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Abstract

In this article, a robust kernel-based model reference adaptive control is proposed for an unstable nonlinear aircraft. The heart of the proposed kernel-based model reference adaptive control scheme comprises an offline neural identifier and an online neural controller. In the offline neural identifier, the kernel-based unified extreme learning machine algorithm is used to identify the aircraft model with the available input–output data in a finite time interval. The finite time interval is selected to avoid the response of the unstable aircraft growing unbounded. In the kernel-based unified extreme learning machine, the hidden layer feature mapping is determined by the kernel matrix. However, the unified extreme learning machine is a batch learning algorithm and is not suitable for the online control learning. To solve the problem, a recursive version of the unified extreme learning machine is developed in this study. Based on a given reference model and the identified model, the recursive version of the unified extreme learning machine algorithm is applied to construct the online control law to compensate for the changes in the aircraft dynamics or characteristics. The performance of the proposed kernel-based model reference adaptive control scheme is validated through the simulation studies of a locally nonlinear longitudinal high-performance aircraft. Simulation studies are also compared with a model reference adaptive control based on the back-propagation algorithm and a model reference adaptive control based on the basic extreme learning machine algorithm in terms of the identification and tracking abilities. The results show that the proposed kernel-based model reference adaptive control can achieve better identification and tracking performance.

Keywords

Unified extreme learning machine kernel-based recursive model reference adaptive control unstable aircraft

Introduction

As most practical systems in the real world are nonlinear, adaptive identification and control for such systems with unknown parameter and unmodeled uncertainties are intense areas of research. Several novel techniques in adaptive control of nonlinear systems are facilitated including feedback linearization methods¹ and backstepping methods.² However, a key assumption in these preceding methods is that the system nonlinearities are known a priori, which increases the complexity of the controller design due to the unknown nonlinearities from the parameter uncertainties and external disturbances. Model reference adaptive control (MRAC) method provides an alternative selection for the nonlinear controller design, which is a powerful method for controlling the systems with unknown parameter and unmodeled uncertainties by means of adaptive learning to compensate for characteristic changes of the systems. Generally, an MRAC consists of a reference model to produce the reference signals, an identifier to identify the system, and a controller to construct the control law based on the information of the identifier. But it is not useful for nonlinearity of the system. In the conventional MRAC scheme, the controller is designed to realize the system output convergence to the reference model output based on the assumption that the system can be linearized. Therefore, the scheme is effective for controlling a linear system with unknown parameters, but it may not be assured to succeed in controlling a nonlinear system with unknown dynamics.

During the past decades, neural control methods have become popular and have been used widely in many practical applications. Neural networks possess an inherent structure suitable for mapping complex characteristics, learning, and optimization. A precise mathematical model to describe the nonlinear system can be avoided. Thus, neural networks have widely been applied for the adaptive control of uncertain nonlinear systems. For adaptive control purposes, neural networks are mostly used as approximation models of unknown nonlinearities. The feasibility of applying neural networks in the MRAC for identification and control of nonlinear systems has been demonstrated through numerous studies.^3–5 Among these research works, neural networks are mostly used to approximate models with unknown nonlinearities, thus removing the need for a priori knowledge of system nonlinearities. Most of these methods mainly apply the back-propagation (BP) learning algorithm for adjusting the network parameters. However, it is clear that BP learning methods are generally very slow due to improper learning steps or may easily converge to local minima. And many iterative learning steps may be required by such learning algorithms in order to obtain better learning performance. Recently, a model reference adaptive neural control based on BP and extreme learning machine (ELM) algorithms has been proposed for nonlinear systems.⁶ In this study, two single-hidden-layer feedforward networks (SLFNs) are used as an identifier to identify the unknown nonlinear system and a controller to construct the control law based on the identified model. Different from the existing technologies where the BP is employed to train the neural identifier and controller, the identifier is trained using the basic ELM algorithm, while the controller is trained using the BP method. Although simulation results show that the proposed approach has faster learning speed and higher tracking performance than the existing methods, the method still applies the BP algorithm to train the neural controller, resulting in slow learning speed.

MRAC methods based on different neural network architectures are widely used in flight control systems.^7–9 In Suresh et al.,¹⁰ a model reference indirect adaptive neural control scheme for an unstable nonlinear aircraft controller design is proposed. Besides, a neural-network-based model following direct adaptive control system design¹¹ is presented for the F-8 fighter to improve damping and also to follow pilot commands accurately. Similarly, these studies mainly apply the BP learning algorithm for adjusting the network parameters, which suffers from the drawbacks of the BP algorithm described above. Recently, ELM algorithm has been proved to generate better generalization performance at extremely high learning speed by many real-world applications than the BP algorithm.^1,12–16 Furthermore, the non-iterative solution of ELMs provides a speedup and better performance compared to multilayer perceptron (MLP), support vector regression (SVR), or support vector machines (SVM) in both regression and classification problems.^17,18 Nevertheless, the network structure, that is, the number of hidden nodes to be used when applying ELM for handling the problems in hand, remains as a trial-and-error process. Considering the feature mappings are unknown to users, kernels can be applied in ELM, namely, unified extreme learning machine (U-ELM)¹⁹ instead of basic ELM, where the randomness does not occur any more.²⁰ However, the U-ELM algorithm only considers the batch learning. In real flight, the aerodynamic parameters of the aircraft are generally perturbing continuously as the flight states change. These perturbations are very complicated and their analytic forms are almost impossible to be obtained. Besides, some unknown disturbances including environmental noise and coupling effects from other subsystems widely exist in the aircraft systems. So the aircraft dynamics is uncertain due to unmodeled dynamics and unknown disturbances, which may cause control performance degradation. The performance of the controller can be enhanced if an online learning process is introduced to compensate for these uncertainties.

To solve the real flight problem, a novel robust kernel-based model reference adaptive control (KMRAC) is proposed to control an unstable aircraft. The proposed KMRAC comprises offline identification and online learning control strategies. Similar to Rong et al.,⁶ two SLFNs are utilized to build the identifier and controller. But different from Rong et al.,⁶ the SLFNs are trained based on the U-ELM algorithm. To suit to the online control learning, a recursive version of unified extreme learning machine (RU-ELM) algorithm, that is, the sequential modification, is developed in this study. Besides, a linear proportional-derivative (PD)-like controller is applied as a robust term to guarantee the stability of the control system. The advantage of the proposed control scheme is demonstrated via different simulation studies.

Aircraft model

The longitudinal dynamics of a high-performance fighter²¹ is considered in this study, which is powered by an afterburning turbofan jet engine. The system states include velocity $V_{t}$ , angle of attack $α$ , pith rate q, pitch attitude $θ$ , the elevator deflection $δ_{e}$ , and the control input is the elevator input u. Considering the level-flight condition ( $V_{t} = 150 ft / s$ , $α = 15 \circ$ , and $h = 2000 ft$ ), the local nonlinear perturbed equations of the aircraft are described as follows¹⁰

$\begin{matrix} \frac{dx}{dt} = Ax + Bu + g (x), x (0) = x_{0} \\ y = Cx + Du \end{matrix}$ (1)

where

$\begin{matrix} A = [\begin{matrix} - 0.0376 & - 0.22 & - 0.0246 & - 9.81 & - 0.1116 \\ - 0.03 & - 3.2797 & 0.9188 & 0 & 0.0129 \\ 1.219 & - 0.5117 & - 2.2150 & 0 & - 3.2529 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 20 \end{matrix}] \\ B = {[\begin{matrix} 0 & 0 & 0 & 0 & 20 \end{matrix}]}^{T} \\ C = [\begin{matrix} 0 & 0 & 1 & 0 & 0 \end{matrix}] \\ D = 0 \end{matrix}$

where $x = [V_{t}, α, q, θ, δ_{e}] \in R^{5}$ , $u \in R$ is the elevator input, and $y \in R$ is the output q. The function $g (\cdot)$ contains the second- and higher-order terms of the state variables x²¹ and is a smooth and bounded nonlinear function given as

$g (x) = [\begin{matrix} 0.009 x_{1} x_{2} + 0.00212 x_{2}^{2} - 0.009 x_{1} x_{3} - 0.12 x_{1} x_{2} x_{3} - 0.0012 x_{2}^{3} \\ - 0.1914 x_{2}^{2} - 0.0271 x_{2} x_{3} + 0.00017 x_{3}^{2} \\ - 0.2712 x_{3}^{2} - 0.0145 x_{3} x_{2} + 0.00071 x_{1} x_{2}^{2} - 0.00721 x_{2}^{3} \\ 0 \\ 0 \end{matrix}]$ (2)

The elevator input u belongs to a class of bounded signals U, where

$u (t) = {u : ∥ u (t) ∥ \leq δ \forall t \geq 0}, δ \in R^{+}$

it is assumed that the constant $δ \in R^{+}$ is known.

As in the work by Shim and Sawan,²² the phugoid mode of the aircraft is statically unstable and consequently cruising flight would be subject to attitude and velocity divergence that would require stabilizing feedback control. The control system is generally designed to accept, interpret, and properly respond to pilot input commands and hence a simple regulator design is not sufficient. In particular, the system should provide decoupled response to pitch rate commands, with responses compatible with level 1 handling requirements.^23,24 Using the compact notation

$\frac{K (s + \frac{1}{τ})}{s^{2} + 2 ξ ω_{n} s + ω_{n}^{2}} \equiv \frac{K (\frac{1}{τ})}{[ξ; ω_{n}]}$ (3)

the desired response transfer function as per level 1 handling qualities is given as

$\frac{q_{c}}{δ_{s}} = \frac{35.12 (0.5)}{[0.89; 2.24]}$ (4)

where the upper and lower limits on the stick deflection $δ_{s}$ are defined as

$- 0.5 \leq δ_{s} \leq 0.5$

The control design problem is to design the control input u that stabilizes the aircraft and forces the output y to follow the pilot pitch rate command signals at all flight conditions.

KMRAC

To achieve the control goal, a KMRAC scheme as shown in Figure 1 is proposed, which includes two parts. One part is the offline identification where an identifier is constructed to approximate the input–output relationship of the aircraft dynamics. The other part is the online control where a unique controller that forces the aircraft output to follow the reference model output accurately is designed, given the identifier model. The convergence of the controller depends on the accurate modeling of the identifier. Considering the merits of the kernel-based U-ELM algorithm, it is employed for the purpose of the identification and control. The design details are presented in the following.

Figure 1.

Diagram of proposed KMRAC.

Offline identification

In this section, we present the design process of the identifier where the unstable aircraft dynamics is approximated using the U-ELM algorithm. Before describing the design details, a brief review of U-ELM algorithm is given in the following.

Brief review of Kernel-based U-ELM

U-ELM is a kernelized version of ELM where the feature mapping $h (x)$ of the hidden nodes need not be known and instead one may use its corresponding kernel. While in the traditional ELM, the hidden nodes’ mapping is randomly generated and known to users finally. As in ELM, the output of U-ELM with L hidden nodes is expressed as

$f (x) = \sum_{i = 1}^{L} β_{i} h_{i} (x) = h (x) β$ (5)

where $β = [β_{1}, \dots, β_{L}]^{T}$ is the output weights and $h (x) = [h_{1} (x), \dots, h_{L} (x)]$ is the hidden nodes’ output with respect to the input $x$ .

Different types of constraints which are application dependent may exist for the optimization objective function. From the standard optimization theory point of view, the objective of U-ELM in minimizing both the training errors and the output weights can be written as

$\begin{matrix} Minimize : L_{P_{ELM}} = \frac{1}{2} ∥ β ∥^{2} + \frac{1}{2} C \sum_{i = 1}^{N} ξ_{i}^{2} \\ Subject to : h (x_{i}) β = d_{i} - ξ_{i}, i = 1, \dots, N \end{matrix}$ (6)

where $ξ_{i}$ is the training error with respect to the training sample $x_{i}$ . $d_{i}$ is the ith target output value and N is the number of sample data.

Based on the Karush–Kuhn–Tucker (KKT) theorem, training of U-ELM is equivalent to solving the following dual optimization problem

$\begin{matrix} L_{D_{ELM}} = \frac{1}{2} ∥ β ∥^{2} + C \frac{1}{2} \sum_{i = 1}^{N} ξ_{i}^{2} - \sum_{i = 1}^{N} α_{i} (h (x_{i}) β - d_{i} + ξ_{i}) \end{matrix}$ (7)

where each Lagrange multiplier $α_{i}$ corresponds to the ith training sample. We can have KKT optimality conditions of equation (7) as follows

$\frac{\partial L_{D_{ELM}}}{\partial β} = 0 \to β = \sum_{i = 1}^{N} β_{i} h (x_{i})^{T} = H^{T} α$ (8)

$\frac{\partial L_{D_{ELM}}}{\partial ξ_{i}} = 0 \to α_{i} = C ξ_{i}, i = 1, \dots, N$ (9)

$\frac{\partial L_{D_{ELM}}}{\partial α_{i}} = 0 \to h (x_{i}) β - d_{i} + ξ_{i} = 0, i = 1, \dots, N$ (10)

where $α = [α_{1}, \dots, α_{N}]$ .

By substituting equations (8) and (9) in equation (10), the aforementioned equations can be equivalently written as

$(\frac{I}{C} + H H^{T}) α = D$ (11)

where

$D = [\begin{matrix} d_{1} \\ ⋮ \\ d_{N} \end{matrix}]$

From equations (8) and (11), we have

$β = H^{T} {(\frac{I}{C} + H H^{T})}^{- 1} D$ (12)

The output function of U-ELM is

$f (x) = h (x) β = h (x) H^{T} {(\frac{I}{C} + H H^{T})}^{- 1} D$ (13)

In U-ELM, the feature mapping $h (x)$ is unknown to users and based on Mercer’s conditions, a kernel matrix is given as

$Ω_{ELM} = H H^{T} : Ω_{EL M_{i, j}} = h (x_{i}) \cdot h (x_{j}) = K (x_{i}, x_{j})$ (14)

Then, the output function of U-ELM can be written compactly as

$\begin{matrix} f (x) = h (x) H^{T} {(\frac{I}{C} + H H^{T})}^{- 1} D \\ = {[\begin{matrix} K (x, x_{1}) \\ ⋮ \\ K (x, x_{N}) \end{matrix}]}^{T} {(\frac{I}{C} + Ω_{ELM})}^{- 1} D \end{matrix}$ (15)

where $K (x, x_{i}) = \exp (- γ ∥ x - x_{i} ∥^{2})$ , $γ > 0$ is a constant selected based on trial-and-error and experimental comparison to achieve a satisfying generalization error level. The number of hidden nodes L is equal to the number of sample data, that is, $L = N$ . The kernel matrix $Ω_{ELM} = H H^{T}$ is only related to the input data $x_{i}$ and the number of training samples.

Next, we present how U-ELM has been implemented in the identifier design setup.

U-ELM identifier design

The universal approximation capability of U-ELM has been proved by Huang et al.¹⁹ and also guarantees that the input–output response of the unstable aircraft can be approximated. Generally, a wide class of nonlinear dynamic systems can be represented by the nonlinear model with an input–output description form

$\begin{matrix} y (t) = & F [u (t), u (t - T), \dots, u (t - pT), \\ y (t - T), \dots, y (t - nT)] \end{matrix}$ (16)

where y is the system output, u is the system input, $F (.)$ is a nonlinear function, and n and p are the maximum lags of the output and input, respectively. T is the sampling time.

Based on the preceding equation, we can construct a U-ELM identifier to approximate the aircraft dynamics as shown in Figure 2. The inputs of the identifier are the present input and p past key inputs and n past key outputs of the aircraft. Selecting $[u (t), u (t - T), \dots, u (t - pT), y (t - T), \dots, y (t - nT)]$ and $y (t)$ as the identifier’s input–output $x_{t}, y_{t}$ at time t, the above equation can be put as

$y_{t} = F (x_{t})$ (17)

Figure 2.

Diagram of U-ELM identifier.

The aim of the U-ELM algorithm is to approximate F and based on equation (15) the output of the identifier equals as

${\hat{y}}_{t} = \hat{F} (x_{t}) = K (x_{t}) ω_{t}$ (18)

where $K (x_{t}) = [K (x_{t}, x_{1}), K (x_{t}, x_{2}), \dots, K (x_{t}, x_{N})]$ , $ω_{t} = {(I / C + ω_{ELM})}^{- 1} D$ , and ${\hat{y}}_{t}$ is the output of U-ELM. This means that the identification objective is to minimize the error between the system output and the output of U-ELM, $∥ y - {\hat{y}}_{t} ∥$ .

The universal approximation property of U-ELM requires that the inputs and outputs belong to the compact sets and then the U-ELM is possible to approximate any function to desired accuracy. But the aircraft considered in this study is unstable, thus the response of the aircraft can grow unbounded although the elevator input is bounded. This means that the input–output data are beyond a compact set, which will affect the convergence of the identifier model.

For the purpose of identification of unstable aircraft dynamics, we make a mild assumption on the boundedness on aircraft state/output responses as in the work by Suresh et al.¹⁰

Assumption 1. Let us assume that for a given class of bounded input u and finite initial conditions $x_{0}$ , the aircraft response does not escape to infinity in finite interval of time

$sup_{t \in [0, T_{c}]} ∥ y (t) ∥ \leq Δ$ (19)

where $Δ$ is a known real positive number. The time $T_{c}$ is referred to as the critical time.

The critical time $T_{c}$ depends on the system dynamics and also on the class of bounded input signals. Using the preceding statement, we assume that the response of the aircraft is less than $Δ$ within the time interval $[0, T_{c}]$ , for all bounded input u. This depicts the aircraft input–output data in the time interval $[0, T_{c}]$ belong to a compact set. Hence, the proposed U-ELM identifier is constructed based on the input–output data in the time interval $[0, T_{c}]$ and thus assures that it can approximate the unstable aircraft dynamics through its universal approximation capability. This provides a good guarantee for the design of the online control law using the identified model, which is described below.

Online learning control

In this section, we consider a strategy to design an online controller to stabilize the unstable aircraft and also follow the arbitrary reference output signals generated from the reference model. Similarly, the U-ELM is used to build the controller model. But it is noteworthy that the U-ELM algorithm in equation (15) is a batch learning and is not suitable to the online controller design. To solve the problem, an online RU-ELM is proposed here. Actually, some researchers^25–27 have devoted to develop the kernel recursive least-square algorithms but they are utilized in the adaptive filter fields. Furthermore, an online sparse kernel-based method in reproducing kernel Hilbert spaces (RKHS)²⁸ is presented for the classification. An online incremental learning algorithm based on the batch learning ELM²⁹ is developed for classification and regression problems. Its performance is verified through some benchmark problems and a real critical dimension (CD) prediction problem of semiconductor production line. Inspired by these ideas, the RU-ELM is proposed for the online control. Before presenting the controller design details, the proposed RU-ELM is first given below.

Recursive version of Kernel-based U-ELM

With a sequence of training data ${x_{j}, d_{j}}_{j = 1}^{t}$ up to time t, the kernel vector $K (x_{t})$ is denoted as $[K (x_{t}, x_{1}), K (x_{t}, x_{2}), \dots, K (x_{t}, x_{t})]$ .

Referring to the output function in equation (15), we introduce

$Ω_{t} = {(\frac{I}{C} + Ω_{ELM, t})}^{- 1} D_{t}$ (20)

where

$D_{t} = [\begin{matrix} d_{1} \\ ⋮ \\ d_{t} \end{matrix}]$

Denoting

$Q_{t} = {(\frac{I}{C} + Ω_{ELM, t})}^{- 1}$ (21)

one has

$Q_{t}^{- 1} = [\begin{matrix} Q_{t - 1}^{- 1} & K {(x_{t - 1})}^{T} \\ K (x_{t - 1}) & \frac{1}{C} + K (x_{t}, x_{t}) \end{matrix}]$ (22)

where

$K (x_{t - 1}) = [K (x_{t}, x_{1}), K (x_{t}, x_{2}), \dots, K (x_{t}, x_{t - 1})]$ , and $Ω_{ELM, t}$ is computable by the kernel trick (equation (14)), in which $Ω_{ELM, t} : Ω_{EL M_{i, j}} = K (x_{i}, x_{j}), i = 1, \dots, t, j = 1, \dots, t$ .

Hence, the inverse of this growing matrix is updated as³⁰

$\begin{matrix} Q_{t} = \\ r^{- 1} [\begin{matrix} r Q_{t - 1} + Q_{t - 1} K {(x_{t - 1})}^{T} K (x_{t - 1}) Q_{t - 1}^{T} & - Q_{t - 1} K {(x_{t - 1})}^{T} \\ - K (x_{t - 1}) Q_{t - 1}^{T} & 1 \end{matrix}] \end{matrix}$ (23)

where $r = 1 / C + K (x_{t}, x_{t}) - K (x_{t - 1}) Q_{t - 1} K (x_{t - 1})^{T}$ .

Therefore, the expansion coefficients of the weight are shown in equation (24),

$\begin{matrix} ω_{t} = Q_{t} D_{t} \\ = r^{- 1} [\begin{matrix} r Q_{t - 1} + Q_{t - T} K {(x_{t - 1})}^{T} K (x_{t - 1}) Q_{t - 1}^{T} & - Q_{t - 1} K {(x_{t - 1})}^{T} \\ - K (x_{t - 1}) Q_{t - 1}^{T} & 1 \end{matrix}] \times [\begin{matrix} D_{t - 1} \\ d_{t} \end{matrix}] \\ = r^{- 1} [\begin{matrix} r Q_{t - T} D_{t - 1} + Q_{t - 1} K {(x_{t - 1})}^{T} K (x_{t - 1}) Q_{t - 1}^{T} D_{t - 1} - Q_{t - 1} K {(x_{t - 1})}^{T} d_{t} \\ - K (x_{t - 1}) Q_{t - 1}^{T} D_{t - 1} + d_{t} \end{matrix}] \end{matrix}$ (24)

where $d_{t}$ is the desired output at time t.

Then, we defined $e_{t}$ is the prediction error computed by the difference between the desired signal and the prediction

$e_{t} = d_{t} - K (x_{t - 1}) Q_{t - 1}^{T} D_{t - 1}$ (25)

Equation (24) becomes

$\begin{matrix} ω_{t} = [\begin{matrix} ω_{t - 1} + r^{- 1} Q_{t - 1} K {(x_{t - 1})}^{T} e_{t} \\ - r^{- 1} e_{t} \end{matrix}] \end{matrix}$ (26)

As we can see, the structure of RU-ELM is similar to the U-ELM at any time t. $ω_{t, j}$ is the jth component of $ω_{t}$ and all the previous data serve as the centers. The coefficients $ω_{t - 1}$ should be stored in the computer during training. The updates needed for RU-ELM at time t is

$ω_{t, t} = r^{- 1} e_{t}$ (27)

$ω_{t, j} = ω_{t - 1, j} + r^{- 1} Q_{t - 1, j} K (x_{t - 1})^{T} e_{t}, j = 1, \dots, t - 1$ (28)

The proposed RU-ELM algorithm is summarized as follows:

Initialize

Q_{t} = {(\frac{1}{C} + K (x_{1}, x_{1}))}^{- 1}

ω_{t} = Q_{t} d_{1}

for

t > 1

K (x_{t - 1}) = [K (x_{t}, x_{1}), K (x_{t}, x_{2}), \dots, K (x_{t}, x_{t - 1})]

r = \frac{1}{C} + K (x_{t}, x_{t}) - K (x_{t - 1}) Q_{t - 1} K (x_{t - 1})^{T}

Calculate

Q_{t}

from equation (23)

e_{t} = d_{t} - K (x_{t - 1}) Q_{t - 1}^{T} D_{t - 1}

ω_{t} = [\begin{matrix} ω_{t - 1} + r^{- 1} Q_{t - 1} K {(x_{t - 1})}^{T} e_{t} \\ - r^{- 1} e_{t} \end{matrix}]

end for

The RU-ELM starts with an empty representation, in which all parameters vanish, then gradually inputs samples, and the weight estimate $ω_{t}$ can be updated recursively from the previous estimate $ω_{t - 1}$ without solving directly following the arrival of each new sample. And we build the kernel-based approximation by allocating a new kernel node for every new data sample as a kernel center. The recursive algorithm distributes the computational burden evenly in each iteration, which is very appealing in online adaptive control where the samples are available sequentially over time.

Next, we present how RU-ELM has been implemented in the online controller design setup.

RU-ELM controller design

The controller design objective is to find the control input u such that the aircraft output $y (t)$ tracks any arbitrary desired output sequence $y^{*} (t)$ generated from the reference model $R_{m}$ . The reference model is selected based on the flying quality requirements of the fighter aircraft. Because the reference model is also a class of dynamic system, the output $y^{*} (t)$ can be represented by its past inputs and outputs as in equation (16)

$\begin{matrix} y^{*} (t) = & \bar{F} [y^{*} (t - T), \dots, y^{*} (t - mT), \\ r (t), r (t - T), \dots, r (t - lT)] \end{matrix}$ (29)

where $\bar{F} [\cdot]$ is a smooth continuous function. m and l are the maximum lags of the desired output and reference input, respectively.

If the asymptotic stability of the zero dynamic together with a well-defined T ensures the existence of a control input that can make the aircraft follow any arbitrary $y^{*} (t)$ , the controller has the form

$\begin{matrix} u (t) = & G [y (t - T), \dots, y (t - nT), \\ u (t - T), \dots, u (t - pT), y^{*} (t)] \end{matrix}$ (30)

where function map $G [\cdot]$ exists and it is unique. Hence, the aim of the controller is to approximate the function map $G [\cdot]$ .

Substituting equation (29) in equation (30)

$\begin{matrix} u (t) = G [y (t - T), \dots, y (t - nT), \\ u (t - T), \dots, u (t - pT), \\ \bar{F} [y^{*} (t - T), \dots, y^{*} (t - mT), r (t), \\ r (t - T), \dots, r (t - lT)]] \end{matrix}$ (31)

The input u depends on the past aircraft response and reference signals y, $y^{*}$ , r, and if the aircraft is able to track any arbitrary sequence, then $y = y^{*}$ . Hence, we can replace $y^{*}$ by y in the preceding equation

$\begin{matrix} u (t) = & R [y (t - T), \dots, y (t - mT), \\ r (t), r (t - T), \dots, r (t - lT)] \end{matrix}$ (32)

where $R [\cdot]$ is a smooth function.

Selecting $[y (t - T), \dots, y (t - mT), r (t), r (t - T), \dots, r (t - lT)]$ and $u (t)$ as the controller’s input–output $x_{t}, u_{t}$ at time t, the above equation can be put as

$u_{t} = R (x_{t})$ (33)

The mapping $R [\cdot]$ is not known. The RU-ELM is used to approximate the unknown $R [\cdot]$ and constructed as the online controller to calculate the target $u (t)$ , as shown in Figure (3). The inputs to the controller are the present and past reference inputs and past responses of the aircraft. The output of the controller equals as

${\hat{u}}_{t} = K (x_{t}) ω_{t}$ (34)

where ${\hat{u}}_{t}$ is the output of RU-ELM at time t. This means that the controller design objective is to minimize the error between the system output and the output of RU-ELM, $∥ u - {\hat{u}}_{t} ∥$ . One problem encountered here is that the desired control output u is not known, as required to obtain $e_{t}$ in equation (25). In the RU-ELM controller, the tracking error $e_{t}$ is obtained by the U-ELM identifier when the identified error is negligibly small. From equation (18)

$\frac{\partial {\hat{y}}_{t}}{\partial u} = \sum_{i = 1}^{N} \frac{\partial {\hat{y}}_{t}}{\partial K (x_{t}, x_{i})} \frac{\partial K (x_{t}, x_{i})}{\partial u}$ (35)

We can obtain

$Δ u = \frac{\partial {\hat{y}}_{t}}{\partial u} e_{c}$ (36)

where $Δ u$ corresponds to $e_{t}$ in equation (26) and $e_{c} = y^{*} - y$ .

Figure 3.

Diagram of RU-ELM controller.

A linear PD-like controller is employed to control the linear dynamics (equation (1)) around a certain operating region and stabilize the aircraft. Thus, the control input applied to the aircraft is the sum of linear PD-like and RU-ELM controller signals

$u = {\hat{u}}_{t} + K_{1} e_{c} + K_{2} {\overset{\cdot}{e}}_{c}$ (37)

The gain $K_{1}$ is selected such that the controller is able to stabilize the aircraft, and the gain $K_{2}$ term provides the improvement to suppress the vibration, especially with atmospheric disturbance.

Simulation results

In this section, the performance of the proposed KMRAC scheme including the offline identification and online control is evaluated. The differential equations describing the aircraft dynamics (equation (1)) are solved using Runge–Kutta higher-order method with the sampling time $T = 0.01 s$ . For the purpose of comparison, the performance of the proposed KMRAC is compared with the MRAC based on the BP algorithm (BPMRAC) and the MRAC based on the ELM algorithm (ELMMRAC).

Simulation results of offline identification

We present the simulation studies for the longitudinal dynamics identification of the unstable aircraft. In the experiments, $γ$ is selected as 50. Figure 4 shows the effect of $γ$ on identified error using U-ELM identifier. We know that $γ$ should be chosen as big as possible in order to increase the separability between different classes in feature space. But if $γ$ is too big ( $γ > γ_{0}$ ), the sample farthest to other class will become support vector, which leads to severe over-fitting. So the kernel parameter is manly comprised by that boundary point $γ_{0}$ . In this work, $γ_{0}$ is about 50 when the identification achieves the least identified error.

Figure 4.

The effect of $γ$ on identified error.

Figure 5 shows the input signals to the elevator, and Figure 6 shows the actual pitch rate q response of the aircraft model. We can observe that the fluctuating response tend to infinity after some time. The first clear peak value is appeared at 10.2 s approximately, which is about $- 1.1 rad / s$ . To ensure the response of the system is always less than $Δ$ defined as $1 rad / s$ , the critical time $T_{c}$ is selected as 10 s in this study.

Figure 5.

Elevator input u.

Figure 6.

Output of pitch rate q.

To verify the performance of the proposed U-ELM identifier, the BPMRAC, ELMMRAC, and KMRAC schemes are separately applied to identify the aircraft longitudinal dynamics between [0,10 s]. The identified results using the three methods are illustrated in Figure 7. And the identification errors of the three methods are shown in Figure 8. From the two figures, it can be found that the KMRAC has better approximation capability than the BPMRAC and ELMMRAC. The comparison results among the three algorithms are also demonstrated in Table 1 including the root-mean-square error (RMSE) value of the identified error and training time. The RMSE value of the ELMMRAC method is given statistically through repeated 10 times experiments, and the value of BPMRAC and KMRAC is not necessary to be replicated because of the non-existence of randomness. The KMRAC achieves less identified error than the BPMRAC and ELMMRAC. The training speed of KMRAC is faster than BPMRAC. And it is worth noting that ELMMRAC costs the least time obviously because of a trade-off that the number of hidden nodes to be used in ELMMRAC remains a trial-and-error process which is selected as 8 fixed in this work, far less than the number in KMMRAC equaling to the number of samples.

Figure 7.

Identified values using BPMRAC, ELMMRAC, and KMRAC.

Figure 8.

Identified errors from BPMRAC, ELMMRAC, and KMRAC.

Table 1.

The comparison results among the three algorithms.

	BPMRAC	ELMMRAC	KMRAC
RMSE value	0.0218	0.0120	0.0024
Training time (s)	1.98	0.21	1.85

MRAC: model reference adaptive control; ELM: extreme learning machine; BP: back-propagation; BPMRAC: MRAC based on the BP algorithm; ELMMRAC: MRAC based on the ELM algorithm; KMRAC: kernel-based model reference adaptive control; RMSE: root-mean-square error.

Simulation results of online control

Aircraft response under nominal conditions

The online controller is designed for forcing the actual pitch rate q to follow the desired pitch rate values generated from the reference model compatible with level 1 handling requirements. In the experiments, $γ$ is selected as 50, $K_{1}$ is selected as 70, and $K_{2}$ is selected as −50. According to equation (4), the pilot inputs and reference pitch rate responses at level-flight condition ( $V_{T} = 150 ft / s$ ) are shown in Figure 9.

Figure 9.

Pilot input $δ_{s}$ and desired pitch rate values.

The simulation results of tracking the pitch rate q based on the KMRAC, BPMRAC, and ELMMRAC schemes are illustrated in Figure 10. From the figure, we can observe that the three schemes stabilize the aircraft and follow the pitch rate accurately from the initial leaning. But it is obvious to see that the tracking performance of KMRAC scheme is better than those of BPMRAC and ELMMRAC schemes. The simulation studies indicate that the proposed online learning control strategy is able to stabilize the unstable aircraft and also provide good tracking performance. The elevator input signals from the three methods are given in Figure 11. As shown in the figure, the control inputs change continuously during [2 s,5 s] although the controlled states have reached the steady conditions. Such responses mainly are to adapt to the change of the pitch attitude.

Figure 10.

Response of q under nominal conditions.

Figure 11.

Control input u under nominal conditions.

Response under different uncertainties

To study the robustness of the proposed RU-ELM controller, modeling uncertainties and partial control surface loss conditions are considered. We consider the 20% modeling uncertainties and control surface loss, that is, the elements of the system matrix A, B given in equation (1) are modified as $A = 0.8 A$ , $B = 0.8 B$ . Besides, the aircraft is highly susceptible to atmospheric turbulences that commonly occur during flight. And the sensor used for measuring pitch rate suffers from the noise inevitably. The atmospheric disturbance is imitated by $d_{1} = 0.01 \cos (2 t + 0.1)$ and the sensor noise is imitated by random noise with the range $d_{2} \in [- 0.01, 0.01]$ .

Considering all these uncertain factors, the three schemes are applied to design the online control law such that the aircraft follows the reference pitch rate commands accurately. The responses of pitch rate using the three methods are shown in Figure 12. The elevator control inputs under these uncertainties are shown in Figure 13. From these figures, we can again see that the proposed KMRAC scheme has better tracking performance than BPMRAC and ELMMRAC schemes. These results demonstrate that the proposed controller compensated by the online learning of the RU-ELM works well even when there are uncertainties.

Figure 12.

Response of q under uncertainties.

Figure 13.

Control input u under uncertainties.

Conclusion

A KMRAC scheme that incorporates an offline neural identifier and an online neural controller is presented for an unstable nonlinear aircraft model. The neural identifier is constructed to identify the model of the unstable aircraft and is trained offline based on the U-ELM algorithm. The neural controller is designed online using the proposed RU-ELM algorithm to compensate for the control performance degradation caused by the unknown uncertainties the aircraft suffers during the real flight. Simulation studies via the BPMRAC algorithm and the ELMMRAC algorithm are provided in this article for the purpose of comparison. The simulation results show that the proposed KMRAC achieves better identification and tracking performance. The robustness of the proposed KMRAC method is tested under different uncertainties and the simulation results indicate that the proposed control scheme rejects the uncertainties very well.

Footnotes

Academic Editor: Chi-man Vong

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was funded,in part,by National Science Council of Shaanxi Province (Grant No. 2014JM8337) and the Fundamental Research Funds for the Central Universities.

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