**1. Introduction**

Mechanical vibrations are problematical phenomena concerning slender structures such as towers, masts, chimneys, bridges [1,2], skyscrapers [3–5], wind turbines [6–10], as well as plate structures [11,12], etc. Most of them are fitted with dedicated solutions for vibration minimisation and fatigue reduction, such as tuned vibration absorbers/tuned mass dampers (TVAs/TMDs), tuned liquid column dampers (TLCDs), bracing systems, etc. [13–18]. TVAs are more and more widely spread vibration reduction systems. A standard (passive) TVA is built as an additional mass connected with the protected structure by a spring and a viscous damper (in parallel), the parameters of which are tuned to the selected mode of the vibration [19]. Passive TVAs cope reasonably well with the vibration of a single frequency but cannot adapt to a broader spectrum [8]. During the structure exploitation lifespan, its frequency response may vary due to i.a. temperature fluctuations, icing, or external loading, apart from the defects that may arise. Thus, more advanced TVA systems have been investigated. Hybrid TVAs (H-TVAs), being the parallel connection of a passive TVA with active actuators [5], are the most dependable systems used in civil

**Citation:** Martynowicz, P. Experimental Study on the Optimal-Based Vibration Control of a Wind Turbine Tower Using a Small-Scale Electric Drive with MR Damper Support. *Energies* **2022**, *15*, 9530. https://doi.org/10.3390/ en15249530

Academic Editor: Mohamed Benbouzid

Received: 15 October 2022 Accepted: 12 December 2022 Published: 15 December 2022

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**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

engineering [15,20–23]. The active force of the H-TVA increases the vibration attenuation efficiency and robustness of the TVA, while its energy, power, and force requirements are much lower than those of an active TVA (i.e., a TVA in which an active actuator replaces a viscous damper) of comparable performance. The devices used for the structural vibration control include active electro-hydraulic/-pneumatic/-magnetic actuators, semiactive magnetorheological (MR) or changeable-orifice dampers, or passive devices such as viscoelastic/hydraulic/friction/granular dampers, etc. [13–15,24].

Most of the (active or semi-active) actuator real-time control solutions are based on the bang-bang strategy [25], fuzzy logic, or two-stage algorithms that consist of the determination of an actuator's required force and its accurate tracking [2,26,27]. The latter concepts experience an inability to produce the required (by the first stage algorithm) force pattern due to the actuator nonlinearities/constraints (including the ever-present force and stroke constraints), the impossibility of generating active forces for semi-active actuators, etc. The stroke limitation of the real-world vibration reduction system, specifically the TVA, is frequently addressed by the use of end-stop bumpers or spring-damper buffer systems [28], which prevent the collision with the primary structure but compromise the control quality at the same time (the presence of the additional springs also alters the TVA tuning frequency within the buffer ranges). As a result, the force pattern determined to be optimal (at the first stage) is not the same as the actuator's output. Some sophisticated algorithms require real-time frequency determination, which may be problematic for polyperiodic or random vibrations.

The idea of a concurrent, parallel operation of the MR damper and the active actuator has only occasionally been investigated, to mention just a few references. Kim et al. [29] used a parallel, concentric connection of the three actuators: a passive air spring, an MR damper, and an electromagnetic coil actuator in a precision machine mount to isolate it from unwanted vibrations. The pneumatic forces constantly supported the heavy weight of an upper structure, the MR damper handled the transient response, while the electromagnetic actuator reduced the resonance response, which was switched mutually with regard to the velocity threshold (the control signal was applied either to the MR damper or the active device). Switching logic was implemented to resolve the problem of interference between the MR damper and active actuator control forces. A simple proportional controller was applied to the MR damper, while a proportional-derivative circuit was applied to the electromagnetic and pneumatic devices. Sophisticated hardware was required to treat the high sampling rate. The parallel combination of the MR damper and active air spring in a hybrid mount system designed for vibration isolation was investigated in [30]. The air spring was used to support the precision stage and to isolate the large loads by controlling the spring coefficient. Additionally, the MR damper force was produced to control the extensive vibrations. The isolation performance was investigated by the utilisation of the simple PID controller for the air servo valve and the MR damper current. The vibration was reduced at low frequencies, ye<sup>t</sup> the response amplitude at frequencies above 30 Hz was slightly increased. In turn, the simultaneous operation of the MR damper arranged in series with the hydraulic cylinder in a bracing system of a shear frame model representing a monopile wind turbine structure was investigated in [31], adopting the linear control theory for the linearised system and actuator—the MR damper was used to emulate the behaviour of the idealised linear dash-pot. The hydraulic cylinder provided the correct displacement across the MR damper, while the bang-bang force tracking algorithm determined the MR damper control current. The controller activated high-frequency modes and generated drift in the actuator displacement, though. Thus, only a fraction of the measured damper force could be used as input to the designed integral force feedback in the real-time hybrid simulations.

With regard to the above considerations, the current study concerns a complex, hybrid MR TVA system (H-MR-TVA) utilising a parallel, concurrent operation of a small-scale electric servo drive [13,22,28,32–34] and an MR damper [2,4,35–41]. An MR damper exhibits a wide range of resistance forces compared with a viscous damper, as well as a millisecond

response time and only signal-level energy requirements, although it cannot add energy to the system when necessary (being a semi-active device); it also suffers from a nonzero remanent force [27,41–45]. On the other hand, a small-scale electric actuator may be used to generate the active forces and cancel the MR damper remanent force while using a modest amount of energy. Simulations and experiments have shown that implementing both an MR damper and a small electric drive in the TVA system may lead to state-of-the-art vibration reduction efficiency. The experimental implementation of a parallel operation of the simultaneously controlled MR damper and electric drive in the TVA system may be considered the originality of this work.

To cope with the control limitations (discussed above), the author devised a concept [34,40,41] to embed the nonlinearities of the actuators (i.e., the MR damper and the electric servo in the current study), including their force constraints, into a control problem formulation, removing efficiency and robustness issues that arise when a determined optimal-based control is imprecisely mapped or beyond the permissible actuator range. This requires the use of nonlinear control methods, which include maximum-principlebased methods [9,12,46], Lyapunov function-based methods [1,3,32,39,44], linearisation methods with linear optimal control theory (LQR/LQG/ *H*2/ *H* ∞) [3,13,28,33,37,47,48], etc. Each method group has advantages and disadvantages, including the high computational load necessary for real-time operation and control authority degradation due to disturbances or unmodelled dynamics. The deployment of a nonlinear maximum-principle-based control method that incorporates actuators' nonlinearities while providing relatively simple real-time hardware implementation is the major contribution of this study.

The concept of the maximum-principle-based nonlinear optimal (or optimal-based) control was previously investigated by the author [40], considering, in particular, a scaled wind turbine tower-nacelle model [41,49–54]. Wind turbines experience varying external loads, such as wind variations, wind shear, Karman vortices, blade passing, changeable inflow conditions for the blades, sea waves, and ice load, etc. Additionally, internal factors, such as rotating machinery unbalance, contribute to the structural vibration and fatigue wear of towers and blades. In order to investigate the problem of wind turbine tower vibrations, a scaled tower-nacelle laboratory model that exhibits partial dynamic similarity (i.e., similarity of motions of tower tips) with a real-world 1.5 MW Vensys82 structure was developed and built. It was assumed that a nacelle, a hub, a shaft, a generator, blades, and a gearbox would all be represented by the rigid body fixed to the top of the rod, modelling a tower. A horizontal force produced by a modal shaker may be applied to the rod modelling a tower or to the rigid body, representing a nacelle assembly. The laboratory model enabled the analysis of two initial bending modes of the tower-nacelle system; however, only the first mode was investigated in the current study, and H-MR-TVA was tuned to its frequency. Current research results may be transferred to a real-world wind turbine thanks to previously determined time, length, and force scale factors [53,54]. The approximate power scale of the laboratory model was 340 W.

The paper is organised as follows. In the next section, a regarded system is described. Subsequently, the optimal vibration control problem is formulated and solved, covering simultaneous MR damper and actuator control. Then, the implementation procedure, the experimental setup, and the test conditions are discussed. This is followed by the key section covering real-time control results. The paper is summarised with several conclusions.

#### **2. A Regarded System**

A scaled wind turbine tower-nacelle model is regarded as a protected structure whose first bending mode modal parameters are: mass *m*1, stiffness *k*1, and damping *c*1. An H-MR-TVA of absorber mass *m*2 and spring stiffness *k*2 is considered (Figure 1). The movement of both *m*1 and *m*2 was constrained to be linear displacement *x*1 and *x*2 (accordingly) along the common axis (horizontal in Figure 1) of an applied excitation force *Fe*, modelling the resultant load applied to the nacelle. An MR damper and a force actuator (*Fa*) were both built parallel to the spring. Although the force actuator was carefully selected to exhibit the

lowest motion damping for the required output nominal force *Fnom*, its influence (excess damping) on the TVA operation was significant; thus, mass *m*2 was increased with regard to the previous research [41] to 14.1% of the mass *m*1 to obtain two local maxima of the primary system displacement (*x*1) frequency response (see Section 7) [19], while stiffness *k*2 was tuned accordingly. Both the MR damper and force actuator were used for control purposes. The values of the adopted system parameters are presented in Table 1.

**Figure 1.** Two-body diagram of a regarded system with an H-MR-TVA.

**Table 1.** The adopted system parameters.


Figure 2 presents the diagram of the theoretical passive TVA efficiency (vs. mass ratio, i.e., *m*2/*m*1), according to the authors of [19], represented by maximum nacelle displacement amplitudes *<sup>A</sup>*(*x*1) for the regarded primary structure parameters (Table 1); *<sup>A</sup>*(•) states for the amplitude. Two dots indicate mass ratios assumed in the current (14.1%) vs. previous research (7.7%) [41]. As it is commonly known, the TVA efficiency characteristic is nonlinear. The reduction in *<sup>A</sup>*(*x*1) due to the mass ratio increase to 14.1% was expected to be 23% only (concerning TVA of 7.7% mass ratio). However, a larger mass ratio contributes to lower TVA sensitivity to detuning [55].

**Figure 2.** Maximum nacelle horizontal displacement amplitude *<sup>A</sup>*(*x*1) vs. passive TVA mass ratio (tuned according to [19]).

#### **3. Control Problem Formulation and Solution**

.

Consider the equation of motion of a vibrating structure with an H-MR-TVA:

$$\dot{z}(t) = f(z(t), u(t), t), \ t \in [t\_0, t\_1] \tag{1}$$

where *z*(*t*) is a state vector:

$$z(t) = \begin{bmatrix} z\_1(t) \ z\_2(t) \ z\_3(t) \ z\_4(t) \end{bmatrix}^T,\tag{2}$$

*u*(*t*) = *u*1 (*t*) *<sup>u</sup>*2 (*t*)*<sup>T</sup>* ∈ *U* (*U* = *R*2) is a piecewise-continuous control vector, and a quality index to be minimised is:

$$G(z, \mu) = \int\_{t\_0}^{t\_1} g(z(t), \mu(t), t)dt. \tag{3}$$

Following Section 2, assume: *z*1 = *x*1, *z*2 = .*x*1, *z*3 = *x*2, *z*4 = .*x*2, thus:

$$f(z(t), u(t), t) = \begin{bmatrix} z\_2(t) \\ \frac{1}{m\_1}(-(k\_1 + k\_2)z\_1(t) - c\_1 z\_2(t) + k\_2 z\_3(t) + F\_{\text{nr}}(z(t), u(t), t) + F\_a(u(t)) + F\_e(t)) \\ z\_4(t) \\ \frac{1}{m\_2}(k\_2 z\_1(t) - k\_2 z\_3(t) - F\_{\text{nr}}(z(t), u(t), t) - F\_a(u(t))) \end{bmatrix} \tag{4}$$

where:

$$\begin{array}{l} F\_{mr}(z(t), u(t), t) = (\mathbb{C}\_1 i\_{mr}(u(t), t) + \mathbb{C}\_2) \tanh\{\nu[(z\_4(t) - z\_2(t)) + (z\_3(t) - z\_1(t))]\} + \\ \quad + (\mathbb{C}\_3 i\_{mr}(u(t), t) + \mathbb{C}\_4)[(z\_4(t) - z\_2(t)) + (z\_3(t) - z\_1(t))] \end{array} \tag{5}$$

is the MR damper force represented by the hyperbolic tangent model with the parameters: *C*1, *C*2, *C*3, *C*4, *ν* [40]; *imr*(*u*(*t*), *t*) is the MR damper control current, *Fa*(*t*) is the actuator force, and *Fe*(*t*) is the excitation force applied to the protected structure. To include the MR damper current restriction to [0, *imax*] range (*imax* > <sup>0</sup>), it was further assumed:

$$i\_{mr}(u(t),t) = i\_{max} \sin^2(u\_1(t)).\tag{6}$$

To include the actuator output (static) nonlinearity, i.e., the nominal force limitation to [−*Fnom*, *Fnom*] range (see the adopted *Fnom* design values in Table 1), it was assumed:

$$F\_{\mathbf{f}}(t) = F\_{\text{nom}} \sin(\mu\_2(t)). \tag{7}$$

*Remark*

For an electric servo drive with a ball screw slide mechanism used as the force actuator in this study (see Section 5), the output was considered to be linear within limits of [−*Fnom*, *Fnom*] (contrary to the electro-hydraulic actuator considered in [34]); the corresponding motor driving torque range was −*rFnom π* − *M*0, *rFnom π* + *<sup>M</sup>*0, where *M*0 is no-load driving torque of the slide unit, while *r* is the spindle radius. In addition to the actuator's static non-linearity (force limitation), its linear dynamics were not considered in this study. The regarded quality function was:

$$\begin{aligned} \mathbf{g}(\mathbf{z}(t), \mathbf{u}(t), t) &= \mathbf{g}\_{11} \mathbf{z}\_1^2(t) + \mathbf{g}\_{12} \mathbf{z}\_2^2(t) + \mathbf{g}\_{13} (\mathbf{z}\_1(t) - \mathbf{z}\_3(t))^2 + \mathbf{g}\_{14} (\mathbf{z}\_2(t) - \mathbf{z}\_4(t))^2 + \mathbf{g}\_{21} \mathbf{i}\_{\text{nr}}{}^2(\mathbf{u}(t), t) + \\ &\quad \mathbf{g}\_{221} \mathbf{F}\_{\text{nr}}^{-2}(\mathbf{z}(t), \mathbf{u}(t), t) + \mathbf{g}\_{222} \mathbf{F}\_{\text{a}}^{-2}(\mathbf{u}(t)) + \mathbf{g}\_{23} \mathbf{P}\_{\text{a}}^{-2}(\mathbf{u}(t)) \end{aligned} \tag{8}$$

to account for the protected structure displacement *z*1 and velocity *z*2 minimisation, the TVA stroke z1 − z3 minimisation, the MR damper coil current *imr* and resistance force *Fmr* minimisation, and the actuator force *Fa* and power *Pa* minimisation, where:

$$P\_{\mathfrak{a}}(t) = F\_{\mathfrak{a}}(t)(z\_2(t) - z\_4(t)).$$

*f* ∗*Tz*


Assume the Hamiltonian in the form:

$$H(\mathcal{J}(t), z(t), u(t), t) = -\mathcal{g}(z(t), u(t), t) + \mathcal{\zeta}^T(t) f(z(t), u(t), t). \tag{9}$$

If (*z*<sup>∗</sup>(*t*), *u*<sup>∗</sup>(*t*)) is an optimal control process, there exists a co-state vector function *ξ* satisfying the equation:

$$\dot{\xi}(t) = -f\_z^{\*T}(z^\*(t), u^\*(t), t)\xi(t) + \mathcal{g}\_z^T(z^\*(t), u^\*(t), t), \ t \in [t\_0, t\_1] \tag{10}$$

with a terminal (transversality) condition:

$$
\xi(t\_1) = 0\tag{11}
$$

so that *u*<sup>∗</sup>(*t*) maximises the Hamiltonian over the set *U* for almost all *t* ∈ [*<sup>t</sup>*0, *t*1] (*fz* and *gz* are *f* and *g* derivatives with respect to *z*; *f* and *g* are continuously differentiable with respect to state and continuous with respect to time and control) [56]. For the analysed system, the co-state vector was:

$$\boldsymbol{\xi}^{\boldsymbol{x}}(t) = \begin{bmatrix} \boldsymbol{\xi}\_{1}(t) \ \boldsymbol{\xi}\_{2}(t) \ \boldsymbol{\xi}\_{3}(t) \ \boldsymbol{\xi}\_{4}(t) \end{bmatrix}^{\boldsymbol{\Gamma}},\tag{12}$$

while:

$$\begin{pmatrix} z^\*(t), u^\*(t), t \end{pmatrix} = \begin{pmatrix} 0 & -\frac{1}{m\_1} \left( k\_1 + k\_2 + \tilde{F}\_{\text{nr}}(z^\*(t), u^\*(t), t) \right) & 0 & \frac{1}{m\_2} \left( k\_2 + \tilde{F}\_{\text{nr}}(z^\*(t), u^\*(t), t) \right) \\ 0 & -\frac{1}{m\_1} \left( c\_1 + \tilde{F}\_{\text{nr}}(z^\*(t), u^\*(t), t) \right) & 0 & \frac{1}{m\_2} \tilde{F}\_{\text{nr}}(z^\*(t), u^\*(t), t) \\ 0 & \frac{1}{m\_1} \left( k\_2 + \tilde{F}\_{\text{nr}}(z^\*(t), u^\*(t), t) \right) & 0 & -\frac{1}{m\_2} \left( k\_2 + \tilde{F}\_{\text{nr}}(z^\*(t), u^\*(t), t) \right) \\ 0 & \frac{1}{m\_1} \tilde{F}\_{\text{nr}}(z^\*(t), u^\*(t), t) & 1 & -\frac{1}{m\_2} \tilde{F}\_{\text{nr}}(z^\*(t), u^\*(t), t) \end{pmatrix} \tag{13}$$

with:

$$\begin{aligned} \bar{F}\_{\text{mr}}(z^\*(t), u^\*(t), t) &= \\ \nu(\mathcal{C}\_1 i\_{\text{mr}}(u^\*(t), t) + \mathcal{C}\_2) \left\{ 1 - \tanh^2 \left[ \nu \left( z\_4^\*(t) + z\_3^\*(t) - z\_2^\*(t) - z\_1^\*(t) \right) \right] \right\} + \left( \mathcal{C}\_3 i\_{\text{mr}}(u^\*(t), t) + \mathcal{C}\_4 \right) \end{aligned} \tag{14}$$

thus:

$$\frac{\partial F\_{\text{nrr}}(z^\*(t), u^\*(t), t)}{\partial z^\*\_3(t)} = \frac{\partial F\_{\text{nrr}}(z^\*(t), u^\*(t), t)}{\partial z^\*\_4(t)} = -\frac{\partial F\_{\text{nrr}}(z^\*(t), u^\*(t), t)}{\partial z^\*\_1(t)} = -\frac{\partial F\_{\text{nrr}}(z^\*(t), u^\*(t), t)}{\partial z^\*\_2(t)}\tag{15}$$

and:

$$\begin{aligned} &g\_z^\*(z^\*(t), u^\*(t), t) \\ &= \begin{bmatrix} 2g\_{11}z\_1^\*(t) + 2g\_{13}(z\_1^\*(t) - z\_3^\*(t)) - 2g\_{221}F\_{mr}'(z^\*(t), u^\*(t), t) \\ 2g\_{12}z\_2^\*(t) + 2g\_{14}(z\_2^\*(t) - z\_4^\*(t)) - 2g\_{221}F\_{mr}'(z^\*(t), u^\*(t), t) + 2g\_{23}F\_a^2(t)(z\_2^\*(t) - z\_4^\*(t)) \\ - 2g\_{13}(z\_1^\*(t) - z\_3^\*(t)) + 2g\_{221}F\_{mr}'(z^\*(t), u^\*(t), t) \\ - 2g\_{14}(z\_2^\*(t) - z\_4^\*(t)) + 2g\_{221}F\_{mr}'(z^\*(t), u^\*(t), t) - 2g\_{23}F\_a^2(t)(z\_2^\*(t) - z\_4^\*(t)) \end{bmatrix} \end{aligned} \tag{16}$$

where:

$$F\_{mr}'(z^\*(t), \mathfrak{u}^\*(t), t) = F\_{mr}(z^\*(t), \mathfrak{u}^\*(t), t)\hat{F}\_{mr}(z^\*(t), \mathfrak{u}^\*(t), t).$$

Thus, Hamiltonian (9) takes a form:

$$\begin{array}{c} H(\boldsymbol{\xi}(t),\boldsymbol{z}(t),\boldsymbol{u}(t),t) = -\boldsymbol{g}\_{11}\boldsymbol{z}\_{1}^{2}(t) - \boldsymbol{g}\_{12}\boldsymbol{z}\_{2}^{2}(t) - \boldsymbol{g}\_{13}(\boldsymbol{z}\_{1}(t) - \boldsymbol{z}\_{3}(t))^{2} - \boldsymbol{g}\_{14}(\boldsymbol{z}\_{2}(t) - \boldsymbol{z}\_{4}(t))^{2} - \boldsymbol{g}\_{15}\boldsymbol{z}\_{3}^{2}(t) \\ \boldsymbol{g}\_{21}\boldsymbol{i}\_{\mathrm{nr}}^{\mathrm{tr}}(\boldsymbol{u}(t),t) - \boldsymbol{g}\_{221}\boldsymbol{F}\_{\mathrm{nr}}^{\mathrm{tr}}(\boldsymbol{z}(t),\boldsymbol{u}(t),t) - \boldsymbol{g}\_{222}\boldsymbol{F}\_{\mathrm{a}}^{\mathrm{tr}}(\boldsymbol{u}(t)) - \boldsymbol{g}\_{23}\boldsymbol{F}\_{\mathrm{a}}^{\mathrm{tr}}(\boldsymbol{u}(t))(\boldsymbol{z}\_{2}(t) - \boldsymbol{z}\_{4}(t))^{2} + \boldsymbol{g}\_{25}\boldsymbol{F}\_{\mathrm{a}}^{\mathrm{tr}}(\boldsymbol{t})f(\boldsymbol{z}(t),\boldsymbol{u}(t),t), \\ \boldsymbol{\xi}^{\mathrm{T}}(t)f(\boldsymbol{z}(t),\boldsymbol{u}(t),t), \end{array}$$

where:

*ξ<sup>T</sup>*(*t*)*f*(*z*(*t*), *<sup>u</sup>*(*t*), *t*) = *ξ*1(*t*) *ξ*2(*t*) *ξ*3(*t*) *ξ*4(*t*) ⎡ ⎢⎢⎢⎣ *<sup>z</sup>*4(*t*) 1 *m*1 (−(*k*<sup>1</sup> + *k*2)*<sup>z</sup>*1(*t*) − *<sup>c</sup>*1*z*2(*t*) + *k*2*z*3(*t*) + *Fmr*(*z*(*t*), *<sup>u</sup>*(*t*), *t*) + *Fa*(*u*(*t*)) + *Fe*(*t*)) *<sup>z</sup>*4(*t*) 1 *m*2 (*k*2*z*1(*t*) − *k*2*z*3(*t*) − *Fmr*(*z*(*t*), *<sup>u</sup>*(*t*), *t*) − *Fa*(*u*(*t*))) ⎤ ⎥⎥⎥⎦

The Hamiltonian maximisation conditions [56] are:

$$\begin{cases} \frac{\partial H(\tilde{z}(t), z^\*(t), \mu(t), t)}{\partial u\_1(t)} = \\\\ \left\{ \left( \frac{1}{m\_1} \mathbb{E} \left( t \right) - \frac{1}{m\_2} \mathbb{E}\_4 (t) - 2 \mathbb{E}\_{\Sigma 21} \mathbb{E}\_{\text{HU}} (z^\*(t), \mu(t), t) \right) \frac{\partial \mathbb{E}\_{\text{tr}} (z^\*(t), \mu(t), t)}{\partial u\_{\text{tr}\uparrow} (\mu(t), t)} - 2 \dot{i}\_{\text{max}} \text{sign}^2 (u\_1(t)) \right\} \sin(2u\_1(t)) \dot{i}\_{\text{max}} = 0 \end{cases} \tag{17}$$

$$\frac{\partial H(\underline{\xi}(t), z^\*(t), u(t), t)}{\partial u\_2(t)} = \left\{ \frac{1}{m\_1} \mathfrak{F}\_2(t) - \frac{1}{m\_2} \mathfrak{F}\_4(t) - 2F\_{\text{non}} \left[ \mathfrak{g}\_{222} + \mathfrak{g}\_{23} (z\_2(t) - z\_4(t))^2 \right] \sin(\mathfrak{u}\_2(t)) \right\} \cos(\mathfrak{u}\_2(t)) \\ F\_{\text{non}} = 0 \tag{18}$$

with the appropriate sign change regimes, where:

$$\frac{\partial F\_{\text{nr}}(\mathbf{z}^\*(t), u(t), t)}{\partial i\_{\text{nr}}(u(t), t)} = \mathbf{C}\_1 \tanh[\mathbf{v}(\mathbf{z}\_4^\*(t) + \mathbf{z}\_3^\*(t) - \mathbf{z}\_2^\*(t) - \mathbf{z}\_1^\*(t))] + \mathbf{C}\_3(\mathbf{z}\_4^\*(t) + \mathbf{z}\_3^\*(t) - \mathbf{z}\_2^\*(t) - \mathbf{z}\_1^\*(t))$$

Let us fix an attention on *<sup>u</sup>*1(*t*) range of [0, *π*]; thus, Equation (17) results in (*g*21 = 0 is assumed):

$$\sin(2u\_1(t)) = 0$$

or:

$$\sin^2(u\_1(t)) = \frac{1}{2i\_{\text{max}\geq 1}} \left( \frac{1}{m\_1} \mathbb{J}\_2(t) - \frac{1}{m\_2} \mathbb{J}\_4(t) - 2g\_{221} F\_{\text{urr}}(z^\*(t), u(t), t) \right) \frac{\partial F\_{\text{urr}}(z^\*(t), u(t), t)}{\partial i\_{\text{mr}}(u(t), t)} \mathbf{a} \tag{19}$$

Analogically to [31]:

$$\mathrm{id}\_{mr}^{\*}\left(\boldsymbol{u}^{\*}\left(\boldsymbol{t}\right),\boldsymbol{t}\right) = \begin{cases} \frac{1}{2\xi\_{21}}\left(\frac{1}{m\_{1}}\mathbb{Z}\_{2}\left(\boldsymbol{t}\right) - \frac{1}{m\_{2}}\mathbb{Z}\_{4}\left(\boldsymbol{t}\right) - 2\xi\_{221}\mathrm{F}\_{\mathrm{mr}}\left(\boldsymbol{z}^{\*}\left(\boldsymbol{t}\right),\boldsymbol{\mu}\left(\boldsymbol{t}\right),\boldsymbol{t}\right)\right) \frac{\partial\mathrm{F}\_{\mathrm{mr}}\left(\boldsymbol{z}^{\*}\left(\boldsymbol{t}\right),\boldsymbol{\mu}\left(\boldsymbol{t}\right),\boldsymbol{t}\right)}{\partial\mathrm{ic}\_{\mathrm{mr}}\left(\boldsymbol{u}\left(\boldsymbol{t}\right),\boldsymbol{t}\right)}, & \text{if } RHS(19) \in \left[0,1\right) \\\\ i\_{\mathrm{max}}\text{ if } RHS(19) \in \geq 1 \end{cases} \tag{20}$$

where *RHS*(19) is the right-hand side of Equation (19). Condition (18) yields (*g*222 = 0 is assumed):

$$\frac{\partial H(\tilde{\xi}(t), z^\*(t), u(t), t)}{\partial u\_2(t)} = \left\{ \frac{1}{2F\_{\text{non}} \left[ g\_{222} + g\_{23} (z\_2(t) - z\_4(t))^2 \right]} \left( \frac{1}{m\_1} \tilde{\xi}\_2(t) - \frac{1}{m\_2} \tilde{\xi}\_4(t) \right) - \sin(\mu\_2(t)) \right\} \cos(\mu\_2(t)) = 0. \tag{21}$$

To analyse Hamiltonian derivative (21) sign change conditions, let us fix an attention on *<sup>u</sup>*2(*t*) range of [−*π*, *π* ], regarding the period of both sin(*<sup>u</sup>*2(*t*)) and cos(*<sup>u</sup>*2(*t*)). This analysis, with the help of Figure 3, yielded proposition (22)(23)(24), covering three disjoint and complementary cases:

$$\begin{aligned} \text{(1)} \quad &if \left\{ \frac{1}{2F\_{\text{norm}} \left[ \mathcal{G}\_{222} + \mathcal{G}\_{23} (z\_2(t) - z\_4(t))^2 \right]} \left( \frac{1}{m\_1} \mathbb{Z}(t) - \frac{1}{m\_2} \mathbb{Z}\_4(t) \right) \right\} \le -1, \text{ then (21) is} \\ &f\_{\text{norm}} \left[ \mathcal{G}(t), z^\*(t), u(t), t \right]\_{1 \le 2, \dots, \omega\_1} \dots \left( \begin{array}{cccc} \omega\_1 & \omega\_2 & \omega\_3 & \omega\_4 & \omega\_5 \end{array} \right) \end{aligned}$$

fulfilled and *∂<sup>u</sup>*2(*t*) exhibits +/− sign change (Hamiltonian maximisation) for: *<sup>u</sup>*2<sup>∗</sup>(*t*) = −*<sup>π</sup>*2only (see Figure 3); *thus*:

$$F\_a^\*(t) = -F\_{\text{now}}.\tag{22}$$

$$\begin{aligned} \text{(2)} \quad &if \left\{ \frac{1}{2F\_{\text{non}} \left[ \mathcal{G}\_{222} + \mathcal{G}\_{23} (z\_2(t) - z\_4(t))^2 \right]} \left( \frac{1}{m\_1} \xi\_2(t) - \frac{1}{m\_2} \xi\_4(t) \right) \right\} \ge 1, \quad \text{then (21) is full-order and } \frac{1}{\sqrt{n}} \xi\_2(t) \text{ is full-order and } \frac{1}{\sqrt{n}} \xi\_2(t) \text{ is full-order, and } \xi\_2(t) \text{ is full-order, respectively.}\\ \text{(2)} \quad &\stackrel{\text{\"}}{=} \frac{1}{\sqrt{n}} (\xi\_2(t) - \xi\_2(t)) \text{ is } \frac{1}{\sqrt{n}} \xi\_2(t) \text{-} (\xi\_2(t) - \xi\_2(t))\\ \text{(3)} \quad &\stackrel{\text{\"}}{=} \frac{1}{2} \text{ only (see Figure 3); } \hbar \text{us} \text{:} \end{aligned}$$

$$F\_a^\*(t) = F\_{nom}.\tag{23}$$

$$\begin{aligned} \text{(3)} \quad &if \left\{ \frac{1}{2\mathbb{F}\_{\text{non}}\left[ \mathcal{G}\_{222} + \mathcal{G}\_{23}(z\_2(t) - z\_4(t))^2 \right]} \left( \frac{1}{m\_1} \mathbb{F}\_2(t) - \frac{1}{m\_2} \mathbb{F}\_4(t) \right) \right\} \in (-1, 1), \quad \text{then (21)}\\ &\text{ is } \mathfrak{c}\text{-field and } \mathfrak{c} \text{ and } \mathfrak{c}\text{-field as } \mathfrak{c} \end{aligned} \tag{21}$$

$$\begin{aligned} \text{is fulfilled and } \frac{\text{var}(\mathcal{G}\_{2}(\cdot))^{2} - (\mathcal{G}\_{1}(\cdot))^{2} - (\mathcal{G}\_{1}(\cdot))^{2}}{\partial u\_{2}(t)} \text{ exhibits } + / - \text{sign change (Hamiltonian maximisation) for } u\_{2}(t) \text{ and } \mathcal{G}\_{1}(\cdot) \text{ are } \pm 1 \text{ and } \pm 2 \text{ respectively.}\\ \text{(a) for: } u\_{2}^{\*} \left(t\right) &= \arcsin\left\{\frac{1}{2F\_{\text{norm}}\left[\mathcal{G}\_{222} + \mathcal{G}\_{23}\left(z\_{2}^{\*}(t) - z\_{4}^{\*}(t)\right)\right]^{2}}{2F\_{\text{norm}}\left[\mathcal{G}\_{222} + \mathcal{G}\_{23}\left(z\_{2}^{\*}(t) - z\_{4}^{\*}(t)\right)\right]^{2}}\right\} \\ \text{ (b) for: } \text{Fixing: } \mathcal{G} \text{ has } \mathcal{G} \end{aligned}$$

only (see Figure 3); thus:

$$F\_a^\*(t) = \frac{1}{2\left[\mathcal{g}\_{222} + \mathcal{g}\_{23}\left(z\_2^\*(t) - z\_4^\*(t)\right)^2\right]} \left(\frac{1}{m\_1}\zeta\_2(t) - \frac{1}{m\_2}\zeta\_4(t)\right) \tag{24}$$

**Figure 3.** Hamiltonian derivative (21) sign analysis.
