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Decoupling and Model Following Control in Variable Structure Systems with Linear Plant
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DECOUPLING AND MODEL FOLLOWING CONTROL IN VARIABLE STRUCTURE SYSTEMS WITH LINEAR PLANT M. E. Penati and G. Bertoni EIt'rtmIllCl. COlllplller Sril'll(/' 111111 .\\.11"111 Thl'Ol1' f) /'p llrllll,' //I. L'llIi'l'nily \ 'illl/, RllIIlgilll/'lI lo 2. -/0136. H olo,!!; //({. II({ly .
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Abstract. One of the most important features of variable structure systems (V.S.S.) is their invariance property when sliding modes occur, that is when the state point slides on the intersection of the switching surfaces. When the plant of the system is linear, it can be very useful to exploit the so-called geometric approach to system and control theory. The aim of this paper is to show how, by means of this approach, the V.S.S. features become more apparent so as to allow the achievement of new results and new applications. In particular, the paper shows that the solution of the model following problem using sliding mode control can be used to implement an interesting procedure which, taking into account the invariance conditions, allows to decouple slightly interconnected systems. This procedure turns out to be very robust and particularly suitable for large scale systems. Keywords . sliding modes, robustness, decoupling, variable structure systems
framework of the geometric approach to system and control theory. In Section 4 the influence of the switching hyperplanes on system dynamics is examined and in Section 5 the switching hyperplane assignment problem is dealt with . In Section 6 the model following problem using a sliding mode control is reviewed by means of the geometric approach and the results are used in Section 7 for decoupling slightly interconnected systems. The invariance conditions of V.S.S. and the perfect model following conditions are reviewed, respectively, in Appendix A and in Appendix B.
INTRODUCTION As it is well known, variable structure systems (V.S .S.) are a special class of systems characterized by a discontinuous control action obtained by changing the structure of the system when it reaches a set of switching surfaces. One of the most important features of V.S.S. is their invariance property when sliding modes occur, that is when the state point (i.e. the point representing the state vector) slides on the intersection of the switching surfaces . In this case, the system behaves as though linear and it turns out to be independent of certai n classes of parameter variations and disturbances. In other words, the sliding mode control of a V.S.S. is basically an adaptive control which forces the response to slide along a predefined trajectory. The robust nature of the control is very useful in many fields of applications and mainly in robotics (particularly in space robotics) where the moment of inertia may vary widely depending on the robot arm and the load it carries. The mathematical treatment of this kind of system is quite complicated and has been mainly dealt with by Russian authors [9, 13]. The analysis becomes easier only when plants are assumed to be linear; in this case it can be very useful to exploit the geometric approach to system and control theory [11 , 15]. The aim of this paper is to show how, by means of this approach, the V.S.S. features become more apparent so as to allow the achievement of new results and applications. In particular, the paper shows that the solution of the model following problem using sliding mode control can be used to implement an interesting procedure which, taking into account the invariance conditions of Drazenovi6 [3] , allows to decouple slightly interconnected systems; this decoupling turns out to be very robust and particularly suitable for large scale systems. The outline of the paper is as follows . In Section 2 the problem of V.S.S. model is dealt with in the general case while in Section 3 the particular case of systems with linear plant is considered in the
2
THE MODEL OF V.S.S.
As it is well known from literature [9, 13], in the general case, the mathematical model of the plant of a V.S.S. is given by: = f(x ,t,u) (1 )
x
where x and f are two (n x 1) vectors and U is a (m x 1) vector whose components Uj (X,t) (i = 1, ...... , m) coincide with one of the two continuous functions Uj +(x,t) or Uj-(x,t) depending on the position (with respect to the switching surface Sj(x)) of the state point. More precisely it can be written : Uj(x,t) = Uj+(x,~ for Sj(x) > 0 i= 1, ...... , m (2) Uj(x,t) = Uj-(x,t) for Sj(x) < 0 i= 1, ...... , m In a certain sense relations (1) and (2) can be considered a grey box type model of a V.S.S.. An interesting characteristic of these systems is that the surfaces Sj(x) and the functions u/(x,t) and Uj-(x,n can be chosen in such a way that, once the intersection surface s(x) = 0 of the switching surfaces Sj(x) = 0 is reached, the state vector remains in the neighborhood OSCillating about it with a high frequency and very small amplitude; infinite frequency and zero amplitude in the ideal case. As mentioned in the Introduction, this is the so-called sliding mode type of mo-
lili
:'\1. E. Pe nali a nd G. Berto ni
188
tion and the surface s(x) = 0 is called sliding surface. However. as it can be seen from (2). in corrispondence of the state points lying on the switching surfaces s; (x) = 0 the control functions are not defined so that these can be considered discontinuity points of the function representing the second member of (1) . Consequently. the Cauchy conditions for the existence of the solution of the differential equation (1) are not satisfied. This means that the model (1). (2) is not suitable to represent the system in all conditions and. in particular. during sliding modes. A way to circumvent this drawback could be to change part (2) of the grey box type model taking into account that . actually. the system does not switch instantaneously from a control function to another. but. indeed. due to the so-called non-idealities in switching devices. the control functions change in a very short (~ut not nUll) interval so that they do not undergo dlscontinuities . but only sharp changes and . hence. they are always defined. Unfortunately. these models. often called real models . assume different mathematical structures because of the different types of switching devices used by the actual systems . This turns out to be a heavy shortcoming because it prevents studying these systems in the same mathematical framework. A more convenient method for determining the model when equations (1). (2) cannot describe the whole behavior of the system. is to use the model given by the so-called mathematical continuation method. that is to use a black box type model. As it is well known. these models are only mathematical descriptions of the link between input-output data of the system so that. in general. the state variables involved in the equations do not have any physical meaning . Fortunately . it can be demonstrated [9] that one of these models. the equivalent control model. has a very interesting property since it tends to coincide with every real model when the effect of the non-idealities goes to zero . This means that the equivalent control model not only gives a mathematical description of the input-output data link. but also represents the physical behavior of the system ; consequently. the state variables involved in this black box model get a precise phYSical meaning. As to the implementation of this model. it is made up of the model of the system plant controlled by a continuous feedback function which keeps the state point on the sliding surface . In conclusion. let us note that. in the general case. determ ining the equivalent control model is quite cumbersome ; this model becomes simple enough only with linear plant and its analysis can be easily accomplished mainly when USing . as mentioned above . the framework of the geometric approach to system and control theory. This is what will be done in the next section .
3
SLIDING MODE EQUATIONS: METRIC APPROACH
THE GEO·
Let us consider a V.S.S. with a linear stationary plant given by : le = Ax + Bu (3) where x is a (n x 1) vector. u a (m x 1) vector. A and B are two matrices of proper dimensions. Let us assume that the switching surfaces s/..x) = 0 (for i = 1•.. .• m) are hyperplanes containing the origin of the space ; consequently the sliding surface is given by : s(x) = Cx = 0
(4)
which shows that the sliding modes take place in the subspace defined by kerC . As to the model of the system when sliding modes occur. let us determine the equivalent control model defined in the previous section . Since during sliding mo-
tion it must hold :
(5)
the continuous control which keeps the state point on the sliding surface (that is the equivalent control u'!~ can be obtained substituting (3) in (5) and solVing Wit respect to u; we get:
ueq = -Kx
(6)
where : K = (CBr'CA Substituting (6) in (3) . the equivalent control model of the system turns out to be :
le = Aeq x = [1- B (CBr'C] Ax = (A - BK) x
(7)
This is the so-called sliding mode matrix equation. Using th is equation we can make some preliminary remarks on the features of V.S.S. when sliding modes occur. Remark 1 (sliding mode dynamics depends only on the sliding surface) From equation (7) it follows that for the equivalent control to exist it must be IC B I "'" O. On the other hand . ICBI "'" 0 means that ker(CB) has null dimension. that is no vector x exists such that CBx = O. Consequently it cannot be neither y = Bx = 0 nor Cy = 0 ; that is y = Bx must belong to imB (i.e . also kerB must have null dimension) and y cannot belong to the subspace kerC. Summarizing it can be written : kerC (") imB = ()
(8)
Relation (8) shows one of the most important properties of V.S.S. when sliding modes occur. i.e. the behavior of these systems does not depend on the controls. but only on the sliding surface. Remark 2 (during sliding motion a reduction of the model dynamic order takes place) When sliding modes occur. a reduction of the model dynamic order takes place from n to (n - m) . wh ich means that only (n - m) equations of system (7) are linearly independent. To show this property. let us remember that . during sliding motion . relation (5) holds so that. substituting le with (7) . we get :
C (A- BK) x = 0 or: kerC::> im(A - BK)
(9)
Since the subspace kerC has (n - m) dimensions. re lation (7) cannot have more than (n - m) linearly independent equations. that is. the system matrix Aeq has m null eigenvalues at least. Remark 3 (the (n - m) eigenvectors corresponding to the non zero eigenvalues form a basis for kerC) Let us label the eigenvalues of Aeq with A; (i = 1•. .. . n) and the correspondent eigenvectors with v; (i = 1 . ... . n) . For any eigenvalue A; it can be written : C [(A - BK) - A;l v;= C (A - BK) v ; - A;Cv; that is. keeping in mind (9) : A; Cv; = 0
(10)
Obviously. relation (10) holds only if A;= 0 or CV; = o. This means that the (n -m) eigenvectors corresponding to the (n -m) non zero eigenvalues must belong to the subspace kerC. If. for the sake of simplicity. we assume that the (n -m) non zero eigenvalues are distinct
" ariabl e Structure Syste m s ,,·ith Lin ear Pla nt and labelled by A.; (i = 1, .... n -m), the correspondent eigenvectors v; (i = 1, ... . n -m) turn out to be linearly independent so that they can make up a basis for kerC . Summarizing, we can conclude that, when sliding modes occur, the model of a V.S.S. with linear plant changes in a very radical manner; that is, in spite of the variable structure of the physical system, the model turns out to be linear and undergoes an order reduction as large as the vector control dimension. Moreover, the behavior of the system is dependent on the structure of matrix C (i.e . on the switching hyperplanes) , but not on the controls.
4 INFLUENCE OF SWITCHING NES ON SYSTEM DYNAMICS
HYPERPLA-
In order to point out the influence of matrix C on V.S.S. dynamics when sliding modes occur, it can be useful to use a proper linear transformation:
z =Tx
(11 )
such as to transform system (3) as follows :
2:, = A" z, + A'2Z2
(12)
2:2 = A 2, Z, + A22 Z2 + B2 U where z, and z2 are, respectively, two [(n - m) x 1] and (mx 1) vectors while A", A'2, A 2" A 22 are matrices of proper dimensions and B2 is a (m x m) non singular matrix . In order to show how (12) can be obtained, let us write the transformation matrix T in the form : (13)
where T2 is a (m x n) matrix and T, is a [(n - m) x n] matrix such that it holds :
T, B = 0
(14)
Substituting (11) in (3) and (4) and defining :
z = Tx =
I :: I (15)
n-m
TAr' =
I
A" A2 ,
B2 = T2 B
then system (3) assumes the form (12) and relation (4) becomes : s = C, z, + C 2 Z2 = 0 (16) where C, and C 2 are, respectively, [m x (n-m)] and
(m x m) matrices.
Taking into account (14) and (15), the equivalent control (6) for system (12) becomes :
Since this equation is a linear combination of the first matrix equation of (12), we can conclude that the V.S.S. model during sliding motion is given by the first matrix equation of (12), that is, in agreement with remark 2, by a reduced order system. On the other hand, since, during sliding motion relation (16) holds, the first one of (12) becomes : 2:, = (An - A'2 F) z,
F=C2-'C,
Using equations (12) and (20) we are now able to make some interesting remarks about the influence of matrix C on the dynamics of V.S.S. when sliding modes occur.
Remark 4 (pole assignment conditions for sliding mode equations) If the couple (A", A(2) of system (20) is controllable the poles of (20) can be arbitrarily assigned by choosing properly the elements of F which acts like a feedback matrix. Obviously this choice (i .e. the choice of the switching hyperplanes) must be done so as to guarantee that relation (18) holds. Let us note that the controllability of the couple (A, B) is a sufficient, but not necessary condition for the pole assignment of equation (20) . Indeed, the couple (A" , A(2) turns out to be controllable (and, hence, the poles of (20) can be arbitrarily assignable) even if the couple (A, B) is not. In fact, if the couple (A", A(2) is controllable and the couple (A22' B 2) is not functionally controllable, but only controllable (because of the non singularity of matrix B 2) then the couple (A, B) results to be not completely controllable.
5
SWITCHING HYPERPLANE ASSIGNMENT
As shown in the previous sections, the switching hyperplanes determine not only the sliding surface, but also the sliding mode dynamics. In this section we will examine how to assign the switching hyperplanes in order to satisfy proper constraints on V.S.S. dynamics. A first constraint which can be chosen is that the (A" - A'2 F) matrix assumes some given eigenvalues A." .•. . A. n- m. Of course this can be done by means of any pole assignment method . However, in .this way, matrix F (and, hence, matrix C) turns out to be only partially determined; this means that, in order to cam· pletely determine matrix F, we must choose some additional constraints [15] . To this aim, let us remember that the dynamics of a system is affected not only by the eigenvalues of its model but also by the eigenvectors which mainly affect the mode amplitudes. However, assigning the (n - m) eigenvectors of system (20), the elements of matrix C still remain partially undetermined and m 2 elements must be chosen in some other way. To show how how m2 elements still rema in undetermined let us define the eigenvector matrix V :
Keeping in mind Remark 3 we can write :
Let us define :
*0
(18)
Substituting (17) in the second relation of (12) we get: (19)
(22)
CV = 0
* 0 and IB21 * 0 it must also be
2:2 = -C 2-'C, (A" z, + A'2 Z2)
(21)
V =[v" .... vn-ml (17)
IC 211B 21= IC2 B21 = ICr'TBI = ICBI
(20)
where F is a [m x (n-m)] matrix given by:
Ueq = - (C2 B 2)-' [(Cl A" + C2 A 2,) z, +
Obviously , given ICBI IC21* O. In fact:
IS9
TV=W=
IW
W,
(23)
2
where W, is a [(n - m) x n - m)] matrix and W 2 is a [m x (n - m)] matrix.
7'.1. E. Pe n ati and G. Bertoni
190
Consequently, relation (22) can be written in the form : CT-'TV =
I C,
C2
Upeq = (CBpr' (CAm Xm - CAp Xp + CBm Um) substituting (28) in (27) we get:
I I :: I =
it
= C2
IF
Im
I I :: 1= C2 (FW, + W2 ) =0
(24)
(2S)
We can conclude that, assigning matrix V and, hence, matrices W, and W 2, we can solve the matrix equation (2S) ; in particular, if W, is non singular, it is very simple to determine the [m x (n - m)] unknown elements of F from relation (2S) . Therefore , from (21) we can determine, for example , C, leaving undetermined the m2 elements of C 2 which can be used , hypothetically, to assign the sliding surtace . Unfortunately, the method we have just outlined not only is unsuitable to completely determine the elements of matrix C , but it also suffers from another severe shortcoming. In fact, as it can be seen in [S], it is not possible to assign no more than m components of each eigenvector in a completely arbitrary manner. Consequently a trial and error procedure must be followed by assigning, for example , m components according to some scheme and then by checking whether the behavior of the system is suitable.
[1- Bp (CBpr'Cj (Am Xm - Ap Xp + Bm Um)
(29)
= [1- Bp (CBpr'Cj
Am e +
[I -
Bp (CB pr' cj (Am-
- Ap) Xp + [1- Bp (CBpr'Cj Bm Um (30) As it follows from (A.2) , equation (30) can be considered as the sliding mode equation of a V.S.S. with a plant having Am as a state matrix, Bp as a control matrix and affected by two disturbances xp and u m. Consequently, keeping in mind (A.4), for the system to be insensitive to these disturbances the following conditions must be satisfied : p[Bp (Am - Ap)]
= p[Bp
Bm]
= p[Bp]
(31)
On the other hand, since the equivalent control (28) is of type (B.11), (31) can also be interpreted as the condition which gives the pertect model following of (26) , that is, the condition which imposes the error equations to become :
Et = [1- Bp (CBpr'Cj Am e
(32)
or, substituting in (29) xp = xm - e:
SLIDING MODE MODEL FOLLOWING
An interesting application of the results outlined in the previous sections is the solution of the model following problem by means of a sliding mode control. From now on we shall refer to this control as sliding mode model
following control. In order to solve this problem let us consider a plant and a model that are both linear and stationary and then represented by the following matrix equations : Xp = Ap xp + Bp up (26) where xm and xp are the two (n x 1) state vectors while u m is the (m' x 1) control vector of the model and up is the (m x 1) control vector of the plant . Without loss of generality, we can assume that the couple (A p ' Bp) and (Am ' Bm) are completely controllable and that Am has eigenvalues with negative real parts. Moreover, let us suppose that all the components of xp are accessible and directly measurable . As shown in Appendix B, the error equation is given by : Et = Am Xm - Ap Xp + Bm Um- Bp up (27) As it is well known from literature [8, 10]. one way to annul the error e, i.e . to force the plant to follow the model , is to determine a proper set of hyperplanes such that they can behave as switching surtaces for a sliding mode control. Let us assume that these hyperplanes contain the origin of the space so that the sliding surtace is given by :
S = Ca As shown in Section 3 the equivalent control up eq is Ihe control which keeps the state point on the sliding surtace, that is the control which imposes 5 = O. Therefore we can write: C
=
Substituting in (29) Xm = e + xp we get:
it
so that we obtain :
6
(28)
e= CAm Xm -
that is :
CAp Xp + CBm Um - CBp up eq = 0
In conclusion , the implementation of the model following of system (26) by means of a sliding mode control turns out to give the same sliding mode equations of a V.S.S. with a plant of the type :
Et = Ame + Bp U
(33)
or of the type :
Et = Ape - Bp U Obviously, in order to determine the matrix C which gives an assigned error dynamics, one of the methods shown in Section 4 can be used.
7
DECOUPLlNG SLIGHTLY INTERCONNECTED SUBSYSTEM
As shown in the literature [17], one way to decouple a system made up of slightly interconnected subsystems is to use a sliding mode control for each subsystem to be decoupled. In order to show how this can be done , let us consider a plant made up of two subsystems given by :
x, = A"
x, + A'2X2+ B, U, (34)
X2 = A 2, x, + A22 X2 + B2 U2
where x, is a (r x 1) vector, X2 is a [(n - I) x 1)] vector, u, and u 2 are two (m x 1) vectors and A" , A'2 ' A 2" A 22 , B, and B2 are matrices of proper dimensions. Obviously, the decoupling of the two subsystems (34) by means of sliding mode controls implies that the two terms A'2 x2 and A 2, x, can be considered as ex1ernal disturbances and that it holds : p[B, Ad = p[B,]
(3S)
Unfortunately, both of these conditions (3S) are not easy to be satisfied. Moreover, it must be noted that the decoupling is pertormed by the same controls which drive the subsystems. Less restrictive conditions can be found using the
\'ariabl e Structure Svste llls with Lin ear Plant
sliding mode model following control with a properly decoupled model. To this aim let us consider again system (34) and assume that the model is made up of two decoupled subsystems given by:
19 1
matrix 0 given by (A.7) in (A .3) it can be verified that (A.4) is a condition for (A.3) to hold. We can conclude that, during the sliding motion, V.S.S. are invariant to the external disturbances only if : imB
(36) X2m = A22 x2m + B 2m u2m where x'm, x2m, u,m and u2m are vectors of the same dimensions of x" X2 ' u, and U2' respectively, Keeping in mind conditions (31) we can say that, when sliding modes occur, system (34) behaves like model (36), i.e. it can be considered decoupled if it holds :
~imD
(A.S)
Invariance to parameter variations Let us consider again system (3) and suppose the matrix A can be represented as : (A.9) were Ac is the (n x n) state matrix and Av is the (n x n) matrix which takes into account the parameter variations. Consequently, the sliding modes equation is given by :
(37) (A.10) As it easy to see, if both conditions (35) are satisfied so are conditions (37), but not vice versa, that is conditions (37) are less restrictive than those of (35) . This new method has an additional advantage due to the fact that there is a clear separation among the input functions U" U2 which implement the decoupling (by means of the sliding mode model following control) and the input functions u'm ' u2m which control the overall decoupled system. The method seems to be quite promising since , mainly in large scale systems, the parameters interconnecting the different subsystems which are to be decoupled are often time varying ; this method assures that, within the boundary values for which conditions (37) are satisfied, decoupling is always garanteed , APPENDIX A: invariance conditions for V.S.S. One of the most important features of V.S .S. during sliding modes is their invariance to some classes of external disturbances and parameter variations [3] .
Invariance to external disturbances Let us consider some external disturbances f entering the system (3) through a coefficient matrix 0 :
x= Ax + Bu + Of
(A.1)
where f is a (r x 1) vector and 0 is a (n x I') input matrix. If the sliding surface is given by (4) the equivalent control is given by :
x= [1 -
In order for the effect of parameter variations to be nUll, it must be : (A.11 ) Comparing relation (A.11) to (A.3) and (A.4), we can write : p[B Av] = p[B] that is : (A.12) imB ~imAv APPENDIX B: perfect model following Let us consider a plant and a model represented by equations (26) . As it is well known the model following problem requires the determination of an optimal con trol which keeps the behavior of the plant as close as possible to the one of the model. In order to solve this problem two kinds of approach are usually adopted [2, 4].
Implicit model following (the model in the performance index) In this case the optimal control is determined minimizing the following performance index : ~
J=~
j [(x m - xp)T Q (xm - Xp) + u TRu] dt
where Q and R are weighing matrices of proper dimensions. In this case, the optimal control turns out to be of the type :
B (CBr'C] (Ax + Of) =
= [1- B (CBr'C] Ax + [1- B (CBr'C] Of
(B.2) (A.2)
In order the effect of disturbances to be null for any f, it must be : (A.3) that is : p[B 0]
= p[B]
(A.4)
This result can be proved taking into account that a matrix equation of the type KX = P
(A.5)
has a solution if it holds : p[K P]
= p[K]
where K, and K2 are the feedback matrices which depend on the parameters of the plant and of the model and on the weighing matrices Q and R. Obviously, the optimality of the control does not imply that, whatever equations (26) are , the way the plant follows the model is acceptable. If the dynamics of the model is very different from the one of the plant the feedback matrix cannot have enough fle xibility to keep the system acceptably close to the model. Let us examine what relations must exist between the parameters of the plant and the ones of the model so that the plant, after suitable transient, behaves exactly like the model. Defining the error among the model and the plant as : e
(A.6)
= x m-
(B .3)
xp
the error equation becomes :
Indeed (A.4) implies that the equation : BX = 0
(B .1)
(A.7)
has a solution, that is a matrix X satisfying (A.3) can always be found . On the other hand, substituting the
e = Am xm -
Ap xp + Bm u m-
Bp Up
(B.4)
In order to verify if a control of type (B.2) is able to annul the steady state error, let us substitute (B .2) in
\1. L Pe nal i and C; . i3 e noni
I'l:!
In this case, the error dynamics can be arbitrarily changed since its depends not only on plant dynamics, but also on the feedback matrices K1 and K 2.
(B.4) so we get: El = Am Xm -
Ap Xp + Bm U m- Bp K 1 Xp - Bp k2 um
substituting xm = e + xp we have:
REFERENCES
El = Ame + [(Am- Ap) - BpK1] xp+ (Bm- Bpk2) um Since A is supposed to have eigenvalues with negative realmparts, the model following has a steady null error if it holds: [(Am - Ap) - Bp K 1] xp + (Bm - Bp K 2) u m = 0
Bm = Bp K2
[2]
H. Erzberger : Analysis and Design of Model Following Systems by State Space Techniques. Proceedings Joint Automatic Control Conference, pp . 572+581 , 1968.
[3]
D. DrazenoviC : The Invariance Conditions in Variable Structure Systems. Automatica, Vol. 5, pp . 287+295, 1969.
[4]
Y. T. Chan : Perfect Model Following with a Real Model. Proceedings Joint Automatic Control Conference , pp . 287+293, 1973.
[5]
S.L. Shah , D.G. Fisher, D.E. Seborg : Eigenvaluel Eigenvector Assignment for Multivariable Systems and Further Results for Output Feedback Control. Electronics Letters, Vol. 11 , N° 16, 1975.
[6]
K.K.D. Young : Asymptotic Stability of Model Reference Systems with Variable Structure Control. IEEE Transaction on A.C., Vol. AC-22, N° 2, pp . 279+281 , April 1977.
[7]
V.1. Utkin : Variable Structure Systems with Sliding Modes. IEEE Transaction on A.C., Vol. AC-22 , N° 2, pp. 212+222, April 1977.
[8]
K.K.D . Young: Design of Variable Structure Model Following Control System. IEEE Transaction on AC. , Vol. AC-23, pp. 1079+1085, 1978.
[9]
V.1. Utkin : Sliding Modes and their Application in Variable Structure Systems. Mir, Moscow, 1978.
(B.6)
in the unknowns K1 and K 2, respectively, must be solvable . Keeping in mind what we said for equation (A.5), in order for equations (B.6) to be solvable , it must be: p[Bp (Am - Ap)] = p[Bp]
(B.7)
p[Bp Bm] = p[Bp]
(B.8)
Real model following (the model in the system) In this case the control is obtained minimizing the performance index:
I. Flugge-Lotz : Discontinuous and Optimal Con-
trol. MacGrow-Hill, New York, 1968.
(B.5)
Obviously, in order for (B .5) to be satisfied for any value of xp and u m' the following equations: Am - Ap = Bp K1
[1]
~
J=~
j [(X m - Xp)TQ(Xm - Xp) +
U TRu]
(B.9)
dt
The control turns out to be of the type : u p = K1 xp + K 2 x m+ K3 u m
(B.10)
where K 1, K2 e K3 are the feedback matrices which depend on the parameters of the plant and of the model and on the weighing matrices Q and R. In order to verify if a control of type (B.1 0) is suitable to annul the steady state error, let us substitute (B.10) in (B.4) ; we get: El = (Am - Bp K 2) xm - (Ap + Bp K 1) xp + (B.11)
+ (Bm- Bp K 3) um Substituting in (B .11) xm = xp + e we obtain:
El = (Am - Bp K 2) e + [(Am - Ap) - Bp (K1 + K 2)] xp + + (Bm - Bp K 3) um
(B.12)
Also in this case , in order for (B .12) to be satisfied for any value of xm and of u m' equations : (B.13)
Am- Ap = BpK where : K = K1 + K2
must be solvable, that is (B .7) and (B.8) must still hold. Obviously, in order for the steady state error to be nUll , K2 must be chosen so that matrix (Am - Bp K 2) has eigenvalues with negative real parts.
Remark (model following dynamics) For implicit model following , the error equation is: e=Ame
(B.14)
hence its dynamics is not modifiable since it depends on the model matrix. For real model following the error equation is given by: (B.15)
[10] AS .1. Zinober, O.M.E. EI-Ghezawi, SA Billings : Multivariable Variable Structure Adaptive Model Following Control Systems. Proceedings of lEE, Vol. 129, N° 1, pp. 6+12, January 1982. [11] O.M.E . EI-Ghezawi , AS . Zinober, SA Billings: Analysis and Design of Variable Structure Systems Using a Geometric Approach. International Journal of Control, Vol . 38, W 3, pp. 657+671 , 1983. [12] V.1. Utkin: Variable Structure Systems: Present and Future. Automation and Remote Control, Vol. 44 , N° 9, pp. 1105+1120, 1983. [13] S.V. Emelyanov : Binary Control Systems. Moscow, 1985. [14] Hansruedi Buhler: Reglage par mode de glissement. Presses polytecnic romandes; Lausanne , 1986. [15] C. M. Dorling , A.S.1. Zinober: Two Approaches to Hyperplane Design in Multivariable Variable Structure Control Systems. International Journal of Control, Vol. 44 , N° 1, pp. 65+82 , 1986. [16] M.E. Penati, G. Bertoni: 11 controllo dei sistemi a struttura variabile mediante sliding modes. Convegno ICES, Centro di cultura scientifica A Volta, Villa Olmo, Como , 10+11 June 1987. [17] V.1. Utkin: Discontinuous Control Systems: State of Art in Theory and Application. 10° World Congress IFAC on Automatic Control , Munich, 27+31 July 1987.
DECOUPLING AND MODEL FOLLOWING CONTROL IN VARIABLE STRUCTURE SYSTEMS WITH LINEAR PLANT M. E. Penati and G. Bertoni EIt'rtmIllCl. COlllplller Sril'll(/' 111111 .\\.11"111 Thl'Ol1' f) /'p llrllll,' //I. L'llIi'l'nily \ 'illl/, RllIIlgilll/'lI lo 2. -/0136. H olo,!!; //({. II({ly .
'1
Holog//II .
Abstract. One of the most important features of variable structure systems (V.S.S.) is their invariance property when sliding modes occur, that is when the state point slides on the intersection of the switching surfaces. When the plant of the system is linear, it can be very useful to exploit the so-called geometric approach to system and control theory. The aim of this paper is to show how, by means of this approach, the V.S.S. features become more apparent so as to allow the achievement of new results and new applications. In particular, the paper shows that the solution of the model following problem using sliding mode control can be used to implement an interesting procedure which, taking into account the invariance conditions, allows to decouple slightly interconnected systems. This procedure turns out to be very robust and particularly suitable for large scale systems. Keywords . sliding modes, robustness, decoupling, variable structure systems
framework of the geometric approach to system and control theory. In Section 4 the influence of the switching hyperplanes on system dynamics is examined and in Section 5 the switching hyperplane assignment problem is dealt with . In Section 6 the model following problem using a sliding mode control is reviewed by means of the geometric approach and the results are used in Section 7 for decoupling slightly interconnected systems. The invariance conditions of V.S.S. and the perfect model following conditions are reviewed, respectively, in Appendix A and in Appendix B.
INTRODUCTION As it is well known, variable structure systems (V.S .S.) are a special class of systems characterized by a discontinuous control action obtained by changing the structure of the system when it reaches a set of switching surfaces. One of the most important features of V.S.S. is their invariance property when sliding modes occur, that is when the state point (i.e. the point representing the state vector) slides on the intersection of the switching surfaces . In this case, the system behaves as though linear and it turns out to be independent of certai n classes of parameter variations and disturbances. In other words, the sliding mode control of a V.S.S. is basically an adaptive control which forces the response to slide along a predefined trajectory. The robust nature of the control is very useful in many fields of applications and mainly in robotics (particularly in space robotics) where the moment of inertia may vary widely depending on the robot arm and the load it carries. The mathematical treatment of this kind of system is quite complicated and has been mainly dealt with by Russian authors [9, 13]. The analysis becomes easier only when plants are assumed to be linear; in this case it can be very useful to exploit the geometric approach to system and control theory [11 , 15]. The aim of this paper is to show how, by means of this approach, the V.S.S. features become more apparent so as to allow the achievement of new results and applications. In particular, the paper shows that the solution of the model following problem using sliding mode control can be used to implement an interesting procedure which, taking into account the invariance conditions of Drazenovi6 [3] , allows to decouple slightly interconnected systems; this decoupling turns out to be very robust and particularly suitable for large scale systems. The outline of the paper is as follows . In Section 2 the problem of V.S.S. model is dealt with in the general case while in Section 3 the particular case of systems with linear plant is considered in the
2
THE MODEL OF V.S.S.
As it is well known from literature [9, 13], in the general case, the mathematical model of the plant of a V.S.S. is given by: = f(x ,t,u) (1 )
x
where x and f are two (n x 1) vectors and U is a (m x 1) vector whose components Uj (X,t) (i = 1, ...... , m) coincide with one of the two continuous functions Uj +(x,t) or Uj-(x,t) depending on the position (with respect to the switching surface Sj(x)) of the state point. More precisely it can be written : Uj(x,t) = Uj+(x,~ for Sj(x) > 0 i= 1, ...... , m (2) Uj(x,t) = Uj-(x,t) for Sj(x) < 0 i= 1, ...... , m In a certain sense relations (1) and (2) can be considered a grey box type model of a V.S.S.. An interesting characteristic of these systems is that the surfaces Sj(x) and the functions u/(x,t) and Uj-(x,n can be chosen in such a way that, once the intersection surface s(x) = 0 of the switching surfaces Sj(x) = 0 is reached, the state vector remains in the neighborhood OSCillating about it with a high frequency and very small amplitude; infinite frequency and zero amplitude in the ideal case. As mentioned in the Introduction, this is the so-called sliding mode type of mo-
lili
:'\1. E. Pe nali a nd G. Berto ni
188
tion and the surface s(x) = 0 is called sliding surface. However. as it can be seen from (2). in corrispondence of the state points lying on the switching surfaces s; (x) = 0 the control functions are not defined so that these can be considered discontinuity points of the function representing the second member of (1) . Consequently. the Cauchy conditions for the existence of the solution of the differential equation (1) are not satisfied. This means that the model (1). (2) is not suitable to represent the system in all conditions and. in particular. during sliding modes. A way to circumvent this drawback could be to change part (2) of the grey box type model taking into account that . actually. the system does not switch instantaneously from a control function to another. but. indeed. due to the so-called non-idealities in switching devices. the control functions change in a very short (~ut not nUll) interval so that they do not undergo dlscontinuities . but only sharp changes and . hence. they are always defined. Unfortunately. these models. often called real models . assume different mathematical structures because of the different types of switching devices used by the actual systems . This turns out to be a heavy shortcoming because it prevents studying these systems in the same mathematical framework. A more convenient method for determining the model when equations (1). (2) cannot describe the whole behavior of the system. is to use the model given by the so-called mathematical continuation method. that is to use a black box type model. As it is well known. these models are only mathematical descriptions of the link between input-output data of the system so that. in general. the state variables involved in the equations do not have any physical meaning . Fortunately . it can be demonstrated [9] that one of these models. the equivalent control model. has a very interesting property since it tends to coincide with every real model when the effect of the non-idealities goes to zero . This means that the equivalent control model not only gives a mathematical description of the input-output data link. but also represents the physical behavior of the system ; consequently. the state variables involved in this black box model get a precise phYSical meaning. As to the implementation of this model. it is made up of the model of the system plant controlled by a continuous feedback function which keeps the state point on the sliding surface . In conclusion. let us note that. in the general case. determ ining the equivalent control model is quite cumbersome ; this model becomes simple enough only with linear plant and its analysis can be easily accomplished mainly when USing . as mentioned above . the framework of the geometric approach to system and control theory. This is what will be done in the next section .
3
SLIDING MODE EQUATIONS: METRIC APPROACH
THE GEO·
Let us consider a V.S.S. with a linear stationary plant given by : le = Ax + Bu (3) where x is a (n x 1) vector. u a (m x 1) vector. A and B are two matrices of proper dimensions. Let us assume that the switching surfaces s/..x) = 0 (for i = 1•.. .• m) are hyperplanes containing the origin of the space ; consequently the sliding surface is given by : s(x) = Cx = 0
(4)
which shows that the sliding modes take place in the subspace defined by kerC . As to the model of the system when sliding modes occur. let us determine the equivalent control model defined in the previous section . Since during sliding mo-
tion it must hold :
(5)
the continuous control which keeps the state point on the sliding surface (that is the equivalent control u'!~ can be obtained substituting (3) in (5) and solVing Wit respect to u; we get:
ueq = -Kx
(6)
where : K = (CBr'CA Substituting (6) in (3) . the equivalent control model of the system turns out to be :
le = Aeq x = [1- B (CBr'C] Ax = (A - BK) x
(7)
This is the so-called sliding mode matrix equation. Using th is equation we can make some preliminary remarks on the features of V.S.S. when sliding modes occur. Remark 1 (sliding mode dynamics depends only on the sliding surface) From equation (7) it follows that for the equivalent control to exist it must be IC B I "'" O. On the other hand . ICBI "'" 0 means that ker(CB) has null dimension. that is no vector x exists such that CBx = O. Consequently it cannot be neither y = Bx = 0 nor Cy = 0 ; that is y = Bx must belong to imB (i.e . also kerB must have null dimension) and y cannot belong to the subspace kerC. Summarizing it can be written : kerC (") imB = ()
(8)
Relation (8) shows one of the most important properties of V.S.S. when sliding modes occur. i.e. the behavior of these systems does not depend on the controls. but only on the sliding surface. Remark 2 (during sliding motion a reduction of the model dynamic order takes place) When sliding modes occur. a reduction of the model dynamic order takes place from n to (n - m) . wh ich means that only (n - m) equations of system (7) are linearly independent. To show this property. let us remember that . during sliding motion . relation (5) holds so that. substituting le with (7) . we get :
C (A- BK) x = 0 or: kerC::> im(A - BK)
(9)
Since the subspace kerC has (n - m) dimensions. re lation (7) cannot have more than (n - m) linearly independent equations. that is. the system matrix Aeq has m null eigenvalues at least. Remark 3 (the (n - m) eigenvectors corresponding to the non zero eigenvalues form a basis for kerC) Let us label the eigenvalues of Aeq with A; (i = 1•. .. . n) and the correspondent eigenvectors with v; (i = 1 . ... . n) . For any eigenvalue A; it can be written : C [(A - BK) - A;l v;= C (A - BK) v ; - A;Cv; that is. keeping in mind (9) : A; Cv; = 0
(10)
Obviously. relation (10) holds only if A;= 0 or CV; = o. This means that the (n -m) eigenvectors corresponding to the (n -m) non zero eigenvalues must belong to the subspace kerC. If. for the sake of simplicity. we assume that the (n -m) non zero eigenvalues are distinct
" ariabl e Structure Syste m s ,,·ith Lin ear Pla nt and labelled by A.; (i = 1, .... n -m), the correspondent eigenvectors v; (i = 1, ... . n -m) turn out to be linearly independent so that they can make up a basis for kerC . Summarizing, we can conclude that, when sliding modes occur, the model of a V.S.S. with linear plant changes in a very radical manner; that is, in spite of the variable structure of the physical system, the model turns out to be linear and undergoes an order reduction as large as the vector control dimension. Moreover, the behavior of the system is dependent on the structure of matrix C (i.e . on the switching hyperplanes) , but not on the controls.
4 INFLUENCE OF SWITCHING NES ON SYSTEM DYNAMICS
HYPERPLA-
In order to point out the influence of matrix C on V.S.S. dynamics when sliding modes occur, it can be useful to use a proper linear transformation:
z =Tx
(11 )
such as to transform system (3) as follows :
2:, = A" z, + A'2Z2
(12)
2:2 = A 2, Z, + A22 Z2 + B2 U where z, and z2 are, respectively, two [(n - m) x 1] and (mx 1) vectors while A", A'2, A 2" A 22 are matrices of proper dimensions and B2 is a (m x m) non singular matrix . In order to show how (12) can be obtained, let us write the transformation matrix T in the form : (13)
where T2 is a (m x n) matrix and T, is a [(n - m) x n] matrix such that it holds :
T, B = 0
(14)
Substituting (11) in (3) and (4) and defining :
z = Tx =
I :: I (15)
n-m
TAr' =
I
A" A2 ,
B2 = T2 B
then system (3) assumes the form (12) and relation (4) becomes : s = C, z, + C 2 Z2 = 0 (16) where C, and C 2 are, respectively, [m x (n-m)] and
(m x m) matrices.
Taking into account (14) and (15), the equivalent control (6) for system (12) becomes :
Since this equation is a linear combination of the first matrix equation of (12), we can conclude that the V.S.S. model during sliding motion is given by the first matrix equation of (12), that is, in agreement with remark 2, by a reduced order system. On the other hand, since, during sliding motion relation (16) holds, the first one of (12) becomes : 2:, = (An - A'2 F) z,
F=C2-'C,
Using equations (12) and (20) we are now able to make some interesting remarks about the influence of matrix C on the dynamics of V.S.S. when sliding modes occur.
Remark 4 (pole assignment conditions for sliding mode equations) If the couple (A", A(2) of system (20) is controllable the poles of (20) can be arbitrarily assigned by choosing properly the elements of F which acts like a feedback matrix. Obviously this choice (i .e. the choice of the switching hyperplanes) must be done so as to guarantee that relation (18) holds. Let us note that the controllability of the couple (A, B) is a sufficient, but not necessary condition for the pole assignment of equation (20) . Indeed, the couple (A" , A(2) turns out to be controllable (and, hence, the poles of (20) can be arbitrarily assignable) even if the couple (A, B) is not. In fact, if the couple (A", A(2) is controllable and the couple (A22' B 2) is not functionally controllable, but only controllable (because of the non singularity of matrix B 2) then the couple (A, B) results to be not completely controllable.
5
SWITCHING HYPERPLANE ASSIGNMENT
As shown in the previous sections, the switching hyperplanes determine not only the sliding surface, but also the sliding mode dynamics. In this section we will examine how to assign the switching hyperplanes in order to satisfy proper constraints on V.S.S. dynamics. A first constraint which can be chosen is that the (A" - A'2 F) matrix assumes some given eigenvalues A." .•. . A. n- m. Of course this can be done by means of any pole assignment method . However, in .this way, matrix F (and, hence, matrix C) turns out to be only partially determined; this means that, in order to cam· pletely determine matrix F, we must choose some additional constraints [15] . To this aim, let us remember that the dynamics of a system is affected not only by the eigenvalues of its model but also by the eigenvectors which mainly affect the mode amplitudes. However, assigning the (n - m) eigenvectors of system (20), the elements of matrix C still remain partially undetermined and m 2 elements must be chosen in some other way. To show how how m2 elements still rema in undetermined let us define the eigenvector matrix V :
Keeping in mind Remark 3 we can write :
Let us define :
*0
(18)
Substituting (17) in the second relation of (12) we get: (19)
(22)
CV = 0
* 0 and IB21 * 0 it must also be
2:2 = -C 2-'C, (A" z, + A'2 Z2)
(21)
V =[v" .... vn-ml (17)
IC 211B 21= IC2 B21 = ICr'TBI = ICBI
(20)
where F is a [m x (n-m)] matrix given by:
Ueq = - (C2 B 2)-' [(Cl A" + C2 A 2,) z, +
Obviously , given ICBI IC21* O. In fact:
IS9
TV=W=
IW
W,
(23)
2
where W, is a [(n - m) x n - m)] matrix and W 2 is a [m x (n - m)] matrix.
7'.1. E. Pe n ati and G. Bertoni
190
Consequently, relation (22) can be written in the form : CT-'TV =
I C,
C2
Upeq = (CBpr' (CAm Xm - CAp Xp + CBm Um) substituting (28) in (27) we get:
I I :: I =
it
= C2
IF
Im
I I :: 1= C2 (FW, + W2 ) =0
(24)
(2S)
We can conclude that, assigning matrix V and, hence, matrices W, and W 2, we can solve the matrix equation (2S) ; in particular, if W, is non singular, it is very simple to determine the [m x (n - m)] unknown elements of F from relation (2S) . Therefore , from (21) we can determine, for example , C, leaving undetermined the m2 elements of C 2 which can be used , hypothetically, to assign the sliding surtace . Unfortunately, the method we have just outlined not only is unsuitable to completely determine the elements of matrix C , but it also suffers from another severe shortcoming. In fact, as it can be seen in [S], it is not possible to assign no more than m components of each eigenvector in a completely arbitrary manner. Consequently a trial and error procedure must be followed by assigning, for example , m components according to some scheme and then by checking whether the behavior of the system is suitable.
[1- Bp (CBpr'Cj (Am Xm - Ap Xp + Bm Um)
(29)
= [1- Bp (CBpr'Cj
Am e +
[I -
Bp (CB pr' cj (Am-
- Ap) Xp + [1- Bp (CBpr'Cj Bm Um (30) As it follows from (A.2) , equation (30) can be considered as the sliding mode equation of a V.S.S. with a plant having Am as a state matrix, Bp as a control matrix and affected by two disturbances xp and u m. Consequently, keeping in mind (A.4), for the system to be insensitive to these disturbances the following conditions must be satisfied : p[Bp (Am - Ap)]
= p[Bp
Bm]
= p[Bp]
(31)
On the other hand, since the equivalent control (28) is of type (B.11), (31) can also be interpreted as the condition which gives the pertect model following of (26) , that is, the condition which imposes the error equations to become :
Et = [1- Bp (CBpr'Cj Am e
(32)
or, substituting in (29) xp = xm - e:
SLIDING MODE MODEL FOLLOWING
An interesting application of the results outlined in the previous sections is the solution of the model following problem by means of a sliding mode control. From now on we shall refer to this control as sliding mode model
following control. In order to solve this problem let us consider a plant and a model that are both linear and stationary and then represented by the following matrix equations : Xp = Ap xp + Bp up (26) where xm and xp are the two (n x 1) state vectors while u m is the (m' x 1) control vector of the model and up is the (m x 1) control vector of the plant . Without loss of generality, we can assume that the couple (A p ' Bp) and (Am ' Bm) are completely controllable and that Am has eigenvalues with negative real parts. Moreover, let us suppose that all the components of xp are accessible and directly measurable . As shown in Appendix B, the error equation is given by : Et = Am Xm - Ap Xp + Bm Um- Bp up (27) As it is well known from literature [8, 10]. one way to annul the error e, i.e . to force the plant to follow the model , is to determine a proper set of hyperplanes such that they can behave as switching surtaces for a sliding mode control. Let us assume that these hyperplanes contain the origin of the space so that the sliding surtace is given by :
S = Ca As shown in Section 3 the equivalent control up eq is Ihe control which keeps the state point on the sliding surtace, that is the control which imposes 5 = O. Therefore we can write: C
=
Substituting in (29) Xm = e + xp we get:
it
so that we obtain :
6
(28)
e= CAm Xm -
that is :
CAp Xp + CBm Um - CBp up eq = 0
In conclusion , the implementation of the model following of system (26) by means of a sliding mode control turns out to give the same sliding mode equations of a V.S.S. with a plant of the type :
Et = Ame + Bp U
(33)
or of the type :
Et = Ape - Bp U Obviously, in order to determine the matrix C which gives an assigned error dynamics, one of the methods shown in Section 4 can be used.
7
DECOUPLlNG SLIGHTLY INTERCONNECTED SUBSYSTEM
As shown in the literature [17], one way to decouple a system made up of slightly interconnected subsystems is to use a sliding mode control for each subsystem to be decoupled. In order to show how this can be done , let us consider a plant made up of two subsystems given by :
x, = A"
x, + A'2X2+ B, U, (34)
X2 = A 2, x, + A22 X2 + B2 U2
where x, is a (r x 1) vector, X2 is a [(n - I) x 1)] vector, u, and u 2 are two (m x 1) vectors and A" , A'2 ' A 2" A 22 , B, and B2 are matrices of proper dimensions. Obviously, the decoupling of the two subsystems (34) by means of sliding mode controls implies that the two terms A'2 x2 and A 2, x, can be considered as ex1ernal disturbances and that it holds : p[B, Ad = p[B,]
(3S)
Unfortunately, both of these conditions (3S) are not easy to be satisfied. Moreover, it must be noted that the decoupling is pertormed by the same controls which drive the subsystems. Less restrictive conditions can be found using the
\'ariabl e Structure Svste llls with Lin ear Plant
sliding mode model following control with a properly decoupled model. To this aim let us consider again system (34) and assume that the model is made up of two decoupled subsystems given by:
19 1
matrix 0 given by (A.7) in (A .3) it can be verified that (A.4) is a condition for (A.3) to hold. We can conclude that, during the sliding motion, V.S.S. are invariant to the external disturbances only if : imB
(36) X2m = A22 x2m + B 2m u2m where x'm, x2m, u,m and u2m are vectors of the same dimensions of x" X2 ' u, and U2' respectively, Keeping in mind conditions (31) we can say that, when sliding modes occur, system (34) behaves like model (36), i.e. it can be considered decoupled if it holds :
~imD
(A.S)
Invariance to parameter variations Let us consider again system (3) and suppose the matrix A can be represented as : (A.9) were Ac is the (n x n) state matrix and Av is the (n x n) matrix which takes into account the parameter variations. Consequently, the sliding modes equation is given by :
(37) (A.10) As it easy to see, if both conditions (35) are satisfied so are conditions (37), but not vice versa, that is conditions (37) are less restrictive than those of (35) . This new method has an additional advantage due to the fact that there is a clear separation among the input functions U" U2 which implement the decoupling (by means of the sliding mode model following control) and the input functions u'm ' u2m which control the overall decoupled system. The method seems to be quite promising since , mainly in large scale systems, the parameters interconnecting the different subsystems which are to be decoupled are often time varying ; this method assures that, within the boundary values for which conditions (37) are satisfied, decoupling is always garanteed , APPENDIX A: invariance conditions for V.S.S. One of the most important features of V.S .S. during sliding modes is their invariance to some classes of external disturbances and parameter variations [3] .
Invariance to external disturbances Let us consider some external disturbances f entering the system (3) through a coefficient matrix 0 :
x= Ax + Bu + Of
(A.1)
where f is a (r x 1) vector and 0 is a (n x I') input matrix. If the sliding surface is given by (4) the equivalent control is given by :
x= [1 -
In order for the effect of parameter variations to be nUll, it must be : (A.11 ) Comparing relation (A.11) to (A.3) and (A.4), we can write : p[B Av] = p[B] that is : (A.12) imB ~imAv APPENDIX B: perfect model following Let us consider a plant and a model represented by equations (26) . As it is well known the model following problem requires the determination of an optimal con trol which keeps the behavior of the plant as close as possible to the one of the model. In order to solve this problem two kinds of approach are usually adopted [2, 4].
Implicit model following (the model in the performance index) In this case the optimal control is determined minimizing the following performance index : ~
J=~
j [(x m - xp)T Q (xm - Xp) + u TRu] dt
where Q and R are weighing matrices of proper dimensions. In this case, the optimal control turns out to be of the type :
B (CBr'C] (Ax + Of) =
= [1- B (CBr'C] Ax + [1- B (CBr'C] Of
(B.2) (A.2)
In order the effect of disturbances to be null for any f, it must be : (A.3) that is : p[B 0]
= p[B]
(A.4)
This result can be proved taking into account that a matrix equation of the type KX = P
(A.5)
has a solution if it holds : p[K P]
= p[K]
where K, and K2 are the feedback matrices which depend on the parameters of the plant and of the model and on the weighing matrices Q and R. Obviously, the optimality of the control does not imply that, whatever equations (26) are , the way the plant follows the model is acceptable. If the dynamics of the model is very different from the one of the plant the feedback matrix cannot have enough fle xibility to keep the system acceptably close to the model. Let us examine what relations must exist between the parameters of the plant and the ones of the model so that the plant, after suitable transient, behaves exactly like the model. Defining the error among the model and the plant as : e
(A.6)
= x m-
(B .3)
xp
the error equation becomes :
Indeed (A.4) implies that the equation : BX = 0
(B .1)
(A.7)
has a solution, that is a matrix X satisfying (A.3) can always be found . On the other hand, substituting the
e = Am xm -
Ap xp + Bm u m-
Bp Up
(B.4)
In order to verify if a control of type (B.2) is able to annul the steady state error, let us substitute (B .2) in
\1. L Pe nal i and C; . i3 e noni
I'l:!
In this case, the error dynamics can be arbitrarily changed since its depends not only on plant dynamics, but also on the feedback matrices K1 and K 2.
(B.4) so we get: El = Am Xm -
Ap Xp + Bm U m- Bp K 1 Xp - Bp k2 um
substituting xm = e + xp we have:
REFERENCES
El = Ame + [(Am- Ap) - BpK1] xp+ (Bm- Bpk2) um Since A is supposed to have eigenvalues with negative realmparts, the model following has a steady null error if it holds: [(Am - Ap) - Bp K 1] xp + (Bm - Bp K 2) u m = 0
Bm = Bp K2
[2]
H. Erzberger : Analysis and Design of Model Following Systems by State Space Techniques. Proceedings Joint Automatic Control Conference, pp . 572+581 , 1968.
[3]
D. DrazenoviC : The Invariance Conditions in Variable Structure Systems. Automatica, Vol. 5, pp . 287+295, 1969.
[4]
Y. T. Chan : Perfect Model Following with a Real Model. Proceedings Joint Automatic Control Conference , pp . 287+293, 1973.
[5]
S.L. Shah , D.G. Fisher, D.E. Seborg : Eigenvaluel Eigenvector Assignment for Multivariable Systems and Further Results for Output Feedback Control. Electronics Letters, Vol. 11 , N° 16, 1975.
[6]
K.K.D. Young : Asymptotic Stability of Model Reference Systems with Variable Structure Control. IEEE Transaction on A.C., Vol. AC-22, N° 2, pp . 279+281 , April 1977.
[7]
V.1. Utkin : Variable Structure Systems with Sliding Modes. IEEE Transaction on A.C., Vol. AC-22 , N° 2, pp. 212+222, April 1977.
[8]
K.K.D . Young: Design of Variable Structure Model Following Control System. IEEE Transaction on AC. , Vol. AC-23, pp. 1079+1085, 1978.
[9]
V.1. Utkin : Sliding Modes and their Application in Variable Structure Systems. Mir, Moscow, 1978.
(B.6)
in the unknowns K1 and K 2, respectively, must be solvable . Keeping in mind what we said for equation (A.5), in order for equations (B.6) to be solvable , it must be: p[Bp (Am - Ap)] = p[Bp]
(B.7)
p[Bp Bm] = p[Bp]
(B.8)
Real model following (the model in the system) In this case the control is obtained minimizing the performance index:
I. Flugge-Lotz : Discontinuous and Optimal Con-
trol. MacGrow-Hill, New York, 1968.
(B.5)
Obviously, in order for (B .5) to be satisfied for any value of xp and u m' the following equations: Am - Ap = Bp K1
[1]
~
J=~
j [(X m - Xp)TQ(Xm - Xp) +
U TRu]
(B.9)
dt
The control turns out to be of the type : u p = K1 xp + K 2 x m+ K3 u m
(B.10)
where K 1, K2 e K3 are the feedback matrices which depend on the parameters of the plant and of the model and on the weighing matrices Q and R. In order to verify if a control of type (B.1 0) is suitable to annul the steady state error, let us substitute (B.10) in (B.4) ; we get: El = (Am - Bp K 2) xm - (Ap + Bp K 1) xp + (B.11)
+ (Bm- Bp K 3) um Substituting in (B .11) xm = xp + e we obtain:
El = (Am - Bp K 2) e + [(Am - Ap) - Bp (K1 + K 2)] xp + + (Bm - Bp K 3) um
(B.12)
Also in this case , in order for (B .12) to be satisfied for any value of xm and of u m' equations : (B.13)
Am- Ap = BpK where : K = K1 + K2
must be solvable, that is (B .7) and (B.8) must still hold. Obviously, in order for the steady state error to be nUll , K2 must be chosen so that matrix (Am - Bp K 2) has eigenvalues with negative real parts.
Remark (model following dynamics) For implicit model following , the error equation is: e=Ame
(B.14)
hence its dynamics is not modifiable since it depends on the model matrix. For real model following the error equation is given by: (B.15)
[10] AS .1. Zinober, O.M.E. EI-Ghezawi, SA Billings : Multivariable Variable Structure Adaptive Model Following Control Systems. Proceedings of lEE, Vol. 129, N° 1, pp. 6+12, January 1982. [11] O.M.E . EI-Ghezawi , AS . Zinober, SA Billings: Analysis and Design of Variable Structure Systems Using a Geometric Approach. International Journal of Control, Vol . 38, W 3, pp. 657+671 , 1983. [12] V.1. Utkin: Variable Structure Systems: Present and Future. Automation and Remote Control, Vol. 44 , N° 9, pp. 1105+1120, 1983. [13] S.V. Emelyanov : Binary Control Systems. Moscow, 1985. [14] Hansruedi Buhler: Reglage par mode de glissement. Presses polytecnic romandes; Lausanne , 1986. [15] C. M. Dorling , A.S.1. Zinober: Two Approaches to Hyperplane Design in Multivariable Variable Structure Control Systems. International Journal of Control, Vol. 44 , N° 1, pp. 65+82 , 1986. [16] M.E. Penati, G. Bertoni: 11 controllo dei sistemi a struttura variabile mediante sliding modes. Convegno ICES, Centro di cultura scientifica A Volta, Villa Olmo, Como , 10+11 June 1987. [17] V.1. Utkin: Discontinuous Control Systems: State of Art in Theory and Application. 10° World Congress IFAC on Automatic Control , Munich, 27+31 July 1987.