We prove that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior. This is then applied to obtain conditions under which the state trajectory of a state representation spans the whole state space. The related question of when the matrix formed from a state sequence has linearly independent rows from the matrix formed from an input sequence and a finite number of its shifts is of central importance in subspace system identification.
Classical linear time-invariant system simulation methods are based on a transfer function, impulse response, or input/state/output representation. We present a method for computing the response of a system to a given input and initial conditions directly from a trajectory of the system, without explicitly identifying the system from the data. Similarly to the classical approach for simulation, the classical approach for control is model-based: first a model representation is derived from given data of the plant and then a control law is synthesised using the model and the control specifications. We present an approach for computing a linear quadratic tracking control signal that circumvents the identification step. The results are derived assuming exact data and the simulated response or control input is constructed off-line.
We prove that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior. This is then applied to obtain conditions under which the state trajectory of a state representation spans the whole state space. The related question of when the matrix formed from a state sequence has linearly independent rows from the matrix formed from an input sequence and a finite number of its shifts is of central importance in subspace system identification.
Abstract. Modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential-algebraic equations. As such, state-space models are not a natural starting point for modeling, while they have utmost importance in the simulation and control phase. The paper addresses the problem of computing state variables for systems of linear differential-algebraic equations of various forms. The point of view from which the problem is considered is the behavioral one, as put forward in [J. C. Willems, Automatica J. IFAC, 22 (1986) 1. Introduction. The usual procedure in modeling consists of tearing and zooming: a system is viewed as an interconnection of subsystems, and modeling consists of describing the subsystems and the interconnection laws. This procedure is often executed hierarchically, with the subsystems in turn viewed as an interconnection. The net result of such a modeling procedure will be a model which involves manifest variables (often called external variables), which are the variables whose behavior we try to model, and latent variables (often called internal variables), which are the variables describing the subsystems. The formalization of this modeling procedure is the philosophy underlying the behavioral approach to systems theory. These ideas have been explained in detail in [8,9,10].As should be apparent, the resulting model will typically involve many algebraic relations (for example, interconnection constraints, resistors laws, spring and damper characteristics, kinematic constraints), combined with differential equations. These may be first-order (for example, inductors, capacitors, the dynamics of dampers), second-order (for example, the dynamics of masses), or higher-order (for example, subsystems whose dynamic laws have been obtained from an identification procedure).A state-space model is hence not the natural end result of the modeling phase, while its importance for simulation or for control design is undisputed. This is one of the reasons why the notion of state is one of the most investigated ones in system theory and why its characterization and construction have been the subject of many papers since the beginning of this discipline. The problem that we deal with in this paper is that of computing state variables, from which a state-space model is easily recovered, starting from an arbitrary set of linear differential-algebraic equations.The paper is organized as follows. In section 2 a set of definitions and results pertaining to the behavioral framework is introduced. In section 3 the consequences of the property of Markovianity, the key to the notion of state, are worked out. In
In his 1986 Automatica paper Willems introduced the influential behavioural approach to control theory with an investigation of linear time-invariant (LTI) discrete dynamical systems. The behavioural approach places open systems at its centre, modelling by tearing, zooming, and linking. In this paper, we show that these ideas are naturally expressed in the language of symmetric monoidal categories.Our main result implies an intuitive sound and fully complete string diagram algebra for reasoning about LTI systems. These string diagrams are closely related to the classical notion of signal flow graphs, but in contrast to previous work, they endowed with semantics as multi-input multi-output transducers that process streams with an infinite past as well as an infinite future. At the categorical level, the algebraic characterisation is that of the prop of corelations, instead of a pushout of the props of spans and cospans. Using this framework, we then derive a novel structural characterisation of controllability, and consequently provide a methodology for analysing controllability of networked and interconnected systems. We argue that this methodology has the potential of providing elegant, simple, and efficient solutions to problems arising in the analysis of systems over networks, a vibrant research area at the crossing of control theory and computer science.
The classical approach for solving control problems is model based: first a model representation is derived from given data of the plant and then a control law is synthesized using the model and the control specifications. We present an alternative approach that circumvents the explicit identification of a model representation. The considered control problem is finite horizon linear quadratic tracking. The results are derived assuming exact data and the optimal trajectory is constructed off-line.
New algorithms for identification of a balanced state space representation are proposed. They are based on a procedure for the estimation of impulse response and sequential zero input responses directly from data. The proposed algorithms are more efficient than the existing alternatives that compute the whole Hankel matrix of Markov parameters. It is shown that the computations can be performed on Hankel matrices of the input-output data of various dimensions. By choosing wider matrices, we need persistency of excitation of smaller order. Moreover, this leads to computational savings and improved statistical accuracy when the data is noisy. Using a finite amount of input-output data, the existing algorithms compute finite time balanced representation and the identified models have a lower bound on the distance to an exact balanced representation. The proposed algorithm can approximate arbitrarily closely an exact balanced representation. Moreover, the finite time balancing parameter can be selected automatically by monitoring the decay of the impulse response. We show what is the optimal in terms of minimal identifiability condition partition of the data into "past" and "future".
In this paper partial discharge (PD) is investigated inside a spherical air lled void at atmospheric pressure using a drift diusion model. Discharge dynamics consisted of an electron avalanche transitioning into positive streamer, in agreement with earlier work on dielectric barrier discharges. Dierent model congurations were utilised to test many of the concepts employed in semi-analytical PD activity models, which use simplistic descriptions of the discharge dynamics. The results showed that many of these concepts may be erroneous, with signicant discrepancies between the canonical reasoning and the simulation results. For example, the residual electric eld, the electric eld after a discharge, is signicantly lower than the estimates used by classical PD activity models in the literature.
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