Identification for control: Optimal input intended to identify a minimum variance controller

It is well known that the quality of the parameters identified during an identification experiment depends on the applied excitation signal. Prediction error identification using full order parametric models delivers an ellipsoidal region in which the true parameters lie with some prescribed probability level. This ellipsoidal region is determined by the covariance matrix of the parameters. Input design strategies aim at the minimization of some measure of this covariance matrix. We show that it is possible to optimize the input in an identification experiment with respect to a performance cost function of a closed-loop system involving explicitly the dependence of the designed controller on the identified model. In the present contribution we focus on finding the optimal input for the estimation of the parameters of a minimum variance controller, without the intermediate step of first minimizing some measure of the model parameter accuracy. We do this in conjunction with using covariance formulas which are not asymptotic in the model order, which is rather new in the domain of optimal input design. The identification procedure is performed in closed-loop. Besides optimizing the input power spectrum for the identification experiment, we also address the question of optimality of the controller. It is a wide belief that the minimum variance controller should be the optimal choice, since we perform an experiment for designing a minimum variance controller. However, we show that this may not always be the case, but rather depends on the model structure.     

Identification of ARX systems with non-stationary inputs — asymptotic analysis with application to adaptive input design

A key problem in optimal input design is that the solution depends on system parameters to be identified. In this contribution we provide formal results for convergence and asymptotic optimality of an adaptive input design method based on the certainty equivalence principle, i.e. for each time step an optimal input design problem is solved exactly using the present parameter estimate and one sample of this input is applied to the system. The results apply to stable ARX systems with the input restricted to be generated by white noise filtered through a finite impulse response filter, or a binary signal obtained from the latter by a static nonlinearity.     

Recursive identification for multi-input–multi-output Hammerstein–Wiener system

In this paper, an identification method based on the recursive auxiliary variable least squares algorithm is proposed for a multi-input–multi-output Hammerstein–Wiener system with process noise. In the proposed identification method, the system is converted into the multivariate regression form under the condition that the nonlinear block in the output part is invertible. Then, the auxiliary variable is constructed, the parameters of the regression equations are identified, and the system parameter matrices can be obtained by matrix composition of the parameter product matrix. A theoretical analysis showed that the proposed method has uniform convergence when the process noise is white and has a finite variance. The effectiveness of the proposed method is validated through the experiments.     

Robust M-ary detection filters and smoothers for continuous-time jump Markov systems

In this paper, we consider a dynamic M-ary detection problem when Markov chains are observed through a Wiener process. These systems are fully specified by a candidate set of parameters, whose elements are, a rate matrix for the Markov chain and a parameter for the observation model. Further, we suppose these parameter sets can switch according to the state of an unobserved Markov chain and thereby produce an observation process generated by time varying (jump stochastic) parameter sets. Given such an observation process and a specified collection of models, we estimate the probabilities of each model parameter set explaining the observation. By defining a new augmented state process, then applying the method of reference probability, we compute matrix-valued dynamics, whose solutions estimate joint probabilities for all combinations of candidate model parameter sets and values taken by the indirectly observed state process. These matrix-valued dynamics satisfy a stochastic integral equation with a Wiener process integrator. Using the gauge transformation techniques introduced by Clark and a pointwise matrix product, we compute robust matrix-valued dynamics for the joint probabilities on the augmented state space. In these new dynamics, the observation Wiener process appears as a parameter matrix in a linear ordinary differential equation, rather than an integrator in a stochastic integral equation. It is shown that these robust dynamics, when discretised, enjoy a deterministic upper bound which ensures nonnegative probabilities for any observation sample path. In contrast, no such upper bounds can be computed for Taylor expansion approximations, such as the Euler-Maryauana and Milstein schemes. Finally, by exploiting a duality between causal and anticausal robust detector dynamics, we develop an algorithm to compute smoothed mode probability estimates without stochastic integrations. A computer simulation demonstrating performance is included.     

Bootstrap control

In this paper, we present a new way to control linear stochastic systems. The method is based on statistical bootstrap techniques. The optimal future control signal is derived in such a way that unknown noise distribution and uncertainties in parameter estimates are taken into account. This is achieved by resampling from existing data when calculating statistical distributions of future process values. The bootstrap algorithm takes care of arbitrary loss functions and unknown noise distribution even for small estimation sets. The efficient way of utilizing data implies that the method is also well suited for slowly time-varying stochastic systems.     

Optimization of user experience in interaction design through a Taguchi‐based hybrid approach

User experience (UX), which encompasses all aspects of a user’s interaction with an interactive product, has been recognized as central to interaction design. This paper proposes a Taguchi‐based hybrid approach to realize the optimal UX design. In this approach, design analysis is first used to identify design patterns and UX characteristics. According to the results, a Taguchi experiment is conducted and the signal‐to‐noise ratios are computed. Subsequently, the preference weights for the UX characteristics are obtained by using analytical hierarchy process‐based group decision making. A multiperformance characteristic index is then defined based on the gray relational grade (GRG) obtained through gray relational analysis. On the basis of the GRG, the optimal design can be obtained. A mobile health application design was used to demonstrate the proposed approach. The results show that this approach can effectively enhance UX quality and be used as a universal design approach for optimizing UX.     

Design of experiments for bivariate binary responses modelled by Copula functions

Optimal design for generalized linear models has primarily focused on univariate data. Often experiments are performed that have multiple dependent responses described by regression type models, and it is of interest and of value to design the experiment for all these responses. This requires a multivariate distribution underlying a pre-chosen model for the data. Here, we consider the design of experiments for bivariate binary data which are dependent. We explore Copula functions which provide a rich and flexible class of structures to derive joint distributions for bivariate binary data. We present methods for deriving optimal experimental designs for dependent bivariate binary data using Copulas, and demonstrate that, by including the dependence between responses in the design process, more efficient parameter estimates are obtained than by the usual practice of simply designing for a single variable only. Further, we investigate the robustness of designs with respect to initial parameter estimates and Copula function, and also show the performance of compound criteria within this bivariate binary setting.     

Closed loop experiment design for linear time invariant dynamical systems via LMIs

All stationary experimental conditions corresponding to a discrete-time linear time-invariant causal internally stable closed loop with real rational system and feedback controller are characterized using the Youla-Kucera parametrization. Finite dimensional parametrizations of the input spectrum and the Youla-Kucera parameter allow a wide range of closed loop experiment design problems, based on the asymptotic (in the sample size) covariance matrix for the estimated parameters, to be recast as computationally tractable convex optimization problems such as semi-definite programs. In particular, for Box-Jenkins models, a finite dimensional parametrization is provided which is able to generate all possible asymptotic covariance matrices. As a special case, the very common situation of a fixed controller during the identification experiment can be handled and optimal reference signal spectra can be computed subject to closed loop signal constraints. Finally, a brief numerical comparison with closed loop experiment design based on a high model order variance expression is presented.     

Identification and the Information Matrix: How to Get Just Sufficiently Rich?

In prediction error identification, the information matrix plays a central role. Specifically, when the system is in the model set, the covariance matrix of the parameter estimates converges asymptotically, up to a scaling factor, to the inverse of the information matrix. The existence of a finite covariance matrix thus depends on the positive definiteness of the information matrix, and the rate of convergence of the parameter estimate depends on its “size”. The information matrix is also the key tool in the solution of optimal experiment design procedures, which have become a focus of recent attention. Introducing a geometric framework, we provide a complete analysis, for arbitrary model structures, of the minimum degree of richness required to guarantee the nonsingularity of the information matrix. We then particularize these results to all commonly used model structures, both in open loop and in closed loop. In a closed-loop setup, our results provide an unexpected and precisely quantifiable trade-off between controller degree and required degree of external excitation.      

Optimal ensemble control of stochastic time-varying linear systems

We consider the optimal guidance of an ensemble of independent, structurally identical, finite-dimensional stochastic linear systems with variation in system parameters between initial and target states of interest by applying a common control function without the use of feedback. Our exploration of such ensemble control systems is motivated by practical control design problems in which variation in system parameters and stochastic effects must be compensated for when state feedback is unavailable, such as in pulse design for nuclear magnetic resonance spectroscopy and imaging. In this paper, we extend the notion of ensemble control to stochastic linear systems with additive noise and jumps, which we model using white Gaussian noise and Poisson counters, respectively, and investigate the optimal steering problem. In our main result, we prove that the minimum norm solution to a Fredholm integral equation of the first kind provides the optimal control that simultaneously minimizes the mean square error (MSE) and the error in the mean of the terminal state. The optimal controls are generated numerically for several example ensemble control problems, and Monte Carlo simulations are used to illustrate their performance. This work has immediate applications to the control of dynamical systems with parameter dispersion or uncertainty that are subject to additive noise, which are of interest in quantum control, neuroscience, and sensorless robotic manipulation.     

Getting the most out of it: Optimal experiments for parameter estimation of microalgae growth models

Mathematical models are expected to play a pivotal role for driving microalgal production towards a profitable process of renewable energy generation. To render models of microalgae growth useful tools for prediction and process optimization, reliable parameters need to be provided. This reliability implies a careful design of experiments that can be exploited for parameter estimation. In this paper, we provide guidelines for the design of experiments with high informative content based on optimal experiment techniques to attain an accurate parameter estimation. We study a real experimental device devoted to evaluate the effect of temperature and light on microalgae growth. On the basis of a mathematical model of the experimental system, the optimal experiment design problem was formulated and solved with both static (constant light and temperature) and dynamic (time varying light and temperature) approaches. Simulation results indicated that the optimal experiment design allows for a more accurate parameter estimation than that provided by the existing experimental protocol. For its efficacy in terms of the maximum likelihood properties and its practical aspects of implementation, the dynamic approach is recommended over the static approach.     

Sliding mode control based on a modified linear control input

In this study, we explain and demonstrate a design method of sliding mode control based on a modified linear control input. In the proposed method, the optimal gain matrix is derived such that it does not depend on the plant parameters. We confirmed the robustness of the proposed method by applying input-side disturbances and plant parameter deviations to plants and the effectiveness of the proposed method by performing a DC motor position control experiment     

Optimal experiment design for linear systems with input-output constraints

The problem of optimal experiment design for parameter estimation in linear dynamic systems is studied. Results relating to both constrained input and output variances are established. For the case of constrained input variance, it is shown that a D-optimal experiment exists in which the system input is generated externally provided the system and noise transfer functions have no common parameters. For the case of constrained output variance, it is shown that an experiment in which the system input is generated by a combination of a minimum variance control law together with an external set point perturbation is D-optimal for certain classes of systems. Other related results are also presented which illustrate the role of feedback in optimal experiment design.     

压缩感知雷达感知矩阵优化

压缩感知雷达(Compressive sensing radar,CSR)的场景恢复性能要求感知矩阵相关系数尽可能小。针对感知矩阵相关系数的最小化问题,提出了基于模拟退火的感知矩阵优化算法,建立了基于随机滤波结构的CSR模型,给出了优化目标函数,采用模拟退火实现了发射波形、测量矩阵的优化以及联合优化。仿真结果表明该算法可以提高场景恢复精度,提升抗噪能力,增大可观测目标个数上限,且联合优化的性能优于波形和测量矩阵的单独优化。     

Optimal inputs for linear system identification

This paper considers the design of optimal inputs for identifying parameters in linear dynamic systems. The criterion used for optimization is the sensitivity of the system output to the unknown parameters as expressed by the weighted trace of the Fisher information matrix. It is shown that the optimal energy constrained input is an eigenfunction of a positive self-adjoint operator corresponding to its largest eigenvalue. Several different representations of the optimal input and several methods for its numerical computation are considered. The results are extended to systems with process noise, and the relationship to other criteria for input design are brought out. Three analytical examples are solved in closed form which show that the optimal input is a sum of sine and cosine functions at appropriate frequencies. Optimal elevator deflections for identifying the short period parameters of C-8 aircraft are computed numerically and compared with the doublet input currently in use.     

Artificial neural network based approach for dynamic parameter design

The Taguchi parameter design method has been recognized as an important tool for improving the quality of a product or a process. However, the statistical methods and optimization procedures proposed by Taguchi have much room for improvement. For instance, the two-step procedure proposed by Taguchi may fail to identify an optimum design condition if an adjustment parameter does not exist, the optimal setting of a design parameter is determined only among the levels included in the parameter design experiment, and, for the dynamic parameter design, the signal parameter is assumed to follow a uniform rather than a general distribution. This paper develops an artificial neural network based dynamic parameter design approach to overcome the shortcomings of the Taguchi and existing alternative approaches. First, an artificial neural network is trained to map the relationship between the characteristic, design, noise and signal parameters. Second, Latin hypercube samples of the signal and noise parameters are obtained and used to estimate the slope between the signal parameter and characteristic as well as the variance of the characteristic at each set of design parameter settings. Then, the dynamic parameter design problem is formulated as a nonlinear optimization problem and solved to find the optimal settings of the design parameters using sequential quadratic programming. The effectiveness of the proposed approach is illustrated with an example.     

Steady-state optimal controller with actuator noise variancelinearly related to actuator signal variance

The linear-quadratic-Gaussian (LQG) method assumes that the covariance matrix of the actuator noise is given before the design of the feedback matrix. However, in practice, the actuator-noise covariance matrix may depend on the actuator signal energy, which depends on the feedback. Consequently, the feedback from LQG theory degrades the system performance. The authors investigate the steady-state optimal controller when the noise variance of an actuator is linearly related to the variance of an actuator signal. This control system could be much more precise and/or spend much less control energy than the one obtained through the use of the ordinary LQG method     

Variational Bayesian sparse additive matrix factorization

Principal component analysis (PCA) approximates a data matrix with a low-rank one by imposing sparsity on its singular values. Its robust variant can cope with spiky noise by introducing an element-wise sparse term. In this paper, we extend such sparse matrix learning methods, and propose a novel framework called sparse additive matrix factorization (SAMF). SAMF systematically induces various types of sparsity by a Bayesian regularization effect, called model-induced regularization. Although group LASSO also allows us to design arbitrary types of sparsity on a matrix, SAMF, which is based on the Bayesian framework, provides inference without any requirement for manual parameter tuning. We propose an efficient iterative algorithm called the mean update (MU) for the variational Bayesian approximation to SAMF, which gives the global optimal solution for a large subset of parameters in each step. We demonstrate the usefulness of our method on benchmark datasets and a foreground/background video separation problem.     

Designing robust optimal dynamic experiments

The quality of experiments designed using standard optimisation-based techniques can be adversely affected by poor starting values of the parameters. Thus, there is a necessity for design methods that are insensitive (“robust”) to these starting values. Here, two novel, robust criteria are presented for computing optimal dynamic inputs for experiments aiming at improving parameter precision, based on previously proposed methods for providing design robustness — the expected value criterion and the max–min criterion. In this paper, both criteria are extended by use of the information matrix derived for dynamic systems. The experiment design problem is cast as an optimal control problem, with experiment decision variables such as sampling times of response variables, time-varying and time-invariant external controls, experiment duration, and initial conditions. Comparisons are made of experiment designs obtained using the conventional approach and the newly proposed criteria. A typical semi-continuous bioreactor application is presented.     

On using multivariate polynomial regression model with spectral difference for statistical model-based speech enhancement

In this paper, we propose a statistical model-based speech enhancement technique using a multivariate polynomial regression (MPR) based on spectral difference scheme. In the analyzing step, three principal parameters, the weighting parameter in the decision-directed (DD) method, the long-term smoothing parameter for the noise estimation, and the control parameter of the minimum gain value are estimated as optimal operating points technique by using to the spectral difference under various noise conditions. These optimal operating points, which are specific according to different spectral differences, are estimated based on the composite measure, which is a relevant criterion in terms of speech quality. Thus, we apply the MPR technique by incorporating the spectral differences as independent variables in order to estimate the optimal operating points. The MPR technique offers an effective scheme to represent complex nonlinear input-output relationship between the optimal operating points and spectral differences so that operating points can be determined according to various noise conditions in the off-line step. In the on-line speech enhancement step, different parameters are chosen on a frame-by-frame basis through the regression according to the spectral difference. The performance of the proposed method is evaluated using objective and subjective speech quality measures in various noise environments. Our experimental results show that the proposed algorithm yields better performances than conventional algorithms.