Chiang Mai J. Sci. 2013; 40(3) 471

Chiang Mai J. Sci. 2013; 40(3) : 471-484http://it.science.cmu.ac.th/ejournal/Contributed Paper

Global Stability by Output Feedback Control for a Classof Nondifferentiable Uncertain Nonlinear SystemsTeerapap Kuptarat [a], Chulin Likasiri [a,c] and Radom Pongvuthithum*[b][a] Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand.[b] Department of Mechanical Engineering, Faculty of Engineering, Chiang Mai University, Chiang Mai 50200,

Thailand.[c] Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand.*Author for correspondence; e-mail: radom@chiangmai.ac.th

Received: 21 June 2012Accepted: 18 October 2012

ABSTRACTIn this paper, we study the problem of global stabilization by output feedback

for a class of nondifferentiable uncertain nonlinear systems of a planar system whoseJacobian linearization is uncontrollable and unobservable, and might not exist. Hence,the stabilization problem cannot be solved by any linear feedback control approacheven locally. A truly nonlinear control design must be used. To solve this difficultproblem, we propose a Co non-Lipschitz control law and an observer based on theconcept of homogeneity and domination design approaches. Our control design doesnot base the separation principle. Both control law and observer are simultaneouslyconstructed in a step-by-step design manner.

Keywords: homogeneity, non-Lipschitz control, global stabilization, output feedback

1. INTRODUCTIONIn this paper, we consider a single-

input-single-output (SISO) planar systemdescribed by equations of the form

(1.1)

where x = (x1, x2)∈ 2, u ∈ and y ∈

are the system state, input and outputrespectively, p is a positive odd integer. Fori = 1,2, φ

i are uncertain nonlinear

continuous functions of (x, t) and vanish

at x = 0, i.e. φi (0,t) = 0. The problem of

output feedback stabilization for (1.1) is tofind under which sufficient conditions acontinuous output feedback control law

z• = η(z, y), z ∈ u = u(z, y) (1.2)

rendering the closed-loop system (1.1)-(1.2)globally asymptotically stable (GAS). Thechallenge of global stabilization by outputfeedback of nonlinear systems is that wecannot use the separation principle. The

472 Chiang Mai J. Sci. 2013; 40(3)

planar system (1.1) is a subclass of genuinelynonlinear systems,

(1.3)

that has been widely studied, [1]-[10] . Whenp=1 and all φ

i(⋅ )′s have triangular

structures, the system (1.3) becomes well-known lower-triangular and upper-triangular systems. When p>1, linearizedsystem of (1.3) can be uncontrollable andunobservable. For example, we consider aplanar system

(1.4)

(1.4) is also a special case of (1.1). Whenp ≥ 3, the subsystem x

1 of the linearized

system of (1.4) is completely decoupledfrom the subsystem x

2. Hence, we cannot

control x1 nor observe x

2 from x

1. In other

words, the linearized system isuncontrollable and unobservable. Thisrenders all linear control designsinapplicable. More sophisticated controlapproaches are needed.

For over the years, researchers havestudied stabilization problem of thesystems (1.3) under various growthconditions of φi(⋅) and obtained someinteresting results. The existing results forsystem (1.3) can be classified into two cases.In the first case, when p = 1, the globalstabilization by output feedback have beenstudied in [1]-[3] under different growthconditions in φi(⋅)′s. [1] showed how toconstruct output feedback for (1.3) whenall φi(⋅)′s are bounded by linear growthcondition of states. The system (1.4)satisfies the growth condition in [1] when

q = 1. Using the concept of homogeneity,the paper [2] proved that the globalstabilization problem by output feedbackof (1.3) is solvable under the assumptionthat the uncertain nonlinear functions arecontinuously differentiable and boundedby homogeneous function of state. In thecase of (1.4), this means q ≥ 1.

In the case when p ≥ 1, as mentionedearlier the stabilization is much moredifficult than the first case since thesystem (1.3) might be unobservable anduncontrollable. Using homogeneity, [6,7]and [9] provided constructive proofs ofoutput feedback control laws under variousgrowth conditions. For a planar system,[6] achieves global stabilization when thepower of the states in the bounds of φi(⋅)are equal to p and the bounding constantcan depend on x1. Next, [7] extended theresult of [6] to n dimension but thebounding parameter was only constant.The results of [6] and [7] only consideredthe case when q = p. The paper [9] usedconditions similar to [2] which greatlyenlarged the class of φi(⋅).

All of the above results only considerthe case when φi(⋅)′s are differentiable, i.e.q ≥ 1. [4,5,8], and [10] show that it ispossible to design an output feedback fornon-differentiable system of the form (1.3).[4] showed how to construct a continuouscontrol law when p = 1 and the growthcondition is bounded by lower orderfunctions, i.e. q ≤ 1. [8] showed aconstruction of continuous control lawfor nondifferentiable systems when p ≥ 1.The growth conditions in [8] are morerestrictive than those in [4] and φi(⋅) mustbe differentiable with respect to theunmeasurable state. The paper [10]combines the growth condition in [8] and[9] together and call the growth conditionin [8] lower order andin [9] higher order.

Chiang Mai J. Sci. 2013; 40(3) 473

[5] uses the growth condition which are thecombination of those in [2] and [4]together.When p ≥ 1, there is no result toaddress the global stabilization problemof the planar system(1.1) with bothmeasurable or unmeasurable states beingnondifferentiable. To solve this problemwe propose similar growth conditions usedin [4] but extend the system to a moregeneral class when p ≥ 1.

This paper organized as follows. Insection 2, we design the dynamic outputfeedback control law of system (1.1). Insection 3, we provide anexample andsimulation results. The last section containsconclusions and proofs of propositions arein the Appendix.

2. MAIN RESULTS:In this section, we will present the

construction of the output feedbackcontrol law for the system (1.1). Thecontrol design consists of three steps. First,in Section 2.1, we assume that all of thestate are available and construct a Lyapunovfunction and a state feedback control lawfor (1.1). Then, in Section 2.2, based onthe state feedback control law, we showhow to choose an observer for a so-callednominal system of (1.1) which is the system(1.1) without the uncertain functions, φi(⋅).Finally, we solve problem of outputfeedback of (1.1) by introducing a changeof coordinates to scale (1.1) into anappropriated form and applying theoutput feedback control law of the nominalsystem to the scaled system.

2.1 Stabilization by Homogeneous StateFeedback

In this section, we present a constructivemethod for a state feedback stabilizer for(1.1) under the following assumption:

Assumption 2.1: There exist a

negative constant τ, satisfying − < τ ≤ 0and a positive constant such that

(2.1)

(2.2)

with(2.3)

Note that form (2.3), 1 ≥ m ≥ m + τ > 0.For simplicity, we assume τ = − withpositive even integers q and positive oddintegers d. As a result, m, m + τ ∈ oodwhere Rood = {x|x= , a, b are odd integers}.

Lemma 2.1: By Assumption 2.1, thereexists a homogeneous state feedbackcontroller such that the nonlinear system(1.1) is globally asymptotically stable.

Proof. The proof itself is a two-stepprocess which relies on simultaneousconstructions of a C1 Lyapunov functionand a C 0 control law.

Step 1. We define

V1 = .The time derivative of V1 along thetrajectory of (1.1) is

(2.4)By Assumption 2.1,

Then, the virtual controller x2*p defined by

yields(2.5)

Step 2. We define the following changesof coordinates:

(2.6)

and the Lypunov function, V2 : 2 → ,

V2 (x1, x2) = V1 (x1) + W2(x1, x2)

474 Chiang Mai J. Sci. 2013; 40(3)

where W2 = (2.7)

which can be proven to be C1 using asimilar method as in [12]. The derivativeof V2 along the trajectories of (1.1) is

(2.8)

Next, we estimate the terms in the righthand side of (2.8). First, it follows frompm = 1 + τ ≤ 1 and Lemma A.1 in theAppendix that

(2.9)

and by Lemma A.2, it can be seen thatthere exists a constant C1 >0 such that,

(2.10)

Using Lemma A.1 and the equations (2.6),(2.2) can be rewritten as

(2.11)

with a constant C~

2 ≥ 0. From Lemma A.2and (2.11), we can show that

(2.12)

for a constant C2 >0.The third term in (2.8) can be estimatedwith the help of Proposition 2.1 whoseproof is included in the Appendix.

Proposition 2.1: There is a constantC3 >0 such that

(2.13)

Substituting the estimates, (2.10), (2.12) and(2.13) into (2.8), we arrive at

for a constant

Choosing an intermediate controller

yields

(2.14)

If the state x2 are available for feedback,the control law, u*, can be implemented andu can be set to equal to u*. Then, the lastterm of V

.2 will disappear and we can

conclude that V.2

Chiang Mai J. Sci. 2013; 40(3) 475

2.2 Stabilization of (1.1) by OutputFeedback

In this section, we show that underAssumption 2.1, the problem of globaloutput feedback stabilization for system(1.1) is solvable. We will first construct ahomogeneous output feedback controllerfor the nominal chain of power integrator,i.e. φ1 (⋅) = φ2 (⋅) = 0:

(2.15)

with p is positive odd integer number.Then, based on this output feedbackcontroller, we develop a scaled observerand controller to render the system (1.1)globally asymptotically stable under thegrowth condition (2.1)-(2.2).

2.2 Output Feedback Control of NominalNonlinear System

Theorem 2.1: Given a real number − < τ ≤ 0, there is a homogeneous outputfeedback controller of degree τ renderingthe nonlinear systems (2.15) globallyasymptotically stable.

Proof. The construction of the homo-geneous output feedback controller isaccomplished in 3 steps. First, by Lemma2.1, a homogeneous state feedback stabilizeris constructed. Then, a homogeneousobserver is designed, and lastly, we replacethe unmeasurable state with its estimate.The closed-loop system can then be provenglobally asymptotically stable by choosingan appropriate observer gain.

State Feedback Controller: For nonlinearsystems (2.15), Assumption 2.1 isautomatically satisfied since φ1(⋅),φ2(⋅) aretrivial. Hence, by Lemma 2.1, there is ahomogeneous (with respect to the weight(2.3)) state feedback controller that globally

stabilizes (2.15). Therefore, there exists aLyapunov function of the form

a homogeneous control law

with

(2.16)

and constants β1,β2 > 0 that renders

(2.17)

Homogeneous Observer Design: Next,similar to [9] and [12], a homogeneousobserver is constructed as follows

(2.18)

where z1 = ẑ 1 and l1 > 0 is the gains to bedetermined in a later step. Based on theestimated state ẑ 2, we design an outputfeedback controller

(2.19)

We choose the Lyapunov function for theobserver (2.18) as follows

where It can be verified that U2 is C

1. In addition,with a constant b, we have the followingrelationships

476 Chiang Mai J. Sci. 2013; 40(3)

Hence, the time derivative of U2 along thetrajectories of (2.15)-(2.18) is

From the definition of γ, we can rearrangethe terms in the above equation as follows:

(2.20)

Let . We estimate the lastterm in (2.20). By Lemma A.1, withconstant m̂

(2.21)

The first terms in (2.20) can be estimatedusing the following Proposition 2.2 whoseproofsare in the Appendix.

Proposition 2.2: For controller ν (ẑ ). Thereis a constant C4 ≥ 0 such that

(2.22)

With the help of the previous propositionand the estimates (2.21), the derivative of

becomes

(2.23)

Determination of Observer Gain l1: Tochoose the gain l1, we combine theLyapunov function of the nominal system(2.15) and the observer (2.18).

T = V2 + U2whose derivative is the combination of

(2.17) and (2.23). Due to the unmeasurablestate, the controller ν = ν (ẑ ) gives aredundant term in (2.17). To deal with thisterm, we use the following proposition.

Proposition 2.3: There is a constant C5>0such that

(2.24)

Combining (2.17), (2.23) and (2.24) togetheryields

(2.25)

Clearly, by choosing ,becomes

(2.26)

Note that from the construction of T, it iseasy to verify that is positive definite andproper with respect to

(2.27)

In addition, the right hand side of (2.26) isnegative definite with respect to Z. Therefore,the closed-loop system (2.15)-(2.16)-(2.18)is globally asymptotically stable. Denotingf3 = η⋅2, it is straightforward to verify thatthe closed-loop system (2.15)-(2.16)-(2.18)can be rewritten in the following form

(2.28)

andis homogeneous. By choosing thedilation weight

(2.29)

it can be shown that (2.28) is homogeneousof degree τ. In addition, T is homogeneousof degree 2−τ and the right hand side of(2.26) is homogeneous of degree .

Chiang Mai J. Sci. 2013; 40(3) 477

Remark 2.1: The right hand side of (2.26)is negative definite and homogenous ofdegree . From Lemma A.5, it can beshown that there is a constant C1 > 0 suchthat

where

2.2.2 Global Output FeedbackStabilization for System (1.1)

Utilizing the homogeneous controller andobserver established in the previoussections, we are ready to construct theoutput feedback for (1.1).

Theorem 2.2: Under Assumption 2.1, thesystem (1.1) can be globally stabilized byoutput feedback.

Proof: Under the new coordinates

(2.30)

with , the system (1.1) can berewritten as

(2.31)

with the scaling gain, L ≥ 1. Next, weconstruct an observer with the scalinggain .

(2.32)

where ẑ 1= z1 and l1 is the gains selected by(2.25) in Theorem 2.1.From (2.27) and(2.28), the closed-loop system (2.31)-(2.32)-(2.19) can be written

(2.33)

F(Z) in (2.33) is the exact same structure as(2.28). Hence, using the same Lyapunovfunction T(Z ) in preceding section, it canbe concluded from Remark 2.1 that

(2.34)

Using the change of coordinates (2.30) andthe fact that L≥1, we deduce fromAssumption 2.1 that for some constants ν1and ν2>0,

(2.35)

Recall that T is homogeneous of degree 2−τ.

Therefore, and are homogeneousof degree 1−τ and 2−τ−m respectively.|z1|

1+τ is homogeneous of degree 1+τ and

+ is homogeneous of degreem + τ. Then,

and

(2.36)

are homogeneous of degree . With (2.35)and (2.36) in mind, we can find a constantρi such that

(2.37)

Substituting (2.37) into (2.34) yields

(2.38)

Obviously, if L is large enough then theright hand side of the (2.38) is negativedefinite. Clearly, the closed-loop system of(1.1) is globally asymptotically stable.

478 Chiang Mai J. Sci. 2013; 40(3)

Note that when τ = 0, Assumption 2.1reduces to the bound described in [12] forplanar systems, where r1 = 1, r2 = 1/p.Thus, the method presented here can beused to globally asymptotically stabilizeany planar system studied in [12].

3. DISCUSSIONS AND EXAMPLESWe now provide simulation results to

discuss characteristics of our control law.Consider the planar system

(3.1)When 1 ≤ q < 4, the system (3.1) can beglobally asymptotically stabilized byoutput feedback under the growthconditions in [9], [10]. For the system (3.1)the methods in [9] and [10] cannot be usedsince they require that φ2 is differentiablewith respect to state x2. In this example,we choose q = .

(3.2)

Clearly, with τ = − , m = , Assumption

2.1 holds, that is ≤ + ,for some ĉ >0. Therefore, using Theorem

2.2, the system (3.2) can be globallyasymptotically stabilized by outputfeedback by the homogeneous dynamicoutput feedback control law,

(3.3)

where β1, β2, l1, and L are large enoughpositive constants. From Theorem 2.2, wecan estimate value of (β1, β2, l1,L) =(5,310,200,1). For simplicity, we fix L = 1since it can be included with the observergain l1. We vary the gains, l1, β1 and β2, toillustrate the effects of these gain to theclosed-loop trajectories. All simulationsare performed with the initial condition(x1(0), x2(0), η(0))=(1,−1,0).From Figures 1a, 2a and 3a, increasing thegain l1 reduces the oscillation in the statesx1 and x2 but increasesthe peak in η. Statetrajectories in Figures 1b, 2b and 3b showthat increasing the gain β2 reduces theconvergence times to zero of all statetrajectories without promoting too muchoscillation. Increasing the gain β1, shownin Figures 1c, 2c and 3c, also reduces theconvergence times of all state trajectoriesbut promotes oscillations especially in x2and η.

time(sec)Figure 1a. State trajectories x1 of the closed-loop system (3.2) -

(3.3) with (x1(0),x2(0),ηηηηη(0))=(1,-1,0) and (βββββ1,βββββ2)=(5,310)

time(sec)Figure 1b. State trajectories x1 of the closed-loop system (3.2) -

(3.3) with (x1(0),x2(0),ηηηηη(0))=(1,-1,0) and (βββββ1,|||||1)=(5,200)

Chiang Mai J. Sci. 2013; 40(3) 479

time(sec)Figure 1c. State trajectories x1 of the closed-loop system (3.2) -

(3.3) with (x1(0),x2(0),ηηηηη(0))=(1,-1,0) and (βββββ1,|||||1)=(310,200)

time(sec)Figure 2b. State trajectories x2 of the closed-loop system (4.2) -

(3.3) with (x1(0),x2(0),ηηηηη(0))=(1,-1,0) and (βββββ1,|||||1)=(5,200)

time(sec)Figure 2c. State trajectories x2 of the closed-loop system (4.2) -

(3.3) with (x1(0),x2(0),ηηηηη(0))=(1,-1,0) and (βββββ2,|||||1)=(310,200)

time(sec)Figure 3a. State trajectories ηηηηη of the closed-loop system (3.2) -

(3.3) with (x1(0),x2(0),ηηηηη(0))=(1,-1,0) and (βββββ1,βββββ2)=(5,310)

time(sec)Figure 3b. State trajectories ηηηηη of the closed-loop system (3.2) -

(3.3) with (x1(0),x2(0),ηηηηη(0))=(1,-1,0) and (βββββ1,βββββ1)=(5,200)

time(sec)Figure 2a. State trajectories x2 of the closed-loop system (3.2) -

(3.3) with (x1(0),x2(0),ηηηηη(0))=(1,-1,0) and (βββββ1,βββββ2)=(5,310)

480 Chiang Mai J. Sci. 2013; 40(3)

time(sec)Figure 3c. State trajectories ηηηηη of the closed-loop system (3.2) -

(3.3) with (x1(0),x2(0),ηηηηη(0))=(1,-1,0) and (βββββ2,βββββ1)=(310,200)

4. CONCLUSIONSWe introduced the new result of global

stabilization by output feedback problemfor a family of planar systems. This classof uncertain nonlinear systems is genuinelynonlinear.The system consists of a chainof power integrators and uncertainnonlinear functions, φi which might notbe differentiable. Therefore, the Jacobianlinearization of this class of systems isneither controllable nor observable ormight not exist. We present constructiveproofs in which global asymptoticstabilization can be achieved by ahomogeneous dynamic output feedbackcontrol law. An example and simulationresults are also provided to show thecharacteristics of our control design.

ACKNOWLEDGEMENTSThis research has received the full

financial support by the Centre ofExcellence in Mathematics, CHE, SiAyutthaya Rd., Bangkok 10400, Thailand.

APPENDIXA. Useful Inequalities

The next three lemmas whose proofscan be found in [9,10] were used in ourpaper.

Lemma A.1: For x, y ∈ , p ≥ 1 is aconstant, the following inequalities hold:

(A1)

(A2)If p ∈ odd, p ≥ 1 then

(A3)

Lemma A.2: Let c, d be positive constant.Given any positive number γ > 0, thefollowing inequality holds:

(A4)

Lemma A.3: Let p ∈ odd, p ≥ 1 and x, ybe real-valued function. Then, for aconstant c>0

(A5)

(A6)

Weighted Homogeneity: (refer to [2, 11,13, 14,15])

For fixed coordinates (x1,..., xn)T ∈ n and

real numbers ri > 0 for i = 1,..., n,

• the dilation Δε(x) is defined by Δε(x) =>0, with ri being

called as the weights of the coordinates(For simplicity of notation , we definedilation weight Δ=(r1,..., rn)).

• a function V ∈C ( n, ) is said to behomogeneous of degree τ if there is areal number τ ∈ such that x∈ n\{0}, ε>0, V(Δε(x)) = ετ V(x1,..., xn).

• a vector field f ∈ C ( n, n) is said to behomogeneous of degree τ if thereis a real number τ ∈ such that fori =1,..., n x ∈ n\{0}, ε>0, fi(Δε(x))=

fi(x1,..., xn).

Chiang Mai J. Sci. 2013; 40(3) 481

• a homogeneous p-norm is defined as

= , x ∈ n, for a

constant p ≥ 1. For the simplicity, in thispaper, we choose p = 2 and write for .

Lemma A.4: Given a dilation weight Δ =(r1,..., rn ), suppose V1(x) and V2(x) arehomogenous functions of degree τ1 and τ2,respectively. Then V1(x)V2(x) is alsohomogeneous with respect to the samedilation weight Δ.

Thus, the new homogeneous degree ofV1.V2 is τ1 + τ2.

Lemma A.5: Suppose V: n → is ahomogenous function of degree τ withrespect to the dilation weight Δ. Then thefollowing holds:

1. ∂V/∂xi is still homogeneous of degreeτ − ri with ri being the homogeneousweight of xi.

2. There is a constant c such thatV(x) ≤ c||x|| .

Moreover, if V(x) is positive definite,||x|| ≤ V(x), for a positive constant > 0.

B. Proof of PropositionsThis section contains the technical

details of the proofs. Here in we use ageneric constant c which represents anyfinite positive constant value and may beimplicitly changed in various places.Nevertheless, the constant c is alwaysindependent of l1.

Proof of Proposition 2.1: First,

(B.1)

Using (A3) with the fact that m ≤ 1,

(B.2)

By the definition of x*2,

(B.3)

where β−1>0, This, together with (2.1), gives

(B.4)

Clearly, Proposition 2.1 follows from (B.2)and (B.4) with Lemma A.2.

482 Chiang Mai J. Sci. 2013; 40(3)

Proof of Proposition 2.2: we have, withq2 := 1 − τ − m

(B.5)

By the homogeneity of v,

≤ , where = ,

Δ=(1,m). From the definition of thehomogeneous norm and lemma A.1, wehave

(B.6)

So, by Lemma A.1

(B.7)

Using (2.16) to ξ2 for z2, we have

(B.8)

Finally, we estimate the last term on theright hand size of (B.5). By Lemma A.3,we can rewrite ẑ 2 into e2,

(B.9)

Applying (B.7)-(B.9) into (B.5) yields,

(B.10)

for a constant c4 ≥ 0. The last relation isobtained by applying Lemma A.2 to eachterm in above inequality.

Proof of Proposition 2.3:We let w(⋅) =

v(⋅) . By the last inequity in (A3) ofLemma A.1, we have

(B.11)

Now, because w is at least C1, let χ2 = z2 −, we expand this function as

(B.12)

By homogeneity of w(⋅) whose degree is 1,

So is homogeneous of degree

1 − m. Hence,

Chiang Mai J. Sci. 2013; 40(3) 483

(B.13)

Since (B.11)-(B.13), we have

for a constant c5 ≥ 0.

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