Information Recovery in Composite Model Reference Adaptive Control

Abstract

This study introduces a new adaptive control approach incorporating frequency-limited estimation of matched uncertainty. While many existing adaptive parameter adjustment laws aim to mitigate uncertainty effects solely through tracking error, it is well-documented that integrating uncertainty estimation error into the adaptation process significantly improves transient performance. Our method incorporates low-frequency uncertainty estimation with a time-varying learning rate structure. Unlike conventional filter-based approaches, our approach also compensates for information loss during signal filtering to suppress high-frequency content. Additionally, we include a regulation term in the standard adaptive weight update law, acting as stability enhancement in the adaptive system. We demonstrate the closed-loop stability of the proposed method using Lyapunov’s stability theorem and highlight its efficacy through numerical examples and software-in-the-loop simulations with the X-plane flight simulator.

1. Introduction

Model Reference Adaptive Control (MRAC) frameworks achieve desired levels of closed-loop stability and performance for uncertain dynamical systems through online adaptive weight update laws. In the absence of restrictive persistent excitation (PE) of system signals, the standard MRAC framework cannot assure closed-loop stability when facing bounded perturbations using only instantaneous data [1]. To enhance the robustness of standard MRAC or ensure stability without PE, robust modifications are introduced in literature [2, 3, 4, 5, 6, 7]. While these modifications guarantee the robustness of the adaptive controller, they generally yield limited enhancements in the transient response, primarily due to their focus on incorporating only the tracking error in the adaptation. Nonetheless, there exist a few additional modifications, such as those detailed in Refs. [8, 9], which notably improve transient performance by extracting additional information on uncertainty estimation error. Inspired by these studies, various modifications have been proposed in the literature, including but not limited to [10, 11, 12, 13, 14, 15, 16]. In these studies, information regarding either uncertainty estimation error or parameter convergence error is extracted using low-pass filters. However, filtering system signals may lead to information loss and degradation of adaptation performance.

To improve the tracking performance and achieve fast adaptation, it is desired to set the adaptive gains as high as possible. However, it is also well-practiced that the high learning rates can excite (possibly unmodeled) high-frequency modes of the system and result in system instability due to high-frequency oscillations. Numerous studies in the literature aim to facilitate the utilization of high gains while ensuring stability, such as the optimal control modification [5] and the low-frequency learning modification [17]. Yucelen and Haddad [17] introduced a modification term in the standard adaptive law to constrain the estimated parameters within bounds around their low-pass filtered signals. In the optimal control modification [5], high adaptive gains are permitted, albeit at the cost of reduced performance concerning robustness. Unlike these modifications, where the learning rate remains constant, there are studies where the adaptation gain varies over time [16, 18]. However, the learning rate vanishes in these methods if the basis function is not sufficiently and/or persistently exciting, causing the termination of adaptation.

So, our motivation becomes to combine these approaches in a way that the modification term allows the use of high adaptation gains without causing any high-frequency oscillations, and the learning rate of the proposed modification is adjusted to prevent undesired behavior during the adaptation.

In this study, we propose a new adaptive law that offers a modification term for information recovery in filter-based adaptive controllers. The proposed modification achieves fast adaptation by adding a new direction in adaptation using high-frequency content of the filtered signals. In addition, a time-varying learning rate is introduced to suppress the undesired effects of high-frequency signals on the system behavior. In this way, the proposed modification term behaves like a stability augmentation system to standard adaptive law. It should be noted that the proposed study builds upon the foundation established in the author’s earlier work [19] by presenting more rigorous theoretical results and introducing additional simulations that highlight the effectiveness of the proposed method. Furthermore, the contributions are extended to cover a wider class of uncertain systems where limited information exists on uncertainty. These improvements result in a clearer and more precise presentation of the research, while expanding the domain of applications with new theoretical results through unstructured uncertainty parameterization.

2. Notation and definitions

The notation used in this study is fairly standard. Specifically, $\mathbb{R}$ denotes the set of real numbers, ${\mathbb{R}}^n$ denotes the set of $n\times 1$ real column vectors, ${\mathbb{R}}^{n\times m}$ denotes the set of $n\times m$ real matrices, ${\mathbb{R}}_{+}$ denotes the set of strictly positive real numbers, ${\left(\cdot \right)}^{-1}$ denotes inverse, ${\left(\cdot \right)}^T$ denotes transpose, ‘$\triangleq$’ denotes equality by definition, $A\succ 0$ denotes that A is a positive definite matrix, ${\left(\cdot \right)}^{†}$ denotes the Moore–Penrose inverse, ‘$\mathrm{vec}\left(\cdot \right)$’ denotes the column stacking operator, ‘$\mathrm{tr}\left(\cdot \right)$’ denotes the trace operator, ${\lambda }_{\min }\left(A\right)$$\left({\lambda }_{\max }\left(A\right)\right)$ returns the minimum eigenvalue (the maximum eigenvalue) of matrix A, ${\sigma }_{\max }\left(A\right)$$\left({\sigma }_{\min }\left(A\right)\right)$ returns the maximum (minimum) singular value of matrix A, and $\mid a\mid$ denotes the absolute value of scalar $a\in \mathbb{R}$. Furthermore, for the vector $x\in {\mathbb{R}}^n$ and matrix $A\in {\mathbb{R}}^{m\times n}$, the Euclidean vector norm $‖x‖$, induced matrix norm $‖A‖$, and Frobenius matrix norm ${‖A‖}_{\mathrm{F}}$ are defined as:

$$‖x‖\triangleq \sqrt{\sum _{i=1}^n{x}_i^2},‖A‖\triangleq \sqrt{{\lambda }_{\max }\left(A^{T}A\right)}={\sigma }_{\max }\left(A\right),{‖A‖}_{\mathrm{F}}\triangleq \sqrt{\mathrm{tr}\left(A^{T}A\right)}.$$E1

Kronecker product is denoted by ‘$\otimes$’. For the vector $\theta \in {\mathbb{R}}^k$ and convex function $f:{\mathbb{R}}^k\to \mathbb{R}$, the gradient operator $\nabla f\left(\theta \right)$ is $\nabla f\left(\theta \right)={\left[\begin{array}{ccc}\frac{\partial f\left(\theta \right)}{\partial {\theta }_1}& \cdots & \frac{\partial f\left(\theta \right)}{\partial {\theta }_k}\end{array}\right]}^T$. Lastly, ‘$■$’ indicates the completion of a mathematical proof.

Definition 1.1 (Persistent Excitation (PE)) The signal$\omega \left(t\right)\in {\mathbb{R}}^n$is said to be persistently exciting over an excitation period of$\tau$if and only if there exist positive constants$\tau ,\beta \in {\mathbb{R}}_{+}$such that the following inequality holds:

$${\int }_{t-\tau }^t\omega \left(s\right){\omega }^T\left(s\right)\mathit{ds}\succcurlyeq \beta I_{n\times n},\forall t\in {\mathbb{R}}_{+}$$E2

for all time$\forall t\in {\mathbb{R}}_{+}$where$I_{n\times n}$is the identity matrix of dimension n.

Definition 1.2 (Interval Excitation (IE)) The signal$\omega \left(t\right)\in {\mathbb{R}}^n$is said to be interval exciting at time$t=T_e$over an excitation period of$\tau$if there exist positive constants$\tau ,\beta \in {\mathbb{R}}_{+}$such that the following inequality holds:

$${\int }_{t-\tau }^t\omega \left(s\right){\omega }^T\left(s\right)\mathit{ds}\succcurlyeq \beta I_{n\times n},t=T_e\in {\mathbb{R}}_{+}$$E3

Remark 1.1. One can realize that the interval excitation (IE) is far less demanding condition than the persistent excitation (PE) as IE condition must be satisfied for all time$\forall t\in {\mathbb{R}}_{+}$to satisfy the PE condition. That is, persistency of excitation is uniform in time, whereas the interval excitation just holds for a particular time interval.

3. Problem definition

In this section, the control system description is provided first. Then, standard MRAC and composite/combined MRAC solutions are briefly reviewed for the completeness of the chapter.

3.1 System description

Consider the following nonlinear uncertain dynamical system:

$$\dot{x}\left(t\right)=\mathit{Ax}\left(t\right)+B\left[u\left(t\right)+\mathrm{\Delta }\left(x\right)\right],x\left(t_0\right)=x_0$$E4

where $x\in {\mathcal{D}}_x\subset {\mathbb{R}}^n$ is the state vector with ${\mathcal{D}}_x$ being a sufficiently large compact set, $u\in {\mathcal{D}}_u\subset {\mathbb{R}}^m$ is the m-dimensional control input that lies in the admissible control set ${\mathcal{D}}_u$, $\mathrm{\Delta }:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is the map representing the unknown uncertainty, $A\in {\mathbb{R}}^{n\times n}$ is known system matrix. Furthermore, input matrix $B\in {\mathbb{R}}^{n\times m}$ has full column rank and is assumed to be known (consistent with the literature; for example [6, 7, 14, 20, 21]), which is necessary to compute low-frequency content of the uncertainty using its pseudo-inverse (detailed later in Section 3.3). The fundamental assumptions are: (i) the pair (A, B) is controllable, (ii) full state measurement is available for feedback.

Assumption 1.1 (Structured Uncertainty). The uncertainty$\mathrm{\Delta }\left(x\right)$can be expressed as linear combination of known basis functions$\varphi :{\mathcal{D}}_x\to {\mathbb{R}}^s$as follows:

$$\mathrm{\Delta }\left(x\right)=W^T\varphi \left(x\right)$$E5

where$W\in {\mathbb{R}}^{s\times m}$is the unknown constant weight matrix with$\parallel W\parallel \le \overline{w}$, $\overline{w}>0$, and known function$\varphi \left(x\right)$is Lipschitz continuous over${\mathcal{D}}_x$.

Assumption 1.2 (Unstructured Uncertainty). Assumption 1.1 can be relaxed using universal approximators such as Single Hidden Layer Neural Networks (SHL-NN) [22] or Radial Basis Functions (RBF) [23] as follows:

$$\mathrm{\Delta }\left(x\right)=W^T\varphi \left(x\right)+\epsilon \left(x\right)$$E6

where the residual$\epsilon \left(x\right)$with$\parallel \epsilon \left(x\right)\parallel \le {\epsilon }_0$, $\forall x\in {\mathcal{D}}_x$.

Remark 1.2. The basis function$\varphi \left(x\right)$in Assumption 1.2 can be chosen such that the residual$\epsilon \left(x\right)$can be made arbitrarily small on a compact set${\mathcal{D}}_x$with a positive scalar upper bound${\epsilon }_0$. Various methods exist for selecting such appropriate basis functions as universal approximators. One such method is to employ Radial Basis Functions (RBF) as suggested by the universal approximation theorem [23,24] while minimizing the approximation error$\epsilon \left(x\right)$. Due to its simplicity and extensive use in the literature (for example [25–27]), we parameterize the unstructured uncertainty inEq. (6)using RBFs given by:

$${\varphi }_i\left(x\right)=\left\{\begin{array}{ll}1& ,i=1\\ \mathit{exp}\left(-\frac{{‖x\left(t\right)-{\overline{c}}_i‖}^2}{2{\overline{\mu }}_i^2}\right)& ,i=2,\cdots ,s\end{array}\right.$$E7

with ${\overline{c}}_i\in {\mathbb{R}}^n$ and ${\overline{\mu }}_i\in {\mathbb{R}}_{+}$ being the center and width of the i^th node, respectively [23]. It should be noted that in the structured uncertainty case, the basis $\varphi$ could be any known function and does not necessarily consist of Gaussian kernels.

The primary goal in model reference adaptive control (MRAC) is to design a control input to ensure that the system trajectories track those of the reference model given by:

$${\dot{x}}_r\left(t\right)=A_r{x}_r\left(t\right)+B_{r}r\left(t\right),x_r\left(t_0\right)=x_{r0}$$E8

with $A_r\in {\mathbb{R}}^{n\times n}$ being Hurwitz reference system matrix and $B_r\in {\mathbb{R}}^{n\times k}$ being reference input matrix. Positive-definite matrix $P\in {\mathbb{R}}^{n\times n}$ satisfies the following Lyapunov equation

$$A_r^{T}P+{\mathit{PA}}_r=-Q,P=P^T\succ 0$$E9

for any user-defined positive definite matrix $Q=Q^T\succ 0$. The reference state vector and reference command are denoted by $x_r\in {\mathcal{D}}_x$ and $r\in {\mathbb{R}}^k$, respectively.

Assumption 1.3 (Matching Conditions). As a standard assumption in adaptive control [28–30], the reference model satisfies the matching conditions. That is, there exist matrices$K_x$and$K_r$such that the following conditions hold:

$$\begin{array}{lll}{A}_r& =& A-{\mathit{BK}}_x\\ B_r& =& {\mathit{BK}}_r\end{array}$$E10

The nominal control input $u_n\left(t\right)$, consisting of feedback and feedforward components, is designed by setting $\mathrm{\Delta }\left(x\right)=0$ in the plant dynamics in Eq. (4) as follows:

$$u_n\left(t\right)=-K_{x}x\left(t\right)+K_{r}r\left(t\right)$$E11

The adaptive controller augmented control input is given by:

$$\begin{array}{ll}u\left(t\right)& =u_n\left(t\right)+u_a\left(t\right)=-K_{x}x\left(t\right)+K_{r}r\left(t\right)-{\widehat{W}}^T\left(t\right)\varphi \left(x\right)\end{array}$$E12

with $\widehat{W}\left(t\right)$ being the online estimation of unknown weight W.

3.2 Standard MRAC review

Let the tracking error be defined as $e\left(t\right)\triangleq x_r\left(t\right)-x\left(t\right)$. Combining the control in Eq. (12), matched uncertainty in Eq. (5), reference model in Eq. (8), and uncertain system dynamics in Eq. (4) with structured uncertainty parameterization yields the following tracking error dynamics:

$$\dot{e}\left(t\right)=A_{r}e\left(t\right)-B{\tilde{W}}^T\left(t\right)\varphi \left(x\right)$$E13

with $\tilde{W}\triangleq W-\widehat{W}$ being the weight estimation error. In the standard adaptive control formulation, the baseline weight update law is given by:

$$\text{Standard\hspace{0.17em}\hspace{0.17em}MRAC}\mathrm{Law}:\dot{\widehat{W}}=-\mathrm{\Gamma }\varphi e^T\mathit{PB}$$E14

with $\mathrm{\Gamma }\succ 0$ being positive definite learning rate. The closed-loop stability of the standard adaptive control system can be shown using radially unbounded Lyapunov function $\mathrm{V}\left(e,\tilde{W}\right)=e^T\mathit{Pe}+\mathrm{tr}\left({\tilde{W}}^T{\mathrm{\Gamma }}^{-1}\tilde{W}\right)>0$ with $\dot{\mathrm{V}}\left(e,\tilde{W}\right)=-e^T\mathit{Qe}\le 0$. Note that $\mathrm{V}\left(0,0\right)=0$ and $\mathrm{V}\left(e,\tilde{W}\right)>0$, for all $\left(e,\tilde{W}\right)\ne \left(0,0\right)$, $\forall t\in {\mathbb{R}}_{+}$. Since $\mathrm{V}\left(e,\tilde{W}\right)$ is lower-bounded by zero, and its derivative $\dot{\mathrm{V}}\left(e,\tilde{W}\right)\le 0$ is less than or equal to zero, Lyapunov function $\mathrm{V}\left(e,\tilde{W}\right)$ approaches to a finite limit as $t\to \infty$. Hence, the boundedness of tracking error e(t) and weight estimation error $\tilde{W}\left(t\right)$ is guaranteed. Since e(t) and $x_r\left(t\right)$ are bounded, system states x(t) and basis vector-function $\varphi \left(x\right)$ immediately become bounded. With bounded $e\left(t\right),x\left(t\right),\varphi \left(x\right)$, and $\tilde{W}\left(t\right)$, time derivative of tracking error dynamics $\dot{e}\left(t\right)$ becomes bounded, which ensures the boundedness of $\ddot{\mathrm{V}}\left(e,\tilde{W}\right)=-2e^{T}Q\dot{e}$, $\forall t\in {\mathbb{R}}_{+}$. It then follows from Barbalat’s Lemma [31] that ${lim}_{t\to \infty }\dot{\mathrm{V}}\left(e\left(t\right),\tilde{W}\left(t\right)\right)=0$ which implies the asymptotic stability of tracking error e(t); that is, ${lim}_{t\to \infty }e\left(t\right)=0$. If the basis vector-function $\varphi \left(x\right)$ is persistently exciting, then the estimated parameters converge to their ideal values; that is, $\widehat{W}\left(t\right)\to W$ as $t\to \infty$. See Ref. [29] for details.

Remark 1.3. If the uncertainty$\mathrm{\Delta }\left(x\right)$is characterized as unstructured uncertainty as in Assumption 1.2, the tracking error dynamics become:

$$\dot{e}\left(t\right)=A_{r}e\left(t\right)-B{\tilde{W}}^T\left(t\right)\varphi \left(x\right)-\mathit{B\epsilon }\left(x\right)$$E15

In this case, the boundedness of the estimated parameters is not guaranteed by the standard MRAC without persistently exciting$\varphi \left(x\right)$. Several robust modifications have been introduced in the literature to enhance the robustness, including but not limited to$\sigma$-mod [2], e-mod [3], optimal control-based modification [5], Kalman filter-based modification [7], and q-modification [9]. Furthermore, it is common practice to include the projection operator [32] to bound the parameters within the prescribed convex set.

Remark 1.4. Consider the adaptive law with$\sigma$-modification as a solution to the stability issue outlined in Remark 1.3:

$$\dot{\widehat{W}}=-\mathrm{\Gamma }\left[\varphi e^T\mathit{PB}+\sigma \widehat{W}\right]$$E16

where $\sigma \in {\mathbb{R}}_{+}$ being constant scalar design parameter. It can be shown using Lyapunov function $\mathrm{V}=\frac{1}{2}{e}^T\mathit{Pe}+\frac{1}{2}\mathit{tr}\left({\tilde{W}}^T{\mathrm{\Gamma }}^{-1}\tilde{W}\right)$ that the closed-loop system is stable in the sense of Lyapunov, even if $\epsilon \left(x\right)\ne 0$ without requiring persistent excitation of the basis function $\varphi$. Specifically, the error signals $e$ and $\tilde{W}$ are ensured to be inside the compact set $D_{\eta }$ which is defined as ${\mathcal{D}}_{\eta }\triangleq \left\{\eta \left(t\right):\parallel \eta \parallel <\sqrt{\frac{{\mu }_2}{{\mu }_1}}\right\}$ with ${\mu }_1,{\mu }_2,\eta$ being as the followings:

$$\begin{array}{lll}{\mu }_1& \triangleq & min\left\{\left(\frac{1}{2}{\lambda }_{\min }\left(Q\right)-c_1\right),\sigma \left(1-c_2\right)\right\}\\ {\mu }_2& \triangleq & \frac{\parallel \mathit{PB}{\parallel }^2{\epsilon }_0^2}{4c_1}+\frac{\sigma \overline{w}}{4c_2}\\ c_1& \in & \left(0,\frac{1}{2}{\lambda }_{\min }\left(Q\right)\right),c_2\in \left(0,1\right)\\ \eta & \triangleq & {\left[e^T,\mathit{vec}\left({\tilde{W}}^T\right)\right]}^T\end{array}$$E17

3.3 Composite/combined MRAC review

Although high gains enable fast adaptation in standard MRAC, it may degrade the transient performance by inducing high-frequency oscillations in the system and even cause instabilities in the presence of large uncertainties. Improving the transient performance in MRAC has always been an attractive problem [5, 11, 17, 20, 21, 33, 34]. As seen from these studies and references therein, including the uncertainty estimation error in the adaptation enhances transient performance significantly. This fact was first utilized in [8], called ‘Composite Model Reference Adaptive Control’. Nearly at the same time, a quite similar approach was introduced [35] as ‘Combined Model Reference Adaptive Control’. Later in [20], these two approaches are generalized to cover the multi-input multi-output systems. These methods are referred in this study as composite/combined model reference adaptive control (CMRAC) to acknowledge both studies. Basically, CMRAC starts with filtering the system dynamics in Eq. (4) as:

$${\dot{x}}_f={\mathit{Ax}}_f+B\left[u_f+{\mathrm{\Delta }}_f\right]$$E18

with subscript ‘f’ indicating that the signal is filtered through the following low-pass filters:

$$\begin{array}{lll}{\dot{x}}_f& =& {\omega }_f\left(x-x_f\right),x_f\left(t_0\right)=x_0\\ {\dot{u}}_f& =& {\omega }_f\left(u-u_f\right),u_f\left(t_0\right)=u_0\\ {\dot{\varphi }}_f& =& {\omega }_f\left(\varphi -{\varphi }_f\right),{\varphi }_f\left(t_0\right)=\varphi \left(x_0\right)\end{array}$$E19

where ${\omega }_f\in {\mathbb{R}}_{+}$ is the cut-off frequency that determines the bandwidth of low-pass filters. Rearranging Eq. (18) yields:

$${\mathrm{\Delta }}_f=B^{†}\left[{\dot{x}}_f-{\mathit{Ax}}_f\right]-u_f$$E20

Note that the signals $x_f$, $u_f$, ${\varphi }_f$, ${\dot{x}}_f$ and ${\dot{\varphi }}_f$ are all accessible through Eq. (19). It should also be noted that ${\mathrm{\Delta }}_f$ can be expressed as ${\mathrm{\Delta }}_f=W^T{\varphi }_f+{\epsilon }_f$ from Eq. (5). Hence, we get

$$W^T{\varphi }_f+{\epsilon }_f=B^{†}\left[{\dot{x}}_f-{\mathit{Ax}}_f\right]-u_f$$E21

Once the low-pass filters are applied to the system dynamics, the low-frequency content of uncertainty, ${\mathrm{\Delta }}_f$, becomes an available signal for control purposes, as all the signals on the right-hand side of Eq. (20) are available. Frequency-limited estimation of the uncertainty is defined as:

$$\begin{array}{l}{\widehat{\mathrm{\Delta }}}_f={\widehat{W}}^T{\varphi }_f\end{array}$$E22

Next, one can define the low-frequency uncertainty estimation error as:

$${\tilde{\mathrm{\Delta }}}_f\triangleq {\mathrm{\Delta }}_f-{\widehat{\mathrm{\Delta }}}_f$$E23

$$=B^{†}\left[{\dot{x}}_f-{\mathit{Ax}}_f\right]-u_f-{\widehat{W}}^T{\varphi }_f$$E24

Then, the modified update law in CMRAC is given by

$$\text{CMRAC}\mathrm{Law}:\dot{\widehat{W}}=\mathrm{\Gamma }\left[-\varphi e^T\mathit{PB}+{\gamma }_c{\varphi }_f{\tilde{\mathrm{\Delta }}}_f^T\right]$$E25

where the error signal ${\tilde{\mathrm{\Delta }}}_f$ is available for feedback in control through Eq. (24) and ${\gamma }_c\in {\mathbb{R}}_{+}$ is a positive scalar learning rate for CMRAC. It should be noted that CMRAC modification term is gradient descent-based minimizing solution of the following optimization problem:

$$\begin{array}{l}\underset{\widehat{W}}{min}\mathcal{J}\\ \mathcal{J}\left(\widehat{W}\right)=\frac{1}{2}\parallel {\mathrm{\Delta }}_f-{\widehat{\mathrm{\Delta }}}_f{\parallel }^2=\frac{1}{2}\parallel {\tilde{\mathrm{\Delta }}}_f{\parallel }^2\end{array}$$E26

where the gradient of the cost function in Eq. (26) is given by:

$$\nabla {\mathcal{J}}_{\widehat{W}}\left(\widehat{W}\right)=-{\varphi }_f{\left[{\mathrm{\Delta }}_f\left(x\right)-{\widehat{W}}^T\left(t\right){\varphi }_f\left(x\right)\right]}^T=-{\varphi }_f{\tilde{\mathrm{\Delta }}}_f^T$$E27

Hence, the CMRAC weight update law can also be written as follows:

$$\text{CMRAC}\mathrm{Law}:\dot{\widehat{W}}=\mathrm{\Gamma }\left[-\varphi e^T\mathit{PB}-{\gamma }_c\nabla {\mathcal{J}}_{\widehat{W}}\left(\widehat{W}\right)\right]$$E28

In this section, we revisit the CMRAC formulation assuming structured uncertainty definition as this is the case in contributing reference studies [8, 20, 36], which implies $\epsilon \left(x\right)={\epsilon }_f\left(x,t\right)=0$. Then, the asymptotic stability of the closed-loop system with CMRAC weight update law in Eq. (25) can be shown with radially unbounded Lyapunov function $\mathrm{V}\left(e,\tilde{W}\right)=\frac{1}{2}{e}^T\mathit{Pe}+\frac{1}{2}\mathrm{tr}\left({\tilde{W}}^T{\mathrm{\Gamma }}^{-1}\tilde{W}\right)$. Its time-derivative along the system trajectories in Eq. (13) and Eq. (25) is given by $\dot{\mathrm{V}}\left(e,\tilde{W}\right)=-\frac{1}{2}{e}^T\mathit{Qe}-{\gamma }_c\mathrm{tr}\left({\tilde{W}}^T{\varphi }_f{\varphi }_f^T\tilde{W}\right)$ which is also equal to $\dot{\mathrm{V}}\left(e,\tilde{W}\right)=-\frac{1}{2}{e}^T\mathit{Qe}-{\gamma }_c\parallel {\tilde{\mathrm{\Delta }}}_f{\parallel }^2$. With this result, boundedness of the Lyapunov function $\mathrm{V}$ is guaranteed, which implies the boundedness of the state tracking error e(t) and weight estimation error $\tilde{W}\left(t\right)$. Next, following the similar discussions made in MRAC, one can show the boundedness of $\ddot{\mathrm{V}}\left(e,\tilde{W}\right)=-\frac{1}{2}{e}^{T}Q\dot{e}-{\gamma }_c{\tilde{\mathrm{\Delta }}}_f^T\left(-{\dot{\widehat{W}}}^T{\varphi }_f+{\tilde{W}}^T{\dot{\varphi }}_f\right)$ using Eqs. (19) and (25). It then follows from Barbalat’s Lemma [31] that ${lim}_{t\to \infty }\dot{\mathrm{V}}\left(e\left(t\right),\tilde{W}\left(t\right)\right)=0$ which implies the asymptotic stability of the state tracking error e(t) and filtered uncertainty estimation error ${\tilde{\mathrm{\Delta }}}_f\left(t\right)$; that is, ${lim}_{t\to \infty }e\left(t\right)=0$ and ${lim}_{t\to \infty }{\tilde{\mathrm{\Delta }}}_f\left(t\right)=0$. It should be emphasized that the parameter convergence is still not guaranteed with CMRAC since the asymptotic stability of ${\tilde{\mathrm{\Delta }}}_f$ does not imply the asymptotic stability of weight estimation error $\tilde{W}$. Furthermore, in case of unstructured uncertainty with nonzero residual $\epsilon \left(x\right)\ne 0$, the boundedness of adaptive weights is not guaranteed without persistent excitation of $\varphi \left(x\right)$ as in standard MRAC formulation, and the comments in Remark 1.3 apply to the CMRAC, as well.

Remark 1.5. Cut-off frequency,${\omega }_f$, inEq. (19)should be chosen carefully to reject the measurement noise and high-frequency content in the uncertainty. Although too small${\omega }_f$successfully filters the high-frequency signals in the adaptation, it may degrade the adaptation performance since$\widehat{W}$response becomes significantly sluggish [36].

4. Main result

Although CMRAC and its further modifications (for example [10, 12, 13, 14, 21, 37]) contributed a lot to the robustness and performance of the standard MRAC, all these filtering-based solutions suffer from losing information during filtering. As emphasized in Remark 1.5, the performance degradation due to lost information may reach significant levels due to a poor selection of filter bandwidth. If the system experiences high-frequency variations (for example, payload drop for an aircraft, actuator failures), sluggish evolution of the adaptive weights may not guarantee the desired tracking performance. Furthermore, a relatively small cut-off frequency ${\omega }_f$ may delay the adaptation due to time-delay nature of low-pass filters. In this study, we address these issues and propose a new model reference adaptive control architecture that compensates for the information lost during filtering by including the high-frequency content of the filtered signals. In addition, a time-varying learning rate is also proposed to suppress the undesired effects of high-frequency signals in the adaptation.

Assumption 1.4 (Interval Excitation). Basis$\varphi \in {\mathbb{R}}^s$is an interval exciting signal. Hence, there exist constant scalars$\tau ,\alpha ,T_e\in {\mathbb{R}}_{+}$such that the following inequality holds:

$${\int }_{t-\tau }^t\varphi \left(s\right){\varphi }^T\left(s\right)\mathit{ds}\succcurlyeq \alpha I_{s\times s},t=T_e$$E29

Remark 1.6. Assumption 1.4 ensures that the basis function$\varphi$contains as many spectral lines as the number of unknown parameters within time interval$t\in \left[T_e-{\tau }_w,T_e\right]$. Note that the basis function$\varphi$is an exogenous signal to the low-pass filter system inEq. (19). Then, the filtered basis function${\varphi }_f$also has the same number of spectral lines with less energy than that of the original basis$\varphi$. This implies the following:

$${\int }_{t-\tau }^t{\varphi }_f\left(s\right){\varphi }_f^T\left(s\right)\mathit{ds}\succcurlyeq \beta I_{n\times n},t=T_e$$E30

with $0<\beta \le \alpha$. Let the transfer function representing the linear filter dynamics be $\mathcal{G}\left(s\right)\triangleq \frac{{\omega }_f}{s+{\omega }_f}$. Then, the degradation in the richness of the signal (which is equivalent to how small is $\beta$ than $\alpha$) depends on the cut-off frequency ${\omega }_f$ and spectrum of the basis $\varphi$. Let the spectral measure of $\varphi$ is given by $S_{\varphi }\left(\left[{\omega }_1{\omega }_2\right]\right)$ with ${\omega }_1<{\omega }_2$. Choosing the filter design parameter ${\omega }_f$ to be larger than ${\omega }_2$ leads to relatively small degradation in the richness of ${\varphi }_f$ with $\beta$ being closer to $\alpha$.

4.1 Integrated uncertainty estimation error

We further manipulate the low-frequency content of the matched uncertainty ${\mathrm{\Delta }}_f$ by defining the following integrals over a moving window with a length of ${\tau }_{\omega }$ seconds:

$$\begin{array}{lll}{\mathcal{M}}_{\omega }& \triangleq & {\int }_{t-{\tau }_{\omega }}^t{\varphi }_f{\mathrm{\Delta }}_f^T\mathit{d\xi }=\left({\int }_{t-{\tau }_{\omega }}^t{\varphi }_f{\varphi }_f^T\mathit{d\xi }\right)W+{\int }_{t-{\tau }_{\omega }}^t{\varphi }_f{\epsilon }_f^T\mathit{d\xi }\\ {\widehat{\mathcal{M}}}_{\omega }& \triangleq & \left({\int }_{t-{\tau }_w}^t{\varphi }_f{\varphi }_f^T\mathit{d\xi }\right)\widehat{\mathrm{\Theta }}={\mathrm{\Phi }}_{\omega }\widehat{\mathrm{\Theta }}\end{array}$$E31

where $\widehat{\mathrm{\Theta }}\in {\mathbb{R}}^{s\times m}$ is an auxiliary weight, which will be discussed in Section 4.2.

Although ideal weight matrix W and filtered residual ${\epsilon }_f$ are unknown, the low-frequency content of the uncertainty, ${\mathrm{\Delta }}_f$, is accessible from Eq. (22). Hence, the integrated signal ${\mathcal{M}}_{\omega }$ can be computed during online operations. Next, the corresponding error ${\tilde{\mathcal{M}}}_{\omega }$ is defined as the following:

$$\begin{array}{l}{\tilde{\mathcal{M}}}_{\omega }\triangleq {\mathcal{M}}_{\omega }-{\widehat{\mathcal{M}}}_{\omega }=\left({\int }_{t-{\tau }_{\omega }}^t{\varphi }_f{\varphi }_f^T\mathit{d\xi }\right)\tilde{\mathrm{\Theta }}+{\int }_{t-{\tau }_{\omega }}^t{\epsilon }_f{\varphi }_f^T\mathit{d\xi }={\mathrm{\Phi }}_w\tilde{\mathrm{\Theta }}+{\int }_{t-{\tau }_{\omega }}^t{\epsilon }_f{\varphi }_f^T\mathit{d\xi }\end{array}$$E32

with $\tilde{\mathrm{\Theta }}\triangleq W-\widehat{\mathrm{\Theta }}$ being an auxiliary weight estimation error. It should also be noted that the signal ${\tilde{\mathcal{M}}}_{\omega }$ is also available since both ${\mathcal{M}}_{\omega }$ and ${\widehat{\mathcal{M}}}_{\omega }$ are accessible online. In order to preserve the information in excited basis functions, temporarily frozen variables $\mathcal{M}$ and $\widehat{\mathcal{M}}$ are defined as follows:

$$\begin{array}{l}\left\{\tilde{\mathcal{M}}\left(t\right),\mathrm{\Phi }\left(t\right)\right\}\triangleq \left\{\begin{array}{cc}\left\{{\tilde{\mathcal{M}}}_{\omega }\left(t\right),{\mathrm{\Phi }}_{\omega }\left(t\right)\right\},& \text{if}\kappa \left({\mathrm{\Phi }}_{\omega }\left(t\right)\right)<\kappa \left(\mathrm{\Phi }\left(t^{-}\right)\right)\\ \left\{\tilde{\mathcal{M}}\left(t^{-}\right),\mathrm{\Phi }\left(t^{-}\right)\right\},& \text{otherwise}\end{array}\right.\end{array}$$E33

where $\tilde{\mathcal{M}}\left(t^{-}\right)$ and $\varphi \left(t^{-}\right)$ have zero initial conditions of appropriate dimensions. Additionally, $\kappa \left(\cdot \right)$ returns the condition number of its argument. With the definition in Eq. (33), it is ensured that the condition number of $\mathrm{\Phi }\left(t\right)$ is monotonically non-increasing.

4.2 Auxiliary weight update law

In this section, we construct the first layer of dual adaptive law using the integrated signals in Section 4.1. The main purpose of the auxiliary adaptive law is to increase the stiffness of primary update law (which will be detailed in Section 4.3) and ensure the boundedness of adaptive weights $\widehat{W}$ when $\mathrm{\Delta }\left(x\right)$ is parameterized as unstructured uncertainty; i.e., $\epsilon \left(x\right)\ne 0$. Additionally, the auxiliary adaptation guarantees the parameter convergence without PE if the uncertainty is structured and Assumption 1.4 on interval excitation holds; i.e., $\tilde{W}\to 0$ as $t\to \infty$ when $\epsilon \left(x\right)=0$. In order to construct the auxiliary weight update law, we begin with the following optimization problem:

$$\begin{array}{l}\underset{\widehat{W}}{min}\mathcal{C},\mathcal{C}\left(\widehat{W}\right)=\frac{1}{2}\parallel \mathcal{M}-\widehat{\mathcal{M}}{\parallel }^2\end{array}$$E34

where the corresponding gradient becomes:

$$\nabla {\mathcal{C}}_{\widehat{W}}\left(\widehat{W}\right)=-\mathrm{\Phi }\left(\mathcal{M}-\widehat{\mathcal{M}}\right)=-\mathrm{\Phi }\left(\mathrm{\Phi }\tilde{\mathrm{\Theta }}+ϵ\right)$$E35

with $ϵ=\mathcal{M}-\mathrm{\Phi }W$. Then, the adaptive law for the auxiliary weights $\widehat{\theta }$ is constructed using gradient descent-based optimization as:

$$\dot{\widehat{\mathrm{\Theta }}}={\mathrm{\Gamma }}_{\mathrm{\Theta }}\mathrm{\Phi }\tilde{\mathcal{M}},\widehat{\mathrm{\Theta }}\left(t_0\right)={\widehat{\mathrm{\Theta }}}_0$$E36

with ${\mathrm{\Gamma }}_{\mathrm{\Theta }}\succ 0$ being positive definite gain matrix of appropriate dimensions.

Theorem 1.1. Consider the uncertain dynamical system inEq. (4)with unstructured uncertainty characterization inEq. (6), auxiliary weight update law inEq. (36), and integrated frozen signals inEq. (33). Then, auxiliary weight estimation error$\tilde{\mathrm{\Theta }}$is uniformly ultimately bounded. Furthermore, if Assumption 1.4 on interval excitation of the basis function$\varphi$is satisfied, the estimated auxiliary weights converge to a closer neighborhood of ideal weight W.

Proof: Consider the following Lyapunov function:

$${\mathrm{V}}_{\mathrm{\Theta }}=\frac{1}{2}\mathrm{tr}\left({\tilde{\mathrm{\Theta }}}^T{\mathrm{\Gamma }}_{\mathrm{\Theta }}^{-1}\tilde{\mathrm{\Theta }}\right)$$E37

Its time-derivative along the trajectory in Eq. (36) is obtained as:

$${\dot{\mathrm{V}}}_{\mathrm{\Theta }}=-\mathrm{tr}\left({\tilde{\mathrm{\Theta }}}^T\mathrm{\Phi }\tilde{\mathcal{M}}\right)=-\mathrm{tr}\left({\tilde{\mathrm{\Theta }}}^T\mathrm{\Phi \Phi }\tilde{\mathrm{\Theta }}\right)-\mathrm{tr}\left({\tilde{\mathrm{\Theta }}}^T\mathrm{\Phi }ϵ\right)=-\parallel \mathrm{\Phi }\tilde{\mathrm{\Theta }}{\parallel }_F^2-\mathrm{tr}\left({\tilde{\mathrm{\Theta }}}^T\mathrm{\Phi }ϵ\right)$$E38

If Assumption 1.4 is not satisfied, $\mathrm{\Phi }\left(t\right)=0$ holds due to Eq. (33) with zero initial condition $\mathrm{\Phi }\left(t_0\right)=0$, which results in ${\dot{\mathrm{V}}}_{\mathrm{\Theta }}=0$. Thus, the adaptation for the auxiliary weights vanishes and $\widehat{\mathrm{\Theta }}$ stays at its initial condition, i.e. $\dot{\widehat{\mathrm{\Theta }}}=0$, $\widehat{\mathrm{\Theta }}\left(t\right)={\widehat{\mathrm{\Theta }}}_0$. Hence, the boundedness of auxiliary parameters is guaranteed if Assumption 1.4 does not hold. From this point forward, let us investigate the case where the interval excitation assumption holds. Note that, at any time t, the integrated residual error $ϵ$ for the unstructured uncertainty case can be bounded from above as:

$$\begin{array}{lll}‖{\int }_{t-{\tau }_w}^t{\epsilon }_f{\varphi }_f^T\mathit{d\xi }‖& \le & {\int }_{t-{\tau }_w}^t‖{\epsilon }_f{\varphi }_f^T‖\mathit{d\xi }\le {\int }_{t-{\tau }_w}^t‖{\epsilon }_f‖‖{\varphi }_f^T‖\mathit{d\xi }\\ & \le & {\int }_{t-{\tau }_w}^t{\epsilon }_0\sqrt{s}\mathit{d\xi }\le {\epsilon }_0{\tau }_w\sqrt{s},\forall t\in {\mathbb{R}}_{+}\end{array}$$E39

by utilizing the Gaussian kernels as radial basis functions for $\varphi \in {\mathbb{R}}^s$. Then, the time-derivative of the Lyapunov function can be bounded as

$${\dot{\mathrm{V}}}_{\mathrm{\Theta }}\le -\parallel \mathrm{\Phi }\tilde{\mathrm{\Theta }}{\parallel }_F^2+c_1\parallel \mathrm{\Phi }\tilde{\theta }{\parallel }_F^2+\frac{1}{4c_1}\parallel ϵ{\parallel }^2=-\left(1-c_1\right)\parallel \mathrm{\Phi }\tilde{\mathrm{\Theta }}{\parallel }_F^2+c_2^2$$E40

with $c_2\triangleq \frac{{\epsilon }_0{\tau }_w\sqrt{s}}{2\sqrt{c_1}}$. Then, its is clear that for any $c_1\in \left(0,1\right)$, the matrix product $\mathrm{\Phi }\tilde{\mathrm{\Theta }}$ is bounded since ${\dot{\mathrm{V}}}_{\mathrm{\Theta }}<0$ if $\parallel \mathrm{\Phi }\tilde{\mathrm{\Theta }}{\parallel }_F>c_2\sqrt{1/\left(1-c_1\right)}$. As Assumption 1.4 is satisfied, the matrix $\mathrm{\Phi }$ is positive definite with ${\lambda }_{\min }\left(\mathrm{\Phi }\right)\ge \beta$ from Remark 1.6. The boundedness of $\mathrm{\Phi }\tilde{\mathrm{\Theta }}$ with an invertible $\mathrm{\Phi }$ guarantees the boundedness of auxiliary parameter estimation error $\tilde{\mathrm{\Theta }}$, since $\tilde{\mathrm{\Theta }}={\mathrm{\Phi }}^{-1}\left(\mathrm{\Phi }\tilde{\mathrm{\Theta }}\right)$, that is, $\parallel \tilde{\mathrm{\Theta }}\parallel \le d_1$ with $d_1>0$ being unknown positive constant that depends on the excitation level $\beta$, residual bound ${\epsilon }_0$, and number of RBF neurons s. With constant unknown weight matrix W and $\tilde{\mathrm{\Theta }}\in {\mathcal{L}}_{\infty }$, the boundedness of auxiliary weight estimation $\widehat{\mathrm{\Theta }}$ is ensured, as well. ■

Corollary 1.1. Consider the control system outlined in Theorem 1.1. With structured uncertainty parameterization inEq. (5), auxiliary weights$\widehat{\mathrm{\Theta }}$converge exponentially to the ideal weights W if Assumption 1.4 holds.

Proof: Structured uncertainty parameterization implies $\epsilon \left(x\right)=ϵ=0$. Hence, Lyapunov derivative becomes:

$${\dot{\mathrm{V}}}_{\mathrm{\Theta }}=-\parallel \mathrm{\Phi }\tilde{\mathrm{\Theta }}{\parallel }_F^2\le -{\lambda }_{\min }\left({\mathrm{\Phi }}^2\right)\parallel \tilde{\mathrm{\Theta }}{\parallel }^2\le -{\beta }^2\parallel \tilde{\mathrm{\Theta }}{\parallel }^2<0$$E41

with $\beta$ being excitation level in Remark 1.6. Quadratic positive definite Lyapunov function with quadratic negative definite derivative ensures the exponential convergence of the auxiliary weight convergence error $\tilde{\mathrm{\Theta }}$ at a rate of ${\beta }^2$; that is, $\parallel \tilde{\mathrm{\Theta }}\left(t\right)\parallel \le \parallel \tilde{\mathrm{\Theta }}\left(t_0\right)\parallel e^{-{\beta }^2\left(t-t_0\right)}\triangleq d_2\left(t\right)\le {\overline{d}}_2$ with ${\overline{d}}_2\in {\mathbb{R}}_{+}$ being an unknown constant scalar. Since the Lyapunov function is radially unbounded, this result holds globally.

Remark 1.7. Unlike the minimum singular value maximization algorithm [38], the condition number minimization is utilized inEq. (33). The main reason is that the deviation from the ideal solution W in least-squares regression due to perturbation$\epsilon$is directly proportional to the condition number of the regressor matrix$\mathrm{\Phi }$. Therefore, minimizing the condition number of$\mathrm{\Phi }$improves the robustness against residual error$\epsilon \left(x\right)$. Furthermore, maximizing the minimum singular value is inherently embedded in condition number minimization due to the definition$\kappa \left(\cdot \right)=\frac{\lambda max\left(\cdot \right)}{\lambda min\left(\cdot \right)}$. It should be emphasized that the condition number minimization also introduces additional information of${\lambda }_{\max }\left(\mathrm{\Phi }\right)$on temporarily frozen matrices, which results in more frequent update, more robust adaptation, and faster development of weight estimations. Further discussion on condition number minimizing algorithm can be found in [39].

4.3 Primary layer of adaptation

In this section, the primary weight update law is introduced for the dual adaptive controller. We start by decomposing the weight update law as:

$$\dot{\widehat{W}}={\dot{\widehat{W}}}_b+{\dot{\widehat{W}}}_m,\text{with}{\dot{\widehat{W}}}_b=-\mathrm{\Gamma }\varphi e^T\mathit{PB}$$E42

where ${\dot{\widehat{W}}}_b$ is the standard MRAC law in Eq. (14) and ${\dot{\widehat{W}}}_m$ is the proposed modification term given as the following:

$${\dot{\widehat{W}}}_m={\left({\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\right)}^{-1}\left[{\gamma }_m{\varphi }_f{\varphi }_f^T\mathrm{\Gamma }\varphi e^T\mathit{PB}+{\gamma }_m{\dot{\varphi }}_f{\tilde{\mathrm{\Delta }}}_f^T+{\gamma }_1{\varphi }_f{\tilde{\mathrm{\Delta }}}_f^T+{\gamma }_2\left(\widehat{\mathrm{\Theta }}-\widehat{W}\right)\right]$$E43

where ${\gamma }_1,{\gamma }_2,{\gamma }_m\in {\mathbb{R}}_{+}$ are user-defined design variables. Then, the proposed weight update law can be written as

$$\dot{\widehat{W}}=-\mathbb{X}\varphi e^T\mathit{PB}+\mathbb{X}\left[{\gamma }_1{\varphi }_f{\tilde{\mathrm{\Delta }}}_f^T+{\gamma }_2\left(\widehat{\mathrm{\Theta }}-\widehat{W}\right)+{\gamma }_m{\dot{\varphi }}_f{\tilde{\mathrm{\Delta }}}_f^T\right]$$E44

where the time-varying learning rate $\mathbb{X}$ is given by

$$\mathbb{X}\triangleq {\left({\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\right)}^{-1}$$E45

Lemma 1.1. The Learning rate$\mathbb{X}$is a positive definite matrix.

Proof:

$$\begin{array}{lll}\mathbb{X}\succ 0& \Rightarrow & {\left({\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\right)}^{-1}\succ 0\\ & \Rightarrow & {\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\succ 0\\ \left(sinc{\mathrm{e}}^{\prime }{\varphi }_f{\varphi }_f^T\succcurlyeq 0^{\text{'}}\text{and}{\mathrm{\gamma }}_{\mathrm{m}}>0\right)& \Rightarrow & {\mathrm{\Gamma }}^{-1}\succ 0\Rightarrow \mathrm{\Gamma }\succ 0\end{array}$$E46

Since $\mathrm{\Gamma }\succ 0$ by the design, the claim $\mathbb{X}\succ 0$ holds, as well.

Remark 1.8. The term${\gamma }_2\left(\widehat{\mathrm{\Theta }}-\widehat{W}\right)$is essentially a$\sigma$-modification term introduced to the adaptive law, causing the estimated weight$\widehat{W}$to remain bounded around auxiliary weight$\widehat{\mathrm{\Theta }}$. Recall that$\widehat{\mathrm{\Theta }}$is bounded from Theorem 1.1. Hence, the stability results from standard MRAC with$\sigma$-modification in Remark 1.4 can directly be applied to ensure the Lyapunov stability of all the closed-loop signals with the proposed adaptive controller, as will be shown by Theorem 1.2 and Theorem 1.3.

4.4 Stability results for unstructured uncertainty

Theorem 1.2. Consider the uncertain system dynamics given inEq. (4), uncertainty parametrization in Assumption 1.2, reference model inEq. (8), control input inEq. (12), filter states inEq. (19), and adaptive law inEq. (44). Then, all the signals in closed-loop system are uniformly ultimately bounded.

Proof: Consider the following radially unbounded Lyapunov function:

$$\begin{array}{l}\mathrm{V}=\frac{1}{2}{e}^T\mathit{Pe}+\frac{1}{2}\mathrm{tr}\left({\tilde{W}}^T{\mathbb{X}}^{-1}\tilde{W}\right)\le \frac{1}{2}{\lambda }_{\max }\left(P\right)\parallel e{\parallel }^2+\frac{1}{2}\left({\lambda }_{\max }\left({\mathrm{\Gamma }}^{-1}\right)+{\gamma }_{m}s\right)\parallel \tilde{W}{\parallel }^2\end{array}$$E47

Note that $\mathrm{V}\left(0,0\right)=0$ and $\mathrm{V}\left(e,\tilde{W}\right)\ne 0$ for all $\left(e,\tilde{W}\right)\ne \left(0,0\right)$, $\forall t\in {\mathbb{R}}_{+}$. Time derivative of the Lyapunov function along the system trajectories in Eqs. (13) and (44) is given as follows:

$$\begin{array}{lll}\dot{\mathrm{V}}& =& e^{T}P\dot{e}+\mathrm{tr}\left({\tilde{W}}^T{\mathbb{X}}^{-1}\dot{\tilde{W}}\right)+\frac{1}{2}\mathrm{tr}\left({\tilde{W}}^T{\mathbb{X}}^{-1}\tilde{W}\right)\\ & =& -\frac{1}{2}{e}^T\mathit{Qe}-\mathrm{tr}\left({\tilde{W}}^T{\mathbb{X}}^{-1}\dot{\widehat{W}}\right)+{\gamma }_m\mathrm{tr}\left({\tilde{W}}^T{\dot{\varphi }}_f{\varphi }_f^T\tilde{W}\right)\end{array}$$E48

Substituting the weight update law in Eq. (44) into Eq. (48) gives:

$$\begin{array}{lll}\dot{\mathrm{V}}& =& -\frac{1}{2}{e}^T\mathit{Qe}-{\gamma }_1\mathrm{tr}\left({\tilde{W}}^T{\varphi }_f{\varphi }_f^T\tilde{W}\right)-{\gamma }_2\mathrm{tr}\left({\tilde{W}}^T\tilde{W}\right)\\ & & -{\gamma }_1\mathrm{tr}\left({\tilde{W}}^T{\varphi }_f{\epsilon }_f^T\right)-{\gamma }_m\mathrm{tr}\left({\tilde{W}}^T{\dot{\varphi }}_f{\epsilon }_f^T\right)-{\gamma }_2\mathrm{tr}\left[{\tilde{W}}^T\left(\widehat{\mathrm{\Theta }}-W\right)\right]\\ & =& -\frac{1}{2}{e}^T\mathit{Qe}-{\gamma }_1\parallel {\tilde{W}}^T{\varphi }_f{\parallel }_F^2-{\gamma }_2\parallel \tilde{W}{\parallel }_F^2\\ & & -{\gamma }_1\mathrm{tr}\left({\tilde{W}}^T{\varphi }_f{\epsilon }_f^T\right)-{\gamma }_m\mathrm{tr}\left({\tilde{W}}^T{\dot{\varphi }}_f{\epsilon }_f^T\right)-{\gamma }_2\mathrm{tr}\left[{\tilde{W}}^T\left(\widehat{\mathrm{\Theta }}-W\right)\right]\\ & \le & -\frac{1}{2}{\lambda }_{\min }\left(Q\right)\parallel e{\parallel }^2-{\gamma }_1\parallel {\tilde{W}}^T{\varphi }_f{\parallel }^2-{\gamma }_2\parallel \tilde{W}{\parallel }^2+\parallel \tilde{W}\parallel d_4\\ & =& -\frac{1}{2}{\lambda }_{\min }\left(Q\right)\parallel e{\parallel }^2-{\gamma }_1\parallel {\tilde{W}}^T{\varphi }_f{\parallel }^2-{\gamma }_2\parallel \tilde{W}\parallel \left(\parallel \tilde{W}\parallel -d_4\right)\end{array}$$E49

where $d_4\triangleq {\gamma }_1{\epsilon }_0\sqrt{s}+{\gamma }_m{\epsilon }_0{d}_3+{\gamma }_2{d}_1$, $\parallel \widehat{\mathrm{\Theta }}-W\parallel \le d_1$ is guaranteed from Theorem 1.1, $\parallel {\dot{\varphi }}_f\parallel \le d_3$ holds for $\varphi \left(x\right)$ consisting of smooth Gaussian kernels, with $d_3>0$ being an unknown positive constant. It is obvious that the time derivative of Lyapunov function is negative definite for $\parallel \tilde{W}\parallel >d_4$. Thus, the error signals e and $\tilde{W}$ asymptotically converge to a compact set, ensuring $e,\tilde{W}\in {\mathcal{L}}_{\infty }$. Since the reference model state $x_r$ is bounded and the unknown weight matrix W is constant, boundedness of system state vector x and adaptive weights $\widehat{W}$ is guaranteed.

4.5 Stability results for structured uncertainty

Theorem 1.3. Consider the uncertain system dynamics given inEq. (4), uncertainty parametrization in Assumption 1.1, reference model inEq. (8), control input inEq. (12), filter states inEq. (19), and adaptive law inEq. (44). Then, the state tracking error$e\left(t\right)$is asymptotically stable. Furthermore, all the signals in a closed-loop system are bounded. If the basis function$\varphi$is interval exciting, then the zero-solution$\left(e,\tilde{W},{\tilde{\mathrm{\Delta }}}_f\right)=\left(\mathrm{0,0,0}\right)$is asymptotically stable.

Proof: Recall the Lyapunov function in Theorem 1.2:

$$\begin{array}{l}\mathrm{V}=\frac{1}{2}{e}^T\mathit{Pe}+\frac{1}{2}\mathrm{tr}\left({\tilde{W}}^T{\mathbb{X}}^{-1}\tilde{W}\right)\le \frac{1}{2}{\lambda }_{\max }\left(P\right)\parallel e{\parallel }^2+\frac{1}{2}\left(\frac{1}{{\lambda }_{\min }\left(\mathrm{\Gamma }\right)}+{\gamma }_{m}s\right)\parallel \tilde{W}{\parallel }^2\end{array}$$E50

with its time derivative being

$$\dot{\mathrm{V}}=-\frac{1}{2}{\lambda }_{\min }\left(Q\right)\parallel e{\parallel }^2-{\gamma }_1\parallel {\tilde{W}}^T{\varphi }_f{\parallel }^2-{\gamma }_2\parallel \tilde{W}\parallel \left(\parallel \tilde{W}\parallel -d_4\right)$$E51

where $d_4={\gamma }_1{\epsilon }_0\sqrt{s}+{\gamma }_m{\epsilon }_0{d}_3+{\gamma }_2{d}_1$. With structured uncertainty parameterization, we have $\epsilon ={\epsilon }_f=ϵ=0$, yielding $d_4={\gamma }_2{d}_1$ and ${\tilde{\mathrm{\Delta }}}_f={\tilde{W}}^T{\varphi }_f$. The discussions for boundedness of all system signals in Theorem 1.2 also hold in structured uncertainty cases with a smaller bound on $\tilde{W}$. It can also be shown that the error signals $e,{\tilde{\mathrm{\Delta }}}_f,\tilde{W}$ are asymptotically stable by incorporating the results from Corollary 1.1. Specifically, upper bound $d_1$ can be replaced with $d_2\left(t\right)$, i.e., $d_4={\gamma }_2{d}_2\left(t\right)$ where $d_2\left(t\right)\to 0$ as $t\to \infty$ exponentially at a rate of ${\beta }^2$. Thus,

$$\dot{\mathrm{V}}\left(t\right)\to -\frac{1}{2}{\lambda }_{\min }\left(Q\right)\parallel e{\parallel }^2-{\gamma }_1\parallel {\tilde{\mathrm{\Delta }}}_f{\parallel }^2-{\gamma }_2\parallel \tilde{W}{\parallel }^2<0,\mathrm{as}t\to \infty$$E52

Therefore, the tracking error e low-frequency uncertainty estimation error ${\tilde{\mathrm{\Delta }}}_f$, and weight estimation error $\tilde{W}$ asymptotically converge to zero, ensuring the asymptotic stability of the zero-solution $\left(e,\tilde{W},{\tilde{\mathrm{\Delta }}}_f\right)=\left(\mathrm{0,0,0}\right)$.

4.6 Discussion

The proposed information recovery-based composite model reference adaptive controller (IR-CMRAC) acquires all the benefits of CMRAC as IR-CMRAC update law already contains the minimizing solution in Eq. (27). Additionally, the proposed method exhibits significant improvements as the following:

• As the design parameter ${\gamma }_m$ tends to zero, the learning rate $\mathbb{X}$ in Eq. (45) tend to $\mathrm{\Gamma }$, furthermore, with ${\gamma }_2=0$, the entire proposed modification term results in CMRAC law in Eq. (25). Hence, the proposed law can be considered as a generalized form of CMRAC.

• Time derivative of gradient of the cost function in Eq. (27) is given by

$$\frac{d}{\mathit{dt}}\left(\nabla {\mathcal{J}}_{\widehat{W}}\right)=2{\dot{\varphi }}_f{\varphi }_f^T\tilde{W}-{\varphi }_f{\varphi }_f^T{\dot{\widehat{W}}}_b-{\varphi }_f{\varphi }_f^T{\dot{\widehat{W}}}_m$$E53

It can be observed that the proposed update law in Eq. (44) contains the derivative information given in Eq. (53). Incorporating the derivative into traditional gradient descent-based optimization allows the shape of the transient behavior. In that respect, it can be considered analogous to the derivative action in traditional proportional/integral/derivative (PID) controllers.

• Each term in Eq. (53) contributes to the adaptation performance in different manners. Specifically, the first term, ‘${\dot{\varphi }}_f{\tilde{\mathrm{\Delta }}}_f$’, compensates for the information lost while filtering the basis function $\varphi$ by including the high-frequency content of basis $\varphi$. Recall that ${\dot{\varphi }}_f$ is given by ${\dot{\varphi }}_f={\omega }_f\left(\varphi -{\varphi }_f\right)$; that is, ${\dot{\varphi }}_f$ actually contains only the high-frequency components of basis $\varphi$. It should also be noted that the product of ${\dot{\varphi }}_f$ and ${\varphi }_f$ is still a high-frequency signal.

• The second term in Eq. (53), ‘${\varphi }_f{\varphi }_f^T{\dot{\widehat{W}}}_b$’, regulates the standard MRAC through gain ${\gamma }_m$. In this context, this term can be considered analogous to proportional feedback control for the standard adaptive law, acting akin to a stability augmentation system. Essentially, it entails readjusting the learning rate $\mathrm{\Gamma }$ to prevent adverse effects on adaptation from large values in the basis function $\varphi$.

• The last term in Eq. (53), ‘${\varphi }_f{\varphi }_f^T{\dot{\widehat{W}}}_m$’, regulates the learning rate of the modification term in a similar way to the standard adaptive law regulation explained in previous item.

• Last but not the least, term ‘${\gamma }_2\left(\widehat{\mathrm{\Theta }}-\widehat{W}\right)$’ in Eq. (44) plays a vital role in ensuring the parameter convergence in structured uncertainty parameterization; i.e. $\widehat{W}\left(t\right)\to W$ as $t\to \infty$. In that respect, this term acts like an integral action in a traditional PID controllers as it analogously allows to remove the steady state error in the weight estimations. Specifically, the auxiliary weight $\widehat{\mathrm{\Theta }}$ evolves in the direction of minimizing the steady-state error in the parameter estimation, and the term ‘${\gamma }_2\left(\widehat{\mathrm{\Theta }}-\widehat{W}\right)$’ ensures that the adaptive weights $\widehat{W}$ stay bounded around $\widehat{\mathrm{\Theta }}$, which inherently results in zero steady-state error in $\tilde{W}$. For unstructured uncertainty parameterization, boundedness of adaptive weights around close neighborhood of the ideal parameters are achieved (instead of parameter convergence).

4.7 Inverse-free weight update law

One might observe that the updated law proposed in Eq. (44) necessitates computing the inverse of a (potentially large matrix) during online operations. This would undoubtedly incur additional computational costs and practical challenges. To address these drawbacks, we proceed to manipulate this term to achieve an inverse-free learning rate.

Lemma 1.2. Given that$\mathrm{\Gamma }={\mathrm{\Gamma }}^T\succ 0$is positive definite matrix,${\gamma }_m\in {\mathbb{R}}_{+}$is a positive constant scalar, and${\varphi }_f\in {\mathbb{R}}^s$, the matrix$\left({\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\right)$is invertible and its inverse is given by

$${\left({\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\right)}^{-1}=\mathrm{\Gamma }-\frac{1}{1+\mathit{tr}\left({\gamma }_m{\varphi }_f{\varphi }_f^T\mathrm{\Gamma }\right)}\mathrm{\Gamma }{\gamma }_m{\varphi }_f{\varphi }_f^T\mathrm{\Gamma }\succ 0$$E54

Proof: Note that $\mathrm{\Gamma }={\mathrm{\Gamma }}^T\succ 0$ and ${\gamma }_m{\varphi }_f{\varphi }_f^T\succcurlyeq 0$. This fact implies $\left({\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\right)$ is also a symmetric positive definite matrix, i.e. $\left({\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\right)\succ 0$. Then, $\left({\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\right)$ is always invertible. Let us define the following for clarity:

$$\begin{array}{lll}E\triangleq {\gamma }_m{\varphi }_f{\varphi }_f^T& =& \left({\varphi }_f\sqrt{{\gamma }_m}\right){\left({\varphi }_f\sqrt{{\gamma }_m}\right)}^T\\ G\triangleq {\mathrm{\Gamma }}^{-1}& \to & G^{-1}=\mathrm{\Gamma }\end{array}$$E55

Then,

$$\begin{array}{lll}{\mathit{EG}}^{-1}E& =& \left({\varphi }_f\sqrt{{\gamma }_m}\right){\left({\varphi }_f\sqrt{{\gamma }_m}\right)}^T\mathrm{\Gamma }\left({\varphi }_f\sqrt{{\gamma }_m}\right){\left({\varphi }_f\sqrt{{\gamma }_m}\right)}^T\\ & =& \left[{\left({\varphi }_f\sqrt{{\gamma }_m}\right)}^T\mathrm{\Gamma }\left({\varphi }_f\sqrt{{\gamma }_m}\right)\right]\left({\varphi }_f\sqrt{{\gamma }_m}\right){\left({\varphi }_f\sqrt{{\gamma }_m}\right)}^T\\ & =& \mathrm{tr}\left[\left({\varphi }_f\sqrt{{\gamma }_m}\right){\left({\varphi }_f\sqrt{{\gamma }_m}\right)}^T\mathrm{\Gamma }\right]\left({\varphi }_f\sqrt{{\gamma }_m}\right){\left({\varphi }_f\sqrt{{\gamma }_m}\right)}^T\\ & =& \mathrm{tr}\left({\mathit{EG}}^{-1}\right)E\end{array}$$E56

Le ${\left(G+E\right)}^{-1}$ be in the form of ${\left(G+E\right)}^{-1}=\left(G^{-1}-\nu G^{-1}{\mathit{EG}}^{-1}\right)$. Then, their product

$$\left(G+E\right)\left(G^{-1}-\nu G^{-1}{\mathit{EG}}^{-1}\right)=I-\nu {\mathit{EG}}^{-1}+{\mathit{EG}}^{-1}-\nu {\mathit{EG}}^{-1}{\mathit{EG}}^{-1}$$E57

gives identity matrix if the equality $\left(\nu {\mathit{EG}}^{-1}-{\mathit{EG}}^{-1}+\nu {\mathit{EG}}^{-1}{\mathit{EG}}^{-1}\right)=0$ is satisfied. Then, combining this inequality with the result in Eq. (56) yields:

$$\nu {\mathit{EG}}^{-1}-{\mathit{EG}}^{-1}+\mathit{\nu tr}\left({\mathit{EG}}^{-1}\right){\mathit{EG}}^{-1}=\left[\nu -1+\mathit{\nu tr}\left({\mathit{EG}}^{-1}\right)\right]{\mathit{EG}}^{-1}=0$$E58

This equality holds for any square matrix E with $rank\left(E\right)=1$ and invertible matrix G if the parameter v is chosen as

$$\nu =\frac{1}{1+\mathrm{tr}\left({\mathit{EG}}^{-1}\right)}=\frac{1}{1+\mathrm{tr}\left({\gamma }_m{\varphi }_f{\varphi }_f^T\mathrm{\Gamma }\right)}$$E59

Substituting v into ${\left(G+E\right)}^{-1}=\left(G^{-1}-\nu G^{-1}{\mathit{EG}}^{-1}\right)$ results in

$${\left({\mathrm{\Gamma }}^{-1}+{\gamma }_m{\varphi }_f{\varphi }_f^T\right)}^{-1}=\mathrm{\Gamma }-\frac{1}{1+\mathrm{tr}\left({\gamma }_m{\varphi }_f{\varphi }_f^T\mathrm{\Gamma }\right)}{\gamma }_m\mathrm{\Gamma }{\varphi }_f{\varphi }_f^T\mathrm{\Gamma }$$E60

That completes the proof. Readers may refer to [40] for details.

5. Illustrative numerical examples

5.1 Systems with integrator dynamics

In this example, the first order roll dynamics of an aircraft is considered:

$$\dot{p}\left(t\right)=L_{p}p\left(t\right)+L_{\delta }\left[{\delta }_a\left(t\right)+\mathrm{\Delta }\left(p\right)\right]$$E61

where $p\left(t\right)$ is roll rate, ${\delta }_a\left(t\right)$ is aileron input, dynamic stability derivative $L_p$ denoting the roll damping is $L_p=-1$, control derivative is $L_{\delta }=1$, and the uncertainty $\mathrm{\Delta }\left(p\right)$ is given by

$$\mathrm{\Delta }\left(p\right)=0.2+p\mid p\mid$$E62

This numerical example can be regarded as an incorrect characterization of the roll mode time constant and the existence of constant wind disturbance trying to roll the aircraft. To improve the tracking performance robustness, an integrator state $x_i$ is augmented to the system as follows:

$$\begin{array}{l}\left[\begin{array}{c}\dot{p}\left(t\right)\\ {\dot{x}}_i\left(t\right)\end{array}\right]=\left[\begin{array}{cc}{L}_p& 0\\ -1& 0\end{array}\right]\left[\begin{array}{c}p\left(t\right)\\ x_i\left(t\right)\end{array}\right]+\left[\begin{array}{c}{L}_{\delta }\\ 0\end{array}\right]\left[{\delta }_a\left(t\right)+\mathrm{\Delta }\left(p\right)\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]p_c\left(t\right)\end{array}$$E63

where $p_c$ is the commanded roll rate.

It is important to note that standard MRAC suffers from degraded performance in systems with augmented integrator dynamics if the learning rate is relatively high. As clearly seen in Figure 1, increasing the learning rate induces high-frequency oscillations in the system. In this example, the efficacy of the proposed method will also be highlighted for the systems with augmented integrator dynamics.

Figure 1.

Performance comparison for standard MRAC in the presence of high gains.

The reference model is chosen as:

$${\dot{x}}_r\left(t\right)=\left[\begin{array}{cc}-2.5& 3\\ -1& 0\end{array}\right]x_r\left(t\right)+\left[\begin{array}{c}0\\ 1\end{array}\right]p_c\left(t\right)$$E64

Thus, the nominal controller becomes:

$$u_n\left(t\right)=-1.5p\left(t\right)+3x_i\left(t\right)$$E65

5.1.1 Without integral action in the adaptation (${\gamma }_2=0$)

Numerical simulations are carried out with sampling frequency of 100 Hz, positive definite solution P is determined using $Q=I_{2\times 2}$, and low-pass filter bandwidth is ${\omega }_f=4$ rad/s. In addition, adaptive gains are ${\gamma }_1={\gamma }_c={\gamma }_m=1$, $\mathrm{\Gamma }=I_{2\times 2}$, ${\mathrm{\Gamma }}_{\mathrm{\Theta }}=2I_{2\times 2}$, ${\tau }_w=20$ s, and ${\gamma }_2=0$. That is, the integral action in adaptation is not activated for results shown in Figures 2 and 3. It should be noted that the common adaptive gains are kept the same to make a fair comparison; that is, ${\gamma }_1={\gamma }_c$ and $\mathrm{\Gamma }$ is constant for all simulations. Lastly, the commanded roll rate $p_c$ is a square wave with an amplitude of 1 rad/s and period of 10 s (see Figure 1).

Figure 2.

Tracking performance comparison of adaptive controllers (${\gamma }_2=0$ in IR-CMRAC).

Figure 3.

Weight estimation performance comparison of adaptive controllers (${\gamma }_2=0$ in IR-CMRAC).

In Figure 2, state tracking performance for different adaptive controllers is illustrated. As clearly seen, the nominal controller results in oscillatory roll response due to insufficient damping. Among all the adaptive controllers, the best tracking performance is achieved by the IR-CMRAC with almost zero state tracking error. Additionally, the desired tracking response is realized by the aileron control input with no oscillations.

Figure 3 indicates the evolution of the weight estimation error $\tilde{W}$. Although the integral action is disabled with ${\gamma }_2=0$, the parameter convergence is achieved within 15 seconds. It should be noted that the exciting signal $p_c\left(t\right)$ is persistently exciting; hence, the parameter convergence is an expected result. However, the important point is that the fastest convergence is achieved with IR-CMRAC as more information is included in the adaptation, which adds new directions to update the adaptive weights.

Figure 4 illustrates the evolution of time-varying learning rate $\mathbb{X}$. One can observe that the variation of diagonal elements ${\mathbb{X}}_{11}$ and ${\mathbb{X}}_{22}$ is more apparent during the transients. This is because the stability augmentation system becomes active especially during the high-frequency variations in the signals. In this way, the undesired effects of high-frequency signals used in the adaptation are suppressed.

Figure 4.

Evolution of time-varying learning rates in IR-CMRAC (${\gamma }_2=0$ in IR-CMRAC).

5.1.2 With integral action in the adaptation (${\gamma }_2=10$)

In this scenario, the integral action in IR-CMRAC is activated by setting the learning rate ${\gamma }_2=10$ in Eq. (44). The state tracking performance, evolution of time-varying learning rate $\mathbb{X}$, and aileron control input are not illustrated for this case since the differences between simulation results with ${\gamma }_2=0$ and ${\gamma }_2=10$ are indistinguishable. However, the weight estimation performance is worth mentioning and presented in Figure 5. Unlike IR-CMRAC with ${\gamma }_2=0$, the adaptive weights keep evolving with ${\gamma }_2=10$ even if the systems states are not excited (for example on the interval of $t\in \left[5,10\right]$). This is because, by the time $t=5$ seconds, the basis function $\varphi$ has become sufficiently exciting, thereby fulfilling Assumption 1.4. As a result, faster convergence of the adaptive weights is achieved, as claimed by Theorem 1.3.

Figure 5.

Weight estimation performance comparison of adaptive controllers.

5.2 Presence of measurement noise and external disturbance

In this example, an external disturbance is added to the uncertainty as:

$$\mathrm{\Delta }\left(p,t\right)=0.2+0.23e^{-0.13t}+0.1sin\left(0.2t\right)+0.13sin\left(0.37t\right)+p\mid p\mid$$E66

Newly introduced sinusoidal signals can be considered as the effects of variable wind speed and direction, whereas the exponential term is used to simulate the wind shear. With this definition of aggregated uncertainty, the unknown weight matrix becomes time-varying:

$$W\left(t\right)=\left[\begin{array}{c}0.2-0.23e^{-0.13t}+0.1sin\left(0.2t\right)+0.13sin\left(0.37t\right)\\ 1\end{array}\right],\varphi \left(p\right)=\left[\begin{array}{c}1\\ p\mid p\mid \end{array}\right]$$E67

Furthermore, the measurement noise is added to the roll rate p as

$$p\left(t\right)\leftarrow p\left(t\right)+\mathrm{\upsilon }\left(t\right)$$E68

where $\mathrm{\upsilon }\left(t\right)$ is the Gaussian white noise with standard deviation of 2 $\mathit{deg}/s$, which practically ensures $\mathrm{\upsilon }\left(t\right)\in \left[-6,6\right]\mathit{deg}/s$, $\forall t\ge t_0$. All the simulation parameters are the same as in Section 5.1.1, except for the learning rate ${\gamma }_2=1$.

Figure 6 illustrates the state tracking performance of the proposed adaptive controller for the outlined simulation scenario. Due to the low-pass filter nature of integration, the integrator state $x_2$ is not affected drastically by the roll rate measurement noise. Furthermore, IR-CMRAC is mainly driven by the low-pass filtered signals (for example ${\varphi }_f$, ${\tilde{\mathrm{\Delta }}}_f$, $u_f$ in Eq. (44)); the effects of measurement noise are suppressed successfully, thereby resulting in smooth aileron control input. This result can also be observed in the time-varying learning rate components ${\mathbb{X}}_{\mathit{ij}}$ given in Figure 7.

Figure 6.

State tracking performance for IR-CMRAC in the presence of noise and disturbance.

Figure 7.

Variation of components of the learning rate $\mathbb{X}$ for IR-CMRAC.

It is important to remember that the convergence results of Theorem 1.3 are valid if the unknown weight matrix W is constant. However, it can readily be shown that all the system signals are uniformly ultimately bounded if the unknown weight matrix is bounded and time-varying, i.e. $\parallel W\left(t\right){\parallel }_2\le {\delta }_w$ with ${\delta }_w\in {\mathbb{R}}_{+}$ being unknown upper bound. In Figure 8, the evolution of the adaptive weights is illustrated in the presence of external disturbance and measurement noise. As seen in the figure, the adaptive parameters stay bounded around their actual values and are practically noise-free.

Figure 8.

Evolution of the adaptive weights for IR-CMRAC in the presence of measurement noise and disturbance (dashed line: Actual weights W(t), solid line: Estimated weights $\widehat{W}\left(t\right)$.)

5.3 Investigation of information recovery

In this example, we conduct numerical simulations to highlight the main contribution of the proposed method: information recovery. In order to make a fair comparison with CMRAC, the integral action in the adaptation is deactivated with ${\gamma }_2=0$. All the other simulation parameters are as in Section 5.1.1. Measurement noise and disturbance are not applied to the system dynamics. Eventually, the following update laws are compared for the plant dynamics in Eq. (61) with the uncertainty in Eq. (62):

$$\begin{array}{lll}\text{CMRAC}\mathrm{Law}& :& \dot{\widehat{W}}=-\mathrm{\Gamma }\varphi e^T\mathit{PB}+\mathrm{\Gamma }{\gamma }_1{\varphi }_f{\tilde{\mathrm{\Delta }}}_f^T\\ \mathrm{IR}‐\text{CMRAC}\mathrm{Law}\left({\gamma }_2=0\right)& :& \dot{\widehat{W}}=-\mathbb{X}\varphi e^T\mathit{PB}+\mathbb{X}\left({\gamma }_1{\varphi }_f{\tilde{\mathrm{\Delta }}}_f^T+{\gamma }_m{\dot{\varphi }}_f{\tilde{\mathrm{\Delta }}}_f^T\right)\end{array}$$E69

Simulation results for several values of ${\gamma }_m$ are shown in Figure 9. The lost information in CMRAC (due to low-pass filtering) is included in the adaptation with nonzero ${\gamma }_m$. Hence, the weight estimation error becomes smaller with increasing ${\gamma }_m$ up to 1. However, further increase in ${\gamma }_m$ results in overshoot and oscillatory response, causing degraded weight estimation performance. This can be considered analogous to the excessive derivative gain in a traditional PID controller, where the unnecessarily large derivative gains might cause instability. It is clear from the figure that judicious tuning of the gain ${\gamma }_m$ results in significantly improved adaptive weight estimation and hence, the desired state tracking performance.

Figure 9.

The effect of ${\gamma }_m$ on the weight estimation error for IR-CMRAC.

5.4 Adaptation with non-PE signals

The parameter convergence in both standard MRAC and CMRAC is dependent on the stringent requirement of persistent excitation of basis function $\varphi \left(x\right)$. Note that PE condition of $\varphi \left(x\right)$ depends on the persistent excitation of system states x [41]. Furthermore, states x are persistently exciting provided that the exogenous reference input r(t) meets PE condition [4]. In this section, we provide simulation results with non-PE reference signal to illustrate the efficacy of the proposed method under interval excitation (IE) condition, which is much weaker than PE as described in Remark 1.1.

Figure 10 compares the state-tracking performances of MRAC, CMRAC, and IR-CMRAC. In the simulated flight case, the aircraft performs a right bank maneuver and returns back to wings-level flight condition exponentially.

Figure 10.

State tracking performance with non-PE reference signal.

As seen clearly from Figure 10, the reference signal is not persistently exciting but satisfies the interval excitation condition. The parameter convergence is achieved without persistent excitation, as suggested by Corollary 1.1 and Theorem 1.3 (see Figure 11).

Figure 11.

Weight estimation error with non-PE reference signal.

6. Software-in-the-loop (SITL) simulations for lateral flight control

6.1 X-plane flight simulator and lateral dynamics

X-plane flight simulator operates according to the blade element theory, where the physics engine calculates velocity components for each blade element. These components stem from various sources like rotational motion, free-stream velocity, propeller inflow, and downwash and wake caused by aerodynamic surfaces and fuselage. Thus, the aircraft model realized by the X-plane flight simulator is highly nonlinear and realistic, which makes X-plane a Federal Aviation Administration (FAA) certified flight simulator [42].

Linear equations of motion for the lateral dynamics of F-4 Phantom II at steady wings-level trim conditions with true airspeed of $V_{\infty }=360$ knots and altitude of $h=24000$ ft. are obtained using ‘Athena Vortex Lattice’ [43] as:

$$\begin{array}{l}\left[\begin{array}{c}\dot{\beta }\\ \dot{p}\\ \dot{r}\\ \dot{\varphi }\end{array}\right]=\underset{\triangleq A_p}{\underbrace{\left[\begin{array}{cccc}-1.47& 0.0& -1.0& 0.075\\ -18.85& -7.52& 2.89& 0.0\\ 15.15& 0.56& -7.61& 0.0\\ 0.0& 1.0& 0.0& 0.0\end{array}\right]}}\underset{\triangleq x_p}{\underbrace{\left[\begin{array}{c}\beta \\ p\\ r\\ \varphi \end{array}\right]}}+\underset{\triangleq B_p}{\underbrace{\left[\begin{array}{cc}0.0& -0.15\\ 8.48& -0.96\\ -0.18& 3.94\\ 0.0& 0.0\end{array}\right]}}\underset{\triangleq u}{\underbrace{\left[\begin{array}{c}{\delta }_a\\ {\delta }_r\end{array}\right]}}\\ y_p=\left[\begin{array}{c}\beta \\ \varphi \end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]x_p=E_p{x}_p\end{array}$$E70

where $\beta$ is sideslip angle [rad], p and r are roll and yaw rates $\left[\mathit{rad}/s\right]$ (resp.), and $\varphi$ is bank angle [rad]. Aileron and rudder inputs are normalized with maximum deflection of 20 degrees to be ${\delta }_a\in \left[-1,1\right]$ and ${\delta }_r\in \left[-1,1\right]$, respectively.

6.2 Nominal controller design

In order to increase the robustness and improve the tracking performance, integrator states are introduced as follows:

$$\begin{array}{l}\dot{z}\left(t\right)=\left[\begin{array}{c}{\beta }_{\mathit{cmd}}\left(t\right)\\ {\varphi }_{\mathit{cmd}}\left(t\right)\end{array}\right]-\left[\begin{array}{c}\beta \left(t\right)\\ \varphi \left(t\right)\end{array}\right]\\ \begin{array}{c}\dot{z}\left(t\right)=r\left(t\right)-y_p\left(t\right)=r\left(t\right)-E_p{x}_p\left(t\right)\end{array}\end{array}$$E71

Then, augmented system dynamics can be expressed as follows:

$$\begin{array}{l}\left[\begin{array}{c}{\dot{x}}_p\\ \dot{z}\end{array}\right]=\underset{\triangleq A}{\underbrace{\left[\begin{array}{cc}{A}_p& 0\\ -E_p& 0\end{array}\right]}}\underset{\triangleq x}{\underbrace{\left[\begin{array}{c}{x}_p\\ z\end{array}\right]}}+\underset{\triangleq B}{\underbrace{\left[\begin{array}{c}{B}_p\\ 0\end{array}\right]}}\left[\begin{array}{c}{\delta }_a\\ {\delta }_r\end{array}\right]+\underset{\triangleq B_r}{\underbrace{\left[\begin{array}{c}0\\ I\end{array}\right]}}\left[\begin{array}{c}{\beta }_{\mathit{cmd}}\\ {\varphi }_{\mathit{cmd}}\end{array}\right]\\ \dot{x}=\mathit{Ax}+\mathit{Bu}+B_{r}r\end{array}$$E72

Reference model satisfying $A_r=A-\mathit{BK}$ is given as

$${\dot{x}}_r=A_r{x}_r+B_{r}r$$E73

The nominal controller $u_n\left(t\right)=-\mathit{Kx}$ is designed using feedback gain K:

$$K=\left[\begin{array}{cccccc}1.14& 1.15& -0.81& 4.14& -3.25& -3.02\\ -0.58& -0.69& 0.09& -1.32& 4.76& 0.97\end{array}\right]$$E74

The fastest mode is the roll subsidence with frequency of 7.66 rad/s. Hence, the cut-off frequency of the low-pass filter is chosen to be ${\omega }_f=8\mathrm{rad}/\mathrm{s}$.

6.3 SITL simulation results

In this section, we demonstrate SITL simulation results of the F-4 Phantom fighter jet’s lateral dynamics. The objective is to execute consecutive left and right bank maneuvers (see Figure 12) while ensuring minimal sideslip to uphold coordinated flight. Initially, adaptation remains inactive for 70 seconds, and then the proposed adaptive controller is activated. The communication between the controller and the X-plane flight simulator is established through the user datagram protocol (UDP) port. Hence, an unknown communication delay is present in the simulations.

Figure 12.

Left and right Bank Maneuvers as the flight scenario.

In Figure 13, state tracking performance is illustrated. As the proposed adaptation is activated, the sideslip angle gets smaller, resulting in better-coordinated bank maneuvers. Furthermore, the oscillations, especially in the roll channel, are significantly reduced with the adaptation.

Figure 13.

State tracking performance for SITL simulation with $\mathit{\Gamma }={\mathit{\Gamma }}_{\mathit{\Theta }}=2I$, ${\gamma }_m=10$, ${\tau }_w=30$, ${\gamma }_1=1$, and ${\gamma }_2=10$.

Figures 14 and 15 illustrate the commanded and actuated control surface commands. It is important to highlight that the improvements in state tracking performance are not because of the excessive use of the controls but due to the effective cancelation of the uncertainties. This is clear as the commanded control surface deflections significantly decrease with the activation of proposed adaptive controller.

Figure 14.

Aileron control for SITL simulation with $\mathit{\Gamma }={\mathit{\Gamma }}_{\mathit{\Theta }}=2I$, ${\gamma }_m=10$, ${\tau }_w=30$, ${\gamma }_1=1$, and ${\gamma }_2=10$.

Figure 15.

Rudder control for SITL simulation with $\mathit{\Gamma }={\mathit{\Gamma }}_{\mathit{\Theta }}=2I$, ${\gamma }_m=10$, ${\tau }_w=30$, ${\gamma }_1=1$, and ${\gamma }_2=10$.

Note that the uncertainty in SITL flight simulation is not guaranteed to be matched. That is, some portion of the uncertainty may not lie in the range space of input matrix B. Corresponding plant dynamics can be expressed as:

$$\dot{x}=\mathit{Ax}+B\left[u+\mathrm{\Delta }\left(x\right)\right]+D{\mathrm{\Delta }}_u\left(x\right)$$E75

with $B^{T}D=0$ and ${\mathrm{\Delta }}_u\left(x\right)$ being unmatched uncertainty. In this case, the reference model state tracking cannot be achieved since it is not possible to suppress all the effects of uncertainties with the control input. Instead, controlled output tracking can be achieved by introducing the command governor controls for the unmatched uncertainties [27].

7. Conclusions

In this chapter, a new model reference adaptive control architecture is proposed to recover the lost information in filter-based adaptive controllers such as composite/combined MRAC. In CMRAC-like formulations, the uncertainty estimation is achieved by filtering the system dynamics, which significantly improves the transient performance. However, filtering the high-frequency content in the system signals may degrade the adaptation due to lost information. This adverse effect can become even more crucial if the low-pass filter bandwidth is remarkably small. With the proposed method, the high-frequency content of the uncertainty estimation is integrated in a frequency-selective manner into the adaptive law to recover the information lost during filtering. Furthermore, learning rates in the weight update law become time-varying so that the proposed modification term behaves as a stability augmentation system. In that regard, the bandwidth of the closed-loop system is adjusted online. The efficacy of the proposed solution is illustrated through numerical simulations of aircraft roll dynamics and software-in-the-loop simulations for lateral dynamics of F-4 Phantom II fighter aircraft through the X-plane flight simulator. Furthermore, rigorous stability proof is provided by Lyapunov’s stability theorem.