Laboratoire A CS/ENIT B.P. 37 1002 Tunis BELVEDERE -Tu~'v1SL4. + Laboratoire LPVMS/INRST HP. 95 2050 Hammam-Lif - TU\1SL4. Abstract: The focus of ...
Copyright © IFAC Control of Industrial Systems. Belfort. France. 1997
V~LESTRUCTURECONTROLANDPERFORMANCECOMPAIDSON
A. Sellami*+, R M'hirf, O. Boubaker', M. Ksourf, and J. Zrida'
+
Laboratoire A CS/ENIT B.P. 37 1002 Tunis BELVEDERE -Tu~'v1SL4 Laboratoire LPVMS/INRST HP. 95 2050 Hammam-Lif - TU\1SL4
Abstract: The focus of this paper is to present a synthesised procedure for the design of state feedback and static output feedback sliding mode controller for a linear timeinvariant system. The simulation of numerical examples illustrates the robustness of the derived control scheme with respect to matched uncertainties. Despite the interesting performances of Variable Structure Control (VSC), the chattering phenomenon due to high frequency switching in the VSC controller is highly undesirable in most process applications. This paper also presents the reaching law method for the design of VSC systems which reduces chattering. Two kinds of reaching laws are considered: the constant plus proportional rate law, and the power rate law. The boundary layer controller is also investigated. In each case, experimental results with a high-accuracy positioning system driven by a DC motor are presented. Keywords: Sliding mode, Output Feedback, Reaching Law Method, Boundary Layer
I. INTRODUCTION
(1995): Seungrohkoh and Khalil(1993». The focus of this paper. is to present a synthesised procedure from bibliographic approaches for the design of state feedback and static output feedback sliding mode controller for a linear time-invariant system. The two steps of the design approach. namely the existence step in which one chooses the sliding surface that gives good behaviour during the sliding mode, and the reaching step in which one chooses the control to ensure that the reaching condition is met are investigated. The simulation of numerical examples illustrates the robustness of the derived control scheme with respect to the so-called matched uncertainties.
Variable Structure Control (VSC) is a robust nonlinear control strategy emplOying feedback of discontinuous signal. It has long been known. that during a certain mode of the transient motion, the response of a yariable structure control system is unaffected by a class of parameter variations and disturbances. However, most design approaches of variable structure control (ltkis (1976): Utkin (1993): Btihler (1986): Dorling and Zinober (1986): Lee and Xu (1994» use full state information. In practice. it is not always possible to have full information regarding the state. In this case, state estimates are instead used (Utkin (1993): Furuhashi et af. (1992». However. this increases the complexity of the implementation, and only asymptotic tending of the state to the sliding manifold can be achieved. One easy to implement alternative is to use static output feedback (Finney and Heck (1995): EI-Khazali (1995): Zohdy et af.
Despite the interesting performances of Variable Structure Control. the chattering phenomenon due to high frequency switching is highly undesirable in most process applications. The reaching law method that could reduce the unwanted chattering is considered in 1\\'0 forms: the constant plus 789
proportional rate law, and the power rate law. The boundary layer controller is also investigated. For each case, experimental results with a high-accuracy positioning system driven by a DC motor are presented.
Z'
z; E mn-m ; z~ E mm
then
fz~ = AlIz] +A 12 z 2
(2)
1
z;=A21Z]+A22Z2+B2U
2. DESIGN PROCEDURE OF VARIABLE STRUCTURE CONTROLLER
and
F]
Consider the following linear time-invariant system: x'== Ax+Bu
= [z; z~];
(3)
(Z]-ZIR)+F2 (Z2 -Z2R)=O
with
(I)
TAT1=[A 1 ]
y==Cx
AA
A 21
Rmis the
where x(t) E Rn is the state vector, u(t) E control input and yet) E RP represents the output vector. The constants matrices A, B and C are of appropriate dimensions. We suppose that the system satisfies the following assumption: A.I) The pair (A,B) is controllable, B has full rank m, and n>m. The two steps of the design approach, namely the existence step and the reaching step, will now be presented.
l2
]
;
FT'=[F j
22
F2 ]
The assumption that the product matrix FB is nonsingular implies that F2 must also be non-singular and the condition defining the sliding mode becomes z2 = -
Fi' F]
1
z] + F2- Fl ZIR + z2R
Z2 = -Kz l
+
KZJR + z2R
with K being an mx(n-m) matrix. The sliding mode is then governed by the equations
2.1. Design ofthe sliding surface
z7 =AllZ] +A]2 Z2 {Z2 = -Kz] + Kz]R + z2R
State Feedback case
representing an (n_m)th order system with ~ playing the role of a state feedback control. The closed loop system will then have the following dynamics:
The switching function has the fonn S = FX, where F is an (mxm) matrix, and X = X - X R is the error between the system's state vector and the reference (constant) trajectory. The equation S = 0 defines the m-dimensional sliding surface. Differentiating with respect to time and inserting (I) gives Ueq= - (FB)" I FAx, when (FB)-1 exists. As a result, the
dynamics
z~ =(A lI -A'2K)ZI +A 12 Kz]R +A 12 Z2R . This indicates that the design of a stable sliding mode (z ~ 0 as t ~ co) requires the selection of the matrix K such that (A u -A 12K) has (n-m) left-halfplane eigenvalues. This may be achieved by using any standard design method giving a linear feedback controller for a linear dynamical system, including the quadratic performance approach, eigenstructure assignment and assignment of eigenvalues in certain regions (Woodham and Zinober (1986».
x = (I-(FB)-'F)Ax== Aeqx
describes the motion on the sliding surface which is independent of the actual value of the control and depends only on the choice of the matrix F. The hyperplane design methods used by Dorling and Zinober (1986) and Utkin (1992) can be used to select the gain matrix F which gives a good and stable motion of the system during the sliding mode.
Upon determining K, the matrix F can be calculated using the fact that K = Fi l F1 . This leads to F=KRBi1[K Im]T (5) where KR is an (mxm) design matrix, and Im is the (mxm) identity matrix (Utkin (1992».
By assumption, the matrix B has full rank m, as a result, there exists an (nxn) transfonnation matrix T such that: TB == [
:2],
The simplest approach to avoid the amount of calculation in the selection of the matrix KR is to let KR=B 2, which is equivalent to specify F2=I m , and
where B2 is (mxm) and non-
singular. Note that the choice of an orthogonal matrix T avoids inverting T when transfonning back to the original system. The transfonned state variable vector is defmed as z=Tx, in tenn of which the state
gives F = [K
I m ]T .
Static Output Feedback case
Z == TATt z + TB u and the sliding condition FX == FTtz == FT t (z- ZR) = O.
equation
(4)
becomes
The switching function has the form S = Gy = GCX: , with S = 0 being the sliding surface. Computing
If the transfonned state is partitioned as
S = GCX: = GC x = GC(Ax + Bu) = 0,
790
(6)
if (GCB)"' exists then u eq = - (GCB)"' (GCA) x, and x = (1- (GCE) -1 GC)Ax = Aeqx
represents
ajj
\jI(x)= [ lj/"
the
l)
]
= { fJij
when Sj(x)x j > 0
(8)
when s;(x)x j < 0
motion on the sliding surface. for
{i.~ 1,
,m
J -l, ,n
The procedure for the design of the switching using only the output variable is function S
= GY
(iii) Reaching law control:
related to the design of a switching surface S = FX for the state variable x via the output equation y = Cx. In fact, once the state switching surface F is detennined, we need to characterise F and C for which the output switching surface G is defmed, i.e., the equation GC= F is solvable. The necessary and sufficient conditions for this to happen are given in Zak and Hui (1993).
B) -I [ : Ax + Qsgn(S) + Rh(S)]
u(x)= - ( :
2.2. Control scheme design
where
sgn(S) = [Sgn(SI)
and
h(S) = [hl(sl)
...
...
(9)
sgn(sm)f,
hm(sm)f.
hi satisfies the condition sj#O, and h;(O) O. The gains Q and Rare (mxm) diagonal matrices with positive elements. Note that the (mxm) matrix
The
scalar function
=
s;h;(s;) > 0, when
Once the existence problem has been solved that is the matrix F (respectively the matrix G in the case of static output feedback control strategy) has been detennined, attention must be turned to solving the reachability problem. This involves the selection of an output feedback control function u(x) (respectively u(y)) which ensures that trajectories are directed towards the switching surface from any
(~
B)
must be non-singular (Hung et al. (1993)).
(iV) Unit vector control:
point in the state space. Therefore, ifV(x,t)=sts is a Lyapunov function for system (I), a suitable control u(x) (respectively u(y)) must be chosen to guarantee
u(x)
=
Lx +
II~II S
(10)
where L is an (mxn) matrix, and p is a positive scalar. The tenn Lx represents a state feedback controller, and it serves only to asymptotically drive the state to zero (Dorling and Zihnober (1986)).
that the reaching condition, namely, S'S· < 0, is satisfied. State Feedback case
Static Output Feedback case The main requirement in the design of the control law is that it should satisfy the reaching condition, which guarantees the existence of the sliding mode on the switching manifold. Four types of control law are given here:
The control structure to be described here is based on that of Heck et al. (1995) and has the form u(y) = -(GCB)-1 Ny - a(GCB)-1 sgn(S)
= GY,
and Cl is a scalar. The gain matrix where S N is chosen to satisfy the reaching condition
(i) Relay Control:
uj(x) =
{ki~ ki
(11)
stS· = xICGt(GCA -
NC)x- allSl1 1 < 0 (12)
m
when Sj(x»O when Si (x) < 0
i=I ... m
where
(7)
;=1
Letting
The relay gain may be either constant or state dependent.
L(N) represent the symmetric part of
C' G ' (GCA - NC), L(N) = CtGt(GCA - NC) + (GCA - NC)tGC
The boundary layer control is obtined by replacing the relay control by: ks· u(x) = - ' - ' -
,
IISIII = Lls;l.
(13)
2 If Cl > 0 then the reaching condition is
Amk L k
(16)
A=[: r]
hI
where the L.'s are (nxn) real symmetric matrices and the nk's are the elements ofN. Specifically, the value of n k in L(N) is defmed as the kth element of the vector fonned by stacking the columns of the N matrix [n 11 ,n 21 ,·· .,n mI;n 12,n 22 ,·· .,n m2 ;·····;n Ip,n 2p ,·· .,nmpJ'. Lo and L k are defmed as: La
= CIGIGCA+(GCA/GC
B=[ ~;] x=[~]
and Km are, respectively, the time constant and satic gain of the system.
(17)
x I = El is the angular position of the motor shaft. x2
and k -
CIG I N C+C I N 'GC k k 2
In
I
F=[f1
f2]=[kRK~ km
-k R
_1_] km
(18)
where KR is a positive design scalar to be freely selected. The values ofKR and BR are chosen to be KR=O.I
.. AT-bus ..J
I
and BR = 1t
II
The reaching law approach, used by Gao and Hung (1993), is considered here, and leeds a control law of the following general fonn:
Host Computer (PC-Compatible)
~
.
The control law: reaching law method
I Coupling
current.~
Amlifier
example,
Xi = -Kx 1 + KEl R The eigenvaIue of this first-order system can be conveniently selected. To this end, we choose to set K = 2, yielding a stable closed loop pole at -2. Now, from Equation (5), it follows that
The application considered is a high-accuracy positioning system (Fig. I) driven by a DC motor used for the selection of solar cells.
1
control
=x2 x 2 = -Kx J + KEl R The closed loop systems dynamics are then described by the equation
3.1. VSC with state feedback
DAC
position
XI·
3. APPLICAnONS
Decoder
this
x R = [X 1R X2Rr = [El R Or defines the reference input. The design procedure of the sliding surface S = FX = 0 is as follows. As B has the right fonn, there is no need for the transfonnation, i.e., take T to be the identity matrix. Here, Equation (4) becomes
Other numerical algorithms can be used for solving the non-smooth convex optimisation problem, such as interior point methods which have received a great deal of attention recently due to their fast convergence proprieties and for which the LMI Control Toolbox is available in Matlab.
,
= n is the angular speed of the motor shaft.
The switching surface
(18)
where N k is an (mxp) matrix with zeros everywhere except a '1' in one position. For k= I, the 1.1 element of N I is 1. As k is increased, the '1 ' moves down the first column, then the second column and so on. To fmd the elements of the matrix N, the cutting plane algorithm of Heck et al. (1995) can be used. It can easily be programmed into Matlab with the optimisation toolbox. IfN does not exist, that results in L(N) being negative semi-defmite, then the gain CL can be appropriately chosen to satisfy the reaching condition in a region of the state space.
110 Industrial Card
01
1
2 L -
C=(1
I
u(x) = -
DC Motor
r
(~ B
1 [:
Ax + Qsgn(S) + Rh(S)]
(19)
This control . law approach is derived from the following differential equation which specifies the dynamics of the ·switching function S(x):
Encode
Fig.]: Experzmental posltlomng system S(x) = -Qsgn(S)- Rh(S)
792
(20)
where Q and F are generally diagonal matrices with positive elements. The choice of Q and R detennines the dynamics and the perfonnances of the controlled system. The reaching condition is guaranteed by the differential equation of an asymptotically stable Sex). Since a single-input system is considered, Q and R are positive scalars, noted q and r.
signal is not realistic, as it reaches excessive values (> 1000). (a) Control input (voll)
(b) Angular position (rd)
1600 ,---.,---.,.--...., 14oof·········,··········;·········~
3.5,----,--_-,--_--, 3 f······· "/' .. /.-.-:-;+:::-:=
1200M·········:··········,·········~
2.5 1000f\
Now computing
.:.
.
j
;
aOOH·········;··········;·········~
2
,
; ..
'~44s
_ 1.5
: : .\\
200 .. \\
:
;
~
-200
o
.
' ... '=,a s
~
:
0"-
and
_
j
0.5
0.02
004
006
o~--=-_...,..-_.....,
0
4
t(s)
6
I (s)
Flg2:Response ofan Ideal system (q-O.OOI.r=IOO)
gives then the following control law:
u( x) = --'-[( f1 k m F2
In Fig.3, r and q are chosen so that the maximum value of the control signal is reduced. This case shows also that increasing q leads to a high switching control signal.
f2 )0 + q sgn(S) - rh(S)] "'
Various choices of q, r and h(s) specify different rates for S and yield different structures in the reaching law. Two fonns ofh(s) are considered:
(a) Control input (volt)
(b) Angular position (rd)
150,----,.---,----,
(i) Constant plus proportional rate reaching law
3.5,----.,.---,--_-,
50
In this case, h(S)=S, and the resulting reaching law is defined as
o -50
S· = -q sgn(S) - rS
(21)
-100
The tenn (-rS) forces the state to approach the switching manifold faster when S gets larger.
t (s)
t (s)
Fig3' System response for q=IO and r=2.5
The resulting control law has the fonn:
u
=.tn+rsgn(S)+oS
In Fig.4, one could see simulation results for a more realistic case where a saturation (±5) is considered for the control signal. Here, the variation of, leads to a different transient response, but the control signal is relatively smooth and has realistic values.
(22)
where
(b) Angular position (rd)
(a) Control,nput (voll)
6,----..,----,---, 5
and
4
3.5,--.,.---,-----,
3f·········,···· .. _
.
25
..,...r,;
..
/
,:
1- .:•••••••••• ~ •••••••••
,,:
: .. ~~
1.5
In a first stage, numerical simulation was done with nominal parameter values of the real system (k m=0.46 and '=4.4s). Parameter variation is simulated by taking, = 8s then, =4.4s.
, / ,,
j'
1
0.5. ,'..
: ._.
';44s
~ -r ~8 s
--:
.
; .
"
.
I
2
Fig.2 shows the simulation results in which the output response remains the same in spite the variation of time constant (,). However, the control
4
6
t (s)
Fig 4: System response for q=O.OOI and r=2.5
793
Fig.S shows the response of the real system. Here the saturation of the command is taken into account in all cases. The coefficients q and r have the same values considered in the simulation stage. It is easy to see the concordance between simulation results and practical results even with parameter variation (T=4.4s and T=8s).
In Fig.6, the response of the real system for given values of r and a is shown.
(a) Angular position (rd) 4
3
(a) Angular position (rd) 35 f (t 3.0l-
= 4,4
s) .----".
2
_
~
25 [
1
2.0 rf
Time(s)
~
0
1.5l f
0
2
1
4
3
5
6
10 ~ ~
05
Time (s)
r
(b) Control input (volt) 6
0.00~'---_:__--;;_2--;;-3- - 7--7'------;!6 4
4
0
~n 3
-4
o
-6
-4 -5
c~
-2
2 1 -1 -2 -3
!
2
(b) Control input (volt)
V
Time(s)
0
1
2
3
4
5
Time(s)
6
Fig,6: Real system response with a=O.3 and r=2.5
0~-~---:2:---3~--74-~5:----,6;:---:!7
Fig.7 shows that an appropriate choice of a could give a chattering free response for the real system.
Fig,5: Real system response with q=O.OOJ and r=2.5
(if) Power rate reaching law
(a) Angular position (rd) 3.5
In this case, the dynamics of the switching function is specified by
3.0 2.5
s· = _rIS!a sgn(S)
2.0
(23)
1.
5
1.
The resulting control law is of the form:
U =
In + o1Sla sgn(S)
0 05 00
(24)
Time(s) 0
1
where
3
2
4
5
6
5
6
(b) Control input (volt) 6 5 4
3 2
and
\ \
1
0 -1
-2 -3 -4 -5 -6
The main effect of this form of control is that it increases the reaching speed when the state is far away from the switching surface, but reduces the rate when the state is near the manifold.. The result is fast reaching and low chattering.
r
V
0
1
2
Time (s) 3
4
Fig. 7.' Real system response with a=O.9 and r=2.5
794
Concerning the power rate reaching law method, the response of the real system, as depicted in Fig.6 and Fig.7, shows that chattering can be eliminated by increasing the value of parameter a . As to the boundary layer control strategy, Fig.8 shows that chattering free and relatively robust response can be obtained with the real system by selecting an appropriate value for the design parameter f;. This technique turns out to be very interesting as saturation is inherently present in real systems, and the control law is generally easily designed and implemented. Furthermore, the obtained responses are chattering free as well as robust with respect to the considered uncertainties.
3.2 VSC with static output feedback The experimental positioning system model does not satisfy a necessary assumption to apply the VSC output feedback technique defined as: A.2) The pair (A,C) is observable, C has full rank pzm, and rank (CB)=m (Zak and Hui (1993)). Therefore, the numerical example used to illustrate he static output feedback sliding mode controller design procedure is concerned with the stabilisation problem of an L-I 0 11 aircraft (Hech et aI., 1995). The inputs of this system are the aileron rudder commands. The outputs are the yaw and roll accelerations, the bank angle, and the wash out filter state. The system's state equations are described by the following matrices:
Fig.8: Real system response to boundary layer control
0
0
I
0
0
0
0
0
-0.154
-0.0042
154
0
-0.744
-0.032
0
0249
-1
-52
0
0.337
-L12
0
0.02
0
0 0
-4
0
0 -20
0
0
0
A= 0.0386 -0.996 -0.0003 -2.177
Performance comparison
0
Let us now try to give an interpretation of all these simulation results. Concerning the results obtained with the constant plus proportional rate reaching law method, it is quite obvious that the design parameters q and r heavily affect the system's performance. It is thus necessary to assess, at least qualitatively, how these parameters affect the behaviour of the system. Looking at Equation (22), it is clear that parameter y (or proportionaly q) is responsible for the chattering if large, while parameter cr (or proportionaly r) governs the amplitude of the control input. This explains why going from Fig.2 to Fig.3 (by increasing q and decreasing r), robustness is conserved, the amplitude of the control input is reduced, but chattering has appeared. FigA shows that by limiting control input magnitude, chattering and control input excessiveness can both be eliminated, but robustness is slightly affected. The real system's response depicted in Fig.5 totally agrees with the simulation findings.
0
05 0
0 0
0
0
0 0
c·r~
LO
0 0 -25
0
0
0
0
0
B= 0 0
0 0
20
0
0
25
1
-0.154 -0.0042 1.54 0 -0.744 032 -0. -1 -52 0 0337 0249 -1.12 0
0
0
0
0 0
0
0
I
0
0 0
J
The following matrix G gives good sliding mode dynamics: G
= [-0.0067 0.0167
0.0167 -0.0333
0.0033 0
0] 0.0033
The cutting plane algorithm was programmed into Matlab to obtain the gain matrix N used in the
795
control law. For a weighting of p=10·7, the maximum x· = (A + M)x+{B+ M)u where
eigenvalue of L(N) }'ma:c {L( N)} = 6.910- 3 and the resulting matrix gain N was the following: N = [1.D427 5.0974
8.1]]2
0.93
0.0698
2.7883
where
0.3399]
A = [1
13.3822
-I I]
1 1 1 1
1 1 1 1 1
The value of the maximum eigenvalue of L(N) Amax {L( N)} is very small in magnitude, but it is positive. The gain a can be appropriately chosen to
1
and jj = -0.5.
-I
The response of the side slip angle (x4) to an initial condition of [0 0 0 1 0 0 0]' is shown in Fig.10(a). The aileron commands ul and u2 are respectively shown in Fig.l O(b) and Fig.l O(c), and finally, the bank angle output (y3) is presented in Fig.10(d).
satisfy the reaching condition Sf s· < 0 in a region of the state space defined by
Ilxll
