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Abstract

The computational burden could be an issue of traditional model predictive control (MPC) in real-time applications. There are different procedures to cope with this problem such as using explicit MPC or approximation of the control inputs using orthogonal-basis functions. This paper presents the design procedure and experimental validation of a generalized discrete-time Malmquist orthogonal functions-based model predictive control (MbMPC) whose usage can decrease calculation concerns. To the authors’ knowledge, this is the first time the Malmquist functions are introduced into the MPC design. The discrete-time orthogonal Malmquist functions properties are used within this approach for approximating the future control input increments with a smaller number of parameters than in traditional MPC approaches. It is shown that the satisfactory controller and closed-loop system performances can be achieved using smaller number of tuning parameters. The performance of the proposed MbMPC is shown and compared with discrete-time Laguerre orthogonal functions based MPC (LbMPC) applied to a DC motor servo system. The simulation and experimental results demonstrate better IAE, ITAE, ISE and ITSE controller performance indices as well as faster average calculation time in comparison with the LbMPC approach. Additionally, the advantage of using the proposed method is emphasized by showing how the length of the prediction horizon affects the computational burden in comparison to the traditional MPC approaches.

Keywords

Malmquist orthogonal functions Laguerre orthogonal functions orthogonal polynomials discrete-time model predictive control DC motor servo system

Introduction

Model predictive control (MPC) belongs to a set of advanced control methods that use a process predictive model to generate future control inputs. MPC calculates the control signal trajectory by solving constrained optimization problem.^1–5 The main issue related to the traditional linear/nonlinear MPC applications is computational complexity for online solving of the optimization problem. Traditionally, the optimal MPC signal is obtained over the control horizon $N_{c}$ by employing the predicted output over the prediction horizon $N_{p}$ starting from the current system state. MPC problem formulation incorporates input and output constraints that also influence the calculation time of the optimal solution. Solving the optimal problem is a procedure that is repeated at each sequential time step, and the computational burden is greater using longer $N_{p}$ and $N_{c}$ . To reduce calculation burden, one can choose shorter control and prediction horizons, but this may affect the closed-loop system performances in terms of stability and feasibility. One way to cope with this problem is to use an explicit MPC approach, which does not calculate optimal control online.^6–8 This method utilizes pre-computed control law obtained offline as an “explicit” function of the system states and reference inputs. It is usually represented in a lookup table form and employs only basic arithmetic operations for the realization of the control algorithm.

Another approach to simplify linear MPC design and reduce the computational load is to optimize the future control trajectory, that is, the predicted trajectory of the control input increments, by employing a set of orthogonal polynomial functions. The use of the orthogonal functions within the MPC approach provides a long control and prediction horizons realization without working with a large number of parameters. Orthogonal basis functions have been first applied in the identification, parameterization and modeling of the dynamic systems.^9–16 The orthogonal properties of these functions are used for approximation of the discrete-time impulse response of dynamic systems. Obtaining the optimal values of the coefficients of the orthogonal polynomials are based on the least squares polynomial approximations. A similar idea is used for the controller design purposes.^17–20 The most frequently utilized orthogonal basis functions for MPC design are Laguerre functions.^21,22 The robust predictive controller design procedure using Laguerre functions is proposed in Ghabi and Dhouibi.²³ These functions are also implemented in linear SISO system design to lower the model complexity and reduce the optimization problem by the worst-case approach. In Rossiter et al.²⁴ and Rossiter and Wang²⁵ the authors have used Laguerre functions to achieve feasibility improvements to the standard MPC algorithm. In order to reduce the computational complexity and improve the sensitivity of MPC, Laguerre functions based MPC (LbMPC) with introduced exponential weights is proposed and applied to fully actuated-by-wire electrical vehicles in Zhang et al.²⁶ The exponential weighted LbMPC is also used for solving issues related to numerical ill-conditioning with an experimental demonstration of the method applied to the Twin Rotor MIMO system.²⁷

There are different methods for tuning MPC parameters to get optimal global solution. The usage of effective intelligent algorithms with fewer adjustable factors is presented in Mohamed et al.²⁸ Similar approaches have been already implemented in tuning PID controllers,²⁹ fuzzy controllers,³⁰ and self-adaptive state-space predictive functional PID control.³¹ In Essa et al.³² bat-inspired algorithm is proposed to find the best parameter values of traditional MPC by using two different performance indices including cross-correlation function and integral time absolute error. The adaptive MPC based on the improved gray wolf optimizer³³ with Kalman filter³⁴ is suggested for control of autonomous vehicles considering system vision uncertainty. The performance index, which depends on the response maximum overshoot, steady-state error, and settling time is minimized to give the optimal values of a sampling time, prediction and control horizons, as well as weighting factors of MPC cost function. In order to optimally tune LbMPC parameters, authors propose the implementation of a new intelligent algorithm named Dandelion Optimizer³⁵ that provides fast convergence rate with the same performance index as in.³³ Other applications of the LbMPC can be found.^36–43

This paper presents the implementation of Malmquist orthogonal functions in the design of MPC. To the authors knowledge, these functions have not been used earlier for the predictive controller design. The initial idea comes from their usage in the identification of dynamic systems.^44–48 The Malmquist orthogonal systems and connection between generalized Malmquist functions and Müntz polynomials are presented in Heuberger et al⁴⁵ and Milovanović et al.⁴⁶ The procedure for mapping poles to zeros and vice versa is performed by using generalized bilinear transformations.⁴⁷ The design procedures of the orthogonal Malmquist filters are presented in continuous- and discrete-time domains.^49–51 The obtained discrete-time Malmquist network is utilized in this paper for developing discrete-time Malmquist functions. Their orthogonal properties represent the basis for developing Malmquist functions based MPC (MbMPC). This class of functions employment has the advantage of having only a few tuning parameters that are independent of the sampling time $T$ , which makes the closed-loop tuning process easier. The smaller number of parameter $N$ means a smaller number of decision variables, which affects the computational burden. By using a smaller number of parameters than the traditional MPC algorithms, the long control horizon can be achieved without using large matrices for online calculations, whose size depends on the length of the control and prediction horizons. The omission of the huge system matrices also does not require large computer memory for its storing.⁵² Another advantage is in yielding better controller tuning performances thanks to the possibility of choosing different rational poles compared to Laguerre functions, where the poles are identical. The parameterized approach, where the predictive control problem is reformulated, provides a simplified solution which is another benefit of the proposed MbMPC approach.

In the sections that follow, the derivation procedure of the Malmquist network shown in Danković et al.⁴⁹ is recalled first, and the new method of the MbMPC design is presented afterward. The simulation and experimental results obtained by applying the proposed MbMPC in DC motor servo system control are also given in the sequel. The main contributions of this paper are:

(1) introducing the novel MbMPC method for the predictive controller design using discrete Malmquist functions,

(2) this parametarized design method enables realization of longer horizons with small number of parameters without affecting the computational burden,

(3) the results emphasis the reduced computational time in comparison with time needed in the traditional MPC approach,

(4) this approach provides better fine-tuning of system performances thanks to non-identical Malmquist orthogonal functions parameters,

(5) the effectiveness of the realized approach is validated on the DC motor servo system using IAE, ITAE, ISE, and ITSE performance indices,

(6) the controller performance of the proposed algorithm is compared with performances of LbMPC approach,

(7) the constrained optimization problem is solved by Hildreth’s QP procedure, which also reduces online calculation time.

Discrete-time Malmquist functions

To obtain discrete-time Malmquist functions, the following rational functions sequence is introduced in Danković et al.⁴⁹

W_{N} (s) = \frac{\prod_{k = 0}^{N - 1} (s - a_{k}^{*})}{\prod_{k = 0}^{N} (s - a_{k})}, N = 1, 2, 3, \dots

(1)

The symmetric transformations $a_{k}^{*} = f (a_{k})$ and $a_{k} = f (a_{k}^{*})$ are used for mapping the zeroes and poles as $a_{k}^{*} = \frac{1}{a_{k}}$ , $a_{k} = \frac{1}{a_{k}^{*}}$ , respectively. Using the given transformations and their substitution to (1) yields

\bar{W_{M} (s)} = \frac{\prod_{k = 0}^{M - 1} (s - \frac{1}{a_{k}})}{\prod_{k = 0}^{M} (s - \frac{1}{a_{k}^{*}})}, M = 1, 2, 3, \dots

(2)

The generalized Malmquist functions, represented by (1) and (2) are orthogonal concerning the inner product defined in⁴⁹

J_{N, M} = (W_{N} (s), \bar{W_{M} (s)}) = \frac{1}{2 π i} \oint_{C_{p}} W_{N} (s) \bar{W_{M} (s)} d s

(3)

where the contour $C_{p}$ comprises all the poles of $W_{N} (s)$ . The Malmquist polynomials can be derived from the orthogonal sequence, determined by $W_{N} (s)$ and $\bar{W_{M} (s)}$ , in the following way

P_{N} (x) = \frac{1}{2 π i} \oint_{C_{p}} W_{N} (s) x^{s} ds

(4)

that yields

P_{N} (x) = \sum_{k = 0}^{N} A_{N, k} x^{a_{k}}

(5)

where

A_{N, k} = \frac{Π_{j = 0}^{N - 1} (a_{k} - \frac{1}{\bar{a_{k}}})}{Π_{j = 0, j \neq k}^{M} (a_{k} - a_{j})}, k = 1, 2, . . . N .

(6)

The discrete-time orthogonal Malmquist functions are obtained by using transformation $S = \frac{\ln (z)}{T}$ . These functions are orthogonal in the complex domain as well, whereas the outputs in the discrete-time domain are orthogonal in the classical sense. The discrete Malmquist network can be realized based on the modified scheme by using the following transfer function

\begin{matrix} W_{N} (z) = \frac{z}{z - a_{1}} Π_{k = 2}^{N} \frac{z - a_{k - 1}^{*}}{z - a_{k}}; a_{k - 1}^{*} = \frac{1}{a_{k}}, \\ a_{k} \in R, a_{k} = [0, 1] \end{matrix}

(7)

as depicted in Figure 1.

Figure 1.

Discrete Malmquist network.

The inverse z-transform of the given Malmquist network does not yield a straightforward calculation of the Malmquist functions in the discrete-time domain. A more practical methodology for determining these functions entails employing a state-space representation of the networks thanks to their cascade form (Figure 1). Accordingly, the set of discrete-time Malmquist functions $φ_{k}^{1}, φ_{k}^{2}, \dots, φ_{k}^{N}$ can be obtained as

Φ_{k + 1} = A_{Φ} Φ_{k}

(8)

where the matrix $A_{Φ}$ is a square matrix of $N^{th}$ order, and $Φ_{k} = [φ_{k}^{1} φ_{k}^{2} \dots φ_{k}^{N}]^{T}$ is a vector of discrete-time Malmquist functions. The initial conditions are defined by a vector $Φ_{0} = [1 1 \dots 1]^{T}$ of dimension N, and the matrix $A_{Φ}$ is formed in the following manner

A_{Φ} = [\begin{matrix} a_{1} & 0 & 0 & \dots & 0 \\ a_{1} - \frac{1}{a_{1}} & a_{2} & 0 & \dots & 0 \\ a_{1} - \frac{1}{a_{1}} & a_{2} - \frac{1}{a_{2}} & a_{3} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & 0 \\ a_{1} - \frac{1}{a_{1}} & a_{2} - \frac{1}{a_{2}} & \dots & a_{N - 1} - \frac{1}{a_{N - 1}} & a_{N} \end{matrix}]

(8)

Detailed explanations and proof of the orthogonality of the Malmquist functions can be found in Danković et al.⁴⁹ The next section provides the design procedure of MbMPC.

Predictive controller design

To design a predictive controller, the basic idea proposed herein is to approximate future control input trajectories by using the Malmquist functions and to optimize them by minimizing the cost function subject to the constraints. To accomplish that, a control input trajectory should have the form of an impulse signal, and that is why the augmented process model with embedded integrator has to be derived resulting in the control input increment $Δ u$ as the decision variable. As $Δ u$ is realized by Malmquist functions, the optimization problem is deduced then to the calculation of the Malmquist coefficients.

Plant model with embedded integrator

To design the MbMPC, the integral action is introduced into the linear discrete-time state-space SISO plant model described by

\begin{matrix} x_{k + 1} = A x_{k} + B u_{k} + E w_{k}, \\ y_{k} = C x_{k}, \end{matrix}

(9)

where $x \in R^{n_{x}}$ is the system state, $u \in R$ is the control input signal, and $w \in R^{n_{x}}$ is the disturbance. This results in the augmented state-space model

\begin{matrix} [\begin{matrix} Δ x_{k + 1} \\ y_{k} \end{matrix}] = [\begin{matrix} A & 0_{n_{x}}^{T} \\ CA & 1 \end{matrix}] [\begin{matrix} Δ x_{k} \\ y_{k} \end{matrix}] + [\begin{matrix} B \\ CB \end{matrix}] Δ u_{k} + w_{k}, \\ y_{k} = [\begin{matrix} 0_{n_{x}} & 1 \end{matrix}] [\begin{matrix} Δ x_{k} \\ y_{k} \end{matrix}] \end{matrix}

(10)

where $Δ x_{k + 1} = x_{k + 1} - x_{k}$ and $Δ u_{k} = u_{k} - u_{k - 1}$ are the state and control input increments, respectively. It is assumed that state $x_{k}$ and output $y_{k}$ are both measurable in each sample $k$ . The compact representation of (10) that will be used for the MPC design is

\begin{matrix} x_{k + 1} = A x_{k} + B u_{k} + w_{k}, \\ y_{k} = C x_{k}, \end{matrix}

(11)

where $x_{k + 1} = [\begin{matrix} Δ x_{k + 1} \\ y_{k} \end{matrix}]$ , $A = [\begin{matrix} A & 0_{n_{x}}^{T} \\ CA & 1 \end{matrix}]$ , $B = [\begin{matrix} B \\ CB \end{matrix}]$ , $C = [\begin{matrix} 0_{n_{x}} & 1 \end{matrix}]$ , and $w_{k} = [\begin{matrix} I \\ 0 \end{matrix}] w_{k}$ . It is assumed that the pair $(A, B)$ is controllable.

Unconstrained MbMPC design

The proposed MbMPC method is based on the approximation of the control input increments $Δ u_{k + i}, i = 1, N$ , using a set of discrete-time Malmquist functions $φ_{k}^{1}, φ_{k}^{2}, \dots, φ_{k}^{N}$ . The future control input increment trajectory is approximated by

Δ u_{k + i} = \sum_{j = 1}^{N} c_{k}^{j} φ_{i}^{j}

(12)

where $τ_{k}$ is the initial sample instant, $i$ represents the future sampling instant, $N$ is the number of Malmquist network terms, and $c_{k}^{j}$ are Malmquist coefficients. The control input increment $Δ u_{k + i}$ can also be written in the following form

Δ u_{k + i} = Φ_{i}^{T} τ_{k}

(13)

where $τ_{k} = [c_{k}^{1} c_{k}^{2} \dots c_{k}^{N}]$ is a vector of Malmquist coefficients that has to be optimized and calculated. In order to obtain optimal values of these coefficients, the following cost function

J = \sum_{m = 1}^{N_{p}} {(r_{k + m} - y_{k + m})}^{T} (r_{k + m} - y_{k + m}) + τ_{k}^{T} R_{M} τ_{k}

(14)

is minimized with respect to $τ_{k}$ , where $R_{M} = diag {r_{w}}_{NxN}, r_{w} \geq 0$ is the weighting matrix, $r$ and $y$ are reference and output signals, respectively. Taking into account the future state and output trajectories

x_{k + j} = A^{j} x_{k} + \sum_{i = 0}^{j - 1} A^{j - i - 1} B Φ_{i}^{T} τ_{k}

(15)

y_{k + j} = C^{j} x_{k} + \sum_{i = 0}^{j - 1} C A^{j - i - 1} B Φ_{i}^{T} τ_{k}

(16)

the cost function (14) is reformulated as

J = \sum_{j = 1}^{Np} x_{k + j}^{T} Q x_{k + j} + τ_{k}^{T} R_{M} τ_{k}

(17)

where $Q = C^{T} C, Q \geq 0$ and $R_{M} \geq 0$ . The minimization of (17) over $τ_{k}$ , gives

τ_{k} = - {(\sum_{j = 1}^{N_{p}} γ_{I} Q γ_{i}^{T} + R_{M})}^{- 1} (\sum_{j = 1}^{N_{p}} γ_{I} Q γ_{i} Q A^{j}) x_{k}

(18)

where $γ_{i} = \sum_{i = 0}^{j - 1} C A^{j - i - 1} B Φ_{i}^{T}$ . This form represents the unconstrained solution of the MbMPC problem. In the next section, the constrained MbMPC problem will be defined and the design procedure will be discussed.

Constrained MbMPC design

In this case, the optimization problem is to find the optimal values of the Malmquist coefficients vector $τ_{k}$ by minimizing the cost function (17) with accounted constraints to obtain the control input increment $Δ u_{k}$ according to (13). The set of constraints on the control input increment and amplitude for the MbMPC problem can be defined by

Ψ τ_{k} \leq Ξ

(19)

where

Ψ = [\begin{matrix} Φ_{i}^{T} \\ - Φ_{i}^{T} \\ \sum_{j = 0}^{i - 1} Φ_{j}^{T} \\ \sum_{j = 0}^{i - 1} Φ_{j}^{T} \end{matrix}], Ξ = [\begin{matrix} Δ u^{\max} \\ - Δ u^{\min} \\ u^{\max} - u_{k - 1} \\ - u^{\min} + u_{k - 1} \end{matrix}] .

(20)

The defined constraints are linear in the decision variables providing the convexity of the cost function. This type of optimization can be handled by many optimization routines. Herein is used the Hildreth’s QP procedure.⁵² This procedure first identifies active constraints and then introduce them in calculation of optimal decision variable $τ_{k}$ .

Tuning procedure of the proposed MbMPC

Based on the known approach,⁵² the proposed MbMPC parameters may be tuned as follows.

Set the weighting matrices $Q = C^{T} C$ and $R_{M}$ first. The matrix $Q$ , when the augmented model is used, does not have weights on the state $Δ x$ , and does have the identity matrix as weights to the error between the reference signal and the system output. The weight matrix $R_{M}$ affects the system response speed, where small values correspond to faster system response.

Select values for $N$ and $a_{i}, (i = 1, N)$ then, to fine tune the controller to get satisfactory closed-loop performance with fixed $Q$ and $R_{M}$ matrices. The values of $a_{i}$ have no significant impact to system responses with greater values of $N$ , but with a smaller $N$ , the change of parameters $a_{i}$ has more effects on the closed-loop response.

Finally, the value of parameter $N_{p}$ has to be chosen sufficiently large to provide decaying of the Malmquist functions to a very small value, that is, that $Δ u_{k + N_{p}} \approx 0$ .

Simulation and experimental results

The proposed MbMPC method is verified by using the modular DC motor servo system,⁵³ shown in Figure 2. The system’s components include a tachogenerator, a DC motor, an encoder, and an inertia load of 2 kg connected to the motor. This modular experimental setup enables real-time design and implementation of advanced control algorithms. It is connected to MATLAB/Simulink via a specific RT-DAC4/USB board, which transfers measured signals from the tachogenerator and encoder to the controller, as well as control signals to the power interface unit. The scaled input voltage is used to control the DC motor via a PWM signal, and the goal of our work is to control the angular position of the DC motor shaft.

Figure 2.

DC motor servo system setup.

This system is nonlinear, and the nonlinearity is expressed by Coulomb’s friction, described as $w = F_{c} sign (x_{2})$ , which represents unmodeled disturbance. The control signal is also affected by the dead-zone in the range of $- 0.15$ to $+ 0.15$ . DC motor servo system parameters are identified using a toolbox provided by Inteco real-time software which is also run in MATLAB/Simulink. By neglecting the saturation, and the mentioned friction, the linear transfer function is obtained as

G (s) = \frac{K_{s}}{s (T_{s} s + 1)}

(21)

where $K_{s}$ and $T_{s}$ are a gain and a time constant of the servo system respectively. As the DC motor is PWM controlled and the input is scaled, the input voltage is $U (t) = V (t) / V_{\max}, - 1 \leq U (t) \leq 1, V_{\max} = 12$ V. The state-space model of the DC motor servo system is obtained from (21) by choosing the state variables $x_{1} = θ$ measured in [ $rad$ ], and $x_{2} = \overset{\cdot}{θ} = ω$ measured in [ $rad / s$ ], and the following differential equations are determined as

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{X}}_{2} = - \frac{1}{T_{s}} x_{2} + \frac{K_{s}}{T_{s}} u \end{matrix}

(22)

where $T_{s} = 0.9 s$ , $K_{s} = 184.73$ . The sampling period is chosen as $T = 0.01 s$ in all simulations and experiments. Using this sample period and the zero-order hold circuit, the linear discrete-time state-space model of DC motor servo system is obtained in the following form

\begin{matrix} x_{k + 1} = [\begin{matrix} 1 & 0.0099 \\ 0 & 0.9890 \end{matrix}] x_{k} + [\begin{matrix} 0.0102 \\ 2.0412 \end{matrix}] u_{k}, \\ y_{k} = [1 0] x_{k} . \end{matrix}

(23)

For design purposes of MbMPC, the following parameters are used following the tuning procedure given in the previous section. The weighting matrices are $Q = C^{T} C$ and $R_{M} = 100$ . The number of terms of the Malmquist network, which present the number of decision variables, is $N = 4$ , and the Malmquist functions parameters are $a_{1} = 1 / 2, a_{2} = 1 / 3, a_{3} = 1 / 4$ , and $a_{4} = 1 / 5$ . To provide enough time for decaying of the Malmquist functions, the prediction horizon is set to $N_{p} = 100$ . These parameters are selected and used in digital simulations of the proposed controller subject to the following constraints on the control input signal and its increment

\begin{matrix} | u_{k} | \leq 1 \\ | Δ u_{k} | \leq 0.5 . \end{matrix}

(24)

The control signal constraints are defined by the system requirements, and we defined the constraints on control increment by our choice. The output signal is illustrated in Figure 3, representing the angular position of the DC motor servo system shaft. The system response to the reference signal is satisfactory, as the output signal effectively follows the reference signal. Figures 4 and 5 depict the control input increment and amplitude responses, respectively, showing that both signals satisfy the constraints (24).

Figure 3.

The angular position θ for proposed MbMPC (simulation results).

Figure 4.

MbMPC input increment $Δ u$ (simulation results).

Figure 5.

MbMPC input $u$ (simulation results).

Two sets of experiments are conducted, where the implementation of novel MbMPC is compared with the usage of LbMPC²² in DC motor servo system design. In the first set, the number of terms for both control methods (MbMPC and LbMPC) is $N = 4$ , and the prediction horizon is $N_{p} = 100$ . The parameter of the LbMPC is $a = 1 / 2$ , and parameters of the MbMPC are the same ones used in digital simulation: $a_{1} = 1 / 2, a_{2} = 1 / 3, a_{3} = 1 / 4$ , and $a_{4} = 1 / 5$ . Figure 6 shows that the proposed MbMPC gives a slightly better response of the controlled system (dashed black line for MbMPC, and dashed green line for the LbMPC).

Figure 6.

The angular position $θ$ for applied control methods (MbMPC, $N = 4$ – dashed black line; LbMPC, $N = 4$ – dashed green line; LbMPC, $N = 3$ – dashed blue line; MbMPC, $N = 3$ – solid black line).

For the second set of experiments, the number of terms is chosen to be $N = 3$ for both control approaches to additionally reduce the size of the optimal problem. The parameter $a = 1 / 2$ for LbMPC stays unchanged, whereas the following values of the MbMPC parameters are selected as $a_{1} = 1 / 2, a_{2} = 1 / 4$ , and $a_{3} = 1 / 8$ in order to get faster system response. Figure 6 also illustrates that the system response for LbMPC (dashed blue line) is even slower than both responses from the previous experimental set, whereas MbMPC, with only three terms of the Malmquist network used, gives the best results in all presented experiments. These tuning parameter values of the MbMPC enable better tuning of the system performances attaining satisfactory system responses presented in Figures 6 –8 (solid black line).

Figure 7.

The control input increment $Δ u$ of applied methods (MbMPC, $N = 4$ – dashed black line; LbMPC, $N = 4$ – dashed green line; LbMPC, $N = 3$ – dashed blue line; MbMPC, $N = 3$ – solid black line).

Figure 8.

The control input $u$ of applied methods (MbMPC, $N = 4$ – dashed black line; LbMPC, $N = 4$ – dashed green line; LbMPC, $N = 3$ – dashed blue line; MbMPC, $N = 3$ – solid black line).

In order to show the improvement in reducing the calculation burden, the execution time of the online optimization routine has been measured for LbMPC and MbMPC approaches. The results obtained for $N = 3$ demonstrate that the proposed MbMPC has a lower average computational time of $1.3461 \times 10^{- 3} s$ compared to $2.5845 \times 10^{- 3} s$ of LbMPC needed to reach the optimal solution.

How prediction horizon $N_{p}$ alters the computational load of traditional MPC and MbMPC, is shown by the experiments performed with $N_{p}$ ranging from 10 to 150 steps for MbMPC and from 10 to 50 steps for the traditional MPC approach. The results are depicted in Figure 9. It is obvious that using a large number for $N_{p}$ in the traditional MPC is time-consuming due to huge matrices determined by control and prediction horizons, even for this second-order SISO system.

Figure 9.

The average computational time (MbMPC, $N = 3, a_{1} = 1 / 2, a_{2} = 1 / 4, a_{3} = 1 / 8$ – solid blue line; traditional MPC, $N_{c} = N_{p}$ – doted black line).

Table 1 presents the closed-loop eigenvalues of the DC motor servo system with both controllers, for the different values of MbMPC and LbMPC parameters. It shows how different and non-identical controller $a_{i}$ parameter values of the MbMPC compared to the identical ones of LbMPC affect the dynamics of the closed-loop system response.

Table 1.

Closed-loop eigenvalues of MbMPC and LbMPC.

MbMPC parameters	LbMPC parameters	Closed-loop eigenvalues
MbMPC parameters	LbMPC parameters	MbMPC	LbMPC
$a_{1} = 1 / 2, a_{2} = 1 / 3$	$N = 4$ ,	$0.8687 + 0.1430 i,$	$0.8689 + 0.1072 i,$
$a_{3} = 1 / 4, a_{4} = 1 / 5$	$a = 1 / 2$ ,	$0.8687 - 0.1430 i,$	$0.8689 - 0.1072 i,$
		$0.9696$	$0.9708$
$a_{1} = 1 / 2, a_{2} = 1 / 4,$	$N = 3$	$0.9401 + 0.0687 i,$	$0.8711 + 0.1024 i,$
$a_{3} = 1 / 8$	$a = 1 / 2$	$0.9401 - 0.0687 i,$	$0.8711 - 0.1024 i,$
		$0.9619$	$0.9705$

To examine the performance of the proposed MbMPC, several performance indices for simulation and experimental environment are calculated. These results are compared with the obtained LbMPC performance indices and summary is given in Table 2. The best performance indices are highlighted in black. It can be seen that simulation results for MbMPC when $N = 3$ and $a_{1} = 1 / 2$ , $a_{2} = 1 / 4$ , $a_{3} = 1 / 8$ gives the best performance as expected. Also, comparing the experimental results, it can be clearly observed that the MbMPC performance indices (highlighted in green) are better than experimental results of the LbMPC performance (highlighted in red).

Table 2.

Controller performance indices.

MbMPC						LbMPC
Case	N	a₁–a_N	IAE	ITAE	ISE	ITSE	a	IAE	ITAE	ISE	ITSE
Simulation	4	a ₁ = 1/2, a₂ = 1/3, a₃ = 1/4, a₄ = 1/5	43.15	134.1	1209	3607	a = 1/2	42.82	143.5	1209	3909
Simulation	3	a ₁ = 1/2, a₂ = 1/4, a₃ = 1/8	41.68	128.4	1199	3573	a = 1/2	46.03	155.1	1315	4279
Experiment	4	a ₁ = 1/2, a₂ = 1/3, a₃ = 1/4, a₄ = 1/5	51.37	163.1	1495	4569	a = 1/2	51.21	175	1495	4941
Experiment	3	a ₁ = 1/2, a₂ = 1/4, a₃ = 1/8	50.42	159.2	1489	4548	a = 1/2	53.91	185.2	1581	5247

Conclusion

This paper describes the use of discrete-time Malmquist orthogonal functions, generated from the discrete-time Malmquist network, in the design of model predictive control (MPC). This results in a novel Malmquist functions based MPC (MbMPC). Thanks to the orthogonal properties of these functions, it is possible to approximate future control input increments over the prediction horizon. MbMPC uses a smaller number of terms for control input increment calculations, which directly affects calculation time which is critical for obtaining optimal solution of the constrained traditional MPC problem. It also improves the fine-tuning of system performances compared to Laguerre functions based MPC (LbMPC) due to different and non-identical set of controller parameters $a_{i}$ in comparison with the only one parameter $a$ in LbMPC. The proposed MbMPC is applied to the experimental DC motor servo system setup subject to its real disturbances and control input constraints. Controller performance indices of the proposed MbMPC are better than ones of LbMPC. The comparative analysis of average computational time of traditional and proposed MPC algorithms demonstrates much better results of MbMPC.

The future work would be oriented to:

introducing some controller parameters tuning methods, such as the ones mentioned in the introductory section,

examination of the numerical stability of the algorithms such as developing MbMPC with a prescribed degree of closed-loop stability especially for a high number of states and MIMO systems,

examination of limitations on practical implementation in the scope of robust issues of the proposed MbMPC. To cope with this problem, the use of robust MPC, such as tube MPC with auxiliary sliding mode controller would be considered,⁵⁴

examination of the tube MPC methodology, where traditional nominal MPC is replaced by MbMPC as well as other robust MPC approaches.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia [grant number 451-03-66/2024-03/200102].

ORCID iDs

Miodrag Spasić

Darko Mitić

Dragan Antić

Nikola Danković

Saša S Nikolić

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Generalized Malmquist orthogonal functions based model predictive control

Abstract

Keywords

Introduction

Discrete-time Malmquist functions

Predictive controller design

Plant model with embedded integrator

Unconstrained MbMPC design

Constrained MbMPC design

Tuning procedure of the proposed MbMPC

Simulation and experimental results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Data availability statement

References