2.1. AUV kinematics

AUVs are capable of moving in 6 DOF. According to SNAME notation [Reference Fossen41], these motions are labelled as follows: linear motion along the X-axis is referred to as surge ( $x$ ), while the corresponding rotational motion is termed roll ( $\varphi$ ). For the Y-axis, linear motion is identified as sway ( $y$ ), with the rotational motion being called pitch ( $\theta$ ). Finally, along the Z-axis, linear motion is designated as heave ( $z$ ), while the associated rotational motion is known as yaw ( $\psi$ ).

The vector $\eta =[\eta _{1}^{T}, \eta _{2}^{T}]^{{\textrm T}}$ comprises linear displacements $\eta _{1}=[x, y, z]^{{\textrm T}}$ and angular displacements $\eta _{2}= [\varphi , \theta , \psi ]^{{\textrm T}}$ , all in relation to the Earth-fixed frame.

In the same manner, the vector $\nu =[\nu _{1}^{T}, \nu _{2}^{T}]^{{\textrm T}}$ represents both linear and angular velocities in relation to the body-fixed frame, where $\nu _{1}=[u, v, w]^{{\textrm T}}$ and $\nu _{2}= [p, q, r]^{{\textrm T}}$ .

Therefore, the kinematics of the AUV can be derived from these definitions as:

(1) \begin{equation} \dot {\eta } =J(\eta )\nu \end{equation}

where $J(\eta )$ is the transformation matrix that converts the velocity from the body-fixed frame to the Earth-fixed frame, expressed as follows:

(2) \begin{equation} J(\eta )=\begin{bmatrix} J_{1}(\eta _{2}) &\quad 0_{3\times 3}\\ 0_{3\times 3} &\quad J_{2}(\eta _{2}) \end{bmatrix} \end{equation}

(3) \begin{equation} J_{1}(\eta _{2})=\small {\begin{bmatrix} c\psi c\theta &\quad -s\psi c\varphi +c \psi s\theta s\varphi &\quad s\psi s\varphi +c\psi c\varphi s\theta \\ s\psi c\theta &\quad c\psi c\varphi +s\varphi s\theta s\psi &\quad -c\psi s\varphi +s\theta s\psi c\varphi \\ -s\theta &\quad c\theta s\varphi &\quad c\theta c\varphi \end{bmatrix} } \end{equation}

(4) \begin{equation} J_{2}(\eta _{2}) = \small {\begin{bmatrix} 1&\quad s\varphi t\theta &\quad c\varphi t\theta \\ 0&\quad c\varphi &\quad -s\varphi \\ 0&\quad s\varphi /c\theta &\quad c\varphi /c\theta \end{bmatrix} } \end{equation}

$s$ , $c$ , and $t$ represent the sine, cosine, and tangent of angles, respectively, and will be used as such in all subsequent references throughout the paper.

2.2. AUV dynamics

The hydrodynamic model of an AUV takes into account various factors, including added mass, viscous damping, and gravitational and restoring forces, as well as the forces and moments generated by the thrusters that control the vehicle’s movement. The interplay of these forces and the resulting motion of the AUV can be represented by the following nonlinear equation:

(5) \begin{equation} {\mathcal{M}}\ddot {\eta } + {\mathcal{C}}(\eta , \dot {\eta })\dot {\eta } + {\mathcal{D}}(\eta , \dot {\eta })\dot {\eta } + g(\eta ) = {\mathcal{T}}+{\mathcal{T}}_{E} \end{equation}

The mass matrix is denoted as ${\mathcal{M}} \in {\mathcal{R}}^{6\times 6}$ , while the Coriolis and centripetal matrix is represented by ${\mathcal{C}}(\eta , \dot {\eta }) \in {\mathcal{R}}^{6 \times 6}$ . The damping matrix is indicated as ${\mathcal{D}}(\eta , \dot {\eta }) \in {\mathcal{R}}^{6 \times 6}$ , and the gravitational matrix is given by $ g(\eta ) \in {\mathcal{R}}^{6}$ . The control input forces ${\mathcal{T}} \in {\mathcal{R}}^{6}$ encompasses the thruster forces and torques, while ${\mathcal{T}}_{E} \in {\mathcal{R}}^{6}$ accounts for disturbances arising from environmental influences.

2.2.1 Properties

P1: ${\mathcal{M}}$ is symmetric and strictly positive definite, satisfying the condition:

(6) \begin{equation} \delta _{m}I_{n} \leq {\mathcal{M}} \leq \delta _{M}I_{n} \end{equation}

$\delta _{m}$ , $\delta _{M}$ are positive constants, and $n$ is the degree of freedom.

P2: Since ${\mathcal{M}}\gt 0$ , therefore, ${\mathcal{M}}^{-1} \neq 0$ . The system parameters are defined as follows [Reference Ma, Tang, Wang, Yan, Liao and Jiang42]:

(7) \begin{equation} {\mathcal{M}}=\begin{bmatrix} m_{11} &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0\\ 0 &\quad m_{22} &\quad 0 &\quad 0 &\quad 0 &\quad 0\\ 0 &\quad 0 &\quad m_{33} &\quad 0 &\quad 0 &\quad 0\\ 0 &\quad 0 &\quad 0 &\quad m_{44} &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad m_{55} &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad m_{66} \end{bmatrix} \end{equation}

(8) \begin{equation} {\mathcal{C}}=\begin{bmatrix} 0_{3\times 3}&\quad {\mathcal{C}}_{12} &\quad \\[2pt] {\mathcal{C}}_{21}&\quad {\mathcal{C}}_{22} \end{bmatrix} \end{equation}

(9) \begin{equation} {\mathcal{C}}_{12}={\mathcal{C}}_{21}=\begin{bmatrix} 0 &\quad m_{33}\dot {z} &\quad -m_{22}\dot {y} \\[2pt] -m_{33}\dot {z} &\quad 0&\quad m_{11}\dot {x} \\[2pt] m_{22}\dot {y} &\quad -m_{11}\dot {x}&\quad 0 \end{bmatrix} \end{equation}

(10) \begin{equation} {\mathcal{C}}_{22}=\begin{bmatrix} 0 &\quad m_{66}\dot {\psi } &\quad -m_{55}\dot {\theta } \\[2pt] -m_{66}\dot {\psi } &\quad 0 &\quad m_{44}\dot {\varphi }\\[2pt] m_{55}\dot {\theta } &\quad -m_{44}\dot {\varphi } &\quad 0 \end{bmatrix} \end{equation}

(11) \begin{equation} {\mathcal{D}}=\begin{bmatrix} d_{11}|\dot {x}| &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0\\[2pt] 0 &\quad d_{22}|\dot {y}| &\quad 0 &\quad 0 &\quad 0 &\quad 0\\[2pt] 0 &\quad 0 &\quad d_{33}|\dot {z}| &\quad 0 &\quad 0 &\quad 0\\[2pt] 0 &\quad 0 &\quad 0 &\quad d_{44}|\dot {\varphi }| &\quad 0 &\quad 0\\[2pt] 0 &\quad 0 &\quad 0 &\quad 0 &\quad d_{55}|\dot {\theta }| &\quad 0 \\[2pt] 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad d_{66}|\dot {\psi }| \end{bmatrix} \end{equation}

(12) \begin{equation} g=\begin{bmatrix} (W-B)s \theta \\[2pt] -(W-B)c \theta s \varphi \\[2pt] -(W-B)c \theta c \varphi \\[2pt] -(y_{G}W-y_{B}B)c \theta c \varphi + (z_{G}W-z_{B}B)c \theta s \varphi \\[2pt] (z_{G}W-z_{B}B)s \theta + (x_{G}W-x_{B}B)c \theta c \varphi \\[2pt] -(x_{G}W-x_{B}B)c \theta s \varphi - (y_{G}W-y_{B}B) s \theta \end{bmatrix}^{{\textrm T}} \end{equation}

where $(x_{G},y_{G},z_{G})$ is the centre of gravity and $(x_{B},y_{B},z_{B})$ is the centre of buoyancy and

(13) \begin{equation} {\mathcal{T}}=\begin{bmatrix} {\mathcal{T}}_{x}&\quad {\mathcal{T}}_{y}&\quad {\mathcal{T}}_{z} &\quad {\mathcal{T}}_{\varphi } &\quad {\mathcal{T}}_{\theta } &\quad {\mathcal{T}}_{\psi } \end{bmatrix}^{{\textrm T}} \end{equation}

2.3. Disturbances affecting an AUV

Disturbances can be present in any system, potentially resulting in adverse effects on its operation and overall stability. The existence of these disturbances and uncertainties leads to poor performance of the controller designed to control the system. These disturbances may be internal, such as sensor noise, or external in nature like environmental disturbance. The two main types of environmental disturbances that pose a threat to the control of AUVs are [Reference Fossen43]:

• • Waves

• • Ocean currents

In the dynamic equations of AUV (5), the environmental disturbance ${\mathcal{T}}_{E}$ can be expressed as

(14) \begin{equation} {\mathcal{T}}_{E} = {\mathcal{T}}_{W} + {\mathcal{T}}_{O} \end{equation}

where ${\mathcal{T}}_{W}$ is a periodic signal caused by waves and ${\mathcal{T}}_{O}$ is a constant signal caused by ocean currents.

Using Eq. (14) in Eq. (5), we have

(15) \begin{equation} {\mathcal{M}}\ddot {\eta } + {\mathcal{C}}(\eta , \dot {\eta })\dot {\eta } + {\mathcal{D}}(\eta , \dot {\eta })\dot {\eta } + g(\eta ) = {\mathcal{T}}+{\mathcal{T}}_{W} + {\mathcal{T}}_{O} \end{equation}

2.4. AUV system with unknown parameters

Let the system described in Eq. (15) have uncertain parameters and be expressed in linear parametric form. Thus, the system dynamics becomes

(16) \begin{equation} Y(\eta , \dot {\eta }, \ddot {\eta })\Phi ={\mathcal{T}}+{\mathcal{T}}_{W} \end{equation}

where $ Y(\eta , \dot {\eta }, \ddot {\eta }) \in {\mathcal{R}}^{6 \times p}$ is a known regressor matrix and $\Phi =[\phi _{1},\ldots , \phi _{p}]^{{\textrm T}}$ is a vector of $p$ unknown parameters.

The regressor matrix $ Y(\eta , \dot {\eta }, \ddot {\eta })$ is an appended matrix given by:

(17) \begin{equation} Y(\eta , \dot {\eta }, \ddot {\eta })\Phi =\begin{bmatrix} Y_{MC}&\quad Y_{D}&\quad Y_{g} &\quad Y_{O} \end{bmatrix}\begin{bmatrix} \phi _{MC}\\ \phi _{D}\\ \phi _{g}\\ \phi _{O} \end{bmatrix} \end{equation}

where

(18) \begin{equation} Y_{MC}= \begin{bmatrix} \ddot {x} &\quad -\dot {y}\dot {\psi } &\quad \dot {z}\dot {\theta }&\quad 0 &\quad 0 &\quad 0\\[2pt] \dot {x}\dot {\psi }&\quad \ddot {y} &\quad -\dot {z}\dot {\varphi }&\quad 0 &\quad 0 &\quad 0\\[2pt] -\dot {x}\dot {\theta } &\quad \dot {y}\dot {\varphi }&\quad \ddot {z} &\quad 0 &\quad 0 &\quad 0\\[2pt] 0 &\quad -\dot {y}\dot {z}&\quad \dot {z}\dot {y}&\quad \ddot {\varphi } &\quad -\dot {\theta }\dot {\psi }&\quad -\dot {\psi }\dot {\theta }\\[2pt] \dot {x}\dot {z}&\quad 0 &\quad -\dot {z}\dot {x}&\quad \dot {\varphi }\dot {\psi }&\quad \ddot {\theta } &\quad -\dot {\psi }\dot {\varphi }\\[2pt] -\dot {x}\dot {y}&\quad \dot {y}\dot {x}&\quad 0 &\quad -\dot {\varphi }\dot {\theta }&\quad \dot {\theta }\dot {\varphi }&\quad \ddot {\psi } \end{bmatrix} \end{equation}

(19) \begin{equation} Y_{D}= \begin{bmatrix} |\dot {x}|\dot {x} &\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\[2pt] 0&\quad |\dot {y}|\dot {y} &\quad 0&\quad 0&\quad 0&\quad 0\\[2pt] 0&\quad 0&\quad |\dot {z}|\dot {z} &\quad 0&\quad 0&\quad 0\\[2pt] 0&\quad 0&\quad 0&\quad |\dot {\varphi }|\dot {\varphi } &\quad 0&\quad 0\\[2pt] 0&\quad 0&\quad 0&\quad 0&\quad |\dot {\theta }|\dot {\theta } &\quad 0\\[2pt] 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad |\dot {\psi }|\dot {\psi } \end{bmatrix} \end{equation}

(20) \begin{equation} Y_{g}= \begin{bmatrix} -s \theta &\quad 0&\quad 0&\quad 0\\[2pt] c \theta s \varphi &\quad 0&\quad 0&\quad 0\\[2pt] c \theta c \varphi &\quad 0&\quad 0&\quad 0\\[2pt] 0&\quad 0&\quad -c \theta c \varphi &\quad c \theta s \varphi \\[2pt] 0&\quad c \theta c \varphi &\quad 0&\quad s \theta \\[2pt] 0&\quad -c \theta s \varphi &\quad - s \theta &\quad 0 \end{bmatrix} \end{equation}

and

(21) \begin{equation} Y_{O}=\begin{bmatrix} J_{1}^{T}(\eta _{2}) &\quad 0_{3\times 3}\\ 0_{3\times 3} &\quad J_{1}^{T}(\eta _{2}) \end{bmatrix} \end{equation}

The unknown parameter matrix $\phi _{MC}$ consists of the mass parameters given by

(22) \begin{equation} \phi _{MC}=\begin{bmatrix} m_{11} &\quad m_{22}&\quad m_{33}&\quad m_{44}&\quad m_{55}&\quad m_{66} \end{bmatrix}^{{\textrm T}} \end{equation}

and $\phi _{D}$ consists of the damping parameters.

(23) \begin{equation} \phi _{D}=\begin{bmatrix} d_{11}&\quad d_{22}&\quad d_{33}&\quad d_{44} &\quad d_{55}&\quad d_{66} \end{bmatrix}^{{\textrm T}} \end{equation}

$\phi _{g}$ consists of the gravitational matrix parameters

(24) \begin{equation} \phi _{g}=\begin{bmatrix} -(W-B) \\ x_{G}W-x_{B}B\\ y_{G}W-y_{B}B\\ z_{G}W-z_{B}B \end{bmatrix} \end{equation}

And lastly, $\phi _{O}$ is a constant matrix of ocean current terms.

(25) \begin{equation} \phi _{O}=\begin{bmatrix} {\mathcal{T}}_{o1} &\quad {\mathcal{T}}_{o2}&\quad {\mathcal{T}}_{o3}&\quad {\mathcal{T}}_{o4}&\quad {\mathcal{T}}_{o5}&\quad {\mathcal{T}}_{o6} \end{bmatrix}^{{\textrm T}} \end{equation}

3.1 Assumption on measured data

A1: The system’s input-output data, which include the control input torque ( ${\mathcal{T}}$ ) as well as the position, velocity, and acceleration vectors ( $\Upsilon =[\eta ,\dot {\eta },\ddot {\eta }]$ ), are not only theoretical but also directly measurable using standard sensors and instrumentation available in many practical systems.

Note: The control input torque ( ${\mathcal{T}}$ ) can be measured through torque sensors or current sensors in electric motors, which directly relate to the torque being applied by the control system. Position, velocity, and acceleration vectors ( $\Upsilon =[\eta ,\dot {\eta },\ddot {\eta }]$ ) can be obtained through position sensors (e.g., encoders, GPS), velocity sensors, and accelerometers, which are widely used in robotic systems like AUVs, automotive applications, and aerospace systems.

3.2 Problem statement:

The AUV system described by Eq. (5) is simulated to generate output data $\Upsilon _{T}=\Upsilon (t)_{t=0}^{\infty }$ from the input torque ${\mathcal{T}}_{T}={\mathcal{T}}(t)_{t=0}^{\infty }$ . This data set $D_{T}=\{{\mathcal{T}}_{T},\Upsilon _{T}\}$ is truncated over the time interval $t \in [0, \tau ]$ to form the resulting set $D_{\tau }=\{{\mathcal{T}},\Upsilon \}$ such that $D_{\tau } \in \mathfrak{L}_{2e}$ . This data set is employed to lay out an adaptive controller assembly.

Given that the system (5) contains uncertain parameters, it is reformulated in a linear parametric form as shown in Eq. (16). The objective is to derive an approximation for the unknown parameter vector $\Phi$ in Eq. (16). Using this parameter estimate, design a controller bank ${\mathcal{C}}_{B}=[{\mathcal{C}}_{1}, {\mathcal{C}}_{2}, \ldots , {\mathcal{C}}_{n}]$ consisting of a fixed number of candidate controllers, which can switch based on performance requirements. This controller assembly will guide the AUV along a desired trajectory while accounting for waves and ocean currents. Additionally, to minimise the tracking errors and control torque, update the candidates in the controller bank ${\mathcal{C}}_{B}$ using a real-time optimisation technique, to achieve a globally optimal solution.

4.1. Unfalsified control

UC is a data-driven supervisory approach that combines the strengths of both robust and adaptive control techniques. AUVs, as nonlinear systems, are significantly influenced by parametric uncertainties and environmental disturbances. Therefore, employing an adaptive control approach that effectively manages disturbances and nonlinearities is advantageous and is independent of specific models. In this iterative procedure, an algorithm is devised using the system’s measured data to find a controller, from a set of a finite number of candidate controllers, that can adapt to the changing environment and fulfil the desired performance criteria. The selected controller that satisfies these criteria is termed “unfalsified.” Conversely, if the available data indicate that a controller would fail to meet performance specifications in a feedback loop, it is considered falsified. The UC structure requires two important constituents – a fictitious reference generator and a cost function involving the fictitious reference. These are defined as follows.

4.1.1. Fictitious reference

A fictitious reference is a virtual signal that would generate specific input-output values for a controller ( ${\mathcal{C}}$ ), assuming it were part of a closed-loop system. Given the input-output data $({\mathcal{T}},\Upsilon )$ , the fictitious reference $\hat {\eta }_{des}$ can be determined using the equation:

\begin{equation*}\hat {\eta }_{des}= {\mathcal{C}}^{-1}{\mathcal{T}}+\Upsilon. \end{equation*}

4.1.2. Cost function

In UC, the cost function typically depends on the system’s input, output, and fictitious reference, and governs the system’s performance and stability. The cost detectability property ensures the stability of the unfalsified controller [Reference Safonov44].

Let,

(26) \begin{equation} \mathfrak{J}^{*}({\mathcal{T}},\eta ,\hat {\eta }_{des},t)= -\iota _{1}+\int _{0}^{t}\left\{\frac {W_{1}\|(\hat {\eta }_{des}(\tau )-\eta (\tau ))\|^{2}+ W_{2}\|{\mathcal{T}}(\tau )\|^{2}}{\iota _{2} + W_{3}\|\hat {\eta }_{des} (\tau )\|^{2}}\right\}\,{\textrm d}\tau \end{equation}

where $\hat {\eta }_{des}(\tau )$ , $\eta (\tau )$ , and ${\mathcal{T}}(\tau )$ are, respectively, the fictitious reference, output, and input of the system truncated over a time period $\tau$ . $\iota _{1}$ is the initial value, $\iota _{2}$ is a small positive number, and $W_{1}$ , $W_{2}$ , and $W_{3}$ are weighting coefficients.

The supremum of $\mathfrak{J}^{*}$ over $t \leq \tau$ is given by

(27) \begin{equation} \sup _{t \leq \tau }\mathfrak{J}^{*}({\mathcal{T}},\eta ,\hat {\eta }_{des},t) \end{equation}

The cost function $\mathfrak{J}$ is the maximum value of $\mathfrak{J}^{*}$ up to a certain time $\tau$ . Therefore,

(28) \begin{equation} \mathfrak{J}({\mathcal{T}},\eta ,\hat {\eta }_{des},\tau ) = \sup _{t \leq \tau }\mathfrak{J}^{*}({\mathcal{T}},\eta ,\hat {\eta }_{des},t) \end{equation}

Hence, $\mathfrak{J}({\mathcal{T}},\eta ,\hat {\eta }_{des},\tau )$ represents the maximum value of the cost function $\mathfrak{J}^{*}$ evaluated upto any time $t$ within the interval $[0, \tau ]$ . For mathematical brevity, $\mathfrak{J}({\mathcal{T}},\eta ,\hat {\eta }_{des},\tau )$ shall be represented as $\mathfrak{J}$ in subsequent discussions.

Figure 2 illustrates the block diagram of the controller design. The closed loop structure includes the AUV system, whose parameters are unknown, and a controller bank ${\mathcal{C}}_{B}= [{\mathcal{C}}_{1},{\mathcal{C}}_{2},\ldots ,{\mathcal{C}}_{n}]$ , consisting of $n$ nonlinear controllers, each designed with parameter values randomly selected from a finite, bounded set. Both the controller input and the system output are available for measurement. The UC method iteratively uses this measured data to identify the most appropriate controller from the bank. An algorithm is employed to switch out controllers that fail to meet the desired criteria and introduce one that does into the closed loop. Fictitious reference signals are generated to mathematically evaluate the candidate controllers without requiring them to be in the closed loop. Ultimately, only the selected controller is integrated into the closed-loop system.

Figure 2.

Adaptive unfalsified control with GA applied to AUV system.

4.2. Genetic algorithm-based adaptive unfalsified control

The literature on UC theory states that an unfalsified controller can be selected from a predetermined set of candidates or updated online using optimisation techniques. To enhance the capabilities of the UC method, this chapter introduces the concept of the GA into the technique of unfalsification to vary the candidate controller set. GA is a well-known evolutionary optimisation technique grounded in Darwinian principles of “Survival of the Fittest.” It is employed to address a broad range of optimisation challenges across various engineering fields [Reference Wright45]. The GA method provides the system with access to a diversified set of candidate controllers and selects the unfalsified controller from an optimised set of values. This approach also considers the optimisation of energy resources in AUV applications. The uncertain parameters of the AUV system are estimated using the method of unfalsification, from which a controller is selected to manage the AUV. This controller enables the AUV to follow a desired trajectory despite disturbances such as waves and ocean currents. Furthermore, GA is applied to update the candidates in the controller bank, generating versatile and optimised options for the controller bank.

4.3. Genetic algorithm implementation

The steps to implement the GA are presented in a flow chart given in Figure A1 of Appendix A.

To initiate the GA, an initial starting population consisting of $n$ potential solutions to the problem is generated using random numbers. A real number representing each solution is referred to as a chromosome. The fitness of the chromosomes is assessed, and a pair of chromosomes is selected as parents by the Roulette wheel selection. These parent chromosomes undergo crossover at randomly chosen positions, where their gene values are exchanged, resulting in the creation of two new chromosomes, referred to as offspring. This process continues until $n$ offspring are produced. Subsequently, a mutation process introduces and maintains diversity within the population by randomly altering values within any chromosome. The operations of crossover, mutation, and fitness evaluation are iteratively performed, with offspring replacing their parent chromosomes after each cycle to form a new population. Due to the random nature of the initial population, each simulation run yields slightly different results, which benefits the proposed design where the controller bank candidates are updated using the GA procedure.

4.4. Adaptive unfalsified controller design with GA

The adaptive unfalsified controller designed for an unknown AUV system is combined with GA to update the controller bank in real time. The nonlinear controllers in the bank adapt to changing performance specifications by switching to the most appropriate candidate within the bank. Incorporating the GA enhances the versatility of the unfalsified controller, thereby improving its efficiency by optimising costs.

The controller design consists of four stages as shown in the flow chart presented in Figure 3.

Figure 3.

Flow chart of the adaptive unfalsified control with GA scheme.

Stage-1

The first stage involves designing the nonlinear computed torque controller (CTC), presuming the system parameters are known. To achieve this, the error, defined as the difference between the desired and actual positions, is calculated as

(29) \begin{equation} e_{\eta } = \eta _{des} - \eta \end{equation}

The CTC generates the control torque, given by

(30) \begin{equation} {\mathcal{T}}= {\mathcal{M}}[\ddot {e}_{\eta }+\Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }] + {\mathcal{M}}\ddot {\eta } + {\mathcal{C}}(\eta ,\dot {\eta })\dot {\eta } + {\mathcal{D}}(\eta ,\dot {\eta })\dot {\eta } + g(\eta ) \end{equation}

$\Lambda _{1}$ and $\Lambda _{2}$ are positive gains that determine the rate at which the tracking error converges to zero. Substituting the value of ${\mathcal{T}}$ in Eq. (5), we get

(31) \begin{eqnarray} {\mathcal{M}}\ddot {\eta } &+& {\mathcal{C}}(\eta ,\dot {\eta })\dot {\eta } + {\mathcal{D}}(\eta ,\dot {\eta })\dot {\eta } + g(\eta ) = {\mathcal{M}}[\ddot {e}_{\eta }+\Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }] \nonumber\\[3pt] && +\, {\mathcal{M}}\ddot {\eta } + {\mathcal{C}}(\eta ,\dot {\eta })\dot {\eta } + {\mathcal{D}}(\eta ,\dot {\eta })\dot {\eta } + g(\eta ) + {\mathcal{T}}_{E} \end{eqnarray}

Using Eq. (14) and Eq. (16), the system dynamics in Eq. (31) can thus be written as

(32) \begin{eqnarray} {\mathcal{M}}\ddot {\eta } &+& {\mathcal{C}}(\eta ,\dot {\eta })\dot {\eta } + {\mathcal{D}}(\eta ,\dot {\eta })\dot {\eta } + g(\eta ) = {\mathcal{M}}[\ddot {e}_{\eta }+\Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }] \nonumber\\[3pt] && +\, {\mathcal{M}}\ddot {\eta } + {\mathcal{C}}(\eta ,\dot {\eta })\dot {\eta } + {\mathcal{D}}(\eta ,\dot {\eta })\dot {\eta } + g(\eta ) + {\mathcal{T}}_{O} + {\mathcal{T}}_{W} \end{eqnarray}

Stage-2

Using the linear parametric form of the 6 DoF model as given in Eq. (16), a controller bank is constituted by considering n random values for each of the unknown parameters.

(33) \begin{align} Y(\eta , \dot {\eta }, \ddot {\eta })\Phi = {\mathcal{M}}[\ddot {e}_{\eta }+\Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }] + Y(\eta , \dot {\eta }, \ddot {\eta })\Phi _{i} + {\mathcal{T}}_{W}, \quad \quad i & =1,2,\ldots , n \end{align}

(34) \begin{align} {\mathcal{M}}[\ddot {e}_{\eta }+\Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }]+ Y(\eta , \dot {\eta }, \ddot {\eta })(\Phi _{i}-\Phi )+{\mathcal{T}}_{W} & =0 \end{align}

Thus, $\Phi _{i}$ , $i=1,2,\ldots , n$ in the Eq. (34) constitutes the controller bank of n candidate controllers.

Stage-3

By the method of unfalsification, near-optimal values are estimated for the unknown parameters, and the CTC with the estimated parameters is inserted into the feedback loop. The CTC takes care of the trajectory tracking by AUV. By property (6), ${\mathcal{M}}^{-1}$ exists and

(35) \begin{equation} 0 \lt {\mathcal{M}}^{-1} \leq \Delta _{m} \end{equation}

where $\Delta _{m}$ is a positive value.

4.5 Assumption on unknown mass matrix

A2: Let the unknown mass matrix be approximated as $\hat {{\mathcal{M}}}$ and $\hat {{\mathcal{M}}}^{-1}$ be bounded as

(36) \begin{equation} 0 \lt \hat {{\mathcal{M}}}^{-1} \leq \delta _{m} \end{equation}

where $\delta _{m}$ is a positive value.

Note: This assumption is particularly relevant for ensuring the system’s stability and feasibility in practical applications. In real-world systems, mass matrices often represent the inertia of mechanical components or robots, and it is crucial for $M^{-1}$ to remain bounded to guarantee the system’s physical viability. The condition (36) ensures that the system avoids scenarios where the mass matrix or its inverse could become unbounded, which might lead to numerical instability or ill-conditioning in control algorithms. This is important for maintaining smooth, stable operation in systems like AUVs and other robotic systems.

In many practical applications, the mass matrix is not always known exactly and must be approximated. The boundedness condition reflects real-world constraints, such as the need to limit variations in mass/inertia due to uncertainties or variations in the system components (e.g., wear, temperature changes, etc.). The upper bound represents a safe margin, ensuring that the inverse of the mass matrix stays within a practical range for control purposes. It also allows for the use of standard control techniques that assume a stable and well-behaved mass/inertia matrix, ensuring that the controller can handle real-world dynamics without significant performance degradation due to model inaccuracies or perturbations.

By assumption (36), the unfalsified controller is obtained with the parameter estimate as

(37) \begin{equation} {\mathcal{T}} = \hat {{\mathcal{M}}}[\ddot {e}_{\eta }+\Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }] + Y(\eta , \dot {\eta }, \ddot {\eta })\hat {\Phi } + {\mathcal{T}}_{W} \end{equation}

where $\hat {\Phi }$ is estimated from the controller bank by the unfalsification method. This estimate is obtained by minimising the cost function $\mathfrak{J}$ defined in 4.1.2. Eq. (37) defines the control law. Using Eq. (16) and Eq. (37), the system dynamics thus become

(38) \begin{equation} Y(\eta , \dot {\eta }, \ddot {\eta })\Phi = \hat {{\mathcal{M}}}[\ddot {e}_{\eta }+\Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }] + Y(\eta , \dot {\eta }, \ddot {\eta })\hat {\Phi } + {\mathcal{T}}_{W} \end{equation}

which implies

(39) \begin{align} \hat {{\mathcal{M}}}[\ddot {e}_{\eta }+\Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }] + Y(\eta , \dot {\eta }, \ddot {\eta })\hat {\Phi } - Y(\eta , \dot {\eta }, \ddot {\eta })\Phi + {\mathcal{T}}_{W} &= 0 \end{align}

(40) \begin{align} \hat {{\mathcal{M}}}[\ddot {e}_{\eta }+\Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }] + Y(\eta , \dot {\eta }, \ddot {\eta }) e_{\Phi } + {\mathcal{T}}_{W} &= 0 \end{align}

where $e_{\Phi }=\hat {\Phi }-\Phi$ is the estimation error. Further modification gives

(41) \begin{align} & [\ddot {e}_{\eta } + \Lambda _{1}\dot {e}_{\eta } + \Lambda _{2}e_{\eta }] + \hat {{\mathcal{M}}}^{-1}Y(\eta , \dot {\eta }, \ddot {\eta }) e_{\Phi } + \hat {{\mathcal{M}}}^{-1}{\mathcal{T}}_{W} = 0 \end{align}

(42) \begin{align} & \ddot {e}_{\eta } =-\Lambda _{1}\dot {e}_{\eta } - \Lambda _{2}e_{\eta } - \hat {{\mathcal{M}}}^{-1}Y(\eta , \dot {\eta }, \ddot {\eta }) e_{\Phi } - {\mathcal{M}}^{-1}{\mathcal{T}}_{W} \end{align}

Eq. (42) in matrix form can be represented as

(43) \begin{equation} \begin{bmatrix} \dot {e}_{\eta }\\ \ddot {e}_{\eta } \end{bmatrix}=\begin{bmatrix} 0 &\quad 1 \\ -\Lambda _{2} &\quad -\Lambda _{1} \end{bmatrix}\begin{bmatrix} e_{\eta }\\ \dot {e}_{\eta } \end{bmatrix} + \begin{bmatrix} 0 \\ -\hat {{\mathcal{M}}}^{-1}Y(\eta , \dot {\eta }, \ddot {\eta }) e_{\Phi } - \hat {{\mathcal{M}}}^{-1}{\mathcal{T}}_{W} \end{bmatrix} \end{equation}

Eq. (43) constitutes the error dynamics of the system. Once the control law is designed according to (37), a fictitious reference is computed for each controller of the bank as

(44) \begin{eqnarray} \ddot {\hat {\eta }}_{des} &+& \Lambda _{1} \dot {\hat {\eta }}_{des} + \Lambda _{2} \hat {\eta }_{des} = \hat {{\mathcal{M}}}^{-1} [{\mathcal{T}} + \hat {{\mathcal{M}}} (\Lambda _{1} \dot {\eta } + \Lambda _{2} \eta ) \\[2pt] && -\, \hat {{\mathcal{C}}} (\Phi ,\dot {\eta },\eta ) \dot {\eta } - \hat {{\mathcal{D}}} (\Phi ,\dot {\eta },\eta ) \dot {\eta }-\hat {g}(\Phi ,\eta )]\nonumber \end{eqnarray}

$\hat {{\mathcal{M}}}$ , $\hat {{\mathcal{C}}}$ , $\hat {{\mathcal{D}}}$ , and $\hat {g}$ represent the estimates of mass matrix, Coriolis and centripetal matrix, damping matrix, and gravitational matrix, respectively. This fictitious reference is used to evaluate the cost function that determines the selection of controllers.

Note: The controller inverse, in this case, requires the inverse of the estimated mass matrix, which is invertible and bounded as given in Eq. (36).

Stage-4: GA for candidate set variation

To begin with, the fitness function is considered to be the cost function $\mathfrak{J}$ defined in Section 4.1.2. Next, the search space is initialised by defining a maximum and a minimum value for each of the unknown parameters of the system. A chromosome, therefore, represents a set of real values for as many unknown parameters available. Let the set of minimum values for the unknown parameters be defined as $\Phi _{min}=[\phi _{1min}, \phi _{2min}, \ldots , \phi _{pmin}]$ and the set of maximum values for the unknown parameters be defined as $\Phi _{max}=[\phi _{1max}, \phi _{2max}, \ldots , \phi _{pmax}]$ , $p$ being the number of unknown parameters. The population size and total number of iterations are predetermined. Random values within the specified range for each of the unknown parameters in each chromosome are generated to create the initial population. This population forms the starting set of candidate controllers.

Let $ \varpi _{1}=\{\varpi _{11}, \varpi _{21}, \ldots , \varpi _{p1}\}$ and $\varpi _{2}=\{\varpi _{12}, \varpi _{22}, \ldots , \varpi _{p2}\}$ be 2 parent chromosomes selected by Roulette Wheel selection from the initial population. By uniform crossover, a new set of offspring $\varepsilon _{1}=\{\varepsilon _{11}, \varepsilon _{21}, \ldots , \varepsilon _{p1}\}$ and $\varepsilon _{2}=\{\varepsilon _{12}, \varepsilon _{22}, \ldots , \varepsilon _{p2}\}$ is generated given by

(45) \begin{align} \varepsilon _{i1} &= \varrho _{i}\varpi _{i1}+(1-\varrho _{i})\varpi _{i2} \end{align}

(46) \begin{align} \varepsilon _{i2} &= \varrho _{i}\varpi _{i2}+(1-\varrho _{i})\varpi _{i1} \end{align}

$\varrho _{i}\in [{-}\varsigma , 1+\varsigma ]$ where $\varsigma$ is a small positive number (say 0.1). Next, the mutation is applied to change any of the chromosome sets, say $\varpi _{j}$ selected at random to $\varpi '_{j}$ . Thus,

(47) \begin{equation} \varpi '_{j} = \varpi _{j} + \beta \end{equation}

where $\beta$ is a random number having normally distributed probability with a mean of $\mu$ and variance of $\sigma$ . The new set of values forms a new set of candidates for the controller bank. Thus, GA is employed to generate a new bank of candidate controllers, from which the controller that minimises the required cost function is selected.

5.1. Stability theorem

For stability analysis of the adaptive unfalsified controller, the following lemmas with proof are stated:

5.2. Lemma 1

Feasibility: The control problem is considered feasible if at any time for a system ${\mathbb{S}}_{\zeta }$ there exists at least one stabilising controller in the bank of candidate controllers ${\mathcal{C}}_{B}$ .

Proof: Let the system (5) lie in the space ${\mathbb{S}}_{\zeta }=\{\zeta \in {\mathbb{R}}:\|\zeta \| \leq \frac {2 \Gamma \|P\|}{\kappa _{min}(Q)}\}$ , where $\zeta =\begin{bmatrix} e_{\eta }\\ \dot {e}_{\eta } \end{bmatrix}$ represents the error matrix and $\dot {\zeta }=\begin{bmatrix} \dot {e}_{\eta }\\ \ddot {e}_{\eta } \end{bmatrix}$ .

The error dynamics (43) can be written as

(48) \begin{equation} \dot {\zeta }=A_{C} \zeta + \chi \end{equation}

where $A_{C}=\begin{bmatrix} 0 & 1 \\ -\Lambda _{2} & -\Lambda _{1} \end{bmatrix}$ , and $\chi = \begin{bmatrix} 0 \\ -\hat {\mathrm{M}}^{-1}Ye_{\Phi }-\hat {\mathrm{M}}^{-1}{\mathcal{T}}_{W} \end{bmatrix}$ .

A quadratic positive Lyapunov function candidate is considered as:

(49) \begin{equation} \mathrm{V}(\zeta )= \frac {1}{2}\zeta ^{T}P\zeta \end{equation}

where P is a symmetric positive definite matrix.

(50) \begin{equation} \kappa _{min} (P) \| \zeta \|^{2} \leq \mathrm{V}(\zeta ) \leq \kappa _{max} (P) \| \zeta \|^{2} \end{equation}

Then

(51) \begin{align} \dot {\mathrm{V}}(\zeta )&= \frac {1}{2} \zeta ^{T}P\dot {\zeta } + \dot {\zeta }^{T}P\zeta \\ \nonumber &= \zeta ^{T}P(A_{C} \zeta + \chi ) + (A_{C} \zeta + \chi )^{T}P\zeta \\ \nonumber &=\zeta ^{T}PA_{C}\zeta +\zeta ^{T}P\chi +\zeta ^{T}A_{C}^{T}P\zeta + \chi ^{T}P\zeta \\ \nonumber &=\zeta ^{T}(PA_{C}+A_{C}^{T}P)\zeta +2\chi ^{T}P\zeta \end{align}

By design, we can ensure that

(52) \begin{equation} PA_{C}+A_{C}^{T}P=-Q \end{equation}

Then Eq. (51) becomes

(53) \begin{align} \dot {\mathrm{V}}&=-\zeta ^{T}Q\zeta +2\chi P \zeta \\ \nonumber & \leq -\kappa _{min} (Q) + 2\|\chi \| \|P\| \|\zeta \| \end{align}

Now,

(54) \begin{align} \|\chi \|=&\| \begin{bmatrix} 0 \\[2pt] -\hat {\mathrm{M}}^{-1}Ye_{\Phi }-\hat {\mathrm{M}}^{-1}{\mathcal{T}}_{W} \end{bmatrix} \| \\[2pt] \nonumber & \leq \|\hat {\mathrm{M}}^{-1}\|(\| Y e_{\Phi } \| + {\mathcal{T}}_{W_{max}}) \\[2pt] \nonumber & \leq \Gamma (\| Y e_{\Phi } \| + {\mathcal{T}}_{W_{max}}) \\[2pt] \nonumber & \leq -\kappa _{min} (Q) \| \zeta \|^{2} + 2 \Gamma \|P\| \|\zeta \| \\[2pt] \nonumber & \leq -\kappa _{min} (Q) \| \zeta \|(\| \zeta \| - \frac {2 \Gamma \|P\|}{\kappa _{min} (Q)}) \end{align}

Hence,

(55) \begin{equation} \| \zeta \| \leq \frac {2 \Gamma \|P\|}{\kappa _{min} (Q)} \end{equation}

Next, as stated in the stability Theorem5.1, the proof of stability requires the knowledge of cost detectability defined as follows:

Therefore, according to Lyapunov stability theory, the system is ultimately bounded within the bounds specified by Eq. (55). This confirms the feasibility (by Definition 5.2) condition of Theorem5.1.

Next, as stated in the stability Theorem5.1, the proof of stability requires the knowledge of cost detectability, defined as follows:

5.3. Lemma 2

Cost Detectability: A cost function $\mathfrak{J}$ is considered cost detectable if it fulfils the conditions given as follows [Reference Safonov and Cheong26]:

1. 1. The cost function $\mathfrak{J}_{i}$ increases monotonically over time $t$ for all $({\mathcal{C}}_{i}, {\mathcal{T}}, \Upsilon , \hat {\eta }_{des_{i}})$ , $i= 1, 2,3 \ldots$ .

2. 2. $\lim _{t\to \infty }\mathfrak{J}_{i}$ remains uniformly bounded for all $({\mathcal{C}}_{i}, {\mathcal{T}}, \Upsilon , \hat {\eta }_{des})$ , $i= 1, 2,3 \ldots$ when ${\mathcal{C}}_{i}$ is stabilising.

Proof: The following cost function is considered for minimisation:

(56) \begin{equation} \mathfrak{J} = \frac {\|\hat {e}_{\eta }\|^{2} + \alpha \|{\mathcal{T}}\|^{2}}{\|\hat {\eta }_{des}\|^{2}+\rho } \end{equation}

$\alpha ,\rho \gt 0$ are positive constants. $\hat {\eta }_{des}$ is the fictitious reference signal and $\hat {e}_{\eta }=\hat {\eta }_{des}-\eta$ represents the fictitious error, which quantifies the difference between the actual output and the fictitious reference. The cost function $\mathfrak{J}$ features a denominator that includes the norm of the fictitious reference $\hat {\eta }_{des}$ . This design choice normalises the numerator, making the cost function independent of variations in the fictitious reference across different controllers. Consequently, it eliminates the need to insert each candidate controller into the loop for input-output data collection, thereby avoiding potentially hazardous experimentation. By utilising fictitious references, data can be gathered without the risks associated with real-time controller deployment. To mitigate potential problems if the fictitious reference causes the denominator to approach zero, a constant $\rho$ is introduced to the denominator.

According to the cost detectability proof, the cost function must be shown to increase over time but remain bounded for all time $t\lt \infty$ . To prove that $\mathfrak{J}$ is monotonically increasing in time, Eq. (56) is differentiated w.r.t time $t$ as follows:

(57) \begin{align} \dot {\mathfrak{J}} &= \frac {d}{dt}\left\{\frac {\|\hat {e}_{\eta }\|^{2} + \alpha \|{\mathcal{T}}\|^{2}}{\|\hat {\eta }_{des}\|^{2}+\rho }\right \} \end{align}

(58) \begin{align} \nonumber &=\frac {d}{dt}\left\{ \frac {\{ (\int _{0}^{\infty } |\hat {e}_{\eta }|^{2} dt)^{\frac {1}{2}}\}^{2} + \alpha \{(\int _{0}^{\infty } |{\mathcal{T}}|^{2} dt)^{\frac {1}{2}}\}^{2}} {\{(\int _{0}^{\infty } |\hat {\eta }_{des}|^{2}dt)^{\frac {1}{2}}\}^{2}+\rho }\right \} \\[6pt] \nonumber &=\frac {d}{dt} \left\{ \frac {(\int _{0}^{\infty } |\hat {e}_{\eta }|^{2} dt) + \alpha (\int _{0}^{\infty } |{\mathcal{T}}|^{2} dt)} {\int _{0}^{\infty } |\hat {\eta }_{des}|^{2}dt+\rho }\right \} \\[6pt] \nonumber &=\displaystyle\frac {\substack {\mbox{$(\int _{0}^{\infty } |\hat {\eta }_{des}|^{2}dt+\rho ) \frac {d}{dt} \{ (\int _{0}^{\infty } |\hat {e}_{\eta }|^{2} dt$}\\[4pt]\mbox{$ +\, \alpha \int _{0}^{\infty } |{\mathcal{T}}|^{2} dt) \} - (\int _{0}^{\infty } |\hat {e}_{\eta }|^{2} dt)$}\\[4pt]\mbox{$\qquad \qquad +\, \alpha (\int _{0}^{\infty } |{\mathcal{T}}|^{2} dt) \frac {d}{dt}(\int _{0}^{\infty }|\hat {\eta }_{des}|^{2}dt+\rho )$}}}{(\int _{0}^{\infty } |\hat {\eta }_{des}|^{2}dt+\rho )^{2}} \\[6pt] \nonumber &= \frac {\substack {\mbox{$(\int _{0}^{\infty } |\hat {\eta }_{des}|^{2}dt+\rho ) (|\hat {e}_{\eta }|^{2} + \alpha |{\mathcal{T}}|^{2})$}\\[6pt] \mbox{$\qquad -\, (\int _{0}^{\infty } |\hat {e}_{\eta }|^{2} dt + \alpha \int _{0}^{\infty } |{\mathcal{T}}|^{2} dt) |\hat {\eta }_{des}|^{2}$}}}{(\int _{0}^{\infty } |\hat {\eta }_{des}|^{2}dt+\rho )^{2}} \\[6pt] \nonumber &= \frac {\substack {\mbox{$(\|\hat {\eta }_{des}\|^{2}+\rho ) (|\hat {e}_{\eta }|^{2} + \alpha |{\mathcal{T}}|^{2})$}\\[2pt] \mbox{$\qquad -\, (\|\hat {e}_{\eta }\|^{2} + \alpha \|{\mathcal{T}}\|^{2}) |\hat {\eta }_{des}|^{2}$}}}{(\|\hat {\eta }_{des}\|^{2}+\rho )^{2}} \\[6pt] \nonumber & = \frac {(\|\hat {\eta }_{des}\|^{2}+\rho ) (|\hat {e}_{\eta }|^{2} + \alpha |{\mathcal{T}}|^{2})}{(\|\hat {\eta }_{des}\|^{2}+\rho )^{2}}-\frac {\|\hat {e}_{\eta }\|^{2}|\hat {\eta }_{des}|^{2}}{(\|\hat {\eta }_{des}\|^{2}+\rho )^{2}} -\frac {\alpha \|{\mathcal{T}}\|^{2} |\hat {\eta }_{des}|^{2}}{(\|\hat {\eta }_{des}\|^{2}+\rho )^{2}} \\[6pt] &=\frac {|\hat {e}_{\eta }|^{2} + \alpha |{\mathcal{T}}|^{2}}{\|\hat {\eta }_{des}\|^{2}+\rho }-\frac {\|\hat {e}_{\eta }\|^{2}|\hat {\eta }_{des}|^{2}}{(\|\hat {\eta }_{des}\|^{2}+\rho )^{2}}-\frac {\alpha \|{\mathcal{T}}\|^{2} |\hat {\eta }_{des}|^{2}}{(\|\hat {\eta }_{des}\|^{2}+\rho )^{2}} \end{align}

Since, $\frac {\|\hat {e}_{\eta }\|^{2}|\hat {\eta }_{des}|^{2}}{(\|\hat {\eta }_{des}\|^{2 }+\rho )^{2}} \geq 0$ and $ \frac {\alpha \|{\mathcal{T}}\|^{2} |\hat {\eta }_{des}|^{2}}{(\|\hat {\eta }_{des}\|^{2}+\rho )^{2}} \geq 0$ .

Therefore,

(59) \begin{equation} \dot {\mathfrak{J}} \leq \frac {|\hat {e}_{\eta }|^{2} + \alpha |{\mathcal{T}}|^{2}}{\|\hat {\eta }_{des}\|^{2}+\rho } \quad \end{equation}

Hence, $0 \leq \dot {\mathfrak{J}} \leq \frac {|\hat {e}_{\eta }|^{2} + \alpha |{\mathcal{T}}|^{2}}{\|\hat {\eta }_{des}\|^{2}+\rho }$ for all $({\mathcal{C}}_{i}, \hat {\eta }_{des}, {\mathcal{T}}, \Upsilon )$ , $i= 1, 2,3 \ldots$ .

This confirms the cost detectability of $\mathfrak{J}$ , which by Theorem5.1 fulfils the other condition of stability, besides feasibility. The candidate controllers aim to minimise the cost function, which depends on the system’s fictitious error, as well as its input and output.

5.4. Lemma 3

Unfalsified Stability: The stability of the system ${\mathbb{S}}_{\zeta }$ is considered falsified by the truncated input-output pair $({\mathcal{T}},\Upsilon )$ if

(60) \begin{equation} \sup _{\tau \in R_{+}, \|{\mathcal{T}}\| \neq 0}{\frac {\|\Upsilon \|_{\tau }}{\|{\mathcal{T}}\|_{\tau }}} \longrightarrow \infty \end{equation}

If this condition is not met, then the stability of the system ${\mathbb{S}}_{\zeta }$ is regarded as unfalsified by the data $({\mathcal{T}},\Upsilon )$ [Reference Manuelli, Cheong, Mosca and Safonov28].

At $t=\tau _{f}$ , the controller with minimum $\mathfrak{J}$ remains unfalsified. Thus, ${\mathcal{C}}={\mathcal{C}}_{f}$ with fictitious reference $\hat {\eta }_{des_f}$ results in

(61) \begin{equation} \lim _{t \to \tau _{f}} \|\hat {\eta }_{des_f}-\eta _{des}\|=0 \end{equation}

This ensures the convergence of the error. Additionally, the finite set of candidate controllers generated with a specified range for the unknown parameters guarantees the boundedness of the estimates. Hence, by Eq. (61) and all the arguments stated above, it is concluded that the control system is stable.

The switching of controllers depending on the attainment of the minimum cost function is guaranteed to be finite by the following algorithm:

5.5. Lemma 4

Hysteresis Switching Algorithm: Let $\Gamma _{k} \in {\mathbb{R}}$ represent a continuous logic input signal, and $\delta$ denote the output switching signal, with both $\delta$ and $k$ belonging to a normed vector space $\mathbb{K}$ . At $t=t_{0}$ , if $\delta (0)=$ be chosen as arg min ${}_{k\in \mathbb{K}} \{\Gamma _{k}(0)\}$ . Then at some later time $t=t_{m}$ , $\delta$ switches to a new value $l \in \mathbb{K}$ . In such a case, $\delta$ will remain constant for all subsequent time $t_{m+1} \gt t_{m}$ provided that $(1 + n)$ min $\{\Gamma _{k}(t_{m+1})\} \leq \Gamma _{l}(t_{m})$ , where $n$ is the hysteresis constant. This process results in a piece-wise constant signal for $\delta$ . If $\Gamma _{k}$ is uniformly bounded and $\Gamma _{k} \geq \omega$ , where $\omega \gt 0$ , $\forall k\in \mathbb{K}$ , and all $t \geq 0$ , then the switching remains finite [Reference Hespanha46].

Proof: Thus, the fact that the set of unfalsified controllers is monotonically decreasing and bounded below by an empty set ensures a finite number of switching amongst controllers. If there are $n$ candidate controllers and there is at least one stable controller that satisfies the desired performance criteria, then by the Definition 5.5 stated above, the convergence of the switching of unfalsified controllers after at most $n-1$ switches can be assured.