A convex relaxation approach to set-membership identification
Set-membership identification of dynamical systems is dealt with in this thesis. Differently from the stochastic framework, in the set-membership context the statistical description of the measurement noise is not available and the only information on such an error is that its amplitude or energy is bounded. In the framework of Set-membership identification, the result of the estimation process is the set of all system parameter values consistent with measured data, assumed model structure and a-priori assumptions on the measurement error. The problem of evaluating bounds on system parameters belonging to the feasible parameter set can be formulated in terms of polynomial optimization problems, where the number of decision variables increases with the length of the experimental data sequence. Such problems are generally nonconvex and NP-hard. Therefore, standard nonlinear optimization tools can not be used to compute parameter bounds, since they can trap in local minima and, as a consequence, the computed bounds are not guaranteed to contain the true values of parameters, which is a key requirement in set-membership identification. In order to overcome such a problem, convex relaxation procedures based on the theory of moments are proposed to efficiently compute relaxed bounds which are guaranteed to contain the true values of system parameters. Unfortunately, a direct application of the theory of moments in relaxing set-membership identification problems leads to semidefinite programming problems with high computational burden, thus limiting, in practice, the use of such relaxation procedures to solve identification problems with a small number of measurements. The aim of the thesis is to derive a number of convex-relaxation based algorithms that, exploiting the peculiar properties of the considered identification problems, make it possible to perform bound computation also when the number of measurements is large. In particular, errors-in-variables (EIV) identification of linear models, concerning identification of linear-time-invariant (LTI) systems based on noise-corrupted measurements of both input and output signals, is tackled through two different relaxation approaches. The first method, which is referred to as dynamic-EIV approach, exploits the sparse structure of EIV problems in order to reduce the computational complexity of the semidefinite programming problems arising from theory-of-moment relaxations. The second technique, referred to as semi-static-EIV approach, is based on a suitable handling of the constraints defining the feasible parameter set, and leads to polynomial optimization problems where the number of decision variables does not depend on the size of the measurement sequence. Thanks to that problem reformulation, theory-of-moment relaxations can be efficiently applied to compute bounds on system parameters also from large data set. Identification of block-oriented nonlinear systems is also addressed. The considered model structures are: Hammerstein-Wiener systems; Hammerstein-like and Wiener-like structures with backlash nonlinearity and block-structured nonlinear feedback systems. The semi-static-EIV approach is extended with suitable modifications to estimate the parameters of Hammerstein-Wiener models with static blocks described by polynomial functions. Then, a unified approach for set-membership identification of Hammerstein and Wiener models with backlash is discussed. By properly selecting a sequence of input/output measurements, the evaluation of parameter bounds is formulated in terms of polynomial optimization problems and the structured sparsity of the formulated problems is exploited to reduce the computational complexity of theory-of-moment based relaxations. Finally, a two-stage method for identification of block-structured nonlinear feedback systems is presented. Nonlinear block parameter bounds are first computed by using input/output data collected from the response of the system to square wave inputs. Then, by stimulating the system with a persistently exciting input signal, bounds on the unmeasurable inner-signal are evaluated, which are used, together with noise-corrupted measurements of the output signal, to formulate the identification of linear block parameters in terms of EIV problems that can be solved either through the dynamic or the semi-static-EIV approach. Then, an "ad hoc" convex relaxation scheme is presented to compute guaranteed bounds on the parameters of linear-parameter-varying (LPV) models in input/output (I/O) form, under the assumption that both the output and the scheduling parameter measurements are affected by bounded noise. The developed set-membership identification algorithms are used to derive an LPV model describing vehicle lateral dynamics based on a set of experimental data, and an LPV model to describe glucose-insulin dynamics for patients affected by Type I diabetes. Finally, the problem of identifying systems a-priori known to be stable is discussed. In particular, suitable relaxation-based algorithms are proposed to enforce BIBO stability and quadratic stability constraints for the cases of LTI and LPV systems, respectively. Applicability of the proposed techniques both in the stochastic and in the set-membership framework is discussed.
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