Comparative Analysis of H2 and H∞ Robust Control Design Approaches for Dynamic Control Systems

T his paper discusses using H2 and H∞ robust control approaches for designing control systems. These approaches are applied to elementary control system designs, and their respective implementation and pros and cons are introduced. The H∞ control synthesis mainly enforces closed-loop stability, covering some physical constraints and limitations. While noise rejection and disturbance attenuation are more naturally expressed in performance optimization, which can represent the H 2 control synthesis problem. The paper also applies these two methodologies to multi-plant systems to study the stability and performance of the designed controllers. Simulation results show that the H2 controller tracks a desirable closed-loop performance, while the H∞ controller guarantees robust stability for the closed-loop system. The validation of the techniques is demonstrated through the robust and performance gamma index, where the H∞ controller achieved a robust gamma index of 0.8591, indicating good robustness and the H2 controller achieved a performance gamma index of 2.1972, indicating a desirable performance. The robust control toolbox of MATLAB is used for simulation purposes. Overall, the paper shows that selecting a suitable, robust control strategy is crucial for designing effective control systems, and the H2 and H∞ robust control approaches are viable options for achieving this goal


INTRODUCTION
The basic feature of robust control is that it ensures that the feedback system achieves stability and works well, even if the controlled process has uncertain parameters or is affected by noise or disturbances in the measurement (Petersen, 2009;Vasičkaninová and Bakošová, 2015). A robust design controller is an approach to control the uncertainties and system-changing. These controllers can be coupled with several control criteria, including places for pole position, worst-case frequency response values, transient response decay rate limitations, and boundaries for the two robust controllers, H∞ and H2 norms (Mehta et al., 2023). It has been demonstrated that the H∞ optimization strategy, created in the last two decades and currently under investigation, is a reliable and effective way to design linear, timeinvariant control systems Many researchers have addressed and studied the UAV system; for instance, (Ibraheem, 2019) proposed an Anti-Disturbance Compensator to stabilize a 6-DoF quadrotor Unmanned Aerial Vehicle (UAV) system. The designed Control Scheme rejects the disturbances subjected to this system and eliminates the effect of the uncertainties that the quadrotor system exhibits. In (Saud and Hasan, 2018), the Integral Backstepping Controller (IBC) is designed and optimized for full control of rotational and translational dynamics of an unmanned Quadcopter (QC), To improve controller capability in the steady state against disturbances; an integral action is used with the BC. To determine the optimal values of the IBC parameters, Particle Swarm Optimization (PSO) is used. Many researchers employed the H∞ and H2 controllers in their research. For instance, in (Lee et al., 2021), a large-scale unmanned aerial vehicle (UAV) is studied, and a controller is proposed. A reliable H∞ decentralized tracking control strategy is suggested to address a non-mountable impractical design issue brought on by the centralized control's excessive processing complexity. In (Jiang et al., 2022), a robust constrained H∞ control method is created, considering the in-wheel active suspension system's control limitation and the sprung mass variation's effects. Since Zames first introduced the concept of H∞ control synthesis, the area has seen significant development. The approach offers many theoretical advantages, some of which are strong disturbance rejection, great stability, and many more, where the H2 norm is used as a performance index. They have been extensively utilized to address several theoretical and practical issues. H∞ loop shaping is one of the strategies commonly recognized among those reviewed in this research paper's literature for the best H∞ controller. This is because performance criteria may be included in the design stage as performance weights (Bansal and Sharma, 2013). Robust controllers can provide better results using a methodical approach to choosing suitable fractional weights ( This work emphasized the validity of H∞ and H2 robustness indices via addressed research problems named UAV and suspension system. The mathematical modeling and state space equations are derived, and an automatic weight selection algorithm is used to design control systems for UAV and active suspension systems. All simulations and validations are done using the Robust Control Toolbox of MATLAB.

PRELIMINARIES
The robust closed-loop (CL) control system can be written as robust H∞ and H2 optimization problems in many ways. Therefore, it is very helpful to have a standard way of putting a problem that can be changed to fit any problem. This generic formulation is made possible by the general arrangement shown in Fig. 1 (Aghaie and Amirifar, 2007; Sojoodi and Majd, 2010), where P and K are generalizations for plant and controller, respectively. The overall control goal is to reduce a specific transfer function (TF) norm from w to z, such as the H∞ norm. Thus, the issue with controller design is to track down a controller K, using the data in v, that provides a control signal u that offsets the impact of w on z, hence reducing the CL norm from w to z. Eq.
(1) represents a matrix of the robust control system, the w is the so-called external signals, such as disruptions and commands, and z is the "error" signals that must be reduced in some way to achieve the controller's design goals.

H∞ CONTROL
In the design of model-based controllers, system uncertainty is essential. A robust control method based on H∞ is suggested to address issues like amplifier delay and sensor offset, which are common examples of uncertainties that can be effectively mitigated using H∞ control (Si et al., 2022). Robust control theory, which Zames first put out. When G (s) is the open TF of the plant P(s), and K(s) is the designed controller, this ensures the robustness and effective operation of the CL system. If the controller meets the following three criteria, controller K(s) may be derived to allow the experiments to be repeated (Vasičkaninová and Bakošová, 2015).
The stability of the system can be indicated through the roots of the characteristic equation (CE), which is the denominator of the TF; the roots should be placed on the left side of the complex s-Plane, and the sensitivity functions are kept small for all frequencies, and the performance criterion is met despite the large disturbances and operating point changes. Finally, the robustness criteria stipulate. Stability and performance must be maintained for the nominal and nearby plant models resulting from modeling errors. To provide high resilience of linear systems, robust controllers are built. Typically, the highest value of a TF's H∞ norm, G, spanning the whole spectrum is denoted by: Here, σ is defined as the largest singular value of a Tcontroller's goal to ensure constraint. Several controller design methods include the two-and three-TF approaches (Bansal and Sharma, 2013). The former can be chosen over the latter for H controller synthesis since it has less computational complexity. A complicated control issue is traditionally split into two halves, one dealing with stability and the other with performance, using two transfer functions, which is the traditional method used in H∞ controller synthesis. The complementary sensitivity function, T, and the sensitivity function, S, provided in Eqs. (5) and (6) are necessary for the controller synthesis.
The size of the complementary sensitivity function is constrained by the robustness weighting function Wt, while the sensitivity function's size is constrained by the performance weighting function Ws, and control energy is constrained by the complementary sensitivity function WKS. The loop shaping approach is the most popular method for selecting the weight functions for the controller's synthesis. In general, the robust control design is constructed in such a way as to bring the H∞ norm of the plant down to its lowest possible value. To accomplish this condition and meet the state's requirements, three weight functions are introduced to the plant to shape the loop. In their most fundamental form, weight functions are lead and lag compensators transfer functions that adjust the system's frequency response according to user preferences. Loop shaping is used with the WF to achieve the required frequency response from the plant. Loop shaping may be accomplished in several different ways. Adjusting the values of the weight functions' parameters is necessary to bring the entire system's frequency response within the acceptable range of values. Fig. 2 shows the mixed Sensitivity problem addressed by (Sojoodi and Majd, 2010; Vasičkaninová and Bakošová, 2015).  The state-space equations that represent the generalized system P in Fig. 1 are given in Eq. where x(t) ∈ R n is the state vector, w(t) ∈ R m1 the exogenous input vector, u(t) ∈ R m2 the control input vector, z(t) ∈ R p1 the error (output) vector, and y(t) ∈ R p2 the measurement vector, with p1 ≥ m2 and p2 ≤ m1. P(s) may be further denoted as:

Journal of Engineering
From Eq. (10) and Eq. (11), the mixed sensitivity function can be written as: Finding a rational function controller K(s) to solve a mixed sensitivity problem and stabilizing the CL system while meeting the following equation: where P is the TF from input W to output Z. By applying the minimum gain theorem, change the H∞ to be the norm of | | less than unity, therefore, Consequently, we can create a stabilizing controller. The algebraic Riccati equations must be solved to obtain K(s), which minimizes the cost function. The robust control theory indicates that choosing two weight functions is necessary for the controller's synthesis. For choosing weights, there are several approaches described in the literature. The H∞ controller is then synthesized using the loop shaping methodology in most of these design methods, and the weighting functions are chosen by trial and error. However, a trial-and-error method could not result in a stabilizing controller, which is the fundamental disadvantage of this kind of synthesis. The tuning parameters are the weights, and it usually takes multiple iterations to achieve weights, resulting in a satisfactory controller. Having stated that, a smart first step is to choose: The value of A should be less than one for maximum steady state equipoise (A = 0.01), defines as desired bandwidth, and M is the sensitivity peak (M = 2). The 1/Ws is an upper bound on the desired sensitivity loop shape, and Wks -1 will affect the controller output u which is identical to Ws boundary to the line = .
The values of A = 0.01(= −40dB), M = 2(= 6dB) and ω o =1rad/sec. To test the H∞ controller design process, a Plant-derived as: = 39 25s 2 + 7 − 4 005 With the weight functions as, Solving the given plant, as a model shown in Fig. 3 Figs. 4, 5, and 6, respectively. The 1/W1 and 1/W3 constraints and the singular values of S and T are shown in Figs. 4 and 5, respectively. The 1/W1 constraint is considered the performance boundary region for the designed controller, as shown in Fig. 4, and 1/W3 constraint is the boundary region of robustness for the designed controller, as shown in Fig. 5. It can be seen that the maximum singular values of S and T are both below the magnitude response of 1/W1 and 1/W3, respectively. This indicates that the performance and robustness specifications outlined by the weighting functions W1 and W3 are met. Fig. 6 depicts the open-loop system (L)'s singular values concerning the performance constraint W1 and robustness bound 1/W3. The lowest singular value of L is above the bound W1 in the low-frequency band, while the greatest singular value of L is below the bound 1/W3 in the high-frequency range as a consequence of meeting performance and robustness criteria.

H2 CONTROL
The H2 controller design method extends the LQG design approach, which may create a state space controller. In (Khayat et al., 2017; Szabolcsi, 2018), the design procedure and solution of the H2 controller design problem are presented. The generalization of the LQG issue to the 'standard problem' and the elimination of random interpretations are well known. The CL control system's standard setup can be seen in Fig. 7. Generally, the H2 norm of a TF, G, is its maximum value over the complete spectrum and is represented as the square of the second norm of the CL control system stable transfer matrix Tzw .
For further discussion, it is assumed that the plant dynamics p(s) is described with the state space representation elaborated in (6), and the plant dynamics p(s) are represented in (8).
The goal of the H2-optimization method was to meet the following criteria: closed-loop system stability, CL dynamic performance, and CL system robustness. To concurrently meet the above-described system characteristics, the controller must guarantee the following criteria: at low and high frequencies, the controller should achieve large and small open loop gains, respectively, and should stay far from the critical point of -1+ 0j at the crossover frequencies.
A Numerical Example for the Unmanned aerial vehicles flights UAV H2 Optimal Control System Design. There is a wider variety of unmanned aircraft systems (UAS) available. It is feasible to access the application's potential on a larger scale. As a component of the UAS, the UAV is getting increasingly complicated. The automatic flight control system, often known as the autopilot, must be installed within the UAV to assure flight safety that is at least as good as that of human aircraft. MIMO or the SISO approach can be used to represent the spatial mobility of a UAV. The Boomerang 60 small UAV is addressed, and a multi-input, multi-output dynamical model is studied (Szabolcsi, 2018): The TF of the Boomerang 60 UAV is given below: where A=1.1895, and T=0.04902 The roll rate stability augmentation mechanism frequently functions as an internal loop in the roll angle stabilization of the UAV, as seen in Fig. 8

ϒ=2.1972
The step response of the plant system with roll rate stability is shown in Fig. 8. TF's UAV CL roll rate stability has a fast response to a unit step function in the roll rate. Applying the normalized step response gives no overshoot indicating that the complete CL control system is non-oscillatory. Although good performance can be observed from the time domain response, the obtained performance index ϒ is greater than one, which means that the controller has feasible robustness but is not optimum.

Figure 8.
Step response of the plant UAV.
ϒ =0.8591 The obtained step response of the UAV plant is presented in Fig. 9. It can be noticed that the H∞ controller has a lower ϒ than the 4th-order controller. Since the obtained ϒ is lower than one, the obtained H∞ controller is more robust than the H2 controller. The outcomes of the H2 and H∞ designed controllers are summarized in Table 1.

Figure 9.
Step response of the UAV with Ki Fast and nonoscillatory Ensures robustness and good performance but may not provide optimal performance

CONCLUSIONS
This paper introduces two methods for designing controllers and analyzing their behavior for unmanned aerial vehicles (UAV) and active suspension systems (ASS). These are the H∞ and H2 control design approaches. According to the results of the simulations, the selection of the controller is highly concerned with the system operations and uncertainties. The analysis generally emphasized that the H∞ controller ensures robustness, superior sensitivity performance, and high disturbance rejection, delivering great stability for any operating condition. In cases of multiple uncertainties and changing operational parameters, the H2 schemes are used intensively to guarantee stabilization and enhance the performance of the UAV. Finally, we concluded that choosing a proper robust controller depends on many parameters, such as the system behavior and the number of uncertainties due to their effects on the system's operation, to achieve the best performance and stability. Our future work will focus on the proposed controllers' order reduction, an attractive issue for industrial applications.