Andrey Polyakov
email: andrey.polyakov@inria.fr.

Miroslav Krstic
email: krstic@ucsd.edu

Keywords:

Finite-and Fixed-Time Nonovershooting Stabilizers and Safety Filters by Homogeneous Feedback Andrey Polyakov and Miroslav Krstic

Abstract-Non-overshooting stabilization is a form of safe control where the setpoint chosen by the user is at the boundary of the safe set. Exponential non-overshooting stabilization, including suitable extensions to systems with deterministic and stochastic disturbances, has been solved by the second author and his coauthors. In this paper we develop homogeneous feedback laws for fixed-time nonovershooting stabilization for nonlinear systems that are input-output linearizable with a full relative degree, i.e., for systems that are diffeomorphically equivalent to the chain of integrators. These homogeneous feedback laws can also assume the secondary role of 'fixed-time safety filters' (FxTSf filters) which keep the system within the closed safe set for all time but, in the case where the user's nominal control commands approach to the unsafe set, allow the system to reach the boundary of the safe set no later than a desired time that is independent of nominal control and independent of the value of the state at the time the nominal control begins to be overridden.

I. INTRODUCTION

Nonovershooting control is aimed at solving a tracking problem under certain state/output constrains (see, e.g. [START_REF] Wieland | Constructive safety using control barrier functions[END_REF], [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF]). Such a control can be employed in a "safety filter" framework to override a potentially unsafe nominal controller [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF]. Nonovershooting and safety controllers have been designed for both linear [START_REF] Phillips | Conditions that guarantee no overshoot for linear syste[END_REF], [START_REF] Longchamp | Influence of zero locations on the number of step-response extre[END_REF], [START_REF] Darbha | On the synthesis of controllers for a nonovershooting step respon[END_REF] and nonlinear plants [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF], [START_REF] Lindemann | Control barrier functions for multi-agent systems under conflicting local signal temporal logic tasks[END_REF], [START_REF] Garg | Multi-rate control design under input constraints via fixed-time barrier functions[END_REF]. Inspired by robust non-overshooting control under deterministic disturbances in [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF], i.e., by stabilization to an equilibrium at the barrier along with input-to-state safety (ISSf), mean-square stabilization of stochastic nonlinear systems to an equilibrium at the barrier, along with a guarantee of non-violation of the barrier in the mean sense, is solved in [START_REF] Li | Mean-nonovershooting control of stochastic nonlinear systems[END_REF].

In many cases, the corresponding synthesis was based on the so-called control barrier functions [START_REF] Ames | Ccontrol barrier function based quadratic programs for safety critical systems[END_REF], [START_REF] Jankovic | Robust control barrier functions for constrained stabilization of nonlinear systems[END_REF], which have been extensively used in various control applications such as automotive systems [START_REF] Ames | Control barrier function based quadratic programs with application to adaptive cruise contro[END_REF], [START_REF] Rahman | Driver intent prediction with barrier functions[END_REF] and multi-agent robotics [START_REF] Wang | Safety barrier certificates for collisions-free multirobot systems[END_REF], [START_REF] Santillo | Collision free navigation with interacting, non-communicating obstacles[END_REF]. The exponential control barrier functions were reported in [START_REF] Nguyen | Exponential control barrier functions for enforcing high relative-degree safety-critical constraints[END_REF] and allowed the use of simple linear tools for a control design. The nonovershooting prescribed-time control has been developed recently [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF] for the integrator chain based on a novel time-varying backstepping procedure [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], [START_REF] Krishnamurthy | A dynamic high-gain design for prescribed-time regulation of nonlinear systems[END_REF]. Such a control allows safety requirements to be fulfilled for a prescribed time interval [0, T ] with T > 0. Prescribed-time controllers [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time[END_REF], which have been developed even for PDEs [START_REF] Espitia | Boundary time-varying feedbacks for fixed-time stabilization of constant-parameter reaction-diffusion systems[END_REF], form a special subset of both finite-time [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] and fixedtime [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF] control algorithms. The concepts of finite-time and Inspired by the mentioned results, this paper deals with generalized homogeneous nonovershooting controller design. The goal is to synthesize feedback laws that ensure finitetime stabilization (FnTS) and fixed-time stabilization (FxTS) without the overshoot of the system's output.

Homogeneity is a dilation symmetry studied in many branches of pure and applied mathematics. For instance, homogeneous differential equations and homogeneous control systems are well studied in the literature [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF], [START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF], [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF]. Being similar (in some sense) to linear control systems, they may demonstrate faster convergence [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], better robustness [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] and smaller "peaking effect" [27, Chapter 1]. To the best of the authors' knowledge, a homogeneous nonovershooting control has never been designed even for the integrator chain.

Finite-time stabilization (FnTS) without an overshoot is formulated as the problem of constructing a feedback law (for example, of a homogeneous type) such that, given an initial state x 0 ∈ R n , the feedback guarantees that the trajectory of the closed-loop system initiated in x 0 at the time instant t = 0 reaches the origin no later than a time instant T (x 0 ) without overshoot in the first coordinate. If, given arbitrary T > 0, the origin is reached by the time T for any x 0 , such a nonovershooting feedback is referred to as fixed-time stabilizing (FxTS).

B. Homogeneous safety filters

To prevent overshoots that may result from the user's nominal control, we also design what are commonly referred to as 'safety filters' and have a particularly simple structure for integrator chains [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF].

a) Finite-and fixed-time safety: Since the notion of fixed-time barrier functions has already appeared in the literature [START_REF] Garg | Multi-rate control design under input constraints via fixed-time barrier functions[END_REF], let us explain what fixed-time safety (FxTSf) means to us. While the precise meaning of this property and the weaker finite-time safety (FnTSf) property will become clear in the statements of Theorems 3 and 2, respectively, here we only give a descriptive, intuitive definition. We consider control systems subject to a nominal control u nom and equipped with a "safety filter" feedback law u = F(u nom , x). The safety filter is said to ensure FxTSf with fixed time T if it ensures safety and the following holds for all initial conditions inside the safe set: whenever the applied control F(u nom (t), x(t)) is different from the nominal control u nom (t) over an entire time interval of duration of T , the solution reaches the boundary of the safe set no later than at the end of that time interval.

Clearly, FnTSf and FxTSf impart less safety on the system than the conventional exponential and asymptotic safety properties, under which the trajectories remain away from the boundary for all finite times. This is precisely the point as FnTSf/FxTSf filters are less conservative. They allow the nominal performance to be achieved sooner, with less distortion. In the language of the "safety vs. agility" tradeoff [START_REF] Ames | Ccontrol barrier function based quadratic programs for safety critical systems[END_REF], the FxTSf filters are the most agile filters available.

A comparison with the prescribed-time safety (PTSf) filters in [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF] is in order. Both the PTSf and FxTSf filters ensure safety while allowing convergence to the safety boundary in an amount of time that is independent of the initial condition. The difference is that FxTSf are applicable in infinite-time safety operation, whereas PTSf are ideal when a safety prohibition is only of finite duration.

b) Release time and restraint time: The time T figures in both the PTSf and FnTSf/FxTSf approaches. It is important to understand the different roles that T plays in the two approaches. In PTSf, T is the time after which the prohibition of entering the unsafe set is lifted. Hence, T should be understood in PTSf as release time (from the safety constraint). In FnTSf/FxTSf, the safety constraint applies in perpetuitythe unsafe set is prohibited for all time. So, in FnTSf/FxTSf, T is the length of a period that can commence at any moment and, if u nom causes itself to be overridden by the safety filter F over the entire interval, the trajectory will necessarily be permitted to reach the boundary of the unsafe set, but not enter it. Hence in FnTSf/FxTSf, T should be understood as restraint time since the safety filter restrains the trajectory away from the boundary for no longer than T . The release time is a property bestowed upon the system by a PTSf safety filter. Likewise, the restraint time is a property bestowed upon the system by the FnTSf/FxTSf safety filter F. In FnTSf the restraint time T may depend on the value of x at the moment the safety override kicks in and on the nominal control u nom , whereas in FxTSf the restraint time can be arbitrarily assigned with the filter and is independent of x and u nom .

C. Contributions and organization

To summarize, this paper's contributions are the design and the associated stability and safety analyses for 1) homogeneous finite/fixed-time nonovershooting stabilizing controllers; 2) homogeneous finite/fixed-time safety filters.

The paper is organized as follows. Section II presents the problem statement. Short preliminaries about the homogeneity are given in Section III. A homogeneous nonovershooting controller design and issues of its tuning are discussed in Section IV. Section V is devoted to homogeneous safety filters. As an example, linear, homogeneous and prescribedtime safety filters for the double integrator are compared in Section VI. Section VII presents concluding remarks. Some auxiliary results are given in Section VIII.

D. Notation

• R is the field of reals and R + = {α ∈ R : α > 0}.

• e i = (0, . . . , 0, 1, 0, . . . , 0) T ∈ R n is the i-th element of the canonical Euclidean basis in R n .

• Σ -= {x ∈ R n : e T 1
x ≤ 0} is the half-space in R n that specifies the safety zone of the system.

• intS denotes an interior of the set S ⊂ R n .

• P 0 (resp. P ≺ 0) means that the symmetric matrix P ∈ R n×n is positive (resp. negative) definite.

• The matrix P

1 2 = M is such that M 2 = P . By definition, P -1 2 = P 1 2 -1
. Notice, for any matrix P 0 there always exists a unique matrix P

ẋ = Ax + Bu, t > 0, x(0) = x 0 , (1)
where x ∈ R n , u ∈ R and

A =          0 1 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . 1 0 • • • • • • • • • 0          , B =      0 . . . 0 1      . (2)
Given x 0 ∈ intΣ -and T > 0, we need to design a state feedback control u = u h (x) such that 1) u h ∈ C(R n \{0}, R) is a uniformly bounded function of the state; 2) the closed-loop system is globally uniformly finite-time stable 1 with

x(t, x 0 ) = 0, ∀t ≥ T ; (3)
3) x(t, x 0 ) ∈ Σ -for all t ≥ 0.

Such a controller (if it exists) steers the state vector at zero without overshoot in the first component, i.e., e T 1 x(t, x 0 ) ≤ 0, ∀t ≥ 0. In this paper we design a homogeneous nonovershooting finite-time controller u h allowing also a stabilization in a fixed-time T > 0 independently of x 0 . Our aim also is to design a 'safety filter' [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF] which applies the user's nominal control u nom while the system operates in the safe set and overrides u nom with the nonovershooting homogeneous control u h in the domain where the user's nominal control commands operation in the unsafe set.

III. PRELIMINARIES: HOMOGENEITY A. Dilation symmetry of functions and vector fields

Homogeneity is a symmetry of a set or a mapping with respect to a group of transformations called dilation (see, e.g. [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]). In this paper we deal only with the following weighted dilations:

d(s) = e sG d , G d = diag{n -i + 1} n i=1 , (4)
In a more general case, a linear dilation [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] in R n is defined by an arbitrary anti-Hurwitz matrix G d ∈ R n×n . Recall [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF] that a vector field f :

R n → R n (a function h : R n → R) is said to be d-homogeneous of degree µ ∈ R n if f (d(s)x) = e µs d(s)f (x), ∀x ∈ R n , ∀s ∈ R (5)
(resp. h(d(s)x) = e µs h(x), ∀x ∈ R n , ∀s ∈ R). ( 6
)
Simple computations show that the linear vector field x → Ax is d-homogeneous of degree -1:

Ad(s) = e -s d(s)A, ∀s ∈ R (7)
where the dilation d is given by (4) and the matrix A is defined by (2). Homogeneous control system are similar (in some sense) to linear control systems, but they may have some additional useful properties such as better robustness, faster convergence and smaller overshoot [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Chapter 1]. It is well known [START_REF] Zubov | Methods of A.M. Lyapunov and Their Applications[END_REF], [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] that any asymptotically stable homogeneous system

ẋ = f (x), t > 0, x(0) = x 0 , (8)
of negative (positive) degree is globally uniformly finitetime (nearly fixed-time 2 ) stable, where f ∈ C(R n \{0}). To guarantee a finite-time stabilization of the system (1), [START_REF] Ames | Control barrier function based quadratic programs with application to adaptive cruise contro[END_REF], it is sufficient to design an asymptotically stabilizing homogeneous control u : R n → R such that u(d(s)x) = u(x), ∀x ∈ R n , ∀s ∈ R. Several schemes of homogeneous and locally homogeneous control design for the chain of integrators are presented in the literature [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. In this paper we utilize the technique proposed in [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF], since it allows simple rules for control parameters tuning and the settling time adjustment to be applied.

B. Canonical Homogeneous Norm

Inspired by [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF] let us define the so-called canonical homogeneous norm • d : R n → [0, +∞) as follows 0 d = 0 and

x d = e sx : d(-s x )x = 1, x = 0, (9)
where x = √ x T P x is the weighted Euclidean norm,

P = P T ∈ R n×n : P G d + G d P 0, P 0. ( 10
)
The latter matrix inequalities guarantee that the linear dilation d is monotone, i.e., the function s → d(s)x is monotone for any x ∈ R n . One can be shown [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF] that

• d ∈ C(R n ) ∩ C 1 (R n \{0}
) and 2 The system is globally uniformly nearly fixed-time stable if it is globally uniformly asymptotically stable and for any r > 0 there exists Tr > 0 such that x(t, x 0 ) ≤ r, ∀t ≥ T, ∀x 0 ∈ R n .

• d(s)x d = e s x d for all x ∈ R n and all s ∈ R.

• x = 1 ⇔ x d = 1 and x ≤ 1 ⇔ x d ≤ 1.
Moreover, for x = 0 we have

∂ x d ∂x = x d x T d T (-ln x d )P d(-ln x d ) x T d T (-ln x d )P G d d(-ln x d )x . (11)
Below we utilize the homogeneous norm x d for both control design and stability analysis (as a Lyapunov function). Since the homogeneous norm is defined implicitly, then, to compute x d for a given x = 0, the equation d(-s x )x = 1 needs to be solved with respect to s x ∈ R.

Example 1. For G d = ( 2 0 0 1 ), by definition, x d = V is a positive definite solution ( x1 x2 ) T V -2 0 0 V -1 P V -2 0 0 V -1 ( x1 x2 ) = 1, (12)
where P = p 11 p 12 p 12 p 22 0.

For x = 0 the solution always exists and unique if

P 0, P G d + G d P 0. (13)
Hence, the homogeneous norm x d is a unique positive root of the quartic equation

V 4 -p 22 x 2 2 V 2 -2p 12 x 1 x 2 V -x 2 1 p 11 = 0 (14)
which can be found using the Ferrari formulas (see Appendix).

Therefore, for n = 2 the canonical homogeneous norm x d can be calculated explicitly. In other cases, a special numerical algorithm may be utilized in order to compute x d in practice, (see, e.g. [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF], [START_REF] Polyakov | Consistent discretization of finite-time and fixed-time stable systems[END_REF], [START_REF] Wang | On generalized homogenization of linear quadrotor controller[END_REF] for more details).

IV. HOMOGENEOUS STABILIZATION WITHOUT OVERSHOOT

This section presents a two step procedure for a homogeneous nonovershooting control design. First, inspired by [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF] we design a linear controller which stabilizes the system exponentially at zero without overshoot. Next, inspired by [27, Chapter 9] we transform ("upgrade") a linear controller to a homogeneous one such that an upper estimate of the settling time for the system with the given initial state x 0 ∈ intΣ - can be assigned a priori.

A. Linear nonovershoting control design

For λ > 0, i = 1, . . . , n-1, let us introduce the row vectors

h 1 = -e T 1 (15) h i = h i-1 A + λh i-1 , i = 2, . . . , n, (16)
namely,

h i = -e T 1 (A + λI) i-1 , i = 1, . . . , n, (17)
and consider the linear cone

Ω = {x ∈ R n : h i x ≥ 0, i = 1, . . . , n} ⊂ Σ -. (18)
The following lemma is a particular case of the results obtained in [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF].

Lemma 1. The linear feedback

u lin = Kx, K = h n (A + λI) = -e T 1 (A + λI) n , (19
) stabilizes exponentially the system (1) at x = 0 and renders the set Ω strictly positively invariant for the closed-loop system (1), [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF]. Moreover, given initial value

x 0 ∈ intΣ -, choosing λ > 0 such that i-1 j=0 C i-1 j e T i-j x 0 λ j ≤ 0, C i-1 j = (i-1)! (i-j-1)!j! , i = 2, . . . , n, (20)
guarantees that x 0 ∈ Ω.

Proof: Inspired by [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF] let us consider the barrier functions ϕ i : R n → R defined as

ϕ 1 = -x 1 , (21)
ϕ i = φi-1 + λϕ i-1 , i = 2, . . . , n, (22)
namely,

ϕ i = h i-1 ẋ + λh i-1 x = h i-1 Ax + λh i-1 x = h i x := -e T 1 (A + λI) i-1 x , i = 2, . . . , n, (23)
which is compactly represented as

Φ =    ϕ 1 . . . ϕ n    = Hx , (24)
H =    h 1 . . . h n    =      -e T 1 -e T 1 (A + λI) . . . -e T 1 (A + λI) n-1      . (25)
The positive orthant {Φ ∈ R n : ϕ i ≥ 0, i = 1, . . . , n} is the same as the set Ω. Furthermore, H is the observability matrix of the pair (A + λI, -e T 1 ). This pair is completely observable by direct verification, using the definition of observability (that the scalar output of this system being identically zero implies the vector state being identically zero). Hence, H is invertible and x = H -1 Φ. Noting that

φn =h n ẋ = h n-1 (Ax + Bu) = -u lin + h n Ax = -λh n x = -λϕ n , (26)
the closed-loop system in the coordinates ϕ 1 , . . . , ϕ n-1 has the form

         φ1 = -λϕ 1 + ϕ 2 , φ2 = -λϕ 2 + ϕ 3 , . . . φn = -λϕ n , (27)
namely, Φ = (-λI + A)Φ. Since the matrix -λI + A is Metzler, the Φ-system is positive and the set {Φ ∈ R n : ϕ i ≥ 0, i = 1, . . . , n} = Ω is strictly positively invariant.

Let us show next that the inequalities [START_REF] Li | Formally correct composition of coordinated behaviors using control barrier certifi-cates[END_REF] are fulfilled for some λ > 0. Indeed, the inclusion

x 0 ∈ intΣ -ensures ϕ 1 (x 0 ) = h 1 x 0 > 0 and, obviously, i-1 j=0 C i-1 j e T i-j x 0 λ j = -h 1 x 0 λ i-1 + i-2 j=0 C i-1 j e T i-j x 0 λ j ≤ 0 (28)
for a sufficiently large λ > 0.

Let us prove now that ϕ i (x 0 ) ≥ 0, i = 1, 2, . . . , n. On the one hand, for any i ≥ 2 we derive

ϕ i (x 0 ) = h 1 (A + λI n ) i-1 x 0 = h 1 i-1 j=0 C i-1 j A i-1-j λ j x 0 .
(29) On the other hand, for j = i -1 we have A i-1-j = A 0 = I n and h 1 A i-1-j = -e T 1 ; for j = i -2 we have A i-1-j = A and h 1 A = -e T 2 ; etc. Hence, we conclude

ϕ i (x 0 ) = h 1 i-1 j=0 C i-1 j A i-1-j λ j x 0 = - i-1 j=0 C i-1 j
e T i-j x 0 λ j (30) and ϕ i (x 0 ) ≥ 0 provided that the inequalities [START_REF] Li | Formally correct composition of coordinated behaviors using control barrier certifi-cates[END_REF] hold. This means that x 0 ∈ Ω.

Finally, since λ > 0, the Φ-system [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] is exponentially stable at Φ = 0. Given that x(t) = H -1 Φ(t) and Φ 0 = Hx 0 , the closed-loop system (1), [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF], which is governed by ẋ = A -e n e T 1 (A + λI) n x, has the solution x(t) = H -1 e (-λI+A)t Hx 0 , and this system is exponentially stable at x = 0.

The parameter λ > 0 in Lemma 1 can be selected using a simpler but more conservative condition.

Corollary 1. If λ > 0 satisfies the inequalities λ ≥ 1 - e T k+1 x0 e T 1 x0 C i-1 i-k-1 , k = 1, . . . , i -1, i = 2, . . . , n, (31)
then the inequalities (20) hold.

Proof: Indeed, for any i ≥ 2 we derive

i-1 j=0 C i-1 j e T i-j x 0 λ j = C i-1 0 e T i x 0 + λ(C i-1 1 e T i-1 x 0 +λ(. . . λ(C i-1 i-2 e T 2 x 0 +λC i-1 i-1 e T 1 x 0 ))).
Since e T 1 x 0 < 0 then the inequalities [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF] imply that

C i-1 i-1-k e T k+1 x 0 + λe 1 x 0 ≤ e T 1 x 0 , k = 1, . . . , i -1. (32)
Hence, taking into account C i-1 i-1 = 1 we conclude

C i-1 i-2 e T 2 x 0 + λC i-1 i-1 e T 1 x 0 ≤ e T 1 x 0 , (33)
C i-1 i-3 e T 3 x 0 + λ(C i-1 i-2 e T 2 x 0 + λC i-1 i-1 e T 1 x 0 ) ≤ C i-1 i-3 e T 3 x 0 + λe T 1 x 0 ≤ e T 1 x 0 , (34)
. . .

C i-1 0 e T i x 0 + λ(C i-1 1 e T i-1 x 0 + λ(... + λ(C i-1 i-2 e T 2 x 0 +λC i-1 i-1 e T 1 x 0 ))) ≤ e T 1 x 0 . ( 35
)
The proof is complete.

B. Upgrading linear controller to a homogeneous one

In this subsection we design a finite-time nonovershooting control by means of an "upgrade" (a transformation) of a linear "nonovershooting" feedback to a homogeneous one. Theorem 1. Let K ∈ R 1×n be the gain of the linear controller [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF] defined by the formula [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF]. Then

• the system of linear matrix inequalities (LMIs)

P (A + BK) + (A + BK) T P ≺ 0, P G d + G d P 0, P 0, (36)
has a solution P = P T ∈ R n×n ; • for any given T > 0 the homogeneous controller

u h (x) = Kd (-ln x/r d ) x, K = Kd(s), (37)
s ≥ ln max 1 ρT , 1 , r > 0 ( 38
)
stabilizes the system (1) at x = 0 in a finite time such that

x 0 ≤ r ⇒ x(t, x 0 ) = 0, ∀t ≥ T, (39)
where d(s) is given by (4),

ρ = -λ max Q 1 2 ZQ -1 2 > 0, (40)
Z = P (A + BK) + (A + BK) T P ≺ 0, (41)
Q = P G d + G d P 0, (42)
the homogeneous norm x d is induced by the weighted Euclidean norm

x = x T d(s)P d(s)x, x ∈ R n ; (43)
• the homogeneous cone

Ω r = x ∈ R n : hi d(-ln x/r d )x ≥ 0, i = 1, ..., n (44
) is a positively invariant set for the closed-loop system (1), [START_REF] Wieland | Constructive safety using control barrier functions[END_REF] and

x 0 ∈ Ω ⇒ x 0 ∈ Ω r ⊂ Σ - ( 45
)
provided that x 0 /r d ≤ e s, where Ω ⊂ Σ -is given by [START_REF] Krishnamurthy | A dynamic high-gain design for prescribed-time regulation of nonlinear systems[END_REF], hi = h i d(s) and the row vectors h i are defined by [START_REF] Kolmanovski | Applied Theory of Functional Differential Equations[END_REF].

Proof: 1) Let P = H T P H, where H is defined by ( 25) and P = P T ∈ R n×n is a symmetric matrix to be defined below. By Corollary 2 we have

H(A + BK) = (-λI n + A)H and HG d = G d + λ(nI n -G d )A T H. Since P (A + BK) + (A + BK) T P = H T P H(A + BK) + (A + BK) T H T P H = H T P (A -λI n ) + (A -λI n ) T H (46)
and

P G d + G d P = H T P HG d + G d H T P H = H T P G d + λ(nI n -G d )A T + G d + λ(nI n -G d )A T T P H (47)
then, the system of LMIs [START_REF] Wang | On generalized homogenization of linear quadrotor controller[END_REF] becomes

   P (A -λI n ) + (A -λI n ) T P ≺ 0 P M + M T P 0, P 0, (48)
where

M = G d + λ(nI n -G d )A T . (49)
If P = diag{p 1 , . . . , p n } with p i > 0 then P 0 and the latter system of LMIs becomes (51) Let us fix an arbitrary p n > 0, e.g., p n = 1. Since λ > 0 then there exists a sufficiently large p n-1 > 0 such that

   2λp1 -p2 0 0 ... 0 0 -p2 2λp2 -p3 0 ... 0 0 0 -p3 2λp3 -p4 ... 0 0 ... ... ... ... ... ... ... 0 0 0 0 ... 2λpn-1 -pn 0 0 0 0 ... -2pn λpn    0, (50)
2λp n-1 -p n -p n 2λp n 0, ( 52
)
4p n-1 (n-1)λp n (n-1)λp n 2p n 0. (53)
simultaneously. Hence, using Schur complement we conclude that there exists a sufficiently large p n-2 > 0 such that the inequalities

  2λp n-2 p n-1 0 p n-1 2λp n-1 -p n 0 -p n 2λp n   0, (54)
  6p n-2 (n-2)λp n-1 0 (n-2)λp n-1 4p n-1 (n-1)λp n 0 (n-1)λp n 2p n   0, (55)
are fulfilled simultaneously. Repeating the same consideration subsequently for p n-3 ,. . . ,p 1 we conclude that the system of LMIs ( 36) is always feasible.

2) Let us show that the homogeneous norm • d is a Lyapunov function of the closed-loop system (1), [START_REF] Wieland | Constructive safety using control barrier functions[END_REF]. Indeed, using the formula [START_REF] Garg | Multi-rate control design under input constraints via fixed-time barrier functions[END_REF] and the identities d(s)A = e s d(s)A, d(s)B = e s B, ∀s ∈ R we derive

d x/r d dt = x/r d (x/r) T d(-ln x/r d )d(s)P d(s)d(-ln x/r d )( ẋ/r) (x/r) T d(-ln x/r d )d(s)P d(s)G d d(-ln x/r d )(x/r) = e s x T d(-ln x/r d )d(s)P (A+BK)d(s)d(-ln x/r d )x x T d(-ln x/r d )d(s)P G d d(s)d(-ln x/r d )x ≤ -ρe s ≤ -max 1 T , ρ ≤ -1/T. ( 56
)
where the linear matrix inequality [START_REF] Wang | On generalized homogenization of linear quadrotor controller[END_REF] and the identity (40) are utilized on the last step. The latter means that the closedloop system (1), ( 37) is globally uniformly finite-time stable and

x(t, x 0 )/r d ≤ x 0 /r d - t T (57)
as long as x(t, x 0 ) = 0. If x ≤ r then x/r d ≤ 1 and x(t, x 0 ) = 0 for all t ≥ T .

3) Let us consider the vector-valued homogeneous barrier functions φ : R n → R n defined as follows φ(x) = Hd(s)d(-ln x/r d )x. We have φ = -Hd(s)

d x/r d dt x/r d G d d(-ln x/r d )x +Hd(s)d -ln x r d Ax+B Kd -ln x r d x . (58)
where Π = -λI n +A+γG d +γλ(nI n -G d )A T . Since γ > 0 and the diagonal matrix nI n -G d is non-negative then Π is a three diagonal Metzler matrix and the system (60) is positive. Therefore, the homogeneous cone Ω r is a strictly positively invariant set of the the closed-loop system (1), [START_REF] Wieland | Constructive safety using control barrier functions[END_REF] and, by construction,

Ω r ⊂ Σ -. Finally, since s ≥ 0 then d(-s)x 0 /r d ≤ e -s x 0 /r d ≤ 1 provided that x 0 /r d ≤ e s. If x 0 ∈ Ω then h i x 0 ≥ 0, i = 1, . . . , n. Taking into account the representation φ i (x 0 ) = h i e -Di ln d(-s)x0 d x 0 we derive φ i (x 0 ) ≥ 0 due to Corollary 2, i.e., x 0 ∈ Ω r .
The feedback ( 37) is discontinuous at zero and smooth away from the origin [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF], so the solution of the system for the time instances greater than the setting time is defined in the sense of Filippov [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF].

Remark 1 (Fixed-time nonovershooting control). To guarantee fixed-time convergence of any trajectory of the system (1), [START_REF] Wieland | Constructive safety using control barrier functions[END_REF] to zero, it is sufficient to select r in [START_REF] Wieland | Constructive safety using control barrier functions[END_REF] in a manner dependent on the initial state, for example, as r = x 0 for x 0 = 0. Theorem 1 presents a scheme for a transformation ("upgrade") of a linear nonovershooting control a homogeneous one. The homogeneous controller guarantees a finite-time stabilization of the system state at zero without overshoot in the first component. The main advantage of the finite-time controller [START_REF] Wieland | Constructive safety using control barrier functions[END_REF] is the simplicity of the parameters tuning. The parameter r in [START_REF] Wieland | Constructive safety using control barrier functions[END_REF] defines the maximum norm of the initial state x 0 for which the settling time of the system is bounded by the arbitrary fixed number T > 0.

Remark 2. The controller [START_REF] Wieland | Constructive safety using control barrier functions[END_REF] is globally uniformly bounded as follows

|u h (x)| 2 =x T d(-ln x/r d ) KT Kd(-ln x/r d )x ≤r 2 λ max P -1 2 K T KP -1 2 for all x ∈ R n , ( 61
)
where the identity

x T d(-ln x/r d )d(s)P d(s)d(-ln x/r d )x = r 2
is utilized on the last step.

The positive cone3 Ω ⊂ Σ -defines a set of initial states of the system (1) for which the linear control (19) stabilizes the system without overshoot in the first coordinate. In the case of the homogeneous nonovershooting control (37) such a positively invariant set is the d-homogeneous cone4 Ω r ⊂ Σ -. Since |u lin (x)| = |u h (x)| for x/r d = e s then linear and homogeneous controllers have the same maximum magnitude on the homogeneous ball

B r = {x ∈ R n : x/r d ≤ e s}. ( 62
)
Theorem 1 implies that the positively invariant compact set Ω r ∩ B r of the homogeneous control system is larger than the positively invariant set Ω ∩ B r of the linear control system despite that both controllers have the same magnitude on B r .

Remark 3. The simple combination of the linear and homogeneous feedbacks

u = u h if x/r d ≤ e s, Kx if x/r d > e s ( 63
)
gives a nonovershooting control u ∈ C(R n \{0}) as well. The closed-loop system (1), ( 63) is globally finite-time stable with the positively invariant set Ω ∪ Ω r . Indeed, let us consider the non-smooth Lyapunov function

V = x/r d if x/r d ≤ e s, d(-s)x/r if x/r d > e s. (64)
By construction, V is continuous on R n , locally Lipschitz continuous on R n \{0}, continuously differentiable on the set R n \{x ∈ R n : x/r d = e s}\{0} and

V ≤ -1 T if 0 < V < e s, -ρV if V > e s, (65)
where ρ = -λ max (P -1/2 ZP -1/2 ) > 0. For V = e s using the chain rule for the Clarke's gradient we derive V ≤ -min{1/T, ρe s} < 0. The continuous barrier functions φi , i = 1, . . . , n for this system

φi = φ i if V ≤ e s, ϕ i if V > e s. ( 66
)
combine smooth linear and homogeneous barrier functions ϕ and φ, respectively. Since V < 0 then the vector φ = ( φ1 , . . . , φn ) T satisfies the switched positive system

d φ dt = e s x/r d Π φ if V ≤ e s, (-λI n +A) φ if V > e s, (67)
having a unique switching isolated in time.

The differential equations ( 27) and (60) for linear and homogeneous barrier functions are slightly different. In the linear case, the equation has the constant Metzler matrix -λI n + A, while the tridiagonal Metzler matrix Π in the homogeneous case depends on the time derivative of the Lyapunov function x/r d . This dependence complicates the analysis and design of the homogeneous safety filters.

Despite the obvious difference of positively invariant sets of linear and homogeneous controllers, their mathematical descriptions in polar homogeneous coordinates are identical provided that we do not care about tuning of the settling time of the homogeneous system, namely, if s = 0 and r = 1. Indeed, the polar homogeneous coordinates [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF] in R n are given by the polar radius ρ = x d and the unit vector y = d(ln x d )x, which defines the so-called homogeneous projection on the unit sphere. In the linear case we deal with the standard dilation d(s) := e Ins , so the polar radius is just the Euclidean norm ρ := x d = x and the homogeneous projection is ỹ := d(ln x d)x = x/ x . Hence, in the polar coordinates we have Ω = {(ρ, ỹ) : h i ỹ ≥ 0, ∀i} and Ω r = {(ρ, y) : h i y ≥ 0, ∀i} provided that r = 1 and s = 0.

The barrier functions as well as the structure of the system (60) depend on the particular control law. Some interesting homogeneous control algorithms for the integrator chain are introduced in [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. However, a design of barrier functions and a selection of control parameters allowing a nonovershooting finite-time/fixed-time stabilization based on these algorithms are unclear yet.

V. HOMOGENEOUS SAFETY FILTERS

A nonovershooting controller can be implemented in practice as the so-called 'safety filter' [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF]:

u = min(u nom , u lin ) (68)
where u nom is a nominal controller which may have overshoots and u lin is a non-overshooting linear controller whose gain can be time-invariant as in Lemma 1 or time-varying as in [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF]. Such an implementation allows overshoots of the nominal controller u nom to be eliminated. For n ≤ 2 the same safety filter is applicable in the case of the homogeneous nonovershooting controller [START_REF] Wieland | Constructive safety using control barrier functions[END_REF]. For n ≥ 3 we show that the overshoots of the nominal controller can be eliminated by the homogeneous controller using a modified filter. Let us define ∆ r = +∞ for n ≤ 2 and

∆ r (x) = γ r γ u + min i=2,...,n-1 ciφi+φi+1 λ(i-1)γuφi-1 , (69)
γ r = φ T P (λIn-A)φ φ T P HG d H -1 φ > 0, ∀x ∈ R n (70) γ u = φ T P B φ T P HG d H -1 φ ≥ 0, ∀x ∈ Ω r (71)
where n ≥ 3, x ∈ int Ω r , c i > 0 are arbitrary constants, φ i = hi d(ln x/r d )x, i = 1, ..., n are the homogeneous barrier functions and P 0 is a diagonal matrix defined in the proof of Theorem 1. The positive function ∆ r may have infinite values on the boundary of the set Ω r :

∆ r (x) := lim inf z→x ∆ r (z) ∈ (0, +∞], z ∈ Ω r , x ∈ ∂Ω r . (72)
Finally, let us define ∆ r (x) = +∞ for x ∈ R n \Ω r . Theorem 2 (Finite-time Safety Filter). Let u nom : [0, +∞) → R be a continuous nominal control signal and u h : R n → R be a nonovershooting homogeneous feedback defined by Theorem 1 with P = H T P H, P = diag{p 1 , ..., p n }, p n = 1.

Let the safety filter be defined on R n \{0} by the formula

u = max{u h -∆ r , min{u nom , u h }}. (73)
• The d-homogeneous cone Ω r is a strictly positively invariant set of the closed-loop system (1), (73).

• If x(τ, x 0 ) = 0 then u nom (t) < 0, ∀t ∈ (τ, τ +ε] ⇒ -x(t, x 0 ) ∈ R n + , (74) u nom (t) ≥ 0, ∀t ∈ [τ, τ +ε] ⇒ x(t, x 0 ) = 0, ( 75
)
where ε > 0 is a positive number. • For any values of time τ > 0 for which

u nom (t) ≥ u h (x(t, x 0 )), ∀t ∈ [τ, τ + T 0 ] : x(t, x 0 ) = 0, ( 76
) it holds that x(τ + T 0 , x 0 ) = 0, ( 77
)
where T 0 = x(τ, x 0 )/r d T .

Proof: 1) Repeating the derivation of the formula (60) in the proof of Theorem 1 we obtain

φ = e s (-λIn+A+γ f HG d H -1 )φ+B(u h -u) x/r d , (78)
as long as x(t, x 0 ) = 0, where γ f = -e -s d dt x/r d and

HG d H -1 = G d +λ(nI n -G d )A T by Corollary 2.
Using the formula [START_REF] Garg | Multi-rate control design under input constraints via fixed-time barrier functions[END_REF] and the identities

H(A + BK) = (-λI n + A)H, d(s)A = e s d(s)A, d(s)B = e s B, ∀s ∈ R we derive γ f = γ r -γ u (u h -u). ( 79
)
Notice that φ T P B = φ n . The equation ( 78) can be rewritten in a component-wise form as follows

φ1 = -(λ -nγ f )φ 1 + φ 2 , φi = -a i φ i + b i , i = 2, . . . , n -1, φn =λγ r (n-1)φ n-1 -λ-γ f + φn-1(u h -u) φ T P HG d H -1 φ φ n +u h -u, (80)
where

a i = λ -(n -i + 1)γ f + c i and b i = φ i+1 + c i φ i + λ(i -1)(γ r -γ u (u h -u))φ i-1 .
For the the safety filter (73) we have

u h -∆ ≤ u ≤ u h , ∀x ∈ Ω r \{0} (81)
Hence, b i ≥ 0 as long as x ∈ Ω r \{0}. Consequently, the system (78) is positive and any trajectory x(t, x 0 ) of the closed-loop system (1), (73) with x 0 ∈ Ω r belongs to Ω r as long as x(t, x 0 ) = 0.

2) Since u h is discontinuous at x = 0 ∈ Ω r then the origin may be a sliding set of the closed-loop system. In this case, solutions are defined in sense of Filippov [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF] and

ẋn ∈ [u min , u max ] for x = 0, (82)
where, due to lim z→0 ∆ r (z) = +∞, we have

u min = min u nom , lim inf z→0 u h (z) ( 83
)
u max = min u nom , lim sup z→0 u h (z) (84)
If u nom ≥ 0 then u min < 0 ≤ u max and the system has a sliding mode at x = 0 as long as u max ≥ 0. The inequality u nom < 0 implies u min ≤ u max < 0, so the system leaves the origin, enters the set {x ∈ R n : x i < 0, i = 1, ..., n} ⊂ int Ω r and stays there, at least, till u nom < 0. Therefore, the implications (74), (75) hold and Ω r is a positively invariant set of the system (1) with the safety filter (73).

3) If u nom (t) ≥ u h (x(t, x 0 )) for all t ∈ [τ, τ + T 0 ] : x(t, x 0 ) = 0 then u(t) = u h (t) for all t ∈ [τ, τ + T 0 ] : x(t, x 0 ) = 0. Using the inequality (56) we derive

x(t, x 0 )/r d ≤ x(τ, x 0 )/r d - t -τ T ( 85
)
as long as x(t, x 0 ) = 0. This means that there exists The key result of this theorem is (76), ( 77), which guarantees that the safety boundary is reached by the "release time" T 0 . Note that τ is not necessarily a single time but it may represent multiple times. Likewise, τ + T 0 may represent multiple times. Depending on the nominal control, the safety filter may kick in and out intermittently and, if the safety override persists, the safety boundary will always be reached in some finite time T 0 , which is not the same at each of the recurrent occasions of the boundary being reached.

t * ∈ [τ, τ + T 0 ] such that x(t * , x 0 )/r d = 0. Since u h (x) > 0 for x ∈ {x ∈ R n : x i < 0, i = 1, ...,
Theorem 2 proves that the safety-filter (68) keeps the system within the set Ω r ⊂ Σ -for all time but, in the case where the user's nominal control u nom commands operation in the unsafe set, allows the system to reach the boundary of the safe set Σ -no later than a finite-time T 0 > 0. The time T 0 =

x(τ, x 0 )/r d T is unknown a-priori. It depends implicitly on the user's nominal control u nom and the initial state x 0 ∈ Ω r . To specify the settling time independently of u nom and x 0 , we slightly modify the homogeneous controller (37) making the parameter r dependent of the system trajectory. • If

x 0 ∈ Ω then x(t, x 0 ) ⊂ Θ r(t) ⊂ Σ -, ∀t ≥ 0, where Θ r = Ω r ∩ B r ( 87
)
and Ω r , B r are given by (44), (62), respectively . • If x(τ, x 0 ) = 0 then the implications (74), (75) hold.

• For any values of time τ > 0 for which x(τ, x 0 ) ∈ Θ r(τ ) and

u nom (t) ≥ u h (x(t, x 0 )), ∀t ∈ [τ, τ + T ] : x(t, x 0 ) = 0, (88)
it holds that x(τ + T, x 0 ) = 0, where T > 0 is a parameter of the homogeneous control [START_REF] Wieland | Constructive safety using control barrier functions[END_REF].

Proof: Notice that by definition of the monotone function r we have d(-s)x(t, x 0 )/r(t) ≤ 1, ∀t ≥ 0.

1) Since Ω r ⊂ Σ -for any r ∈ (0, +∞) (see, Theorem 1) then Θ r ⊂ Σ -for any r ∈ (0, +∞).

Let us show that

r 1 ≤ r 2 ⇒ Θ r1 ⊂ Θ r2 . Indeed, if x ∈ Ω r1 and r 1 ≤ r 2 then h i d(s)d(-ln x/r 1 d )x = h i d(-ln d(-s)x/r 1 d )x ≥ 0, i = 1, . . . , n and d(-s)x/r 2 d ≤ d(-s)x/r 1 d ≤ 1.
Hence, using Corollary 2 we derive

h i d(s)d(-ln x/r 2 d )x = h i d ln d(-s)x/r 1 d d(-s)x/r 2 d d(s)d(-ln x/r 1 d )x ≥ 0. (89)
The system (1), ( 73), ( 86) is a functional differential equation. The existence of its solution x(t, x 0 ) for any x 0 ∈ R n follows, for example, from [START_REF] Kolmanovski | Applied Theory of Functional Differential Equations[END_REF]Theorem 4.1]. Notice also x(t, x 0 ) ∈ Ω r(t) ⇒ Θ r(t) since x(t, x 0 )/r(t) ≤ e s, ∀t ≥ 0. Moreover,

x 0 ∈ Θ r(0) provided that x 0 ∈ Ω.
Let us show that x(t, x 0 ) ∈ Θ r(t) for all t ≥ 0. First, we derive the equation ( 78) with γ = -e -s d dt x/r d and φ i (t 0 ) ≥ 0, i = 1, . . . , n. Next, using the formula [START_REF] Garg | Multi-rate control design under input constraints via fixed-time barrier functions[END_REF] and the identities

H(A + BK) = (-λI n + A)H, d(s)A = e s d(s)A, d(s)B = e s B, ∀s ∈ R we obtain γ = γ r -γ u (u h -u) + ṙφ T P φ r 2 φ T P HG d H -1 φ . (90)
Since t → r(t) is a continuous non-decreasing function then r is differentiable almost everywhere and ṙ ≥ 0.

2) Repeating considerations of the proof of Theorem 2 we conclude x 0 ∈ Ω ⇒ x(t, x 0 ) ∈ Ω r(t) for all t ≥ 0, and the implications (74), (75) hold.

3) If 0 ≤ t 1 < t 2 are such that u nom (t) ≥ u h (x(t, x 0 )) and x(t, x 0 ) = 0 for all ∀t ∈ [t 1 , t 2 ] then repeating the derivation of (56) we obtain

d dt x(t, x 0 )/r(t 1 ) d t=t1 ≤ - 1 T (91)
this means that the function t → e -s x(t, x 0 )/r(t

1 ) d = d(-s)x(t, x 0 )/r(t 1 ) d is strictly decreasing at t = t 1 and r(t) = r(t 1 ), ∀t ∈ [t 1 , t 2 ]. Indeed, otherwise there exists t ∈ (t 1 , t 2 ] such that r(t) = r(t 1 ) for all t ∈ [t 1 , t ] and d(-s)x(t , x 0 )/r(t 1 ) ≥ 1, but this is impossible due to d dt x(t, x 0 )/r(t 1 ) d ≤ - 1 T , ∀t ∈ [t 1 , t ]. (92)
Hence, repeating the proof of Theorem 2 we derive x(τ + T, x 0 )/r(τ ) = 0 provided that u nom (t) ≥ u h (x(t, x 0 )) for all t ∈ [τ, τ + T ] such that x(t, x 0 ) = 0.

The settling time estimate T > 0 of the safety filter (73), [START_REF] Wieland | Constructive safety using control barrier functions[END_REF], (86) is independent of the nominal control u nom and the initial state x 0 ∈ R n . Such a 'fixed-time safety' is guaranteed by means of an adaptation of the parameter r of the homogeneous controller [START_REF] Wieland | Constructive safety using control barrier functions[END_REF] (see, the formula (86)). The parameter r depends implicitly on the user's nominal controller u nom and on the initial state x 0 ∈ Ω of the system. However, this parameter also specifies the maximum magnitude of the homogeneous controller (see, the formula (61)) as well as a positively invariant set of the closed-loop system (see, the formula (44)). In practice, the maximum value of r should be bounded by some r max > 0 due to physical restrictions to admissible control signals. In this case, the safety filtering with the fixed settling time estimate T > 0 may be ensured only in the zone Θ rmax ⊂ Σ -.

If the safety override for a duration of time T recurs, on each occasion the safety boundary will be reached after T time units. The adaptive parameter r is to be different on each of those occasions.

VI. EXAMPLE: DOUBLE INTEGRATOR

A. Linear nonovershooting controller

Let n = 2. According to Lemma 1, given x 0 ∈ intΣ -⊂ R 2 a linear nonovershooting controller can be defined as follows

u lin (x) = Kx, K = h 2 A + λh 2 = (-λ 2 -2λ) (93)
where

h 1 = (-1 0), h 2 = (-λ -1), λ ≥ 1 - h 1 Ax 0 h 1 x 0 = 1 - e T 2 x 0 e T 1 x 0 . ( 94
) Indeed, if ϕ 1 = -x 1 and ϕ 2 = -λx 1 -x 2 , where x = (x 1 , x 2 ) T ∈ R 2 , then φ1 = -λϕ 1 + ϕ 2 , (95)
and

φ2 = -λx 2 -u lin = -λ 2 x 1 -λx 2 = -λϕ 2 . (96)
The system (95), ( 96) is globally asymptotically stable and positive. Taking into account λ > -e T 2 x0 e T 1 x0 we derive φ 1 (0) > 0 and φ 2 (0) > 0, so the controller stabilizes the state vector x at zero without overshoot in the first component:

e T 1 x(t, x 0 ) ≤ 0, ∀t ≥ 0. (97)
The set

Ω = {x ∈ R 2 : h 1 x ≥ 0, h 2 x ≥ 0} (98)
is positively invariant for the closed-loop linear system.

B. Upgrading a linear to a homogeneous controller 1) Homogeneous stabilization. Since for n = 2 one has

G d = ( 2 0 0 1 ) then h 1 h 2 G d = h 1 h 2 + h 1 h 2 1 0 0 0 = 2 0 λ 1 h 1 h 2 (99)
and

h 1 h 2 (A + BK) = 0 -1 λ 2 λ = -λ 1 0 -λ h 1 h 2 . ( 100
)
Hence, for α > 0 and

P = h 1 h 2 T α 0 0 1 h 1 h 2 , (101)
the system of LMIs (36) becomes

h 1 h 2 T -2λα 1 1 -2λ h 1 h 2 ≺0, h 1 h 2 T 4α λ λ 2 h 1 h 2 0, (102)
or, equivalently,

α > max 1 4λ 2 , λ 2 8 . (103)
Selecting s = 1, by Theorem 1, the controller

u h (x) = K 1 x/r 2 d 0 0 1 x/r d x = - λ x/r d 2 x 1 -2 λ x/r d x 2 (104)
stabilizes the double integrator to zero in a finite-time and

x 0 ≤ r ⇒ x(t, x 0 ) = 0, t ≥ T := 1 ρ , (105)
where r > 0 is an arbitrary parameter, the homogeneous norm x d is induced by the norm x = √ x T P x and

ρ = -λ max 4α λ λ 2 -1 2 -2λα 1 1 -2λ 4α λ λ 2 -1 2 (106)
The fixed-time stabilization can be guaranteed by selection r = max{r min , x 0 }, where r min > 0.

2) Homogeneous barrier functions. The time-derivatives of the functions

φ 1 = -1 x/γ 2 d x 1 , φ 2 = -λ x/r 2 d x 1 -1
x/r d x 2 (107) along the trajectories of the closed-loop homogeneous system are φ1 =

2 d dt x/r d x/r 3 d x 1 -x2 x/r 2 d = - λ+2 d dt x/r d x/r d φ 1 + 1 x/r d φ 2 , (108) φ2
= 2λ d dt x/r d x/r 3 d x 1 + -λ+ d dt x/r d x/r 2 d x 2 -1 x/r d u = 2λ(-d dt x/r d ) x/r d φ 1 + λ 2 x/r 3 d x 1 + λ+ d dt x/r d x/r 2 d x 2 = λ(-d dt x/r d ) x/r d φ 1 - λ+ d dt x/r d x/r d φ 2 . ( 109
)
Since γ = -d dt x/r d > 0 and x/r d > 0 as long as x = 0 then the system

φ1 φ2 = 1 x/r d -λ+2γ 1 λγ -λ+γ φ1 φ2 (110)
is positive and the set

Ω r = x ∈ R 2 : φ 1 (x) ≥ 0, φ 2 (x) ≥ 0 . ( 111
)
is a strictly positively invariant set of the closed loop homogeneous system.

3) Tuning of the settling time. Let us optimize the settling time estimate (105) maximizing ρ. From (106) we conclude that ρ = -max{η 1 , η 2 }, where η 1 and η 2 are real negative roots of the algebraic equation

det -2λα 1 1 -2λ -η 4α λ λ 2 = 0, (112)
which can be rewritten as follows

(8α -λ 2 )η 2 + 2(3 + 4α)λη + 4λ 2 -1 = 0. ( 113
) Since D = (3+4α) 2 λ 2 -(8α-λ 2 )(4λ 2 -1) > 0 for α > λ 2 8 (114) then ρ = (3 + 4α)λ -(3 + 4α) 2 λ 2 -(8α -λ 2 )(4λ 2 -1) (8α -λ 2 ) , (115)
or, equivalently, = λ 2 8 and the smallest upper estimate of the settling time x(t, x 0 ) = 0, ∀t ≥ T * corresponds to α * = λ 2 8 and

ρ = 4λ 2 -1 (3 + 4α)λ + (3 + 4α) 2 λ 2 -(8α -λ 2 )(4λ 2 -1) (
T * = 1 ρ * = 6 + λ 2 4λ 2 -1 λ. (117)
By Theorem 1, any settling time estimate T > 0 can be assigned by means of the modification of the feedback gain K = Kd(s), s ≥ ln max{1/(ρT ), 1}. Without such a rescaling of K, the upper estimate of the setting time cannot be less than T * , since the parameter α > 0 must satisfy the inequality α > λ 2 8 to guarantee the feasibility of LMIs [START_REF] Wang | On generalized homogenization of linear quadrotor controller[END_REF].

C. Numerical simulations and the use of the homogeneous controller as a "safety filter"

For x 0 = (-4, 2) we select

λ = 2 ≥ 1 - e T 2
x 0 e T 1 x 0 and K = (-4 -4).

Then

h 1 = (-1 0), h 2 = (-2 -1), (118)
Selecting α = λ 2 +0.01 8 = 0.50125 we derive ρ ≈ 0.7495, so the closed-loop system with the homogeneous nonovershooting controller [START_REF] Wieland | Constructive safety using control barrier functions[END_REF] has the following settling time estimat: T = 1 ρ ≈ 1.3342, i.e., x 0 ≤ r ⇒ x(t, x 0 ) = 0, ∀t ≥ T . A nonovershooting controller for the chain of integrators can assume the role of a "safety filter" of a nominal controller u nom by being implemented as [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF] u = min(u nom , u s )

where the nominal control u nom may demand overshoots, while the override of such an "unsafe" nominal control is performed using a "safe" control u s , which may be either of the linear kind, denoted as u s = u lin , or of the homogeneous kind, denoted as u s = u h ). To illustrate such "safety filtering" through a simulation, we take a nominal control given by

u nom = -4 x 1 + sin πt 2 + 0.8 -4 x 2 + π 2 cos πt 2 , (
120) which periodically and persistently attempts to violate the safety condition x 1 ≤ 0, while also periodically retreating from such an attempt. In this simulation we compare three safety filters: a linear one based on using [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF] with (119), a homogeneous one based on the results of this paper, and a prescribed-time (PT) safety filter introduced in [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF]. The simulation results are shown in Figures 1 and2.

But let us first examine Figure 3, which presents the positively invariant sets Ω and Ω r for, respectively, the linear nonovershooting controller and homogeneous nonovershooting controller with r = x 0 = x T 0 P x 0 ≈ 6.6348. The sets define the "safety zones" outside of which the corresponding controllers cannot guarantee the absence of overshoots. In the case of the fixed-time safety filter this zone is adaptive and depends of max τ ∈[0,t] r(τ ) . The larger a zone, the less conservative the override of the nominal controller. Obviously, the homogeneous nonovershooting controller has a larger positively invariant set than the linear controller (at least close to the origin). The PT controller is a time-varying linear feedback [START_REF] Abel | Prescribedtime safety design for a chain of integrators[END_REF], so its positively invariant set is not defined. The numerical simulations in Figures 1 and2 show that, in comparison with the linear safety filter, both the homogeneous and the PT safety filters perform highly unconservative overrides of the nominal controller. The main difference between the PT and homogeneous safety filters is that the operation time of the PT safety filter terminates after the prohibition on x 1 being positive ends, which is at T = 4 seconds in our case, whereas the homogeneous safety filter continues to restrict the operation to x 1 ≤ 0 for all time.

However, it is remarkable, and evident from Figure 1, that the homogeneous safety filter keeps x 1 (t) strictly negative for no longer than the fixed time T = 4 sec (and, in fact, for little more than 1 sec), in response to the attempts of the nominal control to make x 1 positive, whereas the more conservative linear safety filter keeps x 1 (t) negative all the time, greatly distorting the the resulting (dashed) solution x 1 (t) compared to the nominal solution (red).

It is evident from Figure 1 that the safety override with the homogeneous safety filter recurs (twice during the time shown; note the two flat tops of the blue curve). This is the result of the fact that the periodic nominal control keeps attempting to violate the safety boundary and keeps retreating.

VII. CONCLUSIONS

The paper proposes a two-step procedure for a nonovershooting homogeneous control design for the integrator chain. First, we construct a linear nonovershooting controller using a backstepping procedure [START_REF] Krstic | Nonovershooting control of strictfeedback nonlinear systems[END_REF]. Next, we transform ("upgrade") the linear feedback law to a generalized homogeneous one. The obtained homogeneous controller is globally uniformly bounded a number dependent on control parameters T and r, which defines a convergence time of the system initiated in the ball of radius r. The main advantage of the proposed scheme is the simplicity of the control design and parameter tuning. Moreover, the proposed procedure allows a simple upgrade of an existing linear nonovershooting controller to a homogeneous ones. The main disadvantage is the necessity to use a special computational procedure (see, e.g. [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF], [START_REF] Wang | On generalized homogenization of linear quadrotor controller[END_REF]) to implement the homogeneous controller in practice for high order systems (n ≥ 3). The positively invariant set of the homogeneous control system is larger than the positively invariant set of the linear control system utilized for the "upgrade". This allows a better (less conservative) overriding of a potentially unsafe nominal controller for the safety filter design.

VIII. APPENDIX A. Cubic equation (Cardano formula)

Consider the cubic equation

z 3 + pz 2 + qz + r = 0, p, q, r ∈ R. (121)
Introduce the following numbers

C 1 = 3 ∆1- √ ∆ 2 1 -4∆ 3 0 2 , C 2 = 3 ∆1+ √ ∆ 2 1 -4∆ 3 0 2 (122)
where ∆ 0 = p 2 -3q, ∆ 1 = 2p 3 -9pq + 27r.

C. Auxiliary results

The following lemma studies some properties of the vectors h i , which are utilized below for a non-overshooting control design.

Lemma 2. If the diagonal matrix D i ∈ R n×n is given by

D i = D i-1 + Ii-
2) Using D i-1 A = AD i we obtain

Ae Dis = ∞ k=0 AD k i s k k! = ∞ k=0 D i-1 AD k-1 i s k k! = • • • = ∞ k=0 D k i-1 As k k! = e Di-1s A. (129)
3) By construction (see, the formula (17)), only first i components of the vector h i are nonzero. The latter means that h i Ii 0 0 0 = h i and

h i e s Ii 0 0 0 = ∞ k=0 h i Ii 0 0 0 k s k k! = ∞ k=0 h i s k k! = e s h i . (130)
4) Since h 2 D 2 = 0, . . . , 0) = (i -1)λh 1 for i = 2 then, by induction, for i ≥ 3 we derive

h i D i =h i-1 AD i + λh i-1 D i =h i-1 D i-1 A + λh i-1 D i-1 + Ii-1 0 0 0 =(i -2)λh i-2 A + (i -2)λ 2 h i-2 + λh i-1
=(i -2)λh i-1 + λh i-1 = (i -1)λh i-1 .

(131) 5) Since only first i elements of h i are nonzero then

h i D n =h i (I n-1 + D n-1 ) = h i + h i D n-1 = • • • = (n -i)h i + h i D i , (132)
where h i D i = (i -1)λh i-1 as shown above.

Other useful properties of the vectors h i and the matrices D i are given by the following corollary. Proof: i). Let us consider the functions q i : R + → R defined as follows q i (s) = h i e sDi x, s ≥ 0.

(136) Hence, we have q i (0) ≥ h i x. Let us show, by induction, that these functions are non-decreasing and, consequently, non-negative on R + . Indeed, since D 1 = 0 then q 1 (s) = h 1 e sD1 x = h 1 x is non-decreasing and non-negative. For i ≥ 2 we derive d dt q i (s) =h i D i e sDi x = (i -1)λh i-1 e sDi x =(i -1)λe s q i-1 (s),

where the identities h i D i = (i -1)λh i-1 and

h i-1 e
s Ii-1 0 0 0 = e s h i-1 (see, Lemma 2) are utilized. Since λ > 0 then d ds q i (s) ≥ 0 and, consequently, q i (s) ≥ q i (0), ∀s ≥ 0 provided that q i-1 (s) ≥ 0, ∀s ≥ 0. Hence, taking into account h i d(s) = e (n-i+1)s h i e Dis we conclude that h i d(s)x ≥ 0 for all s ≥ 0 provided that h i x ≥ 0 for all i = 1, 2..., n.

ii). The identity H(A + BK)H -1 = -λI n + A is proven in Lemma 1 by means of the coordinate transformation ϕ = Hx.

iii). Since G d = D n + I n then

HG d = H +    h1Dn h2Dn h3Dn
. . .

  Denoting γ = -e -s d x/r d dt and using the identities d(s)A = e s d(s)A, d(s)B = e s B, ∀s ∈ R we derive φ = e s x/r d H(A + BK + γG d )d(s)d(-ln x d )x. (59) Using Corollary 2 we obtain φ = e s x/r d Πφ,

Theorem 3 (

 3 Fixed-time Safety Filter). If r = r(t) := max r min , max τ ∈[0,t] d(-s)x(τ ) , r min > 0 (86) then under conditions of Theorem 2 the following holds.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Trajectories in (t, x 1 )-plane for x 0 = (-4 2) T .

Fig. 3 .

 3 Fig. 3. Positively invariant sets Ω and Ωr for linear and homogeneous nonovershooting controllers, respectively

  root of the cubic equation.B. Quartic equation (Ferrari formula)Let us consider the quartic equationV 4 + aV 2 + bV + c = 0, a, b, c ∈ R(124)and the adjoint cubic equationz 3 + 2az 2 + (a 2 -4c)z -b 2 = 0.(125)If z 0 ∈ R is a real root of the cubic equation then the roots of the quartic one are

2 )= h i and h i e s Ii 0 0 0 =

 200 0 ... ... ... ... 0 0 i-2 ... ... ... ... 0 ... ... ... ... ... ... ... 0 ... ... 1 0 ... 0 0 ... ... 0 0 ... 0 ... ... .. ... ... ... ... 0 ... ... 0 0 ... 0    , i = 2, . . . , n, D 1 = 0, (127)then for i = 2, . . . , n one has 1) D i-1 A = AD i ; Ae Dis = e Di-1s A, ∀s ∈ R; = e s h i , ∀s ∈ R; 4) h i D i = (i -1)λh i-1 ; 5) h i D n = (n -i)h i + (i -1)λh i-1 .Proof: 1) Simple calculations showAD i =01 0 ... 0 0 0 1 ... 0 ... ... ... ... ... 0 ... ... ... ... 0 0 i-2 ... ... ... ... 0 ... ... ... ... ... ... ... 0 ... ... 1 0 ... 0 0 ... ... 0 0 ... 0 ... ... .. ... ... ... ... 0 ... ... 0 0 ... D i-1 A.

Corollary 2 .

 2 i) If λ > 0 then h i d(s)x ≥ 0, ∀s ≥ 0 (133) provided that h i x ≥ 0 for all i = 1, . . . , n; ii) H(A + BK) = (A -λI n )H, where K ∈ R 1×n is given by (∈ R n×n ;(134)iii)HG d = G d + λ(nI n -G d )A T H(135)where G d = I n + D n .

==

  G d H + λ(nI n -G d ) G d H + λ(nI n -G d )A T H. (142)

  diag{p 1 , . . . , p n } ∈ R n×n denotes a diagonal matrix with the elements p 1 , p 2 ,. . . ,p n on the main diagonal.

	1 2	0.
	• λ max (P ) (resp. λ min (P )) denotes a maximal (resp. min-
	imal) eigenvalue of the symmetric matrix P .
	II. PROBLEM STATEMENT
	Consider the system

•

  116)Hence, to maximize ρ the parameter α should be minimized.

	For λ ≥ 4 √	2 we have α > max 1 4λ 2 , λ 2 8

A system ẋ = f (x), t > 0, x(0) = x 0 is said to be globally uniformly finite-time stable if it is Lyapunov stable and there exists a locally bounded function T : R n → [0, +∞) such that x(t, x 0 ) = 0, ∀t ≥ T (x 0 ) for any solution x(t, x 0 ) of the system.

A set D ⊂ R n is said to be a positive cone if x ∈ D ⇒ e s x ∈ D, ∀s ∈ R.

A set D ⊂ R n is said to be a d-homogeneous cone if x ∈ D ⇒ d(s)x ∈ D, ∀s ∈ R.