Design and Implementation of an ${\mathcal{L}}_{\mathbf{1}}$ Adaptive Proportional Output Feedback Controller

Abstract

A new approach for output feedback ${\mathcal{L}}_1$ adaptive control based on a proportional adaptation law is presented. The effectiveness of this design is assessed in simulation and validated through real-time testing on an airfoil pitch control wind tunnel experimental rig. Experimental evaluation of the robustness of the controllers, assessed by introducing various disturbances into the control signals, shows that the adaptive control has a better performance compared to PID control, particularly in scenarios with reduced control effectiveness and time-varying disturbances. The experimental results demonstrate the efficacy of the proposed method in practical applications.

1. Introduction

Control systems have played a crucial role in aerospace, employing a variety of methods to enhance performance and ensure the resilience of aerospace systems [1]. Traditional controllers were designed to handle small perturbations about a particular operating point, and if there were uncertainties or large variations present, the system could become unstable or perform poorly. As a result, the concept of adaptive control theory was introduced, which involved modifying controller parameters in response to unknown plant variations [2].

One commonly used approach is model reference adaptive control (MRAC); this calculates the controller parameters in order to make an unknown dynamic system follow the desired reference model behaviour [3]. There are two ways to put this into action: direct and indirect [2]. In the indirect approach, the first step involves estimating the parameters of the plant. Once these parameters are known, they are used to compute the appropriate controller parameters. The direct approach, on the other hand, involves directly determining the controller gains needed to achieve the desired system behaviour.

From MRAC systems analysis, it is straightforward to show that improving the response to system variations can be achieved with high adaptation gains [4]. However, it was identified that doing so would lead to a reduction in the time-delay margin and the introduction of high-frequency oscillations in the control signal [4]. Furthermore, the absence of proper guidelines for determining an adequate adaptation rate means a greater dependence on extensive simulations or trial-and-error methods. The challenge was to select the right adaptation gain and ensure good performance bounds in transient regime. Hence, ${\mathcal{L}}_1$ adaptive control architecture was developed to decouple the adaptation rate from the system robustness [5]. ${\mathcal{L}}_1$ adaptive control includes a low-pass filter in the control channel, thus limiting its bandwidth. Inclusion of the low-pass filter helps compensate for the uncertainty within the bandwidth of control channel.

${\mathcal{L}}_1$ adaptive control has been applied for various practical control systems—[6,7,8,9,10,11,12,13], to cite a few. However, relatively few methods have been presented for output feedback ${\mathcal{L}}_1$ adaptive control. The first approach employed a projection-based adaptive law [14], but was constrained to first-order reference systems. Another option, applied to some aerospace problems, caters to non-strictly positive real reference systems [15], utilising a piecewise-adaptive law to meet desired performance bounds. In [16], an ${\mathcal{L}}_1$ adaptive output feedback controller was applied to manage disturbances at the system output, successfully demonstrated in managed pressure drilling experiments. Similarly, Lee et al. [17] devised an ${\mathcal{L}}_1$ adaptive output feedback controller for underactuated multi-input–multi-output (MIMO) systems, augmenting an existing three-loop autopilot. This methodology found application in temperature control, as detailed in [18], while Ma and Cao [19] showcased its integration with model predictive control for multivariable nonlinear systems subject to constraints. Additionally, Ma and Cao [20] explored the application of ${\mathcal{L}}_1$ adaptive output feedback control to general partial differential equation (PDE) systems, and Zhou et al. [21] addressed uncertain nonlinear systems with unmodelled dynamics and actuator faults utilising a similar design strategy. The primary limitation of these approaches lies in their reliance on transfer function formulation, which makes the interpretation of uncertainties and reference models less intuitive compared to a state-space model formulation [22,23]. Indeed, state variables offer a more comprehensive reflection of the internal characteristics of a system [21,24]. However, the main limitation of the approach presented in [22,24] is that it is restricted to non-minimum phase systems with a relative degree of one.

Furthermore, although ${\mathcal{L}}_1$ adaptive control has been successfully demonstrated in various applications in simulation environments, its practical implementations in real aerospace systems are few.

However, several devices have been created for the purpose of demonstrating the working of feedback control on experimental laboratory rigs. For example, Mansor et al. [25] conducted an experiment to test the performance of a standard PID control algorithm using the 3-DOF helicopter rig from [26]. Subramanian et al. [27] implemented model reference adaptive control on a 2-DOF helicopter rig from [28] by incorporating a baseline LQR controller. They were able to improve the stability characteristics by achieving faster convergence. For fixed-wing aircraft, Villarreal-Valderrama et al. [29] built a wind tunnel test bed to study aircraft pitch control. They tested a PID controller and a loop shaping compensator on the test bed and found that both produced responses that were similar to the theoretical response. This demonstrated the suitability of their test unit for teaching or educational purposes. This is similar to the flight control demonstrator unit used in this paper [30,31], which was designed to as a demonstrator of some of the principles of automatic flight control. The equipment consists of a basic aerofoil in a wind tunnel, as shown in the schematic in Figure 1. The aerofoil is a NACA 0012 and is free to pitch about a horizontal axis located at the centre of gravity. It is controlled by an elevator that is driven by a servomotor controlled by an Arduino board that receives the pivot axis angle measurement from a potentiometer.

This paper demonstrates the real-world applicability of an output feedback ${\mathcal{L}}_1$ adaptive control approach via its implementation on the automatic flight control demonstrator unit. The approach is based on the proportional adaptation law that directly relates the adaptation to the estimate error [32]. This method reduces computational processing time by making the error bounds proportional to the adaptation gain, in contrast to the gradient descent approach where the relation is with the square root of the adaptation gain. Additionally, this architecture introduces phase margin in the estimation loop, enabling the state predictor to incorporate first-order sensor dynamics. Although the initial approach relies on complete state feedback, here it is extended to the output feedback control case. Another advantage of using this approach is that it can be applied to non-semi-positive real (SPR) systems, i.e., it relaxes the assumption of non-minimum phase systems with a relative degree of one [22,24] while maintaining the state-space formulation of the systems.

The remainder of this paper is organised as follows. Section 2 describes the proportional ${\mathcal{L}}_1$ adaptive controller design. In Section 3, the analysis of the ${\mathcal{L}}_1$ adaptive control is presented. Simulation results are presented in Section 4. In Section 5, the implementation of the designed controller for the airfoil system is discussed. Finally, some conclusions are drawn.

2. Output Feedback Proportional ${\mathcal{L}}_1$ Adaptive Controller

In this section, the theory and design of the output feedback proportional adaptive law are provided. The design mostly builds upon the original proportional adaptive law introduced by Vanness et al. [32,33]. The significant change made is to the adaptation law itself. As a result, the overall structure and analysis remain quite similar. The proofs have been recreated, with the main difference appearing in how the estimation error is bounded.

Problem Formulation

The controlled system (plant) dynamics are represented in a state space form:

$$\begin{array}{cc}\hfill \dot{x}\left(t\right)& =A_{m}x\left(t\right)+B_m\omega u\left(t\right)+f(t,x\left(t\right))\hfill \\ \hfill y\left(t\right)& =C^{\top }x\left(t\right),x\left(0\right)=x_0,\hfill \end{array}$$

(1)

where $x\left(t\right)$ is the plant state vector, $u\left(t\right)$ is the control signal, $B_m$ is a known constant vector, $A_m$ is a known Hurwitz matrix representing the desired model dynamics, $\omega$ is unknown input gain and f is an unknown continuous nonlinear map representing uncertainties. The control objective is to design an output feedback controller such that the output $y\left(t\right)$ tracks a bounded continuous reference input $r\left(t\right)$.

Assumption 1. 

The given control design assumes that there exists positive constants, denoted as L and B, such that

$$\begin{array}{cc}\hfill & \parallel f(t,x_1)-f(t,x_2){\parallel }_{\infty }\le L{\parallel x_1-x_2\parallel }_{\infty }\hfill \\ \hfill & {\parallel f(t,x)\parallel }_{\infty }\le L{\parallel x\parallel }_{\infty }+B\hfill \end{array}$$

where the numbers L and B can be arbitrarily large.

Assumption 2. 

The unknown input gain ω is bounded, i.e., $0<{\omega }_l\le \omega \le {\omega }_u$.

Control Architecture

The proposed control design involves a state predictor, an adaptation law and a control law, as shown in Figure 2.

State Predictor: Consider the following state predictor:

$$\begin{array}{cc}\hfill \dot{\widehat{x}}\left(t\right)& =A_m\widehat{x}\left(t\right)+B_m{\omega }_{0}u\left(t\right)+\widehat{\eta }\left(t\right),\widehat{x}\left(0\right)=x_0\hfill \\ \hfill \widehat{y}\left(t\right)& =C^{\top }\widehat{x}\left(t\right),\hfill \end{array}$$

(2)

where ${\omega }_0$ is the appropriate guess for unknown input gain $\omega$, and overall uncertainties are then combined together to form the vector of estimates $\widehat{\eta }\left(t\right)$.

Adaptation Law: The proposed adaptation law extends the state feedback case [32,33] to output feedback. The adaptation is determined by

$$\begin{array}{c}\hfill \widehat{\eta }\left(t\right)=-\Gamma W\tilde{y}\left(t\right)\end{array}$$

(3)

where $\tilde{y}\left(t\right)=\widehat{y}\left(t\right)-y\left(t\right)$ is the output prediction error, $\Gamma$ is the adaptation gain and W is a $n\times m$ arbitrary matrix such that $W{C}^{\top }\ne 0$.

Control Law: The control law is defined by

$$\begin{array}{cc}\hfill u\left(s\right)& =-KD\left(s\right)({\omega }_{0}u\left(s\right)+{\widehat{\eta }}_1\left(s\right)-k_{g}r\left(s\right)),\hfill \\ \hfill {\widehat{\eta }}_1\left(s\right)& =\varphi \left(s\right)\widehat{\eta }\left(s\right),\varphi \left(s\right)=\frac{C^{\top }{(s\mathbb{I}-A_m)}^{-1}}{C^{\top }{(s\mathbb{I}-A_m)}^{-1}{B}_m},\hfill \end{array}$$

(4)

where $r\left(s\right)$ and $\widehat{\eta }\left(s\right)$ are the frequency domain representations of $r\left(t\right)$ and $\widehat{\eta }\left(t\right)$, respectively; $k_g=1/\left(C^{\top }{A}_m^{-1}{B}_m\right)$ is the feedforward gain meant to eliminate steady-state error; $D\left(s\right)$ is a strictly proper transfer function; and K is the feedback gain, which gives

$$C\left(s\right)=\frac{\omega kD\left(s\right)}{1+\omega kD\left(s\right)}$$

which is a strictly proper transfer function with a steady-state gain $C\left(0\right)=1$. The above described controller is subjected to the following ${\mathcal{L}}_1$-norm condition:

$${\parallel G\left(s\right)\parallel }_{{\mathcal{L}}_1}L<1$$

where

$$\begin{array}{cc}\hfill & G\left(s\right)=({\mathbb{I}}_n-H_{xm}\left(s\right)C\left(s\right)H_m^{-1}\left(s\right)C^{\top })H_{xum}\left(s\right),\hfill \\ \hfill & H_{xum}\left(s\right)={(s\mathbb{I}-A_m)}^{-1},H_{xm}\left(s\right)={(s\mathbb{I}-A_m)}^{-1}{B}_m,\hfill \\ \hfill & H_m\left(s\right)=C^{\top }{(s\mathbb{I}-A_m)}^{-1}{B}_m\hfill \end{array}$$

3. Controller Analysis

In this section, the performance of the ${\mathcal{L}}_1$ adaptive controller is analysed. More specifically, the following are shown:

• The reference model resulting from perfect knowledge of the uncertainties and a corresponding non-adaptive controller is stable, subject to some conditions involving the filter.
• The prediction error, i.e., the error between the state of the plant and that of the state predictors, is bounded.
• The difference between the state/input of the system and those of the reference system is proportional to the prediction error.

3.1. Reference System Analysis

Consider the following closed-loop reference system:

$$\begin{array}{cc}\hfill & {\dot{x}}_{\mathrm{ref}}\left(t\right)=A_m{x}_{\mathrm{ref}}\left(t\right)+B_m\omega u_{\mathrm{ref}}\left(t\right)+{\eta }_{\mathrm{ref}}\left(t\right),x_{\mathrm{ref}}\left(0\right)=x_0\hfill \\ \hfill & y_{\mathrm{ref}}\left(t\right)=C^{\top }{x}_{\mathrm{ref}}\left(t\right)\hfill \\ \hfill & {\eta }_{\mathrm{ref}}\left(t\right)=f(t,x_{\mathrm{ref}}\left(t\right))\hfill \\ \hfill & u_{\mathrm{ref}}\left(s\right)=-\frac{C\left(s\right)}{\omega }(\varphi \left(s\right){\eta }_{\mathrm{ref}}\left(s\right)-k_{g}r\left(s\right))\hfill \end{array}$$

(5)

Lemma 1. 

If K and $D\left(s\right)$ satisfy the ${\mathcal{L}}_1$-norm condition, then the closed-loop reference system is BIBS stable with respect to reference $r\left(t\right)$ and initial state vector $x_0$.

Proof. 

It follows from (5) that

$$\begin{array}{c}\hfill x_{\mathrm{ref}}\left(s\right)=G\left(s\right){\eta }_{\mathrm{ref}}\left(s\right)+H_{xm}\left(s\right)C\left(s\right)k_{g}r\left(s\right)+x_{in}\left(s\right)\end{array}$$

(6)

where $x_{in}\left(s\right)={(s\mathbb{I}-A_m)}^{-1}{x}_0$ is bounded since $A_m$ is Hurwitz. Then, it follows that

$$\begin{array}{c}\hfill \parallel x_{\mathrm{ref}\tau }{\parallel }_{{\mathcal{L}}_{\infty }}\le {\parallel G\left(s\right)\parallel }_{{\mathcal{L}}_1}\parallel {\eta }_{\mathrm{ref}\tau }{\parallel }_{{\mathcal{L}}_{\infty }}+\parallel H_{xm}\left(s\right)C\left(s\right)k_g{\parallel }_{{\mathcal{L}}_1}{\parallel r\parallel }_{{\mathcal{L}}_{\infty }}+{\parallel x_{in}\parallel }_{{\mathcal{L}}_{\infty }}\end{array}$$

(7)

and using Assumption 1, the upper bound for ${\eta }_{\mathrm{ref}}\left(t\right)$ is

$$\begin{array}{c}\hfill \parallel {\eta }_{\mathrm{ref}\tau }{\parallel }_{{\mathcal{L}}_{\infty }}\le L{\parallel x_{\mathrm{ref}\tau }\parallel }_{{\mathcal{L}}_{\infty }}+B\end{array}$$

Substituting and solving for $\parallel x_{\mathrm{ref}\tau }{\parallel }_{{\mathcal{L}}_{\infty }}$,

$$\begin{array}{c}\hfill \parallel x_{\mathrm{ref}\tau }{\parallel }_{{\mathcal{L}}_{\infty }}\le \frac{{\parallel G\left(s\right)\parallel }_{{\mathcal{L}}_1}B+\parallel H_{xm}\left(s\right)C\left(s\right)k_g{\parallel }_{{\mathcal{L}}_1}{\parallel r\parallel }_{{\mathcal{L}}_{\infty }}+{\parallel x_{in}\parallel }_{{\mathcal{L}}_{\infty }}}{1-{\parallel G\left(s\right)\parallel }_{{\mathcal{L}}_1}L}\end{array}$$

(8)

Hence, $\parallel x_{\mathrm{ref}\tau }{\parallel }_{{\mathcal{L}}_{\infty }}$ is bounded if K and $D\left(s\right)$ verify the ${\mathcal{L}}_1$-norm condition. □

3.2. Estimation Error Analysis

The following lemma states that the bound of the state estimation error $\tilde{x}\left(t\right)=\widehat{x}\left(t\right)-x\left(t\right)$ is uniformly bounded.

Lemma 2. 

If the closed-loop system is stable, then the estimation error is uniformly bounded, and its bound is defined by

$$\begin{array}{c}\hfill \parallel {\tilde{x}}_{\tau }{\parallel }_{{\mathcal{L}}_{\infty }}\le {\gamma }_0={\displaystyle \frac{{\Delta }_m}{\Gamma \parallel WC\parallel }}{\left(\frac{{\lambda }_{max}\left(P\right)}{{\lambda }_{min}\left(P\right)}\right)}^{3/2}\end{array}$$

(9)

where ${\Delta }_m$ is defined below.

Proof. 

From (1) and (2), the prediction error can be written as

$$\begin{array}{cc}\hfill & \dot{\tilde{x}}\left(t\right)=A_m\tilde{x}\left(t\right)+\widehat{\eta }\left(t\right)-\eta \left(t\right)+B_m({\omega }_0-\omega )u\left(t\right)\hfill \\ \hfill & \tilde{y}\left(t\right)=C^{\top }\tilde{x}\left(t\right),\tilde{x}\left(0\right)={\tilde{x}}_0=0\hfill \end{array}$$

(10)

Rewriting the prediction error dynamics in frequency domain, we have

$$\begin{array}{c}\hfill C^{\top }\tilde{x}\left(s\right)=H_m\left(s\right)\tilde{\eta }\left(s\right)\end{array}$$

where

$$\begin{array}{c}\hfill \tilde{\eta }\left(s\right)=({\omega }_0-\omega )u\left(s\right)+{\widehat{\eta }}_1\left(s\right)-{\eta }_1\left(s\right)\end{array}$$

and ${\eta }_1\left(s\right)=\varphi \left(s\right)\eta \left(s\right)$.

Now, we consider the Lyapunov function candidate

$$\begin{array}{c}\hfill V\left(t\right)=\frac{1}{2\Gamma }{\tilde{x}}^{\top }\left(t\right)P\tilde{x}\left(t\right)\end{array}$$

(11)

where P is the solution of the Lyapunov equation

$$\begin{array}{c}\hfill A_m^{\top }P+P{A}_m=-Q\end{array}$$

where $Q=Q^{\top }>0$ is an arbitrary positive definite matrix. Hence, including the adaptation law, it can be written

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)=& -\frac{1}{2\Gamma }{\tilde{x}}^{\top }\left(t\right)Q\tilde{x}\left(t\right)-{\tilde{x}}^{\top }\left(t\right)PW{C}^{\top }\tilde{x}\left(t\right)\hfill \\ \hfill & +\frac{1}{\Gamma }{\tilde{x}}^{\top }\left(t\right)P(B_m({\omega }_0-\omega )u\left(t\right)-\eta \left(t\right))\hfill \end{array}$$

(12)

Hence, an upper bound can be derived for $t\in [0,\tau ]$

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & -{\lambda }_{min}\left(P\right)\parallel WC\parallel \parallel \tilde{x}{\left(t\right)\parallel }^2+\frac{1}{\Gamma }\parallel \tilde{x}\left(t\right)\parallel {\lambda }_{max}\left(P\right){\Delta }_m\hfill \end{array}$$

(13)

where ${\Delta }_m$ is bounded as

$$\begin{array}{c}\hfill \parallel B_m({\omega }_0-\omega )u\left(t\right)-\eta \left(t\right)\parallel \le {\Delta }_m=({\omega }_u-{\omega }_l)\parallel B_m\parallel {\rho }_u+\sqrt{n}(L\rho +B)\end{array}$$

(14)

where $\rho$ and ${\rho }_u$ are the upper bounds for state and control variable, respectively. Hence,

$$\begin{array}{c}\hfill \dot{V}\left(t\right)<0\mathrm{if}\parallel \tilde{x}\left(t\right)\parallel >\frac{{\lambda }_{max}\left(P\right){\Delta }_m}{{\lambda }_{min}\left(P\right)\parallel WC\parallel \Gamma }\end{array}$$

(15)

and since $\frac{1}{2\Gamma }{\lambda }_{min}\left(P\right){\parallel \tilde{x}\left(t\right)\parallel }^2\le V\left(t\right)$, we obtain

$$\begin{array}{c}\hfill \parallel {\tilde{x}}_{\tau }{\parallel }_{{\mathcal{L}}_{\infty }}\le {\displaystyle \frac{{\Delta }_m}{\Gamma \parallel WC\parallel }}{\left(\frac{{\lambda }_{max}\left(P\right)}{{\lambda }_{min}\left(P\right)}\right)}^{3/2}\end{array}$$

(16)

□

3.3. Performance Bounds Analysis

In the following theorem, the performance bounds of ${\mathcal{L}}_1$ proportional adaptive law are shown.

Theorem 1. 

Given the system (1), the reference system (5) and the ${\mathcal{L}}_1$ adaptive controller (2) to (4), we have

$$\begin{array}{c}\hfill \parallel x_{\mathrm{ref}}{-x\parallel }_{{\mathcal{L}}_{\infty }}\le {\gamma }_1\\ \hfill \parallel u_{\mathrm{ref}}{-u\parallel }_{{\mathcal{L}}_{\infty }}\le {\gamma }_2\end{array}$$

(17)

where

$$\begin{array}{cc}\hfill & {\gamma }_1=\frac{{∥H_{xm}\left(s\right)C\left(s\right)H_m^{-1}\left(s\right)C^{\top }∥}_{{\mathcal{L}}_1}}{1-{\parallel G\left(s\right)\parallel }_{{\mathcal{L}}_1}L}{\gamma }_0\hfill \\ \hfill & {\gamma }_2={∥\frac{C\left(s\right)}{\omega }\varphi \left(s\right)∥}_{{\mathcal{L}}_1}L{\gamma }_1+{∥\frac{C\left(s\right)H_m^{-1}\left(s\right)C^{\top }}{\omega }∥}_{{\mathcal{L}}_1}{\gamma }_0\hfill \end{array}$$

Proof. 

The control law (4) can be written as

$$u\left(s\right)=-\frac{C\left(s\right)}{\omega }({\eta }_1\left(s\right)+\tilde{\eta }\left(s\right)-k_{g}r\left(s\right))$$

(18)

Converting the plant dynamics to frequency domain and substituting the given control law, we obtain

$$\begin{array}{c}\hfill x\left(s\right)=H_{xm}\left(s\right)C\left(s\right)k_{g}r\left(s\right)+G\left(s\right)\eta \left(s\right)-H_{xm}\left(s\right)C\left(s\right)\tilde{\eta }\left(s\right)+x_{in}\left(s\right)\end{array}$$

(19)

Taking the difference between (6) and (19),

$$\begin{array}{cc}\hfill & x_{\mathrm{ref}}\left(s\right)-x\left(s\right)=G\left(s\right)({\eta }_{\mathrm{ref}}\left(s\right)-\eta \left(s\right))+H_{xm}\left(s\right)C\left(s\right)\tilde{\eta }\left(s\right)\hfill \\ \hfill & x_{\mathrm{ref}}\left(s\right)-x\left(s\right)=G\left(s\right)({\eta }_{\mathrm{ref}}\left(s\right)-\eta \left(s\right))+H_{xm}\left(s\right)C\left(s\right)H_m^{-1}\left(s\right)C^{\top }\tilde{x}\left(s\right)\hfill \end{array}$$

(20)

Then, the following bound holds for $t\in [0,\tau ]$:

$$\begin{array}{cc}\hfill \parallel x_{\mathrm{ref}}{\left(s\right)-x\left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\le {\parallel G\left(s\right)\parallel }_{{\mathcal{L}}_1}& \parallel {\eta }_{\mathrm{ref}}{\left(s\right)-\eta \left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\hfill \\ & +\parallel H_{xm}\left(s\right)C\left(s\right)H_m^{-1}\left(s\right)C^{\top }{\parallel }_{{\mathcal{L}}_1}{\parallel {\tilde{x}}_{\tau }\left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\hfill \end{array}$$

(21)

From Assumption 1, it follows that

$$\begin{array}{cc}\hfill \parallel x_{\mathrm{ref}}{\left(s\right)-x\left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\le {\parallel G\left(s\right)\parallel }_{{\mathcal{L}}_1}L& \parallel x_{\mathrm{ref}}{\left(s\right)-x\left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\hfill \\ & +\parallel H_{xm}\left(s\right)C\left(s\right)H_m^{-1}\left(s\right)C^{\top }{\parallel }_{{\mathcal{L}}_1}{\gamma }_0\hfill \end{array}$$

(22)

which leads to the bound

$$\begin{array}{c}\hfill \parallel x_{\mathrm{ref}}{\left(s\right)-x\left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\le \frac{\parallel H_{xm}\left(s\right)C\left(s\right)H_m^{-1}\left(s\right)C^{\top }{\parallel }_{{\mathcal{L}}_1}}{1-{\parallel G\left(s\right)\parallel }_{{\mathcal{L}}_1}L}{\gamma }_0\end{array}$$

(23)

Now, taking the difference between the reference system and plant control inputs,

$$\begin{array}{cc}\hfill u_{\mathrm{ref}}\left(s\right)-u\left(s\right)=-\frac{C\left(s\right)}{\omega }& (\varphi \left(s\right){\eta }_{\mathrm{ref}}\left(s\right)-k_{g}r\left(s\right))\hfill \\ & +\frac{C\left(s\right)}{\omega }(\varphi \left(s\right)\eta \left(s\right)+\tilde{\eta }\left(s\right)-k_{g}r\left(s\right))\hfill \end{array}$$

(24)

Hence,

$$\begin{array}{c}\hfill u_{\mathrm{ref}}\left(s\right)-u\left(s\right)=-\frac{C\left(s\right)}{\omega }\varphi \left(s\right)({\eta }_{\mathrm{ref}}\left(s\right)-\eta \left(s\right))+\frac{C\left(s\right)}{\omega }\tilde{\eta }\left(s\right)\end{array}$$

(25)

and the following bound holds for $t\in [0,\tau ]$:

$$\begin{array}{cc}\hfill \parallel u_{\mathrm{ref}}{\left(s\right)-u\left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\le {∥\frac{C\left(s\right)}{\omega }\varphi \left(s\right)∥}_{{\mathcal{L}}_1}L& \parallel x_{\mathrm{ref}}{\left(s\right)-x\left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\hfill \\ & +{∥\frac{C\left(s\right)H_m^{-1}\left(s\right)C^{\top }}{\omega }∥}_{{\mathcal{L}}_1}{\parallel {\tilde{x}}_{\tau }\left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\hfill \end{array}$$

(26)

which leads to the bound on the input,

$$\begin{array}{c}\hfill \parallel u_{\mathrm{ref}}{\left(s\right)-u\left(s\right)\parallel }_{{\mathcal{L}}_{\infty }}\le {∥\frac{C\left(s\right)}{\omega }\varphi \left(s\right)∥}_{{\mathcal{L}}_1}L{\gamma }_1+{∥\frac{C\left(s\right)H_m^{-1}\left(s\right)C^{\top }}{\omega }∥}_{{\mathcal{L}}_1}{\gamma }_0\end{array}$$

(27)

□

4. Simulation Results

In this section, simulation results of the proposed output feedback proportional ${\mathcal{L}}_1$ adaptive controller are presented. The proportional ${\mathcal{L}}_1$ adaptive controller is compared with both a PID controller and the piecewise adaptation output feedback controller presented in [15]. The piecewise controller design is detailed in Appendix A.

The dynamics of the aerofoil can be represented by a second-order differential equation:

$$I\ddot{\alpha }\left(t\right)=-\overline{q}S{l}_0{a}_1\alpha \left(t\right)+\overline{q}S(l_0{a}_2+c{m}_0)\delta \left(t\right)-b\dot{\alpha }\left(t\right)$$

(28)

where i is the moment of inertia of the aerofoil; $\alpha$ is the angle of attack, also equal to the pitch angle; $\overline{q}$ is the dynamic pressure; S is the wing area; $l_0$ is the length between aerodynamic centre and pivot point; c is the chord length; $a_1$, $a_2$, $m_0$, b are aerodynamic derivatives; and $\delta$ is the elevator deflection angle, as illustrated in Figure 3.

Equation (28) is linear for angles of attack below stall, so a Laplace transfer function representation could be obtained by determining the coefficient values in the differential equation. However, Duran [30] used second-order identification techniques to model the aerofoil as

$$\alpha \left(s\right)=\frac{44.94}{s^2+5.12s+89.87}\delta \left(s\right)$$

(29)

The initial step in the process was to design a PID controller with two degrees of freedom, allowing for separate weightings to be adjusted for both the proportional and derivative terms. This specific controller, in parallel form, is expressed by

$$\begin{array}{c}\hfill K\left(s\right)=K_p(b{\alpha }_{ref}-\alpha )+K_i\frac{({\alpha }_{ref}-\alpha )}{s}+K_d\frac{N}{1+N/s}(c{\alpha }_{ref}-\alpha )\end{array}$$

(30)

where ${\alpha }_{ref}$ is the output reference angle; $K_i$, $K_p$ and $K_d$ are the integral, proportional and derivative gain, respectively; N is the filter coefficient; and b and c are weightings for the proportional and derivative terms, respectively, that assign the closed-loop system zeros.

The PID parameters were tuned to achieve the desired performance with values $K_p=0.435$, $K_i=4.427$, $K_d=0.0395$, $N=52.68$, $b=1$ and $c=0$.

Figure 4 depicts the obtained system response for the PID controller. Similarly, the system response is displayed for both the adaptive controllers in Figure 5. All the responses exhibit zero overshoot, accompanied by a short rise time. Moreover, the control signals display smooth variation, devoid of any noise components. This characteristic ensures the prevention of any potential actuator damage.

5. Implementation Results

The implementation of the proportional and the piecewise ${\mathcal{L}}_1$ adaptive controllers on the flight demonstrator unit are presented in this section. All the controllers are implemented using the Arduino IDE [34] in the C++ programming language. The PID algorithm was previously coded by [31] and is now utilised to obtain real-time responses by incorporating the PID parameters from the simulation.

In this ${\mathcal{L}}_1$ adaptive controller implementation, the desired model was chosen to achieve second-order dynamics with a frequency of 0.7: $\mathrm{rad}/\mathrm{s}$ and a damping ratio of $0.7$. The transfer function $D\left(s\right)$ of the ${\mathcal{L}}_1$ adaptive controller was set as $D\left(s\right)=1/s(s+9.8)$, and $k=36$, resulting in a filter $C\left(s\right)=36/s^2+9.6s+36$. The adaptation rate of the proportional controller was set at $\Gamma =1000$, given that the sampling time of the test bed is $T_s=0.0001$ s. This choice is based on the guideline that $\Gamma =1/T_s$ [35].

5.1. Nominal Performance

Figure 6a illustrates the response to a command angle of 6° using the PID parameters obtained from the simulation stage. The control history is also depicted in Figure 6b.

The results are good, with overshoot being 8.67% and no steady-state error observed in the system. Moreover, the system effectively rejects disturbances, and the response is satisfactory, characterised by smooth variations in the control signal. Comparing the real-time response to the simulation depicted in Figure 6c, it is evident that the real-time system shows higher overshoot and more pronounced oscillations. However, it demonstrates a faster response with a rise time of 0.51 s, in contrast to the simulation rise time of 0.92 s. The disturbances are mostly in the airflow; the wind tunnel has not been specifically designed to create a smooth, steady laminar flow so that the disturbance rejection properties of the controller can be tested.

Likewise, Figure 7a illustrates the response of the proportional adaptive law. The control design demonstrates a good response in the form of a minimal overshoot of 6%. In comparison to the PID control history, the proportional adaptive law response appears noisier, which could be attributed to the utilisation of a larger bandwidth for the low-pass filter. However, it boasts a faster rise time of 0.64 s, outperforming the simulated rise time of 1 s.

Lastly, Figure 8 presents the results of the piecewise algorithm. Among all the controller designs, it demonstrates the most favourable performance with the lowest overshoot of 3.17%. Additionally, it achieves a rapid rise time of 0.56 s. However, like the proportional law, the control signal is quite noisy in comparison to the smoother control signal observed in the PID controller. Note that when the reference is zero, greater oscillation is evident. The NACA0012 aerofoil is relatively thick; hence, at low Reynolds numbers (as in this experiment), there is a nonlinearity in the control surface effectiveness, which results from flow separation that occurs on both the upper and lower surfaces and from the formation of a laminar separation bubble [36]. This also causes a reduction in the elevator effectiveness at low angles of attack.

5.2. Robustness Analysis

The key advantage of ${\mathcal{L}}_1$ adaptive control lies in its robustness without compromising the adaptation gain. To demonstrate the robustness, various types of disturbances were introduced in the control channel at around 25 s.

In the first test, the control effectiveness is decreased by thirty percent. The PID Control response decreases to approximately 3.24°, whereas both adaptive controller responses drop to 4.64°, before eventually recovering to the reference command. This behaviour is illustrated in Figure 9a, where it is clear that both adaptive control designs outperform the PID control, exhibiting better performance despite the reduction in control effectiveness. The adaptive controllers adapt more effectively to the changes in the system and maintain better control throughout the process. Figure 9b shows that the reduced control effectiveness leads to an increased control effort requirement for compensation. This heightened control effort is sustained until the end of the operation.

In the second comparison, a time-varying disturbance is introduced into the control channel, in the form of a sine wave with a frequency of 10 rad/s. The response of the control system is analysed, and the results are shown in Figure 10a. It can be observed that with PID control, the system experiences a sudden overshoot due to the disturbance. However, both adaptive control designs handle the time-varying disturbance effectively, as they do not exhibit significant overshoot. The compensation effects are depicted in Figure 10b, where the control signal of the adaptive designs shows the development of high-frequency content. This indicates that the adaptive controllers are actively adjusting to the changing disturbance and effectively maintaining control over the system.

Another robustness check is conducted by introducing a 2° bias in the control channel. Interestingly, there is no significant difference in the response when this bias is introduced for all the controllers. However, when the reference is set back to zero, large oscillations are observed, as shown in Figure 11a. Figure 11b shows the control history, revealing a constant offset in the control signal intended to compensate for the introduced bias.

To further evaluate the robustness of the control systems, different forms of disturbances are combined. The first combination includes a thirty percent reduction in control effectiveness along with a time-varying disturbance represented as a sine wave with a frequency of 10 rad/s. The results, as shown in Figure 12a, indicate that the PID control performs poorly in handling the time-varying disturbance compared to both adaptive controllers. On the other hand, the ${\mathcal{L}}_1$ adaptive control demonstrates effective compensation for both types of disturbances, as depicted in Figure 12b. However, after the introduction of the disturbances, the control signal becomes noisier and exhibits a simultaneous increase in control effort.

The impact of introducing a two-degree bias and thirty percent control reduction can be observed in Figure 13. Like the scenario with only control reduction, the PID response shows a lower performance compared to both adaptive controllers. Conversely, both adaptive controllers handle the combined disturbances effectively, resulting in better control responses. The control history of the adaptive controllers demonstrates the necessary variations required to compensate for the reference command appropriately, ensuring a more robust and stable control performance.

6. Conclusions

An output feedback ${\mathcal{L}}_1$ adaptive control approach employing a proportional adaptation law is introduced. This method enhances the phase margin within the estimation loop, facilitating the integration of first-order sensor dynamics into the state predictor. Moreover, the reduction in error bounds is directly proportional to the adaptation gain, rather than its square root, leading to decreased computational resource requirements.

The design was implemented and tested on a benchtop flight control demonstrator, that being an airfoil pitch control system. To evaluate its robustness, different types of disturbances were introduced. The results demonstrate that the proposed method, along with the piecewise adaptive controller, outperforms PID Control when dealing with reduced control effectiveness and time-varying disturbances.

In ${\mathcal{L}}_1$ adaptive control, a key challenge lies in selecting the appropriate structure and parameters for the low-pass filter, especially in output feedback designs. Currently, no established principles guide this selection, leading to a trial-and-error approach in choosing the filter structure. This results in an ongoing challenge to achieve an optimal filter structure that balances performance and robustness effectively.

The implications of the nonlinearities of the control surface effectiveness on the control system performance remains to be fully analysed. An accurate model of the aerodynamics requires experiment in a high quality wind tunnel. The effect of the nonlinearities on the closed loop performance could then be analysed and simulated. This remains for further work.