On Homogeneous Descriptor Systems and Homogeneity-Based Finite-Time Control of Linear Descriptor Systems

A new definition of homogeneity for descriptor systems is introduced. The homogeneity property can be verified algebraically, and, as for systems described by ordinary differential equations only, it implies scalability of solutions. A finite-time stabilizing feedback controller is designed for linear descriptor systems. The proposed control contains a homogenizing linear feedback term and a nonlinear homogeneous stabilizing term. The parameters tuning is presented in the form of linear matrix equations and inequalities. Performance of the approach is illustrated by numerical examples.


I. INTRODUCTION
The homogeneity property is widely used in automatic control theory for system analysis, control and observer design (see, for example, [2], [3], [4], [5], [6], [7], [8], [9] and references therein). Some of the useful features of homogeneous systems are scalability of trajectories (local properties of homogeneous systems can be translated into global ones); the convergence rate of homogeneous systems can be assessed by its homogeneity degree; robustness with respect to external perturbations and measurement noises; etc. For example, the homogeneity property is widely used for finitetime (ensuring the completion of all transients in a finite time) control design: if a homogeneous system with negative degree is asymptotically stable, then it is finite-time stable (see, for example, [6], [7], [10], [11], etc.).
All mentioned works are focused on the study of systems described by Ordinary Differential Equations (ODEs) only. Descriptor (also referred to as singular or differentialalgebraic) systems, in turn, are of great importance in control systems theory. For example, representation of systems in the descriptor form is relevant for a number of applications, The associate editor coordinating the review of this manuscript and approving it for publication was Qi Zhou. e.g., electrical circuits [12], [13], [14], mechanical systems with phase constraints [15], [16], chemistry and biology [17], [18], [19], economics [20], etc. In the present paper a new definition of homogeneity for descriptor systems (we refer to it as d E -homogeneity) is introduced. As well as for systems in the form of ODEs, the homogeneity property can be verified algebraically, and it implies that a dilation of initial conditions leads to a scaling of trajectories. Necessary and sufficient criterion for a linear descriptor system to be homogeneous with nonzero degree are given.
The finite-time control design problem is addressed in the paper as well. This problem has received much less attention in the literature due to difficulty involved, and as a result, there are few related works (e.g., [21], [22], [23]). Based on the homogeneity property a finite-time stabilizing control is proposed for linear descriptor systems. The control law contains two terms that homogenize (of given negative degree) and stabilize the closed-loop system, respectively. Note that the proposed approach does not require a transition to the ODE form, block decomposition or coordinate transformation, which may be accompanied by computational errors. These advantages and Linear Matrix Inequalities (LMI)-based parameters tuning make the proposed control easier to use in practice. VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ With respect to the preliminary results of [1], the key differences are as follows: • a simplified definition of homogeneity for descriptor systems is given; • a finite-time control for linear descriptor systems with accelerated convergence is proposed; • the proofs of all claims are given; • new examples are considered. Notation: R is the field of real numbers; R + = {x ∈ R : x > 0}; C is the field of complex numbers; · denotes the Euclidean norm on R n ; I n denotes the identity matrix of order n; λ(E, A) denotes λ(E, A) = {s|s ∈ C, s is finite, det(sE − A) = 0} for E, A ∈ R n×n , and λ(A) = λ(I n , A) is a spectrum of the matrix A; the eigenvalues of a matrix G ∈ R n×n are denoted by λ i (G), i = 1, . . . , n.

Consider a descriptor system
where x ∈ R n is the state vector, E : R n → R n×n , 0 < rankE ≤ n, f : R n → R n is a continuous vector field, f (0) = 0. It is assumed that the initial condition x 0 ∈ R n is consistent, and the system (1) has a unique solution x 0 (t).
Definition 1 [39]: The system (1) is said to be globally finite-time stable if it is globally asymptotically stable and any solution x 0 (t) reaches the origin in a finite time, i.e., x 0 (t) = 0 for all t ≥ T (x 0 ) and x 0 (t) = 0, ∀t ∈ [0, T (x 0 )), Consider a control descriptor system where x ∈ R n is the state vector, u ∈ R m is a control input, f : R n × R m → R n , E : R n → R n×n . Definition 2 [26]: A control law u(x) is called admissible, if for any consistent initial condition x 0 , the system (2) has no impulsive solution. The system (2) is impulse controllable if there exists an admissible control law.
Let us recall some basics on linear descriptor systems. Consider the system (1) in the form where E, A ∈ R n×n , rankE = n 1 ≤ n. The solution behavior (regularity) of (3) depends on the properties of the pair (E, A).
Definition 3 [12]: The pair (E, A) is said to be regular if det(sE−A) = 0 for some s ∈ C.
The following lemma gives a necessary and sufficient condition of regularity for linear descriptor systems.
Lemma 1 [12]: The pair of matrices (E, A) is regular (the system (3) has a unique solution x 0 (t)) iff there exist nonsingular matrices Q, P ∈ R n×n such that where N ∈ R n 2 ×n 2 is a nilpotent matrix, A 1 ∈ R n 1 ×n 1 , n 1 + n 2 = n.

III. MAIN RESULT A. HOMOGENEOUS DESCRIPTOR SYSTEMS
Before stating the definition of d E -homogeneity, let us recall the usual definition of linear geometric (generalized) homogeneity. Definition 4 [6]: Let d(s) : R → R n×n denotes the family of dilations given by d( where G d ∈ R n×n is an anti-Hurwitz matrix (i.e. −G d is Hurwitz) called the generator [27]. Thus, where One of the most important properties of homogeneous systems is the scalability of solutions [2], [5], [28], [29]. The scalability of solutions implies a number of properties useful for qualitative analysis (e.g., local stability implies the global one; the existence of strictly invariant (in forward time) compact set implies asymptotic stability [2], [28], etc.). The following theorem provides an analogous result for d E -homogeneous descriptor systems.
Theorem 1: is a solution to (1) with the initial condition Making the change of time t = e νs t new , we complete the proof.
In order to consider an example of a homogeneous descriptor system let us refer to the paper [30], where the following average-consensus reaching algorithm is proposeḋ for i = 1, . . . , n agents, and µ ∈ (0, 1), a ij = a ji ≥ 0. Following [30], let us make the change of variables , where x * is the equilibrium position of the system. Then average-consensus problem will be implied by stability of the following systeṁ that is d E -homogeneous of degree −µ with the generator G d = e s I n . Simulation results (Fig. 2) for the system (8) with the structure given in Fig. 1 (the coupling coefficients a ij are 0 or 1 according to the structure) are presented for different initial conditions to show the scalability of the solutions.

B. LINEAR HOMOGENEOUS DESCRIPTOR SYSTEMS
Now consider the following linear regular descriptor system where x ∈ R n , A, E ∈ R n×n and rankE = n 1 < n. Let the pair (E, A) is regular, i.e., by Lemma 1 it admits the canonical formẋ where The following result gives the criteria for a linear descriptor system (9) to be d E -homogeneous of nonzero degree.
Lemma 2: The next statements are equivalent.
(1) The system (9) is is regular one can rewrite the system (9) in the form Assume that the matrix A 1 is nilpotent. According to [10] the equations are feasible for any ν = 0 and some anti-Hurwitz matrices G 1 ∈ R n 1 ×n 1 , G 2 ∈ R n 2 ×n 2 , and by [10, Lemma 3] for all s ∈ R. Then choosing the generator in the form G d = and i.e., (5) and (6) are satisfied. VOLUME 10, 2022 (1)⇒(2) Let the system (9) is d E -homogeneous of degree According to (7) the system solutions are scalable as follows i.e., Due to ν = 0 we have that λ(A 1 ) = 0 and the matrix A 1 is nilpotent.
(2)⇔(3) Equivalence is straightforward due to Remark 2: Note that if the system (9) is d E -homogeneous with the generator matrix G d , then the corresponding canonical form (10) isd E -homogeneous with the generator G d = P −1 G d P.

C. FINITE-TIME CONTROL OF LINEAR DESCRIPTOR SYSTEMS
In this section we propose a finite-time control design method for linear descriptor systems based on d E -homogeneity.
Firstly, let us introduce the notions of d E -homogenization and d E -homogeneous stabilization.
Definition 6: The descriptor control system (2) is (2) is d E -homogeneous of degree ν; • d E -homogeneously stabilizable with degree ν ∈ R if there exists a feedback control u(x) such that the closed-loop system is d E -homogeneous of degree ν and globally asymptotically stable. Consider the linear descriptor system where x ∈ R n is the measurable state vector, u ∈ R m is the vector of control inputs, E, A ∈ R n×n , B ∈ R n×m (the pair (E, A) is regular, the triplet (E, A, B) is completely controllable, rank B = m ≤ n).
The main goal is to propose a constructive (i.e., equipped with reliable tuning rules) d E -homogenizing and stabilizing in a finite time control for the system (14).
The following lemma gives the necessary and sufficient condition of d E -homogenization of the system (14) via linear feedback control.
Lemma 3: The system (14) is d E -homogenizable via linear feedback control u = K E x, K E ∈ R m×n if and only if K E is such that λ(E, (A + BK E )) = 0.
Proof: The proof is straightforward consequence of Lemma 2.
According to [31] there exists an invertible matrix X ∈ R n×n such that X T E = E T X ≥ 0 and x T X T Ex = 0 iff Note that for E = I n the defined function · d E becomes the canonical d-homogeneous norm as in [11].
The following theorem proposes a d E -homogeneity based stabilizing control for the system (14).
Theorem 2: Let the control be chosen in the form where • K E ∈ R m×n is chosen such that λ(E, (A + BK E )) = 0; • for some ν ∈ [−1, 0) the system of matrix equations and inequalities is feasible for some M , L ∈ R n×n , a ∈ R; • K ∈ R m×n , β ∈ R + are chosen such that for some R, ∈ R n×n , Y ∈ R m×n with K = YX , X = R −1 and G d = M + aI n . Then the closed-loop system (14), (16) is impulse controllable and finite-time stable with Proof: I. Firstly, let us show that the closed-loop system (14), (16) is d E -homogeneous.
Consider the linear part of the closed-loop system. The pair (E, (A + BK E )x) is d E -homogeneous of degree ν with G d = M + aI n , where a ∈ R is sufficiently big for the generator to be anti-Hurwiz. Indeed, choosing (s) = Q −1 I n 1 0 0 e −νs I n 2 e Gs Q, G = G 1 0 0 G 2 , G 1 ∈ R n 1 ×n 1 , G 2 ∈ R n 2 ×n 2 (according to the proof of Lemma 2) one can show that (5), (6) are equivalent to and combining the terms of the same power we obtain that the sufficient condition for the pair (E, for any nonnegative integer i. Similarly to [10,Lemma 3] it is easy to show that this sufficient condition is satisfied if The condition (27) is satisfied if the equations (17), (18) are satisfied, where G d = M + aI n ,L = L + aI n and a ∈ R is sufficiently big for the generator to be anti-Hurwiz according to (20).
is also d E -homogeneous of the degree ν with the same generator G d due to Indeed, on the one hand we have and on the other hand one can show that where the equality was used taking into account (19). II. Due to the homogeneity property, on R n \ {0} we have According to [32] the inequalities (21), (24) imply that the pair (E, (A + BK E + BK )) is impulse free, i.e., for someÃ ∈ R n 1 ×n 1 and invertibleQ,P ∈ R n×n . Then the control (16) is admissible (i.e., the closed-loop system is impulse free).
III. Since the system is impulse free, then for x 0 ∈ G (for Ex 0 = 0 the solution is trivial x 0 (t) = 0) we can consider the implicitly defined Lyapunov candidate function in the form V (x) := x d E that is positive and radially unbounded on the set G and V (0) = 0. Since (by means of Implicit Function Theorem [33]) we obtaiṅ where VOLUME 10, 2022 Since G d = M + aI n then (22), (23) imply for R = X −1 and some > 0. Due to (5) we have Returning to (29) and taking into account that for K = YR −1 the inequality (24) implies Finally, applying the comparison lemma the last inequality guarantees (see, e.g., [24]) the system (14), (16) is finite-time stable and Note, that if ν → 0 the proposed control (16) becomes a linear one.
Remark 3: The presented control can be considered as an extension of the results in [10] for finite-time stabilization of linear ODEs. Indeed, in the case the matrix E is nonsingular (see Remark 1) the control (16) coincides with the given in [10].
Remark 4: The feedback matrix K E can be chosen using pole placement methods.
Note that in the control (16) the linear term K E x homogenizes the closed-loop system with the negative degree ν, while the role of the term x is to stabilize the system.

Remark 5:
The function x d E is defined implicitly by (15). In order to realize the control (16) and find an appropriate value of x i d E at the time instant t i the following simple numerical procedure can be used (see, e.g., [34]): If STEP is applied recurrently many times to the same vector x i then it allows to localize the unique positive root of the equation In practice, there are often time restrictions on transients. The parameters ν, β affect convergence time and allow the upper bound of the settling time function (25) to be adjusted (e.g., the larger β, the smaller bound of T (x 0 )). However, for some parameters (e.g., for sufficiently big β), the inequalities (21)-(24) may become unfeasible. In this case, if the control (16) calculated via Theorem 2 provides settling time estimation (25) greater than the desired transient time, the convergence rate of the proposed control can be accelerated via time rescaling. To accelerate the convergence rate of systems in the form of ODEs with homogeneous control, one can refer to the results [35], [36], [37]. The following corollary provides an extension of Theorem 2 in order to accelerate the convergence rate.
Corollary 1: Let all conditions of Theorem 2 be satisfied and matrix equations be feasible for some M 2 , L 2 ∈ R n×n , a 2 ∈ R. For λ > 1 and = e −M 2 ln λ the control stabilizes system (14) in finite time with Proof: Analogously to the part I of the proof of Theorem 2 it is easy to show that Ee M 2 s = e L 2 s E, (A+BK E )e M 2 s = e (L 2 +I n )s (A + BK E ) and e (L 2 +I n )s B = e s B, ∀s ∈ R. Consider the system (14) with the control u λ (x): Taking into account e −L 2 ln λ E = Ee −M 2 ln λ , for z = x, we obtain The system (33) is finite-time stable since λ z 0 (t) = z 0 (λt), where λ z 0 (t) is a solution of (33) and z 0 (λt) is a solution of the system Eż = Az + bu(z). Then, the settling-time function is bounded by Remark 6: Note that the change of a control law in some cases may lead to a change of the initial conditions x 0 associated with the static part and inputs of a descriptor system (system may have solutions only for a special set of initial values consistent with inputs [40]). In this case, this change does not affect on the value of x 0 d E and the control (31) provides an accelerated convergence according to (32).

IV. EXAMPLE
Consider the approximated linear descriptor submarine model with a dive control system, presented in [38], where The state vector variables x ∈ R 4 correspond to the depth of immersion, vertical velocity, vertical acceleration and selfrotation velocity. According to Theorem 2, by means of (17)-(24) solution, the following parameters of the control algorithm (16) Fig. 3, Fig. 4. The results for x d E (Fig. 4) are shown with the use of the logarithmic scale in order to demonstrate finite-time convergence rate. The settling time estimate is T (x 0 ) ≤ 9.16 according to (25). Fig. 5 and Fig. 6 demonstrate plots of the state vector in the presence of additive disturbances d(t) = 0 1 4 sin(2t)) 1 4 sin(t)) 0 T (in the system under consideration, disturbances can be caused by the presence of compression of the hull, changes in the density of seawater, sea currents, etc.) and measurement band limited noise, respectively. A detailed study of the presented control algorithms on robustness analysis with respect to uncertainties, disturbances, extension of these results on a wider class of systems and experimental approbation goes beyond the scope of this paper providing the subjects for a future research.   In order to provide faster convergence the control (31) has been used. The simulation results for λ = 2,

V. CONCLUSION
In this paper, the concept of d E -homogeneity was introduced for descriptor systems. The homogeneity property can be verified algebraically, and as for the ODE case it implies that a dilation of initial conditions leads to a scaling of trajectories. The homogenizing and stabilizing in a finite-time control is proposed. The parameters tuning procedure is in the form of linear matrix equations and inequalities solution.
The directions of future research include the following: robustness (e.g., Input-to-State Stability) analysis of d E -homogeneous systems, introduction of d E -homogeneous approximations, d E -homogeneous observers and control design, etc. In particular, the proposed approach can be further used to develop existing control algorithms for distributed systems with constraints (e.g., [41]) in order to provide fast convergence.