- Email: [email protected]
Fuzzy Gain-Scheduling Control and its Application to Flight Control
FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
Copyright © 1999 IF AC 14th Triennial WorJd Congress, llcijing, P.R. China
14th World Congress oflFAC
P-8a-02-2
283
FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION TO FLIGHT CONTROL A. Fujilnori· and P. N. Nikiforukf
* Department of M~echanical Engineering, Shizuoka University, 3-5-1, Johoku, Hamamatsu 432-8561, Japan Fax: +81-53-478-106./, Email: [email protected] t Department of Mechanical Engineering University of Saskatchewan, Saskatoon, Saskatchewan S7N SA 9, Canada Fax: +1-306-966-5427
Abstract: This paper presents a gain-scheduling control using fuzzy logic called Fuzzy Gain-Scheduling (FGS) controL A Linear Parameter-Varying (LPV) system is first represented by a fuzzy model which consists of linear time-invariant systems at selected operating points. A FGS controller is constructed as a fullorder observer-based controller. The proposed method was applied to a longitudinal flight conirol problem in which the plant was regarded as a LPV system; that is, the altitude and the flight velocity varied. In simulation studies, c.ompared to a conventional LQG controller, the proposed FGS controller showed better control performance over the entire flight region. Copyright © /999IFAC Keywords: syst.em
1.
Fuzzy gain-scheduling, flight control, LMI, linear parameter-varying
INTRODUCTION
Gain-scheduling is one of the useful control approaches for Linear Parameter- Varying (LPV) and nonlinear systems (YIeressi and Paden, 1994; Shamma and A thans , 1990). A typical design procedure for a gain-scheduling controller is as follows: for selected opera.ting points or equilibrium points, multiple LTI plant models are first constructed, and for each LTl model a linear controller is next designed so as to stabilize the closedloop and to satisfy the control specifications. ,"Vhen the designed controllers are applied to the plant, the controllers are switched or interpolated between the operating points according to the scheduling paramet.ers. A disadvantage of gain-scheduling control is that it 1S necessary to guarantee the global stability of the closed-loop system over the
entire operating range. Although some basic analysis has been presented (Shamma and Athans, 1990; Chen, et al., 1993), a numerical simulation is generally used to verify the global stability. Another disadvantage is that the interpolation increases in complexity as the number of scheduling parameters increases. As an improvement, this paper proposes a new gain-scheduling control technique, called Fuzzy Ga.in-Scheduling (FGS) control, in which fuzzy logic is used to construct a model approximating an LPV or a nonlinear plant and to perform a control law (Tanaka and Sugeno, 1992). In the design of a controller for each operating point, Linear Matrix Inequalities (LMls) (Boyd, et al.) are used to obtain a control law which guarantees the global stability of the closed-loop system over the entire
7961
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
operat.ing range of the fuzzy model. There are many LMI formulations in control design (Iwa.saki and Skelton, 1994; Cahinet and Apkarian, 1994). Moreover, gain-scheduling H""" control has been discussed by other authors (Apkarian, et al., 1995; Apkarian and Cahinet, 1995; Packard, 1995). The contribution of this paper is that it proposes a new LMI formulation for FCS controllers which are based on a full-order observer-based controller.
2.
o
L-L-____
Fig. 1. functions.
[:::;] [~'
T/g
= N gi ,
[~:~:][ :~:n
~~
____
~
______
~
_______
Fuzzy set using triangle membership
~8,
To evaluate the suitability of a given set of T/j (j = 1 J • • • , g), the following variable is introduced £,
(i = 1", 0, r), (2)
where 1\ is the minimum operator in fuzzy logic. That is, the larger hi is, the more suitable is the i-th operating point LTI model for the LPV plant. Moreover, define eti as
hi
Er
£==1
h. i
(i = 1 " , r),
Qi ::-
,=1
(3
)
(4)
l.
It is called a fuzzy model (Tanaka and Sugeno, 1992) and is also one of the poly topic systems (Boyd, et al., 1994).
Fuzzy gain-schednling controller
This section presents a formulation of the FGS controller which is basically derived from a fullorder observer-based controller. For the i-th operating point, a full-order observer-based controller for Eq. (1) is given as follows:
If 1]1 = N1i and··· and Then
where u(t), yet) and x(t) are input, output and state vectors, respectively. The pair of (Ai, Bi, Cd is assumed to be stabilizable and detectable, and wet) and vet) are disturbance and noise vectors whose covariance matrices are WC> 0) and Z(> 0). The variable 't' represent.s time and may be omitted hereafter as the ease may be. N Ji is characterized by the membership function. Figure 1 shows an example using triangle membership functions f1,ji E [0,1] for Nji.
hi = J..lli(1/d 1\ ... 1\ jigi(T/g)
L
Then, a model which corresponds to the entire operating range is given by
2.2
(1)
;,] [ :::; ]
(i=l,···,r) x(t) E »in, 1.I(t) E »im , w(t) E yet), vet) E a?P
=
____
r
2: 0,
Qi
If 1]1 = NIi and ... and Then
£:.
~~
Njiol
Fuzzy model
+
Jlji
where
A fuzzy model which approximates a LPV or a nonlinear controlled plant is presented in this section. Let us select r points of a LPV plant or r equilibrium points of a non linear plant.. These points are called operating points in this paper. For each operating point, a LTI model is constructed. Let?]j (j = 1"", g) be premise variables which recognize the operating points and Nj; (i = 1,···, r) be fuzzy sets of T/j. For the i-th operating point (i = 1, ... ,r), the plant is modeled by the following:
Qi
Jlji-I
FUZZY GAIN-SCHEDULING
CONTROL 2.1
14th World Congress ofTFAC
i = { iJ = U
A;x + Biu Cix
"lg
= lVgi ,
+ Ki(Y-
ii)
+ DiU
(6)
== -FiX
(i=l J ··.,r)
where Fi and [{i (i = 1"", r) are m x n control gains and n x p filter gains, respectively. Similar to the deviation of the fuzzy model Eq. (5), using eti defined in Eq. (3), a control law which corresponds to the entire operating range is given by [
~]
=
tti>.;aJa i=l ;=1
[ A; - BiF;
l
x
t=l
-_~;(CI -
DIF))
: , ] [ : ] . (7)
Equation (7) is called a FGS controller, where Fi and K; are similar to a regulator and an observer gain for a LTI plant. Equation (7) is thus
7962
Copyright 1999 IF AC
ISBN: 008 0432484
FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
14th World Congress ofTFAC
(12)
constructed by interpolating r observer-based controllers among the operating points.
x,
Defining an error vector as e(t) ~ x(t) - x(t) and using Eqs. (5) and (7), the error system is given by r
r
LL
e=
0:; (X/{(A,;
- KiCI)e - G;w + K,;v}.(8)
;=1 [=1
Combining Eq. (8) with Eq. (5), the closed-loop system is given by r
r
T
!cL = LLLQ;QjQdALij,XL
+ GL,wd
(9)
;=1 j=1 1=1
where
[
x(t) ]
WL(t)
e(t)
~
[ wet) ]
= HT Hi
Qi
Y,
L:. 6,,)./ =
~ 0, rankHi Wo, Zi >0
Ri, {
Since QL,
(i
01
=g (i=l, .. ·,r)
= j = I)
(13)
(otherwise ).
~
0, Eq.
(11) is a sufficient condi-
tion that Eq. (9) is quadratically stable. In Eq. (12), Q. + Ft R;Fi is a weighting matrix which resembles a quadratic form of the optimal regulator, while Y(GiWiGT KiZiK{)Y is a covariance matrix which resembles a qua.dratic form of the Kalman filter. Oi,j,1 defined by Eq. (13) means that the weights a.re imposed on r operating points_ A parameterization of control and filter gains Fi and Xi (i :::: 1"", r) in Eq. (7) is given as follows.
+
Theorem 1 Equation (11) holds only if there exist n x n. positive definite matrices X and Y, m x n matrices M; and n x p matrices Ni (i = 1"", r) which satisfy the following r(r + 1) LMls
vet)
o
+ XAT
A,X
[
o ]. Ki
- BiM, - M,T BT
XH!
H,X
-Iq
M.
0
-~.-, MT
<0
3.
CONTROLLER DESIGN USING
The parameters to be designed are the control and filter gains F, and J{i (i = 1,···,r) which guarantee the global stability of the closed-loop system Eq. (9) over the entire operating range. In this paper, the global stability is discussed within the frame of quadratic stability with a Lyapunov function. LMls are then derived to guarantee the global stability and to obtain Fi and K i . Many researchers have proposed LMI formulations for several purposes; Hoo bound and pole-region constraint amongst others (Boyd, et al., 1994; Gahinet, et al., 1995). This section puts forth a new LMI formulation for 1'i and Ki. The dosed-loop system Eq. (9) is quadratically
stable only if there exists a positive definite function V
+ Aj)X + X(Ai + A 1Y + BJM.)T < 0 YA, + ATy - NiG, - eT N?, (Ai
(BiM)
+ BJM;)
[
GTY
(15) N,
YG i
_W,-l
0
0
_Z,-l
"iT " ,
(16)
<0
YeA, +Al) + (Ai + Al)Ty -(NiGI
]
+ N,Ci)T <
(NiC/
+ N/C.) (17)
0
(i=l, ... ,r, l=i+l, .. ·,r).
Then, Fi and Xi in Eq. (7) are given by Fi Ki
= MiX- 1
(18)
=
(19)
y-1Ni
(i=l, ... ,r) To prove Theorem I, the following Lemma is used.
Lenlllla 1 Let E and P be matrices defined as
[XO-I
6. xLPLXL T = = XLT
yo]
XL
(10)
such that
~~
< -
tt t
(11)
Cli
,,,,1 J=1 1",1
where
d
[
E~'
E12] , En
o ] (20) )1P2
where Ell and En are n. x n stable matrices, PI and P2 are n. x n positive definite matrices, and f.l is a positive scaJar valUE. Then, the following statements are equivalent
(i) there exists a positive riefinite matrix P such
QL, ~
[
(14)
-(BiMj
UI/II
]
that
~+For&~
0
Y(G,
wie; + K,Z,K'{)Y
]
PE+ETp
(21)
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Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress ofTFAC
FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
(ii) there exist positive definite matrices PI and P 2 such that
PIE ll P 2 E 22 l~t
Now,
+ EflH < 0 + E:;2P2 < O.
(22) (23)
In summary, the design procedure for the proposed FGS controller is as follows.
us prove Theorem 1. Substituting Eq. (9) into
Proof of Theorem 1: Eq. (11) gives
Using the Schur Complement (Gahinet, et aL, 1995), Eqs. (32) and (34) are transformed into Eqs. (14) and (16). Equations (18) and (19) follow from the definition of M, and Ni Eq. (31). 0
Step 1: Construct a fuzzy plant model Eq. (5) which represents a LPV or a nonlinear plant. Step 2: Select weights and covariance Qi, Ri, W, and Zi (i = 1"", r) in Eq. (12).
r
< -
r
r
LLL
(24)
CliOjCllXIbi,jlQLiXL.
i=1 j=l 1=1
Using P L given by Eq. (10) to Eq. (24), a sufficient condition for the quadratic stability of Eq. (9) is to satisfy the following matrix inequalities [
Step 4: Obtain Fi and Ki (i = 1,,,,, r) from Eqs. (18) and (19).
X-I AF.. + Aj;.. X- ' + Q. + Ft RiF.
-Ft BTx- '
4.
-X- 1 BiF,
<
O.
(25)
,,+ ATFi,.X- 1
X-I A
F'J
-FI BT X-I
_X- 1 B·P
]
'J
Y AKi'
LONGITUDINAL FLIGHT CONTROL PROBLEM
]
YAK,; + A~ .. Y + Y(G. W,G! + K,Z,Kf)Y
[
Step 3: Find X > 0, Y > 0, 1'vii and Ni (i = 1, ... ,r) satisfying Eqs. (14) to (17). If they are not found, go back to Step 2 and re-select Q i, R;., W i and Zi· Otherwise, go to Step 4.
+ AT" Y
< O. (26)
(i=l,"',r, j=i+l.···,r, l=i+l,· .. ,r)
To illustrate the proposed FGS control, simulation studies were carried out the longitudinal flight control of the Lockheed P2V -7 aircraft. In this example, the flight conditions, the altitude and the flight velocity vary; that is, the plant is given as a LPV system.
w}lere
A pij ~ Ai - BiFj,
AKiI ~ Ai - KiC!.
4.1
From Lemma I, Eqs. (25) and (26) are equivalent to X-I AFii + AE,X- 1 + Qi + X-I (AF"
YAK"
Fr RtFi < 0
+ Apj,) + (Aft", + AF,i)T X-I < 0
(27) (28)
+ A~i' Y
+Y(G,WiG[
+ KiZiKT)Y <
0
(29)
+ AK,,) + (AKii + AK,,)Ty < O. (30) (i = 1"" ,T, j = i + 1,···, T, 1= i + 1"", r)
Y(AKiI
To make Eqs. (27) to (30) linea. with respect to the unknown variables, the following variables are introduced
Mi
D.
= FiX,
Ni
= }T}"\i· D.
Equation of {ongit'udinalmotion of ai.craft
A state-space equation for the longitudinal motion of an aircraft is
[~l [~:][:] where
A=
[
Mu
(31)
(i= 1, ... ,.)
+ XA;
+XQ.X
}(a.
ZulU
ZajU
0
0
+ M",Z,,/U
< 0
1
1
Mq
0
(32)
+ AfoZ"jU
+ ZqjU
0
+ AJ)X + X(A; + A-)7' -(Bdlfj + BjAIi) - (B. Ai) + BjMif <
Mc,
0
0
- BiM; - (B;M.)T
+ Mt R i M,
X"
-g
Then, Eqs. (27) to (30) are written as .4, X
(36)
+ MaP + Zq/U)
1
(Ai
0 (33) YA i +ATy - N,G. - (N,Cif +YG. WiGty + l'{iZiN,r < 0 (.34) Y(Ai + Ad + (Ai + AI)Ty -(NiG, + NlC,) - (N,C, + N!G,)T < O. (35) (i=l, ... ,r, i=i+l, ... ,r, i=i+l, ... ,r)
B=
l
c= [0
,~aZ8IU 1
Z6/U
0
Mo
+
'
-1 1 0].
x ~ [u a
{J
q]T E 3?"
b E 3?1
7964
Copyright 1999 IF AC
ISBN: 008 0432484
FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
14th World Congress ofTFAC
H=4000 [m], U=100 [m/s]
6
--
4
:~
0>
'"
~
~
- -
FGS
LOG Open-loop
.'
I
11 -2 \ -4
Fig. 2.
Membership functions of Hand U.
In the state vector x, u is the incremental velocity in the flight direction, a the angle of attack, o t.he pitch angle and q the angular velocity of the pitch angle. , is the angle of the flight path, and fj is the deflection angle of an elevator. In the matrices A and B, X", Za, ... are stability derivatives in the frame of the stability axis, and they are changeable with respect to the altitude, flight velocity, angle of attack, Mach number, etc. In this numerical example, Eq. (36) was regarded as a LPV system whose varying-parameters are the altitude H and the flight velocity U. Their ranges were respectively given as HI < H < H", and U/ ::; U ~ U" Fuzzy rules were given at the following four operating points; (H, U) = (Hr, UI), (HI, Uu ), (Hu, U/), and CH... , UtA)· A LTI model was constructed at each operating point. Membership functions, PH and ~u, of Hand U were given as Fig. 2. Then. hi was given by
hi=J1-H(H)l\l-Iu(U) 4.2
(i=1, .. ·,4).
(;17)
Numerical simulation
~ umerical data of the Lockheed P2V -7 aircraft were used in the simulation (Isozaki, et al., 1980). The ranges of Hand U were given as 1000 < H < 7000 [m] and 50 :::; U ::; 150 [m/s]. Since the m': trix C in Eq. (36) was constant, the LMIs given by Eq. (17) were not needed. The number of LMls to be solved was then 4 x (4 + 1)/2 + 4 = 20. To find matrices X > 0, Y > 0, Mo and Ni (i = 1, .. ·,4) which satisfy the 20 LMIs, the LMI Control Toolbox in Matlab (Gahinet95, et al., 1995) was used for the calculation. After finding X I Y, M; and Ni, the controJ and the filter gains Fi and K j were obtained from Eqs. (lS) and (19). A FGS cont.roller was then constructed as Eq. (7). To compare the FGS controller with a conventional fixed-structure controller, a. LQG controller was designed at (H, U) = (4000, 100).
Figures 3 and 4 show the responses of let) in which the flight eondition was (H, U) (4000,100) and (4000,135), respectively. At (H, U) = (4000,100), the response using the LQG controller was superior to the response using the FGS controller
0
10
20 30 t [sec)
40
50
Fig. 3. Time responses comparing FGS controller to LQG controller at. (H, U) = (4000,100). H=4000 [m]. U=135 [m/s] 6r---~--------~----~--~
FGS
LOG Open-loop
4
Fig. 4. Time responses comparing FGS controller to LQG controller at (H. U) = (4000,135). from t.he viewpoint of the settling time. While at (H, U) (4000,135), the FGS controller showed better performance than the LQG one.
=
To evaluate the control performance over the entire flight range using the time response during t E [0,50], the following index Cost was introduced.
Cost ~
1
50
0
(38)
th(t)ldt
Figures.) and 6 show Cost usiug the FGS and LQG controllers, where Cast was limited to 10000 if Cost was greater than 10000. Cost using the LQG controller showed high values in the low and high flight velocity regions. In particular, the dosedloop system using the LQG controller was unstable in the regions around (H, U) = (7000,50) and (1000, 150). On the other hand, Cost using the FGS controller was less than 600 over the entire flight range. Thus, it is concluded that the FGS controller performed bet.ter than the LQG controller over the entire range of the varyingparameters.
=
5.
CONCLUDING REMARKS
A design of gain-scheduling control using fuzzy logic, called Fuzzy Gain-Scheduling (FGS) COll-
7965
Copyright 1999 IF AC
ISBN: 008 0432484
FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
FGS
14th World Congress ofTFAC
Apkarian, P., and P. Gabinet (1995). A Convex Characterization of Gain-Scheduled H 00 Controllers. IEEE Transactions on Automatic Control, 40, 853-864.
10000 9000
Boyd, S., L. E. Ghaoui, E. Feron and V. Balakrishnan (1994). Linear Matrix Inequailties in System and Control Theory, (SIAM Studies in Ap-
plied Mathematics, Vo1.l5), Philadelphia.
'000
H[r'I"I]
Fig. 5. controller.
00
Chen, Z., Y. Hayakawa and S. Fujii (1993). On the Stability of Plants with Variable Operating Conditions. Proc. o/th.e 22nd SlCE Symposium on Control Theory, 55-60.
U (mf:5;)
Performance index Cost using FGS LOG
Gahinet, P., and P. Apkarian (1994). A Linear Matrix Inequality Approach to Hoo Control. Int. J. Robust and Nonlinear Control, 4, 421-448. Gahinet, P., P. Apkarian and M. Chilali (1994). Affine Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty. Proc. CDC, 2026-2031.
tSO
Fig. 6. con troller.
Performance index Cost using LQG
trol, has been presented in this paper. Compared to gain-scheduling Hoo control (Apkarian et al., 1995; Apkarian and Gahinet, 1995; Packard, 1995), an advantage of the FGS controller is that the only two parameters of the cont.roller have to be obtained. They are the control and filter gains, which are easily parameterized by the LMls solutions. The proposed method was applied to a numerica.l example of a longitudinal flight control problem in which the plant WM regarded as a LPV system; that is, the altitude and the flight velocity were varying. Compared to a conventional LQG controller, the FGS controller showed better cont.rol performance over the entire flight region. The sta.bility criterion used in this paper was a Lyapunov function with a fixed positive definite matrix. To obtain a FGS controller which stabilizes a LPV plant over a larger operating range, it may be possible to use a parameter-dependent Lyapunov function (Gahinet, et al., 1994).
Gahinet, P., A. Nemirovski, A. J. Laub and M. Chilali (1995). LM1 Control Toolbox, The Math Works Inc., Natick. Isozaki, K., K. MMuda, A. Taniuchi and M. Watari (1980). Flight Test Evaluation of Variable Stability Airplane. KH.J. Tech.nical Review, 75, 50-58. Iwasaki, T., and R. E. Skelton (1994). All Controllers for the General He;:, Control Problem: LMI Existence Conditions and State Space Formulas. Auiomaiica, 3D, 1307-1317. Meressi, T. and B. Paden (1994). Gain Scheduled H co Controllers for a Two Link Flexible Manipulator. Journal Guidance, Control, and Dynamlcs, 17,537-543. Packard, A. (1995). Gain Scheduling via Linear Fractional Transformations. Systems and Control Letters, 22, 79-92. Shamma, J, S. and M. Athans (1990), Analysis of K onlinear Gain Scheduled Control. IEEE Transactions on Automatlc Control. AC-35,
898-907. Tanaka, K., and S, Sugeno (1992). Stability Analysis and Design of Fuzzy Control Systems. Fuzzy Sets and Systems, 45, 135-156.
REFERENCES Apkarian,
P.,
P.
Gahinet
and
G.
Becker
(1995). Self-Scheduled H cc Control of Linear Parameter-Varying Systems: A Design Example. Automatica, 31, 1251-1261.
7966
Copyright 1999 IF AC
ISBN: 008 0432484
Copyright © 1999 IF AC 14th Triennial WorJd Congress, llcijing, P.R. China
14th World Congress oflFAC
P-8a-02-2
283
FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION TO FLIGHT CONTROL A. Fujilnori· and P. N. Nikiforukf
* Department of M~echanical Engineering, Shizuoka University, 3-5-1, Johoku, Hamamatsu 432-8561, Japan Fax: +81-53-478-106./, Email: [email protected] t Department of Mechanical Engineering University of Saskatchewan, Saskatoon, Saskatchewan S7N SA 9, Canada Fax: +1-306-966-5427
Abstract: This paper presents a gain-scheduling control using fuzzy logic called Fuzzy Gain-Scheduling (FGS) controL A Linear Parameter-Varying (LPV) system is first represented by a fuzzy model which consists of linear time-invariant systems at selected operating points. A FGS controller is constructed as a fullorder observer-based controller. The proposed method was applied to a longitudinal flight conirol problem in which the plant was regarded as a LPV system; that is, the altitude and the flight velocity varied. In simulation studies, c.ompared to a conventional LQG controller, the proposed FGS controller showed better control performance over the entire flight region. Copyright © /999IFAC Keywords: syst.em
1.
Fuzzy gain-scheduling, flight control, LMI, linear parameter-varying
INTRODUCTION
Gain-scheduling is one of the useful control approaches for Linear Parameter- Varying (LPV) and nonlinear systems (YIeressi and Paden, 1994; Shamma and A thans , 1990). A typical design procedure for a gain-scheduling controller is as follows: for selected opera.ting points or equilibrium points, multiple LTI plant models are first constructed, and for each LTl model a linear controller is next designed so as to stabilize the closedloop and to satisfy the control specifications. ,"Vhen the designed controllers are applied to the plant, the controllers are switched or interpolated between the operating points according to the scheduling paramet.ers. A disadvantage of gain-scheduling control is that it 1S necessary to guarantee the global stability of the closed-loop system over the
entire operating range. Although some basic analysis has been presented (Shamma and Athans, 1990; Chen, et al., 1993), a numerical simulation is generally used to verify the global stability. Another disadvantage is that the interpolation increases in complexity as the number of scheduling parameters increases. As an improvement, this paper proposes a new gain-scheduling control technique, called Fuzzy Ga.in-Scheduling (FGS) control, in which fuzzy logic is used to construct a model approximating an LPV or a nonlinear plant and to perform a control law (Tanaka and Sugeno, 1992). In the design of a controller for each operating point, Linear Matrix Inequalities (LMls) (Boyd, et al.) are used to obtain a control law which guarantees the global stability of the closed-loop system over the entire
7961
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
operat.ing range of the fuzzy model. There are many LMI formulations in control design (Iwa.saki and Skelton, 1994; Cahinet and Apkarian, 1994). Moreover, gain-scheduling H""" control has been discussed by other authors (Apkarian, et al., 1995; Apkarian and Cahinet, 1995; Packard, 1995). The contribution of this paper is that it proposes a new LMI formulation for FCS controllers which are based on a full-order observer-based controller.
2.
o
L-L-____
Fig. 1. functions.
[:::;] [~'
T/g
= N gi ,
[~:~:][ :~:n
~~
____
~
______
~
_______
Fuzzy set using triangle membership
~8,
To evaluate the suitability of a given set of T/j (j = 1 J • • • , g), the following variable is introduced £,
(i = 1", 0, r), (2)
where 1\ is the minimum operator in fuzzy logic. That is, the larger hi is, the more suitable is the i-th operating point LTI model for the LPV plant. Moreover, define eti as
hi
Er
£==1
h. i
(i = 1 " , r),
Qi ::-
,=1
(3
)
(4)
l.
It is called a fuzzy model (Tanaka and Sugeno, 1992) and is also one of the poly topic systems (Boyd, et al., 1994).
Fuzzy gain-schednling controller
This section presents a formulation of the FGS controller which is basically derived from a fullorder observer-based controller. For the i-th operating point, a full-order observer-based controller for Eq. (1) is given as follows:
If 1]1 = N1i and··· and Then
where u(t), yet) and x(t) are input, output and state vectors, respectively. The pair of (Ai, Bi, Cd is assumed to be stabilizable and detectable, and wet) and vet) are disturbance and noise vectors whose covariance matrices are WC> 0) and Z(> 0). The variable 't' represent.s time and may be omitted hereafter as the ease may be. N Ji is characterized by the membership function. Figure 1 shows an example using triangle membership functions f1,ji E [0,1] for Nji.
hi = J..lli(1/d 1\ ... 1\ jigi(T/g)
L
Then, a model which corresponds to the entire operating range is given by
2.2
(1)
;,] [ :::; ]
(i=l,···,r) x(t) E »in, 1.I(t) E »im , w(t) E yet), vet) E a?P
=
____
r
2: 0,
Qi
If 1]1 = NIi and ... and Then
£:.
~~
Njiol
Fuzzy model
+
Jlji
where
A fuzzy model which approximates a LPV or a nonlinear controlled plant is presented in this section. Let us select r points of a LPV plant or r equilibrium points of a non linear plant.. These points are called operating points in this paper. For each operating point, a LTI model is constructed. Let?]j (j = 1"", g) be premise variables which recognize the operating points and Nj; (i = 1,···, r) be fuzzy sets of T/j. For the i-th operating point (i = 1, ... ,r), the plant is modeled by the following:
Qi
Jlji-I
FUZZY GAIN-SCHEDULING
CONTROL 2.1
14th World Congress ofTFAC
i = { iJ = U
A;x + Biu Cix
"lg
= lVgi ,
+ Ki(Y-
ii)
+ DiU
(6)
== -FiX
(i=l J ··.,r)
where Fi and [{i (i = 1"", r) are m x n control gains and n x p filter gains, respectively. Similar to the deviation of the fuzzy model Eq. (5), using eti defined in Eq. (3), a control law which corresponds to the entire operating range is given by [
~]
=
tti>.;aJa i=l ;=1
[ A; - BiF;
l
x
t=l
-_~;(CI -
DIF))
: , ] [ : ] . (7)
Equation (7) is called a FGS controller, where Fi and K; are similar to a regulator and an observer gain for a LTI plant. Equation (7) is thus
7962
Copyright 1999 IF AC
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FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
14th World Congress ofTFAC
(12)
constructed by interpolating r observer-based controllers among the operating points.
x,
Defining an error vector as e(t) ~ x(t) - x(t) and using Eqs. (5) and (7), the error system is given by r
r
LL
e=
0:; (X/{(A,;
- KiCI)e - G;w + K,;v}.(8)
;=1 [=1
Combining Eq. (8) with Eq. (5), the closed-loop system is given by r
r
T
!cL = LLLQ;QjQdALij,XL
+ GL,wd
(9)
;=1 j=1 1=1
where
[
x(t) ]
WL(t)
e(t)
~
[ wet) ]
= HT Hi
Qi
Y,
L:. 6,,)./ =
~ 0, rankHi Wo, Zi >0
Ri, {
Since QL,
(i
01
=g (i=l, .. ·,r)
= j = I)
(13)
(otherwise ).
~
0, Eq.
(11) is a sufficient condi-
tion that Eq. (9) is quadratically stable. In Eq. (12), Q. + Ft R;Fi is a weighting matrix which resembles a quadratic form of the optimal regulator, while Y(GiWiGT KiZiK{)Y is a covariance matrix which resembles a qua.dratic form of the Kalman filter. Oi,j,1 defined by Eq. (13) means that the weights a.re imposed on r operating points_ A parameterization of control and filter gains Fi and Xi (i :::: 1"", r) in Eq. (7) is given as follows.
+
Theorem 1 Equation (11) holds only if there exist n x n. positive definite matrices X and Y, m x n matrices M; and n x p matrices Ni (i = 1"", r) which satisfy the following r(r + 1) LMls
vet)
o
+ XAT
A,X
[
o ]. Ki
- BiM, - M,T BT
XH!
H,X
-Iq
M.
0
-~.-, MT
<0
3.
CONTROLLER DESIGN USING
The parameters to be designed are the control and filter gains F, and J{i (i = 1,···,r) which guarantee the global stability of the closed-loop system Eq. (9) over the entire operating range. In this paper, the global stability is discussed within the frame of quadratic stability with a Lyapunov function. LMls are then derived to guarantee the global stability and to obtain Fi and K i . Many researchers have proposed LMI formulations for several purposes; Hoo bound and pole-region constraint amongst others (Boyd, et al., 1994; Gahinet, et al., 1995). This section puts forth a new LMI formulation for 1'i and Ki. The dosed-loop system Eq. (9) is quadratically
stable only if there exists a positive definite function V
+ Aj)X + X(Ai + A 1Y + BJM.)T < 0 YA, + ATy - NiG, - eT N?, (Ai
(BiM)
+ BJM;)
[
GTY
(15) N,
YG i
_W,-l
0
0
_Z,-l
"iT " ,
(16)
<0
YeA, +Al) + (Ai + Al)Ty -(NiGI
]
+ N,Ci)T <
(NiC/
+ N/C.) (17)
0
(i=l, ... ,r, l=i+l, .. ·,r).
Then, Fi and Xi in Eq. (7) are given by Fi Ki
= MiX- 1
(18)
=
(19)
y-1Ni
(i=l, ... ,r) To prove Theorem I, the following Lemma is used.
Lenlllla 1 Let E and P be matrices defined as
[XO-I
6. xLPLXL T = = XLT
yo]
XL
(10)
such that
~~
< -
tt t
(11)
Cli
,,,,1 J=1 1",1
where
d
[
E~'
E12] , En
o ] (20) )1P2
where Ell and En are n. x n stable matrices, PI and P2 are n. x n positive definite matrices, and f.l is a positive scaJar valUE. Then, the following statements are equivalent
(i) there exists a positive riefinite matrix P such
QL, ~
[
(14)
-(BiMj
UI/II
]
that
~+For&~
0
Y(G,
wie; + K,Z,K'{)Y
]
PE+ETp
(21)
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FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
(ii) there exist positive definite matrices PI and P 2 such that
PIE ll P 2 E 22 l~t
Now,
+ EflH < 0 + E:;2P2 < O.
(22) (23)
In summary, the design procedure for the proposed FGS controller is as follows.
us prove Theorem 1. Substituting Eq. (9) into
Proof of Theorem 1: Eq. (11) gives
Using the Schur Complement (Gahinet, et aL, 1995), Eqs. (32) and (34) are transformed into Eqs. (14) and (16). Equations (18) and (19) follow from the definition of M, and Ni Eq. (31). 0
Step 1: Construct a fuzzy plant model Eq. (5) which represents a LPV or a nonlinear plant. Step 2: Select weights and covariance Qi, Ri, W, and Zi (i = 1"", r) in Eq. (12).
r
< -
r
r
LLL
(24)
CliOjCllXIbi,jlQLiXL.
i=1 j=l 1=1
Using P L given by Eq. (10) to Eq. (24), a sufficient condition for the quadratic stability of Eq. (9) is to satisfy the following matrix inequalities [
Step 4: Obtain Fi and Ki (i = 1,,,,, r) from Eqs. (18) and (19).
X-I AF.. + Aj;.. X- ' + Q. + Ft RiF.
-Ft BTx- '
4.
-X- 1 BiF,
<
O.
(25)
,,+ ATFi,.X- 1
X-I A
F'J
-FI BT X-I
_X- 1 B·P
]
'J
Y AKi'
LONGITUDINAL FLIGHT CONTROL PROBLEM
]
YAK,; + A~ .. Y + Y(G. W,G! + K,Z,Kf)Y
[
Step 3: Find X > 0, Y > 0, 1'vii and Ni (i = 1, ... ,r) satisfying Eqs. (14) to (17). If they are not found, go back to Step 2 and re-select Q i, R;., W i and Zi· Otherwise, go to Step 4.
+ AT" Y
< O. (26)
(i=l,"',r, j=i+l.···,r, l=i+l,· .. ,r)
To illustrate the proposed FGS control, simulation studies were carried out the longitudinal flight control of the Lockheed P2V -7 aircraft. In this example, the flight conditions, the altitude and the flight velocity vary; that is, the plant is given as a LPV system.
w}lere
A pij ~ Ai - BiFj,
AKiI ~ Ai - KiC!.
4.1
From Lemma I, Eqs. (25) and (26) are equivalent to X-I AFii + AE,X- 1 + Qi + X-I (AF"
YAK"
Fr RtFi < 0
+ Apj,) + (Aft", + AF,i)T X-I < 0
(27) (28)
+ A~i' Y
+Y(G,WiG[
+ KiZiKT)Y <
0
(29)
+ AK,,) + (AKii + AK,,)Ty < O. (30) (i = 1"" ,T, j = i + 1,···, T, 1= i + 1"", r)
Y(AKiI
To make Eqs. (27) to (30) linea. with respect to the unknown variables, the following variables are introduced
Mi
D.
= FiX,
Ni
= }T}"\i· D.
Equation of {ongit'udinalmotion of ai.craft
A state-space equation for the longitudinal motion of an aircraft is
[~l [~:][:] where
A=
[
Mu
(31)
(i= 1, ... ,.)
+ XA;
+XQ.X
}(a.
ZulU
ZajU
0
0
+ M",Z,,/U
< 0
1
1
Mq
0
(32)
+ AfoZ"jU
+ ZqjU
0
+ AJ)X + X(A; + A-)7' -(Bdlfj + BjAIi) - (B. Ai) + BjMif <
Mc,
0
0
- BiM; - (B;M.)T
+ Mt R i M,
X"
-g
Then, Eqs. (27) to (30) are written as .4, X
(36)
+ MaP + Zq/U)
1
(Ai
0 (33) YA i +ATy - N,G. - (N,Cif +YG. WiGty + l'{iZiN,r < 0 (.34) Y(Ai + Ad + (Ai + AI)Ty -(NiG, + NlC,) - (N,C, + N!G,)T < O. (35) (i=l, ... ,r, i=i+l, ... ,r, i=i+l, ... ,r)
B=
l
c= [0
,~aZ8IU 1
Z6/U
0
Mo
+
'
-1 1 0].
x ~ [u a
{J
q]T E 3?"
b E 3?1
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Copyright 1999 IF AC
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FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
14th World Congress ofTFAC
H=4000 [m], U=100 [m/s]
6
--
4
:~
0>
'"
~
~
- -
FGS
LOG Open-loop
.'
I
11 -2 \ -4
Fig. 2.
Membership functions of Hand U.
In the state vector x, u is the incremental velocity in the flight direction, a the angle of attack, o t.he pitch angle and q the angular velocity of the pitch angle. , is the angle of the flight path, and fj is the deflection angle of an elevator. In the matrices A and B, X", Za, ... are stability derivatives in the frame of the stability axis, and they are changeable with respect to the altitude, flight velocity, angle of attack, Mach number, etc. In this numerical example, Eq. (36) was regarded as a LPV system whose varying-parameters are the altitude H and the flight velocity U. Their ranges were respectively given as HI < H < H", and U/ ::; U ~ U" Fuzzy rules were given at the following four operating points; (H, U) = (Hr, UI), (HI, Uu ), (Hu, U/), and CH... , UtA)· A LTI model was constructed at each operating point. Membership functions, PH and ~u, of Hand U were given as Fig. 2. Then. hi was given by
hi=J1-H(H)l\l-Iu(U) 4.2
(i=1, .. ·,4).
(;17)
Numerical simulation
~ umerical data of the Lockheed P2V -7 aircraft were used in the simulation (Isozaki, et al., 1980). The ranges of Hand U were given as 1000 < H < 7000 [m] and 50 :::; U ::; 150 [m/s]. Since the m': trix C in Eq. (36) was constant, the LMIs given by Eq. (17) were not needed. The number of LMls to be solved was then 4 x (4 + 1)/2 + 4 = 20. To find matrices X > 0, Y > 0, Mo and Ni (i = 1, .. ·,4) which satisfy the 20 LMIs, the LMI Control Toolbox in Matlab (Gahinet95, et al., 1995) was used for the calculation. After finding X I Y, M; and Ni, the controJ and the filter gains Fi and K j were obtained from Eqs. (lS) and (19). A FGS cont.roller was then constructed as Eq. (7). To compare the FGS controller with a conventional fixed-structure controller, a. LQG controller was designed at (H, U) = (4000, 100).
Figures 3 and 4 show the responses of let) in which the flight eondition was (H, U) (4000,100) and (4000,135), respectively. At (H, U) = (4000,100), the response using the LQG controller was superior to the response using the FGS controller
0
10
20 30 t [sec)
40
50
Fig. 3. Time responses comparing FGS controller to LQG controller at. (H, U) = (4000,100). H=4000 [m]. U=135 [m/s] 6r---~--------~----~--~
FGS
LOG Open-loop
4
Fig. 4. Time responses comparing FGS controller to LQG controller at (H. U) = (4000,135). from t.he viewpoint of the settling time. While at (H, U) (4000,135), the FGS controller showed better performance than the LQG one.
=
To evaluate the control performance over the entire flight range using the time response during t E [0,50], the following index Cost was introduced.
Cost ~
1
50
0
(38)
th(t)ldt
Figures.) and 6 show Cost usiug the FGS and LQG controllers, where Cast was limited to 10000 if Cost was greater than 10000. Cost using the LQG controller showed high values in the low and high flight velocity regions. In particular, the dosedloop system using the LQG controller was unstable in the regions around (H, U) = (7000,50) and (1000, 150). On the other hand, Cost using the FGS controller was less than 600 over the entire flight range. Thus, it is concluded that the FGS controller performed bet.ter than the LQG controller over the entire range of the varyingparameters.
=
5.
CONCLUDING REMARKS
A design of gain-scheduling control using fuzzy logic, called Fuzzy Gain-Scheduling (FGS) COll-
7965
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FUZZY GAIN-SCHEDULING CONTROL AND ITS APPLICATION ...
FGS
14th World Congress ofTFAC
Apkarian, P., and P. Gabinet (1995). A Convex Characterization of Gain-Scheduled H 00 Controllers. IEEE Transactions on Automatic Control, 40, 853-864.
10000 9000
Boyd, S., L. E. Ghaoui, E. Feron and V. Balakrishnan (1994). Linear Matrix Inequailties in System and Control Theory, (SIAM Studies in Ap-
plied Mathematics, Vo1.l5), Philadelphia.
'000
H[r'I"I]
Fig. 5. controller.
00
Chen, Z., Y. Hayakawa and S. Fujii (1993). On the Stability of Plants with Variable Operating Conditions. Proc. o/th.e 22nd SlCE Symposium on Control Theory, 55-60.
U (mf:5;)
Performance index Cost using FGS LOG
Gahinet, P., and P. Apkarian (1994). A Linear Matrix Inequality Approach to Hoo Control. Int. J. Robust and Nonlinear Control, 4, 421-448. Gahinet, P., P. Apkarian and M. Chilali (1994). Affine Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty. Proc. CDC, 2026-2031.
tSO
Fig. 6. con troller.
Performance index Cost using LQG
trol, has been presented in this paper. Compared to gain-scheduling Hoo control (Apkarian et al., 1995; Apkarian and Gahinet, 1995; Packard, 1995), an advantage of the FGS controller is that the only two parameters of the cont.roller have to be obtained. They are the control and filter gains, which are easily parameterized by the LMls solutions. The proposed method was applied to a numerica.l example of a longitudinal flight control problem in which the plant WM regarded as a LPV system; that is, the altitude and the flight velocity were varying. Compared to a conventional LQG controller, the FGS controller showed better cont.rol performance over the entire flight region. The sta.bility criterion used in this paper was a Lyapunov function with a fixed positive definite matrix. To obtain a FGS controller which stabilizes a LPV plant over a larger operating range, it may be possible to use a parameter-dependent Lyapunov function (Gahinet, et al., 1994).
Gahinet, P., A. Nemirovski, A. J. Laub and M. Chilali (1995). LM1 Control Toolbox, The Math Works Inc., Natick. Isozaki, K., K. MMuda, A. Taniuchi and M. Watari (1980). Flight Test Evaluation of Variable Stability Airplane. KH.J. Tech.nical Review, 75, 50-58. Iwasaki, T., and R. E. Skelton (1994). All Controllers for the General He;:, Control Problem: LMI Existence Conditions and State Space Formulas. Auiomaiica, 3D, 1307-1317. Meressi, T. and B. Paden (1994). Gain Scheduled H co Controllers for a Two Link Flexible Manipulator. Journal Guidance, Control, and Dynamlcs, 17,537-543. Packard, A. (1995). Gain Scheduling via Linear Fractional Transformations. Systems and Control Letters, 22, 79-92. Shamma, J, S. and M. Athans (1990), Analysis of K onlinear Gain Scheduled Control. IEEE Transactions on Automatlc Control. AC-35,
898-907. Tanaka, K., and S, Sugeno (1992). Stability Analysis and Design of Fuzzy Control Systems. Fuzzy Sets and Systems, 45, 135-156.
REFERENCES Apkarian,
P.,
P.
Gahinet
and
G.
Becker
(1995). Self-Scheduled H cc Control of Linear Parameter-Varying Systems: A Design Example. Automatica, 31, 1251-1261.
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