We analyze the observability of the continuous and discrete states of a class of linear hybrid systems. We derive rank conditions that the structural parameters of the model must satisfy in order for filtering and smoothing algorithms to operate correctly. We also study the identifiability of the model parameters by characterizing the set of models that produce the same output measurements. Finally, when the data are generated by a model in the class, we give conditions under which the true model can be identified.

We analyze the observability of the continuous and discrete states of continuous-time linear hybrid systems. For the class of jumplinear systems, we derive necessary and su#cient conditions that the structural parameters of the model must satisfy in order for filtering and smoothing algorithms to operate correctly. Our conditions are simple rank tests that exploit the geometry of the observability subspaces. For linear hybrid systems, we derive weaker rank conditions that are su#cient to guarantee the uniqueness of the reconstruction of the state trajectory, even when the individual linear systems are unobservable.

We propose an algebraic geometric solution to the identification of a class of linear hybrid systems. We show that the identification of the model parameters can be decoupled from the inference of the hybrid state and the switching mechanism generating the transitions, hence we do not constraint the switches to be separated by a minimum dwell time. The decoupling is obtained from the so-called hybrid decoupling constraint, which establishes a connection between linear hybrid system identification, polynomial factorization and hyperplane clustering. In essence, we represent the number of discrete states n as the degree of a homogeneous polynomial p and the model parameters as factors of p. We then show that one can estimate n from a rank constraint on the data, the coe#cients of p from a linear system, and the model parameters from the derivatives of p. The solution is closed form if and only if n 4. Once the model parameters have been identified, the estimation of the hybrid state becomes a simpler problem. Although our algorithm is designed for noiseless data, we also present simulation results with noisy data. 1

Abstract. We study continuous-time switched linear systems with unobserved and exogenous mode signals. We analyze the observability of the initial state and initial mode under arbitrary switching, and characterize both properties in both the autonomous and non-autonomous cases. 1

Abstract: We extend previous works on model-free control to switched nonlinear SISO systems. Our contribution, which is utilizing new algebraic methods for numerical differentiations, yields PID-like regulators which ensure practical stability. Several academic examples, with convincing computer simulations, are illustrating our approach.

This paper investigates the switching stabilizability problem for a class of continuous-time switched linear systems with time-variant parametric uncertainties. First, a necessary and sufficient condition for the asymptotic stabilizability of such uncertain switched linear system is derived, under the assumption that the closed-loop switched system does not generate sliding motions. Then, an additional condition is introduced to exclude the possibility of unstable sliding motions. Finally, a necessary and sufficient for the asymptotic stabilizability of such continuous-time uncertain switched linear systems is presented. This result improves upon conditions found in the literature which are either sufficient only or necessary only.

Abstract — We analyze the observability of the continuous and discrete states of discrete-time jump linear systems (JLSs) with deterministic inputs. We consider several definitions of observability for JLSs depending on whether some or all inputs are considered. Unfortunately, checking these definitions can involve an exponential number of rank tests on the parameters of the JLS when the discrete state sequence is arbitrary. Our key contribution is to demonstrate that when there is a minimum separation between consecutive switches, one can verify observability by checking a number of rank tests that is only quadratic in the number of discrete states. The observability conditions we derive are natural generalizations of the well known rank test for linear systems. Moreover, they can be related to the Markov parameters of the individual linear systems. I.

Abstract — This paper aims at estimating the switching time for linear switched systems, i.e. the time instant when a sub-model is switched on while another one is switched off. Assuming that the state of the active sub-models is known, a distribution point of view is adopted to get a real-time estimation of these switching times. Real-time means that an explicit algorithm computes on-line these time instants in a fast and efficient way. Simulations illustrate the proposed techniques which is easily implementable. Keyword—Linear systems, switched systems, hybrid systems, switching time estimation, distribution theory. I.

%e analyze the observability of the continuous and discrete states of a class of linear hybrid systems. We derive rank conditions that the structural parameters of the model must satisfy in order for filtering and smoothing algorithms to operate correctly. We also study the identifiability of the model parameters by characterizing the set of models that produce the same output measurements. Finally, when the data are generated by a model in the class, we give conditions under which the true model can he identified. 1