Output Feedback Stabilization of Nonlinear Systems with Delayed Output

Transcription

1 5 American Control Conference June Portlan OR USA FrC15 Output Feeback Stabilization of Nonlinear Systems with Delaye Output Xianfu Zhang Zhaolin Cheng an Xing-Ping Wang Abstract: It is propose a novel an simple esign scheme of output feeback controllers for a class of nonlinear systems with elaye output The esigne controllers have a very simple structure an o not involve any saturations or recursive computations Moreover the nonlinear systems consiere here are more general than feeforwar systems an they coul be viewe as generalize feeforwar systems By constructing appropriate yapunov- Krasovskii functional KF an solving linear matrix inequalitiesmis the elay-epenent controller making the close-loop system globally asymptotically stable GAS is explicitly constructe A simulation example is given to emonstrate the effectiveness of the propose esign proceure Key wors: Feeforwar nonlinear systems; time-elay systems; output feeback; yapunov-krasovskii functionals; linear matrix inequalities I INTRODUCTION In many engineering applications a process to be controlle or simply monitore is locate far from the computing unit an the measure ate are transmitte through a low-rate communication systemegin aerospace application In the above cases the measure outputs are available for computations after a non negligible time elay In some applicationseg in biochemical reactors the measurement process intrinsically provies an out-of-ate output In [1] the isturbance ecoupling problem for nonlinear systems with elaye output is consiere In [] it is presente a esign approach for the construction of a state observer for nonlinear systems when the output measurements are available for computations after a non negligible time elay On the other han feeforwar systems are nonlinear systems escribe by equations having a specific triangular structure The problem of the asymptotic stabilization by state feeback of these triangular equations in the absence of elay has been stuie by many researchers see [4][5][13] uring the last ecae mainly for two reasons: First the problem of the global asymptotic stabilization of This work was supporte by Project 973 of China uner Grant G19983 the NSF of China uner Grant an the NSF of Shanong Province uner Grant Y4A1 Xianfu Zhang is with Dept of Math an physics Shanong Institute of Architecture an Engineering Jinan 514 PR China zhangxianfu@saieucn Zhaolin Cheng is with School of Mathematics an System Science Shanong University Jinan 51 PR China Xing-Ping Wang is with Institute of Applie Mathematics Naval Aeronautical Engineering Institute Yantai 641 PR China wwwxpnm@sohucom /5/$5 5 AACC 4886 these systems is challenging from a theoretical point of view Note in particular that they are in general not feeback linearizable an cant not be stabilize by applying the backstepping metho Secon a number of physical evices such as the system ball an beam with a friction term see [1] the TORA system see [1] are escribe after a preliminary change of feeback by equations with the feeforwar structure In recent paper [11] the problem of globally uniformly asymptotically an locally exponentially stabilizing a family of nonlinear feeforwar systems when there is a elay in the input is solve Up to now few paper has consiere the problem of output feeback controllers esign for nonlinear feeforwar systems with elaye output Motivate by [6] an [7] in this paper by constructing the appropriate yapunov- Krasovskii functionals KF we investigate the output feeback controllers esign for a class of nonlinear systems with elaye output Inee the nonlinear systems consiere in this paper are more general than feeforwar systems upper-triangular form Our metho is simpler because the most work of our esign proceure can be complete by MATAB toolbox Our stabilizing controller is elay-epenent an hence the propose controller is less conservative II PREIMINARIES We consier the nonlinear systems of imension n escribe by the equations: ẋ 1 t = x t+φ 1 t xtut ẋ t = x 3 t+φ t xtut ẋ n t = x n 1 t+φ n t xtut ẋ n 1 t = x n t+φ n 1 t xtut ẋ n t = ut yt = x 1 t 1 x = [x 1 x n ] T R n is the state u R is the input y R is the output constant is the elay The argument of the functions will be omitte or simplifie whenever no confusion can arise from the context For example we may enote x i t by x i The mappings φ i : R R n R R i =1 n 1 are continuous an satisfy the following growth conition: Assumption 1 For k = 1 n 1 l = 1 N k j = 1 n +1 there exist constants c kl an α jkl satisfying n+1 j=1 α jkl =1 such

2 that φ k t x u N k l=1 c kl x 1 α 1kl x n α nkl u α n+1kl with c kl = for n+1 i=1 i α ikl <k+ N k k = 1 n 1 are positive integers Remark 1: When some α ikl = it can be viewe that there is no x i in the term c kl x 1 α 1kl x n α nkl u α n+1kl When k = n 1 we will get α 1n 1l = = α nn 1l = α n+1n 1l =1 an consequently we have N n 1 φ n 1 t x u u l=1 c n 1l Comparing the conition with the following linear growth conition: φ i t xtut c x i+ + + x n + ut i =1 n φ n 1 t xtut c ut c is a constant It is easy to see that the system 1 satisfying conition which is inee a feeforwar nonlinear system obviously satisfies the conition Now we give an example to illustrate that the nonlinear systems satisfying conition are more general than feeforwar upper-triangular form systems an hence we can view the systems satisfying conition as generalize feeforwar systems See system ẋ 1 = x x 3 sin x 1 ẋ = x 3 ln1 + ln1 + x 1 u 3 ẋ 3 = x e u t ẋ 4 = x 5 + ln1 + u ẋ 5 = u yt = x 1 t 1 3 Since there is a x 1 in the right sie of the secon equation of the system 3 the system 3 is not a feeforwar systems Using the inequality ln1 + s s for s it is easy to see that the system 3 satisfies the conition Up to now to my best knowlege there is no efficient metho to esign the output feeback controller for this system In this paper for the vector ξ =ξ 1 ξ ξ n or matrix M =[m ij ] m n we enote vector ξ 1 ξ ξ n or matrix[ m ij ] m n by ξ or M The matrix M = [m ij ] m n is sai to be a nonnegative matrix if m ij for 1 i m 1 j n The property about the nonnegative matrix can be foun in [9] we let enote the Eucliean norm for vector or the inuce Eucliean norm for matrix For any real matrix A A T enotes the transpose I is use to represent an ientity matrix of appropriate imension The following lemma is useful in the proof of our main result emma 1 see [14] For any y i i =1 k an r i > i =1 k satisfying k i=1 r i =1 the 4887 following inequality hols: y r 1 1 yr yr k k r 1y 1 + r y + + r k y k III OUTPUT FEEDBACK CONTROER In practice when some state variables are unavailable the output variables are neee to control the system Uner the Assumption 1 we will esign the linear output ynamic compensator żt = Mzt+ Mzt +Nyt M R n n M R n n N R n 1 ut = Fzt F R 1 n 4 such that the close-loop system 1 an 4 is GAS at the equilibrium x z = Theorem 1 Uner the Assumption 1 for any given elay there exists a linear output feeback controller of the form 4 which is epenent on the elay such that the close-loop system 1 an 4 is globally asymptotically stable GAS Proof : We begin with by esigning the following linear observer of the system 1 ż 1 = z + a 1 yt z 1 t ż = z 3 + a yt z 1 t ż n 1 = z n + a n 1 yt z n 1 1 t ż n = ut+ a n yt z n 1 t 5 >1 is a gain parameter to be etermine later an a j > j =1 n are coefficients of the Hurwitz polynomial ps =s n + a 1 s n a n 1 s + a n Now we efine ε i = i 1 x i z i x i = i 1 x i z i = i 1 z i It is easy to fin that i =1 n ε 1 t = ε 1 t t t ε 1ss = ε 1 t t 1 t ε s+φ 1 s a 1 ε 1 s s From 1 an 5 a simple calculation gives ε = 1 Aε +Φ + 1 A t 1 1 t ε s+φ 1 s a1 ε 1s s 6 z = 1 J z + n 1 Gut+ 1 A 1ε 1 t 7 ε =[ε 1 ε ε n ] T z =[ z 1 z z n ] T G =[ 1] T

3 A = a 1 1 a 1 A 1 = a n 1 1 a n 1 1 J = Φ= 1 a 1 a a n 1 a n φ 1 φ n φ n 1 et b j > j = 1 n are coefficients of the Hurwitz polynomial qs =s n + b n s n b s + b 1 we can get k 1 φ k k 1 N k l=1 c kl x 1 α 1kl x α kl x n α nkl ut α n+1kl = k 1 N k l=1 c kl x 1 α 1kl x α kl x n ut α n+1kl N k l=1 c kl x 1 α 1kl x n α nkl b 1 b b n zt α n+1kl n 1 α nkl 1 t k1 x 1 + t k x + + t kn x n +t kn+1 b 1 b b n zt k =1 n n φ n 1 n Nn 1 l=1 c n 1l ut 1 t n 1n+1 b 1 b b n zt Next we will choose an appreciate >1 such that the trivial solution of the close-loop system 67 an u = 1 b 1 z n 1 + b z + + b n z n 8 is asymptotically stable From the efinitions of z i i =1 n an 8 we can get u = 1 n b1 z 1 + b z + + b n n 1 z n 9 From 7 an 8 we have B = z = 1 B z + 1 A 1ε 1 t b 1 b b 3 b n Since A is a stable matrix there exists a positive efinite matrix P such that PA+ A T P = I Choosing V 1 = ε T Pεwehave V 1 6 = [ 1 Aε +Φ + A1 t 1 t ε s+φ 1 s a1 ε 1s s] T Pε +ε T P [ 1 Aε +Φ + 1 A t 1 1 t ε s+φ 1 s a1 ε 1s s] = 1 ε +ε T P Φ+ εt PA 1 t 1 t ε s+φ 1 s a1 ε 1s s 11 Noticing Assumption 1 ie c kl = for n+1 i=1 i α ikl < k + the efinitions of ε i an z i >1 an emma t kj k =1 n 1 j=1 n+1 are nonnegative real numbers epenent on c kl α jkl an N k So we have ε T P Φ ε T P T x + T 1 zt ε T P T x + ε T P T 1 zt 1 εt PA 1 t 1 t ε s+φ 1 s a1 ε 1s s κ 1 ε T PA 1 A T 1 Pε + κ 1 t t ε ss t t ε ss + κ ε T PA 1 A T 1 Pε + κ t t φ 1ss t t φ 1ss + a 1 κ 3 ε T PA 1 A T 1 Pε + a 1κ 3 t t ε 1s s t t ε 1s s κ 1 ε T PA 1 A T 1 Pε+ κ 1 t t ε ss + κ ε T PA 1 A T t 1 Pε+ κ t φ 1ss + a 1 κ 3 ε T PA 1 A T 1 Pε+ κ 3a 1 t t ε 1s s 13 t 11 t 1 t 1n t 1 t t n T = t n 1 t n t n n T 1 = t 1n+1 t n+1 t n n+1 t n 1n+1 b 1 b n κ 1 κ an κ 3 are positive numbers to be etermine later

4 From 11 1 an 13 we can get V ε + ε T P T x + ε T P T 1 zt + Choosing KF + κ 1 t t +κ + κ 3a 1 t ε ss + t φ 1ss + a 1 t t ε 1s s Ṽ 1 = ε T Pε+ κ1 t +κ θ+t φ 1ssθ + κ3 a1 κ 1 ε T PA 1 A T 1 Pε κ ε T PA 1 A T 1 Pε κ 3 ε T PA 1 A T 1 Pε t θ+t ε ssθ an noticing that >1 we have t θ+t ε 1ssθ Ṽ ε + ε T P T x + ε T P T 1 zt + κ 1 ε T PA 1 A T 1 Pε + κ1 ε t+ κ ε T PA 1 A T 1 Pε+ κ φ 1t + a 1 κ 3 ε T PA 1 A T 1 Pε+ κ 3 a 1 ε 1t 1 ε + ε T P T x + ε T P T 1 zt + κ 1 + κ + a1 κ 3 ε T PA 1 A T 1 Pε + κ 1 ε t+ κ 3 a 1 ε 1t + κ x T T T T x + z T T3 T T 3 z + x T T T T 3 z T =t 11 t 1 t 1n T 3 = t 1n+1 b 1 b b n Because B is a stable matrix there exists a Q> such that QB + B T Q = I Choosing V = z T Q z wehave V 1 = 1 B z + 1 A 1ε 1 t T Q z + z T Q 1 B z + 1 A 1ε 1 t 1 z + zt QA 1 ε 1 t 1 z + λ κ 4 z T QA 1 A T 1 Q z + κ 4 λ ε 1t κ 4 an λ are positive numbers to be etermine later Choosing an letting V = Ṽ1 + λ δ 1 + δ < V + κ t 4 ε λ 1ss t δ 3 + δ 4 <λ 4889 δ i i =1 3 4 are positive numbers to be etermine later we can get V 61 1 ε + ε T P T ε + z + ε T P T 1 zt + κ 1 + κ + a1 κ 3 ε T PA 1 A T 1 Pε + κ1 ε κ3 a1 t+ ε 1t + κ ε + z T T T T ε + z + z T T3 T T 3 z + ε + z T T T T 3 z λ z + λ κ 4 z T QA 1 A T 1 Q z + κ 4 ε 1 1 ε T z T ε T z T Ωε T z T ε T z T T Ω= W 11 W W 33 W 34 W34 T W 44 W 11 = 1 δ 1 I + κ κ + a 1 κ 3 PA 1 A T 1 P +κ 1 J 1 + κ 3 a 1 J + κ 4 J W = δ 3 I + λ QA 1 A T 1 Q κ 4 W 33 = δ I + κ T T T + P T + T T P W 34 = P T 1 + P T + κ T T T + T 3 W 44 = δ 4 I + κ T + T 3 T T + T 3 J 1 = 1 T 1 J =1 T 1 If we choose > 1 satisfying Ω < we shall get V 61 < which inicate that 61 is asymptotically stable at ε =an z =see[1] The trivial solution of the close-loop system 6 7 an 8 is also asymptotically stable Consequently the close-loop system 1 5 an 9 is also asymptotically stable at x =an z = Therefore we can conclue that 59 is the linear output ynamic compensator of the system 1 Since the inequality Ω < coul not be solve by MATAB we give the following consieration From Schur complements we know that fining >1 as small as possible satisfying Ω < is equivalent to solving the following optimization problem with linear matrix inequalities MIs constraints: min κ 1 κ κ 3 κ 4 δ 1 δ δ 3 δ 4 λ st > 1 δ 1 + δ < δ 3 + δ 4 <λ an Ψ 11 Ψ 13 Ψ < 14 Ψ T 13 Ψ 33 Ψ 11 = δ1 I + κ 1 J 1 + κ 3 a 1 J + κ 4 J δ 3 I

5 W33 W Ψ = 34 W34 T W 44 P A1 P A1 a1 PA Ψ 13 = 1 λqa 1 Ψ 33 = iag[ κ 1 κ κ 3 κ 4 ] MI 14 can be easily solve by MI toolbox in MAT- AB Our MI 14 always has a solution an our esign metho is always efficient for the system 1 satisfying the conition an hence our esign metho propose here is inee a constructive metho However the MIs in [3] [8] maybe have no solution an when the MIs have no solution the esign metho propose there will not work Hence our esign metho is better The proof of the Theorem is complete Remark : For any given elay > we can always fin a constant satisfying 14 The purpose of fining a small >1 satisfying 14 is that we woul like get a bigger control gain There may be many sets a j b j j = 1 n satisfying our esign requirements but we always choose those such that >1 is as small as possible Consequently the gain of the controller is as big as possible an the time of stabilizing system will be as short as possible Remark 3: The control gain we get is epenent on the elay From a theoretical point of view for any elay > we can always fin a stabilizing controller But when the elay > is bigger the control gain of system will be much less The time of stabilizing system will be very long an hence this will lower efficacy of the controller in practical So when we esign a stabilizing controller we shoul ecrease the elay as far as possible Remark 4: Our controller has goo robust performance that is a stabilizing controller esigne for the system 1 with elay > will be also efficient for the system 1 with elay satisfying < IV EXAMPE Example Consier the following nonlinear system: ẋ 1 = x ln1 + x u 1 ẋ = x 3 1+1e u t ẋ 3 = u yt = x 1 t 15 For the system 15 satisfying Assumption 1 it is easy to see that t 1 = 1 16 t 14 = 1 16 t 11 = t 13 = t 1 = t = t 3 = t 4 = 1 1 We can construct the output feeback controller for the system 15 with =1by the following proceures Choose a 1 =1 a =8 a 3 =1 ; b 1 =8 b =1 b 3 =65 Solving the yapunov equations PA+ A T P = I an QB + B T Q = I 489 we can get P = Q = > > By solving 14 we can get =68799 So the output feeback controller of the system 15 is ut = 1 3 b1 z 1 + b z + b 3 z 3 16 z 1 z an z 3 are the states of the following system ż 1 = z + a 1 yt z 1 t ż = z 3 + a yt z 1 t ż 3 = 1 3 b1 z 1 + b z + b 3 z 3 + a3 yt z 3 1 t 17 Fig1 Fig an Fig3 show the state response of the close-loop system an 17 with the initial conition [x 1 t x t x 3 t] = [ ] [z 1 t z t z 3 t] = [ ] for t [ 1 ] time [s] Fig1: Trajectories of x 1 anz 1 x1 z1

6 5 x z time [s] Fig: Trajectories of x an z 3 x3 z3 1 1 [4] Ye Xuong Universal stabilization of feeforwar nonlinear systems Automatica 3 Vol39: [5] Chunjiang Qian an Wei in Using small feeback to stabilize a wier class of feeforwar systems 1999 Proceeings of the IFAC Worl Congress VolE Beijing China [6] Chunjiang Qian an Wei in Output feeback control of a class of nonlinear systems: a nonseparation principle paraigm IEEE Transaction Automatic Control 4711: [7] Praly Asymptotic stabilization via output feeback for lower triangular systems with output epenent incremental rate IEEE Transaction Automatic Control 3 486: [8] Shengyuan Xu an Paul Van Dooren Robust H filtering for a class of nonlinear systems with state elay an parameter uncertainty IntJControl 751: [9] Roger AHorn an Charles RJohnson Matrix analysis Cambrige University press 1985 [1] Jack KHale an Sjoer MVeruyn unel Introuction to functional ifferential equations New York: Springer-Verlag1993 [11] Freeric Mazenc Sabine Monie an Rogelio FranciscoGlobal Asymptotic Stabilization of Feeforwar Systems with Delay in the Input Proc 4st IEEE Conf on Decision an Control MauiHawaii USA [1] RSepulchre MJanković PVKokotovic:Constructive Nonlinear ControlSpringer-Verlag onon 1997 [13] Xoolphe Sepulchre Mrajan Jankovic an Petar VKokotovic Integrator forwaring:a new recursive nonlinear robust esign Automatica 1997 Vol335: [14] Ewin beckenbach an Richar bellman An introuction to inequalities Yale University 1961 [15] Xianfu Zhang Qingrong iu an Zhaolin Cheng Output feeback control of a class of time-elay nonlinear systems Proceeings of the 5 th Worl Congress on Inteligent Control an Automation June Hangzhou PRChina time [s] Fig3: Trajectories of x 3 an z 3 Remark 5: From this example it is easy to fin that the time of stabilizing the system 15 is much longer this is mainly because the system 15 has higher orer an nonlinear terms Inee the gains of stabilizing controller for the strict feeforwar system are in general lower see [5] [11] [13] an the time of stabilizing such system will be longer On the other han the gains of stabilizing controller for the strict feeback lower-triangular form system are in general very high see[6] [7] [15] V CONCUSION In this paper we have stuie the problem of global stabilization by output feeback for a class of nonlinear system with elaye output The metho use here is maybe feasible for the more general nonlinear elay systems REFERENCES [1] CHmoog RCastro-inares Mvelasco-Villa an AMarquez- Martiez The isturbance ecoupling problem for time-elay nonlinear systems IEEE Transaction Automatic Control 45: [] AGermani CManes an PPepez A new approach to state observation of nonlinear systems with elaye output IEEE Transaction Automatic Control 471: [3] Xii an Carlos E e Souza Delay-epenent robust stability an systems: a linear matrix inequality approachieee Transaction Automatic Control :