Polynomial and Relaxed Models for Linear Switched Dynamic Systems

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Computational Aspects of the tdCLF Method

The main computational difficulty to obtain the control signal for a switched system using the tdCLF approach is the solution of the SDP in Eq. (4.26). It is known that SDPs are a subset of convex optimization problems that can be efficiently solved in polynomial time using interior point methods [51], [52]. Also, it has been recognized that many problems in control systems can be solved efficiently, sometimes in real time, using interior point methods [53]. In our case the SDP was modeled using YALMIP [54] version R20190425.
This is a Matlab toolbox that is useful for easily constructing the semidefi-nite constraints. YALMIP translates the problem into a matrix structure suit-able for computation using numerical solvers. We used MOSEK [55] version 9.0.86 to solve the resulting numerical problem. This solver implements a ho-mogeneous interior point algorithm characterized by a robust performance.
Table 4.2 shows, for each of the examples presented in this chapter, the size of the numerical problem and the time taken by the solver to compute the solution. The numbers shown in the problem size column are reported by the interior-point optimizer log, and the execution times are reported by YALMIP’s solvertime property. The results were obtained by running a simulation for each example, all with a test duration of 0.04 s. The solver time is recorded for each of the N iterations of the algorithm, and at the end the mean and standard deviation are computed. It can be observed from these tests that the main contributing factor to the performance is the number of constraints. These results show that the execution time is not small enough to achieve real-time performance. This time has to be reduced at least one or-der of magnitude to make an experimental implementation feasible. This is the main reason why the results presented in this chapter are only for simu-lation. The tests described in this section were carried out in a Linux/64-X86 machine with an Intel Core i7-4700MQ CPU running at 2.4 GHz and with 16 GB of RAM.
More tests are still needed to determine possible ways to improve the performance. A starting point would be a detailed comparison with other solvers like SEDUMI or SDPT3. Also, some implementations of interior-point methods using graphical processing units (GPU) have shown impor-tant performance improvements. The main computational task in these al-gorithms is the solution of a linear system to compute the Newton direction in each iteration. This step uses a method like the Cholesky factorization, which has been shown to be suitable for efficient implementation in a GPU. For instance in [56] this approach was tested for solving a model predictive control (MPC) problem represented as a linear programming problem. Fi-nally, the optimality criterion in problem (4.26) could be dropped and only solve the feasibility problem. This should decrease the time required to find a solution. However, the obtained tdCLF could have a faster decay rate than the one set by the ρ parameter. In this case, it would be necessary to check the impact of these solutions on the control performance.


Lyapunov’s stability theory is used to synthesize stabilizing control laws for switched systems. The general approach relies on the formulation of a pa-rameterized Lyapunov function, considering two cases: fixed and varying parameters.
In the first part it is shown that a proper choice for the fixed parameters can improve the tracking performance with respect to other similar methods. The shape of the decision surfaces that emerge from the parameterization define the trajectories followed by the system. Simulation and experimental results are presented for the case of a three-cell converter, although this ap-proach can be applied to a multicellular converter with an arbitrary number of cells, as well as other converter topologies. It can be observed that the proposed control law ensures good closed loop performance in terms of ref-erence tracking, presenting improvements during transient conditions with a proper choice in the control Lyapunov parameterization. The main advan-tages are that the control law is simple to implement and is computationally cheap.
However, this approach has several drawbacks that limit its performance and applicability. This is the main reason for developing an alternative ap-proach based on trajectory-dependent CLFs. The Lyapunov functions con-sidered in this case depend on parameters that are computed on-line. This method allows to improve upon the fixed parameter case, because adding varying parameters to the Lyapunov function makes them less conservative, improving performance while guaranteeing the closed-loop stability.
The main contribution in this chapter is in the extension of the tdCLF approach originally developed for nonlinear control affine systems to the case of switched systems. To the best of our knowledge, it is the first time that tdCLFs are used in the context of switched systems and power convert-ers. This is achieved by using the method of moments to obtain equivalent linear models that can be plugged into a semidefinite problem suitable to be solved efficiently using numerical methods. The control signal for the switched system is synthesized from the solution of the numerical problem, yielding good tracking performance of the closed-loop system. Several ex-amples are presented for different converter topologies, and the performance of the approach is explored in simulation.
There are still issues that need to be addressed in the proposed tdCLF approach. A practical implementation able to execute at a fast enough rate to run in real-time requires improvements in the solution of the SDP. Tests including a comparison with other solvers or a GPU implementation are pos-sible ways to solve this problem. Also, it would be interesting to compare the Lagrange polynomial basis to other alternatives such as splines or the sinc function. These bases could reduce the error introduced by the approx-imation done when the polynomial basis is evaluated in the control signal synthesis procedure.

Direct Filtering for State Estimation in Power Converters

In the well known pulse width modulated (PWM) converter topologies (buck, boost, buck-boost, Cuk, SEPIC) a pair of switching devices (usually a transis-tor and a diode) are used. The normal operation consists in the complemen-tary activation and deactivation of these devices, in what is known as the con-tinuous conduction mode (CCM). However, under particular circumstances it may occur that both devices are conducting or blocking at the same time, yielding what is known as discontinuous conduction modes (DCM). This condition is a consequence of the unidirectionality properties of the semicon-ductor devices [1], which may cause an autonomous switching event before the end of the PWM periodic cycle. Under DCM operation, properties like the power converter’s dynamic behavior and conversion ratio change drasti-cally with respect to the CCM operation.
Power converters operating in DCM have properties that make them ideal for different applications, like soft switching on the boundary between CCM and DCM [57],[58], higher efficiency on low voltage/battery powered sys-tems under light load conditions [59],[60] and avoidance of inductor core saturation [61],[62]. Also, these converters are typically used for power factor correction (PFC) [63],[58],[64] and power factor preregulation (PFP) [65],[63].
Discontinuous conduction modes have been studied extensively, begin-ning with classical results [22],[66],[67], where the discontinuous inductor current mode is presented and averaging methods are used to obtain small signal linear models for the buck, boost and buck-boost topologies. In [23], the different discontinuous modes present in basic PWM converter topolo-gies are discussed: discontinuous inductor current mode (DICM), discon-tinuous capacitor voltage mode (DCVM) and discontinuous quasi-resonant mode (DQRM). It also introduces averaged small signal models for each of these modes. In [68],[69],[63] averaged models for SEPIC, Cuk and flyback topologies in DCM are presented.
Transitions between CCM and DCM happen when a parameter (induc-tance, capacitance) or operating condition (output load, duty cycle) exceed a critical value. These transitions may be imposed by the designer or can happen when an external condition is present. An example of the former case is [70], where the energy efficiency of a battery-powered application is improved by adaptively driving the converter into DCM operation when the system is operating in light load conditions on stand-by mode. Under these conditions, the current ripple is decreased, minimizing the switching and conduction losses.
Classic approaches to modeling and control of power converters consider only one operation mode (either CCM or DCM), and can not provide perfor-mance and stability guarantees when transitions to another operation mode happen. However, alternatives have been proposed recently, based on av-erage and hybrid models which are valid in both CCM and DCM. For in-stance, in [71] average models are formulated by including numerical cor-rection terms, obtained from simulations of the detailed switching model, in order to account for the fast dynamics present in the discontinuous mode. In [72] the converter operating in CCM and DCM is represented as a hybrid system using the mixed logical dynamical (MLD) framework. This allows to formulate the predictive control problem by including the system and state constraints and minimizing a cost dependent on the tracking error. How-ever, this approach requires the use of mixed-integer quadratic program-ming (MIQP) solvers, and is limited to short prediction horizons to achieve a practical implementation. In [73] an alternative approach to hybrid modeling of a boost converter operating in CCM and DCM is presented.
Recent approaches to control of power converters operating in CCM and´DCM include [74], where a control law for a Cuk converter is signed based on a common quadratic Lyapunov function, with guaranteed stability when it enters DCM. In this work DCVM and DICM are considered. Also, in [75] a switched and adaptive control method with global asymptotic stabil-ity is proposed for power converter operating in both CCM and DCM. An observer for estimating the state and the unknown parameters is designed, making this approach robust with respect to changes in the load resistance and input voltage.
There are not many works in the literature regarding the problem of ob-servation in power converters operating in CCM and DCM. The existing works rely on the existence of a model for the power converter and require the use of an estimator to obtain the operation mode at each time instant. For example, [76] presents the design of a current observer for the PFC boost converter, based on an average model that represents CCM and DCM. In [77] an observer-based control for a buck converter operating in CCM and DCM is considered. The observer is based on the discrete time LC-filter model in CCM, and an integral compensation loop is included to correct the estimates when the system enters DCM. In [78] Luenberger-type switched observers are considered, and the observer gains for the different operation modes are obtained by solving LMIs.
In the present work we consider the problem of current estimation in PWM converters operating in CCM and DCM. We propose an approach based on the set membership framework [36] for identification of nonlinear systems. This method allows to formulate a discrete-time data-based state estimator without knowledge of the system model equations, but assuming the existence of bounds to the model functions gradients and the availability of a sufficiently informative dataset. The structure of the obtained estimator is a causal nonlinear finite impulse response (NFIR) filter, where the output depends on the history of past measured inputs and outputs of the observed system. The resulting estimate includes not only the estimate but also the tightest error bounds, giving a measure of uncertainty in the estimation pro-cess. This approach has several advantages with respect to other estimation methods:
1. It does not require to formulate an explicit model for the power con-verter.
2. It does not require to implement an estimator for the operation mode.
3. It represents the system dynamics in the complete operating range, in-cluding CCM and DCM.
4. It gives a measure of uncertainty of the estimate at each sample time.
5. The estimator is BIBO stable, because of the filter structure.
6. The resulting algorithm is inherently parallelizable.
The SEPIC converter is used as an application example to illustrate the results obtained with the proposed approach.
The main original contributions in this chapter are:
1. Definition of the direct filter design procedure for power converters operating in CCM and DCM (Section 5.4).
2. Parallel implementation of the direct filter estimator procedure using the CUDA framework (Section 5.5).
3. Use of the principal component analysis (PCA) dimensionality reduc-tion technique for compressing the regressor datasets in the direct filter implementation (Section 5.6).
4. Extension of the direct filter approach to full-state estimation (Section 5.7.2).

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Current Estimation for Power Converters in CCM and DCM

In switched power converter applications, having accurate measurements of currents and voltages is an important requirement for achieving high perfor-mance in control and monitoring tasks. In particular, current measurement is critical in several feedback control strategies, where the outer voltage loop produces a current reference for the inner current loop. However, accurate current measurements require special sensing circuits [79] which in some applications may increase complexity and be cost prohibitive. Furthermore, these measurements are contaminated by switching noise. An alternative is not to measure the currents directly, but to estimate them using observers. The usual approach relies on the availability of other measurable signals (in-puts and outputs), and on knowledge of a mathematical model for the sys-tem. With this information, the observer is able to make predictions on the system behavior, providing estimates of the unmeasurable signals.

Table of contents :

1 General Introduction 
1.1 Outline
1.2 Organization
1.3 Contribution
1.4 Publications
2 Background 
2.1 Power Converters
2.1.1 Control and Observation of Power Converters
2.1.2 Continuous and Discontinuous Modes in Power Converters
2.1.3 Models for Power Converters
2.1.4 DC-DC Converter Topologies
2.2 Lyapunov Stability
2.2.1 Parameterized Control Lyapunov Functions (pCLF)
2.3 Method of Moments
2.4 Varieties, Ideals and Groebner Basis
2.5 Direct Filtering for State Estimation of Unknown Systems
2.6 Conclusion
3 Moment Relaxations of Switched Systems 
3.1 Introduction
3.2 Contribution
3.3 Linear Switched Dynamic Systems
3.4 Polynomial and Relaxed Models for Linear Switched Dynamic Systems
3.5 Equilibrium Points for Average, Polynomial and Relaxed Models
3.6 Conclusion
4 Parameterized Control Lyapunov Functions 
4.1 Introduction
4.2 Contribution
4.3 Parameterized Control Lyapunov Functions with Fixed Parameters
4.3.1 Three-Cell Multicellular Converter
4.3.2 Simulation and Experimental Results
4.3.3 Discussion on the Fixed-Parameter Approach
4.4 Trajectory Dependent Control Lyapunov Functions (tdCLF) for Switched Systems
4.4.1 Polynomial Model for the Switched System
4.4.2 Control Lyapunov Function for the Polynomial System
4.4.3 Measure recovery
4.4.4 Synthesis of the Control Signal for the Switched System
4.4.5 tdCLF Algorithm for Switched Systems
4.4.6 Application Examples
4.5 Computational Aspects of the tdCLF Method
4.6 Conclusion
5 Direct Filtering for State Estimation in Power Converters
5.1 Introduction
5.2 Contribution
5.3 Current Estimation for Power Converters in CCM and DCM
5.4 Direct Filter Design for Power Converters Operating in CCM and DCM
5.5 Direct Filter Parallel Implementation using CUDA
5.6 PCA Dimensionality Reduction in Regressor Datasets
5.7 Application Examples
5.7.1 Current Estimation for SEPIC Converter
5.7.2 Observer-based Control for SEPIC Converter
5.7.3 Implementation Details
5.8 Conclusion
6 Conclusion and Perspectives 
6.1 General Conclusion
6.2 Perspectives


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