J: oment of inerti r dm α: Angulr ccelertion There re lso ngulr spring/dmper 9 K y (t) D y (t) u(t) How to describe the system? How mny junctions re there? D(y 'y ') (reltive motion) y (t) u(t) ( y& & ) & y u D y ( y& ) Ky D y& Ky y (t) D(y 'y ') Number of stte vribles? How to select the stte vribles? x y; x y& ; x y x & y& x x & && y x K K x x& x x D ( u Dx D& ) u x& y y y K x u K D x u 5
Exmple: n xil rtificil hert pump AB AB PB P otor P PB Inducer Diffuser AB PB otor PB AB P P Illustrtion The forces cting on the rotor: F F F I P P y y y I P P F : the ctive force tht cn be generted s desired (F k I / (cy ) ki /(cy ) F F : pssive forces F k y F k y similr to springs 6
odeling The motion of the rotor: y && c F F F yc ( ly ly)/ l J&& α l F l F l F α ( y y )/ l l F F F y y α F nd F depend on y nd y. Eqution cn be expressed in terms of y nd y && y y y bf && y y y b F Let x y x y& x y x y& l y c l Center of mss && y y y bf && y y y bf Let x y x y& x y x y& x& y& x x& && y x x bf x& y& x x& && y x x b F In mtrix form? & b b x x F Ax Bu 7
Integrtor/Differentitor eliztion Elements: Amplifiers differentitors nd integrtors f(t) Amplifier y(t) f(t) Differentitor f(t) s y(t) df/dt Integrtor f(t) t y(t) /s f ( τ)dτ y(t) τ t Are they LTI elements? Yes Which one hs memory? Wht re their dimensions? Integrtor hs memory. Dimensions: nd respectively They cn be connected in vrious wys to form LTI systems Number of stte vribles number of integrtors Liner differentil equtions with constnt coefficients 5 u(t) x & x /s /s x & x y(t) Wht re the stte vribles? Select output of integrtors s SVs x & x x u x & y x Wht re the stte nd output equtions? x& x x u x x y & x A B C D [ ] u Liner differentil equtions with constnt coefficients 6 8
Steps to obtin stte nd output equtions: Step : Select outputs of integrtors s stte vribles Step : Express inputs of integrtors in terms of stte vribles nd input bsed on the interconnection of the block digrm Step : Put in mtrix form Step : Do the sme thing for y in terms of stte vribles nd input nd put in mtrix form 7 Exercise: derive stte equtions for the following sys. u b x /s /s x y 8 9
Opertionl Amplifiers (Op Amps) Noninverting terminl v i i b Inverting terminl v b V cc 5V V cc 5V Output v o i o v A (v v b ) with V CC v V CC V cc Slope A V o V cc v v b Usully A > Idel Op Amp: A ~ Implying tht (v v b ) or v v b i nd i b Problem: How to nlyze circuit with idel Op Amps 9 u (t) i u u (t) i i i u u y u y y u u Delinete the reltionship between input nd output Input/Output description Pure gin no SVs Key ides: ke effective use of i i b nd v v b Do not pply the node eqution to output terminls of op mps nd ground nodes since the output current nd power supply current re generlly unknown
u(t) i C v v v C i c i c v v Stte nd output equtions y Wht re the stte vribles? Stte vribles: v nd v dv dt C y v ( v v ) dv dv dt dt C v v dv C dt u C v u v C C v v [ ] u ( v v ) v Tody: odeling of Selected Systems Continuoustime systems (.5) Electricl circuits echnicl systems Integrtor/Differentitor reliztion Opertionl mplifiers DiscreteTime systems (.6): Derive sttespce equtions difference equtions Two simple finncil systems Liner Algebr Chpter Liner spces over field Liner dependence
.6 DiscreteTime Systems Thus fr we hve considered continuoustime systems nd signls y(t) y[k] t T In mny cses signls re defined only t discrete instnts of time T: Smpling period No derivtive nd no differentil equtions The corresponding signl or system is described by set of difference equtions k Elements: Amplifiers dely elements sources (inputs) Amplifiers: u[k] u[k] y[k] u[k] k ~ LTI nd memoryless Dely Element: u[k] z y[k] u[k] y[k] u[k] k ~ LTI with memory ( initil condition) They cn be interconnected to form n LTI system
Exmple y[k] u[k] Dely Dely K y[k] How to describe the bove mthemticlly? I/O description: y[k y[k ] y[k] u[k] ( u[k ] Ky[k ] ) or ] y[k] Ky[k ] u[k] u[k ] A liner difference eqution with constnt coefficient 5 Stte spce description: Select output of dely elements s stte vribles x [k] x [k] x u[k] [k] x [k] y[k] Dely Dely K x [k ] x[k] x[k] u[k] x [k ] K x[k] u[k] y[k] x[k] 6
Exercise: u[k] x [k] x [k] z z x [k] y[k] b x [k] Derive stte equtions for the discretetime system. Two stte vribles x [k] x [k] x [k] u[k] x [k] x [k] bx [k] x [k] y[k] x [k] x[k ] b x[k] u[k] y [k] [ ] x[k] 7 Exmple : Blnce in your bnk ccount A bnk offers interest r compounded every dy t m u[k]: The mount of deposit during dy k (u[k] < for withdrwl) y[k]: The mount in the ccount t the beginning of dy k Wht is y[k]? y[k] ( r) y[k] u[k] 8
Exmple : Amortiztion How to describe pying bck cr lon over four yers with initil debt D interest r nd monthly pyment p? Let x[k] be the mount you owe t the beginning of the kth month. Then x[k] ( r) x[k] p Initil nd terminl conditions: x[] D nd finl condition x[8] How to find p? 9 The system: x[k] ( r) x[k] () p Solution: A B u k x[k] A x[] ( r) ( r) k k k x[] D A m k m k m k m ( r) ( r) Bu[m] k m k m ( )p p ( r) Given D; r.; x[8]; k ( r) D r k p Your monthly pyment 8 8 (.) (.) p. p58.776 5
Tody: odeling of Selected Systems Continuoustime systems (.5) Electricl circuits echnicl systems Integrtor/Differentitor reliztion Opertionl mplifiers DiscreteTime systems (.6): Derive sttespce equtions difference equtions Two simple finncil systems Liner Algebr Chpter Liner spces over field Liner dependence Liner Algebr: Tools for System Anlysis nd Design Our modeling efforts led to sttespce description of LTI system x(t) & Ax(t) Bu(t) y(t) Cx(t) Du(t) x[k ] Ax[k] Bu[k] y[k] Cx[k] Du[k] Anlysis problems: stbility; trnsient performnces; potentil for improvement by feedbck control 6
Consider n LTI continuoustime system x(t) & Ax(t) Bu(t) y(t) Cx(t) Du(t) For prcticl system usully there is nturl wy to choose the stte vribles e.g. x i x L x v c However the nturl stte selection my not be the best for nlysis. There my exist other selection to mke the structure of ABCD simple for nlysis If T is nonsingulr mtrix then z Tx is lso the stte nd stisfies z(t) & TAT z(t) TBu(t) z(t) & A ~ y(t) CT z(t) Du(t) y(t) C ~ z(t) B ~ z(t) D ~ u(t) u(t) Two descriptions x(t) & Ax(t) Bu(t) y(t) Cx(t) Du(t) z(t) & A ~ y(t) C ~ z(t) B ~ z(t) D ~ u(t) u(t) re equivlent when I/O reltion is concerned. For prticulr nlysis problem specil form of A ~ B ~ C ~ D ~ my be the most convenient e.g. λ A ~ A ~ λ C ~ λ [ ] B ~ We need to use tools from Liner Algebr to get desirble description. 7
The opertion x z Tx is clled liner trnsformtion. It plys the essentil role in obtining desired sttespce description z(t) & A ~ y(t) C ~ z(t) B ~ z(t) D ~ u(t) u(t) Liner lgebr will be needed for the trnsformtion nd nlysis of the system Liner spces over vector field eltionship mong set of vectors: LD nd LI epresenttions of vector in terms of bsis The concept of perpendiculrity: Orthogonlity Liner Opertors nd epresenttions 5 Nottion:. Liner Vector Spces nd Liner Opertors n : ndimensionl rel liner vector spce C n : ndimensionl complex liner vector spce n m : the set of n m rel mtrices (lso vector spce) C n m : the set of n m complex mtrices ( vector spce) A mtrix T n m represents liner opertion from m to n : x m Tx n. All the mtrices ABCD in the stte spce eqution re rel 6 8
Liner Vector Spces n nd C n : The set of rel numbers; C: The set of complex numbers If x is rel number we sy x ; If x is complex number we sy x C n : ndimensionl rel vector spce C n : ndimensionl complex vector spce x x x x x x x n n : L n x x x x x x x n n C : L n C If xy n b then x by n n is liner spce. If xy C n b C then x by C n C n is liner spce. 7 Subspce Consider Y n. Y is subspce of n iff Y itself is liner spce Y is subspce iff α y α y Y for ll y y Y nd α α (linerity condition) Subspce of C n cn be defined similrly Exmple: Consider. The set of (x x ) stisfying x x cn be written s x : Y x x x x x Is the linerity condition stisfied? 8 9
Then how bout the set of (x x ) stisfying x x? x Y: xx x x : Y x x x x x Yes. In fct ny stright line pssing through form subspce Wht would be subspce for? Any plne or stright line pssing through {(x x x ): x bx cx } for constnts bc denote plne. How to represent line in the spce? The set of solutions to system of homogeneous eqution is subspce: {x n : Ax}. How bout {x n : Axc}? 9 Consider n Given ny set of vectors {x i } i to n x i n. Form the set of liner combintions n Y αix i i : αi Then Y is liner spce nd is subspce of n. It is the spce spnned by {x i } i to n
Liner Independence eltionship mong set of vectors. A set of vectors {x x.. x m } in n is linerly dependent (LD) iff {α α.. α m } in not ll zero s.t. α x α x.. α n x m (*) If (*) holds nd ssume for exmple tht α then x [α x.. α n x m ]/α i.e. x is liner combintion of {α i } i to m If the only set of {α i } i to m s.t. the bove holds is α α.. α m then {x i } i to m is sid to be linerly independent (LI ) None of x i cn be expressed s liner combintion of the rest A linerly dependent set ~ Some redundncy in the set Exmple. Consider the following vectors: x x x x For the following sets re they linerly dependent (LD) or independent (LI)? {x x } {x x } {x x x } {x x x x } If you hve LD set {x x x m } then {x x x m y} is LD for ny y.
Given set of vectors {x x x m } n how to find out if there re LD or LI? A generl wy to detect LD or LI: {x x.. x m } re LD iff {α α.. α m } not ll zero s.t. α x α x.. α m x m α x α x L α x A n m α α m α m α : m [ x x... x ] m m Aα {x x.. x m }re LD iffaα hs nonzero solution Need to understnd the solution to homogeneous eqution. There is lwys solution α. Question: under wht condition is the solution unique? Detecting LD nd LI through solutions to liner equtions Given {x x.. x m } form A [ x x... x ] m Consider the eqution Aα If the eqution hs unique solution LI; If the eqution hs nonunique solution LD. This is relted to the rnk of A. If rnk(a)m (A hs full column rnk) the solution is unique; If rnk(a)<m the solution is not unique. If nm nd A is nonsingulr det(a) rnk(a)m only α stisfies. Aα hence LI If nm nd A is singulr det(a) rnk(a) < m α s.t. Aα hence LD
5 Are the following vectors LD or LI? 7 x 5 x x LI 7 5 A) det( How bout 6 5 x x x 6 5 A) det( 6 8 5 6 5 6 LD det(a)? 6 All depends on the uniqueness of solution for Aα If m> n A is wide mtrix rnk(a)<m lwys hs nonzero solution e.g. A x x x If m < n nd rnk(a) m LI; If m < n nd rnk(a)<m LD; A A rnk(a )m LI rnk(a )<m LD A m n m n A
Exmples: determine the LD/LI for the following group of vectors sinθ cosθ cosθ sinθ sinθ cosθ cosθ sinθ 5 7 Dimension For liner vector spce the mximum number of LI vectors is clled the dimension of the spce denoted s D Consider {x x x m } n If m>n they re lwys dependent D n For mn there exist x x x n such tht with A[x x x n ] A x i s LI D n Hence D n 8
Tody: odeling of Selected Systems Continuoustime systems (.5) Electricl circuits echnicl systems Integrtor/Differentitor reliztion Opertionl mplifiers DiscreteTime systems (.6): Derive sttespce equtions difference equtions Two simple finncil systems Liner Algebr Chpter Liner spces over field Liner dependence Next time: Liner lgebr continued. 9 Homework Set #. Derive sttespce description for the circuit: Input uv in v c C i L L Output:. Derive sttespce description for the digrm: u /s /s /s b c y V o y 5 5
6 5. Are the following sets subspce of?. : : : b b Y b b Y b b Y 5 5. Are the following groups of vectors LD or LI? cos sin sin cos ) ) ) ) θ θ θ θ b 6) 5)