1   Introduction

Unmanned underwater vehicles (UUVs) are being increasingly used in underwater operations, replacing or supplementing divers, driven by the demand from the offshore oil industry, heightened maritime security concerns, and the need for comprehensive ocean data collection and ocean floor mapping (Brun 2012). Over the years, continuous developments have led to some form of autonomous control of many UUVs. Such autonomous control is challenging, mainly due to model uncertainty, highly nonlinear and time-varying hydrodynamic effects, and non-deterministic external disturbances. During operations, UUVs are consistently subjected to various parameter changes that affect the vehicle motion such as changes in the weight due to different payloads (Cavalletti et al. 2011), changes in buoyancy due to variations in the pressure, temperature, and salinity (Wu et al. 2014), change in the control effectiveness due to partial loss of thrust (Pivano 2008), and changes in the hydrodynamic load near the free surface (Sayer 1996). The mitigation of the effects of such changes on the motion of the vehicle is a crucial factor in complex UUV applications that require precise maneuvers. These include semi-autonomous remotely operated vehicles (ROVs) used in applications such as tidal energy infrastructure servicing under high-flow conditions (Proctor et al. 2015), autonomous underwater vehicles (AUVs) used for assisting divers to carry out underwater task (Stilinović et al. 2015), and launching and recovering of torpedo-shaped AUVs from submarines for military purposes (Rodgers et al. 2008). To enable these applications, it is essential that UUVs have good tracking performance throughout their entire mission. Good tracking performance in UUVs is fundamentally dependent on the motion control system (MCS) which mainly consist of the control system and guidance system supported by the navigation system. (Fossen 2011). Therefore, the MCS used in UUVs should adapt to the changes and ensure good tracking in both steady state and, more importantly, transient time.

Guidance system generates feasible desired reference path/trajectories as input to the control system, thus providing information on where the craft should go and how it should get there (Fossen 2011). Line-of-sight (LOS) is a popular and effective guidance law used extensively for autonomous marine vehicles due to its simplicity and intuitiveness. Traditional LOS methods usually steer the vessel toward a point lying at a constant distance ahead of the vehicle (look-ahead distance) along the desired path (Fossen et al. 2003). Lekkas and Fossen (2012) proposed a revised version of LOS, using a time-varying look-ahead distance, while Borhaug et al. (2008) has proposed an addition of integral actions to LOS (ILOS) to overcome disturbances. An extension of ILOS based on adaptive sideslip observe has been proposed by Fossen and Lekkas (2017) that can compensate effectively for the drift forces. Another guidance law that has gained traction recently for guidance of marine vehicles is vector field–based approach. A path-following controller for an autonomous marine vehicle using a vector field guidance law was developed in Xu and Guedes Soares (2016). This was extended to a time-varying vector field guidance law with a proof of uniform semiglobal exponential stability (USGES) in Xu et al. (2019) and Xu et al. (2020). A comparison between the vector field and the ILOS guidance laws for a UUV is given in Caharija et al. (2015).

Control system enables the vehicle to achieve a certain control objective despite environmental forces by determining the required control forces and moments to be provided by the actuators (Fossen 2011). Numerous advanced solutions for control system of UUVs have been proposed, with more recent literature covering robust control techniques such as higher order sliding mode control (Guerrero et al. 2019), model-based optimization techniques such as model predictive control (Zhang et al. 2019), intelligent control techniques such as fuzzy control (Yu et al. 2017), neural networks (Elhaki and Shojaei 2020), and probabilistic inference learning (Ariza Ramirez et al. 2020). Adaptive control is another important set of control techniques that have a unique advantage in UUV applications due to their inherent ability to adapt to changes that affect the vehicle behavior (von Ellenrieder 2021). This is further highlighted by the fact that some form of adaptation has been built into all of the aforementioned control solutions.

Model Reference Adaptive Control (MRAC) is one subset of adaptive control in which the desired characteristics of the system are represented usually by a reference model (Ioannou and Fidan 2006). The basic MRAC architecture has been modified over the years to yield; reduced parameter drift and increased robustness to unmodeled dynamics (Narendra and Annaswamy 1987; Macnab 2019), improved transient performance (Duarte-Mermoud and Narendra 1989; Yang et al. 2020), and increased robustness to time delay (Dydek et al. 2010). In addition, MRAC has been proposed as an outer loop for improved disturbance rejection of a classical PID (Alagoz et al. 2020) and fractional order PID control systems (Tepljakov et al. 2018).

Even though adaptive control has been proposed as a promising solution for UUVs (Antonelli et al. 2001; Fossen and Fjellstad 1996; McFarland and Whitcomb 2014; Valladarez and Toit 2015; Yuh et al. 1999), there are certain drawbacks that prevent their widespread use in advanced applications. One of the major drawbacks is the trade-off between transient tracking performance and adaptation gains. High adaptation gains are known to achieve accurate transient tracking, which in turn leads to oscillations in the control signal (Stepanyan and Krishnakumar 2012), reduced robustness to noise and time delay, and instability (Crespo et al. 2010). On the other hand, low adaptation gain mitigates the above issues, but it leads to poor reference tracking in the transient region (Zang and Bitmead 1994) that can be dangerous in cluttered environments. Several solutions (Cao and Hovakimyan 2006; Stepanyan and Krishnakumar 2010; Yucelen and Haddad 2012; Yucelen and Johnson 2012a) to this conundrum have been proposed in the past decade including L1 adaptive control (Cao and Hovakimyan 2006) which has been applied to UUVs by Maalouf (2013) and Valladarez (2015) with encouraging results. This method uses a modified MRAC architecture that places a low-pass filter in a unique position that subverts the high-frequency signals and decouples adaptation from robustness (Cao and Hovakimyan 2006). This decoupling theoretically enables the use of high adaptation gains to increase transient tracking but concern has been expressed by several researches that high adaptation gains could lead to numerical instability (Campbell et al. 2010; Ioannou et al. 2014) and parameter freezing (Ortega and Panteley 2014). Some of the other solutions (Stepanyan and Krishnakumar 2010; Yucelen and Haddad 2012), although not widely applied, also use some form of filtering with high adaptation gains and could face the same questions as L1 adaptive control.

Therefore, the authors have focused on modifications to MRAC that uses low adaptive gains, which provide an emphasis on stability and smooth control signals while improving transient performance (Makavita 2018). One such method is Command Governor Adaptive Control (CGAC) (Yucelen and Johnson 2012a) which uses an additional linear dynamical system, driven by the system error, named command governor to modify the command signal. This in turn leads to improved transient performance at low adaptation gains and an inherent disturbance rejection capability (Yucelen and Johnson 2012b). The authors initially applied CGAC to a UUV in simulation to verify the tracking and disturbance rejection improvements in Makavita et al. (2015) and confirmed that CGAC disturbance rejection ability allowed it to overcome a significant actuator dead-zone without using an additional dead-zone inverse in Makavita et al. (2016a). The authors validated through experiments the tracking improvement, disturbance rejection, and dead-zone overcoming effect in Makavita et al. (2019a) for heading control of a UUV, i.e., the AMC ROV.

A possible drawback of CGAC is that the command governor has the tendency to amplify measurement noise (Yucelen and Johnson 2012b) into the control signal that adversely affect the robustness properties of CGAC. A solution to this was provided in Yucelen and Johnson (2012b) that uses a low-pass filter named the command filter to filter out noise from the command governor signal. The authors confirmed through simulations the efficacy of this solution for UUV operations in Makavita et al. (2016a) and showed that at high noise levels the command filter by itself was insufficient and some input filtering of sensor measurements was also required. In addition, it was shown that the command filter also increases robustness to time delay that can cause instability. A further experimental study (Makavita et al. 2019b) of depth control provided an opportunity to show the effect on CGAC of high input noise and time delay due to depth rate estimation and input filtering, respectively. It was shown that although a command filter can be designed to reduce the measurement noise in tandem with input filtering and concurrently overcome instability from time delay, it causes an initial period of very poor reference tracking and the control signal noise level remain much higher than standard MRAC.

Even though CGAC outperformed MRAC in all performance indicators after a sufficient time has passed, it was apparent that a solution was required for this initial period of poor performance as well as further reducing noise levels without sacrificing tracking performance or robustness if CGAC was to achieve its full potential for UUV operations. To this end, this paper presents a possible solution by removing the command filter and combining CGAC with a weight filter and state predictor based on Yucelen and Haddad (2013) and Lavretsky et al. (2010) respectively to improve the overall tracking performance while retaining an improved robustness to noise and time delay without incurring an initial period of poor tracking. The final control system with these modifications is termed modified CGAC (M-CGAC), which is tested using experiments for depth control and compared with previous results derived in Makavita et al. (2019b) for depth control using the CGAC.

2   Adaptive Control Architecture

This section gives a brief introduction to MRAC, CGAC, and modified CGAC.

2.1   Model Reference Adaptive Control

As described in Yucelen and Johnson (2013), consider the nonlinear uncertain dynamical system given by,

where x(t) ∈ ℝ^p is the state vector, u(t) ∈ ℝ^q is the control input, δ :  ℝ^p → ℝ^q is an uncertainty, A ∈ ℝ^p × p is a known system matrix, B ∈ ℝ^p × q is an unknown control input matrix, H ∈ ℝ^p × q is a known uncertainty input matrix, and the pair (A,  B) is controllable. It is also assumed that δ(x) is parameterized as δ(x) = W^Tσ(x), where W ∈ ℝ^s × q is an unknown weight matrix, and σ : ℝ^q → ℝ^s is a known basis function of the form σ(x) = [σ₁(x),  σ₂(x),  ….,  σ_s(x)]^T. It is further assumed that B is parameterized as B = HΛ, where det(H^TH) ≠ 0, and Λ ∈ ℝ^q × q is an unknown control effectiveness matrix.

The ideal reference model that specifies a desired closed-loop dynamical system performance is given by

where x_m(t) ∈ ℝ^p is the reference state vector, c(t) ∈ ℝ^q is the given uniformly continuous bounded command, A_m ∈ ℝ^p × p is the Hurwitz reference system matrix, and B_m ∈ ℝ^p × q is the command input matrix.

The objective of MRAC is to design a feedback control law u(t) such that x(t) asymptotically follows x_m(t), i.e., $ \underset{t\to \infty }{\lim}\left\Vert {\boldsymbol{e}}_m\right\Vert =0 $, where e_m ≙ x − x_m is the system error. Let u(t) be given by

where u_n(t) ∈ ℝ^q is the nominal feedback control law and u_a(t) ∈ ℝ^q is the adaptive feedback control law. The nominal control law is given by

where K₁ ∈ ℝ^q × p is the nominal feedback gain and K₂ ∈ ℝ^q × q is the nominal feedforward gain, such that the following matching condition holds.

Applying the control law defined in Eq. (3) into Eq. (1) and simplifying yields

where W_un ≜ 1 − Λ⁻¹ ∈ ℝ^q × q and W_σ ≜ WΛ⁻¹ ∈ ℝ^s × q. Furthermore, the adaptive feedback law is selected as

where $ {\hat{\boldsymbol{W}}}_{un}(t)\in {\mathbb{R}}^{q\times q} $ and $ {\hat{\boldsymbol{W}}}_{\sigma }(t)\in {\mathbb{R}}^{s\times q} $ are estimates of W_un and W_σ, satisfying the update laws given by

where Γ_un ∈ ℝ^q × q and Γ_σ ∈ ℝ^s × s are learning rates and P = P^T  >  0 is the solution of the Lyapunov equation $ \bf{0}={\boldsymbol{A}}_m^{\mathrm{T}}\boldsymbol{P}+\boldsymbol{P}{\boldsymbol{A}}_m+\boldsymbol{Q} $ for some Q = Q^T  >  0. Now using Eq. (7) in Eq. (6) yields

where $ {\overset{\sim }{\boldsymbol{W}}}_{un}(t)\triangleq {\hat{\boldsymbol{W}}}_{un}(t)-{\boldsymbol{W}}_{un}\in {\mathbb{R}}^{q\times q} $ and $ {\overset{\sim }{\boldsymbol{W}}}_{\sigma }(t)\triangleq {\hat{\boldsymbol{W}}}_{\sigma }(t)-{\boldsymbol{W}}_{\sigma}\in {\mathbb{R}}^{s\times q} $. The system error dynamics is derived by subtracting Eq. (2) from Eq. (10) to give

It is shown by the Lyapunov analysis in Yucelen and Johnson (2013) that for the update laws Eq. (8) and Eq. (9), the system is asymptotically stable, i.e., $ \underset{t\to \infty }{\lim}\left\Vert {\boldsymbol{e}}_m\right\Vert =0 $.

2.2   Command Governor Adaptive Control

Let the command signal in Eqs. (2) and (4) be given by

where c_d(t) ∈ ℝ^q is now the given uniformly continuous bounded command and Gg_f(t) ∈ ℝ^q × q is the command governor signal with G ∈ ℝ^q × p being, the matrix defined by

and g_f(t) ∈ ℝ^p × q is the low-pass-filtered command governor output, g(t) ∈ ℝ^p × q generated by

where f(t) is the command governor state vector, λ = ℝ₊ is the command governor gain, and κ = ℝ₊ is the command governor filter gain that should be selected sufficiently small to ensure efficient low-pass filtering.

Due to the command governor output in Eq. (12), Eqs. (2) and (10) are respectively modified as

where P_H = H(H^TH)⁻¹H^T. However, this does not change the system error dynamics given by Eq. (11) as seen by subtracting Eq. (17) from Eq. (18). Therefore, the update laws in Eqs. (8) and (9) also remain the same.

It has been shown in Yucelen and Johnson (2013) using Lyapunov analysis that the system with the command governor is also asymptotically stable, i.e., $ \underset{t\to \infty }{\lim}\left\Vert {\boldsymbol{e}}_m\right\Vert =0, $ and that $ \underset{t\to \infty }{\lim }{\boldsymbol{g}}_f(t)=0 $. From this it can be shown that the modified reference model given in Eq. (17) asymptotically converge to the ideal reference model given by

where x_l(t) ∈ ℝ^p is the ideal reference vector. Therefore, the uncertain dynamical system in Eq. (1) approaches the ideal reference model in Eq. (19) in steady state.

In addition, from Proposition 7.1 in Yucelen and Johnson (2013), if λ is sufficiently large, the system approximates the ideal reference model in Eq. (19) modified by a term P_H(g_f(t) − g(t)), in transient time without using high learning rates. Although it is shown that $ \underset{t\to \infty }{\lim }{\boldsymbol{P}}_H\left({\boldsymbol{g}}_f(t)-\boldsymbol{g}(t)\right)=0 $, it is expected that there will be deviations from the ideal reference model initially until the modification term has died down. The deviation in transient time depends on the magnitude of the term g_f(t) − g(t). A small command governor filter rate κ as required for increased robustness under high noise and time delay conditions also increase the magnitude of the term thus adversely affecting the initial tracking performance of CGAC.

2.3   Modified Command Governor Adaptive Control

The CGAC given in Section 2.3.1 is modified in three stages to derive M-CGAC. These three stages and expected advantage of each modification are given below.

2.3.1   Removal of Command Filter

If the command filter given by Eq. (16) is removed, then Eqs. (14) and (15) still holds and Eqs. (17) and (18) are modified to

However, this does not change the system error dynamics given by Eq. (11) as seen by subtracting Eq. (20) from Eq. (21). Therefore, the update laws in Eqs. (8) and (9) also remain the same. In addition, the term P_H(g_f(t) − g(t)) will no longer modify ideal reference model in Eq. (19) thus from Proposition 6.1 in Yucelen and Johnson (2013), if λ is sufficiently large, the uncertainties $ \boldsymbol{H}\boldsymbol{\varLambda } \left[{\overset{\sim }{\boldsymbol{W}}}_{un}^{\mathrm{T}}{\boldsymbol{u}}_n(t)+{\overset{\sim }{\boldsymbol{W}}}_{\sigma}^{\mathrm{T}}\boldsymbol{\sigma} \left(\boldsymbol{x}\right)\right] $ in Eq. (21) are rapidly suppressed in transient time through P_Hg(t), and the system approximates the ideal reference model of Eq. (19) in transient time without using high learning rates.

Expected advantage: Improve initial transient performance compared to CGAC due to removal of modification term P_H(g_f(t) − g(t)).

2.3.2   Addition of Weight Filter Modification

As proposed by Yucelen and Haddad (2013) taking $ \hat{\boldsymbol{W}}(t)\in {\mathbb{R}}^{a\times b} $ as a general weight estimate that can represent both $ {\hat{\boldsymbol{W}}}_{un}(t) $ and $ {\hat{\boldsymbol{W}}}_{\sigma }(t) $, a low-pass-filtered weight estimate $ {\hat{\boldsymbol{W}}}_f(t)\in {\mathbb{R}}^{a\times b} $ of $ \hat{\boldsymbol{W}}(t) $ is given by

where Γ_f ∈ ℝ^a × a is a positive definite filter gain matrix chosen such that λ_max(Γ_f) ≤ γ_{f, max}, and where γ_{f, max} ≥ 0 is a design parameter that needs to be small enough to cut off high frequencies from $ \hat{\boldsymbol{W}}(t) $.

For clarity, both Eqs. (8) and (9) are represented by a general update law given by

where Γ = Γ_un or Γ_σ and β(t) = σ(x(t)) or u_n(t). A modification term was added to the update law Eq. (23) to enforce a distance condition between the trajectories of $ \hat{\boldsymbol{W}}(t) $ and $ {\hat{\boldsymbol{W}}}_f(t). $ This leads to a minimization problem of the cost function J given by

with a negative gradient with respect to $ \hat{\boldsymbol{W}}(t) $ given by

which is also the structure of the proposed modification term. This leads to the modified update law of Eq. (26) given by

which yield the following modified update laws for Eqs. (8) and (9) respectively,

where α  >  0 is a modification gain, and $ {\hat{\boldsymbol{W}}}_{unf}(t) $ and $ {\hat{\boldsymbol{W}}}_{\sigma f}(t) $ are the low-pass-filtered weight estimate of $ {\hat{\boldsymbol{W}}}_{un}(t) $ and $ {\hat{\boldsymbol{W}}}_{\sigma }(t) $, respectively.

Expected advantage: Significant reduction of measurement noise due to high-frequency filtering effect of weight filter and additional input filtering made possible by the increased robustness to time delay. This increase robustness to time delay stems from improved phase margin of the closed-loop system due to the weight filter (Yucelen and Haddad 2013).

Furthermore, CGAC with modifications given in Sections 2.3.1 and 2.3.2 is named robust CGAC (R-CGAC) in this paper due to its increased robustness compared to original CGAC.

2.3.3   Addition of Closed-Loop State Predictor

A closed-loop state predictor is introduced to add a prediction error in tandem with the system error for weight estimation. As given in Lavretsky et al. (2010),

where A_prd ∈ ℝ^p × p is a Hurwitz matrix, and $ \hat{\boldsymbol{x}}(t)\in {\mathbb{R}}^p $ is the predictor states vector. When added to CGAC, it will modify Eq. (29) as

The prediction error is defined as $ \hat{\boldsymbol{e}}(t)\stackrel{\wedge}{=}\hat{\boldsymbol{x}}(t)-\boldsymbol{x}(t) $. The predictor error dynamics can be derived by subtracting Eq. (21) from Eq. (30) as

Now, if the update laws are given as shown in Eqs. (32) and (33),

where $ {\boldsymbol{P}}_{prd}={\boldsymbol{P}}_{prd}^{\mathrm{T}} \gt 0 $ is the solution of the Lyapunov equation $ {\bf{0}}={\boldsymbol{A}}_{prd}^{\mathrm{T}}{\boldsymbol{P}}_{prd}+{\boldsymbol{P}}_{prd}{\boldsymbol{A}}_{prd}+{\boldsymbol{Q}}_{prd} $ for some $ {\boldsymbol{Q}}_{prd}={\boldsymbol{Q}}_{prd}^{\mathrm{T}} \gt 0 $, then, it can be shown by a Lyapunov analysis that

1) The system error is uniformly ultimately bounded, square integrable, and globally asymptotically stable, i.e., $ \underset{t\to \infty }{\lim}\left\Vert {\boldsymbol{e}}_m\right\Vert =0 $;

2) The prediction error is uniformly ultimately bounded, square integrable, and globally asymptotically stable, i.e., $ \underset{t\to \infty }{\lim}\left\Vert \hat{\boldsymbol{e}}\right\Vert =0 $.

The proof is given in the Appendix.

Expected advantage: Improved overall reference tracking due to composite adaptation using both tracking and prediction errors based on the CMRAC conjecture (Lavretsky et al. 2010) and experimental results (Makavita et al. 2018).

The resulting adaptive control architecture with both weight filter and state predictor is shown in Figure 1 and is referred in this paper as modified CGAC (M-CGAC).

Figure  1  Visualization of the proposed M-CGAC architecture

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3   Control Plant Model

For the purpose of marine control system design, a simplified model of the complex 6-DOF kinematics and dynamics must be developed. This Control Plant Model (CPM) is also used as a basis for analytical stability analysis and should capture only the essential features of the system. This section describes the CPM used in this study.

3.1   Reference Frames

It is convenient to use two reference frames to model the dynamics of a UUV as shown in Figure 2.

Figure  2  The three thrusters AMC ROV showing the Earth-fixed {E} and body-fixed {B} reference frames

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3.2   Thruster Model

In the actual vehicle, the vertical movement is achieved by a vertical Seabotix© thruster (Le et al. 2013; SeaBotix 2015). The thrust force of the Seabotix© thruster depends on the input voltage, which is approximately represented by a simple thruster model including a dead-zone as given in Makavita et al. (2016a).

If it is assumed that the thruster dead-zone is overcome by a dead-zone inverse, then the thrust F produced by the thruster, which is also the vertical control effort (τ_w), is approximately related to the thruster input voltageV_i linearly by the multiplication factor K_vf given as

The input voltage to a thruster is determined by the appropriate input to the motor controller. The latter input is specified using discrete unitless values between 0 and 255, where zero voltage is specified by 128. Therefore, there is a conversion from the value specified to the motor controller to the input voltage applied to the thruster. This conversion consists of a multiplication factor and a constant bias term. For simplicity, it is assumed that the motor control input denoted by V_m is continuous, although in practice it can easily be converted to discrete values. The multiplication factor is denoted by K_iv while the bias value is 128. Thus,

Inserting Eq. (35) into Eq. (34) yields

By denoting the normalized force as

Equation (36) is written as

The multiplicative factors can be assumed unknown in adaptive controller design and estimated as part of the control effectiveness. Therefore, the control algorithms will generate the normalized force directly, which will then be converted back to continuous motor inputs using Eq. (39) and then rounded to get the discrete motor inputs.

3.3   Control Plant Model for Depth

Several CPMs with varying complexity have been proposed in the literature (Antonelli et al. 2001; Fossen 1994; Healey and Lienard 1993; Smallwood and Whitcomb 2004; Yoerger and Slotine 1991) and their advantages and disadvantages are concisely reported by Refsnes (2007). Of those discussed, a simple and popular CPM is the 1-DOF CPM given in Smallwood and Whitcomb (2004). This CPM has the following general dynamic equation for each DOF_i,

where m_i  >  0 is the effective mass, d_Li  >  0 and d_Qi  >  0 are the linear and quadratic hydrodynamic drag coefficients, υ_i(t) is the velocity, b_i is the buoyancy, and τ_i is the net control force.

It is derived based on the following assumptions that are not theoretically justified (Smallwood and Whitcomb 2004) but are accepted to apply quite well for low-speed underwater vehicles (Fossen 1994; Caccia et al. 2000).

Assumption 1 O_b is chosen to coincide with the center of gravity, CG (i.e., x_g = 0, y_g = 0, z_g = 0) with the body axes, X_b, Y_b, and Z_b, coinciding with the principal axes of inertia (Fossen 1994).

Assumption 2 Off-diagonal components of the added mass matrix can be neglected (Fossen 1994) and the diagonal terms are constant (Caccia et al. 2000).

Assumption 3 Damping terms higher than 2nd order and off-diagonal terms can be neglected (Fossen 1994).

Assumption 4 Linear and nonlinear stabilizing damping dominate the influence of the Coriolis forces and moments (Refsnes 2007).

In addition, due to the structural design, the following assumptions hold for the AMC ROV.

Assumption 5 CG and the center of buoyancy (CB) are offset only in the Z_b direction denoted by z_b.

Assumption 6 Uncontrolled DOFs of pitch angle (θ) and roll angle (ϕ) are assumed to be negligible.

This CPM had been successfully validated by several researchers for different UUVs. In Smallwood and Whitcomb (2004), the CPM for surge, sway, heave, and yaw is validated using experimental data from the Johns Hopkins University Remotely Operated Underwater Vehicle (JHUROV). They derived the CPM parameters using system identification and compared the simulated results with experimental results and showed that the CPM predicted the actual behavior with a mean absolute error of 0.016m/s for surge, sway, and heave and 0.0288rad/s for yaw. Complete details of the identification methods and procedure used to find the plant parameters are given in Smallwood and Whitcomb (2003). In Caccia et al. (2000), the models for surge, sway, and yaw were validated using the UUV designed and developed by the Institute for Ship Automation of the Italian National Research Council named ROMEO, while in Ridao et al. (2001), the surge, heave, and yaw models were validated, for the UUV designed at the University of Girona named GARBI.

The CPM for depth (heave) in this study was developed as follows. From Eq. (40), considering the depth DOF the following equation is obtained,

where m_w, Z_w, and Z_w|w| are the effective mass, linear drag, and quadratic drag in depth DOF, respectively, w is the velocity in the Z_b direction, F_W is the weight of the vehicle, and F_B is the buoyancy force. Rearranging Eq. (41) for $ \dot{w} $ yields,

Replacing τ_w with the normalized moment using Eq. (37) yields $ \dot w = \left( {\frac{{{Z_w}}}{{{m_w}}}} \right)r + \left( {\frac{{{Z_{w\left| w \right|}}}}{{{m_w}}}} \right) + \frac{{{F_W} - {F_B}}}{{{m_w}}} + \left( {\frac{{{K_{vf}}{K_{iv}}}}{{{m_w}}}} \right){{\tilde \tau _w}} $

or

where $ {\theta}_1=\left(\frac{Z_w}{m_w}\right) $, $ {\theta}_2=\left( {\frac{{{Z_{w\left| w \right|}}}}{{{m_w}}}} \right) $, $ {\theta}_3=\frac{F_W-{F}_B}{m_w} $, and $ {\theta}_4=\left(\frac{K_{vf}{K}_{iv}}{m_w}\right) $.

The general kinematic equation for depth of an underwater vehicle is given by

where d is the position in the Z_e direction. Simplifying Eq. (44) using Assumptions 5 and 6 give

From Eqs. (43) and (45), the state space form of the depth CPM is given as

Equation (46) has the general state space form of Eq. (1) with p = 2 and q = 1 where $ \boldsymbol{x}=\left(\begin{array}{l}d\\ {}w\end{array}\right),\kern0.75em \boldsymbol{A}=\left(\begin{array}{ll}0& 1\\ {}0& 0\end{array}\right),\kern0.75em \boldsymbol{H}=\left(\begin{array}{l}0\\ {}1\end{array}\right) $, $ =\boldsymbol{H}\Lambda =\left(\begin{array}{l}0\\ {}{\theta}_4\end{array}\right), $ Λ = θ₄,   δ(x) = θ₁x₂ + θ₂x₂x₂ + θ₃, and $ u={\overset{\sim }{\tau}}_w $.

3.4   Reference Model

A reference model that generates a feasible reference signal must be developed for MRAC applications. There are numerous reference model design methods proposed in the literature (Fossen 2011; Fernandes et al. 2012; Fjellstad et al. 1992) out of which the filter-based method in Fossen (2011) is the simplest. As the CPM is a 2nd order system, the filter is selected to be in the same order. The standard 2nd order filter with the desired natural frequency (ω_n) and damping ratio (ζ) is written as

Equation (47) has the general state space form of (2) with p = 2 and q = 1 where $ {\boldsymbol{x}}_m=\left(\begin{array}{l}{d}_m\\ {}{w}_m\end{array}\right) $, $ {\boldsymbol{B}}_m=\left(\begin{array}{l}0\\ {}{\omega}_n^2\end{array}\right) $, and $ {\boldsymbol{A}}_m=\left(\begin{array}{ll}0& 1\\ {}-{\omega}_n^2& -2\zeta {\omega}_n\end{array}\right) $.

Applying the matching condition in (5) with $ \boldsymbol{A}=\left(\begin{array}{ll}0& 1\\ {}0& 0\end{array}\right) $, $ \boldsymbol{H}=\left(\begin{array}{l}0\\ {}1\end{array}\right) $, K₁ = [k₁₁  k₁₂], and K₂ = k₂ yields

with the solutions $ {k}_{11}=-{\omega}_n^2,{k}_{12}=-2\zeta {\omega}_n $, and $ {k}_2={\omega}_n^2 $. Thus, $ {K}_1=\left[-{\omega}_n^2\kern0.5em -2\zeta {\omega}_n\right] $ and $ {K}_2={\omega}_n^2 $.

Since Eqs. (46) and (47) are second-order representations of Eqs. (1) and (2) the derivations in Section 2 can be directly applied for this CPM.

4   Experimental Setup and Test Cases

The experimental program was carried out using the ROV shown in Figure 2 which was designed and built at the Australian Maritime College (AMC). Its mass is approximately 20 kg and has dimensions of 83 cm × 45 cm × 27 cm in length, width, and height, respectively (Nguyen et al. 2011). The vehicle consists of two horizontal thrusters for horizontal motion and a vertical thruster for vertical motion. The thrusters are Seabotix© BTD-150, which can deliver a maximum thrust of 22 N (SeaBotix 2015). Each thruster operates with an input voltage up to 19V and has a measured thruster dead-zone of approximately 20% of the input. The thrusters are controlled by two MD22 motor controllers.

The depth measurement is obtained using a depth sensor that uses a Measurement Specialties MS5837-30BA pressure sensor (TE Connectivity 2015) which is pre-installed in a waterproof aluminum body (BlueRobotics 2016). The sensor can measure up to 300m in depth, with a resolution of 2mm (BlueRobotics 2016). The precision of the sensor is mainly affected by noise and drift and it has been shown by Gilooly (2018) that the noise rms is only 2.5 mm. Although the drift can be up to 1 cm per hour (Gilooly 2018), as the experiments were conducted for around 150 s and initial depth was reset to zero at beginning of each run, this led to a negligible effect on precision due to drift. An ATmega2560 microcontroller is used to interface sensors and motor controller boards to the control program running in the host computer. The motors and electronics are powered by three 18.5V-15Ah Li-Po batteries. A 40 m tether is used to connect the UUV to the host computer. Signals to and from the microcontroller were communicated through the tether cable using the RS485 communication protocol at a baud rate of 38400 bps. Although a tether was used, operations were autonomous as there was no direct human control of the vehicle.

The control algorithm was implemented as a continuous algorithm using the MATLAB/Simulink^TM platform on the host computer. In addition, an input filter, dead-zone inverse and thrust allocation block was also implemented. The AMC ROV sensor outputs were relayed through the microcontroller, to the computer and captured by the stream input block. The inputs to the motor controller on the AMC ROV were received from the microcontroller through the stream output block from the computer. To enable real-time operation, the Simulink model was run using Simulink Desktop Real-Time^TM (SDRT) in the external mode. The solver used was the ode5 (Dormand-Prince, RK5) with a fixed step size of 0.01 s. A schematic view summarizing the system's hardware architecture is given in Figure 3.

Figure  3  Depth control hardware architecture

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According to Eq. (38), the normalized control force has a range of −128 to 128, with 0 representing the zero thrust. The unit of measurement is considered being dimensionless. The tests were carried out over several weeks in the Survival Centre pool at AMC, which has a length, width, and maximum depth of 25m, 12m, and 4.2m, respectively.

4.1   Parameter Values

The adaptive control parameters were set as follows. For simplicity, all learning rates were taken as dependent on a single positive constant γ such that Γ_σ = γI₃ and Γ_un = γ. Unless otherwise specified, all controllers used γ = 1. The command governor gain λ was set to 100 as done in both Makavita et al. (2015) and Makavita et al. (2019b). The command filter gain κ was set to 3 as done in Makavita et al. (2019b). For simplicity, weight filter gains were taken as dependent on a single positive constant γ_f such that Γ_unf = γ_f and Γ_σf = γ_fI₃. γ_fwas set to 1 as a compromise between low-pass filtering (smallγ_f) and avoiding performance degradation (largeγ_f) (Fravolini et al. 2014). The modification gain α was set to 10 as a compromise between enforcing learning through low frequencies (high α) and maintaining the benefits of adaptation (low α) (Yucelen and Haddad 2013). For the state predictor from Lavretsky et al. (2010), it is proposed that A_prd = μA_m and P_prd = μP where μ is a positive scalar. The value for μ is set to 10 as done in both Makavita et al. (2016b) and Makavita et al. (2018).

All the initial values of the CPM parameters were set to zero ($ {\hat{W}}_{un}=0 $ and $ {\hat{\boldsymbol{W}}}_{\sigma }=\left[0\kern0.5em 0\kern0.5em 0\right] $), thus assuming no a priori knowledge. While this is an extreme assumption considering that some values are known, albeit approximately (e.g., mass), it provides a good basis to test the ability of the controller under severe uncertainty. The reference model parameters were set to ω_n = 0.3 rad/s and ζ = 1, which yields $ {\boldsymbol{K}}_1=\left[-0.09\kern0.5em -0.6\right] $ and K₂ = 0.09. The reference model parameters were selected for a critically damped response with an approximate rise time of t_r = 10 s, settling time of t_s = 20 s by testing the feasibility of the reference signal.

4.2   Experimental Scenario

The experiments were conducted for CGAC, R-CGAC, and M-CGAC under three different phases. The first phase was the comparison between CGAC and R-CGAC for a normal depth change command. The second phase was the comparison of M-CGAC with both R-CGAC and CGAC for a normal depth change command. The final phase was the evaluation of M-CGAC with CGAC for performance under a sudden parameter change represented by the change in control effectiveness due to thrust loss. More details on the experimental scenarios are given below:

4.2.1   CGAC vs R-CGAC

CGAC and R-CGAC were applied to a depth change maneuver of 150 s duration, and their performances were compared, with the main objective of comparing tracking performance in the initial 50 s. In addition, tracking performance during the next 100 s (after the initial 50 s), and control signal noise levels and frequency content was also analyzed.

4.2.2   M-CGAC

The vehicle was tested for depth change for M-CGAC and compared with R-CGAC and CGAC. The main objective was to counteract the negative effect of weight filtering on tracking and to further improve tracking over CGAC. Furthermore, the learning rate was increased slightly with the objective of improving the tracking performance to meet the design specification of having rms depth tracking error of 0.02 m or lower for the entire run. In addition, control signal noise levels and frequency content were also analyzed.

4.2.3   Sudden Parameter Variation

This was represented by a 50% loss of thrust in the vertical thruster during operation. This type of partial failure can occur due to an electrical or mechanical malfunction. This situation was created by halving the voltage to the motor controllers. The partial failure was activated at 85 s after the start at a depth of 1 m. The objective was to ascertain the ability of M-CGAC to overcome such a failure and maintain the depth.

5   Experimental Results

For the purpose of measuring system performance, the system states were compared with the ideal reference states. Thus, the tracking error is defined as e_l ≙ x − x_l. Performance of control methods was measured using six performance indices shown in Table 1. The first four indices were based on the tracking errors in depth (e_d) and depth rate (e_w), where $ {\boldsymbol{e}}_l^{\mathrm{T}}=\left[{e}_d\kern0.5em {e}_w\right] $, while the last two are based on the control effort. These performance indices were designed based on the work of Fossen and Fjellstad (1996).

In addition, other performance indices such as settling time were used as required. The vertical thruster force, when given numerically or graphically, is the value before the dead-zone inverse value is added.

5.1   CGAC vs R-CGAC

The experiments were conducted as mentioned in Section 4.2. Initially CGAC (with the command filter) was compared with R-CGAC (with the weight filter). The results are given in Table 2, Figure 4, and Figure 5. In addition, the evolution of the estimated weight parameters for both CGAC and R-CGAC is given in Figure 6. As evident in Figure 4, under CGAC the vehicle depth and depth rate have a significant deviation in the initial 50 s. It then settles to a reasonably acceptable tracking performance in the next 100s after the modification term due to the filter has died down. In contrast for R-CGAC, tracking in the initial 50 s has significantly improved (Figure 5). A more quantitative analysis can be carried out using the performance metrics presented in Table 2, in which the CGAC and R-CGAC performance indices are given in three parts: full run, first 50 s, and last 100 s. Thus, this analysis will look at the first 50 s and next 100 s separately and compare the performances to capture the clear distinction in performance between first 50 s and next 100 s.

Figure  4  CGAC

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Figure  5  R-CGAC

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Figure  6  CPM parameter evolution

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In the first 50 s, the first four indices that represent tracking errors, d_{e_rms}, d_{e_max}, w_{e_rms}, and w_{e_max} of R-CGAC are lower than those of CGAC by 79% (a factor of 4.8), 72% (a factor of 3.5), 64% (a factor of 2.8), and 60% (a factor of 2.5), respectively. The last two indices that represent control effort, $ {\overset{\sim }{\tau}}_{w\_\mathrm{rms}} $ and $ {\overset{\sim }{\tau}}_{w\_\max } $, of R-CGAC are lower than those of CGAC by 79% (a factor of 4.8) and 66% (a factor of 3), respectively. Thus, there is a clear improvement in all performance metrics for R-CGAC over CGAC.

In the next 100 s although w_{e_rms} and w_{e_max} of R-CGAC are still lower than those of CGAC by 23% (a factor of 1.3), and 27% (a factor of 1.4), respectively, d_{e_rms} and d_{e_max} of R-CGAC are higher than those of CGAC by 55% (a factor of 1.5) and 62% (a factor of 1.6), respectively. Furthermore, $ {\overset{\sim }{\tau}}_{w\_\mathrm{rms}} $ and $ {\overset{\sim }{\tau}}_{w\_\max } $ of R-CGAC remains lower than CGAC by 27% (a factor of 1.4) and 44% (a factor of 1.8), respectively. Thus, although R-CGAC improves on depth rate tracking and control effort over CGAC, it underperforms in the crucial depth tracking metric.

For further analysis of the control signal, the discrete rate of change of the control signal $ \left(\frac{\varDelta u}{\varDelta t}\right) $ versus time is provided in Figure 7 and the frequency spectrum is provided in Figure 8. It is clear from these figures that there is a significant reduction in noise levels and high frequencies in R-CGAC compared to CGAC. This is due to (a) the inherent filtering effect of weight filter and (b) the decrease of the cut-off frequency of the input filter from 12 to 6 rad/s without instability made possible by the increased robustness to time delay of the weight filter.

Figure  7  Discrete derivative ($ \frac{\varDelta u}{\varDelta t} $) of CGAC and R-CGAC

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Figure  8  Frequency spectrum of CGAC and R-CGAC

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Although, R-CGAC has several advantages over CGAC, the reduced performance of R-CGAC in depth tracking after the first 50 s should be remedied as it affects the long-term tracking performance. An additional concern is that the performance in the first 50 s, although much improved, is still much lower than that of the next 100 s.

5.2   M-CGAC

As explained in Section 2.3.3, R-CGAC was combined with closed-loop state predictor to produce modified CGAC (M-CGAC). The performance of M-CGAC at γ = 1 (denoted by M-CGAC₁) is given in Figure 9 and Table 3. Furthermore, the estimated weight parameter variation for M-CGAC₁ is given in Figure 10a.

Figure  9  M-CGAC₁

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Figure  10  a, b CPM parameter evolution

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An immediate improvement is seen with the addition of the prediction error at a learning rate γ = 1. In the first 50 s, the first four indices that represent tracking errors, d_{e_rms}, d_{e_max}, w_{e_rms}, and w_{e_max} of M-CGAC₁ are lower than those of R-CGAC by 49% (a factor of 2), 41% (a factor of 1.7), 26% (a factor of 1.4), and 24% (a factor of 1.3), respectively. The last two indices that represent control effort, $ {\overset{\sim }{\tau}}_{w\_\mathrm{rms}}\ and\ {\overset{\sim }{\tau}}_{w\_\max } $, of M-CGAC₁ are increased in comparison with those of R-CGAC by 4% (a factor of ~1) and 11% (a factor of 1.1), respectively. Thus, M-CGAC₁ improves its tracking performance with only a slight increase in control effort.

In the next 100 s, d_{e_rms}, d_{e_max}, w_{e_rms}, and w_{e_max} of M-CGAC₁ are lower than those of R-CGAC by 46% (a factor of 1.9), 42% (a factor of 1.7), 15% (a factor of 1.2), and 4% (a factor of ~1), respectively. In addition, they are also lower than those of CGAC by 16% (a factor of 1.2), 6% (a factor of 1.1), 35% (a factor of 1.5), and 30% (a factor of 1.4), respectively. Furthermore, $ {\overset{\sim }{\tau}}_{w\_\mathrm{rms}} $ and $ {\overset{\sim }{\tau}}_{w\_\max } $ of M-CGAC₁ while increased from R-CGAC by 8% (a factor of 1.1) and 33% (a factor of 1.5), respectively, are lower than CGAC by 21% (a factor of 1.3) and 26% (a factor of 1.4), respectively.

Thus, M-CGAC has better tracking than R-CGAC and remedies the reduced depth tracking performance of R-CGAC for a marginal increase in control effort. Furthermore, this analysis clearly shows that M-CGAC outperforms CGAC in every single performance index. In addition, if we compare the performances of each individual method in the first 50 s with the next 100 s, M-CGAC has the more homogeneous response compared to CGAC.

While the performance of M-CGAC at γ = 1 was quite satisfactory, it was decided to see if any additional improvements can be made by increasing the learning rate to improve depth tracking such that d_{e_rms} ≤ 0.02 m for both the first 50 s and the next 100 s of the run. This specification was achieved by a small increase in learning rate to γ = 3, denoted by M-CGAC₃. The performance indices for this condition are also given in Table 3 and its parameter evolution is given in Figure 10b.

Comparing these with M-CGAC₁ in the first 50s, d_{e_rms}, d_{e_max}, w_{e_rms}, and w_{e_max} of M-CGAC₃ are lower than those M-CGAC₁ by 29% (a factor of 1.4), 28% (a factor of 1.4), 30% (a factor of 1.4), and 33% (a factor of 1.5), respectively. In addition, the control effort indices of $ {\overset{\sim }{\tau}}_{w\_\mathrm{rms}} $ and $ {\overset{\sim }{\tau}}_{w\_\max } $ of M-CGAC₃ are lower than those of M-CGAC₁ by 32% and 63%, respectively.

In the next 100s, the tracking performance indices of M-CGAC₃ are approximately equal to the performance indices of M-CGAC₁ while the control effort indices have reduced slightly.

It is important to ensure that these performance improvements are not at the expense of noise or high frequencies in the control signal. To verify this, the discrete derivative and the frequency spectrum of the control signal for R-CGAC, M-CGAC₁, and M-CGAC₃ are given in Figures 11 and 12. From both figures, it is clearly seen that the noise levels and frequency distributions are approximately the same. A closer analysis shows, though there is a slight increase in high frequencies and noise for M-CGAC₁ over R-CGAC, this decreases at M-CGAC₃.

Figure  11  Discrete derivative of the control signal for R-CGAC, M-CGAC₁, and M-CGAC₃

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Figure  12  Frequency spectrum of the control signal for R-CGAC, M-CGAC₁, and M-CGAC₃

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5.3   Sudden Parameter Variation

A further experiment was carried out to verify the capability of M-CGAC₃ by comparing it with CGAC for a thrust loss anomaly. A thrust loss manifests itself as a sudden variation of the control effectiveness parameter and is a good candidate to check the ability of the controller to perform under such a variation. The results for 50% thrust loss while holding constant depth are given in Table 4, while the results for changing depth after the thrust loss is given in Table 5 and Figure 13. The parameter evolution for the duration of the run is given in Figure 14.

Figure  13  Depth response of CGAC and M-CGAC₃ for a 50% thrust loss at 85 s

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Figure  14  CPM parameter evolution of CGAC and M-CGAC₃ for 50% thrust loss

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From Table 4, it is seen that both methods have similar performances in tracking before and after thrust loss. M-CGAC₃ has an advantage in terms of maximum deviation which is 40% (a factor of 1.7) less than CGAC and only 0.001 m outside the 2% settling time band of ±0.02 m. As this difference is within the resolution of the depth sensor, it is negligible and thus settling time is not applicable for M-CGAC₃. In addition, the advantage of having a reduced control effort is carried through even after thrust loss, although the control magnitudes have increased to accommodate the reduced thrust.

From Table 5, it is seen that when a depth change is done after thrust loss at 120s, M-CGAC₃ has better performance in all performance indices other than the maximum thrust which is equal for the two. Further insight can be had by observing the plots in Figure 13. It is seen that M-CGAC₃ has increased oscillations in comparison to CGAC just after thrust loss. In addition, CGAC in contrast to M-CGAC₃ undershoots the command in first down step with a peak undershoot of 8.6% and in the second step it is prevented from undershooting only by the physical constraint of hitting the water surface.

Thus, after partial thrust loss both CGAC and M-CGAC₃ perform well, but M-CGAC₃ does have an advantage in both maintaining and changing depth.

5.4   Summary of Results

Overall, the results indicate that the proposed method of M-CGAC, which is an extension of CGAC by replacing the command filter with the weight filter and adding the closed-loop state predictor, has the following advantages over CGAC:

a) Resolves the initial poor tracking problem

b) Overall better tracking performance

c) Less energy consumption due to lower control effort

d) Less noise and high frequencies in the control signal

e) Improved handling of sudden parameter changes

6   Conclusion

This paper proposes an extension to the command governor adaptive control to enhance the initial tracking performance of UUVs intended to use in advanced applications that require precise maneuvring. The proposed M-CGAC replaces the command filter with a weight filter and adds a closed-loop state predictor. Experimental results and analysis indicate that the weight filter alone produces better tracking performance at the start with substantial improvement in reducing control effort, control signal noise, and high frequencies. However, its overall depth tracking performance is reduced compared to that of CGAC. The subsequent addition of the closed-loop state predictor has resolved this issue and improved the overall tracking performance while retaining lower control effort, lower control signal noise, and lower high frequencies. A further increase of the learning rate from 1 to 3 enabled the achievement of a specific design specification for depth tracking. In addition, M-CGAC outperformed CGAC under a 50% partial thruster failure in the vertical thruster.

Thus, M-CGAC has an overall improvement over CGAC and has highly promising performance metrics without using high learning rates. Therefore, it is concluded that M-CGAC is a viable candidate for underwater missions that require precise maneuvres.

Appendix: Lyapunov stability proof of M-CGAC

In order to derive the stable adaptive laws, consider the following Lyapunov function candidate

where $ {\overset{\sim }{\boldsymbol{W}}}_{un}(t)\triangleq {\hat{\boldsymbol{W}}}_{un}(t)-{\boldsymbol{W}}_{un}\in {\mathbb{R}}^{q\times q} $ and $ {\overset{\sim }{\boldsymbol{W}}}_{\sigma }(t)\triangleq {\hat{\boldsymbol{W}}}_{\sigma }(t)-{\boldsymbol{W}}_{\sigma}\in {\mathbb{R}}^{s\times q} $ are weight estimation errors, $ {\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}(t)\triangleq {\hat{\boldsymbol{W}}}_{u{n}_f}(t)-{\boldsymbol{W}}_{un}\in {\mathbb{R}}^{q\times q} $and $ {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}(t)\triangleq {\hat{\boldsymbol{W}}}_{\sigma }(t)-{\boldsymbol{W}}_{\sigma}\in {\mathbb{R}}^{s\times q} $are low-pass-filtered weights estimation errors.

Note that V(0,   0,   0,   0,  0,   0) = 0 and $ V\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right) \gt 0 $ for all $ \left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)\ne \left(0,0,0,0,0,0\right) $. In addition, $ V\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right) $ is radially unbounded.

Now, differentiating Eq. (49) yields

Substituting from error dynamics of Eqs. (11) and (31) in Eq. (50) yields

Simplyfing Eq. (51) yields

Substituting in Eq. (52) from the Lyapunov equations $ \bf{0}={\boldsymbol{A}}_m^{\mathrm{T}}\boldsymbol{P}+\boldsymbol{P}{\boldsymbol{A}}_m+\boldsymbol{Q} $ for some Q = Q^T  >  0 and$ \bf{0}={\boldsymbol{A}}_{prd}^{\mathrm{T}}{\boldsymbol{P}}_{prd}+{\boldsymbol{P}}_{prd}{\boldsymbol{A}}_{prd}+{\boldsymbol{Q}}_{prd} $ for some $ {\boldsymbol{Q}}_{prd}={\boldsymbol{Q}}_{prd}^{\mathrm{T}} \gt 0 $ yields

Defining $ \overline{\boldsymbol{e}}=\left({\hat{\boldsymbol{e}}}^{\mathrm{T}}{\boldsymbol{P}}_{prd}-{\boldsymbol{e}}_m^{\mathrm{T}}\boldsymbol{P}\right)\boldsymbol{H} $ Eq. (53) simplify to

Using the trace identity a^Tb = trace(ba^T) in Eq. (54) gives

Substituting from the filtered weights Eq. (22) and proposed M-CGAC weight update laws given by (32) and (33) in Eq. (55) yields

Further simplifying Eq. (56)

By definition

$ {\displaystyle \begin{array}{l}\mathrm{trace}\left[\left({\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)}^{\mathrm{T}}\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)\right)\Lambda \right]=\kern0.5em \sum \limits_{i=1}^N\sum \limits_{j=1}^M{\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)}_{ij}^2{\Lambda}_{ii}\ge {\left\Vert {\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right\Vert}_F^2{\Lambda}_{\mathrm{min}}\\ {}\kern0.5em \end{array}} $ $ \mathrm{where}\ {\left\Vert {\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right\Vert}_F^2=\sum \limits_{i=1}^N\sum \limits_{j=1}^M{\left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right)}_{ij}^2 $ is the Forbenius norm of $ \left({\overset{\sim }{\boldsymbol{W}}}_{un}-{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f}\right) $ and Λ_min is the minimum diagonal element of Λ. A similar definition is applicable to $ \mathrm{trace}\left[\left({\left({{\overset{\sim }{\boldsymbol{W}}}_{\sigma}}^{\mathrm{T}}-{{\overset{\sim }{\boldsymbol{W}}}^T}_{\sigma_f}\right)}^{\mathrm{T}}\left({\overset{\sim }{\boldsymbol{W}}}_{\sigma }-{\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)\right)\Lambda \right] $.

Thus, Eq. (57) reduces to

Therefore, as Λ_min is positive and the Forbenius norm is positive

Hence, this proves that the closed-loop system is Lyapunov stable and that $ {\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em $ and $ {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f} $ are uniformly ultimately bounded. Since c(t)is bounded and A_mis Hurwitz, then x_m(t) and $ {\dot{\boldsymbol{x}}}_m(t) $ are bounded. Hence, the system state x(t) is bounded. This implies $ \hat{\boldsymbol{x}}(t) $, u_n(t), and σ(x) are bounded. Since the weight estimation errors $ {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma } $ are bounded and the ideal weights W_un, W_σ are constant, the weight estimations $ {\hat{\boldsymbol{W}}}_{un},{\hat{\boldsymbol{W}}}_{\sigma } $ are also bounded. Thus, it follows from Eqs. (11) and (31) that $ {\dot{\boldsymbol{e}}}_m,\kern0.5em \dot{\hat{\boldsymbol{e}}} $ are also bounded. Therefore, $ \ddot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right) $ is bounded. Now, it follows from Barbalat's lemma (Ioannou and Fidan 2006) that $ \underset{t\to \infty }{\lim}\dot{V}\left({\boldsymbol{e}}_m,\kern0.5em \hat{\boldsymbol{e}},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{un},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma },{\overset{\sim }{\boldsymbol{W}}}_{u{n}_f},\kern0.5em {\overset{\sim }{\boldsymbol{W}}}_{\sigma_f}\right)=0 $, which consequently shows that e_m(t) and $ \hat{\boldsymbol{e}}(t) $ asymptotically converge to zero as t → ∞. Moreover, since the Lyapunov function V is radially unbounded, this convergence is global.

Thus, the system error and prediction error are globally, uniformly asymptotically stable, i.e., $ \underset{t\to \infty }{\lim}\left\Vert {\boldsymbol{e}}_m\right\Vert =0 $ and $ \underset{t\to \infty }{\lim}\left\Vert \hat{\boldsymbol{e}}\right\Vert =0 $. This completes the proof.