for buck synchronous DC-DC converter has been designed. The proposed ..... on-line, predicting future behavior until a horizon in time, thus it is applicable only ...
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Explicit Sensorless Model Predictive Control of Synchronous Buck Converter Gionata Cimini, Gianluca Ippoliti, Giuseppe Orlando, Matteo Pirro Dipartimento di Ingegneria dell’Informazione Universit`a Politecnica delle Marche Via Brecce Bianche, 60131 Ancona, Italy Email: {g.cimini,g.ippoliti}@univpm.it {g.orlando,m.pirro}@univpm.it Abstract—In this paper a Model Predictive Control (MPC) for buck synchronous DC-DC converter has been designed. The proposed solution considers an off-line explicit MPC formulation resulting in a PieceWise Affine (PWA) controller structure. A model based observer for inductor current has been taken into account to overcome the several problems related to sensors in high current ripple DC converters. Reported numerical results show that the proposed control scheme enhances the overall performances of common averaged current mode control with linear regulators.
I.
I NTRODUCTION
Nowadays, power electronics are widely used and the industry drives toward smaller, lighter and more efficient electronics in paricular in renewable energies field. For these reasons Switching Mode Power Supply (SMPS), employing high switching frequencies, is used as replacements for linear regulators when higher efficiency, smaller size or lighter weight are required: SMPS incorporates a switching regulator to convert electrical power efficiently and desired voltage regulation is achieved by varying the ratio of on-to-off time. In particular the buck topology is a specific type of DC-DC power electronic converter whose goal is to step down DC voltage to a lower level with minimal ripple; practical applications of buck converter rely on medium and low power systems. It is widely used in portable devices, especially in noise-sensitive circuitry to provide a stable and low-noise supply voltage [1]; it is also used to regulate the charging of a battery in a PhotoVoltaic (PV) system [2], [3]. In many of these applications, the current limitation is of utmost concern to protect both the load and the converter and to optimize the power circuit size. Input current in DC-DC converters is often limited by reducing the controller aggressiveness; this results in decreasing the system performance during transients especially in startup phase or against load and supply variations and, due to the inaccuracy of this process, it is difficult to determine the optimal controller specifications suitable for all operating points and possible transients. The most popular control method for switching circuits in power electronics, such as DC-DC converters, is a Pulse Width Modulator (PWM) together with a linear controller structure driven by output voltage error [4]. Usually, the design for these linear controllers (such as PI) is based on a continuous or discrete state-space averaged model of the converter [5]. With the development of powerful microprocessors, new and non linear control schemes have been adopted and implemented
for power electronics, such as Variable Structure Control (VSC) method [6] or Passivity Based Control (PBC) [7], [8]. Such control strategies have been successfully implemented for both electrical [9] and electromechanical systems [10], [11]; in particular in [12], [13] the buck regulation problem is addressed. Among these new control schemes, MPC [14], [15] appears to be a very interesting alternative for the power electronics control. MPC theory doesn’t rely on a specific control structure but rather all the controllers belonging to MPC family are characterized by the same control strategy. Generally, industrial processes are nonlinear, whereas most MPC applications are based on the use of linear plants. MPC requires the explicit use of a model to predict process outputs at future time instants and those predicted values are used to calculate a control sequence which minimize a designed objective function; the first control signal of the sequence is applied at each step. Over the years several MPC algorithms have been proposed and some of them have been widely used in industry such as Dynamic Matrix Control (DMC) [16] and Model Algorithmic Control (MAC) [17] which belong to non parametric methods category. Furthermore Generalized Predictive Control (GPC) [18] basic idea is to calculate a sequence of future control signals in such a way that it minimizes a multistage cost function defined over a prediction horizon. The main disadvantage of MPC is at the computation time required which considerably limits the bandwidth of processes to which it can be applied. In spite of this in [14], [15] is demonstrated that MPC algorithm in presence of constraints results in a QP problem if objective function to minimize is quadratic; if a 1-norm type of objective function is taken into account, the system solution can be reformulated as an LP problem. Although very efficient algorithms exist for solving QP or LP problems, computation time is a critical feature in real DSP-based control of fast processes, such as motor drives and can be too high to achieve satisfying performances. In [19] has been shown that for constrained MPC the problem could be considered as a multiparametric quadratic problem or a linear programming one [20]; thus MPC solution turns out to be a easy-to-implement PieceWise Affine (PWA) controller: in [21] the proposed method gives evaluation times which are logarithmic in the number of regions in the PWA function, while the storage required by the data structure is polynomial in the number of regions. Due to the increasing demands in performance and efficiency for drives and power electronics, new control schemes must be taken into account and MPC algorithms are particularly suited for various reasons: accurate linear models can be obtained for
power converters, several constraints inherent to the system are usually required, such as voltage and current limitations. Nevertheless, in literature, some model predictive controllers have been applied in power converters or electrical drives [22]. For what concern buck converter regulation problem, in [23] a current sensorless explicit MPC with both input and output voltage sensors has been proposed; in [24] a PWA model for the whole operating range has been adopted and finally in [25] an MPC-PI approach has been evaluated. In this paper, a model predictive control for synchronous DC-DC buck converter is presented. Explicit PWA formulation with look-up table control laws and binary search tree has been used to overcome computational time problem of a real implementation. A model based observer for inductor current has been used; besides obvious economic benefits offered by sensors reduction, sensorless approach has several other advantages such as elimination of sensors offsets, insensitivity to noise and size reduction of converter system [26]–[29]. Proposed solution is verified on a simulation platform and algorithm performances are compared to results obtained with common PI-based average current mode control in a simulation scenario with input voltage and load variations. The paper is organized as follows. Synchronous buck converter model is presented in an averaged formulation in first part of Section II; in same section a brief overview of common linear current mode control approach has been given. In Section III MPC solution for the buck converter is explored, starting from the general problem formulation. Section IV presents the design of a current observer based on average model obtained in Section II. Finally, in Section IV, the numerical results obtained in different control scenarios and comparison between linear and MPC controllers are treated. The paper ends with conclusions in Section V. II.
Figure 1.
Synchronous Buck Converter.
Following state-space averaging method [4], the variables are averaged over one switching period Ts and are expressed with following notation hx(t)iTs
1 = Ts
t+T Z s
x(τ )dτ.
(1)
t
Since synchronous buck converter is mostly designed with two identical switching devices (Q1 and Q2), the averaged model is obtained as follows: ∂hiL (t)i d(t)Vg − (RDSon + RL )hiL (t)i − hv(t)i = ∂t L hvi hiL i − ∂hvc i R = ∂t C
(2)
(3)
where hiL (t)i and hv(t)i are respectively the averaged inductor current and the averaged output voltage, L and C are the inductor and the capacitance values respectively, Vg is the input voltage, RDSon and RL are respectively the switches onresistance and the inductor series resistance. Averaged model (2), (3) is obtained considering several losses as a successfully implementation of MPC controller and model based observer deeply rely on accurate system modeling.
S YSTEM M ODELING AND L INEAR C ONTROL
In this section an averaged model of synchronous buck converter is proposed and the common linear average current mode control is briefly summarized. A. Buck Converter Model Synchronous buck converter scheme is presented in Fig. 1. The asynchronous topology uses just one high frequency switched MOSFET (Q1) for the up-side as the control switch, and a diode at Q2 place. In this topology losses in the freewheeling diode are high due to large forward voltage drop and, consequently, it reduces the overall efficiency of the converter systems (typically less than 90%). In the synchronous topology the freewheeling diode is replaced by a MOSFET (Q2) acting as a rectifier, which lower resistance from drain to source (RDSON ) helps reduce losses significantly and therefore optimizes the overall conversion [30]. In addition Q2 provides current to flow in reverse direction and thus, synchronous buck converter operates in the Continuous Conduction Mode (CCM) even when inductor current ripple is larger than it’s mean value avoiding the need of additional model for Discontinous Condutcion Mode (DCM). However, all of this demands a more complicated drive circuitry to control both the switches. Care has to be taken to ensure both Q1 and Q2 are not turned on at the same time, avoiding the shoot-through issue.
B. Averaged Current Mode Control The standard cascade averaged current mode control scheme is presented in Fig. 2. Outer loop is driven by load voltage error and the output of controller provides the reference signal of inner current control; the duty cycle of PWM gating signal is the control variable. The two controller Gcv (s) and Gc (s) are commonly designed with linear considerations starting from buck synchronous model in (2), (3). The design of linear controllers Gcv (s) and Gv (s) has been based on the following transfer function respectively: R (4) 1 + sRC Vin (1 + sRC) Gc (s) = 2 s LRC + (L + RC(RDSon + RL )s + R + RDSon + RL ) (5) Gcv (s) =
III.
MPC C ONTROL D ESIGN
In this section the formulation for the constrained LQ tracking problem has been provided and the explicit control law has been obtained in a look-up table formulation.
Figure 2.
where the matrices G, W and E are derived from Q, R and P ones in Eq. (7). In order to exploit previous formulation and to obtain the optimal actuation, a QP solution has to be computed on-line, predicting future behavior until a horizon in time, thus it is applicable only to slow processes. In [19] has been shown that the optimization problem of the form in (12) and (13) can be solved off-line considering it as a Multi Parametric Quadratic Program (MP-QP) problem and treating x(k) as a parameter vector. This approach results into a partition of the whole state space in several convex polyhedral regions and for each region the static control matrices is calculated as the result of the MP-QP problem. The overall formulation results in a continuous PWA control law of the state
Sensorless averaged current mode control scheme.
ui (k) = Fi x(k) + Gi
A. Solution to the constrained LQ regulation problem Principal feature of MPC is the use of system linear model to predict the future behavior of the controlled variables. Thus the following Linear Time Invariant (LTI) system is considered x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k)
(6)
where x ∈ X ⊂
