Systems & Control Letters

Published by Elsevier
Print ISSN: 0167-6911
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.
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.
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.
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.
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
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.
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
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
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
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
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.< >
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.
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
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
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).
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
The error manifold showing six distinct regions and the boundaries.
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.
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.
A theorem that settles a conjecture formulated by Tusnady is presented. The proof uses a uniqueness theorem on the asymptotic likelihood equation
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
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.
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
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.
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
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
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.
Exclusion-inclusion pairs involving fixed regions 
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
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
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
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.
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
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
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.
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).
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
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.
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.
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.
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>
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
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
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
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
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
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.
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
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
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.
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
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
Top-cited authors
Eduardo D Sontag
  • Northeastern University
Yuan Wang
  • Florida Atlantic University
L. Xiao
  • Xi'an Jiaotong University
Lihua Xie
  • Nanyang Technological University
Zongli Lin
  • University of Virginia