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Analysis of Some Classes of Nonlinear Discrete Systems Under Bounded Disturbances
Copyright © IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998
ANALYSIS OF SOME CLASSES OF NONLINEAR DISCRETE SYSTEMS UNDER BOUNDED DISTURBANCES Vsevolod M. KUNTSEVICH* Boris N. PSHENITCHNYI **
* Space Research Institute of N AS and NSA of Ukraine 40 Prsp. Akad. Glushkov, 252650 MSP 252022 Kiev-22, Ukraine Phone: +38044 2664124 Fax: +38 044 2664124 E-mail: [email protected] ** Institute for Applied System Analysis of NAS and ME of Ukraine 37 Prospekt Peremogy, 252056 Kiev-56, Ukraine Phone: +38 044 2662297 Fax: +38 044 2664124
Abstract: The constructive solution of the calculating minimal invariant sets problem for some classes of autonomous and non autonomous discrete systems has been obtained. For the nonautonomous discrete systems the problem is solved on the assumption that only set-valued estimates for additive bounded disturbances are available. The class of systems with bounded nonlinear parts is considered. This class contains systems with saturation, relay systems having a dead zone and others. An example illustrates the proposed method. Copyright © 1998 IFAC Keywords: discrete dynamic systems, bounded disturbances, invariant sets
prevented developing the hard-bound methods of analysis. These methods have been attracting the researchers anew last years.
1. INTRODUCTION
Analysis of dynamic systems under bounded disturbances seems to become a problem among those most actual ones of the modern theory of dynamic systems. Recently, making use of the hard bounds on disturbances, meaning the use of setvalued estimates (see (Kurzhanski, 1977; di Masi et al., 1991; Kurzhanski and Veliov, 1994; Kuntsevich and Lychak, 1992; Milanese et al., 1996; Kurzhanski and Valyi, 1997) in particular) , is becoming as much popular as the use of stochastic models when applying for analysis of dynamic systems under uncertainty.
The early results on calculating minimal invariant sets that provide the most complete (in certain sense) characteristics of the analyzed systems can be found in the preceding papers (Kuntsevich and Pshenitchnyi, 1996b; Kuntsevich and Pshenitchnyi, 1996a; Kuntsevich and Pshenitchnyi, 1997) published by the authors. In the present paper, these results are widened to a class of nonlinear dynamic systems found in many practical applications.
It is worth to note that the problem of analyzing linear dynamic systems under additive bounded disturbances was first stated and discussed in (Bulgakov, 1946) . However, a wide expansion of the methods for statistic dynamics
273
2. MINIMAL INVARlANT SETS OF LINEAR SYSTEMS UNDER BOUNDED DISTURBANCES
Theorem 1. At the assumptions made on A and
J, The set WL(A,J) defined by (4) is a compact in lR m , i.e., a closed bounded set.
Although the paper devoted to non linear systems analysis we starts with the consideration of linear systems. This is done because of the fact that the methods applicable to linear systems are used further for analyzing nonlinear systems of certain classes as well.
One can find the proof for the following theorem in (Kuntsevich and Pshenitchnyi, 1997) once again. Theorem 2. WL(A, J) is a minimal invariant set, meaning WL(A, J) is invariant and WL(A, J) ~ WL as soon as WL is also closed invariant. Furthermore, a system trajectory which starts at an arbitrary point Xo E lRm at any € > 0 remains in the cneighborhood of WL(A , J) at sufficiently large n.
Consider a dynamic system Xn+l = AXn
+ Fn ,
n = 0, 1, 2, .. . ,
(1)
where Xn E lRm is a state vector, A is an m x m real matrix, Fn E lRm is a sequence of vectorvalued disturbances taking independent values from a given set at time instances,
Theorems 1 and 2 define the minimal invariant set of system (I), ~hich is also convex as soon as J is convex compact in (!Rm . Consider, J is a convex polytope given by
(2)
J
Here J is a given bounded (non necessarily convex) set. Assume, a matrix A satisfies the condition
{F i } ,
(5)
i=I ,N
where F i is the i-th vertex and N is the total number of vertices. In this case, significant in view of practical applications, the set WL(A, J) is calculated as
(3)
where >'i(A) is the i-th eigen value of the matrix A.
WL(A,J)
Due to the mentioned properties of a matrix A and to the boundedness of disturbances F n , the reaction of a system (1) to these disturbances (the dissipativity) is obviously bounded. However, despite of an illusory simplicity of calculating the set which contains all the states Xn in asymptotic, the problem has appeared nontrivial and the solution to it has been found only recently.
=
conv_{Fi
+ AFj +
i,j ,k ... =I ,N
The minimal invariant sets defined this way provide exhausting characteristics for the considered dynamic systems under bounded disturbances. However, these definitions are poorly applicable for comparing the properties of distinct systems. In view of the latter, it is needed to have a numeric measure, for instance a scalar function defined on these sets. This could be the diameter D(WL) of a set WL(A , J)
For more convenience, the essential results found in (Kuntsevich and Pshenitchnyi, 1996b; Kuntsevich and Pshenitchnyi, 1996a; Kuntsevich and Pshenitchnyi, 1997) are presented below in brief. These will be used further when analysing nonlinear systems. Introduce a set
WL(A,J) =~+AJ+A2J+ . .. ,
= conv
D(WL)
= XE!m max
IIXII,
(7)
(4)
where IIXII is the Euclidean norm of X. where the sum of sets is a Minkovski sum, that is WLI
+ WL2 = {Xl + X 2
:
The diameter of an invariant set, when it is used as a measure of the dynamic precision of system (1)-(2), is similar in certain sense to the dispersion of the process Xn at Fn being a stochastic process.
Xl E WL I , X 2 E WL2 } .
Recall, a closed set WL is called an invariant for system (I), if the inclusion X E WLleads to AX + FE WL at any F E J , meaning AWL + J ~ WL.
Assume, J is a convex polytope given by (5). One can calculate the diameter D(WL) at any required accuracy by use of the formula
Definition 1. A compact set is called minimal invariant for system (I), if it is invariant and has no (other but itself) compact invariant subset.
(8)
where pI and Pq are the i-th and q-th vertices respectively of the estimating set for WL, which has been calculated by use of a finite number
The following theorem has been proven in (Kuntsevich and Pshenitchnyi, 1997).
274
of items in the series (4). In (Kuntsevich and Pshenitchnyi, 1997), the following upper bound estimat e for D(VJt) was obtained.
In the same paper (Kuntsevich and Pshenitchnyi, 1997), the estima te for the diamet er of the invariant set of system (10), (11) and (2)
.
(9)
D(VJt) ::; - C1 D(J) . -q
Here D(J) is the diamet er of the set J and c is a constan t value determined by the known linear algebraic inequality
~
D[!JJt(A , Je)] ::; D[VJt(A, J)] + 1 _ q c ~
::; -1-(D(J ) -q
+ c)
(15)
has been obtaine d.
n=0, 1, 2, . . .. 3. MINIMAL INVARIANT SETS OF SYSTEMS WITH BOUNDED NONLINEAR PARTS
2.1 Invaria nt sets of nonstationary systems Making use of the results found in (Kuntsevich and Psheni tchnyi, 1997), one can calculate the minimal invaria nt sets of nonsta tionary linear systems under additiv e bounded disturbances. Define the class of linear nonsta tionary systems (10) where the notatio ns are the same as above, except the matrix 0
_
-
(11 )
A =A +An·
3.1 Autono mous systems with bounded nonlinear
parts Consider a system
where W and
IRm
-+
IRm is a continuous function (17)
o
Here A is a given real matrix and time-varying coefficients of the matrix An can take arbitra ry indlPe ndent (from each other) values from a given set 11, i.e.,
A E U = {A: IIAII ::; ~} . o
(12)
-
In this case, the inequality 11 A +Anll < 1 may not take place because of the given definitions for o matrices A and An. o
Assume, the matrix A satisfies the conditions like (3) . Introduce a set
VJtC4, JE)
= VJt(A, J) + cS + cAS •2
+c A S + . . . ,
(13)
where c > 0, S is a unit solid sphere with the center in the origin and Je = J + cS is an cneighborhood of J. In (Kuntsevich and Pshenitchnyi, 1997) , the following theorem has been proven. Theorem 3. If at some c > 0'
l-q ~
The analysis of the inner structu re of the invariant sets under consideration, the analysis of the types of trajectories that belong to these sets by means of w-limit points of the trajectories, is given in (Kuntsevich and Pshenitchnyi, 1997) and hence is omitted here . Next, consider a significant subclass of systems having bounded, in terms of (17), nonlinear parts. These are relay systems described by the equations X n+ 1
°
c D(J)+ c '
< - - ....-:-..,..--
is a compact set. Here cl(x) means closing for the set x. Expressions (16) and (17) lead to the fact that at w(Xn) given by (17), every trajecto ry of a system (16) is one of those feasible trajecto ries of a system (1) . And due to Theorem 1 the set VJt(A , J) is invariant for system (16), therefore system (16) has a compact invariant set.
= AXn + BsignO'n, an = eT X n ,
(18) (19)
where
+1 ,
(14)
signO'n = {
then the set VJt(A,Je) is invariant for a system given by (10), (11) and (2).
if an
> 0,
an < 0, [-1 , +1], 1f an = 0,
-1,
~
(20)
and B E IRm is an m-dimensional real vector.
275
Without the loss of generality, assume that IICII = 1 and consider a subclass of systems (18), (19) and (20) determined by the condition C E (!:
=
{C : IICII
(24) where nonlinear vector-valued function 'l1 (.) satisfies the condition (17) and J is a given compact set.
= I}.
As it is led from (20), the class of dynamic systems defined by (18), (19) and (20) has a multi-valued right hand side. Due to the definitions (18), (19) and (20) the condition (17) is fulfilled, therefore the set given by formula (4) is a minimal invariant set for the class of systems (18), (19) and (20).
Introduce the set (25) Obviously, the problem of calculating the minimal invariant set for a system (23), (24) and (25) is reduced to the problem solved previously to within the notations. In particular, the class of systems (23) includes relay systems under disturbances:
Introduce the following
Definition 2. A trajectory T(Xo) of a system (18), (19) and (20) is called regular, if there is such £ > 0 that
ICT Xnl > £,
n = 0,1, . . ..
Lemma 1. Assume IICII = 1. If IIX - YII < £/2, then
IcT XI 2:
X n+ 1
(26)
mi~~~
Calculating the invariant set for these systems does not differ from the way it has been made above.
(21) £ and
= sign c TX, IcT XI > £/2.
= AXn + B sign C T Xn + Fn.
(22)
3.3 Invariant sets of relay systems with unstable linear part
On the basis of this lemma, the following theorems have been proven in (Kuntsevich and Pshenitchnyi, 1997).
In practical control system design, the following case is met frequently: system (26) does not satisfy the strict conditions (3) , however, the weakened conditions on A,
sign CTy
Theorem 4. If T(Xo) is a regular trajectory, then for every point Yo sufficiently close to X o, the respective trajectory is also regular and formulae (22) are correct.
P'i(A)1 = 1,
are fulfilled. This takes place when the linear part of a system (26) contains an integrator (summation element) . As it is known, at certain conditions determined in particular by the familiar frequency inequality and at Fn 0 autonomous system (26) may appear to be dissipative due to the boundedness of disturbances Fn . However, since condition (3) is not fulfilled, one cannot formally make use of Theorem 2 for calculating a minimal invariant set. Despite this, calculating the invariant set of a system (14) at the only condition (23) fulfilled is possible by reformulating the problem. First of all, one has to give up with analysis of system (26), (27) over all state space, but a bounded set in IRm :
Theorem 5. If T(Xo) is a regular trajectory, then there is such periodic trajectory that the conditions (22) are fulfilled starting at sufficiently large time instance n.
=
Theorem 5 can be reformulated as follows: at n -+ 00, every regular trajectory asymptotically tends to a periodic trajectory, i.e., to a stable limit cycle. Note that this theorem does not claim the unique limit cycle and does not provide a period value for the limit cycles.
n = {X:
The class of systems with bounded (see (17)) nonlinear parts contains some other significant subclasses: dynamic systems with saturation, relay systems having a dead zone, etc. The same systems under external bounded disturbances will be considered next.
IIXII
= i=l,m max IXil·
(28)
Here Xi is the i-th coefficient of a vector X and p is some, arbitrarily large in general, number to be chosen later. By taking into account the identity signXl ,n
Consider a system
= AXn + 'l1(Xn) + Fn ,
IIXII ~ p},
where
3.2 Nonautonomous systems with bounded nonlinear parts
X n +1
l'xICA) I < 1 Vi = 2, rn, (27)
= £Xl. n + (1 -
£Ixl,nl) signxl,n, £ > 0,
expression (20) at C T = (1,0, ... ,0) can be rewritten in the equivalent form
(23)
276
CP(Xl,n)
= -kcXl ,n -
= Signxl,n
kO:(Xl,n) SignXl,n,
where IIXII is determined by (28). Checking the latter inequality requires solution of the standard convex programming problem. If inequalities (36) are not fulfilled, i.e. the set n is chosen insufficiently "wide", one has to enlarge the value p by reducing (if necessary) the value c.
(29)
where (30)
Next, equation (26) takes the form
Xn +1
3.4 Invariant sets of nonlinear systems with saturation
= AXn + BO:(Xl,n)CP(Xl,n) + BFn, Fn E ~, (31)
The class of systems with bounded (in sense described above) nonlinear part contains also such important for application subclass as systems with saturation for which nonlinear function w(Xn) given by
where
A=
o Am
I
= Am -
ekB.
(32)
where an = eT X n . The calculation of minimal invariant set for a system (23), (24), (25) and (38) is performed by the same method. Consider minimal invariant sets for the well-studied nonlinear systems of Lurie type under bounded additive disturbances that are described by the difference equation
By choosing an arbitrarily small c > 0 (at A satisfying the condition (27)), one can ever make the matrix A to satisfy the conditions
(33) Next, choose p the way that p < c- 1 . Then the nonlinear function
where the notations are the same as used above, a = C T X, e is a given real vector and the nonlinear function W(X) = w(O') is bounded by means of
(39)
Corollary 1. If the number p: p < is chosen sufficiently large, then for a system (31) satisfying the condition (17), a minimal bounded invariant set M: MEn = {X: IIXII ~ p} exists and is calculated by formula (4).
c- 1
Here 1£, k > 0 are given numbers. Without loss of generality, assume that the matrix A satisfies the conditions (4). In fact, if in contrary, A does not satisfy these conditions, one can easily meet the requirements for the matrix A = A + BST by choosing a proper S . With this aim, add and subtract BST Xn in the right hand side of equations (38) and introduce a new function
Assume, the first N items are summarized in the series (4) and this way the estimate M(N)(.) for the set 9Jl(.) is obtained. It is necessary to check the inclusion
W(Xn)
(34)
Change the setting of the problem and impose additional constraint on function W(0'). Instead of function ~( . ) introduce new function -q,(.) in the form
since this is crucial for calculating the minimal invariant set of a system (31). It is easy to prove that inclusion (34) is fulfilled, if IXi(M(N»)1 ~ p Vi
= I,m,
-q,(a) = ~(a), §. ~ -q,(a) ~ 8,
(35)
-8 where Xi(M(N») is the projection of the set M(N») onto the axe Xi' Inequalities (35) are equivalent to the one, max
XEM(N}
IIXII
~ p,
= -ST Xn + W(Xn) .
if 10'1 ~ a* , > 0'*, ~ ~(a) ~ -§., if 0' < -0'*. if 0'
(40)
Defined by such way function -q,(0') is bounded in the sense above-mentioned definition and therefore system (23), (24), (25) and (39) - (40) satisfies the conditions of Theorem 2.
(36)
277
4. EXAMPLE
between the two diameters of successive estimates is less than the given a priori admissible error (in terms of absolute or relative values), terminate the process.
Consider a system (18), (19) and (20) at m = 2 and A
= IIl~2
As it is easily seen, the methods for calculating minimal invariant sets of a wide class of nonlinear dynamic systems are applicable at the same time for calculating stationary sets of the respective evolutionary set equations.
1~211'
B=II~II' c=II~II·
(41)
In this case, equations (18) take the form 6. REFERENCES
1
Xl,n+l
= '2Xl,n + X2,n,
X2,n+l
= ~X2,n + sign(xl,n) .
Bulgakov, B.V. (1946). On the accumulation of disturbances in linear oscillatory systems with constant parameters. Doklady Acad. Sci. USSR 51, 343-g4~~ (in Russian). di Masi, G., A. Gomo8.Jli and A.B. Kurzhanski (1991). Modelling, estimation and control with uncertainty. Progress in Systems and Control Theory. Kuntsevich, V.M. and B.N. Pshenitchnyi (1997). Minimal invariant sets of discrete systems with stable linear parts. Problemy upravleniya i informatiki (Problems of Control and Informatics) (1), 81-91. (in Russian). Kuntsevich, V.M. and B.N. Pshenitchnyi (1996a). Invariant and stationary sets of nonlinear discrete systems under bounded disturbances. Problemy upravleniya i informatiki (Problems of Control and Informatics) (1-2),35-45. (in Russian). Kuntsevich, V.M. and B.N. Pshenitchnyi (1996b). Minimal invariant sets for dynamic systems under additive bounded perturbations. Kibernetika i sistemnyi analiz (Cybernetics and System Analysis) (1), 74-81. (in Russian). Kuntsevich, V.M. and M.M. Lychak (1992) . Guaranteed estimates, adaptation and robustness in control systems. Lecture notes in control and information sciences. Kurzhanski, A.B. (1977). Control and observation under uncertainty. Nauka. Moscow. (in Russian). Kurzhanski, A.B. and I. Valyi (1997). Ellipsoidal calculus for estimation and control. Birkhauser. Boston. Kurzhanski, A.B. and V.M. Veliov (1994) . Modelling techniques for uncertain systems. In: Progress in Systems and Control Theory (V.M. Veliov, Ed.). Vol. 18. Birkhauser. Boston. Milanese, M., Norton, J., Piet-Lahanier, H. and Walter, E., Eds.) (1996). Bounding approaches to system identification. Plenum Press. New-York and London.
(42)
These lead to the fact that Xl = 4, X2 = 2,
Xl
= -4,
X2
=-2
are stable points of the system, i.e. they make up a cycle with the period T = O. The point (4/9, -2/3) is repeated in one period, hence if Xo = (4/9, -2/3) , then X2 = Xo . Therefore, the system (42) has three cyclic trajectory: the two with a trivial period and the one with a unit period. In the case, theorem 4 claims only that the set B[-I, +1] + AB[-l, +1] + A2 B[-I, +1] + . . (43)
is invariant for the system (42). A detailed description of set (43) is nontrivial, however this is not the point. According to Theorem 5, every regular trajectory of the system (42) tends to a cycle. Set (43) decomposes into several isolated points. The trajectories starting from these points tend to the mentioned cycles and other trajectories either tend to other cycles or appear to be irregular. Thus, the minimal invariant set (43) of the illustrating system (42) has been determined. This set contains the three periodic regular trajectories. However, the question, how do the other (possibly existing) regular or irregular trajectories fill out the space in between the set (43) and the attraction regions of those three periodic trajectories, has not been answered yet.
5. CONCLUSIONS When applying the presented method for calculating minimal invariant sets, one obviously faces the problem of the infinite series (4) termination criterion. One of those possible solutions to this problem is the following . At every step of the iterative procedure for calculating an invariant set, one calculates the diameter of the currently obtained estimate. If at the next step the difference
278
ANALYSIS OF SOME CLASSES OF NONLINEAR DISCRETE SYSTEMS UNDER BOUNDED DISTURBANCES Vsevolod M. KUNTSEVICH* Boris N. PSHENITCHNYI **
* Space Research Institute of N AS and NSA of Ukraine 40 Prsp. Akad. Glushkov, 252650 MSP 252022 Kiev-22, Ukraine Phone: +38044 2664124 Fax: +38 044 2664124 E-mail: [email protected] ** Institute for Applied System Analysis of NAS and ME of Ukraine 37 Prospekt Peremogy, 252056 Kiev-56, Ukraine Phone: +38 044 2662297 Fax: +38 044 2664124
Abstract: The constructive solution of the calculating minimal invariant sets problem for some classes of autonomous and non autonomous discrete systems has been obtained. For the nonautonomous discrete systems the problem is solved on the assumption that only set-valued estimates for additive bounded disturbances are available. The class of systems with bounded nonlinear parts is considered. This class contains systems with saturation, relay systems having a dead zone and others. An example illustrates the proposed method. Copyright © 1998 IFAC Keywords: discrete dynamic systems, bounded disturbances, invariant sets
prevented developing the hard-bound methods of analysis. These methods have been attracting the researchers anew last years.
1. INTRODUCTION
Analysis of dynamic systems under bounded disturbances seems to become a problem among those most actual ones of the modern theory of dynamic systems. Recently, making use of the hard bounds on disturbances, meaning the use of setvalued estimates (see (Kurzhanski, 1977; di Masi et al., 1991; Kurzhanski and Veliov, 1994; Kuntsevich and Lychak, 1992; Milanese et al., 1996; Kurzhanski and Valyi, 1997) in particular) , is becoming as much popular as the use of stochastic models when applying for analysis of dynamic systems under uncertainty.
The early results on calculating minimal invariant sets that provide the most complete (in certain sense) characteristics of the analyzed systems can be found in the preceding papers (Kuntsevich and Pshenitchnyi, 1996b; Kuntsevich and Pshenitchnyi, 1996a; Kuntsevich and Pshenitchnyi, 1997) published by the authors. In the present paper, these results are widened to a class of nonlinear dynamic systems found in many practical applications.
It is worth to note that the problem of analyzing linear dynamic systems under additive bounded disturbances was first stated and discussed in (Bulgakov, 1946) . However, a wide expansion of the methods for statistic dynamics
273
2. MINIMAL INVARlANT SETS OF LINEAR SYSTEMS UNDER BOUNDED DISTURBANCES
Theorem 1. At the assumptions made on A and
J, The set WL(A,J) defined by (4) is a compact in lR m , i.e., a closed bounded set.
Although the paper devoted to non linear systems analysis we starts with the consideration of linear systems. This is done because of the fact that the methods applicable to linear systems are used further for analyzing nonlinear systems of certain classes as well.
One can find the proof for the following theorem in (Kuntsevich and Pshenitchnyi, 1997) once again. Theorem 2. WL(A, J) is a minimal invariant set, meaning WL(A, J) is invariant and WL(A, J) ~ WL as soon as WL is also closed invariant. Furthermore, a system trajectory which starts at an arbitrary point Xo E lRm at any € > 0 remains in the cneighborhood of WL(A , J) at sufficiently large n.
Consider a dynamic system Xn+l = AXn
+ Fn ,
n = 0, 1, 2, .. . ,
(1)
where Xn E lRm is a state vector, A is an m x m real matrix, Fn E lRm is a sequence of vectorvalued disturbances taking independent values from a given set at time instances,
Theorems 1 and 2 define the minimal invariant set of system (I), ~hich is also convex as soon as J is convex compact in (!Rm . Consider, J is a convex polytope given by
(2)
J
Here J is a given bounded (non necessarily convex) set. Assume, a matrix A satisfies the condition
{F i } ,
(5)
i=I ,N
where F i is the i-th vertex and N is the total number of vertices. In this case, significant in view of practical applications, the set WL(A, J) is calculated as
(3)
where >'i(A) is the i-th eigen value of the matrix A.
WL(A,J)
Due to the mentioned properties of a matrix A and to the boundedness of disturbances F n , the reaction of a system (1) to these disturbances (the dissipativity) is obviously bounded. However, despite of an illusory simplicity of calculating the set which contains all the states Xn in asymptotic, the problem has appeared nontrivial and the solution to it has been found only recently.
=
conv_{Fi
+ AFj +
i,j ,k ... =I ,N
The minimal invariant sets defined this way provide exhausting characteristics for the considered dynamic systems under bounded disturbances. However, these definitions are poorly applicable for comparing the properties of distinct systems. In view of the latter, it is needed to have a numeric measure, for instance a scalar function defined on these sets. This could be the diameter D(WL) of a set WL(A , J)
For more convenience, the essential results found in (Kuntsevich and Pshenitchnyi, 1996b; Kuntsevich and Pshenitchnyi, 1996a; Kuntsevich and Pshenitchnyi, 1997) are presented below in brief. These will be used further when analysing nonlinear systems. Introduce a set
WL(A,J) =~+AJ+A2J+ . .. ,
= conv
D(WL)
= XE!m max
IIXII,
(7)
(4)
where IIXII is the Euclidean norm of X. where the sum of sets is a Minkovski sum, that is WLI
+ WL2 = {Xl + X 2
:
The diameter of an invariant set, when it is used as a measure of the dynamic precision of system (1)-(2), is similar in certain sense to the dispersion of the process Xn at Fn being a stochastic process.
Xl E WL I , X 2 E WL2 } .
Recall, a closed set WL is called an invariant for system (I), if the inclusion X E WLleads to AX + FE WL at any F E J , meaning AWL + J ~ WL.
Assume, J is a convex polytope given by (5). One can calculate the diameter D(WL) at any required accuracy by use of the formula
Definition 1. A compact set is called minimal invariant for system (I), if it is invariant and has no (other but itself) compact invariant subset.
(8)
where pI and Pq are the i-th and q-th vertices respectively of the estimating set for WL, which has been calculated by use of a finite number
The following theorem has been proven in (Kuntsevich and Pshenitchnyi, 1997).
274
of items in the series (4). In (Kuntsevich and Pshenitchnyi, 1997), the following upper bound estimat e for D(VJt) was obtained.
In the same paper (Kuntsevich and Pshenitchnyi, 1997), the estima te for the diamet er of the invariant set of system (10), (11) and (2)
.
(9)
D(VJt) ::; - C1 D(J) . -q
Here D(J) is the diamet er of the set J and c is a constan t value determined by the known linear algebraic inequality
~
D[!JJt(A , Je)] ::; D[VJt(A, J)] + 1 _ q c ~
::; -1-(D(J ) -q
+ c)
(15)
has been obtaine d.
n=0, 1, 2, . . .. 3. MINIMAL INVARIANT SETS OF SYSTEMS WITH BOUNDED NONLINEAR PARTS
2.1 Invaria nt sets of nonstationary systems Making use of the results found in (Kuntsevich and Psheni tchnyi, 1997), one can calculate the minimal invaria nt sets of nonsta tionary linear systems under additiv e bounded disturbances. Define the class of linear nonsta tionary systems (10) where the notatio ns are the same as above, except the matrix 0
_
-
(11 )
A =A +An·
3.1 Autono mous systems with bounded nonlinear
parts Consider a system
where W and
IRm
-+
IRm is a continuous function (17)
o
Here A is a given real matrix and time-varying coefficients of the matrix An can take arbitra ry indlPe ndent (from each other) values from a given set 11, i.e.,
A E U = {A: IIAII ::; ~} . o
(12)
-
In this case, the inequality 11 A +Anll < 1 may not take place because of the given definitions for o matrices A and An. o
Assume, the matrix A satisfies the conditions like (3) . Introduce a set
VJtC4, JE)
= VJt(A, J) + cS + cAS •2
+c A S + . . . ,
(13)
where c > 0, S is a unit solid sphere with the center in the origin and Je = J + cS is an cneighborhood of J. In (Kuntsevich and Pshenitchnyi, 1997) , the following theorem has been proven. Theorem 3. If at some c > 0'
l-q ~
The analysis of the inner structu re of the invariant sets under consideration, the analysis of the types of trajectories that belong to these sets by means of w-limit points of the trajectories, is given in (Kuntsevich and Pshenitchnyi, 1997) and hence is omitted here . Next, consider a significant subclass of systems having bounded, in terms of (17), nonlinear parts. These are relay systems described by the equations X n+ 1
°
c D(J)+ c '
< - - ....-:-..,..--
is a compact set. Here cl(x) means closing for the set x. Expressions (16) and (17) lead to the fact that at w(Xn) given by (17), every trajecto ry of a system (16) is one of those feasible trajecto ries of a system (1) . And due to Theorem 1 the set VJt(A , J) is invariant for system (16), therefore system (16) has a compact invariant set.
= AXn + BsignO'n, an = eT X n ,
(18) (19)
where
+1 ,
(14)
signO'n = {
then the set VJt(A,Je) is invariant for a system given by (10), (11) and (2).
if an
> 0,
an < 0, [-1 , +1], 1f an = 0,
-1,
~
(20)
and B E IRm is an m-dimensional real vector.
275
Without the loss of generality, assume that IICII = 1 and consider a subclass of systems (18), (19) and (20) determined by the condition C E (!:
=
{C : IICII
(24) where nonlinear vector-valued function 'l1 (.) satisfies the condition (17) and J is a given compact set.
= I}.
As it is led from (20), the class of dynamic systems defined by (18), (19) and (20) has a multi-valued right hand side. Due to the definitions (18), (19) and (20) the condition (17) is fulfilled, therefore the set given by formula (4) is a minimal invariant set for the class of systems (18), (19) and (20).
Introduce the set (25) Obviously, the problem of calculating the minimal invariant set for a system (23), (24) and (25) is reduced to the problem solved previously to within the notations. In particular, the class of systems (23) includes relay systems under disturbances:
Introduce the following
Definition 2. A trajectory T(Xo) of a system (18), (19) and (20) is called regular, if there is such £ > 0 that
ICT Xnl > £,
n = 0,1, . . ..
Lemma 1. Assume IICII = 1. If IIX - YII < £/2, then
IcT XI 2:
X n+ 1
(26)
mi~~~
Calculating the invariant set for these systems does not differ from the way it has been made above.
(21) £ and
= sign c TX, IcT XI > £/2.
= AXn + B sign C T Xn + Fn.
(22)
3.3 Invariant sets of relay systems with unstable linear part
On the basis of this lemma, the following theorems have been proven in (Kuntsevich and Pshenitchnyi, 1997).
In practical control system design, the following case is met frequently: system (26) does not satisfy the strict conditions (3) , however, the weakened conditions on A,
sign CTy
Theorem 4. If T(Xo) is a regular trajectory, then for every point Yo sufficiently close to X o, the respective trajectory is also regular and formulae (22) are correct.
P'i(A)1 = 1,
are fulfilled. This takes place when the linear part of a system (26) contains an integrator (summation element) . As it is known, at certain conditions determined in particular by the familiar frequency inequality and at Fn 0 autonomous system (26) may appear to be dissipative due to the boundedness of disturbances Fn . However, since condition (3) is not fulfilled, one cannot formally make use of Theorem 2 for calculating a minimal invariant set. Despite this, calculating the invariant set of a system (14) at the only condition (23) fulfilled is possible by reformulating the problem. First of all, one has to give up with analysis of system (26), (27) over all state space, but a bounded set in IRm :
Theorem 5. If T(Xo) is a regular trajectory, then there is such periodic trajectory that the conditions (22) are fulfilled starting at sufficiently large time instance n.
=
Theorem 5 can be reformulated as follows: at n -+ 00, every regular trajectory asymptotically tends to a periodic trajectory, i.e., to a stable limit cycle. Note that this theorem does not claim the unique limit cycle and does not provide a period value for the limit cycles.
n = {X:
The class of systems with bounded (see (17)) nonlinear parts contains some other significant subclasses: dynamic systems with saturation, relay systems having a dead zone, etc. The same systems under external bounded disturbances will be considered next.
IIXII
= i=l,m max IXil·
(28)
Here Xi is the i-th coefficient of a vector X and p is some, arbitrarily large in general, number to be chosen later. By taking into account the identity signXl ,n
Consider a system
= AXn + 'l1(Xn) + Fn ,
IIXII ~ p},
where
3.2 Nonautonomous systems with bounded nonlinear parts
X n +1
l'xICA) I < 1 Vi = 2, rn, (27)
= £Xl. n + (1 -
£Ixl,nl) signxl,n, £ > 0,
expression (20) at C T = (1,0, ... ,0) can be rewritten in the equivalent form
(23)
276
CP(Xl,n)
= -kcXl ,n -
= Signxl,n
kO:(Xl,n) SignXl,n,
where IIXII is determined by (28). Checking the latter inequality requires solution of the standard convex programming problem. If inequalities (36) are not fulfilled, i.e. the set n is chosen insufficiently "wide", one has to enlarge the value p by reducing (if necessary) the value c.
(29)
where (30)
Next, equation (26) takes the form
Xn +1
3.4 Invariant sets of nonlinear systems with saturation
= AXn + BO:(Xl,n)CP(Xl,n) + BFn, Fn E ~, (31)
The class of systems with bounded (in sense described above) nonlinear part contains also such important for application subclass as systems with saturation for which nonlinear function w(Xn) given by
where
A=
o Am
I
= Am -
ekB.
(32)
where an = eT X n . The calculation of minimal invariant set for a system (23), (24), (25) and (38) is performed by the same method. Consider minimal invariant sets for the well-studied nonlinear systems of Lurie type under bounded additive disturbances that are described by the difference equation
By choosing an arbitrarily small c > 0 (at A satisfying the condition (27)), one can ever make the matrix A to satisfy the conditions
(33) Next, choose p the way that p < c- 1 . Then the nonlinear function
where the notations are the same as used above, a = C T X, e is a given real vector and the nonlinear function W(X) = w(O') is bounded by means of
(39)
Corollary 1. If the number p: p < is chosen sufficiently large, then for a system (31) satisfying the condition (17), a minimal bounded invariant set M: MEn = {X: IIXII ~ p} exists and is calculated by formula (4).
c- 1
Here 1£, k > 0 are given numbers. Without loss of generality, assume that the matrix A satisfies the conditions (4). In fact, if in contrary, A does not satisfy these conditions, one can easily meet the requirements for the matrix A = A + BST by choosing a proper S . With this aim, add and subtract BST Xn in the right hand side of equations (38) and introduce a new function
Assume, the first N items are summarized in the series (4) and this way the estimate M(N)(.) for the set 9Jl(.) is obtained. It is necessary to check the inclusion
W(Xn)
(34)
Change the setting of the problem and impose additional constraint on function W(0'). Instead of function ~( . ) introduce new function -q,(.) in the form
since this is crucial for calculating the minimal invariant set of a system (31). It is easy to prove that inclusion (34) is fulfilled, if IXi(M(N»)1 ~ p Vi
= I,m,
-q,(a) = ~(a), §. ~ -q,(a) ~ 8,
(35)
-8 where Xi(M(N») is the projection of the set M(N») onto the axe Xi' Inequalities (35) are equivalent to the one, max
XEM(N}
IIXII
~ p,
= -ST Xn + W(Xn) .
if 10'1 ~ a* , > 0'*, ~ ~(a) ~ -§., if 0' < -0'*. if 0'
(40)
Defined by such way function -q,(0') is bounded in the sense above-mentioned definition and therefore system (23), (24), (25) and (39) - (40) satisfies the conditions of Theorem 2.
(36)
277
4. EXAMPLE
between the two diameters of successive estimates is less than the given a priori admissible error (in terms of absolute or relative values), terminate the process.
Consider a system (18), (19) and (20) at m = 2 and A
= IIl~2
As it is easily seen, the methods for calculating minimal invariant sets of a wide class of nonlinear dynamic systems are applicable at the same time for calculating stationary sets of the respective evolutionary set equations.
1~211'
B=II~II' c=II~II·
(41)
In this case, equations (18) take the form 6. REFERENCES
1
Xl,n+l
= '2Xl,n + X2,n,
X2,n+l
= ~X2,n + sign(xl,n) .
Bulgakov, B.V. (1946). On the accumulation of disturbances in linear oscillatory systems with constant parameters. Doklady Acad. Sci. USSR 51, 343-g4~~ (in Russian). di Masi, G., A. Gomo8.Jli and A.B. Kurzhanski (1991). Modelling, estimation and control with uncertainty. Progress in Systems and Control Theory. Kuntsevich, V.M. and B.N. Pshenitchnyi (1997). Minimal invariant sets of discrete systems with stable linear parts. Problemy upravleniya i informatiki (Problems of Control and Informatics) (1), 81-91. (in Russian). Kuntsevich, V.M. and B.N. Pshenitchnyi (1996a). Invariant and stationary sets of nonlinear discrete systems under bounded disturbances. Problemy upravleniya i informatiki (Problems of Control and Informatics) (1-2),35-45. (in Russian). Kuntsevich, V.M. and B.N. Pshenitchnyi (1996b). Minimal invariant sets for dynamic systems under additive bounded perturbations. Kibernetika i sistemnyi analiz (Cybernetics and System Analysis) (1), 74-81. (in Russian). Kuntsevich, V.M. and M.M. Lychak (1992) . Guaranteed estimates, adaptation and robustness in control systems. Lecture notes in control and information sciences. Kurzhanski, A.B. (1977). Control and observation under uncertainty. Nauka. Moscow. (in Russian). Kurzhanski, A.B. and I. Valyi (1997). Ellipsoidal calculus for estimation and control. Birkhauser. Boston. Kurzhanski, A.B. and V.M. Veliov (1994) . Modelling techniques for uncertain systems. In: Progress in Systems and Control Theory (V.M. Veliov, Ed.). Vol. 18. Birkhauser. Boston. Milanese, M., Norton, J., Piet-Lahanier, H. and Walter, E., Eds.) (1996). Bounding approaches to system identification. Plenum Press. New-York and London.
(42)
These lead to the fact that Xl = 4, X2 = 2,
Xl
= -4,
X2
=-2
are stable points of the system, i.e. they make up a cycle with the period T = O. The point (4/9, -2/3) is repeated in one period, hence if Xo = (4/9, -2/3) , then X2 = Xo . Therefore, the system (42) has three cyclic trajectory: the two with a trivial period and the one with a unit period. In the case, theorem 4 claims only that the set B[-I, +1] + AB[-l, +1] + A2 B[-I, +1] + . . (43)
is invariant for the system (42). A detailed description of set (43) is nontrivial, however this is not the point. According to Theorem 5, every regular trajectory of the system (42) tends to a cycle. Set (43) decomposes into several isolated points. The trajectories starting from these points tend to the mentioned cycles and other trajectories either tend to other cycles or appear to be irregular. Thus, the minimal invariant set (43) of the illustrating system (42) has been determined. This set contains the three periodic regular trajectories. However, the question, how do the other (possibly existing) regular or irregular trajectories fill out the space in between the set (43) and the attraction regions of those three periodic trajectories, has not been answered yet.
5. CONCLUSIONS When applying the presented method for calculating minimal invariant sets, one obviously faces the problem of the infinite series (4) termination criterion. One of those possible solutions to this problem is the following . At every step of the iterative procedure for calculating an invariant set, one calculates the diameter of the currently obtained estimate. If at the next step the difference
278