Data-Driven Control Design for Power Converters Approximated as Switched Affine Systems and Experimental Validation

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by IEEE: M. C. Merchán-Riveros, C. Albea and A. Seuret, "Data-Driven Control Design for Power Converters Approximated as Switched Affine Systems and Experimental Validation," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 71, no. 9, pp. 4236-4240, Sept. 2024, doi: 10.1109/TCSII.2024.3379193. “© 2024 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.” IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II CLASS FILES, VOL. XX, NO. X, DECEMBER 2023 1 Data-driven control design for power converters approximated as switched affine systems and experimental validation Mar´ ıa Camila Merch´ an-Riveros, Carolina Albea and Alexandre Seuret Abstract—This paper introduces a data-driven control design approach for power converters modelled as a switched affine system, that guarantees the global stability. Unlike many existing approaches, our contribution does not require a precise identification of a nonlinear model but it rather relies on prior random experimental data, which allows the design of a stabilizing control law in a direct manner. The proposed method is then evaluated on an Imperix power test bench. The experimental results not only confirm the viability of the data-driven approach for nonlinear systems but also demonstrate its applicability in real-world scenarios. Index Terms—Data-driven control, switched affine system, power converter. I. INTRODUCTION Power converters play a pivotal role in modern electrical systems, ranging from ubiquitous devices like smartphones and laptops to expansive renewable energy systems [5], [12]. Designing efficient control strategies for these systems has garnered significant attention at the intersection of power electronics and Automatic Control. One of the particularities of these systems is the presence of uncertainties in the parameters. When uncertainties are substantial, it becomes pertinent to develop a control strategy directly based on data, rather than initially attempting to identify a precise but approximate model, followed by employing classical control strategies with fixed operating parameters [2], [22]. Recently, a notable interest in data-driven control design has surged within the field of Automatic Control due to the potential of this approach to handle systems with significant uncertainties, as it does not assume complete knowledge of the plant [8], [21]. To bridge this knowledge gap, the idea of conducting multiple random experiments prior to the control design phases has gained traction. The collected data must be sufficiently rich to represent the plant dynamics with minimal error. Many methods have been developed assuming that the plant can be approximated as a linear system [3], [7], [8], [13], [16]. Other works were focused on plants modelled as nonlinear systems [14], and on switched affine systems, [11], [18], [19], [2]. Some data-driven methods were applied to power converters as [6], [15], considering that these systems can be approximated as linear systems. The data-driven approaches presented Mar´ ıa Camila Merch´ an-Riveros, C. Albea and A. Seuret are with the Departamento de Ingenier´ ıa de Sistemas y Autom´ atica de la Universidad de Sevilla, Spain. Email: mmriveros,albea,[email protected] This work was supported by grant PID2019-105890RJ-I00 and PID2019- 109071RB-I00, funded by MCIN/ AEI /10.13039/501100011033/. in [23], [15] were experimentally evaluated. The first one relies on an iterative learning predictive control [23] applied to a Neutral-Point-Clamped (NPC) converter. The second one [15] provides a data-driven approach using a frequency response for power converters.The bottom line is that all these methods are based on a linear model of power converters. However, power converters follow a nonlinear behavior, which can be efficiently captured using switched affine models [9], [10], [20], [1], where the control input is the switching signal, in contract to a linear feedback, which is less realistic. The difficulty of such models refers to the fact that they only can be stabilized in a set around of the operating point. In this paper, our objective is to deploy a data-driven control design for power converters modelled as a class of nonlinear system: a switched affine system and subsequently evaluate it experimentally. This work focuses on power converters controlled by two functioning modes, which refers to many architectures such as buck, boost, quadratic-boost converters, half-bridge, boost inverters, etc. To achieve our goal, the remainder of the paper is divided into two parts. The next section presents the theoretical contributions, encompassing data acquisition to control design. The second part describes how this theoretical framework is adapted to align with the experimental setup. The control design is then evaluated on the experimental setup, thereby experimentally validating our theoretical contribution. Notations: Throughout the paper, N,R,Rn×mand Sn denote the set of natural and real numbers, of n-dimensional Euclidean space, of real n×mmatrices and symmetric matrices in Rn×n. For any nand min N, matrices In, 1nand 0stand for the identity matrix of Rn×n, the vector in Rn, whose entries are all equal to 1and the null matrix of suited dimension, respectively. When no confusion is possible, the subscripts of these matrices that precise the dimension will be omitted. For any matrix Mof Rn×n, the notation M≻0means that Mis symmetric positive definite. For any matrices A=A⊤ , B, C =C⊤of appropriate dimensions, matrix [A B ∗C]denotes the symmetric matrix A B B⊤C. For any matrix M≻0and any vector hof appropriate dimension, we denote the shifted ellipsoid E(M, h) := x∈Rn,(x−h)⊤ M−1(x−h)≤1. The set of coefficients generating convex combinations is denoted by ΛK:= {λ∈[0,1]N\Pi∈Kλi= 1}. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II CLASS FILES, VOL. XX, NO. X, DECEMBER 2023 2 II. DATA COLLECTION AND DATA-DRIVEN CONTROL DESIGN FOR POWER CONVERTERS The objective of this paper is to propose a simple and direct procedure for the design of a stabilizing control law of a class of power converters with two functioning modes towards a desired output voltage, without any a priori knowledge of the system dynamics. A. Data acquisition The first step is to conduct experiments to implement the data-driven control design described in the following sections. The basic approach involves generating a sequence of random values for σand using this signal as a periodically sampled input with a period T > 0, while measuring the state xat each sampling instant. Subsequently, the experimental data is organized into matrices as follows X:= x0x1. . . xp,Σ := σ0σ1. . . σp,(1) where pstands for the total number of samplings obtained from the experiments, and where xℓand σℓstand for the ℓ’s experiment, for any integer ℓ≤p. A preliminary data pre-processing step involves segregating the data corresponding to each activated mode σℓ∈K. This is achieved through a straightforward algorithm that stacks xℓ+1 and xℓinto matrices X+ jand Xjin Rn×pj, respectively, where σℓ=j, for all j∈K. Here, pjdenotes the number of experiments in which mode jwas active. It is assumed that the experiments satisfy rank(hXj 1⊤ ji) = 3 for all jin K. B. Data processing and main assumptions Once the data X+ jand Xjhave been collected for j∈K, we are in position to make the first step towards the control design. First, we estimate a possible and acceptable operating point xeto the converter from a desired output voltage vCe, only based on the data. To do so, let us first introduce the common discrete-time model for power converters that is given by the following switched affine dynamics xk+1 =Aσkxk+Bσk,(2) where xkin Rnis the state vector and σkis the control action and we assume that can take values in K:= {1,2}depending on the mode of the converter. A component of the state xk is the output voltage of the converter vc. It is important to recall that matrices Aσkand Bσkare unknown and possibly time-varying. However, the data experiments allow to provide an approximation using a least square optimization. This is performed by solving the following problem, consisting of minimizing a cost function min [AjBj]∈Rn×(n+1)(X+ j− AjXj− Bj)(X+ j− AjXj− Bj)⊤, whose well-known solution is given by AjBj=X+ jhXj 1⊤ ji⊤hXj 1⊤ jihXj 1⊤ ji⊤−1 ,(3) and the minimum of the cost function is given by ∆j:=(X+ j−AjXj−Bj)(X+ j−AjXj−Bj)⊤=X+ jPjX+⊤ j, where Pjis a projection matrix (P2 j=Pj) given by Pj:= Ipj−hXj 1⊤ pji⊤hXj 1 ⊤ pjihXj 1 ⊤ pji⊤−1 hXj 1 ⊤ pji, j ∈K. This expression of matrices Ajand Bjis only an approximation, and we recall that the real matrices Ajand Bjare unknown and possibly time-varying. So, to relax the constraint of a constant approximation and to account for the variability of the matrices of the systems, we propose the following assumption on the real discrete-time system’s matrices Assumption 1: The unknown matrices of the real discretetime matrices Aj, Bjin (2) are assumed to belong to the set Ωj, a subset of Rn×(n+1), given by Ωj:=(A⊤ j B⊤ j⊤ ,A⊤ j−A⊤ j B⊤ j−B⊤ j⊤ hXj 1⊤ jihXj 1⊤ ji⊤ A⊤ j−A⊤ j B⊤ j−B⊤ j⪯αj∆j), where αj≥0are known parameters, that represent the magnitude of the allowable uncertainties in the system’s matrices, which is proportional to the least square approximation error. Note that the sets Ωj, for j= 1,2, represent the set of matrices Ajand Bjthat are close enough to the least squares approximation Ajand Bj. The magnitude of the approximation error is captured by the least squares error ∆j. In particular, if the data strictly follow dynamics (2) with constant matrices Ajand Bj, then matrix ∆jis zeros and Ωjonly contain matrices Aj=Aj=Ajand Bj=Bj=Bj. C. Data driven control design In this section, the objective is to apply the model-based control design method conducted in [17], which ensures the convergence of the closed-loop trajectories to a robust limit cycle. However, contrary to the context of this paper, the next steps therein require knowledge of the system matrices, making this method not directly applicable. Consequently, it becomes necessary to adapt this design phase to accommodate scenarios where only the data and assumptions about the system’s structure, following a switched affine model given in (2), are available. The method presented in [17] relies on a priori knowledge of a cycle denoted as ν, which is a periodic function of modes corresponding to the control objectives. More precisely, this implies the existence of Nin N\{0}such that ν(ℓ+N) = ν(ℓ),∀ℓ∈N. The minimum period and the minimum domain of ν, respectively are Nν:=min N∈N\{0}s.t. ν(ℓ+N) = ν(ℓ)∈K,∀ℓ∈N, Dν:={1,2, . . . , Nν}. Finally, ⌊i⌋ν=i, for any i= 1,. . ., Nνand ⌊Nν+1⌋ν= 1. The next step is to identify the cycle that fits with the control objective consisting in ensuring that the closed-loop system converges to a neighborhood of a given and desired voltage vCe, which is a component of xe. To do so, we need to verify some assumptions related with the existence of the operating point since power converters can operate in a limited set of operating points. Hence, we include the following assumption. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II CLASS FILES, VOL. XX, NO. X, DECEMBER 2023 3 Assumption 2: For a given desired operating voltage vCe, there exists a corresponding state xeand weights λe= [λ1λ2] such that λ1in [0,1] (and λ2= 1 −λ1), and 2 X j=1 λj¯ Bj:= 2 X j=1 λj((Aj−I)xe+Bj)=0.(4) This assumption is based on the computation of the solutions Ajand Bjto the least squares optimization problem. The value of λ1in this assumption gives an information about the duty cycle that corresponds to the desired voltage vCe. To translate this information into a cycle ν, we propose the following procedure to build the corresponding cycle. For a given Nν∈N, define the cycle νgiven by ∀ℓ∈[1, Nν], ν(ℓ) = 1if ℓ≤Round(Nνλ1), 2if ℓ > Round(Nνλ1).(5) In practice, one has to select Nνsuch that Round(Nνλ1) is sufficiently close to Nνλ1. We are now in position to state our main result. Theorem 1: Given data experiments Xj,X+ j, a desired operating voltage vCe, associated with corresponding operating vector xeand cycle ν(i.e. λ1) such that Assumptions 1 and 2 are satisfied, and for a given a parameter µ∈(0,1), consider {(Wi, ρi, ηi)}i∈Dν in S2×R2×(R>0)2as the solution to the following problem. min {(Wi,ρi,ηi)}i∈Dν ε s.t. Φi(X+ ν(i), Xν(i), xe)≻0, ηi>0, εI ⪰Wi≻0,∀i∈Dν, (6) where matrices Φiare given by Φi(X+ ν(i), Xν(i), xe) :=        (1−µ)Wi0WiA⊤ ν(i)Wi0 ∗µΦ(23) i(ρi−xe)⊤1 ∗ ∗ Φ(33) i00 ∗ ∗ ∗ ηiXν(i)X⊤ ν(i)ηXν(i)1ν(i) ∗ ∗ ∗ ∗ ηipν(i)        , (7) where Φ(23) i=ρ⊤ iA⊤ ν(i)+¯ B⊤ ν(i)−ρ⊤ ⌊i+1⌋ν,Φ(33) i=W⌊i+1⌋ν− ηiαν(i)∆ν(i). Then, under Assumption 1, the set Sν:= [ i∈Dν E(W−1 i, ρi+xe)(8) is robustly globally exponentially stable for system (2) with ˜x=x−xeand the switching control law u(x) := ν(θ), θ ∈argmin i∈Dν (˜x−ρi)⊤W−1 i(˜x−ρi)⊂K. (9) Proof 1: Following [17, Th.2], the conclusion of the previous theorem holds if the following conditions 0≺Ψi(Aν(i), Bν(i), xe) :=   (1−µ)Wi0WiA⊤ ν(i) ∗µ(ρi−xe)⊤ A⊤ ν(i)+B⊤ ν(i)−(ρ⌊i+1⌋ν−xe)⊤ ∗ ∗ W⌊i+1⌋ν  . are ensured for all uncertain matrices Aν(i), Bν(i). Then, replacing Aν(i)and Bν(i)by Aj−(Aν(i)−Aν(i))and Bν(i)−(Bν(i)−Bν(i)), respectively, the previous LMI can be rewritten as Ψi(Aν(i),¯ Bν(i))+He"h Wi0 (ρi −xe)⊤1i 0#03×2A⊤ ν(i)−A⊤ ν(i) B⊤ ν(i)−B⊤ ν(i)≻0, (10) where we recall that ¯ Bν(i)= (Aj−I)xe+Bj. Then, Young’s inequality ensures that the previous inequality is equivalent to the existence of ηi>0such that Ψi(Aν(i),¯ Bν(i)) −η−1 i"h Wi0 (ρi −xe)⊤1i 0#hXj 1⊤ jihXj 1⊤ ji⊤−1"h Wi0 (ρi −xe)⊤1i 0#⊤ −ηi  0 0 0A⊤ ν(i)−A⊤ ν(i) B⊤ ν(i)−B⊤ ν(i)⊤hXj 1⊤ jihXj 1⊤ ji⊤A⊤ ν(i)−A⊤ ν(i) B⊤ ν(i)−B⊤ ν(i) ≻0. The proof is concluded by using Assumption 1, stating that matrices Ajand Bjare in Ωj, so that the term in αj∆jappears naturally and by application of the Schur Complement to the second term of the previous inequality. Remark 1: If matrices Ajand Bjare constant, the least squares approximation Ajand ¯ Bjare equal to the corresponding matrices Ajand Bj. In this situation, Proposition 2 in [17] guarantees that the solution to the LMI problem is equivalent to the condition A⊤ ν(i)W−1 ⌊i+1⌋ν Aν(i)−W−1 i≺0, which according to [4], is equivalent to having a Schur stable monodromy matrix Φν:= ARound(Nνλ1) 1ANν−Round(Nνλ1) 2. In other words, the eigenvalues of Φνare strictly inside the unit circle. A consequence of the previous statement is that, in the case of time-varying matrices Ajand Bj, a necessary condition for the solvability of the LMI is that Φνis Schur. Remark 2: Problem (6) is convex if the scalar parameter µ∈(0,1) is fixed, and therefore has a unique global minima. A grid search on µcan be performed to optimize the result. D. Summary of the data-driven method To get a better understanding of the previous developments, we propose in this section a summary of the procedure for the data-driven control design, formulated as Algorithm 1. In this algorithm, we specify the mechanism of choosing controller parameters ρiand Wi,∀i∈Dν, from a selecting output voltage, vCe. First, we have a while loop to select the appropriate amount of data such that rank hXj 1⊤ ji= 3, to identify the matrices from the data acquisition. This ensures that a desired operating point xeexists, along with its associated λe=λ1, which satisfies Assumption 2. Then, a cycle that approximates λeis selected according to (5), i.e., ensuring that Round(Nνλ1)≈1. Finally, for a given µ, the optimal solution to (6) provides control matrices and parameters. III. EXPERIMENTAL VALIDATION This section is devoted to the experimental validation of the data-driven design on a boost converter, considering that it can be approximated by a switched affine model (2) with two IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II CLASS FILES, VOL. XX, NO. X, DECEMBER 2023 4 Algorithm 1 Control matrices and parameter repeat Take: Extract Xj, X+ j∈Rn×pjfrom Xand Σ. Identify: Ajand Bjusing Xj, X+ j,∀j∈ {1,2}. Test: for a desired vCe,∃any xeand λe. until Assumption 2 is satisfied Select : ν(ℓ)∀ℓ∈N, according to (5) and µ∈(0,1),αj. Compute : ρi,ηiand Wi,∀i∈Dν, s.t. (6) are satisfied. Fig. 1. Configured Imperix unit. functioning modes. The power converter is implemented in an Imperix power test bench, as shown in Fig. 1 and its general architecture of the experimental setup in Fig. 2. More precisely, the boost converter is configured from a PEB8038 half-bridge power module with silicon carbide (SiC) MOSFETs rated for 800V and 38A. Moreover, a power inductanceLand a film capacitor C0are embedded in a passive filter box. Furthermore, a DC source Elektro-Automtik PSB 9360-40 is employed to provide the converter input voltage, vin. The control algorithm is implemented in a B-Box rapid prototyping controller (RCP), which sends gate drive signals via optical channels and acquires sensors signals through analog RJ45 connectors, running with an internal sampling frequency of 50 kHz. The inductance current, iL, and capacitor voltage, vC, are measured using galvanically isolated onboard sensors in the PEB8038 module, which are LEM CKSR 50-P and Avago ACP-C87B, respectively. The B-BOX RCP is programmed using the Simulink blockset of the Imperix ACG software development kit (SDK). Control tests, debugging, data logging, and signal monitoring are acquired from the Imperix Cockpit real-time monitoring software. The electrical block diagram illustrates two different control programs configured, namely, data acquisition and closed-loop control. Both programs use an ADC and a DO-PWM (Direct output PMW) modules, where it is necessary to convert σinto 1and 0for the on/off state of the MOSFET, respectively. A. Data generation Some data were collected by feeding the system by a random input signal, σc∈ {1,2}. The selected data set is depicted in Fig. 3. The selected frequency of the random signal σcis 50KHz, and the data sample has a frequency also of 50KHz. Fig. 2. Block diagram of the experimental setup. 0 0.005 0.01 0.015 0.02 0.025 0.03 0 10 20 0 0.005 0.01 0.015 0.02 0.025 0.03 50 100 Fig. 3. Experimental data of iLand vCobtained with a random input σ. B. Closed-loop control validation The objective is to drive the output voltage to a neighbourhood of vCe= 66V. We highlight that the identified system and proposed control law depend on the sampling frequency of 50 KHz. Following the Algorithm 1, the experimental data delivers the following desired operating point xe= [1.2 65.9]⊤ with associated λ1= 0.13 and identified matrices A1=[ 1.08 0 0 1.01 ],A2=0.91 0 −0.01 0.97 ,¯ B1=[ 0.38 1.81 ],¯ B2=−0.44 −2.13 , which satisfy Assumption 2. Then, we select the cycle ν=[1222222], which is consistent with the required λ1, i.e. it has been built so that Round(Nνλ1) = 0.91 ≈1. The robust limit cycle associated to this cycle, which is solution to the optimization problem (6) with α1=α2= 1, is obtained by the composition of the centers ρithat are stacked in the following matrix ρ=−0.09 0.07 0.03 0.00 −0.03 −0.06 −0.07 0.06 0.94 0.77 0.61 0.45 0.30 0.15 , and the ellipsoids obtained with the following symmetric positive definite matrices W1=210.45 −3.22 −3.22 171.53, W2=170.1−4.9 −4.9 158.9, W3=173.58 −4.79 −4.79 161.69, W4=176.53 −4.70 −4.70 163.92, W5=178.39 −4.56 −4.56 165.69, W6=180.67 −4.37 −4.37 167.37, W7=189.74 −4.1 −4.1 169.29. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II CLASS FILES, VOL. XX, NO. X, DECEMBER 2023 5 Fig. 4. Imperix Cockpit real-time monitoring software. The computational time to solve the LMIs by using CVX- Sedumi in Matlab on a computer with a 2 GHz Intel Core i5 processor and 4 cores is 1.19s. We have implemented an external loop through a PI control such that the output voltage is regulated in vCe. Fig. 4 shows the voltage and current evolution of the boost converter configured in the Imperix unit controlled with the presented data-driven approach, including the predictor and the voltage output regulation. The load in the boost converter has been changed from 69Ωto 34.5Ω. The figure also shows the voltage error. The experiment validates the data-driven approach. IV. CONCLUSIONS In this paper, we present a theoretical contribution addressing the data-driven control design for power converters modelled as a switched affine system. The distinctive feature of this control design lies in the unknown dynamics of the system. However, this lack of knowledge is mitigated by leveraging prior experimental data. Building upon several non-restrictive assumptions, we validate the theoretical results using an experimental setup, specifically an Imperix boost converter. Furthermore, we incorporate an external control loop that regulates the output voltage, which is validated through experiments. This contribution marks a significant milestone for future research. We have already identified several crucial issues that merit further investigation. For example, the method relies on the satisfaction of Assumption 1. It would be of interest to devise a methodology for evaluating the values of parameters αjdirectly from the data. Another important concern lies in mode identification, accounting for additional power converter features such as the discontinuous conduction operation mode. 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