Fuzzy set theory for optimal design in controlling underactuated dynamical systems with uncertainty

Abstract

The fuzzy set theory for optimal control design of underactuated dynamical systems with uncertainty is proposed. The uncertainty is (possibly fast) time varying whose bound is unknown and lies in a prescribed set expressed by a fuzzy set. Control goals of uncertain dynamical systems are formulated as a series of constraints, which may be holonomic and nonholonomic. From the view of constraint-following control, we design an adaptive robust control based on a creative uncertainty decomposition, which divides the uncertainty into matched and mismatched terms. This decomposition renders the mismatched term to “disappear” in the stability analysis. The proposed control is able to drive the system to approximately follow the constraints and guarantee system performance (uniform boundedness and uniform ultimate boundedness), in the presence of the uncertainty. It is in deterministic form and not “if–then” fuzzy rules based. Because of the expressed fuzzy set of the uncertainty bounds, a performance index, which balances the conflict between system performance and control cost, can be constructed. An optimal design gain can be obtained by minimizing the performance index whose solution is guaranteed to always exist. Numerical results on a planar vertical takeoff and landing aircraft are presented for demonstration.

Keywords

Underactuated dynamical systems uncertainty fuzzy set theory constraint-following control adaptive robust control optimal design gain

1. Introduction

Because underactuated systems extensively exist in the engineering practice, servo control design for underactuated systems has been a hot spot issue in recent years (Chen and Sun, 2019; Kant et al., 2019; Xu and Ümit, 2008; Yin et al., 2020). Research studies on the motion control of underactuated systems, such as planar vertical takeoff and landing aircraft (Castañeda et al., 2018), active suspension system (Pan and Sun, 2019), two-wheeled mobile robot (Silva and Sup, 2017), and so on, have been widely attracted by academia and industry. From the perspective of the motion control design, a salient feature of underactuated systems is that it has fewer control inputs than the degrees of freedom to be controlled. Therefore, this is a challenge problem usually encountered by researchers. Despite of the complexity, there are various control approaches considering this issue from different angles (Liu et al., 2019; Mobayen, 2019). However, the problem of the control design for underactuated systems still needs new development and more insight. The purpose of this study is to contribute to addressing this problem.

In real-world systems, uncertainty always exist in practical underactuated dynamic systems (Azimi and Koofigar, 2015; Hwang et al., 2013; Roy and Baldi, 2020), such as parameters, external disturbance, and so on, which may significantly influence the system performance. Therefore, it is necessary to consider the uncertainty when designers design control strategies. In general, if the bound of uncertainty in underactuated systems is known, a robust control can be designed to ensure the system performance (Petersen and Tempo, 2014; Yang et al., 2019). During the design process of robust control, the uncertainty set must cover all the possible values, and the bound of uncertainty can be defined. Therefore, the defined bound may be too large and thus be rather conservative, which may lead to obtain an unnecessary high control gain. On the contrary, an adaptive robust control does not need the known bound of the uncertainty (Chen et al., 2019; Sun et al., 2018a, 2018b). Generally, the bound can be emulated by constructing an adaptive law (e.g., leakage law), which aims to avoid generating a high control gain. The adaptive robust control can effectively deal with the unknown bound of the time-varying uncertainty in underactuated systems.

As for the unknown bound of time-varying uncertainty, researchers can collect a certain amount of data to design the special distribution function for describing the unknown bound. The distribution function can be used for optimal design of an adaptive robust control and constructed by a particular uncertainty theory. The probability theory is famous for handling this issue (Durrett, 2010). It expresses the bound of uncertainty by the frequency occurrence. From the 1950s, the integration between the probability theory and the dynamic systems occurs, which is called stochastic dynamical system. Many researchers have worked for its development and achieved great success (Haddad et al., 2019; Rajpurohit and Haddad, 2016; Rajpurohit and Haddad, 2017). However, there exist some criticisms about the probability theory. The main contention is that the probability theory is precisely constructed on the basis of obtaining a large amount of data, whereas it is very hard to complete such task in the engineering practice (Adhikari and Khodaparast, 2014). To solve the above problem, the fuzzy set theory can be selected and used for describing the unknown bound of uncertainty.

In 1965, Zadeh (1965) proposed the fuzzy set theory, which is not a fuzzy logic theory. The fuzzy set theory is different from the probability theory. It expresses the uncertainty by the degree of occurrence and focus on indicating the vague information such as “Jim’s age is close to 18 years old.” Therefore, the fuzzy theory does not involve collecting a large amount of data but is determined by the professional experience. Until now, the fuzzy theory has been widely used for expressing the uncertainty and successfully applied in various fields (Maues et al., 2020; Taylan et al., 2016). For the control field, the fuzzy set theory is less well known than the fuzzy logic control theory which is based on the “if–then” fuzzy rules (Gu et al., 2019). Fortunately, many researchers have carried out some constructive work about applying the set theory for the full actuated dynamical systems. For example, Chen (2001), (2011) pioneered to investigate the merger between the fuzzy set theory and the full actuated dynamical systems; Huang et al. (2015) explored the integration between an adaptive robust control and the fuzzy set theory for full actuated systems from the avenue of the constraint-following control.

The constraint-following control initiated based on the Udwadia–Kalaba (U–K) approach is well known for its capability to handle both holonomic and nonholonomic constraints in mechanical systems (Chen, 2009). It is also accompanied with advantages, such as not requiring any system linearizations or nonlinear cancellations, or any auxiliary variables (such as Lagrange multiplier), or pseudo variables (such as generalized speeds), rendering the control force to satisfy Gauss’s minimum principle and the Lagrange’s form of d’Alembert’s principle (hence exhibiting a modest control magnitude in practice). These advantages inspire some researchers to keep exploring an adaptive robust control for full actuated systems, in which uncertainty bound is described by the fuzzy set, from the avenue of constraint-following control (Xu et al., 2015; Zhao et al., 2016). However, there are few research studies on applying the fuzzy set theory in underactuated dynamical systems for optimal control gain from the view of the constraint-following control. Therefore, it is promising to call for more explorations.

The main contribution of this study is to apply the fuzzy set theory to design an adaptive robust control based on the constraint-following control and obtain optimal control gain for underactuated dynamical systems with uncertainty. The uncertainty is (possibly fast) time varying whose bound is unknown and lies in a prescribed set expressed by a fuzzy set. It should be emphasized that the fuzzy numbers in the fuzzy set are all crisp. The control design is implemented in three steps. First, we formulate the control goals of underactuated dynamical systems as a class of constraints, which may be holonomic and nonholonomic. According to constraints and the structure of systems, a creative uncertainty decomposition is proposed to ensure the mismatched portion to “invisible,” which can be taken advantage of the stability analysis. Second, based on the constraint-following control, an adaptive robust control for underactuated dynamical systems is designed, in which a smooth-zone adaptive law of leakage type is constructed. The control is deterministic and is not “if–then” fuzzy rules based. The uniform boundedness (UB) and uniform ultimate boundedness (UUB) of underactuated dynamical systems under the proposed control are rigorously proved, which indicates that the approximate constraint-following is guaranteed. Third, the optimal control design can be achieved by using the fuzzy set numbers that describe the bound of uncertainty. Specifically, the combined cost, including the average system performance described by fuzzy numbers and control effort, is designed to be minimized by an appropriate choice of the control gain. This is formulated as a constrained optimal design problem, whose solution is guaranteed to always exist. Numerical simulation results on a planar vertical takeoff and landing aircraft are presented for verification.

2. Basic of fuzzy set theory

The fuzzy set can be regarded as a generalization of the conventional set (Chen, 2011). In the scenario of the conventional set, an element is either “totally belonging to” or “totally not belonging to” the set. However, an element can be “possibly belonging to” the set in the scenario of the fuzzy set. In general, a fuzzy set $J$ is described as

J = {(v, μ_{N} (v)) | v \in N, μ_{N} (v) \in [0,1]}

(1)

where N is known and compact; μ_N(v) denotes the membership function and quantifies possibly that v belongs to

J

. The “0” and “1” values of μ_N(v) show the least possible v to be belonging to

J

and the most possible v to be belonging to

J

. If the value of μ_N(v) equals either “0” or “1” for all v ∈ N, then the fuzzy set

J

degenerates into a conventional set. This indicates that the fuzzy set can be viewed as a generalization of the conventional set.

In general, the fuzzy set $J$ can represent any kind of fuzzy set. However, we just use it for describing uncertain parameters of system. The fuzzy set $J$ is regarded as a fuzzy number with the following properties: (1) $J$ is normal; (2) $J$ is convex; (3) the support of $J$ is bounded; and (4) all α-cuts are closed intervals in R. In this study, the universe of discourse N of the fuzzy number $J$ is assumed to be its 0-cut. If the value of μ_N(v) equals one for all v ∈ N, then the fuzzy number $J$ will degenerate into a closed interval set. The $E$ -operation is introduced to take averages of functions of fuzzy numbers as follows.

Definition 1

Consider a fuzzy number described as (1). For any function f: N → R, the $E$ -operation $E [f (v)]$ can be expressed as

E [f (v)] = \frac{\int_{N} f (v) μ_{N} (v) d v}{\int_{N} μ_{N} (v) d v}

(2)

Lemma 1For any crisp constant a ∈ R

E [a f (v)] = a E [f (v)]

(3)

The proof of Lemma 1 and some other properties of fuzzy numbers are presented in Appendix (Chen, 2011).

3. Uncertain underactuated system subject to constraints

Consider the dynamical model of underactuated systems with uncertainty as follows

M (q (t), \dot{q} (t), σ (t), t) \ddot{q} (t) = N (q (t), \dot{q} (t), σ (t), t) + B (q (t), \dot{q} (t), σ (t), t) τ (t)

(4)

Here, t ∈ R is the time, q ∈Rⁿ is the displacement, $\dot{q} \in R^{n}$ is the velocity, and $\ddot{q} \in R^{n}$ is the acceleration. $M (q (t), \dot{q} (t), σ (t), t) \in R^{n \times n}$ is the inertia matrix and σ ∈ Σ ⊆ R^p is the uncertain parameter vector with the compact set Σ representing the possible bound. $N (q (t), \dot{q} (t), σ (t), t) \in R^{n}$ denotes the force vector. $τ (q (t), \dot{q} (t), t) \in R^{m}$ (m < n) is the control input and $B (q (t), \dot{q} (t), σ (t), t) \in R^{n \times m}$ denotes the input matrix.

Remark 1

The displacement q can be selected as generalized coordinate. The matrices N(⋅) and B(⋅) are continuous.

Now, we formulate control goals of dynamical systems as the following first-order form constraints (Yin et al., 2020)

\sum_{i = 1}^{n} A_{r i}^{'} (q, \dot{q}, t) {\dot{q}}_{i} = c_{r}^{'} (q, \dot{q}, t) r = 1, \dots, m

(5)

where

{\dot{q}}_{i}

is the ith component of

\dot{q}

, A_ri^′ (⋅) and c_r^′ (⋅) are continuous. The constraints (5) may be holonomic or nonholonomic.

To obtain the second-order form of constraints (5), we take derivative of (5) with respective to t, and the following is obtained

\sum_{i = 1}^{n} (\frac{d}{d t} A_{r i}^{'} (q, \dot{q}, t)) {\dot{q}}_{i} + \sum_{i = 1}^{n} A_{r i}^{'} (q, \dot{q}, t) {\ddot{q}}_{i} = \frac{d}{d t} c_{r}^{'} (q, \dot{q}, t)

(6)

where

\frac{d}{d t} A_{r i}^{'} (q, \dot{q}, t) = \sum_{k = 1}^{n} \frac{\partial A_{r i}^{'} (q, \dot{q}, t)}{\partial q_{k}} {\dot{q}}_{k} + \sum_{k = 1}^{n} \frac{\partial A_{r i}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{k}} {\ddot{q}}_{k} + \frac{\partial A_{r i}^{'} (q, \dot{q}, t)}{\partial t}

(7)

\frac{d}{d t} c_{r}^{'} (q, \dot{q}, t) = \sum_{k = 1}^{n} \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial q_{k}} {\dot{q}}_{k} + \sum_{k = 1}^{n} \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{k}} {\ddot{q}}_{k} + \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial t}

(8)

Because

\sum_{i = 1}^{n} (\sum_{k = 1}^{n} \frac{\partial A_{r i}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{k}} {\ddot{q}}_{k}) {\dot{q}}_{i} = \sum_{i = 1}^{n} (\sum_{k = 1}^{n} \frac{\partial A_{r k}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}} {\dot{q}}_{k}) {\ddot{q}}_{i}

(9)

\begin{array}{l} \sum_{k = 1}^{n} \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{k}} {\ddot{q}}_{k} = \sum_{i = 1}^{n} \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}} {\ddot{q}}_{i} \\ \sum_{k = 1}^{n} \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial q_{k}} {\dot{q}}_{k} = \sum_{i = 1}^{n} \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial q_{i}} {\dot{q}}_{i} \end{array}

(10)

the second-order form of constraints, that is, (6), can be rewritten as

\begin{array}{l} \sum_{i = 1}^{n} (A_{r i}^{'} (q, \dot{q}, t) + \sum_{k = 1}^{n} \frac{\partial A_{r k}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}} {\dot{q}}_{k} - \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}}) {\ddot{q}}_{i} \\ = - \sum_{i = 1}^{n} (\sum_{k = 1}^{n} \frac{\partial A_{r i}^{'} (q, \dot{q}, t)}{\partial q_{k}} {\dot{q}}_{k} + \frac{\partial A_{r i}^{'} (q, \dot{q}, t)}{\partial t} - \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial q_{i}}) {\dot{q}}_{i} \\ + \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial t} = : b_{r} (q, \dot{q}, t) r = 1, \dots, m \end{array}

(11)

or in the matrix form

A (q, \dot{q}, t) \ddot{q} = b (q, \dot{q}, t)

(12)

where

A = {[A_{r i}]}_{m \times n}

b = {[b_{1}, b_{2}, \dots, b_{m}]}^{T}

and

A_{r i} : = A_{r i}^{'} (q, \dot{q}, t) + \sum_{k = 1}^{n} \frac{\partial A_{r k}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}} {\dot{q}}_{k} - \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}}

(13)

Furthermore, the first-order form of constraints, that is, (5), can be rewritten as

\begin{matrix} \sum_{i = 1}^{n} (A_{r i}^{'} (q, \dot{q}, t) + \sum_{k = 1}^{n} \frac{\partial A_{r k}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}} {\dot{q}}_{k} - \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}}) {\dot{q}}_{i} \\ = c_{r}^{'} (q, \dot{q}, t) + \sum_{i = 1}^{n} (\sum_{k = 1}^{n} \frac{\partial A_{r k}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}} {\dot{q}}_{k} - \frac{\partial c_{r}^{'} (q, \dot{q}, t)}{\partial {\dot{q}}_{i}}) {\dot{q}}_{i} \\ = : c_{r} (q, \dot{q}, t) r = 1, \dots, m \end{matrix}

(14)

or in the matrix form

A (q, \dot{q}, t) \dot{q} = c (q, \dot{q}, t)

(15)

where

c = {[c_{1}, c_{2}, \dots, c_{m}]}^{T}

Remark 2

The proposed constraints in (15) are different from those in Chen (2009), which cover the velocity vector $\dot{q}$ as an argument in the functions “A(⋅)”and “c(⋅)” in their first-order form. This indicates that constraint proposed in this study can be linear or nonlinear with respect to $\dot{q}$ . Furthermore, the proposed constraints can be holonomic or nonholonomic. In addition, they also formulate as other standard control goals, such as trajectory following, stabilization, and optimality. Therefore, the constraints (15) include a rather wide range of practical system needs.

Definition 1For given Γ and ϒ, the equation ΓΛ = ϒ is called consistent if there exists at least one solution Λ.

Assumption 1

Constraint (12) is consistent with respect to $\ddot{q}$ .

Lemma 1. By Noble and Daniel (1997), a necessary and sufficient condition for the constraint (12) to be consistent is AA⁺b = b. Here, “+” denotes the Moore-Penrose (MP) generalized inverse.

3.1. Adaptive robust servo control design

Now, we decompose the matrices M, N, and B in (4) as follows

\begin{matrix} M (q, \dot{q}, σ, t) = \bar{M} (q, \dot{q}, t) + Δ_{M} (q, \dot{q}, σ, t) \\ N (q, \dot{q}, σ, t) = \bar{N} (q, \dot{q}, t) + Δ_{N} (q, \dot{q}, σ, t) \\ B (q, \dot{q}, σ, t) = \bar{B} (q, \dot{q}, t) + Δ_{B} (q, \dot{q}, σ, t) \end{matrix}

(16)

where

\bar{M}

\bar{N}

, and

\bar{B}

denote “nominal” portions; Δ_M, Δ_N, and Δ_B denote the uncertain portions. Here, the functions

\bar{M} (\cdot)

\bar{N} (\cdot)

\bar{B} (\cdot)

, Δ_M(⋅), Δ_N(⋅), and Δ_B(⋅) are all continuous. In addition, (

\bar{\cdot}

) always represents the nominal portion throughout the study.

Assumption 2

For all $(q, \dot{q}, t) \in Ω_{q} \times Ω_{\dot{q}} \times Ω_{t}$ , the m × m matrix $A (q, \dot{q}, t) D (q, \dot{q}, t) \bar{B} (q, \dot{q}, t)$ is invertible.

Let

D (q, \dot{q}, t) = {\bar{M}}^{- 1} (q, \dot{q}, t) Δ_{D} (q, \dot{q}, σ, t) = M^{- 1} (q, \dot{q}, σ, t) - {\bar{M}}^{- 1} (q, \dot{q}, t) E (q, \dot{q}, σ, t) : = \bar{M} (q, \dot{q}, t) M^{- 1} (q, \dot{q}, σ, t) - I

(17)

We obtain

Δ_{D} (q, \dot{q}, σ, t) = D (q, \dot{q}, t) E (q, \dot{q}, σ, t)

(18)

We decompose E based on $\bar{B}$ . Let

E (q, \dot{q}, σ, t) = \bar{B} (q, \dot{q}, t) \hat{E} (q, \dot{q}, σ, t) + {\tilde{Δ}}_{E} (q, \dot{q}, σ, t)

(19)

E is decomposed into two portions: the matched portion $(\bar{B} \hat{E})$ and the mismatched portion $({\tilde{Δ}}_{E})$ . The matched portion is within the range space of $\bar{B}$ , and the mismatched portion is out of the range space of $\bar{B}$ . For notational consistency, ( $\hat{\cdot}$ ) and ( $\tilde{\cdot}$ ) always represent the matched portion and the mismatched portion, respectively. Theoretically, choices for the matched and mismatched portions are infinite. For instance, a direct choice is an arbitrary $\hat{E}$ paired with ${\tilde{Δ}}_{E} : = E - \bar{B} \hat{E}$ . In this study, we specifically choose them as

\hat{E} = {(A D \bar{B})}^{- 1} A D E

(20)

{\tilde{Δ}}_{E} = E - \bar{B} {(A D \bar{B})}^{- 1} A D E

(21)

The uncertain portion Δ_B is decomposed based on $\bar{B}$ in a similar form as (19) whose matched and mismatched portions ( ${\hat{Δ}}_{B}$ , ${\hat{Δ}}_{B}$ ) have the similar form as (20) and (21), respectively. For simplicity, the expression for the decomposition of is omitted here.

Theorem 1

Let ${\tilde{Δ}}_{U}$ denote the mismatched portions ${\tilde{Δ}}_{E}$ and ${\tilde{Δ}}_{B}$ . Each column of the matrix ${\tilde{Δ}}_{U} (q, \dot{q}, σ, t)$ is in the null space (or kernel) of $A (q, \dot{q}, t) D (q, \dot{q}, t)$ . This can be described as

A (q, \dot{q}, t) D (q, \dot{q}, t) {\tilde{Δ}}_{U} (q, \dot{q}, σ, t) \equiv 0

(22)

Proof

To prove (22), we take the case ${\tilde{Δ}}_{U} = {\tilde{Δ}}_{E}$ as an example. By using (19)–(21) yields

\begin{matrix} A D {\tilde{Δ}}_{E} \\ = A D (E - \bar{B} \hat{E}) \\ = A D E - A D \bar{B} \hat{E} \\ = A D E - A D \bar{B} {(A D \bar{B})}^{- 1} A D E \\ = A D E - A D E \\ = 0 \end{matrix}

(23)

for all q ∈ Ω_q,

\dot{q} \in Ω_{\dot{q}}

, σ ∈ Σ, t ∈ Ω_t. The other case, that is,

{\tilde{Δ}}_{U} = {\tilde{Δ}}_{B}

, can be proved in a similar way Q.E.D.

Remark 3

By (22), the above uncertainty decomposition ((20) and (21)) creatively assigns the mismatched uncertain portions to be within the null space of AD. Consequently, the mismatched portions are always invisible, and only the matched portions in the range space of AD are in question. As A and D are, respectively, related to the constraints and the system, this decomposition in a sense is based on the structure of the system and constraints. It will be used in later development.

Assumption 3

Based on Assumption 2, for given positive definite matrix G ∈R^m×m, let

W = G A D \bar{B} (\hat{B} + \hat{E} B) {(A D \bar{B})}^{- 1} G^{- 1}

(24)

There exists a constant ρ_E ≥ −1 such that for all q ∈ Ω_q, $\dot{q} \in Ω_{\dot{q}}$ , σ ∈ Σ, t ∈ Ω_t

\frac{1}{2} \min_{σ \in Σ} λ_{m} (W (q, \dot{q}, σ, t) + W^{T} (q, \dot{q}, σ, t)) \geq ρ_{E}

(25)

where λ_m(λ_M) denotes the minimum (maximum) eigenvalue of the matrix.

Remark 3

If there is no uncertainty in M, that is, $M = \bar{M}$ , then $\hat{E} = 0$ . If, furthermore, there is no uncertainty in B, then $\hat{B} = 0$ and W = 0. We can choose ρ_E = 0 to satisfy Assumption 3. Therefore, this assumption imposes the effect of uncertainty σ on the possible deviation of M from $\bar{M}$ to be within a certain threshold. Note that only the lower bound of ρ_E is restricted, and no upper bound of ρ_E is imposed.

Assumption 4

Let, $β : = A \dot{q} - c$ . There exist a constant vector α ∈ (0,∞)^k and a known function $Π (\cdot) : {(0, \infty)}^{k} \times R \times R^{n} \times R^{n} \times R \to R_{+}^{z}$ such that for all q ∈ Ω_q, $\dot{q} \in Ω_{\dot{q}}$ , σ ∈ Σ, t ∈ Ω_t

\begin{array}{l} {(1 + ρ_{E})}^{- 1} \max_{σ \in Σ} ‖ G A [Δ_{D} (N + B p_{1} + B p_{2}) + D Δ_{B} (p_{1} + p_{2}) \\ + D Δ_{N}] ‖ \leq Π (α, κ, q, \dot{q}, t) \end{array}

(26)

where

p_{1} = {(A D \bar{B})}^{- 1} [b - A D \bar{N}]

(27)

p_{2} = - κ {(A D \bar{B})}^{- 1} G^{- 1} β

(28)

and κ ≥ 0 is a scalar constant. Note that 1 + ρ_E > 0 in (26) because Assumption three imposes ρ_E > −1. The function Π(⋅) can be linearized with respect to α for each

(α, κ, q, \dot{q}, t)

. That is, there exists a function

\tilde{Π} (\cdot) : R_{+} \times R^{n} \times R^{n} \times R \to R_{+}^{z}

such that

Π (α, κ, q, \dot{q}, t) = α^{T} \tilde{Π} (κ, q, \dot{q}, t)

(29)

We now construct the control as

τ (t) = p_{1} (q, \dot{q}, t) + p_{2} (q, \dot{q}, t) + p_{3} (\hat{α}, q, \dot{q}, t)

(30)

with

p_{3} = - {(A D \bar{B})}^{- 1} G^{- 1} γ (\hat{α}, κ, q, \dot{q}, t) μ (\hat{α}, κ, q, \dot{q}, t) Π (\hat{α}, κ, q, \dot{q}, t)

(31)

Here

γ (\hat{α}, κ, q, \dot{q}, t) = {\begin{matrix} \frac{1}{‖ μ (\hat{α}, κ, q, \dot{q}, t) ‖} & if & ‖ μ (\hat{α}, κ, q, \dot{q}, t) ‖ > ε \\ \frac{1}{ε} & if & ‖ μ (\hat{α}, κ, q, \dot{q}, t) ‖ \leq ε \end{matrix}

(32)

with ɛ > 0 a scalar constant, and

μ (\hat{α}, κ, q, \dot{q}, t) = β (q, \dot{q}, t) Π (\hat{α}, κ, q, \dot{q}, t)

(33)

The estimated parameter $\hat{α}$ is a time-varying vector with same dimension as α. It is governed by the following adaptive law

\dot{\hat{α}} = {\begin{array}{l} \begin{array}{l} κ [k_{1} \tilde{Π} (κ, q, \dot{q}, t) ‖ β (q, \dot{q}, t) ‖ - k_{2} \hat{α}] \\ if ‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ ‖ β (q, \dot{q}, t) ‖ > ζ \end{array} \\ \begin{array}{l} κ [\begin{array}{l} k_{1} \tilde{Π} (κ, q, \dot{q}, t) \frac{‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ {‖ β (q, \dot{q}, t) ‖}^{2}}{ζ} \\ - k_{2} \hat{α} \end{array}] \\ if ‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ ‖ β (q, \dot{q}, t) ‖ \leq ζ \end{array} \end{array}

(34)

where k_1,2and ζ are positive constants.

Remark 4

Compared with the dead-zone adaptive law, the proposed adaptive law is continuous at the critical point. Therefore, we call it the smooth-zone adaptive law. The first terms on the RHS of (34) intend to emulate the increase of $\hat{α}$ to compensate the uncertainty. The second terms on the RHS of (34) are of leakage type, which intend to emulate the decrease of $\hat{α}$ in case of over compensation. The leakage amount is tuned by k₂.

By using (27)–(29), (31), and (33), the control input (30) can be rewritten as

\begin{matrix} τ = {(A D \bar{B})}^{- 1} [b - A D \bar{N}] - κ {(A D \bar{B})}^{- 1} G^{- 1} β \\ - {(A D \bar{B})}^{- 1} G^{- 1} γ β {(α^{T} \tilde{Π})}^{2} \end{matrix}

(36)

By (36), the proposed control is a model-based state feedback control. The control input τ is determined by the constraint matrix (i.e., A) and the dynamical model (i.e., $D, \bar{B}, \bar{N}$ ).

Theorem 2

Consider the dynamical systems (4) and let $δ = {[β^{T} {(\hat{α} - α)}^{T}]}^{T}$ . Subject to Assumptions 1–4, the control (30) renders the ASS (4) to satisfy the following performances:

Uniform boundedness: For any r > 0, there is a d(r) < ∞ such that if ||δ(t₀)|| < r, then ||δ (t)|| ≤ d(r) for all t ≥ t₀;

Uniform ultimate boundedness: For any r > 0 with ||δ(t₀)|| < r, there exists a $\underline{d} > 0$ such that $‖ δ (t) ‖ \leq \bar{d}$ for any $\bar{d} > \underline{d}$ as $t \geq t_{0} + T (\bar{d}, r)$ with $0 \leq T (\bar{d}, r) \leq \infty$ .

Proof

See the Appendix 1.

Remark 5

From the analysis in the proof (see the Appendix 1), we can find that the proposed control can enable the uniform boundedness and the uniform ultimate boundedness of dynamical systems subject to uncertainties and constraints, which may be holonomic or nonholonomic. From (90)–(93), it can be found that the control parameter κ affects the size of the uniform boundedness and the uniform ultimate boundedness, which is associated with the system performance. On the other hand, κ also influences the control cost. This indicates a trade-off between the system performance and the control cost. Because the fuzzy sets describing the uncertainty bounds are available, it is possible to address the trade-off by optimal design considerations with utilization of them.

4. Optimal gain design

4.1. Performance index for the optimal gain design

By Rayleigh’s principle, we yield

λ_{m} (G) {‖ β ‖}^{2} \leq β^{T} G β \leq λ_{M} (G) {‖ β ‖}^{2}

(37)

Because λ_m(G) > 0, we can obtain

- 2 κ \frac{λ_{m} (G)}{λ_{M} (G)} {‖ β ‖}^{2} \geq \frac{- 2 κ}{λ_{M} (G)} β^{T} G β \geq - 2 κ {‖ β ‖}^{2}

(38)

From (86)–(88), we have

\begin{matrix} \dot{V} \leq - 2 κ {‖ β ‖}^{2} - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} \\ + 2 (1 + ρ_{E}) (ζ + k_{1}^{- 1} k_{2} ‖ α ‖) ‖ \hat{α} - α ‖ + (1 + ρ_{E}) \frac{ε}{2} \\ \leq - 2 κ {‖ β ‖}^{2} - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} + (1 + ρ_{E}) \frac{ε}{2} \\ + (1 + ρ_{E}) {(ζ + k_{1}^{- 1} k_{2} ‖ α ‖)}^{2} + (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} \\ = - 2 κ {‖ β ‖}^{2} - κ (2 k_{2} - k_{1}) {(κ k_{1})}^{- 1} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} + ψ \end{matrix}

(39)

where

ψ = (1 + ρ_{E}) {(ζ + k_{1}^{- 1} k_{2} ‖ α ‖)}^{2} + (1 + ρ_{E}) ε / 2

. Invoking (70) yields

\begin{matrix} - 2 κ {‖ β ‖}^{2} - κ (2 k_{2} - k_{1}) {(κ k_{1})}^{- 1} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} \\ \leq \frac{- 2 κ}{λ_{M} (G)} β^{T} G β - κ (2 k_{2} - k_{1}) {(κ k_{1})}^{- 1} \\ \times (1 + ρ_{E}) {(\hat{α} - α)}^{T} (\hat{α} - α) \\ \leq - κ \hat{ρ} V \end{matrix}

(40)

where

\hat{ρ} = min {2 / λ_{M} (G), (2 k_{2} - k_{1})}

(here, we always choose the values of k_1,2 such that 2k₂ − k₁ > 0). By (39) and (40), we have

\dot{V} (t) \leq - κ \hat{ρ} V (t) + ψ

(41)

where

V (t_{0}) = V_{0} = β {(t_{0})}^{T} G β (t_{0}) + {(κ k_{1})}^{- 1} (1 + ρ_{E}) {(\hat{α} (t_{0}) - α)}^{T} (\hat{α} (t_{0}) - α)

(42)

By Hale (1969), the corresponding differential equation of (41) is

\dot{ω} (t) = - κ \hat{ρ} ω (t) + ψ, ω (t_{0}) = V_{0}

(43)

The solution of (43) can be described as

ω (t) = (V_{0} - \frac{ψ}{κ \hat{ρ}}) \exp [- κ \hat{ρ} (t - t_{0})] + \frac{ψ}{κ \hat{ρ}}

(44)

Based on the theory of differential inequality, we can get

V (t) \leq ω (t) = (V_{0} - \frac{ψ}{κ \hat{ρ}}) \exp [- κ \hat{ρ} (t - t_{0})] + \frac{ψ}{κ \hat{ρ}}

(45)

for all t > t₀. Similarly, we can obtain that for any t_s and any τ > t_s

V (τ) \leq (V_{s} - \frac{ψ}{κ \hat{ρ}}) \exp [- κ \hat{ρ} (τ - t_{s})] + \frac{ψ}{κ \hat{ρ}}

(46)

with

V_{s} = V (t_{s}) = β {(t_{s})}^{T} G β (t_{s}) + {(κ k_{1})}^{- 1} (1 + ρ_{E}) {(\hat{α} (t_{s}) - α)}^{T} (\hat{α} (t_{s}) - α)

. The time t_s may not be the initial time t₀, while the control in (30) starts to act on the system. If

V_{s 1} = β {(t_{s})}^{T} G β (t_{s})

and

V_{s 2} = k_{1}^{- 1} (1 + ρ_{E}) {(\hat{α} (t_{s}) - α)}^{T} (\hat{α} (t_{s}) - α)

, then

V_{s} = V_{s 1} + V_{s 2} / κ

By (37), we can find $V (τ) \geq λ_{m} (G) {‖ β ‖}^{2} + {(κ k_{1})}^{- 1} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2}$ . Therefore, the RHS of (46) gives an upper bound of $λ_{m} (G) {‖ β ‖}^{2} + {(κ k_{1})}^{- 1} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2}$ , which leads to an upper bound of ||δ||². For each τ ≥ t_s, let

η (κ, τ, t_{s}) = (V_{s} - \frac{ψ}{κ \hat{ρ}}) \exp [- κ \hat{ρ} (τ - t_{s})]

(47)

η_{\infty} (κ) = \frac{ψ}{κ \hat{ρ}}

(48)

It can be found that for any given κ and t_s, η(κ, τ, t_s) → 0 as τ → ∞. We propose the performance index as follows: for any t_s, let

J (κ, t_{s}) : = ϕ_{1} E [\int_{t_{s}}^{\infty} η^{2} (κ, τ, t_{s}) d τ] + ϕ_{2} E [η_{\infty}^{2} (κ)] + ϕ_{3} κ = ϕ_{1} J_{1} (κ, t_{s}) + ϕ_{2} J_{2} (κ) + ϕ_{3} J_{3} (κ)

(49)

Considering the first term on the RHS of (49), we have

\begin{matrix} \int_{t_{s}}^{\infty} η^{2} (κ, τ, t_{s}) d τ \\ = {(V_{s} - \frac{ψ}{κ \hat{ρ}})}^{2} \int_{t_{s}}^{\infty} \exp [- 2 κ \hat{ρ} (τ - t_{s})] d τ \\ = {(V_{s} - \frac{ψ}{κ \hat{ρ}})}^{2} (- \frac{1}{2 κ \hat{ρ}}) \exp [- 2 κ \hat{ρ} (τ - t_{s})] |_{t_{s}}^{\infty} \\ = (\frac{1}{2 κ \hat{ρ}}) {(V_{s 1} + \frac{V_{s 2}}{κ} - \frac{ψ}{κ \hat{ρ}})}^{2} \\ = \frac{1}{2 κ \hat{ρ}} V_{s 1}^{2} + \frac{1}{κ^{2} \hat{ρ}} V_{s 1} (V_{s 2} - \frac{ψ}{\hat{ρ}}) + \frac{1}{2 κ^{3} \hat{ρ}} {(V_{s 2} - \frac{ψ}{\hat{ρ}})}^{2} \end{matrix}

(50)

Therefore

\begin{matrix} E [\int_{t_{s}}^{\infty} η^{2} (κ, τ, t_{s}) d τ] \\ = \frac{1}{2 κ \hat{ρ}} E [V_{s 1}^{2}] + \frac{1}{κ^{2} \hat{ρ}} E [V_{s 1} (V_{s 2} - \frac{ψ}{\hat{ρ}})] \\ + \frac{1}{2 κ^{3} \hat{ρ}} E [{(V_{s 2} - \frac{ψ}{\hat{ρ}})}^{2}] \end{matrix}

(51)

Considering the second term on the RHS of (49), we have

E [η_{\infty}^{2} (κ)] = E [\frac{ψ^{2}}{κ^{2} {\hat{ρ}}^{2}}] = \frac{1}{κ^{2} {\hat{ρ}}^{2}} E [ψ^{2}]

(52)

Based on (49), (51), and (52), we obtain

\begin{matrix} J = ϕ_{1} \frac{1}{2 κ \hat{ρ}} E [V_{s 1}^{2}] + ϕ_{1} \frac{1}{κ^{2} \hat{ρ}} E [V_{s 1} (V_{s 2} - \frac{ψ}{\hat{ρ}})] \\ + ϕ_{1} \frac{1}{2 κ^{3} \hat{ρ}} E [{(V_{s 2} - \frac{ψ}{\hat{ρ}})}^{2}] + ϕ_{2} \frac{1}{κ^{2} {\hat{ρ}}^{2}} E [ψ^{2}] + ϕ_{3} κ \\ = ξ_{1} κ^{- 1} + ξ_{2} κ^{- 2} + ξ_{3} κ^{- 3} + ϕ_{3} κ \end{matrix}

(53)

where

\begin{matrix} ξ_{1} = \frac{ϕ_{1}}{2 \hat{ρ}} E [V_{s 1}^{2}], ξ_{2} = \frac{ϕ_{1}}{\hat{ρ}} E [V_{s 1} (V_{s 2} - \frac{ψ}{\hat{ρ}})] + \frac{ϕ_{2}}{{\hat{ρ}}^{2}} E [ψ^{2}] \\ ξ_{3} = \frac{ϕ_{1}}{2 \hat{ρ}} E [{(V_{s 2} - \frac{ψ}{\hat{ρ}})}^{2}] \end{matrix}

(54)

The optimal gain design problem can be described as the following constrained optimization problem: for any t_s

\min_{κ} J (κ, t_{s}) subject to κ > 0

(55)

By (52), it can be found that (1) J can be viewed as a continuous function of κ on (0; + ∞); (2) J > 0; and (3) J → + ∞ as κ → 0⁺ or κ → + ∞. Therefore, we can conclude that the minimum of J always exists by the property of continuous functions, that is, the solution of (55) always exists. How to obtain the solution of (55) will be presented as follows.

4.2. Solution of the optimal gain design

The solution of the constrained optimization problem (55) belongs to the set of roots of the following the equation

\frac{\partial J}{\partial κ} = 0

(56)

which is equivalent to

κ^{4} + ω_{1} κ^{2} + ω_{2} κ + ω_{3} = 0

(57)

with

ω_{1} = - ξ_{1} / ϕ_{3}

ω_{2} = - 2 ξ_{2} / ϕ_{3}

, and

ω_{3} = - 3 ξ_{3} / ϕ_{3}

. This is a quartic equation. In general, it can be solved by using the Ferrari method (Brookfield, 2007) presented as follows. Therefore, the four roots of (57) are

κ = \frac{\pm_{1} \sqrt{2 \bar{ω}} \pm_{2} \sqrt{- (2 \bar{ω} + 2 ω_{1} \pm_{1} \frac{\sqrt{2} ω_{2}}{\sqrt{\bar{ω}}})}}{2}

(58)

where

{(- ω_{2})}^{2} - 4 (2 \bar{ω}) ({\bar{ω}}^{2} + \bar{ω} ω_{1} + \frac{ω_{1}^{2}}{4} - ω_{3}) = 0

(59)

\bar{ω}

is a root of (59); “±₁” and “±₂” stand for either “+” or “−”; and the two occurrences of “±₁” must stand for the same sign.

Denote the four roots of (55) in (58) as κ_1,2,3,4. Based on the discussions in the last paragraph, we can conclude that there is at least one positive real root among κ_1,2,3,4. As ω₃ < 0, we can further conclude that there may be one or three positive real roots among κ_1,2,3,4. By the property of the roots of quartic equations, we make the following discussions.

In the case that there is only one positive real root among κ_1,2,3,4, the other three roots may be all negative real numbers or one negative real number with two conjugate complex numbers. Denote the only one positive real root as κ_opt+; then, the solution of the constrained optimization problem (55) is

κ_{opt} = κ_{opt +}

(60)

2. In the case that there are three positive real roots among κ_1,2,3,4, the other root is a negative real number. Among the three positive real roots, there exists at least one root satisfying

\frac{\partial^{2} J}{\partial^{2} κ} = 4 κ^{3} + 2 ω_{1} κ + ω_{2} > 0

(61)

Suppose that there are z roots with 1 ≤ z ≤ 3 satisfying (61) and denote them as κ_opti (i = 1, …, z). Then, the solution of the constrained optimization problem (55) is

κ_{opt} = \underset{κ \in {κ_{opt 1}, \dots, κ_{opt z}}}{arg min} J (κ, t_{s})

(62)

which is readily to be solved because 1 ≤ z ≤ 3.

5. Illustrative example

A planar vertical takeoff and landing aircraft, which is shown in Figure 1, is used for demonstrating the proposed control approach. The dynamical equation of normalization parameter can be expressed as (Hauser et al., 1992)

m \ddot{x} = - \sin (θ) μ_{1} + ð \cos (θ) μ_{2} m \ddot{y} = \cos (θ) μ_{1} + ð \sin (θ) μ_{2} - 1 J_{z} \overset{..}{θ} = μ_{2}

(63)

where x and y are the horizontal and vertical positions, m and J_z are the mass of the aircraft and the mass moment of the inertial around axis, θ is the angle with respect to the imaginary horizontal line, ð characterizes the coupling between the rolling moment and the lateral acceleration and is set as one in this study, μ₁ is the thrust force for the elevation of the aircraft, and μ₂ is the rolling moment about its corresponding center of mass.

Figure 1.

Planar vertical takeoff and landing aircraft.

Equation (63) can be cast into the form of (4) by letting

\begin{matrix} M = [\begin{matrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & J_{z} \end{matrix}] \ddot{q} = [\begin{matrix} \ddot{x} \\ \ddot{y} \\ \overset{..}{θ} \end{matrix}] \\ N = [\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix}] B = [\begin{matrix} - \sin θ & ð \cos θ \\ \cos θ & ð \sin θ \\ 0 & 1 \end{matrix}] τ = [\begin{matrix} μ_{1} \\ μ_{2} \end{matrix}] \end{matrix}

(64)

Suppose that there exist uncertainties in m and J_z, and the nominal values of them are 1 kg and 1 kg ⋅m². Then, m and J_z can be decomposed as m = 1 + B₁ and J_z = 1 + B₂, where B₁ and B₂ are the time-varying uncertainties. These uncertainties are bounded as |B₁| ≤ B_1m and |B₂| ≤ B_2m with B_1m and B_2m unknown constants but lying within the prescribed fuzzy numbers. M, N, and B can be decomposed into the form of (16) by letting

\begin{matrix} \bar{M} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] Δ_{M} = [\begin{matrix} B_{1} & 0 & 0 \\ 0 & B_{1} & 0 \\ 0 & 0 & B_{2} \end{matrix}] \bar{N} = [\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix}] \\ Δ_{N} = 0 \bar{B} = [\begin{matrix} - \sin θ & ð \cos θ \\ \cos θ & ð \sin θ \\ 0 & 1 \end{matrix}] Δ_{B} = 0 \end{matrix}

(65)

Suppose that the following constraints are imposed on the motion of a planar vertical takeoff and landing aircraft

y = x, θ = χ

(66)

where χ are reference values. Equation (66) can be cast into the form of (15) and (12) by letting

A = [\begin{matrix} - 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] c = [\begin{matrix} - {\hat{k}}_{1} (y - x) \\ - {\hat{k}}_{2} (θ - χ) \end{matrix}] b = [\begin{matrix} - {\hat{k}}_{1} (\dot{y} - \dot{x}) \\ - {\hat{k}}_{2} \dot{θ} \end{matrix}]

(67)

in which

{\hat{k}}_{1}

and

{\hat{k}}_{2}

are positive constants.

The unknown bounds of B_1m and B_2m lie within the following fuzzy numbers. The fuzzy (linguistic) descriptions of B_1m and B_2m are expressed as “B_1m and B_2m are close to 0.2.” The corresponding membership functions for B_1m and B_2m are (all triangular)

B_{1 m} (v) = {\begin{array}{l} 25 v - 4 & 0.16 \leq v \leq 0.2 \\ - 25 v + 6 & 0.2 \leq v \leq 0.24 \end{array}

(68)

B_{2 m} (v) = {\begin{array}{l} 20 v - 3 & 0.15 \leq v \leq 0.2 \\ - 20 v + 5 & 0.2 \leq v \leq 0.25 \end{array}

(69)

For simulation, we select G = I, ζ = ɛ = 0.5, k₁ = k₂ = 1, ${\hat{k}}_{1} = {\hat{k}}_{2} = 3$ , χ = 0.5 rad. The initial conditions are as follows: $x (0) = \dot{x} (0) = 0$ , y(0) = 50 m, $\dot{y} (0) = 0$ , θ(0) = 0.6 rad, $\dot{θ} (0) = 0$ . The time-varying uncertainties in m and J_z are B₁ = 0.2 sin(20πt) kg and B₂ = 0.2 sin(10πt) kg. Simulation results are follows.

Figure 2 and Figure 3 show the horizontal position, the vertical position, and the angle with respect to the time under the proposed control. It can be found from Figure 2 and Figure 3 that the system of the planar vertical takeoff and landing aircraft meets the constraint (66) in finite time under the effect of the uncertainty.

Figure 2.

Horizontal and vertical positions with respect to the time under the proposed control.

Figure 3.

Angle with respect to the time under the proposed control.

To show the merit of the proposed control, the sliding mode control (Aguilar-Tbanez, 2016) is selected for comparison, which is well known for the capability of handling the model error and parametric uncertainties. It is designed and based on a simple feedback linearization procedure in combination with a saturation function. The selected control parameters for the sliding mode control only guarantee the locally asymptotically stable, which is not obtained by the optimal design.

The constraint-following error ||β(t)|| is used for indicating the system performance. Figure 4 shows the ||β(t)|| histories under no control, the sliding mode control, and the proposed control (under κ_opt = 4.5835 when using ϕ₁ = ϕ₂ = 1, ϕ₃ = 0.25). It can be found that ||β(t)|| history under no control is far from 0, whereas ||β(t)|| histories under the proposed control and the sliding mode control converge to 0 in finite time. Furthermore, it can be found that the accumulation error of ||β(t)|| (i.e., $\int_{0}^{2} ‖ β (t) ‖ d t$ ) under the proposed control is much smaller than that under the sliding mode control. Figure 5 and Figure 6 compare the time histories of control inputs τ₁ and τ₂. It can found that the control inputs of the proposed control are milder than those of the sliding mode control. Therefore, the proposed control can achieve better system performance. Figure 7 plots the time history of the adaptive parameter $\hat{α}$ , which converges to 0 in finite time.

Figure 4.

Constraint-following error histories under no control, the proposed control, and the sliding mode control.

Figure 5.

Comparison of control input τ₁ under the sling mode control and the proposed control.

Figure 6.

Comparison of control input τ₂ under the sling mode control and the proposed control.

Figure 7.

Adaptive parameter history.

Figure 8 shows the performance index J as a function of the parameter κ under three groups of weighing factors (ϕ₁, ϕ₂, ϕ₃). It can be found that κ_opt always exists and makes the performance index J minimize under three different groups of weighing factor (ϕ₁, ϕ₂, ϕ₃). Figure 9 shows the ||β(t)|| histories under three different κ_opt. Figure 10 shows accumulation errors under different κ_opt. It can be found that the larger the κ_opt value, the smaller the accumulation error is. Figure 11 plots the time histories of the control efforts ∥τ∥ under three different κ_opt.

Figure 8.

Performance index J.

Figure 9.

Comparison of constraint-following errors under different κ_opt.

Figure 10.

Comparison of accumulation errors under different κ_opt.

Figure 11.

Comparison of control efforts under different κ_opt.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The article is supported by the National Natural Science Foundation of China under Grant 51806066 and the Natural Science Foundation for Young Scholars of Jiangxi Province under Grant 20181BAB216023.

ORCID iD

Wu Qin

Appendix.1

Choose the Lyapunov function candidate as (70)

V = β^{T} G β + {(κ k_{1})}^{- 1} (1 + ρ_{E}) {(\hat{α} - α)}^{T} (\hat{α} - α)

Taking derivative of V with respect to t yields (for simplicity, most of arguments of functions in this proof are omitted except for some critical ones) (71)

\dot{V} = 2 β^{T} G \dot{β} + 2 {(κ k_{1})}^{- 1} (1 + ρ_{E}) {(\hat{α} - α)}^{T} \dot{\hat{α}}

By using the decompositions M⁻¹ = D + Δ_D and $N = \bar{N} + Δ_{N}$ , we have (72)

\begin{matrix} 2 β^{T} G \dot{β} \\ = 2 β^{T} G (A \ddot{q} - b) \\ = 2 β^{T} G [A (D + Δ_{D}) (N + B (p_{1} + p_{2} + p_{3})) - b] \\ = 2 β^{T} G [(A D + A Δ_{D}) (\bar{N} + Δ_{N} + B (p_{1} + p_{2} + p_{3})) - b] \\ = 2 β^{T} G A [D \bar{N} + D B (p_{1} + p_{2}) + D Δ_{N} \\ + Δ_{D} (N + B p_{1} + B p_{2}) + (D + Δ_{D}) B p_{3}] - b \\ = 2 β^{T} G A [D \bar{N} + D (\bar{B} + Δ_{B}) p_{1} + D (\bar{B} + Δ_{B}) p_{2} \\ + D Δ_{N} + Δ_{D} (N + B p_{1} + B p_{2}) + (D + Δ_{D}) B p_{3}] - b \end{matrix}

Notice that $p_{1} = {(A D \bar{B})}^{- 1} (b - A D \bar{N})$ , we have (73)

\begin{matrix} A (D \bar{N} + D \bar{B} p_{1}) - b \\ = A (D \bar{N} + D \bar{B} {(A D \bar{B})}^{- 1} (b - A D \bar{N})) - b \\ = 0 \end{matrix}

Based on $p_{2} = - κ {(A D \bar{B})}^{- 1} G^{- 1} β$ , we yield (74)

2 β^{T} G A D \bar{B} p_{2} = - 2 κ {‖ β ‖}^{2}

By applying Assumption 4 yields (75)

\begin{matrix} 2 β^{T} G A [D Δ_{N} + Δ_{D} (N + B p_{1} + B p_{2}) + D Δ_{B} (p_{1} + p_{2})] \\ \leq 2 ‖ β ‖ ‖ G A [Δ_{D} (N + B p_{1} + B p_{2}) + D Δ_{B} (p_{1} + p_{2}) + D Δ_{N}] ‖ \\ \leq 2 ‖ β ‖ (1 + ρ_{E}) Π (α, κ, q, \dot{q}, t) \end{matrix}

Next, by (72), we have (76)

\begin{array}{l} 2 β^{T} G A (D + Δ_{D}) B p_{3} \\ = 2 β^{T} G A D B p_{3} + 2 β^{T} G A Δ_{D} B p_{3} \end{array}

Using $B = \bar{B} + Δ_{B}$ and (22), we yield (77)

\begin{matrix} 2 β^{T} G A D B p_{3} \\ = 2 β^{T} G A D (\bar{B} + Δ_{B}) p_{3} \\ = 2 β^{T} G A D (\bar{B} + \bar{B} \hat{B} + {\tilde{Δ}}_{B}) p_{3} \\ = 2 β^{T} G A D (\bar{B} + \bar{B} \hat{B}) p_{3} \\ = 2 β^{T} G A D \bar{B} (I + \hat{B}) p_{3} \end{matrix}

By applying Δ_D = DE and (22), we have (78)

\begin{matrix} 2 β^{T} G A Δ_{D} B p_{3} \\ = 2 β^{T} G A D E B p_{3} \\ = 2 β^{T} G A D (\bar{B} \hat{E} + {\tilde{Δ}}_{E}) B p_{3} \\ = 2 β^{T} G A D \bar{B} \hat{E} B p_{3} \end{matrix}

According to (76)–(78), Assumption 3, $μ = β Π (\hat{α}, κ, q, \dot{q}, t)$ , and $p_{3} = - {(A D \bar{B})}^{- 1} G^{- 1} γ μ Π (\hat{α}, κ, q, \dot{q}, t)$ , we can obtain (79)

\begin{matrix} 2 β^{T} G A (D + Δ_{D}) B p_{3} \\ = 2 β^{T} G A D \bar{B} (I + \hat{B}) p_{3} + 2 β^{T} G A D \bar{B} \hat{E} B p_{3} \\ = 2 β^{T} G A D \bar{B} (I + \hat{B} + \hat{E} B) p_{3} \\ = 2 β^{T} G A D \bar{B} p_{3} + 2 β^{T} G A D \bar{B} (\hat{B} + \hat{E} B) p_{3} \\ = - 2 β^{T} G A D \bar{B} {(A D \bar{B})}^{- 1} G^{- 1} γ μ Π (\hat{α}, κ, q, \dot{q}, t) \\ - 2 β^{T} G A D \bar{B} (\hat{B} + \hat{E} B) {(A D \bar{B})}^{- 1} G^{- 1} γ μ Π (\hat{α}, κ, q, \dot{q}, t) \\ = - 2 β^{T} γ μ Π (\hat{α}, κ, q, \dot{q}, t) \\ - 2 μ^{T} [G A D \bar{B} (\hat{B} + \hat{E} B) {(A D \bar{B})}^{- 1} G^{- 1} γ μ] \\ = - 2 μ^{T} γ μ - 2 μ^{T} [G A D \bar{B} (\hat{B} + \hat{E} B) {(A D \bar{B})}^{- 1} G^{- 1} γ μ] \\ = - 2 γ {‖ μ ‖}^{2} - 2 γ \frac{1}{2} μ^{T} [G A D \bar{B} (\hat{B} + \hat{E} B) {(A D \bar{B})}^{- 1} G^{- 1} \\ + G^{- 1} {(A D \bar{B})}^{- T} {(\hat{B} + \hat{E} B)}^{T} {\bar{B}}^{T} D A^{T} G] μ \\ \leq - 2 γ {‖ μ ‖}^{2} - 2 γ \frac{1}{2} λ_{m} (W + W^{T}) {‖ μ ‖}^{2} \\ \leq - 2 γ {‖ μ ‖}^{2} - 2 γ ρ_{E} {‖ μ ‖}^{2} \\ \leq - 2 γ (1 + ρ_{E}) {‖ μ ‖}^{2} \end{matrix}

With (32), if ||μ|| > ɛ, then (80)

- 2 γ (1 + ρ_{E}) {‖ μ ‖}^{2} = - 2 (1 + ρ_{E}) ‖ μ ‖

if ||μ|| ≤ ɛ, then (81)

- 2 γ (1 + ρ_{E}) {‖ μ ‖}^{2} = - 2 (1 + ρ_{E}) \frac{{‖ μ ‖}^{2}}{ε}

Based on (72)–(75) and (79)–(81), we can obtain, for ||μ|| > ɛ (82)

\begin{matrix} 2 β^{T} G \dot{β} \\ \leq - 2 κ {‖ β ‖}^{2} - 2 (1 + ρ_{E}) ‖ μ ‖ + 2 (1 + ρ_{E}) ‖ β ‖ Π (α, κ, q, \dot{q}, t) \\ = - 2 κ {‖ β ‖}^{2} + (1 + ρ_{E}) [- 2 ‖ β ‖ Π (\hat{α}, κ, q, \dot{q}, t) \\ + 2 ‖ β ‖ Π (α, κ, q, \dot{q}, t)] \end{matrix}

for ||μ|| ≤ ɛ (83)

\begin{matrix} 2 β^{T} G \dot{β} \\ \leq - 2 κ {‖ β ‖}^{2} - 2 (1 + ρ_{E}) \frac{{‖ μ ‖}^{2}}{ε} \\ + 2 (1 + ρ_{E}) ‖ β ‖ Π (α, κ, q, \dot{q}, t) \\ = - 2 κ {‖ β ‖}^{2} + (1 + ρ_{E}) [- 2 \frac{{‖ μ ‖}^{2}}{ε} + 2 ‖ β ‖ Π (\hat{α}, κ, q, \dot{q}, t)] \\ + (1 + ρ_{E}) [- 2 ‖ β ‖ Π (\hat{α}, κ, q, \dot{q}, t) + 2 ‖ β ‖ Π (α, κ, q, \dot{q}, t)] \\ \leq - 2 κ {‖ β ‖}^{2} + (1 + ρ_{E}) \frac{ε}{2} + (1 + ρ_{E}) [- 2 ‖ β ‖ Π (\hat{α}, κ, q, \dot{q}, t) \\ + 2 ‖ β ‖ Π (α, κ, q, \dot{q}, t)] \end{matrix}

with (82)–(83) and

Π (α, κ, q, \dot{q}, t) = α^{T} \tilde{Π} (κ, q, \dot{q}, t)

, we can obtain that for all ||μ|| (84)

\begin{matrix} 2 β^{T} G \dot{β} \\ \leq - 2 κ {‖ β ‖}^{2} + (1 + ρ_{E}) \frac{ε}{2} \\ + 2 (1 + ρ_{E}) ‖ β ‖ {(α - \hat{α})}^{T} \tilde{Π} (κ, q, \dot{q}, t) \end{matrix}

Now, we deal with the second term on the RHS of (71) as follows. With the adaptive law ((34) and (35)), we have that if $‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ ‖ β ‖ > ζ$ , then (85)

\begin{matrix} 2 {(κ k_{1})}^{- 1} (1 + ρ_{E}) {(\hat{α} - α)}^{T} \dot{\hat{α}} \\ = 2 k_{1}^{- 1} (1 + ρ_{E}) {(\hat{α} - α)}^{T} (k_{1} \tilde{Π} (κ, q, \dot{q}, t) ‖ β ‖ - k_{2} \hat{α}) \\ = 2 (1 + ρ_{E}) {(\hat{α} - α)}^{T} \tilde{Π} (κ, q, \dot{q}, t) ‖ β ‖ \\ - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {(\hat{α} - α)}^{T} \hat{α} \\ = 2 (1 + ρ_{E}) {(\hat{α} - α)}^{T} \tilde{Π} (κ, q, \dot{q}, t) ‖ β ‖ \\ - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {(\hat{α} - α)}^{T} (\hat{α} - α + α) \\ = 2 (1 + ρ_{E}) {(\hat{α} - α)}^{T} \tilde{Π} (κ, q, \dot{q}, t) l ‖ β ‖ \\ - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {(\hat{α} - α)}^{T} (\hat{α} - α) \\ - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {(\hat{α} - α)}^{T} α \\ \leq 2 (1 + ρ_{E}) {(\hat{α} - α)}^{T} \tilde{Π} (κ, q, \dot{q}, t) ‖ β ‖ \\ - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} + 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) ‖ \hat{α} - α ‖ ‖ α ‖ \end{matrix}

Based on (71), (84), and (85), we have (86)

\begin{matrix} \dot{V} \leq - 2 κ {‖ β ‖}^{2} + (1 + ρ_{E}) \frac{ε}{2} - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} \\ + 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) ‖ \hat{α} - α ‖ ‖ α ‖ \\ \leq - {\underline{k}}_{1} {‖ δ ‖}^{2} + {\underline{k}}_{2} ‖ δ ‖ + {\underline{k}}_{3} \end{matrix}

with

{\underline{k}}_{1} = \min {2 κ, 2 k_{1}^{- 1} k_{2} (1 + ρ_{E})}

{\underline{k}}_{2} = 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) ‖ α ‖

, and

{\underline{k}}_{3} = (1 + ρ_{E}) ε / 2

. If

‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ ‖ β ‖ \leq ζ

, then (87)

\begin{array}{l} 2 {(κ k_{1})}^{- 1} (1 + ρ_{E}) {(\hat{α} - α)}^{T} \dot{\hat{α}} \\ = 2 k_{1}^{- 1} (1 + ρ_{E}) {(\hat{α} - α)}^{T} (k_{1} \tilde{Π} (κ, q, \dot{q}, t) \frac{‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ {‖ β (q, \dot{q}, t) ‖}^{2}}{ζ} - k_{2} \hat{α}) \\ = 2 (1 + ρ_{E}) {(\hat{α} - α)}^{T} \tilde{Π} (κ, q, \dot{q}, t) \frac{‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ {‖ β (q, \dot{q}, t) ‖}^{2}}{ζ} \\ - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {(\hat{α} - α)}^{T} (\hat{α} - α + α) \\ \leq 2 (1 + ρ_{E}) ‖ \hat{α} - α ‖ \frac{{(‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ ‖ β ‖)}^{2}}{ζ} - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) \\ \times {(\hat{α} - α)}^{T} (\hat{α} - α) - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {(\hat{α} - α)}^{T} α \\ \leq 2 (1 + ρ_{E}) ‖ \hat{α} - α ‖ ζ - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} \\ + 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) ‖ \hat{α} - α ‖ ‖ α ‖ \end{array}

Based on (71), (84), and (87), we have (88)

\begin{matrix} \dot{V} \\ \leq - 2 κ {‖ β ‖}^{2} + (1 + ρ_{E}) \frac{ε}{2} + 2 (1 + ρ_{E}) ‖ β ‖ {(α - \hat{α})}^{T} \tilde{Π} (κ, q, \dot{q}, t) \\ + 2 (1 + ρ_{E}) {(\hat{α} - α)}^{T} \tilde{Π} (κ, q, \dot{q}, t) \frac{‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ {‖ β ‖}^{2}}{ζ} \\ - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} + 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) ‖ \hat{α} - α ‖ ‖ α ‖ \\ \leq - 2 κ {‖ β ‖}^{2} + (1 + ρ_{E}) \frac{ε}{2} + 2 (1 + ρ_{E}) {(α - \hat{α})}^{T} ‖ β ‖ \tilde{Π} (κ, q, \dot{q}, t) \\ \times (1 - \frac{‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ ‖ β ‖}{ζ}) - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} \\ + 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) ‖ \hat{α} - α ‖ ‖ α ‖ \\ \leq - 2 κ {‖ β ‖}^{2} + (1 + ρ_{E}) \frac{ε}{2} + 2 (1 + ρ_{E}) ‖ β ‖ ‖ α - \hat{α} ‖ ‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ \\ - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} + 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) ‖ \hat{α} - α ‖ ‖ α ‖ \\ \leq - 2 κ {‖ β ‖}^{2} - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} + 2 (1 + ρ_{E}) ‖ α - \hat{α} ‖ ζ \\ + 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) ‖ \hat{α} - α ‖ ‖ α ‖ + (1 + ρ_{E}) \frac{ε}{2} \\ \leq - 2 κ {‖ β ‖}^{2} - 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) {‖ \hat{α} - α ‖}^{2} + 2 (1 + ρ_{E}) \\ \times (ζ + k_{1}^{- 1} k_{2} ‖ α ‖) ‖ \hat{α} - α ‖ + (1 + ρ_{E}) \frac{ε}{2} \\ \leq - {\underline{k}}_{1} {‖ δ ‖}^{2} + {\underline{k}}_{2}^{'} ‖ δ ‖ + {\underline{k}}_{3} \end{matrix}

with

{\underline{k}}_{2}^{'} = 2 ζ (1 + ρ_{E}) + 2 k_{1}^{- 1} k_{2} (1 + ρ_{E}) ‖ α ‖

. Because

{\underline{k}}_{2} < {\underline{k}}_{2}^{'}

, we can conclude from (86) and (88) that for all

‖ \tilde{Π} (κ, q, \dot{q}, t) ‖ ‖ β ‖

(89)

\dot{V} \leq - {\underline{k}}_{1} {‖ δ ‖}^{2} + {\underline{k}}_{2}^{'} ‖ δ ‖ + {\underline{k}}_{3}

By Khalil (2002), we can obtain uniform boundedness with (90)

d (r) = {\begin{array}{l} \sqrt{\frac{η_{2}}{η_{1}}} R & if r \leq R \\ \sqrt{\frac{η_{2}}{η_{1}}} r & if r > R \end{array}

(91)

R = \frac{{\underline{k}}_{2}^{'} + \sqrt{{\underline{k}}_{2}^{' 2} + 4 {\underline{k}}_{1} {\underline{k}}_{3}}}{2 {\underline{k}}_{1}}

where

η_{1} = \min {λ_{m} (G), k_{1}^{- 1} (1 + ρ_{E})}

and

η_{2} = \max {λ_{M} (G), k_{1}^{- 1} (1 + ρ_{E})}

Furthermore, uniform ultimate boundedness follows with (92)

\underline{d} = \sqrt{\frac{η_{1}}{η_{2}}} R

(93)

T (\bar{d}, r) = {\begin{array}{l} 0 & if r \leq \bar{d} \sqrt{\frac{η_{1}}{η_{2}}} \\ \frac{η_{2} r^{2} - (η_{1}^{2} / η_{2}) {\bar{d}}^{2}}{{\underline{k}}_{1} {\bar{d}}^{2} (η_{1} / η_{2}) - {\underline{k}}_{2}^{'} \bar{d} {(η_{1} / η_{2})}^{1 / 2} - {\underline{k}}_{3}} & if r > \bar{d} \sqrt{\frac{η_{1}}{η_{2}}} \\ Q . E . D . \end{array}

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