As a first step towards developing efficient building energy management techniques, in this paper, we first study the energy consumption patterns of heating, ventilation and cooling (HVAC) systems across the KTH Royal Institute of Technology campus and we identify some possible areas where energy consumption can be made less wasteful. Later, we describe a test-bed where wireless sensor networks are used to collect data and eventually control the HVAC system in a distributed way. We present some of the data, temperature, humidity, and CO2 measurements, that are collected by the aforementioned network and compare them with the measurements collected by the legacy sensors already in place. In the end we present a preliminary result on modelling the dynamics of the temperature, humidity, and CO2 using the data gather by the sensor network. We check the validity of the model via comparing the out put of the system with measured data. As a future work we identify the possibility of using the models obtained here for model based control, and fault detection and isolation techniques.

A control scheme is proposed for stabilization of backward driving along simple paths for a miniaturized vehicle composed of a truck and a two-axle trailer. The paths chosen are straight lines and arcs of circles. When reversing, the truck and trailer under examination can be modeled as an unstable nonlinear system with state and input saturations. The simplified goal of stabilizing along a trajectory (instead of a point) allows us to consider a system with controllable linearization. Still, the combination of instability and saturations makes the task impossible with a single controller. In fact, the system cannot be driven backward from all initial states because of the jack-knife effects between the parts of the multibody vehicle; it is sometimes necessary to drive forward to enter into a specific region of attraction. This leads to the use of hybrid controllers. The scheme has been implemented and successfully used to reverse the radio-controlled vehicle.

The use of adaptive antenna techniques to increase the channel capacity is discussed. Directional sensitivity is obtained by using an antenna array at the base station, possibly both in receiving and transmitting modes. A scheme for separating several signals at the same frequency is proposed. The method is based on high-resolution direction-finding followed by optimal combination of the antenna outputs. Comparison with a method based on reference signals is made. Computer simulations are carried out to test the applicability of the technique to scattering scenarios that typically arise in urban areas. The proposed scheme is found to have great potential in rejecting cochannel interference, albeit at the expense of high computational requirements.

The application of adaptive antenna techniques to increase the channel capacity in mobile radio communication is discussed. Directional sensitivity is obtained by using an antenna array at the base station, possibly both in receiving and transmitting mode. A scheme for separating several signals at the same frequency is proposed. The method is based on high-resolution direction finding following by optimal combination of the antenna outputs. Comparisons to a method based on reference signals are made. Computer simulations are carried out to test the applicability of the technique to scattering scenarios that typically arise in urban areas. The proposed scheme is found to have great potential in rejecting cochannel interference, albeit at the expense of high computational requirements.

We present an Alternating Direction Method of Multipliers (ADMM) algorithm for solving optimization problems with an ℓ1 regularized least-squares cost function subject to recursive equality constraints. The considered optimization problem has applications in control, for example in ℓ1 regularized MPC. The ADMM algorithm is easy to implement, converges fast to a solution of moderate accuracy, and enables separation of the optimization problem into sub-problems that may be solved in parallel. We show that the most costly step of the proposed ADMM algorithm is equivalent to solving an LQ regulator problem with an extra linear term in the cost function, a problem that can be solved efficiently using a Riccati recursion. We apply the ADMM algorithm to an example of ℓ1 regularized MPC. The numerical examples confirm fast convergence to sufficient accuracy and a linear complexity in the MPC prediction horizon.

Consider a Gaussian stationary stochastic vector process with the property that designated pairs of components are conditionally independent given the rest of the components. Such processes can be represented on a graph where the components are nodes and the lack of a connecting link between two nodes signifies conditional independence. This leads to a sparsity pattern in the inverse of the matrix-valued spectral density. Such graphical models find applications in speech, bioinformatics, image processing, econometrics and many other fields, where the problem to fit an autoregressive (AR) model to such a process has been considered. In this paper we take this problem one step further, namely to fit an autoregressive moving-average (ARMA) model to the same data. We develop a theoretical framework and an optimization procedure which also spreads further light on previous approaches and results. This procedure is then applied to the identification problem of estimating the ARMA parameters as well as the topology of the graph from statistical data.

Consider a Gaussian stationary stochastic vector process with the property that designated pairs of components are conditionally independent given the rest of the components. Such processes can be represented on a graph where the components are nodes and the lack of a connecting link between two nodes signifies conditional independence. This leads to a sparsity pattern in the inverse of the matrix-valued spectral density. Such graphical models find applications in speech, bioinformatics, image processing, econometrics and many other fields, where the problem to fit an autoregressive (AR) model to such a process has been considered. In this paper we take this problem one step further, namely to fit an autoregressive moving-average (ARMA) model to the same data. We develop a theoretical framework which also spreads further light on previous approaches and results.

In this paper, we discuss and compare three different approaches for L2- gain estimation of Hammerstein systems. The objective is to find the input signal that maximizes the gain. A fundamental difference between two of the approaches is the class, or structure, of the input signals. The first approach involves describing functions and therefore the class of input signals is sinusoids. In this case we assume that we have a model of the system and we search for the amplitude and frequency that give the largest gain. In the second approach, no structure on the input signal is assumed in advance and the system does not have to be modelled first. The maximizing input is found using an iterative procedure called power iterations. In the last approach, a new iterative procedure tailored for memoryless nonlinearities is used to find the maximizing input for the unmodelled nonlinear part of the Hammerstein system. The approaches are illustrated by numerical examples.

This work is an extension of the paper (Mosskull et al., 2003), in which the modelling, identification and stability of an nonlinear induction machine drive is studied. The validation of the stability margins of the system is refined by an improved estimate of the induced L2 loop gain of the system. This is done with a procedure called power iterations where input sequences suitable for estimating the gain are generated iteratively through experiments on the system. The power iterations result in higher gain estimates compared to the experiments previously presented. This implies that more accurate estimates are obtained as, in general, only lower bounds can be obtained as estimates for the gain. The new gain estimates are well below one, which suggests that the feedback system is stable. The experiments are performed on an industrial hardware/software simulation platform. in this paper we also discuss the power iterations from a more general point of view. The usefulness of the method for gain estimation of nonlinear systems is illustrated through simulation examples. The basic principles of the method are provided.

In this paper we present and discuss some data-driven methods for estimation of the L2-gain of dynamical systems. Partial results on convergence and statistical properties are provided. The methods are based on multiple experiments on the system. The main idea is to directly estimate the maximizing input signal by using iterative experiments on the true system. We study such a data-driven method based on a stochastic gradient method. We show that this method is very closely related to the so-called power iteration method based on the power method in numerical analysis. Furthermore, it is shown that this method is applicable for linear systems with noisy measurements. We will also study L2-gain estimation of Hammerstein systems. The stochastic gradient method and the power iteration method are evaluated and compared in simulation examples. © 2009 IFAC.

This paper concerns model reduction of dynamical systems using the nuclear norm of the Hankel matrix to make a trade-off between model fit and model complexity. This results in a convex optimization problem where this tradeoff is determined by one crucial design parameter. The main contribution is a methodology to approximately calculate all solutions up to a certain tolerance to the model reduction problem as a function of the design parameter. This is called the regularization path in sparse estimation and is a very important tool in order to find the appropriate balance between fit and complexity. We extend this to the more complicated nuclear norm case. The key idea is to determine when to exactly calculate the optimal solution using an upper bound based on the so-called duality gap. Hence, by solving a fixed number of optimization problems the whole regularization path up to a given tolerance can be efficiently computed. We illustrate this approach on some numerical examples.

We consider a class of weighted nuclear norm optimization problems with important applications in signal processing, system identification, and model order reduction. The nuclear norm is commonly used as a convex heuristic for matrix rank constraints. Our objective is to minimize a quadratic cost subject to a nuclear norm constraint on a linear function of the decision variables, where the trade-off between the fit and the constraint is governed by a regularization parameter. The main contribution is an algorithm to determine the so-called approximate regularization path, which is the optimal solution up to a given error tolerance as a function of the regularization parameter. The advantage is that we only have to solve the optimization problem for a fixed number of values of the regularization parameter, with guaranteed error tolerance. The algorithm is exemplified on a weighted Hankel matrix model order reduction problem.

The widely used nuclear norm heuristic for rank minimizationproblems introduces a regularization parameter which isdifficult to tune. We have recently proposed a method to approximatethe regularization path, i.e., the optimal solution asa function of the parameter, which requires solving the problemonly for a sparse set of points. In this paper, we extendthe algorithm to provide error bounds for the singular valuesof the approximation. We exemplify the algorithms on largescale benchmark examples in model order reduction. Here,the order of a dynamical system is reduced by means of constrainedminimization of the nuclear norm of a Hankel matrix.

This paper treats the problem of approximating a complex stochastic process in a given frequency region by an estimated autoregressive (AR) model. Two frequency domain approaches are discussed: a weighted frequency domain maximum likelihood method and a prefiltered covariance extension method based on the theory of Lindquist and co-workers. It is shown that these two approaches are very closely related and can both be formulated as convex optimization problems. An examples illustrating the methods and the effect of prefiltering/weighting is provided. The results show that these methods are capable of tuning the AR model fit to a specified frequency region.

The aim of this correspondence is to study the connection between weighted frequency-domain maximum-likelihood power spectral estimation and the time-domain prefiltered covariance extension approach. Weighting and prefiltering are introduced to emphasize the model fit in a certain frequency range. The main result is that these two methods are very closely related for the case of autoregressive (AR) model estimation, which implies that both can be formulated as convex optimization problems. Examples illustrating the methods and the effect of prefiltering/weighting are provided.

This contribution deals with the problem of structure determination for generalized orthonormal basis models used in system identification. The model structure is parameterized by a prespecified set of poles representing a finite-dimensional subspace of H2. Given this structure and experimental data, a model can be estimated using linear regression techniques. Since the variance of the estimated model increases with the number of estimated parameters, one objective is to find coordinates, or a basis, for the finite-dimensional subspace giving as compact or parsimonious a system representation as possible. In this paper, a best basis algorithm and a coefficient decomposition scheme are derived for the generalized orthonormal rational bases. Combined with linear regression and thresholding this leads to compact transfer function representations. The methods are demonstrated with several examples.

This contribution deals with the problem of structure determination for generalized orthonormal basis models used in system identification. The model structure is parameterized by a pre-specified set of poles. Given this structure and experimental data a model can be estimated using linear regression techniques. Since the variance of the estimated model increases with the number of estimated parameters, the objective is to find structures that are as compact/parsimonious as possible. A natural approach would be to estimate the poles, but this leads to nonlinear optimization with possible local minima. In this paper, a best basis algorithm is derived for the generalized orthonormal rational bases. Combined with linear regression and thresholding this leads to compact transfer function representations.

This paper deals with the problem of structure determination for generalized orthonormal basis models used in system identification. The model structure is parametrized by a pre-specified set of poles. Given this structure and experimental data a model can be estimated using linear regression techniques. Since the variance of the estimated model increases with the number of estimated parameters, the objective is to find structures that are as compact/parsimonious as possible. A natural approach would be to estimate the poles, but this leads to nonlinear optimization with possible local minima. In this paper, a best basis algorithm is derived for the generalized orthonormal rational bases. Combined with linear regression and thresholding this leads to compact transfer function representations

A standard approach for estimating the frequency function of a linear dynamical system is to use spectral estimation. Traditionally, this is done by smoothing the noisy frequency data using linear filters. The method has proved to be successful in most cases and is widely used. However, if the frequency response has fine details appearing only locally in frequency, the loss of resolution caused by smoothing might result in unacceptable errors. In this paper, a different method for frequency response estimation is suggested. The method utilizes recently proposed wavelet-based denoising schemes combined with traditional smoothing techniques. The wavelet transform is applied in the frequency domain in order to provide a suitable frequency window. Tested through simulations, this approach provides an alternative when traditional methods fail. © 1998 John Wiley & Sons, Ltd.

A problem in prediction error system identification methods is estimation of pole locations. Typically, iterative numerical optimization methods are used. Reliable initial values are then necessary for good results. The parameterization is often done in the coefficients of transfer function polynomials or some canonical form. In this contribution we discuss a couple of issues related to the above problem. First, we study how all-pass systems can be used to generate suitable model structures. This analysis is based on the relation between balanced realizations of all-pass filters and orthonormal basis transfer functions. Next, we investigate the effects of a priori fixed pole locations, such as in Laguerre and Kautz models. One idea is to use very flexible high-order models. However, the corresponding estimation problem has to be regularized in order to reduce the variance errors due to noise. We will discuss how this can be done by using thresholding of the estimated coefficients

This paper studies the input design problem for system identification where time domain constraints have to be considered. A finite Markov chain is used to model the input of the system. This allows to directly include input amplitude constraints in the input model by properly choosing the state space of the Markov chain, which is defined so that the Markov chain generates a multi-level sequence. The probability distribution of the Markov chain is shaped in order to minimize the cost function considered in the input design problem. Stochastic approximation is used to minimize that cost function. With this approach, the input signal to apply to the system can be easily generated by extracting samples from the optimal distribution. A numerical example shows how this method can improve estimation with respect to other input realization techniques.

Presents algorithms for time-delay estimation based on a parametric approach. The suggested methods combine an exhaustive search with a low complexity since they do not require filtering or correlation computations. Consequently, the problems with local minima of gradient search algorithms are avoided. The proposed detection schemes are experimentally verified by way of computer simulations. Furthermore, receiver operator characteristics are presented

The problem of detection and discrimination of double talk and change in the echo path in a telephone channel is considered. The phenomenon echo path change requires fast adaptation of the channel model to be able to equalize the echo dynamics. On the other hand, the adaption rate should be reduced when double talk occurs. Thus, it is critical to quickly detect a change in the echo path while not confusing it with double talk, which gives a similar effect. The proposed likelihood based approach compares a global channel model with a local one over a sliding window, both estimated with the recursive least squares algorithm

This paper addresses the problem of time-delay estimation. Two new algorithms for time-delay estimation are developed and analyzed. The suggested methods combine an exhaustive search with a low complexity. Consequently, the problems with local minima of gradient search algorithms are avoided. The receiver operating characteristics (ROC) are computed, and together with simulation results these verify the performance of the estimation schemes.

This contribution considers one central aspect of experiment design in system identification, namely application set approximation. When a control design is based on an estimated model, the achievable performance is related to the quality of the estimate. The degradation in control performance due to plant-modeling missmatch is quantified by an application cost function. A convex approximation of the set of models that satisfy the control specification is typically required in optimal input design. The standard approach is to use a quadratic approximation of the application cost function, where the main computational effort is to find the corresponding Hessian matrix. Our main contribution is an alternative approach for this problem, which uses the structure of the underlying optimal control problem to considerably reduce the computations needed to find the application set. This technique allows the use of applications oriented input design for MPC on much more complex plants. The approach is numerically evaluated on a distillation control problem.

We consider the problem of estimating the occupancylevel in buildings using indirect information such as CO2 concentrations and ventilation levels. We assume that oneof the rooms is temporarily equipped with a device measuringthe occupancy. Using the collected data, we identify a gray-boxmodel whose parameters carry information about the structuralcharacteristics of the room. Exploiting the knowledge of thesame type of structural characteristics of the other rooms inthe building, we adjust the gray-box model to capture the CO2dynamics of the other rooms. Then the occupancy estimatorsare designed using a regularized deconvolution approach whichaims at estimating the occupancy pattern that best explainsthe observed CO2 dynamics. We evaluate the proposed schemethrough extensive simulation using a commercial software tool,IDA-ICE, for dynamic building simulation.

We propose and test on real data a two-tier estimation strategy for inferring occupancy levels from measurements of CO2 concentration and temperature levels. The first tier is a blind identification step, based either on a frequentist Maximum Likelihood method, implemented using non-linear optimization, or on a Bayesian marginal likelihood method, implemented using a dedicated Expectation-Maximization algorithm. The second tier resolves the ambiguity of the unknown multiplicative factor, and returns the final estimate of the occupancy levels. The overall procedure addresses some practical issues of existing occupancy estimation strategies. More specifically, first it does not require the installation of special hardware, since it uses measurements that are typically available in many buildings. Second, it does not require apriori knowledge on the physical parameters of the building, since it performs system identification steps. Third, it does not require pilot data containing measured real occupancy patterns (i.e., physically counting people for some periods, a typically expensive and time consuming step), since the identification steps are blind.

We address the problem of estimating the occupancy levelsin rooms using the information available in standardHVAC systems. Instead of employing dedicated devices, weexploit the significant statistical correlations between the occupancylevels and the CO2 concentration, room temperature,and ventilation actuation signals in order to identify adynamic model. The building occupancy estimation problemis formulated as a regularized deconvolution problem, wherethe estimated occupancy is the input that, when injected intothe identified model, best explains the currently measuredCO2 levels. Since occupancy levels are piecewise constant,the zero norm of occupancy is plugged into the cost functionto penalize non-piecewise constant inputs. The problemthen is seen as a particular case of fused-lasso estimator byrelaxing the zero norm into the `1 norm. We propose bothonline and offline estimators; the latter is shown to performfavorably compared to other data-based building occupancyestimators. Results on a real testbed show that the MSE ofthe proposed scheme, trained on a one-week-long dataset, is half the MSE of equivalent Neural Network (NN) or SupportVector Machine (SVM) estimation strategies.

We address the problem of estimating the number of people in a room using information available in standard HVAC systems. We propose an estimation scheme based on two phases. In the first phase, we assume the availability of pilot data and identify a model for the dynamic relations occurring between occupancy levels, CO2 concentration and room temperature. In the second phase, we make use of the identified model to formulate the occupancy estimation task as a deconvolution problem. In particular, we aim at obtaining an estimated occupancy pattern by trading off between adherence to the current measurements and regularity of the pattern. To achieve this goal, we employ a special instance of the so-called fused lasso estimator, which promotes piecewise constant estimates by including an l(1) norm-dependent term in the associated cost function. We extend the proposed estimator to include different sources of information, such as actuation of the ventilation system and door opening/closing events. We also provide conditions under which the occupancy estimator provides correct estimates within a guaranteed probability. We test the estimator running experiments on a real testbed, in order to compare it with other occupancy estimation techniques and assess the value of having additional information sources.

A new approach to experimental design for identification of closed-loop models is presented. The method considers the design of an experiment by minimizing experimental cost, subject to probabilistic bounds on the input and output signals, and quality constraints on the identified model. The input and output bounds are common in many industrial processes due to physical limitations of actuators. The aforementioned constraints make the problem non-convex. By assuming that the experiment is a realization of a stationary process with finite memory and finite alphabet, we use results from graph-theory to relax the problem. The key feature of this approach is that the problem becomes convex even for non-linear feedback systems. A numerical example shows that the proposed technique is an attractive alternative for closed-loop system identification.

We present a new approach to Model Predictive Control (MPC) oriented experiment design for the identification of systems operating in closed-loop. The method considers the design of an experiment by minimizing the experimental cost, subject to probabilistic bounds on the input and output signals due to physical limitations of actuators, and quality constraints on the identified model. The excitation is done by intentionally adding a disturbance to the loop. We then design the external excitation to achieve the minimum experimental effort while we are also taking care of the tracking performance of MPC. The stability of the closed-loop system is guaranteed by employing robust MPC during the experiment. The problem is then defined as an optimization problem. However, the aforementioned constraints result in a non-convex optimization which is relaxed by using results from graph theory. The proposed technique is evaluated through a numerical example showing that it is an attractive alternative for closed-loop experiment design.

We consider the problem of occupancy estimation in buildings using available environmental information. In particular, we study the problem of how to collect data that is informative enough for occupancy estimation purposes. We thus propose an application-oriented input design approach for designing the ventilation signal to be used while collecting the system identification datasets. The main goal of the method is to guarantee a certain accuracy in the estimated occupancy levels while minimizing the experimental time and effort. To take into account potential limitations on the actuation signals we moreover formulate the problem as a recursive nonlinear and nonconvex optimization problem, solved then using exhaustive search methods. We finally corroborate the theoretical findings with some numerical examples, which results show that computing ventilation signals using experiment design concepts leads to occupancy estimator performing 4 times better in terms of Mean Square Error (MSE).

In this paper we propose a method for applications oriented input design for linear systems in open-loop under time-domain constraints on the amplitude of input and output signals. The method guarantees a desired control performance for the estimated model in minimum time, by imposing some lower bound on the information matrix. The problem is formulated as a time-domain optimization problem, which is non-convex. This is addressed through an alternating method, where we separate the problem into two steps and at each step we optimize the cost function with respect to one of two variables. We alternate between these two steps until convergence. A time recursive input design algorithm is performed, which enables us to use the algorithm with control. Therefore, a receding horizon framework is used to solve each optimization problem. Finally, we illustrate the method with a numerical example which shows the good ability of the proposed approach in generating an optimal input signal.

In this article, we approximate bounded operators by orthogonal expansion. The rate of convergence depends on the choice of basis functions. Markov-Laguerre functions give rapid convergence for open-loop stable systems with long delay. The Markov-Kautz model can be used for lightly damped systems, and a more general orthogonal expansion is developed for modeling multivariable systems with widely scattered poles. The finite impulse response model is a special case of these models. A-priori knowledge about dominant time constants, time delay and oscillatory modes is used to reduce the model complexity and to improve conditioning of the parameter estimation algorithm. Algorithms for predictive control are developed, as well as conditions for constraint compatibility, closed-loop stability and constraint satisfaction for the ideal case. An H8-like design technique proposed guarantees robust stability in the presence of input constraints; output constraints may give ￜchatter.ￜ A chatter-free algorithm is proposed.

This paper deals with recursive identification of time-varying systems using Laguerre models. Laguerre models generalize finite impulse response (FIR) models by using a priori information about the dominating time constants of the system to be identified. Three recursive algorithms are considered: the stochastic gradient algorithm, the recursive least squares algorithm and a Kalman-filter-like recursive identification algorithm. Simple and explicit expressions for the model quality are derived under the assumptions that the system varies slowly, that the model is updated slowly and that the model order is high. The derived expressions show how the use of Laguerre models affects the model quality with respect to tracking capability and disturbance rejection.

Frequency domain expressions for the quality of recursively identified Laguerre models are presented. These models generalize finite impulse response (FIR) models by using a priori information about the dominating time constants of the system to be identified. Expressions for the model quality are derived under the assumptions that the system varies slowly, that the model is updated slowly, and that the model order is high. The model quality is evaluated by investigating the properties of the estimated transfer function, and explicit expressions for the mean square error (MSE) of the transfer function, and explicit expressions for the mean square error of the transfer function estimate are derived.

The authors’ approach to blind equalization examines the possible input sequences directly by using a bank of filters and, in contrast to common approaches, does not try to find an approximative inverse of the channel dynamics. The identifiability question of a noise-free finite impulse response (FIR) model is investigated. A sufficient condition for the input sequence (persistently exciting of a certain order) is given which guarantees that both the channel model and the input sequence can be determined exactly in finite time. A recursive algorithm is given for a time-varying infinite impulse response (IIR) channel model with additive noise, which does not require a training sequence. The estimated sequence is an arbitrarily good approximation of the maximum a posteriori estimate. The proposed method is evaluated on a Rayleigh fading communication channel. It shows fast convergence properties and good tracking ability

This paper presents a novel approach to blind equalization (deconvolution), which is based on direct examination of possible input sequences. In contrast to many other approaches, it does not rely on a model of the approximative inverse of the channel dynamics. To start with, the blind equalization identifiability problem for a noise-free finite impulse response channel model is investigated. A necessary condition for the input, which is algorithm independent, for blind deconvolution is derived. This condition is expressed in an information measure of the input sequence. A sufficient condition for identifiability is also inferred, which imposes a constraint on the true channel dynamics. The analysis motivates a recursive algorithm where all permissible input sequences are examined. The exact solution is guaranteed to be found as soon as it is possible. An upper bound on the computational complexity of the algorithm is given. This algorithm is then generalized to cope with time-varying infinite impulse response channel models with additive noise. The estimated sequence is an arbitrary good approximation of the maximum a posteriori estimate. The proposed method is evaluated on a Rayleigh fading communication channel. The simulation results indicate fast convergence properties and good tracking abilities.

Equalization is concerned with estimation of the input sequence of a linear system given noisy measurements of the output signal. In case the system description is unknown we have the problem of blind equalization. A scheme for blind equalization which is based on the assumption that the input signal belongs to a finite alphabet is proposed. A finite impulse response model can be directly estimated by the least-squares method if the input sequence is known. Since we know that the number of possible input sequences is limited, we can associate one system estimate to each possible input sequence. This allows us to determine the a posteriori probability of an input sequence given output observations. The maximum a posteriori (MAP) input sequence estimate is then taken as the most probable input sequence. Sufficient conditions for identifiability of the input signal and the system are given. The complexity of this scheme increases exponentially with time. A recursive approximate MAP estimator of fixed complexity is obtained by, at each time update, only keeping the K most probable input sequences. This method is evaluated on a Rayleigh fading communication channel.