Lyapunov Stability Analysis for Feedback Control Design

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1 Copyright F.L. Lewis 008 All rights reserved Updated: uesday, November, 008 Lyapuov Stability Aalysis for Feedbac Cotrol Desig Lyapuov heorems Lyapuov Aalysis allows oe to aalyze the stability of cotiuous-time dyamical systems ad discrete-time dyamical systems of the form: Cotiuous-time oliear system f( ), (0) with state ( t) R ad f(0)0 so the origi is a equilibrium poit. We assume that f(.) is cotiuous so there eist solutios, ad that f() is Lipschitz so there eists a uique solutio. with Discrete-time oliear system f( ), + 0 R, f(.) cotiuous ad Lipschitz, ad f(0)0. Defiitio. Lyapuov Fuctio Cadidate (LFC). A scalar fuctio V( ) : R R is said to be a Lyapuov Fuctio Cadidate (LFC) if:. V() is a cotiuous real-valued fuctio. V( ) > 0, i.e. V() is positive defiite A Lyapuov fuctio is a LFC that is oicreasig with time ad hece bouded. Defiitio: Cotiuous-time (C) Lyapuov Fuctio V( ): R R is said to be a C Lyapuov Fuctio if: V() is a LFC ad 3. dv 0, i.e. ( ) is egative semi-defiite dt Defiitio: Discrete-ime (D) Lyapuov Fuctio

2 V( ): R R is said to be a D Lyapuov Fuctio if: V() is a LFC ad 3. Δ V( ) V( + ) V( ) 0, i.e. the first differece Δ V( ) is egative semidefiite he followig results were prove by A.M. Lyapuov. Stability for C systems refers to the jω-ais stability boudary i the s-plae. Stability for D systems refers to the uit circle stability boudary i the s-plae. Lyapuov heorem. SISL. Suppose that for a give system there eists a Lyapuov fuctio. he the system is SISL. Lyapuov heorem. AS. Suppose that for a give system there eists a Lyapuov fuctio which also satisfies the stroger third coditio: For C systems: dv < 0, i.e. ( ) is egative defiite. dt For D systems: Δ V( ) V( + ) V( ) < 0, i.e. Δ V( ) is egative defiite. he the system is AS. Epoetial stability ca be verified by etra aalysis to relate V() ad ( ) ad show that the Lyapuov fuctio decreases epoetially. Liear-ime Ivariat Autoomous Systems- Recap We have already cosidered liear time ivariat (LI) autoomous systems i the followig form. Cotiuous-time (C) LI autoomous systems A, (0) with state ( t) R. Discrete-time (D) (LI) autoomous system + A, 0 with state R. he et results hold.

3 heorem. SISL for LI C Systems. Let Q be a symmetric positive semidefiite matri. he the system A is SISL (e.g. margially stable) if ad oly if the (symmetric) matri P which solves the C Lyapuov equatio AP+ PA Q is positive defiite. heorem. SISL for LI D Systems. Let Q be a symmetric positive semidefiite matri. he the system + A is SISL (e.g. margially stable) if ad oly if the (symmetric) matri P which solves the D Lyapuov equatio APA P Q is positive defiite. he proof relies o the fact that, if the Lyapuov equatios have solutios as specified, the V( ) P serves as a Lyapuov fuctio, with costat erel matri P symmetric ad positive defiite, i.e. P P > 0. If Q Q is i fact positive defiite, the theorems yields AS. Note that the solutio properties of the C Lyapuov equatio AP+ PA Q refer to the locatios of the poles of system matri A with respect to the left half of the comple plae. O the other had, the solutio properties of the D Lyapuov equatio APA P Q refer to the locatios of the poles of system matri A with respect to the uit circle of the comple plae. Lyapuov Aalysis for Cotrolled Systems We ow wat to use Lyapuov Aalysis to study the stability of systems with cotrol iputs. We shall see that istead of the Lyapuov equatios AP+ PA Q for C systems APA P Q for D systems we obtai Riccati equatios, amely the equatios above but with etra terms quadratic i P. 3

4 Cotrol Desig for Noliear C Systems Let us see how direct Lyapuov techiques are by cofrotig a oliear C system. HIS SECION IS NO ON HE EE 5307 Homewor or eams. It is for fu oly. Cosider the oliear system be give by f ( ) + g( ) u y h( ) with state ( t) R ad f(0)0 so the origi is a equilibrium poit. We assume that f(.) is cotiuous so there eist solutios, ad that f() is Lipschitz so there eists a uique solutio. he for ay scalar C fuctio V() oe has dv V V V f + gu V f + V gu dt V with V the gradiet, which is assumed here to be a colum vector. Completig the squares for ay matri R R > 0 yields V f V gu V f V gr u R R g V u ) + + ( + ) ( + V gr g V Now suppose that V()>0, V(0)0, ad satisfies the Hamilto-Jacobi (HJ) iequality V f + h h V gr g V 0. Assume the system is locally iput/output detectable i the sese that there eists a eighborhood such that u( t) 0 ad y( t) 0 t implies that ( t) 0. his is implied by i/o observability. he the closed-loop system is asymptotically stable if oe selects the cotrol iput as u Ru. u R g ( ) V. o prove this, ote that, accordig to the HJ equatio ( V gr + u ) R( R g V + u ) h h u Ru ad accordig to the cotrol selectio h h u Ru y y u Ru which is egative defiite uder the i/o detectable assumptio. herefore V() is a Lyapuov fuctio with V < 0. Result. 4

5 Based o this aalysis oe has the et theorem. heorem for Cotrol of Noliear C Systems. Suppose that V( ) : R R is cotiuous, V()>0, V(0)0, ad V() satisfies the Hamilto-Jacobi-Bellma (HJB) equatio V f + h h V gr g V. 0 he the closed-loop system is stable usig the cotrol iput u R g ( ) V. o prove this, ote that if V() satisfies the HJB equatio, it solves the HJ iequality. HIS MEANS that to desig a feedbac cotrol for a oliear system, oe may first solve the HJB equatio for the value V(), the compute the cotrol usig the formula above. Cotrol Desig for Liear ime Ivariat C Systems Here we specialize the previous developmet to the case of LI C systems of the form A + Bu We wat to fid a SVFB u K to stabilize the system. For ay scalar C fuctio V() oe has dv V V V A + Bu V A + V Bu dt Completig the squares for ay matri R R > 0 yields ( V A V Bu V A V BR u ) R( R B V u) V BR B V u Ru. Now suppose that V()>0, V(0)0, ad satisfies the Hamilto-Jacobi (HJ) iequality V A+ C C V BR B V 0 for ay matri C such that (A,C) is observable. Note that the selectio of C i the HJ effectively defies a output y C hrough which the state is observable. his output is NO used for cotrol purposes, but oly to obtai a suitable value V() through solutio of the HJ. 5

6 Let us mae the HJ symmetric. Note that the first term is a scalar so that it equals its ow traspose, i.e. V A A V. hus, write the HJ iequality equivaletly i symmetric form ( ) A V + V A + y y V BR B V 0 (o se that this is symmetric, traspose it ad you will get the same equatio.) Now select the cotrol iput as u R B V. he the system is AS. o prove this, ote that, accordig to the HJ equatio ( V BR + u ) R( R B V + u) y y u Ru ad accordig to the cotrol selectio h h u Ru y y u Ru which is egative defiite uder the observabilityassumptio. herefore V() is a Lyapuov fuctio with V < 0. C Riccati Equatio he et results mae it easy to use this machiery to desig stabilizig cotrollers for C LI systems. It turs out that V() for liear systems ca always be selected i the quadratic form V( ) P with P P > 0. herefore V ( P) P ad the HJ becomes ( A P + PA) + C C PBR B P ( A P+ PA+ C C PBR B P) 0 Sice this must hold for every state (t) it is equivalet to A P + PA + C C PBR B P 0 he stabilizig cotrol is selected accordig to u R B P K heorem. Stabilizig Cotrol Desig for C LI Systems. 6

7 Let Q Q > 0, R R > 0 be symmetric positive defiite matrices. he the system A + Bu is AS if there eists a positive defiite solutio P to the C algebraic Riccati equatio (ARE) A P + PA + Q PBR B P 0 he, the state feedbac K R B P maes the closed-loop system stable. o prove this theorem, ote that if Q is positive defiite the (A,C) is observable for ay square root C of Q, Q C C, for the C is osigular. Moreover, if P is a solutio to the C ARE the the HJ iequality holds. Compare the C ARE to the C Lyapuov equatio AP+ PA+ Q 0 he theorem shows the followig desig method for C LI SVFB cotrollers. Select desig matrices Q Q > 0, R R > 0. Solve the C ARE A P + PA + Q PBR B P 0 3. Compute the SVFB K R B P he ARE is easily solved usig may routies, amog them the MALAB routie [ KP, ] lqrabqr (,,, ) Optimal Cotrol for C LI Systems Now we see that the above costructios actually yield the OPIMAL C cotroller. Cosider the LI C system A + Bu We wat to fid a SVFB u K to miimize the performace ide 0 ( (0), ) ( J u Q+ u Ru) dt heorem. Optimal Cotrol Desig for C LI Systems. 7

8 Let Q Q > 0, R R > 0 be symmetric positive defiite matrices. Suppose there eists a positive defiite solutio P to the C algebraic Riccati equatio (ARE) A P + PA + Q PBR B P 0 he, the state feedbac K R B P miimizes J((0),u) ad maes the closed-loop system stable. Proof: Select the Lyapuov fuctio V( ) P he it was show above that if P satisfies the ARE ad oe selects the give SVFB, oe has ( ) Q+ u Ru Itegratig both sides yields ( Vdt ) Q+ u Ru dt t t lim V( ()) t V( ()) t J( (), t u) t However, the system has already bee show AS, so that lim t ( ) 0 t which implies lim V( ( t)) 0 t or V( ()) t () t P() t J( (), t u) with u K R B P. It remais to show that V((t)) is the optimal value of J((0),u). Ca you do it? his theorem shows how to select desig matrices Q, R. amely, as discussed i the otes o LQR. Cotrol Desig for Liear ime Ivariat D Systems Give the LI D system A + + Bu it is desired to fid a stabilizig SVFB cotrol u K 8

9 Note that for ay scalar quadratic form V() oe has the first forward differece evaluated alog the system trajectories give as Δ V ( ) V+ V + P+ P ( A + Bu) P( A + Bu) P A PA P + A PBu + u B PBu where lac of a subscript o time fuctios idicates their values at time. Note that the first differece of V(.) is QUADRAIC i the state dyamics A+Bu. By cotrast, i the C LI case the differetial of V(.) is LINEAR i the state dyamics A+Bu. his maes epressios for D systems more comple tha for C systems. Complete the square to see that, for ay osigular matri R oe has ( A PBR + u ) R( R B PA + u) A PBR B PA + A PBu + u R u So that oe writes Δ V( ) as Δ V ( ) A PA P + u B PBu + ( APBR + u) R( R BPA+ u) APBR BPA uru Now, to get rid of the third term o the right-had side ad itroduce a positive defiite quadratic term i u, defie R B PB + R. he, Δ V ( ) A PA P u Ru + ( A PBR + u ) R( R B PA + u) A PBR B PA Now suppose V( ) > 0, V(0) 0 ad it satisfies the D HJ iequality APA P+ Q APBBPB ( + R) BPA 0 he ΔV Q u Ru + A PBR + u R R B PA + u. ( ) ( ) ( ) Selectig the cotrol as ( u B PB + R) B PA yields ΔV( ) Q u Ru. so that V() is a Lyapuov fuctio. Now assume i/o detectability with the output y C with Q Q Q C C, ad the system is AS. It is ot hard to show that i fact this choice of cotrol also miimizes the performace measure 9

10 + i J(, u) ( Q u Ru ) where u deotes the cotrol sequece { uu +, }. hat is, for the above V( ) oe has + u u i V( ) mi J(, u) mi ( Q u Ru ) Oe calls V( ) the VALUE FUNCION. Summary of Desig procedure for D Systems:. Select desig matrices Q Q > 0, R R > 0. Solve the D ARE APA P+ Q APBBPB ( + R) BPA 0 3. Compute the SVFB ( K B PB + R) B PA he ARE is easily solved usig may routies, amog them the MALAB routie [ K, P] dlqr( A, B, Q, R) 0