Extremum Seeking for Constrained Inputs

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 9, SEPTEMBER 2013 2405

[10] M. Pavone and E. Frazzoli, Decentralized policies for geometric pat-tern formation and path coverage, J. Dyn. Syst., Meas., Control, vol.129, pp. 633643, 2007.

[11] J. Ramirez, New Decentralized Algorithms for Spacecraft FormationControl Based on Cyclic Pursuit Approach, Ph.D. dissertation, MIT,Cambridge, MA, USA, 2010.

[12] A. Sinha and D. Ghose, Generalization of linear cyclic pursuit with ap-plication to rendezvous of multiple autonomous agents, IEEE Trans.Autom. Control, vol. 51, no. 11, pp. 18191824, Nov. 2006.

Extremum Seeking for Constrained Inputs

Ying Tan, Senior Member, IEEE, Yuping Li, Member, IEEE, andIven M. Y. Mareels, Fellow, IEEE

AbstractExtremum seeking control (ESC) is an adaptive controlscheme that locates an extremum of an input-output map, without anyexplicit knowledge of this map apart from its existence. As is typicalin adaptive control an integrator is used to drive the parameters thatare being adapted. Due to this integrator, it is possible that the adaptedparameters wander outside their physically relevant domain as theunderlying adaptation technique is blind to this constraint. As theseconstraints may represent realistic operational limits it is important todesign ESC to respect them. Two such ESC schemes are proposed. Oneis based on a constrained optimization approach in which some penaltyfunction is used to adapt the search so as not to violate the constraints. Theother technique uses an anti-windup scheme, a widely used mechanismin engineering to prevent windup of integral action in a controller. It isdemonstrated that both methods are essentially equivalent. In that for anypenalty-function based ESC, there exists an equivalent anti-windup ESCwhose phase portrait is a global approximate of the penalty-function ESC.Some simulations are presented to illustrate the main results.

Index Terms Extremum seeking control (ESC), input constraints.

I. INTRODUCTION

Extremum seeking control (ESC) seeks an optimal input for a gen-erally unknown objective function (or input-to-output). This technicalnote focuses on adaptive ESC, which has been widely used in var-ious applications [2], [10], [16]. ESC uses a dither signal to probe theinput-output map so as to estimate an approximate gradient of this map,and then uses this gradient estimate to drive the input-output map to adesired extremum. As indicated in [16], the simplest possible adaptiveESC schemewhen applied to a static plant is shown in Fig. 1.One of the key components of ESC is the integrator. As discussed in[16], together with the dither signal , the integral action canextract the gradient of the unknown static mapping , provided thatthemapping is continuous and differentiable so that the gradientis well-defined.Typically, the input variable must live within a particular domain,

as restricted by the operational limits of the plant . Given that (in

Manuscript received August 19, 2012; revised August 22, 2012; acceptedJanuary 09, 2013. Date of publication March 26, 2013; date of current versionAugust 15, 2013. This work is supported by the Australian Research Council(ARC) under Future Fellow Project: FT0991385. Recommended by AssociateEditor L. Zaccarian.The authors are with Department of Electrical and Electronic Engineering,

The University of Melbourne, VIC 3010, Australia (e-mail: [email protected]; [email protected]; [email protected]).Color versions of one or more of the figures in this technical note are available

online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2013.2254638

Fig. 1. Simplest possible adaptive ESC scheme.

Fig. 1) is driven via an integrator it is quite feasible that this input vari-able will wander outside this acceptable domain, and hence the actualinput may get saturated on the boundary of the domain, say as a conse-quence of actuator limits and will differ from the presumed input. Thisdifference is unfortunate, as with a fixed input the output will not cor-relate with the dither. As a consequence, the integrator will equally sat-urate, i.e., and get stuck at this inappropriate point on the boundaryof the operational domain. This is very similar to integrator wind-upin classic control [3].In order to solve such a problem in ESC, two approaches have been

proposed. From an optimization point of view, the input saturationcan be viewed as a constraint. Thus ESC with input saturation can betreated as a constrained optimization problem. A penalty function be-comes a natural way to handle input saturation. There is the rich liter-ature on penalty functions ([4], [13], [14]), which may be used to ad-vantage to propose penalty-function-based ESC. The first result showsthat the penalty-function-based ESC converges semi-globally practi-cally asymptotically to the optimal value as long as the optimal inputis indeed found within the saturation limits.On the other hand, froma traditional control perspective, anti-windup

techniques (see for example [1], [5], [7], [17], [19] and referencestherein) are of a natural choice to deal with integrator wind-up like be-havior described above. As indicated in [6], an anti-windup mechanismmay be added in an adaptive ESC loop to avoid this integral windup.In particular, several anti-windup ESC schemes were proposed andused in various applications to handle integral windup [9], [11], [15].Although these anti-windup compensators work in those applications,they lack a theoretical analysis. Most existing anti-windup mechanismswere developed in the context of linear systems. Generally it is hard todemonstrate how anti-windup mechanisms can work well for generalnonlinear (static/dynamic) systems.A simple static anti-windup mechanism [1] is added to ESC to com-

pensate for the saturation effects. It is shown that this anti-windup ESCis very similar to a special case of penalty-function-based ESC, i.e., thetrajectories of this anti-windup ESC are close to those of the specialpenalty-function-based ESC. The convergence of the penalty-function-based ESC and the closeness of two solutions ensure that the outputof this anti-windup ESC converges to the a small neighborhood. Thesize of which depends on the particular design parameters in the ESCalgorithm.In order to show that the anti-windup ESC works, we exploit the

link (closeness of solutions) between the anti-windup ESC and thepenalty-function-based ESC. It is not surprising that these twomethodsare closely related to each other as the key ideas of these two methodsare the same, though they come from different perspectives. When sat-uration happens (or input constraints being violated), both the penaltyfunction and the anti-windup component will drive the system backto unsaturated mode. Both methods use some penalty to preventor overcome saturation. By exploring the similarities between thesetwo methods, we uncover some links between the anti-windup mech-anism and the penalty function in constrained optimization. That is,for any penalty-function-based ESC, there exists an equivalent anti-

0018-9286 2013 IEEE

2406 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 9, SEPTEMBER 2013

windup ESC which approximates the behavior of the penalty-func-tion-based ESC. This link provides new ways to design anti-windupcontrol schemes (not limited to ESC) using the rich literature of penaltyfunctions in constrained optimization, and vice versa. It also providesa mathematical characterization of anti-windup mechanisms used incontrol engineering.The technical note is organized as follows. Section II provides the

problem formulation. The main results are presented in Section III, in-cluding penalty-function-based ESC, anti-windup ESC and their links.Simulation results are provided in Section IV, followed by conclu-sions in Section V. Brief proofs of the main results are provided inthe Appendix.

II. PROBLEM FORMULATION

In this note, the set of real numbers is denoted by . A functionis of class if for each fixed the

function is continuous, zero at zero and strictly increasing andfor each the function is strictly decreasing to zero. Thenotation is the order of magnitude notation.In order to illustrate the main ideas of this technical note, a

simple static single-input-single-output (SISO) plant with inputsaturation is considered. A similar result can be extended tomulti-input-multi-output dynamic systems by using multiple ESCloops and singular perturbation techniques.1

(1)

where is a smooth function and the saturation function isdefined as follows:

ififif

(2)

where and and are the upper limitand the lower limit of the input respectively.For simplicity, the following assumption is used as in [16, Assump-

tion 3]. This assumption can be relaxed if the derivative of is de-fined almost everywhere (a.e.), again using a smooth approximationtechnique.Assumption 1: There exists a unique such that

the following holds:

(3)

Assumption 1 indicates that even though is unknown, it is knownthat there exists a unique maximum . The control objec-tive of the extremum seeking is to find the appropriate control inputsuch that the output of the system (1) satisfies

(4)

III. MAIN RESULTS

A. Penalty-Function-Based ESC

The problem of interests (1) can be treated as some constrained op-timization problem

(5)

where only the output signal is measurable.

1When a dynamic plant is considered, by replacing by andby in Fig. 1, where is a small positive constant. The dynamic systembecomes a faster system compared with the slow updating law of . By usingsingular perturbation techniques [8], the reduced system in time be-comes a static system (1). With some appropriate assumptions on the boundarylayer system, it can be shown that ESC can work for dynamic systems providedthat it works for static systems, see [16] for more details.

This constrained optimization problem can be tackled by using somepenalty function methods. The penalty function usually is some mea-sure of violation of the constraints. The measure of violation is nonzerowhen the constraints are violated and is zero in the region where con-straints are not violated. The penalty function approach, instead ofsolving the original optimization with constraints, generates an aux-iliary problem (without constraints) by augmenting the objective func-tion with a penalty function. In the sequel, the following auxiliary ob-jective function is introduced:

(6)

where is some penalty function, satisfying the followingassumption:Assumption 2: The penalty function is absolutely con-

tinuous. Moreover, the following inequalities hold:

(7)

Some penalty functions satisfying Assumption 2 (see also [13], [14])are listed.1) The logarithmic penalty function:

(8)

2) The inverse penalty function:

3) The polynomial penalty function:

4) The absolute-value penalty function

where is a positive constant.5) The Courant-Beltrami penalty function

where is a positive constant.Remark 1: When the input constraints are violated, the derivative of

with respect to is zero. The derivative ofwill be dominated by that of the penalty function. Assumption 2 showsthat when the constraints are violated, if gradient search methods(for example, ESC) are used, the output of will converge to theboundary of feasible region. Once the ESC converges to the boundaryof feasible region in which the penalty function is a constantalmost everywhere, the derivative of is dominated by theoriginal objective function whose extremum can be found byESC.Remark 2: Observe that knowledge of is assumed in the con-

struction of the constrained function. In particular this implies that itis known in advance in which interval of the real line to search for themaximum.The diagram of a penalty-function-based ESC scheme is shown in

Fig. 2.The first result is stated as follows.Theorem 1: For any and satisfying Assumption 1, if

there exists satisfying Assumption 2, then the output of thepenalty-function-based ESC in Fig. 2 semi-globally practically asymp-totically converges to the optimal value , uniformly in .

Proof: See Appendix VI-A.


Fig. 2. Penalty-function-based ESC scheme.

Fig. 3. Anti-windup ESC scheme.

Remark 3: As discussed in [16], the averaged system coming fromaveraging technique2 of the ESC approximates the gradient of theunknown function .As is not smooth, the gradient of theobjective function is not well-defined, though it is intuitivelyclear that the averaging technique will smooth out any non-smoothcomponent from the integrator. Smooth approximation techniques arethus employed in the proof to facilitate the convergence analysis.Remark 4: In principle it would suffice to establish the theorem just

for all initial conditions in , but the theorem is truefor all initial conditions. This is important when generalizing the resultto higher dimensions, as only in the 1D case can it be guaranteed thatthe search trajectory will stay within the desired domain at all times.

B. Anti-Windup ESC

Instead of treating the input saturation as a constraint in the ex-tremum seeking, anti-windup technique can tackle the windup-likebehavior due to the existence of the integrator and input saturation.Anti-windup controller design has been an active research topic for

over several decades. Many anti-windup techniques have been pro-posed in the literature (see for example [1], [5], [7], [17], [19] andreferences therein). Most anti-windup mechanisms in literature are de-signed for linear systems. In the context of ESC, the plant of inter-ests is always nonlinear. Usually it is hard to show how anti-windupmechanism can work rigorously for general nonlinear systems. Thistechnical note explores the similarities between anti-windup ESC andpenalty-function-based ESC. These similarities are utilized in the con-vergence analysis of anti-wind ESC.Start with a simple static anti-windup mechanism [9] as shown in

Fig. 3, where it is assumed that the output of the saturation of the inputis available for measurement. (This is not strictly necessary, an estimateis sufficient.) The parameter is selected as wherethe positive constant is a design parameter. In the sequel, the systemcan be represented as

(9)

2Averaging is an approximation method for analysis of time-varying systems.As pointed out in [12], an auxiliary time-invariant system , called theaveraged system, is used to investigate properties of a time-varying dynamicalsystem, that depends on a small parameter .

Fig. 4. Special case of penalty-function-based ESC scheme.

where . Introducing the change of the co-ordinates, , the system takes the following form:

(10)

Note that the anti-windup block: is the approx-imated gradient of a quadratic polynomial penalty function:

, when saturation happens. Hence,the way anti-windup works is as if there was a particular penaltyfunction. As with penalty-function-based ESC, an auxiliary objectivefunction is introduced

(11)

which is a special case of (6) in that a quadratic polynomial penaltyfunction is used. The corresponding penalty-function-based ESC isshown in (4). Theorem 1 informs that this penalty-function-based ESCscheme in Fig. 2 works well.Introduce the following averaged system for the system (10) in

co-ordinate and time , it yields

(12)

The following proposition shows that the averaged system of the anti-windup ESC in Fig. 3 is a perturbed averaged system of the penalty-function-based ESC shown in Fig. 4. The perturbation part can be madearbitrarily small by selecting a sufficiently small for a given .Proposition 1: For any , , there exists ,

such that for any , the following equality holds:

(13)

Proof: See Appendix VI-B.The next result follows immediately from Proposition 1 and The-

orem 1 and is stated here without proof.Theorem 2: Assume Assumption 1 holds. Let be a positive

pair and . For any , there exists a positive pairsuch that for any and , the solutions for thesystem (10) satisfy

(14)


Remark 5: Although the penalty-function-based ESC in Fig. 4 isquite similar to anti-windup ESC in Fig. 3, there is an obvious differ-ence: in the penalty-function-based ESC, is treated as if itwere completely unknown and the ESC generates an approximation ofits gradient. In anti-windup ESC, as the second term of isknown, its gradient is used directly. Moreover, with the penalty func-tion approach the output of the saturated input is not required, whereaswith the anti-windup scheme we need to know the saturated input aswell as the input, i.e. the output of the saturation is measurable, that isnot always practical.Remark 6: Note that since the search size of ESC is in the order

of , the selection of is important in the anti-windupESC, though can be a very large number. As the gradient ofin is estimated from the ESC with the coefficient , thesame coefficient is needed for the second term of .The role of penalty function in and the anti-windup

component in Fig. 3 is very similar: when saturation happens, both thepenalty function and the anti-windup component will compensatethe effect coming from input saturation. The literature dealing withconstrained optimization introduces many different penalty functionsas listed above. All of them can be used to design corresponding anti-windup controllers.

C. Links Between Penalty-Function-Based ESC and Anti-WindupESCThe similarities between anti-windup ESC (Fig. 3) and penalty-func-

tion-based ESC (4) are sufficient motivations to explore the links be-tween these two methods further.Introduce the following function for convenience:

(15)

where satisfies Assumption 2. The function hasthe following properties

P1

(16)

P2 For any and given satisfying Assumption 2,there exists such that for any , the followingholds

(17)

Both properties comes from the definition of andAssumption2, thus the proof is omitted.With the introduction of the anti-windup function , it leads

to an anti-windup ESC coming from the penalty-function-based ESC.Fig. 5 shows the structure of such an anti-windup ESC coming fromFig. 2.This leads to the following closed-loop system:

(18)

With the help of Property 2, the following result is a direct outcomeof Theorem 1.Theorem 3: Let be a function satisfying Assumption 2,

as in (15). Let be a positive pair and .

Fig. 5. Anti-windup ESC corresponding to the penalty-function-based ESC inFig. 2.

Assume Assumption 1 holds. Then there exists a positive pairsuch that for any and , the solutions of thesystem (18) satisfy

(19)

Remark 7: Theorem 3 clearly shows a link between the two ESCschemes as presented. That is, for any penalty-function-based ESCscheme, there exists a corresponding anti-windup ESC, which achievesa similar performance. The different choice of the penalty functionwill lead to different (static) anti-windup mechanism. The rich liter-ature on penalty functions provides many new ways to design variousanti-windup mechanisms.Remark 8: It is noted that Property 1 of shows how anti-

windup ESC actually works. When the input is far away from the con-straints, the anti-windup will pull the input back to the nearestsaturation bound. This property can be used to rigorously characterizethe static anti-windup mechanisms. Our future work will explore howto use penalty functions to design various anti-windup mechanism (notlimited to ESC) more systematically.Remark 9: Theorem 3 shows that for any penalty-function-based

ESC scheme, there exists a corresponding anti-windup mechanism. Onthe other hand, for any integrable nonlinear function sat-isfying Property 1 in Fig. 5, the following function:

(20)

always satisfies Assumption 2 and Property 2. This also shows thepossibility to get a penalty-function-based ESC from a given staticanti-windup ESC.

IV. SIMULATION RESULTS AND DISCUSSIONSThis section provides simulation results to demonstrate the effective-

ness of the two proposed ESC schemes. A very simple example is usedto illustrate the main messages of this note

(21)

with , . Let and . Assumption1 holds. Select , and . From Fig. 6, it isclear that when there is no input saturation, the ESC works well. Withinput saturation, the output of the system does not converge to its globalmaximum , but rather sticks at the boundary of the acceptabledomain as input signal is saturated at .

A. Penalty-Function-Based ESC and Anti-Windup ESCIn this case study, the penalty function to cope with the saturation is

set as

(22)

where is a fixed positive number, with which in (22) sat-isfies Assumption 2.


Fig. 6. Performance of the standard ESC with/without input saturation.

Fig. 7. Performance of the penalty-function-based ESC and its correspondinganti-windup ESC.

The corresponding compensator in the anti-windup ESC is.

In this case, the parameter is selected as 0.01. Fig. 7 compares theclosed-loop response with anti-windup ESC (the thick solid line) andthe penalty-function-based ESC (the thick dashed line). In both cases,the output converges to a small neighborhood of the maximum.

B. Anti-Windup Compensation for a PID Controller

It is intuitively clear that the anti-windup function sat-isfying Property P1 can be used to compensate the effect of windupin other integral control actions (other than ESC), provided that theclosed-loop stability is achieved by the nominal feedback controllerwhen the controller output is within the saturation limits.In order to illustrate this idea, we use an example from [18] in

which an anti-windup PID control was proposed. The plant modelis ,where is the Laplace transform of the output and is theLaplace transform of the input. The nominal stabilizing PI controlleris . We implement an anti-windup compen-sator derived from an logarithmic barrier function-based ESC, i.e.

. Thesaturation bounds are set as and . As shownin Fig. 8, the output of the plant with anti-windup mechanism, PIcontroller and saturated input outperforms the PI controller withoutanti-windup mechanism. The performance of this simple anti-windupmechanism is also comparable to the well-tuned anti-windup PIcontroller in [18].

V. CONCLUSION

Two ESC schemes are proposed to deal with input saturation. Inpenalty-function-based ESC, one adds some penalty function to theobjective function to penalize the violation of input constraints. Anti-windup ESC introduces an anti-windup component to the ESC to pre-vent wind-up like behavior. It is interesting to observe that both

Fig. 8. Anti-windup designed from barrier function implemented with PI con-trol.

penalty function and anti-windup mechanism work along very similarlines. In particular, it has been established that for any penalty-func-tion-based ESC, there is a corresponding (static) anti-windup ESC.This link shows the possibility to generate new anti-windup mecha-nism in control engineering and new penalty function in constrainedoptimization in a systematic way.

APPENDIX

A. Proof of Theorem 1

The proof consists of three steps.Step 1 shows the piece-wise continuous function satisfies

Assumption 1 a.e.Proof: It is easy to show that there exists a unique maximum

of and . On the other hand, the derivativeof with respect to its first argument is

or (23)

The result holds by applying Assumption 2.Step 2 finds a smooth approximation of .An approximation of is constructed as

where and are smooth functions and is some constantwhich ensures that within saturation bound.Two smooth functions and satisfy the following conditions:

This construction shows that satisfies Assumption 1 for anygiven and . Moreover, this construction ensures thatand are only different in two small intervals with size .Note the smoothness of and , we can conclude that

for all in some compact set.Step 3 shows convergence.


The closed-loop system shown in Fig. 2 in the new coordinatedis

(24)

for which the averaged system in coordinate in time is

The first term inside the integral is in a standard ESC form. The semi-global practical convergence is ensured as shown in [16, Corollary 1]).The second term can be made arbitrarily small by selecting sufficientlysmall . Therefore, we can show that the trajectories in (24) willconverge to 0 semi-globally practically asymptotically, uniformly in

.

B. Proof of Proposition 1

A simple calculation leads to

(25)

Let , applying Taylor series expansion for function atand performing integration with respect to , it follows that:

(26)

The proof is completed by substituting (26) into (25).

REFERENCES[1] K. J. mstrm and T. Hgglund, PID Controllers. Theory, Design

and Tuning. Research Triangle Park, NC: Instruments Society ofAmerica, 1995.

[2] K. B. Ariyur and M. Krsti, Real-Time Optimization by ExtremumSeeking Control. Hoboken, NJ: Wiley-Interscience, 2003.

[3] C. H. Edwards and D. E. Penney, Calculus With Analytic Geometry,5th ed. Upper Saddle River, NJ: Prentice Hall, 1998.

[4] R. Fletcher, An ideal penalty function for constrained optimization,IMA J. Appl. Math., vol. 15, no. 3, pp. 319342, 1975.

[5] S. Galeania and A. R. Teel, On a performance-robustness trade-offintrinsic to the natural anti-windup problem, Automatica, vol. 42, pp.18491861, 2006.

[6] B. I. Godoy, J. H. Braslavsky, and J. C. Agero, A simulation studyon model predictive control and extremum seeking control for heapbioleaching processes, in Proc. 17th World Congress Int. Fed. Autom.Control, Seoul, Korea, Jul. 611, 2008, pp. 93689373.

[7] G. Grimm, A. R. Teel, and L. Zaccarian, Robust linear anti-windupsynthesis for recovery of unconstrained performance, Int. J. RobustNonlin. Control, vol. 14, no. 1315, pp. 11331168, 2004.

[8] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ:Prentice-Hall, 2002.

[9] N. J. Killingsworth and M. Krsti, PID tuning using extremumseeking: Online, model-free performance optimization, IEEE ControlSyste. Mag., vol. 2, pp. 7079, 2006.

[10] M. Krsti and H. H.Wang, Stability of extremum seeking feedback forgeneral nonlinear dynamic systems,Automatica, vol. 36, pp. 595601,2000.

[11] P. Li and Y. Li, Efficient operation of air-side economizer using ex-tremum seeking control, J. Dyn. Syst., Meas., Control, vol. 132, no.3, pp. 031009031018, 2010.

[12] D. Nesic and P. M. Dower, A note on input to state stability and av-eraging of systems with inputs, IEEE Trans. Autom. Control, vol. 46,no. 11, pp. 17601765, Nov. 2001.

[13] J. Nocedal and S. J. Wright, Numerical Optimization. New York:Springer, 1999, 0-387-98793-2.

[14] J. A. Snyman, Practical Mathematical Optimization. New York:Springer Science + Business Media, Inc., 2005.

[15] E. Schuster, M. L. Walker, D. A. Humphreys, and M. Krsti, Plasmavertical stabilization with actuation constraints in the DIII-D tokamak,Automatica, vol. 41, pp. 11731179, 2005.

[16] Y. Tan, D. Nei, and I. Mareels, On non-local stability properties ofextremum seeking control, Automatica, vol. 42, pp. 889903, 2006.

[17] S. Tarbouriech and M. Turner, Anti-windup design: An overview ofsome recent advances and open problems, IET Control Theory Appl.,vol. 3, no. 1, pp. 119, 2009.

[18] D. Xue, Y. Chen, and D. P. Atherton, Linear Feedback Control:Analysis and Design with MATLAB (Advances in Design andControl). Philadelphia, PA: Society for Industrial and AppliedMathematics, 2009.

[19] L. Zaccarian and A. R. Teel, Modern Anti-Windup Synthesis: Con-trol Augmentation for Actuator Saturation. Princeton, NJ: PrincetonUniv. Press, 2011.

Stability Analysis of Second-Order Sliding Mode ControlSystems With Input-Delay Using Poincar Map

Xiangjun Li, Xinghuo Yu, and Qing-Long Han

AbstractThis note discusses the stability of dynamical systems withinput-delay under the second-order sliding mode control algorithm.Poincar Map is constructed to analyze the switching dynamics and toderive the stability conditions. Different parameter setting options aregiven for ensuring stability. Simulation examples are presented to verifythe theoretical results.

Index TermsHigh-order sliding mode (HOSM), sliding mode control(SMC).

I. INTRODUCTION

Sliding mode control (SMC) systems are known to be simple fordesign and robust in parameter variations and disturbances. However,

Manuscript received March 28, 2012; revised August 29, 2012; acceptedFebruary 28, 2013. Date of publication April 03, 2013; date of current versionAugust 15, 2013. This work was supported in part by the Australian ResearchCouncil Discovery Project DP130104765. Recommended by Associate EditorX. Chen.X. Li and X. Yu are with RMIT University, Melbourne, VIC 3001, Australia

(e-mail: [email protected]; [email protected]).Q.-L. Han is with the Centre for Intelligent and Networked Systems, and the

School of Information and Communication Technology, Central QueenslandUniversity, Rockhampton QLD 4702, Australia (e-mail: [email protected]).Color versions of one or more of the figures in this technical note are available

online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2013.2256673

0018-9286 2013 IEEE

Extremum Seeking for Constrained Inputs

Documents

Transcript of Extremum Seeking for Constrained Inputs