IAENG Iteratioal Joural of Applied Mathematics, 36:, IJAM_36 A Algorithm for Maximizig the Cotrollable Set for Ope-Loop Ustable Systems uder Iput Saturatio We-Liag, Abraham, Wag, ad Ye-Mig Che Chug Hua Uiversity, abewag@michuedutw Natioal Kaohsiug First Uiversity, yjjche@ccmskfustedutw A optimizatio techique is preseted for approximatig the cotrollable set by a ellipsoid for a liear time-ivariat ope-loop ustable system subject to iput saturatio A techique ad algorithms for maximizig the cotrollable set are also preseted I stead of startig from a positive defiite right-had side matrix Q of the Lyapuov equatio as doe i almost all applicatios of the Lyapuov fuctios, we start from a positive defiite Hessia matrix P for the Lyapuov fuctio so that the resultig Lyapuov fuctio will be covex Keywords: iput saturatio, Lyapuov theorem, ellipsoidal cotrollable set INRODUCION he cocept of cotrollable set i cotrol systems was itroduced by Sow [], whe he defied the cotrollable set as the reachable set of the system with time reversed For liear systems, there is complete duality betwee reachability (the ability to reach ay desired fial state from a give iitial state) ad cotrollability (the ability to reach a give fial state from ay iitial state) But this is ot geerally true for oliear systems Determiatio of the reachable set uder iput saturatio has bee widely studied usig a ope-loop approach See Summers [], Sabi ad Summers [3], Summers, Wu ad Sabi [4], ad Qui ad Summers [5] However, oe of these papers covers the cotrollable set of a closed-loop system For this paper, without loss of geerality, we assume that the destiatio state is the origi I other words, the cotrollable set is defied to be the set of the states for which there is at least oe cotrol which will drive it to the origi For a liear time-ivariat system, the cotrollable set is the etire state space if the system is asymptotically stable, ie, if all the eigevalues of the system are located o the ope left-half plae his is because the state will evetually go to the origi with zero iput o matter where it starts For a ope-loop ustable liear system, ie, a system with at least oe ope-loop eigevalue o the ope right-half plae or a system with at least oe multiple ope-loop eigevalue o the imagiary axis, the cotrollable set ca be made the etire state space if the ope-loop system is liear stabilizable his is because with a liear feedback cotrol all the eigevalues of the resultig closed-loop system ca be placed o the ope left-half plae thus rederig the closed-loop system asymptotically stable However, the cotrollable set for the same system may ot be made the etire state space, if the liear feedback is subject to iput saturatio here are two approaches for studyig the cotrollable set of a ope-loop ustable liear system: (i) ope-loop approach, i which we drive the state to the origi with the iput u( restricted to u i (, i =,,, m; (ii) closed-loop approach, i which we drive the state to the origi with liear feedback with saturatio: (Advace olie publicatio: 4 May 7)
u( = sat( v( ); v( = K, where K is a costat matrix ad sat ( ) is the saturatio fuctio I this paper, we take the closed-loop approach ad fid a approximate cotrollable set HEORY Cosider a liear time-ivariat cotiuous-time system with iput saturatio x ( = A + Bu( () where u( = sat( K ), () m A R is a give costat matrix, B R is a give costat matrix, R is m the state vector, u( R is the cotrol vector, with u( = [ u(,, u ( ], ad sat( ) deotes the m saturatio fuctio he oe-dimesioal versio of the saturatio fuctio is defied by, if y sat ( y) = y, if y (,), y R (3), if y ad we compoetwise exted its defiitio to the multi-dimesioal versio: sat( y) sat( y) sat( y) =, y R sat( y3) m (4) Here we assume that A is ot ecessarily asymptotically stable We also assume that the system (A,B) is liearly stabilizable I other words, it is assumed that, without saturatio, the system would be stabilizable Hece there exists at least oe matrix K such that x ( = A BK = ( A BK) is asymptotically stable Actually it is possible to select the locatio of the system eigevalues (ie, the eigevalues of A-BK) arbitrarily Hece we assume that matrix K has bee selected so as to place the system eigevalues i the desired locatio Sice A = A BK is Hurwitz, for every positive defiite matrix Q, there exists a uique P R ad P > satisfyig A P + PA = Q, Our goal is first to fid a ier approximatio Ω (P) of the cotrollable set Ω of our system () ad () based o the quadratic Lyapuov fuctio V ( ξ ) = ξ Pξ, ad the to maximize the approximate cotrollable set Ω (P) by varyig the approximatio parameter P i such a way that the resultig matrix Q = ( A P + PA) remais positive defiite We deote the ith row of matrix K by k i, i =,, m : k k K = k m We ow cosider the case of a sigle iput Defie f ( ξ ) = Αξ Βsat( Kξ ) (5) π { ξ R Kξ < } π { ξ R Kξ } π ξ R Kξ } ( A BK) ξ, ifξ H = = Aξ B, ifξ H = (6) Aξ + B, ifξ H = { + Defie V ( = V ( ) akig derivative of V ( t ) alog the trajectory x (, we obtai the followig cases: Case x H : usaturated case, ie, u( = K ( d V ( = f ( ) Pf ( ) = (( A BK ) ) P( A BK ) (7) = (( A BK ) = Q, P + P( A BK )) Case x H : positively saturated case, ie, u ( = ( +
d V( = f ( ) Pf( ) = ( A + B) P( A + B) where = ( A P+ PA) + B PB = Qxt ( ) + B PB, (8) Q = Δ ( A P + PA) (9) Case 3 x ( H : egatively saturated case, ie, u ( = d V( = f ( ) Pf( ) = ( A B) P( A B) () = ( A P+ PA) B PB = Qxt ( ) B PB, where Q is defied as i (9) Ispired by the right-had sides (7), (8) ad () d for V ( t ), we defie g g g + ( ξ) = ξq ξ, ξ H; ( ξ) = ξq ξ + B Pξ + ξ PB, ξ H+ ; ( ξ) = ξq ξ B Pξ ξ PB, ξ H ; Combiig these three fuctios ito oe fuctio, we obtai g( ξ ) if ξ H, g( ξ ) = g+ ( ξ ) if ξ H +, () g ( ξ ) if ξ H, Observe that d V ( t ) = g ( x ( t )) We ote that, i Case Q > because P is selected so that Q > I other words, because we use oly those P that will make Q = ( A P + PA) >, the right-had side for d V ( t ) is egative: g ( ξ ), ξ Hece g( ξ ) <, ξ H {} herefore, the equilibrium poit is locally asymptotically stable i H However, i Case ad 3, sice the ope-loop system may be ustable, matrix A may ot be Hurwitz Give a positive defiite matrix P that will make Q >, the Q defied by (9) may or may ot be positive defiite I order to satisfy the Lyapuov descet coditio g( ξ ) < for a give ξ, we require that for each ξ, there exists at least oe cotrol value ν satisfyig ν ad g( ξ ) = ξ ± ξ PBν < he the state space R ca be divided ito the followig regios: (a) R = { ξ R ξ > } If ξ R, the g ( ξ ) < (b) { R = ξ R ξ PB < ξ Q } + ξ ξ R +, the set ν = so that g ( ξ ) < (c) { R = ξ R ξ PB > ξ } ξ R, the set ν = so that g ( ξ ) < (d) R { R R R+ } If ξ R { R R R+ }, the it is ot possible to fid ν [, ], such that g ( ξ ) < he approach for fidig the maximal level set { R V ( ξ = ξ P } ) ) L ( c = ξ ξ c which is P cotaied i the uio of the regios (a), (b) ad (c), ie, c = max{ c LP ( c) R R+ R }, ca be foud by the followig maximizatio problem c = subject to ad mi V ( ξ ) = ξ Pξ g ( ξ ) = ξ + B + Kξ + Pξ + ξ PB, see Wag, Che, ad Mukai [] So far we treated matrix P as a give positive defiite, symmetric matrix which will make Q > We ow seek to fid a P which will maximize the volume (area) of the level set L P ( c { ξ ξ P c ( P) } ( P)) = ξ If If
ad hece the volume of the approximate cotrollable set L P ( c( P)) We observe that, as P varies, the resultig Q must remai positive defiite Fially, fidig the maximal approximate cotrollable set (stability regio) for the system is equivalet to maximizig the volume W of the maximal ellipsoid i Step herefore, the origial I MAXIMIZE HE ELLIPSOIDAL CONROLLABLE SE maximizatio problem becomes: We ow itroduce the procedures of fidig the π ( c ( P)) max maximal ellipsoidal cotrollable set for our system () P W = Γ( / + ) det P ad () Step Let P = P( α, β,) be a symmetric matrix which depeds o at least oe parameter α, ad at Q = ( A P + PA) >, ad P > where W is the volume of the ellipsoid, c ( P) is the maximum level defied i Step, Γ ( ) is the Gamma fuctio, ad det(p ) is the determiat of most ( + ) the matrix P parameters α, β, For istace, geeral forms of P for = ad = 3 cases may be Example Cosider the sigle-iput ope-loop ustable system α β x ( = A + Bu(, α, α β α γ δ u( = K, β δ where () A =, Step B = Next, we wat to maximize the level set he ope-loop system is ustable with eigevalues λ = ad λ = Suppose that the desired L p ( c) = { ξ R ξ Pξ c } i which the Lyapuov eigevalues are λ = + i ad λ = i criterio is satisfied I other words, we wat to maximize the value c subject to the costrait herefore, usig a stadard eigevalue placemet method, we may select a feedback matrix L P (c) remais iside the set where the Lyapuov K = [ 4 ], descet criterio g (ξ ) remais egative, ie, which results i c ( P) = max{ c LP ( c) { ξ g( ξ ) } } A = A BK = Rewrite the above problem i the followig way as whose eigevalues are λ = + i ad λ = i before: as desired For a give positive defiite matrix Q, where c ( P) = mi{ ξ Pξ ξ + ξ PB + B Pξ, Kξ }, Q = or We ca fid a uique symmetric, positive defiite c ( P) = mi{ ξ Pξ ξ ξ PB B Pξ, Kξ } solutio P, satisfyig the Lyapuov equatio, We ote that the above two problems are equivalet 5 5 P = due to symmetry 5 375 Step 3 he cotrollable set iside the liear regio foud by
applyig the method of Lee ad Hedrick [4], ad the cotrollable set uder the Lyapuov descet coditio [] are show i Figure as follows 5 8 6 4 5-5 - x - -4-6 -8 - - -5 5 x -5 - - -5 - -5 5 5 Figure : he maximal ellipsoidal cotrollable set Fig : Cotrollable set iside the liear usaturated regio (ier ellipsoid) ad the cotrollable set uder Lyapuov descet coditio (outer ellipsoid) 5 5 x he ier ellipsoid represets the cotrollable set for the system iside the liear usaturated regio, while the outer ellipsoid represets the cotrollable set for the system uder the Lyapuov coditio Fially, to fid the maximal ellipsoidal cotrollable set for the system, we apply the three steps, the maximal cotrollable set is show i Figure as follows Figure 3 shows the compariso of the three cotrollable sets: the maximal cotrollable set, the cotrollable set uder the Lyapuov descet coditio, ad the cotrollable set iside the liear regio As we ca see from Fig 3, the cotrollable set uder the Lyapuov descet coditio ad the cotrollable set iside the liear regio are cotaied i the maximal ellipsoidal cotrollable set foud by our approach -5 - -5 - - -5 - -5 5 5 x Fig 3: Compariso of the cotrollable set iside the liear regio, uder the Lyapuov descet coditio, ad the maximal cotrollable set foud by our approach Fially, we coclude this paper with a practical example, which has bee studied several times i the past; see eg, [5], [6] Example Cosider the double itegrator, a sigle iput plat of the form x = x + u, u Here we ote that the eigevalues of the ope-loop systems are foud as, Sice there are two
ope-loop zero eigevalues for the system, the system is ope-loop ustable Suppose that the desired eigevalues λ = σ + iω ad λ = σ iω for the closed-loop system are as follows: σ =, ω = For a give Q, where Q =, the feedback gai K i ca be selected by the stadard pole placemet techique ad P ca be foud as follows: 5 5 K = [ ], P = 5 375 Solvig the above optimizatio problems, we obtai the volume (area) of the ellipsoidal cotrollable sets as follows: W = 698; W = 685; W = 367 l L M We deote by W l the volume (area) foud by applyig the techique from Lee ad Hedrick, W L the volume (area) foud by applyig Lyapuov descet criterio, ad W M the volume (area) foud by applyig our techique Figure 4 shows the volumes (areas) of the asymptotically stable regio for the three techiques Ideed, the asymptotically stable regios foud by our proposed techique are superior to the other two approaches x 8 6 4 - -4-6 Maximal cotrollable set W M =367 Lyapuov cotrollable set W L =685 Cotrollable set iside liear regio W l = 698-8 -8-6 -4-4 6 8 x Fig 4: Compariso of the cotrollable set iside the liear regio, uder the Lyapuov descet coditio, ad the 3 CONCLUSION I this paper, we preseted a techique for approximatig the cotrollable set for a ope-loop ustable system uder iput saturatio ad maximized this cotrollable set by a ellipsoid Istead of startig from a positive defiite matrix Q as doe i almost all applicatios of the Lyapuov fuctios, we reversed the approach by startig from a positive defiite matrix P so that the Lyapuov fuctio x Px will be positive defiite Also, based o the formula for the volume of a geeral -dimesioal ellipsoid, we developed the algorithm of maximizig the volume of the cotrollable set of the system From our example, the maximal cotrollable set was foud ad is ideed larger tha those of cotrollable sets iside the liear regio ad uder the Lyapuov descet coditio APPENDIX We explai why the matrix P ca be take i the form α α β α β α γ δ, β δ Let Pˆ be a symmetric, positive defiite matrix of the form ˆ ˆ α P = ˆ ˆ α β Sice Pˆ is positive defiite, >, ˆ β >, ˆ β αˆ > Note that, if we multiply both P ad L by a costat c, the area of W will remai the same herefore multiplyig ay costat to the positive defiite matrix P has o effect to the area W of the ellipse so log as L is multiplied by the same costat It ca also be exteded to the -dimesioal case, where > Hece we scale Pˆ so that the first etry is, the maximal cotrollable set foud by our approach
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