# Systems & Control Letters

Published by Elsevier

Print ISSN: 0167-6911

Published by Elsevier

Print ISSN: 0167-6911

Publications

After stimulation by chemoattractant, Dictyostelium cells exhibit a rapid response. The concentrations of several intracellular proteins rise rapidly reaching their maximum levels approximately 5-10 seconds, after which they return to prestimulus levels. This response, which is found in many other chemotaxing cells, is an example of a step disturbance rejection, a process known to biologists as perfect adaptation. Unlike other cells, however, the initial first peak observed in the chemoattractant-induced response of Dictyostelium cells is then followed by a slower, smaller phase peaking approximately one to two minutes after the stimulus. Until recently, the nature of this biphasic response has been poorly understood. Moreover, the origin for the second phase is unknown. In this paper we conjecture the existence of a feedback path between the response and stimulus. Using a mathematical model of the chemoattractant-induced response in cells, and standard tools from control engineering, we show that positive feedback may elicit this second peak.

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In our previous work, we investigated detectability of discrete event systems, which is defined as the ability to determine the current and subsequent states of a system based on observation. For different applications, we defined four types of detectabilities: (weak) detectability, strong detectability, (weak) periodic detectability, and strong periodic detectability. In this paper, we extend our results in three aspects. (1) We extend detectability from deterministic systems to nondeterministic systems. Such a generalization is necessary because there are many systems that need to be modeled as nondeterministic discrete event systems. (2) We develop polynomial algorithms to check strong detectability. The previous algorithms are based on observer whose construction is of exponential complexity, while the new algorithms are based on a new automaton called detector. (3) We extend detectability to D-detectability. While detectability requires determining the exact state of a system, D-detectability relaxes this requirement by asking only to distinguish certain pairs of states. With these extensions, the theory on detectability of discrete event systems becomes more applicable in solving many practical problems.

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Our previous work considers detectability of discrete event systems which is to determine the current state and subsequent states of a system based on event observation. We assume that event observation is static, that is, if an event is observable, then all its occurrences are observable. However, in practical systems such as sensor networks, event observation often needs to be dynamic, that is, the occurrences of same events may or may not be observable, depending on the state of the system. In this paper, we generalize static event observation into dynamic event observation and consider the detectability problem under dynamic event observation. We define four types of detectabilities. To check detectabilities, we construct the observer with exponential complexity. To reduce computational complexity, we can also construct a detector with polynomial complexity to check strong detectabilities. Dynamic event observation can be implemented in two possible ways: a passive observation and an active observation. For the active observation, we discuss how to find minimal event observation policies that preserve four types of detectabilities respectively.

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This paper studies monotone tridiagonal systems with negative feedback. These systems possess the Poincaré-Bendixson property, which implies that, if orbits are bounded, if there is a unique steady state and this unique steady state is asymptotically stable, and if one can rule out periodic orbits, then the steady state is globally asymptotically stable. Two different approaches are discussed to rule out period orbits, one based on direct linearization and another one based on the theory of second additive compound matrices. Among the examples that illustrate the theoretical results is the classical Goldbeter model of the circadian rhythm.

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We present a method for estimating the domain of attraction for a
discrete-time linear system under a saturated linear feedback. A simple
condition is derived in terms of an auxiliary feedback matrix for
determining if a given ellipsoid is contractively invariant. Moreover,
the condition can be expressed as LMI in terms of all the varying
parameters and hence can easily be used for controller synthesis. The
following surprising result is revealed for systems with single input:
suppose that an ellipsoid is made invariant with a linear feedback, then
it is invariant under the saturated linear feedback if and only if it
can be made invariant with any saturated (nonlinear) feedback

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This paper carries out an analysis of the L<sub>2</sub> gain and L<sub>infin</sub> performance for singular linear systems under actuator saturation. The notion of bounded state stability (BSS) with respect to the influence of L<sub>2</sub> or L<sub>infin</sub> disturbances is introduced and conditions under which a system is bounded state stable are established in terms of linear matrix inequalities (LMIs). The disturbance tolerance capability of the system is then measured as the bound on the L<sub>2</sub> or L<sub>infin</sub> norm of the disturbances under which the system remains bounded state stable and the disturbance rejection capability is measured by the restricted L<sub>2</sub> gain from the disturbance to the system output or L<sub>infin</sub> norm of the system output. Based on the BSS conditions, the assessment of the disturbance tolerance and rejection capabilities of the system under a given state feedback law is formulated and solved as LMI constrained optimization problems. By viewing the feedback gain as an additional variable, these optimization problems can be readily adapted for control design. Our analysis and design reduce to the existing results for regular linear systems in the degenerate case where the singular linear system reduces to a regular system, and to the existing results for singular systems in the absence of actuator saturation or when the disturbance is weak enough to not cause saturation.

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It is shown that, if a linear system is asymptotically null
controllable with bounded controls, then, when actuators are arranged in
daisy chains and are subject to both position and rate saturation, it is
semi-globally stabilizable by linear state feedback. If, in addition,
the system is also detectable, then it is semi-globally stabilizable via
linear output feedback

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It is shown via explicit construction of feedback laws that, if a discrete-time linear system is asymptotically null controllable with bounded controls, then, when subject to both actuator position and rate saturation, it is semi-globally stabilizable by linear state feedback. If, in addition, the system is also detectable, then it is semi-globally stabilizable via linear output feedback

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This paper deals with adaptive control of a class of second-order
nonlinear systems with a triangular structure and concave/convex
parametrization. In the paper by Annaswamy et al. (1998), it was shown
that nonlinearly parametrized systems that satisfy certain matching
conditions can be adaptively controlled in a stable manner. In this
paper, we relax these matching conditions and include additional
dynamics between the nonlinearities and the control input. Global
boundedness and convergence to within a desired precision ε is
established. No over-parametrization of the adaptive controller is
required

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A solution to the problem of designing a globally convergent truly
decentralized adaptive controller for systems of arbitrary relative
degree without any matching assumptions is presented. It is proved that
using Morse's new dynamic certainty equivalent adaptive controller, it
is possible to determine a class of unmodeled interconnections in the
face of which regulation of the plant output to zero with internal
stability is still possible. It is proved that the output regulation
happens with some guaranteed transient performance bounds, that the
L <sub>2</sub> norm of the output is uniformly bounded by the
initial parameter estimation error and that, if the latter is
sufficiently small, a bound for the L <sub>∞</sub> norm of
the output

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The theoretical issues linked to the development of indirect adaptive robot controllers are discussed, and some possible solutions are proposed. After a review of the prediction models used for robotic parameter estimation, a variety of parameter estimation methods are discussed under a common framework based on an exact solution approach. A novel indirect adaptive controller structure, which consists of a modified computed torque using parameters obtained from any of the estimators discussed, is presented. It is shown that a critical difficulty in using indirect adaptive control is the necessity to explicitly guarantee that the estimated inertia matrix remains positive definite in the course of adaptation, a requirement avoided by both the direct and the composite adaptive controllers. A practical solution to this difficulty is proposed.< >

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This paper presents new stability conditions for closed-loop piecewise-affine (PWA) systems. The result is based on controlled invariant sets for PWA systems, which are defined by extending the notion of semi-ellipsoidal invariant sets for constrained linear systems reported in previous research. The paper shows that by proper use of the control input, concatenations of semi-ellipsoidal sets can be made invariant for the trajectories of PWA systems. Furthermore, based on these controlled invariant sets, the paper presents a result for stability of a closed-loop PWA system which is less conservative than existing approaches in the literature. In this result, it is shown that a PWA system is stable inside the intersection of any level set for a local Lyapunov function and the design set where the function is defined, provided the flow points inwards at the boundaries of the intersection. This result is less conservative than previous approaches and it enables the designer to have an estimation of a much larger region of exponential stability then it would be possible using previous results. A numerical example is presented, in which it is made clear by comparison with previous approaches that the estimated region of stability can be made significantly larger using the new stability conditions developed in this paper.

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We introduce the notion of a “meaningful” average of a
collection of dynamical systems as distinct from an
“ensemble” average. Such a notion is useful for the study of
a variety of dynamical systems such as traffic flow, power systems, and
econometric systems. We also address the associated issue of the
existence and computation of such an average for a class of
interconnected, linear, time invariant dynamical systems. Such an
“average” dynamical system is not only attractive from a
computational perspective, but also represents the average behavior of
the interconnected dynamical system. The problem of analysis and control
of hierarchical, large scale control systems can be simplified by
approximating the lower level dynamics of such systems with such an
average dynamical system

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We show that the sequences of the solutions of the decoupled
algebraic Lyapunov equations used for finding the positive semidefinite
stabilizing solutions of the coupled algebraic Riccati equations of the
optimal control problem of jump parameter linear systems are monotonic
under proper initialization

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An input-output, frequency domain characterization of decentralized fixed modes is given in this paper, using only standard block-diagram algebra, well-known determinantal expansions and the Binet-Cauchy formula. Using this characterization, an algebraic proof is presented of the fact that, in a strongly connected system, spectrum assignment (stabilization) is possible using decentralized dynamic compensation if and only if the system has no fixed modes (only stable fixed modes).

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The authors address the problem of estimating the size of the
perturbations on the stabilizing solution of the continuous-time
algebraic Riccati equation when its coefficient matrices are subject to
small perturbations. Upper bounds on the norm of the perturbations on
the stabilizing solution are presented. Moreover, is is shown that these
perturbations can be determined as a single-valued continuous function
of the perturbations in the coefficients of the Riccati equation

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This paper considers the coupled formation control of three mobile agents moving in the plane. Each agent has only local inter-agent bearing knowledge and is required to maintain a specified angular separation relative to its neighbors. The problem considered in this paper differs from similar problems in the literature since no inter-agent distance measurements are employed and the desired formation is specified entirely by the internal triangle angles. Each agent's control law is distributed and based only on its locally measured bearings. A convergence result is established which guarantees global convergence of the formation to the desired formation shape.

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The problem of robust control for the angular velocity of a rigid body subject to external disturbances is addressed. It is shown that if the disturbances are matched there exists a control law attenuating the effect of the disturbances, whereas in the case of non-matched disturbances no such a feedback law generically exists. Hence, a new concept of disturbance attenuation is introduced and it is proved that the aforementioned problem is solvable in this weaker sense. Explicit expressions of the control laws solving the proposed robust stabilization problems are given.

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A theorem that settles a conjecture formulated by Tusnady is presented. The proof uses a uniqueness theorem on the asymptotic likelihood equation

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We show that a continuous dynamical system on a state space that
has the structure of a vector bundle on a compact manifold possesses no
globally asymptotically stable equilibrium. This result is directly
applicable to mechanical systems having rotational degrees of freedom.
In particular, the result applies to the attitude motion of a rigid
body. In light of this result, we explain how attitude stabilizing
controllers appearing in the literature lead to unwinding instead of
global asymptotic stability

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Rigid body models with two controls cannot be locally asymptotically stabilized by continuous state feedbacks. Existence of a locally stabilizing smooth time-varying feedback has however been proved. Such a feedback is explicitly derived.

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We consider the problem of enlarging the basin of attraction for a linear system under saturated linear feedback. An LMI based approach to this problem is developed. For discrete-time system, this approach is enhanced by the lifting technique, which leads to further enlargement of the basin of attraction. The low convergence rate inherent with the large invariant set (hence, the large basin of attraction) is prevented by the construction of a sequence of invariant ellipsoids nested within the large one obtained

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In this article, we give a complete characterization of the reachable set at all times for a class of bilinear control systems with time varying drift and unbounded control amplitude. These results are of fundamental interest in geometric control theory and have important applications to control of coupled spins in solid state NMR spectroscopy.

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The problem of regulating the output voltage of the Boost DC-to-DC
power converter has attracted the attention of many control researchers
for several years now. Besides its practical relevance, the system is an
interesting theoretical case study because it is a switched device whose
averaged dynamics are described by a bilinear second order non-minimum
phase system with saturated input, partial state measurement and a
highly uncertain parameter-the load resistance. In this paper we provide
a solution to the problem of designing an output-feedback saturated
controller which ensures regulation of the desired output voltage and
is, at the same time, insensitive to uncertainty in the load resistance.
Furthermore, bounds on this parameter can be used to tune the controller
so as to (locally) ensure robust performance, e.g., that the transient
has no (under)over-shoot. The controller, which is designed following
the energy-balancing methodology recently proposed by Ortega, van der
Schaft and Maschke, is a simple static nonlinear output feedback, hence
it is computationally less demanding than the industry standard lead-lag
filters

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For nonlinear control systems, the authors consider the problem of dynamic feedback linearization. In particular, for a restricted class of dynamic compensators that correspond to adding chains of integrators to the inputs, the authors give an upper bound for the order of the compensator that needs to be considered. Moreover, in the case of 2-input systems, it is shown that this bound is sharp

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This paper considers frequency weighted model reduction. The explicit lower and upper approximation error bounds are derived for certain classes of weighted model reduction problems. The approximation is based on some unweighted approximations and the error bounds are given in terms of the Hankel singular values of the weighted model.

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In this paper we introduce new bounds for the real structured singular value. The approach is based on absolute stability criteria with plant-dependent multipliers that exclude the Nyquist plot from fixed plane curve shapes containing the critical point -1+j0. Unlike half-plane and circle-based bounds the critical feature of the fixed curve bounds is their ability to differentiate between the real and imaginary components of the uncertainty. Since the plant-dependent multipliers have the same functional form at all frequencies, the resulting graphical interpretation of the absolute stability criteria are frequency independent in contrast to the frequency-dependent off-axis circles that arise in standard real-μ bounds

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In this paper, we study robustness analysis of control systems
affected by bounded uncertainty. Motivated by the difficulty to perform
this analysis when the uncertainty enters into the plant coefficients in
a nonlinear fashion, we study a probabilistic approach. In this setting,
the uncertain parameters q are random variables bounded in a set Q and
described by a multivariate density function f(q). We then ask the
following question: Given a performance level, what is the probability
that this level is attained? The main content of this paper is to derive
explicit bounds for the number of samples required to estimate this
probability with a certain accuracy and confidence apriori specified. It
is shown that the number obtained is inversely proportional to these
thresholds and it is much smaller than that of classical results.
Finally, we remark that the same approach can be used to study several
problems in a control system context. For example, we can evaluate the
worst-case H<sup>∞</sup> norm of the sensitivity function or
compute μ when the robustness margin is of concern

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The problem of noninteracting control with stability via dynamic
feedback is considered for nonlinear systems with two inputs and two
outputs. Given a system which may not be capable of decoupling via
static state feedback but is capable of decoupling via dynamic feedback,
a canonical dynamic extension is constructed. It is shown that every
dynamic extension contains this canonical one. Hence, the
Δ<sub>mix</sub> dynamics as defined by K. Wagner (1989) of the
canonical extension is contained in every other dynamic extension.
Following Wagner's result, the stability of this Δ<sub>mix</sub>
dynamics is necessary for noninteraction with internal stability. The
results obtained show that in the investigation of noninteraction with
stability one can assume without loss of generality that the system is
capable of decoupling via static state feedback

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Global robust stabilization of nonlinear cascaded systems is a challenging problem when the zero-dynamics is not exponentially stable. Recently, some recursive procedure has been developed for handling this problem utilizing the small gain theorem. However, the success of the procedure depends on the satisfaction of some conditions which arise at each step of the recursion. In this paper, we will show that, for the important class of cascaded polynomial systems, the solvability conditions can be made satisfied by appropriately implementing the recursive procedure. This result leads to an explicit construction of the control law.

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This paper gives a control law for stabilizing multiple input
chained form control systems. This extends an earlier result of Teel,
Murray, and Walsh (1992) on stabilizing the above class of systems which
have two inputs. In addition, the authors generalize this law to control
dynamical control systems and construct a transformation from general
chained form systems with multiple generators to a power form. A control
law which stabilizes the origin of a three-input control system that
models the kinematics of a fire truck is simulated, confirming the
theoretical results

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The problem of state and output feedback stabilization of
nonholonomic multiple chained systems is addressed and solved using a
particular class of discontinuous control laws. The obtained control
laws are relatively simple, compared with others existing in the current
literature, and guarantee exponential convergence of the closed loop
system. A simulation example, showing the main features of the proposed
controllers, is enclosed

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Develops a method for the synthesis of linear parameter-varying (LPV) controllers in discrete time. LPV plants under consideration have linear fractional transformation (LFT) representation and specifications consist of a set of H<sub>2</sub>/H<sub>∞</sub> conditions that can be defined channel-wise. In contrast to earlier results, the proposed method involves a specific transformation of both the Lyapunov and scaling/multiplier variables which renders possible the use of different Lyapunov functions and of different scaling variables for each channel/specification. Appropriate linearizing transformations on the controller data and on the scheduling function are then established to finally recast the problem as an easily tractable LMI program.

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This paper concerns the characterization of positive real matrices generated by substitutions (of the Laplace variable s) in scalar rational transfer function by matrix positive real functions. Our main results are restricted to both strongly strictly positive real matrices arid strictly bounded real matrices. As a way to illustrate our main results, we also include here a partial extension of both the Kalman-Yakubovich-Popov lemma (for strongly strictly positive real systems of zero relative degree) and the circle criterion (for strictly positive real systems of zero relative degree).

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This paper deals with regional stabilization of linear
time-invariant systems by dynamic output feedback controllers subject to
known bounds on the magnitudes of the control inputs. Specifically, we
consider the achievable region of attraction, i.e., the set of vectors
with the following property: there exists a (nonlinear) controller such
that any closed-loop state trajectory converges to the origin as long as
the initial state belongs to the set. Two subsets of such set are
characterized: one is derived from the linear analysis that considers
the behavior of the states in the linear (nonsaturated) region only;
while the other is based on the nonlinear analysis using the multi-loop
circle criterion. The main result obtained shows that the two sets are
exactly the same. Thus we conclude that the circle criterion does not
help, within our framework, to increase the size of the region of
attraction in saturating control synthesis when compared with that
resulting from the linear analysis

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We study the invariance of the convex hull of an invariant set for a class of nonlinear systems satisfying a generalized sector condition. The generalized sector is bounded by two symmetric functions which are convex/concave in the right half plane. In a previous paper, we showed that, for this class of systems, the convex hull of a group of invariant level sets (ellipsoids) of a group of quadratic Lyapunov functions is invariant. This paper shows that the convex hull of a general invariant set needn't be invariant, and that the convex hull of a contractively invariant set is, however, invariant.

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Recently, it was shown [1] how to reduce the order of the decentralized control problem for interconnected systems by using local dynamic feedback. The analysis was carried out with polynomial matrices. In this letter, these results are reproduced in state space using the Generalized Hessenberg Representation (GHR). This analysis shows how the interaction between the interconnection structure and the local system structure generates the result above. In addition, this analysis leads directly to numerical algorithms.

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This paper presents a new inverse scattering method for reconstructing the reflectivity function of symmetric two-component wave equations, or the potential of a Schrodinger equation, when the reflection coefficient is rational. This method relies on the so-called Chandrasekhar equations which implement the Kalman filter associated to a stationary state-space model. These equations are derived by using first a general layer stripping principle to obtain some differential equations for reconstructing a general scattering medium, and by specializing these recursions to the case when the probing waves have a state-space model.

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Considers the problem of simultaneous H<sup>∞</sup> control
of a finite collection of linear time-invariant systems via a nonlinear
digital output feedback controller. The main result is given in terms of
the existence of suitable solutions to Riccati algebraic equations and a
dynamic programming equation. Our main result shows that if the
simultaneous H<sup>∞</sup> control problem for k linear
time-invariant plants of orders n<sub>1</sub>,
n<sub>2</sub>,···,n<sub>k</sub> can be solved, then
this problem can be solved via a nonlinear time-invariant controller of
order
n<sub>=</sub><sup>Δ</sup>n<sub>1</sub>+n<sub>2
</sub>+···+n<sub>k</sub>

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Develops an observation control method for refining the
Kalman-Bucy estimates, which is based on impulsive modeling of the
transition matrix in an observation equation, thus engaging
discrete-continuous observations. The impulse observation control
generates online computable jumps of the estimate variance from its
current position towards zero and, as a result, enables us to
instantaneously obtain the estimate, whose variance is closer to zero.
The filtering equations over impulse-controlled observations are
obtained in the Kalman-Bucy filtering problem. The method for feedback
design of control of the estimate variance is developed. First, the pure
impulse control is used, and, next, the combination of the impulse and
continuous control components is employed. The considered examples allow
us to compare the properties of these control and filtering
methodologies

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Conditions for nonlocal existence of a continuous storage function
for nonlinear dissipative system are presented. More precisely, it is
shown that under the local ω-uniform reachability assumption at
one point x<sub>*</sub> the required supply function is continuous on
the set of points reachable from x. Conditions for the local
ω-uniform reachability based on the local controllability
properties of the system are provided

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The purpose of the paper is to generalize the concept of
contractibility of decentralized control laws in the inclusion
principle. After a system with overlapping subsystems is expanded into a
larger space, and decentralized control laws are formulated for the
disjoint subsystems, the laws need to be contracted for implementation
in the original space. We propose broader definitions of restriction and
aggregation in the framework of inclusion, which provide more
flexibility in the contraction phase of the expansion-contraction
process. In particular, we discuss contractibility conditions for
dynamic output controllers including state observers which have been of
special interest in applications

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An attempt is made to give a general formalism for synchronization
in dynamical systems encompassing most of the known definitions and
applications. The proposed set-up describes synchronization of
interconnected systems with respect to a set of functionals and captures
peculiarities of both self-synchronization and controlled
synchronization. Various illustrative examples are given

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This paper considers the problem of robust tracking for a
non-minimum phase system under a sensitivity constraint. Because of
non-minimum phase zeros, the exact tracking of arbitrary reference input
is impossible. However, with an introduction of some delay, tracking is
possible within any given tolerance ε1>0. It is shown in the
paper that the minimum delay required to satisfy the tracking
performance is decoupled from the sensitivity constraint. Moreover, it
is shown that calculation of the minimum delay is simply a binary search
problem

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A parametrization of all decentralized stabilizing controllers is provided in terms of a single parameter Q that is constrained to satisfy a finite number of quadratic equations. The results are presented in a general algebraic framework and apply to linear time-invariant systems. Implications of this parametrization for simultaneously stabilizing control are also discussed.

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The problem of optimally fitting controllers to data is formulated
and solved in the behavioral framework of Willems (1991) expanded to
accommodate partial information about signals. This formulation is
applied to the case in which the class of controllers considered is the
same as the one used in model reference adaptive control problem and in
which we minimize the induced norm of the control error. The solution to
this problem, besides illustrating an use of the saying “let the
data speak” to its extremes, gives a robustness-oriented
perspective on model reference adaptive control

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The problem of finding the set of all stabilizing full order
controllers is equivalent to a problem of finding all matrices which can
be assigned to the closed loop system as a state covariance. Necessary
and sufficient conditions for a given matrix to be assignable as a
covariance are given, and all controllers (of plant order) which assign
a specified covariance are parametrized explicitly. The structure of
covariance controllers is shown for the first time to be observer based
(state estimator plus estimated-state feedback). The
“central” state estimator of the covariance controller is
shown to be the Kalman filter. Unlike the traditional estimator-based
controller, the separation principle does not hold, but one can design a
controller by assigning the estimation error covariance and the plant
state covariance simultaneously

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This paper demonstrates that, provided the system input is persistently exciting, the recursive least squares estimation algorithm with exponential forgetting factor is exponentially convergent. Further, it is shown that the incorporation of the exponential forgetting factor is necessary to attain this convergence and that the persistence of excitation is virtually necessary. The result holds for stable finite-dimensional, linear, time-invariant systems but has its chief implications to the robustness of the parameter estimator when these conditions fail.

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In this paper, we present a technique for constructing a class of
quadratic Lyapunov functions for exponentially stable periodic orbits.
This construction is facilitated by the use of a special set of local
coordinates that serve to highlight the tangential and transverse
dynamics of the system

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In the direct white noise theory of nonlinear filtering, the state
process is still modeled as a Markov process satisfying an Ito
stochastic differential equation, while a finitely additive white noise
is used to model the observation noise. In the present work, this
asymmetry is removed by modeling the state process as the solution of a
(stochastic) differential equation with a finitely additive white noise
as the input. This makes it possible to introduce correlation between
the state and observation noise, and to obtain robust nonlinear
filtering equations in the correlated noise case

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