Proof of Lemma 6

First consider the case of (2.17) where the proof is relatively easy. If a symmetric matrix is positive definite, than all the block diagonal terms are positive definite. This immediately results in the inequality of D^TD ≻ β²I corresponding to the lower right diagonal block of (2.17). This can only be satisfied if D^TD has full rank, i.e., D has full column rank, and n_y ≥n_u. Consequently, if n_u > n_y then the dual representation has to be used for computing theH− index of the system.

Next, the case of (2.18) is investigated. The application of Lemma 5 resolves necessity of the direct feedthrough, and the H− index can be calculated over a finite frequency range when D = 0. However, the constraint to the relation of n_y ≥n_u still holds. The proof relies on the fact (discussed in [19] Section 6.3.2) that the norm of a system and its dual is the same, but the dual representation possess a different Lyapunov function.

The matrix multiplications in (2.18) can be expanded as M =

M₁₁ M₁₂ M₁₂^T M₂₂

≺0, with M₁₁=−A^T_cQ_cA_c+A^T_cP_c+jA^T_cω˜

2Q_c+P_cA_c−jω˜

2Q_cA_c−¯ωωQ¯ _c−C_c^TC_c, M12=−A^T_cQcBc+PcBc−jω˜

2QcBc−C^TD, M22=−B_c^TQcBc−D^TD+β²I,

(B.9)

where M ∈C^{(nx+nu)×(nx+nu)} and M is partitioned as

n_x×n_x n_x×n_u n_u×n_x n_u×n_u

.

Also carry out the multiplications in (2.18) for the dual representation of the system given

by (2.3) to obtain M˜ =

M˜₁₁ M˜₁₂ M˜₁₂^T M˜22

≺0, with M˜₁₁=−A_cQ_cA^T_c +A_cP_c+jA_cω¯

2Q_c+P_cA^T_c −jω¯

2Q_cA^T_c −¯ωωQ¯ _c−B_cB^T_c, M˜12=−AcQcC_c^T +PcC_c^T −jω¯

2QcC_c^T −BD^T, M˜₂₂=−C_cQ_cC_c^T −DD^T +β²I,

(B.10)

where ˜M ∈C^{(nx+ny)×(nx+ny)}, and ˜M is partitioned as

n_x×n_x n_x×n_y n_y×n_x n_y×n_y

.

Take the case when D = 0. Note again the fact that if a symmetric matrix is positive definite, than its diagonal terms are also positive definite. From (B.9) and (B.10) we have M₂₂ = −B_c^TQ_cB_c+β²I ≺ 0 and ˜M₂₂ = −C_cQ_cC_c^T +β²I ≺ 0. Since Q_c ∈ C^(nx)×(nx), these inequalities can only be satisfied ifn_u ≤n_x and n_y ≤n_x respectively.

Next we continue with the more general case, when D ̸= 0. Suppose that ny > nu. The proof is based on the following two rank identities

rank(X+Y)≤rank(X) + rank(Y), (B.11) rank(XY)≤min(rank(X),rank(Y)). (B.12) The input-output norm equality implies that the term β² is the same for both (B.9) and (B.10), i.e., M₂₂ and ˜M₂₂ have to satisfy

β²I ≺B_c^TQcBc+D^TD, (B.13)

β²I ≺C_cQ_cC_c^T +DD^T. (B.14)

The left hand side of (B.13) and (B.14) are full rank, with ranks n_u and n_y respectively. Ac-cording to (B.12) and the fact that Q_c∈C^nx×nx:

rank(B^T_cQcBc)≤nx, rank(CcQcC_c^T)≤nx. (B.15) At the same time, it was assumed that n_y ≥n_u, which implies:

rank(D^TD) = rank(DD^T)≤n_u. (B.16)

So according to (B.11) maximal rank of the right hand sides of (B.13) and (B.14) is n_x+n_u. Furthermore, the right hand sides of (B.13) and (B.14) has to be full rank, in order to satisfy the inequalities. However, since they have the same upper bound, it means that only the smaller dimensional can hold and the other one on the right hand side will have zero eigenvalues. If n_y ≥n_u than then_u dimensional equation (B.13) is solvable, and so LMI (B.9) provides theH−

index.

It can be easily seen that if ny < nu, then based on the same reasoning (B.14) and the (B.10) LMI corresponding to the dual representation become solvable, where the latter is tall. If n_y =n_u than (B.9) and (B.10) yields the sameH− index. Therefore by the given formulations, theH− index can only be calculated for tall or square systems.

Uncertain system modeling

C.1 On the selection of static IQC multipliers

This section discusses the selection of a Π =

Q S S^T R

multiplier for describing various sources of uncertainties in the plant model. Various types of uncertainties may be described by various forms of Π. The following provides some examples for the selection of Π relying on Section 6.4.3 of [109]. The relevant set of multipliers may be defined by

Π:=

(

Π∈R^{(nw+nv)×(nw+nv)} : Π = Π^T, ∆

I T

Π ∆

I

>0 for all ∆∈∆ )

, (C.1) with Π multiplier partitioned comformably to

∆ I

. In the thesis structured uncertainty is considered, and so ∆ has a blockdiagonal structure, with n_w =n_v.

ˆ Structured nonlinear uncertainties [109]. This type of uncertainty has the form of

∆







v1

...

v_m







 =



∆1(v1) ...

∆_m(v_m)



 with causal ∆_j that satisfy ||∆_j||2 ≤ 1. The corresponding class of multiplier is given as

Π:=

Q 0

0 R

, Q= diag(−r₁I, ...−r_mI), R= diag(r₁I, ..., r_mI)>0

. (C.2)

ˆ Repeated structured uncertainty [109]. It is described by an uncertainty block

∆(t) = diag(δ1(t)I, ..., δmI(t)I), with |δj(t)| ≤ 1 for t ≥ 0. Here the blocks on the di-agonal of ∆(t) are repeated scalar valued functions. The corresponding class of multiplier is defined as

Π:=

Q S S^T R

, R= diag(R₁, ..., R_m)>0, Q=−R, S = diag(S₁, ..., S_m), S_j+S_j^T = 0

. (C.3)

ˆ Polytopic uncertainty [109]. Let us assume that the uncertainty is defined as w(t) =

∆(t)v(t), where ∆ satisfies

∆(t)∈co

∆₁, ...,∆_np for all t≥0. (C.4) Here ∆_j are fix matrices at the vertex points which generate the convex hull that defines the set of values which can be taken by the time-varying uncertainties. At the selection of Π one needs to assure that

∆_j I

T

Π ∆_j

I

⪰0 for all j= 1, ...n_p. (C.5)

A corresponding class of multiplier is given as Π:=

( Π =

Q S S^T R

: Q≺0, ∆_j

I T

Π ∆_j

I

≻0, forj= 1, ..., n_p )

. (C.6)

C.2 Dual integral quadratic constraints

System duality plays a key role in the proposed decoupling algorithm, therefore the dual IQCs are introduced briefly, based on the discussion in [130].

Definition 20 .: The dual IQC multiplier [130]. Given the strict PN primal IQC multiplier Π = Π^∼ ∈ RL⁽ⁿ∞^{v+nw)×(nv+nw)}, the dual IQC multiplier is denoted by DDD(Π) ∈ RL⁽ⁿ∞^{w+nv)×(nw+nv)} and is defined as

DD D(Π) :=

0 −Inw

I_nv 0

Π^−T

0 −Inv

I_nw 0

. (C.7)

Here, Π^−T is the transpose of the inverse of Π. The definition assumes Π to be a strict PN multiplier with Π₁₁ ≻0 and Π₂₂≺0 ∀ω∈R∪ {∞}, therefore Π⁻¹ and DDD(Π) exist. Definition 20 can also be extended for the case when Π is given by its stable (Ψ, M) factorization. Then by Definition 20, the dual IQC multiplier can be expressed as

DDD(Π) =

0 −I_nw I_nv 0

Ψ^−T∼M⁻¹Ψ^−T

0 −I_nv I_nw 0

, (C.8)

from which it follows that D

DD(Π) =DDD(Ψ)^∼MDDD(Ψ), whereDDD(Ψ) =

0 −I_nw I_nv 0

Ψ^−T

0 −I_nv I_nw 0

. (C.9)

Finally, the connections between the primal and the dual representations are discussed in Lemma 19 and are illustrated in Figure C.1 [130].

M

∆

Ψ

M^T

∆D

DD

⇐⇒ D(Ψ) (Lemma 19)

y u

v w

z

¯

y u¯

¯

v w¯

¯ z

Figure C.1: Relationship between the nominal and dual IQC representations

Lemma 19.: The dual uncertain system [130]. GivenMand Π, the following state-ments hold.

1. Mis quadratically stable if and only if its dual M^T is quadratically stable.

2. Π is a strict PN multiplier if and only ifDDD(Π) is a strict PN multiplier.

3. TheFu(M,∆) and Fu(M^T,∆D) representations of the uncertain system (shown in Fig-ure C.1) have the same maximum sensitivity.

4. Let (Ψ, M) be any stable factorization of Π and (ΨD, MD) be any stable factorization of DDD(Π). Denote the maximum sensitivity analysis condition in (2.53) by LM I_∆∞. Then, ∃P ∈S^nx satisfying the LMI condition LMI_∆∞(M, P, γ,Ψ, M) ≺0 if and only if

∃PD ∈S^nx satisfying LMI∆∞(M^T, PD, γ,ΨD, MD)≺0.

Proof. The proof can be found in [130].

Alternating projections

Define two sets with a possible intersection, as shown in Figure D.1. The Γ_convex set is described by the LMIs (3.5) and (3.6). This is the solution set of Proposition 1, without the rank constraint.

Γ_rank denotes the non-convex rank constraint on Ku. The aim is to find a solution at the intersection of the two sets, denoted byK_u^⋆ in Figure D.1. The alternating projection algorithm has two consecutive steps in a sequence. The first step involves an orthogonal projection from the Γconvex to the Γ_rank set by Lemma 20. This assures that the rank ofKu is reduced by 1. In a second step one needs to project this reduced rankK_u matrix back to the Γ_convex solution set.

This is done based on Lemma 21. Then one needs to iterate these two steps until a solution is found in the intersection of the two sets. If a solution is found, then the rank of Ku has been successfully decreased by 1. In order to satisfy the rank constraint in Proposition 1, the whole projection sequence shown in Figure D.1 has to be evaluated n_u-1 times until rank(K_u^⋆) = 1 is achieved. Lemma 20 and Lemma 21 are as follows.

Γ _rank K _{u 0} Γ _convex

K _u ^?

Figure D.1: An alternating projection sequence, corresponding to the rank reduction of K_u0 by one.

Lemma 20.: Orthogonal projection to a lower dimensional set [47]. LetZ ∈Γ^n×n_rank and let Z = U SV^T be a singular value decomposition of Z. The orthogonal projection, Z^⋆=P_Γ^n−k

rankZ, of Z onto the Γⁿ_rank^−k×n−k dimensional set is given by

Z^⋆ =U S_n−kV^T, (D.1)

where the S_n−k diagonal matrix is obtained by replacing the smallestk singular values by zeros.

In document Optimal Decoupling of Dynamical Systems (Pldal 155-161)