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Dissipative Control and Observation of Linear Systems
with Delays: Part I
Qian Feng
To cite this version:
Dissipative Control and Observation of Linear Systems with
Delays: Part I
Qian Feng
Abstract
The paper consists of two parts. The aim of Part I is to develop optimization-based solutions for the dissipative control and observation of a linear system with general delay structures. Specifically, our system’s model considers multiple pointwise-delay channels with general distributed-delays (DDs) existing at the states, inputs and outputs, where the DD kernels can be any square-integral function over a bounded interval. By applying a novel decomposition approach to the distributed delays, users can choose which square-integral functions inside of the DDs are directly handled and which are approximated by any differentiable function. In addition, the proposed decomposition scheme allows one to construct Krasovskiĭ functionals whose integral kernels are independent of the system’s DD kernels. We then present our proposed methods in several theorems concerning the topics of dissipative state-feedback control, observation and observer-based control design. Sufficient solutions are derived in terms of matrix inequalities which can be solved directly by standard numerical solvers of semidefinite programming if they are convex, or solved by some iterative algorithms. To the best of our knowledge, the dissipative control and observation problem investigated in this paper has not been considered by the existing results in the literature. Numerical examples are presented to demonstrate the effectiveness of the proposed methodologies.
Keywords: Dissipative Systems; General Distributed Delays; Controller Synthesis; Observer design; Observer Based Stabilization.
1. Introduction
Dynamical systems with time delays Richard (2003) can characterize real-time processes affected by transport, propagation or aftereffects Michiels & Niculescu (2014). Mathematically, systems with delays generally can be denoted by functional differential equations Hale & Lunel (1993); Kolmanovskii & Myshkis (1999) or posed as infinite-dimensional systems Curtain & Zwart (1995); Safi et al. (2017). For systems operating in a real-time environment, delays can be introduced by engineering devices such as actuators, sensors, wires or communication channels etc. It has been shown that the presence of delays in a system can lead to positive Ramírez & Sipahi (2018) or negative Silva et al. (2001) consequences. As a result, developing effective methodologies for the analysis and control of time-delay systems is a vital part of the study on control engineering.
In contrast to finite-dimensional systems which are usually denoted by ordinary differential equations, the dimension of the systems with delays is generally infinite. This renders the analysis of time-delay
systems (TDSs) significantly more intricate, even only linear systems are considered. Generally speaking, two different directions, which are based on time and frequency domains, have been proposed to deal with the stabilization and stability analysis of linear delay systems.
To the best of the author’s knowledge, the newest trend of frequency-domain-based methods for the stabilization of linear delay systems is represented by the results in Gumussoy & Michiels (2011); Michiels (2012); Michiels & Niculescu (2014) and Apkarian et al. (2018); Apkarian & Noll (2018)1. These methods are
predominately nourished by the recent development of non-smooth optimization Noll & Apkarian (2005a,b); Lewis (2013); Apkarian et al. (2016). Though stabilizing a linear system with pointwise-delays may be perfectly handled by the frequency-domain based methods, it seems that no such approaches may handle the synthesis problem of a linear system with general distributed-delays (DDs). Specifically, it is not easy to analyze the spectrum of a general DD if its Laplace transform cannot be denoted analytically.
On the other hand, the LMIs-based Krasovskiĭ functional (KF) approach Gu et al. (2003); Kharitonov (2012) is one of the major time-domain-based methods for the stability analysis and controller synthesis (observer design) of linear systems with pointwise-delays, which has produced a significant amount of research outputs Li & de Souza (1995); Choi & Chung (1997); Fridman & Shaked (2001); Fiagbedzi & Cherid (2003); Sename & Briat (2007); Seuret & Gouaisbaut (2013); Feng & Lam (2012); Safi et al. (2017); Peet (2019). For comprehensive collections of the existing literature on this topic, see the monographs in Fridman (2014); Briat (2014). In contrast, research on using the KF approach for systems with general distributed-delays is still underdeveloped, where most existing results have obvious shortcomings. In general, these results often subject to mathematical conservatism and weaknesses, such as imposing restrictions on the DD kernels Fridman & Tsodik (2009); Seuret et al. (2015); Feng & Nguang (2016) or system’s structure Münz et al. (2009). In addition, we found out that no existing methods in the peer-revised literature have dealt with a synthesis problem concerning both multiple pointwise and general DD delays. Based on the aforementioned summary, this clearly motives us to develop non-conservative methodologies for the control and observation of linear systems with general delay structures, ideally imposing minimal restriction on the system’s structure. In this paper, we propose new methods for the control and observation of a linear system with general delay structure based on the LKF approach, where the DD kernels can be anyL2 function over a bounded
interval. Our system considers multiple pointwise and distributed-delays at system’s states, inputs and outputs where an arbitrary number of delays can be considered. The main results of this paper are summa-rized in several theorems and algorithms concerning the topics of dissipative state-feedback design (DSFD), dissipative observer design (DOD) and dissipative observer-based controller design, (DOBC) where all these three problems can be solved via standard numerical solvers of semidefinite programming. Moreover, when no input or output delays are considered, our proposed methods can be further modified to solve variant control and observation problems where the controllers and observers possessing more general forms.
The major contributions of this paper are summarized as follows
• We believe the control and observation problems investigated in this paper has not been considered by existing results in the literature. This is especially true for the problems of observer design and observer-based control. Moreover, the model of the considered system is sufficiently general with the
1The synthesis schemes in Apkarian et al. (2018); Apkarian & Noll (2018) are developed for infinite-dimensional linear
consideration of dissipativity, imposing no constraints on the number of pointwise and distributed-delays. Moreover, all distributed-delays can contain anyL2 function which is clearly sufficient for the
modeling of practical delay systems.
• TheL2distributed-delay kernels are coped with by a novel decomposition approach, which allows users
to determine which DD kernels are directly handled and which are approximated via any continuous function. The proposed decomposition scheme also enables one to construct an LKF whose integral terms can be totally different from the DD kernels. This implies that theoretically the integral terms of the LKF in this paper can contain any function as long as they are differentiable, which significantly indicates the generality of our LKFs.
• For each case of the control or observation or observer-based control, we propose two theorems with an iterative algorithm. Specifically, the second theorem in each case is a convexified version of the first theorem based on the application of Projection Lemma which implies that the structure of the parameters of the LKF is not weakened by the procedure of convexification. Moreover, the first theorem in each case can be solved by the proposed iterative algorithm initiated by a feasible solution of the second theorem in each case. As a result, our proposed methodologies essentially are convex solutions which do not need to be solved by nonlinear optimizations solvers.
The organization of the rest of the paper is outlined as follows. The synthesis problem in this paper is first presented in Section 2. Next, some important lemmas and definition are presented in Section 3 which include a novel integral inequality. The main results on dissipative controller synthesis are presented in Section, whereas the main results on dissipative observer design and dissipative observer-based control are presented in Section 5 and Section 6, respectively. Several Numerical examples are tested in Section 7 prior to the final conclusion.
Notation
YX = {f(·) : f(·) is a function from X onto Y } and R
≥a = {x ∈ R : x ≥ a} and Sn = {X ∈ Rn×n : X = X⊤}. Moreover, C(X # Rn) := nf (·) ∈ (Rn)X : f (·) is continuous on Xo and Ck([a, b]# Rn) := nf (·) ∈ C([a, b] # Rn) : dkf (x)
dxk ∈ C([a, b] # R
n)o in which the derivatives at a and b are one sided.
ML(X )/B(R) X #R=f (·) ∈ XR:∀Y ∈ B(R), f−1(Y) ∈ L(X ) denotes the space of allL (X ) /B(R) mea-surable functions fromX onto R, where L (X ) contains all the subsets of X which are Lebesgue measurable withX ∈ L (R), and B(R) is the Borel σ–algebra on R. In addition,Lp(X # Rn) :={f(·) ∈ M
L(R)/B(Rn) X # Rn:∥f(·)∥ p < +∞} where X ⊆ R n and ∥f(·)∥ p := R X∥f(x)∥ p 2dx 1
p. The notations of universal
quan-tifier∀ and the existential quantifier ∃ are frequently utilized throughout this paper. Sy(X) := X + X⊤ stands for the sum of a matrix with its transpose. Coln
i=1xi:=
Rowni=1x⊤i ⊤ =x⊤1 · · · x⊤i · · · x⊤n⊤denotes a column vector containing a sequence of mathematical objects (scalars, vectors, matrices etc.). The symbol ∗ is used as abbreviations for [∗]Y X = X⊤Y X or X⊤Y [∗] = X⊤Y X or [A B
∗ C] = A B
B⊤C
. On×m stands for a n× m zero matrix which can be abbreviated into On with n = m, whereas 0n denotes a n× 1 vector. The notation⊕ is defined asX⊕ Y =X O
∗ Y
with its n-ary formLνi=1Xi = X1⊕ X2⊕ · · · ⊕ Xν to denote the diagonal sum of matrices. ⊗ stands for the Kronecker product. We use√X to represent2 the unique
square root of X ≻ 0. The order of matrix operations in this paper is matrix (scalars) multiplications >⊗ > ⊕ > +. Finally, empty matrices are applied in this paper, which follow the same definition and rules in Matlab. Note that we define Coln
i=1 = [] when n < 1, where [] is an empty matrix with an appropriate column dimension based on specific contexts.
2. Problem formulations
Consider a linear distributed-delay system ˙ x(t) = ν X i=0 Aix(t− ri) + ν X i=1 Z −ri−1 −ri e Ai(τ )x(t + τ )dτ + ν X i=0 Biu(t− ri) + ν X i=1 Z −ri−1 −ri e Bi(τ )u(t + τ )dτ + D1w(t), z(t) = ν X i=0 Cix(t− ri) + ν X i=1 Z −ri−1 −ri e Ci(τ )x(t + τ )dτ + ν X i=0 Biu(t− ri) + ν X i=1 Z −ri−1 −ri e Bi(τ )u(t + τ )dτ + D2w(t) ∀θ ∈ [−rν, 0], x(t0+ θ) = ψ(τ ), t≥ t0∈ R (1)
to be stabilized, where ψ(·) ∈ C([−rν, 0]# Rn), and rν > rν−1>· · · > r2> r1> r0= 0 are given constants.
Furthermore, x : [t0− rν,∞) → Rn satisfies (1), u(t) ∈ Rp denotes input signals, w(t) ∈ Rq represents disturbance, z(t) ∈ Rm is the regulated output. The size of the given state space parameters in (1) is determined by the values of n∈ N and m; p; q ∈ N0. Note that the distributed-delay integrals in (1) can be
recast into the form of ν X i=1 Z −ri−1 −ri e Ai(τ )x(t + τ )dτ = ν X i=1 Z 0 −1 ´ riAei(´riτ− ri−1) x(t + ´riτ− ri−1)dτ (2) wherer´i= ri− ri−1 and all the integrals at the right side of (2) are defined over [−ri,−ri−1]. Finally, the matrix-valued distributed-delay terms in (1) satisfy
∀i = 1 · · · ν, eAi(·) ∈ L2([−ri,−ri−1]# Rn×n), Cei(·) ∈ L2([−ri,−ri−1]# Rm×n) e
Bi(·) ∈ L2([−ri,−ri−1]# Rn×p), Bei(·) ∈ L2([−ri,−ri−1]# Rm×p).
(3)
Proposition 1. (3) is true if and only if there exists fi(·) ∈ C1([−ri,−ri−1]# Rdi), φi(·) ∈ L2([−ri,−ri−1]# Rδi), ϕ
i(·) ∈ L2([−ri,−ri−1]# Rµi), Mi ∈ Rdi×κi, bAi ∈ Rn×κin, bBi ∈ Rn×κip, bCi ∈ Rm×κin and bBi ∈ Rm×κip with i = 1· · · ν such that
hold for all i = 1· · · ν, where gi(τ ) =
ϕ⊤i (τ ) φ⊤i (τ ) fi⊤(τ )⊤ and κi = di+ δi+ µi, κi = di+ δi with
di; δi; µi ∈ N0 for all i = 1· · · ν. Moreover, the derivatives in (6) at τ = −ri and τ =−ri−1 are one-sided
derivatives.
Proof. First of all, it is straightforward to see that (3) is implied by (4)–(7) since φi(·) ∈ L2([−ri,−ri−1]#Rδi),
fi(·) ∈ C1([−ri,−ri−1]# Rdi)⊂ L2([−ri,−ri−1]# Rdi) and ϕi(·) ∈ L2([−ri,−ri−1]# Rµi) for all i = 1· · · ν. Now we start to prove that the conditions in (3) implies the existence of the parameters in Proposition 1 satisfying (4)–(7). Given any fi(·) ∈ C1([−ri,−ri−1]# Rdi), i = 1· · · ν satisfying
R−ri−1 −ri f
⊤
i (τ )fi(τ )dτ ≻ 0, one can always construct appropriate ϕ⊤i (τ ) and φi(·) ∈ L2([−ri,−ri−1]# Rδi) with Mi ∈ Rdi×κi such that the conditions in (6)–(7) are satisfied with gi(τ ) =
ϕ⊤i (τ ) φ⊤i (τ ) fi⊤(τ )⊤. Note that R−ri−1
−ri f
⊤
i (τ )fi(τ )dτ ≻ 0 is implied by (7) which indicates that the functions in gi(·) in (6) are linearly independent3 in a Lebesgue sense over [−r
i,−ri−1] for each i = 1· · · ν. The aforementioned conclusion is true because dfi(τ )
dτ (·) ∈ C([−ri,−ri−1]# Rdi)⊂ L2([−ri,−ri−1]# Rdi) for all i = 1· · · ν, and the dimensions of φi(τ ) and ϕi(τ ), i = 1· · · ν can be arbitrarily enlarged with more linearly independent functions. Note that if any vector-valued function fi(τ ), φi(τ ), ϕi(τ ) is []0×1, then it can be handled by properties of empty
matrices.
Since the dimensions of gi(τ ) in (6)–(7) can be arbitrarily increased, hence there always exist ´Ai,j∈ Rn×n, ´
Ci,j ∈ Rm×n, ´Bi,j ∈ Rn×p, ´Bi,j ∈ Rm×p and gi(τ ) = Colκj=1i gi,j(τ ) =
ϕ⊤i (τ ) φ⊤i (τ ) fi⊤(τ )⊤ for the distributed delay terms in (3) such that
∀τ ∈ [−ri,−ri−1], eAi(τ ) = κi X j=1 ´ Ai,jgi,j(τ ), Cei(τ ) = κi X j=1 ´ Ci,jgi,j(τ ), (8) ∀τ ∈ [−ri,−ri−1], eBj(τ ) = κi X j=1 ´ Bi,igi,j(τ ), eBi(τ ) = κi X j=1 ´ Bi,jgi,j(τ ) (9)
with κi ∈ N0 for i = 1· · · ν, where φi(·) ∈ L2([−ri,−ri−1]# Rδi), fi(·) ∈ C1([−ri,−ri−1]# Rdi) and
ϕi(·) ∈ L2([−ri,−ri−1]# Rµi) satisfy (6)–(7) for some Mi∈ Rdi×κi, i = 1· · · ν. Now (8)–(9) can be further rewritten as ∀τ ∈ [−ri,−ri−1], eAi(τ ) = κ i Row i=1 ´ Ai,j (g(τ )⊗ In) , Cei(τ ) = κ i Row i=1 ´ Ci,j (g(τ )⊗ In) ∀τ ∈ [−ri,−ri−1], eBi(τ ) = κ i Row i=1 ´ Bi,j (g(τ )⊗ Ip) , Bei(τ ) = κ i Row i=1 ´ Bi,j (g(τ )⊗ Ip) (10)
which are in line with the decompositions in (4)–(5) by letting bAi = Rowκi=1i A´i,j, bCi = Rowκi=1i C´i,j, b
Bi = Rowκi=1i B´i,j and bBi = Rowκi=1i B´i,j for all i = 1· · · ν. Given all the aforementioned statements we
have presented, then Proposition 1 is proved. ■
Remark 1. The decompositions in (4)–(5) provide an effective way to handle the infinite-dimensional
distributed-delay terms in (1) by using groups of “basis” functions to decompose them at each delay interval [−ri,−ri−1]. The potential choice of the functions in (4)–(5) will be further discussed in the next section in light of the construction of a KF related to fi(·).
Remark 2. The method proposed in this paper allows users to decide whichL2functions in the
distributed-delay terms to be approximated and which are handled directly without using approximations. Considering (4)–(6), it indicates that φi(τ ) is handled directly together with fi(τ ) and ϕi(τ ) can be approximated. Note that (4)–(7) are assumed based on the requirements of constructing a KF to derive synthesis conditions. Thus in order to fully understand the rationale behind the mathematical structures in (4)–(7), one should refer to the main results on stabilization in later sections.
2.1. Formulation of closed-loop system
The following property of the Kronecker product will be used throughout the paper.
Lemma 1. ∀X ∈ Rn×m, ∀Y ∈ Rm×p, ∀Z ∈ Rq×r,
(X⊗ Iq)(Y ⊗ Z) = XY ⊗ Z = XY ⊗ ZIr= (X⊗ Z)(Y ⊗ Ir). (11) (X⊗ Iq)(Y ⊗ Z) = XY ⊗ Z = ImXY ⊗ (ZIr) = (Im⊗ Z)(XY ⊗ Ir). (12)
Moreover,∀X ∈ Rn×m, we have A B C D ⊗ X = A⊗ X B ⊗ X C⊗ X D ⊗ X (13) for any A, B, C, D with appropriate dimensions which make the block matrix at the left hand of the equality in (13) to be compatible.
Considering the decompositions in Proposition 1, let χ(t, θ) = Colν
i=1x(t + ´riθ− ri−1) ∈ Rnν with
θ∈ [−ri,−ri−1] and apply a state feedback controller u(t) = Kx(t), K ∈ Rp×n to (1), then the resulting closed loop system can be expressed as
˙ x(t) = (A0+ B0K) x(t) + ν Row i=1 (Ai+ BiK) χ(t,−1) + D1w(t) + ν X i=1 Z −ri−1 −ri b Ai+ bBi(Iκi⊗ K) (gi(τ )⊗ In) x(t + τ )dτ, t≥ t0 z(t) = (C0+ B0K) x(t) + ν Row i=1 (Ci+ BiK) χ(t,−1) + D2w(t) + ν X i=1 Z −ri−1 −ri b Ci+ bBi(Iκi⊗ K) (gi(τ )⊗ In) x(t + τ )dτ ∀θ ∈ [−rν, 0], x(t0+ θ) = ψ(τ ) (14)
where the form of the distributed-delays are obtained via the relations
(gi(τ )⊗ Ip) K = (gi(τ )⊗ Ip) (1⊗ K) = Iκigi(τ )⊗ KIn = (Iκi⊗ K) (gi(τ )⊗ In) , i = 1· · · ν. (15) Remark 3. The use of χ(t, θ) in (14) is inspired by the state variable z(t, θ) of the ODE-PDE coupled
system in Safi et al. (2017). By introducing χ(t, θ) = Colν
i=1x(t + ´riθ− ri−1) with θ∈ [−ri,−ri−1], the expressions in (14) can be denoted by a more compact form. The advantage of utilizing χ(t, θ) will be further illustrated in light of the derivation of the synthesis conditions in the next section.
In this paper, bfi(τ ) is utilized to approximate the function ϕi(τ ) in gi(τ ) based on the application of Hilbert projection theorem. Namely,
ϕi(τ ) = Γif (τ ) + εb i(τ ), i = 1· · · ν, τ ∈ [−ri,−ri−1] (16) where Γi:= Z −ri−1 −ri ϕi(τ ) bfi⊤(τ )dτ𝟋i, 𝟋i= Z −ri−1 −ri b fi(τ ) bfi⊤(τ )dτ i = 1· · · ν (17) and εi(τ ) := ϕi(τ )− Γifbi(τ ) defines the error of approximations. Note that 𝟋−1i in (28) are well defined given the conditions in (7). To measure the error residual of the approximation scheme in (16), we utilize
Ei:=
Z −ri−1 −ri
εi(τ )ε⊤i(τ )dτ ∈ S
where E−1i are well defined given the conclusion of the eq.(18) in Feng et al. (2020). Now by (16) and the definition of bfi(τ ) in Proposition 1, we have
gi(τ ) = h ϕ⊤i (τ ) φ⊤i (τ ) fi⊤(τ ) i⊤ = bΓifbi(τ ) + eIiεi(τ ), bΓi= " Γi Iκi # , eIi = " Iµi Oκi×µi # (19) which further gives the identity
(Iκi⊗ K) (gi(τ )⊗ In) = (Iκi⊗ K) h bΓifbi(τ ) + eIiεi(τ ) ⊗ In i = (Iκi⊗ K) bΓi⊗ In b fi(τ )⊗ In + (Iκi⊗ K) eIi⊗ In (εi(τ )⊗ In) = bΓi⊗ Ip (Iκi⊗ K) b fi(τ )⊗ In + eIi⊗ Ip (Iκi⊗ K) (εi(τ )⊗ In) , i = 1· · · ν (20) considering the relation in (15) with (11). Moreover, we can also obtain the identities
ν Row i=1 h b Ai+ bBi(Iκi⊗ K) (gi(τ )⊗ In) = ν Row i=1 b Ai bΓi⊗ In + bBi bΓi⊗ K i b fi(τ )⊗ In + ν Row i=1 h b Ai eIi⊗ In + bBi eIi⊗ K i (εi(τ )⊗ In) = ν Row i=1 b Ai bΓi √ 𝟋i⊗ In + bBi bΓi √ 𝟋i⊗ K Mν i=1 q 𝟋−1i fbi(τ )⊗ In + ν Row i=1 b Ai eIi p Ei⊗ In + bBi eIi p Ei⊗ K Mν i=1 q E−1i εi(τ )⊗ In (21) ν Row i=1 b Ci+ bBi(Iκi⊗ K) (gi(τ )⊗ In) = ν Row i=1 b Ci bΓi √ 𝟋i⊗ In + bBi bΓi √ 𝟋i⊗ K Mν i=1 q 𝟋−1 i fbi(τ )⊗In + ν Row i=1 b Ci eIi p Ei⊗ In + bBi eIi p Ei⊗ K Mν i=1 q E−1i εi(τ )⊗ In (22) based on (19) and (11) with the properties of block matrices, whereLν
i=1Xi= X1⊕ X2⊕ · · · ⊕ Xν. Now by (19)–(21) with (11), the system in (14) can be further simplified into
˙ x(t) = A + B1[(I1+ν+κ⊗ K) ⊕ Oq] ϑ(t) z(t) = (C + B2[(I1+ν+κ⊗ K) ⊕ Oq]) ϑ(t), t≥ t0 ∀θ ∈ [−rν, 0], x(t0+ θ) = ψ(θ) (23)
with t0 and ψ(·) in (1), where κ =
Pν
i=1κi with κi= di+ δi+ µi and
ϑ(t) = x(t) χ(t,−1) Colνi=1R−r−ri−1
i q 𝟋−1 i fbi(τ ) ⊗ In x(t + τ )dτ
Colνi=1R−r−ri−1 i q E−1i εi(τ ) ⊗ In x(t + τ )dτ w(t) (28)
3. Important lemmas and definition
In this section, some lemmas and definition are presented which are crucial for the derivations of the results in the next section. A novel integral inequality is also derived to handle the presence of multiple delay channels in the context of constructing KFs.
Firstly, we define the following weighted Lebesgue function space L2 ϖ K # R d:=nϕ(·) ∈ M L(R)/B(Rd) K # Rd :∥ϕ(·)∥2,ϖ<∞ o (29) with d∈ N and ∥ϕ(·)∥2,ϖ:= R
Kϖ(τ )ϕ⊤(τ )ϕ(τ )dτ where ϖ(·) ∈ ML(R)/B(R)(K # R≥0) and the function ϖ(·) has only countably infinite or finite number of zero values. Furthermore,K ⊆ R ∪ {±∞} and the Lebesgue measure ofK is non-zero.
Lemma 2. GivenK and ϖ(·) in (29) and U ∈ Sn
⪰0:={X ∈ Sn : X⪰ 0}with n∈ N. Let fi(·) ∈ L2ϖ K # Rli and g(·) ∈ L2
ϖ K # Rλi
with li∈ N and λi∈ N0, i = 1· · · ν, in which the functions fi(·) and gi(·) satisfy Z K ϖ(τ ) gi(τ ) fi(τ ) g⊤i (τ ) f⊤i (τ ) dτ ≻ 0, i = 1 · · · ν. (30)
Then the inequality Z K ϖ(τ )x⊤(τ ) ν M i=1 Ui ! x(τ )dτ ≥ [∗] " ν M i=1 F−1 i ! ⊗ ν M i=1 Ui !# Z K ϖ(τ ) " ν M i=1 fi(τ ) ! ⊗ In # x(τ )dτ ! + [∗] " ν M i=1 E−1 i ! ⊗ ν M i=1 Ui !# Z K ϖ(τ ) " ν M i=1 ei(τ ) ! ⊗ In # x(τ )dτ ! (31) holds for all x(·) ∈ L2
ϖ(K # Rnν), where Fi = R Kϖ(τ )fi(τ )f⊤i (τ )dτ ∈ Sd≻0. In addition, ei(τ ) = gi(τ )− Aifi(τ )∈ Rλi and Ai= R Kϖ(τ )gi(τ )f⊤i (τ )dτFi∈ Rλi×li andEi:= R Kϖ(τ )ei(τ )e⊤i (τ )dτ ∈ Sλi.
Proof. By using the inequality in eq.(10) in ν times, then (31) can be obtained. Note that the definition of
Fi here is different from the definition of the related term in ■
A stability criterion based on the KF approach and the definition of dissipativity are presented as follows.
Lemma 3. Let w(t)≡ 0q in (23) and r2 ≥ r1 ≥ 0, r2 > 0 be given, then the trivial solution x(t)≡ 0n
of (23) is uniformly asymptotically stable in C([−rν, 0]# Rn) if there exist ϵ1; ϵ2; ϵ3> 0 and a differentiable
functional v :C([−rν, 0]# Rn)→ R with v(0n(·)) = 0 such that
for any ϕ(·) ∈ C([−rν, 0]# Rn) in (23), where t0 is given in (23) and ∥ϕ(·)∥ 2 ∞ := sup−rν≤τ≤0∥ϕ(τ)∥ 2 2 and d+ dxf (x) := limsupη↓0 f (x+η)−f(x) η . Furthermore, xt(·) in (33) is defined by ∀t ≥ t0, ∀θ ∈ [−rν, 0], xt(θ) = x(t + θ) in which x : [t0− rν,∞) → Rn satisfies (23) with w(t)≡ 0q.
The following definition of the dissipativity of (23) is based on the general definition of dissipativity in Willems (1972).
Definition 1. The closed-loop system (23) with a supply rate function s(z(t), w(t)) is said to be dissipative
if there exists a differentiable functional v :C([−rν, 0]# Rn)→ R such that
∀et≥ t0, ˙v(xt(·)) − s(z(t), w(t)) ≤ 0 (34)
with t0in (23) and z(t), w(t) in (23). Moreover, xt(·) in (34) is defined by the equality ∀t ≥ t0,∀θ ∈ [−rν, 0], xt(θ) = x(t + θ) with x(t) satisfying (23).
Note that (34) is equivalent to the original definition of dissipativity via the application of the properties of Lebesgue integrations. To characterize dissipativity, a quadratic supply function
s(z(t), w(t)) = " z(t) w(t) #⊤"e J⊤J1−1Je J2 ∗ J3 # " z(t) w(t) # , Sm∋ eJ⊤J1−1Je⪯ 0, Sm∋ J1−1≺ 0, eJ ∈ Rm×m (35) is applied in this paper where the structure of (35) is constructed in this paper based the general quadratic constraints applied in Scherer et al. (1997) together with the idea of factorizing the matrix Uj in Scherer
et al. (1997). Note that (35) is able to characterize numerous performance criteria such as
• L2 gain performance: J
1=−γIm, J = Ie m, J2= Om,q, J3= γIq where γ > 0. • Passivity: J1∈ Sm≺0, J = Oe m, J2= Im, J3= Omwith m = q.
4. Main results on dissipative state feedback controller synthesis
The main results on dissipative controller synthesis are presented in this section, which are summarized in two theorems and an algorithm. Specifically, the second theorem is proposed as a convexification of the bilinear term in a condition of the first theorem which can be further solved iteratively by the proposed algorithm.
Theorem 1. Let all the parameters in Proposition 1 be given, then the closed-loop system (23) with the
supply rate function in (35) is dissipative and the trivial solution of (23) with w(t) ≡ 0q is uniformly
asymptotically stable if there exist K ∈ Rp×n and P
where Σ = C + B2[(I1+ν+κ⊗ K) ⊕ Oq] and Ω = A + B1[(I1+ν+κ⊗ K) ⊕ Oq] with C and B2 in (26) and (27), and
Ψ = Sy
I
nO
n,ϱO
νn×nO
νn×ϱO
κn×nbI
⊤O
µn×nO
µn×ϱO
q×nO
q×ϱ
P
1P
2∗ P
3Ω
bF ⊗ I
nO
ϱ×(nµ+q)−
"
O
(n+nν+nκ)×mJ
2⊤#
Σ
+ Ξ
(39) Ξ = (Q + RΛ)⊕ On⊕ Oκn⊕ Oq − On⊕ Q ⊕ (Iκ⊗ R) ⊕ (Iµ⊗ R) ⊕ J3 , (40) bI = Mν i=1 ´ ri q F−1i Odi×δi Idi √ 𝟋i ! ⊗ In, Λ = ν M i=1 ´ riIn, ´ri= ri− ri−1 (41) bF =hLν i=1 q F−1i fi(0) 0d Od×κ i −hOd Lν i=1 q F−1i fi(−1) Lν i=1 q F−1i Mi √ 𝟋i i (42) with κ = Pν i=1κi, κi = di + δi and µ = Pνi=1µi, where A, B1 are given in (24)–(25) and Fi = R0
−1fi(τ )fi⊤(τ )dτ , i = 1· · · ν. Moreover, the rest of the parameters in (38) is defined as
P = h P1 On×νn P2bI On×nµ On,q On,m i , Π =Ω On,m (43) and Φ = Sy P2 Oνn×ϱ bI⊤P 3 O(nµ+q+m)×ϱ hbF ⊗ In Oϱ×(nµ+q+m) i + O(n+nν+nκ)×m −J⊤ 2 e J Σ Om + Ξ ⊕ (−J1) . (44)
Proof. The proof of Theorem 1 is based on the construction of the KF:
v(xt(·)) = η⊤(t) P1 P2 ∗ P3 η(t) + Z 0 −1 χ⊤(t, τ )QΛ + (1 + τ )RΛ2χ(t, τ )dτ (45) where xt(·) follows the same definition in (34), and P1∈ Sn, P2∈ Rn×ϱ, P3∈ Sϱ, and
Q = ν M i=1 Qi, R = ν M i=1 Ri, Qi∈ Sn, Ri∈ Sn, i = 1· · · ν (46) η(t) := Col x(t), ν Col i=1 Z −ri−1 −ri q F−1i fi(τ ) ⊗ In x(t + τ )dτ (47) with Fi = R−ri−1 −ri fi(τ )f ⊤ i (τ )dτ , i = 1· · · ν. Note that q
F−1i , i = 1· · · ν are well defined and unique due to the conditions in (7).
From the definition of χ(t, θ) = Colν
i=1x(t + ´riθ− ri−1) with (46), one can derive the following relations
ν X i=1 d dt Z −ri−1 −ri (τ + ri)[∗]Rix(t + τ )dτ = ν X i=1 d dt Z −ri−1 −ri (´riτ + ´ri)[∗]Rix(t + ´riτ− ri−1)d(´riτ ) = d dt Z 0 −1 χ⊤(t, τ )(1 + τ )RΛ2χ(t, τ )dτ = 2 Z 0 −1 χ⊤(t, τ )(1 + τ )RΛ2d dtχ(t, τ )dτ = 2 Z 0 −1 χ⊤(t, τ )(1 + τ )RΛ d dτχ(t, τ )dτ = 2χ ⊤(t, 0)RΛχ(t, 0)− 2Z 0 −1 χ⊤(t, τ )RΛχ(t, τ )dτ − 2 Z 0 −1 d dτχ ⊤(t, τ )(1 + τ )RΛχ(t, τ )dτ = χ⊤(t, 0)RΛχ(t, 0)−Z 0 −1 χ⊤(t, τ )RΛχ(t, τ )dτ. (50)
Given t0 ∈ R in (23) with rν >· · · r2 > r1 > r0 = 0 and the relations in (48)–(50), differentiating v(xt(·)) along the trajectory of (23) and consider (35) produces
∀t ≥ t0, ˙v(xt(·)) − s(z(t), w(t)) = ϑ⊤(t) Sy In On,ϱ Oνn×n Oνn×ϱ Oκn×n bI⊤ Oµn×n Oµn×ϱ Oq,n Oq,ϱ P1 P2 ∗ P3 "A + B 1[(I1+ν+κ⊗ K) ⊕ Oq] h bF ⊗ In Oϱ×(nµ+q) i #−"O(n+nν+nκ)×m J2⊤ # Σ ϑ(t) + χ⊤(t, 0) (Q + RΛ) χ(t, 0)− χ⊤(t,−1) Qχ(t, −1) − Z 0 −1 χ⊤(t, τ )RΛχ(t, τ )dτ − w⊤(t)J 3w(t)− ϑ⊤(t)Σ⊤Je⊤J1−1J Σϑ(t)e (51)
where Λ =Lνi=1riInand ϑ(t) is given in (28) and Σ, bI and bF are defined in the statements of Theorem 1. Note that bI and bF in (41)–(42) are obtained by the identities
Z 0 −1 d dτ ν M i=1 q F−1i fi(τ )⊗ In ! χ(t, τ )dτ = ν M i=1 q F−1i r´iMi √ 𝟋i⊗ In ! Z 0 −1 ν M i=1 q 𝟋−1i fbi(τ )⊗ In ! χ(t, τ )dτ = ν M i=1 q F−1i Mi √ 𝟋i⊗ In ! ν Col i=1 Z −ri−1 −ri q 𝟋−1 i fbi(τ ) ⊗ In x(t + τ )dτ . (54)
Note that also the parameters A, B1, C and B2 in (51) are given in (24)–(27).
Given (37) and (48), apply (31) with ϖ(τ ) = 1 and fi(τ ) = bfi(´riτ− ri−1), i = 1· · · ν to the integral termsR−10 χ⊤(t, τ )ΛRχ(t, τ )dτ =Pνi=1R−r−ri−1
i x ⊤(t + τ )Rix(t + τ )dτ in (51). Then we have Z 0 −1 χ⊤(t, τ )ΛRχ(t, τ )dτ = ν X i=1 Z −ri−1 −ri x⊤(t + τ )Rix(t + τ )dτ ≥ ν X i=1 Z −ri−1 −ri x⊤(t + τ ) b fi⊤(τ ) q 𝟋−1 i ⊗ In dτ Iκ⊗ Ri Z −ri−1 −ri q 𝟋−1 i fbi(τ )⊗ In x(t + τ )dτ = [∗] Iκ⊗ R ν Col i=1 Z −ri−1 −ri q 𝟋−1 i fbi(τ ) ⊗ In x(t + τ )dτ (55) through which one can derive the inequality
∀t ≥ t0, ˙v(xt(·)) − s(z(t), w(t)) ≤ ϑ⊤(t)
Ψ− Σ⊤Je⊤J1−1J Σe
ϑ(t) (56)
with Ψ in (39) and ϑ(t) in (28). Now it is obvious to conclude that if (37) and Ψ− Σ⊤Je⊤J−1
1 J Σe ≺ 0 are
satisfied, then
∃ϵ3> 0 : ∀t ≥ t0, ˙v(xt(·)) − s(z(t), w(t)) ≤ −ϵ3∥x(t)∥2. (57)
Moreover, considering the structure of Ψ− Σ⊤Je⊤J−1
1 J Σe ≺ 0, one can conclude that (57) infers
∃ϵ3> 0, d+ dtv(xt(·)) t=t0,xt0(·)=ϕ(·) ≤ −ϵ3∥ϕ(0)∥2 (58)
for any ϕ(·) ∈ C([−rν, 0]# Rn) in (23) with t = t0 and w(t)≡ 0q. Note that xt(·) in (58) is in line with the definition in (33). As a result, it is obvious that the existence of the feasible solutions of (37) and
Ψ− Σ⊤Je⊤J1−1J Σe ≺ 0 infers that (45) satisfies (34) and (33). Finally, applying the Schur complement to
Ψ− Σ⊤Je⊤J1−1J Σe ≺ 0 with (37) and J1−1 ≺ 0 gives (38). Therefore we have proved that the existence of the feasible solutions of (37) and (38) infer the existence of a functional (45) and ϵ3> 0 satisfying (34) and
(33).
Now we start to show that there exist ϵ1> 0 and ϵ2> 0 such that (45) satisfies (32) if (36) and (37) are
satisfied. Let∥ϕ(·)∥2
∞:= sup−rν≤τ≤0∥ϕ(τ)∥ 2
2and consider the structure of (45) with t = t0, it follows that
for any ϕ(·) ∈ C ([−rν, 0]# Rn) in (23), where (59) is derived via the property of quadratic forms: ∀X ∈ Sn,∃λ > 0 : ∀x ∈ Rn\ {0}, x⊤(λI
n− X) x > 0 together with the application of (31) with ϖ(τ) = 1 and appropriate f(τ ). Consequently, the inequality in (59) shows that it is possible to find an upper bound for (45) which satisfies (32) with a ϵ2> 0.
Now we want to prove that the existence of the feasible solutions of (36) and (37) infer that (45) satisfies (32) with certain ϵ1> 0 and ϵ2> 0. Applying (31) to (45) with ϖ(τ ) = 1 and appropriatef(τ ) produces
Z 0 −1 χ⊤(t, τ )ΛQχ(t, τ )dτ = ν X i=1 Z −ri−1 −ri x⊤(t + τ )Qix(t + τ )dτ≥ Z −ri−1 −ri x⊤(t + τ ) fi⊤(τ ) q F−1i ⊗ In dτ Id⊗ Qi Z −ri−1 −ri q F−1i fi(τ )⊗ In x(t + τ )dτ = [∗] Id⊗ Q ν Col i=1 Z −ri−1 −ri q F−1i fi(τ )⊗ In x(t + τ )dτ (60) provided that (37) holds. Moreover, by utilizing (60) to (45) with (37) and (59), it is clear to see that the existence of the feasible solutions of (36) and (37) infer that (45) satisfies (32) with some ϵ1; ϵ2> 0.
In conclusion, we have shown that the existence of the feasible solutions of (36)–(38) infers the existence of a functional (45) and ϵ1; ϵ2 > 0 satisfying the dissipative condition in (34), and the stability criteria in
(32)–(33). As a result, it shows that the existence of the feasible solutions of (36)–(38) infers that the trivial solution of (23) with w(t)≡ 0q is uniformly asymptotically stable, and (23) with (35) is dissipative. ■
Remark 4. Theoretically, fi(τ ) in (47) can be any differentiable function since the decompositions in (4)– (5) are always achievable via the proper choices of φi(τ ) and ϕi(τ ). This gives great flexibility to the structure of the KF (45) in this paper. On the other hand, the functions in fi(τ ) can be selected based on the elements in the distributed-delay terms of (1).
The inequality in (38) is bilinear if a synthesis problem is concerned, where it cannot be solved directly via standard semidefinite programming solvers. In the following theorem, a convex dissipative synthesis condition is derived via the application of Projection Lemma Gahinet & Apkarian (1994) whose feasible solutions infer the existence of the feasible solutions of the conditions in Theorem 1.
Lemma 4 (Projection Lemma). Gahinet & Apkarian (1994) Given n; p; q ∈ N, Π ∈ Sn, P ∈ Rq×n, Q ∈ Rp×n, there exists Θ∈ Rp×q such that the following two propositions are equivalent :
Π + P⊤Θ⊤Q + Q⊤ΘP ≺ 0, (61)
P⊥⊤ΠP⊥≺ 0 and Q⊤⊥ΠQ⊥≺ 0, (62)
where the columns of P⊥ and Q⊥ contain bases of null space of matrix P and Q, respectively, which means that P P⊥= O and QQ⊥= O.
Proof. Refer to Gahinet & Apkarian (1994) and Briat (2014). ■
Theorem 2. Given the functions and parameters in Proposition 1 with{αi}1+ν+κi=1 ⊂ R, then the closed-loop
system (23) with the supply rate function in (35) is dissipative and the trivial solution of (23) with w(t)≡ 0q
is uniformly asymptotically stable if there exists ´P1∈ Sn, ´P2 ∈ Rn×ϱ, ´P3 ∈ Sϱ and ´Qi; ´Ri ∈ Sn, i = 1· · · ν
and V ∈ Rp×n such that
Sy In Col1+ν+κi=1 αiIn O(q+m),n −X ´Π +On P´ ∗ Φ´ ≺ 0 (65) where ´P = h ´ P1 On×νn P´2bI On×(nµ+q+m) i and ´ Π =A [(I1+ν+κ⊗ X) ⊕ Iq] + B1[(I1+ν+κ⊗ V ) ⊕ Oq] On,m with bI in (41) and ´ Φ = Sy ´ P2 Oνn×ϱ bI⊤P´3 O(nµ+q+m)×ϱ h bF ⊗ In Oϱ×(nµ+q+m) i + O(n+nν+nκ)×m −J⊤ 2 e J Σ´ Om − On⊕ ´Q⊕ Iκ⊗ ´R ⊕Iµ⊗ ´R ⊕ J3⊕ (−J1) + ´ Q + Λ ´R ⊕ On⊕ Oκn⊕ Oq+m (66) with bFin (42) and ´Σ = C [(I1+ν+κ⊗ X) ⊕ Iq] + B2[(I1+ν+κ⊗ V ) ⊕ Oq]and A,B1,B2,C in (24)–(27). The
controller gain is calculated via K = V X−1.
Proof. First of all, note that the inequality Sy P⊤Π+ Φ≺ 0 in (38) can be reformulated into
Sy P⊤Π+ Φ = Π In+nν+nκ+q+m ⊤ On P ∗ Φ Π In+nν+nκ+q+m ≺ 0 (67)
which is equivalent to (38). It is easy to observe that the structure of (67) is similar to one of the inequalities in (62) as part of the statements of Projection Lemma. Given the fact that two matrix inequalities are presented in (62), thus a new matrix inequality must be constructed accordingly. Now consider the following inequality Υ⊤ On P ∗ Φ Υ≺ 0 (68) with Υ⊤ := h O(q+m),(2n+nν+κn) Iq+m i
, which can be further simplified into
Υ⊤ On P ∗ Φ Υ = −J3− Sy(D2⊤J2) D⊤2Je ∗ J1 ≺ 0. (69)
Note that (69) is the very matrix produced by extracting the 2× 2 block matrix at the bottom right of the matrices Sy P⊤Π+ Φ or Φ. As a result, it is clear that (69) is automatically satisfied if (67) or (38) holds. Hence the constructed inequality (69) has no impact to the solvability of the original condition in (38). On the other hand, the following identities
−In Π Π In+nν+nκ+q+m = On×(n+nν+nκ+q+m), −In Π ⊥= Π In+nν+nκ+q+m , I2n+nν+κn O(2n+nν+κn),(q+m) O(2n+nν+κn),(q+m) Iq+m =I2n+nν+κn O(2n+nν+κn),(q+m) Υ = O(2n+nν+κn),(q+m), Υ =I2n+nν+κn O(2n+nν+κn),(q+m) ⊥= O(2n+nν+κn),(q+m) Iq+m (70)
which satisfy rank −In Π
= n and rank I2n+nν+κn O(2n+nν+κn),(q+m)
Applying Lemma 4 to (67) and (69) with (70) yields the conclusion that (67) and (69) are true if and only if ∃W ∈ R(2n+nν+κn)×n: Sy I2n+nν+κn O(q+m),(2n+nν+κn) W−In Π + On P ∗ Φ ≺ 0. (71)
Now the inequality in (71) is still bilinear due to the product between W and Π. To convexify (71), consider
W := ColW, Col1+ν+κi=1 αiW
(72) with W ∈ Sn and{α
i}1+ν+κi=1 ⊂ R. With (72), (71) becomes
Θ = Sy W Col1+ν+κi=1 αiW O(q+m),n −In Π +On P ∗ Φ ≺ 0 (73)
which infers (67). Note that having the structural constraints in (72) infers that (73) is no longer an equivalent but only a sufficient condition implying (67) or (38). It is also important to stress that an invertible W is automatically implied by (73) since the expression −2W is the only element at the first diagonal block of Θ.
Let X⊤ = W−1, we apply congruence transformations to the matrix inequalities in (36),(37) and (73) with the fact that an invertible W is implied by (73). Then one can conclude that
(Iν⊗ X) Q (Iν⊗ X) ≻ 0, (Iν⊗ X) R (Iν⊗ X) ≻ 0, I2+ν+κ⊗ X⊤ ⊕ Iq+m Θ [(I2+ν+κ⊗ X) ⊕ Iq+m]≺ 0, [∗] P1 P2 ∗ P3 (I1+ν⊗ X) ≻ 0 (74)
hold if and only if (36),(37) and (73) hold. Moreover, with (11) and ´ P1 P´2 ∗ P´3 := [∗] P1 P2 ∗ P3 (I1+ν⊗ X) , ´Q = ν M i=1 ´ Qi = ν M i=1 XQiX = (Iν⊗ X) Q (Iν⊗ X) , (75) the inequalities in (74) can be rewritten into (63) and (64) and
[∗] Θ [(I2+ν+κ⊗ X) ⊕ Iq+m] = ´Θ = Sy Col1+ν+κIn i=1 αiIn O(q+m),n −X ´Π +On P´ ∗ Φ´ ≺ 0 (76) where ´ P = XP [(I1+ν+κ⊗ X) ⊕ Iq+m] = h ´ P1 On×νn P´2bI On×µn On,q On,m i (77) and ´ Π = Π [(I1+ν+κ⊗ X) ⊕ Iq+m] = A [(I1+ν+κ⊗ X) ⊕ Iq] + B1[(I1+ν+κ⊗ KX) ⊕ Oq] On,m =A [(I1+ν+κ⊗ X) ⊕ Iq] + B1[(I1+ν+κ⊗ V ) ⊕ Oq] On,m (78) with V = KX and ´Φ in (66). Note that (76) is the same as (65), and the form of ´Φ in (66) is derived via
the relations bI (Iκ⊗ X) = (Id⊗ X) bI and h bF ⊗ In Oϱ,(q+m) i [(I1+ν+κ⊗ X) ⊕ Iq+m] = h IdbF ⊗ XIn Oϱ,(q+m) i = (Id⊗ X) h bF ⊗ In Oϱ,(q+m) i , (79)
which are derived from the properties of matrices with (11),(13). Furthermore, since−2X is the only element at the first diagonal block of ´Θ in (65), thus X is invertible if (65) holds. This is consistent with the fact
As a result, we have shown the equivalence between (36)–(37) and (63)–(64). Meanwhile, it has been shown that (65) is equivalent to (73) which infers (38). Consequently, (36)–(38) are satisfied if (63)–(65) hold with some W ∈ Sn and {α
i}1+ν+κi=1 ⊂ R. Thus it demonstrates that the existence of the feasible solutions of (63)–(65) ensures that the trivial solution x(t)≡ 0n of the closed-loop system (23) with w(t) ≡ 0q is
uniformly asymptotically stable and (23) with (35) is dissipative. ■
Remark 5. The use of Projection Lemma in (71) successfully decouples the bilinear term in (38) between
K and P which contains variables which are part of the functional in (45). Although the assumption in (72) can introduce conservatisms compared to the original condition in (38), the structure of (63) is not simplified compared to (36) as the structure of ´P in (65) is identical to P in (38). As a result, it is not
unreasonable to believe that the resulting synthesis condition in Theorem 2 is less conservative than the condition constructed via simplification of the functional parameters P1and P2.
Remark 6. Theorem 2 is specifically derived to solve a synthesis problem. If an open-loop system is
concerned with Bi = eBi(τ ) = On,p and Bi = eBi(τ ) = Om,p, i = 1· · · ν, then Theorem 1 should be applied instead of Theorem 2 as the introduction of the slack variables in Theorem 2 does not introduce extra feasibility with reference to the optimization constraints in 1.
Remark 7. For{αi}1+ν+κi=1 ⊂ R in (65), some values of αi can be more significant than others in terms of their impact on the feasibility of (65). For example, the value of α1 may have a significant impact on the
feasibility of (65) since it may determine the feasibility of the very diagonal block related to A0 in (65). A
simple assignment of{αi}1+ν+κi=1 ⊂ R can be αi = 0 for i = 2· · · 1 + ν + κ which allows one to only adjust the value of α1∈ R to use Theorem 2.
4.1. An inner convex approximation solution of Theorem 1
By fixing the values of {αi}1+ν+κi=1 ⊂ R, Theorem 2 provides a convex synthesis solution of (23). Never-theless, the simplification applied in (72) can render Theorem 2 to be more conservative than Theorem 1. In this subsection, an iterative algorithm is derived based on the method proposed in Dinh et al. (2012) to solve the conditions in Theorem 1 in an iterative fashion, where the algorithm can be initiated by a feasible solution of Theorem 2. Thus the advantage of both Theorem 1 and 2 are combined together in the proposed algorithm without the need to solve nonlinear optimization constraints.
First of all, note that the inequality in (38) is nonconvex in general whereas (36) and (37) remain convex even when a synthesis problem is considered. Now it is obvious that (38) can be rewritten into
U(H, K) := SyP⊤Π+ Φ = Sy P⊤B [(I1+ν+κ⊗ K) ⊕ Op+m] + bΦ≺ 0 (80) with B =hB1 On,m i and bΦ = Sy P⊤ h A On,m i
+ Φ, where P is given in (43), and A and B1are given
in (24)–(25), and H :=hP1 P2
i
with P1 and P2 in Theorem 1. Note that there are no products between
decision variables in bΦ in (80), thus bΦ contains no non-convexities. Considering the results of Example 3
in Dinh et al. (2012), one can conclude that ∆
·
, eG,·
, eΓis a psd-convex overestimate of ´∆(G, Γ) = T + SyG⊤Γwith respect to the parameterization " vec( eG) vec(eΓ) # = " vec(G) vec(Γ) # . (83) Let T = bΦ, G = P = h P1 On×νn P2bI On×nµ On,q On,m i , e G = eP = h e P1 On×νn Pe2bI On×nµ On,q On,m i , H = h P1 P2 i , eH := h e P1 Pe2 i , eP1∈ Sn, eP2∈ Rn×dn Γ = BK, K = [(I1+ν+κ⊗ K) ⊕ Op+m] , eΓ = B eK, K =e h I1+ν+κ⊗ eK ⊕ Op+m i (84)
in (81) with l = n + nν + nκ + q + m and Z⊕ (In− Z) ≻ 0 and bΦ, H and K are in line with the definition in (80). Then one can obtain
U(H, K) = bΦ + SyP⊤B [(I1+ν+κ⊗ K) ⊕ Op+m] ⪯ SH, eH, K, eK := b Φ + Sy eP⊤BK + P⊤B eK− eP⊤B eK+ h P⊤− eP⊤ K⊤B⊤− eK⊤B⊤ i [Z⊕ (In− Z)]−1[∗] (85) by (82), whereS(
·
, eH,·
, eK) is a psd-convex overestimate ofU(H, K) in (80) with respect to theparameter-ization " vec( eH) vec( eK) # = " vec(H) vec(K) # . (86)
From (85), it is obvious thatSH, eH, K, eK
≺ 0 infers (80). Moreover, it is also true that SH, eH, K, eK
≺ 0 holds if and only if
b Φ + Sy eP⊤BK + P⊤B eK− eP⊤B eK P⊤− eP⊤ K⊤B⊤− eK⊤B⊤ ∗ −Z On ∗ ∗ Z− In ≺ 0 (87)
holds based on the application of the Schur complement given Z⊕ (In− Z) ≻ 0. Now (80) is inferred by (87) which can be handled by standard numerical solvers of semidefinite programmings provided that the values of eH and eK are known.
By compiling all the aforementioned procedures according to the expositions in Dinh et al. (2012), an iterative algorithm is constructed in Algorithm 1 where x consists of all the variables in P3, Q1, Q2 R1, R2
in Theorem 1 and Z in (87). Furthermore, H, eH, K and eK in Algorithm 1 are defined in (84) and ρ1, ρ2
and ε are given constants to achieve regularizations and determine error tolerance, respectively.
Remark 8. To initialize the iterative algorithm based on the results in Dinh et al. (2012), one has to obtain
certain initial data which in our case are for eH and eK. Hence a right candidate for the values of eH and eK must be part of a feasible solution of (36)–(38) in Theorem 1. Namely, eP1← P1, eP2← P2and eK← K can
be used for the initial data of eH and eK, if P1, P2 and K are the feasible solutions of (36)–(38).
Remark 9. When a convex objective function is considered in Theorem 1, for instance L2 gain γ > 0
Algorithm 1: An inner convex approximation solution for Theorem 1 begin
solve Theorem 2 with given αi to obtain a controller gain K and then solve Theorem 1 with the aforementioned K to obtain H =P1 P2
. update eH←− H, eK←− K, solve min x,H,Ktr h ρ1[∗] H − eH + ρ2[∗] K − eK i
subject to (36)–(37) and (87) obtain H and K
while vec(H) vec (K) − " vec( eH) vec( eK) # ∞ " vec( eH) vec( eK) # ∞ + 1 ≥ ε do update eH←− H, eK←− K; solve min x,H,Ktr h ρ1[∗] H − eH + ρ2[∗] K − eK i
subject to (36)–(37) and (87) to obtain H and K;
end end
4.2. A variant scheme of controller design
The proposed methodologies in Theorem 1, 2 and Algorithm 1 can be modified to solve a different synthesis problem if no input delays exist at (1).
Specifically, let Bi = On, i = 2· · · ν and eBj(τ ) = On, j = 1· · · ν and Bi = Om×n, i = 2· · · ν and e
Bj(τ ) = Om×n, j = 1· · · ν in (1), which gives a distributed-delay system without input delays. Now assume that the open-loop system considered is stabilized by the following state feedback controller
u(t) = ν X i=0 Kix(t− ri) + ν X i=1 Z −ri −ri−1 e Ki(τ )x(t + τ )dτ (88)
which contains pointwise and distributed-delay terms. Moreover, by using the approximation scheme out-lined in (16)–(19) and by adding the condition
∀τ ∈ [−ri,−ri−1], Kei(τ ) =Ki(gi(τ )⊗ In) , i = 1· · · ν (89) in Proposition 1 with someKi∈ Rp×nκi, i = 1· · · ν, the corresponding closed-loop system stabilized by (88) is denoted by ˙ x(t) = A + B0K ϑ(t), z(t) = (C + B0K) ϑ(t), t≥ t0 ∀θ ∈ [−rν, 0], x(t0+ θ) = ψ(θ) K = ν Row i=0 Ki ν Row i=1 Ki bΓi √ 𝟋i⊗ In ν Row i=1 Ki eIi p Ei⊗ In On,q (90)
with t0 and ψ(·) in (1), where A, C, ϑ(t) follows the same definitions in (24)–(28).
Remark 10. Note that the synthesis scheme in (88) can be considered even if no delays exist at the
By modifying the results in Theorem 1 and 2 in accordance to the structure in (90), then the following two corollaries can be derived.
Corollary 1. Let all the parameters in Proposition 1 and (89) be given. Then the closed-loop system
(90) with the supply rate function in (35) is dissipative and the trivial solution of (90) with w(t) ≡ 0q
is uniformly asymptotically stable if there exist P1 ∈ Sn, P2 ∈ Rn×ϱ, P3 ∈ Sϱ with ϱ = n
Pν
i=1di, and
Qi; Ri ∈ Sn, K0; Ki ∈ Rp×n,Ki ∈ Rp×κin, i = 1· · · ν such that (36)–(38) hold with Ω = A + B0K and Σ = C + B0K where A and C are given in (24),(26) and K is given in (90).
Proof. This corollary can be derived via direction substitution of Ω = A + B0K and Σ = C + B0K onto
(38). ■
Corollary 2. Given the conditions in Proposition 1 with (89) and known parameters{αi}1+ν+κi=1 . Then the
closed-loop system (90) with the supply rate function in (35) is dissipative and the trivial solution of (90) with w(t)≡ 0q is uniformly asymptotically stable if there exist ´P1∈ Sn, ´P2∈ Rn×ϱ, ´P3∈ Sϱ and ´Qi; ´Ri ∈ Sn
and V0; Vi∈ Rp×n,Vi∈ Rp×κin, i = 1· · · ν such that (63)–(65) hold with ´
Π =A [(I1+ν+κ⊗ X) ⊕ Iq] + B1V On,m
, ´Ω = C [(I1+ν+κ⊗ X) ⊕ Iq] + B1V
where A and C are given in (24),(26) and
V = ν Row i=0 Vi ν Row i=1 Vi bΓi √ 𝟋i⊗ In ν Row i=1 Vi eIi p Ei⊗ In On,q . (91)
Finally, the controller gains are calculated via the relations K0 = V0X−1 and Ki = ViX−1 and Ki = Vi(Iκi⊗ X) for i = 1 · · · ν.
Proof. The proof of this corollary is straightforward considering the procedure of proving Theorem 2. Note that the corresponding procedure at (78) is
´ Π =A [(I1+ν+κ⊗ X) ⊕ Iq] + B0K [(I1+ν+κ⊗ X) ⊕ Iq] On,m =A [(I1+ν+κ⊗ X) ⊕ Iq] + B1V On,m (92) with V = K [(I1+ν+κ⊗ X) ⊕ Iq] in (91) where V0= K0X, Vi= KiX andVi=Ki(Iκi⊗ X) for i = 1 · · · ν .
Moreover, the relation K [(I1+ν+κ⊗ X) ⊕ Iq] = V in (91) can be established by the application of (11). ■ Finally, the conditions in Corollary 1 can be solved by a modified version of Algorithm 1 with K in (90) and e K = ν Row i=0 e Ki ν Row i=1 e Ki bΓi √ 𝟋i⊗ In ν Row i=1 e Ki eIi p Ei⊗ In On,q (93) and the substitution B1→ B with more prescribed regularization parameters for the decision variables Ki, Ki and bKi, bKi.
5. Dissipative Observer Design
As we have solved a state feedback problem in the previous section for (1), we will present the corre-sponding solution of the observer dual problem in this section.
Note that this time z(t) in (1) is not considered. We want to construct a observer with the mathematical model4 ˙bx(t) = ν X i=0 Aibx(t − ri) + ν X i=1 Z −ri−1 −ri e Ai(τ )bx(t + τ)dτ + ν X i=0 Biu(t− ri) + ν X i=1 Z −ri−1 −ri e Bi(τ )u(t + τ )dτ − L y(t)− ν X i=0 Cibx(t − ri)− ν X i=1 Z −ri−1 −ri eCi(τ )bx(t + τ)dτ ! +D3w(t) (95)
with the observer gain L∈ Rn×p. Now combine the state equation in (1) with the observer in (95) considering the measurement output in (94), we have
˙ e(t) = ν X i=0 (Ai+ LCi) e(t− ri) + ν X i=1 Z −ri−1 −ri e Ai(τ ) + LeCi(τ ) e(t + τ )dτ + (D1+ LD4−D3)w(t). (96) Remark 11. It is very crucial to stress that the disturbance term D3w(t) is not part of the actual
im-plementation of the observer as this term contributes nothing to the stabilization of (96) with w(t)≡ 0q.
D3w(t) is introduced into the observer equation so that anti-disturbance (dissipativity) properties of the
observer can be secured by the proposed design method. On the other hand, D2w(t) is presented in the
measurement output equation as a realistic assumption.
Similar to Proposition 1, we assume the distributed-delays in (96) satisfy the following assumption:
Assumption 1. ∃fi(·) ∈ C1([−ri,−ri−1]#Rdi), φi(·) ∈ L2([−ri,−ri−1]#Rδi), ϕi(·) ∈ L2([−ri,−ri−1]#Rµi),
Mi∈ Rdi×κi, bAi ∈ Rn×κin, bCi ∈ Rp×κin with i = 1· · · ν such that
∀τ ∈ [−ri,−ri−1], Aei(τ ) = bAi(gi(τ )⊗ In) , (97) ∀τ ∈ [−ri,−ri−1], eCi(τ ) = bCi(gi(τ )⊗ In) , (98) ∀τ ∈ [−ri,−ri−1], dfi(τ ) dτ = Mifbi(τ ), fbi(τ ) = φi(τ ) fi(τ ) , (99) Gi:= Z 0 −1 gi(τ )g⊤i (τ )dτ ≻ 0 (100)
hold for all i = 1· · · ν, where gi(τ ) =
ϕ⊤i (τ ) φ⊤i (τ ) fi⊤(τ )⊤ and κi = di+ δi+ µi, κi = di+ δi with
di; δi; µi ∈ N0 for all i = 1· · · ν. Moreover, the derivatives in (6) at τ = 0 and τ = −rν are one-sided derivatives.
Remark 12. Given the demonstration in Section ??, the conditions in Proposition 1 indicate that eAi(τ ) and eCi(τ ) in (96) can contain anyL2 function defined over [−ri,−ri−1].
Now assume a regulated output z(t) = e(t) for the closed loop system in (96). Considering the approxi-mation scheme outlined in (16)–(19) and by using the decompositions in Proposition 1, the system in (96) with z(t) = e(t) is denoted by
˙ e(t) = (A0+ LC0) e(t) + ν X i=1 (Ai+ LCi) χ(t,−1) + Z 0 −1 ν Row i=1 h b Ai+ LbCi (gi(τ )⊗ In) i χ(t, τ )dτ + (D1+ LD4− D3)w(t), z(t) = e(t), χ(t, θ) = ν Col i=1e(t + ´riθ− ri−1)∈ R nν, θ∈ [−r i,−ri−1] (101)
4This is the mathematical description of the observer taking into account disturbance stem from uncertainty factors such as
