- Email: [email protected]
Adaptive Output Stabilization of Time-Delay Nonlinear System
Adaptive Output Stabilization of Time-Delay Nonline
ar System *
Alexey A. Bobtsov 1, 2 and Anton A. Pyrkin1 1
The Department of Control Systems and Informatics, Saint Petersburg State University of Information Technologies Mechanics and Optics, Kron verkskiy av. 49, Saint Petersburg, 197101 Russia 2
Laboratory "Control of Complex Systems", Institute for Problems of Mechanical Engineering, Bolshoy pr. V.O. 61, St.Petersburg, 199178, Russia E-mail: [email protected], [email protected]
Abstract: This paper deals with the output stabiliz ation of time-delay systems with sectorbounded nonlinearity. In this paper we will consider the pr oblem of absolute stability for a class of timedelay systems which can be represented as a feedback conn ection of a linear dynamical system with unknown parameters and a uncertainty nonlinearity satisfyin gsector a constraint. For a class of output control algorithms a controller providing output asymptotic stability of equilibrium position is designed. Keywords: Absolute stability, time-delay systems, s tabilization methods, decision feedback, nonlinear control, output regulation, single-input/single-out put systems, robust control, robustness.
1. INTRODUCTION
efforts. Many significant results have been present ed in the literature (see for example (Gu et al., 2003; Richard, 2003)). However little attention has been focused on nonlin ear time The absolute stability problem, formulated by Lurie anddelayed systems. The problem of stability for nonli near coworkers in the 40’s, has been a well-studied and fruitful systems with delay has been studied in (Bliman, 200 1; area of research, as presented in (Lurie and Postni kov, 1944). Germani and Manes, 2001; Germani et al., 2003; Ge et al., Since works of Lurie (Lurie and Postnikov, 1944), i nterest of 2003; Gouaisbaut et al., 2001; He and Wu, 2003; Hua et al., researches of control systems has been attracted by structures2002; Marquez-Martinez and Moog, 2004; Moog et al., including linear block and nonlinear feedback stati c block. It2000; Oguchi et al., 2002). In paper (Ge et al., 2003) is possible to allocate big enough number of works devoted adaptive neural control was presented for a class o f strictto solution of problems of nonlinear systems stabil ization forfeedback nonlinear systems with unknown time delays . For a a case when output of the nonlinear block is given as control case of measurability of the state vector (Ge et al., 2003), input of the linear block (see the review (Kokotovi c and using appropriate Lyapunov–Krasovskii functionals, the Arcak, 2001)). It is also possible to allocate a bl ock of works uncertainties of unknown time delays are compensate d for (Byrnes and Isidori, 1991; Khalil and Esfandiari, 1 993; Linsuch that iterative backstepping design can be carr ied out. In and Saberi, 1995) in which nonlinear part and stati c (Germani et al., 2003) the relationships between the internal nonlinearity adjust with an input on control. Howev er,state and input dynamics of a controlled nonlinear delay approaches (Byrnes and Isidori, 1991; Khalil and Es fandiari,system are studied. An interesting result is that a suitable 1993; Lin and Saberi, 1995) are focused on stabiliz ation of stability assumption on the internal state dynamics ensures systems with nonlinearity, resulted to an input of system, and that, when the output is asymptotically driven to z ero, both do not allow one to solve more general problems. To day the state and control variables asymptotically deca y to zero. problems of control of nonlinear systems in which In (Oguchi et al., 2002) the input-output linearization nonlinearity is not coordinated with control are of interest.problem for retarded non-linear systems is consider ed, which Among works devoted to these subjects one can alloc ate have time-delays in the state. By using an extensio n of the papers (Bobtsov, 2005a; Bobtsov and Nikolaev, 2005b ;Lie derivative for functional differential equation s, authors Krstic and Kokotovic, 1996; Praly, 2003; Qian and L in, derive coordinates transformation and a static stat e feedback 2002) in which similar problem is considered. Howev er,to obtain linear input-output behaviour for a class of retarded these results mentioned above are delay-independent . During non-linear systems. The obtained coordinates transf ormation the last two decades, the problem of stability of l inear timeis allowed to contain not only the current value of the state delay systems has been the subject of considerable research variables but also the past values of ones. In (Hua et al., 2002), a robust control problem of a class of nonli near time * This work was supported by the RFBR (projects 09-08 -00139-a), the delay systems is considered under the case when non linear 92P/5: ADTP (project 2.1.2/6326), and the FTP (project NKuncertainties are bounded by first-order function. P498/05.08.2009).
Geometrical method is employed to investigate the c ontrol u (t ) = - m y (t ) + v(t ) , (3) problem of time delay systems in (Germani and Manes , 2001; Marquez-Martinez and Moog, 2004; Moog et al., 2000). where v(t ) is an additional input and parameter m > 0 . Corresponding state feedback controller and output feedback controller are designed, but the strict conditions should beLemma (Bobtsov, 2005; Bobtsov and Nikolaev, 2005; imposed on the investigated systems. Fradkov, 1974; Fradkov, 2003; Kokotovic and Arcak, 2001). In this paper a robust version of the results on hi gh-gain There exists some positive number m 0 , such that for any stabilization of nonlinear systems is extended to a class of the system time-delay nonlinear systems without matching condi tions. m ‡ m 0 b( p ) c( p) The given work, developing approaches obtained in ( Bliman,y (t ) = v(t ) + w (t ) has the SPR 2001; Germani and Manes, 2001; Germani et al., 2003; Ge et a ( p ) + m b( p ) a ( p ) + m b( p ) al., 2003; Gouaisbaut et al., 2001; He and Wu, 2003; Hua et (strictly positive real) transfer function al., 2002; Marquez-Martinez and Moog, 2004; Moog et al., 2000; Oguchi et al., 2002), offers new solution of b( p ) H ( p) = . stabilization problem of nonlinear system consistin g of a ( p ) + m b( p ) structures including a linear block and nonlinear f eedback static block. Assuming, that parameters of the line ar part and Let r > 1 and the derivatives of output y (t ) be measured. delay are unknown, the output is measured (but not its derivatives), and the characteristic of the nonline ar feedbackChoose control in the following form block is unknown, a controller providing asymptotic stability u (t ) = a ( p)u (t ) , (4) of equilibrium position is designed. In this paper an interesting approach is offered, that does not use the procedure of linearization of nonlinear system, des ign ofwhere any polynomial a ( p) of r - 1 degree is Hurwitz and u (t ) is a new variable. observer and iterative backstepping design. 2. STATEMENT OF THE PROBLEM
Then we can rewrite the model (1) in the following
Consider the following nonlinear system
b( p ) c( p) y (t ) = u (t ) + w (t ) , a( p) a( p)
y (t ) = (1)
b( p )a ( p) c( p ) u (t ) + w (t ) , a ( p) a( p)
form: (5)
p = d / dt denotes differential operator; output y = y (t ) is measured, but its derivatives are not measured;
where polynomial b( p )a ( p ) is Hurwitz and the relative b( p )a ( p) degree of the transfer function is r = 1 . a( p)
b( p) = bm p m + ... + b1 p + b0 ,
Choose u (t ) according to the equation (3)
where
r
c( p ) = c r p + c r - 1 p
r- 1
+ ... + c1 p + c 0
and
a ( p) = p n + a n - 1 p n - 1 + ... + a1 p + a 0 are monic coprime polynomials with unknown coefficients; number r £ n - 1 ; b( p ) transfer function has relative degree r = n - m ; a( p) polynomial b( p) is Hurwitz and parameter bm > 0 ; unknown function w (t ) = j ( y (t - t )) is such that:
j ( y (t - t )) £ C0 y (t - t ) for all y (t - t ) ,
(2)
where t > 0 is the unknown constant delay, y (J ) = f (J ) for " J ˛ [- t ,0] and number C0 > 0 is unknown. The purpose of control is to obtain asymptotical st equilibrium position y = 0 .
3. CONTROL DESIGN
u (t ) = - m y (t ) + v(t ) .
(6)
Substituting (6) into the equation (5), we obtain c system
y (t ) =
losed-loop
b( p)a ( p) c( p) v(t ) + w (t ) . a( p) + m b( p)a ( p) a( p) + m b( p)a ( p)
Then by using lemma for some m > m 0 it is easy to see that the following transfer function is SPR
W ( p) =
b( p)a ( p) . a ( p) + m b( p )a ( p)
(7)
ability of However control of the form (4), (6) can not be app lied because it is impossible to measure derivatives of function y (t ) . Choose the control
b( p ) u (t ) = - a ( p )(m + k ) yˆ (t ) , (8) has a( p) relative degree r = 1 and polynomial b( p) is Hurwitz. where number m and polynomial a ( p) are such that the Choose control of the following form (Bobtsov and N ikolaev,transfer function (7) is SPR, the positive paramete k ris used 2005; Fradkov, 1974; Fradkov, 2003):
Let the system (1) be such that transfer function
for compensation of the uncertainty j ( y (t - t )) (see proof of the theorem, inequality (24)) and the function yˆ (t ) is the
AT P + PA = - Q1 , Pb = c ,
(15)
estimation of output y (t ) . The function yˆ (t ) is aclculated according to the following algorithm
where Q1 = Q1T > 0 and parameters of matrix Q1 depend on m and do not depend on k .
x&1 (t ) = sx 2 (t ), x&2 (t ) = sx 3 (t ), ... x& (t ) = s (- k x (t ) - ... - k x 1 1 r -1 r r -1 yˆ (t ) = x 1 (t ) ,
Let us rewrite model (9), (10) in the form
x &(t ) = s (G x (t ) + dk1 y (t )) ,
(9)
yˆ (t ) = h T x (t ) ,
- 1 (t ) + k1 y (t )),
1 0 0 0 where number s > m + k (see proof of the theorem, where G = 0 0 inequality (23)) and parameters ki are calculated for the M M system (9) to be asymptotically stable. - k - k 2 1 It is obvious, that the control (8) – (10) is techn ically possible h T = [1 0 0 ... 0] . as contains known or measurable signals. Substituti ng (8) into equation (1), we obtain Consider vector y (t ) = =
(10)
b( p ) c( p ) [- a ( p)( m + k ) yˆ (t )] + w (t ) a( p) a( p)
...
0 0 0 0 ... 0 , d = 0 O M M 1 ... - k r - 1 ...
0 M - k3
and
h (t ) = hy (t ) - x (t ) ,
b( p ) [- a ( p )(m + k ) y (t ) + a ( p)(m + k )e (t )] a( p) c( p) + w (t ) , (11) a( p)
where the error e (t ) = y (t ) - yˆ (t ) .
then by force of structure of vector h error e (t ) will become
e (t ) = y (t ) - yˆ (t ) = h T hy (t ) - h T x (t ) = hT (hy (t ) - x (t )) = h T h (t ) . For derivative of h (t ) we obtain
After simple transformations, for model (11) we hav
e
a( p) y (t ) + ma ( p)b( p) y (t ) = b( p )a ( p)[(m + k )e (t ) - k y (t )] + c( p)w (t ) and
b( p)a ( p) y (t ) = [- k y (t ) + ( m + k )e (t )] a( p) + m b( p)a ( p) c( p) + w (t ) , (12) a( p) + m b( p)a ( p) where transfer function W ( p ) =
0 1
b( p)a ( p) is SPR a( p) + m b( p)a ( p)
(see equation (7)).
Since dk1 = - G h (can be checked by substitution), then
h & (t ) = hy& (t ) + s G h (t ) , e (t ) = h T h (t ) ,
(16)
where matrix G is Hurwitz by force of calculated parameters ki of system (9) and
G T N + NG = - Q2 ,
(17)
where N = N T > 0 , Q2 = Q2T > 0 . Theorem. Consider the nonlinear system (13), (14), (16). Let number r = n - m ‡ 1 and unknown function
Let us present model (12) in the form
x& (t ) = Ax(t ) + b(- k y (t ) + ( m + k )e (t )) + qw (t ) , (13) y (t ) = c T x(t ) ,
h & (t ) = hy& (t ) - s (G (hy (t ) - h (t )) + dk1 y (t )) = hy& (t ) + s G h (t ) - s (dk1 + G h) y (t ) .
(14)
w (t ) = j ( y (t - t )) be such that: j ( y (t - t )) £ C0 y (t - t ) for all y (t - t ) ,
where t > 0 is the constant delay, y (J ) = f (J ) for where x(t ) ˛ R n is a state vector of system (13); A , b , q " J ˛ [- t ,0] and number C0 > 0 is unknown. Let number and c are appropriate matrices of transition from model (12)s be such that the following ratio is executed to model (13), (14). Since transfer function W ( p ) is SPR then - s Q2 + d - 1 ( m + k )hhT + ( m + k ) Nhc T bbT ch T N
+ ( m + k )hhT + d
-1
Nhc T AAT ch T N
+ ( m + k )h T (t ) Nhc T bbT chT Nh (t )
+ k Nhc T qq T ch T N + d - 1k NhcT bbT chT N £ - Q ,
+ ( m + k )h T (t )hhT h (t )
where the number 0 < d < 0.5 is such that
+d
- Q1 + d I + (dm + 2dk - k ) PbbT P
-1 T
h
(t ) Nhc T AAT chT Nh (t ) + d x T (t ) x(t )
+ kh T (t ) Nhc T qq T chT Nh (t ) + k
+ d Pqq P £ - Q < 0 . T
-1
+d
-1
k ‡ C02 (k
-1
+d
Proof. Choose a Lyapunov-Krasovskii functional candidate as t
V (t ) = xT (t ) Px(t ) + h T (t ) Nh (t ) + k
∫y
2
, we obtain
+d
-1
[w (t )]2 + ( m + k )h T (t ) Nhc T bbT chT Nh (t )
T
+d
Substituting in (19) equations (15), (17) and takin account inequalities
2 x T (t ) PbhT h (t ) £ d x T (t ) PbbT Px(t ) + d T
g into
-1 T
h
(t )hhT h (t ) ,
-1
2
2 x (t ) Pqw (t ) £ d x (t ) Pqq Px(t ) + d
[w (t )] ,
2h T (t ) Nhc T bhT h (t ) £ h T (t ) Nhc T bb T chT Nh (t )
- s Q2 + d
- 2kh
T
(t ) Nhc by (t ) £ d
kh
T
T
T
T
(t ) Nhc bb ch Nh (t )
T
(22)
be such that the following ratio is executed -1
( m + k )hhT + ( m + k ) Nhc T bbT ch T N -1
Nhc T AAT ch T N
k NhcT bbT chT N £ - Q . (23)
-1
Substituting (23) into the inequality (22), we obta
+ (d
-1
+k
2
- k y (t - t ) + (k
T
-1
in
(t )Qh (t ) - k y 2 (t - t )
-1
)[w (t )]2
+d
T
(t )Qh (t )
-1
)C02 y 2 (t - t ) ,
where from (2) [w (t )]2 £ C02 y (t - t ) . Let number k be such that
k ‡ C02 (k
V& (t ) £ - x T (t )Q1 x(t ) - sh T (t )Q2h (t ) - k x T (t ) PbbT Px(t ) - k y 2 (t - t ) + d x T (t ) Pqq T Px(t ) + d
[w (t )]2
(t ) Nhc T bbT chT Nh (t ) .
+ k Nhc T qq T ch T N + d
for the equation (19) we obtain
-1
-1
2
T
+ dk x (t ) Pbb Px(t )
+ d ( m + k ) x T (t ) PbbT Px(t ) + d
T
£ - x T (t )Qx(t ) - h
[w (t )]2 ,
-1
kh
(t ) Nhc T AAT chT Nh (t )
2h T (t ) Nhc T qw (t ) £ kh T (t ) Nhc T qq T chT Nh (t ) T
-1
+ ( m + k )hhT + d
+ d x T (t ) x ( t ) , -1
(t ) Nhc T AAT ch T Nh (t )
V& (t ) £ - x T (t )Qx(t ) - h
+ h T (t )hh T h (t ) ,
h
Let number s
(19)
h
+ kh T (t ) Nhc T qq T chT Nh (t ) + k
N + NG )h (t )
+ 2h T (t ) Nhc T qw (t ) - 2kh T (t ) Nhc T by (t ) .
-1 T
-1 T
+d
(t )s (G
( m + k )h T (t )hhT h (t )
+ ( m + k )h T (t )hhT h (t )
+ 2h T (t ) Nhc T Ax(t ) + 2( m + k )h T (t ) Nhc T bh T h (t )
T
-1
- k y 2 (t - t ) + d
+ k y 2 (t ) - k y 2 (t - t ) T
in
V& £ - x T (t )Qx(t ) - sh T (t )Q2h (t )
V& (t ) = x T (t )( AT P + PA) x(t ) + 2( m + k ) x T (t ) PbhT h (t ) + 2 x T (t ) Pqw (t ) - 2k x T (t ) Pby (t ) + h
(21)
Substituting (21) into the inequality (20), we obta
(q )dq . (18)
Differentiating (18) along the trajectories of (16)
(20)
+ d Pqq T P £ - Q < 0 .
t- t
+k
(t ) Nhc T bbT chT Nh (t )
where the number d > 0 . Let the number 0 < d < 0.5 be such that totic - Q1 + d I + (dm + 2dk - k ) PbbT P
).
Then the nonlinear system (13), (14), (16) is asymp stable.
2h T (t ) Nhc T Ax(t ) £ d
[w (t )]2
+ dk x T (t ) PbbT Px(t ) ,
Let number k be such that
T
T
kh
-1
( m + k )h T (t ) hh T h (t ) -1
[w (t )]2
-1
+ d - 1) .
(24)
Then we have
V& (t ) £ - x T (t )Qx(t ) - h
T
(t )Qh (t ) .
(25)
From the expression (25) follows asymptotic stabili system (13), (14), (16).
ty of
Remark. In some cases, the problem of choosing the 4. SIMULATION RESULTS parameters k , m and s of regulator (8), (9) which satisfy Consider the following nonlinear system theorem can arise (see (21), (23) and (24)). This choice presents no appreciable difficulties for known poly nomials b( p ) c( p) y (t ) = u (t ) + w (t ) , (29) t plant (1) and also for a definite a ( p ) , b( p) and c( p ) of he a( p) a( p) number C0 . However, if the parameters of the plant (1) are unknown, the problem of calculating k , m and s may where polynomials b( p) = b0 , a ( p) = p( p - a1 )( p - a2 ) , prove to be problematic. As was demonstrated by theorem, if c( p) = c0 p - c1 with unknown parameters b0 , a1 , a 2 , c0 , condition (23) is met, then there exists a number c1 , delay h=3 and nonlinear function s > m + k such that lim y (t ) = 0 . A possible variant of w (t ) = j ( y (t - t )) = y (t - t ) sin y (t - t ) . t fi ¥ adjustment of k , m and s lies in increasing them until the Choose the control according to equation (8) following goal condition is met
y (t ) < d
0
for t ‡ t1 ,
u (t ) = - a ( p )(m + k ) yˆ (t ) ,
(26)
where the number d 0 is set by the designer of the control system. This concept may be realized by using the following adjustment algorithm
where for relative degree r = 3 the polynomial a ( p) we choose as
a ( p) = ( p + 1) 2 .
∫
(27)
u (t ) = - ( p + 1) 2 ( m + k ) yˆ (t ) = = - ( m + k )( &yˆ&(t ) + 2 y&ˆ (t ) + yˆ (t )) ,
t0
~ where k = k + m and the function l (t ) is as follows for
0
0
y (t ) > d
~
s = s 0k 2 ,
(28)
where s 0 > 0 . Obviously, for this calculation of s there will be a time instant t1 > t 0 such that the following conditions of theorem are satisfied -1
( m + k )hhT + ( m + k ) Nhc T bbT ch T N
+ ( m + k )hhT + d + k Nhc T qq T ch T N + d
-1
Nhc T AAT ch T N
k NhcT bbT chT N £ - Q ,
-1
where the number 0 < d < 0.5 is chosen so that
- Q1 + d I + (dm + 2dk - k ) PbbT P + d Pqq T P £ - Q < 0 , and the parameters k ‡ C 02 (k - 1 + d - 1 ) .
(32)
where the positive numbers m > 0 , k > 0 and the function yˆ (t ) is calculated by the following algorithm
x&1 (t ) = sx 2 (t ), & x 2 (t ) = s (- k1x 1 (t ) - k 2x 2 (t ) + k1 y (t )), yˆ (t ) = x 1 (t ) ,
where the number l 0 > 0 . We take s as follows
- s Q2 + d
rm
0
y (t ) £ d 0 ,
for
(31)
Then we can rewrite the control in the following fo
t
~ k (t ) = l (q )dq ,
l l (t ) =
(30)
where the number s > m + k
(33) (34)
and parameters ki are k1 = 1 ,
k2 = 2 .
~ k = m + k = 5 and s = 12 . The results of a computer simulation for variable y (t ) for the case b0 = 1 a1 = 1 , a2 = 0.5 , c0 = 1 , c1 = 0.3 , t = 3 , y (0) = 2 and yˆ (0) = 0 are presented in Fig. 1. The results of a computer simulation for variable y (t ) for the case b0 = 1 a1 = - 1 , a 2 = - 0.5 , c0 = 1 , c1 = - 0.3 , t = 3 , y (0) = 5 and yˆ (0) = 0 are presented in Fig. 2. We can see that for two different groups of parameters presented control pr ovides asymptotic stability of equilibrium y = 0 .
Choose the parameters
5. CONCLUSION In the paper the problem of synthesis of control fo r timedelay nonlinear system (1) was considered. This pap er has extended the theory of output feedback control of t ime-delay nonlinear systems. The new control law (8) – (10), providing output asymptotic stability of system (1) was desig ned. Only measurements of the output, but not its derivatives were used. Dimension of the robust controller (8) – (10) is r - 1 and
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7), (28) is
ar System *
Alexey A. Bobtsov 1, 2 and Anton A. Pyrkin1 1
The Department of Control Systems and Informatics, Saint Petersburg State University of Information Technologies Mechanics and Optics, Kron verkskiy av. 49, Saint Petersburg, 197101 Russia 2
Laboratory "Control of Complex Systems", Institute for Problems of Mechanical Engineering, Bolshoy pr. V.O. 61, St.Petersburg, 199178, Russia E-mail: [email protected], [email protected]
Abstract: This paper deals with the output stabiliz ation of time-delay systems with sectorbounded nonlinearity. In this paper we will consider the pr oblem of absolute stability for a class of timedelay systems which can be represented as a feedback conn ection of a linear dynamical system with unknown parameters and a uncertainty nonlinearity satisfyin gsector a constraint. For a class of output control algorithms a controller providing output asymptotic stability of equilibrium position is designed. Keywords: Absolute stability, time-delay systems, s tabilization methods, decision feedback, nonlinear control, output regulation, single-input/single-out put systems, robust control, robustness.
1. INTRODUCTION
efforts. Many significant results have been present ed in the literature (see for example (Gu et al., 2003; Richard, 2003)). However little attention has been focused on nonlin ear time The absolute stability problem, formulated by Lurie anddelayed systems. The problem of stability for nonli near coworkers in the 40’s, has been a well-studied and fruitful systems with delay has been studied in (Bliman, 200 1; area of research, as presented in (Lurie and Postni kov, 1944). Germani and Manes, 2001; Germani et al., 2003; Ge et al., Since works of Lurie (Lurie and Postnikov, 1944), i nterest of 2003; Gouaisbaut et al., 2001; He and Wu, 2003; Hua et al., researches of control systems has been attracted by structures2002; Marquez-Martinez and Moog, 2004; Moog et al., including linear block and nonlinear feedback stati c block. It2000; Oguchi et al., 2002). In paper (Ge et al., 2003) is possible to allocate big enough number of works devoted adaptive neural control was presented for a class o f strictto solution of problems of nonlinear systems stabil ization forfeedback nonlinear systems with unknown time delays . For a a case when output of the nonlinear block is given as control case of measurability of the state vector (Ge et al., 2003), input of the linear block (see the review (Kokotovi c and using appropriate Lyapunov–Krasovskii functionals, the Arcak, 2001)). It is also possible to allocate a bl ock of works uncertainties of unknown time delays are compensate d for (Byrnes and Isidori, 1991; Khalil and Esfandiari, 1 993; Linsuch that iterative backstepping design can be carr ied out. In and Saberi, 1995) in which nonlinear part and stati c (Germani et al., 2003) the relationships between the internal nonlinearity adjust with an input on control. Howev er,state and input dynamics of a controlled nonlinear delay approaches (Byrnes and Isidori, 1991; Khalil and Es fandiari,system are studied. An interesting result is that a suitable 1993; Lin and Saberi, 1995) are focused on stabiliz ation of stability assumption on the internal state dynamics ensures systems with nonlinearity, resulted to an input of system, and that, when the output is asymptotically driven to z ero, both do not allow one to solve more general problems. To day the state and control variables asymptotically deca y to zero. problems of control of nonlinear systems in which In (Oguchi et al., 2002) the input-output linearization nonlinearity is not coordinated with control are of interest.problem for retarded non-linear systems is consider ed, which Among works devoted to these subjects one can alloc ate have time-delays in the state. By using an extensio n of the papers (Bobtsov, 2005a; Bobtsov and Nikolaev, 2005b ;Lie derivative for functional differential equation s, authors Krstic and Kokotovic, 1996; Praly, 2003; Qian and L in, derive coordinates transformation and a static stat e feedback 2002) in which similar problem is considered. Howev er,to obtain linear input-output behaviour for a class of retarded these results mentioned above are delay-independent . During non-linear systems. The obtained coordinates transf ormation the last two decades, the problem of stability of l inear timeis allowed to contain not only the current value of the state delay systems has been the subject of considerable research variables but also the past values of ones. In (Hua et al., 2002), a robust control problem of a class of nonli near time * This work was supported by the RFBR (projects 09-08 -00139-a), the delay systems is considered under the case when non linear 92P/5: ADTP (project 2.1.2/6326), and the FTP (project NKuncertainties are bounded by first-order function. P498/05.08.2009).
Geometrical method is employed to investigate the c ontrol u (t ) = - m y (t ) + v(t ) , (3) problem of time delay systems in (Germani and Manes , 2001; Marquez-Martinez and Moog, 2004; Moog et al., 2000). where v(t ) is an additional input and parameter m > 0 . Corresponding state feedback controller and output feedback controller are designed, but the strict conditions should beLemma (Bobtsov, 2005; Bobtsov and Nikolaev, 2005; imposed on the investigated systems. Fradkov, 1974; Fradkov, 2003; Kokotovic and Arcak, 2001). In this paper a robust version of the results on hi gh-gain There exists some positive number m 0 , such that for any stabilization of nonlinear systems is extended to a class of the system time-delay nonlinear systems without matching condi tions. m ‡ m 0 b( p ) c( p) The given work, developing approaches obtained in ( Bliman,y (t ) = v(t ) + w (t ) has the SPR 2001; Germani and Manes, 2001; Germani et al., 2003; Ge et a ( p ) + m b( p ) a ( p ) + m b( p ) al., 2003; Gouaisbaut et al., 2001; He and Wu, 2003; Hua et (strictly positive real) transfer function al., 2002; Marquez-Martinez and Moog, 2004; Moog et al., 2000; Oguchi et al., 2002), offers new solution of b( p ) H ( p) = . stabilization problem of nonlinear system consistin g of a ( p ) + m b( p ) structures including a linear block and nonlinear f eedback static block. Assuming, that parameters of the line ar part and Let r > 1 and the derivatives of output y (t ) be measured. delay are unknown, the output is measured (but not its derivatives), and the characteristic of the nonline ar feedbackChoose control in the following form block is unknown, a controller providing asymptotic stability u (t ) = a ( p)u (t ) , (4) of equilibrium position is designed. In this paper an interesting approach is offered, that does not use the procedure of linearization of nonlinear system, des ign ofwhere any polynomial a ( p) of r - 1 degree is Hurwitz and u (t ) is a new variable. observer and iterative backstepping design. 2. STATEMENT OF THE PROBLEM
Then we can rewrite the model (1) in the following
Consider the following nonlinear system
b( p ) c( p) y (t ) = u (t ) + w (t ) , a( p) a( p)
y (t ) = (1)
b( p )a ( p) c( p ) u (t ) + w (t ) , a ( p) a( p)
form: (5)
p = d / dt denotes differential operator; output y = y (t ) is measured, but its derivatives are not measured;
where polynomial b( p )a ( p ) is Hurwitz and the relative b( p )a ( p) degree of the transfer function is r = 1 . a( p)
b( p) = bm p m + ... + b1 p + b0 ,
Choose u (t ) according to the equation (3)
where
r
c( p ) = c r p + c r - 1 p
r- 1
+ ... + c1 p + c 0
and
a ( p) = p n + a n - 1 p n - 1 + ... + a1 p + a 0 are monic coprime polynomials with unknown coefficients; number r £ n - 1 ; b( p ) transfer function has relative degree r = n - m ; a( p) polynomial b( p) is Hurwitz and parameter bm > 0 ; unknown function w (t ) = j ( y (t - t )) is such that:
j ( y (t - t )) £ C0 y (t - t ) for all y (t - t ) ,
(2)
where t > 0 is the unknown constant delay, y (J ) = f (J ) for " J ˛ [- t ,0] and number C0 > 0 is unknown. The purpose of control is to obtain asymptotical st equilibrium position y = 0 .
3. CONTROL DESIGN
u (t ) = - m y (t ) + v(t ) .
(6)
Substituting (6) into the equation (5), we obtain c system
y (t ) =
losed-loop
b( p)a ( p) c( p) v(t ) + w (t ) . a( p) + m b( p)a ( p) a( p) + m b( p)a ( p)
Then by using lemma for some m > m 0 it is easy to see that the following transfer function is SPR
W ( p) =
b( p)a ( p) . a ( p) + m b( p )a ( p)
(7)
ability of However control of the form (4), (6) can not be app lied because it is impossible to measure derivatives of function y (t ) . Choose the control
b( p ) u (t ) = - a ( p )(m + k ) yˆ (t ) , (8) has a( p) relative degree r = 1 and polynomial b( p) is Hurwitz. where number m and polynomial a ( p) are such that the Choose control of the following form (Bobtsov and N ikolaev,transfer function (7) is SPR, the positive paramete k ris used 2005; Fradkov, 1974; Fradkov, 2003):
Let the system (1) be such that transfer function
for compensation of the uncertainty j ( y (t - t )) (see proof of the theorem, inequality (24)) and the function yˆ (t ) is the
AT P + PA = - Q1 , Pb = c ,
(15)
estimation of output y (t ) . The function yˆ (t ) is aclculated according to the following algorithm
where Q1 = Q1T > 0 and parameters of matrix Q1 depend on m and do not depend on k .
x&1 (t ) = sx 2 (t ), x&2 (t ) = sx 3 (t ), ... x& (t ) = s (- k x (t ) - ... - k x 1 1 r -1 r r -1 yˆ (t ) = x 1 (t ) ,
Let us rewrite model (9), (10) in the form
x &(t ) = s (G x (t ) + dk1 y (t )) ,
(9)
yˆ (t ) = h T x (t ) ,
- 1 (t ) + k1 y (t )),
1 0 0 0 where number s > m + k (see proof of the theorem, where G = 0 0 inequality (23)) and parameters ki are calculated for the M M system (9) to be asymptotically stable. - k - k 2 1 It is obvious, that the control (8) – (10) is techn ically possible h T = [1 0 0 ... 0] . as contains known or measurable signals. Substituti ng (8) into equation (1), we obtain Consider vector y (t ) = =
(10)
b( p ) c( p ) [- a ( p)( m + k ) yˆ (t )] + w (t ) a( p) a( p)
...
0 0 0 0 ... 0 , d = 0 O M M 1 ... - k r - 1 ...
0 M - k3
and
h (t ) = hy (t ) - x (t ) ,
b( p ) [- a ( p )(m + k ) y (t ) + a ( p)(m + k )e (t )] a( p) c( p) + w (t ) , (11) a( p)
where the error e (t ) = y (t ) - yˆ (t ) .
then by force of structure of vector h error e (t ) will become
e (t ) = y (t ) - yˆ (t ) = h T hy (t ) - h T x (t ) = hT (hy (t ) - x (t )) = h T h (t ) . For derivative of h (t ) we obtain
After simple transformations, for model (11) we hav
e
a( p) y (t ) + ma ( p)b( p) y (t ) = b( p )a ( p)[(m + k )e (t ) - k y (t )] + c( p)w (t ) and
b( p)a ( p) y (t ) = [- k y (t ) + ( m + k )e (t )] a( p) + m b( p)a ( p) c( p) + w (t ) , (12) a( p) + m b( p)a ( p) where transfer function W ( p ) =
0 1
b( p)a ( p) is SPR a( p) + m b( p)a ( p)
(see equation (7)).
Since dk1 = - G h (can be checked by substitution), then
h & (t ) = hy& (t ) + s G h (t ) , e (t ) = h T h (t ) ,
(16)
where matrix G is Hurwitz by force of calculated parameters ki of system (9) and
G T N + NG = - Q2 ,
(17)
where N = N T > 0 , Q2 = Q2T > 0 . Theorem. Consider the nonlinear system (13), (14), (16). Let number r = n - m ‡ 1 and unknown function
Let us present model (12) in the form
x& (t ) = Ax(t ) + b(- k y (t ) + ( m + k )e (t )) + qw (t ) , (13) y (t ) = c T x(t ) ,
h & (t ) = hy& (t ) - s (G (hy (t ) - h (t )) + dk1 y (t )) = hy& (t ) + s G h (t ) - s (dk1 + G h) y (t ) .
(14)
w (t ) = j ( y (t - t )) be such that: j ( y (t - t )) £ C0 y (t - t ) for all y (t - t ) ,
where t > 0 is the constant delay, y (J ) = f (J ) for where x(t ) ˛ R n is a state vector of system (13); A , b , q " J ˛ [- t ,0] and number C0 > 0 is unknown. Let number and c are appropriate matrices of transition from model (12)s be such that the following ratio is executed to model (13), (14). Since transfer function W ( p ) is SPR then - s Q2 + d - 1 ( m + k )hhT + ( m + k ) Nhc T bbT ch T N
+ ( m + k )hhT + d
-1
Nhc T AAT ch T N
+ ( m + k )h T (t ) Nhc T bbT chT Nh (t )
+ k Nhc T qq T ch T N + d - 1k NhcT bbT chT N £ - Q ,
+ ( m + k )h T (t )hhT h (t )
where the number 0 < d < 0.5 is such that
+d
- Q1 + d I + (dm + 2dk - k ) PbbT P
-1 T
h
(t ) Nhc T AAT chT Nh (t ) + d x T (t ) x(t )
+ kh T (t ) Nhc T qq T chT Nh (t ) + k
+ d Pqq P £ - Q < 0 . T
-1
+d
-1
k ‡ C02 (k
-1
+d
Proof. Choose a Lyapunov-Krasovskii functional candidate as t
V (t ) = xT (t ) Px(t ) + h T (t ) Nh (t ) + k
∫y
2
, we obtain
+d
-1
[w (t )]2 + ( m + k )h T (t ) Nhc T bbT chT Nh (t )
T
+d
Substituting in (19) equations (15), (17) and takin account inequalities
2 x T (t ) PbhT h (t ) £ d x T (t ) PbbT Px(t ) + d T
g into
-1 T
h
(t )hhT h (t ) ,
-1
2
2 x (t ) Pqw (t ) £ d x (t ) Pqq Px(t ) + d
[w (t )] ,
2h T (t ) Nhc T bhT h (t ) £ h T (t ) Nhc T bb T chT Nh (t )
- s Q2 + d
- 2kh
T
(t ) Nhc by (t ) £ d
kh
T
T
T
T
(t ) Nhc bb ch Nh (t )
T
(22)
be such that the following ratio is executed -1
( m + k )hhT + ( m + k ) Nhc T bbT ch T N -1
Nhc T AAT ch T N
k NhcT bbT chT N £ - Q . (23)
-1
Substituting (23) into the inequality (22), we obta
+ (d
-1
+k
2
- k y (t - t ) + (k
T
-1
in
(t )Qh (t ) - k y 2 (t - t )
-1
)[w (t )]2
+d
T
(t )Qh (t )
-1
)C02 y 2 (t - t ) ,
where from (2) [w (t )]2 £ C02 y (t - t ) . Let number k be such that
k ‡ C02 (k
V& (t ) £ - x T (t )Q1 x(t ) - sh T (t )Q2h (t ) - k x T (t ) PbbT Px(t ) - k y 2 (t - t ) + d x T (t ) Pqq T Px(t ) + d
[w (t )]2
(t ) Nhc T bbT chT Nh (t ) .
+ k Nhc T qq T ch T N + d
for the equation (19) we obtain
-1
-1
2
T
+ dk x (t ) Pbb Px(t )
+ d ( m + k ) x T (t ) PbbT Px(t ) + d
T
£ - x T (t )Qx(t ) - h
[w (t )]2 ,
-1
kh
(t ) Nhc T AAT chT Nh (t )
2h T (t ) Nhc T qw (t ) £ kh T (t ) Nhc T qq T chT Nh (t ) T
-1
+ ( m + k )hhT + d
+ d x T (t ) x ( t ) , -1
(t ) Nhc T AAT ch T Nh (t )
V& (t ) £ - x T (t )Qx(t ) - h
+ h T (t )hh T h (t ) ,
h
Let number s
(19)
h
+ kh T (t ) Nhc T qq T chT Nh (t ) + k
N + NG )h (t )
+ 2h T (t ) Nhc T qw (t ) - 2kh T (t ) Nhc T by (t ) .
-1 T
-1 T
+d
(t )s (G
( m + k )h T (t )hhT h (t )
+ ( m + k )h T (t )hhT h (t )
+ 2h T (t ) Nhc T Ax(t ) + 2( m + k )h T (t ) Nhc T bh T h (t )
T
-1
- k y 2 (t - t ) + d
+ k y 2 (t ) - k y 2 (t - t ) T
in
V& £ - x T (t )Qx(t ) - sh T (t )Q2h (t )
V& (t ) = x T (t )( AT P + PA) x(t ) + 2( m + k ) x T (t ) PbhT h (t ) + 2 x T (t ) Pqw (t ) - 2k x T (t ) Pby (t ) + h
(21)
Substituting (21) into the inequality (20), we obta
(q )dq . (18)
Differentiating (18) along the trajectories of (16)
(20)
+ d Pqq T P £ - Q < 0 .
t- t
+k
(t ) Nhc T bbT chT Nh (t )
where the number d > 0 . Let the number 0 < d < 0.5 be such that totic - Q1 + d I + (dm + 2dk - k ) PbbT P
).
Then the nonlinear system (13), (14), (16) is asymp stable.
2h T (t ) Nhc T Ax(t ) £ d
[w (t )]2
+ dk x T (t ) PbbT Px(t ) ,
Let number k be such that
T
T
kh
-1
( m + k )h T (t ) hh T h (t ) -1
[w (t )]2
-1
+ d - 1) .
(24)
Then we have
V& (t ) £ - x T (t )Qx(t ) - h
T
(t )Qh (t ) .
(25)
From the expression (25) follows asymptotic stabili system (13), (14), (16).
ty of
Remark. In some cases, the problem of choosing the 4. SIMULATION RESULTS parameters k , m and s of regulator (8), (9) which satisfy Consider the following nonlinear system theorem can arise (see (21), (23) and (24)). This choice presents no appreciable difficulties for known poly nomials b( p ) c( p) y (t ) = u (t ) + w (t ) , (29) t plant (1) and also for a definite a ( p ) , b( p) and c( p ) of he a( p) a( p) number C0 . However, if the parameters of the plant (1) are unknown, the problem of calculating k , m and s may where polynomials b( p) = b0 , a ( p) = p( p - a1 )( p - a2 ) , prove to be problematic. As was demonstrated by theorem, if c( p) = c0 p - c1 with unknown parameters b0 , a1 , a 2 , c0 , condition (23) is met, then there exists a number c1 , delay h=3 and nonlinear function s > m + k such that lim y (t ) = 0 . A possible variant of w (t ) = j ( y (t - t )) = y (t - t ) sin y (t - t ) . t fi ¥ adjustment of k , m and s lies in increasing them until the Choose the control according to equation (8) following goal condition is met
y (t ) < d
0
for t ‡ t1 ,
u (t ) = - a ( p )(m + k ) yˆ (t ) ,
(26)
where the number d 0 is set by the designer of the control system. This concept may be realized by using the following adjustment algorithm
where for relative degree r = 3 the polynomial a ( p) we choose as
a ( p) = ( p + 1) 2 .
∫
(27)
u (t ) = - ( p + 1) 2 ( m + k ) yˆ (t ) = = - ( m + k )( &yˆ&(t ) + 2 y&ˆ (t ) + yˆ (t )) ,
t0
~ where k = k + m and the function l (t ) is as follows for
0
0
y (t ) > d
~
s = s 0k 2 ,
(28)
where s 0 > 0 . Obviously, for this calculation of s there will be a time instant t1 > t 0 such that the following conditions of theorem are satisfied -1
( m + k )hhT + ( m + k ) Nhc T bbT ch T N
+ ( m + k )hhT + d + k Nhc T qq T ch T N + d
-1
Nhc T AAT ch T N
k NhcT bbT chT N £ - Q ,
-1
where the number 0 < d < 0.5 is chosen so that
- Q1 + d I + (dm + 2dk - k ) PbbT P + d Pqq T P £ - Q < 0 , and the parameters k ‡ C 02 (k - 1 + d - 1 ) .
(32)
where the positive numbers m > 0 , k > 0 and the function yˆ (t ) is calculated by the following algorithm
x&1 (t ) = sx 2 (t ), & x 2 (t ) = s (- k1x 1 (t ) - k 2x 2 (t ) + k1 y (t )), yˆ (t ) = x 1 (t ) ,
where the number l 0 > 0 . We take s as follows
- s Q2 + d
rm
0
y (t ) £ d 0 ,
for
(31)
Then we can rewrite the control in the following fo
t
~ k (t ) = l (q )dq ,
l l (t ) =
(30)
where the number s > m + k
(33) (34)
and parameters ki are k1 = 1 ,
k2 = 2 .
~ k = m + k = 5 and s = 12 . The results of a computer simulation for variable y (t ) for the case b0 = 1 a1 = 1 , a2 = 0.5 , c0 = 1 , c1 = 0.3 , t = 3 , y (0) = 2 and yˆ (0) = 0 are presented in Fig. 1. The results of a computer simulation for variable y (t ) for the case b0 = 1 a1 = - 1 , a 2 = - 0.5 , c0 = 1 , c1 = - 0.3 , t = 3 , y (0) = 5 and yˆ (0) = 0 are presented in Fig. 2. We can see that for two different groups of parameters presented control pr ovides asymptotic stability of equilibrium y = 0 .
Choose the parameters
5. CONCLUSION In the paper the problem of synthesis of control fo r timedelay nonlinear system (1) was considered. This pap er has extended the theory of output feedback control of t ime-delay nonlinear systems. The new control law (8) – (10), providing output asymptotic stability of system (1) was desig ned. Only measurements of the output, but not its derivatives were used. Dimension of the robust controller (8) – (10) is r - 1 and
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7), (28) is