Global Output Feedback Stabilization of a Class of Upper-Triangular Nonlinear Systems

Transcription

1 9 American Control Conference Hyatt Regency Riverfront St Louis MO USA June - 9 FrA4 Global Output Feedbac Stabilization of a Class of Upper-Triangular Nonlinear Systems Cunjiang Qian Abstract Tis paper considers te global output feedbac stabilization of a class of upper-triangular nonlinear systems wit a single input and multiple output It is sown tat under a structural condition te output feedbac stabilization problem can be solved for tis class of systems by coupling wit a finite-time convergent observer and a saturated omogeneous stabilizer More general results are also acieved for some complex nonlinear systems witout te omogeneous growt condition I INTRODUCTION Tis paper considers a class of nonlinear systems wit a single input and multiple outputs (SIMO described by te following equations: were ẋ i = x i + φ i (x i+ u ẋ i = x i3 + φ i (x i+ u ẋ ini = x i+ + φ ini (x i+ u y =[x x x m ] T i = m u = x m+ ( x i := [ x i x ini x m x mnm ] i = m and x m+ := Null are system states u IR is te control input and y IR m is te measurable output Te functions φ i j ( j = n i i = m are C nonlinear functions wit φ i j (=and lim φ mn m (u/u = ( u Te objective of tis paper is to design an output feedbac controller of te form η = F(ηy u = u(ηy (3 suc tat te closed-loop system (-(3 is globally asymptotically stable A notable feature of te feedforward system ( is tat te system as only one control input but multiple outputs It is actually not an unrealistic assumption since most of iger-dimensional mecanical systems in te real world ave multiple displacement variables (angular and/or linear wic are usually easily measurable Te autor is wit te Department of Electrical and Computer Engineering University of Texas at San Antonio San Antonio Texas 7849 USA cunjiangqian@utsaedu Tis material is based upon wor supported by te US National Science Foundation under Grant No ECCS-395 Te stabilization of feedforward systems is an interesting problem and as attracted a great deal of attention In te literature a number of state feedbac stabilizers ave been developed using nested-saturation [6] [8] or forwarding [] [5] However wen only partial states are measurable te global output feedbac stabilization of feedforward systems is more callenging and as received very little attention compared to lower-triangular systems In fact as illustrated by a counter-example in [4] te structure of upper-triangular systems leads to an intrinsic obstacle wic maes it impossible to acieve even semi-global output feedbac stabilization of te general feedforward systems In te area of nonlinear control tere are a small number of existing results in dealing wit te problem of output feedbac stabilization for feedforward systems One existing approac employed bounded control to deal wit feedforward systems wit marginally stable free dynamics [7] [] For marginally unstable system wit upper-triangular nonlinearities global output feedbac stabilizers were constructed under te linear growt condition [3] including te Lipscitz condition [] From bot practical or teoretical viewpoints it is too restrictive to require te nonlinearities to satisfy te linear growt condition By developing a omogeneous domination approac te linear growt condition was lifted in [4] were global output feedbac stabilization was acieved for more general nonlinearities under a omogeneous growt condition Later te paper [4] extended te result [4] to a class of upper-triangular nonlinear systems wit uncontrollable linearization Note tat most of tese existing results required te nonlinearities to satisfy te omogeneous growt condition (linear growt is a special case of omogeneous growt Tis is not surprising since even for low-triangular systems wose output feedbac stabilization problem as been well-studied te omogeneous growt condition is also widely assumed even for te most general results Under te omogeneous growt condition te omogeneous domination approac [3] as enabled us to deal wit nonlinear systems wit iger-order nonlinearities eiter in lower or upper triangular forms But on te oter and te omogeneous condition is still a quite restrictive condition wic limits us to solve te output feedbac stabilization problem of some of practical systems wit non-omogeneous nonlinearities In tis paper we sow tat if we ave more tan one measurable outputs we can significantly enlarge te class of nonlinear feedforward systems wose output feedbac stabilization problem is solvable From practical point of view most of iger-dimensional mecanical systems (num /9/$5 9 AACC 3995

2 ber of states 4 usually ave more tan one displacement variables wic can be easily measured Wit te elp of te recently developed finite-time convergent observer [] [5] [6] we will develop a cascaded observer consisting of several sub-observers wic provides us te real states of te system in a finite time Tis new observer togeter wit a new saturated omogeneous state feedbac stabilizer will globally stabilize te feedforward system ( II OUTPUT FEEDBACK STABILIZER DESIGN In tis section a finite-time convergent observer will be first constructed to obtain te real states of te system in a finite time Ten we sow tat te separation principle olds for feedforward system ( wit various state feedbac stabilizers wit bounded magnitude A Observer Design For i = m design te observer ˆx i = ˆx i + φ i (ˆx i+ u+a i [x i ˆx i ] r ˆx i = ˆx i3 + φ i (ˆx i+ u+a i [x i ˆx i ] r 3 ˆx ini = ˆx i+ + φ ini (ˆx i+ u+a ini [x i ˆx i ] r n i + (4 were ˆx m+ = u ˆx i = [ ] ˆx i ˆx ini ˆx m ˆx mnm ˆx m+ = Null r = r j+ = r j + ˆτ j = n i wit ˆτ = p/q ( /n i for an odd integer q and an even integer p First we consider te last subsystem of (4 were i = m ˆx m = ˆx m + φ m (u+a m [x m ˆx m ] r ˆx m = ˆx m3 + φ m (u+a m [x m ˆx m ] r 3 ˆx mnm = u + φ mnm (u+a mnm [x m ˆx m ] r nm+ (5 Defining te errors as e m j = x m j ˆx m j j = n m we ave te following error dynamics based on (-(5 ė m = e m a m [e m ] r ė m = e m3 a m [e m ] r 3 ė mnm = a mnm [e mnm ] r nm+ (6 By Lemma A4 tere exist constants a m j j = n m suc tat te above system is globally finite-time stable Rougly speaing te solutions of a system are said to be finite-time stable if te solutions are stable and will reac te origin in a finite time and stay tere afterwards A more precise definition can be found in te references [] [5] Hence after a finite time T As a consequence ˆx m j = x m j j = n m φ m j (ˆx m u=φ m j (x m u j = n m Hence te dynamics of error e m j = x m j ˆx m j j = n m for (m t sub-system become ė m = e m a m [e m ] r ė m = e m 3 a m [e m ] r 3 ė m nm = a m nm [e m nm ] r n m + t T (7 Similarly we now tat for appropriate constants a m j s tere is a finite time T suc tat ˆx m j = x m j j = n m t T + T Following te same line we can conclude tat after a finite time T + T + + T m := T ˆx i j = x i j i j wic means te estimates obtained from observer (4 ave become te real states of te system ( B Output Feedbac Stabilizers Te finite-time convergent observer presented in te previous subsection can be easily integrated wit several state feedbac controllers First we employ te small state feedbac controllers designed for feedforward systems wit iger-order nonlinearities Teorem : Assume tat for j = n i i = m te following olds φ i j (x i+ u ( x i+ + u ρ i j (x i+ u (8 for a smoot function ρ i j (x i+ u Ten tere is an output feedbac controller under wic te states of system ( are bounded and tend to te origin ultimately Proof First under te condition (8 we can design a small controller u(x x = x using eiter te saturation design [6] or te forwarding design [9] Since te states x are not available for feedbac in te very beginning we will substitute te states x by teir estimates ˆx = ˆx Note tat one common property of bot saturation and forwarding metods is tat te controller is bounded by an arbitrarily small positive number Hence even before te convergence time T all te states of te system will stay bounded After te convergence time T te controller u(ˆx=u(x wic will ultimately render te system globally asymptotically stable Note tat Teorem still requires te iger-order condition (8 However tis requirement can be lifted and more general results can be acieved using a saturated 3996

3 omogeneous controller In wat follows we sow tat global output feedbac stabilization can be acieved witout te iger-order condition (8 In order to design te state feedbac stabilizer we consider te following system for neatness of notation ż = z + f (z z 3 z n u ż = z 3 + f (z 3 z n u ż n = u + f n (u (9 Assumption : Tere is a constant τ suc tat lim f i(ε i+ z i+ ε n z n ε n+ u/ε i+ = ( ε were = i+ = i + τ i = n Lemma : Under Assumption tere are constants b i and saturation functions defined as { sign(s Mi s > M σ i (s= i for a constant M s s M i i suc tat te following controller n u = b n σ n (zn + b n σ n (z n n + + b σ (z + b σ (z ( renders system (9 globally asymptotically stable Proof of Lemma We first sow tat for system ż = z ż = z 3 ż n = z n ż n = u ( tere is a omogeneous controller n u = c n (zn + c n (z n n + + c (z + c z were = i+ = i + τ i = n For te simplicity we assume τ = q p wit q an even integer and p an odd integer We will employ an inductive argument to construct te controller Initial Step Coose V = n + τ z( n+τ/ Clearly te virtual controller z defined by z = z / n := (z n+/ / n+ β yields V (z nz n+/ + z ( n+τ/ [z z ] (3 Inductive Step Suppose at step tere are a C Lyapunov function V :IR IR wic is positive definite and proper and a set of virtual controllers z z defined by z = ξ = z n+/ z n+/ z = ξ / n+ β ξ = z n+/ z n+/ z = ξ / n+ β ξ = z n+/ z n+/ (4 wit constants β > β > suc tat V (n + l= l + ξ + n n+ (z z ξ (5 Obviously (5 reduces to te inequality (3 wen = We claim tat (5 also olds at step To prove te claim we consider te Lyapunov function V :IR IR defined by V (z z =V + z z [ s n+/ z n+/ ] + n n+ Te derivative of te Lyapunov function V along te system (9 is V V + ξ + n n+ (n + ξ l= z + + c ξ n n+ ξ l + ξ + n n+ (z z + ξ + n n+ z + + c ξ n n+ ξ (6 Next we estimate eac term in te rigt and side of (6 First it follows from Lemmas A and A tat ξ + n n+ (z z n c ξ + n+ ξ / n+ ds ξ + c ξ (7 for a constant c > Te last term in (6 namely c ξ n n+ ξ can be estimated as te follows First by te definition of te ξ l we ave c ξ n n+ ξ c ξ n n+ n+ n+ ( z z + + z z c ξ n n+ +τ n+ ( z + + z n+ +τ c ξ n n+ +τ n+ +τ n+ ( ξ n+ + + ξ n+ ( ξ + + ξ + ξ ĉ (8 were ĉ is a constant Note tat te last inequality follows from Lemma A Substituting te estimates (7 and (8 into (6 we ave V (n + + ξ + n n+ ( l= l + ξ + n n+ (z + z + ξ z + +(c + ĉ ξ +/ n+ Observe tat a virtual controller of te form z + = ξ +/ n+ β = ξ +/ n+ [n + + c + ĉ ] (9 renders V (n + ξ l= l + ξ + n n+ (z + z

4 Tis completes te inductive proof Te inductive argument sows tat (5 olds for = n+ wit a set of virtual controllers (4 Hence at te last step coosing u = ξ n+/ n+ n β n n = c n (zn + c n (z yields n n + + c (z + c z ( V n ( ξ + + ξ n < z ( were V n (z z n is a positive definite and proper Lyapunov function As a result ( ( is globally asymptotically stable Based on te omogeneous controller ( following te same line in te wor [7] we are able to coose appropriate saturation constants for functions σ i (s s and adjust controller coefficients b i s suc tat n u = b n σ n (zn + b n σ n (z n n + + b σ (z + b σ (z globally stabilizes te system (9 under Assumption Te procedure to coose te constants is very similar to te subtle procedure [7] tat was introduced for a class of uppertriangular systems wit uncontrollable linearization For te sae of space ere we will not repeat te lengty detailed procedure wic can be recovered from [7] Remar : It can be verified tat system ( satisfies Assumption automatically To sow tis we first denote i x i j = z i + j i = n l φ i j = f i + j(z i +n i + z n u l= Under te new notations togeter wit te fact tat m l= n l = n it is clear tat system ( is in te form of (9 Moreover due to te fact tat φ i j (x i+ u is C and φ i j ( = we now f i + j(z i +n i + z n u=(z i +n i + z n u f i + j( for a continuous function f i + j( As a result one as f i + j(ε i +n i + z i +n i + ε n z n ε n+u ε i + j ε i +n i + i + j ( z i +n i + + ε τ z i +n i ε n i + j z n + ε n+ i + j u f i + j( Note tat j n i Hence i +n i + i + j =(+n i jτ > wic implies f i + j(ε i +n i + z i +n lim i + ε n z n ε n+u ε ε = i + j Tis togeter wit ( implies tat te assumption ( olds for system ( Teorem : Tere is an output feedbac controller under wic te states of system ( are bounded and tend to te origin ultimately Proof: By Remar we now system ( satisfies Assumption automatically Consequently by Lemma we can construct a state feedbac controller u(x bounded by a small positive constant Similar to te proof of Teorem we also construct a finite-time convergent observer (4 and use te observer states ˆx instead of te unmeasurable states x in te small controller Wit te bounded controller u(ˆx all te states of te closed-loop system ( are bounded for any finite time Due to te finite-time convergence of te observer after a finite time T u(ˆx=u(x wic will eventually globally stabilize system ( according to Lemma C An Example In tis subsection computer simulation is conducted for te following feedforward system ẋ = x + x + x + x ẋ = x + sin(x ẋ = x ẋ = u ( were y =[x x ] T is te measurable output Te nonlinearities of system ( apparently are in te feedforward form However te nonlinearities are not iger-order as required in te most existing results Hence even te state feedbac stabilization problem is unsolved On te oter and by coosing τ = it can be verified tat Assumption olds for te system As a result we can design an output feedbac stabilizer for system ( as follows: ˆx = ˆx + x + ˆx + ˆx +[x ˆx ] r ˆx = ˆx + sin( ˆx +[x ˆx ] r ˆx = ˆx +[x ˆx ] r ˆx = u +[x ˆx ] r u = b 4 σ 4 ( ˆx 9/7 + b 3σ 3 (x 9/5 + b σ ( ˆx 9/3 + b σ (x 9/ Fig sows tat te trajectories of te states x and x converge to te zero Fig plots te trajectories of te unmeasurable states x and x compared to teir corresponding estimates ˆx and ˆx Note tat te estimated states converge to te real states in a finite time and te real stats are asymptotically stable III EXTENSIONS In tis section we sow tat te tecniques presented in te previous section can be extended to a class of more 3998

5 x x Fig Trajectories of x and x 5 5 x ˆx ˆx x 5 5 Fig Trajectories of unmeasurable states and teir estimates general nonlinear systems ẋ i = x i + φ i (x i+ yu ẋ i = x i3 + φ i (x i+ yu ẋ ini = x i+ + φ ini (x i+ yu i = m (3 were x i y =[x x m ] T andu = x m+ are defined same as in ( Since te additional variable y in te nonlinear functions is measurable a finite-time observer can be still constructed for te system (3 As a matter of fact we can design te following observer: ˆx i = ˆx i + φ i (ˆx i+ yu+a i [x i ˆx i ] r ˆx i = ˆx i3 + φ i (ˆx i+ yu+a i [x i ˆx i ] r 3 ˆx ini = ˆx i+ + φ ini (ˆx i+ yu+a ini [x i ˆx i ] r n i + (4 for i = m Following te same argument in Section II it is straigtforward to see tat tere is a finite time T after wic all te states of observer (4 will become te real states of ( as long as te real states of ( are still bounded in [T ] In addition we assume tat tere is a state feedbac controller u(x satisfying te following conditions: (H u(x is a global asymptotic stabilizer for (3 (H For any finite time T te state feedbac controller u(ˆx using te estimate from te observer (4 eeps te states of system (3 away from infinity for t [T] Under tese two conditions we can see tat after obtaining te real values of te states of system (3 after a finite time T te controller u(ˆx=u(x wic renders te system ( globally asymptotically stable Remar 3: Clearly for te general feedforward system (3 te main tas to obtain an output feedbac controller is to find state feedbac controller satisfying conditions (H- (H In te literature tere are vast amount of results on ow to design small controllers For example we can use te forwarding tecnique in [9] to easily design a controller u(x for te following system in te form of (3 ẋ = x + x (x + x ẋ = x + x ẋ = x + ln( + x u ẋ = u (5 Te controller u(x constructed in [9] is arbitrarily small and ence we can verify tat (H olds for any finite time Hence te global output feedbac stabilization of (3 can be acieved by combining te finite-time observer and te existing state feedbac stabilizer In te ligt of te above design pilosopy global output feedbac stabilizer can be acieved for te following feedforward system tat is not exactly in te form of ( ẋ = x ẋ = sinx + x x ẋ = x ẋ = u (6 were y =[x x ] T is te measurable output Design ˆx = ˆx +[x ˆx ] r ˆx = sinx +[x ˆx ] r + x ˆx ˆx = ˆx +[x ˆx ] r ˆx = u +[x ˆx ] r (7 Now we design a state feedbac controller based on te one developed in [] We first adopt te virtual controller x = v(x x x Next we design te controller u(x x x x = arctan([x v(x x x ] 7/9 v(x x x x (8 wic guarantees tat x = v(x x x in a finite time Finally we replace te unmeasurable states x and x in u(x x x x wit teir estimate ˆx and ˆx 3999

6 Te simulation results are illustrated in Fig 3 and Fig 4 wic sow tat te all te states converge to zero asymptotically and te estimates converge to te real values in a finite time x x Fig 3 Trajectories of x and x 5 5 ˆx x ˆx x 5 5 Fig 4 Trajectories of unmeasurable states and teir estimates APPENDIX In te appendix we include several useful lemmas Lemma A: For x IRy IR p is a constant te following inequality olds: x + y p p x p + y p (A Lemma A: Let c d be positive constants Given any positive number γ > te following inequality olds: x c y d c c + d γ x c+d + d c + d γ d c y c+d (A Lemma A3: Let p be a real number and x y be realvalued functions Ten for a constant c > x p y p p x y (x p + y p (A3 c x y (x y p + y p (A4 Lemma A4: Consider ė = e a [e ] r ė = e 3 a [e ] r 3 ė N = a N [e ] r N+ (A5 were r = + ˆτ r i+ = r i + ˆτ i = N wit ˆτ = p/q ( /n] for an odd integer q and an even integer p Tere are constants a i suc tat te system (A5 is globally asymptotically stable Wen ˆτ < te solution of (A5 will tend to te origin in a finite time Proof: Te lemma was first proved for two-dimensional version of (A5 in [] [5] Later te paper [6] provided a proof for te general dimensional system (A5 REFERENCES [] C Barbu R Sepulcre W Lin and P Kootovic Global asymptotic stabilization fo te ball and beam system CDC 3: [] S P Bat and D S Bernstein Continuous finite-time stabilization of te translational and rotational double integrators IEEE Transactions on Automatic Control 43(5: [3] H-L Coi and L Jong-Tae Global exponential stabilization of a class of nonlinear systems by output feedbac IEEE Trans Automat Control 5(: [4] M T Frye R Trevino and C Qian Output feedbac stabilization of nonlinear feedforward systems using low gain omogeneous domination In Proc of 7 IEEE International Conference on Control and Automation pages [5] Y Hong J Huang and Y Xu On an output feedbac -time stabilization problem IEEE Trans Automat Control 46(:35 39 [6] J Li C Qian and M Frye A dual observer design for global output feedbac stabilization of nonlinear systems wit low-order and igorder nonlinearities In Proc of 7 IEEE Conference on Decision and Control pages New Orleans LA 7 [7] W Lin Bounded smoot state feedbac and a global separation principle for non-affine nonlinear systems Systems Control Lett 6(: [8] W Lin and X Li Syntesis of upper-triangular non-linear systems wit marginally unstable free dynamics using state-dependent saturation Internat J Control 7(: [9] F Mazenc Stabilization of feedforward systems approximated by a non-linear cain of integrators Systems Control Lett 3(4: [] F Mazenc and L Praly Adding integrations saturated controls and stabilization for feedforward systems IEEE Trans Automat Control 4(: [] F Mazenc and J Vivalda Global asymptotic output feedbac stabilization of feedforward systems Euro J of Control 8:59 53 [] J Polendo and C Scrader Output feedbac stabilization of nonlinear feedforward systems using arbitrarily bounded control In Proceedings of te 5 American Control Conference pages [3] C Qian A omogeneous domination approac for global output feedbac stabilization of a class of nonlinear systems In Proceedings of 5 American Control Conference pages June 5 [4] C Qian and J Li Global output feedbac stabilization of uppertriangular nonlinear systems using a omogeneous domination approac International Journal of Robust and Nonlinear Control 6(9: [5] R Sepulcre M Janovic and P V Kootovic Integrator forwarding: a new recursive nonlinear robust design Automatica J IFAC 33(5: [6] A R Teel Feedbac Stabilization: nonlinear solution to inerently nonlinear problems PD tesis University of California Bereley 99 [7] M Tzamtzi and J Tsinias Explicit formulas of feedbac stabilizers for a class of triangular systems wit uncontrollable linearization Systems Control Lett 38(: