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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 2, FEBRUARY 2013 435
A Framework for Extremum Seeking Control of
Systems With Parameter Uncertainties
Dragan Nešić, Fellow, IEEE, Alireza Mohammadi, and Chris Manzie, Member, IEEE
Abstract—Traditionally, the design of extremum seeking algo-
rithm treats the system as essentially a black-box, which for many
applications means disregarding known information about the
model structure. In contrast to this approach, there have been
recent examples where a known plant structure with uncertain
parameters has been used in the online optimization of plant
operation. However, the results for these approaches have been
restricted to specific classes of plants and optimization algorithms.
This paper seeks to provide general results and a framework for
the design of extremum seekers applied to systems with parameter
uncertainties. General conditions for an optimization method and
a parameter estimator are presented so that their combination
guarantees convergence of the extremum seeker for both static
and dynamic plants. Tuning guidelines for the closed loop scheme
are also presented. The generality and flexibility of the proposed
framework is demonstrated through a number of parameter
estimators and optimization algorithms that can be combined
to obtain extremum seeking. Examples of anti-lock braking and
model reference adaptive control are used to illustrate the effec-
tiveness of the proposed framework.
Index Terms—Extremum seeking, optimization, parameter
estimation.
I. INTRODUCTION
Astanding assumption in extremum seeking is that the
model of the plant is unknown and that the steady-state
relationship between reference input signals and plant outputs
is such that it contains an extremum [19]. The goal is to tune
the system inputs online so that it operates in the vicinity of
this extremum in steady state; see [5]. This situation arises in
a range of classical, as well as certain emerging engineering
applications. Since an extremum seeking controller does not
need the exact model of the plant and also can easily deal with
multi input systems, it has been successfully used in a range of
applications, such as biochemical reactors [6], anti-lock braking
system (ABS) control in automotive brakes [9], variable cam
timing engine operation [24], and mobile sensor networks [22].
Manuscript received October 13, 2010; revised December 21, 2011 and July
02, 2012; accepted July 29, 2012. Date of publication August 24, 2012; date
of current version January 19, 2013. This work was supported by an Australia
Research Council Discovery Grant, DP0985388. A shorter version of this paper
appeared in the Proceedings of the 49th IEEE Conference on Decision and Con-
trolAtlanta, Georgia2010 [20]. Recommended by Associate Editor A. Loria.
D. Nešićis with the Department of Electrical and Electronic Engi-
neering, University of Melbourne, Parkville, 3010 VIC, Australia (e-mail:
dnesic@unimelb.edu.au).
A. Mohammadi and C. Manzie are with the Department of Mechanical En-
gineering, University of Melbourne, Parkville, 3010 VIC, Australia (e-mail:
a.mohammadi@pgrad.unimelb.edu.au; manziec@unimelb.edu.au).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2012.2215270
There are several main methods for design of extremum
seeking controllers. An adaptive control approach for contin-
uous time systems is pursued in [5], [15], [26], [27] whereas
a nonlinear programming approach in [29] and simultaneous
perturbation stochastic approximation in [24] and [25] are
proposed for discrete time systems. Results in [29] are signif-
icant as they show how to combine a large class of nonlinear
programming optimization methods with a gradient estimator
in order to achieve extremum seeking. Also in [21], a frame-
work is proposed for design of extremum seeking controllers
assuming that the reference-to-output map is not known at all
and the estimation of derivatives of this map is done directly.
The power of the results in [21] and [29] is that they provide a
prescriptive framework that can be used to design a large class
of extremum seeking controllers and show their convergence
properties in a unified manner. Within this framework it is
shown how to combine the classical nonlinear programming
optimization algorithms with various estimation algorithms to
obtain a powerful controller design framework for extremum
seeking.
Results in [2], [3], [10], and [11] are derived under subtly
different assumptions. Here the plant is assumedtobeparam-
eterized by an unknown parameter and various parameter esti-
mation based techniques are used to achieve extremum seeking.
While the parameter is unknown, it isassumedthatitisknown
how the plant model depends on this parameter. This slightly
stronger assumption allows a more direct use of classical adap-
tive control methods in the context of extremum seeking. Nev-
ertheless, the results in [2], [3], [10], and [11] are presented
for particular classes of plants and particular optimization al-
gorithms are used to achieve extremum seeking. As far as the
authors are aware, a prescriptive framework that would allow
extremum seeking designers to combine a large class of opti-
mization algorithms with a large class of stable plants (or un-
stable plants with controller, observer and parameter estimator)
that is similar to results in [21] and [29] has not been reported
in the literature.
It is the purpose of this paper to propose a prescriptive ex-
tremum control design framework reminiscent of [21] and [29]
for methods based on parameter estimation. The framework pro-
vides precise conditions under which one can combine a large
class of continuous optimization algorithms with a large class of
controllers, observers and parameter estimators to achieve con-
vergence of the closed loop trajectories to the desired extremum.
Moreover, the results prescribe how the controller parameters
need to be tuned in order to achieve convergence to the desired
extremum.
It is shown that a large class of optimization algorithms and
parameter estimators satisfy the main assumptions and their var-
0018-9286/$31.00 © 2012 IEEE
436 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 2, FEBRUARY 2013
ious combinations can be used to construct various extremum
seeking algorithms; this provides a flexible toolbox previously
unavailable in the literature. The provided examples demon-
strate the power of the proposed framework since a large number
of new extremum seeking algorithms can be easily obtained fol-
lowing the proposed approach and tuning the parameters in the
controller to obtain the desired convergence.
The paper is organized as follows. Section II presents pre-
liminaries. The first main result is presented in Section III for
static plants in order to introduce the ideas and assumptions in
a simple setting. Subsequently, in Section IV, additional results
for stable and unstable dynamic plant models are presented.
Section V provides examples of estimation and optimization
algorithms that satisfy the assumptions of the framework.
Section VI demonstrates the application of the main results
through simulation studies and illustrates the generality of the
design framework.
II. PRELIMINARIES
The set of real numbers is denoted by . The continuous
function is said to belong to class if
it is nondecreasing and . The continuous function
is of class if it is nondecreasing
in its first argument, strictly decreasing to zero in its second ar-
gument and for all .
Consider the static mapping denoted as
(1)
where is a fixed unknown parameter vector,
is the input and is the output of the
static system. This study is carried out under the following basic
assumption on the static map.
Assumption 1: The map is known but the parameter
vector is unknown. In addition, the map is smoothly
differentiable sufficiently many times and for any there exists
an extremum.1
The vector
.
.
.
where
for denotes the iterated derivatives of
with respect to its input arguments.
Now consider an optimization scheme of the form
(2)
which is used to generate an extremum seeking scheme. The
following assumption is placed on the optimization algorithm.
1Without loss of generality only maxima are considered; minima can easily
be treated in the same manner by defining and then applying the theory
to .
Assumption 2: For any given (but unknown) there exists an
equilibrium of system (2) which corresponds to the
extremum of the map .
Remark 1: Note that sometimes all derivatives in the vector
are not required to generate a certain scheme. For instance,
in order to generate the continuous time gradient method, only
the first derivative of is required. However, some other
algorithms require other derivatives and the proofs are provided
for the very general case. In addition, (2) can be more general
where the model depends on extra states or the right-hand side
of (2) is discontinuous. In the latter case, differential equations
should be generalized to differential inclusions which requires
extra assumptions on the model. However, the further general-
ization is omitted in this paper to keep the model simple enough
to state the main idea of the framework.
III. CONVERGENCE CONDITIONS FOR STATIC PLANTS
The first main result is presented in this section for an ex-
tremum seeking scheme applied to a static plant, which leads
to a general class of extremum seeking schemes based on esti-
mation of the parameter vector . In these schemes, by tuning
a parameter in the extremum seeking controller the closed
loop system exhibits a time scale separation. Therefore, the re-
duced system behaves approximately as the given optimization
scheme (2) and under appropriate stability properties of the
parameter estimator and the optimization scheme, convergence
to the extremum of can be achieved.
Consider the following class of extremum seeking schemes:
(3)
(4)
(5a)
(5b)
(6)
where is a controller tuning parameter (typically, a small pos-
itive number) that may be adjusted. Throughout the paper we
assume that all functions are sufficientlysmoothsothatappro-
priate singular perturbation results can be used.
Equation (3) is the static plant model, (4) is the input into the
plant where is a dither signal that is typically chosen so
that appropriate parameter convergence can be achieved and
comes from the optimization algorithm (6). The optimization al-
gorithm uses the estimated parameter that is obtained from the
estimator (5a), (5b). The parameter estimation algorithm (5a)
may contain extra states to widen the class of
estimators considered; the further possible generalizations are
explained in Remark 18. Considering as a non-square matrix
is convenient for the later examples, and does not affect any
subsequent proofs. Fig. 1 shows the relations between different
parts of the closed loop system for a static plant.
Remark 2: By introducing , ,
(where is the equilibrium of (3)–(6)) and writing the
closed loop equations in coordinates and in the time scale
NEŠIĆet al.: A FRAMEWORK FOR EXTREMUM SEEKING CONTROL OF SYSTEMS WITH PARAMETER UNCERTAINTIES 437
Fig. 1. Proposed framework for static plants.
, the model of the systems (5a), (5b), (6) are
obtained in the standard singular perturbation form:
(7a)
(7b)
(8)
While a singular perturbation approach could potentially be
used to analyze the system (7a), (7b) and (8) (and therefore
(3)–(6)), the standard assumptions in the classical singular
perturbation literature on uniform boundedness of the functions
and with small may not hold because of the dither
signal . However, the system (3)–(6) is regularly
perturbed and the standard singular perturbation results hold
under weaker smoothness assumptions than those stated in [14]
for the system (7a), (7b), and (8); this was done, for instance,
in [30] by dealing directly with a class of systems that subsume
(3)–(6).
Next stability assumptions from singular perturbation tech-
niques are used to state the main result for static plants. Con-
sider the following assumption for the parameter estimation al-
gorithm (7a), (7b):
Assumption 3: The origin ,of the boundary layer
(fast) system:
(9a)
(9b)
is uniformly asymptotically stable (UAS) uniformly in
and with a basin of attraction .
Remark 3: From [14] it follows that when the system is UAS,
there exists a function so that the trajectories can be bounded
in an appropriate manner.
Then, it is assumed that optimization algorithm (8) satisfies
the following hypothesis:
Assumption 4: The origin for the reduced (slow) system:
(10)
is UAS with a basin of attraction .
Now, the stability properties of the overall system (3)–(6) are
stated in the following:
Theorem 1: Suppose that Assumptions 1–4 hold. Then, there
exist and such that for any strictly
positive real number , there exists such that for all
the following holds:
(11)
(12)
(13)
for all ,and .
In particular, ,
and .
Proof: The proof is omitted since it follows directly from
[30].
Remark 4: Note that Theorem 1 allows us to combine any
continuous optimization method of the form (2) which satisfies
Assumptions 2 and 4 with a parameter estimation scheme (5a),
(5b) which satisfies Assumption 3 in order to achieve extremum
seeking. Hence, results of Theorem 1 provide a prescriptive
framework for extremum seeking controller design that com-
bines a large class of optimization methods with a large class of
parameter estimation schemes that satisfy conditions of the the-
orem. Section V provides a range of algorithms for parameter
estimator and optimizer and conditions that guarantee satisfac-
tion of the required assumptions.
Remark 5: The conclusions in Theorem 1 are quite intuitive.
There exists a ball of initial conditions such that for any
desired accuracy characterized by , the parameter can be ad-
justed (i.e. reduced) so that for all initial conditions in the ball
:
• The parameter estimate converges to the ball centered
at the true value of the parameter in time scale (see (11)).
• The extra state of the parameter estimator converges to
the ball centered at the equilibrium of (5b) in time
scale (see (12)).
• The optimizer state converges in the slow time scale
to the ball centered at the optimal value (see (13)).
Remark 6: If conditions of Theorem 1 are modified by
requiring that the boundary layer and reduced systems are
uniformly exponentially stable (UES), then stronger UES
stability of the closed loop system can be proved [14]. In
other words, under those stated conditions there are positive
constants such that there exists
so that for all :
(14)
(15)
(16)
for all .
438 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 2, FEBRUARY 2013
Fig. 2. Proposed framework for dynamic stable plants.
Remark 7: Note that the conditions in Theorem 1 can be
strengthened so that global asymptotic stability can be assumed
for the reduced system and the boundary layer system from As-
sumptions 3 and 4. In that case, semi-global practical stability of
the closed loop system is concluded. However, few optimization
algorithms satisfy such strong conditions and so only the local
results are stated here.
IV. CONVERGENCE CONDITIONS FOR DYNAMIC PLANTS
In this section, extremum seeking is considered for dynamic
plants. The extremum seeking controller is parameterized in
such a way to exhibit multiple time scales in the closed-loop
system which allows singular perturbation theory to be applied.
Multi-time-scale singular perturbation results are used to show
that if the parameters are tuned appropriately, the system
achieves practical asymptotical convergence to the extremum.
The following two subsections deal respectively with stable
and unstable plants.
A. Stable Plant
The extremum seeking scheme in the previous section can
be readily modified to deal with dynamic plants. Consider the
following closed loop system with a dynamical plant:
(17)
(18)
(19)
(20a)
(20b)
(21)
where is the plant state, and are tuning
parameters of the estimator and optimization scheme, respec-
tively, and all other variables are the same as before (Fig. 2).
Note that in this case, with appropriate tuning of and ,there
are three time scales, where the plant is the fastest subsystem,
is the medium system and is the slow system. The scalars
are controller parameters that need to be tuned. The
dither signal may be needed to ensure an appropriate per-
sistence of excitation condition that guarantees convergence of
.
Assumption 5: The following equation:
has a unique solution and the map:
has an extremum at .
Remark 8: Assumption 5 says that for every constant the
output converges to , uniformly in . Note that this
problem formulation is different from classical adaptive con-
trol literature where one typically wants to stabilize an unstable
plant. In the case considered in this section, the plant is assumed
to be stable. If the plant is unstable, then one can use any suit-
able controller to stabilize it so that the assumption holds.
Suppose that Assumption 1 holds for the map .This
is a strong assumption since finding the map explicitly
is hard in general. Moreover, it will be assumed that the opti-
mization scheme of the form (21) and a parameter estimation
scheme (20a), (20b) have similar properties to the ones stated
in Assumptions 3 and 4, respectively, but with the above con-
structed map .
Writing the closed loop in coordinates
gives
(22)
(23a)
(23b)
(24)
This system is in singularly perturbed form with tuning con-
troller parameters and . It has three time scales, the plant
is the fastest, then the estimator is the middle and the optimiza-
tion algorithm is the slowest time scale. The stability analysis
is done via three systems, the fast system, the medium system
and the slow system that correspond respectively to the plant,
the estimator and the optimization algorithm. Stability analysis
of this class of systems in standard form was considered in [12]
and [30] which requires Assumptions 3 and 4 to be replaced by
the following three assumptions.
Assumption 6: The origin of the fast system
(25)
is UAS, uniformly in and with a basin of attraction
.
NEŠIĆet al.: A FRAMEWORK FOR EXTREMUM SEEKING CONTROL OF SYSTEMS WITH PARAMETER UNCERTAINTIES 439
Assumption 7: The origin ,of the medium
system
(26a)
(26b)
(in time scale )isUAS,uniformlyin with a basin
of attraction .
Assumption 8: The origin for the slow system
(27)
(in the time scale ) is UAS with a basin of attraction
.
Then, the main result is provided in the following:
Theorem 2: Suppose that Assumptions 1, 2 and 5–8 hold.
Then, there exist and such that for
any strictly positive real number ,thereexists
such that for all and the following
holds:
(28)
(29)
(30)
(31)
for all
and all . In particular, (28), (29), and (31) imply
respectively, , ,
and .
Proof: Stability analysis of this class of systems in stan-
dard singular perturbation form was considered in [12]; a similar
analysis can be carried out under weaker conditions as in [30]
for multiple time scale systems in regular form. These modifi-
cations are straightforward and are omitted for space reasons.
Remark 9: Note that the reason for having a time scale sep-
aration between the and is to be able to use the steady state
map for parameter estimation. Note that this is not nec-
essary and it is sometimes possible to directly use dynamical
equations for the -subsystem to estimate .
Note that in this case, there would be two time scales in the
overall system rather than three. This was done in the next sec-
tion for unstable plants.
Remark 10: If Assumptions 6–8 of the Theorem 2
are modified by requiring uniform exponential stability
of the slow, medium, and fast systems, then the the-
orem will be changed by stating uniformly exponen-
tial stability for the closed loop system (22)–(24) for all
and all
, see [8]. Also, by strengthening the stability
conditions in assumptions to hold globally, semi-global prac-
tical asymptotic stability of the closed loop can be concluded
[20].
Fig. 3. Proposed framework for dynamic unstable plants.
B. Unstable Plant
All of the results in the previous section were stated under the
assumption that the plant is stable. This assumption is relaxed in
this section and it is assumed that the plant is unstable and needs
to be stabilized using the output measurements , see [10] for
specific plants and optimization schemes. Moreover, results in
this section illustrate the point made in Remark 9.
Consider an uncertain dynamic plant
(32)
(33)
where all variables have the same meaning as in the previous
section. Here it may be useful to make a distinction between
the measured output that is used to stabilize the plant by
designing the observer, controller and parameter estimator and a
performance output thatmaybeusedinextremumseeking.
In this section, it will be assumed that a controller, parameter
estimator and state estimator have been designed and they are
of the following form:
(34a)
(34b)
(35a)
(35b)
(36)
where is the estimate of the state, and are extra states
for the state estimator and parameter estimator, respectively, to
widen the class of estimators that may be considered, is the
new “reference” input and all other variables were defined in
the previous section. The relation between the above equations
is shown in Fig. 3.
440 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 2, FEBRUARY 2013
It is convenient to rewrite the above equations by using
, , , (where
is the equilibrium of (34a)–(35b)) which gives
(37)
(38a)
(38b)
(39a)
(39b)
By introducing , (37)–(39b) is rewritten as
follows:
(40)
Now Assumptions 6–8 are replaced by the following three
assumptions.
Assumption 9: There exists such that
and the map
has an extremum (maximum) at .
To state the main result, it is convenient to define
where to obtain
(41)
In order to achieve extremum seeking, the above system is con-
trolled with a slow optimization algorithm
(42)
where ,is the maximum of the map
and .
The relation between plant output, observer, controller, pa-
rameter estimator and optimizer is shown in Fig. 3.
Assumption 10: The origin of the boundary layer
(fast) system:
(43)
is uniformly asymptotically stable (UAS) uniformly in
and with a basin of attraction .
Assumption 11: The origin for the reduced (slow) system
(44)
is UAS with a basin of attraction .
Then, the following result can be stated:
Theorem 3: Suppose that the Assumptions 1, 2, and 9–11
hold. Then, there exist and such that for
any strictly positive real number , there exists such
that for all the following holds:
(45)
(46)
for all and all
. In particular, (45) and (46) imply, respec-
tively: and .
Proof: This result follows directly from the singular per-
turbation result in [30].
Remark 11: Note that typical adaptive control designs do not
produce UAS of the system (41) in general and an appropriate
persistence of excitation condition needs to hold in order to get
UAS. A range of persistency of excitation conditions that can be
used to conclude UAS of (41) can be found in [17]. Note also
that while authors of [2], [3], [10], and [11] do not state their
results at this level of generality, they do require appropriate
persistence of excitation that guarantees UAS of (41).
Remark 12: The results in [2], [3], [10], and [11] are similar
to the main result in Theorem 3 in the sense that in both they
assume that the performance function is explicitly known as a
function of the system states and uncertain parameters from the
dynamic equations. However, the results in [2], [3], [10], [11]
are presented for particular classes of plants and particular op-
timization algorithms are used to achieve extremum seeking.
Remark 13: Time scale separation is only one possible way
to ensure convergence of the extremum seeking algorithm. In
particular, one could use different conditions based on small
gain arguments. Note that (41) and (42) with is a feedback
connection of two systems. If each of these two systems satisfies
an input-to-state stability property and a small gain condition
holds then it can be concluded that converges, which implies
extremum seeking.
Remark 14: If Assumptions 10–11 are modified by requiring
that the the boundary layer and reduced systems are UES,
then the closed loop full system (41), (42) is UES for all
and all .
Remark 15: Note that all of the results can be restated so that
instead of uniform asymptotic stability requirements in the as-
sumptions, uniform global asymptotic stability (UGAS) prop-
erty is used. All of the results of this paper still hold but the
conclusions are then stronger as semi-global stability can be
achieved. These results are presented in [20].
V. OPTIMIZATION AND ESTIMATION ALGORITHMS
This section provides different optimization algorithms of the
form (2) that satisfy Assumptions 4, 8, and 11 in Theorems 1, 2,
and 3 and parameter estimators that satisfy Assumptions 3 and
7 in Theorems 1 and 2, respectively. Designers can choose and
combine the following parameter estimators and optimization
algorithms that are appropriate for their problems. Note that the
NEŠIĆet al.: A FRAMEWORK FOR EXTREMUM SEEKING CONTROL OF SYSTEMS WITH PARAMETER UNCERTAINTIES 441
framework is not restricted to these algorithms and any other
algorithms that satisfy the assumptions can be employed.
A range of existing algorithms are taken off-the-shelf from
the literature and conditions that guarantee satisfaction of our
assumptions are pointed out. In this manner, it is demonstrated
that the results of this paper provide a flexible design framework
in which a large class of optimizers can be combined with a large
class of parameter estimation algorithms to achieve extremum
seeking.
Remark 16: Note that in this section, all conditions are pro-
vided for the local stability of the optimization and estimation
algorithms. By strengthening the conditions, global stability of
the algorithms can be concluded. However, to be consistent with
the main results, only conditions for local stability are stated.
A. Optimization Algorithms (OA)
While the algorithms provided are taken off-the-shelf from
the literature, they demonstrate the utility of the unifying de-
sign framework as any of these algorithms can be combined
with appropriate parameter estimators to obtain novel extremum
seeking algorithms not considered previously in the literature.
It should be noted that the reduced systems in Assumptions
4, 8, and 11 take the same form of optimization algorithms
presented in this subsection but they operate on different time
scales depending on the problem setting considered. It is not
hard to see that modulo the time scale, the dynamics of these
systems is exactly the same as that given by (2). Hence, it is
assumed that the chosen optimization algorithm is stable in an
appropriate sense. In addition, to simplify the notation, the de-
pendence on the unknown parameter is suppressed in this sub-
section.
OA1. Gradient Descent Algorithm: Consider the continuous-
time gradient descent system [1]
(47)
where denotes the gradient of at .
Definition 1: Apoint is a strict local minimum of
if there exists such that for all such
that .
It was shown in [1] that if equilibrium of (47) is a strict
local minimum, then is asymptotically stable.
Proposition 1: If is a strict local extremum, then is a
uniformly asymptotically stable point of the optimization algo-
rithm (47) and therefore this algorithm satisfies Assumptions 4,
8, and 11.
OA2. Continuous Newton Method: Consider the following
continuous Newton method [7]
(48)
where represents the Jacobian of and is an
arbitrary positive definite matrix.
Condition 1 (Nonsingularity): The Jacobian matrix is
invertible in some neighborhood of .
Proposition 2: [7] Suppose that Condition 1 holds, then for
all the Newton method (48) is exponentially
stable and satisfies Assumptions 4, 8, and 11.
OA3. Continuous Jacobian Matrix Transpose: Consider the
continuous Jacobian matrix transpose algorithm as follows [7]:
(49)
Proposition 3: [7] Under Condition 1, the optimization algo-
rithm (49) is asymptotically stable and Assumptions 4, 8, and
11 hold for this algorithm.
OA4. Combination of Newton and Gradient Methods:Con-
sider the following continuous Newton-type differential equa-
tion [32]:
(50)
where
if
if
if
where denotes the smallest eigenvalue of ,
are two predefined positive constants, and and
are set as
(51)
(52)
Condition 2 (Boundedness): has the following
properties:
•is bounded from below by .
•Let
be the level set of ,and denote the connected
subset of that contains the point , then for any ,
is bounded.
Proposition 4: [32] Suppose that Condition 2 holds, then op-
timization algorithm (50) is asymptotically stable and satisfies
Assumptions 4, 8, and 11.
OA5. Levenberg-Marquardt Method: Consider a continuous
analogue of the Levenberg–Marquardt method [28]
(53)
where is a positive number.
Condition 3 (Full Rankness): The Jacobian matrix is
of full rank for an extremum point of .
Proposition 5: [28] If Condition 3 holds, then is an asymp-
totically stable point of the system (53) and therefore satisfies
Assumptions 4, 8, and 11.
OA6. Newton-Raphson-Ben-Israel:Con
sider a continuous
analogue of the Newton-Raphson-Ben-Israel method [28]
(54)
where is the Moore–Penrose inverse of .
442 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 2, FEBRUARY 2013
Proposition 6: [28] If Condition 3 holds, then is an asymp-
totically stable point of the optimization algorithm (54) and
therefore this algorithm satisfiesAssumptions4,8,and11.
B. Estimation Algorithms (EA)
In this subsection, different examples of parameter estimation
algorithms in the form of (5a), (5b) and (20a), (20b), respec-
tively, for Theorems 1 and 2 are presented that satisfy Assump-
tions 3 and 7 under specified conditions. In these two theorems,
parameter estimation algorithms are designed using steady-state
mapping between the input and the output. Since the UAS con-
ditions of Assumptions 3 and 7 may be hard to satisfy in gen-
eral, the estimator design is usually considered for a class of
steady-state maps that are linearly parameterized with as
(55)
AsshowninFigs.1and2, and As-
sumptions 3 and 7 should be verified for ; hence,
(56)
where and
. Therefore, in the following, different examples
of estimation algorithms are presented for the steady-state
mapping in the form of (56). In other cases, any parameter esti-
mation algorithm in the form of (5a), (5b) and (20a), (20b) that
satisfy the required conditions can be employed. In addition, if
the estimator should be designed using plant dynamics instead
of steady-state mapping, then Theorem 3 can be used where
parameter estimation is in the form of (35a), (35b) and should
satisfy Assumption 10.
EA1. Gradient Algorithm: An estimate of the unknown pa-
rameter vector can be obtained via the gradient algorithm [13]
(57)
where is a diagonal matrix of gains.
Condition 4 (Persistency of Excitation): is per-
sistently exciting, i.e., for any there exist such
that for any fixed in a neighborhood of
(58)
Proposition 7: [13] If Condition 4 holds, then the estimation
algorithm (57) is exponentially stable and satisfies Assumptions
3 and 7 and their respective UES conditions.
EA2. Gradient Algorithm with Integral Cost Function:This
parameter estimation algorithm is as follows [13]:
(59)
(60)
(61)
where is a design matrix referred to as the adaptive
gain and is a design constant acting as a forgetting factor,
,and . Note that by introducing
, estimation algorithm (59)–(61) is in the same form of
(5a), (5b) and (20a), (20b).
Remark 17: Note that the asymptotic stability assumption on
can be replaced by a boundedness condition with no conse-
quence for the estimation algorithm. A restatement and proof of
Theorems 1 and 2 with only boundedness of is not undertaken
here so as not to cloud the key concept of the presented frame-
work.
Proposition 8: [13] If Condition 4 holds, the estimation al-
gorithm (59)–(61) is exponentially stable and satisfies Assump-
tions 3 and 7 and their respective UES conditions.
EA3. Pure Least-Squares Algorithm: Consider the following
parameter estimation algorithm [13]:
(62)
(63)
Proposition 9: [13] If Condition 4 holds for ,thenesti-
mation algorithm (62), (63) is asymptotically stable and satisfies
Assumptions 3 and 7.
EA4. Recursive Least-Squares Algorithm with Forgetting
Factor: An alternative approach to estimate the unknown
parameters in (56) is the following continuous-time recursive
least-squares (RLS) algorithm [13]:
(64)
(65)
where and is the error covariance matrix.
Proposition 10: [13] If satisfies the persistency of excita-
tion condition (58), then (64) is uniformly exponentially stable
and is bounded.
Remark 18: It should be noted that there exist other estima-
tion algorithms that do not exactly fit the proposed framework,
thereby introducing a potential for further generalization. For
instance, due to resetting in Pure Least-Squares Algorithm with
Covariance Resetting [13] and due to the discontinuity of the
right-hand side in Modified Least-Squares Algorithm with For-
getting Factor [13], these estimators do not satisfy the smooth-
ness condition on the functions required in the main results.
Nonetheless, the range of parameter estimators provided in this
paper can be combined with optimizationalgorithmstoproduce
a variety of extremum seeking algorithms.
VI. APPLICATIONS OF THE MAIN RESULTS
In this section, to illustrate the generality of the framework, a
given plant with different optimization schemes and parameter
estimators from Section V is considered.
A. A Class of Stable Linear Systems With Output Nonlinearities
Below the general result for stable dynamic plants (Theorem
2) is applied to the situation where the dynamic plant (17) and
the nonlinear output (18) have special structures. Consider a
NEŠIĆet al.: A FRAMEWORK FOR EXTREMUM SEEKING CONTROL OF SYSTEMS WITH PARAMETER UNCERTAINTIES 443
Fig. 4. Friction force coefficient in different road conditions.
class of linear systems with linearly parameterized nonlinear
outputs
(66)
(67)
where is Hurwitz. This class of problems has been investi-
gated in more detail for linear control systems in [3].
By solving
the steady state map is
whichisinformof(56).Therefore, any parameter estimation
algorithm from Section V-B satisfying required conditions
can be combined with any optimization algorithm presented
in Section V-A satisfying appropriate conditions in order to
achieve practical asymptotical convergence to the extremum of
the performance function (67).
Example 1: In this example, first a feedback linearizing con-
troller is designed for the ABS problem in order to obtain a
stable linear system in the form of (66). Then, a linearly param-
eterized nonlinear output in the form of (67) is considered for
the tire-road friction model. The purpose of the ABS is to regu-
late the wheel longitudinal slip at its optimum point in order to
generate the maximum braking force (Fig. 4).
Consider a quarter vehicle model, where the tire dynamics are
as follows [5]:
(68)
(69)
Here, is the longitudinal speed of the vehicle, is its mass,
is the force at the tire, the wheel inertia, the
angular velocity, the bearing friction torque, the radius
of the wheel, the braking friction torque, the tire-road
friction force coefficient and is the wheel slip which is defined
as
(70)
Using (68)–(70) gives
(71)
It is assumed that the longitudinal and angular velocities can
be measured. Thus, by using the feedback linearizing controller
(72)
where is a positive constant, the following linear model is
obtained:
(73)
where is the control input to the new system which is expo-
nentially stable. The objective here is to achieve maximum fric-
tion force by maximizing the friction coefficient .There-
fore, a parameterized model of the is required that has a
maximum at the optimal slip . In this paper, the tire-road
friction model proposed in [31] is used:
(74)
where are unknown parameters to be esti-
mated. It should be noted that although this model depends on
, its optimum value is independent of and only depends on .
After applying logarithm to both sides of (74) and rearranging
it in vector form, the output is
(75)
where and .
Now, the assumptions of Theorem 2 should be verified. The
map in (75) is known and is smoothly differen-
tiable. In addition, has a global maximum at which is
given by the solution to
(76)
which can be seen in Fig. 4, so that Assumption 1 holds.
Next, the continuous Jacobian matrix transpose method
(OA3) is proposed as the optimization scheme
(77)
To verify Assumptions 2 and 5, consider the system (73)
which has an equilibrium at and the map has an
extremum at with respect to formulation (76), which
444 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 2, FEBRUARY 2013
corresponds to the equilibrium point of the optimization scheme
(77) at
(78)
if and so that Assumptions 2 and 5 hold.
To verify UAS of the optimization algorithm, (77) is written
in new coordinate and :
(79)
Since it is assumed that and ,theJa-
cobian matrix is nonsingular so that Condition 1 holds.
Hence, according to Proposition 3, optimization algorithm (79)
satisfies Assumption 8.
To verify the last assumption of Theorem 2 regarding recur-
sive least-squares (EA4) as estimator, Condition 4 should be
hold for in (75). In the ABS problem, Condition 4 holds using
the dither signal in . Hence,
based on Proposition 10 and Remark 17, Assumption 7 holds.
Therefore, all assumptions of Theorem 2 hold and it can be
concluded that by tuning the parameters and in the ex-
tremum seeker, practical asymptotical stability of the closed
loop system w.r.t. the given bounded region can be achieved.
For simulation purposes, the parameter values used are
,,and . The initial conditions are
and , which makes ,
,and . The unknown parameters
are initialized at , identical
tothosein[31].Itisassumedthatthebrakingstartsonadryas-
phalt road and after 5 m the road becomes icy. The ES schemes
estimate values of in different road conditions and then drive
to its optimal value so that friction coefficient converges to
its maximum point at each road condition.
The simulation results in Fig. 5 (solid curves) show that
during braking, maximum value of friction coefficient on dry
asphalt road and icy road are reached
and the vehicle stopped within the shortest distance (31.39 m)
which is highlighted in Table I. The dashed-curve in Fig. 5
shows the case that of dry asphalt road in used for the whole
road. In this case, the vehicle stops after 37.2 m.
Note that the solution of this problem is not restricted to
the above estimator (EA4) and optimization algorithm (OA3)
and any type of methods explained in the previous section can
be employed here provided that the required conditions hold.
Table I shows that some combinations of optimization algo-
rithms (OA) and estimation algorithms (EA) from the previous
section lead to better closed-loop behavior under some condi-
tions, but all are handled by the proposed framework.
B. A Class of Unstable Systems With Output Nonlinearities
Here, an application of the last main result (Theorem 3) is
presented for a class of systems which includes the closed loop
system in Model Reference Adaptive Control (MRAC) of linear
systems, together with an optimization scheme. The reason for
choosing MRAC scheme as an example for application of The-
orem 3 is that the plant and the estimator are in the same time
Fig. 5. ABS design using extremum seeking framework. Dashed-line shows
the case that extremum seeking scheme is not utilized and of dry asphalt
road in used for the whole road.
TAB L E I
SIMULATED STOPPING DISTANCE UNDER ABS FOR DIFFERENT COMBINATIONS
OF OPTIMIZATION AND ESTIMATION ALGORITHMS
scale and a controller also has to be designed to stabilize the
nonlinear system. The objective of this example is to manipulate
the input of the reference model dynamics in order to achieve
the extremum of the performance function appended to the non-
linear system dynamics.
Consider the problem of MRAC with feedback linearizable,
linearly parameterized nonlinear system:
(80)
(81)
where the state , the performance output ,
the input , the unknown parameters and
are smooth functions on .
Since the above system is feedback linearizable, there exists
a dipheomorphism such that the change of variable
and a state feedback controller transforms
(80) and (81) into the form
.
.
.(82)
(83)
NEŠIĆet al.: A FRAMEWORK FOR EXTREMUM SEEKING CONTROL OF SYSTEMS WITH PARAMETER UNCERTAINTIES 445
with locally Lipschitz. Furthermore, there exists a contin-
uous nondecreasing function such that
(84)
for almost all . Then (82) can be rewritten in the following
compact form:
(85)
where
.
.
..
.
..
.
.
Now, consider the following reference model for (85) which is
described in the controllable canonical form
(86)
where is an input to closed-loop system (85) and is Hurwitz
hence there exists such that
(87)
Assuming that full state is available for feedback, an adap-
tive state feedback controller can be designed as follows. Let
and rewrite system (82) as
(88)
Next, consider the Lyapunov function candidate
(89)
where ,and is an estimate of to be determined by
the parameter adaptive rule. The time derivative of along the
trajectories of (88) is given by
Tak ing
(90)
the expression for can be rewritten as
(91)
Therefore, the adaptation rule should be chosen as
(92)
to ensure that . Thus, the system dynamics will be
(93)
(94)
(95)
(96)
where .
Definition 2: [18] A function is said to be uniformly
-persistently exciting with respect to if for each
there exist ,and s.t.
where , , and
.
Condition 5: is with respect to .
By introducing , (93)–(96) can be rewritten in
the following compact form:
(97)
which is in of the form (40). Therefore, in the following, the
Assumptions 9–11 of Theorem 3 are verified for (97).
The equilibrium of (97) is given as
(98)
Thus, the steady-state map between the input and the output is
(99)
which is assumed to be continuously differentiable with an ex-
tremum at . Hence, Assumption 9 holds.
Defining the new coordinates where
gives
(100)
(101)
Thus, the boundary layer (fast) system is
(102)
In order to prove UAS of (102), consider the Lyapunov candi-
date (89) is augmented with ,i.e.,
The time derivative of evaluated along the trajectories of
(102) yields
Hence the system (102) is uniformly stable. Now, to show
asymptotic stability of the system (102) an argument similar
to the one used in Theorem 1 and Example 2 of [23] is used.
Basedonthistheorem, w.r.t. condition on
guarantees uniform asymptotic stability of the fast system
(102), so Assumption 10 holds.
446 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 2, FEBRUARY 2013
Now, in order to locate the extremum of the performance
function (83), consider an optimization algorithm satisfying ap-
propriate conditions from Section V-A.
Since all of the assumptions of Theorem 3 hold, the full
system (82), (83) is practically asymptotically stable and con-
verges to the extremum of the performance function (83).
Remark 19: Since verification of the condition in
general is difficult, Proposition 2 in [23] can be used to relax
this condition. Based on this proposition, if there exists bounded
function satisfying PE condition (58), a unitary vector
and ,suchthat
(103)
then w.r.t. .
Remark 20: The result above is similar to the results pre-
sented in [10] and [4]. It yields a generalization of the results in
these papers, allowing different types of optimization schemes
to achieve extremum of the performance function.
Example 2: Consider the following nonlinear, linearly pa-
rameterized system which is similar to the problem addressed
in [10]:
(104)
(105)
To show practical asymptotical stability of this system, the
conditions mentioned above are verified.
Consider the following change of variables:
(106)
then and and output map satisfy
(107)
(108)
which is in the form of (82), (83) with and
which is continuously differentiable, hence it is
locally Lipschitz. Note that the change of variables is
dipheomorphism since the map is continuously differentiable
and there exists continuously differentiable inverse map
such that . The inverse map is as follows:
(109)
Next, consider the following reference model:
(110)
where is an input to the system and is Hurwitz.
The controller (90) for this example is as follows:
(111)
Before verifying Assumption 9, the full description of the
tracking error and parameter estimate dynamics are established.
According to the analysis before this example, the tracking error
dynamics (93) and the parameter estimate (94) for this example
are
(112)
(113)
where .
Then, by introducing , the dynamics are given
by
(114)
To obtain the equilibrium point, set which results in
Since , it can be concluded that and
. Thus, the steady-state output map
(115)
has an extremum at ,soAssumption9holds.
Then, condition (103) is verified for
(116)
To this end, it should be shown that there exist a unitary vector
,, and persistently exciting function which satisfy the
inequality (103). First, the solution of (110) is obtained as the
following, which is required in PE verification of :
(117)
Next, is considered as a function that is per-
sistently exciting. Therefore, with this persistently exciting
and the unitary vector , the condition (103) holds as
follows:
Finally, to verify Assumption 11, the Levenberg–Marquardt
method (OA5) is used as the optimization scheme
(118)
Since , Condition 3 holds and therefore
based on Proposition 5, optimization algorithm (118) satisfies
Assumption 11.
Since all the assumptions of Theorem 3 hold, it can be con-
cluded that the system (107) is practically asymptotically stable
NEŠIĆet al.: A FRAMEWORK FOR EXTREMUM SEEKING CONTROL OF SYSTEMS WITH PARAMETER UNCERTAINTIES 447
Fig. 6. Reference signal , state , parameters estimation , input ,and
output .
and converges to the extremum of the performance function
(108).
Now, the behavior of the extremum seeking controller of the
system (107) with performance function (108) is illustrated in
simulated conditions. The simulation results are shown in Fig. 6
with the initial conditions
and . The controller (111) regulates the state
to the reference state in reference dynamics (110). In (110),
the reference input is produced by optimization scheme (118).
As mentioned before, the reference input with a bounded dither
signal is appended to provide some richness con-
dition on it. This is necessary to guarantee the convergence of
parameter estimate to its true value .Itisshownin
Fig. 6 that the output converges to its maximum value
at reference input .
Fig. 6 shows that the parameter estimates and converge
faster than optimization signal due to the time-scale separation
with . By increasing the value of the optimization
scheme can converge faster but at the expense of lower accu-
racy. The tuning parameter can even be set to 1 which removes
any time-scale separation between two systems (plant-estimator
and optimizer). However, in such a case, each of the two systems
has to satisfy ISS and small gain conditions as discussed in Re-
mark 13.
VII. CONCLUSION
A framework is proposed to design extremum seeking con-
trollers for a class of uncertain plants which are parameterized
with unknown parameters. This prescriptive framework pro-
vides an attractive way to combine a large class of optimization
methods with a large class of parameter estimation schemes to
achieve convergence of the closed-loop trajectories to the de-
sired extremum. The applications of the framework to ABS de-
sign and MRAC are used to illustrate the feasibility and flexi-
bility of the proposed framework.
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Dragan Nešić(S’96–M’02–SM’02–F’08) received
the B.E. degree in mechanical engineering from
The University of Belgrade, Belgrade, Yugoslavia,
in 1990 and the Ph.D. degree from Systems En-
gineering, RSISE, Australian National University,
Canberra, Australia, in 1997.
He is a Professor in the Department of Electrical
and Electronic Engineering (DEEE), The University
of Melbourne, Melbourne, Australia. Since February
1999, he has been with The University of Melbourne.
His research interests include networked control sys-
tems, discrete-time, sampled-data and continuous-time nonlinear control sys-
tems, input-to-state stability, extremum seeking control, applications of sym-
bolic computation in control theory, hybrid control systems, and so on.
Prof. Nešićwas awarded a Humboldt Research Fellowship (2003) by the
Alexander von Humboldt Foundation, an Australian Professorial Fellowship
(2004–2009), and Future Fellowship (2010–2014) by the Australian Research
Council. He is a Fellow of IEAust. He is currently a Distinguished Lecturer
of CSS, IEEE (2008-). He served as an Associate Editor for the journals Auto-
matica,IEEET
RANSACTIONS ON AUTOMATIC CONTROL,Systems and Control
Letters,andEuropean Journal of Control.
Alireza Mohammadi received the B.S. degree in
mechanical engineering from the Iran University of
Science and Technology, Tehran, Iran. and the M.Sc.
degree also in mechanical engineering from Sharif
University of Technology, Tehran, Iran, in 2005
and 2008, respectively. He is currently pursuing the
Ph.D. degree at the University of Melbourne, Mel-
bourne, Australia, performing research on extremum
seeking and its application to online optimization of
alternative fueled engines.
His research interests include extremum seeking,
engine control, and mechatronics.
Chris Manzie (M’04) received the B.S. degree in
physics and the B.E. degree (with honors) in elec-
trical and electronic engineering and the Ph.D. degree
from the University of Melbourne, Melbourne, Aus-
tralia, in 1996 and 2001, respectively.
Since 2003, he has been affiliated with the Depart-
ment of Mechanical Engineering, University of Mel-
bourne, where he is currently an Associate Professor
and an Australian Research Council Future Fellow.
He was a Visiting Scholar with the University of Cal-
ifornia, San Diego, in 2007. He has industry collabo-
rations with companies including Ford Australia, BAE Systems, ANCA M otion,
and Virtual Sailing. His research interests lie in applications of model-based and
extremum-seeking control in fields including mechatronics and energy systems.
Prof. Manzie is a member of the IFAC Technical Committees on Automotive
Control.
