Improved Model Predictive Speed Control of a PMSM via Laguerre Functions

. This paper proposes a model predictive speed control strategy for a surface-mounted permanent magnet synchronous motor by applying Laguerre functions. The model predictive controller (MPC) incorporates an integrator. A quadratic programming procedure is applied to solve the constrained optimization problem online. The paper also provides a solution for stability. The performance ef ﬁ ciency of the proposed scheme is validated by comparing the results with the performance of an optimal linear quadratic regulator, conventional state-space model predictive control, and a simple MPC algorithm with integral action. Extensive simulation results con ﬁ rm the ef ﬁ cacy of the proposed scheme, showing that it achieves good steady-state performance while maintaining a fast dynamic response.


Introduction
Permanent magnet synchronous motors (PMSMs) have been intensively used in applications, such as industrial robots and electrical machines.Permanent magnet motors are prevalent in everyday machining tools due to their good dynamic properties and compact structure [1][2][3][4].In addition to high efficiency, the advantages of permanent magnet motors include their lightweight and small size, which in turn facilitate installation and maintenance.Among the many techniques used in electric drive control, the proportional integrator (PI), linear quadratic regulator (LQR), and recently developed model predictive controller (MPC) have been employed.A combination of cascaded linear controller structures and PI controllers has been used for the speed regulation of PMSMs due to their simplicity [5].To overcome extensive overshoots and ringing, cascaded linear controllers use a limited bandwidth.Hence, a reasonably good dynamic response can be achieved [6,7].However, during the transient time and in the presence of a load disturbance, the dynamic performance of the PI controller is reduced.Extensive research has focused on speed controller design for adjustable-speed PMSM systems to enhance the transient response, recovery time from a load disturbance, tracking ability, and robustness [8][9][10][11][12][13][14].The LQR method is a contemporary control technique that is efficient but limited to linear systems.To evaluate the performance of the LQR controller, this paper compares it with results obtained from other control techniques.This means designing the same system using another control technique which could be cumbersome.On the other hand, an MPC can be applied to control both linear and nonlinear systems [15,16].In an MPC, the nonlinear system and corresponding linear model can be easily compared with little modifications.
It is known that tuning an MPC is not as difficult as tuning a PI, even for multiple input multiple output (MIMO) systems.Nevertheless, to optimally regulate the response of a system to a reference while respecting the constraints, an MPC requires more computational effort than either a PI or an LQR.Since an optimal solution can be analytically accomplished, the feedback gain can be precalculated offline when the operational constraints are absent or not active.However, if the constraints are available, quadratic programing (QP) can be used to solve the optimization problem online.Hence, an MPC is used intensively in industrial control where fast sampling is not required.In contrast, an MPC has limited applications in the control of electric drives and power converters due to the computational load and fast sampling requirements to solve the QP problem.An enhancement in processor performance has notably surged, and new faster algorithms have been developed, rendering the MPC implementation possible for a power converter and an electric drive [17][18][19].The proposed scheme in this paper focuses mainly on improving the performance of PMSM systems by designing an MPC using orthonormal functions called Laguerre functions (LMPC).
The three notable contributions of the proposed scheme are illustrated below: (1) A controller with the capability to reject disturbances from the load torque of a PMSM system is developed.
To ensure zero steady-state error in the presence of a load torque disturbance, an integrator is incorporated into the design.This modification improves the steady-state performance of the PMSM system.(2) An LMPC is employed as a control technique for a PMSM system.The LMPC method incorporates the advantages of standard MPC algorithms such as constraint handling and online optimization and produces an algorithm with a low online computational burden.Because the LMPC method uses a simpler design, the computational load is reduced.The performance improvements of a PMSM designed using the LMPC are compared with those designed using the algorithm presented in a different studies [20,21].(3) To achieve a lower computational load and minimize the numerical problem, this paper proposes an exponentially decreasing objective function for the LMPC, specifically for a large prediction horizon to control the speed of the PMSM.The proposed LMPC method enhances the controller performance in an organized manner.
The structure of the proposed paper is as follows.Section 2 illustrates the drive model and linearization of the model for MPC design.Section 3 describes the MPC design.Section 4 outlines the simulation results, and Section 5 presents the conclusion.

Drive Model
The permanent magnet synchronous machines considered in this paper are called surface-mounted PMSMs.Models for speed control can be derived from the stator's equation and torque's equilibrium equations [22].The following equations determine the synchronous reference d-q frame [22][23][24]: If the permanent magnets are mounted on the rotor surface, and there is no significant internal asymmetry in the iron parts of the rotor, the direct-axis and quadratureaxis inductances of the machine are approximately equal, L d ¼ L q .Hence, the torque in Equation ( 1) can be simplified to T ¼ 3  2 pψi q allowing the electromagnetic torque to be controlled through i q directly and maintaining i d ¼ 0. Maintaining i d ¼ 0 helps to achieve maximum efficiency of the PMSM, i.e., maximum torque per ampere condition in the whole operation range.ω ¼ pω m is electromechanical speed, ψ is the permanent flux linkage, p is the number of pole pairs, and ω m is the mechanical speed.In addition, T L is the disturbance from the load torque, while J and B v are the motor's moment of inertia and viscous coefficient.In Equation ( 1), the PMSM dynamics are expressed by a nonlinear set of equations because of motional coupling terms ðω: i q and ωi d Þ: ; hence, the equations will be linearized at operating points for use in the MPC design.Using a first-order Taylor series to approximate the nonlinear coupled terms as: Substituting Equations ( 2) and (3) into Equation (1) yields the following equation: where v 2 R 3 is the steady-state parameter and load torque, x p ¼ i d ðtÞ i q ðtÞ ωðtÞ Â Ã T 2 R 3 is the state, y ¼ i d ðtÞ ωðtÞ ½ T 2 R 2 is the output, and The MPC requires a discrete-time model, and Equation ( 3) is thus discretized with a sampling period using 2 Mathematical Problems in Engineering a zero-order hold, resulting in the discrete-time model as follows: where and d v has constant entries.For speed regulation of a PMSM, n u ¼ 2; n y ¼ 2, and n x ¼ 3 will be used, where n u ; n y , and n x correspond to input, output, and state, respectively.
Let Δx p ðkÞ: ¼ x p ðkÞ: − x p ðk − 1Þ: and ΔuðkÞ: ¼ uðkÞ: − uðk − 1Þ: denote increment on state and input variables, respectively, determined from the corresponding variables in Equation (7).The state dynamics in the incremental model are as follows: In a similar manner, the output incremental dynamics are given as follows: By choosing a new state, xðkÞ: ¼ Δx p ðkÞ T yðkÞ T Â Ã : T the augmented state-space model is obtained by combining Equation (8) with Equation ( 9): where A ¼ Using the augmented model in Equation (10) has two advantages.First, the augmented model removes the d v of Equation (7) including any uncertain parameters.Second, it eliminates the unknown torque.

Model Predictive Control Design
To achieve good controller performance while operational constraints are present, use an MPC scheme.The performance of a PMSM speed controller employing Laguerre functions, i.e., an LMPC, is compared with an optimal discrete LQR (DLQR) in this and the following section.In later sections, the performance of an LMPC is compared with the MPC-IA and with a more conventional MPC based on a state-space design (SS-MPC).
3.1.State-Space MPC.An SS-MPC is formulated using the state-space approach.An SS-MPC with a convex quadratic performance index is a QP problem, which is appealing because QP must be solved online [25].The state variable xðkÞ: is estimated through an observer to obtain an optimal solution in the presence of operational constraints [26].Thus, apply an observer of the form: where K ob is the observer gain matrix, it is apparent that the observer gain K ob can be used to manipulate the convergence rate of the error.K ob is used to place the closed-loop eigenvalues of the error system matrix A − K ob C at a desired location of the complex plane.The estimate of the output yields the following equation: Model in Equation ( 10) is used to calculate the future state variables through ΔuðkÞ: where N p is the prediction horizon and N c (control horizon) is a parameter that determines the number of future control inputs to be included in optimization.This, therefore, assumes N c ⩽ N p .In its compact form, the output prediction for the next N p instants is as follows: Consider the cost function: where R s ¼ R s rðkÞ: . Upon minimizing the cost function in Equation ( 17), obtain the optimal control vector ΔU ¼ Mathematical Problems in Engineering . The receding horizon principle to obtain the control law: where k y is the matrix correspond to change in set-point and k mpc is the state feedback gain matrix.Operational constraints in control algorithms come from physical systems.
In PMSM, the input voltages and are limited by the DC bus voltage.Further, space vector pulse width modulator modulates the maximum voltage to V dc = ffiffi ffi 3 p .The cost function in Equation ( 17) is minimized with respect to ΔU: where The constraints in compact form is given as follows: And the optimization problem with constraints is solved by QP [25].
3.2.Laguerre-Based Model Predictive Control.The general procedure for MPC designed using Laguerre function was explained in a previous study [27].The author addressed this design method for single input single output.This paper presents an LMPC design method for MIMO systems specifically for the use to control the speed of PMSM systems.
Design framework: Let ΔuðkÞ: ¼ Δu 1 ðkÞ Δu 2 ðkÞ ⋯ ½ Δu r ðkÞ: T and the partitioned input matrix be B B 1 B 2 ½ ⋯ B r : , where r is the number of inputs and B i is the ith column of the matrix.Use orthonormal basis Laguerre functions to model the control trajectory.The Laguerre function in the ztransform is as follows: where 0 ⩽ a i ⩽ 1 is the pole of the Laguerre network.
The control trajectory can be described using the Laguerre functions: where l i j ðmÞ: is the inverse z-transform of jth term in the discrete Laguerre network, i.e., Γ i j and the coefficients d i j are unknowns and must be obtained from systems data.
The tuning parameters a i and N i are related to N c [26] by the following equation: Now, rewriting Equation ( 23): where ξ i and L i ðkÞ: T and L i ðkÞ: ¼ T .The state prediction can be written as follows: where the vector ξ and the matrix ϕðhÞ T are given by ξ Similarly, the output is described as follows: 3.2.1.Solution without Constraints.The objective is to determine the parameter vector that minimizes the disparity between the predicted output and the desired set point signal.To achieve this objective, a cost function designed for this specific purpose is formulated.This cost function serves as a guiding metric to measure how closely the predicted output aligns with the set point signal.The cost function for this purpose is as follows: where Q L ≥ 0 and R L ≥ 0. Performing the partial derivative on J L , i.e., ∂J L =∂ξ ¼ 0, the optimal Laguerre coefficients vector in absence of constraints is found as follows: After calculating the optimal Laguerre coefficients vector ξ, the receding horizon control (RHC) law is realized as follows: 4 Mathematical Problems in Engineering where o T k ; k ¼ 1; 2; ⋯; h is a row vector with an appropriate dimension.
In linear state feedback control form, the control variable Δuðk i Þ: is written as follows: where the gain is given as follows: 3.2.2.Solution to Constrained LMPC.The optimization procedure is to minimize the cost function J L while ensuring Equations ( 33) and ( 34): where Δu min and Δu max represent the lower and upper boundaries for the change in control input ðΔuÞ: , while u min and u max define the lower and upper constraints for the control input itself ðuÞ: .Here, uðk i − 1Þ: denotes the control input at the previous time step.Additionally, o T m is a zero-row vector, and its dimension corresponds to that of L m ð0Þ T .Therefore, given these conditions and considerations, the constraints originally described in Equations ( 33) and (34) can now be redefined as follows: where M and γ compatible matrices in the QP problem [24].This compatibility signifies that they are structurally suitable for the specific requirements and constraints of the problem, ensuring a harmonious integration within the framework of the QP problem.
3.2.3.Stability.In this context, this paper is broadening the scope of stability analysis for the MPC algorithm, specifically when constraints are in effect and Laguerre polynomials are employed.To recap, in RHC, the control trajectory's first term, denoted as ΔuðkÞ: for time step k, is typically applied.However, this work considers the case where Laguerre polynomials are used, which extends our understanding of the MPC algorithm's stability, especially when constraints are activated.This allows us to explore the broader range of control actions over multiple time steps, Δuðk þ hÞ: ; h ¼ 0; 1; 2; ⋯; N p , rather than just the immediate ΔuðkÞ: action.This extension of stability analysis is crucial for a deeper understanding of how the MPC algorithm performs when constraints are in play, enabling more robust and effective control in various real-world applications.
The assumption made here is that as the prediction horizon becomes sufficiently long, the control trajectory Δuðk þ hÞ: ¼ LðhÞ T ξ tends to converge to zero.This convergence behavior is observed when the terminal state constraints are met, specifically when xðk þ N p Þ: ¼ 0.
Theorem 1 (Stability).Under Assumptions 1 and 2, the PMSM system described in Equation ( 10) exhibits asymptotic stability when operated in a closed-loop configuration.This is achieved through the implementation of a receding horizon controller denoted as ΔuðkÞ: , an objective function defined in Equation (34), and the constraints outlined in Equations ( 33) and (34).
Remark 1.In MPC algorithms that contain integrators, the prediction horizon also affects numeric condition [27].Specifically, for a large control horizon, the MPC algorithm becomes ill-conditioned.A proven technique to solve the problem is to use exponential weighted cost function.In contrast to the more common exponentially increasing weight [28,29], an exponentially decreasing weighting is suggested in this paper.The main purpose of focusing on exponentially decreasing weighting is to enhance the numerical stability of the class of MPC algorithms that contain integrators for multivariable systems.
The exponentially weighted cost function is as follows: Subject to the constraints: With state equation: Lemma 1 (Cost Function Equivalence).The solution of the exponentially weighted objective function (Equation ( 42)) subject to the inequality constraints Equation (44) and state equation constraints Equation (43) can be found by minimizing the following equation: Subject to the following equation: , where M σ is the matrix defined by the following equation: . This leads us to the equivalence of J (Equation (42)) and b J (Equation (45)).
Remark 2. It should be mentioned that the result Lemma 1 shows that an objective function with exponential data weight is similar to unweighted objective function with no weights if proper scaling factor σ is used to achieve a stable model.

Model Predictive Control with an Integral Action (MPC-IA).
Standard MPC algorithms usually do not have an integrator.This is because algorithms employing integral actions may not result in optimal operation.Among the many 6 Mathematical Problems in Engineering techniques to achieve optimality when integrator is in use, apply a simple MPC algorithm in increment form proposed in Di Ruscio's [21] study for our task of controlling speed of the PMSM system and compare with the proposed algorithm.

Simulation Parameters and Results
4.1.Parameters.For simulation purpose, the system parameter values given as follows have been used.Take p ¼ 2, The steady-state value for the q-axis current is i q0 ¼ 1 A.
The performance efficiency of the LMPC control scheme is validated using simulations on a PMSM system.First, begin by examining the responses of LMPC controller analyzed for load disturbance.Next, the stability is checked with those obtained from the optimal discrete linear quadratic regulator (DLQR) for large prediction horizon.Finally, the PMSM system under three control algorithms, i.e., SS-MPC, LMPC, and MPC-IA control, are compared by simulation.

Tuning the LMPC. Tuning using parameters in the weight matrix.
The cost function weights affect the closed-loop performance of the LMPC.The larger the weights, the faster the response.
The weight matrices in the cost function ( 28) are as follows: Q L ¼ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 q 1 0 0 0 0 0 q 2 where q 1 and q 2 will be used as parameters for tuning the controller.Use r w ¼ 0:1.Several simulations on PMSM system show that tuning is not advisable, as the responses are less sensitive to changes in the value of r w .
4.3.1.Tuning Using the Laguerre Parameters.Let us look at the effects of Laguerre parameters (a and N) on the control horizon N c , tuning process, and the overall computational burden of the LMPC algorithm.Using Equation ( 24), i.e., a ≈ e −N=N c for relating the control horizon (N c ) the Laguerre parameters a and N, consider the two cases: (1) Constant N: Choosing a constant parameter N ¼ 5 obtain Table 1.
Thus, the control horizon can be increased by manipulating the parameter a without changing the network order N.
(2) Constant control horizon: Assume the target control horizon is 20.Choose an integrator specifying the Laguerre network order N. Thereafter, Equation (24) to calculate the relevant scaling factor a as shown in Table 2 has been used.
In order to achieve a lower overshoot, the weight coefficients are selected as q 1 ¼ 1 and q 2 ¼ 0:01.In three d-axis current: i d , motor speed, d-axis voltage: v d , q-axis voltage: v q , d-axis incremental voltage: Δv d and q-axis incremental voltage: Δv q cases (N ¼ 4; N ¼ 6, and N ¼ 10), the LMPC tracks the reference speed very well (Figure 1).To check the computing resources required, times to compute the control law while varying the network order have been recorded.Table 2 shows that as N increases the, the computation time also increases.The experiments were carried out on a PC with Windows 10, 64 bits, Intel Core i7-7820X CPU with 3.60 GHz, 16 GB of RAM.Thus, to ensure a lower computation burden a Laguerre network of lower order N is chosen yet achieving the same control horizon.The computational effort is lower when smaller number of parameter is used.For N ¼ 4, only four Laguerre term is used to capture the control input while N ¼ 10 requires 10 Laguerre terms to capture the same control input.The other important thing to remember   Mathematical Problems in Engineering here is that the computational rate is slower than the sampling rate causing computational delay.This computational delay is related to using single MPC controller replacing the cascade control structure and can be solved using encoders [31,32].
To check whether the trajectories in Figure 1 are optimal, the LMPC is compared with optimal controller, i.e., a DLQR [33,34].This optimal controller minimizes the same cost function of Equation (36) particularly when N P → 1.Using the same parameters used for the LMPCs, the eigenvalues of DLQR are 0.7875 AE j0.1661, 0.9021 AE j0.1390, 0.8337 with feedback gain: K lqr ¼ 9:8924 0:1185 −0:1334 2:6522 −0:0125 0:0047 9:0901 2:9696 0:1210 0:2748 As it can be seen from Table 3 as N increases, the control trajectory is trying to converge to the optimal solution generated by using DLQR (for N as large as 10 the LMPC controller is not optimal).To achieve optimality, the exponential data weighting will be used in later sections.

4.4.
Response to Load Disturbances.Figure 2 shows a simulation of the torque disturbance in Equation (1).Choose q 1 ¼ 1 and q 2 ¼ 1 with a Laguerre order of N ¼ 6 to ensure a faster dynamic response.Table 3 shows a comparison of the performance indices on the three Laguerre orders.It is observed that the LMPC is able to reject the disturbance and continue to track the reference speed in 5 ms (Figure 2).4.5.Stability Analysis.In this section, an exponentially decreasing data weighting suggested in Lemma 1 in order to improve the numerical condition while ensuring the closed-loop stability of the LMPC algorithm in the presence of a large prediction horizon has been used.With Q L ⩾ 0; R L >0, and N p → 1, minimizing the cost function in Equation ( 53) is equivalent to solving the DLQR problem using the algebraic Riccati Equation (53).It is assumed that the pair ðA σ ; B σ Þ: is controllable and ðA σ ; DÞ: is observable with Q L ¼ D T D. The state feedback control gain for the stabilization b K is as follows: This makes the closed-loop system stable with all poles inside the unit circle and the closed-loop system being described by the following equation: From Equation (54), the modified system has all its eigenvalues inside the unit circle taking N p → 1. So: Thus, by choosing σ >1, it is possible to make stable.Several simulations on the PMSM system indicate that a choice of σ greater than unity stabilizes the system.To examine the effects of prediction horizon on numeric condition, compute the Hessian matrix, i.e., Π ¼ ∑ N p h¼1 ΘðhÞ: Q L ΘðhÞ T þ R L as the N p vary.Choose the Laguerre order of N ¼ 6, a control horizon of N c ¼ 20 and varying prediction horizon, i.e., N p 0 ¼ 25; N p 1 ¼ 50, and N p 2 ¼ 200.Table 4 shows the effect of scaling on Hessian matrix as the prediction horizon varies.Specifically, when weights are used on the objective function the conditioning number converges to small finite value.On the other hand, for the cost functions without weighting, the conditioning number becomes too big as N p     Mathematical Problems in Engineering inicreases, making the systems unstable and restricting our choose of N P .
In order to examine the stability of the LMPC controller, check if all the eigenvalues appear inside a unit circle on twoplane.To grant such stability, σ ¼ 1:2; N p ¼ 200 and N ¼ 10 are used.Figure 3 illustrates the eigenvalues of the LMPC (blue stars) coinciding with the DLQR (red circles) and all appearing inside unit circle.Figure 3 confirms that both the control gain matrices of LMPC (blue dots) and DLQR (red squares) are identical to each other with conditioning number 14.7619.4.6.Performance of LMPC as Compared to SSMPC and MPC-IA.This section focuses on the performance of LMPC by comparing to MPC-IA and SS-MPC when used to regulate PMSM speed ðω ε Þ: to reference signal ðω * ε Þ: .Here, the integral of absolute error (IAE) is introduced as performance metric in addition to scheduling time and overshoot 12 Mathematical Problems in Engineering to measure the performance of the controllers.It is defined as follows: Next, examine the response of the three control scheme to step load disturbance.Assume a control horizon of 15 and prediction horizon of 50 to be achieved.
As shown in Figure 4, all three control schemes regulate the speed of PMSM to the reference (83.8 rad/s), though MPC-IA and SS-MPC regulation has achieved at the cost of higher computational complexity, because the LMPC requires only seven parameters (N = 7) compared the minimum of 15 parameters in the MPC-IA and SS-MPC (the LMPC uses N = 7 with a ¼ 0:6271 achieving N c ¼ 15).The settling time and overshoot of the three controllers are the same.It can also be seen that when a load torque is applied, both the LMPC and SS-MPC remove the disturbance with shorter recovery time than MPC-IA (Figure 5).However, the LMPC controller removes the disturbance with a lower computational complexity, requiring only seven control parameters compared to 15 in the case of the SS-MPC.

Conclusion
In this paper, the design of Laguerre-based model predictive speed controller for the PMSM system has been extensively investigated.The paper has presented a solution to stability and speed regulation of PMSM systems.Also, exponential data weighing is used to decrease numerical issue, particularly with large prediction horizon.For stability analysis, LMPC has been compared to the optimal DLQR system, whereas for speed tracking LMPC has been compared to a popular statespace MPC and a simple MPC with an integrating action (MPC-IA).Therefore, the proposed controller significantly has better performance compared to MPC-IA controller.The developed LMPC method also exhibited improved steady-state characteristics by successfully eliminating the steady-state error and stability issues.The simulation results also showed that when reference and disturbance changes are encountered, the developed method operates with minimum settling time and overshoot.Therefore, the dynamic properties of the proposed control scheme are similar to SS-MPC controller but with less computational effort while providing more flexibility.By selecting appropriate values of a and N, LMPC reduces the number of parameters required for accurate prediction.Hence, the results obtained in this paper indicate that the control scheme performs very well and is considered feasible.
the zero matrices and I is identity matrices of appropriate dimensions.

15 FIGURE 2 :
FIGURE 2: Rejection of torque disturbance.d-axis current i d , motor speed, d-axis voltage: v d ; q-axis voltage: v q , d-axis incremental voltage: Δv d , and q-axis incremental voltage: Δv q .* Operating values.

FIGURE 4 :FIGURE 5 :
FIGURE 4: Disturbance rejection performance (a) current (i d ) in case of N c ¼ 15, (b) speed ω e in case of N c ¼ 15, (c) current (i d ) in case of N c ¼ 7, (d) speed ω e in case of N c ¼ 7.