Description of Switched Systems by Implicit Representations
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Description of Switched Systems by Implicit Representations M. Bonilla and M. Malabre.
Abstract— This paper shows that a certain class of time–dependent, autonomous switched systems can also be studied using the structural concepts and properties of the linear time-invariant implicit systems theory. We first consider the modeling aspects, and then, the control aspects are tackled. This fact contributes to the enrichment of the switched systems theory and enlarges its action field. In Section I we specify a class of time–dependent, autonomous switched systems which enables us to describe systems with a variable internal structure. In Section II we show how to translate the considered class of time–dependent, autonomous switched systems into an implicit representation and then we recall some important results, of the implicit systems theory, which enable us to control them in such a way that they behave as a specified linear time–invariant system whatever be the position of the switches. In Section IV, we consider some reachability aspects. In Section V, we conclude.
where IIi (·) is the characteristic function of the interval ¯ ×m , A ∈ Rn ¯ ׯ n and C ∈ Rpׯ n , with θ ∈ I . The Ii , B ∈ Rn θ θ signals u and y are the input and output variables, respectively, and s is the switching signal. As usual we assume that B is a monic matrix and the Cθ are epic matrices. In the terminology of van der Schaft and Schumacher [19], there are η locations (or modes) described by the following η state space representations, Σssr (Aθ , B, Cθ ): d x ¯ dt
= Aθ x ¯ + Bu and y = Cθ x ¯
where θ ∈ I , x¯(Ti−1 ) = ci , and t ∈ Ii , for some i ∈ N, some and some Ii ∈ T. We restrict our discussion to matrices, Aθ and Cθ , θ ∈ I , having the following particular structure: Ti−1 ∈ T,
Aθ = A0 + A1 D(θ) and Cθ = C 0 + C 1 D(θ),
I. Linear Time–Dependent Autonomous Switched Systems In this paper, we adopt the dynamic system definition given by Willems [23], which is known as the behavioral approach (see the synthesis reference of Polderman and Willems [16] and also the version of van der Schaft and Schumacher [19] for Hybrid Dynamical Systems). Let us define the sets: T = {Ti ∈ R+ (i ∈ Z+ ) | T0 = 0, Ti−1 < Ti ∀ i ∈ N, with lim Ti = ∞}, I = {Ii = [Ti−1 , Ti ) i→∞ ⊂ R+ | Ti−1 , Ti ∈ T, i ∈ N}, S(I, I ) = {s : I → I | s(Ii ) = θ}, I = {θ1 , . . . , θη }. T is the switching time set, defining a partition on the time axis, R+ . I is the locations invariant set, defining disjoints time intervals covering all the time axis. S is the switching signals set, which elements are functions mapping from I, to an index set, I , of cardinality η (see [11] and [19] for details). Let Σsw = I × R+ , I × Rm × Rp , Bsw be a dynamical system which switches in a time–dependent and autonomous way [11], where the behavior, Bsw , is defined as follows: + m loc + p (s, u, y) ∈ S(I, I ) × Lloc 1 (R , R ) × L1 (R , R ) P ¯ s.t.: y(t) = ∃ ci ∈ Rn IIi (t)Cs(Ii ) exp As(Ii ) (t − Ti−1 ) ci i∈N Z t−Ti−1 + exp As(Ii ) (t − Ti−1 − τ ) Bu(Ti−1 + τ )dτ , 0 t ∈ R+ , Ti−1 ∈ T, Ii ∈ I ,
0 1 1 0 A0 = , A1 = , B= , 1 0 1 1 C 0 = 0 0 , C 1 = −1, D(θ) = α β , θ = (α, β) ∈ I = θ1 , θ2 , θ3 , θ4 , θ1 = (−1, −1), θ2 = (−1, 0), θ3 = (−1, −5), θ4 = (1, 1).
(3) For a given pair, (α, β), we have the transfer function: Fθ (s) = Cθ (sI − Aθ )−1 B =
´ M. Bonilla, CINVESTAV-IPN, CONTROL AUTOMATICO, ´ UMI 3175 CINVESTAV-CNRS. A.P. 14-740. MEXICO 07000,
"[email protected]" . M. Malabre, LUNAM Universit´ e, CNRS, IRCCyN UMR CNRS 6597, 1 Rue de la Noe, F- 44321 Nantes, FRANCE.
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
(2)
where: A0 ∈ Rn¯ ׯn , C 0 ∈ Rpׯn , A1 ∈ Rn¯ ×q , C 1 ∈ Rp×q , and n . We also assume that the matrix C is epic. D(θ) ∈ Rqׯ 1 Note that this structure of variation described by a finite set of matrices Aθ is very similar to the one used by Narendra and Balakrishnan [14] and Shorten and Narendra [20]. As we will see hereafter, the presence of the finite set of matrices Cθ will allow us to play with their unobservable spaces, in order to take into account variations of the internal structure. With this kind of time–dependent, autonomous switched systems one is able to describe, and later to control, systems which internal structure changes among a known set of linear time invariant proper systems. 1) Example (part 1) : Let us consider:
Bsw =
"[email protected]".
(1)
−(βs+α) s2 −(α+β)s−(1+α+β)
(4)
For the possible four selections of θ ∈ I , we nhave the following h four possible behaviors: io ∞ I , R2 ∩ ker −1 ( d + 1) B∞ = (u, y) ∈ C , i θ1 n dt h io ∞ I , R2 ∩ ker −1 ( d + 1)( d ) B∞ = (u, y) ∈ C , j θ2 n dt dt h io d d d ∞ ∞ 2 Bθ3 = (u, y) ∈ C Ik , R ∩ ker −(5 dt + 1) ( dt + 1)( dt + 5) , n io h d Bθ4∞ = (u, y) ∈ C ∞ I` , R2 ∩ 1 ( dt − 3) , for some disjoints Ii , Ij , Ik , I` ∈ I (i, j, k, ` ∈ N).
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As one can see later on, the key idea for modeling linear systems with variable internal structure is to carefully handle the unobservable space. In [4] we have introduced the so-called ladder systems, which are a particular subclass of such systems with internal switches. II. Implicit Representations In this Section we introduce some necessary concepts of the linear implicit theory, which will enable us, in Section III, to model time–dependent autonomous switched system Σsw , represented by the behavior Bsw and the matrices (2), by means of a linear implicit representation. A. Implicit systems We deal with implicit representations which were introduced by Rosenbrock [18] as a generalization of proper linear systems (see for example [10]). An implicit representation, Σimp (E, A, B, C), is a set of differential and algebraic equations of the following form: Edx/dt = Ax + Bu and y = Cx, t ∈ R+ ,
are called the descriptor, the equation, the input and the output spaces, respectively. It is usual to assume that: ker B = {0}
E 0 | {z } E
and Im C = Y . for almost all λ ∈ C.
Im [λE − A] = X eq ,
Assumption H1 states that the input variable, u, has no redundant components and that the set of output equations are linearly independent. And assumption H2 implies that (5) is solvable, namely: “for any u(·) ∈ C ∞ (R+ , U ), there exists at least one trajectory x(·) ∈ C ∞ (R+ , Xd ) solution of (5) for all t ≥ 0” . Definition 1: (Implicit rectangular representation) An implicit rectangular representation, Σirr (E, A, B, C), is an implicit representation (5) such that: (i) dim Xd > dim X eq , and (ii) the following geometric condition holds: Im A + Im B ⊂ Im E. (6) The geometric condition (6) guarantees that for any, u ∈ C ∞ (R+ , U ), there exists at least one descriptor variable trajectory, x ∈ C ∞ (R+ , Xd ), satisfying the implicit rectangular representation, Σirr (E, A, B, C) (see Cor. 11-[5]). Definition 2: (Algebraic constraint) An algebraic constraint is an implicit representation such that: (i) it is only formed by a set of linearly independent algebraic equations, (ii) it is independent of the input variable, (iii) it has no output equation; namely, Σalc (0, D, 0): 0 = Dx. where D : Xd → X alc is a map and the space, X alc , is called the algebraic constraint space. Definition 3: (Implicit global representation) The implicit representation obtained by the union of an implicit rectangular representation, Σirr (E, A, B, C), with an algebraic constraint, Σalc (0, D, 0), is called an implicit
d x dt
=
A B x+ u and y = Cx, D 0 | {z } | {z } A
(7)
B
t∈ . The cartesian product, X g =: X eq × X alc , is called the global equation space. It is usual to assume that: R+
X g = Im E ⊕ Im D . (8) Condition (8) states that the implicit global representation (7) is only composed by independent linear equations (see Corollary 11-[5] and Proposition 8-[5]).
B. Some structural properties Definition 4: (Regularity [7]) A pencil [λE − A] is regular if it is square and det[λE − A] 6= 0. An implicit representation is called regular if [λE − A] is regular. Definition 5: (Internal properness [1]) An implicit representation is internally proper if it is regular and has no infinite elementary divisor of order greater than 1. In Proposition 9-[5], we show that any given implicit global representation, (7), is internally proper iff:
(5)
where: E : Xd → X eq , A : Xd → X eq , B : U → X eq and C : Xeq → Y are maps. The spaces Xd , X eq , U and Y
H1 H2
global representation, Σigr (E, A, B, C):
Xd = ker D ⊕ ker E
(9)
Definition 6: (External equivalence [23], [16]) Two representations are called externally equivalent if the corresponding sets of all possible trajectories for the external variables, expressed in an input/output partition (u, y), are the same. Definition 7: (Algebraic redundancy [3]) A part of the descriptor variable which is a linear combination of other descriptor variable components, and do not contribute to describe the external behavior (set of all possible input-output, (u, y), trajectories) is called an algebraic redundant descriptor variable. Lemma 1: (Localization of the resulting state space representation of a proper implicit global representation; Proposition 10-[5]) Let the implicit global representation (7) satisfying (6), (8) and (9), then the degree of freedom, characterized by the algebraic redundant descriptor variable, is located in ker E . The part of (7) which is restricted to ker D in the domain and to Eker D in the co-domain, contains no algebraic redundant descriptor variables. Moreover, there exist unique maps (E, A, B, C): EV = V E , AV = V A
and CV = C ,
where, V : ker D → Xd and V : Im E → X g , are natural insertions. b A, b B) b = (0, I, 0) are the unique maps: Furthermore, (E, b , P A = Pard A b P alc E = Pard E alc
and P alc B = Bb ,
where, Pard : Xd → ker E , is the natural projection on along ker D, and, Palc : X g → Dker E , is the natural projection on Dker E along Eker D. Finally, (7) is externally equivalent to the reduced state space representation, Σssr (A0 , B0 , C):
ker E
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d¯ x/dt = A0 x ¯ + B0 u
and y = C x¯,
ker E = ker D ∩ ker E = {0}, where: E = P eq EV = EV , Im E = Eker D = Im E , A = P eq AV = AV , B = P eq B = B ,
and B0 = E (−1) B . This means that one way for enhancing the state space representation which is inside the proper implicit global representation, Σigr (E, A, B, C), is to decompose, Xd , as ker D ⊕ ker E , and, X g , as Eker D ⊕ Dker E . 2) Example (part 2) : The time–dependent, autonomous switched system, described by (1), (2) and (3), is also described by the implicit global representation: Eker D
C = CV , A0 = E
(−1)
A
fixed structure of the considered linear time–dependent autonomous switched system. This allows us to take advantage of the available structural tools, of the implicit system theory, for the analysis and the synthesis tasks. A. Implicit global representation In view of Lemma 1 we can algebraically characterize the variable nature of the system. Indeed, if we gather a redundant descriptor variable, i.e. xˆ = 0, in the state space representation, (1) and (2), we can bring them to the following implicit global representations:
ker -E
61 0 ? 0
0 0 0
0 1 0
x ¯ x ˆ
6 ?
α
0 1 β
0 0 , 1
d¯ x dt dˆ x dt
=
A0 + A1 D(θ) −A1 0 I C 0 + C 1 D(θ) −C 1
x ¯ x ˆ
x ¯ x ˆ
+
B 0
u
,
◦
-
t ∈ Ii,
(10) Let us note that: (i) The implicit global representation (10) is externally equivalent to the state space representation, Σssr (Aθ , B, Cθ ), restricted to ker D, in the domain, and to Eker D, in the co-domain, which state space matrices are enclosed by solid line boxes. (ii) Above the solid line of the differential and algebraic equations set, there is an implicit rectirr −1 −1 −1 angular representation, Σ (ET (θ), AT (θ), B, CT (θ)) 1
y=
α (1 + β) −1 0 x ¯ −1 (1 + α) β + 1 u x ˆ ....................... 0 . . . .0. . . . . . . . .0. . . . . . . . 1. . x ¯ 1 − α − β y= x ˆ
with T (θ) = 0
I 0 0 0
=
..
Dker E
..
d dt
ker D
restricted to Eker D in the
co-domain, which behavioral matrices are enclosed by dashed line boxes. This part of the representation takes into account, in an implicit way, the changes of behaviors due to the commutation of the parameters α and β . (iii) Below the solid line of the differential and algebraic equations set, there is an algebraic constraint, Σalc (0, DT −1 (θ), 0), restricted to Dker E in the co-domain, which behavioral matrices are enclosed by doted line boxes. The part restricted to ker E , in the domain, and to Dker E , in the co-domain, is algebraically redundant and includes the descriptor variables which are always null. (iv) Since: Im A + Im B ⊂ Im E , Xg = Eker D ⊕ Dker E = Im E ⊕ Im D and Xd = ker D ⊕ ker E , this implicit global representation is internally proper. III. Implicit Global Representation of a Time–Dependent Autonomous Switched system In this Section we show that the time–dependent autonomous switched system Σsw , with behavior Bsw and represented by the state representations (1) and (2) can also be represented by means of implicit global representations, Σigr (E, Aθ , B, C), θ ∈ I . From these implicit global representations, we will extract an implicit rectangular representation, Σirr (E, A, B, C), which characterizes the
where: limt→T + x¯(t) = x¯(Ti−1 ) = ci , limt→T + xˆ(t) i−1 i−1 = x ˆ(Ti−1 ) = 0, θ ∈ I (fixed for each Ii ), and i ∈ N. By ◦ X we denote the interior set of a set X . In order to get an implicit rectangular representation, describing all the common structure of the time–dependent autonomous switched system, represented by (1) and (2), let us define the descriptor variable: x = xx1 = T −1 (θ) xx¯ˆ , where: 2
I D(θ)
T (θ) =
0 I
, namely, Σigr (E, Aθ , B, C): ◦
d t ∈ I i E dt Cx, x = Aθ x + Bu and y = B A E , B= , Aθ = E= 0 Dθ 0
(11)
where: limt→T + x1 (t) = x1 (Ti−1 ) = ci , limt→T + x2 (t) = i−1 i−1 x2 (Ti−1 ) = −D(θ)ci , θ ∈ I (fixed for each Ii ), and i ∈ N. The maps E : Xd → X eq , A : Xd → X eq , B : U → X eq , Dθ : Xd → X cal , and C : Xd → Y are equal to: E=
I
0
, A = A0 −A1 , C = C 0 Dθ = D(θ) I .
−C 1
,
(12) Let us note that: (i) The fixed structure of Σigr (E, Aθ , B, C), which is active for any Dθ , θ ∈ I , is described by the following implicit rectangular representation, Σirr (E, A, B, C), d E dt x = Ax + Bu and y = Cx,
t ∈ R+ \ T.
(13)
where: limt→T + x1 (t) = x1 (Ti−1 ) and limt→T + x2 (t) = i−1 i−1 x2 (Ti−1 ). (ii) The degree of freedom is characterized by the algebraic constraints, Σalc (0, Dθ , 0), 0 = Dθ x,
t ∈ Ii
(14)
for some θ ∈ I , some i ∈ N, and some Ii ∈ T. (iii) Since dim X eq < dim Xd , there then exists a degree of freedom (see the introductory section of [5]). (iv) The geometric conditions, (6), (8) and (9), are fulfilled. (v) Since Im A + Im B ⊂ Im E = X eq , the implicit rectangular representation has at least one solution. (vi) Since Im Dθi = X cal and dim(X eq × X cal ) = dim(Xd ), for each θi ∈ I , the implicit global representations have unique solutions for each θi ∈ I . (vii) Since ker Dθ ⊕ ker E = Xd ,
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for all θ ∈ I , then all the implicit global representations are proper. (viii) Since Im E = X eq the possible jumps taking place at Ti ∈ T do not generate any impulse (c.f. Theorem 4.5 of [8]). 3) Example (part 3) : In order to enhance the fixed structure of (10), let us define the descriptor variable x ¯ x = T −1 (θ) . Then, we get the implicit global reprex ˆ sentation (11) with:
E
Aθ C
B
=
1 0 0
0 1 0
0 0 0
0 1 α 0
1 0 β 0
−1 −1 1 1
0 1 0
(15)
Above the solid line of the differential and algebraic equation set of the implicit global representation (11) and (15), there is the implicit rectangular representation (13) with:
E
A C
B
=
1 0
0 1
0 0
0 1 0
1 0 0
−1 −1 1
0 1
α
β
1
(16)
(17)
Let us note that: (i) When splitting the implicit global representations (10) (via (11) and (15)) into the rectangular implicit description (13) and (16) and into the algebraic constraints (14) and (17), we get the fixed active structure of the system, represented by (13) and (16). (ii) The Kronecker normal forms [7] of the pencils, [λE − Aθ ], associated to (11) and (15) are: if 1
0 0 , if β 6= −1 and α + β = −2: (λ − α) 0 0 0 (λ + 1) 0 0 , and if β 6= −1 and α + β 6= −2: (λ + 1) 1 0 (λ + 1) 0 0 . (iii) The Kronecker nor(λ − 1 − α − β) 0 0 (λ + 1)
β = −1: 0
1 0 0 1 0 0
mal form of the pencil, [λE − A], associated to (13) and 0 (16) is: λ0 10 (λ + (iv) Comparing the Kronecker 1) normal forms of the pencils associated to (11) and (15), with the Kronecker normal form of the pencil associated to (13) and (16), we can see that the internal variable structure of (11) and (15) is now taken into account by the invariant column minimal index block, λ 1 , of (13) and (16). IV. Reachability and Control of the implicit global representation In [5] has been shown that the reachability of is split into the reachability from the control input and that due to an internal degree of freedom. A Corollary of [5] is the following (see Appendix A, for the proof): Corollary 1: (Reachability subspace of an implicit rectangular representation) Given the implicit global representation, (11) and (12), the reachability subspace, Σigr (E, Aθ , B, C)
, of the implicit rectangular description, (13), and the d ∗ , of the state space representareachability subspace, Ru,θ tion, (1), with θ ∈ I , are related as follows: ∗ ∗ RX = R0∗ + Vθ Ru,θ d
(18)
where Vθ : ker Dθ → Xd = ker Dθ ⊕ ker E is the natural insertion map of ker Dθ in Xd , and R0∗ is the reachability subspace of the autonomous implicit representation, ◦ d Σirr 0 (E, A, 0): E dt x(t) = Ax(t), t ∈ I i , where: limt→T + x1 (t) i−1 = x1 (Ti−1 ) = ci , limt→T + x2 (t) = x2 (Ti−1 ) = −D(θ)ci , θ ∈ I i−1 (fixed for each Ii ), and i ∈ N. The reachability subspaces, ∗ , are equal to:1 R0∗ and Ru,θ
∗ R0∗ = Vθ Aθ Im A1 ⊕ ker E and Ru,θ = Aθ Im B
(19) Moreover, given the following controllability matrices: Cθ =
Below the solid line of the differential and algebraic equation set of the implicit global representation (11) and (15), there is the algebraic constraint (14) with: Dθ =
∗ RX
A1 B
Aθ A1 B
···
¯ −1 , θ ∈ I, An A1 B θ
(20)
∗ ∗ ⇔ then: RX = Xd ⇔ Im E = Aθ Im A1 B = ERX d d rank C θ = n ¯ , for all θ ∈ I . In other words, the reachability of the implicit rectangular representation, Σirr (E, A, B, C), is achieved by both, the input action, through Im B , and the degree of freedom, through Im A1 . Definition 8: ((A, E, B)–invariance [21]) The supremal (A, E, B)–invariant subspace contained in ker C , V ∗ = sup{V ⊂ ker C| AV ⊂ EV + Im B}, is the limit of the following non-increasing geometric algorithm: V 0 = Xd ; V µ+1 = ker C ∩ A−1 (EV µ + Im B) , µ ≥ 0
(21) This subspace characterizes the maximal part of the implicit representation, Σimp (E, A, B, C), which can be made unobservable with a suitable proportional and derivative descriptor variable feedback. A Corollary of Theorems 14-[5], 15-[5] and 16-[5], is the following (see Appendix B, for the proof): Corollary 2: (Internal variable structure decoupling) Let us consider the implicit rectangular representation (13), Σirr (E, A, B, C), which is extracted from the implicit global representation (11), and with the behavioral maps (12). Then: V ∗ = ker C . Moreover, if: p ≤ m, holds, there then exists, a proportional and derivative descriptor variable feedback, u = Fp∗ x + Fd∗ dx/dt, such that the closed loop system, represented by Σirr (EFd∗ , AFp∗ , B, C), where: EFd∗ = E − BFd∗ and AFp∗ = A + BFp∗ , satisfies: (i) Im EFd∗ = Im E , i.e., the existence of solution is conserved. (ii) ker EFd∗ ⊂ V ∗ , i.e., the internal variable structure of the closed loop system is made unobservable. (iii) Σirr (EFd∗ , AFp∗ , B, C) is externally equivalent to the state space representation, Σssr (A∗ , B ∗ , C∗ ); where A∗ , B ∗ and C∗ are the unique maps induced by AFp∗ and EFd∗ , such that: ΠAFp∗ = A∗ Φ, ΠEFp∗ = E∗ Φ, ΠB = B∗ Φ and C = C∗ Φ,
X : V → W and Y : W → W , Y Im X stands for the reachability subspace [22]: Im X + Y Im X + · · · + Y dim(W )−1 Im X.
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1 Given,
d d where Φ : Xd → X and Π : EXd → EEX are canon∗ V∗ F∗V d ∗ ical projections, EFd∗ = E − BFd , AFp∗ = A + BFp∗ , with: (Fp∗ , Fd∗ ) ∈ F(V ∗ ) = (Fp , Fd ) (A + BFp )V ∗ ⊂ (E − BFd )V ∗ , and Im B ∩ (E − BFd∗ )V ∗ = {0}. E∗ is an isomorphism, and (−1) E∗ is its inverse Dmap. E A∗ Im B ∗ = Xd /V ∗ , namely: Furthermore:
rank
h
B∗
A∗ B ∗
···
p−1
A∗
B∗
i
= p
∗ =X Let us note that: (i) Since (6) implies: VX d d ∗ and Im B ⊂ EVXd , p ≤ m, implies the output uniqueness property, item ii) of Theorem 7-[2]. This means that the internal variation structure is made unobservable. (ii) Using the Kronecker theory, Lebret and Loiseau [9] generalized the canonical form of Morse [13]. In such a generalization, they also characterized the parts of the implicit representation which are responsible for the existing internal structure variation, by the so called Lσi blocks, with cardinality dim(V ∗ ∩ ker E), and Lqi blocks, with cardinality dim(ker E/(V ∗ ∩ ker E)); the first ones characterize the unobservable internal structure variation and the second ones characterize the observable internal structure variation. The internal structure variation will then be unobservable, if there is no Lqi blocks, namely, if the closed loop system satisfies: V ∗ ⊃ ker (E − BFd ) (iii) We then realize that, we have to find a proportional and d x + R, with (Fp , Fd ) derivative control law, u = Fp x + Fd dt ∗ ∈ F(V ), such that the internal structure variation of the closed loop system is made unobservable, namely, such that V ∗ ⊃ ker (E − BFd ) is satisfied. (iv) Since the behavior of the implicit rectangular description (13), contains the behavior of the implicit global representation (11), one realizes that: If the internal structure variation is contained in V ∗ , one is able to find a proportional and d derivative descriptor variable feedback, u = Fp x + Fd dt x, such that the internal structure variation of the closed loop system is unobservable. 4) Example (part 4) : Let us continue the example. a) Reachability: From (11) and (15), (2), (3) and (20), we get (note ¯ = 2): that n = 4 and n
rank C θ
=
rk
1 1
0 1
(1 + α + β) (1 + β) (1 + α + β) β ∗ RX = Xd , i.e.
= 2. Then, form Corollary 1: the d implicit rectangular representation (13) and (16) is reachable, whatever be the values of θ = (α, β); c.f. with the illustrative example of Section 4-[5]. b) Decoupling: Applying (21) to (13) and(16),
we
confirm
that:
V∗
= span
1 0 0 , 1 0 0
=
ker C .
Since p = 1 = m, then Corollary 2 holds, so there exists a suitable proportional and derivative descriptor variable feedback, which makes unobservable the internal variable structure. c) Control law: In order to satisfy V ∗ ⊃ ker (E − BFd ), the derivative part of the control law has to contain 0 1 −1 , indeed: ker (E − BFd∗ ) = span the term
0 1 ⊂ V ∗. 0
On the other hand, in order to satisfy
(A + BFp∗ )V ∗ ⊂ (E − BFd∗ )V ∗ ,
the proportional part of the
control i h law has to contain the term
. Since: then the poles of the closed loop transfer function can be placed at will; let for example, the root of the polynomial, π(s) = (s + 1/τ ), where τ is a positive real number. Then the term [ 0 0 (1 − 1/τ ) has to be added to the proportional feedback. Thus, the proportional and derivative feedback d x + τ1 R, is: u∗ = −1 0 (1 − τ1 ) x + 0 1 −1 dt d) Closed loop system: Applying u∗ to (11) and (15) we get the closed loop system described by the implicit global representation (see [6] for the approximation of the P.D. feedback): rank
B∗
0 0 0
1 0 0
A∗ B ∗
0 d 1 dt x 0 y∗
···
p−1
A∗
B∗
−1
0
0
= p,
0 1 −1 0 = 0 0 −1/τ x + 1/τ R 0 α β 1 = 0 0 1 x
(22) V. Conclusion In this paper, we have shown how linear time-invariant implicit systems theory can be efficiently used to model and to control a certain class of time-dependent, autonomous systems with internal switches. For the state space representations Σssr (Aθ , B, Cθ ), (1) and (2), we have found the common fixed structure, represented by the implicit rectangular representation Σirr (E, A, B, C), (13); namely, we have extracted a linear time-invariant implicit representation from the linear switching system Σsw . Indeed, the implicit global representation Σigr (E, Aθ , B, C), (11), is time dependent, but the implicit rectangular representation Σirr (E, A, B, C), (13), is time invariant. As illustrated on a simple example, the structure variations that can be considered in that way are very wide, and include for instance: variable relative degree, variable gain, variable finite zeros. In the particular proposed structure (2), only the matrices Aθ and Cθ depend on θ. If the matrix B would have a θ dependence, we could not isolate any longer the variable structure nature of the system into the algebraic equation (algebraic constraints) (14), and the proposed methodology would be useless. On the other hand, the independence of matrix B , with respect to θ, is not restrictive because of the zeros and the unobservable subspace are characterized by means of the structure of the matrices Aθ and Cθ . Lemma 1 is very important since it enables us to model the considered class of switched system (described by the state space representation Σssr (Aθ , B, Cθ ), (1) and (2), or described by the implicit global representation Σigr (E, Aθ , B, C), (11)) by means of: (i) a linear time invariant rectangular implicit representation Σirr (E, A, B, C), (13), which describes the common fixed structure, and (ii) a set of algebraic constraints Σalc (0, Dθ , 0), (14), which algebraically isolate the variable structure nature of the system. Corollaries 1 and 2 are important because they enable us to have a better understanding, from the classical state
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space point of view, of the reachability of this kind of systems and the variable structure decoupling nature of the proposed ideal P.D. control law. The reader can see [12], [17], for other interesting discussions about the study of switching systems since the behavioral approach. As a future research on this subject, we aim at finding conditions for the admissible changing structure parameters in order to guarantee exponential stability; we believe that this could be possible using the stability results of Narendra [14], [15]. Another interesting research is the one related with finding a common optimal control law, based on the common structure, for these variable structure systems. References
[19] Schaft, A.J. van der, and H. Schumacher (2000). An Introduction to Hybrid Dynamical Systems. New York: Springer–Verlag, Lecture Notes in Control and Information Sciens 251. [20] Shorten, R.N. and K.S. Narendra (2002). Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time–invariant systems. International Journal of Adaptive Control and Signal Processing, 16, 709–728. [21] Verghese, G.C. (1981) Further notes on singular descriptions. JACC TA4, Charlottesville. [22] Wonham, W.M. (1985). Linear Multivariable Control: A Geometric Approach. New York: Springer-Verlag, 3rd ed. [23] Willems, J.C. (1983). Input–output and state space representations of finite–dimensional linear time–invariant systems. Linear Algebra and its Applications, 50, 81–608.
Appendix A. Proof of Corollary 1 ∗ ∗ (i) Let us rewrite the subspaces, RX ,Σ , RX ,Σ0 and ∗ ∗ ∗ , reof Theorem 13-[5] as: RXd , R0 and Ru,θ spectively. Then from (12), (2), (21)-[5], (20a-c)-[5], and Lemma 1, we get (18) and (19). (ii) From (18) and (19), ∗ ∗ we have that, RX = Xd , implies (recall (12)): ERX = d
d A B , for all θ ∈ I . (iii) Now, if: Im E = A θ Im 1
∗ RK ssr , D ,Σ
[1] Bernhard, P. (1982). On singular implicit dynamical systems. SIAM J 20(5), 612–633. [2] Bonilla, M., G. Lebret, et M. Malabre (1994). Output Dynamics Assignment for Implicit Descriptions. Circuits, Systems and Signal Processing, special issue on “Implicit and Robust Systems”. 13(2-3), 349–359. [3] Bonilla, M. and M. Malabre (1997). Structural Matrix Minimization Algorithm for Implicit Descriptions. Automatica, 331(4), 705–710. [4] Bonilla, M. and M. Malabre (2000). More about non square implicit descriptions for modelling and control. 39th IEEE– CDC, 3642–3647. [5] Bonilla, M. and M. Malabre (2003). On the control of linear systems having internal variations. Automatica, 39, 1989– 1996. [6] Bonilla M., J. Pacheco and M. Malabre (2003). Almost Rejection of Internal Structural Variations in Linear Systems. 42nd IEEE–CDC, 116–121. [7] Gantmacher, F.R. (1977). The Theory of Matrices. Vol. II, New York: Chelsea. [8] Geerts, T. (1993). Solvability Conditions, Consistency, and Weak Consistency for Linear Differential-Algebraic Equations and Time-Invariant Singular Systems: The General Case. Linear Algebra and its Applications, 181, 111–130. [9] Lebret, G. and J.J. Loiseau(1994). Proportional and Proportional-derivative Canonical Forms for Descriptor Systems with Outputs. Automatica, 30(5), 847–864. [10] Lewis, F.L. (1992). A tutorial on the geometric analysis of linear time-invariant implicit systems. Automatica, 28(1), 119– 137. [11] Liberzon, D. (2003). Switching in Systems and Control. Boston, MA: Birkhauser, series Systems and Control: Found. and App. [12] Liberzon, D. and S. Trenn (2009). On stability of linear switched differential algebraic equations. Joint 48th IEEE– CDC and 28th CCC, 2156–2161. [13] Morse, A.S. (1996). Supervisory Control of Families of Linear Set-Point Controllers–Part 1: Exact Matching. IEEE–TAC, 41, 1413–1431. [14] Narendra, K.S. and J. Balakrishnan (1994). A Common Lyapunov Function for Stable LTI Systems with Commuting A– Matrices. IEEE–TAC, 39, 2469–2471. [15] Narendra, K.S. (1997). Adaptive Control Using Multi Models. IEEE–TAC, 42, 171–187. [16] Polderman, J.W., and J.C. Willems (1998). Introduction to Mathematical Systems Theory: A Behavioral Approach. New York: Springer–Verlag. [17] Rocha, P., J.C. Willems, P. Rapisarda, and D. Napp (2011). On the stability of switched behavioral systems. 50th IEEE– CDC and ECC (CDC-ECC), 1534–1538. [18] Rosenbrock, H.H. (1970) State–Space and Multivariable Theory. Nelson, London.
∗ RX = Vθ Aθ Im A1 ⊕ ker E + Vθ Aθ Im B 6= Xd = d ¯ ker Dθ ⊕ ker E , for all θ ∈ I , there then exists, θ ∈ I , such that: A θ¯ Im A B = 6 Im E . (iv) The last equivalence, 1 Im E = Aθ Im A1 B ⇔ rank C θ = n ¯ , is standard (see for
example [22] or [16]).
B. Proof of Corollary 2 Let us first note that the geometric conditions (6), (8) and (9) are already satisfied, so the only condition for satisfying Theorems 14-[5] and 15-[5] is: dim (ker E) ≤ dim V ∗ ∩ E −1 Im B . (A1) Applying column elementary operations over matrices, E , A and C , we get (see (12)): r E ≈ I 0 , A ≈ A0 − A1 C 1 C 0 −A1 , C ≈ 0 −C 1 , (A2) r where C 1 is some right inverse of C 1 . Then applying (21) to (A2): V ∗ = ker C . Let us now prove that: dim ker C ∩ E −1 Im B = dim (Ker E) + dim (Im B) − dim (Im C) .
(A3) From (A2) we see that: Xd = ker C + ker E, which implies: dim(ker C +
(A4)
E −1 Im
B) = dim(Xd ); thus: dim(ker C ∩ E −1 Im B) = dim(ker E) + dim(Im B ∩ Im E) − dim(Im C). Since Im B ⊂ Im E , we get (A3). Then since V ∗ = ker C, and from (A3), it follows that the
geometric condition (A1) is equivalent to the dimensional condition p ≤ m,. ∗ Let us note that the geometric condition, RX = Xd , of d ∗ ∗ Theorem 16-[5], can be relaxed to RXd + V = Xd , because Xd is quotiented by V ∗ ; which in our particular case is translated into: ∗ RX + ker C = Xd (A5) d Let us finally prove that (A5) is satisfied. Indeed, from (18) and (19) get: Xd = ker E∗ + ker C =
we
(A4), + ker C . Vθ Aθ Im A1 ⊕ ker E + Aθ Im B + ker C = RX d D E ∗ Then: A∗ Im B ∗ = Xd /V .
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Description of Switched Systems by Implicit Representations M. Bonilla and M. Malabre.
Abstract— This paper shows that a certain class of time–dependent, autonomous switched systems can also be studied using the structural concepts and properties of the linear time-invariant implicit systems theory. We first consider the modeling aspects, and then, the control aspects are tackled. This fact contributes to the enrichment of the switched systems theory and enlarges its action field. In Section I we specify a class of time–dependent, autonomous switched systems which enables us to describe systems with a variable internal structure. In Section II we show how to translate the considered class of time–dependent, autonomous switched systems into an implicit representation and then we recall some important results, of the implicit systems theory, which enable us to control them in such a way that they behave as a specified linear time–invariant system whatever be the position of the switches. In Section IV, we consider some reachability aspects. In Section V, we conclude.
where IIi (·) is the characteristic function of the interval ¯ ×m , A ∈ Rn ¯ ׯ n and C ∈ Rpׯ n , with θ ∈ I . The Ii , B ∈ Rn θ θ signals u and y are the input and output variables, respectively, and s is the switching signal. As usual we assume that B is a monic matrix and the Cθ are epic matrices. In the terminology of van der Schaft and Schumacher [19], there are η locations (or modes) described by the following η state space representations, Σssr (Aθ , B, Cθ ): d x ¯ dt
= Aθ x ¯ + Bu and y = Cθ x ¯
where θ ∈ I , x¯(Ti−1 ) = ci , and t ∈ Ii , for some i ∈ N, some and some Ii ∈ T. We restrict our discussion to matrices, Aθ and Cθ , θ ∈ I , having the following particular structure: Ti−1 ∈ T,
Aθ = A0 + A1 D(θ) and Cθ = C 0 + C 1 D(θ),
I. Linear Time–Dependent Autonomous Switched Systems In this paper, we adopt the dynamic system definition given by Willems [23], which is known as the behavioral approach (see the synthesis reference of Polderman and Willems [16] and also the version of van der Schaft and Schumacher [19] for Hybrid Dynamical Systems). Let us define the sets: T = {Ti ∈ R+ (i ∈ Z+ ) | T0 = 0, Ti−1 < Ti ∀ i ∈ N, with lim Ti = ∞}, I = {Ii = [Ti−1 , Ti ) i→∞ ⊂ R+ | Ti−1 , Ti ∈ T, i ∈ N}, S(I, I ) = {s : I → I | s(Ii ) = θ}, I = {θ1 , . . . , θη }. T is the switching time set, defining a partition on the time axis, R+ . I is the locations invariant set, defining disjoints time intervals covering all the time axis. S is the switching signals set, which elements are functions mapping from I, to an index set, I , of cardinality η (see [11] and [19] for details). Let Σsw = I × R+ , I × Rm × Rp , Bsw be a dynamical system which switches in a time–dependent and autonomous way [11], where the behavior, Bsw , is defined as follows: + m loc + p (s, u, y) ∈ S(I, I ) × Lloc 1 (R , R ) × L1 (R , R ) P ¯ s.t.: y(t) = ∃ ci ∈ Rn IIi (t)Cs(Ii ) exp As(Ii ) (t − Ti−1 ) ci i∈N Z t−Ti−1 + exp As(Ii ) (t − Ti−1 − τ ) Bu(Ti−1 + τ )dτ , 0 t ∈ R+ , Ti−1 ∈ T, Ii ∈ I ,
0 1 1 0 A0 = , A1 = , B= , 1 0 1 1 C 0 = 0 0 , C 1 = −1, D(θ) = α β , θ = (α, β) ∈ I = θ1 , θ2 , θ3 , θ4 , θ1 = (−1, −1), θ2 = (−1, 0), θ3 = (−1, −5), θ4 = (1, 1).
(3) For a given pair, (α, β), we have the transfer function: Fθ (s) = Cθ (sI − Aθ )−1 B =
´ M. Bonilla, CINVESTAV-IPN, CONTROL AUTOMATICO, ´ UMI 3175 CINVESTAV-CNRS. A.P. 14-740. MEXICO 07000,
"[email protected]" . M. Malabre, LUNAM Universit´ e, CNRS, IRCCyN UMR CNRS 6597, 1 Rue de la Noe, F- 44321 Nantes, FRANCE.
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
(2)
where: A0 ∈ Rn¯ ׯn , C 0 ∈ Rpׯn , A1 ∈ Rn¯ ×q , C 1 ∈ Rp×q , and n . We also assume that the matrix C is epic. D(θ) ∈ Rqׯ 1 Note that this structure of variation described by a finite set of matrices Aθ is very similar to the one used by Narendra and Balakrishnan [14] and Shorten and Narendra [20]. As we will see hereafter, the presence of the finite set of matrices Cθ will allow us to play with their unobservable spaces, in order to take into account variations of the internal structure. With this kind of time–dependent, autonomous switched systems one is able to describe, and later to control, systems which internal structure changes among a known set of linear time invariant proper systems. 1) Example (part 1) : Let us consider:
Bsw =
"[email protected]".
(1)
−(βs+α) s2 −(α+β)s−(1+α+β)
(4)
For the possible four selections of θ ∈ I , we nhave the following h four possible behaviors: io ∞ I , R2 ∩ ker −1 ( d + 1) B∞ = (u, y) ∈ C , i θ1 n dt h io ∞ I , R2 ∩ ker −1 ( d + 1)( d ) B∞ = (u, y) ∈ C , j θ2 n dt dt h io d d d ∞ ∞ 2 Bθ3 = (u, y) ∈ C Ik , R ∩ ker −(5 dt + 1) ( dt + 1)( dt + 5) , n io h d Bθ4∞ = (u, y) ∈ C ∞ I` , R2 ∩ 1 ( dt − 3) , for some disjoints Ii , Ij , Ik , I` ∈ I (i, j, k, ` ∈ N).
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As one can see later on, the key idea for modeling linear systems with variable internal structure is to carefully handle the unobservable space. In [4] we have introduced the so-called ladder systems, which are a particular subclass of such systems with internal switches. II. Implicit Representations In this Section we introduce some necessary concepts of the linear implicit theory, which will enable us, in Section III, to model time–dependent autonomous switched system Σsw , represented by the behavior Bsw and the matrices (2), by means of a linear implicit representation. A. Implicit systems We deal with implicit representations which were introduced by Rosenbrock [18] as a generalization of proper linear systems (see for example [10]). An implicit representation, Σimp (E, A, B, C), is a set of differential and algebraic equations of the following form: Edx/dt = Ax + Bu and y = Cx, t ∈ R+ ,
are called the descriptor, the equation, the input and the output spaces, respectively. It is usual to assume that: ker B = {0}
E 0 | {z } E
and Im C = Y . for almost all λ ∈ C.
Im [λE − A] = X eq ,
Assumption H1 states that the input variable, u, has no redundant components and that the set of output equations are linearly independent. And assumption H2 implies that (5) is solvable, namely: “for any u(·) ∈ C ∞ (R+ , U ), there exists at least one trajectory x(·) ∈ C ∞ (R+ , Xd ) solution of (5) for all t ≥ 0” . Definition 1: (Implicit rectangular representation) An implicit rectangular representation, Σirr (E, A, B, C), is an implicit representation (5) such that: (i) dim Xd > dim X eq , and (ii) the following geometric condition holds: Im A + Im B ⊂ Im E. (6) The geometric condition (6) guarantees that for any, u ∈ C ∞ (R+ , U ), there exists at least one descriptor variable trajectory, x ∈ C ∞ (R+ , Xd ), satisfying the implicit rectangular representation, Σirr (E, A, B, C) (see Cor. 11-[5]). Definition 2: (Algebraic constraint) An algebraic constraint is an implicit representation such that: (i) it is only formed by a set of linearly independent algebraic equations, (ii) it is independent of the input variable, (iii) it has no output equation; namely, Σalc (0, D, 0): 0 = Dx. where D : Xd → X alc is a map and the space, X alc , is called the algebraic constraint space. Definition 3: (Implicit global representation) The implicit representation obtained by the union of an implicit rectangular representation, Σirr (E, A, B, C), with an algebraic constraint, Σalc (0, D, 0), is called an implicit
d x dt
=
A B x+ u and y = Cx, D 0 | {z } | {z } A
(7)
B
t∈ . The cartesian product, X g =: X eq × X alc , is called the global equation space. It is usual to assume that: R+
X g = Im E ⊕ Im D . (8) Condition (8) states that the implicit global representation (7) is only composed by independent linear equations (see Corollary 11-[5] and Proposition 8-[5]).
B. Some structural properties Definition 4: (Regularity [7]) A pencil [λE − A] is regular if it is square and det[λE − A] 6= 0. An implicit representation is called regular if [λE − A] is regular. Definition 5: (Internal properness [1]) An implicit representation is internally proper if it is regular and has no infinite elementary divisor of order greater than 1. In Proposition 9-[5], we show that any given implicit global representation, (7), is internally proper iff:
(5)
where: E : Xd → X eq , A : Xd → X eq , B : U → X eq and C : Xeq → Y are maps. The spaces Xd , X eq , U and Y
H1 H2
global representation, Σigr (E, A, B, C):
Xd = ker D ⊕ ker E
(9)
Definition 6: (External equivalence [23], [16]) Two representations are called externally equivalent if the corresponding sets of all possible trajectories for the external variables, expressed in an input/output partition (u, y), are the same. Definition 7: (Algebraic redundancy [3]) A part of the descriptor variable which is a linear combination of other descriptor variable components, and do not contribute to describe the external behavior (set of all possible input-output, (u, y), trajectories) is called an algebraic redundant descriptor variable. Lemma 1: (Localization of the resulting state space representation of a proper implicit global representation; Proposition 10-[5]) Let the implicit global representation (7) satisfying (6), (8) and (9), then the degree of freedom, characterized by the algebraic redundant descriptor variable, is located in ker E . The part of (7) which is restricted to ker D in the domain and to Eker D in the co-domain, contains no algebraic redundant descriptor variables. Moreover, there exist unique maps (E, A, B, C): EV = V E , AV = V A
and CV = C ,
where, V : ker D → Xd and V : Im E → X g , are natural insertions. b A, b B) b = (0, I, 0) are the unique maps: Furthermore, (E, b , P A = Pard A b P alc E = Pard E alc
and P alc B = Bb ,
where, Pard : Xd → ker E , is the natural projection on along ker D, and, Palc : X g → Dker E , is the natural projection on Dker E along Eker D. Finally, (7) is externally equivalent to the reduced state space representation, Σssr (A0 , B0 , C):
ker E
3210
d¯ x/dt = A0 x ¯ + B0 u
and y = C x¯,
ker E = ker D ∩ ker E = {0}, where: E = P eq EV = EV , Im E = Eker D = Im E , A = P eq AV = AV , B = P eq B = B ,
and B0 = E (−1) B . This means that one way for enhancing the state space representation which is inside the proper implicit global representation, Σigr (E, A, B, C), is to decompose, Xd , as ker D ⊕ ker E , and, X g , as Eker D ⊕ Dker E . 2) Example (part 2) : The time–dependent, autonomous switched system, described by (1), (2) and (3), is also described by the implicit global representation: Eker D
C = CV , A0 = E
(−1)
A
fixed structure of the considered linear time–dependent autonomous switched system. This allows us to take advantage of the available structural tools, of the implicit system theory, for the analysis and the synthesis tasks. A. Implicit global representation In view of Lemma 1 we can algebraically characterize the variable nature of the system. Indeed, if we gather a redundant descriptor variable, i.e. xˆ = 0, in the state space representation, (1) and (2), we can bring them to the following implicit global representations:
ker -E
61 0 ? 0
0 0 0
0 1 0
x ¯ x ˆ
6 ?
α
0 1 β
0 0 , 1
d¯ x dt dˆ x dt
=
A0 + A1 D(θ) −A1 0 I C 0 + C 1 D(θ) −C 1
x ¯ x ˆ
x ¯ x ˆ
+
B 0
u
,
◦
-
t ∈ Ii,
(10) Let us note that: (i) The implicit global representation (10) is externally equivalent to the state space representation, Σssr (Aθ , B, Cθ ), restricted to ker D, in the domain, and to Eker D, in the co-domain, which state space matrices are enclosed by solid line boxes. (ii) Above the solid line of the differential and algebraic equations set, there is an implicit rectirr −1 −1 −1 angular representation, Σ (ET (θ), AT (θ), B, CT (θ)) 1
y=
α (1 + β) −1 0 x ¯ −1 (1 + α) β + 1 u x ˆ ....................... 0 . . . .0. . . . . . . . .0. . . . . . . . 1. . x ¯ 1 − α − β y= x ˆ
with T (θ) = 0
I 0 0 0
=
..
Dker E
..
d dt
ker D
restricted to Eker D in the
co-domain, which behavioral matrices are enclosed by dashed line boxes. This part of the representation takes into account, in an implicit way, the changes of behaviors due to the commutation of the parameters α and β . (iii) Below the solid line of the differential and algebraic equations set, there is an algebraic constraint, Σalc (0, DT −1 (θ), 0), restricted to Dker E in the co-domain, which behavioral matrices are enclosed by doted line boxes. The part restricted to ker E , in the domain, and to Dker E , in the co-domain, is algebraically redundant and includes the descriptor variables which are always null. (iv) Since: Im A + Im B ⊂ Im E , Xg = Eker D ⊕ Dker E = Im E ⊕ Im D and Xd = ker D ⊕ ker E , this implicit global representation is internally proper. III. Implicit Global Representation of a Time–Dependent Autonomous Switched system In this Section we show that the time–dependent autonomous switched system Σsw , with behavior Bsw and represented by the state representations (1) and (2) can also be represented by means of implicit global representations, Σigr (E, Aθ , B, C), θ ∈ I . From these implicit global representations, we will extract an implicit rectangular representation, Σirr (E, A, B, C), which characterizes the
where: limt→T + x¯(t) = x¯(Ti−1 ) = ci , limt→T + xˆ(t) i−1 i−1 = x ˆ(Ti−1 ) = 0, θ ∈ I (fixed for each Ii ), and i ∈ N. By ◦ X we denote the interior set of a set X . In order to get an implicit rectangular representation, describing all the common structure of the time–dependent autonomous switched system, represented by (1) and (2), let us define the descriptor variable: x = xx1 = T −1 (θ) xx¯ˆ , where: 2
I D(θ)
T (θ) =
0 I
, namely, Σigr (E, Aθ , B, C): ◦
d t ∈ I i E dt Cx, x = Aθ x + Bu and y = B A E , B= , Aθ = E= 0 Dθ 0
(11)
where: limt→T + x1 (t) = x1 (Ti−1 ) = ci , limt→T + x2 (t) = i−1 i−1 x2 (Ti−1 ) = −D(θ)ci , θ ∈ I (fixed for each Ii ), and i ∈ N. The maps E : Xd → X eq , A : Xd → X eq , B : U → X eq , Dθ : Xd → X cal , and C : Xd → Y are equal to: E=
I
0
, A = A0 −A1 , C = C 0 Dθ = D(θ) I .
−C 1
,
(12) Let us note that: (i) The fixed structure of Σigr (E, Aθ , B, C), which is active for any Dθ , θ ∈ I , is described by the following implicit rectangular representation, Σirr (E, A, B, C), d E dt x = Ax + Bu and y = Cx,
t ∈ R+ \ T.
(13)
where: limt→T + x1 (t) = x1 (Ti−1 ) and limt→T + x2 (t) = i−1 i−1 x2 (Ti−1 ). (ii) The degree of freedom is characterized by the algebraic constraints, Σalc (0, Dθ , 0), 0 = Dθ x,
t ∈ Ii
(14)
for some θ ∈ I , some i ∈ N, and some Ii ∈ T. (iii) Since dim X eq < dim Xd , there then exists a degree of freedom (see the introductory section of [5]). (iv) The geometric conditions, (6), (8) and (9), are fulfilled. (v) Since Im A + Im B ⊂ Im E = X eq , the implicit rectangular representation has at least one solution. (vi) Since Im Dθi = X cal and dim(X eq × X cal ) = dim(Xd ), for each θi ∈ I , the implicit global representations have unique solutions for each θi ∈ I . (vii) Since ker Dθ ⊕ ker E = Xd ,
3211
for all θ ∈ I , then all the implicit global representations are proper. (viii) Since Im E = X eq the possible jumps taking place at Ti ∈ T do not generate any impulse (c.f. Theorem 4.5 of [8]). 3) Example (part 3) : In order to enhance the fixed structure of (10), let us define the descriptor variable x ¯ x = T −1 (θ) . Then, we get the implicit global reprex ˆ sentation (11) with:
E
Aθ C
B
=
1 0 0
0 1 0
0 0 0
0 1 α 0
1 0 β 0
−1 −1 1 1
0 1 0
(15)
Above the solid line of the differential and algebraic equation set of the implicit global representation (11) and (15), there is the implicit rectangular representation (13) with:
E
A C
B
=
1 0
0 1
0 0
0 1 0
1 0 0
−1 −1 1
0 1
α
β
1
(16)
(17)
Let us note that: (i) When splitting the implicit global representations (10) (via (11) and (15)) into the rectangular implicit description (13) and (16) and into the algebraic constraints (14) and (17), we get the fixed active structure of the system, represented by (13) and (16). (ii) The Kronecker normal forms [7] of the pencils, [λE − Aθ ], associated to (11) and (15) are: if 1
0 0 , if β 6= −1 and α + β = −2: (λ − α) 0 0 0 (λ + 1) 0 0 , and if β 6= −1 and α + β 6= −2: (λ + 1) 1 0 (λ + 1) 0 0 . (iii) The Kronecker nor(λ − 1 − α − β) 0 0 (λ + 1)
β = −1: 0
1 0 0 1 0 0
mal form of the pencil, [λE − A], associated to (13) and 0 (16) is: λ0 10 (λ + (iv) Comparing the Kronecker 1) normal forms of the pencils associated to (11) and (15), with the Kronecker normal form of the pencil associated to (13) and (16), we can see that the internal variable structure of (11) and (15) is now taken into account by the invariant column minimal index block, λ 1 , of (13) and (16). IV. Reachability and Control of the implicit global representation In [5] has been shown that the reachability of is split into the reachability from the control input and that due to an internal degree of freedom. A Corollary of [5] is the following (see Appendix A, for the proof): Corollary 1: (Reachability subspace of an implicit rectangular representation) Given the implicit global representation, (11) and (12), the reachability subspace, Σigr (E, Aθ , B, C)
, of the implicit rectangular description, (13), and the d ∗ , of the state space representareachability subspace, Ru,θ tion, (1), with θ ∈ I , are related as follows: ∗ ∗ RX = R0∗ + Vθ Ru,θ d
(18)
where Vθ : ker Dθ → Xd = ker Dθ ⊕ ker E is the natural insertion map of ker Dθ in Xd , and R0∗ is the reachability subspace of the autonomous implicit representation, ◦ d Σirr 0 (E, A, 0): E dt x(t) = Ax(t), t ∈ I i , where: limt→T + x1 (t) i−1 = x1 (Ti−1 ) = ci , limt→T + x2 (t) = x2 (Ti−1 ) = −D(θ)ci , θ ∈ I i−1 (fixed for each Ii ), and i ∈ N. The reachability subspaces, ∗ , are equal to:1 R0∗ and Ru,θ
∗ R0∗ = Vθ Aθ Im A1 ⊕ ker E and Ru,θ = Aθ Im B
(19) Moreover, given the following controllability matrices: Cθ =
Below the solid line of the differential and algebraic equation set of the implicit global representation (11) and (15), there is the algebraic constraint (14) with: Dθ =
∗ RX
A1 B
Aθ A1 B
···
¯ −1 , θ ∈ I, An A1 B θ
(20)
∗ ∗ ⇔ then: RX = Xd ⇔ Im E = Aθ Im A1 B = ERX d d rank C θ = n ¯ , for all θ ∈ I . In other words, the reachability of the implicit rectangular representation, Σirr (E, A, B, C), is achieved by both, the input action, through Im B , and the degree of freedom, through Im A1 . Definition 8: ((A, E, B)–invariance [21]) The supremal (A, E, B)–invariant subspace contained in ker C , V ∗ = sup{V ⊂ ker C| AV ⊂ EV + Im B}, is the limit of the following non-increasing geometric algorithm: V 0 = Xd ; V µ+1 = ker C ∩ A−1 (EV µ + Im B) , µ ≥ 0
(21) This subspace characterizes the maximal part of the implicit representation, Σimp (E, A, B, C), which can be made unobservable with a suitable proportional and derivative descriptor variable feedback. A Corollary of Theorems 14-[5], 15-[5] and 16-[5], is the following (see Appendix B, for the proof): Corollary 2: (Internal variable structure decoupling) Let us consider the implicit rectangular representation (13), Σirr (E, A, B, C), which is extracted from the implicit global representation (11), and with the behavioral maps (12). Then: V ∗ = ker C . Moreover, if: p ≤ m, holds, there then exists, a proportional and derivative descriptor variable feedback, u = Fp∗ x + Fd∗ dx/dt, such that the closed loop system, represented by Σirr (EFd∗ , AFp∗ , B, C), where: EFd∗ = E − BFd∗ and AFp∗ = A + BFp∗ , satisfies: (i) Im EFd∗ = Im E , i.e., the existence of solution is conserved. (ii) ker EFd∗ ⊂ V ∗ , i.e., the internal variable structure of the closed loop system is made unobservable. (iii) Σirr (EFd∗ , AFp∗ , B, C) is externally equivalent to the state space representation, Σssr (A∗ , B ∗ , C∗ ); where A∗ , B ∗ and C∗ are the unique maps induced by AFp∗ and EFd∗ , such that: ΠAFp∗ = A∗ Φ, ΠEFp∗ = E∗ Φ, ΠB = B∗ Φ and C = C∗ Φ,
X : V → W and Y : W → W , Y Im X stands for the reachability subspace [22]: Im X + Y Im X + · · · + Y dim(W )−1 Im X.
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1 Given,
d d where Φ : Xd → X and Π : EXd → EEX are canon∗ V∗ F∗V d ∗ ical projections, EFd∗ = E − BFd , AFp∗ = A + BFp∗ , with: (Fp∗ , Fd∗ ) ∈ F(V ∗ ) = (Fp , Fd ) (A + BFp )V ∗ ⊂ (E − BFd )V ∗ , and Im B ∩ (E − BFd∗ )V ∗ = {0}. E∗ is an isomorphism, and (−1) E∗ is its inverse Dmap. E A∗ Im B ∗ = Xd /V ∗ , namely: Furthermore:
rank
h
B∗
A∗ B ∗
···
p−1
A∗
B∗
i
= p
∗ =X Let us note that: (i) Since (6) implies: VX d d ∗ and Im B ⊂ EVXd , p ≤ m, implies the output uniqueness property, item ii) of Theorem 7-[2]. This means that the internal variation structure is made unobservable. (ii) Using the Kronecker theory, Lebret and Loiseau [9] generalized the canonical form of Morse [13]. In such a generalization, they also characterized the parts of the implicit representation which are responsible for the existing internal structure variation, by the so called Lσi blocks, with cardinality dim(V ∗ ∩ ker E), and Lqi blocks, with cardinality dim(ker E/(V ∗ ∩ ker E)); the first ones characterize the unobservable internal structure variation and the second ones characterize the observable internal structure variation. The internal structure variation will then be unobservable, if there is no Lqi blocks, namely, if the closed loop system satisfies: V ∗ ⊃ ker (E − BFd ) (iii) We then realize that, we have to find a proportional and d x + R, with (Fp , Fd ) derivative control law, u = Fp x + Fd dt ∗ ∈ F(V ), such that the internal structure variation of the closed loop system is made unobservable, namely, such that V ∗ ⊃ ker (E − BFd ) is satisfied. (iv) Since the behavior of the implicit rectangular description (13), contains the behavior of the implicit global representation (11), one realizes that: If the internal structure variation is contained in V ∗ , one is able to find a proportional and d derivative descriptor variable feedback, u = Fp x + Fd dt x, such that the internal structure variation of the closed loop system is unobservable. 4) Example (part 4) : Let us continue the example. a) Reachability: From (11) and (15), (2), (3) and (20), we get (note ¯ = 2): that n = 4 and n
rank C θ
=
rk
1 1
0 1
(1 + α + β) (1 + β) (1 + α + β) β ∗ RX = Xd , i.e.
= 2. Then, form Corollary 1: the d implicit rectangular representation (13) and (16) is reachable, whatever be the values of θ = (α, β); c.f. with the illustrative example of Section 4-[5]. b) Decoupling: Applying (21) to (13) and(16),
we
confirm
that:
V∗
= span
1 0 0 , 1 0 0
=
ker C .
Since p = 1 = m, then Corollary 2 holds, so there exists a suitable proportional and derivative descriptor variable feedback, which makes unobservable the internal variable structure. c) Control law: In order to satisfy V ∗ ⊃ ker (E − BFd ), the derivative part of the control law has to contain 0 1 −1 , indeed: ker (E − BFd∗ ) = span the term
0 1 ⊂ V ∗. 0
On the other hand, in order to satisfy
(A + BFp∗ )V ∗ ⊂ (E − BFd∗ )V ∗ ,
the proportional part of the
control i h law has to contain the term
. Since: then the poles of the closed loop transfer function can be placed at will; let for example, the root of the polynomial, π(s) = (s + 1/τ ), where τ is a positive real number. Then the term [ 0 0 (1 − 1/τ ) has to be added to the proportional feedback. Thus, the proportional and derivative feedback d x + τ1 R, is: u∗ = −1 0 (1 − τ1 ) x + 0 1 −1 dt d) Closed loop system: Applying u∗ to (11) and (15) we get the closed loop system described by the implicit global representation (see [6] for the approximation of the P.D. feedback): rank
B∗
0 0 0
1 0 0
A∗ B ∗
0 d 1 dt x 0 y∗
···
p−1
A∗
B∗
−1
0
0
= p,
0 1 −1 0 = 0 0 −1/τ x + 1/τ R 0 α β 1 = 0 0 1 x
(22) V. Conclusion In this paper, we have shown how linear time-invariant implicit systems theory can be efficiently used to model and to control a certain class of time-dependent, autonomous systems with internal switches. For the state space representations Σssr (Aθ , B, Cθ ), (1) and (2), we have found the common fixed structure, represented by the implicit rectangular representation Σirr (E, A, B, C), (13); namely, we have extracted a linear time-invariant implicit representation from the linear switching system Σsw . Indeed, the implicit global representation Σigr (E, Aθ , B, C), (11), is time dependent, but the implicit rectangular representation Σirr (E, A, B, C), (13), is time invariant. As illustrated on a simple example, the structure variations that can be considered in that way are very wide, and include for instance: variable relative degree, variable gain, variable finite zeros. In the particular proposed structure (2), only the matrices Aθ and Cθ depend on θ. If the matrix B would have a θ dependence, we could not isolate any longer the variable structure nature of the system into the algebraic equation (algebraic constraints) (14), and the proposed methodology would be useless. On the other hand, the independence of matrix B , with respect to θ, is not restrictive because of the zeros and the unobservable subspace are characterized by means of the structure of the matrices Aθ and Cθ . Lemma 1 is very important since it enables us to model the considered class of switched system (described by the state space representation Σssr (Aθ , B, Cθ ), (1) and (2), or described by the implicit global representation Σigr (E, Aθ , B, C), (11)) by means of: (i) a linear time invariant rectangular implicit representation Σirr (E, A, B, C), (13), which describes the common fixed structure, and (ii) a set of algebraic constraints Σalc (0, Dθ , 0), (14), which algebraically isolate the variable structure nature of the system. Corollaries 1 and 2 are important because they enable us to have a better understanding, from the classical state
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space point of view, of the reachability of this kind of systems and the variable structure decoupling nature of the proposed ideal P.D. control law. The reader can see [12], [17], for other interesting discussions about the study of switching systems since the behavioral approach. As a future research on this subject, we aim at finding conditions for the admissible changing structure parameters in order to guarantee exponential stability; we believe that this could be possible using the stability results of Narendra [14], [15]. Another interesting research is the one related with finding a common optimal control law, based on the common structure, for these variable structure systems. References
[19] Schaft, A.J. van der, and H. Schumacher (2000). An Introduction to Hybrid Dynamical Systems. New York: Springer–Verlag, Lecture Notes in Control and Information Sciens 251. [20] Shorten, R.N. and K.S. Narendra (2002). Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time–invariant systems. International Journal of Adaptive Control and Signal Processing, 16, 709–728. [21] Verghese, G.C. (1981) Further notes on singular descriptions. JACC TA4, Charlottesville. [22] Wonham, W.M. (1985). Linear Multivariable Control: A Geometric Approach. New York: Springer-Verlag, 3rd ed. [23] Willems, J.C. (1983). Input–output and state space representations of finite–dimensional linear time–invariant systems. Linear Algebra and its Applications, 50, 81–608.
Appendix A. Proof of Corollary 1 ∗ ∗ (i) Let us rewrite the subspaces, RX ,Σ , RX ,Σ0 and ∗ ∗ ∗ , reof Theorem 13-[5] as: RXd , R0 and Ru,θ spectively. Then from (12), (2), (21)-[5], (20a-c)-[5], and Lemma 1, we get (18) and (19). (ii) From (18) and (19), ∗ ∗ we have that, RX = Xd , implies (recall (12)): ERX = d
d A B , for all θ ∈ I . (iii) Now, if: Im E = A θ Im 1
∗ RK ssr , D ,Σ
[1] Bernhard, P. (1982). On singular implicit dynamical systems. SIAM J 20(5), 612–633. [2] Bonilla, M., G. Lebret, et M. Malabre (1994). Output Dynamics Assignment for Implicit Descriptions. Circuits, Systems and Signal Processing, special issue on “Implicit and Robust Systems”. 13(2-3), 349–359. [3] Bonilla, M. and M. Malabre (1997). Structural Matrix Minimization Algorithm for Implicit Descriptions. Automatica, 331(4), 705–710. [4] Bonilla, M. and M. Malabre (2000). More about non square implicit descriptions for modelling and control. 39th IEEE– CDC, 3642–3647. [5] Bonilla, M. and M. Malabre (2003). On the control of linear systems having internal variations. Automatica, 39, 1989– 1996. [6] Bonilla M., J. Pacheco and M. Malabre (2003). Almost Rejection of Internal Structural Variations in Linear Systems. 42nd IEEE–CDC, 116–121. [7] Gantmacher, F.R. (1977). The Theory of Matrices. Vol. II, New York: Chelsea. [8] Geerts, T. (1993). Solvability Conditions, Consistency, and Weak Consistency for Linear Differential-Algebraic Equations and Time-Invariant Singular Systems: The General Case. Linear Algebra and its Applications, 181, 111–130. [9] Lebret, G. and J.J. Loiseau(1994). Proportional and Proportional-derivative Canonical Forms for Descriptor Systems with Outputs. Automatica, 30(5), 847–864. [10] Lewis, F.L. (1992). A tutorial on the geometric analysis of linear time-invariant implicit systems. Automatica, 28(1), 119– 137. [11] Liberzon, D. (2003). Switching in Systems and Control. Boston, MA: Birkhauser, series Systems and Control: Found. and App. [12] Liberzon, D. and S. Trenn (2009). On stability of linear switched differential algebraic equations. Joint 48th IEEE– CDC and 28th CCC, 2156–2161. [13] Morse, A.S. (1996). Supervisory Control of Families of Linear Set-Point Controllers–Part 1: Exact Matching. IEEE–TAC, 41, 1413–1431. [14] Narendra, K.S. and J. Balakrishnan (1994). A Common Lyapunov Function for Stable LTI Systems with Commuting A– Matrices. IEEE–TAC, 39, 2469–2471. [15] Narendra, K.S. (1997). Adaptive Control Using Multi Models. IEEE–TAC, 42, 171–187. [16] Polderman, J.W., and J.C. Willems (1998). Introduction to Mathematical Systems Theory: A Behavioral Approach. New York: Springer–Verlag. [17] Rocha, P., J.C. Willems, P. Rapisarda, and D. Napp (2011). On the stability of switched behavioral systems. 50th IEEE– CDC and ECC (CDC-ECC), 1534–1538. [18] Rosenbrock, H.H. (1970) State–Space and Multivariable Theory. Nelson, London.
∗ RX = Vθ Aθ Im A1 ⊕ ker E + Vθ Aθ Im B 6= Xd = d ¯ ker Dθ ⊕ ker E , for all θ ∈ I , there then exists, θ ∈ I , such that: A θ¯ Im A B = 6 Im E . (iv) The last equivalence, 1 Im E = Aθ Im A1 B ⇔ rank C θ = n ¯ , is standard (see for
example [22] or [16]).
B. Proof of Corollary 2 Let us first note that the geometric conditions (6), (8) and (9) are already satisfied, so the only condition for satisfying Theorems 14-[5] and 15-[5] is: dim (ker E) ≤ dim V ∗ ∩ E −1 Im B . (A1) Applying column elementary operations over matrices, E , A and C , we get (see (12)): r E ≈ I 0 , A ≈ A0 − A1 C 1 C 0 −A1 , C ≈ 0 −C 1 , (A2) r where C 1 is some right inverse of C 1 . Then applying (21) to (A2): V ∗ = ker C . Let us now prove that: dim ker C ∩ E −1 Im B = dim (Ker E) + dim (Im B) − dim (Im C) .
(A3) From (A2) we see that: Xd = ker C + ker E, which implies: dim(ker C +
(A4)
E −1 Im
B) = dim(Xd ); thus: dim(ker C ∩ E −1 Im B) = dim(ker E) + dim(Im B ∩ Im E) − dim(Im C). Since Im B ⊂ Im E , we get (A3). Then since V ∗ = ker C, and from (A3), it follows that the
geometric condition (A1) is equivalent to the dimensional condition p ≤ m,. ∗ Let us note that the geometric condition, RX = Xd , of d ∗ ∗ Theorem 16-[5], can be relaxed to RXd + V = Xd , because Xd is quotiented by V ∗ ; which in our particular case is translated into: ∗ RX + ker C = Xd (A5) d Let us finally prove that (A5) is satisfied. Indeed, from (18) and (19) get: Xd = ker E∗ + ker C =
we
(A4), + ker C . Vθ Aθ Im A1 ⊕ ker E + Aθ Im B + ker C = RX d D E ∗ Then: A∗ Im B ∗ = Xd /V .
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