QCs characterization of robust stability with simultaneous uncertainties in plant and controller

Systems & Control Letters 133 (2019) 104550

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QCs characterization of robust stability with simultaneous uncertainties in plant and controller✩ Liu Liu School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

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Article history: Received 19 December 2018 Received in revised form 12 September 2019 Accepted 27 September 2019 Available online 9 October 2019 Keywords: Robust stability Quadratic constraint Gap metric Time-varying system Nest algebra

a b s t r a c t Within the frame work of nest algebra, the robust stability of feedback interconnections of timevarying discrete-time systems with combined uncertainties in the plant and controller is derived via quadratic constraint (QC) approach, where the plant and controller are taken from two path connected uncertainty sets with respect to the topology induced by the gap metric. Some sufficient conditions for the existence of complementary QCs for the components of stable uncertain feedback systems are derived. The path-connectedness is established for some uncertainty sets with particular properties. The fundamental robust stability result represents a generalization of the input–output operator robust stability theorem of uncertainties appearing in the plant alone, to include the case of simultaneous uncertainties appearing in the controller. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Robust stability analysis based on the combining of integral quadratic constrains (IQCs) [1] and the ν -gap metric [2] is proposed in [3] for unstable time-invariant distributed parameter transfer functions. Also, the study is extended to the time-varying casual continuous systems by using the generalizing ν -gap in [4, 5], Specifically, Cantoni, Jönsson, Kao and Khong, etc. focus on the stability analysis for the feedback interconnections of a given system and an uncertain system set, where the uncertainty set is described as a path-connected set with respect to topology induced by the ν -gap. It is shown that the feedback stability is preserved along the continuous path of systems as long as a complementary IQC condition, which complements characteristics of a fixed open-loop component known to achieve feedback stability with one point, holds at all other points along the path. The main goal of this paper is to generalize some results on robust feedback stability in [4,6] to the case of feedback systems with simultaneous uncertainties in the plant and controller for casual time-varying discrete-time systems, where the plant and controller are taken from two path-connected sets with respect to the topology induced by the gap metric. Also, the path-connectedness is established for some uncertainty sets with particular properties. Without the requirements of pathconnectedness for the uncertainty set, the existence of the complementary quadratic constraints (QCs) [7] for the components ✩ This research is supported by NSFC, China (Item Number: 11871131, 11671065) and the Fundamental Research Funds for the Central Universities, China (Item Number: DUT18LK17). E-mail address: [email protected]. https://doi.org/10.1016/j.sysconle.2019.104550 0167-6911/© 2019 Elsevier B.V. All rights reserved.

in the stable uncertain feedback systems is studied. From the input–output perspective, the causal time-varying linear system considered here is a lower-triangular linear operator (possibly unbounded) defined on a certain separable Hilbert space, and the stable system algebra is represented by the nest algebra [8]. As a natural combination of the nonlinear small gain theory and the conic sector theory, a robust stability criterion of interconnected nonlinear systems is expressed in terms of coupled conic sector conditions in [9], where the two systems of the interconnection are both varying. In the linear system settings, the robust stability with simultaneous uncertainties characterized by the gap metric balls centered at the nominal plant and nominal controller are studied in various framework [2,10–13]. In [12], the robustness is established for the feedback systems of linear time invariant finite-dimensional continuous-time systems. In [13], a necessary and sufficient condition for the robust stability of feedback interconnection of infinite-dimensional timevarying linear systems is presented by using operator-theoretic and geometric approach. Motivated by the work of [13], it is natural to consider the stability criteria for the feedback system with combined uncertainties to more structured uncertainty sets of infinite-dimensional linear systems. It is shown in [6] that the path-connected set is more structured than a gap metric ball, and a standard gap metric ball robust stability result can be recovered within the blended IQC and gap metric framework, which only requires the existence of gap metric continuous paths within the uncertainty set. In particular, the path connectedness of the sufficiently small ν -gap ball is established by exploiting the existence of a certain J-spectral factorization for the Callier–Desoer class of systems.

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L. Liu / Systems & Control Letters 133 (2019) 104550

Following the idea and methods of the work in [4,6] for continuous-time linear systems, this paper combines the gap metric and quadratic constraint to study the stability of uncertain feedback interconnections for discrete-time linear systems. Firstly, under a requirement that the plant and controller uncertainty sets are path-connected with respect to the gap metric, some sufficient conditions for the robust stability of feedback systems are derived by means of QCs. Note that the proof is independent on the normalized strong graph representations, it differs from that in continuous-time settings [3,4]. In other words, these blended QCs/gap robust stability criteria can be directly used for casual discrete-time case without considering the existence of strong representations. Moreover, some standard gap-metric ball robust stability results with simultaneous uncertainty can be recovered within the QCs framework, which only requires the path-connectedness for the uncertainty set of interest. Secondly, some sufficient conditions to the existence of complementary QCs for the components are explored for stable feedback interconnections. Particularly, a sufficient condition is established as the weak closedness for the union of system graphs. Based on this result, it is derived that the simultaneous stability for finite systems must satisfy the coupled QCs condition. In the last, the gap-metric path-connectedness is studied along lines similar to the work in [6,14]. The sufficiently small gap-metric ball is proved to be path-connected under a certain J-spectral factorization. While, unlike the continuous-time result of [6], the existence of the J-spectral factorization is not known to hold in nest algebra. An advantage is that the gap-metric pathconnectedness can be preserved under the perturbation by an arbitrary stable linear system. This paper evolves along the following line. Some notations, definitions and preliminaries on operator theory and system theory are gathered in Section 2. Section 3 characterizes the feedback stability based on QCs, in which the plant and controller are subject to two path-connected sets with respect to the gap metric. Some sufficient conditions to the coupled QCs are established in Section 4. Section 5 investigates the path connectedness in the topology induced by the gap metric. The paper ends with a conclusion in Section 6. Notations: The symbols C and N denote respectively the complex numbers and natural numbers. Let H and K be two Hilbert spaces, H × K stands for the product space of H and K. Given a linear manifold M of H, the closure of M is denoted by M, and the orthogonal complement by M⊥ . ΠM denotes the orthogonal projection of H onto the closed subspace M. For a linear transformation T : dom(T ) ⊂ H → K, the image and kernel are denoted, respectively, ImT = {y ∈ K : y = Tx, x ∈ H} {[ and ]KerT = {x ∈ H } : Tx = 0}. The graph of T x is G (T ) := , Tx ∈ K ⊂ H×K Tx : x ∈ H {[ ] } , and the inverse Tx : x ∈ H, Tx ∈ K . B (H, K) denotes graph of T is G ′ (T ) := x the Banach space of bounded linear operators from H into K with the usual operator norm. Given T ∈ B(H, K), the minimum modulus and induced operator norm of T are defined respectively as µ(T ) = inf∥x∥H =1 ∥Tx∥K and γ (T ) = sup∥x∥H =1 ∥Tx∥K . It holds that γ (T −1 ) = 1/µ(T ), when T has a bounded inverse. The adjoint of T ∈ B(H, K), denoted as T ∗ ∈ B(K, H), satisfies ⟨Tx, y⟩K = ⟨x, T ∗ y⟩H , x ∈ H, y ∈ K. T is an isometry if T ∗ T = I, equivalent to ∥Tx∥K = ∥x∥H holding [for all ] x ∈ H. T [∈ B(]H) := B(H, H) is self-adjoint if T = T ∗ . J = 0I −0I and J1 = −0I 0I . 2. Preliminaries

2. The set M ⊆ H is weakly sequentially closed if the limit point of each weakly convergent sequence in M belongs to M. 3. The set M ⊆ H is weakly sequentially compact if each sequence in M has a weakly convergent subsequence. Lemma 2.1 ([15]). The closed unit ball of the separable Hilbert space is weakly sequentially compact. Definition 2.2. A family N of closed subspaces of the Hilbert space H is a complete nest if 1. {0}, H ∈ N, 2. For N1 , N2 ∈ N, either N1 ⊂ N2 or N2 ⊂ N1 ,⋂ 3.⋁ If {Nα : α ∈ A} is a ⋁ subfamily in N, then α∈A Nα ∈ N and α∈A Nα ∈ N, where α∈A Nα is the closed linear span of {Nα : α ∈ A⋃ }, i.e., the intersection of all closed subspaces of H that contain α∈A Nα . Since every closed subspace N of H is identifiable to the orthogonal projection ΠN with range N , a complete nest can be seen as a family of projections. The nest algebra determined by the complete nest N is Alg(N) = {T ∈ B(H) : T N ⊂ N , ∀N ∈ N}

= {T ∈ B(H) : (I − ΠN )T ΠN = 0, ∀N ∈ N}. The input–output signal space considered in this paper is the complex separable Hilbert space

{ ℓ = (x1 , x2 , . . .) : 2

1. {xn } weakly converges to x, denoted by xn limn→∞ ⟨xn , y⟩ = ⟨x, y⟩ holds for all y ∈ H,

W

−→ x, if

} |xi | < ∞, xi ∈ C , 2

i=1

∑+∞

¯ where the inner product is defined by ⟨x, y⟩ = i=1 xi yi . Let ℓ2e = {(x1 , x2 , . . .) : xi ∈ C} denote the extended space of ℓ2 . For each n ∈ N, the truncated projection Pn on ℓ2 or ℓ2e is defined by Pn (x1 , x2 , . . . , xn , . . .) = (x1 , x2 , . . . , xn , 0, . . .). Denote P0 = 0 and P∞ = I. The seminorm ∥ · ∥n on ℓ2e is defined by

∥x∥n = ∥Pn x∥,

x ∈ ℓ2e .

{Pn : n ∈ N} is used to define the physical definition of causality. A linear transformation L on ℓ2e is causal if Pn LPn = Pn L for each n ∈ N. L is a (time-varying) linear system if it is a causal linear transformation on ℓ2e and continuous with respect to the standard seminorm topology (Ch 5, [8]). Indeed, any linear system is an infinite-dimensional lower triangular matrix with respect to the standard basis of ℓ2 . A linear system is stable if its restriction to ℓ2 is a bounded operator. Let L be the algebra of linear systems with respect to the standard addition and multiplication. The set of stable ones, denoted by S , is a weakly closed algebra of the Banach algebra B(ℓ2 ) with the usual operator norm, referred to in the operator theory literature as a nest algebra determined by the complete nest {I − Pn : n ∈ N ∪ {∞}} [Ch 3, [8]]. Mm×n (S ) denotes the set of m × n matrices with entries in S . Consider the feedback interconnection of the plant L ∈ L and the controller C ∈ L (see Fig. 1), defined by

[

e1 e2

]

[ =

I L

C I

][

u1 u2

]

.

The feedback interconnection {L, C } is stable [ if]each entry of the e1 operator matrix from external input e = to internal input e2

[ Definition 2.1 ([15]). Let {xn } be a bounded sequence in the Hilbert space H.

∞ ∑

u=

[

I L

u1 u2 C I

] is causal bounded, i.e.,

]−1

[ =

(I − CL)−1 −L(I − CL)−1

−C (I − LC )−1 (I − LC )−1

]

∈ M2×2 (S ).

L. Liu / Systems & Control Letters 133 (2019) 104550

3

(i). {L, C } is stable. (ii). G (L) + G ′ (C ) = ℓ2 × ℓ2 and G (L) ∩ G ′ (C ) = {0}. ˆ Γ is invertible on ℓ2 × ℓ2 . (iii). G { } (iv). δ (G ′ (C )⊥ , G (L)) = max γ (ΠG (L) ΠG ′ (C ) ), γ (ΠG (L)⊥ ΠG ′ (C )⊥ ) < 1. (v). The projection to G ′ (C )⊥ restricted on G (L), denoted by ΠG ′ (C )⊥ |G (L) , is invertible. Define the robust performance margin of the feedback interconnection system {L, C } by

Fig. 1. Feedback interconnection.

{ bL,C := L is stabilizable if there exists a linear system C ∈ L such that {L, C } is stable. In this case, C is called a stabilizing controller for L. The system strong representation plays an important role in the study of feedback stability of time-varying linear system within the frame work of nest algebra. Definition 2.3 ([16]). Given a plant L ∈ L and a controller C ∈ L. (i). G ∈ M2×1 (S ) is a strong right representation of L if G (L) = ImG and G has a left inverse in M1×2 (S ), G is normalized if G∗ G = I. ˆ ∈ M1×2 (S ) is a strong left representation of L if G (L) = (ii). G ˆ and Gˆ has a right inverse in M2×1 (S ), Gˆ is normalized if KerG ˆ Gˆ ∗ = I G (iii). Γ ∈ M2×1 (S ) is a strong right inverse representation of C if G ′ (C ) = ImΓ and Γ has a left inverse in M1×2 (S ), Γ is normalized if Γ ∗ Γ = I. (iv). Γˆ ∈ M1×2 (S ) is a strong left inverse representation of C if G ′ (C ) = KerΓˆ and Γˆ has a left inverse in M2×1 (S ), Γˆ is normalized if Γˆ Γˆ ∗ = I. It is shown in [8,16] that the stabilizability of a causal linear system in L is equivalent to it admits normalized strong left and right representations. By convention, throughout this paper ˆ denote respectively normalized strong right and left G and G representations of a given stabilizable plant L ∈ L, respectively; likewise, Γ and Γˆ denote the inverse ones of a given controller C ∈ L. The gap metric is another important tool in the study of robust stability of time-varying linear systems.

γ ((ΠG ′ (C )⊥ |G (L) )−1 ΠG ′ (C )⊥ )−1 , {L, C } is stable 0, other w ise. (2.1)

and the maximal robustness margin bopt (L) := sup{bL,C : C stabilizes L}. According to Theorem 2 in [10], Theorem 3 in [17] and the commutant lifting theorem of the nest algebra, the maximal 1 robustness margin can be attained and bopt (L) = (1 − γ (HGˆ ∗ )2 ) 2 , HGˆ ∗ is the time-varying Hankel operator from A2 to (C2 ⊕ C2 ) ⊖ (A2 ⊕ A2 ) defined by

ˆ ∗ f , f ∈ A2 , HGˆ ∗ f = Π G where Π is the orthogonal projection onto the subspace (C2 ⊕ C2 ) ⊖ (A2 ⊕ A2 ), C2 is the space of Hilbert–Schmidt operators on ℓ2 and A2 = C2 ∩ S . The robust margin can be characterized in terms of normalized system strong representations or the gap metric between the plant and controller. Lemma 2.4 ([13]). If the feedback system {L, C } is stable, then bL,C = µ(Γˆ G) =

√

ˆ Γ ) = bC ,L . 1 − δ 2 (G ′ (C )⊥ , G (L)) = µ(G

3. Robust stability analysis via quadratic constraints

=

In this section, we focus on the robust stability of feedback system {L, C }, where the plant L and the controller C are taken from two path-connected sets with respect to the gap metric. This approach is directly aimed at the case of robust stability for time-varying linear systems with simultaneous uncertainties in plant and controller. This section generalizes an elegant result of Cantoni, Khong and Jönsson [3,4] on the robust stability analysis combined the ν -gap and integral-quadratic-constrains to the simultaneous plant-controller uncertainty case for discrete-time systems. Before the main results, the definitions of connectedness and QCs for a system set are given as follows.

Lemma 2.3 ([8]). Let M0 , M1 , M2 be closed subspaces of H such that δ 2 (M0 , M1 ) + δ 2 (M0 , M2 ) < 1. Then

Definition 3.1. The path θ ∈ [0, 1] → Lθ is continuous in the topology induced by the gap metric if for any ϵ > 0, there exists δ > 0 such that δ (Lθ1 , Lθ2 ) < ϵ whenever |θ1 − θ2 | ≤ δ .

Definition 2.4 ([8]). Let M1 , M2 be two closed subspaces of H. − → The directed gap from M1 to M2 is δ (M1 , M2 ) = γ (ΠM⊥ ΠM1 ), 2 and the gap between M1 and M2 is defined by

− → − → δ (M1 , M2 ) = max{ δ (M1 , M2 ), δ (M2 , M1 )}. Lemma 2.2 ([8]). If δ (M1 , M2 ) − → − → δ (M1 , M2 ) = δ (M2 , M1 ).

<

1, then δ (M1 , M2 )

√ δ (M1 , M2 ) ≤ δ (M0 , M1 ) 1 − δ 2 (M0 , M2 ) √ + δ (M0 , M2 ) 1 − δ 2 (M0 , M1 ). The gap metric defines a metric on the set of closed operators by the gap metric for the graph of closed operators. In particular, the gap metric between two plants L1 and L2 in L is denoted by

δ (L1 , L2 ) := δ (G (L1 ), G (L2 )) { } = max γ (ΠG (L1 ) ΠG (L2 )⊥ ), γ (ΠG (L1 )⊥ ΠG (L2 ) ) . Theorem 2.1 ([13]). Given two stabilizable systems L ∈ L and C ∈ L. Then the following are equivalent:

Definition 3.2. An uncertainty set Θ ⊂ L is path-connected with respect to the topology induced by the gap metric if for any La , Lb ∈ Θ there exists a continuous path between La and Lb with respect to the gap metric. Definition 3.3. Given a system set Θ ⊂ L and a self-adjoint operator Φ ∈ B(ℓ2 × ℓ2 ). (i). The set Θ satisfies the strictly quadratic constraint determined by Φ , denoted by Θ ∈ SQC(Φ ), if there exists ε > 0 such that

⟨Φ v, v⟩ ≥ ε∥v∥2 , ∀v ∈ G (L), L ∈ Θ ,

4

L. Liu / Systems & Control Letters 133 (2019) 104550

(ii). The set Θ satisfies the complementary SQC, denoted by Θ ∈ SQCc (Φ ), if there exists ε > 0 such that

⟨Φ v, v⟩ ≤ −ε∥v∥2 , ∀v ∈ G ′ (L), L ∈ Θ . (iii). The set Θ satisfies the quadratic constraint determined by Φ , denoted by Θ ∈ QC(Φ ), if

⟨Φ v, v⟩ ≥ 0, ∀v ∈ G (L), L ∈ Θ , (iv). The set Θ satisfies the complementary QC, denoted by Θ ∈ QCc (Φ ), if

⟨Φ v, v⟩ ≤ 0, ∀v ∈ G ′ (L), L ∈ Θ . Note that for a given linear system L, {L} ∈ SQC(Φ ) is conveniently denoted by L ∈ SQC(Φ ), this notation coincides with the IQC condition for one system defined as usual. Theorem 3.1. Given two path connected sets Θ , ∆ ⊂ L. Assume that there exist L0 ∈ Θ and C0 ∈ ∆ such that {L0 , C0 } is stable. If there exists a self-adjoint operator Φ ∈ B(ℓ2 × ℓ2 ) such that (i). Θ ∈ SQC(Φ ), (ii). ∆ ∈ QCc (Φ ), then the following hold: (1). {{L, C } is stable for all}L ∈ Θ and C ∈ ∆. (2). bL,C : L ∈ Θ , C ∈ ∆ is bounded below by a positive constant. Proof. Along the proof of Theorem 7 in [3] and Lemma 5.1 in [4], first note that the coupled QC conditions (i) and (ii) are used to prove the existence of a scalar 0 < α < 1 such that

γ (ΠG (L) ΠG ′ (C ) ) ≤ α,

∀L ∈ Θ , ∀C ∈ ∆ .

(3.1)

Since {L0 , C0 } is stable, according to Lemma 2.2 and Theorem 2.1, one can get

δ (G ′ (C0 )⊥ , G (L0 )) = γ (ΠG (L0 ) ΠG ′ (C0 ) ) = γ (ΠG (L0 )⊥ ΠG ′ (C0 )⊥ ) ≤ α. (3.2) By hypothesis, for any L ∈ Θ and C ∈ ∆, there is a gap continuous path from L0 to L: a ∈ [0, 1] → La , and a gap continuous path from C0 to C : b ∈ [0, 1] → Cb , where L1 := L and C1 := C . So there exists η > 0 such that for any a, b ∈ [0, 1] with |a − b| ≤ η,

δ (La , Lb ) <

1−α 3

and δ (Ca , Cb ) <

1−α

.

3 For any a, b ∈ [0, η], it follows from the above inequalities and (3.2) that

( ) γ (ΠG (La )⊥ ΠG ′ (Cb )⊥ ) = γ (ΠG (La )⊥ ΠG (L0 ) + ΠG (L0 )⊥ ΠG ′ (Cb )⊥ ) ≤ γ (ΠG (La )⊥ ΠG (L0 ) ) + γ (ΠG (L0 )⊥ ΠG ′ (Cb )⊥ ) ≤ δ (La , L0 ) + γ (ΠG (L0 )⊥ ΠG ′ (C0 )⊥ ) + γ (ΠG ′ (C0 ) ΠG ′ (Cb )⊥ ) ≤ δ (La , L0 ) + δ (G ′ (C0 )⊥ , G (L0 )) + δ (C0 , Cb ) 2+α < .

3 Putting these together, ones can conclude that

δ (G ′ (Cb )⊥ , G (La )) = max{γ (ΠG (La ) ΠG ′ (Cb ) ), γ (ΠG (La )⊥ ΠG ′ (Cb )⊥ )} 2+α < max{α, } < 1.

3 It follows from Theorem 2.1 that {La , Cb } is stable for all a, b ∈ [0, η]. If η = 1, the proof is completed. Otherwise, repeat the arguments employed in the proof of the stability of {Lη , Cη }, by considering Lη as L0 and Cη as C0 , respectively. Then, ones can get {La , Cb } is stable for all a, b ∈ [η, 2η]. After [ η1 ]+ 1 steps, we obtain

that {L1 , C1 } is stable and δ (G ′ (C1 )⊥ , G (L1 )) < 2+α . Hence bL1 ,C1 > 3 1−α > 0. In conclusion, the interconnection system {L, C } is stable 3 { } for all L ∈ Θ and C ∈ ∆, and the set bL,C : L ∈ Θ , C ∈ ∆ is . □ bounded below with the lower bound 1−α 3 Remark 3.1. The proof of Theorem 3.1 borrows heavily from the proofs of continuous-time results in [3,4]. The only difference is that the proof of Theorem 3.1 uses the properties of orthogonal projection but not the strong representations, so the QCs robust stability criterion in Theorem 3.1 can be directly used to timevarying discrete-time system without considering the existence of strong representations. Remark 3.2. Note that the stability of {L, C } is equivalent to that of {C , L}, so Theorem 3.1 also holds with conditions (i) and (ii) replaced by Θ ∈ QC(Φ ) and ∆ ∈ SQCc (Φ ), respectively. Corollary 3.1. Given L, C ∈ S . If {rL : r > 0} ∈ SQC(Φ ) and {sC : s > 0} ∈ QCc (Φ ) hold for some Φ = Φ ∗ ∈ B(ℓ2 × ℓ2 ), then {rL, sC } is stable for all s, r ≥ 0. Proof. It is clear that {rL : r ∈ R} and {sC : s ∈ R} are pathconnected within the topology induced by the operator norm. In fact, the gap topology restricted to B(H) gives the norm topology. So {rL : r ∈ R} and {sC : s ∈ R} are path-connected within the gap-metric topology. Using Theorem 3.1 and the stability of the feedback interconnection {0, 0} yields the robust stability. □ Theorem 3.1 means that the closed-loop stability is preserved along the gap continuous paths of the plants and controllers so long as coupled QCs conditions. In what follows, we are concerned with a special case of robust stability, in which the uncertainty appears only in the plant, which can be seen as a corollary of Theorem 3.1. The following results coincides with the wellknown IQC-based robust stability result for linear continuoustime time-varying system obtained in [4]. Theorem 3.2. Let C0 ∈ L and Θ be a path-connected uncertainty set. Suppose that there exists L0 ∈ Θ such that {L0 , C0 } is stable. Then there exists an operator Φ = Φ ∗ ∈ B(ℓ2 × ℓ2 ) such that Θ ∈ QC(Φ ) and C0 ∈ SQCc{(Φ ) if and only } if C0 robustly stabilizes the uncertainty set Θ and bL,C0 : L ∈ Θ is bounded below by a positive constant. Proof. The necessity is clear from Remark 3.2. The sufficiency follows from Remark 1 in [3] by choosing Φ = Γˆ 0∗ Γˆ 0 − β 2 , where β > 0 is a lower bound of {bL,C0 : L ∈ Θ }. □ Next, we give another robust stability criterion for simultaneous uncertainty in the plant and the controller by means of QCs conditions for the plant and the controller, but not the system sets as shown in Theorem 3.1. Theorem 3.3. Given two gap path-connected sets Θ , ∆ ⊂ L, suppose there exist L0 ∈ Θ and C0 ∈ ∆ such that {L0 , C0 } is stable. If there exists an operator Φ = Φ ∗ ∈ B(ℓ2 × ℓ2 ) such that the following complementary QCs hold: (i). L ∈ SQC(Φ ), ∀L ∈ Θ , (ii). C ∈ SQCc (Φ ), ∀C ∈ ∆, then {L, C } is stable for all L ∈ Θ and C ∈ ∆. Proof. Condition (i) implies Θ ∈ QC(Φ ), applying Theorem 3.2 yields the stability of {L, C0 } for all L ∈ Θ . Furthermore, for any given L ∈ Θ , the feedback system {L, C0 } can be seen as the nominal system, the plant set satisfies {L} ∈ SQC(Φ ) and the controller set satisfies ∆ ∈ QCc (Φ ), then {L, C } is stable for all C ∈ Θ by Theorem 3.1. □

L. Liu / Systems & Control Letters 133 (2019) 104550

Remark 3.3. Note that conditions (i) and (ii) in Theorem 3.3 are weaker than conditions (i) and (ii) in Theorem 3.1, that is the use of Theorem 3.3 is wider in determining the robust stability. However, whether conditions (i) and (ii) in Theorem 3.3 are necessary for the robust stability is not known. Compared with the robust stability analysis on single uncertainty in plant, whether the coupled QCs conditions are necessary is still not known The previous theorem presents the following robust stability criterion, and we will revisit some standard gap-ball robust stability criteria as consequences of the following result. Theorem 3.4. Given two path-connected uncertainty sets Θ ⊂ L and ∆ ⊂ L. Suppose there exist L0 ∈ Θ and C0 ∈ ∆ such that {L0 , C0 } is stable. If there exists an α ∈ R such that (i). δ 2 (L0 , L) − b2L,C < α, ∀L ∈ Θ , 0 (ii). δ 2 (C0 , C ) − b2C ,L < −α, ∀C ∈ ∆, 0 then {L, C } is stable for all L ∈ Θ , C ∈ ∆.

ˆ ∗ Gˆ 0 + Γˆ ∗ Γˆ 0 . It remains to prove that L ∈ Proof. Set Φ = α − G 0 0 SQC(Φ ) and C ∈ SQCc (Φ ) hold for all L ∈ Θ and C ∈ ∆. First note that, for any L ∈ Θ , there exists εL > 0 such that

5

Proof. The proof is similar to that of Corollary 3.2. □ 4. Sufficient conditions for the coupled QCs conditions In the previous section, the equivalence between the robust stability and the coupled QCs is explored under the existence of gap-metric continuous path within the plant uncertainty set. In this section, we will describe sufficient conditions for the complementary QCs. The first one expressed in terms of stability margin is obvious from the proof of Theorem 3.2. Theorem 4.1. Given Θ ⊂ L. Suppose that C stabilizes the uncertainty set Θ . If {bL,C : L ∈ Θ } is bounded below by a positive constant, then there exists an operator Φ = Φ ∗ ∈ B(ℓ2 × ℓ2 ) such that (i). Θ ∈ QC(Φ ), (ii). C ∈ SQCc (Φ ). The following sufficient condition for coupled QCs is expressed in terms of weak closedness for the union of plants’ graphs.

This gives that

Theorem 4.2. ⋃ Given Θ ⊂ L. Suppose that C stabilizes the uncertainty set Θ . If L∈Θ G (L) is weakly sequentially closed, then there exists an operator Φ = Φ ∗ ∈ B(ℓ2 × ℓ2 ) such that (i). Θ ∈ QC(Φ ), (ii). C ∈ SQCc (Φ ).

⟨Φ Gx, Gx⟩ = α∥x∥2 − ∥Gˆ0 Gx∥2 + ∥Γˆ0 Gx∥2

Proof. For any given L0 ∈ Θ , since {L0 , C } is stable, we have

α − δ 2 (L0 , L) + b2L,C0 ≥ εL .

∥Gˆ 0 Γ x∥2 > 0, µ(Gˆ 0 Γ )2 = inf x̸ =0 ∥Γ x∥2

≥ α∥x∥2 − γ (Gˆ0 G)2 ∥x∥2 + µ(Γˆ0 G)2 ∥x∥2 ≥ (α − δ 2 (L0 , L) + b2L,C0 )∥x∥2

ˆ 0 Γ x∥2 ≥ ε∥Γ x∥2 . This then there exists ε > 0 such that ∥G implies that

≥ εL ∥Gx∥2 holds for all x ∈ ℓ2 . That is, L ∈ SQC(Φ ). Likewise, the condition (ii) implies C ∈ SQCc (Φ ), ∀C ∈ ∆. Then {L, C } is stable for all L ∈ Θ and C ∈ ∆ by Theorem 3.3. □ Lemma 3.1 (Pro. III.1, [18]). Given C0 , L0 , L ∈ L, suppose that the feedback system {L0 , C0 } is stable. If δ (L, L0 ) < bL0 ,C0 , then the followings hold: (1). {L, C0 } is stable, √ (2). bL,C0 ≥ bL0 ,C0

√

1 − δ 2 (L0 , L) − δ (L0 , L) 1 − b2L

0 ,C0

.

Corollary 3.2. Let {L0 , C0 } be stable and r > 0, s > 0 be fixed numbers such that r < bL0 ,C0 , s < bL0 ,C0 and s2 + r 2 < 1. Suppose that the gap metric balls Br (L0 ) := {L ∈ L : δ (L0 , L) < r } and Bs (C0 ) := {C ∈ L : δ (C0 , C ) < s} are path-connected. If

√

√

1 − s2 + s 1 − r 2 < bL0 ,C0

holds for all α > 0. That is, C ∈ SQCc (Φα ), where Φα = α Γˆ ∗ Γˆ − ˆ ∗ Gˆ 0 . It remains to demonstrate that there exists α0 > 0 such G 0 that Θ ∈ QC(Φα0 ). If not, for any natural number n ∈ N, there exists xn ∈ ℓ2 with ∥xn ∥ = 1 such that

⟨(nΓˆ ∗ Γˆ − Gˆ ∗0 Gˆ 0 )Gn xn , Gn xn ⟩ < 0, hence

∥Γˆ Gn xn ∥ <

Using Lemma 3.1 and Theorem 3.4, the following classical robust stability criteria for feedback systems with combined gapmetric balls uncertainty can be covered via QCs approach.

r

⟨(α Γˆ ∗ Γˆ − Gˆ ∗0 Gˆ 0 )Γ x, Γ x⟩ ≤ −ε∥Γ x∥2 , ∀x ∈ ℓ2

(3.3)

then {L, C } is stable for all L ∈ Br (L0 ) and C ∈ Bs (C0 ). Proof. Using Lemma 3.1 and the condition (3.3) with simple manipulations yields that bL,C0 > s and bL0 ,C > r. These imply that conditions (i) and (ii) in Theorem 3.4 hold for α = r 2 − s2 . □ Corollary 3.3. Let {L0 , C0 } be stable and suppose that the gap metric balls Br (L0 ) and Bs (C0 ) are path-connected. If s + r < bL0 ,C0 , then {L, C } is stable for all L ∈ Br (L0 ) and C ∈ Bs (C0 ).

1 n

∥Gˆ 0 Gn xn ∥ ≤

1 n

, ∀n ∈ N ,

then

∥Γˆ Gn xn ∥ → 0, n → ∞.

(4.1)

Note that ∥Gn xn ∥ = 1, by Lemma 2.1, the sequence {Gn xn }n≥1 has a weakly convergent subsequence {Gnk xnk }k≥1 , assume that W

Gnk xnk −→ y as k → ∞, and ∥y∥ = 1. It follows from the boundedness of the operator Γˆ that W Γˆ Gnk xnk −→ Γˆ y, k → ∞.

(4.2)

Combining (4.1) and (4.2), we obtain that⋃Γˆ y = 0, that is y ∈ KerΓ = G ′ (C ). By the hypothesis ⋃ that L∈Θ G (L) is weakly sequentially closed, we have y ∈ G (L). Since C stabilizes L∈Θ⋂ Θ , according to Theorem 2.1 that G ′ (C ) G(L) = {0}, ∀L ∈ Θ . These conclude that y = 0, which is a contradiction to ∥y∥ = 1. Therefore, C ∈ SQCc (Φα0 ) and Θ ∈ QC(Φα0 ) hold for some α0 > 0. □ Lemma 4.1.

If L ∈ L, then G (L) is weakly sequentially closed.

6

L. Liu / Systems & Control Letters 133 (2019) 104550

Assume that Lxnn ∈ G (L) and Lxnn −→ yx , n → ∞. First [ x ] [e ] [ ] [e ] note that for any j ∈ N, limn→∞ ⟨ Lxnn , ejj ⟩ = ⟨ yx , ejj ⟩, this implies that

Since Θ is a path-connected, there exists a continuous path: λ → Qλ , where Q0 = La − L0 and Q1 = Lb − L0 . It follows from (5.1) that the path λ → Qλ + L0 is a continuous path between La and Lb , so Θ ′ is path-connected. □

lim ⟨xn − x, ej ⟩ = 0,

5.2. Arbitrarily large operator-norm balls in S

Proof. Given the normal orthogonal basis {ej : j ∈ N} for ℓ2 .

[

x

]

[

n→∞

x

]

W

[ ]

lim ⟨Lxn − y, ej ⟩ = 0,

n→∞

Theorem 5.2. For any r > 0, the {L ∈ S : γ (L) < r } is a path-connected set.

then for any m ∈ N, lim ∥xn − x∥m = lim ∥Pm (xn − x)∥

n→∞

n→∞

= lim (

m ∑

n→∞

1

|⟨xn − x, ej ⟩|2 ) 2 = 0,

(4.3)

j=1

[

and

GL :=

lim ∥Lxn − y∥m = lim ∥Pm (Lxn − y)∥

n→∞

Proof. First note that for any L ∈ S with γ (L) < r, it follows from Theorem 2.1.5 in [8] that the positive operator I + L∗ L has an 1 invertible square root operator (I + L∗ L) 2 , the following operators I L

]

1

(I + L∗ L)− 2 and

n→∞

= lim (

m ∑

1

2 12

|⟨Lxn − y, ej ⟩| ) = 0,

(4.4)

ˆ L := (I + LL∗ )− 2 G

[

−L

I

]

Since L ∈ L is continuous with respect to the topology induced by {∥ · ∥m : m ∈ N}, combining Equation (4.3), we have

are isometric and co-isometric, respectively. Using the fact that ˆ L = G (L), we have GL GL ∗ is the orthogonal projection ImGL = KerG ∗ 2 2 ˆ L is the orthogonal projection from from ℓ × ℓ to G (L) and GˆL G 2 2 ⊥ ℓ × ℓ to G (L) . Finally, observe that for any λ1 , λ2 ∈ [0, 1],

lim ∥Lxn − Lx∥m = 0.

− → δ (λ1 L, λ2 L) = γ (Gˆ ∗λ2 L Gˆ λ2 L Gλ1 L Gλ1 L ∗ )

n→∞

j=1

n→∞

[ Observe the above equation and Eq. (4.4), Lx = y, so G (L), then G (L) is weakly sequentially closed.

x y

1

] ∈

1

≤ |λ1 − λ2 |γ ((I + |λ2 |2 LL∗ )− 2 )γ (L)γ ((I

□

Theorem 4.3. Given a finite set Θ . If C ∈ L stabilizes the set Θ , then there exists an operator Φ = Φ ∗ ∈ B(ℓ2 × ℓ2 ) such that (i). Θ ∈ QC(Φ ), (ii). C ∈ SQCc (Φ ). Proof. Since the finite union of closed sets is closed, by Lemma 4.1, the proof is completed. □ Remark 4.1. Theorem 4.3 means that the simultaneous stability for finite plants must satisfy the coupled QCs 5. Path-connectedness with respect to the gap metric topology This section presents some results about the pathconnectedness in the gap metric topology. First, it is shown that the path-connectedness can be preserved under the perturbation by a stable linear system. Secondly, the path-connectedness of two type uncertainty sets is proved: arbitrary large norm ball in S ; and the sufficient small gap metric ball in L under the existence of a certain J −spectral factorization. These results use the ideas and methods developed in [3,6,14,19] 5.1. Path-connectedness preserved under the stable perturbation The path-connectedness is preserved under the perturbation of linear systems by a stable system as follows.

+ |λ1 |2 L∗ L)

− 21

)

< |λ1 − λ2 |γ (L), so δ (λ1 L, λ2 L) < |λ1 − λ2 |γ (L). Then the path λ → λL is a continuous path between 0 and L with respect to the gap metric. □ Remark 5.1. Since the path-connectedness can be preserved by the perturbation of a stable linear system, the arbitrarily large norm ball {L ∈ S : γ (L − L0 ) < r } centered at a given L0 ∈ S , is path-connected in the gap-metric topology. As such, any convex subset of S is path-connected. In fact, the gap-metric topology restricted to B(H) gives the norm topology, so arbitrarily large stable gap-metric balls Bsr (L0 ) := {L ∈ S : δ (L0 , L) < r } are path-connected. 5.3. Sufficient small gap-metric balls in L The aforementioned LFT characterization of gap metric is developed in the following, assuming that a certain J-spectral factorization exists. The fractional characterization of gap metric has been studied in various framework [6,14,19], Ch9 in [2]. Following this, it is shown how to recover the gap metric ball robustness result within the QC based framework. After that, the path connectedness of sufficient small gap-metric ball is provided via LFT characterization. Definition 5.1. Φ ∈ B(ℓ2 × ℓ2 ) has a J-spectral factorization with [ ] R11 R12 respect to S if there exists an operator R = ∈ R21 R22 ∗ M2×2 (S ) such that Φ = R JR and R is invertible in M2×2 (S ).

Lemma 5.1 (Th 9.3.4, [8]). If Li ∈ L and L0 ∈ S , then

δ (L1 + L0 , L2 + L0 ) ≤ 2(1 + γ (L0 )2 )δ (L1 , L2 ). Theorem 5.1. Given L0 ∈ S . If Θ is a path-connected uncertainty set in L, then Θ ′ = {L0 + L : L ∈ Θ } is path-connected. Proof. For any La , Lb ∈ Θ ′ , by Lemma 5.1,

δ (La , Lb ) ≤ 2(1 + γ (L0 )2 )δ (La − L0 , Lb − L0 )

1

= |λ1 − λ2 |γ ((I + |λ2 |2 LL∗ )− 2 L(I + |λ1 |2 L∗ L)− 2 )

(5.1)

Lemma 5.2. Given a stabilizable linear system L0 ∈ L and r ≤ ˆ 0 is a normalized strong left representation bopt (L0 ). Assume that G ∗ˆ 2 ˆ of L0 . If r I − G0 G0 has a J-spectral factorization:

ˆ ∗0 Gˆ 0 = R∗ JR, r 2I − G

L. Liu / Systems & Control Letters 133 (2019) 104550

7

then δ (L0 , L) < r if and only if F (R; L) := (R21 + R22 L)(R11 + R12 L)−1 is stable and γ (F (R; L)) < 1.

to L2 is

Proof. First note that for any L ∈ L,

where Qt = (1 − t)Q1 + tQ2 , Qi = F (R; Li ) = (R21 + R22 Li )(R11 + R12 Li )−1 , i = 1, 2. □

(R11 + R12 L)

−1

dom(F (R; L))

= (R11 + R12 L) {x ∈ ℓ2 : (R11 + R12 L)−1 x ∈ dom((R21 + R22 L))} −1

= (R11 + R12 L)−1 {x ∈ ℓ2 : (R11 + R12 L)−1 x ∈ dom(L)} = dom(L),

[

I L

]

(R11 + R12 L)−1 dom(F (R; L)) = RG (L).

ˆ −1 are strong representations of F (R; L), respecThen RG and GR ˆ are strong representations of L. tively, whenever G and G δ (L0 , L) < r if and only if there exists εL > 0 such that ∥Gˆ 0 Gx∥2 ≤ (r 2 − εL2 )∥Gx∥2 .

(5.2)

ˆ ∗ Gˆ 0 = R∗ JR, (5.2) is equivalent to Since r 2 I − G 0 ⟨R∗ JRGx, Gx⟩ ≥ εL2 ∥Gx∥2 ≥

εL2 γ

(R)2

5.4. Straight Line in S Within the context of uncertainty sets that are path-connected with respect to the topology induced by the gap metric, these uncertainty sets can be more structured than a ball and can be unbounded, such as the arbitrary straight line in S .

it follows that G (F (R; L)) = R

t → Lt := F (R−1 ; Qt )

Lemma 5.3. Given L0 , ∈ S . The system set {rL + L0 : r ∈ R} is a path-connected set. Proof. {rL : r ∈ R} is path-connected within the gap-metric topology. Applying the fact that the path-connectedness can be preserved by the stable perturbation implies that any straight line is path connected with respect to the gap metric. □ 6. Conclusion

∥RGx∥2 .

so ∥F (R; L)|dom(F (R;L)) ∥ < 1. r ≤ bopt (L0 ) means r ≤ bL0 ,C for some C ∈ L stabilizing L0 . Assume that Γ is a normalized strong inverse right representation ˆ 0 Γ ) and Γ ∗ Γ = I that of C , it follows from the facts bL0 ,C = µ(G r ≤ bL0 ,C is equivalent to

This paper establishes the robust stability criteria for discretetime time-varying linear systems with simultaneous uncertainties in the plant and the controller in terms of QCs. The uncertainty is characterized as a path-connected set with respected to the gap metric. Without the requirements of pathconnectedness for the uncertainty set, some sufficient conditions to the existence of the complementary quadratic constraints (QCs) [7] for the components of the stable uncertain feedback systems are derived. Most results still hold with similar arguments to continuous-time linear systems.

∥Gˆ 0 Γ x∥2 ≥ r 2 ∥x∥2 = r 2 ∥Γ x∥2 , ∀x ∈ ℓ2 ,

Declaration of competing interest

That is

∥F (R; L)y∥2 ≤

γ (R)2 − εL2 ∥y∥2 , y ∈ dom(F (R; L)), γ (R)2 + εL2

which can be rewritten as

⟨R∗ JRΓ x, Γ x⟩ ≤ 0.

(5.3)

Define X (R; C ) = (R11 C + R12 )(R21 C + R22 )−1 . It can be checked that (R21 C + R22 )−1 dom(X (R; C )) = dom(C ), thus

[ G (X (R; C )) = R

C I

]

(R21 C + R22 )−1 dom(X (R; C )) = RG ′ (C ).

Then RΓ is a strong right inverse representation for X (R; C ). Combining the inequality (5.3), ones can get ∥X (R; C )|dom(X (R;C )) ∥ ≤ 1. Since {L0 , C } is stable and δ (L0 , L) < r, the interconnecˆ Γ is invertible in S , then tion system {L, C } is stable, hence G −1 ˆ (GR)(R Γ ) is invertible in S , so the interconnection system {F (R; L), X (R; C )} is stable. By Large gain theorem [Cor. 1 [20]], we have dom(F (R; L)) = ℓ2 , so F (R; L) is bounded and ∥F (R; L)|∥ < 1. □ Theorem 5.3. Given a stabilizable linear system L0 ∈ L and ˆ ∗ Gˆ 0 has a J-spectral factorr < bopt (L0 ). If the operator r 2 − G 0 ization, then the gap-metric ball Br (L0 ) = {L ∈ L : δ (L, L0 ) < r } is path-connected. Proof. The proof follows the proof of Theorem 5 in [6]. Assume ˆ ∗ Gˆ 0 has a J-spectral factorization r 2 I − that the operator r 2 − G 0 ∗ ∗ ˆ Gˆ 0 = R JR. For any L1 , L2 ∈ Br (L0 ), the continuous path from L1 G 0

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] A. Megretski, A. Rantzer, System analysis via integral quadratic constraints, IEEE Trans. Automat. Control 42 (1997) 819–830. [2] G. Vinnicombe, Uncertainty and Feedback: H∞ Loop-Shaping and the nu-Gap Metric, Imperial College Press, 2000. [3] M. Cantoni, U. Jönsson, C.Y. Kao, Robustness analysis for feedback interconnections of distributed systems via integral quadratic constraints, IEEE Trans. Automat. Control 57 (2) (2012) 302–317. [4] M. Cantoni, S.Z. Khong, U. Jönsson, Robust stability analysis for feedback interconnection of time-varying linear system, SIAM J. Control Optim. 51 (1) (2013) 353–379. [5] M.S. Akram, M. Cantoni, Gap metrics for linear time-varying system, SIAM J. Control Optim. 56 (2) (2018) 782–800. [6] S.Z. Khong, M. Cantoni, Reconciling ν -gap metric and IQC based robust stability analysis, IEEE Trans. Automat. Control 58 (8) (2013) 2090–2095. [7] L. Liu, Y.F. Lu, Stability analysis for time-varying systems via quadratic constraints, Systems Control Lett. 60 (2011) 832–839. [8] A. Feintuch, Robust Control Theory in Hilbert Space, Springer-Verlag 1998. [9] A.R. Teel, On graphs, conic relations, and input–output stability of nonlinear feedback systems, IEEE Trans. Automat. Control 41 (5) (1996) 702–709. [10] T.T. Georgiou, M.C. Smith, Optimal robustness in the gap metric, IEEE Trans. Automat. Control 35 (1990) 673–686. [11] L. Qiu, E.J. Davison, Feedback stability under simultaneous gap metric uncertainties in plant and controller, Systems Control Lett. 18 (1992) 9–22. [12] L. Qiu, E.J. Davison, Pointwise gap metrics on transfer matrices, IEEE Trans. Automat. Control 37 (1992) 741–758.

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L. Liu / Systems & Control Letters 133 (2019) 104550

[13] C. Foias, T.T. Georgiou, M.C. Smith, Robust stability of feedback systems: a geometric approach using the gap metric, SIAM J. Control Optim. 31 (6) (1993) 1518–1537. [14] M. Cantoni, A characterisation of the gap metric for approximation problems, in: Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, December, 2006, pp. 13–15. [15] J.B. Conway, A Course in Functional Analysis, Springer, 1990. [16] W.N. Dale, M.C. Smith, Stabilizanbility and existence of system representation for discrete-time time-varying system, SIAM J. Control Optim. 31 (6) (1993) 1538–1557.

[17] S.M. Djouadi, Commutant lifting for linear time-varying systems, in: American Control Conference, June, 2009, pp. 4067–4072. [18] M. Cantoni, G. Vinnicombe, Linear feedback systems and the graph topology, IEEE Trans. Automat. Control 47 (5) (2002) 710–719, 2002. [19] M. Cantoni, On model reducing in the ν -gap metric, in: Proceedings the 40h IEEE Conference on Decision and Control Orlando, Florida USA, December, 2001, pp. 3665–3670. [20] T.T. Georgiou, M. Khammash, A. Megretski, On a large-gain theorem, Systems Control Lett. 32 (4) (1997) 231–234.