Linear Parameter-Varying System Identification

The following sections are included:

• Preface

• Contents

• Acronyms

The study of Linear Parameter Varying (LPV) systems was originally motivated by the control design methodology of gain scheduling [Rugh and Shamma (2000)]. The notion of LPV systems, first introduced in [Shamma and Athans (1990)], proved to be very relevant by allowing the automatic control community to overcome some limitations of the gain scheduling approach. Examples of such limitations are the large number of linear models often required to achieve a given performance, and the necessity for slow changes between two operating points. Currently, modeling, identification, and control design of LPV systems form an active research area that contributes to the development of gain scheduling controllers able to deal with fast variation of the operating points and with tight performance bounds…

In the last two decades, LPV and Hybrid Switching models have been extensively studied and identification and control design techniques have been devised. In the present chapter, after an overview of the existing literature, starting from some motivating examples, we introduce the modeling of Hybrid LPV systems. An application example is considered in detail, with reference to the modeling of the traffic flow in wireless ad-hoc networks, together with some possible identification schemes. Then, we highlight some open problems.

In this chapter, we consider the identification of single-input single-output linear-parameter-varying models when both the output and the time-varying parameter measurements are affected by bounded noise. First, the problem of computing exact parameter uncertainty intervals is formulated in terms of semialgebraic optimization. Then, a suitable relaxation technique is presented to compute parameter bounds by means of convex optimization. Advantages of the presented approach with respect to previously published results are discussed.

A Set Membership identification method for state-space Linear Parameter Varying (LPV) systems is proposed in this chapter. The method, based on the assumption of unknown but bounded noise and on a mild regularity assumption on the system to be identified, allows the identification of almost-optimal and optimal state-space LPV models, and the evaluation of tight uncertainty bounds on the elements of the model parameter-varying matrices. A relevant feature of the proposed method is that the obtained model is sparse, i.e., it is described by a linear combination of basis functions, where the vector of linear combination coefficients has only a few non-zero elements. Two simulation examples show the effectiveness of the proposed method: identification of the forced Van der Pol oscillator, and identification and control of a 2-degrees of freedom robot manipulator.

This chapter presents an overview of the available methods for identifying input-output LPV models both in discrete time and continuous time with the main focus on noise modeling issues. First, a least-squares approach and an instrumental variable method are presented for dealing with LPV-ARX models. Then, a refined instrumental variable approach is discussed to address more sophisticated noise models like Box-Jenkins in the LPV context. This latter approach is also introduced in continuous time and efficient solutions are proposed for both the problem of time-derivative approximation and the issue of continuous-time modeling of the noise.

Subspace identification methods for linear parameter-varying state-space systems aim at the reconstruction of the state sequence based on the available measurements of the inputs, outputs and time-varying parameters. Several methods have been proposed in the literature that perform computations with data matrices of which the number of rows grow exponentially with the order of the system. The data matrices for relatively low-order systems with only a few inputs, outputs and time-varying parameters can easily have more than 100 000 rows. Besides computational issues, these methods require a huge amount of measurements (larger than or equal to the number of rows in the data matrices). This severely limits the practical applicability of the LPV subspace methods. In this chapter a method to reduce the dimensions of the data matrices is discussed. It is based on an efficient RQ factorization in which only the rows that have the largest influence on the output-error of the LPV model are used. Also a kernel method is discussed that avoids computations with large dimensional matrices.

In this chapter we present a novel algorithm to identify LPV systems with affine parameter dependence. This algorithm is applicable for data generated in either open or closed-loop. A factorization is introduced that makes it possible to form a predictor, which is based on past inputs, outputs, and scheduling data. The predictor contains the LPV equivalent of the Markov parameters. Using this predictor, ideas from closed-loop LTI identification are developed to estimate the state sequence from which the LPV system matrices can be constructed. A numerically efficient implementation is presented using the kernel method.

LPV model identification has attracted a significant attention in the last few years, in view of the increasing interest for the analysis and the design of gain-scheduled control systems based on parameter-varying models. In particular, a number of reliable algorithms are now available for the estimation of state-space and input-output LPV models both in a local and in a global perspective. Most of the methods proposed so far in the literature, however, deal with the identification of discrete-time models. This can be viewed as a limitation, for the following reasons: first, most LPV control design methods are formulated in continuous-time and therefore assume that a continuous-time model of the plant to be controlled is available; in addition, the conversion of LPV models from discrete-time to continuous-time is not as straightforward and well understood as its LTI counterpart; finally, when it comes to the choice of model structure (i.e., the definition of the parameter vector scheduling the dynamics of the LPV model), the intuition associated with continuous-time variables generally plays a major role and might be lost when performing the modelling exercise in discrete-time. In view of the above issues, in this work the problem of deriving continuous-time LPV models in state-space form from sampled measurements of input, output and scheduling variables is considered, in the framework of a local approach. The proposed method is formulated in the subspace model identification framework and builds on previous, recent results on the subspace-based identification of continuous-time LTI models derived by the Authors.

The successive approximation Linear Parameter Varying systems subspace identification algorithm for discrete-time systems is based on a convergent sequence of linear time invariant deterministic-stochastic state-space approximations. In this chapter, this method is modified to cope with continuous-time LPV state-space models. To do this, the LPV system is discretised, the discrete-time model is identified by the successive approximations algorithm and then converted to a continuous-time model. Since affine dependence is preserved only for fast sampling, a subspace downsampling approach is used to estimate the linear time invariant deterministic-stochastic state-space approximations. A second order simulation example, with complex poles, illustrates the effectiveness of the new algorithm.

In this chapter an Orthonormal Basis Functions (OBFs) model structure is proposed in the LPV context with advantageous properties in terms of estimation and realization. A solid system theoretic basis for the description of LPV systems in terms of these model structures is presented together with a general approach to LPV identification. Data-driven model structure selection is also discussed in this setting and the stochastic properties of the employed identification schemes are analyzed. The introduced approaches are demonstrated on an industrially relevant example.

This chapter examines the estimation of multivariable linear models for which the parameters vary in a time-varying manner that depends in an affine fashion on a known or otherwise measured signal. These locally linear models which depend on a measurable operating point are known as linear parameter varying (LPV) models. The contribution here relative to previous work on the topic is that in the Gaussian case, an expectation-maximisation (EM) algorithm-based solution is derived and profiled.

In this chapter, identification of systems composed by interconnected LTI systems and static nonlinearities is addressed. The interconnection is described by a linear fractional representation in which the nonlinear block is modeled by a piecewise affine map. A two-step iterative identification procedure, with non-increasing cost at each iteration, is proposed. Features and effectiveness of the presented identification procedure are illustrated through a numerical example and an application to the silver-box problem (a popular benchmark in nonlinear system identification).

Identification of switched systems has received considerable attention during the past few years. This chapter addresses the problem of identification and (in)validation of switched autoregressive models with external inputs from experimental data. In the first part of the chapter identification of switched linear systems from noisy measurements is considered. Given a finite number of measurements of input/output and a bound on the measurement noise, the objective is to identify a switching sequence and a set of linear submodels that are compatible with the a priori information, while minimizing the number of submodels. A deterministic approach based on convex optimization is proposed. Namely, by recasting the identification problem as polynomial optimization, we develop deterministic algorithms, in which the inherent sparse structure is exploited. A finite dimensional semi-definite problem is then given which is equivalent to the identification problem. Moreover, to address computational complexity issues, an equivalent rank minimization problem subject to deterministic linear matrix inequality constraints is provided, as efficient convex relaxations for rank minimization are available in the literature.

Since the identification problem is generically NP-Hard, the majority of existing algorithms, including the one we proposed, are based on heuristics or relaxations. Therefore, it is crucial to check the validity of the identified models against additional experimental data. The second part of this chapter addresses the problem of model (in)validation for switched linear autoregressive exogenous systems with unknown switches. Necessary and sufficient conditions are obtained for a given model to be (in)validated by the experimental data. In principle, checking these conditions requires solving a sequence of convex optimization problems involving increasingly large matrices. However, as we show, if in the process of solving these problems either a positive solution is found or the so-called flat extension property holds, then the process terminates with a certificate that either the model has been invalidated or that the experimental data is indeed consistent with the model and a–priori information. By using duality, the proposed approach exploits the inherently sparse structure of the optimization problem to substantially reduce its computational complexity.

The effectiveness of the proposed methods is illustrated using both academic examples and a non-trivial problem arising in computer vision: activity monitoring.

The following sections are included:

• Index