LMS finite memory estimators for discrete-time state space models

Ahlstrom Corp, Finland

Linear regressive realizations of LTI state space models

In this paper, we treat the problem of decentralized implicit adaptive regulation for large-scale stochastic systems composed into a set of interconnected systems that are described by discrete-time state-space mathematical models with unknown parameters. The key idea in the decentralized regulation method is to design local regulator using only local information such that the state of each interconnected system is regulated to a certain constant reference signal. The main contribution is the proposition of a decentralized implicit adaptive regulator based on state-feedback strategy that can be applied to stochastic interconnected systems with unknown parameters. Furthermore, the practical implementation of the proposed decentralized implicit adaptive regulator can be made easily (low-cost implementation of the electronic components, short computation of the decentralized control law, etc.). A theorem is established and proved which gives sufficient stability conditions of the resulting closed-loop interconnected systems by using the Lyapunov method. An example of numerical simulation is treated to test the performance of the proposed decentralized implicit adaptive regulator.

This paper proposes a new approach to the localization of a perspective camera with respect to a known target. The problem is stated as the dual - in the automatic control sense - of visual servoing: while visual servoing would move a camera from visual data so as to stabilize its relative pose with respect to a target, the aim of localization is to estimate/forecast this relative situation from the knowledge of visual data and of the camera command input. The proposed methodology entails a discrete-time rational state space model of interactions. This model is embedded in a structured norm-bounded linear recursive inclusion, so that the forecast of the relative situation at the next sampling time is computed by robust filtering techniques. The solution is obtained via an optimization problem under linear matrix inequalities constraints. Despite its difficulty of implementation, the proposed set-membership method seems quite promising, for it can handle parametric uncertainties as well as dynamic and measurement noises, and encloses the relative situations compatible with the visual data.

In this paper, a reduced-order modeling approach based on computational fluid dynamics is presented for an elastic wing with control surfaces in the transonic regime. To treat the computational fluid dynamics grid around the geometrical discontinuities, due to the deflection of control surfaces, the constant volume tetrahedron method and the transpiration method are combined together without deforming the grid. Based on the input–output data from the computational fluid dynamics solver, one multiple-input/multiple-output discrete-time state-space model for the wing is identified via a robust subspace algorithm. For each control surface, one one-input/multiple-output discrete-time state-space model is identified using the same algorithm. With the precomputed state-space models for a few flight parameters, the generalized aerodynamic forces over a range of flight parameters can be computed by interpolating the output data. The methodology is applied to an elastic wing model with two control surfaces...

Discrete-time state-space models have been extensively used in simulation-based design of dynamical systems. These prediction models may not accurately represent the true physics of a dynamical system due to potentially flawed understanding of the system, missing physics, and/or numerical approximations. To improve the validity of these models at new design locations, this paper proposes a novel dynamic model discrepancy quantification (DMDQ) framework. Time-instantaneous prediction models are constructed for the model discrepancies of “hidden” state variables, and are used to correct the discrete-time prediction models at each time-step. For discrete-time models, the hidden state variables and their discrepancies are coupled over two adjacent time steps. Also, the state variables cannot be directly measured. These factors complicate the construction of the model discrepancy prediction models. The proposed DMDQ framework overcomes these challenges by proposing two discrepancy modeling approaches: an estimation-modeling approach and a modeling-estimation approach. The former first estimates the model discrepancy and then builds a nonparametric prediction model of the model discrepancy; the latter builds a parametric prediction model of the model discrepancy first and then estimates the parameters of the prediction model. A subsampling method is developed to reduce the computational effort in building the two types of prediction models. A mathematical example and an electrical circuit dynamical system demonstrate the effectiveness of the proposed DMDQ framework and highlight the advantages and disadvantages of the proposed approaches.

Discrete-time models with state-dependent parameters are derived for non-linear sampled-data systems. Velocity-based linearization of the continuous-time system followed by integration over the sampling interval is used to represent the sampled-data system by a discrete-time state-space model with state-dependent parameters. By reconstructing the state a quasi-ARX model is obtained, the parameters of which are functions of past inputs and outputs. As the velocity-based linearization procedure does not involve suppression of higher-order terms, the state-dependent state-space model and the quasi-ARX description give exact representations of the non-linear system.

This paper proposes a new fixed-lag smoother for state estimation in discrete-time state space models, which is called the unbiased, fixed-lag, and optimal (UFO) smoother. The proposed UFO smoother is obtained by directly solving an optimization problem with the unbiasedness constraint using only the most recent finite observations. Therefore, the proposed UFO smoother is both unbiased and optimal by design. The proposed UFO smoother also has inherent properties such as deadbeat, time-invariance, and robustness. Via numerical simulations, the performance of the proposed UFO smoother is evaluated by the comparison with that of the existing fixed-lag Kalman smoother, which shows that the proposed UFO smoother could be appropriate for fast estimation of signals that vary relatively quickly.

The mathematical model for a flexible spacecraft that is rotating about a single axis rotation is described by coupled rigid and flexible body degrees-of-freedom, where the equations of motion are modeled by integro-partial differential equations. Beam-like structures are often useful for analyzing boom-like flexible appendages. The equations of motion are analyzed by introducing generalized Fourier series that transform the governing equations into a system of ordinary differential equations. Though technically straightforward, two problems arise with this approach: (1) the model is frequency-truncated because a finite number of series terms are retained in the model, and (2) computationally intense matrix-valued transfer function calculations are required for understanding the frequency domain behavior of the system. Both of these problems are resolved by: (1) computing the Laplace transform of the governing integro-partial differential equation of motion; and (2) introducing a generalized state space (consisting of the deformational coordinate and three spatial partial derivatives, as well as single and double spatial integrals of the deformational coordinate). The resulting math model is cast in the form of a linear state-space differential equation that is solved in terms of a matrix exponential and convolution integral. The structural boundary conditions defined by Hamilton’s principle are enforced on the closed-form solution for the generalized state space. The generalized state space model is then manipulated to provide analytic scalar transfer function models for original integro-partial differential system dynamics. Symbolic methods are used to obtain closed-form eigen decomposition–based solutions for the matrix exponential/convolution integral algorithm. Numerical results are presented that compare the classical series based approach with the generalized state space approach for computing representative spacecraft transfer function models.

An adaptive LQG control with no control cost is considered. In such case the loop transfer recovery (LTR) effect can be obtained. The control problem is handled using discrete-time state-space model and the parameter estimation is performed for corresponding ARMAX model which can be represented in innovation state-space form. Thus the direct estimation of model parameters is possible by means of standard ERLS procedure and the adaptive control is implemented through certainty equivalence principle. In such a situation the problem of solution continuity of Riccati equation can arise for nonminimum-phase systems. Computer simulations of third-order systems modeled by a second-order minimum-phase and nonminimum-phase models are given to illustrate the robustness and performance properties of the adaptive controller, particularly with respect to the modelling error parameter η.

LCL filters installed at converter outputs offer a higher harmonic attenuation than L filters. However, as a three order resonant circuit, it is difficult to stabilize and has a risk of oscillating with the power grid. Therefore, careful design is required to damp LCL resonance. Compared to a passive damping method, an active damping method is a more attractive solution for this problem, since it avoids extra power losses. In this paper, the damping capabilities of capacitor current, capacitor voltage, and grid-side current feedback methods, are analyzed under the discrete-time state-space model. Theoretical analysis shows that the grid-side current feedback method is more suitable for use in active power filters, because it can damp LCL resonance more effectively than the other two methods when the ratio of the resonance and the control frequency is between 0.225 and 0.325. Furthermore, since there is no need for extra sensors for additional states measurements, this method provides a cost-efficient solution. To support the theoretical analysis, the proposed method is tested on a 7-kVA single-phase shunt active power filter.

A single-stage TRIAC (TRIode for alternating current switch) dimmable LED (light emitting diode) driver using a buck converter fed through a capacitor input rectifier provides many advantages such as small-scale implementation, low output current ripple, and stable dimming operation. This circuit topology, however, has a serious issue: A large inrush current to the smoothing capacitor. To solve this issue, this paper focuses on a bleeder circuit in the TRIAC dimmable LED driver. This paper presents the mechanism of the large inrush current issue and its relationship to a conventional bleeder circuit. Moreover, this paper proposes a turn-off delay controlled bleeder circuit. The proposed bleeder circuit is analyzed by using its discrete-time state-space model in order to set proper circuit parameters for inrush current reduction. Experimental results demonstrate the inrush current reduction by the proposed method, which leads to a 37% reduction of maximum input current, as well as proper dimming operation and low output current ripple. The proposed method achieves further small-scale implementation, which enhances the advantages of the intended main circuit topology.

This paper focuses on the parameter estimation of multi-input multi-output (MIMO), linear time-periodic (LTP) systems in the time domain. Discrete-time state-space models are selected for the description of LTP systems and the Covariance Based Realization Algorithm (CoBRA), one branch of subspace methods, is used for the identification of the generalized time-lifted state-space model for an LTP system. The use of the CoBRA method for estimation is motivated by the robustness against output noise with (high-order) noise dynamics and a focus on the estimation of low rank deterministic model for the LTP system. In addition to the use of CoBRA, a novel method is proposed to compute a topologically equivalent realization for the original LTP system. The method includes two different but theoretically equivalent approaches of calculating the state matrices, which can be also used as a measure for the estimation quality. Finally, a simulation is given to illustrate the efficiency of the proposed method.

An optimal tracking control of a heat exchanger with load change is considered. It is assumed that the plant is described by a discrete-time state-space model with time-delay, and a direct link between input and output. First, based on the linear quadratic inverse technique, the authors derive an optimal tracking controller having integral, state feedback plus preview actions. To derive a state-space model of a heat exchanger, an ARX model was fitted to the open-loop data obtained from the plant. The properties of the tracking control algorithm are analyzed by simulation studies using the identified model. The optimal control algorithm with an observer was installed on a process computer to provide the set points for the analog PID (proportional plus integral plus derivative) regulators of a heat exchanger plant. Experimental studies show that the control performance of water outlet temperature in the presence of load change is greatly improved.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Subspace-based state space system identification method is well known for its computational efficiency and robustness. However, when directly applied to nonlinear models, intrinsic drawbacks exist and may lead to less accurate estimation. This paper explains how the error is generated when transforming continuous-time model to discrete-time one and then presents an approach to identify nonlinear systems by choosing two different sample frequencies for different parts of the discrete-time state-space model. The method allows to identify the nonlinear coefficients and the underlying linear system more robustly based on only the input and output of the structural system. Several numerical examples are given to illustrate the correctness of the analysis and the effectiveness of the method.