Active mode observability of switching linear systems

Active mode observability is addressed for a class of discrete-time linear systems that may switch in an unknown and unpredictable way among different modes taken from a finite set. The active mode observation problem consists in determining control sequences (discerning control sequences) that allow to reconstruct the switching sequence on the basis of the observations. The presence of unknown but bounded noises affecting both the system and measurement equations is taken into account. A general condition is derived that characterizes discerning controls in a finite-horizon setting. Such a result is extended to the infinite-horizon case in order to derive “persistently discerning” control sequences. A numerical example is reported to clarify the approach.     

Span-reachability and observability of bilinear hybrid systems

The paper presents an algebraic characterization of observability and span-reachability of bilinear hybrid systems without guards, i.e. hybrid systems whose continuous dynamics are determined by bilinear control systems, and whose discrete transitions are triggered externally. The proposed characterization provides necessary and sufficient conditions for observability and span-reachability in terms of rank conditions for generalized observability and controllability matrices. This characterization is useful for the system theory of bilinear hybrid systems, in particular, it is used in the realization theory of bilinear hybrid systems, (Petreczky, 2006).     

Graphic approach for the determination of the existence of sequences guaranteeing observability of switched linear systems

This paper deals with the design of sequences allowing the observability of the state of a switched linear system. The proposed method, based on a graph-theoretic approach, needs only the knowledge of the system’s structure. The necessary and sufficient conditions for the existence of at least one switching sequence guaranteeing the observability of a switched structured system are provided. Such conditions are particularly intuitive and they can be implemented with classic algorithms of the graph theory.     

Polynomial-time probabilistic observability analysis of sampled-data piecewise affine systems

This paper proposes a polynomial-time probabilistic approach to solve the observability problem of sampled-data piecewise affine systems. First, an algebraic characterization for the system to be observable is derived. Next, based on the characterization, we propose a randomized algorithm that can determine if the system is observable in a probabilistic sense or the system is not observable in a deterministic sense. Finally, it is shown with some examples, for which it is hopeless to check the observability in a deterministic way, that the proposed algorithm is very useful.     

Moving horizon estimation for switching nonlinear systems

This paper is concerned with moving horizon estimation for a class of constrained switching nonlinear systems, where the system mode is regarded as an unknown discrete state to be estimated together with the continuous state. In this work, we establish the observability framework of switching nonlinear systems by proposing a series of concepts about observability and analyzing the properties of such concepts. By fully applying the observability properties, we prove the stability of the proposed moving horizon estimators. Simulation results are reported to verify the derived results.     

Discerning controllers for switching linear systems: Existence and genericity

This paper presents recent developments in the study of non-autonomous switching linear systems. For such systems, we address the issue of how to systematically design linear controllers allowing the active process mode to be observable from closed-loop data. The results are stated formally by introducing the notion of discerning controllers. Both existence and genericity problems are discussed. It is finally shown how a given family of discerning controllers can be implemented as a single hybrid system which preserves the discerning capability of the original controllers. Examples are discussed to substantiate the analysis.     

On analytic and algebraic observability of nonlinear delay systems

The observability of nonlinear delay systems has previously been defined in an algebraic setting by a rank condition on modules over noncommutative rings. We introduce an analytic definition of observability to ensure the local uniqueness of state and initial conditions that correspond to a given input-output behaviour. It is shown that an algebraically observable delay system can be reformulated as a system of ordinary differential equations. Analytic observability is then decided by the local uniqueness of solutions to a boundary value problem for this ODE system.     

On controllability and observability for a class of impulsive systems

Many dynamic systems in physics, chemistry, biology, engineering, and information science have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical processes. These complex dynamic behaviors can be modeled by impulsive differential systems. This paper studies the controllability and observability for a class of time-varying impulsive control systems. Several sufficient and necessary conditions for state controllability and state observability of such systems are established and the corresponding criteria for time-invariant impulsive control systems are also obtained. Meanwhile, several new results associated with variation of parameters for time-varying impulsive control systems are derived.     

Quantitative measure of observability for linear stochastic systems

In this study we define a new observability measure for stochastic systems: the mutual information between the state sequence and the corresponding measurement sequence for a given time horizon. Although the definition is given for a general system representation, the paper focuses on the linear time invariant Gaussian case. Some basic analytical results are derived for this special case. The measure is extended to the observability of a subspace of the state space, specifically an individual state and/or the modes of the system. A single measurement system represented in the observable canonical form is examined in detail. A recursive form of the observability measure for a finite time horizon is derived. The possibility of using this form for designing a sensor selection algorithm is demonstrated by two examples.     

On a switching control scheme for nonlinear systems with ill-defined relative degree

This paper discusses the applicability of a switching control scheme for a nonlinear system with ill-defined relative degree. The control scheme switches between exact and approximate input–output linearisation control laws. Unlike a linear system under a switching control scheme, the equilibria of a nonlinear system may change with the switching. It is pointed out that this is not sufficient to cause instability. When the region of the approximate linearisation control law is attractive to the exact zero dynamics, it is possible that the closed-loop system under the switching control scheme is still stable. The results in this paper shows that the switching control scheme proposed in Tomlin and Sastry (Systems Control Lett. 35(3) (1998) 145) is applicable for a wider class of nonlinear systems.     

Robust observability for a class of time-varying discrete-time uncertain systems

This paper introduces a concept of robust observability for a class of uncertain discrete-time systems. This notion is an extension of the standard notion of observability for discrete-time, linear time-varying systems. The uncertainty and noise are modelled deterministically via a sum quadratic constraint. A necessary and sufficient condition for robust observability is presented in terms of existence of a suitable solution to a Riccati difference equation. The set of possible initial states given noisy measurements over a finite number of sampling periods is shown to be an ellipsoid.     

On the observability of linear motion quantities in navigation systems

Navigation systems are a key element in a large variety of mobile platforms, where the correct knowledge of their position and attitude is essential in most applications. This paper focuses on the observability of linear motion quantities (position, linear velocity, linear acceleration, and accelerometer bias). It presents necessary and sufficient conditions, with a clear physical insight, for the observability of these variables in 3-D. The analysis provided is based on kinematic models, which are exact and intrinsic to the motion of a rigid-body, and different cases are presented depending on the assumptions made on the sensor suite that is available on-board.     

Stability and switching control design issues for a class of discrete time hybrid systems

For the class of systems considered, necessary and sufficient stabilizability conditions are unknown. However, by considering the same systems with unknown but bounded exogenous disturbances, we give finitely computable conditions, sufficient for stabilizability without disturbances, yet necessary for stabilizability with disturbances.     

Exact observability and controllability for linear neutral type systems

The problem of exact observability is analyzed for a wide class of neutral type systems by an infinite dimensional approach. The duality with the exact controllability problem is the main tool. It is based on an explicit expression of a neutral type system which corresponding to the abstract adjoint system. A nontrivial relation is obtained between the initial neutral system and the system obtained via the adjoint abstract state operator. The characterization of the duality between controllability and observability is deduced, and then observability conditions are obtained.     

Synthesis of optimal controllers for piecewise affine systems with sampled-data switching

This paper discusses the optimal control problem of the continuous-time piecewise affine (PWA) systems with sampled-data switching, where the switching action is executed based upon a condition on the state at each sampling time. First, an algebraic characterization for the problem to be feasible is derived. Next, an optimal continuous-time controller is derived for a general class of PWA systems with sampled-data switching, for which the optimal control problem is feasible but whose subsystems in some modes may be uncontrollable in the usual sense. Finally, as an application of the proposed approach, the high-speed and energy-saving control problem of the CPU processing is formulated, and the validity of the proposed methods is shown by numerical simulations.     

On the detectability and observability of discrete-time Markov jump linear systems

This paper presents a new detectability concept for discrete-time Markov jump linear systems with finite Markov state, which generalizes the MS-detectability concept found in the literature. The new sense of detectability can similarly assure that the solution of the coupled algebraic Riccati equation associated to the quadratic control problem is a stabilizing solution. In addition, the paper introduces a related observability concept that also generalizes previous concepts. A test for detectability based on a coupled matrix equation is derived from the definition, and a test for observability is presented, which can be performed in a finite number of steps. The results are illustrated by examples, including one that shows that a system may be detectable in the new sense but not in the MS sense.     

On the controllability of switching linear systems

This note presents a necessary and sufficient condition for small time controllability of a linear switching system (that is, a collection of linear time-invariant control systems, where a trajectory is any concatenation of trajectories of the individual systems). This result extends the controllability condition recently obtained for unconstrained linear switching systems to the case of control which is constrained in a cone.     

On stability of switched linear systems with perturbed switching paths

1 Introduction The study of switched and hybrid dynamical systems has attracted much attention since 1990s. A switched linear sys- tem is a hybrid system which consists of several linear time- invariant subsystems and a switching path that orchestrates the switching among them. The importance of the switched linear systems scheme stems from the facts that i) the frame- work represents a wide class of practical systems, ii) the two-level system structure provides an effective multiple- controll…     

On the preservation of observability under sampling

In this paper, we investigate the problem of the preservation of observability under sampling.