To increase the reliability of systems, fault-tolerant control structures (FTC) usually fix a system with faults so that it can continue its mission with certain limitations of functionality and quality. Considering this, the different approaches were studied in FTC design (see, e.g., Huang et al. (2020); Ding (2021); Du et al. (2021); Lan and Patton (2021) and the references therein).

To eliminate any disconnection of the nominal controller from the control loop and its replacing with a new one, adapted to the really occurred fault conditions, the virtual approach keeps the nominal controller in the reconfigured closed-loop system and “virtually adapts” the faulty plant to the nominal controller. This is done in such a way that the activated virtual reconfiguration block signals, in coincidence with the faulty plant measurements, imitate the fault-free system controller input. Since in healthy conditions the virtual block is not active, the design of the virtual reconfiguration block signals may seem to be independent of the nominal controller. Designated to sensor faults the reconfiguration block is termed virtual sensor (VS), while in the case of actuator faults is named virtual actuator (VA) (Steffen (2005); Richter (2011)). Bearing in mind that reconfiguration after occurrence of an actuator fault can be related to disturbance decoupling, the trend using the terms of a finite set of linear matrix inequalities (LMI) in VA design condition formulation is rather natural (Tabatabaeipour et al. (2015); Krokavec et al. (2016); Bessa et al. (2021); Hu et al. (2021)).

In many control problems exist system parameter limitations. Typical are positive systems, whose state variables are strictly positive quantities (Nikaido (1968); Smith (1995)). Restricting to Metzler structure of the system matrix when dealing with continuous-time positive systems (Berman and Plemmons (1979); Berman et al. (1989); Metzler (2016)), such systems are often denoted as Metzler systems. Most modes for stabilization of Metzler systems use linear memory-free controllers, maintaining its positivity within specific system parameter constraints (De Leenheer and Aeyels (2001); Shen and Lam (2015); Bhattacharyya and Patra (2018)). Although the principle of diagonal stabilization for positive linear systems provides a way to solve partly the considered problems using iteration methods (Shen, 2017), the new approaches mean the constraints representation by LMIs (Krokavec and Filasová, 2018).

Using the concept related to an interval system of equations or inequalities (Moore et al., 2009), then interval observers can be designed for uncertain linear systems, providing intervals to which estimated states are belonging at the evolution time (Gouzé et al., 2000), and specifically in the presence of non-stationary disturbances and uncertainties (Bolajraf et al. (2011); Mazenc and Bernard (2011); Raissi and Efimov (2018)). Unfortunately, the interval representation of the single actuator fault as a loss of gain must be interpreted as a system with polytopic region of uncertainty and the related methods (Prempain and Postlethwaite (2007); Krokavec and Filasová (2021b)) have to be applied with explicit constraint representation in the form of LMIs for this problem.

From the previous overview of methods and principles it can be seen how many different aspects have to be covered in VA design for uncertain Metzler systems. Since the parametric constraints representation is conditioned by the principle of diagonal stabilization, the approach is established to give explicit design conditions in the form of LMIs. This guarantees the strictly positive nominal closed-loop system if the uncertain system model takes the strictly positive VA. Note, the proposed LMI structures cover the system’s structural constraints, the parameter uncertainties, and the diagonal stabilisation principle. The primary goal is to find the stabilizing nominal controller for the Metzler system, being consistent with the positive configuration, as well as to design the VA parameters to retain that the positive continuous-time system properties will be preserved also in the faulty regime after VA activating, when a single actuator fault occurs. Although such defined class of systems prescribes the set of strong structural parametric constraints, the proposed design conditions allow obtaining numerical solutions in the straightforward access. The main contribution is a strict feasibility assumption, mainly for convenience that there will be only modified structural variable constraints in the non-negative formulation of the problem. It is the authors’ belief that further extensions can be done through the given theoretical framework.

The paper is organized as follows. In Section 2 linear Metzler systems formalism and control system strategies are presented. A key theme of Section 3 are the H _∞ norm based methods for nominal controller and VA synthesis. In Section 4 the concept of positive stabilization problem with H _∞ norm characterization analysis is discussed for single actuator fault interpreted as a gain loss. The illustrative numerical example and some concluding remarks are given in Section 5 and Section 6, respectively.

For sake of convenience, throughout the paper used notations reflect usual conventionality so that x ^T, X ^T denotes the transpose of the vector x , and the matrix X , respectively, X ⁻¹, ρ( X ) signifies the inverse and the eigenvalue spectrum of a square matrix X , respectively, for a symmetric square matrix X ≺ 0 means that X is negative definite matrix, diag  [ ⋅ ] marks the elements of a (block) diagonal matrix, ∗ represents the block in a square symmetric matrix that is readily inferred by the matrix symmetry, the symbol I _n indicates the n-th order unit matrix, R ( R + ) qualifies the set of (nonnegative) real numbers, R + + n × r ( R + n × r ) refers to the set of n × r strictly positive (nonnegative) real matrices and M − + n × n , ( M − + ◦ n × n ) is the set of strictly (purely) Metzler matrices.

To describe a linear, time-invariant continuous-time MIMO Metzler system the state equations of the form q ̇ ( t ) = A q ( t ) + B u ( t ) + D d ( t ) , (1) y ( t ) = C q ( t ) , (2) can be used, where q ( t ) ∈ R + n , u ( t ) ∈ R r , y ( t ) ∈ R + m are the system state vector, control input and measurable output, respectively and d ( t ) ∈ R + r d is the system disturbance.

Since there exist different techniques to preserve properties of the continuous-time Metzler linear systems, it is considered that B ∈ R + n × m , C ∈ R + m × n , D ∈ R + n × p are non-negative matrices and A ∈ M − + n × n is strictly Metzler.

Exploiting the Metzler matrix structure notation in the efficient and flexible modeling of positive continuous-time systems, the key features have to be highlighted in the following.

Definition 1

(Cvetković_2020) A square matrix A ∈ M − + ◦ n × n is purely Metzler if its diagonal elements are negative and its off-diagonal elements are nonnegative. A square matrix A ∈ M − + n × n is strictly Metzler if its diagonal elements are negative and its off-diagonal elements are positive. A Metzler matrix is stable if it is Hurwitz. From a strictly Metzler matrix A ∈ M − + n × n imply n ² structural constraints a i i < 0 ∀ i = 1 , … n , a i j , i ≠ j > 0 ∀ i , j = 1 , … n . (3)

The problem gets even significantly more complex if purely Metzler matrices in the control loops are considered.

Remark 1

Since B ∈ R + n × r , C ∈ R + m × n are non-negative, a negative feedback makes smaller (nonnegative or positive) off-diagonal elements and it could destroy the Metzler structure setting one (or more) of the off-diagonal elements to a negative value. This fact also highlights that structural constraints must be included in the synthesis conditions to keep the desired structure Metzler.

While for general linear systems it is possible to work with sign indefinite elements in the matrix inversion of a square system matrix, for Metzler systems it may be difficult, or impossible, to provide general statements if this matrix operation has to be done. Since a square matrix X and its inverse have nonnegative structure if X is positively definite diagonal, to guarantee structural constraints the LMI based design conditions for Metzler systems are formulated using positive definite diagonal matrix variables and the term “diagonal stability” is used (Berman and Hershkowitz, 1983), (Krokavec and Filasová, 2018). If A ∈ R − + n × n is only purely Metzler, the synthesis conditions have to reflect further structural constraints, includable in design by related structured diagonal matrix variables (Krokavec and Filasová, 2019).

Proposition 1

(Farina_and_Rinaldi_2000) A solution q (t) of the disturbance-free state model (1) for t ≥ 0 is asymptotically stable and positive if A ∈ M − + n × n is a stable Metzler matrix, B ∈ R + n × r is a nonnegative matrix and the state vector q ( t ) ∈ R + n for given u ( t ) ∈ R + r and q ( 0 ) ∈ R + n . The linear disturbance free system (1), (2) is asymptotically stable and positive if A ∈ M − + n × n is a stable Metzler matrix, B ∈ R + n × r , C ∈ R + m × n are non-negative matrices and both the state vector q ( t ) ∈ R + n and the output vector y ( t ) ∈ R + m for all u ( t ) ∈ R + r and q ( 0 ) ∈ R + n .

Definition 2

(Horn and Johnson, 1995) A matrix L ∈ R n × n is a permutation matrix if exactly one item in each column and row is equal to 1 and all other elements are equal to 0.

Keeping into account Definition 2 and envisaging a diagonal Y ∈ R n × n such that Y = diag y 1 y 2 ⋯ y n , (4) then it yields L T Y L = diag y 2 ⋯ y n y 1 , (5) if L T ∈ R n × n takes the circulant form L T = 0 I n − 1 1 0 . (6)

Remark 2

(Krokavec and Filasová, 2021a) The diagonal stabilization problem can be reformulated using a rhombic mapping of the square strictly Metzler matrix A ∈ M − + n × n of the form A = a 11 a 12 a 13 ⋯ a 1 n a 21 a 22 a 23 ⋯ a 2 n a 31 a 32 a 33 ⋯ a 3 n ⋮ ⋮ ⋮ ⋮ ⋮ a n 1 a n 2 a n 3 ⋯ a n n , (7) where the rhombic mapping is constructed using circular shifts of rows of A ∈ M − + n × n as A Θ = a 11 a 12 a 13 ⋯ a 1 n a 22 a 23 ⋯ a 2 n a 21 a 33 ⋯ a 3 n a 31 a 32 ⋱ ⋮ ⋮ ⋮ ⋱ a n n a n 1 a n 2 ⋯ a n , n − 1 . (8)

It is evident that generally n ² parametric constraints (3) can be defined by the negativeness of A Θ ( i , i + h ) ∈ R + n × n for h = 0 and by positiveness of (n − 1) diagonal matrices A Θ ( i , i + h ) ∈ R + n × n for h = 1, …, n − 1 with A Θ ( i , i + h ) = diag a 1,1 + h ⋯ a n − h , n a n − h + 1,1 ⋯ a n , h , (9) related to the diagonals of (8).

Definition 3

(Bellman, 1970) Let U ∈ R m × m , O ∈ R n × n then the (mn)-dimensional matrix, called the Kronecker product of U and O , is constructed as U ⊗ O = u i j O i , j = 1 m , U = u i j i , j = 1 m . (10)

It can be underlined at this point that the following Kronecker product properties (Brewer, 1978) will be exploited ( I n ⊗ U ) ( O ⊗ I m ) = ( O ⊗ I m ) ( I n ⊗ U ) , (11) ( U ⊗ O ) − 1 = U − 1 ⊗ O − 1 , (12) ( U ⊗ O ) T = U T ⊗ O T . (13)

Consider the system (Eqs 1, 2), and the properties of diagonals of the mapping (Eqs 8, 9), with the specific relation to Metzler system diagonal stabilization principle. To keep the notation simple, without loss of generality, this principle is briefly formulated in these lemmas.

Lemma 1

(Krokavec and Filasová, 2018) Let the matrix A ∈ M − + n × n be strictly Metzler then it is Hurwitz if and only if there exists a positive definite diagonal matrix Q ∈ R + n × n such that for h = 1, …, n − 1 and the circulant L ∈ R + n × n the following sets of LMIs is feasible Q ≻ 0 , Q A T + A Q ≺ 0 , (14) A Θ ( l , l ) Q ≺ 0 , L h A Θ ( l , l + h ) L h T Q ≻ 0 . (15)

Note, in the context of Lyapunov stability analysis by using the Lyapunov technique, (Eq. 14) results from a quadratic Lyapunov function candidate.

Lemma 2

(Krokavec and Filasová, 2020) Let a square real n × n matrix Λ is partitioned as Λ = A − B K C , (16) where A ∈ M − + n × n , B ∈ R + n × m , C ∈ R + m × n , K ∈ R + m × m , while A is strictly Metzler. Then Λ is strictly Metzler if, equivalently,

(i) a i i − b i T K c i < 0 f o r a l l i = 1 , … , n , a i j − b i T K c j > 0 f o r a l l i , j = 1 , … n , i ≠ j , (17)

(ii) A Θ ( i , i ) − B d K d C d ≺ 0 , A Θ ( i , i + h ) − B d K d C d h ≻ 0 , (18) where B = b 1 T ⋮ b n T , B d = d i a g b 1 T ⋯ b n T , (19) C = c 1 ⋯ c n , C d = d i a g c 1 ⋯ c n , (20) K d = I n ⊗ K , C d h = S h T C d L h , S = L ⊗ I m . (21) while B is separated by its rows and C by its columns.

Moreover, the square matrix representation related to its rhombic diagonals is given as Λ = ∑ h = 0 n − 1 ( A Θ ( i , i + h ) − B d D d C d h ) L h T . (22)

Since the positivity of the systems is defined by a nonnegative system state, nonnegative system input and output matrix parameters and by a Metzler system matrix structure, it is necessary to proceed from these facts also in the synthesis of the static output controller.

Within the main goal of this paper it is necessary to define primarily the problem performed on positiveness base to control the Metzler system in the faulty-free mode and, in particular, to use of the LMI criteria form in the stability guarantee.

The stabilizing approach, referred to as the static output control (Zhang et al. (2020); Gritli et al. (2021)) has to be also adapted in the control law gain computing for positive continuous-time systems. Supposing that the square Metzler system (Eqs 1, 2), is stabilizable by the static output control u ( t ) = − K y ( t ) = − K C q ( t ) , (23) with strictly positive gain K ∈ R + + m × m , this multi-variable problem subject to given parametric constraints can be solved as follows:

Theorem 1

The closed-loop constructed on the square linear Metzler system (Eqs 1, 2), under static output control (Eq. 23) is stable if there exist positive definite diagonal matrices Q ∈ R + n × n , V ∈ R + m × m , a strictly positive matrix R ∈ R + + m × m and a positive scalar γ ∈ I R ₊ such that for h = 1, 2, …n − 1 Q ≻ 0 , V ≻ 0 , γ > 0 , (24) A Q + Q A T − B R C − C T R T B T ∗ ∗ D T − γ I r d ∗ C Q 0 − γ I m < 0 , (25) A Θ ( i , i ) Q − B d R d C d ≺ 0 , (26) L h A Θ ( i , i + h ) L h T Q − L h B d S T h R d C d ≻ 0 , (27) C d Q = V d C d , (28) with the structured matrix variables R , R d = I n ⊗ R , V d = I n ⊗ V (29) and with the enhanced design parameters given in (Eqs 9, 19, 20).

When the above conditions hold the strictly positive K is constructed by using R , V as K = R V − 1 . (30)

Hereafter, ∗ denotes the symmetric item in a symmetric matrix.

Proof

According to the condition (Eq. 14) it must be applied for a positive definite diagonal matrix Q ∈ R + n × n that A c Q + Q A c T ≺ 0 , (31) where A c = A − B K C , (32)

Hence, upon inserting (Eqs 22, 32), it can be obtained an explicit expression A c Q + Q A c T = ∑ h = 0 n − 1 ( A Θ ( i , i + h ) L h T − B d K d C d h L h T ) Q + ∑ h = 0 n − 1 Q ( A Θ ( i , i + h ) L h T − B d K d C d h L h T ) T . (33)

Substituting (Eq. 21) and using (Eq. 11) it can be solved for the above defined specific matrix element (since L ^h L ^hT = I _n ) B d K d C d h L h T Q = B d K d S h T C d h L h L h T Q = B d ( I n ⊗ K ) ( L h T ⊗ I m ) C d Q = B d ( L h T ⊗ I m ) ( I n ⊗ K ) C d Q = B d S h T K d C d Q (34) and the product K _d C _d Q can be rewritten as follows K d C d Q = K ⋱ K C d Q = K V ⋱ K V V − 1 ⋱ V − 1 C d Q = R d V d − 1 C d Q , (35) where R _d , V _d are the structured matrix variables defined in (Eq. 29), V ∈ I R + m × m is a positive definite diagonal matrix, R ∈ R + m × m is strictly positive structured matrix variable and R = K V . (36)

Thus, Prescribing That V d − 1 C d = C d Q − 1 ⇒ C d Q = V d C d , (37) then K d C d Q = R d C d (38) and (Eq. 37) gives the matrix equality (Eq. 28).

Multiplying the right side of (Eq. 18) by Q and using (Eq. 38) then (Eq. 18) implies ( A Θ ( i , i ) − B d K d C d ) Q = A Θ ( i , i ) Q − B d R d C d ≺ 0 , (39) which forces (Eq. 26).

Analogously, pre-multiplying the left side by L ^h and post-multiplying the right side by L ^hT Q it results from (Eq. 18) that L h A Θ ( l , l + h ) L h T Q − L h B d K d C d L h T Q ≻ 0 (40) and using (Eqs 34, 38), (38) then (Eq. 40) implies (Eq. 27).

If the matrix T ∈ R + m × m n is defined as T = I n ⋯ I n (41) then, multiplying the left side of (Eq. 38) by T , T K d C d Q = K C Q = T R d C d = R C (42) and using (Eq. 42) it will be A c Q = ( A − B K C ) Q = A Q − B R C (43)

In the context of Lyapunov stability analysis by using the Lyapunov technique, a Lyapunov function candidate of the form v ( q ( t ) ) = q T ( t ) P q ( t ) + γ − 1 ∫ 0 t ( y T ( τ ) y ( τ ) − γ 2 d T ( τ ) d ( τ ) ) d τ > 0 (44) is chosen, where a positive definite diagonal matrix P ∈ R + n × n is prescribed and γ > 0 is considered as an upper bound of H _∞ norm of the disturbance transfer function matrix (see, e.g., Filasová and Krokavec (2012) for more details).

Simple computations show that the time derivative of this Lyapunov function is defined by v ̇ ( q ( t ) ) = q ̇ T ( t ) P q ( t ) + q T ( t ) P q ̇ ( t ) + γ − 1 y T ( t ) y ( t ) − γ d T ( t ) d ( t ) = q T ( t ) ( A c T P + P A c + γ − 1 C T C ) q ( t ) + + d T ( t ) D T P q ( t ) + q T ( t ) P D d ( t ) − γ d T ( t ) d ( t ) < 0 (45) and Lyapunov’s method states that if such a function exists, then the equilibrium of the system is stable.

Thus, defining q ⋄ T ( t ) = q T ( t ) d T ( t ) , (46)

(Eq. 45) can be written as v ̇ ( q ⋄ ( t ) ) = q ⋄ T ( t ) P ⋄ q ⋄ ( t ) < 0 , (47) where, because P is diagonal, P ⋄ = A c T P + P A c + γ − 1 C T C P D D T P − γ I p ≺ 0 . (48)

Therefore, in the sense of Schur complement, (Eq. 48) admits A c T P + P A c P D C T D T P − γ I r d 0 C 0 − γ I m ≺ 0 , (49) where (Eq. 49) takes the form by sense of the Bounded Real Lemma (Scherer et al. (1997).

Thus, with X = diag Q I r d I m , Q = P − 1 , (50) it can obtain from (Eq. 49), when pre-multiplying the left side and post-multiplying the right side by X , A c Q + Q A c T D Q C T D T − γ I r d 0 C Q 0 − γ I m ≺ 0 . (51)

Hence, substituting (Eq. 43), then (Eq. 51) implies (Eq. 25). This concludes the proof.

In a faulty case with a single actuator fault the control structure is modified by adding the associated VA block that masks the actuator fault, and allows the controller to perceive the system as it was before the fault, i.e., the nominal controller may still be used without it being necessarily readjusted. The synthesis methodology assumes a known system model and its parameters for working in nominal mode and a system model and its parameters for faulty mode. In addition, it is assumed that the active FTC structure contains a part of fault detection and localization (FDI) that will allow the correct activation of the control structure reconfiguration after the fault has occurred.

The state-space description of the system with a single actuator fault is considered as follows q ̇ f a ( t ) = A q f a ( t ) + B f u f a ( t ) + V d ( t ) , (52) y f a ( t ) = C q f a ( t ) , (53) where q f a ( t ) ∈ R n denotes the faulty system state variables vector, u f a ( t ) ∈ R r labels the vector of acting control input variables in the faulty regime, y f a ( t ) ∈ R m identifies the vector of output variables in the faulty regime, while the matrix B f ∈ R n × r and rank( B _f ) ≤ rank( B ). Moreover, it is supposed that the pair ( A , B _f ) is controllable and the input vector u _fa (t) is available for reconfiguration (all inputs to the plant are available as they use the nominal controller, but one associated with the faulty actuator is broken) and the covering of the faulty control input is defined as follows u f a ( t ) = − G e f a ( t ) , (54) where G ∈ R r × n . To obtain the VA state-space description, the following lemma is proven.

Lemma 3

The system dynamics of VA for the system (Eqs 52, 53), with a single actuator fault is given as e ̇ f a ( t ) = A c f e f a ( t ) − B u c ( t ) , (55) where A c f = A − B f G , (56) e f a ( t ) = q f a ( t ) − q ( t ) (57)

and e f a ( t ) ∈ R n , A c f ∈ R n × n .

Proof

Writing (Eq. 1) and (Eq. 52) compactly as q ̇ f a ( t ) q ̇ ( t ) = A 0 0 A q f a ( t ) q ( t ) + B f 0 0 B u f a ( t ) u c ( t ) + D D d ( t ) , (58) then the behavior of the extended system can be fully described also by using q _fa (t) and by the error vector e _fa (t) (Eq. 57). Then, to perform the system coordinate change, the transform matrix T can be defined with respect to (Eq. 57) as follows T = I 0 I − I , T − 1 = I 0 I − I (59) and, accordingly, it can be obtained T q f a ( t ) q ( t ) = I 0 I − I q f a ( t ) q ( t ) = q f a ( t ) e f a ( t ) , (60) I 0 I − I B f 0 0 B = B f 0 B f − B , (61) I 0 I − I D D = D 0 , (62) I 0 I − I A 0 0 A I 0 I − I = A 0 0 A . (63)

Moreover, consistently with these notations, (Eq. 58) can be rewritten in the following form q ̇ f a ( t ) e ̇ f a ( t ) = A 0 0 A q f a ( t ) e f a ( t ) + B f 0 B f − B u f a ( t ) u c ( t ) + D 0 d ( t ) . (64)

The choice of the faulty control input covering, defined by (Eq. 54), solves the problem and provides that the substitution of (Eq. 54) in (Eq. 64) leads to q ̇ f a ( t ) e ̇ f a ( t ) = A − B f G 0 A − B f G q f a ( t ) e f a ( t ) − 0 B u c ( t ) + D 0 d ( t ) . (65)

Using the above argumentations it is possible to state that the second row of the equation (Eq. 65) implies the VA equation (Eq. 55). This concludes the proof.

Corollary 1

Using the nominal controller, defined in analogy with (Eq. 23) as u c ( t ) = − K y ( t ) = − K C q ( t ) , (66) where K ∈ R + + m × m , then q ̇ ( t ) = A c q ( t ) + V d ( t ) , (67) y ( t ) = C q ( t ) , (68) A c = A − B K C (69) and it is evident that K can be designed using Theorem 1.

The first row of the equation (Eq. 65) implies q ̇ f a ( t ) = A q f a ( t ) − B f G e f a ( t ) + D d ( t ) = A q f a ( t ) − B f G ( q f a ( t ) − q ( t ) ) + D d ( t ) = ( A − B f G ) q f a ( t ) + D B f d ( t ) G q ( t ) . (70)

Thus, denoting D f = D B f , d f a T ( t ) = d T ( t ) q T ( t ) G T , (71) the state-space description of the faulty closed-loop system with activated VA is as follows q ̇ f a ( t ) = A c f q f a ( t ) + D f d f a ( t ) , (72) y f a ( t ) = C q f a ( t ) , (73) where D f a ∈ R n × r f a , d f a ( t ) ∈ R r f a , r _fa = r + r _d  

When comparing (Eq. 55) and (Eq. 72) it is necessary to emphasize that with such defined structures, the dynamics of the VA and the dynamics of the faulty system will be the same.

Remark 3

Writing (Eq. 66) as u c ( t ) = − K C ( q f a ( t ) − e f a ( t ) ) = − K ( y f a ( t ) − C e f a ( t ) ) (74) it is evident that the measured faulty output y _fa (t) can be masked on the input of the controller by the value − Ce _fa (t), where e _fa (t) is obtained from the VA equation (Eq. 55).

Practically it means to start reconfiguration regime by including the virtual block (Eq. 55) into the control loop and to initialize it with e _fa (0) .

Remark 4

The possible question to ask is whether (Eq. 65) happens to constitute the classical separation principle invalidity, if the single actuator faults are projected onto the uncertainty of input system matrix B _f , see, e.g., the accompanying tasks in Do et al. (2021); Rauh et al. (2021).

Since, substituting (Eqs 69–71) Into (Eq. 65), it can be obtained q ̇ f a ( t ) e ̇ f a ( t ) = A c f 0 0 A c f q f a ( t ) e f a ( t ) − 0 B u c ( t ) + D f 0 d f a ( t ) , (75) the closed-loop structure with VA should not have such unfavorable response.

To give the design conditions for VA for strictly Metzler systems, it is necessary to repeat a few results on diagonal parameterisations with relation to state control problem.

Remark 5

(compare Krokavec and Filasová (2020)) In contrast to the diagonal representation of the system parameters according to (Eqs 19, 20) in the static output control design, the state control form of (Eq. 56) requires separation of the matrix B _f by its columns and the matrix G by its rows to obtain different structure of the diagonal representation, defined as follows for j = 1, …, r B f = b 1 ⋯ b i f ⋯ b r , B f d j = d i a g b j 1 ⋯ b j n , j ≠ i , B f d i = d i a g 0 ⋯ 0 , (76) G = g 1 T ⋮ g n T , G d j = d i a g g j 1 ⋯ g j n . (77)

The square matrix representation of A _cf from its rhombic diagonals is now given as A c f = ∑ h = 0 n − 1 A Θ ( i , i + h ) − ∑ j = 1 r j B f d j G d j h L h T , (78) where G d j h = L h T G d j L h , (79) while the open form representation of A _cf is A c f = A − ∑ j = 1 r j b f j g j T = A − ∑ j = 1 r j B f d j l n l n T G d j , (80) where l n = 1 1 ⋯ 1 T ∈ R n . (81)

The use of these relationships eliminates introduction of additional structured matrix variables into LMI formulation.

The control structure with VA is standard (see, e.g., Richter (2011), Fig. 4.8fig48, p. 73).

To the above given faulty system model, and persisted parametric constraints, the design condition are imposed in the following form:

Theorem 2

VA (Eq. 55) is Metzler and stable if the faulty system (Eqs 52, 53) is Metzler and there exist positive definite diagonal matrices U , V j ∈ R + n × n and a positive scalar δ ∈ R + such that for h = 1, …, n − 1 , j = 1, …, r, U ≻ 0 , V j , ≻ 0 , δ > 0 , (82) A U + U A T − ∑ j = 1 r ( B f d j l n l n T V j + V j l n l n T B f d j ) ∗ ∗ D f T − δ I r ∗ C U 0 − δ I m < 0 , (83) A Θ ( i , i ) U − ∑ j = 1 r B f d j V j ≺ 0 , (84) L T A Θ ( i , i + h ) L h T U − ∑ j = 1 r L h B f d j T h T V j ≻ 0 . (85) With a feasible solution, the nonnegative VA gain G ∈ R + r × n that solves the design task is G d j = V j U − 1 , g j T = l T G d j , G = g 1 T ⋮ g r T , (86) only implicitly defining the gain non-negativeness.

Proof

If h = 0, then (Eq. 78) impliesr A Θ ( i , i + h ) − ∑ j = 1 r B f d j G d j ≺ 0 . (87)

Since U ∈ R + n × n is a positive definite diagonal matrix, multiplying the right side of (Eq. 88) by U gives A Θ ( i , i + h ) U − ∑ j = 1 r B f d j G d j U ≺ 0 (88) and with V j = G d j U , (89) then (Eq. 88) forces (Eq. 84).

Analogously, for an arbitrary h ∈ ⟨1, n − 1⟩, A Θ ( i , i + h ) − ∑ j = 1 r B f d j G d j h L h T = A Θ ( i , i + h ) L h T − ∑ j = 1 r B f d j L h T G d j L h L h T ≻ 0 (90) and, pre-multiplying the left side by L ^h and post- multiplying the right side by U , then (Eq. 90) implies L h A Θ ( i , i + h ) L h T U − ∑ j = 1 r L h B f d j L h T G d j U ≻ 0 . (91)

Therefore, using (Eq. 89), it comes out that (Eq. 91) gives (Eq. 85).

When considering a positive definite diagonal matrix W ∈ R + n × n , in the Lyapunov sense it can be set v ( q f a ( t ) ) = q f a T ( t ) W q f a ( t ) + δ − 1 ∫ 0 t ( y f a T ( τ ) y f a ( τ ) − δ 2 d f a T ( τ ) d f a ( τ ) ) d τ > 0 (92) and, consequently, it can be expressed the time derivative of (Eq. 91) as v ̇ ( q f a ( t ) ) = q ̇ f a T ( t ) W q f a ( t ) + q f a T ( t ) W q ̇ f a ( t ) + δ − 1 y f a T ( t ) y f a ( t ) − δ d f a T ( t ) d f a ( t ) = q f a T ( t ) ( A c f T W + W A c f + δ − 1 C T C ) q f a ( t ) + + d f a T ( t ) D f T W q f a ( t ) + q f a T ( t ) W D f d f a ( t ) − δ d f a T ( t ) d f a ( t ) < 0 . (93)

Thus, defining q f a ⋄ T ( t ) = q f a T ( t ) d f a T ( t ) , (94)

(Eq. 93) can be written as v ̇ ( q f a ⋄ ( t ) ) = q f a ⋄ T ( t ) W f a ⋄ q f a ⋄ ( t ) < 0 , (95) where, because W is diagonal, W f a ⋄ = A c f T W + W A c f + γ − 1 C T C P D f a D f T P − γ I r ≺ 0 . (96)

Therefore, in the sense of Schur complement, (Eq. 96) admits A c f T W + W A c f W D f C T D f T W − δ I r 0 C 0 − δ I m ≺ 0 (97) and with Z = diag U I r I m , U = W − 1 , (98) it can be obtained from (Eq. 97), when pre-multiplying the left side and post-multiplying the right side by Z that A c f U + U A c f T D f U C T D f T − γ I r 0 C U 0 − γ I m ≺ 0 . (99)

Since multiplying the right side of (Eq. 80) by U gives A c f U = A U − ∑ j = 1 r B f d j l n l n T G d j U , (100) then with (Eq. 89) it yields A c f U = A U − ∑ j = 1 r B f d j l n l n T V d j (101) and substituting (Eq. 101) then (Eq. 99) implies (Eq. 83). This concludes the proof.