3.1. Fixed-Time Adaptive Event-Triggered Guaranteed Performance Controller Design
First, the transformation errors are defined as:
where
and
both satisfy the definition from (
16).
The error system is constructed as:
where
and
represent the virtual controllers of
v and
, respectively.
Moreover, for the purpose of handling asymmetric barrier constraints, the function
is defined as:
Step 1. According to Equations (
3), (
6), (
7), and (
11), the derivatives of
and
can be expressed as follows:
where
.
The ABLFs
and
are selected as follows:
where
represent the designed parameters satisfying
, respectively. It should be noted that the inequalities
and
hold when
and
are bounded, respectively.
By combining (
26), (
28), and (
29), the derivatives of
and
can be given by:
where
and
, which are designed for simplifying the equations.
Then, construct the following virtual controllers:
where
and
are positive-definite gain constants.
Remark 2. According to Lemma 1, to ensure fixed-time convergence of the system, the value range of the designed parameter ξ is . However, ξ must also meet the following inequalities and to prevent the singularity of the terms with ξ raised to the power, such as and . Therefore, the value of ξ must be selected in the range .
By considering (
32) and (
33), new expressions of
and
, respectively, can be obtained as follows:
Step 2. Aimed at conserving the information interchanging resources, the relative threshold ETM is introduced into the channel from the controller to the actuator. The event-triggered control law is designed as follows:
where
,
,
,
;
represents the measurement error;
represents the intermediate control law, which would be designed later;
denotes the triggering moment, and
. The diagram that illustrates how ETM works in the control framework is shown in
Figure 2.
Remark 3. It can be surely acknowledged that it is more practical and beneficial to place the ETM in the sensor-to-controller channel, which could both help alleviate the communication burden and reduce computation cost. However, by utilizing the aforementioned method, it would render data discrete and lead to challenges for handling differential problems in the controller design process. Therefore, one focus spot of our future research would be to construct an ETM implemented in both sensor-to-controller and controller-to-actuator channels.
From the first equation of (
36) and the definition of
, the following inequality can be obtained:
where
and
.
According to Equations (
36) and (
37) and Lemma 2, it can be obtained as follows:
Then, by considering (
3) and (
26), it can be deduced that:
where
and
. Both
and
are matrices with unknown parameters.
In this step, the ABLF is chosen as:
where
represents the error between the ideal unknown matrix
and its estimated value
;
denotes the inverse matrix of
, which is a
symmetric positive-definite gain matrix.
Differentiate Equation (
42) by substituting from (
38) to (
41), we obtain:
The parameter adaptive laws are defined as follows:
According to Lemma 3, it is true that:
From Lemma 4, by letting
,
,
,
,
, one has:
where
.
In the similar way, the following inequality is obtained:
Substituting inequalities from (
46) to (
49) into (
43), one has:
where
.
Next, for the following equation:
By utilizing Lemma 3, one has the following inequality:
where
.
Substitute inequalities (
52) and (
53) into (
51), one gets:
where
.
Further, for
,
where
,
is the maximum eigenvalue of
. Similarly, we obtain:
where
, and
is the maximum eigenvalue of
.
By substituting inequalities (
55) and (
56) into (
50),
can be written as:
where
.
Now, by inserting (
34) and (
35) into (
57), one has:
Subsequently, the intermediate control laws are constructed as follows:
Obviously, it can be deduced that:
For convenience, it can be rewritten as:
where
,
,
,
,
,
, , , , , , , , , and , ; was defined before.
Based on (
63) and Lemma 5, it holds that:
where
,
,
.
3.2. Stability Analysis
Theorem 1. Consider the NMR system (1); by constructing virtual controllers (32), (33) and using parameter adaptive law (44), (45), intermediate control law (59)–(61), and ETM (36), it can be achieved that: - (1)
All the signals are bounded, and the tracking error system (11) converges to prescribed regions and within a bounded settling time of the system. - (2)
Zeno phenomenon will not present.
Proof of Theorem 1 (1). From Lemma 1 and Equation (
64), it can be concluded that the convergence time of the system (
1) is practical fixed-time, and satisfies:
where
.
Furthermore, it can be obtained that the trajectories of transformation errors
and
satisfy:
From (
29) and (
42), the ABLFs are chosen as:
It can be obviously deduced that the last four items on the right side of function
V are entirely semi-positive. Combining from (
66) to (
68), the following inequalities hold:
Case 1. If
, (
71) and (
72) can be simplified as follows:
According to the relationship between
and
, we can obtain that
. Thus, the following equations can be deduced.
It is known that
holds. Therefore, by a simplification operation, it can be obtained as follows:
Case 2. If
, it is similar to the deduction in Case 1. However, it is worth mentioning that, here,
becomes negative. Thus, the consequence in Case 2 is presented as follows:
By further deduction, it yields:
where
.
Thus, combining with (
16), the following inequalities can be obtained:
According to the definition of
V, we can obtain that all the errors
,
,
,
,
, and
are bounded. As
and
are bounded variables, then
, and one can obtain that
is bounded. In the same way, we can get that
is also bounded. Additionally, according to the definition of
and
, it can be deduced that
and
are bounded. Last but not least, the boundness of
,
,
, and
are yielded from (
36), (
59), (
60), and (
61).
Ultimately, by utilizing the proposed method, it can be proved that all the signals are bounded, and the tracking error system (
11) converges to prescribed regions
and
within a bounded settling time of the system. □
Proof of Theorem 1 (2). According to (
36), the derivative of
satisfies the following inequality:
where
represents the upper boundary of
|
|.
As
and
, the following inequality can be obtained:
Further, it can be seen that:
That is, it can be deduced that there exists the minimal interval triggered time between any two continuous triggering to prevent Zeno behavior from happening. Furthermore, increasing the value of the specified parameter gi could help extend the minimal interval triggered time. Additionally, the more the value of the designed parameter σi increases, the more communication resources could be reserved. □
At this point, the proof process is completed.
The proposed theorem has been proved. With the proposed adaptive event-triggered tracking control method, it can be deduced that the NMR system, which has asymmetric relative distance and bearing angle constraints, is capable of converging with guaranteed performance within a bounded settling time of the system.
Remark 4. On the foundation of the aforementioned analysis, it can be deduced that the upper bound of convergence time of the system can be defined, which is independent of different system’s initial conditions. Furthermore, it can be also obtained that by choosing different values of ξ, , , and Θ, different upper bounds of the convergence time can be selected. Then, and can be defined by the designed parameters such as , and .