Adaptive Controller Design for Nonlinear Systems with Time-Varying State Delays
M.Hashemi*, J. Askari, J.Ghaisari and M. Kamali, Member, IEEE Department of Electrical and Computer Engineering, Isfahan University of Technology
Isfahan, 84156-83111, Iran E-mail*: [email protected]
Abstract-This paper focuses on the problem of adaptive tracking control for a class of multiple delays nonlinear systems in strict- feedback form. Time delays are considered to be unknown and time-varying in this system. With the help of a Backstepping design method, a state feedback controller is designed, which does not need the precise knowledge of the time delays. The proposed adaptive controller guarantees that all the closed loop signals are bounded and the system output converges to a small neighborhood of the reference signal. Simulation results are provided to show the effectiveness of the proposed approach.
Keywords: Adaptive control, Nonlinear time-delay systems, Time-varying delay, Backstepping design method.
I. INTRODUCTION
Time delays are frequently encountered in mechanics, physics, applied mathematics, biology and engineering systems.
Because of the effect of time delays, these systems may own instability or poor performance [1]-[2]. One major difficulty lies in the control of time delay systems is that the delays are usually not perfectly known. So far, many researches have been done on time delay systems [3]-[17]. For linear systems with time delays, many results have been achieved in the recent years, which in most of them the controller is designed based on the Lyapunov- Krasovskii functional method and the Razumikhian lemma [2]- [6].
For nonlinear systems with time delays, the system can be linearized at its equilibrium point and then the controller is designed based on the proposed methods for linear time delay systems. However, in this method, the designed controller is only locally effective. The disturbances in the engineering systems may make the system run in a large area; thus, the local controller is not feasible any more.
The backstepping method has proved to be a powerful tool for the controller design of nonlinear systems with strict- feedback form [7]. Many practical industrial systems are found to be in the strict-feedback form. In [8] the state feedback
controller is designed for piezoactuator driven stages via backstepping method.The output feedback backstepping control method is proposed in [9] for the magnetic levitation system.
The adaptive control problem for strict-feedback nonlinear time delay systems is considered in [10]-[15] and they just proved that all the signals in the system are uniformly ultimately bounded. In [16], [17] the adaptive tracking control problem is investigated for a class of nonlinear time delay systems where each state permits to have only one constant delay. The authors proved that all the signals in the closed loop system are bounded and the system output converges to a small neighborhood of the reference signal. In [16], adaptive neural control is presented for a class of strict-feedback nonlinear systems and neural networks are used to approximate the unknown nonlinear functions.
Adaptive backstepping is compared with other control methods in [7], [10] and concluded that the advantages of backstepping design method lie in its flexibility, due to its recursive use of Lyapunov functions and its robustness against unmodelled dynamic of the system.
In this paper, the adaptive tracking problem is presented for parametric strict-feedback nonlinear systems with unknown parameters and multiple unknown time-varying state delays. The adaptive controller is designed based on the backstepping design method. The proposed controller proves that not only all the signals in the closed loop system are bounded, but also the tracking error converges to a small neighborhood of the origin.
The paper is organized as follows. In section 2, the system description is given along with the necessary assumptions. In section 3, the adaptive controller is designed based on the backstepping approach. In section 4, simulation example is studied to illustrate the effectiveness of the proposed control scheme and finally the paper is concluded in section 5.
II. PROBLEM STATEMENT
Consider a class of strict-feedback nonlinear time delay systems of the form
The 22nd Iranian Conference on Electrical Engineering (ICEE 2014), May 20-22, 2014, Shahid Beheshti University
, , , 1 1
, ,
(1) where , , … , , , are the state variables, system input and output respectively. . and , . are known smooth function vectors, and , are unknown constant parameter vectors and , 1, … , are unknown time-varying state delays.
The control objective is to design a state feedback controller for plant (1), without knowledge of plant parameters and delays in order to assure that all the closed loop signals are bounded and the plant output y(t) tracks a reference signal . For this purpose, the following assumptions are considered.
Assumption 1 The reference signal and its first n-th order derivatives 1, … , are known, bounded, and piecewise continuous.
Assumption 2 The unknown time-varying delays , are differentiable functions satisfying
0 , 1, 1 (2)
where and are known positive constants.
III. CONTROLLER DESIGN
In this section, the procedure of designing a controller for the time delay system (1) is explained. Consider the state transformation
. . .
(3)
for system (1). The transformed system is obtained as
, ,
, ,
1
, ,
(4) First, for the subsystem, the Lyapunov function is considered as (5).
1
2 1 , ,
1
2 (5)
where and is the estimate of that will be defined later.
Along system (4), the time derivative of and satisfies
, ,
(6) 1
2 1 , ,
1
2 1 ,
,
(7) Note that based on assumption 2, and 1 , then it follows that
1
2 1 , ,
1
2 , ,
(8) And using Young’s inequality [17], the time derivative of becomes
1
2 , ,
1
2 ,
,
(9) Accordingly, is
, ,
(10) where
, , ,
,1 2
(11)
According to (10), the virtual control input is selected as:
1 1 2
2 1 , ,
(12)
However, it can be seen that at 0, controller singularity will be occurred because the virtual controller is not well-defined at this point. Therefore, the designed virtual controller should be modified at this point. Using function which is defined in lemma1, the problem of singularity in the controllers will be solved.
Lemma 1 [16]: let the function : is defined as, 1 | |
, ,
0, | |
(13)
In which is a positive integer number, constant parameters
, 0 and and are
, , Where
2 1 !
!
then is (n-i)-th differentiable and limited between 0 and 1. Therefore, the virtual controller is rewritten as
1 1 2
2 1 , ,
(14)
and the update law for parameter is selected as
(15)
where 0 is a small positive constant.
In order to clarify, the following sets are considered:
Region 1: | | |
Region2: | | |
Region3: || |
Case 1) For region 1; | | , 1 and the time derivative of becomes
1 2
1
2 (16)
and the update law (15) becomes
(17)
Using the inequality [17],
the time derivative of becomes 1
2
min , , 1
2
(18)
In this region, the time derivative of is dependent on , therefore the boundedness of (t) will be proved in next step when the boundedness of is verified.
Case 2) For region 2; | | , from the boundedness of , it can be deduced that and are bounded. Using (15) and (5), the time derivative of in this region becomes
(19) Applying the inequality
1 2 ′
′
2 , ′ 0
(20) The equation (19) is rewritten as
1 2
1
′
1
2 ′ .
(21) In this region, is a smooth and bounded function and
0,1 . Choosing ′ such that ′ 0, 1
2
1
2 ′
′ (22)
Since | | , then
0 0 (23)
It can be concluded that is bounded too. Eventually is
bounded for | | .
Case 3) For region 3; | | , In this region, is bounded and according to assumption (1), is bounded too. Therefore, is also bounded. In this region, 0 , hence from equation (14)-(15) it can be concluded that 0, 0 and is kept unchanged in bounded values; thus is bounded. For bounded and , it can be concluded that
and are bounded, therefore is bounded.
As can be seen, the singularity problem of the virtual control input (12) are solved. From the stability analysis in the three regions, it can be deduced that the boundedness of all the closed loop signals in region2 and region 3 is guaranteed and independent of signal , but the boundedness of the closed loop signals in region 1 is dependent on the boundedness of the signal
, that will be regulated in the following.
Now, consider the subsystems for 2 :
, ,
(24)
For subsystems, the Lyapunov functions
1
2 1 , ,
1
2 (25) are considered where and is the estimate of
, 1, … , , that will be defined later.
The time derivative of and along subsystem (24) becomes
∑ , ,
(26) 1
2 1 , ,
1
, , (27)
Based on assumption2, and 1 ,
then it follows that 1
2 1 , ,
1
2 ,
,
(28)
Note that the virtual controllers for 1 are functions of , , … , , , … , . Therefore, the time derivative of using Young’s inequality [16], becomes
∑ ∑
∑
∑ ∑ , , (29)
where the unknown parameter , 2, … , , are defined as
, , ,, , , , … , , , ,
, , , , … , , , , (30)
, , , … , , , (31)
2 , … , ,
2
Therefore, the adaptive control laws and virtual controllers are selected as
, 1 (32)
1 2
1 1
2 1 , ,
(33)
Similar as the previous section, the stability analysis are carried out for 3 defined regions.
Case 1) For region 1; | | , Using the inequality [17], the time derivative of with 1 becomes
1
min , 2,
(34)
It can be seen from the above inequality that the stability of subsystems in this region is dependent on .
Case 2) For region 2; | | , from the boundedness of , it can be concluded that and are bounded. Also the boundedness of can be used for previous subsystems which their stabilities are dependent on the boundedness of . Thus all , , … , signals are bounded.
Accordingly all the , , … , , signals and the virtual controllers , , … , are bounded. In addition, the boundedness of can be obtained from the similar analysis carried out in region 2 for subsystem. Therefore is
bounded for | | .
Case 3) For region 3; | | , In this region, is bounded and according to the boundedness of , signals , , … , are bounded. Thus all the , , … , , signals and the virtual controllers , , … , are bounded. In this region 0 , consequently from equation (32)-(33), it can be
concluded that 0, 0 and 0. Hence
is kept unchanged in bounded values, therefore is bounded. Since and are bounded, and are bounded;
consequently, is also bounded.
Now the following theorem is stated.
Theorem 1. Consider the closed loop system (1). Under assumption 1-2, the controller structure (33) with the parameter update law (32), assure that: all the signals of the closed loop system are globally bounded and the signal , , … , converges to the compact set
| . (35)
max 2 ,
∑ ,
Proof: The following Lyapunov function is considered.
(36)
Case 1) For region 1; | | , 1, … . , , the time derivative of is
,
min , , … , , 1
2
(37)
0 (38)
Consequently
2 0 (39)
Eventually
: | | lim
∞
max 2 , .
(40)
Thus it is concluded that (t) is bounded;accordingly, all the closed loop signals are bounded.
Case 2) For region 2; | | , 1, … , , the boundedness of , 1, … , , give the result that , , , , … , and , , … , , are bounded. Furthermore, the boundedness of can be obtained from the similar analysis carried out in region 2 for subsystem. Therefore
is bounded for | | and
: | |
(41)
Case 3) For region 3; | | , 1,2, … , , all ’s are
bounded and 0, thus 0, 0,
0 and all ’s are kept fixed in bounded values and all ’s 1,2, … , are bounded also.
: | | (42)
Case4) Some ’s are satisfying | | , some of them are satisfying | | and some of them are
satisfying | | . For | | , 1, and the
time derivative of ) becomes:
1 2
min , , 1
2
(43)
If | | , the problem becomes similar to case1.
If | | or | | , both and
are bounded and the boundedness of all the closed loop signals in step and 1 will be guaranteed.
IV. SIMULATION RESULTS
In this section, the obtained results are simulated to verify the effectiveness of the proposed method. For this purpose, the following nonlinear plant is considered.
,
u t ,
,
(44) where , , , , , and , are unknown parameters.
The control objective is to track the reference signal 0.1sin 0.1 .
For simulation purpose, the plant parameters 0.3,
, 1, 0.1, , 0.3, , 0.2 and the unknown
time delays 0.5 cos 0.5, 0.9 sin 1
are selected. The simulation parameters are adjusted as:
0 , 0 0.1,0.1 , 1, 2, 3 ,
0.4 , 0.05, 0
0.5,0.5 , 0
0.5; 0.5; 0.5; 0.5; 0.5 , 1, 0.5,
2, 0.9, 0.001,
0.01.
Simulation results are shown in fig. 1 and fig. 2.
Fig.1. (a) The plant and reference signal outputs y(t) and . (b) The tracking error.
Fig. 2. (a) Control input .
(b) Parameter estimates: (Solid), (Dashed).
V. CONCLUSION
In this paper, the adaptive tracking control is presented for the parametric strict-feedback nonlinear system with multiple unknown time-varying delays in states. The proposed method is based on the backstepping method. It can guarantee global boundedness of all the closed loop signals, in addition to convergence of the tracking error of the system to a small neighborhood of the origin. Simulation results have been conducted to verify the effectiveness of the proposed control method.
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-0.2 0 0.2
y,yd
y (a) yd
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-0.02 0
0.02 (b)
Tracking Error
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20 (a)
Control Input
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