applied sciences Article Parametrical Non-Complex Tests to Evaluate Partial Decentralized Linear-Output Feedback Control Stabilization Conditions from Their Centralized Stabilization Counterparts Manuel De la Sen 1, * and Asier Ibeas 2 1 Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, 48940 Leioa (Bizkaia), Spain 2 Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona UAB, 08193 Barcelona, Spain; Asier.Ibeas@uab.cat * Correspondence: manuel.delasen@ehu.eus  Received: 18 March 2019; Accepted: 22 April 2019; Published: 26 April 2019  Abstract: This paper formulates sufficiency-type linear-output feedback decentralized closed-loop stabilization conditions if the continuous-time linear dynamic system can be stabilized under linear output-feedback centralized stabilization. The provided tests are simple to evaluate, while they are based on the quantification of the sufficiently smallness of the parametrical error norms between the control, output, interconnection and open-loop system dynamics matrices and the corresponding control gains in the decentralized case related to the centralized counterpart. The tolerance amounts of the various parametrical matrix errors are described by the maximum allowed tolerance upper-bound of a small positive real parameter that upper-bounds the various parametrical error norms. Such a tolerance is quantified by considering the first or second powers of such a small parameter. The results are seen to be directly extendable to quantify the allowed parametrical errors that guarantee the closed-loop linear output-feedback stabilization of a current system related to its nominal counterpart. Furthermore, several simulated examples are also discussed. Keywords: Output-feedback; centralized control; decentralized control; closed-loop stabilization 1. Introduction Control systems are very important in real world applications and, therefore, they have been investigated exhaustively concerning their properties of stability, stabilization, controllability control strategies etc. See, for instance, [1–4] and references therein. Some extra constraints inherent to some systems, like solution positivity in the case of biological systems or human migrations or the needed behavior robustness against parametrical changes of disturbance actions add additional complexity to the related investigations and need the use of additional mathematical or engineering tools for the research development, [5–7]. A large variety of modeling and design tools have to be invoked and developed in the analysis depending on the concrete systems under study and their potential applications as, for instance, the presence of internal and external delays, discretization, dynamics modeling based on fractional calculus, the existence of complex systems with interconnected subsystems, [8–13], hybrid coupled continuous/digital tandems, nonlinear systems and optimization and estimation techniques [14–19] as well as robotic and fuzzy-logic based systems, [20,21]. In particular, decentralized control is a useful tool for controlling dynamic systems by cutting some links between the dynamics coupling a set of subsystems integrated in the whole system at hand. It is claimed to keep the main properties related to the use of centralized control such as stability, controllability, observability, Appl. Sci. 2019, 9, 1739; doi:10.3390/app9091739 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1739 2 of 19 etc. In summary, a centralized controller keeps all the information on the system and coupling links as available to the control designer while decentralized control ignores some of such information or even cuts on occasions some of coupling signals between the various subsystems integrated in the whole system at hand. It can be pointed out that the stability studies are often performed trough Lyapunov theory which requires to find a Lyapunov function (see [20,21] and some references therein). It turns out that, if the neglected couplings are strong and are not taken into account by the controller, the stabilization and other properties such as the controllability can become lost. The use of decentralized stabilization and control tools is of interest when the whole system has physically separated subsystems that require the implementation of local control actuators but the control has to be global for the whole system. An ad-hoc example provided in [2–4,19] where decentralized control is of a great design interest is the case of several coupled cascade hydroelectric power plants allocated in the same river but separated far away from each other. It has to be pointed out that the term “decentralized control” versus “centralized control” refers to the eventual cut of links of the shared information between tandems of integrated subsystems, or coupling signals between them, to be controlled rather than to the physical disposal of the controller. In other words, if the whole controller keeps and uses all the information on measurable outputs and control components design to implement the control law, such a control is considered to have a centralized nature even if its various sub-control stations are not jointly allocated. It is a common designer´s basic idea in mind for complex control designs to try to minimize the modeling designs and computational loads without significantly losing the system´s performance and its essential properties. For instance, in [8], the dynamic characteristic of a discrete-time system is given as an extended state space description in which state variables and output tracking error are integrated while they are regulated independently. The proposed robust model predictive control is much simpler than the traditional versions since the information of the upper and lower bounds of the time-varying delay are used for design purposes. On the other hand, in [9], a control law might be synthesized for a hydropower plant with six generation units working in an alternation scheme. To assess the behavior of the controlled system, a model of such a nonlinear plant is controlled by a fractional proportional/integral/derivative control device through a linearization of its set points, the fractional part being relevant in the approach on the controller derivative actions. In addition, a set of applied complex control problems are studied, for instance, in [10–16] with the aim of giving different ad hoc simplification tools to deal with the appropriate control methodologies. In particular, a decentralized control approach is proposed in [16]. In this paper, the decentralized control design versus its decentralized control counterpart, under eventual output linear feedback, are studied from the point of view of the amount of information that can be lost or omitted in terms of the total or partial knowledge of the coupled dynamics between subsystems necessary in the decentralized case to keep the closed-loop stability. The study is made by using the information on the worst-case deviation, in terms of norms, between the respective matrices of open-loop dynamics and the respective controller gains under which the closed-loop stability is kept. This paper is organized as follows. The problem statement is given in Section 2 while the main stabilization results of the paper are provided in Section 3. The proofs of some of the results of Section 3, which are technically involved, but conceptually simple, are distributed in various technical auxiliary that are given in Appendices A and B. It is claimed to give a non-complex method to test the feasibility of the implementation of decentralized control and conditions for its design, which be a fast and simpler stability test compared to Lyapunov stability theory [20,21], for instance, under a partial removal of information or physical cuts of links of coupling dynamics between the various subsystems or state, control and output components. Section 4 discusses several examples and, finally, the concluding remarks end the paper. Notation n = {1, 2, · · · , n}, Appl. Sci. 2019, 9, 1739 3 of 19 R+ = { z ∈ R : z > 0} ; R0+ = { z ∈ R : z ≥ 0}, sp(A) and det(A) are the spectrum and determinant of A ∈ Rn×n , respectively. For A ∈ Rq×` , being in general rectangular, kAk denotes any unspecified norm of A, kAk2 denotes the `2 or spectral norm of a matrix A, ρ(A) denotes its spectral radius, and k.k∞ denotes the H√∞ -norm of a stable real rational transfer matrix or function, Iq denotes the qth identity matrix, and i = −1 is the complex imaginary unit. Let A ∈ Rn×n be symmetric. Then, λmax (A) and λmin (A) are, respectively, the maximum and minimum eigenvalues of A. Mn×n E is the set of Metzler matrices (any off-diagonal entry is non-negative) of nth order. Z n×n is the set of Z-matrices (any off-diagonal entry is non-positive) of nth order. Mn×n is the set of M-matrices   (Z-matrices   which are stable or critically stable) of nth order. Assume that A = Aij , B = Bij ∈ Rn×n . Then, the notations AB, A  B and A  B, are, respectively, equivalent to B≺A, B ≺ A and B ≺≺ A, meaning that Aij ≥ Bij , Aij ≥ Bij (and B , A) and Aij > Bij ; ∀i, j ∈ n, respectively. In particular, A0, A  0 and A  0 are reworded as A is non-negative, positive and strictly positive, respectively, and A≺0, A ≺ 0 and A ≺≺ 0 are reworded as A is non-positive, negative and strictly negative, respectively. 2. Problem Statement Consider the following linear and time-invariant system under linear output-feedback centralized control: . xc (t) = Ac xc (t) + Bc uc (t); xc (0) = xc0 (1) yc (t) = Cc xc (t) + Dc uc (t) (2) uc (t) = Kc yc (t) (3) where xc (t) ∈ Rn is the state vector; uc (t) ∈ Rm is the centralized control vector; yc (t) ∈ Rp is the output; Ac , Bc , Cc and Dc are the system, control, output and input–output interconnection matrices, respectively, of orders being compatible with the dimensions of the above vectors; and Kc ∈ Rm×p is the control matrix. If the system runs in a decentralized control context, we have: . xd (t) = Ad xd (t) + Bd ud (t) ; xd (0) = xd0 (4) yd (t) = Cd xd (t) + Dd ud (t) (5) ud (t) = Kd yd (t) (6) where xd (t) ∈ Rn is the state vector; ud (t) ∈ Rm is the centralized control vector; yd (t) ∈ Rp is the output; Ad , Bd , Cd and Dd are the system, control, output and input–output interconnection matrices, respectively, of orders being compatible with the dimensions of the above vectors; and Kd ∈ Rm×p is the control matrix. Basically, the differences between centralized and decentralized controls are as follows: (1) In the centralized control, all control components, or more generally, all subsystems if subsystems are considered in the model, have a complete information on the output available for feedback. This means that all control components or block-control inputs are available for controlling each state component (or each individual substate including several state components in the case of a more generic decomposition structure). Basically, the matrix Kc has a complete non-diagonal or block non-diagonal structure. In the decentralized control, the various input components or block-control inputs are not available for controlling each state component. Thus, Kd does not have a complete free design structure of its non-diagonal part and in some cases (completely decentralized disposal) its diagonal or block diagonal. (2) In a more general context, some control or output links can be cut in the decentralized case for the sake of computational simplicity or a more economic control design. In our case, the Appl. Sci. 2019, 9, 1739 4 of 19 decentralized input, output and interconnection matrices Bd , Cd and Dd can be distinct from the centralized ones and, roughly speaking, to a have a “more diagonal” or “sparser” structure than their centralized counterparts Bc , Cc and Dc . If the parameterization of the system (or dynamics) matrix is available to the designer, then some off-diagonal block matrices of Ac could be zeroed or simply re-disposed in a more sparse structure to construct Ad . (3) The only strictly necessary condition for the system to be subject to partially (or, respectively, fully) decentralized control is that some (or, respectively, all) of the off-diagonal entries of Kd are forced to be zero even if the system, control, output and interconnection matrices remain identical in Equations (4) and (5) with respect to Equations (1) and (2). Assumption 1. The system in Equations (1) and (2) is linear output-feedback stabilizable via some centralized control law (Equation (3)). Note that Assumption 1 does not hold if the open-loop system in Equations (1) and (2) has unstable or critically stable fixed modes that cannot be removed via linear feedback. Proposition 1. If Assumption 1 holds, then there exists a centralized stabilizing controller gain Kc ∈ Rm×p such that the matrices (Im − Kc Dc ) and Ip − Dc Kc are non-singular, thus the closed-loop centralized control system is solvable and given by: .   xc (t) = Ac + Bc (Im − Kc Dc )−1 Kc Cc xc (t); xc (0) = xc0 (7)   yc (t) = Ip − Dc Kc −1 Cc xc (t) (8) and asymptotically stable for any given xc0 ∈ Rn under the generated control law: uc (t) = (Im − Kc Dc )−1 Kc Cc xc (t) (9)   that is, the polynomial p(s) = det sIn − Ac − Bc (Im − Kc Dc )−1 Kc Cc is Hurwitz. Proof. The replacement of Equation (3) into Equations (1) and (2) yields Equation (7)–(9). Since the (1) and (2) is linear output stabilizable, a stabilizing controller gain Kc ∈ Rm×p has to exist such that (7)–(9) are solvable and the closed-loop dynamics is stable.  Assumption 2. The system in Equations (4) and (5) is linear output-feedback stabilizable via some decentralized control law (Equation (6)). In the same way as Proposition 1, we get the following result: Proposition 2. If Assumption 2 holds, then there exists  a decentralized stabilizing controller gain Kd ∈ Rm×p such that the matrices (Im − Kd Dd ) and Ip − Dd Kd are non-singular, thus the closed-loop decentralized control system is solvable and given by: .   xd (t) = Ad + Bd (Im − Kd Dd )−1 Kd Cd xd (t); xd (0) = xd0 (10)   yd (t) = Ip − Dd Kd −1 Cd xd (t) (11) and asymptotically stable for any given xd0 ∈ Rn under the control law: ud (t) = (Im − Kd Dd )−1 Kd Cd xd (t) (12)   that is, the polynomial p(s) = det sIn − Ad − Bd (Im − Kd Dd )−1 Kd Cd is Hurwitz. Appl. Sci. 2019, 9, 1739 5 of 19 Proposition 3. Assume that Ad = Ac , Bd = Bc , Cd = Cc and Dd = Dc , and that the system in Equations (1) and (2) is not linear output-feedback stabilizable via some centralized control law (Equation (3)). Then, it is not stabilizable under any linear output-feedback decentralized control (Equation (6)) either. Proof. Obviously, if there is no completely free-design matrix Kc that stabilizes Equations (1) and (2), then there is no Kd with at least a forced zero off-diagonal entry that stabilizes it since Kd has extra design constraints related to Kc .  It can be pointed out that decentralized control has also been proved to be useful in applications. For instance, an integral-based event-triggered asymptotic stabilization of a continuous-time linear system is studied in [17] by considering actuator saturation and observer-based output feedback are considered. In the proposed scheme, the sensors and actuators are implemented in a decentralized manner and a type of Zeno-free decentralized integral-based event condition is designed to guarantee the asymptotic stability of the closed-loop systems. On the other hand, two decentralized fuzzy logic-based control schemes with a high-penetration non-storage wind-diesel system are studied in [18] for small power system with high-penetration wind farms. In addition, several examples concerning decentralized control are described in [4] to illustrate the theoretical design analysis. A typical described case is that of tandems of electrical power system with a tandem disposal on the same river which are not physically nearly allocated. The next section discusses some simple sufficiency-type conditions which ensure that, provided that the system is stabilizable under linear output-feedback centralized control, it is also stabilizable under decentralized control in two cases: (a) the system matrix remains identical but the other parameterization matrices can eventually vary; and (b) the system matrix can vary as well in the decentralized case with respect to the centralized one. A result elated to the maintenance of the stability of a matrix under an additive matrix perturbation is summarized through a set of sufficiency-type conditions simple to test in Theorem A1. Theorem A2 proves sufficiency-type for the stability of the matrix function A(t) = A0 + A e(t) with A0 stable and Ae(t) being time-varying. Appendix B includes calculations and auxiliary results to quantify the tolerance to cut some dynamics links between subsystems, state components or control centers or components while keeping the closed-loop stability of the whole coupled system. The results of Appendices A and B are used in the proofs of the main results in the next section. 3. Main Results The first set of technical results which follow are concerned with centralized and decentralized control stabilizability. Assertion 1. A necessary and sufficient condition for the system to be linear state-stabilizable via some centralized control law is that rank[sIn − Ac Bc ] = n for all Res ≥ 0. Proof. Assume that rank[sIn − Ac Bc ] = n1 < n for some Res ≥ 0. Then, there is some Laplace h iT h i h i transform x̂T (s) , ûT (s) = Lap xT (t) , uT (t) T , 0 such that [sIn − Ac Bc ] x̂T (s) , ûT (s) T = 0 for " # In some Res ≥ 0 and [sIn − Ac Bc ] x̂(s) = 0 for any Kc ∈ Rn×n and some x̂(s) , 0 with Res ≥ 0 Kc " #! " #! In In since rank [sIn − Ac Bc ] ≤ min rank[sIn − Ac Bc ] , rank ≤ min(n1 , n) ≤ n1 < n for some Kc Kc Res ≥ 0. Therefore, the closed-loop system has some unstable or critically stable solution for any given (centralized) control gain. This proves the necessary part. Sufficiency follows since, if n×n rank[sIn − Ac Bc ] = n, then "x̂(s) ≡#! 0 for all Res ≥ 0 and some Kc ∈ R which can be found so that In min rank[sIn − Ac Bc ] , rank = min(n, n) = n.  Kc Appl. Sci. 2019, 9, 1739 6 of 19 Assertion 1. is a particular adapted ad-hoc test for stabilizability of the celebrated Popov–Belevitch –Hautus rank controllability test [6]. Note that, if there exist unstable or critically stable fixed modes (i.e., those present in the open-loop system that cannot be removed via feedback control), then neither centralized nor decentralized stabilizing control laws can be synthesized. Note that the stabilizability rank test of Assumption 1 can only be evaluated for the critically and unstable eigenvalues of Ac instead for all the open right-hand complex plane. In all the remaining points of such a plane, the test always gives a full rank of the tested matrix. The parallel controllability test should always be applied in the same matrix to any eigenvalues of Ac . Assertion 2. A necessary condition for the system to be linear state-stabilizable via some partially or totally decentralized control is that it be stabilizable via centralized control (i.e., Assertion 1 holds). Proof. It is obvious from Assertion 1 since any gain Kd used for centralized or decentralized is sparser than a centralized gain counterpart so that the proof follows from Assertion 1.  Assertion 3. A necessary condition for the system to be linear output-stabilizable via some partially or totally decentralized control is that it be stabilizable via centralized control (i.e., that Assertion 1 holds). Proof. It is obvious from Assertions 1 and 2 that, when replacing Kc → (Im − Kc Dc )−1 Kc Cc and Kd → (Im − Kd Dd )−1 Kd Cd (see Equations (9) and (12)), the second replacement happens under sparser parameterizations.  Now, consider the closed-loop system matrices from Equations (7) and (10) for the case A = Ac = Ad . Acc = A + Bc (Im − Kc Dc )−1 Kc Cc ; Adc = A + Bd (Im − Kd Dd )−1 Kd Cd with its parametrical error being: edc = Acc − Adc = Bc (Im − Kc Dc )−1 Kc Cc − Bd (Im − Kd Dd )−1 Kd Cd A A first main technical result follows: Theorem 1. If Assumption 1 holds, assume also that Kc ∈ Rm×p is a centralized linear output-feedback stabilizing controller gain such that the resulting closed-loop system matrix Acc ∈ Rn×n has a stability abscissa (−ρcc ) < 0. Then, the following properties hold: (i) Adc ∈ Rn×n is a closed-loop stability matrix under a linear output-feedback stabilizing controller gain Kd ∈ Rm×p if any of the subsequent sufficiency-type conditions holds: (1) The H∞ -norm of (sIn − Acc )−1 A edc k < 1, edc satisfies k(sIn − Acc )−1 A ∞ (2) kAdc k2 < 1/ sup k(iωIn − Acc ) k2 . e −1 ω∈R0+ Other alternative  sufficiency-type conditions to Conditions 1 and 2 for the stability of Adc are: (3) ρ A−1cc edc < 1, A cc Adc k2 < 1, (4) kA−1 e     (5) kAdc k < 1/kA−1 k , that is, λmax A e 2 cc 2 edc < λmin AT Acc , eT A dc cc in the following particular cases: (a) Acc ≺ 0 and A edc  Acc ; and   (b) Acc = Accij ∈ Mn×n E and Aedc = A edcij ≤ Acc ; ∀i, j(, i) ∈ n. edcij fulfills A ij (ii) Assume that Property (i) holds and that the number of inputs and outputs are identical, i.e., p = m, and decompose both the controller gain matrices as sums of their diagonal and off-diagonal parts leading to Kc = Kcd + Kcod and Kd = Kdd + Kdod , thus K e = Kc − Kd = (Kcd − Kdd ) + (Kcod − Kdod ). Then, the system is stabilizable under partially decentralized control linear output-feedback control in the sense that Equations (4) and (5), is asymptotically stable under a control law (Equation (6)), if Kd ∈ Rm×p is such that, if there Appl. Sci. 2019, 9, 1739 7 of 19 is at least one non-diagonal zero entry in at least one of its rows in the off-diagonal controller error matrix eod = Kcod − Kdod . If K K eod = 0, then the system is stabilizable under decentralized control. Proof. Property (i) is a direct translation of the results of Theorem A1 in Appendix A to the closed-loop system matrices. Property (ii) holds if Property (i) holds with an off-diagonal controller error matrix between the centralized a decentralized controller gain that has at least one non-diagonal zero at some row (or its identically zero) so that a feedback from some crossed output to some of the inputs is not provided to the control law for closed-loop stabilization the stabilization.  The following result follows for the time-varying case from Theorem 1 and Theorem A2: Corollary 1. Assume that Adc (t) and then A edc (t) are everywhere piecewise-continuous time-varying. Then, Theorem 1 still holds if Condition 1 is replaced with ρkcccc sup kA edc (τ)k < 1; ∀t ∈ R0+ with kcc ≥ 1 and ρcc > 0 0≤τ p (i.e., there are more inputs than outputs) is irrelevant for the stabilization from the strict algebraic point of view since the (m − p) extra inputs would be redundant. In the case that m ≤ p, Theorem 1 (ii) might be directly generalized to a  T subsystem´s decomposition philosophy if a number q ≤ m of subsystems of inputs and outputs uT1 , uT2 , · · · , uTq  T Pq Pq and yT1 , yT2 , · · · , yTq with ui ∈ Rmi , yi ∈ Rpi ; ∀i ∈ n with p = i=1 pi and m = i=1 mi . Remark 2. Theorem 1 can be easily generalized to cases when some dynamics transmission links between state, input or output components (or subsystems, in general) can be suppressed by manipulation. In more general cases, it is possible to extend Theorem 1 to combinations of the subsequent situations with the matrix decompositions having the same sense (in the various modified contexts) as that of Theorem 1 (ii): • Case 1. Suppression of some transmission links between the coupled open-loop dynamics e = Ac − Ad = by examining the decompositions: Ac = Acd + Acod , Ad = Add + Adod , and A (Acd − Add ) + (Acod − Adod ). (a) If there is at least one non-diagonal zero entry in at least one of its rows in the off-diagonal controller error matrix A eod = Acod − Adod which is not a corresponding zero in Acod ; and (b) if there is at least one non-diagonal zero entry in at least one of its rows in the off-diagonal controller error matrix eod = Kcod − Kdod , then the closed-loop system is stabilizable under a partial decentralized control even K if some links of the dynamics between crossed components are cut if Theorem 1 (ii) holds. If only Condition a is addressed, then the system is stabilizable by centralized control when cutting certain transmission links between coupled dynamics in the open-loop system. This idea can be extended to total decentralized control for a purely diagonal open-loop system´s dynamics under full zeroing of the off-diagonal corresponding error dynamics. It can be also generalized to the “ad hoc” decompositions between subsystems. Other cases with similar interpretations in the new contexts are: • Case 2. Suppression of some crossed entries in the open-loop control matrix by examining the B = Bc − Bd = (Bcd − Bdd ) + (Bcod − Bdod ). decompositions: Bc = Bcd + Bcod , Bd = Bdd + Bdod , and e • Case 3. Suppression of some crossed entries in the open-loop output matrix by examining the e = Cc − Cd = (Ccd − Cdd ) + (Ccod − Ddod ). decompositions: Cc = Ccd + Ccod , Cd = Cdd + Cdod , and C • Case 4. Suppression of some crossed entries in the open-loop interconnection matrix by examining the decompositions: Dc = Dcd + Dcod , Dd = Ddd + Ddod , and D e = Dc − Dd = (Dcd − Ddd ) + (Dcod − Ddod ). • Case 5. Any combinations of Cases 1–4. Appl. Sci. 2019, 9, 1739 8 of 19 Problem 1. Find a stabilizing decentralized family of control gains by assuming that Ad = Ac , such that e = Ac − Ad = 0 and Assumption 1 holds with Kc being a stabilizing centralized controller gain. A The following more general result for the eventual case A e , 0 (that is Ac , Ad eventually), follows from Theorem 1, Theorem A1 and Theorem A2 and Lemmas B1, B2 and B3: Theorem 2. Define the following error matrices between the centralized and decentralized system parameterizations: e = Ac − Ad ; e A B = Bc − Bd ; C e = Cc − Cd ; D e = Dc − Dd ; K e = Kc − Kd (13) such that k Ak e ≤e σA ε, k e σB ε, k Ck Bk ≤ e σc ε, k Dk e ≤e σD ε and k Kk e ≤e σK ε for some ε ∈ R0+ and given e e ≤e σA , e σB , σC , e e σD ∈ R+ . Assume that: (1) Assumption 1 holds; (2) Kc ∈ Rm×p is a centralized linear output-feedback stabilizing controller gain such that the resulting closed-loop system matrix Acc ∈ Rn×n has a stability abscissa (−ρcc ) < 0 and such that kKc Dc k2 < 1 (so that (Im − Kc Dc ) is non-singular); (3) Ac = Ad = A; and (4) Define ε∗ = min (1 , ε, ε1 , ε2 ), where: q (e σK kDc k)2 + 4e σD kKc k + e σK /k(Im − Kc Dc )−1 k − (e σDe σD kKc k + e σK kDc k) ε= , σDe 2e σK   k(Im − Kc Dc )−1 k −1 ε1 = ; ε2 = 1/e   adc sup k(iωIn − Acc ) k2 , σD + εe 2[kKc ke σDeσK + k Dc ke σK ] ω∈R0+ where adc = (1 − kKc Dc k)−1 ×[e e σB kKc Cc k + ( kBc k + kKc Cc k)(kKc ke σC + k Cc ke σK + C [kKc ke σD + k Dc ke σK ])], where the non-negative real constant C is given in Equation (A17). Then, the following properties hold: σA = 0 (that is, Ac = Ad ), then Adc is stable and ε ∈ [0 , ε∗ ] . (ii) If k Ak (i) If e e 2 ≤e σA ε, then Adc is stable  0 0 and ε ∈ [0 , ε0 ] where ε0 = min (1 , ε, ε1 , ε2 ) and ε2 = 1/(adc + e ∗ ∗ σA ) sup k(iωIn − Acc ) k2 . −1  e   ω∈R0+ e(t) is piecewise continuous and bounded, then Property (ii) holds by replacing k Ak (iii) If A σA ε by e 2 ≤e sup k A e(t)k ≤ e 2 σA ε. 0≤t<∞ Remark 3. Some quantified results are given in Lemmas B.2 and B.3 to modify ε2 (and hence ε02 ) in Theorem 2 by considering the second power of ε in the calculations of the disturbed parameterization guaranteeing the closed-loop stability in the decentralized case. Remark 4. If the corresponding parametrical error matrices of Equation (13) have some zero off-diagonal entries (or off-diagonal block matrices in the more general case that the system is described by coupled subsystems), then we have at least a partial closed-loop stabilization under decentralized control or, eventually, cut coupled dynamic links to the light of the various Cases 1–5 described after Remark 2 such that closed-loop stability is preserved. Remark 5. Theorem 2 also applies to the case of state-feedback control by replacing the output matrices σC = 0. Cc , Cd → In and fixing e Appl. Sci. 2019, 9, 1739 9 of 19 Remark 6. Theorem 2 also applies directly to the cases where Equations (1)–(3) are a given nominal asymptotically stable closed-loop system configuration and Equations (4)–(6) are a perturbed one whose closed-loop asymptotic stability maintenance related to its nominal counterpart is a suited objective and which is not necessarily of partial of compete decentralized type. 4. Simulation Examples This Section contains some numerical simulation examples to illustrate the theoretical results introduced in Section 3. Example 1. Consider the interconnected linear system with less inputs than outputs given by, [19]: . x1 (t) = A11 x1 (t) + A12 x2 (t) + B1 u1 (t) . x2 (t) = A21 x1 (t) + A22 x2 (t) + B2 u2 (t) y1 (t) = C1 x1 (t) y2 (t) = C2 x2 (t) with x1 (t)T = h x12 (t) , x2 (t)T = x21 (t) i h i x11 (t) x22 (t) and matrices defined by: " # " # " # " # " # 1 0 −1 0 −1 0 1 2 A11 = , A12 = A21 = , A22 = , B1 = , B2 = 0 2 0 −1 0 −2 2 1 C1 = C2 = I2 This system can be cast into the form of Equations (1) and (2) by composing the matrices:      1 0 −1 0   1 0       0 2 0 −1 , Bc =  2   0  Ac =  , Cc = I4 , Dc = 0  −1 0 −1 0    0  2    0 −1 0 −2 0 1    Note that matrix Ac is unstable with eigenvalues given by {2.36, .1.41, −1.41, −2.23}. A static feedback output controller of the form of Equation (3) can be designed for this system, which leads to the following gain: " # −9.3 8.2 −0.015 0.02 Kc = 0.01 −0.01 0.375 0.25 that places the closed-loop poles at {−0.84, −1.1, −2.8, −3.3} and thus stabilizes the closed-loop system. The static feedback gain Kc corresponds to a centralized controller as it can be readily observed. The question that arises now is whether a decentralized controller defined by: " # −9.3 8.2 0 0 Kd = 0 0 0.375 0.25 is enough to stabilize the system or not. Note that Kd is a block-diagonal matrix with zero off-block-diagonal entries. Theorem 1 enables us to guarantee the asymptotic stability of the above system when the decentralized controller Kd is used. Therefore, Ad = Ac , Bd = Bc and Cd = Cc while the feedback gain Kd is restricted to the proposed particular structure. In this way, consider now Acc = Ac + Bc Kc Cc , Adc = Ac + Bc Kd Cc and A edc = Bc (Kc − Kd )Cc . Condition 2 of Theorem 1 (i) yields: edc k < 1 0.055 = kA 2 = 0.07 sup k(iωI4 − Acc )−1 k2 ω∈R0+ ~ 1 0.055 = Adc  = 0.07 2 sup (iI 4 − Acc )−1 R0 + 2 Appl. Consequently, Sci. 2019, 9, 1739 we can conclude from Theorem 1 that the closed-loop system controlled by the 10 of 19 decentralized static output gain K d is asymptotically stable. Thus, we have been able to easily analyze the stability of Consequently, wethe candecentralized conclude from caseTheorem from the1stability that theproperty of the closed-loop centralized system one. Figure controlled by the 1decentralized shows the trajectories static outputofgain the Kclosed-loop d is system asymptotically when stable. the Thus, gain we K have d is been deployed able to with easily initial analyze the stability of the decentralized case from the stability property of the centralized one. Figure 1 shows conditions given the trajectories of by x1( t )T = − 3system thei closed-loop − 4 , when = gain x2 ( t )Tthe 5 6K. Itiscan be observed deployed in Figure with initial 1 thatgiven conditions all the by d T h T h i x1 (t) = −3 −4 , x2 (t) = 5 6 . It can be observed in Figure 1 that all the states converge to states converge to zero as predicted by Theorem 1. zero as predicted by Theorem 1. 6 x 11 x 12 4 x 21 x 22 2 0 -2 -4 -6 0 1 2 3 4 5 6 7 8 9 10 Time (sec) Figure 1. States evolution when the decentralized controller K is employed. Figure 1. States evolution when the decentralized controller K dd is employed. Example Example2. 2.Consider Consider thethelinear linearsystem systemwith with thethe same same number number of inputs of inputs and outputs and outputs composed composed of two of two identical Appl. Sci. identical pendulums 2019, 9, x pendulums FOR PEER THAT are coupled REVIEW THAT areby acoupled bysubject spring and a spring anddistinct to two subjectinputs, to twoasdistinct inputs, displayed 12 of 23 as displayed in Figure 2, [19]. in Figure 2, [19]. Figure 2. Two inverted pendulums coupled by a spring. Figure 2. Two inverted pendulums coupled by a spring. Themathematical The mathematicalmodel modelof ofsuch suchinterconnected interconnectedsystem systemisisgiven givenby: by: . x ( t ) = A11 x1( t ) + A12 x2 ( t ) + B1u1( t ) x1 (1t) = A11 x1 (t) + A12 x2 (t) + B1 u1 (t) .  ) =AA21xx1((tt) )++ A x2x(2t)( t= + BB2uu2 ((tt)) A22 xx2((tt))+ 21 1 22 2 2 2 y1( t ) = C1 yx1((tt)) = C x (t) 1 1 1 y 2 ( t ) = C 2yx22((t)t )= C2 x2 (t) . .     with x1 (t)T T= θ1 θ1 , x2 (tT)T = θ2 θ2 and matrices defined by: h i h i with x1( t ) = 1 1 , x2 ( t ) =  2  2 and matrices defined by:     " #  0 1   0 0  0 A11 = A22 =  g ka02 , A12 = A21 =  2 1   0ka 00 , B1 = B2 =  0 1 2  l − ml2 2−µ  ,A = A =  ka  , B = B =  1 ml  2 ml2 A11 = A22 = g ka  − 2 −   12 21  2 0 1 2  2  l ml   ml   ml  C1 = C2 = 1 0 where g represents the gravity,  accounts for the friction, m = m = m are the masses of both Appl. Sci. 2019, 9, 1739 11 of 19 h i C1 = C2 = 1 0 where g represents the gravity, µ accounts for the friction, m = m1 = m2 are the masses of both pendulums, k is the spring constant and the meanings of the geometrical parameters are shown in Figure 2. This linear model corresponds to the linearization of the pendulum nonlinear equations around the up-right position equilibrium point. The following values were used in simulation, [19]: g 1 k a = 1, 2 = 1, µ = 1, = 2, = 0.5 l ml m l This system can be cast into the form of Equations (1) and (2) as:      0 1 0 0   0 0  " #      0.5 −1 0.5 0 , Bc =  1   0 , Cc = 1  0 0 0 Ac =  , Dc = 0  0 0 0 1    0  0   0 0 1 0  0.5 0 0.5 −1 0 1    A static output feedback controller can be designed for this system to achieve its asymptotic stability. In this way, the feedback gain " # 2.92 0.65 Kc = 0.64 2.80 places the closed-loop poles at {−0.5 ± 1.5i, −0.5 ± 1.4i} with negative real parts. Now, we implement a decentralized controller with feedback gain given by: " # 2.92 0 Kd = 0 2.80 Theorem 2 is now used to analyze the stability of the closed-loop system when this controller is employed. This case is of practical importance and corresponds to the situation when the local controller has only available for control purposes the information regarding the local output, and not the output of the complete system. Thus, the centralized and decentralized systems are the same and only the static feedback gain changes. Theorem 2 conditions are applied with e σA = e σB = e σC = e σD = 0, σK = kKc − Kd k2 = 0.65 while the stability condition for this special case (see Appendix B) reads: e 0.65 = 0.65 × 1 × 1 = kKc − Kd k2 kBc k2 kCc k2 < 1 Accordingly, the closed-loop system attained with the decentralized controller is asymptotically stable and all the outputs will convergeh to zero asymptotically. hFigure 3 displays the evolution of both T i T i angles from initial conditions x1 (t) = 0.5 −0.5 , x2 (t) = 0.15 0.5 , where it can be observed that both pendulums are stabilized in the up-right position with the decentralized controller. Accordingly, the closed-loop system attained with the decentralized controller is asymptotically stable and all the outputs will converge to zero asymptotically. Figure 3 displays the evolution of both angles from initial conditions x1( t )T = 0.5 − 0.5 , x2 ( t )T = 0.15 0.5 , where it can be observed that both pendulums are stabilized in the up-right position with the decentralized Appl. Sci. 2019, 9, 1739 12 of 19 controller. Figure 3. Evolution of the angles of both pendulums. Figure 3. Evolution of the angles of both pendulums. Example 3. Consider the linear interconnected system given by: Example 3. Consider the linear interconnected . system given by: xc (t) = Ac xc (t) + Bc uc (t) xyc((tt)) ==ACxc x(c (t t))++BDuc u(ct()t) c c c c c with matrices defined by: yc ( t ) = Cc xc ( t ) + Dc uc ( t )        −1 with matrices defined 0.4 0.3   1 0.1 0   1 0.1 0   by:      Ac =  0.2 −2 0.1 , Bc =  0 2 0 , Cc =  0 1 0.1 , Dc = 0.1I3  − 1 0.20.4−30.3 10 0.01 02  1  00.1 00  1        −0.1 Ac =  0.2 − 2 0.1 , Bc = 0 2 0 ,Cc = 0 1 0.1 , Dc = 0.1I 3 − 0.1 0.2 − 3 0 0 2 0 0 1  This system is controlled by the static output feedback gain given by: This system is controlled by the static output feedback gain given  by:  0.19 0.05 0.04  Kc =  00.05 .04     .19 0−0.01 .05 00.03  .03   K c =  0−0.02 .05 − 00.02 .01 0−0.46 − 0.02 0.02 − 0.46 which places the closed-loop poles at {−1.17, −2, −2.1}. The decentralized system is now given by:    1 0 0    Ad = Ac , Bd =  0 2 0 , Cd = I3 , Dd = 0   0 0 2  The decentralized system corresponds to the case when some transmission links have been suppressed from the original open-loop coupled dynamics, as considered in Remark 2. The following decentralized gain iz employed to stabilize the decentralized system in Equations (4)–(6) parameterized by the above matrices:    0.19 0 0    Kd =  0 −0.01 0    0 0 −0.46 Theorem 2 is used to analyze the stability of the decentralized closed-loop system. To this end, we calculate: kAc − Ad k2 = eσA = 0 σB ε = 2.7 × 0.07 = 0.19 0.1 = kBc − Bd k2 ≤ e σC ε = 2.7 × 0.07 = 0.19 0.1 = kCc − Cd k2 ≤ e Appl. Sci. 2019, 9, 1739 13 of 19 σD ε = 2.7 × 0.07 = 0.19 0.1 = kDc − Dd k2 ≤ e σK ε = 2.7 × 0.07 = 0.19 0.07 = kKc − Kd k2 ≤ e With these values, we can compute ε = 0.31, ε1 = 0.24, ε2 = 0.071 so that 0.07 = ε < ε∗ = min(1, ε, ε1 , ε2 ) = 0.071. Since kKc Dc k2 = 0.05 < 1, we are in conditions of applying Theorem 2 (i) and we can conclude that the decentralized closed-loop system is asymptotically stable. In this way, the presented results allow establishing the stability of the decentralized system by a simple method based on the stability and design of h the centralized system. Figure 4 shows the state variables evolution T i Appl. fromSci. the2019, 9, x FOR initial statePEER x(t)REVIEW = 5 −5 1 . As shown in Figure 4, the state variables converge15 to of 23 zero asymptotically, as concluded from Theorem 2. 5 x 1 (t) 4 x 2 (t) x 3 (t) 3 2 1 0 -1 -2 -3 -4 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (sec) Figure 4. Evolution of the state space variables when the decentralized controller Kd is used. Figure 4. Evolution of the state space variables when the decentralized controller K d is used. 5. Concluding Remarks 5. Concluding This paper Remarks is devoted to formulating sufficiency-type linear-output feedback decentralized closed-loop stabilization conditions if the continuous-time linear dynamic system can be stabilized This paper is devoted to formulating sufficiency-type linear-output feedback decentralized under linear output-feedback centralized stabilization. The developed stability tests are conceptually closed-loop stabilization conditions if the continuous-time linear dynamic system can be stabilized simplelinear under to evaluate and they rely output-feedback on the quantification centralized stabilization.inThe terms of worst-case developed norms stability testsofare interconnection conceptually and open-loop simple to evaluatesystem dynamics and they rely onmatrices and the corresponding the quantification control gains in terms of worst-case norms inofthe decentralized interconnection caseopen-loop and comparedsystemto the dynamics centralizedmatrices counterpart. and theThe tolerances ofcontrol corresponding the various parametrical gains in matrix the decentralized case compared to the centralized counterpart. The tolerances of the various parametrical matrixa errors have been quantified by considering the first or second powers of a small parameter. Such parameter errors have is a design been factor quantified byto characterize considering thein theorworst-case first second powers for the of aallowed tolerancesSuch small parameter. to the a perturbed parameterization norms. Simulated examples are discussed to illustrate parameter is a design factor to characterize in the worst-case for the allowed tolerances to the the obtained results. The decentralized perturbed control design parameterization versus norms. its decentralized Simulated examplescontrol counterpart, are discussed under eventual to illustrate output the obtained linear feedback, has been studied from the point of view of the amount of information results. The decentralized control design versus its decentralized control counterpart, under eventual that can be lost or omitted in terms of the total or partial knowledge of the coupled dynamics output linear feedback, has been studied from the point of view of the amount of information that between subsystems necessary can be lostin orthe decentralized omitted in terms case to total of the keep orthepartial closed-loop knowledgestability. A foreseen of the coupled related dynamics future work between relies on the application of the method to some applied control problems such subsystems necessary in the decentralized case to keep the closed-loop stability. A foreseen related as consensus protocols under decentralized future work relies on control and continuous-discrete the application of the method hybrid to somecontroller applied designs. control problems such as consensus protocols under Author Contributions: decentralized Author control and M.D.l.S. contributed continuous-discrete to the manuscript in the hybrid aspects controller designs. of conceptualization; methodology; formal analysis; investigation; writing original draft preparation; writing review and editing; Author Contributions: Author M.S. contributed to the manuscript in the aspects of conceptualization; supervision and fund acquisition. Author A.I. contributed to the manuscript in the various aspects of preparation methodology; formal of the numerical analysis;software experiments; investigation; writing original implementation; draft preparation; visualization and originalwriting review and editing; draft preparation. supervision and fund acquisition. Author A.I. contributed to the manuscript in the various aspects of preparation Funding: This research was co-funded by the Spanish Government and the European Fund of Regional of the numerical experiments; software implementation; visualization and original draft preparation. Development FEDER, grant numbers DPI2015-64766-R and DPI2016-77271-R (MINECO/FEDER, UE) and the UniversityThis Funding: of the Basque was research Country UPV /EHU, co-funded grant by the PGC 17/33. Spanish Government and the European Fund of Regional Development FEDER, grant numbers DPI2015-64766-R and DPI2016-77271-R (MINECO/FEDER, UE) and the University of the Basque Country UPV /EHU, grant PGC 17/33. Acknowledgments The authors are grateful to the Spanish Government for grants DPI2015-64766-R and DPI2016-77271-R (MINECO/FEDER, UE), and to UPV /EHU for grant PGC 17/33. They also thank the referees for their useful comments. Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2019, 9, 1739 14 of 19 Acknowledgments: The authors are grateful to the Spanish Government for grants DPI2015-64766-R and DPI2016-77271-R (MINECO/FEDER, UE), and to UPV /EHU for grant PGC 17/33. They also thank the referees for their useful comments. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Auxiliary Stability Results on Perturbed Matrices under Constant and Time-Varying Perturbations of Stability Matrices Theorem A1. Assume that A0 ∈ Rn×n is a stability matrix with stability abscissa (−ρc ) < 0. Then, A = A0 + A e is a stability matrix if any of the subsequent sufficiency-type conditions holds: (1) The H∞ -norm of (sIn − A0 )−1 A e < 1, e satisfies k(sIn − A0 )−1 Ak ∞ (2) kAk2 < 1/ sup k(iωIn − A0 ) k2 . e −1 ω∈R0+ Other alternative  sufficiency-type conditions to Conditions 1 and 2 for the stability of A are: (3) ρ A0 A < 1, −1 e (4) kA−1 e < 1, Ak 0 2     e 2 < 1/kA−1 k , that is, λmax A kAk e < λmin AT A0 , in the following particular cases: eT A 0 2 0 (a) A0 ≺ 0 andA ≺ −A0 ; and   (b) A0 = A0ij ∈ Mn×n E e= A and A eij ≥ −A0 ; ∀i, j(, i) ∈ n. eij fulfils A ij Proof. Note that      det(sIn − A) = det sIn − A0 − A e = det (sIn − A0 ) In − (sIn − A0 )−1 A e   (A1) = det(sIn − A0 ) det In − (sIn − A0 )−1 A e ; ∀s ∈ C   and then det(sIn − A) = det(sIn − A0 ) det In − (sIn − A0 )−1 A e , 0 Then, for all s < sp(A), and also for all s ∈ sp(A0 ) if the H∞ -norm of (sIn − A0 )−1 A, e which exists since A0 is a stability matrix, satisfies k(sIn − A0 ) Ak < 1, which is guaranteed if kAk −1 e ∞ e 2 < 1/ sup k(iωIn − A0 )−1 k . Then, A is a stability 2 ω∈R0+ matrix if Conditions 1 or 2 holds. On the other hand, if A0 and A are negative (implying that A ≺ −A0 ), or if they are both Metzler-stable (implying for all off-diagonal entries that A eij ≥ −A0 ; ∀i, j(, i) ∈ n), ij then their dominant abscissa (perhaps multiple) eigenvalue is real and negative since A0 being a stability matrix is claimed to guarantee that A is stable. Since A0 is a stability matrix, it is non-singular with eigenvalues with negative  real parts.  Then, by the continuity  of the eigenvalues with respect to the matrix entries, A = A0 In + A−1 Ae is a stability matrix if ρ A−1 Ae ≤ kA−1 Ak e ≤ kA−1 k kAk e2<1 0 0 0 2 0 2 leading to the sufficiency of Conditions 3–5 for the stability of A if is A0 stable. The last, sufficient condition comes  directly  by upper-bounding  Condition  4 by norm  product and it is equivalent to e 2 = λ1/2 A kAk e < 1/kA−1 k = 1/λ1/2 A−1 A−T = λ1/2 AT A0 .  eT A max 0 2 max 0 0 min 0 Theorem A2. Assume that A0 ∈ Rn×n is a stability matrix and that A(t) = A0 + A e : R0+ → Rn×n e(t), where A k e(τ)k < 1; ∀t ∈ R0+ , is piecewise-continuous and bounded. Then, A : R0+ → Rn×n is stable if ρ00 sup kA 0≤τ≤t where (−ρ0 ) < 0 is the stability abscissa of A0 and k0 = k0 (kA0 k) ≥ 1 is a real constant satisfying keA0 t k ≤ k0 e−ρ0 t ;∀t ∈ R0+ . Proof. Consider the linear time-varying system: .   x ( t ) = A0 + A e(t) x(t), x(0) = x0 ; ∀t ∈ R0+ (A2) Appl. Sci. 2019, 9, 1739 15 of 19 where Ae : R0+ → Rn×n is piecewise-continuous and bounded [1]. Such a system is globally asymptotically stable if and only if A : R0+ → Rn×n is a stability matrix. The state-trajectory solution of Equation (A2) satisfies: Rt k0 kx(t)k ≤ keA0 t k kx0 k + e(τ)k kx(τ)k dτ≤ k0 e−ρ0 t kx0 k + eA0 (t−τ) kA sup kx(τ)k 0 ρ0 0≤τ≤t k0   (A3) = K0 e−ρ0 t kx0 k + sup kAe(τ)k kx(t0)k; ∀t ∈ R0+ ρ0 0≤τ≤t   Let t0 = t0(t) be defined for each t ∈ R0+ as t0 = z = max τ : kx(z)k ≥ kx(t)k . Then, 0≤τ≤t k0 kx(t)k ≤ sup kx(τ)k = sup kx(τ)k = kx(t0)k ≤ k0 −ρ0 t0 kx0 k + e(τ)k kx(t0)k; ∀t ∈ R0+ sup kA (A4) 0≤τ≤t 0≤τ≤t0 ρ0 0≤τ≤t k0 Since 1 > ρ0 e(τ)k; ∀t ∈ R0+ , one gets: sup kA 0≤τ≤t !−1 k e(τ)k k0 e−ρ0 t0 kx0 k k x(t)k ≤ k x(t0)k = sup kx(τ)k ≤ 1 − ρ00 sup kA 0≤τ≤t0 0≤τ<∞ !−1 (A5) k0 e(τ)k kx0 k ; ∀t ∈ R0+ ≤ M = k0 1 − ρ0 sup kA 0≤τ<∞ Therefore, x(t) is bounded for any t ∈ R 0+ if x0 is finite. Now, ∞assume the following cases.     Case a: For any Ts ∈ R+ , the sequence  sup kx ( t ) k is strictly decreasing for some      nT ≤t<(n+m )T  s n s n=k finite positive integer k = k(Ts ) and some positive sequence  {mn } of bounded ∞ integer numbers which   satisfies mn+1 > mn − 1 for n ≥ 0. As a result,    sup kx(t)k → 0 as n → ∞ for any    nT ≤t<(n+m )T   s n s n=k given Ts > 0. Then, one gets from Equation (A3) that kx(t)k → 0 , as t → ∞ since kx(nTs + t)k≤ k0 1 + ρ10 sup kx(τ)k; ∀t ∈ (nTs , (n + 1)Ts ). The result is proved for this case.   nTs ≤τ ρ00 sup kA e(τ)k; ∀t ∈ R0+ : 0≤τ≤t sup kx(t)k ≤ sup kx(t)k≤ K0 e−ρ0 (tn+1 −nTs ) kx(nTs )k nTs ≤t<(n+mn )Ts nTs ≤t<(n+mn +mn+mn )Ts + Kρ00   sup e(τ)k kA sup kx(t)k (A6) nTs ≤t<(n+mn +mn+mn )Ts nTs ≤t<(n+mn +mn+mn )T < K0 e−ρ0 (tn+1 −nTs ) kx(nTs )k + sup kx(t)k nTs ≤t<(n+mn +mn+mn )T If n → ∞ , mn → ∞ , then (tn+1 − mn+mn ) → ∞ , thus the following contradiction arises:     0 = lim sup  sup kx(t)k − sup kx(t)k < 0. n→∞ nTs ≤t<(n+mn +mn+mn )Ts nTs ≤t<(n+mn +mn+mn )Ts Appl. Sci. 2019, 9, 1739 16 of 19 Thus, Case b is not possible and the whole result follows from Case a.  Note that the stability abscissa of A0 , that is, (−ρ0 ) < 0 is not smaller than the dominant eigenvalue abscissa. Appendix B Calculations for solving Problem 1 Assume that the matrix (Im − Kc Dc ) is non-singular with kKc Dc k2 < 1 and A = Ac = Ad . Then, one gets from Equations (10) and (13) that: Ad + Bd (Im − Kd Dd )−1 Kd Cd   −1   = A + Bc − e B Im − Kc Dc − e ∆ ∆1 Kc Dc − e 0 B (Im − Kc Dc ) Im − (Im − Kc Dc )−1 e  h i   = A + Bc − e ∆0 −1 Kc Cc − e ∆1 (A7) −1 −1  B Im − (Im − Kc Dc )−1 e    = A + Bc − e ∆0 (Im − Kc Dc ) Kc Cc − e ∆1 ∆2 (Im − Kc Dc )−1 Kc Cc − e      = A + Bc − e B Im + e ∆1   provided that D e and K ∆0 −1 exists (note that this always e are such that (Im − Kd Dd )−1 = Im − Kc Dc − e e = 0p×m and K holds if D e = 0m×p from Assumption 1), where:     ∆0 = Kd Dd − Kc Dc = K e eD e= K e − Dc − Kc D e − Kc De − KD e c (A8)     ∆1 = K e e + Kc C e Cc − C e = Kc − K eC e + KC e c (A9) ∆2 = Im − (Im − Kc Dc )−1 e h i e ∆0 −1 − Im (A10) and note that ∆0 k ≤ ε [kKc ke ke σD + εe σDe σK + k Dc ke σK ] (A11) ∆1 k ≤ ε [kKc ke ke σC + εe σCe σK + k Cc ke σK ] (A12) and one gets from Banach´s Perturbation Lemma [7] that 1 1 ∆2 k ≤ 1 + ke ≤ 1+ (A13) 1 − k(Im − Kc Dc ∆ )−1 e 0k ∆0 k 1 − k(Im − Kc Dc )−1 k ke ∆0 k < e provided that ke δ0 = 1/k(Im − Kc Dc )−1 k. Equivalently, if q(ε) = e σk ε2 + (e σDe σK kDc k)ε − 1/k(Im − Kc Dc )−1 k < 0 σD kKc k + e (A14) since q(ε) is a convex parabola with zeros ε1 < 0 and ε = ε2 > 0, Equation (A14) holds, guaranteeing ∆0 k < e that ke δ0 , if ε ∈ [0 , ε), where: q (e σK kDc k)2 + 4e σD kKc k + e σDe σK /k(Im − Kc Dc )−1 k − (e σD kKc k + e σK kDc k) ε= (A15) σDe 2e σK Before continuing with the calculations, we give the following auxiliary result: Lemma B1. If (Im − Kc Dc ) is non-singular with kKc Dc k2 < 1 and ke ∆0 k < eδ0 = 1/k(Im − Kc Dc )−1 k, equivalently if ε ∈ [0 , ε), with ε defined in Equation (A15), then ke ∆2 k ≤ Cke ∆0 k < 1 with a norm-dependent real constant C ≥ 2δ0 if ε ∈ [0 , ε1 ) with 1 k(Im − Kc Dc )−1 k ε1 = (A16) σD + εe 2[kKc ke σDeσK + k Dc ke σK ] Appl. Sci. 2019, 9, 1739 17 of 19 ∆2 k ≤ Cke Proof. One gets from Equation (A10) and Banach´s Perturbation Lemma [7] that, if ke ∆0 k for some C ∈ R+ , then: 1 −1 ∆0 k ≥ 1 − k(Im − Kc Dc )−1 k ke ∆2 + Im k = kIm − (Im − Kc Dc )−1 e  ≥k e ∆0 k ∆0 k 1 − Cke provided that C < 1/ke ∆0 k. One gets that the above inequality holds if 1/ke ∆0 k > C ≥ 1 2δ0 ≥ k(Im −Kc Dc )−1 k and, one gets from Equation (A11) that ∆0 k k(Im −Kc Dc )−1 k 1+ke ∆2 k ≤ ε C [kKc ke ke σD + εe σDe σK + k Dc ke σK ] < 1 (A17) if ε < ε1 .  Now, rewrite the system matrices of closed-loop dynamics of Equations (7) and (10), equivalently Equation (A7), with A = Ac = Ad and its incremental value as follows: Acc = A + Bc (Im − Kc Dc )−1 Kc Cc (A18) Adc = A + Bd (Im − Kd Dd )−1 Kd Cd (A19) edc = Acc − Adc = Bc (Im − Kc Dc )−1 Kc Cc − Bd (Im − Kd Dd )−1 Kd Cd A    −1   = Bc (Im − Kc Dc )−1 Kc Cc − Bc − e B Im + e ∆2 (Im − Kc Dc ) Kc Cc − e ∆1   −1   ∆2 (Im − Kc Dc ) Kc Cc − e = Bc (Im − Kc Dc )−1 Kc Cc − Bc Im + e ∆1   −1   +eB Im + e∆2 (Im − Kc Dc ) Kc Cc − e ∆1 = Bc (Im − Kc Dc )−1 e ∆1 (A20) ∆2 (Im − Kc Dc )−1 Kc Cc + Bc e −Bc e ∆2 (Im − Kc Dc )−1 e ∆1 −1 −1 +B(Im − Kc Dc ) Kc Cc − B(Im − Kc Dc ) ∆1 e e e −1 −1 +e∆2 (Im − Kc Dc ) Kc Cc − e Be ∆2 (Im − Kc Dc ) e Be ∆1 Now, the following technical result follows directly from Equations (A20), (A11), (A12) and (A17), the norm upper-bounding values of the control, output interconnection and controller matrix errors and Lemma B1: Lemma B2. The following properties hold for any ε ∈ [0 , ε̂) with ε̂ = min(ε , ε1 ) calculated from Equations (A15) and (A16): (i) h   i ∆1 k 1 + ke edc k ≤ kBc k(1 − kKc Dc k)−1 ke kA ∆2 k + kKc Cc kke ∆2 k −1    +ε (1 − kKc Dc k) e σB 1 + ke ∆2 k kKc Cc k + ke∆1 k ≤ εkBc k(1 − kKc Dc k)−1 (A21) σC + εe ×[(kKc ke σCeσK + k Cc ke σK )(1 + kKc Cc k + εC [kKc ke σD + εeσDeσK + k Dc ke σK ])C [kKc ke σD + k Dc ke σK + εe σDe σK ]] −1 +ε (1 − kKc Dc k) e σB (1 + ε C [kKc ke σD + εe σDe σK + k Dc ke σK ])(kKc Cc k + ε [kKc ke σC + εe σCe σK + k Cc ke σK ] ) (ii) If, furthermore, ε ≤ 1, then εr ≤ ε for any real r ≥ 1 so that one gets from Equation (A21) by taking the upper-bound ε2 for ε3 that   edc k ≤ εkBc k(1 − kKc Dc k)−1 ke kA ∆10 k + kKc Cc kke ∆20 k + ε2 ke ∆20 k kBc k(1 − kKc Dc k)−1 −1 −1   +ε (1 − kKc Dc k) e σB kKc Cc k+ε2 (1 − kKc Dc k) e σB ke∆10 k + kKc Cc kke∆20 k (A22) −1 +ε3 (1 − kKc Dc k) e σB ke ∆20 kke ∆10 k≤ uε + vε2 ≤ (u + v)ε Appl. Sci. 2019, 9, 1739 18 of 19 ∆ i = εe where e ∆i0 for i = 1, 2 and it has been used that ε3 ≤ ε2 , with   −1 ∆10 k + kKc Cc kke u = kBc k(1 − kKc Dc k)−1 ke ∆20 k + (1 − kKc Dc k) e σB kKc Cc k −1 −1   (A23) v = (1 − kKc Dc k) σB k∆20 kk∆10 k+k∆20 k kBc k(1 − kKc Dc k) + (1 − kKc Dc k) e e e e e −1 σB ke ∆10 k + kKc Cc kke ∆20 k (iii) If the upper-bound ε is used for ε2 and ε3 , one gets that edc k ≤ εkBc k(1 − kKc Dc k)−1 kA σC + e ×[(kKc ke σCeσK + k Cc keσK )(1 + C [kKc ke σD + e σK + k Dc ke σDe σK ]) + kKc Cc kC [kKc ke σD + e σK + k Dc ke σDe σK ]] (A24) −1 +ε (1 − kKc Dc k) e σB (1 + C [kKc ke σD + e σDe σK + k Dc ke σK ])(kKc Cc k + [kKc ke σC + e σCe σK + k Cc ke σK ] ) edc k ≤ εe which also yields that kA adc + o(ε) in the case that ε < 1 after grouping all the additive contributions of terms in ε for i ≥ 2 into an additive bounded term, which converges to zero as ε → 0 , where i adc = (1 − kKc Dc k)−1 e (A25) ×[eσB kKc Cc k + ( kBc k + kKc Cc k)(kKc ke σC + k Cc ke σK + C [kKc ke σD + k Dc ke σK ])]  Now, the following technical result follows directly from Lemma B2 and Theorem A1 (i):   Lemma B3. Define ε2 = 1/eadc sup k(iωIn − Acc ) k2  from Equation (A25) and assume that kKc Dc k2 < 1 −1    ω∈R0+ and that ε2 < min (ε , ε1 , 1). Then, Adc is stable if Acc is stable and ε ∈ [0 , ε2 ]. By using Equations (A22) and (A23), a√better bound of the maximum allowable kAdc k2 is found as ε ∈ [0 , ε3 ] with ε3 < min (ε , ε20 , 1) and u2 +4vω−u ε20 = 2v . Proof. Note from Theorem A1 (i) that the H∞ -norm of (sIn − Acc )−1 A edc k < 1, edc satisfies k(sIn − Acc )−1 A ∞ which is guaranteed if ε < ε2 , then Adc is stable since Acc is stable. The result follows by taking also into account, in addition, the constraints in Equations (A15) and (A16) of Lemma B.1 by using ε3 ≤ ε2 ≤ ε. If the second power of ε is considered and the third one is upper-bounded as ε3 ≤ ε2 , we examine the stability constraint uε + vε2 < ω = 1/ sup k(iωIn − Acc )−1 k2 by building the convex parabola ω∈R0+ √ −u± u2 +4vω θ(ε) = + uε − ω < 0 whose negative and positive zeros are ε1,2 = vε2 2v . 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