- Email: [email protected]
Robust Adaptive Nonlinear Controller Design for Uncertain Nonlinear Systems
Copyright @ IFAC Adaptive Systems in Control and Signal Processing. Glasgow. Scotland. UK. 1998
ROBUST ADAPTIVE NONLINEAR CONTROLLER DESIGN FOR UNCERTAIN NONLINEAR SYSTEMS
Jae-kwan Lee and Ken-ichi Abe
Department of Electrical and Communication Engineering Graduate School of Engineering, Tohoku University Aoba-yama 05, Sendai 980-8579, Japan Email: lee.abeOabe .ecei.tohoku.ac .Jp
Abstract: Assuming whether input vector field is affected by constant uncertain parameters or not, this paper introduces two robust adaptive output tracking methods for a class of SI SO uncertain nonlinear systems with uncertainties which are due both to constant uncertain parameters entering linearly and to unmodeled dynamics comprised in a boundary range, via input-output feedback linearizations by state feedback. For such methods, while two control schemes to be designed externally follow indirect adaptive control process based on a combined local-domain of state variable vector and parameter estimates without causing no overparameterization for unknown constant parameters, they have two different internal structures, i.e. indirect and direct feedback linearization procedures, owing to two assumptions on input vector field. Copyright @ 19981FAC Keywords: Uncertain nonlinear systems, Output tracking, Robust adaptive control, Feedback linearization, State feedback
1. INTRODUCTION
Adaptive versIOns for nonlinear systems were announced from the mid-1980's (H. K. Khalil, A. Saberi, 1987) (K. -H. Nam, A. Arapostathis, 1988) and have been recently expanded in some works (M. Krstic. 1. Kanellakopoulos, P. V. Kokotovic. 1993) (1. Kanellakopoulos. P . V . Kokotovic, A. S. Morse. 1991). On the other hand, Isidori, Khalil. Marino and Tomei have studied theories of robust versions for nonlinear systems (R. Marino. P . Tomei. 1995) (.-\.. Isidori. 1996) (H. K. Khalil. 1994) (R. Marino. P. Tomei. 1993) .
The fact is that virtually all physical plants. such as aircraft, spacecraft, automobile or robot. are nonlinear in nature, and all control systems are nonlinear to a certain extent. Thus. the topic of nonlinear control design for the output tracking, a controller which forces the plant output to track a time-varying trajectory, has attracted particular attention. At the eighties, generalizations of pole placement and observer design techniques (J. S. :\.. Hepburn. W . ~I. Wonham. 1984) (M. Vidyasagar. 1985) for nonlinear systems were obtained by using differential geometric nonlinear theory which is a procedure to construct the linearizing coordinates. !vIore recently, we were confronted with more realistic problems that were caused by \"arious uncertainties about either plants or disturbances.
However. the nonlinear control algorithms studied in the above paragraph were not available to a class of nonlinear systems affected by both constant uncertain parameters. i.e. unknown constant parameters. and unmodeled dynamics. To solve such a complex situation. in this paper. we shall present two robust adaptive nonlinear control schemes for uncertain nonlinear systems
339
= g(x) . With this assumption, the first robust adaptive nonlinear controller is designed separately with robust adaptive law and inputoutput feedback linearization by state feedback. For this control scheme. let us first illustrate the robust adaptive law to identify e. The explanation can be evolved as the transformed form for (1):
with both constant uncertain parameters and unmodeled dynamics, via indirect and direct inputoutput feedback linearizations, respectively. The control schemes to be developed are designed and analyzed on the following assumptions: 1) the nonlinear vector fields expressing system properties are smooth, 2) the unmodeled dynamics is bounded in a compact set, 3) the nonlinear functions are given linearly with respect to the unknown constant parameters and the control input. 4) the full-state measurable condition is satisfied, and 5) the nonlinear systems have a well-defined relative degree.
y(x , e)
(s
-·T
Xi = W(S)[Bi Wi
x ERn, u E R,B E RP
y=h(x),yER
(1)
= W(s)[e i
Wi
1 S i S n. (5)
+ zo;], 1 Si S n .
Then, the estimation error fi
== Xi -Xi
= -ei + ((PT Wi fi = ej, 1 S i S n
ei
(6)
is got from
~fi)
(7)
with the parameter error cfJi == Bj - Bi. Finally, the robust adaptive law is given in the following theorem, by considering the Lyapunov-like function
(2)
with Coo nonlinear functions 1;, gi : Rn -+ Ri = 1"" ,p, and Coo vector field ~f. Our objective is the design of nonlinear feedback control schemes to force the output y to approximately track the reference trajectory Yrn (t) satisfying
i=O" 1 · ·· r Ilym(ijll
+ ~I; + Zoi],
'!.T
Xi
P
= I: Bdi(X) + ~f(x) i=1
(4)
From (5), the state estimate can be constructed as
where x is the state, u is the control input, B is the unknown constant parameters, y is the output, h : Rn -+ R is the Coo output function, and f, g are two Coo vector fields with g(x,B) "# O. We assume that the unknown constant parameters B are restricted to appearing linearly and the vector field f is affected by the unmodeled dynamics ~f:
f(x,B)
+ ~fi + gi(X)U
where ei is the unknown constant parameter parameterized linearly with the i - th non linear function f;. In (4), we can define the following linear expression with the strictly proper S P R transfer function W ( s) == s~ 1 ' the time invariant parameter vector BiT == (Bi 0 1), the state variable vector Wi == (ji(X) 0 Xi)T and the piecewise continuous signal input Zoj == gi(X) u:
Let us now consider SISO nonlinear systems
+ g(x , B)u,
B;!i(X)
+Xj, 1 Si S n
2. ROBUST ADAPTIVE NONLINEAR OUTPUT TRACKING
.:i: = f(x,B)
+ l)[xi] =
Theorem 2.1. If we select the robust adaptive law as
(3) (9)
where f!l~ > 0 and r is the relative degree to be introduced in a short time. For this, the control law and parameter update law must be designed on a local domain of both the state x and the parameter estimates of B, together with the robustness of all the closed-loop signals including which may be drifted to infinity due to ~f. Let i(x,B) be I:f=1 8;!i(X) in (2) and Xe be an equilibrium point of (1) . Moreover. we assume that .:If(x e ) = 0 and h(x e ) = 0 at Xe.
with the leakage term Wj (t) = Vj Ilfd I, it is clear that \'i, e j . fi, cfJ are bounded for some initial conditions.
e
Proof: This proof is given K. Lee. January 1998). 0
e
111
Reference (J. -
Remark 2. 1. To synthesize all the terms from the i - th to the n - th. we can consider the following
whole Lyapunov-like function and robust adaptive laws 2.1 Robust adaptive output tracking with indirect input-output feedback linearization
(10)
Here. we assume that the input vector field 9 is not affected by the unknown parameters B. i.e.
0= (j = - f . * W 340
W .
* (j
(ll)
with the element-by-element product
.*.
(2
= Lj(Z,~)h(x) +Lg(z,L j(r.iJ)h(x)u ~
~3(Z,iJ,
If the unknown constant parameters B in (1) are estimated by using (9), we can see (1) as a class o~ non linear plants with the knowable parameters B at time t . However, since (1) is affected by ~f, the feedback linearization procedure must be considered with f).f. To overcome the problem. with the definition /(x,O) == Er=l Oi f;(x) and Frobenius's theorem, we can define the following relative degree and the local diffeomorphism which are not dependent on f).f: in U6(B) x U.(x e ),
L g(z)L/(z,8)h(x)
= 0,
Lg(z)L'j~zl,8)h(x)
#0
Since Lg(z)L /(x.9)h(x) is again zero for all x in U 6 (B) x U. (x e), we shall differentiate again and again until the integer r:
~r
+Lt:>IL'j~zl,iJ)h(x) . With the above definitions A, Band Cl4>r, the normal dynamics (14) can be got. 0
0S i S r - 2 (12)
Remark 2.2. With the assumption Clf(xe) = 0 and the knowable value 0 at time t. it is obvious that Clct>(xe, 0, Clf(xe)) = 0 and 11f).4>(x, 0, Cl!)11 S KII~II for K > O.
and
(~,1])
= (~,,,. ,~'j~zl,8)h(x); z,(z,8)
1]r+ 1, . . . ,
...
zr(z,8)
1]n) .
Finally, assuming the locally exponentially stable of the zero dynamics q( 0,1]) in (1) and using the pole placement control input
(13)
Theorem 2.2. If (12) and (13) are assumed, (1) is input-output feedback linearizable into
~ = A/~
= L'j(z,9)h(x) + Lg(z)L'j~zl,9)h(x)u
=
with a Hurwitz polynomial coefficients Qi, i 0"" ,r - 1, the output tracking can be approximately achieved.
+ B/v + f).4>(x, 0, f).f(x»
r, = q(~, 1]) Y=~l
(14) 2.2 Robust adaptive output tracking with direct input-output feedback linearization
where (A/, Bt) is in Brunovsky controller form.
When the input vector field 9 is dependent on the unknown constant parameters B, i.e. g(x, B) = E;=I Bigi(X), B can not be estimated indirectly as the previous method. Thus, to solve the estimation problem, we here replace the state variables ~ with their estimates ~ by replacing the unknown constant parameters B appearing in ~ by their estimates B. Using B. ~ and Frobenius's theorem, let us define the following relative degree and the local diffeomorphism for (1) in U6 (B) x U. (x e ):
with
Proof: Let us differentiate y with respect to time
=
L g(z,9)Lj(z,iJ)h(x) O. 0 r l L 9 (z, 9)L l(z,O) - . h(x):;;i 0
t:
S
is r
-
2 (16)
and Since Lg(z)h(x) = 0 in (12). we have
(E, 1]) = (Ei
= Li7:.9) h(x), i = 1. .... r. ~
il = L j(Z,iJ)h(x) +
.',(z.O)
!-t:>lh(x),
~
In the second step with the knowable value time t.
l/r+I. ·· · .1),,) .
(1 i)
respectively. Assuming ~f(xe) = 0 and defining nx,B) == E;=I iJ i !i(x). the normal dynan1ics can be summarized by the following theorem.
B at
341
Theorem 2.3. If there exists a region U6(0) x Ut (x e ) satisfying (16) and (17), (1) is input-output feedback linearizable into
(21)
After continuing this differentiation operation up to r. we can obtain from Lg(x.8)Lj~xl.iJ)h(x) ;/= 0 in (16) that
(= A/~ + BIL' + W(x.e. ~f(x)) 17=q(~,'1)
-
(18)
y=6 where (AI, BI)
=0 -
-
0, AI
=
ai ~,
p
+ L(Oj - iJj)[L/j(x)L'j~xl.8)h(x)
= [WI"'"
lV
~r = L'j(x ,8)h(x) + Lg(z .8)L'j~xl.8)h(x)u
Brunovsky controller form.
IS In
j=l
wrIT
r-'
+u L gj(z) L /(z.8)
p
= ~)Oj -
W;(X,e)
ej)[L/j(z) Lj(:.8) h(x)],
+ L~/(x)L'j~zl.8)h(x) .
j=l
= 1,···, r
i
-
h()] a~r8: x + aB (22)
,
1
p
wr(x, u,e)
= I)Oj -
6j)[L/j(z)Ll~Z',8)h(x)
L g(x.8)L'j(X .iJ)h(x), B = L'j(z.iJ)h(x) and the input transformation (19), we can have the normal dynamics (18), finally. 0
With the definitions
j=1
+uLgj(z)Ll~z'.iJ) h(x)]
= [~I'''' '~'Y] T le-I ~4>k(X, e, ~f) = L~/L /(z .8)h(x), k = 1,··· ,r
A=
1
-
~(x,O,~f)
u
1
- -
== -_-_-[v - B({, '1)]
Remark 2.3. It is noted that unlike (14) affected by only ~f, (18) is dependent on both and ~f. Since ~f does not affect the determination of (17), it is natural that the zero dynamics q(O, '1) in (18) do not depend on only 8 but also ~f. If there exist two positive constants K\ and K2 such that IIxll Kllltll and 11~4>(x,B, ~f)11 K211xll for all iJ E U6(8),
e
(19)
A({,'1)
with
:s
-
-
A({, '1)
1
= Lg(z.iJ)L'j~z.8)h(x)
:s
B(~, '1) = L'j(z .8)h(x) . Proof: Rewriting (1) into
x=
f(x,O)
Since ~ is the unobservable variables with 8, let us now estimate ~ by the state observer dynamics with ~(O) = ~(O):
+ g(x,8)t/ + f(x.B)
+g(x,B)u - f(x,6) - g(x,e)u and using the assumption (16), the following result is obtained by differentiating y with respect to time t: •
where A. is the bottom companion matrix with the coefficients etj, i = 0" " ,r - 1 chosen above. This implies that we can get the error dynamics
P
~l = L /(z .iJ)h(x) + 2:)e j - ej)[L/)(x)h(x)]
s = .48 -
~j=l
+ L~/(x)h(x) . ,
e(
~2 = L l(x,8) h(x) + "--' ~3
m
2
= 1 + II~W
of (: (26)
P
I) OJ - 8j)[L
(25)
in which 8 == ~ - ( However. from the boundary condition in (23). the stability of (25) can not be previously decided with the exception of tldJ. Thus. we shall present the following normalized dynamics for (25) with the normalizing signal
(20)
.f
In the next step, since t) is the time function. ~2 is dependent on not only the unJ?odeled dynamics ~f but also the time function 8: •
W
/)(x)
L /(x'(l)
where 5 == ..:1... IV == !!::. To design 7n · 1n' ~(J) == ~d>, m a robust adaptive law satisfying the stability of
j=l
342
the closed-loop system with respect to :icl>, the following Lyapunov function is here used: (32)
(2i)
(33) where e x, == Xz - Xz, I = 0.3, <:> == B - (). According to the design procedures in Subsection 2.1, we can have the following normal dynamics for (29):
Theorem 2.4. If the robust adaptive law
8 = -nWTps - nL8
(28)
where P = pT > 0 with IIPI! ::; Amaz, without loss of generality, to the Lyapunov equation AT p+ p A -I, n nT > 0 is the adaptive gain matrix of dimension p, and L is the leakage matrix which is constructed by the diagonal scalar signals w;(t) = v;jlsl! with the design constants Vi > 0, i = 1"" ,p, is used for (26), all the states in (25) and (26) become approximately stable.
=
(34)
=
~z
~
B(z,6)
u
Then, the output tracking can be derived from
= Y~) + Or_l(y~-l) -
{r)
+ ... + OO(Ym
=
1
-
_ (v - B(x,B» .
A(x,B)
Then, using the pole placement control input v = Ym + 2(Ym - 6) + 2(Ym - 6), the output tracking may be approximately established. The parameter estimate 8( t) converges to a bounded range in Figure 1, and Figure 2 indicates the desired result of the output tracking.
Proof: The detailed account of this proof is shown in Reference (J. -K. Lee, January 1998). 0
v
=...(8cos(xd) +
- ~l)
.. --- .. _---
&--
by assuming that the zero dynamics q( 0, TJ) is locally exponentially stable.
3. SIMULATIONS AND RESULTS
. . .. _
To show performance of two nonlinear feedback control schemes presented, we shall consider the single-link rigid robot rotating on a vertical plane modeled by
Fig. 1. Closed-loop system result (1) - O(t)
i
XI =
.
Xz
X2
Y=
Xl
o:~
. .....
~
(29)
1 = Bcos(xd + IU +!:l.f
where B is the unknown constant parameter given by the true value 1 and !:l.f is the unmodeled dynamics caused by the parameter oscillation !:l.f = 0.3 cos(300t) COS(Xl)' The objective of control is to make the output Y track the desired position Fig. 2. Closed-loop system result (2) - y(t) , Ym(t)
(30) which is specified by the motion planning system.
3.2 Robust adaptive control (2) 3.1 Robu.'Jt adaptive control (1)
\Ve shall now use the second control method. Since B can not be separately estimated, the controller may be derived from the normal dynamics
Let us now start with the first robust adaptive nonlinear control scheme. The following terms indicate the state estimator. the parameter estimator and the Lyapunov function for (29), respectively,
~l
343
=
y=
(35 )
~2
=...... (e COS( x d ) +
1 . (I) u + (/J - /J) cos( x d
"~
8(z,6)
u= .1 .
A.(x, /J)
did not lead to any overparameterization for parameter identification. Second. it was shown that the systematic designs for robust adaptive laws were similar to those for uncertain linear systems. Lastlv it was also important to note that these nonli~~ar control methods led to the robustness and approximate output tracking of all the closedloop system with robust adaptive laws. Because the control schemes developed were designed on some restrictive assumptions such as the full-state measurable condition, the local domain of both state variables and parameter estimate, and the single-input, single-output nonlinear systems, we need to study some prospective works like output feedback control, global domain control, and multi-input, multi-output control based on the control schemes developed in this paper.
+~
.4.(z.6)
(v-B(x,8».
It is clear that the nonnal fonn (35) has the unobservable state variables ~ 1 and ~2' Thus, defining ~ as the state estimate and cancelling !:l. in (35), the following state observer is considered to estimate the un observable dynamics (35):
t
= ~2
+ (~2 - ~2) = ~2 ~2 = V + [-2(~1 - ~l) - 2(t2 - ~2)1
(36)
with the robust adaptive law
(37)
5. REFERENCES
where 82 == ~2 - ~2' I 0.8. Since !:l. !:l.f E £00' the nonnalizing signal is not considered here. The output tracking is finally achi_eved by using v Ym + 2(Ym - t2) + 2(Ym - (1). Then, the time responses of (37) and the output tracking are shown in Figure 3 and Figure 4, respectively.
=
=
A. Isidori (1996). Semiglobal robust regulation of nonlinear systems - in Colloquium on automatic control. Springer-Verlag. London. H. K. Khalil (1994). Robust servomechanism output feedback controllers for feedback linearizable systems. Automatica, 30, pp. 1587-1599. H. K. Khalil, A. Saberi (1987). Adaptive stabilization of a class of nonlinear systems using highgain feedback. IEEE Trans. Autom. Contr., 32, pp. 1031-1035. 1. Kanellakopoulos, P. V. Kokotovic, A . S. Morse (1991). Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans. Autom. Contr., 36, pp. 1241-
=
&--
---------------------------------------------_ .. _-_.--_ ... _-----
I
1253. J. -K. Lee (January 1998). Study on robust adaptive control of nonlinear systems. Ph.D. thesis. C.S.E .. Tohoku University, JAPAN. J. S. A. Hepburn, W . M. Wonham (1984). Structurally stable nonlinear regulation with step inputs. Math. Systems Theory, 17, pp. 319-
Fig. 3. Closed-loop system results (1) - e(t)
333. K. -H. Nam, A . Arapostathis (1988) . A modelreference adaptive control scheme for purefeedback nonlinear systems. IEEE Trans. Autom. Contr., 33 . pp. 803-811. M. Krstic. 1. Kanellakopoulos, P. V. Kokotovic (1993). Adapti\'e nonlinear control without overparametrization. Syst. Contr. Lett. , 19 . pp. 177-185. :'.1. Vidyasagar (1985). Control system synthesis : a factorization approach. MIT Press Series. R. Marino, P. Tomei (1993). Global adaptive output-feedback control of nonlinear systems, part i: Linear parameterization. IEEE Trans. A utom. Contr .. 38. pp. 1 i -32. R. :'.Iarino. P. Tomei (1995) . Nonlinear control design - geometnc , adaptive, and robust. Prentice-Hall. Inc .. Englewood Cliffs. N. J.
Fig. 4. Closed-loop system results (2) - y(t), Ym(t)
4. CONCLUSIONS AND PROSPECTS
In this paper, we introduced two robust adaptive nonlinear control schemes for SI S 0 nonlinear systems in the presence of both unknown constant parameters and unmodeled dynamics. Then, the main contributions of this paper could be summarized as shown below. First. it was noted that all the robust adaptive laws developed in this paper
344
ROBUST ADAPTIVE NONLINEAR CONTROLLER DESIGN FOR UNCERTAIN NONLINEAR SYSTEMS
Jae-kwan Lee and Ken-ichi Abe
Department of Electrical and Communication Engineering Graduate School of Engineering, Tohoku University Aoba-yama 05, Sendai 980-8579, Japan Email: lee.abeOabe .ecei.tohoku.ac .Jp
Abstract: Assuming whether input vector field is affected by constant uncertain parameters or not, this paper introduces two robust adaptive output tracking methods for a class of SI SO uncertain nonlinear systems with uncertainties which are due both to constant uncertain parameters entering linearly and to unmodeled dynamics comprised in a boundary range, via input-output feedback linearizations by state feedback. For such methods, while two control schemes to be designed externally follow indirect adaptive control process based on a combined local-domain of state variable vector and parameter estimates without causing no overparameterization for unknown constant parameters, they have two different internal structures, i.e. indirect and direct feedback linearization procedures, owing to two assumptions on input vector field. Copyright @ 19981FAC Keywords: Uncertain nonlinear systems, Output tracking, Robust adaptive control, Feedback linearization, State feedback
1. INTRODUCTION
Adaptive versIOns for nonlinear systems were announced from the mid-1980's (H. K. Khalil, A. Saberi, 1987) (K. -H. Nam, A. Arapostathis, 1988) and have been recently expanded in some works (M. Krstic. 1. Kanellakopoulos, P. V. Kokotovic. 1993) (1. Kanellakopoulos. P . V . Kokotovic, A. S. Morse. 1991). On the other hand, Isidori, Khalil. Marino and Tomei have studied theories of robust versions for nonlinear systems (R. Marino. P . Tomei. 1995) (.-\.. Isidori. 1996) (H. K. Khalil. 1994) (R. Marino. P. Tomei. 1993) .
The fact is that virtually all physical plants. such as aircraft, spacecraft, automobile or robot. are nonlinear in nature, and all control systems are nonlinear to a certain extent. Thus. the topic of nonlinear control design for the output tracking, a controller which forces the plant output to track a time-varying trajectory, has attracted particular attention. At the eighties, generalizations of pole placement and observer design techniques (J. S. :\.. Hepburn. W . ~I. Wonham. 1984) (M. Vidyasagar. 1985) for nonlinear systems were obtained by using differential geometric nonlinear theory which is a procedure to construct the linearizing coordinates. !vIore recently, we were confronted with more realistic problems that were caused by \"arious uncertainties about either plants or disturbances.
However. the nonlinear control algorithms studied in the above paragraph were not available to a class of nonlinear systems affected by both constant uncertain parameters. i.e. unknown constant parameters. and unmodeled dynamics. To solve such a complex situation. in this paper. we shall present two robust adaptive nonlinear control schemes for uncertain nonlinear systems
339
= g(x) . With this assumption, the first robust adaptive nonlinear controller is designed separately with robust adaptive law and inputoutput feedback linearization by state feedback. For this control scheme. let us first illustrate the robust adaptive law to identify e. The explanation can be evolved as the transformed form for (1):
with both constant uncertain parameters and unmodeled dynamics, via indirect and direct inputoutput feedback linearizations, respectively. The control schemes to be developed are designed and analyzed on the following assumptions: 1) the nonlinear vector fields expressing system properties are smooth, 2) the unmodeled dynamics is bounded in a compact set, 3) the nonlinear functions are given linearly with respect to the unknown constant parameters and the control input. 4) the full-state measurable condition is satisfied, and 5) the nonlinear systems have a well-defined relative degree.
y(x , e)
(s
-·T
Xi = W(S)[Bi Wi
x ERn, u E R,B E RP
y=h(x),yER
(1)
= W(s)[e i
Wi
1 S i S n. (5)
+ zo;], 1 Si S n .
Then, the estimation error fi
== Xi -Xi
= -ei + ((PT Wi fi = ej, 1 S i S n
ei
(6)
is got from
~fi)
(7)
with the parameter error cfJi == Bj - Bi. Finally, the robust adaptive law is given in the following theorem, by considering the Lyapunov-like function
(2)
with Coo nonlinear functions 1;, gi : Rn -+ Ri = 1"" ,p, and Coo vector field ~f. Our objective is the design of nonlinear feedback control schemes to force the output y to approximately track the reference trajectory Yrn (t) satisfying
i=O" 1 · ·· r Ilym(ijll
+ ~I; + Zoi],
'!.T
Xi
P
= I: Bdi(X) + ~f(x) i=1
(4)
From (5), the state estimate can be constructed as
where x is the state, u is the control input, B is the unknown constant parameters, y is the output, h : Rn -+ R is the Coo output function, and f, g are two Coo vector fields with g(x,B) "# O. We assume that the unknown constant parameters B are restricted to appearing linearly and the vector field f is affected by the unmodeled dynamics ~f:
f(x,B)
+ ~fi + gi(X)U
where ei is the unknown constant parameter parameterized linearly with the i - th non linear function f;. In (4), we can define the following linear expression with the strictly proper S P R transfer function W ( s) == s~ 1 ' the time invariant parameter vector BiT == (Bi 0 1), the state variable vector Wi == (ji(X) 0 Xi)T and the piecewise continuous signal input Zoj == gi(X) u:
Let us now consider SISO nonlinear systems
+ g(x , B)u,
B;!i(X)
+Xj, 1 Si S n
2. ROBUST ADAPTIVE NONLINEAR OUTPUT TRACKING
.:i: = f(x,B)
+ l)[xi] =
Theorem 2.1. If we select the robust adaptive law as
(3) (9)
where f!l~ > 0 and r is the relative degree to be introduced in a short time. For this, the control law and parameter update law must be designed on a local domain of both the state x and the parameter estimates of B, together with the robustness of all the closed-loop signals including which may be drifted to infinity due to ~f. Let i(x,B) be I:f=1 8;!i(X) in (2) and Xe be an equilibrium point of (1) . Moreover. we assume that .:If(x e ) = 0 and h(x e ) = 0 at Xe.
with the leakage term Wj (t) = Vj Ilfd I, it is clear that \'i, e j . fi, cfJ are bounded for some initial conditions.
e
Proof: This proof is given K. Lee. January 1998). 0
e
111
Reference (J. -
Remark 2. 1. To synthesize all the terms from the i - th to the n - th. we can consider the following
whole Lyapunov-like function and robust adaptive laws 2.1 Robust adaptive output tracking with indirect input-output feedback linearization
(10)
Here. we assume that the input vector field 9 is not affected by the unknown parameters B. i.e.
0= (j = - f . * W 340
W .
* (j
(ll)
with the element-by-element product
.*.
(2
= Lj(Z,~)h(x) +Lg(z,L j(r.iJ)h(x)u ~
~3(Z,iJ,
If the unknown constant parameters B in (1) are estimated by using (9), we can see (1) as a class o~ non linear plants with the knowable parameters B at time t . However, since (1) is affected by ~f, the feedback linearization procedure must be considered with f).f. To overcome the problem. with the definition /(x,O) == Er=l Oi f;(x) and Frobenius's theorem, we can define the following relative degree and the local diffeomorphism which are not dependent on f).f: in U6(B) x U.(x e ),
L g(z)L/(z,8)h(x)
= 0,
Lg(z)L'j~zl,8)h(x)
#0
Since Lg(z)L /(x.9)h(x) is again zero for all x in U 6 (B) x U. (x e), we shall differentiate again and again until the integer r:
~r
+Lt:>IL'j~zl,iJ)h(x) . With the above definitions A, Band Cl4>r, the normal dynamics (14) can be got. 0
0S i S r - 2 (12)
Remark 2.2. With the assumption Clf(xe) = 0 and the knowable value 0 at time t. it is obvious that Clct>(xe, 0, Clf(xe)) = 0 and 11f).4>(x, 0, Cl!)11 S KII~II for K > O.
and
(~,1])
= (~,,,. ,~'j~zl,8)h(x); z,(z,8)
1]r+ 1, . . . ,
...
zr(z,8)
1]n) .
Finally, assuming the locally exponentially stable of the zero dynamics q( 0,1]) in (1) and using the pole placement control input
(13)
Theorem 2.2. If (12) and (13) are assumed, (1) is input-output feedback linearizable into
~ = A/~
= L'j(z,9)h(x) + Lg(z)L'j~zl,9)h(x)u
=
with a Hurwitz polynomial coefficients Qi, i 0"" ,r - 1, the output tracking can be approximately achieved.
+ B/v + f).4>(x, 0, f).f(x»
r, = q(~, 1]) Y=~l
(14) 2.2 Robust adaptive output tracking with direct input-output feedback linearization
where (A/, Bt) is in Brunovsky controller form.
When the input vector field 9 is dependent on the unknown constant parameters B, i.e. g(x, B) = E;=I Bigi(X), B can not be estimated indirectly as the previous method. Thus, to solve the estimation problem, we here replace the state variables ~ with their estimates ~ by replacing the unknown constant parameters B appearing in ~ by their estimates B. Using B. ~ and Frobenius's theorem, let us define the following relative degree and the local diffeomorphism for (1) in U6 (B) x U. (x e ):
with
Proof: Let us differentiate y with respect to time
=
L g(z,9)Lj(z,iJ)h(x) O. 0 r l L 9 (z, 9)L l(z,O) - . h(x):;;i 0
t:
S
is r
-
2 (16)
and Since Lg(z)h(x) = 0 in (12). we have
(E, 1]) = (Ei
= Li7:.9) h(x), i = 1. .... r. ~
il = L j(Z,iJ)h(x) +
.',(z.O)
!-t:>lh(x),
~
In the second step with the knowable value time t.
l/r+I. ·· · .1),,) .
(1 i)
respectively. Assuming ~f(xe) = 0 and defining nx,B) == E;=I iJ i !i(x). the normal dynan1ics can be summarized by the following theorem.
B at
341
Theorem 2.3. If there exists a region U6(0) x Ut (x e ) satisfying (16) and (17), (1) is input-output feedback linearizable into
(21)
After continuing this differentiation operation up to r. we can obtain from Lg(x.8)Lj~xl.iJ)h(x) ;/= 0 in (16) that
(= A/~ + BIL' + W(x.e. ~f(x)) 17=q(~,'1)
-
(18)
y=6 where (AI, BI)
=0 -
-
0, AI
=
ai ~,
p
+ L(Oj - iJj)[L/j(x)L'j~xl.8)h(x)
= [WI"'"
lV
~r = L'j(x ,8)h(x) + Lg(z .8)L'j~xl.8)h(x)u
Brunovsky controller form.
IS In
j=l
wrIT
r-'
+u L gj(z) L /(z.8)
p
= ~)Oj -
W;(X,e)
ej)[L/j(z) Lj(:.8) h(x)],
+ L~/(x)L'j~zl.8)h(x) .
j=l
= 1,···, r
i
-
h()] a~r8: x + aB (22)
,
1
p
wr(x, u,e)
= I)Oj -
6j)[L/j(z)Ll~Z',8)h(x)
L g(x.8)L'j(X .iJ)h(x), B = L'j(z.iJ)h(x) and the input transformation (19), we can have the normal dynamics (18), finally. 0
With the definitions
j=1
+uLgj(z)Ll~z'.iJ) h(x)]
= [~I'''' '~'Y] T le-I ~4>k(X, e, ~f) = L~/L /(z .8)h(x), k = 1,··· ,r
A=
1
-
~(x,O,~f)
u
1
- -
== -_-_-[v - B({, '1)]
Remark 2.3. It is noted that unlike (14) affected by only ~f, (18) is dependent on both and ~f. Since ~f does not affect the determination of (17), it is natural that the zero dynamics q(O, '1) in (18) do not depend on only 8 but also ~f. If there exist two positive constants K\ and K2 such that IIxll Kllltll and 11~4>(x,B, ~f)11 K211xll for all iJ E U6(8),
e
(19)
A({,'1)
with
:s
-
-
A({, '1)
1
= Lg(z.iJ)L'j~z.8)h(x)
:s
B(~, '1) = L'j(z .8)h(x) . Proof: Rewriting (1) into
x=
f(x,O)
Since ~ is the unobservable variables with 8, let us now estimate ~ by the state observer dynamics with ~(O) = ~(O):
+ g(x,8)t/ + f(x.B)
+g(x,B)u - f(x,6) - g(x,e)u and using the assumption (16), the following result is obtained by differentiating y with respect to time t: •
where A. is the bottom companion matrix with the coefficients etj, i = 0" " ,r - 1 chosen above. This implies that we can get the error dynamics
P
~l = L /(z .iJ)h(x) + 2:)e j - ej)[L/)(x)h(x)]
s = .48 -
~j=l
+ L~/(x)h(x) . ,
e(
~2 = L l(x,8) h(x) + "--' ~3
m
2
= 1 + II~W
of (: (26)
P
I) OJ - 8j)[L
(25)
in which 8 == ~ - ( However. from the boundary condition in (23). the stability of (25) can not be previously decided with the exception of tldJ. Thus. we shall present the following normalized dynamics for (25) with the normalizing signal
(20)
.f
In the next step, since t) is the time function. ~2 is dependent on not only the unJ?odeled dynamics ~f but also the time function 8: •
W
/)(x)
L /(x'(l)
where 5 == ..:1... IV == !!::. To design 7n · 1n' ~(J) == ~d>, m a robust adaptive law satisfying the stability of
j=l
342
the closed-loop system with respect to :icl>, the following Lyapunov function is here used: (32)
(2i)
(33) where e x, == Xz - Xz, I = 0.3, <:> == B - (). According to the design procedures in Subsection 2.1, we can have the following normal dynamics for (29):
Theorem 2.4. If the robust adaptive law
8 = -nWTps - nL8
(28)
where P = pT > 0 with IIPI! ::; Amaz, without loss of generality, to the Lyapunov equation AT p+ p A -I, n nT > 0 is the adaptive gain matrix of dimension p, and L is the leakage matrix which is constructed by the diagonal scalar signals w;(t) = v;jlsl! with the design constants Vi > 0, i = 1"" ,p, is used for (26), all the states in (25) and (26) become approximately stable.
=
(34)
=
~z
~
B(z,6)
u
Then, the output tracking can be derived from
= Y~) + Or_l(y~-l) -
{r)
+ ... + OO(Ym
=
1
-
_ (v - B(x,B» .
A(x,B)
Then, using the pole placement control input v = Ym + 2(Ym - 6) + 2(Ym - 6), the output tracking may be approximately established. The parameter estimate 8( t) converges to a bounded range in Figure 1, and Figure 2 indicates the desired result of the output tracking.
Proof: The detailed account of this proof is shown in Reference (J. -K. Lee, January 1998). 0
v
=...(8cos(xd) +
- ~l)
.. --- .. _---
&--
by assuming that the zero dynamics q( 0, TJ) is locally exponentially stable.
3. SIMULATIONS AND RESULTS
. . .. _
To show performance of two nonlinear feedback control schemes presented, we shall consider the single-link rigid robot rotating on a vertical plane modeled by
Fig. 1. Closed-loop system result (1) - O(t)
i
XI =
.
Xz
X2
Y=
Xl
o:~
. .....
~
(29)
1 = Bcos(xd + IU +!:l.f
where B is the unknown constant parameter given by the true value 1 and !:l.f is the unmodeled dynamics caused by the parameter oscillation !:l.f = 0.3 cos(300t) COS(Xl)' The objective of control is to make the output Y track the desired position Fig. 2. Closed-loop system result (2) - y(t) , Ym(t)
(30) which is specified by the motion planning system.
3.2 Robust adaptive control (2) 3.1 Robu.'Jt adaptive control (1)
\Ve shall now use the second control method. Since B can not be separately estimated, the controller may be derived from the normal dynamics
Let us now start with the first robust adaptive nonlinear control scheme. The following terms indicate the state estimator. the parameter estimator and the Lyapunov function for (29), respectively,
~l
343
=
y=
(35 )
~2
=...... (e COS( x d ) +
1 . (I) u + (/J - /J) cos( x d
"~
8(z,6)
u= .1 .
A.(x, /J)
did not lead to any overparameterization for parameter identification. Second. it was shown that the systematic designs for robust adaptive laws were similar to those for uncertain linear systems. Lastlv it was also important to note that these nonli~~ar control methods led to the robustness and approximate output tracking of all the closedloop system with robust adaptive laws. Because the control schemes developed were designed on some restrictive assumptions such as the full-state measurable condition, the local domain of both state variables and parameter estimate, and the single-input, single-output nonlinear systems, we need to study some prospective works like output feedback control, global domain control, and multi-input, multi-output control based on the control schemes developed in this paper.
+~
.4.(z.6)
(v-B(x,8».
It is clear that the nonnal fonn (35) has the unobservable state variables ~ 1 and ~2' Thus, defining ~ as the state estimate and cancelling !:l. in (35), the following state observer is considered to estimate the un observable dynamics (35):
t
= ~2
+ (~2 - ~2) = ~2 ~2 = V + [-2(~1 - ~l) - 2(t2 - ~2)1
(36)
with the robust adaptive law
(37)
5. REFERENCES
where 82 == ~2 - ~2' I 0.8. Since !:l. !:l.f E £00' the nonnalizing signal is not considered here. The output tracking is finally achi_eved by using v Ym + 2(Ym - t2) + 2(Ym - (1). Then, the time responses of (37) and the output tracking are shown in Figure 3 and Figure 4, respectively.
=
=
A. Isidori (1996). Semiglobal robust regulation of nonlinear systems - in Colloquium on automatic control. Springer-Verlag. London. H. K. Khalil (1994). Robust servomechanism output feedback controllers for feedback linearizable systems. Automatica, 30, pp. 1587-1599. H. K. Khalil, A. Saberi (1987). Adaptive stabilization of a class of nonlinear systems using highgain feedback. IEEE Trans. Autom. Contr., 32, pp. 1031-1035. 1. Kanellakopoulos, P. V. Kokotovic, A . S. Morse (1991). Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans. Autom. Contr., 36, pp. 1241-
=
&--
---------------------------------------------_ .. _-_.--_ ... _-----
I
1253. J. -K. Lee (January 1998). Study on robust adaptive control of nonlinear systems. Ph.D. thesis. C.S.E .. Tohoku University, JAPAN. J. S. A. Hepburn, W . M. Wonham (1984). Structurally stable nonlinear regulation with step inputs. Math. Systems Theory, 17, pp. 319-
Fig. 3. Closed-loop system results (1) - e(t)
333. K. -H. Nam, A . Arapostathis (1988) . A modelreference adaptive control scheme for purefeedback nonlinear systems. IEEE Trans. Autom. Contr., 33 . pp. 803-811. M. Krstic. 1. Kanellakopoulos, P. V. Kokotovic (1993). Adapti\'e nonlinear control without overparametrization. Syst. Contr. Lett. , 19 . pp. 177-185. :'.1. Vidyasagar (1985). Control system synthesis : a factorization approach. MIT Press Series. R. Marino, P. Tomei (1993). Global adaptive output-feedback control of nonlinear systems, part i: Linear parameterization. IEEE Trans. A utom. Contr .. 38. pp. 1 i -32. R. :'.Iarino. P. Tomei (1995) . Nonlinear control design - geometnc , adaptive, and robust. Prentice-Hall. Inc .. Englewood Cliffs. N. J.
Fig. 4. Closed-loop system results (2) - y(t), Ym(t)
4. CONCLUSIONS AND PROSPECTS
In this paper, we introduced two robust adaptive nonlinear control schemes for SI S 0 nonlinear systems in the presence of both unknown constant parameters and unmodeled dynamics. Then, the main contributions of this paper could be summarized as shown below. First. it was noted that all the robust adaptive laws developed in this paper
344