Sage Journals: Discover world-class research

Abstract

In this article, the extremum seeking control of a two-dimensional mobile robot with external disturbances is discussed by applying dynamic angular velocity turning method. First, the extremum seeking scheme is proposed to describe the trajectory of the two-dimensional robot and to achieve extreme value optimization through dynamic feedback. Secondly, the method of finite-time control and dynamic feedback is proposed to ensure that the dynamic angular velocity converges to the virtual controller within a finite time. Thirdly, the sliding mode disturbance observer is designed to guarantee that the observer converges to an unknown disturbance in finite time. Furthermore, we allow the averaging method and the results are applied in stability analysis. Finally, our control scheme is feasible by a series of simulations.

Keywords

Extremum seeking dynamic angular velocity turning 2-D mobile robot external disturbances

Introduction

Extremum seeking control (ESC), as an optimization method, has shown its superiority in many fields in the past years and has been widely studied.^{1

–10} From local convergence¹ to semi-global convergence and then to finite-time convergence, it has attracted considerable attention in the field of control. However, with the deepening of research, dynamic angular velocity ES with external disturbances is still a challenge, because how to control it from kinematics design to dynamics is not available currently.¹¹ With the increase of engineering applications, the advantages of dynamic control input will be more prominent, and the input of dynamic control is a controller based on torque, force, or current, making it easier to apply in practical engineering. Under the creative proposal of the feedback and stability convergence model¹ and based on Lyapunov method, the conjunction of the Lie bracket-based averaging method¹² with the conclusion of Moreau and Aeyels¹³ by Dürr et al.¹⁴ made the ES more and more mature. Some of the recent work has applied ES in some unknown practical systems, such as jet noise ESC,² reduce bluff-body drag,³ antilock brake system (ABS) design,¹⁰ maximum power point tracking (MPPT) of photovoltaic systems,⁴ tunable thermoacoustic cooler,⁵ urban traffic signal control,⁶ fed-batch processes,⁷ smart inverters for visual-aural radio (VAR) compensation,⁸ human exercise machine,⁹ and periodic orbits of ESC.¹⁵

For the turning problem of dynamic angular velocity, Sanders et al. have improved the classical averaging method¹⁶ and applied it to the high-frequency oscillation closed-loop system to achieve the convergence of vehicle trajectory. Chen et al.¹⁷ proposed finite-time nonholonomic robot tracking moving targets in polar coordinates¹⁸ based on angle turning analysis. Recent work by Matsuzaki et al.¹⁹ proposed a maneuvering target tracking filter based on constant velocity and angular velocity model, which ensures the inconsistency between the actual target and the tracking target caused by angular velocity changes. Although the above control methods can achieve the stability of the dynamic system in a limited time, there is still much more research space compared with the case of optimizing the angular velocity extremum. Moreover, the classical theories described above always have the trouble of unknown perturbation, because the influence of some external unknown signals makes the system have certain limitations in practical application. Therefore, we propose a virtual disturbance observer method²⁰ and reasonably add Lyapunov function to make it converge to the unknown disturbance signal in a finite time.

This article introduces the application of adaptive controller on the level of external disturbance with dynamic feedback. On the premise of ensuring the stability of the system, we limit the measurable but unknown dynamic input to sine and cosine functions²¹; meanwhile, the averaging method is combined with external disturbances, joined the Lyapunov stability theory to make the two-dimensional (2-D) robot system angular velocity transformation converges to the virtual controller within finite time, finally makes the system trajectory error exponentially converges to zero. The further simulation results also prove this point. Then, we would like to express special thanks to Alexander Scheinker et al. whose tracking controller can be perfectly applied to our dynamic feedback system. The main ideas and contributions can be summarized as follows:

The angular velocity converges in a finite time through the input control of the torque and force, and the addition of external disturbance makes our control system better applied to the actual situation.

The virtual disturbance observer enables the system converge to unknown disturbance in finite time by sliding mode control.

The torque and the input of specific function form a kind of double-control closed-loop system, so that the robot can quickly find the optimal point even if there exists interference.

The organization of this article is as follows. In the second section, we give the background on averaging and stability as well as the main question formulation in this article. In the third section, some optimization results and main conclusions are presented, and the background on averaging and stability is stated to demonstrate our results. In the fourth section, we present the simulation results of the stated scheme, and finally conclusions are proposed in the last section.

Averaging and stability results and problem formulation

Averaging results of Sanders et al.

Theorem 1

For T ∈ (0,∞), and set K $\in R^{n}$ , the following differential equation is considered^16,22
–24

\dot{x} = f (x, t) + \sum_{i = 1}^{n} f_{i} (x, t) φ_{i, k} (t)

where the functions

\begin{array}{l} f (x, t) : R^{n} \times [0, T] \to R^{n} \\ f_{i} (x, t) : R^{n} \times [0, T] \to R^{n} \\ φ_{i, k} (t) : [0, T] \to R \end{array}

are continuous and meet the Lipschitz condition, and $D f_{i}$ , $D^{2} f_{i}, \frac{\partial f_{i}}{\partial t}, \frac{\partial}{\partial t} D f_{i}, and \frac{\partial}{\partial t} D^{2} f_{i}$ are continuous and bounded, where D = $\frac{\partial}{\partial x}$ .

If the function $φ (t)$ are continuous as $k \to \infty$ and its integrals satisfy

ϑ_{i, k} (t) = \int_{0}^{t} φ_{i, k} (τ) \to 0

So that there exists measurable function $γ_{i, j} (t) : [0, T] \to R$ for all $s, t \in [0, T]$

\begin{array}{l} lim_{k \to \infty} \int_{0}^{t} φ_{j, k} (τ) ϑ_{i, k} (t) d τ = \int_{0}^{t} γ_{i, j} (t) d τ \\ | ϑ_{i, k} (t) - ϑ_{i, k} (s) | \leq X_{1} | t - s |^{α}, as 0 < α < \frac{1}{2} \\ {| \int_{0}^{t} φ_{j, k} (τ) [ϑ_{i, k} (t) - ϑ_{i, k} (s)] d τ | \leq X_{2} | t - s |}^{β} \end{array}

where $1 - α < β \leq 1.$

Then, for all $t \in [0, T]$ and $x \in K$ , the solutions (1) can be stated as follows

x_{k} (t) = x_{0} + \int_{0}^{t} {f (x_{k}, τ) + \sum_{i = 1}^{n} f_{i} (x_{k}, τ) φ_{i, k} (τ)} d τ

Converges with respect to k, the solution x(t) satisfying

\dot{x} = f (x, t) + \sum_{i, j = 1}^{n} γ_{i, j} (t) (D f_{i} (x, t)) f_{j} (x, t), x (0) = x_{0}

Stability results

A system is given by reviewing the definitions in Moreau and Aeyels.¹³

\dot{x} = f (t, x)

Let $ψ (t, t_{0}, x_{0})$ be the solution of equation (3) which pass through state $x_{0}$ to t ₀. Union (3), we consider the systems as follows

\dot{x} = f^{ϵ} (t, x)

Similarly whose trajectory is denoted as $φ^{ϵ} (t, t_{0}, x_{0})$ .

Definition 1 (convergence of trajectories)

The systems of (3) and (4) above can be said to satisfy the converging trajectories property if for every $Ť \in (0, \infty)$ and $Κ \subset R^{n}$ , satisfying ${(t, t_{0}, x_{0}) \in R \times R \times R^{n} : t \in [t_{0}, t_{0} + Ť], x_{0} \in Κ} \subset$ Dom $ψ$ , for every $d \in (0, \infty)$ there exists $ϵ^{*}$ such that for all $t_{0} \in R$ , for all $x_{0} \in Κ$ , and for all $ϵ \in (0, ϵ^{*})$

∥ ϕ^{ϵ} (t, t_{0}, x_{0}) - ψ (t, t_{0}, x_{0}) ∥ < d, \forall t \in [t_{0}, t_{0} + Ť]

Theorem 2 (GUAS)

Consider system (3), the origin of equation (4) can be said to be global uniformly asymptotically stable (GUAS) if the following three situations are all satisfied¹³:

Uniform stability: Just for every $c_{2} \in (0, \infty)$ , there exists $c_{1} \in (0, \infty)$ , such that for all $t_{0} \in R$ and for all $x_{0} \in R^{n}$ with $∥ x_{0} ∥ < c_{1}$ and for all $ϵ \in (0, \hat{ϵ})$

∥ ψ^{ϵ} (t, t_{0}, x_{0}) ∥ < c_{2} \forall t \in [t_{0}, \infty]

Uniform ultimate boundedness: For every $c_{1} \in (0, \infty)$ , there exists $c_{2} \in (0, \infty)$ , such that for all $t_{0} \in R$ and for all $x_{0} \in R^{n}$ with $∥ x_{0} ∥ < c_{1}$ and for all $ϵ \in (0, \hat{ϵ})$

∥ ψ^{ϵ} (t, t_{0}, x_{0}) ∥ < c_{2} \forall t \in [t_{0}, \infty]

Global uniform attractivity: For all $c_{1}, c_{2} \in (0, \infty)$ , there exists $T \in (0, \infty)$ 0,∞), such that for all $t_{0} \in R$ and for all $x_{0} \in R^{n}$ with $∥ x_{0} ∥ < c_{1}$ and for all $ϵ \in (0, \hat{ϵ})$

∥ ψ^{ϵ} (t, t_{0}, x_{0}) ∥ < c_{2} \forall t \in [t_{0} + T, \infty]

With this method above, we define the stability laws of system (4) as follows:

Definition 2 (practical globally uniformly ultimate boundedness)

Uniform stability: For every $c_{2} \in (0, \infty)$ , there exists $c_{1} \in (0, \infty)$ and $\hat{ϵ} \in (0, \infty)$ , such that for all $t_{0} \in R$ and for all $x_{0} \in R^{n}$ with $∥ x_{0} ∥ < c_{1}$ and for all $ϵ \in (0, \hat{ϵ})$

∥ ψ^{ϵ} (t, t_{0}, x_{0}) ∥ < c_{2} \forall t \in [t_{0}, \infty]

Uniform ultimate boundedness: For every $c_{1} \in (0, \infty)$ , there exists $c_{2} \in (0, \infty)$ and $\hat{ϵ} \in (0, \infty)$ , such that for all $t_{0} \in R$ and for all $x_{0} \in R^{n}$ with $∥ x_{0} ∥ < c_{1}$ and for all $ϵ \in (0, \hat{ϵ})$

∥ ψ^{ϵ} (t, t_{0}, x_{0}) ∥ < c_{2} \forall t \in [t_{0}, \infty]

Global uniform attractivity: For all $c_{1}, c_{2} \in (0, \infty)$ , there exists $T \in (0, \infty)$ 0,∞) and $\hat{ϵ} \in (0, \infty)$ such that for all $t_{0} \in R$ and for all $x_{0} \in R^{n}$ with $∥ x_{0} ∥ < c_{1}$ and for all $ϵ \in (0, \hat{ϵ})$

∥ ψ^{ϵ} (t, t_{0}, x_{0}) ∥ < c_{2} \forall t \in [t_{0} + T, \infty]

Corollary 1

If functions (3) and (4) satisfy the convergence of trajectories and if the origin is a GUAS equilibrium point of equation (3), then the origin of equation (4) is $ϵ -$ PGUUB.²¹ Similarly if the origin of system (2) is GUAS, then the origin of system (1) is $\frac{1}{k} -$ PGUUB.

Problem formulation

Consider this model of 2-D mobile robot with external disturbances has the following dynamic feedback of angular velocity ω

{\begin{matrix} \dot{x} = \sqrt{α ω} cos (ω t + k V (x, y) + θ_{0}) \\ \dot{y} = \sqrt{α ω} sin (ω t + k V (x, y) + θ_{0}) \\ \dot{θ} = ω + k \frac{\partial V}{\partial t} \\ \dot{ω} = τ + d \end{matrix}

where $θ_{0}$ is an unknown initial orientation.

In this scheme, the position of the mobile robot $(x, y)$ in a global positioning system (GPS)-denied environment is changeable which is determined by the value of the unknown but measurable function $V (x, y)$ . It is not difficult to see the robot velocity can be stated as $υ = \sqrt{{\dot{x}}^{2} + {\dot{y}}^{2}} = \sqrt{α ω}$ , as shown in Figure 1. θ is the angle between the moving forward direction and the positive direction of the X-axis; specially, in this case, speed of change in direction of heading is given as $\dot{θ} = ω + k \frac{\partial V}{\partial t}$ , when the trajectory of the robot changes; if the function $V (x, y)$ increases with time and satisfy $\frac{\partial V}{\partial t} > 0$ , then the rate of change of heading direction is increase; in an approach same to this, the system will retrace back from the wrong direction to the right direction as quickly as possible and finally find the optimal value $\hat{x}, \hat{y}$ . τ is the generalized torque input. And the parameter ω represents the angular velocity of the robot, which has to do with external disturbances d.

Figure 1.

Velocity of mobile robot.

Remark 1

In order to make our analysis clearer, and due to

{sin}^{2} (k V + θ_{0}) + {cos}^{2} (k V + θ_{0}) = 1

It is established whether $θ_{0}$ exists or not; therefore, we set the arbitrary initial orientation $θ_{0}$ = 0 from now on, and it will not affect our results.

We set an arbitrary number g which is less than c ₂, such that

max_{t \in [0, T]} | x (t) - \hat{x} (t) | < g

as well as $y (t) .$

Therefore, it can be seen that the main problem studied in this article is to design a dynamic controller in an environment where GPS is not applicable and exists of unknown disturbance to achieve the purpose of ES by describing the dynamic angular velocity turning and make the following two expressions are established.

\begin{array}{l} lim_{t \to \infty} x (t) = \hat{x} (t) \\ lim_{t \to \infty} y (t) = \hat{y} (t) \end{array}

Main results

This section presents the solution to the problem of vehicle control. Firstly, τ is proposed as a dynamic controller which makes the angular velocity converges to the virtual controller in a finite time through a series of proofs. Then, on the basis of convergence, the unknown disturbance will be added and finally converge to the disturbance signal in the finite time by establishing the disturbance observer. Finally, the convergence proof of the vehicle trajectory in the finite time is completed by the averaging method.

Assumption 1

We definite that the controller error system

$Δ ω = ω - \bar{ω}$
7

The derivative of $Δ ω$ with respect to time is as follows

$\dot{Δ ω} = \dot{ω} - \dot{\bar{ω}}$
8

By combining equations (5) and (8), we can get

$\dot{Δ ω} = τ + d - \dot{\bar{ω}}$
9

The virtual controller can be designed as

$\bar{ω} = \sqrt{α_{i} ω_{i}} cos (ω_{i} + k V (x, y))$
10

Where the letter i represents the ith element of our virtual control, and in this article, we specify that i just represents the control laws of x and y directions; since the analysis in the x and the y directions is exactly consistent, we will not specifically distinguish them in the following paragraphs.

By expanding equation (10), the derivative of $\bar{ω}$ with respect to time is as follows

$\dot{\bar{ω}} = \sqrt{α ω} [- sin (ω t + k V (x, y))] \cdot (ω + k \frac{\partial V}{\partial t})$
11

And rewrite equations (9) and (11) as

$\dot{Δ ω} = τ + d - \sqrt{α ω} [- sin (ω t + k V (x, y))] \cdot (ω + k \frac{\partial V}{\partial t})$
12

Lemma 1

For a first-order system $\dot{ζ} = ϒ,$ and we state $ζ$ is a positive value function with respect to time, there exists a positive letter $ϒ_{0}$ , satisfying | $ϒ | \leq | ϒ_{0}$ |.¹⁷ For a certain number $ζ (0)$ , there always exists a positive letter $Λ$ , satisfying | $ζ (0) | \leq Λ$ . In this article, we set a Lyapunov function $ϒ = - ρ sgn (ζ) | ζ |^{δ}$ , where $ρ and ζ$ are constant and satisfy $ρ \leq \frac{ϒ_{0}}{Λ^{δ}}$ , 0< $δ < 1.$ So there is a finite time $Τ_{0}$ , satisfying $Τ_{0} = \frac{| ζ (0) |^{1 - δ}}{ρ (1 - δ)} \leq \frac{Λ^{1 - δ}}{ρ (1 - δ)}$ such that $lim_{t \to Τ} ζ = 0, ζ \equiv 0 (t \geq Τ_{0 +})$ .

By equation (12), the kinetic variable control laws τ can be designed as

$τ = \sqrt{α ω} [- sin (ω t + k V (x, y))] \cdot (ω + k \frac{\partial V}{\partial t}) - ˇ sgn (Δ ω) | Δ ω |^{ε} - d$
13

Theorem 3

Consider the controller error system (9) and the kinetic controller, the error system converges to zero in a finite time based on lemma 1. So that the angular velocity ω converges to the virtual control $\bar{ω}$ in a finite time.

Proof

Substituting control law (13) into error system (9), it can be obtained that

$\dot{Δ ω} = - ˇ sgn (Δ ω) | Δ ω |^{ε}$

Consider Lyapunov function $W = \frac{1}{2} {(Δ ω)}^{2}$ . The derivative of W with respect to time is

$W = Δ ω \dot{Δ ω} = - ˇ Δ ω sgn (Δ ω) | Δ ω |^{ε} = - ˇ | Δ ω |^{ε + 1} = - ˇ {(Δ ω^{2})}^{\frac{ε + 1}{2}} = - ˇ {(2 W)}^{\frac{ε + 1}{2}}$
14

Equation (14) can be written as

$\frac{d W}{{(W)}^{\frac{ε + 1}{2}}} = - 2^{\frac{ε + 1}{2}} ˇ d t$

Integrate both sides of this equation, we can obtain that

$\begin{array}{l} \int_{W (0)}^{W} W^{- \frac{ε + 1}{2}} d W = \int_{0}^{t} - 2^{\frac{ε + 1}{2}} ˇ d t \\ \frac{1}{1 - \frac{ε + 1}{2}} W^{1 - \frac{ε + 1}{2}} = - 2^{\frac{ε + 1}{2}} ˇ t + \frac{1}{1 - \frac{ε + 1}{2}} {(W (0))}^{1 - \frac{ε + 1}{2}} \end{array}$
15

Eventually, we can get this equation

$W^{\frac{1 - ε}{2}} = - 2^{\frac{ε - 1}{2}} ˇ (1 - ε) t + {(W (0))}^{\frac{1 - ε}{2}}$

Obviously, through a series of derivations, W can exponentially decreases to zero with time in a finite time, so $Δ ω$ can converge to zero in a finite time. Based on the Lyapunov function, the dynamic angular velocity ω of the vehicle can eventually converge to the virtual controller $\bar{ω}$ . This completes the proof.

Lemma 2

We propose a continuous positive definite and bounded function $Γ (t)$ , satisfying these two properties²³

$Γ (t_{0}) \geq 0$ .

$\dot{Γ (t)} \leq - c Γ^{μ} (t), \forall t \geq t_{0}$ .

Where t ₀ is the initial value, and 0 < $μ 〈 1, c 〉 0$ are positive constant, and $Γ (t)$ can converge to zero in finite time.

By system (5), the sliding mode surface of disturbance can be designed as

$a = Φ - ω$
16

$\dot{a} = \dot{Φ} - \dot{ω}$
17

And $Φ = \int (ξ + τ) d a$ , where the sliding mode disturbance observer ξ is designed as

$ξ = - c a - λ sgn (a) - η a^{\frac{m_{1}}{n_{1}}}$
18

$\dot{Φ} = - c a - λ sgn (a) - η a^{\frac{m_{1}}{n_{1}}} + τ$
19

Where these parameters c, $λ, η,$ η, $m_{1,} n_{1}$ satisfy the following conditions

${\begin{matrix} c > 0, \\ λ \geq ξ, \\ η > 0, \\ m_{1,} n_{1} > 0, \frac{m_{1}}{n_{1}} < 1 \end{matrix}$
20

By combining equations (5) and (17) to (19), we can get the conclusion

$\dot{a} = ξ - d = - c a - λ sgn (a) - η a^{\frac{m_{1}}{n_{1}}} - d$
21

Consider the system (5), sliding mode surface (16), and disturbance observer (18), the sliding mode disturbance observer ξ can converge to the disturbance signal d in a finite time.

Proof

With the equation of derivative of a with respect to time, substitute equation (21) into the Lyapunov function $Γ (a) = \frac{1}{2} a^{2}$

$\dot{Γ (a)} = a \dot{a} = - c a^{2} - λ | a | - η a^{\frac{m_{1}}{n_{1}} + 1} - a d$
22

According to those parameters in equation (20), equation (22) can be rewritten as

$\dot{Γ (a)} \leq - c a^{2} - η a^{\frac{m_{1}}{n_{1}} + 1} \leq - 2 c Γ (a) - 2^{\frac{m_{1} + n_{1}}{2 n_{1}}} η Γ {(a)}^{\frac{m_{1} + n_{1}}{2 n_{1}}}$
23

Based on equation (23) and lemma 2, it can be proved that the sliding mode surface can converge to zero in finite time, easily confirm the disturbance observer converge to unknown disturbance in finite time by equation (17).

By lemma 2 and theorem 3, we replace the external disturbance signal with disturbance observer to better illustrate the convergence of the system and the system (13) can be ultimately transformed as

$τ = \sqrt{α ω} [- sin (ω t + k V (x, y))] \cdot (ω + k \frac{\partial V}{\partial t}) - ˇ sgn (Δ ω) | Δ ω |^{ε} - [- c a - λ sgn (a) - η a^{\frac{m_{1}}{n_{1}}}]$

After the above proofs, we have clearly studied the dynamic angular velocity turning problem of 2-D vehicles with disturbance, in order to better illustrate the movement of the mobile robot in the case of unknown orientation; the process of trajectory optimization is described concisely and clearly in Figure 2.

Figure 2.
ES control scheme. ES: extremum seeking.

And we assume a GPS-denied environment, then set up a 2-D robot that has an unknown initial direction. The goal of the robot is to find the minimum or maximum value of the unknown but measurable function $V (x, y)$ that we’ve set.

Lemma 3

By theorems 1 and 2, we state that for all T ∈ (0, $\infty$ ), and any compact set K $\in R^{n}$ , the functions $f (x, t)$ , $f_{i} (x, t)$ satisfy these two theorems; as a result, for any g, ι >0, the trajectory $x (t)$ of the mobile robot is²¹

$\dot{x} = f (x, t) + \sum_{i = 1}^{n} f_{i} (x, t) {(k k_{i})}^{ι} cos (k k_{i})^{2 ι} t - \sum_{i = 1}^{n} g_{i} (x, t) {(k k_{i})}^{ι} sin (k k_{i})^{2 ι} t$

And the trajectory $\hat{x} (t)$ of the robot is

$\hat{x} (t) = f (\hat{x}, t) - \frac{1}{2} \sum_{i \neq j}^{n} [f_{i} (\hat{x}, t), g_{j} (\hat{x}, t)], \hat{x} (0) = x (0) \in K$

With the function (6), the Lie bracket of the functions f_i and g_j is

$[f_{i} (\hat{x}, t), g_{j} (\hat{x}, t)] = \frac{\partial g_{j}}{\partial \hat{x}} f_{i} - \frac{\partial f_{i}}{\partial \hat{x}} g_{j}$

Remark 2

If the function $V (x, y)$ has a global minimum or maximum at ( $x^{Δ}, y^{Δ})$ , conform to that

$\nabla V |_{(x^{Δ}, y^{Δ})} = 0, \nabla V \neq 0, \forall (x, y) \neq (x^{Δ}, y^{Δ})$

So that for any $c_{2} > 0$ , just pick an infinite large choice of $k α$ , and then this point $(x^{Δ}, y^{Δ})$ is going to fit $\frac{1}{ω} -$ PGUUB, as shown in Figure 2

$\begin{array}{l} \dot{x} = \sqrt{α ω} cos (ω t + k V (x, y)) \\ \dot{y} = \sqrt{α ω} sin (ω t + k V (x, y)) \end{array}$
24

And we expand

$\begin{array}{l} cos (ω t + k V) = cos (ω t) cos (k V) - sin (ω t) sin (k V) \\ sin (ω t + k V) = sin (ω t) cos (k V) + sin (k V) cos (ω t) \end{array}$

So equation (24) can be stated as follows

$[\dot{\begin{matrix} x \\ \dot{y} \end{matrix}}] = \sqrt{ω} cos (ω t) [\begin{matrix} \sqrt{α} cos (k V) \\ \sqrt{α} sin (k V) \end{matrix}] + \sqrt{ω} sin (ω t) [\begin{matrix} - \sqrt{α} sin (k V) \\ \sqrt{α} cos (k V) \end{matrix}]$

By theorem 1, the averaging method can be expanded that

$[\begin{matrix} \dot{\hat{x}} \\ \dot{\hat{y}} \end{matrix}] = \frac{α}{2} D ([\begin{matrix} - sin (k V) \\ cos (k V) \end{matrix}]) [\begin{matrix} cos (k V) \\ sin (k V) \end{matrix}] - \frac{α}{2} D ([\begin{matrix} cos (k V) \\ sin (k V) \end{matrix}]) [\begin{matrix} - sin (k V) \\ cos (k V) \end{matrix}]$

By simplifying the right-hand side, we can obtain

$\begin{array}{l} = \frac{k α}{2} [\begin{matrix} - \frac{\partial V}{\partial \hat{x}} cos (k V) & - \frac{\partial V}{\partial \hat{y}} cos (k V) \\ - \frac{\partial V}{\partial \hat{x}} sin (k V) & - \frac{\partial V}{\partial \hat{y}} sin (k V) \end{matrix}] [\begin{matrix} cos (k V) \\ sin (k V) \end{matrix}] \\ - \frac{k α}{2} [\begin{matrix} - \frac{\partial V}{\partial \hat{x}} sin (k V) & - \frac{\partial V}{\partial \hat{y}} sin (k V) \\ \frac{\partial V}{\partial \hat{x}} cos (k V) & \frac{\partial V}{\partial \hat{y}} cos (k V) \end{matrix}] [\begin{matrix} - sin (k V) \\ cos (k V) \end{matrix}] \end{array}$

We have

$\begin{array}{l} \dot{\hat{x}} = - \frac{k α}{2} (\frac{\partial V}{\partial \hat{x}} {cos}^{2} (k V) + \frac{\partial V}{\partial \hat{x}} {sin}^{2} (k V)) \\ \dot{\hat{y}} = - \frac{k α}{2} (\frac{\partial V}{\partial \hat{y}} {sin}^{2} (k V) + \frac{\partial V}{\partial \hat{y}} {cos}^{2} (k V)) \end{array}$
25

Apply the formula ${sin}^{2} (k V + θ_{0}) + {cos}^{2} (k V + θ_{0}) = 1$ to equation (25). Ultimately, we arrive at the average system dynamics

$[\begin{matrix} \dot{\hat{x}} \\ \dot{\hat{y}} \end{matrix}] = - \frac{k α}{2} {(\nabla V (\hat{x}, \hat{y}))}^{T}$

Therefore, the trajectory $x (t), y (t)$ of the system (24) can converge to the trajectory $\hat{x} (t), \hat{y} (t)$ ultimately by expanding theorem 1. Then for any $g > 0$ , there always exist arbitrarily large values $k α$ that can make a g neighborhood of $(x (t), y (t))$ with $(\hat{x} (t), \hat{y} (t))$ . And the inequality (6) is true. Above are all proofs.

Simulations

In this part, the finite-time theory and the virtual controller tracking method will be proved by MATLAB simulation in the case of an unknown robot with external disturbance. Our simulations will be based on the following system

$\begin{array}{l} \dot{x} = \sqrt{α ω} cos (ω t + k V (x, y)) \\ \dot{y} = \sqrt{α ω} sin (ω t + k V (x, y)) \end{array}$

and we assume that the initial position of the robot is (1, −1), and the function $V = x^{2} + y^{2}$ , the goal is to seek the optimal station for the robot. And the parameters are set as $α = \frac{1}{3}, k = 2,$ and $ω = 30$ . According to theorem 3, we design $ε = \frac{1}{2}$ , in order to demonstrate the convergence of $Γ (a)$ , some parameters are stated $n_{1} = 2, m_{1} = 1, c = 1, η = 2$ by lemma 2.

The simulation results are shown as follows. In Figures 3 and 4, showing the trajectory of the system $(x, y)$ and the averaging system $(\hat{x}, \hat{y})$ , it is easy to see that the convergence of the trajectory is proved, so that $(x, y)$ will ultimately converge to $(\hat{x}, \hat{y})$ . In Figure 5, it can be observed that $Δ ω$ can converge to zero in a finite time. Based on the Lyapunov function, the dynamic angular velocity ω of the vehicle can eventually converge to the virtual controller $\bar{ω}$ . In Figure 6, we can obtain that $Γ (a)$ can converge to zero in finite time, easily confirm the disturbance observer converge to unknown disturbance in finite time. Combined with formula (6), we know that our control goal is to make $lim_{t \to \infty} x (t) = \hat{x} (t)$ and $lim_{t \to \infty} y (t) = \hat{y} (t)$ , so when we analyze the convergence in the x or y directions separately, in Figures 7 and 8, we can see that the robot trajectory can converge to the average state we set in a limited time.

Figure 3.
The trajectory of the robot and averaging system.

Figure 4.
The trajectory of x direction.

Figure 5.
The controller error system.

Figure 6.
The difference with observer and disturbance signal.

Figure 7.
System trajectory and average state of x.

Figure 8.
System trajectory and average state of y.

Conclusion

This article studies the problem of dynamic angular velocity turning for ESC of a 2-D mobile robot with external disturbances through a method of averaging and a virtual controller is designed with a torque or force as inputs as well as an unknown disturbance. The virtual controller makes the tracking error system can converge to zero in finite time. And the trajectory $(x, y)$ will ultimately converge to $(\hat{x}, \hat{y})$ and achieve the goal of ES. In the future, this method will stand a good chance to apply to polar coordinates or adaptive delay control scheme.

In addition, although the theory of this article has not been realized in practice, after studying the control characteristics of unknown systems in depth, it is found that the robust control methods applied to uncertain robots mainly include adaptive control, variable structure control, modern Lu. Rod control, and so on; these can well solve the uncertainty of unknown system parameters and external disturbances.

Wang et al.²⁵ proposed an adaptive delay control scheme, which is a scheme that is easy to apply to the actual situation,²⁶ and the time delay estimation (TDE)-based control method can be well applied to the movement of this article’s robot; this is a control method that uses an intentional time delay signal to estimate the dynamics of an unknown system.²⁷ When a 2-D robot is subjected to an unknown external disturbance, it uses TDE to obtain an estimate of the system dynamics, so no system dynamic model information is needed.^28,29 We will combine this method to improve the theory of this article in subsequent research.

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: This article was supported by the Natural Science Foundation of China (61304004, 61503205) and the Fundamental Research Funds for the Central Universities (2019B40114).

ORCID iD

Hua Chen

References

Krstić

Wang

. Stability of extremum seeking feedback for general nonlinear dynamic systems. Automatica 2000; 36(4): 595–601.

Maury

Koenig

Cattafesta

. Extremum-seeking control of jet noise. Int J Aeroacoust 2012; 11(3-4): 459–473.

Beaudoin

Cadot

Aider

. Bluff-body drag reduction by extremum-seeking control. J Fluid Struct 2006; 22(6-7): 973–978.

Leyva

Alonso

Queinnec

. MPPT of photovoltaic systems using extremum-seeking control. IEEE Trans Aero Elec Sys 2006, 42(1): 249–258.

Rotea

Chiu

GTC

. Extremum seeking control of a tunable thermoacoustic cooler. IEEE T Contr Syst T 2005; 13(4): 527–536.

Kutadinata

Moase

Manzie

. Extremum-seeking for adaptation of urban traffic signal control. IFAC Proceedings Volumes 2014; 47(3): 5067–5072.

Martín

Fabricio

Hernán

. Smooth extremum-seeking control for fed-batch processes. IFAC-PapersOnLine 2016; 49(7): 103–108.

Arnold

Negrete-Pincetic

Stewart

. Extremum seeking control of smart inverters for VAR compensation. In: 2015 IEEE Power & Energy Society General Meeting, 2015, pp. 1–5. IEEE. https://ieeexplore.ieee.org/abstract/document/7286378

Zhang

Dawson

Dixon

. Extremum-seeking nonlinear controllers for a human exercise machine. IEEE/ASME Trans Mech 2006; 11(2): 233–240.

10.

Zhang

Ordonez

. Numerical optimization-based extremum seeking control with application to ABS design. IEEE Trans Automat Contr 2007; 52(3): 454–467.

11.

Roy

Pratihar

. Kinematics, dynamics and power consumption analyses for turning motion of a six-legged robot. J Intell Robot Syst 2014; 74(3-4): 663–688.

12.

Gurvits

. Smooth time-periodic solutions for non-holonomic motion planning. Nonholonomic Motion Planning, Kluwer, 1992.

13.

Moreau

Aeyels

. Practical stability and stabilization. IEEE Trans Automat Contr 2000; 45(8): 1554–1558.

14.

Dürr

Stanković

Ebenbauer

. Lie bracket approximation of extremum seeking systems. Automatica 2013; 49(6): 1538–1552.

15.

Guay

Dochain

Perrier

. Extremum-seeking control over periodic orbits. IFAC Proc Volume 2005; 38(1): 206–211.

16.

Sanders

Verhulst

Murdock

. Averaging methods in nonlinear dynamical systems. New York: Springer, 2007. http://doi.org/10.1155/2018/7360643

17.

Chen

Sun

. Robust stabilization of extended nonholonomic chained-form systems with dynamic nonlinear uncertain terms by using active disturbance rejection control[J]. Complexity, 2019. DOI: 10.1155/2019/1365134.

18.

Chen

Chu

. Finite-time switching control of nonholonomic mobile robots for moving target tracking based on polar coordinates. Complexity 2018. https://ieeexplore.ieee.org/abstract/document/7286378

19.

Matsuzaki

Kameda

Tsujimichi

. Maneuvering target tracking using constant velocity and constant angular velocity model. In: SMC 2000 Conference Proceedings. 2000 IEEE International Conference on Systems, Man and Cybernetics. ‘Cybernetics Evolving to Systems, Humans, Organizations, and Their Complex Interactions’(CAT. NO. 0), Vol. 5, 2000, pp. 3230–3234. IEEE.

20.

Bayat

Mobayen

Javadi

. Finite-time tracking control of nth-order chained-form non-holonomic systems in the presence of disturbances. ISA Trans 2016; 63: 78–83.

21.

Scheinker

Krstić

. Extremum seeking with bounded update rates. Syst Control Lett 2014; 63: 25–31.

22.

Kurzweil

Jarník

. Limit processes in ordinary differential equations. Zeitschrift für angewandte Mathematik und Physik ZAMP 1987; 38(2): 241–256.

23.

Mobayen

Tchier

. Design of an adaptive chattering avoidance global sliding mode tracker for uncertain non-linear time-varying systems. Trans Instit Meas Control 2017; 39(10): 1547–1558.

24.

Sussmann

. New differential geometric methods in nonholonomic path finding. Systems, models and feedback: theory and applications. Boston: Birkhäuser, 1992, pp. 365–384.

25.

Wang

Yan

Chen

. A new adaptive time-delay control scheme for cable-driven manipulators. IEEE Trans Ind Inform 2018.

26.

Wang

Yan

Jiang

. Time delay control of cable-driven manipulators with adaptive fractional-order nonsingular terminal sliding mode. Adv Eng Softw 2018; 121: 13–25.

27.

Wang

Jiang

Chen

. A new continuous fractional-order nonsingular terminal sliding mode control for cable-driven manipulators. Adv Eng Softw 2018; 119: 21–29.

28.

Wang

Yan

. Practical adaptive fractional-order nonsingular terminal sliding mode control for a cable-driven manipulator. Int J Robust Nonlin Control 2019; 29(5): 1396–1417.

29.

Wang

Chen

Yan

. Adaptive super-twisting fractional-order nonsingular terminal sliding mode control of cable-driven manipulators. ISA Trans 2019; 86: 163–180.

Dynamic angular velocity turning for extremum seeking control of a two-dimensional mobile robot with external disturbances

Abstract

Keywords

Introduction

Averaging and stability results and problem formulation

Averaging results of Sanders et al.

Theorem 1

Stability results

Definition 1 (convergence of trajectories)

Theorem 2 (GUAS)

Definition 2 (practical globally uniformly ultimate boundedness)

Corollary 1

Problem formulation

Remark 1

Main results

Assumption 1

Lemma 1

Theorem 3

Proof

Lemma 2

Proof

Lemma 3

Remark 2

Simulations

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References