Robust H ∞ Loop Shaping Controller Synthesis for SISO Systems by Complex Modular Functions

: In this paper, a loop shaping controller design methodology for single input and a single output (SISO) system is proposed. The theoretical background for this approach is based on complex elliptic functions which allow a ﬂexible design of a SISO controller considering that elliptic functions have a double periodicity. The gain and phase margins of the closed-loop system can be selected appropriately with this new loop shaping design procedure. The loop shaping design methodology consists of implementing suitable ﬁlters to obtain a desired frequency response of the closed-loop system by selecting appropriate poles and zeros by the Abel theorem that are fundamental in the theory of the elliptic functions. The elliptic function properties are implemented to facilitate the loop shaping controller design along with their fundamental background and contributions from the complex analysis that are very useful in the automatic control ﬁeld. Finally, apart from the ﬁlter design, a PID controller loop shaping synthesis is proposed implementing a similar design procedure as the ﬁrst part of this study.


Introduction
The loop shaping design procedure and synthesis for several kinds of single-input and single-output (SISO) systems have been implemented during recent years due to the selection of the frequency domain characteristics of the open and the closed-loop systems, this means, designing the control strategy based on the gain and phase margin specifications of the open-loop plant and the specifications of the closed-loop system [1][2][3][4][5][6]. The H ∞ loop shaping robust controller design was studied in recent years to synthesize controllers for SISO plants with unmodeled dynamics and/or uncertainties due to the importance of the controller synthesis in the process control field. Robust H-infinity control consists of establishing a performance index based on the transfer function's norm [7]. In the case of single input single output SISO systems, it is done similarly for linear multi input multi output MIMO systems. In the SISO system, the H-infinity controller is found by synthesizing the performance index to find the desired controller meeting the robust stability and robust performance conditions. Meanwhile, the suitable controller and gain matrices are obtained for the MIMO case, similar to in the SISO case, synthesizing the selected performance index [8].
Loop shaping design methodologies have been studied in recent decades for SISO cases and multiple input multiple output (MIMO) cases. In [9], the H-infinity loop shaping design procedure for multivariate systems with constrained inputs is shown, and the controller synthesis is derived from linear matrix inequalities, a subject that has been studied in recent years. A multivariable H-infinity loop shaping design procedure is proposed in [10] to stabilize a chaotic system. In this case, the original system is linearized, which could add losses to the original system. In [11], the stabilization of multivariable systems with delays with a H-infinity procedure is done along with an observer predictive law. The H-infinity loop shaping design procedure in the multivariable cases is the most found in the literature. For example, in [12], an approach has been implemented in the controller synthesis for a switched reluctance motor used in an electrical vehicle, considering the uncertainties and disturbances of the system model. In [13], the H-infinity loop shaping design controller synthesis for a 2DOF platform is shown, and it is a very interesting application for mechanical systems.
In [14], an interesting application of an enhanced H-infinity loop shaping design controller for an industrial motion system is evinced taking into account the performance and robustness requirements. Finally, in [15], a fixed structure 2DOF H-infinity loop shaping controller for an active current mode converter is shown, considering that a genetic algorithm is used for the loop shaping controller design. This loop shaping 2DOF controller consists of adding an extra degree of freedom to the controller to track the system's reference or setpoint along with the tracking of a disturbance input. It is important to notice that in recent years the use of evolutionary algorithms in the design of controllers for SISO and MIMO systems has been increased. In [16], the stabilization of a tele-micromanipulation system is done by a H-infinity loop shaping design procedure, and the implementation of this approach has been extensively used in other applications different than process control such as mechanical and electrical systems. Actually, there are novel loop shaping design strategies, and in some cases, along with H-infinity control. In papers such as [17], a significant study related to this present work evinces the design of an H-infinity loop shaping controller with weighting function optimization by linear matrix inequalities. Then in [18], a data-driven H2-H infinity loop shaping controller is presented. In studies such as [19], an H-infinity loop shaping controller is designed for linear parameter varying plant. Other interesting results that are useful for this study are found in [20] in which a parameterized loop shaping extremum seeking approach is done for controller synthesis. Finally, in [21], an H-infinity loop shaping design procedure is proposed for linear quadratic and PID regulators. In that paper, a novel H-infinity loop shaping controller synthesis is proposed based on complex elliptic functions. Complex elliptic functions are those kinds of functions that have two periods, where a meromorphic function in the plane with two independent periods are called elliptic [22][23][24]. The H-infinity loop shaping design procedure consists of implementing the elliptic functions in order to obtain the desired gain and phase margins specifications of the open-loop plant, shaping the plant with two weighting functions using Abel's theorem [22].
In this paper, first, a robust controller synthesis is obtained by implementing two weighting functions in order to meet the robust H-infinity performance, and then, a robust PID controller synthesis is obtained by implementing an unique weighting function. In both cases, the weighting functions are obtained by the Abel theorem. It is important to remark that the main contribution and opposition to other research studies found in the literature is that in the proposed control approach, the weighting functions for the loop shaping control system consider the maximum advantages of complex modular function by taking into consideration that these functions are double-periodic. So this is an advantage because the gain and frequency crossover frequencies can be selected independently to shape the frequency domain characteristics of the open-loop plant for optimal closed-loop performance. The main advantage of this approach in comparison with other studies found in the literature is due to the double periodicity of elliptic functions, the open-loop gain and phase margins obtained by placing the required poles and zeros, allowing an appropriate selection of the gain margin, phase margin, and cut-off frequency.
This paper is divided into the following sections. In Section 2, a brief review of elliptic functions is shown in order to introduce the theoretical background. In Sections 3 and 4, the derivation of the robust and PID loop shaping controller synthesis are presented respectively. In Section 5, a numerical example of the two controller synthesis are shown and they are compared with PID controller strategies as shown in [3,25].

Brief Review of Complex Elliptic Functions
As explained before, complex elliptic functions are those kinds of double periodic and meromorphic functions as described in the complex analysis field. The mathematical definition of elliptic functions is given in [22]. There are some important research studies in the literature related to this topic of pure mathematics. In papers such as [26], the bounds over modular functions are explained in which it is shown that its Fourier coefficients can obtain a harmonic cochain. Other interesting results are found in [27] in which the periods of Hilbert modular forms are obtained. Another interesting result is found in [28]. Some inequalities for integral and modular functions are presented in which monotonicity and convexity of elliptic integrals are analyzed to obtain some inequalities. In [29], the potential modularity of elliptic curves defined on a real field is presented. Finally, in [30], multiple elliptic gamma functions are developed and evinced. First, it is important to define a lattice L, therefore consider the lattice with two vectors (periods) ω 1 and ω 2 in C which are linearly independent in R. Then, the lattice for an elliptic function is given by: An elliptic function is given by the following definition [22]: An elliptic function for a lattice L is a meromorphic function with the property for ω ∈ L and z ∈ C Apart from this important definition in the complex analysis, considering that the appropriate selection of poles and zeros of the weighting function of the H-infinity loop shaping design procedure proposed in this paper, Abel's theorem for the selection of prescribed poles and zeros of an elliptic function is fundamental. Before explaining this theorem, consider the following [22]: where Q(z) = 0 and f : C → C, the Abel's theorem as explained later is fundamental in the results obtained in this study [22].

Theorem 1.
An elliptic function with prescribed poles b 1 , ..., b n and zeros a 1 , ..., a n can be obtained iff With this definition and theorem 1, the derivation of the H-infinity loop shaping design procedure for SISO systems with a standard or PID controllers can be established.

Standard H ∞ Loop Shaping Controller Synthesis by Elliptic Functions
In this section, a standard H-infinity loop shaping controller synthesis by elliptic functions is shown. In the loop shaping design procedures found in the literature in recent years [13][14][15], one of the main issues that must be solved is the selection of appropriate weighting filters to shape the plant transfer function. It is important to notice that presently this procedure is mostly done by implementing process control software and toolboxes but there are not more efficient methods to design the weighting filters for the shaping procedure. Therefore, considering these disadvantages of previous H-infinity loop shaping controller synthesis procedures, in this paper the loop shaping design approach is improved by a novel feedforward system architecture that is a new approach that improves the robustness of the closed-loop system.
In this paper, the design of weighting filters based on elliptic functions is proposed. As explained in the previous section, elliptic functions provide a useful theoretical background to select functions with prescribed zeros and poles. Considering the double-periodicity of elliptic functions. The selection of the cross-over frequencies for the gain and phase margins, respectively, along with an appropriate cut-off frequency is straightforward. For this reason, the weighting filter design is done in a theoretical and accurate way, and it is not necessary to implement other design methodologies that are not stronger from a theoretical point of view. In this section, the design of a standard H-infinity loop shaping controller synthesis is shown. In Figure 1 the block diagram of the compensated closed-loop system is depicted, where K(s) is the controller obtained by loop shaping and G(s) is the plant.

H ∞ Loop Shaping Design Procedure
The H-infinity loop shaping design procedure is done in the following steps [15]: Find the robust controller K ∞ (s) in order that the following condition is met: where T(s) is the complementary sensitivity closed-loop transfer function and γ min and max are given due to the following robust condition explained later: whereM andÑ are the coprime factors with where ∆ M and ∆ N are the coprime factor uncertainties. Finally, It is important to consider that the gain K(s) shown in the feedforward compensated block diagram, depicted in Figure 1, is found using the equivalent block diagram shown in this figure. This complementary sensitivity function is obtained to derive the K ∞ gain matrix and, later, the gain K(s) is obtained with the loop shaping design procedure as shown in step 2 and 3. Remember that in the robust loop shaping design given in (7), (8), and (9), the max and γ min are found obtaining the coprime factorization of the plant to be shaped G(s) as presented in Figures 2 and 3. One of the novel contributions of this work is that the gain K ∞ is obtained from the complementary function T(s) from Figure 1 to improve the robustness of the closed-loop system surpassing non compensated approaches as shown in [16]. As a conclusion of this robust loop shaping design procedure, first the gain matrix K ∞ is obtained from the sensitivity function obtained from Figure 1 (feedforward compensated system) but considering the values of max and γ min obtained from (7)-(9) and the uncertainties of the diagrams shown in Figures 2 and 3. Then, the loop shaping design procedure is concluded in steps 2 and 3.

2.
The second step is to find the weighting functions W 1 (s) and W 2 (s) in order to obtain a desired open-loop frequency response characteristic for G(s) in the following form G n (s) = W 1 (s)G(s)W 2 (s). This step is crucial in this study because the weighting function selections are done by complex elliptic functions, for the standard and PID controller design.

Weighting Function and Standard H ∞ Controller Design with Complex Elliptic Functions
As explained before, one of the main issues that must be solved in the H-infinity loop shaping controller design is the selection of the weighting function. In this paper, a weighting function design implementing elliptic complex functions is proposed. Due to the double periodicity of elliptic functions the selection of an elliptic function with prescribed poles and zero is done, as explained in Section 2, to select the gain margin and phase margin along with the cut-off frequencies (step 2). Another improvement in comparison with other results is found in the literature that the K ∞ (s) controller is obtained to meet the robustness specifications.
In the following theorem, both issues explained in the previous paragraph are established [22].

Theorem 2. The weighting functions W 1 (s) and W 2 (s) is obtained by using Abel's theorem and the function σ(s) explained later
in order to find the controller K(s) = W 1 (s)K ∞ (s)W 2 (s) Proof. Define the following set: The variable s ∈ D and the following σ function which is defined in Abel's theorem to obtain an elliptic function with prescribed poles and zeros where ω = 0. Consider the following plant transfer function: and the following complementary sensitivity function: Then, according to the step 1 of the H-infinity loop shaping design the following result is obtained: So by defining the transfer function: the supremum of the sensitivity function shown in (15) is found as To simplify the controller design it is necessary to implement a Taylor series expansion of the following function: To derive the weighting filters, consider (12) with the frequencies ω 1 = 1 T and ω 2 = 1 βT where T is the period. The weighting function which is a low pass filter is given by: where now, consider Therefore, for W 1N (s) and W 1M (s) and for W 2N (s) and W 2M (s) Therefore, to avoid the exponential term, W 2 (s) is approximated by a Taylor series-defining Then, implementing only the first terms of the Taylor series expansion of f (s) We obtain W 2 (s) = 1 + 1 27) and to define the numerator and denominator in (20), it is necessary to apply the following change of variables: α = z 0 − a 1 and β = z 0 − b 1 . The equivalent filter is given by: and this completes the proof.
To find the periods of the elliptic functions (ω 1 , ω 2 ) along with the crossover frequencies and cut-off frequencies (ω pi ,ω gi ,ω c ), the following equations must be solved by a nonlinear algebraic equation solver where A i is the gain margin and φ mi is the required phase margin.

H ∞ PID Loop Shaping Controller Synthesis by Elliptic Functions
Proportional-integral-derivative (PID) controllers were implemented in the process control and other applications [31][32][33][34][35]. For these reasons, as explained in the previous section, the H-infinity PID loop shaping controller system, given in Figure 4, is obtained by elliptic functions. The following theorem establishes the theoretical background for this objective.

PID Controller G(s)
Reference y + - where K p is the proportional gain, τ d is the derivative time constant and τ i is the integral time constant.
Proof. The PID controller is given by Consider the α variable shown in the previous section. By using (12), the following filter is obtained: and (30) becomes in which represents the PID controller and this completes the proof.
Using the filter W 2 (s) the following equivalent transfer function is obtained: To find the elliptic function periods ω 1 and ω 2 along with the cross over frequencies ω pi , ω gi and the cutoff frequency ω c , the following nonlinear algebraic equations must be solved numerically: where A i is the gain margin and φ mi is the required phase margin.

Numerical Experiments
In this section, three numerical experiments are shown to test the theoretical results obtained in this study. The first and second experiments are developed considering a step and impulse response respectively, while the third numerical experiment is done by regulating the acid concentration in a water reservoir. The simulation is performed with GNU Octave 4.2.2 in an Intel Core i3 laptop. The plant used in the first and second experiment is:

Experiment 1
In experiment 1, the H-infinity loop shaping controller synthesis is tested in the control of system (36). The frequencies found for the weighting filters are ω 1 = 0.60789 rad/s and ω 2 = 3.8580 rad/s for A = 4 and φ mi = 0.9 rad. Then, the following system response is found comparing the results with the strategies evinced in [36,37] as depicted in Figure 5. The system is tested with a step reference and an impulse reference, as can be noticed there, is less overshoot with the proposed approach in comparison with the approaches shown in [36,37] moreover, the steady-state error is smaller with the proposed strategy in comparison with the compared strategies. Then, in Figure 6 the error variable for both strategies is shown. As can be noticed the error between the reference and the output of the system is smaller with the proposed control strategy in comparison with the strategies shown in [36,37] proving the effectiveness of the controller design. In Figure 7 the root locus of the weighted and shaped plant is shown in which, as can be noticed, the addition of poles and/or zeros by the weighting filters shape the plant so that the system performance is improved by moving the locus region to the desired position. Then in Figure 8, the impulse response of the system is shown corroborating that the response reaches the zero value in a short time in opposition to the impulse response obtained with the approaches shown in [36,37].  The robustness index is given by the following: Table 1 shows the robustness indexes of the three strategies. As can be noticed, the robustness index of the proposed strategy is smaller than [37] and greater than [36], this means that the proposed control strategy provides a moderate robustness index taking into consideration that there is a trade-off between the system performance index and robustness. Table 1. Robustness indexes of the proposed strategy and comparative approaches.

Robustness Index
Proposed Strategy 0.624339 Da Silva et al. [37] 0.643336 Diaz et al. [36] 0.56655 As can be noticed in Table 2 that when the integral squared error ISE increase, the robustness index decrease, or in other words they are inversely proportional, remarking again that our proposed method provides a moderated robustness index and performance in comparison with the other two comparative strategies. For this reason, the proposed control strategy provides its optimal performance. Table 2. ISE of the proposed strategy and comparative approaches.

Experiment 2
In this experiment, the loop shaping PID controller synthesis design is tested to corroborate the theoretical results obtained. The frequencies of the weighting functions are ω 1 = 2.1324 rad/s and ω 2 = −4.2475 rad/s with A = 0.001 and φ mi = 0.7 rad.
In Figure 9, the system response of the proposed strategy in comparison with the standard PID controller and the approaches shown in [38,39] are depicted. The overshoot and steady-state error are smallest than the standard PID controller so the H-infinity PID controller synthesis yields better results. In Figure 10 the error variable between the reference and the system output is presented and the steady-state error with the proposed controller synthesis is significantly smaller in comparison with the results obtained by [38,39].

Experiment 3: Acid Concentration Regulation in a Water Reservoir
For this numerical experiment, consider the following mathematical model of a water reservoir with an acid input flow as appears in Figure 11. The mathematic model of the proposed system is [40]: where V is the volume of the reservoir, C is the acid concentration in mol/m 3 , U is the acid rate input and q is an appropriate constant. Then the transfer function is: This numerical experiment consists of regulating the acid concentration in the water reservoir as 500 mol/m 3 starting from a 0 mol/m 3 concentration until the desired setpoint is reached.
In Figures 12 and 13, it can be noticed how the acid concentration reach the desired setpoint value when a step response is needed. It can be noticed how the acid concentration reaches the desired value faster, and as noticed in Figure 12, there is a fast change from the initial to the final desired setpoint value. It is important to observe that the results obtained with the proposed control strategy provide faster and more accurate results than the strategies proposed in [36,37]. These results are corroborated in Figure 13 in which it is noticed how a faster and smaller output response is obtained with the proposed control strategy in comparison with [36,37] something that proves that the proposed control strategy improves other results found in the literature.

Conclusions
In this article, a novel approach for the H-infinity loop shaping controller synthesis is proposed. Two of the main contributions shown in this article are the robust H-infinity controller derivation and the weighting function synthesis implementing the elliptic function to select the appropriate poles and zeros to meet the gain margin, phase margin, and cutoff frequency. It is proved, theoretically, that one of the main advantages of this approach is that there is a strong mathematical methodology to obtain the weighting filters and the gain matrices. Finally, these results were proved by numerical examples.