An integral extension technique for continuous homogeneous state‐feedback control laws preserving nominal
 performance

This paper proposes a technique to extend a nominal homogeneous state‐feedback control law by continuous or discontinuous integral terms. Compared to pure state feedback, this permits to suppress non‐vanishing perturbations that are either constant or Lipschitz continuous with respect to time. The proposed technique seeks to do this while maintaining nominal performance in the sense that the nominal control signal and closed‐loop behavior is not modified in the unperturbed case. The class of controllers thus obtained is shown to include the well‐known super‐twisting algorithm as a special case. Simulations comparing the technique to other approaches demonstrate its intuitive tuning and show a performance preserving effect also in the perturbed case.


INTRODUCTION
Rejecting disturbances acting on a plant is one of the main goals of feedback control. Linear state feedback can usually attenuate very small disturbances or disturbances that vanish with vanishing state to a satisfactory degree, but fails to suppress large nonvanishing disturbances. Homogeneous state-feedback control laws improve on the disturbance rejection capability of linear controllers due to their increase in (linearized) gain close to the equilibrium. 1 Nevertheless, also they fail to completely reject nonvanishing (even constant) disturbances. Handling such disturbances typically requires techniques such as disturbance observers or integral control. This contribution proposes a technique for adding an integral part to any homogeneous state-feedback controller that stabilizes a nominal, that is, unperturbed plant in finite time. The integral controller is constructed in such a way that the nominal performance is recovered in the unperturbed case. It is shown that global finite-time stability of the closed loop is achieved in the presence of constant or slope bounded, that is, Lipschitz continuous perturbations. For the latter disturbance class in particular, the technique yields integral state-feedback control laws with discontinuous integrands.
In the context of sliding-mode control, 2 the proposed technique may be used to obtain higher order sliding-mode control laws for sliding surfaces of arbitrary relative degree. In contrast to the integral sliding mode technique, 3 which has a similar objective of preserving the performance of a nominal controller, no dynamic extension or additional sliding surface is used. Rather, a whole class of state-feedback controllers with integral part is obtained, which contain the well-known super-twisting algorithm as a special case.
The design of state-feedback controllers with discontinuous integrands has been studied, for example, using integral sliding mode control, 4 passivity-based techniques, 5 explicit Lyapunov functions, [6][7][8] and implicit Lyapunov functions. 9 Thereof, some of the nonpassive Lyapunov-based approaches 6,7 allow for the design of pure output-feedback controllers. They are not designed to extend a given nominal control law, however. The passivity-based approach 5 is able to do so, but applying it requires a Lyapunov function for the nominal closed loop, which is not always available. The implicit Lyapunov function approach 9 remedies this to some extent, but requires a compatible nominal control law. With the technique proposed here, a Lyapunov function for the nominal case is needed only for obtaining quantitative (rather than qualitative) closed-loop stability conditions, and in further contrast to the approaches from the literature, not only homogeneous but also nonhomogeneous closed loops may be obtained using it. Though none of the existing approaches aim to preserve a nominal controller behavior, they have several other interesting features. Some of those are illustrated by comparing two approaches 5,6 from literature with the proposed technique in the course of a simulation.
The paper is structured as follows: After some preliminaries and notational details in Section 2, the considered problem is stated in Section 3. Section 4 then discusses basic assumptions and properties of the nominal homogeneous state-feedback control law, which forms the basis of the proposed technique. The integral extension technique is proposed in Section 5, and its structural and performance preserving properties are shown. Closed-loop stability is analyzed in Section 6 for both constant and Lipschitz continuous perturbations, and conditions for global asymptotic and finite-time stability are derived. Section 7, finally, compares the proposed technique with two approaches found in the literature, and Section 8 gives concluding remarks. An Appendix contains the proofs of all lemmas, whereas theorems and propositions are proven in the main text.

PRELIMINARIES
The abbreviations for y ∈ R and p ≠ 0, as well as ⌊y⌉ 0 = sign(y) are used throughout the paper. Matrices and vectors are written as boldface symbols, and R denotes the field of real numbers. The Euclidian norm of a vector x ∈ R n is written as ||x|| . Some notions of generalized homogeneity of scalar valued functions and vector fields are now discussed. Let a vector r = [r 1 r 2 … r n ] T of weights r i > 0, i = 1, … , n be given. The associated dilation matrix D r is defined as The scalar valued function V ∶ R n → R is called homogeneous of degree m ∈ R with respect to the weights r, denoted by holds for all > 0 and for all x ∈ R n . Furthermore, deg r x i is written for the weight of a single variable x i , that is, deg r x i = r i .
The vector field f ∶ R n → R n is called homogeneous of degree m with respect to the weights r if f(D r x) = m D r f(x) (4) holds for all > 0 and for all x ∈ R n , that is, if deg r f i = r i + m holds for the components of f. By appropriate choice of the weight vector r one can always normalize a vector field's homogeneity degree to either-in case of a negative degree-minus one,-in case of positive degree-plus one, or zero otherwise.

PROBLEM STATEMENT
To simplify considerations, the technique is demonstrated using a perturbed integrator chain as a plant. Such a system is routinely obtained by applying exact linearization techniques to a nonlinear plant, 10 or when considering sliding variable dynamics in the context of sliding mode control. 2 Therefore, the proposed technique may straightforwardly be applied to problems of sliding-mode or other nonlinear control problems with nonlinear plant dynamics. Consider a perturbed nth order integrator chain governed by the differential equation with output , control input u, and a matched disturbance w. The disturbance is assumed to be Lipschitz continuous with its time derivativeẇ bounded by The control task is to steer to zero in finite time and keep it there by means of a continuous control signal u, that is, to achieve (t) = 0 for all t ≥ T after some finite time T depending on the initial conditions. Without the disturbance, that is, for w = 0, the system may be stabilized in finite time by means of continuous homogeneous state-feedback control. One such control law for n = 2, for example, is proposed by Bacciotti and Rosier 1 based on an example due to Bhat and Bernstein 11 as with positive parameters k 1 ,k 2 . Similarly structured control laws exist also for higher plant order. 12 For nonvanishing w, however, such state-feedback control laws can achieve neither finite-time nor asymptotic convergence, because they cannot compensate for nonzero disturbances in equilibrium.
In this paper, the following problem is considered: Given any finite-time stabilizing homogeneous state-feedback control law, extend it by an integral part in such a way that (i) finite-time convergence is achieved in the perturbed case, and (ii) performance is preserved in the sense that, in the unperturbed case, the closed loop behaves as the original state feedback control law.

FINITE-TIME STABILIZING HOMOGENEOUS CONTROL LAWS
In state-space form, the perturbed integrator chain may be written aṡ with state variables x 1 , … , x n aggregated in the vector x ∶= [x 1 … x n ] T and with the bound on the perturbation's slopeẇ. Homogeneous control laws for this system are considered, which stabilize it in finite-time by means of a continuous control signal. This implies that the closed loop's homogeneity degree is negative. In order to simplify the further analysis by removing all redundant degrees of freedom, this homogeneity degree is fixed at m = −1, without restriction of generality. This requires the homogeneity weights to satisfy for i = 1, … , n − 1. Therefore, the weight vector's structure is fully determined as with a single scalar parameter > 1. Since this parameter fully determines the weights in the dilation matrix (2), it is called the dilation generator in the following. Throughout the paper, the short-hand notation deg f ∶= deg r f , with r as defined in (10), is used for the homogeneity degree of some function f with respect to the weights r generated by . Note that this notation is consistent also for n = 1, because then r = .
In the following, a nominal state-feedback control law is assumed to be given by with a continuous homogeneous function k, which stabilizes the unperturbed plant. This is reflected in the following definition.

Definition 1.
A continuous homogeneous function k ∶ R n → R is called a nominal stabilizing feedback, if the origin of the closed-loop system, formed by the control law u = −k(x) and the plant (8) with w = 0, is globally asymptotically stable.
For the overall closed loop to be homogeneous with degree m = −1, the homogeneity degree of this function has to be deg k = − 1 > 0. In this case, asymptotic stability is equivalent to global finite-time stability. One family of such control laws for arbitrary order n is proposed by Bhat and Bernstein. 12 With the introduced notation-the originally proposed controller is parametrized using −1 ∈ (0, 1) rather than -it is given by with positive parameters k 1 , … , k n and > 1. It is shown 12 that if k 1 + k 2 s + … + k n s n − 1 + s n is a Hurwitz polynomial, then the resulting closed loop is globally finite-time stable for sufficiently large values of , that is, for all > * with some sufficiently large * ≥ 1. It may be noted that in this case the exponents in (12) are (sufficiently) close to one.

PERFORMANCE PRESERVING INTEGRAL EXTENSIONS
This section introduces the proposed integral extension technique and discusses the choice of its design parameters. Section 5.1 shows how to construct the proposed controller using a homogeneous function and a scalar positive gain as parameters. This yields an entire family of control laws with continuous and discontinuous integrands. The resulting closed-loop structure and a performance preserving property of the proposed controller is then shown in Section 5.2. Section 5.3 discusses the choice of the homogeneous function parametrizing the family of controllers.

Integral extension
Consider a dilation generator > 1 and a control law u = −k(x) with a nominal stabilizing feedback function k of degree deg k = − 1. An integral extension of this controller is proposed as Therein, k I is a positive parameter, h is a homogeneous function to be chosen, its homogeneity degree = deg h > 1 represents another important parameter, and the homogeneous function g is given by Note that the control law's parameter k I may be interpreted as an integrator gain. The values of and determine the closed loop's structural properties; this will be discussed in more detail later on. Figure 1illustrates the structure of the control law.
For the proposed control law to be well-defined and in order to guarantee closed-loop stability later on, the function h, which generates it, has to have some properties that are summarized in the following definition. Its first item ensures that the control signal is continuous and that the integrand g is locally bounded and continuous on R n ⧵ {0}, which is reasonable from a practical point of view. The second item will be shown to be important for ensuring closed-loop stability with the proposed control law.

Definition 2.
A homogeneous function h ∶ R n → R is called an admissible generating feedback, if it has the following two properties: (i) h is continuous on R n and continuously differentiable on R n ⧵ {0}; (ii) the partial derivative of h with respect to x n is strictly positive on R n ⧵ {0}, that is, h Remark 1. Note that item (i) in particular implies that partial derivatives of h are continuous and thus locally bounded on R n ⧵ {0}.
The generating feedback's homogeneity degree and the dilation generator determine whether the extended control law and thus the overall closed loop are homogeneous and whether the integranḋv is continuous or discontinuous with respect to the state x. This is stated in the following proposition and is also summarized in Table 1. For the second statement, note that all partial derivatives of h and, thus also g, are locally bounded on R n ⧵ {0} due to item (i) of Definition 2. Since g is homogeneous with nonnegative homogeneity degree, it is then locally bounded also at the origin. For the third statement, note that homogeneity implies (Reference 12, theorem 4.1) continuity of g unless deg g ≤ 0, which shows necessity of = 1. To show also sufficiency, note that g being locally bounded with deg g = 0 implies that g is globally bounded and either discontinuous or constant. Assume now that g is constant and equal to G 0 ∈ R. Since h x n is sign definite and h(0) = 0, there exists a vector x 0 ∈ R n satisfying h(x 0 ) ≠ 0 and h(x 0 )G 0 ≥ 0. Then, |h(x(t))| is positive and nondecreasing along the trajectory x(t) of the nominal closed loop with initial condition converges to zero in finite time. This contradicts the fact h(0) = 0; therefore, g is discontinuous. ▪ Example 1 (Controller for the first order integrator-super-twisting algorithm). As an example, consider a first-order integratoṙx = u + w. For a given dilation generator , the only admissible generating feedback with homogeneity degree ≥ 1-up to a scaling, which is however redundant with k I -is h(x) = ⌊x⌉ . One verifies that its derivative h x = |x| and thus the proposed extended control law (13) is One can see that the integrand is discontinuous if = 1, and that in general the control law is not homogeneous. For the special case = 1 + ≥ 2 and with the abbreviations = k I v, k 1 = k S + k I , and k 2 = k S k I one obtains the homogeneous control law For = 1, the integral term is discontinuous and the control law corresponds to the well-known super-twisting algorithm. 13

Closed-loop system and performance preserving property
The proposed state-feedback controller with integral part permits to preserve the performance of the nominal closed loop. In order to show this, and to investigate stability later on, the closed-loop system is considered. Introducing the state variable q = k I v + w, one finds that the closed loop formed by the plant (8) and control law (13) is governed by the differential equationṡx Since g may be discontinuous in the origin if = 1 and sinceẇ may take any value in the interval [− L,L], solutions of this system are defined in the sense of Filippov. 14 In particular, (17c) with |ẇ| ≤ L is to be understood as the differential inclusionq with If > 1 and thus deg g > 0, then g is continuous in the origin and these quantities are G − = G + = g(0) = 0. Otherwise, G − = −inf x∈R n ⧵{0} g(x) and G + = sup x∈R n ⧵{0} g(x) by homogeneity, and the infimum and supremum can be found by computing the minimum and maximum of g(x) on the unit sphere, for example, where g is continuous.
While (17) is a well-defined Filippov inclusion describing the dynamics of the closed-loop system, it does not provide any immediate insight into its properties. A more insightful description is obtained by choosing the state variable which for an admissible generating feedback h is differentiable on R n ⧵ {0}, but not necessarily in the origin. Since the time derivative of h along the closed-loop trajectories satisfies when it is differentiable, one finds that for x ≠ 0 the closed loop is equivalently governed by the differential equationṡ As noted, h x n may not be differentiable for x = 0 and even has a singularity (ie, an essential discontinuity) in the origin if < , because deg h x n = − . Therefore, (21) does not necessarily correspond to a well-defined Filippov inclusion. Nevertheless, the differential equations may be used to compute time derivatives for x ≠ 0 later on.
For constant perturbations, that is, forẇ = 0, relation (21c) suggests that z will decay to zero for k I > 0, because h x n is strictly positive. In particular, if the initial condition z(0) is zero, one may expect that z(t) = 0 will hold indefinitely, thus preserving the nominal performance obtained with u = −k(x). The following proposition and the stability analysis in Section 6 will show that both of these intuitions are correct and, moreover, that x and z even converge to zero in finite time under certain conditions. Remark 2. Although the proposition only considers the unperturbed case, a performance preserving effect is observed also in the perturbed case for sufficiently large values of the integrator gain k I . This is demonstrated in Section 7 in the course of simulation studies with different values of k I . It can also be seen intuitively from the definition of z in (19) and the corresponding differential equation (21c). As z tends to zero, the term k I h(x) − k I v reconstructs (ie, tends to) the perturbation w and the nominal system is recovered from (21). This happens the faster the larger the gain k I is.
Proof. Let x(t) be a trajectory of the nominal closed loop obtained from (8) with w = 0, u = −k(x) and denote by T the corresponding finite convergence time. Then, since x(t) ≠ 0 for all t < T, one can use (21) to verify that this trajectory along with z(t) = 0 satisfies the differential inclusion (17) for t < T. For t ≥ T, finally, one can verify that x(t) = 0 and z(t) = q(t) − k I h(x(t)) = q(t) = 0 also satisfies (17). Therefore, the trajectories of both systems coincide, which concludes the proof. ▪

Design of the generating feedback
For = , the relation deg h = = deg x n suggests to choose the generating feedback h(x) = x n , which is indeed admissible. In the general case, however, it is not immediately obvious how to find a generating feedback h satisfying all conditions of Definition 2. The following lemma, which is proven in the appendix, gives a sufficient condition for such a function: Lemma 1. Let a dilation generator > 1 and two homogeneous functions p, q ∶ R n ⧵ {0} → R be given, whose homogeneity degrees satisfy deg p < deg q. Suppose that p and q are continuously differentiable, that p is strictly positive, and that holds for all x ≠ 0. Then, the function h ∶ R n → R defined by h(0) = 0 and for x ≠ 0 is an admissible generating feedback.
Based on this lemma, the following proposition gives a useful class of generating feedback functions for any given homogeneity degree ≥ 1.

Proposition 3 (Class of admissible generating feedbacks). Let a dilation generator > 1, a desired homogeneity degree
≥ 1, and a real-valued constant > + n − 1 be given. Then, for all positive constants a 1 , … , a n the function h ∶ R n → R given by x n ( a 1 |x 1 | +n−1 + a 2 |x 2 | +n−2 + … + a n−1 |x n−1 | +1 + a n |x n | is an admissible generating feedback with homogeneity degree deg h = . Remark 3. Note that there are no restrictions on the sign of − , and for = in particular, h(x) = x n is obtained.
Proof. Comparing h with (23), one can see that the denominator p and the numerator q are given by the homogeneous functions with degrees deg p = − < = deg q. Therefore, h is homogeneous with degree deg h = deg q − deg p = and it has the form required by Lemma 1. It will be shown that the functions p,q also satisfies the other conditions of that lemma. Obviously, q is continuous and continuously differentiable and p is continuous and strictly positive on R n ⧵ {0}. Furthermore, p is continuously differentiable for all nonzero x, because +n−i > 1 for all i = 1, … , n, that is, all exponents of single variable expressions are greater than one. Finally, one finds that and thus (22) holds for all x ≠ 0. Therefore, h is an admissible generating feedback. ▪ Example 2 (Controller for the second order integrator chain). It is now shown how an integral controller previously studied in a conference paper 15 is obtained. Consider a second order integrator chaiṅx 1 = x 2 ,̇x 2 = u + w with dilation generator = 2 and nominal stabilizing feedback with positive parameters k 1 , k 2 . The corresponding control law u = −k(x 1 ,x 2 ) stabilizes the unperturbed double integrator in finite time for all positive parameter values. 1 Using Proposition 3 with = 4, a 1 = 3k 1 2 , a 2 = 2, the function is found to be an admissible generating feedback with homogeneity degree deg h = ≥ 1. Its partial derivatives are given by (29) One can see that they are locally bounded on R 2 ⧵ {0} and that h x 2 is furthermore strictly positive. The function g in (13c) is computed as and from g(0, x 2 ) = −2 − 5 4 k 2 ⌊x 2 ⌉ −1 one can see that it is discontinuous in the origin if = 1. For this latter value of and after simplifying the above expression, the proposed control law (13) is given by It is worth to point out that, similar to the so-called quasi-continuous 16 sliding-mode controllers, the right-hand side of (31b) is discontinuous only in the origin, but is continuous everywhere else.

STABILITY ANALYSIS
In the following, the stability properties of the perturbed closed loop system (17) are studied. Like the structural properties of the control law in Table 1, also the closed loop's stability and robustness properties depend on the dilation generator and the generating feedback's homogeneity degree . If the function g is continuous, that is, if ≠ 1, stability can only be guaranteed for constant perturbations with L = 0, because the origin is not even an equilibrium otherwise. Section 6.2 discusses this case, showing that choosing < permits to ensure finite-time stability, while only asymptotic convergence can be guaranteed otherwise. For = 1, on the other hand, it will be shown in Section 6.3 that Lipschitz continuous perturbations with arbitrarily large Lipschitz constant L can be handled by appropriate tuning of the parameter k I . Table 2 shows an overview of the stability and robustness properties that will be proven in the following.

TA B L E 2 Global stability and robustness
properties of the closed loop formed by plant (8) with extended control law (13) for different values of the generating feedback's homogeneity degree ≥ 1 and the dilation generator > 1 (with > 0 sufficiently large, see Theorem 2 in Section 6.3.1 and its Corollary 1 in Section 6.3.2)

Nominal Lyapunov function
The stability analysis is based on a Lyapunov function for the nominal closed loop, which is obtained by applying the nominal state-feedback control law u = −k(x) to the unperturbed plant (8). Its dynamics are governed bẏ The existence of a Lyapunov function for this system is guaranteed by standard converse Lyapunov results. 12 This is stated in the following lemma and proven in the appendix.

Lemma 2.
Let a dilation generator > 1, a nominal stabilizing feedback k with deg k = − 1, and a constant ≥ 1 be given. Then, there exist constants c 1 ,c 2 and a continuous, homogeneous, positive definite function V ∶ R n → R with homogeneity degree deg V = , which is continuously differentiable on R n ⧵ {0} and whose time derivativeV along the trajectories of the nominal closed loop (32) and partial derivative V for all x ≠ 0.

Constant perturbation
The case of a constant perturbation, that is, L = 0, is considered first. For this class of perturbations, global asymptotic stability may be guaranteed regardless of and , and global finite-time stability is achieved if < . The former is proven by showing that, with z as in (19) and a nominal Lyapunov function V as in Lemma 2, is a Lyapunov function for the actual closed loop (17) for sufficiently large values of > 0. In order to show finite-time stability, a contraction property of the nonnegative expressions V(x) and |z| is furthermore employed. Proving this requires several technical arguments that also have to deal with the fact that V is not everywhere differentiable. Since these offer little additional insight, the main statements are encapsulated in the following technical lemma, whose proof the interested reader finds in the Appendix.
hold for all x ≠ 0. For any trajectory x(t) and z(t) = q(t) − k I h(x(t)) of the closed-loop system (17) Then, V is nonincreasing and lim t→∞ V(t) = 0. Moreover, if < , then the implication holds for all t 0 and all nonnegative constants D.
Using this lemma, the main stability result for constant perturbations-asymptotic stability for every k I > 0 and, additionally, finite-time stability for < -is now shown. |z| is nonincreasing and converges to zero along every closed-loop trajectory. This implies global asymptotic stability of the origin. Consider now the case < and define Repeatedly applying implication (37) from Lemma 3 then shows that the time T for V and |z| to converge to zero is bounded from above by the sum of two geometric series as to the plant (8) with constant disturbance w yields an asymptotically stable closed loop for every k I > 0. Although finite-time stability is lost compared to the nominal controller, this control law offers an easy way to make the closed loop robust with respect to constant disturbances. At the same time, nominal performance-and thus also the nominal controller's capability to attenuate other disturbances-is preserved in the sense of Proposition 2 by selecting the integrator's initial condition as v(0) = x n (0).

Lipschitz continuous perturbation
The case of (nonconstant) Lipschitz continuous perturbations, that is, L > 0 is now considered. With these perturbations, stability can only be guaranteed for = 1; otherwise, the integrand is not discontinuous and therefore the origin is not an equilibrium of the Filippov inclusion (17). For = 1 and sufficiently large k I , global finite-time stability of the closed loop will be shown using the Lyapunov function (34) with an appropriately selected . For this purpose, the following technical lemma is needed, which is proven in the Appendix.
is obtained.

SIMULATION RESULTS AND COMPARISONS TO EXISTING APPROACHES
This section shows simulation results with the proposed approach and compares them to two other techniques for designing discontinuous integral controllers from the literature. In particular, a passivity based approach proposed by Laghrouche et al and output feedback-based approaches proposed by Moreno et al are investigated. For each approach an integral extension of the nominal state-feedback control law is considered. The corresponding dilation generator is = 2. For the proposed performance preserving technique, the extended control law is derived in Example 2 and is given by (31) in Section 5.3. For the two other approaches, the basic technique and the considered control law are briefly explained in the following.

Passivity-based integral extension
Laghrouche et al 5 propose an integral extension technique for homogeneous state-feedback controllers that is constructed using passivity properties. For a given dilation generator and a given nominal stabilizing feedback k, that control law (with some minor notational adaptations) is given by Therein, k I is a positive design parameter and V ∶ R n → R is a Lyapunov function for the nominal closed loop formed by the unperturbed plant (8) and the nominal control law u = −k(x). It is constructed by feeding back the passive output V x n of the nominal closed loop with storage function V. If the homogeneity degree of V is deg V = , then the integral part's homogeneity degree is given by deg V x n = 0 and the integrand is discontinuous. Like the performance preserving integral extension presented here, this passivity-based integral extension has the advantage that it guarantees finite-time stability in the unperturbed case for any value of k I that is positive (or sufficiently large in the perturbed case).
For comparison purposes, this technique is applied to the nominal state feedback (54) with which is a strict Lyapunov function 15 for 3k 3 4 1 > 2k 2 . Since deg V = 2 = , one obtains the following controller with a discontinuous term in the integral: Similar to (31b), the right-hand side of (57b) is discontinuous in the origin, as can be seen for

Output feedback integral extension
The approaches considered up to now apply a discontinuous function of the full state to the integrator. Moreno et al 6,7 propose and study discontinuous integral controllers that permit to make this function depend only on the plant's output x 1 . This has the advantage of being more useful when the plant's state is not known exactly; the schemes furthermore exploit this to propose pure output feedback integral controllers. These include an observer with continuous right-hand side and unlike the concepts studied in this paper do not require any knowledge of the full state. The observer-based variants are not considered here, because the additional observer parameters limit the comparability with the other approaches. Therefore, an integral controller with full state feedback is selected for comparison purposes, whose integrand can be chosen to depend only on the plant's output. 6 With some minor notational adaptations it is given by where k 3 is arbitrary and k 1 ,k 2 ,k I are positive design parameters. One can see that the discontinuous function in the integrator depends only on x 1 for k 3 = 0, which is selected in the following. Furthermore, compared to the two other control laws in (31) and (57), it is structurally simpler and thus easier to implement. These advantages come at the cost, however, that k I can not be chosen arbitrarily large nor (in the perturbed case) arbitrarily small, which makes tuning more difficult.

Comparisons
For all approaches, the parameters k 1 = 6, k 2 = 3 are chosen, leading to identical nominal behavior with all approaches for k I = 0 and w = 0. Furthermore, k 3 = 0 is selected in (58), such that the integrand depends only on the plant's output x 1 . Figure 2 compares the output x 1 (t) obtained using the three approaches with initial state x(0) = [4 0] T , v(0) = 0 for different values of k I . The smallest value k I = 0.5 is chosen such that all three approaches show a similar behavior. The proposed extension tends toward the nominal behavior with growing k I , thus recovering the performance of the nominal state-feedback controller. The same is true initially for the output feedback approach up to k I = 2, while the performance of the passivity-based approach improves only slightly. Nonetheless, the latter maintains finite-time stability also for arbitrarily large values of k I , whereas the output feedback approach eventually becomes unstable.

CONCLUSION
A technique for extending homogeneous state-feedback control laws by an integrator was proposed. It can feature either a continuous or a discontinuous integrand and consequently is able to reject either constant or Lipschitz continuous perturbations, respectively. It furthermore recovers the behavior of the nominal state-feedback controller in the unperturbed case, thus preserving the nominal performance in some sense. The type and speed of closed-loop convergence was shown be be determined by two tuning parameters, a desired homogeneity degree and the integrator gain k I , respectively. In particular, it was shown that global asymptotic or finite-time stability may be guaranteed by appropriate selection of for every sufficiently large value of the gain k I . Comparisons with two other approaches from literature demonstrated the intuitive tuning of the proposed approach and showed that nominal performance is preserved to some extent also in the perturbed case by increasing the integrator gain. In the future, extending the approach for plants with unknown control coefficient may be studied.