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Approximate Function

The Experts below are selected from a list of 300 Experts worldwide ranked by ideXlab platform

Björn Ottersten – 1st expert on this subject based on the ideXlab platform

  • Successive Convex Approximation Algorithms for Sparse Signal Estimation With Nonconvex Regularizations
    IEEE Journal of Selected Topics in Signal Processing, 2018
    Co-Authors: Yang Yang, Marius Pesavento, Symeon Chatzinotas, Björn Ottersten

    Abstract:

    In this paper, we propose a successive convex approximation framework for sparse optimization where the nonsmooth regularization Function in the objective Function is nonconvex and it can be written as the difference of two convex Functions. The proposed framework is based on a nontrivial combination of the majorization-minimization framework and the successive convex approximation framework proposed in literature for a convex regularization Function. The proposed framework has several attractive features, namely, first, flexibility, as different choices of the Approximate Function lead to different types of algorithms; second, fast convergence, as the problem structure can be better exploited by a proper choice of the Approximate Function and the stepsize is calculated by the line search; third, low complexity, as the Approximate Function is convex and the line search scheme is carried out over a differentiable Function; fourth, guaranteed convergence to a stationary point. We demonstrate these features by two example applications in subspace learning, namely the network anomaly detection problem and the sparse subspace clustering problem. Customizing the proposed framework by adopting the best-response type approximation, we obtain soft-thresholding with exact line search algorithms for which all elements of the unknown parameter are updated in parallel according to closed-form expressions. The attractive features of the proposed algorithms are illustrated numerically.

  • Parallel and Hybrid Soft-Thresholding Algorithms with Line Search for Sparse Nonlinear Regression
    2018 26th European Signal Processing Conference (EUSIPCO), 2018
    Co-Authors: Yang Yang, Marius Pesavento, Symeon Chatzinotas, Björn Ottersten

    Abstract:

    In this paper, we propose a convergent iterative algorithm for nondifferentiable nonconvex nonlinear regression problems. The proposed parallel algorithm consists in optimizing a sequence of successively refined Approximate Functions. Compared with the popular iterative soft-thresholding algorithm commonly known as ISTA, which is the benchmark algorithm for such problems, it has two attractive features which lead to a notable reduction in the algorithm’s complexity: the proposed Approximate Function does not have to be a global upper bound of the original Function, and the stepsize can be efficiently computed by the line search scheme which is carried out over a properly constructed differentiable Function. Furthermore, when the parallel algorithm cannot be fully parallelized due to memory/processor constraints, we propose a hybrid updating scheme that divides the whole set of variables into blocks which are updated sequentially. Since the stepsize is obtained by performing the line search along the coordinate of each block variable, the proposed hybrid algorithm converges faster than state-of-the-art hybrid algorithms based on constant stepsizes and/or decreasing stepsizes. Finally, the proposed algorithms are numerically tested.

Marko Hinkkanen – 2nd expert on this subject based on the ideXlab platform

  • minimizing losses of a synchronous reluctance motor drive taking into account core losses and magnetic saturation
    European Conference on Power Electronics and Applications, 2014
    Co-Authors: Zengcai Qu, Toni Tuovinen, Marko Hinkkanen

    Abstract:

    This paper proposes a loss-minimizing controller for synchronous reluctance motor drives. The proposed method takes core losses and magnetic saturation effects into account. The core-loss model consists of hysteresis losses and eddy-current losses. Magnetic saturation is modeled using two-dimensional power Functions considering cross coupling between the dand q-axes. The efficiency optimal d-axis current is calculated offline using the loss model and motor parameters. Instead of generating a look-up table, an Approximate Function was fitted to the loss-minimizing results. The loss-minimizing method is applied in a motion-sensorless drive and the results are validated by measurements. Introduction In vector control of a synchronous reluctance motor (SyRM), certain speed and torque can be achieved by different combinations of dand q-axes currents. The total losses can be minimized by adjusting the dand q-axes current ratio. Generally, there are two categories of loss-minimizing controllers: the loss-model based controller (LMC) which uses the motor model and parameters to calculate the lossminimizing currents, and the online search controller (SC) which adjusts the current vector online based on the feedback of input power measurement. Since the SC does not require any motor parameters beforehand, the bulk of the work to determine the motor parameters can be avoided. However, the searching process may cause unwanted losses and torque ripples. Furthermore, it might be sensitive to the measurement noise and errors. The main disadvantage of the LMC is its dependancy on the motor parameters. However, the motor parameters are required in many parts of electric drives, e.g., they are required in speed sensorless control. Therefore, the LMC can be a better option in the case of known motor parameters. The total losses in the SyRM can be formulated as a loss Function of control variables and the motor parameters. By minimizing this Function, the efficiency optimal control variable, e.g., the d-axis current, can be found. Simple LMCs assuming constant inductances and constant core-loss resistance can be found in the literature [1, 2]. However, the inductances vary with the stator currents and the core-loss resistance is a Function of the flux and speed. Magnetic saturation effects were ignored for simplification in [3–11], and only a few LMCs have taken magnetic saturation into account [12,13]. However, parameter sensitivity of loss minimization was studied in [14] and it shows that the inductance variations due to magnetic saturation affect the optimal current significantly. The core losses can be modeled using a constant resistance if the hysteresis losses are omitted. In [4], the core losses were modeled as hysteresis losses and eddy-current losses. The stray-load losses were also taken into account in [9] and [10]. The nonlinear magnetic saturation model and core-loss model usually result in a complicated loss Function. An analytical solution of loss minimization is difficult to derive. However, iteration methods can be used. For example, the cross-magnetic saturation was modeled and the optimum current was derived using an iteration method in [13]. Simple core-loss and magnetic saturation models were applied in [8], but the core-loss resistance and the inductances were estimated by the extended Kalman filter. Neural networks (NNs) and fuzzy logic have been used in the LMCs. In [15], a NN was used as an adaptive model of the SyRM. The NN was trained online, and the input is the torque reference and the outputs are the d-q axes current references. In [7], the loss Function with constant inductances and core-loss resistance was applied, and an offline-trained NN was used to map the optimal current. These methods are actually similar to the simple LMCs since they just map the loss Function to neural networks. This paper aims to break the tradeoff between accuracy and complexity of LMCs. Nonlinear core-loss and magnetic saturation models make the loss Function too complicated to get an analytical solution. Iterative methods can be used to find the minimum points of the loss Function. However, it is computation demanding for online utilization. A general idea of the proposed LMC is to calculate the optimal currents iteratively offline and then fit the results to a simple Function for online use. A similar approach can be found in [16], where the core-loss and magnetic saturation parameters were obtained from finite element analysis. The optimal currents were calculated offline. A loss-minimizing look-up table was generated from the off-line calculation results. However, the loss minimization was not validated by experimental measurement. In this paper, both the core losses and magnetic saturation are taken into account. The core-loss model consists of hysteresis losses and eddy current losses, and the magnetic saturation is modeled using twodimensional power Functions taking into account cross coupling between the dand q-axes. The parameters of core losses and magnetic saturation are determined by experimental measurements. An iteration method is used to calculate the optimal d-axis current offline for given operating points. The results are fitted to a simple Function which can be easily implemented online. Hence, extensive computation can be avoided in real-time control. The proposed method is validated by experiments of a 6.7-kW SyRM. SyRM Model Space-vector Model Fig. 1 shows the dynamic space-vector model of an SyRM. The d-axis of the rotating coordinate system is defined as the direction of the maximum inductance. Real space vectors will be used in the model. For example, the stator-current vector is is = [id, iq] T, where id and iq are the components of the vector and the matrix transpose is marked with the superscript T. The magnitude is denoted by is = √ id + i 2 q. The orthogonal rotation matrix is J = [ 0 −1 1 0 ]. Per-unit quantities will be used. ωmJψs is Rs im us Rc ic dψs dt Figure 1: Dynamic space-vector model of a SyRM in rotor coordinates. The voltage equation is dψs dt = us −Rsis − ωmJψs (1) where ψs is the stator-flux vector, us the stator-voltage vector, Rs the stator resistance, is the stator current, and ωm is the angular speed of the rotor. The core-loss current is ic = is − im, where im is the magnetizing currrent which is a nonlinear Function of the flux due to magnetic saturation. The core losses are modeled as a nonlinear resistance Rc. The electromagnetic torque is given by Te = imqψd − imdψq (2) Magnetic Saturation The effects of magnetic saturation are often an important issue in model-based loss minimization. The stator inductances vary with the fluxes (or the currents) of both axes. The inductances can be obtained from finite-element methods or measurements. The look-up table is usually computationally inefficient and needs interpolation. As reviewed in [17], the measured data are often fitted to explicit Functions. The magnetic saturation effects can be modeled as current Functions of the fluxes [17] using two-dimensional power Functions: imd(ψd, ψq)= ψd Ldu [ 1 + (α|ψd|) a + γLdu d+2 |ψd| |ψq| d+2 ] (3a) imq(ψd, ψq)= ψq Lqu [ 1 + (β|ψq|) b + γLqu c+2 |ψd| |ψq| d ] (3b) where Ldu and Lqu are the unsaturated inductances, and α, β, γ, a, b, c, and d are nonnegative constants. The fitted parameters of the 6.7-kW SyRM are shown in Table I and the fitting results are shown in Fig. 2. Table I: Fitted per-unit parameters [17]. Ldu Lqu α β γ a b c d 2.73 0.843 0.847 3.84 2.37 6.61 1.33 0.41 0 0 0.4 0.8 1.2 0 1 2 3 ψd (p.u.) L d (p .u .) (a) 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 ψq (p.u.) L q (p .u .) (b) Figure 2: Results of curve fitting to experimental data [17]: (a) Ld as a Function of ψd for three different values of ψq; (b) Lq as a Function of ψq for three different values of ψd. In (a), the values of ψq are 0.1 p.u. (black line), 0.2 p.u. (blue line) and 0.3 p.u. (red line). In (b), the values of ψd are 0.6 p.u. (black line), 0.8 p.u. (blue line) and 1.0 p.u. (red line). Core Losses The core losses can be divided into two parts: hysteresis losses and classical eddy-current losses. In steady state, the stator core losses are classically modeled as a Function of the rotor frequency ωm and the stator-flux magnitude ψs, PFe = ΛHy|ωm|ψ 2 s +GFtω 2 mψ 2 s (4) where the first term corresponds to the hysteresis losses and the second term corresponds to the eddycurrent losses [18]. The hysteresis losses are proportional to the frequency, while the eddy-current losses are proportional to the square of the frequency. The constants ΛHy and GFt determine the ratio between the loss components at a given stator flux and angular frequency. The core losses are typically modeled using a core-loss resistor Rc in steady state. The core-loss resistance corresponding to (4) becomes Rc = 1 ΛHy/|ωm|+GFt (5) It can be seen that the core-loss resistance Rc is constant if the hysteresis losses are omitted (ΛHy = 0). The parameters can be identified using series of no-load tests at different frequencies. The fitted parameters for the 6.7-kW SyRM are ΛHy = 0.018 p.u. and GFt = 0.042 p.u. Fig. 3 shows the core-loss curves and the measured core losses at different flux levels. 0 0.2 0.4 0.6 0.8 0 0.01 0.02 0.03 0.04 0.05 ωm (p.u.) P F e / P N Figure 3: Core-loss curves as a Function of angular frequency ωm for ΛHy = 0.018 p.u. and GFt = 0.042 p.u. Markers show the measured core losses from no-load tests (different flux levels were applied at each stator frequency). Loss Minimization Conventional LMC Many LMCs are based on constant motor parameters Rc, Ld, and Lq. The per-unit losses are given by Ploss = [ Rs + (Rs +Rc) ω2 mL 2 d R2 c ]

  • Minimizing losses of a synchronous reluctance motor drive taking into account core losses and magnetic saturation
    2014 16th European Conference on Power Electronics and Applications, 2014
    Co-Authors: Zengcai Qu, Toni Tuovinen, Marko Hinkkanen

    Abstract:

    This paper proposes a loss-minimizing controller for synchronous reluctance motor drives. The proposed method takes core losses and magnetic saturation effects into account. The core-loss model consists of hysteresis losses and eddy-current losses. Magnetic saturation is modeled using two-dimensional power Functions considering cross coupling between the d- and q-axes. The efficiency optimal d-axis current is calculated offline using the loss model and motor parameters. Instead of generating a look-up table, an Approximate Function was fitted to the loss-minimizing results. The loss-minimizing method is applied in a motion-sensorless drive and the results are validated by measurements.

Zhenhua Xiong – 3rd expert on this subject based on the ideXlab platform

  • discrete time sliding mode control with improved quasi sliding mode domain
    IEEE Transactions on Industrial Electronics, 2016
    Co-Authors: Jianhua Wu, Zhenhua Xiong

    Abstract:

    In this paper, a new discrete reaching law with improved quasi-sliding-mode domain (QSMD) is proposed and a sliding-mode controller is designed for discrete-time systems with uncertainties. By redefining the change rate as the second-order difference of the system uncertainties and adopting the continuous-Approximate Function, smaller width of the QSMD can be guaranteed. Moreover, the QSMD of the proposed reaching law is obtained and the system dynamics in and out the QSMD are theoretically analyzed. Perturbation estimation technique is employed to estimate the unknown uncertainties. Thus, no prior knowledge of the uncertainty bound is required. Both numerical simulations and experimental results on a piezoelectric actuator are presented to demonstrate the performance of the proposed method.