A piecewise affine PI controller for buck converter generated DC motor

Received Sep 29, 2018 Revised Jan 16, 2019 Accepted Mar13, 2019 This paper presents a new Piecewise Affine Proportional-Integral (PA-PI) controller for angular velocity tracking of a buck converter generated DC motor. A Safe Experimentation Dynamics (SED) algorithm is employed as a data-driven optimization tool to find the optimal PA-PI controller parameters such that the integral square of error and input are reduced. The essential feature of the PA-PI controller is that the parameters of proportional and integral gains are adaptive to the error variations according to the Piecewise Affine (PA) function. Moreover, the proposed PA function is expected to provide better control accuracy than the other existing variable structure PID controller. In order to verify the effectiveness of the PA-PI controller, a widely known buck converter generated DC motor is considered. The performances of the proposed controller are observed in terms of the integral square of error and input, and the responses of the angular velocity and duty ration input. The simulation results verify that the proposed PA-PI controller yields higher control accuracy than the other existing controllers of buck converter generated DC motor.


INTRODUCTION
Recently, various applications, such as robotic manipulator, drones, conveyor belts, and electric cranes, require a high precision DC motors. Meanwhile, there are large numbers of standard pulse width modulation (PWM) applications for DC motor execution. Nevertheless, this technique degrades the precision motion of the DC motor owing to intractable switching strategy that resulting to jerk response in the voltage and current of the DC motor [1]. As a solution to the problem, a DC-DC buck converter is adopted to provide a smooth tracking motion of the DC motor. In particular, it can track the required trajectory of both angular velocity and position trajectory through regulating an input voltage only. In other words, the selection of good control scheme is very important to provide a high precision of motion of the DC motor fed by the DC-DC buck converter.
To the best of our knowledge, a variety of approaches have been proposed in regulating buckconverter generated DC motor. Firstly, a feedback controller with energy shaping and damping injection [2] is used to control a 4 th order model of buck converter generated DC motor. Next, a flatness based method is adopted for angular velocity tracking of the DC motor fed by the DC-DC buck converter in [1], [3] and [4].
Here, the controller design in [4] is derived from the simplified 2 nd order model, while the controller in [1] and [3] are derived from the 4 th order model. Moreover, a sliding mode controller with delta modulation based GPI is reported in [5]. In particular, the simplified mathematical model in [4] is applied. Furthermore, a backstepping controller with non-adaptive and adaptive mechanisms is designed based on 4 th order model  [6]. Their work claims that the adaptive mechanism is significant in improving the control performance even with load torque variations. Other control strategies of DC motor fed by DC-DC buck converter are LQR and PI controllers [7], two-stage control with differential flatness [8], H∞ controller [9], robust control law with active disturbance rejection [10], neural network controller [11], and hybrid PI with Fuzzy controller [12].
Based on the stated control schemes, most of the controller techniques have employed the modelbased control schemes, which are synthesized based on the 2 nd or 4 th order model. Nevertheless, it is hard to employ the controller derived from the model in real experiment of buck-converter generated DC motor plant owing to some justifications. These are the unmodeled dynamic problems, inaccuracy of the simplified model, and massive gap between real applications and control theory. Thence, a data-driven controller, which is derived from the output and input data of the plant in the absence of any knowledge of the plant, may become a promising idea. Meanwhile, the data-driven PID control scheme is mostly preferred and established controller structure owing to its simplicity in design and easy to be implemented [13]. Moreover, the control accuracy of the standard PID control scheme can be enhanced by upgrading it to nonlinear PID [14]- [19]. Therefore, it is significant to assess the effectiveness of the data-driven nonlinear PID controller for input tracking of the angular velocity of DC motor fed by the DC-DC buck-converter.
The paper presents the data-based Piece-wise Affine PI controller for input tracking of angular velocity of DC motor fed by the DC-DC buck converter. Specifically, the Piecewise Affine PI, which is in the family of variable structure PID, is expected to provide better control accuracy than the sigmoid based PID in [19]. This is because our proposed variable structure, which is in a piece-wise affine form, is more general and not restrict to the sigmoid function. Thence, it proffers high possibility to enhance the control accuracy by using the time-varying PI gains according to the changes of angular velocity error with respect to the optimal Piecewise Affine function. Moreover, the study in [19] does not includes an explicit description of data-driven controller tuning, while in this study, a complete data-driven control synthesize is shown. Specifically, a safe experimentation dynamics (SED) algorithm [20] is employed to obtain the Piecewise Affine PI gains that minimize the angular velocity error. The SED algorithm is widely established to be a significant optimization tool owing to its capability in solving variety of optimization problems with fast convergence speed even for large number of parameters tuning [21]- [25]. Next, the proposed data-based Piecewise Affine PI controller design is verified to a widely established model of buck-converter generated DC motor [1]. The control performances of the Piece-wise Affine PI controller based SED are analyzed in terms of duty cycle input energy and angular velocity trajectory tracking. Lastly, a performance comparison between the Piecewise Affine PI controller based SED and the conventional PI [7] and the sigmoid based PI [19] is presented.

BUCK CONVERTER GENERATED DC MOTOR
The complete 4 th order mathematical model of buck converter generated DC motor , which is taken from [1], is given by The detailed of the variables of the plant in (1) is given in Table I. Note that the input u(t) of the plant is a duty ratio , while the angular velocity ω is the output of the plant y(t).  Figure 1 shows the Piecewise Affine PI controller (PA-PI) for buck converter generated DC motor. Here, the symbols r(t), u(t), y(t) and e(t) are the desired angular velocity trajectory, the control input, the actual angular velocity response, and the error between desire and actual angular velocity, respectively. The plant H is defined as the buck converter generated DC motor system as presented in the previous section. The PA-PI controller is expressed by

PROBLEM FORMULATION
are the given input points for the gain P(t) satisfying and connecting the line segments, and are the output points corresponding to each input point. For simple notation, let define . The illustration of the piecewise affine functions for the gain P(t) is shown in Figure   2. In this study, it is assumed that P(t) is one-to-one map to e(t) and P(t) = 0 if e(t) = 0. Thence, Note that a similar piece-wise affine function is applied to integral gain I(t). Next, the control performance assessment for the control system block diagram in Figure 1 is given by Where t0 ∈ 0 ∪ R , and tf ∈ R+. Note that the time interval [t0, tf] corresponds to the period for the performance evaluation. The symbols ꞷ1 and ꞷ2 correspond to weighting coefficients, which are given by the designer. The first term in (3) corresponds to the tracking error, while the second means the control input energy. Here, the values of ꞷ1 and ꞷ2 are selected in a similar way to the standard Linear Quadratic Regulator (LQR) problem. Finally, the problem statement is described by: Problem 3.1. Consider the control system block diagram in Figure 1. Then, find the PA-PI controller C(s) in (2) such that the control performance assessment Q(P(t),I(t)) is minimized based on the u(t) and y(t) data. Remark 3.1. The proposed PA-PI controller is expected to produce better control performance from our previous version of sigmoid based PI in [19]. This is because the nonlinear PI gains in the proposed PA-PI are not restricted to only sigmoid-based function. Though it looks that we can handle general class of nonlinear subsystems PI gains by adopting so many basis functions (such as piece-wise affine functions).

PA-PI CONTROLLER BASED SAFE EXPERIMENTATION DYNAMICS
In this section, the main solution of the Problem 3.1 is explained. Firstly, an overview of the Safe Experimentation Dynamics (SED) algorithm is presented. Secondly, it is shown on how to apply the SED for tuning the parameters of PA-PI such that the control performances index in (4) is reduced.

Overview of safe experimentation dynamics
Safe Experimentation Dynamics (SED) is a game theoretic method that defines each element of the design variable as a player [20]. Here, the random motion of each player is decided based on pre-determined probability such that an optimal goal or design variable is achieved, which corresponds to the minimum value of the loss function. In particular, let the optimization problem is given by where x ∈ R n is the design variable, and L :R n →R is denoted by the loss function. The SED algorithm iteratively updates x ∈ R n using the updated law for τ = 0, 1, … where r1 and r2 are independent random numbers which uniformly distributed between the ranges of [0, 1], E is a scalar that defines the probability of using the new updated design variable, λ represents the step size gain of the design variable, xi is the i-th element of x ∈ R n , x̄i is the i-th element of x̄ ∈ R n , and x̄ ∈ R n ,is denoted as the current best design variable vector. In (6), the function f is given by where xup and x1o are pre-defines upper and lower bound values of design variable, respectively. The important feature of SED is that it can provide a stable convergence due to its capability to keep the best design variable during the tuning process. Moreover, the SED uses a fixed interval step size, which is independent of the number of iterations. Therefore, it will be useful data-driven optimization tool for the case when disturbances or uncertainties or stochastic delay occur during the tuning process. In this study, the proposed SED algorithm terminates using pre-specified stopping criterion τmax with the solution

Application of SED to tune PA-PI controller
In this section, it is described on how to employ the SED algorithm in Section 4.1 for tuning the proposed PA-PI controller of buck converter generated DC motor. Initially, the design variable with respect to the PA-PI controller gains is defined by  . Next, a logarithmic function is utilized for every element of ρ R  x to provide a fast design variable searching. In particular, the new setting of the design variable is . Lastly, the stepby-step procedure to apply the SED algorithm to PA-PI controller is stated as follow: Step 1: Set the initial design variable x(0) with pre-specified τmax and Step 2: Execute the SED algorithm in (6) by setting the design variable with υ in (8) and L := Q.
Step 3: After τmax iterations, obtain the optimal design variable ρ * R  x and apply to C(s) in (2).

NUMERICAL RESULTS
In this section, the effectiveness of the proposed PA-PI controller based SED is presented. The model of the buck-converter generated DC motor in (9) is used. The desired angular velocity used in this numerical example is based on the tangent hyperbolic function from [19]. The design variable of Piecewise Affine PI controller C(s) is given in the second column of Table II Table II is selected based on the values of conventional PI controller in [7]. For example, the value of the proportional gain from [7], which is 0.0069, is multiplied with w P to obtain the first five values of vector υ(0) in the column 4 of Table II.
The optimal design variable ρ * R  υ is tabulated in Table 2. Based on the obtained , the resulting Piecewise Affine function for the proportional and integral gains can be observed in Figures 3 and  4, respectively. Note that, the proportional gain requires high magnitude of gain for the angular velocity error ranges 0 -3 and 13 -15. Meanwhile, the integral gain requires high magnitude of gain for the angular velocity error between 0 and 3. It shows that by employing time varying P(t) and I(t) gains according to the changes of error, it will offer more rooms for the controller improvement. Moreover, Figures 5 and 6 show the responses of the angular velocity and the duty ratio for the PA-PI controller in comparison the conventional PI and Sigmoid-PI controllers. Note that, the reference is represented by the dot black line, the PI controller is represented by thick grey line, the sigmoid-based PI controller is represented by the dash black line and the PA-PI controller is represented by the thick black line. In general, the responses clarify that the proposed PA-PI based SED yields better angular velocity trajectory tracking with slightly minimum duty ratio than other existing methods. Specifically, from the magnified picture, the actual response of y(t) is much closer to the reference r(t) than other controllers. Similarly, the response of duty ratio of the PA-PI controller is slightly lower than the recent sigmoid based PI controller. This is supported by the numerical analysis of the integral square of error and input as tabulated in Table 3. It is shown that the values of the integral square of error and input for PA-PI controller are much lower than other controllers.

CONCLUSION
This paper presents a Piecewise Affine Proportional-Integral (PA-PI) controller based SED for buck converter generated DC motor. It justifies that the Piecewise Affine Proportional-Integral (PA-PI) controller has improved the control accuracy, in terms of the responses of the angular velocity and duty ratio input, and the integral square of error and input. In particular, the PA-PI controller based SED is able to closely track the desired angular velocity with small duty ratio input. This fact is also supported by the analysis of the integral square of error and input, where it produces slightly smaller values than the conventional PI and the sigmoid based PI controllers. Thence, this finding justifies the effectiveness of the PA-PI controller in offering more generic class of variable structure PID, resulting in more prospects of controller improvement.