- Email: [email protected]
Fixed poles of almost disturbance decoupling by state feedback
4th IFAC Symposium on System, Structure and Control Università Politecnica delle Marche Ancona, Italy, Sept 15-17, 2010
Fixed poles of almost disturbance decoupling by state feedback Runmin Zou ∗ , Michel Malabre ∗∗ ∗
School of Information Science and Engineering, Central South University, Changsha 410083, P.R.China. Email: [email protected] ∗∗ IRCCyN, CNRS, UMR 6597, Ecole Centrale de Nantes, 1 rue de la No¨e, 44321 Nantes Cedex 03, France. Phone: (+)33240376912, Fax : (+)33240376930. Email: [email protected] Abstract: When the almost disturbance decoupling problem (ADDP) is solvable, some poles of the closed loop system are fixed for any state feedback solution. In Zou and Malabre (2009c), the authors have characterized them by means of some invariant zeros but in the particular case of (A, B)-controllable systems. As a complement, here, we give the characterization of the fixed poles of ADDP for the general case, namely without any controllability assumption. We also propose a method to solve the ADDP for this general case, as well as a numerical example to illustrate our contributions. Keywords: Linear control systems; geometric approach; fixed pole; almost disturbance decoupling. 1. INTRODUCTION The Almost Disturbance Decoupling Problem by static state feedback (ADDP) aims at designing a suitable state feedback such that the system controlled output is almost (in a precise sense) decoupled from the disturbance input while guaranteeing some performances (at least the internal stability of the resulting closed-loop feedback system and, possibly, pole placement in some desired region). It is well accepted that this problem occupies a central part in classical as well as modern control theory; several important problems, such as robust, decentralized, noninteracting, model reference or tracking control etc., are linked to (almost) disturbance decoupling, see for instance Saberi et al. (2006). Since ADDP was first introduced in Willems (1981), it has been intensively studied in the 1980’s Willems (1981); Schumacher (1984); Trentelman (1985); Weiland and Willems (1989), but all the contributions without exception, focused on the solvability of the problem with stability. In Zou and Malabre (2009c), the authors have proposed a general procedure to solve ADDP, by introducing the key concept of almost self-bounded controlled invariant subspace and characterized the so-called fixed poles of ADDP, but just in the particular case of (A, B)controllable system. The present contribution aims at giving an answer to some remaining issues. To get our results, we use the so-called “Geometric Approach”. We prove that the (A, [B, D])-controllable subsystem is the key reduced sub-system, and on the basis of a commutative diagram between the original system and the reduced sub-system, we also show that there exists a one to one correspondence between the solutions of the ADDP of these two systems, which enables us to get the pole placement abilities of the solutions of the ADDP 978-3-902661-83-8/10/$20.00 © 2010 IFAC
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for the general case, on the basis of the similar solution for an (A, B)-controllable system as described in Zou and Malabre (2009c). 2. NOTATION AND GEOMETRIC PRELIMINARIES We shall consider linear time-invariant disturbed systems Σ(A, B, D, E) described by: x(t) ˙ = Ax(t) + Bu(t) + Dq(t) Σ: (1) z(t) = Ex(t) where x, u, q, and z are respectively the state, control input, disturbance input, and output to be controlled. These signals belong to the vector spaces X , U , Q, and Z , respectively. Vectors are denoted by lower case letters, matrices/maps by capitals and subspaces by script capitals. If A is a square matrix, then σ(A) denotes its spectrum. If A : X 7→ Y and V ⊆ X , the restriction of the map A to V is denoted by A|V . If V1 and V2 are A-invariant subspaces and V2 ⊆ V1 , the map induced by A in the quotient space V1 /V2 is denoted by A|V1 /V2 . To simplify, we sometimes use B in place of ImB, the image of B and K in place of KerE, the kernel of E. ⊕ denotes direct sum of subspace, ⊎ denotes union of sets with common elements repeated. Let us denote Σ(A,B)x := {x(t) : [0, ∞) → X ; x(t) is a.c.(absolutely continuous), and x(t) ˙ − Ax(t) ∈ ImB a.e.(almost everywhere)}, and similarly Σ(A,[B|D])x := {x(t) : [0, ∞) → X ; x(t) is a.c., and x(t)−Ax(t) ˙ ∈ ImB + ImD a.e.}. If X is a normed vector space, with norm k.k, and L a subspace of X , then for any x ∈ X , its distance to L is denoted as: d(x, L ) := inf y∈L kx − yk. 10.3182/20100915-3-IT-2017.00039
SSSC 2010 Ancona, Italy, Sept 15-17, 2010
For any measurable function, say W : [0, ∞) → X , we say that W ∈ Lp [0, ∞) if kW kLp < +∞, where: Z 1/p ∞ p kW (t)k dt for 1 ≤ p < ∞ kW kLp := 0 for p = ∞ ess sup kW (t)k t≥0
The reachable space of Σ (by the control u) is denoted by hA|Bi := B + AB + A2 B + · · · + An−1 B, where n is the dimension of X . A subspace V ⊂ X is called (A, B) (or controlled)invariant if there exists F : X → U such that (A + BF )V ⊂ V . F is called a friend of V and we denote F (V ) the set of all such F . A subspace R ⊂ X is called an (A, B) controllability subspace if there exist F : X → U , and G : Y → U , with Y ⊂ U , such that: R := hA + BF |Im(BG)i. A subspace Va ⊂ X is called an almost (A, B) (or controlled)-invariant subspace if for any x0 ∈ Va and for any ǫ > 0 there exists a state trajectory xǫ ∈ Σ(A,B)x with the properties that xǫ (0) = x0 and d(xǫ (t), Va ) ≤ ǫ, for any t ≥ 0. A subspace Ra ⊂ X is called an almost (A, B) controllability subspace if for any x0 ∈ Ra , and any x1 ∈ Ra there exists T > 0 such that, for any ǫ > 0 there exists a state trajectory xǫ ∈ Σ(A,B)x with the properties that xǫ (0) = x0 , xǫ (T ) = x1 and d(xǫ (t), Ra ) ≤ ǫ, ∀t ≥ 0. The supremal (A, B) (or controlled)-invariant subspace contained in K is denoted by V ∗ , or by V ∗ (K ). It is the limit of the following non increasing algorithm, see Basile and Marro (1992); Wonham (1985): ( V0=X (2) V i+1 = K ∩ A−1 ImB + V i Similarly, R ∗ , or R ∗ (K ), the supremal (A, B) controllability subspace contained in K , is the limit of the following non decreasing algorithm, see Wonham (1985): ( R0 = 0 (3) R i+1 = V ∗ ∩ AR i + ImB Ra∗ , or Ra∗ (K ), the supremal almost (A, B) controllability subspace contained in K , is the limit of the following non decreasing algorithm, see Willems (1981): ( Ra0 = 0 (4) Rai+1 = K ∩ ARai + ImB Va∗ , or Va∗ (K ), the supremal almost (A, B) controlledinvariant subspace contained in K , satisfies: Va∗ = V ∗ + Ra∗ , see Willems (1981): Let us denote by S ∗ the limit of the following algorithm: ( S0 = 0 S i+1 = ImB + A KerE ∩ S i ∗
S is usually introduced in the context of (K , A) invariance (dual to (A, B) invariance). In our present context, 106
we prefer to handle it through its almost controllability properties, as established in Willems (1981), namely: S ∗ = ARa∗ + ImB and Ra∗ = K ∩ S ∗ . Note that all these notions of exact/almost controlled invariance or controllability properties, can easily be defined, similarly, for the “composite” system (let Bc := [B|D]) , say Σ(A, Bc , 0, E), i.e. with U ⊕ Q in place of U . They ∗ will be noted, respectively, Vc∗ , Rc∗ , Rca , Sc∗ . They are, respectively, the limits of the following algorithms: ( Vc0 = X Vc∗ : Vci+1 = KerE ∩ A−1 ImBc + Vci ( Rc0 = 0 ∗ Rc : Rci+1 = Vc∗ ∩ ARci + ImBc ( 0 Rca =0 ∗ Rca : i+1 i Rca = KerE ∩ ARca + ImBc ( Sc0 = 0 Sc∗ : Sci+1 = ImBc + A KerE ∩ Sci Definition 1. ADDP, the almost disturbance decoupling problem, is said to be solvable if the following holds: ∀ǫ > 0, ∃F : X 7→ U such that in the closed loop system with x(0) = 0, kz(t)kLq ≤ ǫ kq(t)kLp for all Lp measurable disturbance input q(t) and for all 1 ≤ p ≤ q ≤ ∞, see Willems (1981). Equivalently, as shown in Trentelman (1985), ADDP is solvable if: ∀ǫ > 0, there exists a
ǫ→0 sequence {Fǫ } such that Ee(A+BFǫ )t D L −→ 0 for p = 1 p and ∞. It is well known Willems (1981) that ADDP is solvable by static state feedback if and only if: ImD ⊂ Va∗ (5) Some particular system structures play a key role in the solution of control problems, among them are the invariant zeros. The finite invariant zeros of Σ(A, B, 0, E), i.e. from u to z, are equal to the dynamics 1 of the system in the quotient space V ∗ /R ∗ : Z(A, B, E) := σ(A + BF |(V ∗ /R ∗ )), for any F ∈ F (V ∗ ).
3. FIXED POLES OF ADDP When the ADDP is solvable, some fixed poles are present for any state feedback solution. In Zou and Malabre (2009c), the authors have characterized them by means of some invariant zeros but in the particular case of (A, B)controllable systems. As a complement, in this section, we will give the characterization of the fixed poles of ADDP for the general case, namely without any controllability assumption. 1
Which indeed are fixed, after having applied any state feedback, say F , i.e. replacing A by A + BF . They are also invariant after any change of basis in X , U and Z as well as any output injection L : Z → X , i.e. when replacing A by A + LE.
SSSC 2010 Ancona, Italy, Sept 15-17, 2010
Lemma 2. The following properties always hold: fixed poles, σfADDP ixed (A, B, D, E). ∗ ∗ i. Sc ∩ V is an (A, B)-invariant subspace ; - The finite fixed poles of ADDP are characterized ∗ X ii. if ImD ⊆ Va∗ , then ImD ⊆ Sc∗ ∩ V ∗ + Ra∗ ; as σfADDP (A, B, D, E) = σ A + BΠ ixed Sc ∩V ∗ +hA|Bi ⊎ S ∗ ∩V ∗ iii. if ImD ⊆ V ∗ + S ∗ , then ImD ⊆ Sc∗ ∩ V ∗ + S ∗ . , where Π is any map which makes σ A + BΠ cR∗ ∗ ∗ Proof. i. See Zou and Malabre (2009c). Sc ∩ V (A + BΠ) invariant. finite fixed poles of ii. & iii. Before proving these two properties, let’s - When (A, B) is controllable, theADDP ADDP, are characterized as : σ recall the direct conclusion from the Property 4.2.1 and f ixed (A, B, D, E) = S ∗ ∩V ∗ ∗ c Equation (4.2.8) in Basile and Marro (1992): if ImD ⊆ V σ A + BΠ R∗ . (or ImD ⊆ V ∗ + ImB), then ImD ⊆ Sc∗ ∩ V ∗ (or ImD ⊆ Sc∗ ∩ V ∗ + ImB). Now, we can give the first important result of this contriIn Zou and Malabre (2009c), the authors dealt with ˆ E) and Σ(A, B ˆc , E), where two virtual systems Σ(A, B, ∗ ˆ ˆ ˆ ImB := S and Bc := [B|D], and proved that Vˆ∗ = Va∗ , Rˆ∗ = Ra∗ and Sˆc∗ = Sc∗ always hold. So ImD ⊆ Va∗ (or ˆ ImD ⊆ V ∗ +S ∗ ) means ImD ⊆ Vˆ∗ (or ImD ⊆ Vˆ∗ + B), based on the previous conclusion, obviously follows that ˆ namely ImD ⊆ Sˆc∗ ∩ Vˆ∗ (or ImD ⊆ Sˆc∗ ∩ Vˆ∗ + ImB), ImD ⊆ Sc∗ ∩ Va∗ = Sc∗ ∩ (V ∗ + Ra∗ ) = Sc∗ ∩ V ∗ + Ra∗ (or ˆ = S ∗ ∩V ∗ +S ∗ ), which ends ImD ⊆ Sc∗ ∩V ∗ +Ra∗ +ImB c this proof. Lemma 3. If ImD ⊆ Va∗ (or ImD ⊆ V ∗ + S ∗ ), then hA|ImB + ImDi = hA|ImBi + Sc∗ ∩ V ∗ Proof. Because Va∗ ⊆ S ∗ , obviously, here we just need to show that if ImD ⊆ V ∗ + S ∗ then hA|ImB + ImDi = hA|ImBi + Sc∗ ∩ V ∗ . First, hA|ImBi + Sc∗ ∩ V ∗ ⊆ hA|ImB + ImDi is an obvious fact thanks to Sc∗ ⊆ hA|ImB + ImDi and hA|ImBi ⊆ hA|ImB + ImDi. So, we just have to prove that hA|ImB + ImDi ⊆ hA|ImBi + Sc∗ ∩ V ∗ . Assuming ImD ⊆ V ∗ + S ∗ , from Lemma 2, we have AImD ⊆ A(Sc∗ ∩ V ∗ + S ∗ ) = A(Sc∗ ∩ V ∗ ) + AS ∗ ⊆ Sc∗ ∩ V ∗ + ImB + AS ∗ ⊆ Sc∗ ∩ V ∗ + ImB + AhA|ImBi ⊆ Sc∗ ∩ V ∗ + hA|ImBi
1 2 σfADDP ixed (A, B, D, E) =σf ixed ⊎ σf ixed
(6)
σf1 ixed
where gathers the uncontrollable poles of the pair (A, [B|D]), and σf2 ixed is given by the following equation : Z(A, B, E) =σf2 ixed ⊎ Z(A, [B|D], E) Proof. Based on Lemma 3 and Lemma 4, we conclude immediately that X ADDP σf ixed (A, B, D, E) = σ A + BΠ hA|ImB + ImDi ∗ S ∩V∗ ⊎ σ A + BΠ c ∗ R X = σ A + BΠ hA|Im[B|D]i ∗ Sc ∩ V ∗ ⊎ σ A + BΠ R∗ ∗ Sc ∩ V ∗ 1 = σf ixed ⊎ σ A + BΠ R∗ ∗ ∗ where Π is any map which makes Sc ∩ V (A + BΠ) invariant. To complete this proof, we have to prove that σf2 ixed = S ∗ ∩V ∗ . From Lemma 18 in Zou and Malσ A + BΠ cR∗ abre (2009c), it can be easily proved that, when ADDP is solvable, the quotient spaces Vc∗ /Rc∗ and V ∗ /(Sc∗ ∩ ∗ /Ra∗ and V ∗ ) are isomorphic and the quotient spaces Rca ∗ ∗ ∗ (Sc ∩ V )/R are isomorphic. Thanks to the definition of the zeros, we conclude that Z(A, B, E) = finite invariant S ∗ ∩V ∗ σ A + BΠ cR∗ ⊎ Z(A, [B|D], E), namely σf2 ixed = S ∗ ∩V ∗ σ A + BΠ cR∗ , the result obviously follows, which ends the proof.
Then taking i = 2, 3, · · · , we have Ai ImD ⊆ Ai−1 (Sc∗ ∩ V ∗ + hA|ImBi) = Ai−1 (Sc∗ ∩ V ∗ ) + hA|ImBi ··· ⊆ Sc∗ ∩ V ∗ + hA|ImBi So, we get hA|ImB + ImDi = hA|ImBi + ImD + AImD + A2 ImD + · · · + An−1 ImD ⊆ Sc∗ ∩ V ∗ + hA|ImBi which ends the proof.
bution. Theorem 5. Assume that ADDP is solvable, then the finite fixed poles of ADDP, are characterized as :
When ADDP is solvable, the fixed poles of ADDP are just the union of the uncontrollable poles of the pair (A, [B|D]) with the invariant zeros of Σ(A, B, 0, E) which are not invariant zeros of Σ(A, [B|D], 0, E).
Before characterizing the fixed poles of ADDP for the general case, let us recall some results in Zou and Malabre (2009c). Lemma 4. Assume that ADDP is solvable, - Any feedback solution of ADDP contains a set of finite 107
4. STATE FEEDBACK SOLUTION OF ADDP Theorem 5 characterizes the fixed poles of ADDP for the general case, which provides another method that gives the solvability of the problem with stability: indeed, just
SSSC 2010 Ancona, Italy, Sept 15-17, 2010
X * 6 [B, D] Π U ⊕Q HH HH H [B, D] j RBD
A
V
- X 6 HHHE Hj H Π Z * E A RBD
From linear state space theory, it is well known that, for any given system Σ(A, B, D, E), if we keep the trajectories which are controllable or get rid of those which are uncontrollable, we can get a reduced controllable system. Furthermore, there exists a commutative relationship between this reduced system and the original one. For the composite system Σ(A, [B|D], 0, E), thanks to the fact ImD ⊂ RBD , we get the following commutative relationship between the reduced system and the original one, see Fig.1. The terms commutative (which is commonly used in the geometric approach, see e.g. Wonham (1985)) simply means that the following equations are satisfied:
(7)
where Π denotes the insertion map of RBD into X . It is obvious that the maps A, B, D and E are unique since Π is monic. ∗
Theorem 6. Let K := Ker(E), and let us denote by V the supremal (A, B)-invariant subspace contained in K , ∗ Ra the supremal almost (A, B) controllability subspace ∗ contained in K , and Va the supremal almost (A, B)invariant subspace contained in K . Then the following properties always hold: V
∗
= Π−1 V ∗
Ra = Π−1 Ra∗
iii.
Va = Π−1 Va∗
ImB + V
i
= Π−1 V i+1
ii: For the proof, similarly we first show by induction that i Ra = Π−1 Rai , ∀i = 0, 1, · · · , with: Ra 0 = 0 Ra i+1 = K ∩ ARa i + ImB where Rai , ∀i = 0, 1, · · · are the elements of Algorithm.(4). Equality is clearly true for i = 0. Now suppose it is true i for i ≥ 0, i.e. Ra = Π−1 Rai , then i+1 i Ra = K ∩ ARa + ImB = Π−1 Rai+1 i
Since Ra = Π−1 Rai , ∀i = 0, 1, · · · , of course, for their ∗ limits, we also have Ra = Π−1 Ra∗ . ∗
∗
∗
iii: Based on the facts that Va = V + Ra , Va∗ = V ∗ + ∗ Ra∗ , and the conclusions of i and ii, we get Va = Π−1 Va∗ , which ends the proof. Now it is rather easy to conclude that the ADDP is solvable for the initial system Σ(A, B, D, E) if and only if it is solvable for the reduced realisation Σ(A, B, D, E), thanks to Theorem 6 and Equation (7). Moreover, we can also show that there exists a one-to-one correspondence between the solutions of the ADDP of the initial system Σ(A, B, D, E) and of the reduced realization Σ(A, B, D, E). Theorem 7. For the reduced system Σ(A, B, D, E), suppose that ADDP is solved by the state feedback sequence {Fǫ }, then the sequences {Fǫ } with Fǫ Π = Fǫ solve ADDP for the system Σ(A, B, D, E). Proof. First, we should note that ADDP is solved by the state feedback sequence {Fǫ } for the system Σ(A, B, D, E) means that, in a precise sense, the norm of the transfer function matrix E{sI − (A + BFǫ )}−1 D tends to zero as ǫ −→ 0. We can easily prove that, with Fǫ defined by Fǫ Π = Fǫ , there is indeed an equality between the transfer function matrices of both closed-loop systems, namely : E{sI − (A + BFǫ )}−1 D ≡ E{sI − (A + BFǫ )}−1 D This is a direct consequence of (7) and of the fact that :
∗
ii.
i
write that the fixed poles are stable. It also confirms the importance of the composite system Σ(A, [B|D], 0, E). In the following, we will show that this composite system also plays a critical role in the the state feedback solution of ADDP. Moreover, there is a key subspace for the design of the state feedback solution, namely RBD := hA|[B|D]i, the reachability subspace with respect to all the input signals for the system Σ(A, B, D, E).
i.
−1
=K ∩A
Since V = Π−1 V i , ∀i = 0, 1, · · · , of course, for their ∗ limits, we also have V = Π−1 V ∗ .
Fig. 1. Commutative diagrams
[B, D] = Π B, D EΠ = E AΠ = ΠA
i+1
(A + BFǫ )i D = Π(A + BFǫ )i D ,
∗
Proof. i: For the proof, we first show by induction that i V = Π−1 V i , ∀i = 0, 1, · · · , with: V 0 = RBD V i+1 = K ∩ A−1 ImB + V i
This ends the proof.
∀i ≥ 0 .
Now it is possible to give the algorithm for the numerical computation of state feedback maps that solve the ADDP for the general case, namely without any controllability assumption.
where V i , ∀i = 0, 1, · · · are the elements of Algorithm.(2).
Step 1 . Compute the involved subspaces and check whether ADDP is solvable. If not, stop.
Equality is clearly true for i = 0 since RBD = Π−1 X . i Now suppose it is true for i ≥ 0, i.e. V = Π−1 V i , then
Step 2 . Check whether the system is (A, B)-controllable, if yes, follow the procedure provided in Zou and Malabre
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Step 3 . Compute the required fixed poles, and follow the procedures provided in Zou and Malabre (2009b) (for the single control input system) or that in Zou and Malabre (2009a) (for the system with multiple control inputs) to compute the required infinite poles, then follow the procedure provided in Zou and Malabre (2009c) to solve the ADDP for the reduced subsystem Σ(A, B, D, E), and get the corresponding solution sequences {Fǫ }.
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 = hA|Im[B, D]i = Im 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
Step 4 . Obtain the solution sequence {Fǫ } of ADDP for the original system Σ(A, B, D, E), with Fǫ Π = Fǫ .
Let us consider the system Σ(A, B, D, E) with: 1 0 0 A= 0 0 0
0 0 0 0 0
0
0 0 0 0 0
0
1 0 0 0 0
0
0 1 0 0 0
0
0 0 1 0 0
0
0 0 0 1 −3 0 0 0 0 0 0 −1
0 0 0 0 0 0
1
1
0 0 −2 0 0 0
0
0 1 0 0 0 0 1 0 0 B = D = E = 0 0 0 0 1 0 0 1 0
T
0
0 0 0 0 0
Then we obtain immediately the reduced subsystem Σ(A, B, D, E), thanks to Equations (7).
Some subspaces of interest: 1 0 1 0 0 0 0 0 0 0 0
0 0 0 ∗ Ra = Im 0 0
Va∗
0 0 0 0 = Im 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 1 0 0 ∗ V = Im 0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
The insertion map Π can be chosen as: 100 0000 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 Π= 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 000 0000
0 1
0
0
0
RBD
0 0 0 0 0 1 0 0
We can easily compute the uncontrollable poles of the pair (A, [B|D]) σf1 ixed = {−2}. From Z(A, B, E) = {−1, −3, 0} and Z(A, Bc , E) = {0}, we get immediately σf2 ixed = {−1, −3}, namely: to solve this ADDP, in the closed-loop system, we must have a fixed pole set 1 2 σfADDP ixed (A, B, D, E) = σf ixed ⊎ σf ixed = {−1, −2, −3}.
−2
0 −2
Since, ImD 6⊂ V ∗ + ImB, but ImD ⊂ Va∗ , exact DDP is not solvable, while ADDP is solvable by high gain state feedback. It is obvious that the system Σ(A, B, D, E) is neither (A, B)-controllable, nor (A, [B|D])-controllable.
5. ILLUSTRATIVE EXAMPLE
0
hA|ImBi =
1 0 0 0 Im 0 0 0 0
0 1 0 0 0 0 0 0
(2009c) to solve the ADDP; if not, compute the reduced subsystem Σ(A, B, D, E) and the insertion map Π.
0 0 0 0 0 0
0
0 0 0 0 0
0
1 0 0 0 0
0
0 1 0 0 0
0
0 0 1 0 0
0
1 0 A=0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 −3 0
0 0 0 0 0 0 −1
1
0 0 1 0 0 0 0 0
1
0 0
0 1 0 0 0 0 B = 0 D = 0 E = 1 0 0 0 0 1 0 0
109
1
T
0 0 0 0 0
0 0
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For this reduced subsystem, we can also get its subspaces of interest : 1 0 0 0 0 0 Ra
∗
0 0 = Im 0 0
1 0 0 0 0 0 0 0
0
V
∗ a
0 0 = Im 0 1 0 0
0 0 0 0 0 1 0
0 0 0 1 ∗ 0 V = Im 0 1 0 0 0 0 0
0 0 0 0 0 0 1
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 1 0 0 0 0
We can verify that the system Σ(A, B, D, E) is (A, [B|D])controllable, and its ADDP is solvable. Following Zou and Malabre (2009b), we conclude that, at least 2 infinite poles are required to solve ADDP, and our maximal freedom in pole placement while solving this ADDP is dim(RBD ) − n2f ixed − 2 = 3, where n2f ixed denotes the number of fixed poles of ADDP, i.e. the number of elements contained in σf2 ixed . Now let us arbitrarily chose three finite poles, e.g. {−4, −5, −6}, following the procedure provided in Zou and Malabre (2009c), we can get a sequence of state feedback solutions of ADDP for this reduced system, namely: T −15 + 2 p −74 + 30 p − p2 −120 + 148 p − 15 p2 Fp = −74 p2 + 240 p 2 −120 p 0 0 Thanks to Theorem 7, we obtain T −15 + 2 p 2 −74 + 30 p − p −120 + 148 p − 15 p2 2 −74 p + 240 p Fǫ = 2 −120 p 0 0 0 We can verify that the poles of the closed-loop system are {−6, −5, −4, −3, −2, −1, p, p} with u = Fǫ x. With R MAPLE , we can verify that lim kG(t)kL1 = 0 and p→−∞
lim kG(t)kL∞ = 0, where p→−∞ G(t) := L −1 E(sI − (A + BFǫ ))−1 D , namely, the sequence {Fǫ } solves ADDP. See also Fig.2.
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Fig. 2. Disturbance impulse response G(t) 6. CONCLUSION We have characterized the fixed poles of ADDP by state feedback for a general linear time invariant system, possibly non complete controllable using the so-called “Geometric Approach”, and proposed a procedure to solve it. REFERENCES G. Basile and G. Marro. Controlled Invariants and Conditioned Invariants in Linear System Theory. PrenticeHall, Englewood Cliffs, New Jersey, 1992. ISBN 0-13172974-8. Ali Saberi, Anton A. Stoorvogel, and Peddapullaiah Sannuti. Filtering Theory: With Applications to Fault Detection, Isolation, and Estimation. Birkh¨auser Boston, 2006. ISBN 978-0817643010. J.M. Schumacher. Almost stabilizability subspaces and high gain feedback. IEEE Transactions on Automatic Control, AC-29:620–627, 1984. H.L. Trentelman. Almost Invariant Subspaces and High Gain Feedback. PhD thesis, Rijksuniversiteit Groningen, 1985. S. Weiland and J.C. Willems. Almost disturbance decoupling with internal stability. IEEE Transactions on Automatic Control, AC-34:277–285, 1989. J.C. Willems. Almost invariant subspaces: an approach to high gain feedback design,part i:almost controlled invariant subspaces. IEEE Transactions on Automatic Control, AC-26:235–252, 1981. W.M. Wonham. Linear multivariable control: a geometric approach (3rd ed.). Springer-Verlag, New York, 1985. Runmin Zou and Michel Malabre. Almost disturbance decoupling via static state feedback with pole placement: a geometric approach. In European Control Conference 2009, Budapest, August 2009a. Runmin Zou and Michel Malabre. Almost disturbance decoupling: static state feedback solutions with maximal pole assignment. In 2009 American Control Conference, St.Louis, USA, pages 463–468, June 2009b. Runmin Zou and Michel Malabre. Almost disturbance decoupling and pole placement. Automatica, 45:2685– 2691, November 2009c.
Fixed poles of almost disturbance decoupling by state feedback Runmin Zou ∗ , Michel Malabre ∗∗ ∗
School of Information Science and Engineering, Central South University, Changsha 410083, P.R.China. Email: [email protected] ∗∗ IRCCyN, CNRS, UMR 6597, Ecole Centrale de Nantes, 1 rue de la No¨e, 44321 Nantes Cedex 03, France. Phone: (+)33240376912, Fax : (+)33240376930. Email: [email protected] Abstract: When the almost disturbance decoupling problem (ADDP) is solvable, some poles of the closed loop system are fixed for any state feedback solution. In Zou and Malabre (2009c), the authors have characterized them by means of some invariant zeros but in the particular case of (A, B)-controllable systems. As a complement, here, we give the characterization of the fixed poles of ADDP for the general case, namely without any controllability assumption. We also propose a method to solve the ADDP for this general case, as well as a numerical example to illustrate our contributions. Keywords: Linear control systems; geometric approach; fixed pole; almost disturbance decoupling. 1. INTRODUCTION The Almost Disturbance Decoupling Problem by static state feedback (ADDP) aims at designing a suitable state feedback such that the system controlled output is almost (in a precise sense) decoupled from the disturbance input while guaranteeing some performances (at least the internal stability of the resulting closed-loop feedback system and, possibly, pole placement in some desired region). It is well accepted that this problem occupies a central part in classical as well as modern control theory; several important problems, such as robust, decentralized, noninteracting, model reference or tracking control etc., are linked to (almost) disturbance decoupling, see for instance Saberi et al. (2006). Since ADDP was first introduced in Willems (1981), it has been intensively studied in the 1980’s Willems (1981); Schumacher (1984); Trentelman (1985); Weiland and Willems (1989), but all the contributions without exception, focused on the solvability of the problem with stability. In Zou and Malabre (2009c), the authors have proposed a general procedure to solve ADDP, by introducing the key concept of almost self-bounded controlled invariant subspace and characterized the so-called fixed poles of ADDP, but just in the particular case of (A, B)controllable system. The present contribution aims at giving an answer to some remaining issues. To get our results, we use the so-called “Geometric Approach”. We prove that the (A, [B, D])-controllable subsystem is the key reduced sub-system, and on the basis of a commutative diagram between the original system and the reduced sub-system, we also show that there exists a one to one correspondence between the solutions of the ADDP of these two systems, which enables us to get the pole placement abilities of the solutions of the ADDP 978-3-902661-83-8/10/$20.00 © 2010 IFAC
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for the general case, on the basis of the similar solution for an (A, B)-controllable system as described in Zou and Malabre (2009c). 2. NOTATION AND GEOMETRIC PRELIMINARIES We shall consider linear time-invariant disturbed systems Σ(A, B, D, E) described by: x(t) ˙ = Ax(t) + Bu(t) + Dq(t) Σ: (1) z(t) = Ex(t) where x, u, q, and z are respectively the state, control input, disturbance input, and output to be controlled. These signals belong to the vector spaces X , U , Q, and Z , respectively. Vectors are denoted by lower case letters, matrices/maps by capitals and subspaces by script capitals. If A is a square matrix, then σ(A) denotes its spectrum. If A : X 7→ Y and V ⊆ X , the restriction of the map A to V is denoted by A|V . If V1 and V2 are A-invariant subspaces and V2 ⊆ V1 , the map induced by A in the quotient space V1 /V2 is denoted by A|V1 /V2 . To simplify, we sometimes use B in place of ImB, the image of B and K in place of KerE, the kernel of E. ⊕ denotes direct sum of subspace, ⊎ denotes union of sets with common elements repeated. Let us denote Σ(A,B)x := {x(t) : [0, ∞) → X ; x(t) is a.c.(absolutely continuous), and x(t) ˙ − Ax(t) ∈ ImB a.e.(almost everywhere)}, and similarly Σ(A,[B|D])x := {x(t) : [0, ∞) → X ; x(t) is a.c., and x(t)−Ax(t) ˙ ∈ ImB + ImD a.e.}. If X is a normed vector space, with norm k.k, and L a subspace of X , then for any x ∈ X , its distance to L is denoted as: d(x, L ) := inf y∈L kx − yk. 10.3182/20100915-3-IT-2017.00039
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For any measurable function, say W : [0, ∞) → X , we say that W ∈ Lp [0, ∞) if kW kLp < +∞, where: Z 1/p ∞ p kW (t)k dt for 1 ≤ p < ∞ kW kLp := 0 for p = ∞ ess sup kW (t)k t≥0
The reachable space of Σ (by the control u) is denoted by hA|Bi := B + AB + A2 B + · · · + An−1 B, where n is the dimension of X . A subspace V ⊂ X is called (A, B) (or controlled)invariant if there exists F : X → U such that (A + BF )V ⊂ V . F is called a friend of V and we denote F (V ) the set of all such F . A subspace R ⊂ X is called an (A, B) controllability subspace if there exist F : X → U , and G : Y → U , with Y ⊂ U , such that: R := hA + BF |Im(BG)i. A subspace Va ⊂ X is called an almost (A, B) (or controlled)-invariant subspace if for any x0 ∈ Va and for any ǫ > 0 there exists a state trajectory xǫ ∈ Σ(A,B)x with the properties that xǫ (0) = x0 and d(xǫ (t), Va ) ≤ ǫ, for any t ≥ 0. A subspace Ra ⊂ X is called an almost (A, B) controllability subspace if for any x0 ∈ Ra , and any x1 ∈ Ra there exists T > 0 such that, for any ǫ > 0 there exists a state trajectory xǫ ∈ Σ(A,B)x with the properties that xǫ (0) = x0 , xǫ (T ) = x1 and d(xǫ (t), Ra ) ≤ ǫ, ∀t ≥ 0. The supremal (A, B) (or controlled)-invariant subspace contained in K is denoted by V ∗ , or by V ∗ (K ). It is the limit of the following non increasing algorithm, see Basile and Marro (1992); Wonham (1985): ( V0=X (2) V i+1 = K ∩ A−1 ImB + V i Similarly, R ∗ , or R ∗ (K ), the supremal (A, B) controllability subspace contained in K , is the limit of the following non decreasing algorithm, see Wonham (1985): ( R0 = 0 (3) R i+1 = V ∗ ∩ AR i + ImB Ra∗ , or Ra∗ (K ), the supremal almost (A, B) controllability subspace contained in K , is the limit of the following non decreasing algorithm, see Willems (1981): ( Ra0 = 0 (4) Rai+1 = K ∩ ARai + ImB Va∗ , or Va∗ (K ), the supremal almost (A, B) controlledinvariant subspace contained in K , satisfies: Va∗ = V ∗ + Ra∗ , see Willems (1981): Let us denote by S ∗ the limit of the following algorithm: ( S0 = 0 S i+1 = ImB + A KerE ∩ S i ∗
S is usually introduced in the context of (K , A) invariance (dual to (A, B) invariance). In our present context, 106
we prefer to handle it through its almost controllability properties, as established in Willems (1981), namely: S ∗ = ARa∗ + ImB and Ra∗ = K ∩ S ∗ . Note that all these notions of exact/almost controlled invariance or controllability properties, can easily be defined, similarly, for the “composite” system (let Bc := [B|D]) , say Σ(A, Bc , 0, E), i.e. with U ⊕ Q in place of U . They ∗ will be noted, respectively, Vc∗ , Rc∗ , Rca , Sc∗ . They are, respectively, the limits of the following algorithms: ( Vc0 = X Vc∗ : Vci+1 = KerE ∩ A−1 ImBc + Vci ( Rc0 = 0 ∗ Rc : Rci+1 = Vc∗ ∩ ARci + ImBc ( 0 Rca =0 ∗ Rca : i+1 i Rca = KerE ∩ ARca + ImBc ( Sc0 = 0 Sc∗ : Sci+1 = ImBc + A KerE ∩ Sci Definition 1. ADDP, the almost disturbance decoupling problem, is said to be solvable if the following holds: ∀ǫ > 0, ∃F : X 7→ U such that in the closed loop system with x(0) = 0, kz(t)kLq ≤ ǫ kq(t)kLp for all Lp measurable disturbance input q(t) and for all 1 ≤ p ≤ q ≤ ∞, see Willems (1981). Equivalently, as shown in Trentelman (1985), ADDP is solvable if: ∀ǫ > 0, there exists a
ǫ→0 sequence {Fǫ } such that Ee(A+BFǫ )t D L −→ 0 for p = 1 p and ∞. It is well known Willems (1981) that ADDP is solvable by static state feedback if and only if: ImD ⊂ Va∗ (5) Some particular system structures play a key role in the solution of control problems, among them are the invariant zeros. The finite invariant zeros of Σ(A, B, 0, E), i.e. from u to z, are equal to the dynamics 1 of the system in the quotient space V ∗ /R ∗ : Z(A, B, E) := σ(A + BF |(V ∗ /R ∗ )), for any F ∈ F (V ∗ ).
3. FIXED POLES OF ADDP When the ADDP is solvable, some fixed poles are present for any state feedback solution. In Zou and Malabre (2009c), the authors have characterized them by means of some invariant zeros but in the particular case of (A, B)controllable systems. As a complement, in this section, we will give the characterization of the fixed poles of ADDP for the general case, namely without any controllability assumption. 1
Which indeed are fixed, after having applied any state feedback, say F , i.e. replacing A by A + BF . They are also invariant after any change of basis in X , U and Z as well as any output injection L : Z → X , i.e. when replacing A by A + LE.
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Lemma 2. The following properties always hold: fixed poles, σfADDP ixed (A, B, D, E). ∗ ∗ i. Sc ∩ V is an (A, B)-invariant subspace ; - The finite fixed poles of ADDP are characterized ∗ X ii. if ImD ⊆ Va∗ , then ImD ⊆ Sc∗ ∩ V ∗ + Ra∗ ; as σfADDP (A, B, D, E) = σ A + BΠ ixed Sc ∩V ∗ +hA|Bi ⊎ S ∗ ∩V ∗ iii. if ImD ⊆ V ∗ + S ∗ , then ImD ⊆ Sc∗ ∩ V ∗ + S ∗ . , where Π is any map which makes σ A + BΠ cR∗ ∗ ∗ Proof. i. See Zou and Malabre (2009c). Sc ∩ V (A + BΠ) invariant. finite fixed poles of ii. & iii. Before proving these two properties, let’s - When (A, B) is controllable, theADDP ADDP, are characterized as : σ recall the direct conclusion from the Property 4.2.1 and f ixed (A, B, D, E) = S ∗ ∩V ∗ ∗ c Equation (4.2.8) in Basile and Marro (1992): if ImD ⊆ V σ A + BΠ R∗ . (or ImD ⊆ V ∗ + ImB), then ImD ⊆ Sc∗ ∩ V ∗ (or ImD ⊆ Sc∗ ∩ V ∗ + ImB). Now, we can give the first important result of this contriIn Zou and Malabre (2009c), the authors dealt with ˆ E) and Σ(A, B ˆc , E), where two virtual systems Σ(A, B, ∗ ˆ ˆ ˆ ImB := S and Bc := [B|D], and proved that Vˆ∗ = Va∗ , Rˆ∗ = Ra∗ and Sˆc∗ = Sc∗ always hold. So ImD ⊆ Va∗ (or ˆ ImD ⊆ V ∗ +S ∗ ) means ImD ⊆ Vˆ∗ (or ImD ⊆ Vˆ∗ + B), based on the previous conclusion, obviously follows that ˆ namely ImD ⊆ Sˆc∗ ∩ Vˆ∗ (or ImD ⊆ Sˆc∗ ∩ Vˆ∗ + ImB), ImD ⊆ Sc∗ ∩ Va∗ = Sc∗ ∩ (V ∗ + Ra∗ ) = Sc∗ ∩ V ∗ + Ra∗ (or ˆ = S ∗ ∩V ∗ +S ∗ ), which ends ImD ⊆ Sc∗ ∩V ∗ +Ra∗ +ImB c this proof. Lemma 3. If ImD ⊆ Va∗ (or ImD ⊆ V ∗ + S ∗ ), then hA|ImB + ImDi = hA|ImBi + Sc∗ ∩ V ∗ Proof. Because Va∗ ⊆ S ∗ , obviously, here we just need to show that if ImD ⊆ V ∗ + S ∗ then hA|ImB + ImDi = hA|ImBi + Sc∗ ∩ V ∗ . First, hA|ImBi + Sc∗ ∩ V ∗ ⊆ hA|ImB + ImDi is an obvious fact thanks to Sc∗ ⊆ hA|ImB + ImDi and hA|ImBi ⊆ hA|ImB + ImDi. So, we just have to prove that hA|ImB + ImDi ⊆ hA|ImBi + Sc∗ ∩ V ∗ . Assuming ImD ⊆ V ∗ + S ∗ , from Lemma 2, we have AImD ⊆ A(Sc∗ ∩ V ∗ + S ∗ ) = A(Sc∗ ∩ V ∗ ) + AS ∗ ⊆ Sc∗ ∩ V ∗ + ImB + AS ∗ ⊆ Sc∗ ∩ V ∗ + ImB + AhA|ImBi ⊆ Sc∗ ∩ V ∗ + hA|ImBi
1 2 σfADDP ixed (A, B, D, E) =σf ixed ⊎ σf ixed
(6)
σf1 ixed
where gathers the uncontrollable poles of the pair (A, [B|D]), and σf2 ixed is given by the following equation : Z(A, B, E) =σf2 ixed ⊎ Z(A, [B|D], E) Proof. Based on Lemma 3 and Lemma 4, we conclude immediately that X ADDP σf ixed (A, B, D, E) = σ A + BΠ hA|ImB + ImDi ∗ S ∩V∗ ⊎ σ A + BΠ c ∗ R X = σ A + BΠ hA|Im[B|D]i ∗ Sc ∩ V ∗ ⊎ σ A + BΠ R∗ ∗ Sc ∩ V ∗ 1 = σf ixed ⊎ σ A + BΠ R∗ ∗ ∗ where Π is any map which makes Sc ∩ V (A + BΠ) invariant. To complete this proof, we have to prove that σf2 ixed = S ∗ ∩V ∗ . From Lemma 18 in Zou and Malσ A + BΠ cR∗ abre (2009c), it can be easily proved that, when ADDP is solvable, the quotient spaces Vc∗ /Rc∗ and V ∗ /(Sc∗ ∩ ∗ /Ra∗ and V ∗ ) are isomorphic and the quotient spaces Rca ∗ ∗ ∗ (Sc ∩ V )/R are isomorphic. Thanks to the definition of the zeros, we conclude that Z(A, B, E) = finite invariant S ∗ ∩V ∗ σ A + BΠ cR∗ ⊎ Z(A, [B|D], E), namely σf2 ixed = S ∗ ∩V ∗ σ A + BΠ cR∗ , the result obviously follows, which ends the proof.
Then taking i = 2, 3, · · · , we have Ai ImD ⊆ Ai−1 (Sc∗ ∩ V ∗ + hA|ImBi) = Ai−1 (Sc∗ ∩ V ∗ ) + hA|ImBi ··· ⊆ Sc∗ ∩ V ∗ + hA|ImBi So, we get hA|ImB + ImDi = hA|ImBi + ImD + AImD + A2 ImD + · · · + An−1 ImD ⊆ Sc∗ ∩ V ∗ + hA|ImBi which ends the proof.
bution. Theorem 5. Assume that ADDP is solvable, then the finite fixed poles of ADDP, are characterized as :
When ADDP is solvable, the fixed poles of ADDP are just the union of the uncontrollable poles of the pair (A, [B|D]) with the invariant zeros of Σ(A, B, 0, E) which are not invariant zeros of Σ(A, [B|D], 0, E).
Before characterizing the fixed poles of ADDP for the general case, let us recall some results in Zou and Malabre (2009c). Lemma 4. Assume that ADDP is solvable, - Any feedback solution of ADDP contains a set of finite 107
4. STATE FEEDBACK SOLUTION OF ADDP Theorem 5 characterizes the fixed poles of ADDP for the general case, which provides another method that gives the solvability of the problem with stability: indeed, just
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X * 6 [B, D] Π U ⊕Q HH HH H [B, D] j RBD
A
V
- X 6 HHHE Hj H Π Z * E A RBD
From linear state space theory, it is well known that, for any given system Σ(A, B, D, E), if we keep the trajectories which are controllable or get rid of those which are uncontrollable, we can get a reduced controllable system. Furthermore, there exists a commutative relationship between this reduced system and the original one. For the composite system Σ(A, [B|D], 0, E), thanks to the fact ImD ⊂ RBD , we get the following commutative relationship between the reduced system and the original one, see Fig.1. The terms commutative (which is commonly used in the geometric approach, see e.g. Wonham (1985)) simply means that the following equations are satisfied:
(7)
where Π denotes the insertion map of RBD into X . It is obvious that the maps A, B, D and E are unique since Π is monic. ∗
Theorem 6. Let K := Ker(E), and let us denote by V the supremal (A, B)-invariant subspace contained in K , ∗ Ra the supremal almost (A, B) controllability subspace ∗ contained in K , and Va the supremal almost (A, B)invariant subspace contained in K . Then the following properties always hold: V
∗
= Π−1 V ∗
Ra = Π−1 Ra∗
iii.
Va = Π−1 Va∗
ImB + V
i
= Π−1 V i+1
ii: For the proof, similarly we first show by induction that i Ra = Π−1 Rai , ∀i = 0, 1, · · · , with: Ra 0 = 0 Ra i+1 = K ∩ ARa i + ImB where Rai , ∀i = 0, 1, · · · are the elements of Algorithm.(4). Equality is clearly true for i = 0. Now suppose it is true i for i ≥ 0, i.e. Ra = Π−1 Rai , then i+1 i Ra = K ∩ ARa + ImB = Π−1 Rai+1 i
Since Ra = Π−1 Rai , ∀i = 0, 1, · · · , of course, for their ∗ limits, we also have Ra = Π−1 Ra∗ . ∗
∗
∗
iii: Based on the facts that Va = V + Ra , Va∗ = V ∗ + ∗ Ra∗ , and the conclusions of i and ii, we get Va = Π−1 Va∗ , which ends the proof. Now it is rather easy to conclude that the ADDP is solvable for the initial system Σ(A, B, D, E) if and only if it is solvable for the reduced realisation Σ(A, B, D, E), thanks to Theorem 6 and Equation (7). Moreover, we can also show that there exists a one-to-one correspondence between the solutions of the ADDP of the initial system Σ(A, B, D, E) and of the reduced realization Σ(A, B, D, E). Theorem 7. For the reduced system Σ(A, B, D, E), suppose that ADDP is solved by the state feedback sequence {Fǫ }, then the sequences {Fǫ } with Fǫ Π = Fǫ solve ADDP for the system Σ(A, B, D, E). Proof. First, we should note that ADDP is solved by the state feedback sequence {Fǫ } for the system Σ(A, B, D, E) means that, in a precise sense, the norm of the transfer function matrix E{sI − (A + BFǫ )}−1 D tends to zero as ǫ −→ 0. We can easily prove that, with Fǫ defined by Fǫ Π = Fǫ , there is indeed an equality between the transfer function matrices of both closed-loop systems, namely : E{sI − (A + BFǫ )}−1 D ≡ E{sI − (A + BFǫ )}−1 D This is a direct consequence of (7) and of the fact that :
∗
ii.
i
write that the fixed poles are stable. It also confirms the importance of the composite system Σ(A, [B|D], 0, E). In the following, we will show that this composite system also plays a critical role in the the state feedback solution of ADDP. Moreover, there is a key subspace for the design of the state feedback solution, namely RBD := hA|[B|D]i, the reachability subspace with respect to all the input signals for the system Σ(A, B, D, E).
i.
−1
=K ∩A
Since V = Π−1 V i , ∀i = 0, 1, · · · , of course, for their ∗ limits, we also have V = Π−1 V ∗ .
Fig. 1. Commutative diagrams
[B, D] = Π B, D EΠ = E AΠ = ΠA
i+1
(A + BFǫ )i D = Π(A + BFǫ )i D ,
∗
Proof. i: For the proof, we first show by induction that i V = Π−1 V i , ∀i = 0, 1, · · · , with: V 0 = RBD V i+1 = K ∩ A−1 ImB + V i
This ends the proof.
∀i ≥ 0 .
Now it is possible to give the algorithm for the numerical computation of state feedback maps that solve the ADDP for the general case, namely without any controllability assumption.
where V i , ∀i = 0, 1, · · · are the elements of Algorithm.(2).
Step 1 . Compute the involved subspaces and check whether ADDP is solvable. If not, stop.
Equality is clearly true for i = 0 since RBD = Π−1 X . i Now suppose it is true for i ≥ 0, i.e. V = Π−1 V i , then
Step 2 . Check whether the system is (A, B)-controllable, if yes, follow the procedure provided in Zou and Malabre
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Step 3 . Compute the required fixed poles, and follow the procedures provided in Zou and Malabre (2009b) (for the single control input system) or that in Zou and Malabre (2009a) (for the system with multiple control inputs) to compute the required infinite poles, then follow the procedure provided in Zou and Malabre (2009c) to solve the ADDP for the reduced subsystem Σ(A, B, D, E), and get the corresponding solution sequences {Fǫ }.
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 = hA|Im[B, D]i = Im 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
Step 4 . Obtain the solution sequence {Fǫ } of ADDP for the original system Σ(A, B, D, E), with Fǫ Π = Fǫ .
Let us consider the system Σ(A, B, D, E) with: 1 0 0 A= 0 0 0
0 0 0 0 0
0
0 0 0 0 0
0
1 0 0 0 0
0
0 1 0 0 0
0
0 0 1 0 0
0
0 0 0 1 −3 0 0 0 0 0 0 −1
0 0 0 0 0 0
1
1
0 0 −2 0 0 0
0
0 1 0 0 0 0 1 0 0 B = D = E = 0 0 0 0 1 0 0 1 0
T
0
0 0 0 0 0
Then we obtain immediately the reduced subsystem Σ(A, B, D, E), thanks to Equations (7).
Some subspaces of interest: 1 0 1 0 0 0 0 0 0 0 0
0 0 0 ∗ Ra = Im 0 0
Va∗
0 0 0 0 = Im 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 1 0 0 ∗ V = Im 0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
The insertion map Π can be chosen as: 100 0000 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 Π= 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 000 0000
0 1
0
0
0
RBD
0 0 0 0 0 1 0 0
We can easily compute the uncontrollable poles of the pair (A, [B|D]) σf1 ixed = {−2}. From Z(A, B, E) = {−1, −3, 0} and Z(A, Bc , E) = {0}, we get immediately σf2 ixed = {−1, −3}, namely: to solve this ADDP, in the closed-loop system, we must have a fixed pole set 1 2 σfADDP ixed (A, B, D, E) = σf ixed ⊎ σf ixed = {−1, −2, −3}.
−2
0 −2
Since, ImD 6⊂ V ∗ + ImB, but ImD ⊂ Va∗ , exact DDP is not solvable, while ADDP is solvable by high gain state feedback. It is obvious that the system Σ(A, B, D, E) is neither (A, B)-controllable, nor (A, [B|D])-controllable.
5. ILLUSTRATIVE EXAMPLE
0
hA|ImBi =
1 0 0 0 Im 0 0 0 0
0 1 0 0 0 0 0 0
(2009c) to solve the ADDP; if not, compute the reduced subsystem Σ(A, B, D, E) and the insertion map Π.
0 0 0 0 0 0
0
0 0 0 0 0
0
1 0 0 0 0
0
0 1 0 0 0
0
0 0 1 0 0
0
1 0 A=0 0 0
0 0 0 0 0 0 1 0
0 0 0 1 −3 0
0 0 0 0 0 0 −1
1
0 0 1 0 0 0 0 0
1
0 0
0 1 0 0 0 0 B = 0 D = 0 E = 1 0 0 0 0 1 0 0
109
1
T
0 0 0 0 0
0 0
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For this reduced subsystem, we can also get its subspaces of interest : 1 0 0 0 0 0 Ra
∗
0 0 = Im 0 0
1 0 0 0 0 0 0 0
0
V
∗ a
0 0 = Im 0 1 0 0
0 0 0 0 0 1 0
0 0 0 1 ∗ 0 V = Im 0 1 0 0 0 0 0
0 0 0 0 0 0 1
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 1 0 0 0 0
We can verify that the system Σ(A, B, D, E) is (A, [B|D])controllable, and its ADDP is solvable. Following Zou and Malabre (2009b), we conclude that, at least 2 infinite poles are required to solve ADDP, and our maximal freedom in pole placement while solving this ADDP is dim(RBD ) − n2f ixed − 2 = 3, where n2f ixed denotes the number of fixed poles of ADDP, i.e. the number of elements contained in σf2 ixed . Now let us arbitrarily chose three finite poles, e.g. {−4, −5, −6}, following the procedure provided in Zou and Malabre (2009c), we can get a sequence of state feedback solutions of ADDP for this reduced system, namely: T −15 + 2 p −74 + 30 p − p2 −120 + 148 p − 15 p2 Fp = −74 p2 + 240 p 2 −120 p 0 0 Thanks to Theorem 7, we obtain T −15 + 2 p 2 −74 + 30 p − p −120 + 148 p − 15 p2 2 −74 p + 240 p Fǫ = 2 −120 p 0 0 0 We can verify that the poles of the closed-loop system are {−6, −5, −4, −3, −2, −1, p, p} with u = Fǫ x. With R MAPLE , we can verify that lim kG(t)kL1 = 0 and p→−∞
lim kG(t)kL∞ = 0, where p→−∞ G(t) := L −1 E(sI − (A + BFǫ ))−1 D , namely, the sequence {Fǫ } solves ADDP. See also Fig.2.
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Fig. 2. Disturbance impulse response G(t) 6. CONCLUSION We have characterized the fixed poles of ADDP by state feedback for a general linear time invariant system, possibly non complete controllable using the so-called “Geometric Approach”, and proposed a procedure to solve it. REFERENCES G. Basile and G. Marro. Controlled Invariants and Conditioned Invariants in Linear System Theory. PrenticeHall, Englewood Cliffs, New Jersey, 1992. ISBN 0-13172974-8. Ali Saberi, Anton A. Stoorvogel, and Peddapullaiah Sannuti. Filtering Theory: With Applications to Fault Detection, Isolation, and Estimation. Birkh¨auser Boston, 2006. ISBN 978-0817643010. J.M. Schumacher. Almost stabilizability subspaces and high gain feedback. IEEE Transactions on Automatic Control, AC-29:620–627, 1984. H.L. Trentelman. Almost Invariant Subspaces and High Gain Feedback. PhD thesis, Rijksuniversiteit Groningen, 1985. S. Weiland and J.C. Willems. Almost disturbance decoupling with internal stability. IEEE Transactions on Automatic Control, AC-34:277–285, 1989. J.C. Willems. Almost invariant subspaces: an approach to high gain feedback design,part i:almost controlled invariant subspaces. IEEE Transactions on Automatic Control, AC-26:235–252, 1981. W.M. Wonham. Linear multivariable control: a geometric approach (3rd ed.). Springer-Verlag, New York, 1985. Runmin Zou and Michel Malabre. Almost disturbance decoupling via static state feedback with pole placement: a geometric approach. In European Control Conference 2009, Budapest, August 2009a. Runmin Zou and Michel Malabre. Almost disturbance decoupling: static state feedback solutions with maximal pole assignment. In 2009 American Control Conference, St.Louis, USA, pages 463–468, June 2009b. Runmin Zou and Michel Malabre. Almost disturbance decoupling and pole placement. Automatica, 45:2685– 2691, November 2009c.