Parametric Robust Generalized Minimum Variance Control

This work presents a hybridization of the Generalized Minimum Variance control (GMVC) with Parametric Robust Control (RPC) for Linear Time Invariant (LTI) Uncertain Systems. A linear parametric uncertainties model is used to describe the system dynamic behavior. The controller synthesis is based on PID controller with a low-pass filter and formulated as a convex optimization problem which takes in account the desired closed-loop performance and the uncertainties on the model parameters. Then, the robust controller parameters, that represent the best solution for the all possible models and assign the closed-loop poles within the desired region in the s-plane, are transferred to GMVC by Tustin’s method, resulting in a parametric robust generalized minimum variance (PRGMV) controller, which was compared to two other approaches, carrying out several simulation essays in Matlab environment. In order to emphasize its performances and its efficiency, the performance index and sensitivity analysis were assessed. The results confirmed that the proposed controller ensures better robustness and performance for reference tracking and disturbance rejection.


I. INTRODUCTION
I N 1997 it was established that high order controllers were acutely sensitive to plants parameter perturbations, leading to attention to the synthesis and design of low order controllers, such as Proportional-Integral-Derivative (PID) [1], [2]. Indeed, in the last years, owing to their clear functionality and ease of use and tuning, the majority of industries still has employed such controllers, although many control practitioners pointed out that a PID tuned via conventional approach was not robust. From there, arose the need of robust control techniques for industrial applications employing PID with modifications in its original structure in order to guarantee the robustness of the control loop [3].
At the first sight, such techniques proved to be better than the PID in its original form, but subsequently it was observed that these controllers became more complex in an uncertain environment [1], [4]. This made the researchers feel that there was the need to combine the simplicity of this controller with advanced control techniques [1]. In this context, GMVC (Generalized Minimum Variance Control) became attractive due to its properties to put different closedloop characteristics under consideration into its optimization problem [5], [6].
Proposed by [7], the GMVC uses a generalized output that is applied to a wider class of plants, being one of the simplest member of the Stochastic Control, differing to the most common Model-based Predictive Control (MPC) for being based on ARMAX (Auto-Regressive Moving Average with eXternal inputs) models while MPC on ARX (Auto-Regressive with eXternal inputs), i.e.., the GMVC considers a full description of system's deterministic and stochastic parts [8]. This important features enables the controller to distinguish between noise and uncertainty from deterministic cause and effect, in addition, the controlled plant presents a more stable behavior with less sensitivity to measurement and modelling uncertainty noises [9].
In this context, several works about GMVC have been published. For example, we have the work of [10], that presented a review of the controller design not only to eliminate the steady state offset error but also to stabilize a nonlinear monovariable system. Similarly, in [11] it was proposed a study on self-tuning control strategies with GMVC in a fixed two degree of freedom structure. This way, a considerable number of successful design studies have been carried out to hybridize PID and GMVC over the last years, like the one developed in [12] and [13].
Other applications can be seen in [14], in which presented a hybrid GMVC design with PID tuning, whose excessive actuator movements could be avoided and non-minimum phase plants were controlled, and in [15], that presented a new self-tuning PID controller design, in which the parameters were calculated online based on the relationship between the PID and GMVC laws by considering the pole assignment in the control system. Latest applications employing GMVC control were presented in [16], [17] and [18].
In control system design the primary requirement is the asymptotic stability of controlled systems and this can be guaranteed through the design of an appropriate controller based on the process nominal model. However, in reality there may be several sources of uncertainties which make the nominal model inaccurate, leading to controller performance degradation [19]. This limitation is a drawback of designing GMVC based on the process nominal model.
Robust control deals explicitly with system uncertainties and guarantees the controlled system performance, primarily asymptotic stability, when there are uncertainties [19], [20]. This design method has been studied extensively in [21], [22] and [23]. In addition, recent studies developed in [24], [25], [26], [27] and [28] addressed an outstanding contribution for the current state-of-the-art on the study of parametric uncertainties. These works focused on robust controller synthesis and analysis for converters by using Parametric Robust Control (PRC), whose controller's parameters were obtained by solving a linear programming problem.
The work proposed in [29] presents a design of a fixed order controller for an uncertain system that guarantees closed loop stability and a suitably adequate level of performance under a stated restricted frequency range. Then, an LMI condition is given to ensure the robust stability of the closed-loop system in the presence of the bounded parametric uncertainties.
Given the state-of-the-art in GMVC and PRC, it is possible to observe that lack a study on the hybridization of these two cited areas for parametric uncertainty plants. Furthermore, most studies in GMVC essentially still focus on inflicting a scalar λ-tuning weight over the control signal, such as in the positional case, with Q(z −1 ) = λ, as was made in [30], and the incremental case for inclusion of a polynomial weight, in [11]. In this cases, the calibration of λ was mostly related to energy constraints regarding control effort in order to have good disturbance rejection and reference tracking, but without any consideration being made on how this parameter would affect the relative stability of the control loop [11], [12].
In this context, this paper presents a hybridization of the GMVC with PRC, resulting in a parametric robust predictive control law which utilizes the predictive nature of the GMVC for parametric uncertainty plant. The proposed design is based on PID controller with a low-pass filter, whose parameters are obtained by means of the robust pole placement technique and solved by formulating a convex optimization problem integrated to the Chebyshev theorem, which incorporates additional constraints on the system and desired performance, allowing the designer to find the controller parameters that place closed-loop poles within desired intervals. PID gains are transferred to GMVC by Tustin's method, resulting in a parametric robust generalized minimum variance (PRGMV) controller.
The PRGMV is compared to two other controllers, one proposed in [22] and another in [31]. It is important to highlight that the PID gains and the robust stability analysis are obtained in the continuous frequency domain while the sensitivity and robustness analysis of the PRGMV are realized in the discrete frequency domain, whose results confirm that the PRGMV outperforms the other controllers. Hence, the design method proposed uses both continuous and discrete domains approaches, being one of the contributions in this paper.
The major contribution of this work lies in the nonexistence of a methodology directed to the hybridization of the stochastic control theory and parametric robust control for parametric uncertainty plants. Furthermore, the PRGMV provides a non-scalar Q(z −1 ), which filters the control signal and reduces its variance, avoiding the need to tune this parameter based on trial and error, and still ensures better robustness and performance for reference tracking and disturbance rejection, allowing to figure out the closed-loop performance in terms of robustness analysis using sensitivity functions by means of magnitude plots in the frequency domain. From those functions, important indices can be obtained to quantify the trade-off between robustness and performance to guarantee a suitable well tuned controller.
In addition, a Parametric Robust Control approach, of the type proposed in this paper, represents a competitive option when dealing with some safety critical plants, such as biomedical and electric power system. In such systems, it is not recommendable, for safety reasons, to use adaptive control approaches demanding the injection of disturbance test signals into the plant, a fact that may not be allowed in such safety critical applications.
Therefore, beyond this introductory section, this paper is organized as follows: in Section 2 it is presented the Parametric Robust control (PRC) background, addressing the stability analysis, the Kharitonov theorem and the interval pole placement design; in Section 3 it is shown the Incremental GMVC Control; the synthesis of the PRGMV is developed in Section 4, followed by its robustness analysis in terms of sensitivity functions. In order to illustrate the effectiveness of the proposed controller, simulation results are given in Section 5. Finally, conclusions and prospects are given in Section 6.

II. PARAMETRIC ROBUST CONTROL (PRC)
Much of modern control theory addresses problems involving mathematical model uncertainty. When it involves various physical parameters specified within given bounds, it is called parametric or structured uncertainty, but when uncertainty concerns model structure or incorrect modelling, it is called non parametric or unstructured uncertainty [20].
Theses uncertainties occur owing to real systems cannot be accurately represented by a mathematical model, which compromises the system stability [19], [32]. In this case, it is of fundamental importance in control systems that the model uncertainty be considered, so that the asymptotic stability of the controlled systems can be guaranteed.
Parametric Robust Control deals explicitly with uncertainties clarified above, providing satisfactory performance in a closed-loop system for all uncertainties of the process and bounded disturbances. As regards stability, the robust control system should ensure closed-loop stability as under nominal conditions and as under uncertainties in the model [33], [34]. It is highlighted that this paper focuses on robust controllers design for systems with parametric uncertainties.

A. PROBLEM STATEMENT
A Single Input Single Output (SISO) system, in the continuous frequency domain, can be represented by a transfer function G(s). The basic structure for uncertain systems is generally described by uncertain polynomials N (s, n) and D(s, d), restricted within pre-specified closed real intervals [35], [36]. Thus, the transfer function of the uncertain system is given by: where n i and d i are bounded by intervals n − i , n + i and d − i , d + i , respectively. Such a set of polynomials is called an interval polynomial family [37]. Then, once the minimum and maximum values of the various coefficients are assigned, the parametric uncertainty is completely characterized by letting n i and d i lie in a box defined by So, G(s) can be represented by

B. ROBUST POLE ASSIGNMENT
Consider the closed-loop system as depicted in Fig. 1.The controller C(s) is given by: In the case of a low-order controller, we have r ≤ t. By using (3) and (4) the closed-loop transfer function will be given by Let us now introduce a desired characteristic polynomial δ(s) of degree l + t which is Hurwitz and which has the desired set of closed-loop characteristic roots: In order to attain the desired characteristic polynomial by choosing the controller, it is necessary and sufficient to solve a set of liner equations [21], [38], but it is known in the literature that when t = l − 1 the linear equations admit a solution for the controller coefficients, for arbitrary δ(s) whenever the plant has no common pole zero pairs, and for t < l − 1 the exact attainment of the desired polynomial is impossible [22]. So, for t ≤ l − 1, (7) is enlarged into a box in the coefficient space, containing the point representing the original desired characteristic polynomial. This corresponds to choosing an interval desired polynomial family given by (8), which facilitates a solution from a root space point of view, where δ − n ≤ δ n ≤ δ + n for n = 0, 1, ..., l + t. [δ(s)] = δ n s n + δ n−1 s n−1 + ... + δ 0 To guarantee robust stability and robust performance, the roots clusters of [∆(s)] must be inside the roots clusters of [δ(s)], i.e., where ℜ(.) depict the roots clusters of [∆(s)] and [δ(s)] [22]. From (9), one can formulate the set of linear inequalities given in (10) that poses constraints to the controller's coefficients so that (9) can be satisfied.
Then, the robust design problem is summarized in the choice of a parameters vector of the controller to be optimized (x c ), if possible, so that the set of inequalities in (10) is satisfied for all The aforementioned robust performance control design problem, given in (10), can be rewritten as the following optimization problem [21], [39]: The cost function f (x c ) must be built and minimized according to the control goals. The solution of the problem given in (11) can be idealized as a solution to a linear programming (LP) problem although different techniques .
can be used to solve it. However, its standard solution is sometimes efficient and fast, so that this problem can be rewritten as a problem of minimization or maximization, subject to restrictions [26].

1) Linear Programming Based on Chebyshev's Theorem
From Fig. 2, the Chebyshev's theorem guarantees to be possible to find the largest ball B of center x c and maximum radius r, which is contained in the politope P , described by the set of linear inequalities constraints, whose norm is Euclidean. The ball center x c is called the Chebyshev Center and is a point inside P that is farthest from the exterior of P [40]. When the set P is convex, the computation of x c become a convex optimization problem. More specifically, suppose P ⊆ R n is defined by a set of convex inequalities, i.e., P = {a i x c ≤ b i , i = 1, ..., n}. Then, one can find x c by solving the LP given in (12) for r ≥ 0, [40]: Comparing (11) and (12), the robust pole assignment technique reduces to a LP combined to the Chebyshev's theorem as follows: Equivalently, where The vector x contains the controller gains and the radius of the Chebyshev sphere, f (x) is a linear cost function that must be built and maximized according to the control goals, ∥a i ∥ 2 is the Euclidean norm of the coefficients of a i , i.e., ∥a i ∥ 2 = (a i T a i ) [41]. In this study, the cost function f (x) has been chosen to be the sum of the elements of x [21], [38]. Then, when a solution exists, (14) provides the robust parameters of C(s) and ensures a maximum stability region for uncertain systems.
The Kharitonov theorem states that the stability of the four Kharitonov polynomials, given by is a necessary and sufficient condition to guarantee the stability of the whole interval polynomial family [42]. Therefore, the closed-loop characteristic polynomial, given in (6), is robustly stable if and only if the four Kharitonov polynomials in (16) are stable [35], [43].

III. INCREMENTAL GMV CONTROL
The GMVC was developed from the Minimum Variance Regulator (MVR) first introduced by Karl Johan Åström in the 1970s. According to [12], its original structure does not guarantee reference tracking and disturbance rejection for plants without integrators [11], [12], being necessary to add a filter ∆ = 1 − z −1 at the control weight of the generalized output [44], being The polynomials P (z −1 ), T (z −1 ) and Q(z −1 ) are weighting filters for system's output, reference and control signal, respectively. The generalized output ϕ(k) is posed into a stochastic optimization problem of minimizing the GMV cost function [30], [45], with respect to the control signal increment, i.e., where E[·] denotes the mathematical expectation operator. Let's expand GMVC's equations from ARIMAX (Auto-Regressive Integrated Moving Average with eXogenous Input) to generalized stochastic plant models of the following form: where y(k) is the system's output, u(k) is the input, d is the discrete time delay and ξ(k) represents a white noise disturbance. By inspection of the right hand side of the generalized output in (17), y(k + d) is a future measurement not available at instant k, so it must be predicted in order to work with ϕ(k + d) [11]. Then, shifting (20) d-steps ahead and remembering that P (z −1 ) multiplies the output y(k + d) in (17), one obtains the following: The future term of the noise, ξ(k + d), is unknown at a present time k, therefore it can be represented by present and future parts, as it follows: whereas these two auxiliary polynomials, E(z −1 ) and F (z −1 ), can be determined by solving the following Diophantine equation: Using only the present data of (22), the predicted output y(k + d|k) is By considering the elimination of the part E(z −1 )ξ(k + d) of (22) it introduces an error in the prediction, given by Thus, the present stochastic signal ξ(k), obtained from the prediction error is Substituting (26) into (24) and after some algebraic manipulations and using (22), the d-steps ahead Minimum Variance Predictor (MVP) turns to, By substitution of (27) into the generalized output in (17), the GMVC control law is thus obtained by solving (19) [9], [45], resulting in VOLUME 4, 2016 which guarantee in steady-state that the variance of the generalized output, σ 2 ϕ , is a minimum. At last, considering C ξ (z −1 ) = 1, the closed-loop transfer function is obtained through the substitution of (28) and (23) into (20). After some algebraic manipulations, one finds that which considers the closed-loop dynamics from the reference, y r (k), to the output, and from the white noise type sequence, ξ (k), to the output.

IV. PRGMV CONTROLLER DESIGN
This section presents the hybridization of the GMVC with PRC, whose design is based on PID controller with a lowpass filter, resulting in a PRGMV control law which utilizes the predictive nature of the GMVC for parametric uncertainty plant. The PID controller parameters are obtained by solving the convex optimization problem given in (14) and are then transferred to the GMVC by using Tustin's method.

A. GMVC BASED ON PID CONTROLLER
The controller C(s) is considered to be a PID with a low-pass filter, represented in the Laplace complex domain by with U (s), E r (s) = Y r (s) -Y(s), α, k p , k i and k d being the controller's output signal, error signal, high frequency lag, proportional, integral and derivative gains, respectively. The low-pass filter is included on the control law to reduce the influence of non-minimum phase (undershoot) and to assess further enhancements on the performance of the control-loop, particularly if the error is measured in a noisy environment [2]. Then, the problem of stabilization comes down to determine the values of C(s), i.e., x c := [k d , k p , k i , α], for which the closed-loop characteristic polynomial family [∆(s)] in (6) is Hurwitz [46]- [48]. It must be emphasized that due to the presence of integral action on the error, any controller with this feature already provides asymptotic tracking and disturbance rejection for step-like references [2], [32]. On the other hand, integrating plants might impose more difficulties to be stabilized by PID controllers and sometimes it might be necessary to exclude the integral part of the controller. Defining the Tustin's method for converting (30) to its corresponding discrete-time control law, being t s the sampling time, the discrete PID control law is obtained after some algebraic manipulations, which results in the following RST structure [31]: being The addition of ∆ = 1 − z −1 to both (28) and (32) will guarantee step-like reference tracking and disturbance rejection for the closed-loop systems. In such a form these controllers are generating the control increment which is then applied to the process as follows: From (17), when ϕ(k + d) = 0, we have By comparing (35) to (32), the polynomials P (z −1 ), T (z −1 ) and Q(z −1 ) become By looking at (36) and (33), note that parameters of the polynomials P (z −1 ), T (z −1 ) and Q(z −1 ) strongly depend on the PID controller parameters, which will be computed by solving the convex optimization problem given in (14), making them robust. Therefore, since the PID controller parameters given in (33) are robust, the GMVC control law in (28) becomes which represents the PRGMV controller control law and guarantee in steady-state that the variance of the generalized output, σ 2 ϕ , is a minimum whereas the Diophantine equation, given by (23), then becomes where

VOLUME 4, 2016
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3185020 Note that the PRGMV control law in (37) is also defined into the RST structure. Thus, the control loop of the PRGMV is depicted in Fig. 3 [49], where all (z −1 ) were hidden for the sake of simplicity and d(k) can represent an output disturbance or a noise added to the measured output. At last, considering C ξ (z −1 ) = 1, the closed-loop transfer function is given by Note that the transfer function that characterizes the closed loop dynamic behavior of y(k) from y r (k) in (40) is explicitly represented by the polynomials that weight the optimization problem in the GMVC, i.e., P prgmv (z −1 ), T prgmv (z −1 ) and Q prgmv (z −1 ). Therefore, theses polynomials guarantee robust stability and robust performance for parametric uncertainties model, satisfying (9), since its parameters strongly depend on the PID controller parameters, which when computed by solving the problem given in (14) is what makes it robustly stable.
Finally, by considering the PRGMV control loop in Fig.  3, it is important to remark that, since the PRGMV is a stochastic controller, the sensitivity function of the controlled systems might differ from the closed-loop sensitivity of the PID case, leading to better results regarding the PRGMV that enhances to a more stable behavior with less sensitivity to the measurement noise. This occurs due to presence of the E(z −1 ) polynomial in both (29) and (40), that is obtained from the solution of GMV's Diophantine equation and is related to the noise prediction [9].

B. ROBUSTNESS ANALYSIS
This section covers robustness analysis using magnitude plots of the sensitivity functions S sen (z −1 ) and T sen (z −1 ) in the frequency domain. From those functions, important indices, such as gain margin (GM), phase margin (PM) and maximum amplitude ratios, can be obtained to quantify the trade-off between robustness and performance to guarantee a suitable well tuned controller. Thus, according to [50], (40) is rewritten to where S sen (z −1 ) is the transfer function between the disturbance d(k) and the output y(k) (output sensitivity function) and characterizes the system performance from the point of view of disturbance rejection, and T sen (z −1 ) is the sensitivity function between y r (k) and y(k) (also called complementary sensitivity function) and characterizes the closed loop system performance [11], [49]. (37), since the PRGMV controller is based on the PID structure, the sensitivity functions have the property S sen (z −1 ) + T sen (z −1 ) = 1 and reflect many interesting properties of the closed-loop system, particularly, robustness to the model's uncertainties. The maximum amplitude ratios of S sen (z −1 ) and T sen (z −1 ), respectively M S and M T , are given by these quantifies the sensibility of the control loop to the excitation signal under consideration, e.g., small M S values make the system less sensible to d(k), whereas M T considers the influence of the reference signal, y r (k), and is equivalent to the amplitude of the resonant peak as well, that in general, is desirable to be kept small [11]. In [50], it is suggested that M S and M T values be respectively in the range of 1.2-2 and 1-1.5 to give a suitable trade-off between robustness and performance. According to [51], [52], gain and phase margins have been vastly used to characterize robustness of control loops, and it is common sense, that GM → ∞ and P M ≥ 60 • characterize robust stability. To obtain a softer compromise between robustness and performance, [50] suggests that GM and P M should be kept in the range of 1.7 dB-4dB and 30 • to 45 • , respectively, and that both indices can be obtained using the following equations: Therefore, from (43), it is possible to characterize the "stability margin" and "robustness" of the closed-loop system in relation to variations of the system parameters (or uncertainties in parameter values).

V. SIMULATIONS RESULTS
In this section two simulation examples are presented to illustrate the effectiveness of the design presented in Section IV. The PRGMV performance is compared to two other controllers, one designed by a robust pole placement (RPP) method, presented in [22] and the other by a classic pole placement (CPP) method, presented in [31]. In addition, the performance of the three controllers is quantified by using the integral of the squared error (ISE) [24]. VOLUME 4, 2016 A. EXAMPLE 1: SECOND ORDER SYSTEM Consider the following process to represent a DC motor position control problem with intervals parameters [39]: Since (44) is an integrating process, it is not considered the integral part of the controller given in (30). So, the design objective is to find x c := [k d , k p , α] by solving the LP problem in (14).

1) Robust Stability Analysis
The robust stability analysis of the three control-loops was made by means of the four Kharitonov polynomials, by plotting its roots in the s-plane, as shown in Fig. 4. From Fig. 4, note that the four Kharitonov polynomials are stable. This means that the three controllers can stabilize all the interval polynomial family given in (44). Therefore, this feature will be transferred to GMV, which will stabilize all the interval polynomial family as well.
2) Reference Tracking Fig. 5 depicts the simulated output behavior and the control signal by using the three controllers. During the tests, at first, a step reference is applied for three operating points: lower, nominal and upper. After the system achieves the steady state, the reference is varied from 1 pu to 2 pu at t = 10 sec and at t = 20 sec it is reduced again to 1 pu.
According to Fig. 5, both controllers can compensate for the reference variation and the output could asymptotically track the step-like sequence in the face of parameter uncertainty. It can be seen that the CPP and RPP have similar performance with severe control signal spikes and take a longer time to restore the output values to their respective set points. The PRGMV outperforms them both, since it quickly compensates the oscillations and provides a faster response while requiring less control effort.
In Fig. 6 it is shown the test result when a load disturbance of magnitude 0.5 pu is applied to the nominal system output at t = 5 sec. Observe that the PRGMV presents the best performance among all, outperforming the other approaches by minimizing the disturbance effect and reducing the recovery time of the system output to the desired set point.
In Fig. 7 it is presented the results when a white noise signal ξ(k), with a relatively small variance of σ 2 ξ = 0.01, is applied to the system output. Once again the PRGMV presents the best results, outperforming the other approaches by mitigating the influence of the white noise signal on the system, minimizing more the variance of the control and the output signals. Fig. 8 shows a cost comparison between the assessed controllers by means of the ISE cost function. This index evaluates the impact of the step reference on the controllers' performances. It can be observed that the ISE is almost constant regardless of the operating point. The PRGMV presents the smallest ISE, confirming its improved performance with the lowest cost. Fig. 9 presents the robustness analysis of the three controllers by plotting the sensitivity functions. Based on Section IV-B, the PRGMV again outperforms the other controllers, because it makes the system less sensible to the output disturbance d(k), it guarantees a larger stability margin and consequent closed-loop robustness to disturbances and modeling uncertainties. Gain and phase margins shown in Table 2 ratify the effectiveness of the PRGMV in the three operating points, because this controller, at least in this studied example, presented minimal control effort and energy consumption to track the reference signal for the assessed operating points.
By considering that the three controllers can stabilize all the intervalar system given in (44), the same procedure is repeated for a sinusoidal reference sequence at a frequency f o = 0.1 Hz ≈ 0.628 rad/s, which lays within the lower frequencies region where the complementary sensitivity is still close to 0 dB. It must be remarked, however, that since integral control action is included in all three design methods investigated, then a phase lag of 90 • is introduced into the control-loop which eventually reflects in the results shown in        According to [53], [54], if a stable, linear, time-invariant system is submitted to a sinusoidal input, it will have a steady state output sinusoidal with the same frequency as the input. In Fig. 10, it can be seen that the steady-state response of the system for the sinusoidal input is also sinusoidal, at the same input frequency and differing slightly only in amplitude and phase angle, as expected. It is also noticeable that the PRGMV outperforms the other controllers by minimizing more the variances of the controlled system.

B. EXAMPLE 2: NON-MINIMUM PHASE PROCESS
The following example represents a SISO linearized model with a variation of 10% in its parameters: [55]. The main design purpose is to obtain x c := [k d , k p , k i , α] by solving the LP problem given in (14). Table 3 presents the PID gains by considering the PRGMV and RPP methods only, whose values in the discrete domain were obtained using a sampling time t s = 0.25 sec, since for CPP it was not possible to find a solution, because the PID is designed by a classic pole placement that works only with plants of order n a ≤ 2 [56]. Then it is clear that the order of a control system that can be designed by pole assignment is considerably limited when employing traditional controllers due to their limited number of tuning parameters [57].  Fig. 11 shows the robust stability analysis of the closedloop system by means of the four Kharitonov polynomials, by plotting its roots in the complex plane. Note that the four Kharitonov polynomials are stable, meaning that the two controllers stabilize all the interval polynomial family given in (45). Therefore, this feature will be transferred to GMV, which will also stabilize the whole interval polynomial family.

2) Reference Tracking
The reference tracking performance in closed-loop for three operating points (lower, nominal and upper) is presented in Fig. 12. After the system achieves the steady state, the reference is varied from 1 pu to 2 pu at t = 10 sec and at t = 20 sec it is reduced again to 1 pu. Fig. 12 shows the simulated output behavior and the control signal by using RPP and PRGMV. Observe that both 10 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and controllers can compensate the step-like reference change and the output converges asymptotically in the face of parameter uncertainty. However, the RPP presents output oscillations, control signal spikes and takes a slightly longer time to restore the output values to the desired set points; the PRGMV, on the other hand, quickly tracks the reference without oscillations, providing a better performance with faster response and less control effort. Fig. 13 presents a load disturbance essay. The load magnitude of 0.5 was applied to the nominal system output at t = 50 sec. Note that the PRGMV presents the best recovery performance, outperforming the RPP by minimizing the load effect, reducing the recovery time. Fig. 14 presents the simulation results when a white noise sequence is applied to the system's output. The noise signal ξ, with a relatively small variance of σ 2 = 0.01, was assessed. Once again the PRGMV presented the best performance when compared to the RPP, since the control and output variances in the PRGMV case were smaller, confirming that PRGMV is more cost-effective than RPP. Fig. 15 shows the performance comparison using the ISE cost function. This index evaluates the impact of the steplike reference on the control-loop. It can be observed that the ISE is almost constant regardless of the operating point value for step input variation. However, for PRGMV the ISE was smaller than for RPP, confirming PRGMV's performance improvement while requiring less control energy to do so. Fig. 16 presents the robustness analysis of the two controllers by plotting the sensitivity functions. Based on Section IV-B, the PRGMV once again outperforms the RPP, because it makes the system less sensible to d(k) and exhibits a larger stability margin and robustness of the closed-loop system, as shown in Table 4, whose values of GM and PM ratify the effectiveness of the proposed controller for reference tracking and can confirm, at least for this studied example, the good performance of the PRGMV for parameter uncertainty plants with minimum control effort and minimum energy for   tracking the reference signal.
According to Fig. 11, the RPP and PRGMV stabilize all VOLUME 4, 2016 11 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.  the system given in (45), so, the reference tracking perfor-   From Fig. 17, it can be seen that the steady-state response of the system for the sinusoidal input is also sinusoidal, at the same input frequency and differing in amplitude and phase angle [53], [54]. The results highlight the good performance of the PRGMV, since it outperforms the other controllers by minimizing variance of system output and control signal.
12 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3185020

C. GENERAL ASSESSMENT OF RESULTS
Based on the presented results of the previous section it is observed that PRGMV copes better with the requirements of disturbance rejection and asymptotic reference tracking while keeping a smooth control signal and faster response with considerable variance minimization in the output and control signals and therefore provoking little stress on the controlled system, whereas the other controllers presented control signal spikes and chattering.
The PRGMV outperforms the other controllers thanks to its formulation structure with polynomial filters P prgmv (z −1 ), T prgmv (z −1 ) and Q prgmv (z −1 ) which robustly weight the output, the reference and the control signals, respectively. The non-scalar polynomial Q prgmv (z −1 ) = r 0 + r 1 z −1 filters out the control signal variations caused by the feedback of noisy measurements or disturbances and reduces the aggressiveness of the controller, leading to a smooth behavior with minimum control effort and energy consumption in the task of tracking the reference signal, according to Fig. 7 and Fig. 14 and confirmed by the variance minimization, i.e.., the linear power minimization.
The weighting polynomials P prgmv (z −1 ) and T prgmv (z −1 ) assign robust performance characteristics for the PRGMV, producing a better desired closed-loop response for reference tracking and disturbance rejection, as depicted in the figures 5, 6, 10, 12, 13 and 17. It also assigns greater stability margins and minor sensitivity to disturbances, as presented in figures 8, 9, 15 and 16, as well as in Table 2 and Table 4. The presented data confirmed, for the studied cases, that PRGMV can guarantee the stability of the system for all the vertices of the uncertain polytopic P domain as depicted in Fig. 2. It also achieves a desired maximum stability region thanks to Chebyshev's theorem that minimizes the overall deviation from the desired performance for the closed-loop system, maintaining an adequate compromise between robustness and performance with lower sensitivity to parametric uncertainties.

VI. CONCLUSION
This paper presented a novel methodology for designing robust predictive controllers that can ensure robust stability and performance for a predefined uncertainty region. The design procedure is inspired in [22] and [25]. The proposed method was compared to two other control methods, assessing their performances in terms of reference tracking and robustness to modeling uncertainties, load and noise. The results clearly showed that the proposed method was able to maintain the desired performance while compensating for load and stochastic disturbances, outperforming the other controllers.
The performance indicators obtained for the proposed method provides an improved performance in comparison with the other methods that were unable to adequately compensate the parametric variations to avoid several performance degradation. The results indicate that the PRGMV is justified and present relevant improvements for parametric uncertain systems, offering robust performance and stability.
Finally, taking into account the lack of research regarding the hybridization of the two areas mentioned, of plants with parametric uncertainty and predictive minimum variance control, glimpses the development of the proposed methodology directly in the discrete frequency domain, i.e, obtaining the PID gains by pole placement and conducting the Kharitonov's test directly in the discrete domain, which is currently an open topic for investigation that might pay off in the end, as the presented results in this work suggest, glimpses the development of the proposed methodology for: MIMO system; systems with transport delay; other predictive controllers, such as GPC.
WALTER BARRA, JR. received the B.E., M.Sc., and Ph.D. degrees in electrical engineering from the Federal University of Para (UFPA), Brazil, in 1991, 1997, and 2001, respectively. He is currently an Associate Professor with the Department of Electrical Engineering, UFPA.
He has experience in electrical engineering with an emphasis on automation and control of industrial and electrical power systems. His main research topics include modelling and robust control of industrial, and electric power systems.