switching control

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2012 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 9, OCTOBER 2008

Safe Adaptive Switching Control:Stability and Convergence

Margareta Stefanovic, Member, IEEE, and Michael G. Safonov, Fellow, IEEE

AbstractA formal theoretical explanation of the model-mis-match instability problem associated with certain adaptive controldesign schemes is proposed, and a solution is provided. To addressthe model-mismatch problem, a primary task of adaptive controlis formulated as finding an asymptotically optimal, stabilizingcontroller, given the feasibility of adaptive control problem. Aclass of data-driven cost functions called cost-detectable is intro-duced that detect evidence of instability without reference to priorplant models or plant assumptions. The problem of designingadaptive systems that are robustly immune to mismatch instabilityproblems is thus placed in a setting of a standard optimizationproblem. We call the result safe adaptive controlbecause it robustly

achieves adaptive stabilization goals whenever feasible, withoutprior assumptions on the plant model and, hence, without the riskof model-mismatch instability. The result improves the robustnessof previous results in hysteresis switching control, both for discreteand for continuously-parameterized candidate controller sets.Examples are provided.

Index TermsAdaptive control, convergence, robustness, sta-bility, switching.

I. INTRODUCTION

T

HE book Adaptive Control [2] begins in the followingway: In everyday language, to adapt means to change a

behavior to conform to new circumstances. Intuitively, an adap-tive controller is thus a controller that can modify its behavior inresponse to changes in the dynamics of the plant and the char-acter of the disturbances. Whether it is conventional, contin-uous adaptive tuning or more recent adaptive switching, adap-tive control has an inherent property that it orders controllersbased on evidence found in data. Any adaptive algorithm canthus be associated with a cost function, dependent on availabledata, that it minimizes, though this may not be explicitly present.The differences among adaptive algorithms arise in part dueto the specific algorithms employed to approximately computecost-minimizing controllers. And, major differences arise due

to the extent to which additional assumptions are tied with thiscost function. The cost function needs to be chosen to reflect

ManuscriptreceivedFebruary 6, 2007; revised January 30,2008. Current ver-sion published October 8, 2008. This work was supported in part by the AirForce Office of Scientific Research under Grant F49620-01-1-0302. Recom-mended by Associate Editor J. Hespanha.

M. Stefanovic is with the Department of Electrical and Computer En-gineering, University of Wyoming, Laramie, WY 82071 USA (e-mail:[email protected]).

M. G. Safonov is with the Department of Electrical Engineering-Systems,University of Southern California, Los Angeles, CA 90089-2563 USA (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2008.929395

control goals. The perspective adopted in this paper hinges onthe notion of feasibility of adaptive control. An adaptive con-trol problem is said to be feasible if the plant is stabilizable andat least one (a priori unknown) stabilizing controller exists inthe candidate controller set that achieves the specified controlgoal for the given plant. Given feasibility, our view of a pri-mary goal of adaptive control is to recognize when the accumu-lated experimental data shows that a controller fails to achievedesired stability and performance objectives. If a destabilizingcontroller happens to be the currently active one, adaptive con-

trol should eventually switch it out of the loop, and replace itwith an optimal, stabilizing one. An optimal controller is onethat optimizes the controller ordering criterion (cost function)given the currently available evidence. This perspective rendersthe adaptive control problem in a form of a standard constrainedoptimization. A concept similar to this feasibility notion can befound in [15].

To address the emerging need for robustness for larger uncer-tainties or achieve tighter performance specifications, severalrecent important advances have emerged, such as [13] andmulti-model controller switching formulations of the adaptivecontrol problem, e.g., supervisory based control design in [8],

[10], [18], and [19], or data-driven unfalsified adaptive controlmethods of [23] (based on criteria of falsifiability [20], [31],[32]), which exploit evidence in the plant output data to switcha controller out of the loop when the evidence proves that thecontroller is failing to achieve the stated goal. In both cases,the outer supervisory loop introduced to the baseline adaptivesystem allows fast discontinuous adaptation in highly uncer-tain nonlinear systems, and thus leads to improved performanceand overcomes some limitations of classical adaptive control.These formulations have led to improved optimization-basedadaptive control theories and, most importantly, significantlyweaker assumptions of prior knowledge. Both indirect [18],[19], [33] and direct [6], [15], [33] switching methods have

been proposed for the adaptive supervisory loop. Recently,performance issues in switching adaptive control have beenaddressed in robust multiple model adaptive control schemes(RMMAC) [3]. The results of this paper build on the result ofMorse et al. [17], [18] and Hespanha et al. [9][11], and widenthe theoretical ground in our paper [25] by allowing the class ofcandidate controllers to be infinite so as to allow considerationof continuously parameterized adaptive controllers, in additionto finite sets of controllers. It is shown that, under some mild ad-ditional assumptions on the cost function (designer-based, notplant-dependent), stability of the closed-loop switched systemis assured, as well as the convergence to a stabilizing controller

in finitely many steps. We show, via an example, that, when0018-9286/$25.00 2008 IEEE


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STEFANOVIC AND SAFONOV: SAFE ADAPTIVE SWITCHING CONTROL: STABILITY AND CONVERGENCE 2013

Fig. 1. Switching adaptive control system .

there is a mismatch between the true plant and the assumptions

on the plant (priors), a wrong ordering of the controllers canin some cases give preference to destabilizing controllers. This

phenomenon is called model mismatch instability. However,

prior knowledge and plant models, when they can be found,can be incorporated into the design of the candidate controllers,

which could be used together with the safe switching algorithm.

In certain ways, the paradigm of adaptive control problem

cast as a constrained optimization problem bears similarities

with the ideas found in machine learning algorithms [16],

[26]. As a direct, data-driven switching adaptive method, it

is more similar to the reinforcement learning algorithms than

supervised/unsupervised learning. In reinforcement learning,

the algorithm learns a policy of how to act given an observation

of the world. Every action has some impact in the environment,

and the environment provides feedback that guides the learning

algorithm.

The organization of the paper is as follows. Preliminary facts

are given in Section II. Section III contains the main results as

well as an example of the cost function satisfying sufficient con-ditions for stability and finiteness of switches. An illustrative ex-ample is presented in Section IV. Section V provides some finalconclusions and directions for future work.

II. PRELIMINARIES

Consider the adaptive system shown in

Fig. 1, where and are the plant input and output vector sig-

nals, and is the linear vector space of functions whose

norm, defined as , exists forany finite . For any , a truncation operatoris a linear projection operator that truncates the signal at .

The symbol will be used for the truncated signal [22].

The adaptive controller switches the currently active controller

at times with . For brevity,

we also denote the controller switched in the loop

during the time interval . If finite, the total numberof switches is denoted by , so the final switching time isand the final controller is .

We define the setwhere is an unknown plant. Unknown disturbances and noises

may also affect the plant relation . Let

represent the output data signals measured in one experiment,defined on the time-interval .

We consider a possibly infinite set (e.g., containing a con-tinuum) of the candidate controllers. The finite controller setcase is included as a special case. The parametrization of ,

denoted , will initially be taken to be a subset of ; the

treatment of the infinite dimensional spaces will be discussed inthe Comment 5.

Recall some familiar definitions from the stability theory. Afunction belongs in class if iscontinuous, strictly increasing and . The -norm of

a truncated signal is given as .

The Euclidean norm of the parameterization of the

controller is denoted . A functional is said

to be coercive [4] if when .

Definition 1: A system with input and

output is said to be stable if for every such input

there exist constants such that

(1)

Otherwise, is said to be unstable. Furthermore, if (1) holdswith a single pair for all , then the system

is said to be finite-gain stable, in which case the gain of is the

least such .

Comment 1: In general, can depend on the initial state.

Specializing to the system in Fig. 1, and (without loss of gen-

erality) disregarding , stability of the closed-loop system

means that for every , there exist such that

.

Definition 2: The system is said to be incrementally stable

if, for every pair of inputs and outputs

, there exist constants such that

(2)

and the incremental gain of , when it exists, is the least

satisfying (2) for some and all .

Definition 3: The adaptive control problem is said to be fea-

sible if a candidate controller set contains at least one con-

troller such that the system is stable. A controller

is said to be a feasible controller if the system is

stable.

Safe adaptive control problem goal is then formulated as

finding an asymptotically optimal, stabilizing controller, giventhe feasibility of the adaptive control problem. Under this con-

dition, safe adaptive control should recognize a destabilizingcontroller currently present in the loop, and replace it with an

as-yet-unfalsified controller. Hence, we have the following (andonly) assumption on the plant that we will use in this paper.

Assumption 1: The adaptive control problem is feasible.

A similar notion of the safe adaptive control appears in [1],

where Anderson et al. define the safe adaptive switching controlas one that always yields stable frozen closed-loop; the solution

to this problem is achieved using the -gap metric. Limitations

of the -gap metric are discussed in [12].

Prevention from inserting a destabilizing controller in the

loop is not assured (since the adaptive switching system cannot

identify with certainty a destabilizing controller beforehand,

based on the past data), but if such a controller is selected, itwill quickly be switched out as soon as the unstable modes


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are excited. Under the feasibility condition, an unfalsifiedcontroller will always be found and placed in the loop. Whether

the optimal, robustly stabilizing controller will eventually be

found and connected, depends on whether the unstable modes

are sufficiently excited.It follows from the above definition of the adaptive control

goal that feasibility is a necessary condition for the existenceof a particular that robustly solves the safe adaptive

control problem. This paper aims to show that feasibility is also

a sufficient condition to design a robustly stable adaptive systemthat converges to a , even when it is not known a priori

which controllers in the set are stabilizing.

Definition 4: Stability of the system is said to

be unfalsified by the data if there exist such

that (1) holds; otherwise, we say that stability of the system

is falsified by .

Unfalsified stability is determined from (1) based on the datafrom one experiment for one input, while stability requiresadditionally that (1) hold for the data from every possible input.

Any adaptive control scheme has a cost index inherently tiedto it, which orders controllers based on evidence found in data.

This index is taken here to be a cost functional , de-

fined as a causal-in-time mapping

(3)

An example of the cost function according to the above defi-nition, which satisfies the desired properties introduced later inthe text, is given in Section III-A. Various examples of the cost

are discussed in [24], [29], among others.

The switched system comprised of the plant and the cur-

rently active controller , where is de-

noted (Fig. 1). For the switched system inFig. 1, the is defined as

, where .

Definition 5: Given the pair , a controller

is said to be falsified at time by the past measurement if. Otherwise it is said to

be unfalsified at time by .Then, a robust optimal controller is one that stabilizes

(in the sense of the Def. 1) the given plant and minimizes the

true cost .

Therefore, (and is not nec-

essarily unique). Due to the feasibility assumption, at least one

such exists, and .Definition 6: [23] For every , a fictitious reference

signal is defined to be an element of

In other words, is a hypothetical reference signal that

would have exactly reproduced the measured data had the

controller been in the loop for the entire time period over

which the data was collected.

Definition 7: [28] When for each and there is a unique

, then we say is causally left invertible (CLI)

and we denote by the induced causal map . The

causal left inverse is called the fictitious reference signalgenerator(FRS) for the controller . When is incremen-

tally stable, is called stably causally left invertible controller

(SCLI).

Definition 8: Let denote the input and

denote the resulting plant data collected with as the current

controller. Consider the adaptive control system of

Fig. 1 with input and output . The pair is said to

be cost-detectable if, without any assumption on the plantand for every with finitely many switching times, thefollowing statements are equivalent:

1) is bounded as increases to infinity;2) stability of the system is unfalsified by the input-

output pair .

Comment 2: With cost-detectability satisfied, we can use thecost to reliably detect any instability exhibited by

the adaptive system, even when initially the plant is completely

unknown.

Comment 3: Cost-detectability is different from the plant de-

tectability. Cost-detectability is determined from the knowledge

of the cost function and candidate controllers, without refer-

ence to the plant. In [9], a problem similar to ours is approachedusing the following assumptions: 1) the plant itself is detectable

and 2) the candidate plant models are stabilized by the corre-

sponding candidate controllers. The difference between the ap-

proach in [9], [7] and this paper lies in the definition of cost-de-tectability introduced in this paper, which is the property of the

cost function/candidate-controller-set pair, but is independent of

the plant.

In the following, we use the notation

for a family of functionals

with the common domain , with .

Let

denote the level set in the controller space corresponding tothe cost at the first switching time instant. With the family offunctionals with a common domain , a restriction to the

set is associated, defined as a family of functionalswith a common domain .

Thus, is identical to on , and is equal to

outside .

Consider now the cost minimization hysteresis switching

algorithm reported in [17], together with the cost functional

. The algorithm returns, at each , a controller

which is the active controller in the loop:

-HYSTERESIS SWITCHING ALGORITHM A1 [17]

where is the Kroneckers , and is the limit of frombelow as .

The switch occurs only when the current unfalsified cost re-lated to the currently active controller exceeds the minimum

(over ) of the current unfalsified cost by at least (Fig. 2).The hysteresis step serves to limit the number of switches on

any finite time interval to a finite number, and so prevents thepossibility of the limit cycle type of instability. It also ensures a

nonzero dwell time between switches.

The hysteresis switching lemma of [17] implies that a

switched sequence of controllers that min-imize (over ) the current unfalsified cost at each


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Fig. 2. Cost versus control gain time snapshots.

switch-time , will also stabilize the plant if the cost related to

each fixed controller has the following properties: first, it isa monotone increasing function of time and second, it is uni-

formly bounded above if and only if is stabilizing. But, these

properties were demonstrated for the cost functions

in [17] only by introducing prior assumptions on the plant,

thereby also introducing the possibility of model-mismatch

instability.

Definition 9: [30]. Let be a topological space. A family

of complex functionals with a common

domain is said to be equicontinuous at a point if for

every there exists an open neighborhood such that

. The family is

said to be equicontinuous on if it is equicontinuous at each

. is said to be uniformly equicontinuous on if

such that

, where denotes an open neighborhoodof size .

In a metric space with a metric , uniform equiconti-

nuity means that

.

Lemma 1: If isa compactmetricspace, then any family

that is equicontinuous on is uniformly

equicontinuous on .

Proof: In [21].

III. MAIN RESULT

The results on stability and finiteness of switches are devel-

oped in the sequel.Lemma 2: Consider the feedback adaptive control system

inFig. 1 withinput (generality is not lostif istakeninsteadof

the input ) and output , together with

the hysteresis switching algorithm A1. Suppose there are finitelymany switches. If the adaptive control problem is feasible (Def.

3), candidate controllers are SCLI, and the following properties

are satisfied: is cost-detectable (Def. 8) is monotone increasing in time

then the final switched controller is stabilizing. Moreover, thesystem response with the final controller satisfies the perfor-mance inequality

Proof: It suffices to consider the final controller . De-note the final switching time instant . Then, by the definitionof , and feasibility of the control problem (Def. 3), it

follows that for all

(4)

(5)

Further, by monotonicity in of , it follows that (5)

holds for all . Due to the cost-detectability, stability of

with is not falsified by , that is, there exist constantscorresponding to the given such that

(6)

According to Lemma 6 in Appendix A, there exist

such that . This, along with (6),

implies , for some .

Lemma 3: Let be a continuous and coercive

function on . Then for any scalar , the levelset

is compact.

Proof: Since , we show that is closed and

bounded: Let be a convergent sequence, and

. Since is continuous, .

Also, . Then,

, so . Hence, is closed. To

show that is is bounded, proceed by contradiction. As-

sume that is not bounded; then there exists a sequence

such that . Since is co-

ercive, ; in particular, such that

, for any fixed . Then, ,

which contradicts the above assumption. Thus, is closedand bounded in , therefore compact.

Lemma 4: Consider the feedback-adaptive control system in

Fig. 1, together with the switching algorithm A1. If the adap-

tive control problem is feasible (Def. 3), and the associated

cost functional/controller set pair is cost-detectable,

is monotone increasing in time and, in addition:

For all , the cost functional iscoercive on

(i.e., ), and

The family of restricted cost functionals with a common domain

is equicontinuous on ,

then the number of switches is uniformly bounded above for all

by some .

Proof: Our proof is similar to the convergence lemmas of

[10], [17]. A graphical representation of the switching process,

giving insight to the derivation presented below, is shown in

Fig. 3. Due to Lemma 3, the level set is compact. Then,

the family is uniformly

equicontinuous on (Lemma 1), i.e., for a hysteresis step

such that for all

(i.e.,

is common to all and all ). Since is

compact, there exists a finite open cover ,with such that ,


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Fig. 3. Derivation of the upper bound on switches in the continuum case.

where depends on the chosen hysteresis step (this is a di-

rect consequence of the definition of a compact set). Let bethe controller switched into the loop at the time , and the corre-sponding minimum cost achieved is .

Consider that at the time a switch occurs at the

same cost level , i.e., where

. Therefore,

is falsified, and so are all the controllers . Letbe the index set of the as-yet-unfalsified -balls of controllersat the time . Since , for some

( is not necessarily a singleton as may belong to more

than one balls , but it suf fices for the proof that thereis at least one such index ), also falsified are all the controllers

, so that . Thus, is up-

dated according to the following algorithm ( is the index of the

switching time ).

Unfalsified index set algorithm:

1) Initialize: Let

2) . If : Set //

Optimal cost increases

Else ,where is such that

.

3) go to (2)

Therefore, the number of possible switches to a single

cost level is upper-bounded by , the number of -balls

in the cover of . The next switch (the very first after theth one), if any, must occur to a cost level higher than

, due to the monotonicity of . But then, according

to Algorithm 1, ,

with and

. Combining the two bounds, the overall

number of switches is upper-bounded by

Equicontinuity assures that the cost functionals in the said

family have associated -balls offinite, non-zero radii, which isused to upper bound the number of switches. If holds,

the set is compact; otherwise, an additional requirement that

the set is compact is needed. The finite controller set caseis obtained as a special case of the Lemma 4, with being the

number of candidate controllers instead of the number of -ballsin the cover of . The main result follows.

Theorem 1: Consider the feedback adaptive control system

in Fig. 1, together with the hysteresis switching algorithm A1.

Suppose that the adaptive control problem is feasible, the asso-

ciated cost functional is monotone in time, the pair

is cost-detectable, candidate controllers are SCLI, and

the conditions of Lemma 4 hold. Then, the switched closed-loop

system is stable, according to Def. 1. In addition, for each , the

system converges after finitely many switches to the controllerthat satisfies the performance inequality

(7)

Proof: Invoking Lemma 4 proves that there are finitelymany switches. Then, Lemma 2 shows that the adaptive con-

troller stabilizes, according to Def. 1, and that (7) holds.

Comment 4: Note that, due to the coerciveness of

is bounded below (by a nonnegative

number, if the range of is a subset of ), for all .

Comment 5: The parametrization of the candidate controller

set can be more general than ; in fact, it can belong

to an arbitrary infinite dimensional space; however has to becompact in that case, in order to ensure uniform equi-continuity

property.

Note that the switching ceases after finitely many steps for all. If the system input is sufficiently rich so as to increase

the cost more than above the level at the time of the latest

switch, a switch to a new controller that minimizes the current

cost will eventually occur at some later time. The values of these

cost minima at any time are monotone increasing and bounded

above by . Thus, suf ficient richness of the systeminput (external reference signal, disturbance or noise signals)

will affect the cost to approach .

Comment 6: The minimization of the cost functional over the

infiniteset istractableif the compact set can berepresentedas a finite union of convex sets, i.e., the cost minimization is aconvex programming problem.

A. Cost Function Example

An example of the cost function and the conditions under

which it ensures stability and finiteness of switches accordingto Theorem 1 may be constructed as follows. Consider (a not

necessarily zero-input zero-output) system in

Fig. 1. Choose as a cost functional

(8)

where are arbitrary positive numbers. The constant is

used to prevent division by zero when (unless

has zero-input zero-output property), ensures evenwhen , and ensures coerciveness of .


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Fig. 4. MFD ofa controller inclosed loopwiththe plant [14]:(a) feedbackloop

with the current controller written in MFD form and (b) fictitious feedbackloop associated with the candidate controller written in MFD form (in bothcases, are the actually recorded data).

Alternatively, in order to avoid the restriction to the minimum

phase (SCLI) controllers (which would assure causality and in-

cremental stability of the map ), the denominator of

(8) can contain instead of [14], [5], where is definedvia the matrix fraction description (MFD) form of the controller

, as and

(Appendix B), where the mentioned signals are shown in Fig. 4

(9)

Both (8) and (9) satisfy the required properties of Theorem 2,

i.e.,: monotonicity in time, coerciveness on , equicontinuity

of the restricted cost family , and cost detectability. The firsttwo properties are evident by inspection of (8) and (9). The jus-

tification for the last two properties is as follows.Since in (8) and (9) is continuous in , then ,

defined as the family of the cost functionals restrictedto the level set , is equicon-

tinuous (since for any and are either equal to , or

clamped at ).

Lemma 5: Consider the cost functions (8) and (9) with

. For to be cost-detectable, it is sufficientthat the candidate controllers in the set are SCLI, or that

they admit matrix fraction description (MFD) form considered

in [14].

Proof: Cost-detectability of with in (8) follows

from the following: 1) the fact that is bounded asif and only if stability is unfalsified by the input-output

pair [28]; 2) SCLI property of the controllers; 3) sta-

bility of the mapping (Lemma 6 in Appendix A); and

4) unfalsified stability by the data (see Appendix C).These results can be elaborated further using [14] for the class

of non-SCLI controllers and the cost function (9), which also

ensure internal stability of the adaptive system designed usingcost-detectable cost-functions of the forms (8) or (9).

IV. SIMULATION EXAMPLE

The algorithm A1 in Section II originated as the hysteresis

switching algorithm in [17]. We emphasized that the power ofthe hysteresis switching lemma was clouded in the cited work by

imposing unnecessary assumptions on the plant in the demon-

strations of the algorithm functionality. One of the plant prop-

erties required in [17] for ensuring tunability was the min-imum phase property of the plant. We have shown in theory

that the cost detectability is assured by properly choosing a cost

function, and is not dependent on the plant or exogenous sig-

nals. In the following, we present a simulation example thatdemonstrates these findings. Assume that a true, unknown planttransfer function is given by .

Given is the set of three candidate controllers:

and , each of which

stabilizes a different possible plant model. The task of the

adaptive control is to select one of these controllers, based on

observed data. The problem is complicated by the fact that in

this case, as is often the case in practice, the true plant is

not in the model set, i.e., there exists a model mismatch.A simple analysis of the nonswitched system (true plant in

feedback with each of the controllers separately) shows that

is stabilizing (yielding a nonminimum phase but stable closed

loop) while and are destabilizing. Next, a simulation was

performed of a switched system, where A1 was used to select

the optimal controller, and the cost function was chosen to be a

combination of the instantaneous error and a weighted accumu-

lated error

(10)

where is the fictitious error of the th controller, defined as

(11)

and and , where is astable, minimum phase reference model. This is the same cost

function used in the multiple model switching adaptive control

scheme [19], with replaced by , the identification error ofthe th plant model [for the special case of the candidate con-

trollers designed based on the MRAC method, is equivalent

to the control error and to the fictitious error (11)].The simulations assume a band-limited white noise at the

plant output and the unit-magnitude square reference signal. The

stabilizing controller has initially been placed in the loop,

and the switching, which would normally occur as soon as the

logic condition of Algorithm A1 is met, is suppressed during the

initial 5 s of the simulation. That is, the adaptive control loop isnot closed until time s. The reason for waiting some pe-

riod of time before engaging the switching algorithm will be

explained shortly. The forgetting factor is chosen to be 0.05.

Figs. 5(a) and 6(a) show the cost dynamics and the reference

and plant outputs, respectively.

Soon after the switching was allowed (after ), the al-

gorithm using cost function (10) discarded the stabilizing con-

troller initially placed into the loop and latched onto a desta-

bilizing one, despite the evidence of instability found in data.

Even though the stabilizing controller was initially placed in the

loop, and forced to stay there for some time (5 s in this case), as

soon as the switching according to (10) was allowed, it promptly

switched to a destabilizing one. This model-match instabilityhappens because the cost function (10) is not cost-detectable.


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Fig. 5. Without cost-detectability, the optimal cost-minimizing controller

destabilizes the system. (a) Non-cost-detectable case; cost (10). (b) Cost-de-tectable case; cost (12).

Note that the initial idle period of 5s is used only to empha-

size that, even when the data are accumulated with the stabi-

lizing controller, the switching algorithm based on (10) can dis-

regard this data, and latch onto a destabilizing controller. Thisidle period is not the dwell time in the same sense used in the

dwell-time switching control literature.

Next, a simulation was performed of the same system, but

using a cost-detectable cost function (i.e., one that satisfies con-ditions of Theorem 1)

(12)

viz., an -gain type cost function (factor added for co-

erciveness is not necessary, since the set of candidate controllers

is finite in this example). Cost-detectability of followsfrom Lemma 5. The modified fictitious reference signal isused, instead of , due to the presence of the non-minimum

Fig. 6. Switching using cost function (a) (10) and (b) (12). Reference and plantoutputs. (a) Non-cost-detectable case; (10), (b) Cost-detectable case; (10).

phase controller . It is calculated from the on-line data as

[according to (9)], where

and for ; and

for ; and and for .

The corresponding simulation results are shown in Figs. 5(b)

and 6(b). The initial controller was chosen to be (a desta-

bilizing one). The constant was chosen to be 0.01. The fore-

going example shows that closing an adaptive loop that is de-signed using a non-cost-detectable cost function like (10) can

destabilize, even when the initial controller is stabilizing. In the

example, this happens because there is a large mismatch be-

tween the true plant and the plant models used to design the

candidate controllers. On the other hand, Theorem 1 implies,

and the example confirms, that such model-mismatch instabilitycannot occur when the adaptive control loop is designed using

the cost-detectable -gain type cost function (12).

V. CONCLUSION

The goal of stabilizing an uncertain plant by means of

switching through an infinite candidate controller set is solvedin the paper, provided that feasibility (existence of at least


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Fig. 7. Generators of the true and fictitious reference signals.

one stabilizing solution in the candidate controller set) holds.

Noting that every adaptive scheme is optimal with respect to

some data-driven controller-ordering cost function, sufficientconditions are derived on the data-driven cost function to ensure

stability and performance. To this end, we have re-examined

the Morse-Mayne-Goodwin hysteresis algorithm for adaptivecontrol from the perspective of unfalsified control theory. Anupper bound on the number of switches for a general continuum

controller set case is calculated. The result is a solution to the

problem of model mismatch instability. Our cost- detectability

approach to the adaptivecontrollerswitching robustly eliminates

the possibility of model-mismatch instability that inevitably

accompanies the introduction of any prior assumption on the

plant. It does so by replacing the plant-dependent/controller-

independent tunability requirements with the plant-independent/

controller-dependent cost-detectability requirements.

In the present paper, our focus has been on how a given can-

didate controller set should be pruned based on data in orderto adaptively converge to a controller in the candidate set that

achieves and maintains stable behavior and acceptable perfor-

mance. Though this data driven process requires no prior knowl-

edge of the plant model, this does not mean that our formulation

is model-free. Plant models, when available, play an essential

role in determining and designing the candidate controller set

. The issue of how one might optimally utilize data for con-

tinuously and adaptively generating new candidate controllers

on the fly remains an open question.

APPENDIX A

RELATION BETWEEN AND

Lemma 6: Consider the switching feedback adaptive control

system (Fig. 1), where uniformly bounded reference input

, as well as the output are given. Suppose there

are finitely many switches. Let and denote the finalswitching instant and the final switched controller, respectively.Suppose that the final controller is SCLI (i.e., the fictitiousreference signal is unique and incrementally stable).

Then

(13)

Proof: By the assumption there are finitely many switches.Consider the control configuration in Fig. 7. The top branch gen-

erates the fictitious reference signal of the controller . Its in-puts are the measured data , and its output is . The

output is generated by the fictitious reference signal generatorfor the controller , denoted . Inthe middle interconnec-

tion, the signal , generated asthe outputof the final controllerexcited by the actual applied signals and , is simply in-

verted by passing through the causal left inverse . Finally,the bottom interconnection has the identical structure as the top

interconnection (series connection of and ), except that

it should generate the actually applied reference signal . To this

end, another input to the bottom interconnection is added (de-

noted ), as shown in Fig. 7. This additional input can be

thought of as a compensating (bias) signal, that accounts for the

difference between the subsystems generating and before

the time of the last switch. In particular, it can be shown (as seen

in Fig. 7) that (due to the fact that

).

By definition, is incrementally stable. Thus, there existconstants such that

Whence by the triangle inequality for norms, inequality (13)

holds with

APPENDIX BDefinition 10: [27]. The ordered pair is a left ma-

trix fraction description (MFD) of a controller if and

are stable, is invertible, and .

Comment 7: To avoid restricting our attention to only those

controllers that are stably causally left invertible. i.e., controllers

whose FRS generator is stable, we can use the MFD represen-

tation of the controllers and write a modifiedfictitious reference

signal as and [14].

Similarly, would represent the modified applied ref-erence signal , that is related to the active controller in the loop

. Thus, although may not be stable, which is the case with

nonminimum phase controllers, is, by construction.

APPENDIX C

UNFALSIFIED STABILITY VERIFICATION FOR

THE COST FUNCTION (8) IN SECTION III-A

Recall that the controller switched in the loop at time is

denoted ; and are the

indices of the switching instants. When , we have

. Let the controller switched

at time be denoted (so, ). Then, due to the

cost minimization property of the switching algorithm,

, and .

Denote by the time of the final switch, and the cor-responding controller . Consider the time interval .


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During this time period, the active controller in the loop is

since .

Since ris uniformly bounded, .

At , the cost of the current controller exceeds the cur-

rent minimum by

(14)

and so, according to the hysteresis switching algorithm, a switch

occurs to the controller . Expres-sion in (14) is finite since is finite and

where is finite due to the feasibility assumption.Denoting the sum by , we have

Now considerthe nextswitchingperiod, . The activecon-

troller in the loop is . We have:

where the second inequality from the left follows from the

monotone increasing property of .

Therefore, . Now

Thus, is finite, and so are . By induction,we conclude that

where is the final switching time. Since was the finalswitching time

Thus, is finite for any finite

From here we conclude that stability of the closed-loop switched

system with the final controller is unfalsified by .

ACKNOWLEDGMENT

The authors would like to thank Prof. J. Hespanha for

insightful comments and referral to the hysteresis switching

lemma of Morse et al. [17] and the related references. They

also thank anonymous reviewers for their valuable comments

that helped improve the manuscript quality.

REFERENCES

[1] B. D. O. Anderson, T. S. Brinsmead, D. Liberzon, and A. S. Morse,

Multiple model adaptive control with safe switching, Int. J. Adapt.Control and Signal Process., vol. 15, pp. 445470, 2001.

[2] K. J. strm and B. Wittenmark, Adaptive Control. Reading, MA:Addison-Wesley, 1995 .

[3] M. Athans et al., Issues, progress and new results in robust adaptivecontrol, Int. J. Adapt. Control Signal Process., vol. 20, pp. 519579,Dec. 2006.

[4] D. P. Bertsekas, Nonlinear Programming, 2nd ed. Nashua, NH:Athena, 1999.

[5] A. Dehghani, B. D. O. Anderson, and A. Lanzon, Unfalsified adap-tive control: A new controller implementation and some remarks, in

European Control Conf. ECC07, Kos, Greece, Jul. 2007.[6] M. Fu and B. R. Barmish, Adaptive stabilization of linear systems via

switching control, IEEE Trans. Automat. Control, vol. AC-31, no. 12,pp. 10971103, Dec. 1986.

[7] J. P. Hespanha and A. S. Morse, Certainty equivalence implies de-tectability, Syst. Contr. Lettr., vol. 1, no. 1, pp. 113, Jan. 1999.

[8] J. P. Hespanha, Logic-Based Switching Algorithms in Control, Ph.D.dissertation, Dept. Elect. Eng., Yale University, New Haven, CT, 1998.

[9] J.P. Hespanha, D.Liberzon, and A.S. Morse, Supervision of integral-input-to-state stabilizing controllers, Automatica, vol. 38, no. 8, pp.13271335, Aug. 2002.

[10] J. P. Hespanha, D. Liberzon, and A. S. Morse, Hysteresis-basedswitching algorithms for supervisory control of uncertain systems,

Automatica, vol. 39, no. 2, Feb. 2003.[11] J. P. Hespanha, D. Liberzon, and A. S. Morse, Overcoming the lim-

itations of adaptive control by means of logic-based switching, Syst.Control Lett., vol. 49, pp. 4965, 2003.

[12] G. C. Hsieh and M. G. Safonov, Conservatism of the gap metric,IEEE Trans. Automat. Control, vol. 38, no. 4, pp. 594598, Apr. 1993.

[13] E. Kosmatopoulos and P. A. Ioannou, Robust switching adaptive con-trol of multi-input nonlinear systems, IEEE Trans. Automat. Control,vol. 47, no. 4, pp. 610624, Apr. 2002.


10/10


[14] C. Manuelli, S. G. Cheong, E. Mosca, and M. G. Safonov, Stability ofunfalsified adaptive control with non SCLI controllers and related per-formance under different prior knowledge, in Proc. European ControlConf. (ECC07), Kos, Greece, Jul. 2007.

[15] B. Mrtensson, The order of any stabilizing regulator is sufficient in-formation for adaptive stabilization, Syst. Control Lettr., vol. 6, no. 2,pp. 8791, 1985.

[16] T. Mitchell, Machine Learning. New York: McGraw-Hill, 1997.

[17] A. S. Morse, D. Q. Mayne, and G. C. Goodwin, Applications of hys-teresis switching in parameter adaptive control, IEEE Trans. Automat.Control, vol. 37, no. 9, pp. 13431354, Sep. 1992.

[18] A. S. Morse, Supervisory control of families of linear set-point con-trollersPart I: Exact matching, IEEE Trans. Automat. Control, vol.41, no. 10, pp. 14131431, Oct. 1996.

[19] K. S. Narendra and J. Balakrishnan, Adaptive control using multiplemodels, IEEE Trans. Automat. Control, vol. 42, no. 2, pp. 171187,Feb. 1997.

[20] K. R. Popper, Logic of Scientific Discovery 1934.[21] W. Rudin, Principles of Mathematical Analysis, 3rd ed. New York:

McGraw-Hill, 1976.[22] M. G. Safonov, Stability and Robustness in Multivariable Feedback

Systems. Cambridge, MA: MIT Press, 1980, Signal Processing, Op-timization and Control.

[23] M. G. Safonov and T. Tsao, The unfalsified control concept andlearning, IEEE Trans. Automat. Control, vol. 42, no. 6, pp. 843847,

Jun. 1997.[24] M. Stefanovic, Safe Switching Adaptive Control: Stability and Con-

vergence, Ph.D. dissertation, Univ. Southern California, Los Angeles,2005.

[25] M. Stefanovic, R. Wang, and M. G. Safonov, Stability and conver-gence in adaptive systems, in Proc. American Control Conf., Boston,MA, 2004.

[26] R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduc-tion. Cambridge, MA: MIT Press, 1998.

[27] M. Vidyasagar, Control System Synthesis: A Factorization Ap-proach. Cambridge, MA: The MIT Press, Sep. 1988.

[28] R. Wang, A. Paul, M. Stefanovic, and M. G. Safonov, Cost-de-tectability and stability of adaptive control systems, in Proc. 44th

IEEE Conf. Decision and Control, Seville, Spain, Dec. 2005.[29] R. Wang, Cost Detectability and Safe MCAC, Ph.D. dissertation,

Univ. Southern California, Los Angeles, 2005.[30] R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduc-

tion to Real Analysis. New York: Marcel Dekker, 1977.[31] J. C. Willems, Paradigms and puzzles in the theory of dynamical sys-

tems, IEEE Trans. Automat. Control, vol. 36, no. 3, pp.259294, Mar.1991.

[32] G. Zames, On the input-output stability of nonlinear time-varyingfeedback systems, part 1: Conditions derived using concepts of loopgain, conicity and positivity, IEEE Trans. Automat. Control, vol.AC-11, pp. 465476, Apr. 1966.

[33] P. Zhivoglyadov, R. H. Middleton, and M. Fu, Localization basedswitching adaptive control for time-varying discrete-time systems,

IEEE Trans. Automat. Control, vol. 45, no. 4, pp. 5275, Apr. 2000.

Margareta Stefanovic (S03M05) received theDipl.-Ing. degree from the University of Nis, Yu-goslavia, in 1996, and the M.Sc. and Ph.D. degreesin electrical engineering from the University ofSouthern California, Los Angeles, in 2002 and 2005,respectively.

Shejoined theUniversityof Wyoming, Laramie, in2005 as an Assistant Professor in the Electrical and

Computer Engineering Department. Her current re-search interests include switching adaptive control ofuncertain systems and coordinated control and deci-

sion making in hyper-spectral imaging nano-satellite networks for Space Situ-ational Awareness.

Michael G. Safonov (F89) was born in Pasadena,CA, on November 1, 1948. He received the B.S.,M.S., Eng., and Ph.D. degrees in electrical en-gineering from the Massachusetts Institute ofTechnology, Cambridge, in 1971, 1971, 1976, and1977, respectively.

From 1972 to 1975, he served with the U.S. Navyas an Electronics Division Officer aboard the aircraftcarrier U.S.S. Franklin D. Roosevelt (CVA-42).Since1977, he has been with the University of SouthernCalifornia, Los Angeles, where he is presently a

Professor of Electrical Engineering. He has been a Consultant to The AnalyticSciences Corp., Honeywell Systems and Research Center, Systems Control,Systems Control Technology, Scientific Systems, United Technologies, TRW,Northrop Aircraft, Hughes Aircraft, and others. His consulting and universityresearch activities have involved flight control system design studies in whichmodern robust multivariable control techniques were applied to a variety ofaircraft including the CH-47 Chinook helicopter (Analytic Sciences Corp.,1976), the NASA HiMAT aircraft (Honeywell/USC, 1980) and the F/A-18Hornet (Northrop, 19871991). During the academic year 19831984, he wasa Senior Visiting Fellow with the Department of Engineering, Cambridge Uni-versity, Cambridge, U.K., and in summer 1987 he held a similar appointment atImperial College of Science and Technology, London, U.K. During 1990 1991,he was a Visiting Faculty Member at Caltech, Pasadena, CA. He has authoredor co-authored more than 150 journal and conference papers and the book

Stability and Robustness of Multivariable Feedback Systems (Cambridge, MA:MIT Press, 1980). Additionally, he is co-author of the MATLAB Robust ControlToolbox (Natick, MA: MathWorks), a software package for use with MATLAB.He is presently an Editor of the International Journal of Robust and NonlinearControl and Systems and Control Letters. His research interests include robustcontrol, infinity-norm optimal control theory, and nonlinear system theory withapplications to aerospace control design problems.

Dr. Safonov served as an Associate Editor of the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL from 1985 to 1987.He was elected IFAC Fellowin 2008.From 1993 to1995, he was Chair of the AACC Awards Committee of the Amer-ican Automatic Control Council.

switching control

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