Theory of Control: I. Overview. Types of Structural Control Active Control Systems. Types of Structural Control

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1 Thory of Cotrol: I Asia Pacific Summr School o Smart s Tchology Richard Christso Uivrsity of Cocticut Ovrviw Itroductio to structural cotrol Cotrol thory Basic fdback cotrol Optimal cotrol stat fdback cotrol Obsrvrs ad LQG cotrollrs Typs of Structural Cotrol Passiv Cotrol Systms Passiv Dvic Typs of Structural Cotrol Activ Cotrol Systms Excitatio Rspos Excitatio Rspos m Ssor m Ssor Passiv Dampr Bas Isolatio Tud ass Dampr 3 Activ Bracig Activ ass Dampr 4 Typs of Structural Cotrol Activ Cotrol Systms Typs of Structural Cotrol Hybrid Cotrol Systms Passiv Dvic Excitatio Rspos Excitatio Rspos Ssor fdforward fdback fdforward fdback Ssors Cotrollr Ssors Actuator Ssors Cotrollr Ssors Activ Bas Isolatio 5 6

2 Typs of Structural Cotrol Smiactiv Cotrol Systms Typs of Structural Cotrol Fuctioally Upgradd Passiv Systms Excitatio Rspos Excitatio Rspos Passiv Dvic fdforward fdback fdforward Passiv Dvic fdback Ssors Cotrollr Ssors Ssors Cotrollr Ssors 7 8 Typs of Structural Cotrol Our focus today Excitatio Rspos fdforward fdback Ssors Cotrollr Ssors Itroductio to Structural Cotrol Itroductio to thory bhid automatic cotrol systms closd loop cotrol Cotrol is usd primarily for: 1. Rduc ssitivity to variatios. Rduc ssitivity to disturbac 3. Ability to cotrol systm badwidth 4. Stabilizatio of a ustabl systm 5. Cotrol systm trasit rspos Sgway 9 Itroductio to Structural Cotrol Exampl: Fillig a bathtub with watr* Cotrollig th tmpratur of fluid i a tak Op loop cotrol I op loop cotrol th commad sigal alo is slctd to achiv th dsird rspos r( rfrc iput Cotrollr G( u( cotrol iput Plat H( Itroductio to Structural Cotrol Exampl: Fillig a bathtub with watr Op loop cotrol Op hot watr tap spcifid amout Op cold watr tap spcifid amout If you hav do this may tim bfor, you might kow rathr wll th cssary sttigs Howvr, a umbr of factors might affct th cotrol of th *tak from Liar Cotrol Systms, (Rohrs( Rohrs,, t al.)

3 Itroductio to Structural Cotrol Exampl: Fillig a bathtub with watr Closd loop cotrol I closd loop cotrol, fdback masurmts ar icludd to achiv th dsird rspos r( rfrc iput Cotrollr G( u( cotrol iput Plat H( Itroductio to Structural Cotrol Exampl: Fillig a bathtub with watr Closd loop cotrol I closd loop cotrol fdback masurmts ar icludd to achiv th dsird rspos Fl th watr at svral itrvals whil th tub is fillig If watr is ot at right tmpratur, adjust hot or cold watr faucts I this mar, th systm affcts th cotrol of th systm Itroductio to Structural Cotrol Closd loop cotrol Us of th stat of th is trmd fdback or masurmts (tmp. of ach fauct, rat of chag of tmp.) ca achiv bttr rsults Closd loop may b mor complx tha op loop, but ca provid bttr prformac Compromis btw stability ad prformac Itroductio to Structural Cotrol Closd loop cotrol Compromis btw stability ad prformac Cotrollig oly hot watr (cold prdtrmid lvl) by turig fully o or fully off; Our slow rspos tim with th dramatic rspos may caus oscillatios i tmpratur Commo causs of istability i automatic cotrol systms: (1) dlay; ad () high gai Itroductio to Structural Cotrol Tak th huma out of th closd loop cotrol Automatic closd loop cotrol Ssor to masur th rquird variabls Actuator to adjust cotrol valvs Cotrollr to itrprt ssors ad sd cotrol sigal (which would th b amplifid) to actuator Itroductio to Structural Cotrol odlig th systm is a crucial stp i th dsig of a cotrollr Th quality of th cotrollr is likd to th quality of th modl usd i th cotrol dsig Sic o systm ca b prfctly modld, car must b tak i dsigig th cotrollr Paramtr iaccuracis Umodld dyamics Noliaritis

Ovrviw Itroductio to structural cotrol Cotrol thory Basic fdback cotrol Optimal cotrol stat fdback cotrol Obsrvrs ad LQG cotrollrs Gral form of th closd loop cotrol systm r( rfrc iput Cotrollr G( u(

4 Ovrviw Itroductio to structural cotrol Cotrol thory Basic fdback cotrol Optimal cotrol stat fdback cotrol Obsrvrs ad LQG cotrollrs Gral form of th closd loop cotrol systm r( rfrc iput Cotrollr G( u( cotrol iput Plat H( Fdback cotrol ca tak may forms Lt s bgi with a xampl xamiig th ffct of cotrol gais i th forward path: r( rfrc iput +- ( rror Cotrollr G( u( cotrol iput Plat H( Lt s bgi with a xampl xamiig th ffct of cotrol gais i th forward path: r( rfrc iput Clos loop systm H cl ( +- ( rror K u( cotrol iput H( Wh G( K, this is calld a proportioal cotrollr with uity gai fdback Wh G( K, this is calld a proportioal cotrollr with uity gai fdback r( ( +- K u( H( Th goal is to choos th cotrol gai (K) to stabiliz th systm ad improv rspos tim Usig th block diagram, w ca writ Y ( H( U( Y ( H( KE( [ ( Y( )] Y( H( K R s Y( KH( H cl ( R( + KH( Y( R( [ 1 KH( ] [ 1+ KH( ] Cosidr a simpl xampl of a dyamic systm I th Laplac domai, th trasfr fuctio of th plat is 1 H( s( s +τ ) Simpl modl of a lctric motor or hydraulic actuator with th commad/voltag as th iput ad th positio/disp disp.. as th Not that this systm is margially stabl bcaus o pol is at th origi

5 Th clos loop systm is: KH( K H cl ( 1+ KH( s + sτ + tao 1; K 1; um K; d [1 tao K]; sys tf(um,d); [y,t]stp(sys,; plot(t,y) [ ] K ovrshoot ucotrolld systm H( K10 K1 improvd rspos tim K0.1 Lt s look at th closd loop pols K H cl ( τ ± s + sτ + K has pols at p1, Not: Th systm is stabl wh Th systm is udrdampd wh τ 4K < τ 4K > τ τ 4K W ca us th pol placmt approach to assig K valus to achiv th spcific bhavior K1 K10 K0.1 K10 K1 K0.1 This systm ca quivaltly b cosidrd i stat spac Y( 1 H( U( s( s +τ ) Y ( ( s + τ U( y & ( + τy& ( u( y & ( τy& ( + u( This systm ca quivaltly b cosidrd i stat spac y & ( τy& ( + u( y& ( u( && τ & [ 1 0] + [ 0] u( & z( Az( Cz( + Du( 0 A 0 1 τ This systm ca quivaltly b cosidrd i stat spac 0 1 A 0 τ λ λ 1 dt( λi A) 0 λ 0 τ 0 λ + τ λ ( λ + τ ) ( 1)(0) λ + λτ Pols of th trasfr fuctio ar qual to th igvalus of th stat spac A matrix

6 Lt s look at th pols ad th stp rspos of a scod ordr diffrtial quatio y&& ( + ω y& ( + ω ω r( Th Laplac Trasform (zro IC) is Y( ω R( s + ω s + ω Cosidr th pols of th systm s ω ± ( ) ω 1 Assumig th systm is udrdampd ω ω s ω ± jω ral ω imagiary agitud: ( ω ) ( ω ) + ( ω ) ω ( ) ω + + ω ω ω Assumig th systm is udrdampd ω s siθ ω ω ± jω ral ω imagiary Agl: ω si( θ ) ω Cosidr th rspos of th systm ω Y( R( s + ωs + ω 1 To a stp iput R( s Th stp rspos ca b dtrmid as y 1 ω t () t si( ω t + Ψ) Ψ arcta Th pak rspos occurs at t ω π Th sttlig tim (dfid as tim rquird for rspos to rmai withi 5% of fial valu) is Th pak valu of y is th Ad th ovrshoot is π Not: ovrshoot is oly a fuctio of dampig tmax ) 1+ π t ω 0.05 ω t 3 t s 3 ω Not: icrasig w dcrass th ris tim

Lt s look at th pols ad th stp rspos Thory of Cotrol: II θ π ovrshoot sttlig tim t s 3 ω Asia Pacific Summr School o Smart s Tchology mag ω Richard Christso Uivrsity of Cocticut Optimal pols mov away

7 Lt s look at th pols ad th stp rspos Thory of Cotrol: II θ π ovrshoot sttlig tim t s 3 ω Asia Pacific Summr School o Smart s Tchology mag ω Richard Christso Uivrsity of Cocticut Optimal pols mov away from origi at dsird dampig Ovrviw Itroductio to structural cotrol Cotrol thory Basic fdback cotrol Optimal cotrol stat fdback cotrol Obsrvrs ad LQG cotrollrs Optimal Cotrol odr cotrol thory uss th approach that a optimal cotrollr ca b obtaid for a plat takig th form w( xcitatio u( cotrol Plat ( Az( + Ew( Cz( + Du( + Fv( Cotrollr G( v( Stat Fdback Cotrol Assum that all stats ar masurd ad a full stat fdback cotrol law taks th form u( Kz( Th closd loop dyamics ar giv by ( ( A BK ) z( Aclz( Th pols of this systm may b placd arbitrarily if th systm is cotrollabl Howvr, optimal placmt is possibl with a proprly chos cost fuctio Stat Fdback Cotrol Cosidr th Liar Quadratic Rgulator (LQR) W sk a stat fdback cotrollr (K) that miimizs th cost fuctio tf Whr Q is positiv smidfiat, R is positiv dfiit, ad subjct to T T J ( z Qz + u Ru) dt 0 z & Az + Bu z( 0) z 0

8 Stat Fdback Cotrol Th solutio to th LQR problm is giv by u( Kz( Whr th cotrol gai matrix K is giv by 1 T K R B P Whr P is th Riccati matrix which is govrd by th Riccati quatio T P & ( A P( + P( A + R P( BR 1 T B P, P( t ) 0 f Stat Fdback Cotrol As t f gos to ifiity, w s that P bcoms costat ad ca b dtrmid by solvig th algbraic Riccati quatio (ARE) T 1 T 0 A P( + P( A + R P( BR B P W ca us ATLAB to radily obtai this solutio Th matrics Q ad R provid th mchaisms to dsig a ffctiv cotrollr Stat Fdback Cotrol Lt s cosidr a xampl of a sdof buildig with activ bracig Activ Bracig Ssor w 1**pi; % rad/sc xsi 5/100; % dampig 5% 100; K *w^; C *xsi*w*; % Stat Spac Systm % dx Ac*x + Bc*u + Ec*w % y Cc*x + Dc*u + Fc*w Ac [0 1;-iv()*K -iv()*c]; Bc [0;1]; Ec [0;iv()*1]; Cc [y();-iv()*k -iv()*c]; Dc [1]; Fc [0]; Stat Fdback Cotrol Plot th ucotrolld systm s rspos du to Kob arthquak [w(] sys ss(ac,ec,cc,fc); t lispac(0,30,1000); load kob w itrp1(k(1,:),k(,:),; y lsim(sys,w,; figur(3); subplot(311);plot(t,:,1),'g'); subplot(31);plot(t,:,),'g'); subplot(313);plot(t,:,3),'g'); Stat Fdback Cotrol Plot th ucotrolld systm s rspos du to Kob arthquak [w(] Stat Fdback Cotrol Dsig a LQR cotrollr to wight displacmt ad vlocity qually sys ss(ac,ec,cc,fc); t lispac(0,30,1000); load kob w itrp1(k(1,:),k(,:),; y lsim(sys,w,; figur(3); subplot(311);plot(t,:,1),'g'); subplot(31);plot(t,:,),'g'); subplot(313);plot(t,:,3),'g'); Q diag([1 1]); R 1-; Klqr lqr(ac,bc,q,r,[]); sys ss(ac-bc*klqr,ec,cc-dc*klqr,fc); y lsim(sys,w,; figur(3); subplot(311);hold o;plot(t,:,1),'b'); subplot(31);hold o;plot(t,:,),'b'); subplot(313);hold o;plot(t,:,3),'b');

Stat Fdback Cotrol Dsig a LQR cotrollr to wight displacmt tf ad vlocity qually Q diag([1 1]); R 1-; Klqr lqr(ac,bc,q,r,[]); T T J ( z Qz + u Ru) dt 0 Stat Fdback Cotrol Lt s look at th pols Q

9 Stat Fdback Cotrol Dsig a LQR cotrollr to wight displacmt tf ad vlocity qually Q diag([1 1]); R 1-; Klqr lqr(ac,bc,q,r,[]); T T J ( z Qz + u Ru) dt 0 Stat Fdback Cotrol Lt s look at th pols Q displacmt wightig R dcrass tf T T J ( z Qz + u Ru) dt 0 Q vlocity wightig R dcrass sys ss(ac-bc*klqr,ec,cc-dc*klqr,fc); y lsim(sys,w,; figur(3); subplot(311);hold o;plot(t,:,1),'b'); subplot(31);hold o;plot(t,:,),'b'); subplot(313);hold o;plot(t,:,3),'b'); Stat Obsrvrs I practic, it is ot fasibl or practical to masur all of th stats of th systm Thus, fdback cotrol dsig oft rquirs that o stimat th stat variabls w( xcitatio u( cotrol Plat ( Az( + Ew( Cz( + Du( + Fv( Cotrollr G( v( Stat Obsrvrs A obsrvr is a dyamic systm with iputs u (cotrol ipu ad y (masurd rspos, ad that stimats th stat vctor (calld xha Obsrvr (liar, cotiuous tim) catgory: Op loop obsrvr Full ordr obsrvr Kalma (Bucy) filtr Op Loop Obsrvrs Th objctiv is: lim x( xˆ( 0 t Obsrvr (liar, cotiuous tim) catgory: Liar ivariat systm: x &( Ax( Cx( x( t ) x 0 0 Auxiliary dyamical systm: x & ˆ( Axˆ( Estimatio rror: ( x( xˆ( & ( Axˆ( ( Axˆ( ) A( t t0 If A is stabl, th ( approachs zro Drawbacks: Uboudd rror for ustabl stat matrix Fails i th prsc of modlig rrors ad disturbacs Stat Obsrvrs Full ordr obsrvr Lubrgr obsrvr Liar ivariat systm: 0 0 Auxiliary dyamical systm: x& ˆ( Axˆ( + L( Cxˆ( ) Estimatio rror: ( x( xˆ( & ( ( A LC) ( t t0 Obsrvr fdback If (A LC) is stabl, th ( approachs zro Drawbacks: Still fails i th prsc of modlig rrors ad disturbacs

10 Stat Obsrvrs Stochastic stat obsrvr, Kalma Bucy filtr Liar ivariat systm: x &( Ax( + w( Cx( + v( Procss ois with covariac Q( Disturbac with covariac R( Both ois trms ar assumd whit, Gaussia ad mutually idpdat Auxiliary dyamical systm: x& ˆ( Axˆ( + K( Cxˆ( ) T T Riccati quatio: P& ( AP( + P( A + Q( K( R( K ( T Kalma Gai: K( P( C ( R 1 ( Stat Obsrvrs Th Kalma filtr provids th bst stimat of th stats basd o currt availabl oisy iformatio Th solutio P( to th associatd diffrtial Riccati quatio (DRE) is also th covariac of stimatio rror If oly th stady stat bhavior is of itrst, th tim drivativ is limiatd i th DRE ad ad rsults i a algbraic Riccati quatio (ARE) LQG Cotrol Kalma filtr is oft kow as liar quadratic stimatio (LQE) Wh w combi optimal stat fdback with stimator dsig, w raliz a liar quadratic Gaussia (LQG) cotrollr ˆ( Azˆ( + L u( Kzˆ( ( y Czˆ ) LQG Cotrol Th closd loop systm is thus ˆ( Azˆ( + L ˆ( A zˆ( + B w( xcitatio u( cotrol u( Kzˆ( C zˆ( ˆ( u( Plat ( Az( + Ew( Cz( + Du( + Fv( Cotrollr Azˆ( + Kzˆ( Bu( v( + L( y Czˆ ) ( y Czˆ ) ( A LC BK ) zˆ( + L LQG Cotrol Th closd loop systm is thus ( Az( + BCzˆ( + Ew( Cz( + DCzˆ( ˆ( Azˆ( + B u( Kzˆ( C zˆ( ( A z & ˆ( BC [ C DC ] BC z( E + + w( A BDC zˆ( 0 z( zˆ( Stat Fdback Cotrol Dsig a LQG cotrollr Eww 0.1; Evv 4-5; Lgai lq(ac,ec,cc(3,:),eww,evv); Ak Ac-Bc*Klqr-Lgai*Cc(3,:); Bk Lgai; Ck -Klqr; Dk 0; Acl [Ac Bc*Ck;Bk*Cc(3,:) Ak+Bk*Dc(3,:)*Ck]; Bcl [Ec;zros(,1)]; Ccl [Cc Dc*Ck]; Dcl zros(3,1); sys ss(acl,bcl,ccl,dcl); y lsim(sys,w,;

11 Stat Fdback Cotrol Dsig a LQG cotrollr Stats (actual blu; stimatd gr) Rspos (lqr blu; lqg rd)

CDS 101: Lecture 5.1 Reachability and State Space Feedback

CDS 101: Lecture 5.1 Reachability and State Space Feedback CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray ad Hido Mabuchi 5 Octobr 4 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls