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International audience; This paper presents a computational method to test the stabilizability of discrete-time switched linear systems. The existence of a conic cover of the space on whose elements a convex condition holds is proved to be necessary and sufficient for stabilizability. An algorithm for computing a conic partition that satisfies the new necessary and sufficient condition is given. The algorithm, that allows also to determine bounds on the exponential convergence rate, is proved to overcome the conservatism of conditions equivalent to periodic stabilizability and is applied to a four dimensional system.

International audience; In this paper we consider the-stabilization of th-order linear time-invariant (LTI) dynamical systems using Multiplicity-Induced-Dominancy (MID)-based controller design in the presence of delays in the input or the output channels. A sufficient condition is given for the dominancy of a real root with multiplicity at least n + 1 and at least using an integral factorization of the corresponding characteristic function. A necessary condition for-stabilizability is analyzed utilizing the property that the derivative of \alpha-stable quasipolynomial is also-stable under certain conditions. Sufficient and necessary conditions are given for systems with real-rooted open loop characteristic function: the delay intervals are determined where the conditions for dominancy and \alpha-stabilizability are satisfied. The efficiency of the proposed controller design is shown in the case of a multi-link inverted pendulum.

This paper studies controllability and stabilizability of linear complementarity systems that can be cast as continuous piecewise affine dynamical systems. Under a certain right-invertibility assumption, we present a la Hautus necessary and sufficient conditions for both controllability and stabilizability.

In this paper stabilization of switched differential algebraic equations is considered, where Dirac impulses in both the input and the state trajectory are to be avoided during the stabilization process. First it is shown that stabilizability of a switched DAE and the existence of impulse-free solutions are merely necessary conditions for impulse-free stabilizability. Then necessary and sufficient conditions for the existence of impulse-free solutions are given, which motivate the definition of (impulse-free) interval-stabilization on a finite interval. Under a uniformity assumption, which can be verified for a broad class of switched systems, stabilizability on an infinite interval can be concluded based on interval-stabilizability. As a result a characterization of impulse-free interval stabilizability is given and as a corollary we provide a novel impulse-free null-controllability characterization. Finally, the results are compared to results on interval-stabilizability where Dirac impulses are allowed in the input and state trajectory.

International audience; We study control properties of a linearized fluid-structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to prove this, we prove a general result for this kind of systems that generalizes in particular the case without structure. The exponential stabilization of the system is obtained with a finite-dimensional feedback control acting only on the momentum equation on a subset of the fluid domain and up to some rate that depends on the coefficients of the system. We also show that, as in the case without structure, the system is not exactly null-controllable in finite time.

This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the \begin{document}$ N $\end{document}-dimensional linear reaction-convection-diffusion equation with \begin{document}$ N\geq 1 $\end{document} and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state.

We investigate the stabilizability of discrete-time linear switched systems, when the sole control action of the controller is the switching signal, and when the controller has access to the state of the system in real time. Despite their apparent simplicity, determining if such systems are stabilizable appears to be a very challenging problem, and basic examples have been known for long, for which the stabilizability question is open. We provide new results allowing us to bound the so-called stabilizability radius, which characterizes the stabilizability property of discrete-time linear switched systems. These results allow us to compute significantly improved explicit lower bounds on the stabilizability radius for the above-mentioned examples. As a by-product, we exhibit a discontinuity property for this problem, which brings theoretical understanding of its complexity., 15 pages; numerical results in lower row of Tab. 1 corrected, not affecting any of the discussion or conclusions

International audience; Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of α-null controllability. We comment on the relationships between those various concepts, at the light of the observability inequalities that characterize them.

For the computation of the lower spectral radius of a finite family of matrices that shares an invariant cone, two recent papers by Guglielmi and Protasov [8] and Guglielmi and Zennaro [9] make use of so-called antinorms. Antinorms are continuous, nonnegative, positively homogeneous and superadditive functions defined on the cone and turn out to be related to the lower spectral radius of the family in a similar way as norms are related to the joint spectral radius. In this paper, we revisit the theory of antinorms in a systematic way, filling in some theoretical holes, correcting a common mistake present in the literature and adding some new properties and results. In particular, we prove that, under suitable assumptions, the lower spectral radius is characterized by a Gelfand type limit computed on an antinorm.

International audience; In this paper, the stabilizability of discrete-time switched linear systems subject to constraints on the switching law is considered. The admissible switching sequences are given by the language generated by a nondeterministic finite state automaton. Constructive necessary and sufficient conditions for recurrent stabilizability are given and the exact relations with the existence of control Lyapunov functions and with general stabilizability are provided. The dependence of stabilizability on the automaton initial state is also proved.

A short survey on delayed feedback stabilization is given. The Huijberts–Michiels–Nijmeijer problem on the delayed feedback stabilization of unstable equilibria of two- and three-dimensional dynamical systems is considered. It is shown that the methods of delayed feedback stabilization of unstable periodic orbits can be used with advantage for the stabilization of unstable equilibria. An analytical study based on the D-decomposition method is given. Efficient necessary and/or sufficient conditions for the stabilizability of the systems in question are obtained in the form of explicit analytic expressions. These conditions define the boundaries of stabilizability domains in terms of system parameters. It follows from these conditions that the introduction of a delayed feedback control generally extends the possibilities of stationary stabilization of linear systems with delay-free feedback.

a b s t r a c t Article history: Received 2 July 2014 Available online xxxx MSC: 35L10 65M60 93B05 93B40 93D15 Keywords: Controllability Stabilizability Fluid-particle System Burgers equation In this paper we study, from a control theoretic view point, a 1D model of fluid-particle interaction. More precisely, we consider a point mass moving in a pipe filled with a fluid. The fluid is modelled by the viscous Burgers equation whereas the point mass obeys Newton's second law. The control variable is a force acting on the mass point. The main result of the paper asserts that for any initial data there exist a time T> 0a nd a control such that, at the end of the control process, the particle reaches a point arbitrarily close to a given target, whereas the velocities of the fluid and of the point mass are driven exactly to zero. Therefore, within this simplified model, we can control simultaneously the fluid and the particle, by using inputs acting on the moving point only. Moreover, the main result holds without any smallness assumptions on the initial data. Alternatively, we can see our results as yielding controllability of the viscous Burgers equation by a moving internal boundary.

International audience; The paper deals with the co-design of a control policy, composed by both the state feedback and the switching control law, for discrete-time switched linear systems. Constructive conditions are given that are necessary and sufficient for the stabilizability of systems which are periodic stabilizable. The conditions are in form of a Linear Matrix Inequality (LMI) problem whose solution provides the switching law and a family of state feedback gains stabilizing the system as well as a bound on the exponential decreasing rate. The effectiveness of the proposed technique is illustrated by comparison with results from the literature.

International audience; In this paper, we consider the well-known Fattorini's criterion for approximate controllability of infinite dimensional linear systems of type $y'=A y+Bu$. We precise the result proved by H. O. Fattorini in \cite{Fattorini1966} for bounded input $B$, in the case where $B$ can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini's criterion is satisfied and if the set of geometric multiplicities of $A$ is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini's criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier-Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini's criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems.

This paper studies controllability and stabilizability of continuous piecewise affine dynamical systems which can be considered as a collection of ordinary finite-dimensional linear input/state/output systems, together with a partition of the product of the state space and input space into (full-dimensional) polyhedral regions. Each of these regions is associated with one particular linear system from the collection. The main results of the paper are Popov-Belevitch-Hautus-type necessary and sufficient conditions for both controllability and stabilizability of such systems.

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in Hs(𝕋), s > 2. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.

This article deals with stability of discrete-time switched systems. Given a family of nonlinear systems and the admissible switches among the systems in the family, we first propose a class of switching signals under which the resulting switched system is globally asymptotically stable. We allow unstable systems in the family and our stability condition depends solely on asymptotic behaviour of the switching signals. We then discuss algorithmic construction of the above class of switching signals, and show that in the presence of exogenous inputs and outputs, a switching signal so constructed also ensures input/output-to-state stability for discrete-time switched nonlinear systems. We finally show that our class of switching signals that ensures global asymptotic stability also extends to the continuous-time setting with minor modifications under standard assumptions. We employ multiple Lyapunov-like functions and graph theoretic tools as the main apparatuses for our analysis. (C) 2016 Elsevier Ltd. All rights reserved.

International audience; In this paper we deal with the stabilizability property for discrete-time switched linear systems. A recent necessary and sufficient characterization of stabilizability, based on set theory, is considered as the reference for comparing the computation-oriented sufficient conditions. The classical BMI conditions based on Lyapunov-Metzler inequalities are considered and extended. Novel LMI conditions for stabilizability, derived from the geometric ones, are presented that permit to combine generality with computational affordability. For the different conditions, the geometrical interpretations are provided and the induced stabilizing switching laws are given. The relations and the implications between the stabilizability conditions are analyzed to infer and compare their conservatism and their complexity.

In this article, we give a complete characterization (geared towards a parametrization) of all the static feedbacks of a given LTI system triplet, in terms of the coefficients of some reduced order Riccati equation, with simultaneously stabilizing solutions and structural constraints – on the coefficients.

The problem of L-2-gain analysis and control synthesis is studied for a class of uncertain discrete-time switched linear systems with time delay and saturating actuators by using the multiple Lyapunov functions method. With the state feedback controllers adopted beforehand, an analysis condition on disturbance tolerance is derived under which the operating state trajectory with zero initial condition will remain inside a bounded set in its proximity. Then with this condition at hand, the largest disturbance tolerance level is determined via the solution of a constrained optimisation problem. Then, a sufficient condition for the restricted L-2-gain over the set of tolerable disturbances is derived. The smallest upper bound on the restricted L-2-gain is also obtained by solving a constrained optimisation problem. Furthermore, when controller gain matrices are design variables, these optimisation problems are adjusted for solving the control design task. An illustrative numerical example is given to demonstrate the feasibility and effectiveness of the proposed method.

This paper deals with the problem of reliable stabilization and H ∞ control for a class of continuous-time switched Lipschitz nonlinear systems with actuator failures. We consider the case that actuators suffer “serious failure”—the never failed actuators cannot stabilize the given system. The differential mean value theorem (DMVT) allows transforming the switched Lipschitz nonlinear systems into switched linear parameter varying (LPV) systems. Based on average dwell time scheme and under the condition that activation time ratio between stabilizable subsystems and unstabilizable ones is not less than a specified constant, sufficient conditions for reliable exponential stabilization of the switched systems are derived by hybrid observer-based output feedback control. The result is also extended to the reliable H ∞ control problem.

It is well known that a finite-dimensional output space implies limitations on the systems properties, like observability and detectability. In this paper we extend this result for infinite-dimensional output spaces, under the condition that the output operator is relatively compact. We show that if this holds, and the system is exactly observable in finite-time, then the inverse of the infinitesimal generator must be compact. By means of an example we show that this result does not hold for exact observability in infinite-time. Using the Hautus test, we obtain spectral properties of the generator for this case. A consequence of this result is that if the system is exponentially detectable, then the unstable part of the spectrum consists of only point spectrum with finite multiplicity.

This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are well known. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the first main result of this paper, we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors. Secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices. We will also establish conditions for the equivalence of rational image representations. Finally, we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation.

In this paper, we study the steady-state of a semilinear pulse-width sampler controlled system on infinite dimensional spaces. Firstly, by virtue of Schauder's fixed point theorem, the existence of periodic solutions is given. Secondly, utilizing a generalized Gronwall inequality given by us and the Banach fixed point theorem, the existence and stabilizability of a steady-state for the semilinear control system with pulse-width sampler is also obtained. At last, an example is given for demonstration.

Let X , U be two Banach spaces, let Θ be a metric space and let σ be a flow on Θ . For A ∈ l ∞ ( Θ , B ( X ) ) and B ∈ l ∞ ( Θ , B ( U , X ) ) , we consider the variational discrete system with control ( A , B ) x ( θ ) ( n + 1 ) = A ( σ ( θ , n ) ) x ( θ ) ( n ) + B ( σ ( θ , n ) ) u ( n ) , ∀ ( θ , n ) ∈ Θ × N , where x : Θ → S ( X ) and u ∈ l p ( N , U ) . We prove that if the discrete cocycle associated with the system ( A , B ) is surjective and the variational discrete system ( A , B ) is completely stabilizable, then ( A , B ) is exactly controllable. By illustrative examples we show that our hypotheses cannot be dropped and also we study the validity of the converse implication.

The discrete component of the hybrid state of a discrete-time linear switching system is assumed to be uncontrolled and unobserved. Conditions of stabilizability for this class of systems are given in terms of a new definition of control invariance, based on the realization of a discrete observer that permits reconstruction of the discrete-state in certain intervals of the time basis. This paper highlights the relationship between the minimum dwell time of the system and its stabilizability. An almost complete characterization of stabilizability is offered in terms of certain subsets of the continuous-state space. These sets are amenable to tractable parametric procedures for controller synthesis. Copyright © 2008 John Wiley & Sons, Ltd.

This paper presents a sum of squares (SOS) approach for modeling and control of nonlinear dynamical systems using polynomial fuzzy systems. The proposed SOS-based framework provides a number of innovations and improvements over the existing linear matrix inequality (LMI)-based approaches to Takagi-Sugeno (T-S) fuzzy modeling and control. First, we propose a polynomial fuzzy modeling and control framework that is more general and effective than the well-known T--S fuzzy modeling and control. Secondly, we obtain stability and stabilizability conditions of the polynomial fuzzy systems based on polynomial Lyapunov functions that contain quadratic Lyapunov functions as a special case. Hence, the stability and stabilizability conditions presented in this paper are more general and relaxed than those of the existing LMI-based approaches to T-S fuzzy modeling and control. Moreover, the derived stability and stabilizability conditions are represented in terms of SOS and can be numerically (partially symbolically) solved via the recently developed SOSTOOLS. To illustrate the validity and applicability of the proposed approach, a number of analysis and design examples are provided. The first example shows that the SOS approach renders more relaxed stability results than those of both the LMI-based approaches and a polynomial system approach. The second example presents an extensive application of the SOS approach in comparison with a piecewise Lyapunov function approach. The last example is a design exercise that demonstrates the viability of the SOS-based approach to synthesizing a stabilizing controller.

We show that Zhang's sandpile model (N,[a,b]) on N sites and with uniform additions on [a,b] has a unique stationary measure for all 0 = 1, in which topplings occur according to a Markov toppling process, and we study the stabilizability of initial configurations chosen according to some measure ��. We show that for a stationary ergodic measure ��with density ��, for all ��< 1/2, ��is stabilizable; for all ��>= 1, ��is not stabilizable; for 1/2, 22 pages

In this article behaviors defined by ‘differential equations’ involving matrices of rational functions are introduced. Conditions in terms of controllability, observability, and stabilizability for the existence of rational representations that are prime over various subrings of the field of rational functions are derived. Elimination of latent variables, (observable) image-like representations of controllable systems, and the structure of the rational annihilators of a behavior are discussed.

In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system $$\frac{{\partial w\left( {x,t} \right)}}{{\partial t}} = A(D_x )w(x,t) - A(D_x )u(x,t), x \in \mathbb{R}^n , t > h, $$ where D x =(-i∂/∂x 1,...-i∂/∂x n ), A(σ) and B(σ) are polynomial matrices (m×m), det B(σ)≡0 on ℝ n , w is an unknown function, u(·,t)=P(D x )w(·,t−h) is a control, h>0. Here P is an infinite differentiable matrix (m×m), and the norm of each of its derivatives does not exceed Γ(1+|σ|2)γ for some Γ, γ∈ℝ depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the Schrodinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski—Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det {Iλ-A(σ)+B(σ)P(σ)e -hλ.

In this note, we obtain a perfect characterization of generator of the perturbation semigroup for Pritchard-Salamon systems. From this we give a much simpler and more straight proof of the theorem on nest of feedback loop and answer the open problem posed by Curtain, Logemann, Townley and Zwart. More precisely, we establish that the smooth condition is unnecessary in the constructive argument for Pritchard-Salmon systems.

We derive formulas for normalized doubly coprime factorizations over H-infinity for a class of nonexponentially stabilizable systems. Using these results, we solve the problem of robust stabilization with respect to normalized coprime factor perturbations for a class of infinite-dimensional systems Sigma(A, B, B*, D), with finite-rank inputs and outputs and dissipative operator A; these systems are not exponentially stabilizable. We give a parameterization of controllers that achieve a given robustness margin. (C) 2000 Elsevier Science Ltd. All rights reserved.

This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable.

A generalization of a class of PD controllers is developed to stabilize a class of systems with two point delays in their state.

International audience; A criterion of exact controllabilty using the resolvent of the state space operator is given for linear control system in Hilbert space . Only surjectivity of the semi-group operators is assumed. This condition is necessary for exact controllability, so the criterion is quite general. Relations between exact controllability and complete stabilizability are specified.

Mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system, as well as formulae for such a factorization. The results are illustrated by two examples of delay systems, one of which has infinitely many unstable poles. Copyright (C) 1996 Elsevier Science Ltd.

In this paper we present an approach to linear dynamical systems which combines the positive features of two well known formulations, namely, standard state space theory (see for e.g., Wonham, 1978 [9]) and behavioural systems theory (see Polderman and Willems, 1997 [4]). Our development is also 'geometric' in the tradition of Wonham and others. But, instead of using explicit linear maps, we work with linear relations implicitly which amounts to working with subspaces. One of our primary motivations is computational efficiency all our computations can be performed on the system as it is without elimination of variables and further (unlike the 'behaviourists' who manipulate matrices with polynomial entries) we work only with real matrices. Using our formulation we derive the standard vector space results on controlled and conditioned invariant subspaces of linear dynamical systems. Duality, which is a distinctive feature of state space theory but not of the behavioural view point, comes out naturally in our approach too through the use of the adjoint. We illustrate our ideas for an important class of dynamical systems viz., electrical networks. The theory proposed in this paper gives a unified description of both the standard linear dynamical systems and the linear singular systems (or the linear descriptor systems) (see for e.g., F. Gantmacher 1959 [1] and F. L Lewis 1986 [2]). Therefore, the algorithms described for the invariant spaces in this paper are also applicable to linear singular systems. (C) 2013 Elsevier Inc. All rights reserved.

Most of the work in multiple model adaptive control with various forms of switching focused on continuous-time systems. The purpose of this technical note is to extend the results of one approach, the adaptive mixing control (AMC), to discrete-time systems. Further, the technical note solves the tracking problem which has not been addressed in most schemes of this class. Stability and robustness properties of the AMC scheme for discrete-time systems are analyzed. It is shown that in the ideal case, when no disturbances or unmodeled dynamics are present, the tracking error converges to zero. In the non ideal case, the mean-square tracking error is of the order of magnitude of the modeling error provided the unmodeled dynamics satisfy a norm-bound condition. While these robustness results are conceptually similar to those of traditional robust adaptive control, the proposed scheme does not suffer from the drawback of stabilizability of the estimated plant and in addition performs much better in simulation studies. Furthermore, it allows well developed results from robust control to be incorporated in the design. © 2011 IEEE. 57 4 1040 1045 Cited By :20

In this paper, we deal with the stabilizability of homogeneous polynomials of degree three by means of homogeneous feedbacks of degree two.

Quadratic control problems for integrodifferential equations of parabolic type are considered. A state-space representation of the system is obtained by choosing an appropriate product space. By using the standard method based on Riccati equation, a unique optimal control over a finite horizon and under a stabilizability condition is obtained and the quadratic problem over an infinite horizon is solved. It is shown that the approach is also valid for some integrodifferential equations of different types. Two examples covered by the model are given.

We analyze the stability and stabilizability properties of mixed retarded-neutral type systems when the neutral term is allowed to be singular. Considering an operator model of the system in a Hilbert space we are interesting in the critical case when there exists a sequence of eigenvalues with real parts approaching to zero. In this case the exponential stability is not possible and we are studying the strong asymptotic stability property. The behavior of spectra of mixed retarded-neutral type systems does not allow to apply directly neither methods of retarded system nor the approach of neutral type systems for analysis of stability. In this paper two technics are combined to get the conditions of asymptotic non-exponential stability: the existence of a Riesz basis of invariant finite-dimensional subspaces and the boundedness of the resolvent in some subspaces of a special decomposition of the state space. For unstable systems the technics introduced allow to analyze the concept of regular strong stabilizability for mixed retarded-neutral type systems. The present paper extends the results by R. Rabah, G.M. Sklyar, A.V. Rezounenko on stability obtained in [J. Diff. Equat., 214(2005), No. 2, 391-428] and on stabilizability from [J. Diff. Equat., 245(2008), No. 3, 569-593]., 46 pages