一类非线性系统的自适应时延观测器设计(英文)
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can be obtained
X = AX + Bf ( X , t , u) , y = CX ,
(2)
where X = ( x1 , x2 , …, xn) T.
Suppose that system (2) is bounded- input and bound2
ed-state (BIBS) stable , let τ be a positive cons2
the estimated state error equation can be written
·
X = AG X + B (ε(τ) - Ksgn ( y) ) , y = CX . (9)
Choose the Lyapunov candidate as
V = X T PX .
(10)
Leabharlann Baidu
Differentiate V along with Eq. (9) , we obtain
| ε(τ) | ≤ K1τ,
(6)
where K is a positive constant .
3 M ai n r es ults
We now present the time- delay observer for system
(2) as
·
X^ = AX^ +B ( f ( X , t , u) + Ksgn ( y - CX^ ) ) + G( y - CX^ ) ,
f oll ows
Dτi y ( t) = τ- 1 ( Dτi - 1y ( t) - Dτi - 1y ( t - τ) ) =
i
∑ τ- i ( - 1) jCji y ( t - τj ) , i = 2 , …, n - 1 , j =0
(4) where Cji represents a number of combinations of selec2
·
·
V = X T PX + X T PX =
X
T
(
A
T G
P
+
PA G)
X
+2 (ε(τ)
-
Ksgn
( y T) ) B T PX =
- X T QX + 2 (εT(τ) - Ksgn ( y T) ) y .
(11)
Using (8) and (9) , we have V = - X T QX + 2 (εT (τ) - Ksgn ( y T) ) y ≤
Ke y wor ds : nonlinear system ; difference approach ; delayed measurement ; state observer ; sliding mode CL C n u m be r : TP273 Doc u me nt code : A
tions j from total i .
For a sufficiently small τ, the following approxima2
tion formulations are valid.
xi = xi - 1 ≈ Dτi - 1y ( t) , i = 2 , …, n.
(5)
Thus , the function f ( X , t , u) can be written as :
methods [1] , sliding- mode control [2 ] , and adaptive con2 trol [3] . Unfortunately , some states are often very diffi 2
cult to measure , even if they are , the increased number
Received date :2001 - 09 - 10 ; Revised date :2002 - 09 - 09. Foundation item :supported by the National Natural Science Foundation of China (60174019) .
fectiveness of the proposed observers .
2 Pr o ble m s t a t e m e nt
Let us consider a nonlinear SISO systems in the form
of the following differential equation
x ( n) = f ( x , x (1) , …, x ( n - 1) , t , u) ,
(1)
where x , u ∈R denote the output and the input , respec2
tively , x ( i) ( i = 1 ,2 , …, n) stand for the ith- order
关键词 : 非线性系统 ; 差分方法 ; 延迟测量 ; 状态观测器 ; 滑模
1 I nt r o d uctio n
Several popular control techniques for nonlinear plants
requires state feedback , such as feedback linearization
derivative of the output x , f ( x , x (1) , …, x ( n- 1) , t , u) is
smooth nonlinear function.
Given xi +1 = x ( i) , the following state variables form
一类非线性系统的自适应时延观测器设计
向峥嵘1 , 朱瑞军2 , 胡维礼1
(1. 南京理工大学 自动化系 , 江苏 南京 210094 ; 21 大连理工大学 自动化系 , 辽宁 大连 116023) 摘要 : 利用输出差分及滑模方法研究一类非线性系统的自适应时延状态观测器设计问题. 首先将系统输出的 微分及高阶微分用输出和其延迟测量值的差商来表示 , 然后在此基础上构造了一种时延状态观测器 , 估计误差的 影响可由滑模项来消除. 在系统满足线性参数化的条件下给出了一种自适应时延状态观测器设计. 最后给出仿真 研究说明设计方法的有效性.
and its delayed measurement y ( t - τ) , which is given by
Dτy ( t) = τ- 1 ( y ( t) - y ( t - τ) ) .
(3)
The higher- order difference quotient in turn is defined as
Vol . 20 No. 2
Apr. 2003
XIANG Zheng- rong1 , ZHU Rui- jun2 , HU Wei- li1
(1. Department of Automation , Nanjing University of Science and Technology , Jiangsu Nanjing 210094 , China ; 2. Department of Automation , Dalian University of Technology , Liaoning Dalian 116023 , China)
© 1995-2007 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
278
Control Theory & Applications
Vol . 20
tant , Dτy denote the difference quotient between output y
In this paper , a time- delay observer is proposed by
combining difference approach with sliding mode
method , in which the nonlinear function is more gener2 al , and no output- matching and bound condition is re 2 quired. Then an adaptive time- delay observer is present 2 ed. Finally , simulation results are given to show the ef 2
(7)
where G is constant vector which makes AG = A - GC
asymptotically stable , constant K ≥ K1τ. Let P be a positive definite symmetric matrix satisfying
of sensors may make the overall system more complex or
expensive . As a solution to this difficulty , the use of an observer may be an attractive alternative [4～6] .
A bs t ract : An adaptive time- delay observer (A TDO) technique has been proposed for a class of systems including un2 known constants by combining difference approach with sliding mode. Firstly , the differential and its higher- order differentials of the output are represented by the difference quotients between the output and its delayed measurement , and a novel time- delay observer is presented by using the estimation of nonlinear function , then an adaptive time- delay observer (A TDO) is established under the assumption of linear-parameterization. For the proposed observers , the convergence property has been analyzed. Fi2 nally , simulation results are given to demonstrate the effectiveness of the proposed observers .
f ( X , t , u) =
f ( y , Dτy ( t) , …, Dτn- 1y ( t) , t , u) +ε(τ) ,
where X = [ y ( t) , Dτy ( t) , …, Dτn- 1y ( t) ] ,ε(τ) has
the same order as τ, that is
2第00230年卷4第月2 期
控制理论与应用
Control Theory & Applications
Article ID : 1000 - 8152 (2003) 02 - 0277 - 03
A dap tive ti me- dela y obs e r ve r design f or a cl ass of nonli ne a r s ys t e ms
( A - GC) T P + P ( A - GC) = - Q , B T P = C , (8)
where Q is a given positive definite symmetric matrix.
Define X = X - X^ as the estimated state error , then
X = AX + Bf ( X , t , u) , y = CX ,
(2)
where X = ( x1 , x2 , …, xn) T.
Suppose that system (2) is bounded- input and bound2
ed-state (BIBS) stable , let τ be a positive cons2
the estimated state error equation can be written
·
X = AG X + B (ε(τ) - Ksgn ( y) ) , y = CX . (9)
Choose the Lyapunov candidate as
V = X T PX .
(10)
Leabharlann Baidu
Differentiate V along with Eq. (9) , we obtain
| ε(τ) | ≤ K1τ,
(6)
where K is a positive constant .
3 M ai n r es ults
We now present the time- delay observer for system
(2) as
·
X^ = AX^ +B ( f ( X , t , u) + Ksgn ( y - CX^ ) ) + G( y - CX^ ) ,
f oll ows
Dτi y ( t) = τ- 1 ( Dτi - 1y ( t) - Dτi - 1y ( t - τ) ) =
i
∑ τ- i ( - 1) jCji y ( t - τj ) , i = 2 , …, n - 1 , j =0
(4) where Cji represents a number of combinations of selec2
·
·
V = X T PX + X T PX =
X
T
(
A
T G
P
+
PA G)
X
+2 (ε(τ)
-
Ksgn
( y T) ) B T PX =
- X T QX + 2 (εT(τ) - Ksgn ( y T) ) y .
(11)
Using (8) and (9) , we have V = - X T QX + 2 (εT (τ) - Ksgn ( y T) ) y ≤
Ke y wor ds : nonlinear system ; difference approach ; delayed measurement ; state observer ; sliding mode CL C n u m be r : TP273 Doc u me nt code : A
tions j from total i .
For a sufficiently small τ, the following approxima2
tion formulations are valid.
xi = xi - 1 ≈ Dτi - 1y ( t) , i = 2 , …, n.
(5)
Thus , the function f ( X , t , u) can be written as :
methods [1] , sliding- mode control [2 ] , and adaptive con2 trol [3] . Unfortunately , some states are often very diffi 2
cult to measure , even if they are , the increased number
Received date :2001 - 09 - 10 ; Revised date :2002 - 09 - 09. Foundation item :supported by the National Natural Science Foundation of China (60174019) .
fectiveness of the proposed observers .
2 Pr o ble m s t a t e m e nt
Let us consider a nonlinear SISO systems in the form
of the following differential equation
x ( n) = f ( x , x (1) , …, x ( n - 1) , t , u) ,
(1)
where x , u ∈R denote the output and the input , respec2
tively , x ( i) ( i = 1 ,2 , …, n) stand for the ith- order
关键词 : 非线性系统 ; 差分方法 ; 延迟测量 ; 状态观测器 ; 滑模
1 I nt r o d uctio n
Several popular control techniques for nonlinear plants
requires state feedback , such as feedback linearization
derivative of the output x , f ( x , x (1) , …, x ( n- 1) , t , u) is
smooth nonlinear function.
Given xi +1 = x ( i) , the following state variables form
一类非线性系统的自适应时延观测器设计
向峥嵘1 , 朱瑞军2 , 胡维礼1
(1. 南京理工大学 自动化系 , 江苏 南京 210094 ; 21 大连理工大学 自动化系 , 辽宁 大连 116023) 摘要 : 利用输出差分及滑模方法研究一类非线性系统的自适应时延状态观测器设计问题. 首先将系统输出的 微分及高阶微分用输出和其延迟测量值的差商来表示 , 然后在此基础上构造了一种时延状态观测器 , 估计误差的 影响可由滑模项来消除. 在系统满足线性参数化的条件下给出了一种自适应时延状态观测器设计. 最后给出仿真 研究说明设计方法的有效性.
and its delayed measurement y ( t - τ) , which is given by
Dτy ( t) = τ- 1 ( y ( t) - y ( t - τ) ) .
(3)
The higher- order difference quotient in turn is defined as
Vol . 20 No. 2
Apr. 2003
XIANG Zheng- rong1 , ZHU Rui- jun2 , HU Wei- li1
(1. Department of Automation , Nanjing University of Science and Technology , Jiangsu Nanjing 210094 , China ; 2. Department of Automation , Dalian University of Technology , Liaoning Dalian 116023 , China)
© 1995-2007 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
278
Control Theory & Applications
Vol . 20
tant , Dτy denote the difference quotient between output y
In this paper , a time- delay observer is proposed by
combining difference approach with sliding mode
method , in which the nonlinear function is more gener2 al , and no output- matching and bound condition is re 2 quired. Then an adaptive time- delay observer is present 2 ed. Finally , simulation results are given to show the ef 2
(7)
where G is constant vector which makes AG = A - GC
asymptotically stable , constant K ≥ K1τ. Let P be a positive definite symmetric matrix satisfying
of sensors may make the overall system more complex or
expensive . As a solution to this difficulty , the use of an observer may be an attractive alternative [4～6] .
A bs t ract : An adaptive time- delay observer (A TDO) technique has been proposed for a class of systems including un2 known constants by combining difference approach with sliding mode. Firstly , the differential and its higher- order differentials of the output are represented by the difference quotients between the output and its delayed measurement , and a novel time- delay observer is presented by using the estimation of nonlinear function , then an adaptive time- delay observer (A TDO) is established under the assumption of linear-parameterization. For the proposed observers , the convergence property has been analyzed. Fi2 nally , simulation results are given to demonstrate the effectiveness of the proposed observers .
f ( X , t , u) =
f ( y , Dτy ( t) , …, Dτn- 1y ( t) , t , u) +ε(τ) ,
where X = [ y ( t) , Dτy ( t) , …, Dτn- 1y ( t) ] ,ε(τ) has
the same order as τ, that is
2第00230年卷4第月2 期
控制理论与应用
Control Theory & Applications
Article ID : 1000 - 8152 (2003) 02 - 0277 - 03
A dap tive ti me- dela y obs e r ve r design f or a cl ass of nonli ne a r s ys t e ms
( A - GC) T P + P ( A - GC) = - Q , B T P = C , (8)
where Q is a given positive definite symmetric matrix.
Define X = X - X^ as the estimated state error , then