2.1 UAS-S4 linear local models

By considering the aircraft differential equations of motion [Reference Caughey23], UAS-S4 state variables can be linearly modeled about its several equilibrium points. The UAS control problem can be solved for both its lateral and longitudinal motions. In this paper, the state variables of the UAS-S4 both lateral and longitudinal motions are controlled.

The state variables of the longitudinal flight dynamics are represented by ${X_{lon}} = {\left[ {u\quad w\quad q\quad \theta } \right]^T}$ , with the axial velocity $u$ , vertical velocity $\;w$ , pitch rate $q$ , and pitch angle $\theta $ while the control input is ${\delta_{lon}} = {\left[ {\begin{array}{cccccc}{{\delta_e}}\quad {{\delta_T}}\end{array}} \right]^T}$ . Even though the control vector is formed by the elevator deflection ${\delta_e}$ and thrust ${\delta_T}$ , the former plays the key role for the pitch control. The lateral flight dynamics state variables represented by ${X_{lat}} = {\left[ {v\quad p\quad \eta\quad \varphi } \right]^T}$ , with the side velocity $v$ , roll rate $p$ , yaw rate $\eta $ , and roll angle $\varphi $ . Based on the aileron and ruder deflections, ${\delta_{lat}} = {\left[ {\begin{array}{cccccc}{{\delta_a}}\quad {{\delta_r}}\end{array}} \right]^T}\;$ is in charge of lateral controls input.

Knowing that the linearised state-space representation of the model around an equilibrium pointis [Reference Nelson24]:

(1) \begin{align}\dot X(t) = A{\rm{\;}}X(t) + B{\rm{\;}}\delta (t)\end{align}

where the longitudinal state-space matrices are:

(2) \begin{align} {A_{lon}} & = \left[ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} {G_u} & {{G_w}} & 0 & - {g}\cos {\theta_0}\\ {{H_u}} & {{H_w}} & {{u_0}} & - {g}\sin {\theta_0}\\ {{M_u} + {M_{\dot w}}{H_u}} & {{M_w} + {M_{\dot w}}{H_w}} & {{M_q} + {u_0}{H_{\dot w}}} & 0\\ 0 & 0 & 1 & 0 \\ \end{array} \right]\nonumber\\[10pt] {B_{lon}} & = \left[ \begin{array}{c@{\quad}c} {{G_{{\delta_e}}}} & {{G_{{\delta_T}}}}\\ {{H_{{\delta_e}}}} & {{H_{{\delta_T}}}}\\ {{{\delta_e}} + {M_{\dot w}}{H_{{\delta_e}}}} & {{M_{{\delta_T}}} + {M_{\dot w}}{H_{{\delta_T}}}}\\ 0 & 0 \end{array} \right]\end{align}

and the lateral state-space matrices are:

(3) \begin{align}{A_{lat}} = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} {{Y_v}} & {{Y_p}} & { - \left( {{u_0} - {Y_r}} \right)} & {{g}\cos {\theta_0}}\\[5pt] {{L_v}} & {{L_p}} & {{L_r}} & 0\\[5pt] {{N_v}} & {{N_p}} & {{N_r}} & 0\\[5pt] 0 & 1 & 0 & 0 \end{array}} \right],\ {B_{lat}} = \left[ {\begin{array}{c@{\quad}c} 0 & {{Y_{{\delta_r}}}}\\[5pt] {{L_{{\delta_a}}}} & {{L_{{\delta_r}}}}\\[5pt] {{N_{{\delta_a}}}} & {{N_{{\delta_r}}}}\\[5pt] 0 & 0 \end{array}} \right]\end{align}

where ${G_u},\;{G_w},{H_u},{H_w},{M_u},{M_w},{M_q}\;$ are the UAS-S4 longitudinal state matric dimensional stability derivatives, and ${G_\delta },{H_\delta },{M_\delta }\;$ are its longitudinal control matric dimensional stability derivatives. In addition, ${Y_v},\;{Y_p},\;{Y_r},\;{L_v},\;{L_p},\;{L_r},\;{N_v},\;{N_p},\;{N_r}$ are the UAS-S4 lateral state matric dimensional stability derivatives, and $\;{Y_\delta },{L_\delta },{N_\delta }$ are its lateral control matric dimensional stability derivatives.

In order to obtain the UAS-S4 state-space matrices’ elements, it is needed to compute the dimensional aerodynamic coefficients and their derivatives. While several research projects on aircraft modeling have been conducted at the LARCASE [Reference Botez, Hamel, Ghazi, Boughari, Theel and Mendoza25–Reference Bardela and Botez27], the most comprehensive study on the UAS-S4 modeling was detailed in [Reference Kuitche and Botez3]. The UAS-S4 model was obtained at the LARCASE using four sub-models representatives of aerodynamics, actuator, propulsion, and mass and inertia.

The first sub-model (aerodynamics) was set up according to the Fderivatives in-house code; this code was based on new aerodynamics methodologies added to DATCOM [Reference Anton, Botez and Popescu28]. The second sub-model (propulsion) was built using a two-stroke engine integration model relying on the operation of an internal combustion engine (Otto Cycle), and on the propeller analysis (Blade Element Theory) [Reference Kuitche, Botez, Viso, Maunand and Moyao29, Reference Romeo, Cestino, Pacino, Borello and Correa30]. Raymer and DATCOM techniques were used to implement the third sub-model (mass and inertia) [Reference Tondji and Botez31]. Finally, the fourth sub-model (a control surface actuation system) was designed using the servomotors’ characteristics, and the final UAS-S4 model was obtained by the sub-models integration [Reference Kuitche and Botez3].

In this way, the UAS-S4 flight dynamics related to both longitudinal and lateral motions was represented using several linear state-space models. Each state-space model expresses the linearised state variables about a specific equilibrium point corresponding to a certain range of altitudes and speeds. However, by increasing the operational range about an equilibrium point, the modeling error due to the linearisation also increases. In order to enhance the models’ accuracy, several equilibrium points can be considered, and consequently, several local linear models can be better fitted into the actual flight dynamics model. Therefore, a fuzzy logic approach is utilised for the UAS-S4 modeling.

2.2 UAS-S4 Fuzzy Logic Model

Basically, an aircraft nonlinear Flight Dynamics Model (FDM) can be represented through its affine system formulation [Reference Lin32] by the equation $ \dot{X} = \mathbb{F}(X) + \mathbb{G}(X) \delta $ , where the control input vector $\delta $ is adjusting the state vector variables $X$ using $\mathbb{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} $ and $\mathbb{G}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} $ functions, that are unknown. A simple nonlinear FDM was found to be more efficient than a complex nonlinear system for the design of a model-based controller, which was our main objective. The higher efficiency of the simple nonlinear FDM was due to its reduced computational complexity, while providing fast control signal calculations in real-time operations [Reference Ying33]. Therefore, the fuzzy logic approach was chosen, as it provided this procedure for approximating affine nonlinear systems [Reference Zeng, Keane and Wang34].

Fuzzy logic offers the type of models that can be used to support the impression of partial truths, where the truth concept may range between completely true and entirely false [Reference Zadeh35]. Fuzzy logic provides a tool for assembling several local linear models, relying on membership functions, with the objective of approximating a nonlinear model. The Takagi-Sugeno Fuzzy Logic modeling method is known as a practical and user-friendly technique for modeling real physical systems [Reference Takagi and Sugeno36] and was chosen in this study.

The Takagi-Sugeno Fuzzy Logic Model (T-S FLM) consists of a set of models that have been locally linearised about their equilibrium points. Based on the expert-defined fuzzy rules in Equation (4), the association of local models can approximate the actual nonlinear continuous-time flight dynamics model. According to the T-S procedure for generating rules, the $i$ ^th rule of the fuzzy model is defined as the following [Reference Takagi and Sugeno36].

(4) \begin{align}{\rm{Rul}}{{\rm{e}}^i}:\;\left\{ {\begin{array}{cccccc}{{\rm{\;if}}\;\;\;\;\;\;{x_1}\;{\rm{is}}\;\;\Gamma_1^i\;{\rm{and}} \ldots {\rm{and}}\;{x_n}\;{\rm{is}}\;\Gamma_n^i}\\[5pt] {\;{\rm{then}}\;\;\dot X(t) = {A_i}X(t) + {B_i}\delta (t)}\\[5pt] {{\rm{where}}\;i = 1, \ldots ,j}\end{array}} \right.\;\end{align}

where the state variables vector $X(t) \in {R^n}$ is controlled by the input $\delta (t) \in R$ for a $j$ number of defined rules. The state-space matrices for the UAS-S4 model should then be converted into their controllable Canonical form, as shown in Equation. (5).

(5) \begin{align}{A_{{i_{n \times n}}}} = \left[ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 && 1 & \ldots & & 0\\[5pt] 0 & & 0 &&&\\[5pt] & \vdots & &\ddots &&\vdots\\[5pt] 0 & & 0 & & 0 & 1\\[5pt] {a_i^n} && {a_i^{n - 1}}& \cdots & {a_i^2} & {a_i^1} \end{array} \right]\end{align}

The fuzzy logic model representation based on the first-order models relying on $j$ rules is [Reference Takagi and Sugeno36]:

(6) \begin{align}\dot X(t) = \frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t)\left( {{A_i}X(t) + {B_i}\delta (t)} \right)}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}}\end{align}

It should be mentioned that ${\phi_i}(t) = \prod\nolimits_{h = 1}^n \Gamma_h^i\left( {X(t)} \right)$ activates the $i$ ^th rule by considering the collected grades $\Gamma_h^i\left( {X(t)} \right)$ that are associated with the membership of $X(t)$ in $\Gamma_h^i$ . An appropriate algorithm is further designed for flight dynamics control by utilising the fuzzy model presented in this section.

3.1 Reference model

Basically, a reference model should define the desirable response of the controlled system to the input command. The design of the reference model is one of the basic aspects of an adaptive control strategy. In addition to offering performance index values (whether for frequency-domain or time-domain characteristics), the reference model should also satisfy its constraints, such as its relative degree and order.

According to the above-mentioned concerns regarding the reference model definition, the desired reference model specifications are determined using the Linear Quadratic Regulator (LQR) procedure applied around the equilibrium point. An LQR controls the state variables using an optimal state-feedback law, that is computed while minimiSing a fine-tuned cost function [Reference Boughari and Botez37]. The design of an LQR is based on the linear state-space model representation, as given in Equation (1). The LQR algorithm calculates the control signal while minimising the following energy-based cost function:

(7) \begin{align}J = \frac{1}{2}\smallint\nolimits_0^\infty {X^T}(t)Q\;X(t) + {\delta ^T}(t)\;R\;\delta (t)\;dt\end{align}

where $Q$ and $R$ are the weight matrices (positive-semi-definite or positive-definite), that clarify the importance of cost function related to the state vector and the control vector, respectively.

Consequently, the LQR control law is:

(8) \begin{align}\delta (t) = - {\rm{{\rm K}}}\;X(t)\end{align}

Following the state feedback gain ${\rm{{\rm K}}}$ and state variables vector $X$ values, the LQR procedure stabilises the flight dynamics of the closed-loop model with respect to the state-space variables usingEquation (9):

(9) \begin{align}\dot X(t) = \left( {A - B{\rm{{\rm K}}}} \right)\;X(t) + B{\rm{{\rm K}}}\;\delta (t)\end{align}

The feedback gain ${\rm{{\rm K}}}$ is computed by:

(10) \begin{align}{\rm{{\rm K}}} = {R^{ - 1}}{B^T}{\mathcal{P}}\end{align}

where matrix ${\mathcal{P}}$ is obtained by solving the following algebraic Riccati equation:

(11) \begin{align}{A^T}{\mathcal{P}} + {\mathcal{P}}A + {\mathcal{Q}} - {\mathcal{P}}PB{R^{ - 1}}{B^T}{\mathcal{P}} = 0\end{align}

Next, the control block of the UAS-S4 model needs to be designed by taking into account the controlled reference model. The Fuzzy Logic Control (FLC) approach is employed in order to solve the challenge of model nonlinearities, as well as to outperform linear controllers.

3.2. Fuzzy Logic Controller (FLC)

Over the past two decades, the use of fuzzy logic for systems control has been developed for a variety of industrial applications. In most comparison studies, the FLC outperforms classical controllers in solving the challenges of nonlinearities, mathematical complexities, and in uncertainties removal [Reference Lin, Zhou, Lu, Wang and Yi38–Reference Radhakrishnan and Swarup40]. In fact, FLC allowed obtaining accurate inputs from approximate inputs through an intuitive converting process [Reference Babaei, Mortazavi and Moradi39].

Basically, the FLC implementation is done in three fundamental steps: fuzzification, fuzzy interface, and defuzzification [Reference Mehrjerdi, Saad and Ghommam41]. The fuzzification block converts crisp data into fuzzy data using proper membership functions. The prepared data is then fed to the Fuzzy Inference System (FIS), which processes the fuzzy data and performs the control tasks according to the IF-THEN rules. Finally, the computed fuzzy control signal is converted into its real signal values through the defuzzification block. The FLC signal is applied to the UAS-S4 flight dynamics, which is modeled using FLM. This control signal is computed as function of the error (the difference between the measured and the desired flight dynamics values). Figure 3 shows the concept of FLC utilised in the closed-loop architecture in charge of the UAS-S4 flight dynamics.

Figure 3.

The fuzzy logic controller utilised for the UAS-S4 flight dynamics.

Regarding the Takagi-Sugeno Fuzzy Logic Model (T-S FLM) described in subsection 2.2, the UAS-S4 FLM should be controlled by use of a compatible FLC. Hence, the T-S Fuzzy Logic Controller (T-S FLC) is needed to be designed.

3.3. T-S Fuzzy Logic Controller

Takagi-Sugeno Fuzzy Logic Control (T-S FLC) method can manage nonlinearities and time-varying parameters while avoiding control algorithm complexity [Reference Tseng, Chen and Uang42]. T-S FLC proved its efficiency on nonlinear systems in terms of state variables regulation and reference model tracking [Reference Kamalasadan and Ghandakly43]. The T-S FLC is structured based on the classical feedback compensator theory [Reference Doyle, Francis and Tannenbaum44], that is established for each local model. The rule-based control law can be mathematically represented by Equation (12) [Reference Takagi and Sugeno36].

(12) \begin{align} {\rm{Rul}}{{\rm{e}}^i}:\;\left\{ \begin{array}{c} {\rm{if}}\ {x_1}\ is\ \acute{\Gamma}_1^i\ {\rm{and}}\ \ldots\ {\rm{and}}\ {x_n}\ {\rm{is}}\ \acute{\Gamma}_n^i \\[4pt] {\;{\rm{then}}\;\;\delta (t) = - {K_i}X(t) + {Z_i}r(t)} \\[4pt] {{\rm{where}}\;i = 1, \ldots ,j} \end{array}\right.\end{align}

where the state variables are controlled by $\delta (t)$ , and rely on the reference signal $r(t)\;$ and adjustable gains denoted by ${K_i}_{1 \times n}$ and ${Z_i}_{1 \times 1}$ .

The T-S FLC output is given by Equation (13):

(13) \begin{align}\delta (t) = {{\sum\nolimits_{i = 1}^j {{\acute{\phi}}_i}(t)\left( { - {K_i}X(t) + {Z_i}r(t)} \right)} \over {\sum\nolimits_{i = 1}^j {{\acute{\phi}}_i}(t)}}\end{align}

By considering $\acute{\phi}_{i}(t) = \prod_{h=1}^{n} \acute{\Gamma}_h^{i} (X(t)) $ , which activates the $i^{\text{th}}$ rule of the fuzzy controller based on the collected grades $\acute{\Gamma}_h^{i}(X(t))$ associated with the membership of $X(t)$ in $\acute{\Gamma}_h^{i}$ . With the aim of obtaining a zero-value tracking error $\acute{\phi}_i(t) = \phi_i(t)|Z_i^{-1}|$ , should be determined in order to formulate Lyapunov function for the system to become asymptotically stable; when the gain of the reference signal value was 1, the controller could fires the proper rule with the same collected grade in the fuzzy model.

The T-S control law can be reproduced, as shown in Equation (14):

(14) \begin{align}\sum \limits_{i = 1}^j {\phi_i}(t)Z_i^{ - 1}\delta (t) - \sum \limits_{i = 1}^j {\phi_i}(t)Z_i^{ - 1}\left( { - {K_i}X(t) + {Z_i}r(t)} \right) = 0\end{align}

Even though the FLC handles nonlinearities, it is affected by the adverse effects of parameters uncertainties. Since the concept of adjustable gains is supposed to overcome these problems, the modified Adaptive Fuzzy Logic Controller (AFLC) is employed, as it relies on adjustable gains. Additionally, we consider the other two main sources of uncertainties, namely external disturbance, and model imperfection. The robust adaptive configuration of the T-S FLC is our solution.

3.4. Adaptive T-S Fuzzy Logic Controller

In Control Systems Engineering, uncertainty is an issue that may appear due to a variety of reasons, and it can adversely affect controller performance. Uncertainty presence may reduce controller robustness, and may lead to systems dynamics instabilities. Therefore, an algorithm should control the nonlinear flight dynamics model while remaining efficient in the presence of uncertainties. To fulfill this objective, a reference model is defined by applying the T-S FLC. Then, the errors are measured by subtracting the UAS-S4 state variables values from the reference model’s state variables values [Reference Cho, Seo and Lee22]. Finally, using a Lyapunov function (which relies on the measured error) for guaranteeing the flight dynamics asymptotic stability, the adaptation laws for gain tuning are calculated. Equation (15) defines the reference model containing the desired state variables, as follows:

(15) \begin{align}{\dot X_r}(t) = {A_r}{X_r}(t) + {B_r}r(t)\end{align}

If ${k_i}_{1 \times n}$ and ${z_i}_{1 \times 1}$ are assumed to be the gains of the desired compensator corresponding to each fuzzy rule, which can regulate the closed-loop response, such that the UAS-S4 state variables exactly follow the reference model state variables, then ${A_r} = {A_i} - {B_i}{k_i}$ and ${B_r} = {B_i}{z_i}$ need to be satisfied. By rearrangingthese last formulations as ${A_i} = {A_r} + {B_i}{k_i}$ and ${B_r} = {B_i}{z_{}}$ , and then, by substituting them into Equation (6), the aircraft’s T-S fuzzy logic representation using the reference model is given in Equation (16).

(16) \begin{align}{\dot{X}}(t) = \left( {{A_r} + \frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}z_i^{ - 1}{k_i}}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){X}(t) + \left( {\frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}z_i^{ - 1}}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){\delta}(t)\end{align}

The error is defined as $E(t) = X(t) - {X_r}(t)$ . This error is further obtained by subtracting Equation (16) from Equation (15). Therefore, the next Equation (17) represents this error.

(17) \begin{align}{\dot E_j}(t) = {A_r}{E_j}(t) + \left( {\frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}z_i^{ - 1}{k_i}}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){X}(t) + \left( {\frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}z_i^{ - 1}}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){\delta}(t) - \left( {\frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){r}(t)\end{align}

By replying Equation (14) into Equation (17), the error can be obtained using next Equation (18):

(18) \begin{align}{\dot E_j}(t) = {A_r}{E_j}(t) + \left( {\frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}\!\left( {{k_i}z_i^{ - 1} - {K_i}Z_i^{ - 1}} \right)}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){X}(t) + \left( {\frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}\!\left( {z_i^{ - 1} - Z_i^{ - 1}} \right)}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){\delta}(t)\end{align}

In order to converge the error to zero, the following Lyapunov function for the stabilisation analysis and reference signal tracking was employed:

(19) \begin{align}V = E_j^TP{E_j} + \sum \limits_{i = 1}^j \left( {\frac{1}{{{\gamma_1}}}{{\left( {{k_i} - {K_i}} \right)}^T}\left| {z_i^{ - 1}} \right|\left( {{k_i} - {K_i}} \right) + \frac{1}{{{\gamma_2}}}{{\left( {{z_i} - {Z_i}} \right)}^T}\left| {z_i^{ - 1}} \right|\left( {{z_i} - {Z_i}} \right)} \right)\end{align}

where $P = {P^T} \gt 0$ is positive-definite matrices and ${A_r}$ stability assumption is guaranteed by use of $A_{{r_i}}^TP + PA_{{r_i}} \lt - Q_{{i}} $ for all matrices $Q_{{i}} = Q_i^T \gt 0$ . In addition, ${\gamma_1}$ and ${\gamma_2}$ are positive constant parameters that are used to finely tune the gains. The gains of the fuzzy controller in Equation (13) can be adjusted via the following adaptation laws (based on FLC gains and their derivatives), obtained by solvingEquation (19) [Reference Cho, Seo and Lee22].

(20) \begin{align}{\dot K_i} = {\gamma_1}sign\!\left( {z_i } \right)\frac{{{\phi_i}B_r^TP{E_j}{X^T}}}{{\sum\nolimits_{i = 1}^j {\phi_i}}},\;\;{\dot Z_i} = - {\gamma_2}sgn\!\left( {z_i } \right)\frac{{{\phi_i}B_r^TP{E_j}\!\left( {\delta + {K_i}X} \right)}}{{{Z_i}\sum\nolimits_{i = 1}^j {\phi_i}}}\end{align}

The stability theorem of adaptive gains is given in [Reference Cho, Seo and Lee22]. Uncertainties dues to the unknown controller’s parameters could affect the adaptive gain ${Z_i}$ , and may approach it to zero value. Since adaptive gain ${Z_i}$ appears in the denominator of Equation (20), in order to guarantee the model stability, the adaptation laws should be modified in cases when the denominator approaches to zero. Therefore, the modified tuning law for an adaptive fuzzy controller is represented in Equation (21) [Reference Hashemi, Botez and Grigorie45]:

(21) \begin{align} & {\dot Z_i} = \left\{\begin{array}{c@{\quad}c} {{w_i},} & {{\rm{if}}\ \left| {{Z_i}} \right| \gt {Z_i}_0\ \ or\ \ {Z_i} = {Z_i}_0\ {\rm{and}}\ {w_i}\ {\rm{sign}}\!\left( {{Z_i}}\right) \lt 0}\\[5pt] {0}, & {\rm{otherwise}} \end{array} \right.\nonumber\\[5pt] & {\rm{where}}\qquad\qquad\qquad\qquad {w_i} = - {\gamma_2}\ {\rm{sign}}\!\left( {z_i } \right)\frac{{B_r^TP{E_j}\!\left( {\delta + {K_i}X} \right)}}{{{Z_i}\sum\nolimits_{i = 1}^j {\phi_i}}}\end{align}

With respect to the stability proof given in [Reference Cho, Seo and Lee22], by assuming a uniformly bounded reference input while analysing the stable reference model, the control law $(K_{i}, Z_{i}, \acute{\phi}_i)$ and tracking error $E$ were guaranteed bounded for all $j\;$ fuzzy logic rules. The convergence of the reference model was ensured, such that $lim_{t \to \infty } {E_j}(t) = 0$ , as the tracking error $E$ converges to zero. This assumption is clarified in the mathematical proof of the general theorem formulated for the designed robust adaptive fuzzy logic laws after Equation (26). Although the presented Adaptive Fuzzy Logic Controller (AFLC) can control nonlinear flight dynamics in the presence of uncertainties, that are dues to unknown controller’s parameters, it remains sensitive against other sources of uncertainties. Model imperfection and external disturbances are the two main causes of uncertainties that adversely affect controller performance, and both of them can be solved using robust control theory.

3.5 Robust Adaptive T-S Fuzzy Logic Controller

Robust control is a static approach that deals explicitly with uncertain parameters and disturbances. In other words, it is utilised to guarantee stability and to obtain robust performance while taking into account disturbances and modeling errors (both of which are assumed to be bounded) [Reference Chabir, Boukhnifer, Bouteraa, Chaibet and Ghommam46].

The uncertainties dues to external disturbances, such as wind shear, gust, and turbulence can be considered mathematically as bounded functions $d\left( {X,t} \right)$ , in which ${D_{{{n \times 1}}}} = {\left[ {0\;\;0\; \ldots 0\;\;1} \right]^T}$ .

(22) \begin{align}\dot X(t) = \frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t)\left( {_i\;X(t) + {B_i}\;\delta (t)} \right)\;}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}} + Dd\!\left( {X,t} \right)\end{align}

Additionally, even if an aircraft is modeled by a skilled expert, relying on perfect aircraft data, uncertainties in modeling may be dues to other causes:

1. – Time-varying parameters, where a fixed controller can not always stabilise its state variables

2. – Ignoring high-order dynamics for the nominal model simplification

3. – Nonlinearities, where systems contain nonlinear dynamics, and models are represented approximately (such as our aircraft nonlinear dynamics, which is approximated using Fuzzy Logic modeling)

Eventually, uncertainties associated with modeling errors of system dynamics can be added mathematically into the state-space matrices of a T-S fuzzy model, as shown in Equation (23):

(23) \begin{align}\dot X(t) = {{\sum\nolimits_{i = 1}^j {\phi_i}(t)\left( {[{A_i} + {\epsilon_{{A_i}}}\left] {\;X(t) + [{B_i} + {\epsilon_{{B_i}}}} \right]\;\delta (t)} \right)\;} \over {\sum\nolimits_{i = 1}^j {\phi_i}(t)}} + Dd\!\left( {X,t} \right)\end{align}

where the errors are bounded, such as $\|{\epsilon_A}_{i}\|_{\infty} \lt \varepsilon$ and $\|{\epsilon_{B}}_{i}\|_{\infty} \lt \varepsilon$ for $i = 1, \ldots ,j$ .

Equation (23) can be written under the following form:

(24) \begin{align}{\dot X}(t) = \left( {{A_r} + {{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}z_i^{ - 1}{k_i}} \over {\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){X}(t) + \left( {{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}z_i^{ - 1}} \over {\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){\delta}(t) + \epsilon \left( {X,\delta } \right) + Dd\!\left( {X,t} \right)\end{align}

and then by considering the uncertainties defined as $f(\epsilon,d) = \epsilon(X,\delta) + Dd(X,t)$ the model would be:

(25) \begin{align}{\dot X}(t) = \left( {{A_r} + {{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}z_i^{ - 1}{k_i}} \over {\sum\nolimits_{i = 1}^j {\phi_i}(t)}}} \right){X}(t) + \left( {{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}z_i^{ - 1}} \over {\sum\nolimits_{ = 1}^j {\phi_i}(t)}}} \right){\delta}(t) + f\!\left( {\epsilon ,d} \right)\end{align}

Following a procedure for computing adaptive gains similar to the ones used in the previous subsection, and based on a robust control strategy [Reference Dullerud and Paganini47], the modified adaptation laws are:

\begin{align*}{\dot K_i} = {\gamma_1}\;\left[ {{\rm{sign}}\!\left( {z_i } \right)\frac{{{\phi_i}B_r^TP{E_j}{X^T}}}{{\sum\nolimits_{i = 1}^j {\phi_i}}} - \vartheta {K_i}\|{E_j}}\| \right],\end{align*}

(26) \begin{align} {\dot Z_i} = \left\{ \begin{array}{c@{\quad}c} {{w_i}}, & {{\rm{if}}\ \left| {{Z_i}} \right| \gt {Z_i}_0\ \ {\rm{or}}\ \ {Z_i} = {Z_i}_0\ \ {\rm{and}}\ \ {w_i}\ {\rm{sign}}\!\left( {{Z_i}} \right) \lt 0}\\[5pt] 0, & {\rm{otherwise}} \end{array}\right.\nonumber\\[5pt] {w_i} = - {\gamma_2}\left[ {\rm{sign}}\!\left( {z_i } \right)\frac{{B_r^TP{E_j}\!\left( {\delta + {K_i}X} \right)}}{{{Z_i}\sum\nolimits_{i = 1}^j {\phi_i}}} - \vartheta {Z_i}\|{E_j}\| \right]\end{align}

Proof. The stability analysis is done based on the designed adaptive laws, as seen in Equations (20) and (21), by use of the Lyapunov function, described in Equation (19). The conditions $\left| {{Z_i}} \right| \gt {Z_i}_0\;\;{\rm{or}}\;\;{Z_i} = {Z_i}_0\;\;{\rm{and}}\;{w_i}{\rm{\;sign}}\!\left( {{Z_i}} \right) \lt 0$ were considered, and $\dot V = - E_j^TQ{E_j}$ was obtained.

In the condition expressed by ${Z_i} = {Z_i}_0$ , when the Lyapunov function is represented with Equation (19), its time derivative is given by:

(27) \begin{align}\dot V = - E_j^TQ{E_j} + 2E_j^TP\frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}\!\left( {z_i^{ - 1} - Z_0^{ - 1}} \right)\left( {{K_i}X + \delta } \right)}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}}\end{align}

As $\left| {{z_i}} \right| \gt {Z_0},\;$ so that $\left( {z_i^{ - 1} - Z_0^{ - 1}} \right)sign\!\left( {z_i^{ - 1}} \right) \lt 0$ , therefore:

(28) \begin{align}E_j^TP\frac{{\sum\nolimits_{i = 1}^j {\phi_i}(t){B_r}\!\left( {z_i^{ - 1} - Z_0^{ - 1}} \right)\left( {{K_i}X + \delta } \right)}}{{\sum\nolimits_{i = 1}^j {\phi_i}(t)}} \lt 0\end{align}

which means that $\dot V \lt 0$ . Hence, for both conditions shown in Equation (21):

\begin{align*}{\dot Z_i} = \left\{\begin{array}{c@{\quad}c} {{w_i},} & {{\rm{if}}\;\left| {{Z_i}} \right| \gt {Z_i}_0\;\;\;\;or\;\;\;\;\;{Z_i} = {Z_i}_0\;\;{\rm{and}}\;{w_i}{\rm{\;sign}}\!\left( {{Z_i}} \right) \lt 0}\\[5pt] {0,} & {{\rm{otherwise}}} \end{array} \right.\end{align*}

(21) \begin{align}{\rm{where}}\;\;\;\;\;\;\;\;\;{w_i} = - {\gamma_2}{\rm{\;sign}}\!\left( {z_i } \right)\frac{{B_r^TP{E_j}\!\left( {\delta + {K_i}X} \right)}}{{{Z_i}\sum\nolimits_{i = 1}^j {\phi_i}}}\end{align}

it obtains:

(29) \begin{align}\dot V \gt - E_j^TQ{E_j}\end{align}

Therefore:

(30) \begin{align}\smallint\nolimits_0^\infty E_j^T{E_j}\; \le \;\frac{{V\!\left( 0 \right) - V\!\left( \infty \right)}}{{{\lambda_{min}}(Q)}}\end{align}

while relying on the Barbalat’s lemma, $lim_{t \to \infty } {E_j}(t) = 0.$

Then, by considering that the adaptive laws contain a robust term represented by Equation (26) in the conditions expressed by $\;\left| {{Z_i}} \right| \gt {Z_i}_0\;\;{\rm{or}}\;\;{Z_i} = {Z_i}_0\;\;{\rm{and}}\;{w_i}{\rm{\;sign}}\!\left( {{Z_i}} \right) \lt 0$ , the Lyapunov function is expressed by Equation (19), therefore, its time derivative becomes:

(31) \begin{align}\dot{V} \le & - {\lambda_{min}}\;(Q)\left\|{E_j}\right\|^2 + 3{\lambda_{max}}\;(P)\varepsilon \!\left\|{E_j}\right\|^2 + {\lambda_{max}}\;(P)\varepsilon\! \left\|{X_r}\right\|^2 + 2{\lambda_{max}}\;(P)\varepsilon \sum \nolimits_{i = 1}^j \left\|{K_i}\right\|\left\|{E_j}\right\|^2\nonumber\\[3pt] & + 2{\lambda_{max}}(P)\varepsilon\! \left\|{E_j}\right\| \sum \nolimits_{i = 1}^j \left[ {\left| {{Z_i}r} \right| + \left\|{K_i}\right\|\left\|{E_j}\right\|} \right] - \vartheta \sum \nolimits_{i = 1}^j {\left( {{k_i} - {K_i}} \right)^T}\left| {z_i^{ - 1}} \right|\left( {{k_i} - {K_i}} \right)\nonumber\\[3pt] & - \vartheta \sum \nolimits_{i = 1}^j \left| {z_i^{ - 1}} \right|{\left( {{z_i} - {Z_i}} \right)^2} + \vartheta \sum \nolimits_{i = 1}^j \left\|\left( {{k_i} - {K_i}} \right)\left| {z_i^{ - 1}} \right| - {K_i}\left\|{E_j}\right\|\right\|^2\nonumber\\[3pt] & + \vartheta \sum \nolimits_{i = 1}^j \left\|\left( {{z_i} - {Z_i}} \right)\left| {z_i^{ - 1}} \right| - {Z_i}\left\|{E_j}\right\|\right\|^2\end{align}

We can determine, so that $6{\lambda_{max}}(P)\varepsilon \lt {\lambda_{min}}(Q)$ . Therefore:

(32) \begin{align} & \dot{V} \le - \frac{1}{2}{\lambda_{min}}\;(Q)\left\|{E_j}\right\|^2 + {\lambda_{max}}\;(P)\varepsilon \left\|{X_r}\right\|^2 + 2{\lambda_{max}}\;(P)\varepsilon \sum \nolimits_{i = 1}^j \left\|{K_i}\right\|\left\|{E_j}\right\|^2\nonumber\\ & + 2{\lambda_{max}}\;(P)\varepsilon \left\|{E_j}\right\| \sum \nolimits_{i = 1}^j \left[ \left| {{Z_i}r} \right| + \left\|{K_i}\right\|\left\|{E_j}\right\| \right] - \vartheta \sum \nolimits_{i = 1}^j {\left( {{k_i} - {K_i}} \right)^T}\left| {z_i^{ - 1}} \right|\left( {{k_i} - {K_i}} \right)\nonumber\\ & - \vartheta \sum \nolimits_{i = 1}^j \left| {z_i^{ - 1}} \right|{\left( {{z_i} - {Z_i}} \right)^2} + \vartheta \sum \nolimits_{i = 1}^j \left\|\left( {{k_i} - {K_i}} \right)\left| {z_i^{ - 1}} \right| - {K_i}\!\left\|{E_j}\right\|\right\|^2 + \vartheta \sum \nolimits_{i = 1}^j \left\|\left( {{z_i} - {Z_i}} \right)\left| {z_i^{ - 1}} \right| - {Z_i}\!\left\|{E_j}\right\|\right\|^2\end{align}

If we consider that:

(33) \begin{align}\dot V \le - \alpha V + \beta \end{align}

where:

(34) \begin{align}\alpha = \frac{{\min\limits \left\{ {\dfrac{1}{2}{\lambda_{min}}(Q),\vartheta } \right\}}}{{\max\limits \left\{ {{\lambda_{min}}(Q),\gamma_1^{ - 1},\gamma_2^{ - 1}} \right\}}}\end{align}

and

(35) \begin{align}\beta & = {\lambda_{max}}\;(P)\varepsilon\!\left\|{X_r}\right\|^2 + 2{\lambda_{max}}\;(P)\varepsilon \sum \nolimits_{i = 1}^j \left\|{K_i}\right\|\left\|{E_j}\right\|^2 + 2{\lambda_{max}}\;(P)\varepsilon \!\left\|{E_j}\right\| \sum \nolimits_{i = 1}^j \left[ {\left| {{Z_i}r} \right| + \left\|{K_i}\right\|\left\|{E_j}\right\|} \right]\nonumber\\[4pt] & + \vartheta \sum \nolimits_{i = 1}^j \left\|\left( {{k_i} - {K_i}} \right)\left| {z_i^{ - 1}} \right| - {K_i}\left\|{E_j}\right\|\right\|^2 + \vartheta \sum \nolimits_{i = 1}^j \left\|\left( {{z_i} - {Z_i}} \right)\left| {z_i^{ - 1}} \right| - {Z_i}\left\|{E_j}\right\|\right\|^2\end{align}

Therefore, $V \le \dfrac{\beta }{\alpha }$ causes exponentially convergence of the Lyapunov function, and feasible stable region in order to guarantee the flight dynamics stability is:

(36) \begin{align}{\mathbb {O}} = \left\{ {x\left| {\;{\beta \over \alpha }} \right. < V} \right\}\end{align}

In other words, adaptive gains guarantee the flight dynamics stability, as long as the amount of bounded uncertainties respect the threshold (the border of Equation (36) as the feasible stable region). Additionally, the leakage factor $\vartheta $ in the robust term should be carefully tuned based on a trade-off; a larger value for $\vartheta $ improves the controller robustness, while a smaller value provides more accurate reference model state variables tracking [Reference Blažič, Matko and Škrjanc48]. The mechanism of our designed T-S-based Robust Adaptive Fuzzy Logic Controller (RAFLC) block diagram is depicted in Fig. 4.

Figure 4.

The designed Robust Adaptive T-S Fuzzy Logic Controller (RAFLC) mechanism.