Martin Enqvist

Linear approximations of nonlinear FIR systems for separable input processes

Nonlinear systems can be approximated by linear time-invariant (LTI) models in many ways. Here, LTI models that are optimal approximations in the mean-square error sense are analyzed. A necessary and sufficient condition on the input signal for the optimal LTI approximation of an arbitrary nonlinear finite impulse response (NFIR) system to be a linear finite impulse response (FIR) model is presented. This condition says that the input should be separable of a certain order, i.e., that certain conditional expectations should be linear. For the special case of Gaussian input signals, this condition is closely related to a generalized version of Bussgangs classic theorem about static nonlinearities. It is shown that this generalized theorem can be used for structure identification and for the identification of generalized Wiener-Hammerstein systems.

Handling Certain Structure Information in Subspace Identification

The prediction-error approach to parameter estimation of linear models often involves solving a non-convex optimization problem. In some cases, it is therefore difficult to guarantee that the global optimum will be found. A common way to handle this problem is to find an initial estimate, hopefully lying in the region of attraction of the global optimum, using some other method. The prediction-error estimate can then be obtained by a local search starting at the initial estimate. In this paper, a new approach for finding an initial estimate of certain linear models utilizing structure and the subspace method is presented. The polynomial models are first written on the observer canonical state-space form, where the specific structure is later utilized, rendering least-squares estimation problems with linear equality constraints.

Study of the LTI relations between the outputs of two coupled Wiener systems and its application to the generation of initial estimates for Wiener-Hammerstein systems

This paper consists of two parts. In the first, more theoretic part, two Wiener systems driven by the same Gaussian noise excitation are considered. For each of these systems, the best linear approximation (BLA) of the output (in mean square sense) is calculated, and the residuals, defined as the difference between the actual output and the linearly simulated output is considered for both outputs. The paper is focused on the study of the linear relations that exist between these residuals. Explicit expressions are given as a function of the dynamic blocks of both systems, generalizing earlier results obtained by Brillinger [Brillinger, D. R. (1977). The identification of a particular nonlinear time series system. Biometrika, 64(3), 509-515] and Billings and Fakhouri [Billings, S. A., & Fakhouri, S. Y. (1982). Identification of systems containing linear dynamic and static nonlinear elements. Automatica, 18(1), 15-26]. Compared to these earlier results, a much wider class of static nonlinear blocks is allowed, and the efficiency of the estimate of the linear approximation between the residuals is considerably improved. In the second, more practical, part of the paper, this new theoretical result is used to generate initial estimates for the transfer function of the dynamic blocks of a Wiener-Hammerstein system. This method is illustrated on experimental data.

Order and structural dependence selection of LPV-ARX models using a nonnegative garrote approach

In order to accurately identify Linear Parameter-Varying (LPV) systems, order selection of LPV linear regression models has prime importance. Existing identification approaches in this context suffer from the drawback that a set of functional dependencies needs to be chosen a priori for the parametrization of the model coefficients. However in a black-box setting, it has not been possible so far to decide which functions from a given set are required for the parametrization and which are not. To provide a practical solution, a nonnegative garrote approach is applied. It is shown that using only a measured data record of the plant, both the order selection and the selection of structural coefficient dependence can be solved by the proposed method.

A Nonlinear Block Structure Identification Procedure Using Frequency Response Function Measurements

Based on simple frequency response function (FRF) measurements, we give the user some guidance in the selection of an appropriate nonlinear block structure for the system to be modeled. The method consists in measuring the FRF using a Gaussian-like input signal and varying in a first experiment the root-mean-square (rms) value of this signal while maintaining the coloring of the power spectrum. Next, in a second experiment, the coloring of the power spectrum is varied while keeping the rms value constant. Based on the resulting behavior of the FRF, an appropriate nonlinear block structure can be selected to approximate the real system. The identification of the selected block-oriented model itself is not addressed in this paper. A theoretical analysis and two practical applications of this structure identification method are presented for some nonlinear block structures.

Identification of Block-oriented Systems Using the Invariance Property

Identification of systems that can be written as interconnected linear time-invariant (LTI) dynamical subsystems and static nonlinearities has been an active research area for several decades. These systems are often referred to as block-oriented systems since their structures can be characterised using linear dynamical and static nonlinear blocks. In particular, block-oriented systems where the blocks are connected in series have received special attention. For example, Wiener and Hammerstein systems are common examples of series connected block-oriented systems.

Least-Squares Phase Predistortion of a +30 dBm Class-D Outphasing RF PA in 65 nm CMOS

This paper presents a model-based phase-only predistortion method suitable for outphasing radio frequency (RF) power amplifiers (PA). The predistortion method is based on a model of the amplifier with a constant gain factor and phase rotation for each outphasing signal, and a predistorter with phase rotation only. Exploring the structure of the outphasing PA, the model estimation problem can be reformulated from a nonconvex problem into a convex least-squares problem, and the predistorter can be calculated analytically. The method has been evaluated for 5 MHz Wideband Code-Division Multiple Access (WCDMA) and Long Term Evolution (LTE) uplink signals with Peak-to-Average Power Ratio (PAPR) of 3.5 dB and 6.2 dB, respectively, applied to one of the first fully integrated +30 dBm Class-D outphasing RF PAs in 65 nm CMOS. At 1.95 GHz for a 5.5 V (6.0 V) supply voltage, the measured output power of the PA was +29.7 dBm (+30.5 dBm) with a power-added efficiency (PAE) of 27%. For the WCDMA signal with +26.0 dBm of channel power, the measured Adjacent Channel Leakage Ratio (ACLR) at 5 MHz and 10 MHz offsets were - 46.3 dBc and - 55.6 dBc with predistortion, compared to -35.5 dBc and -48.1 dBc without predistortion. For the LTE signal with +23.3 dBm of channel power, the measured ACLR at 5 MHz offset was - 43.5 dBc with predistortion, compared to -34.1 dBc without predistortion.

Detection of roof load for automotive safety systems

The performance and design of lateral stability systems in cars depend on the ratio between the height of the center of gravity and the wheel base. This ratio is car specific, but a roof load can affect this and decrease the stability margins. We investigate the use of vehicle roll dynamics to detect and estimate changes in the overall sprung mass as well as the load positioned on the roof. It is assumed that the vehicle is equipped with a lateral accelerometer and a roll gyro, and a second order physical model is derived. The parameters in this model are partly unknown, and here estimated with a greybox and an ARMAX approach. The changes in load distribution can be detected and the approach is supported by experimental data in a lab environment.

A convex relaxation of a dimension reduction problem using the nuclear norm

The estimation of nonlinear models can be a challenging problem, in particular when the number of available data points is small or when the dimension of the regressor space is high. To meet these challenges, several dimension reduction methods have been proposed in the literature, where a majority of the methods are based on the framework of inverse regression. This allows for the use of standard tools when analyzing the statistical properties of an approach and often enables computationally efficient implementations. The main limitation of the inverse regression approach to dimension reduction is the dependence on a design criterion which restricts the possible distributions of the regressors. This limitation can be avoided by using a forward approach, which will be the topic of this paper. One drawback with the forward approach to dimension reduction is the need to solve nonconvex optimization problems. In this paper, a reformulation of a well established dimension reduction method is presented, which reveals the structure of the optimization problem, and a convex relaxation is derived.

Inverse Regression for the Wiener Class of Systems

The concept of inverse regression has turned out to be quite useful for dimension reduction in regression analysis problems. Using methods like sliced inverse regression (SIR) and directional regression (DR), some high-dimensional nonlinear regression problems can be turned into more tractable low-dimensional problems. Here, the usefulness of inverse regression for identification of nonlinear dynamical systems will be discussed. In particular, it will be shown that the inverse regression methods can be used for identification of systems of the Wiener class, that is, systems consisting of a number of parallel linear subsystems followed by a static multiple-input single-output nonlinearity. For a particular class of input signals, including Gaussian signals, the inverse regression approach makes it possible to estimate the linear subsystems without knowing or estimating the nonlinearity.