Projection-based integrators for improved motion control: Formalization, well-posedness and stability of hybrid integrator-gain systems ✩

In this paper we formally describe the hybrid integrator-gain system (HIGS), which is a nonlinear integrator designed to avoid the limitations typically associated with linear integrators. The HIGS keeps the sign of its input and output equal, thereby inducing less phase lag than a linear integrator, much like the famous Clegg integrator. The HIGS achieves the reduced phase lag by projection of the controller dynamics instead of using resets of the integrator state, which forms a potential benefit of this control element. To formally analyze HIGS-controlled systems, we present an appropriate mathematical framework for describing these novel systems. Based on this framework, HIGS-controlled systems are proven to be well-posed in the sense of existence and forward completeness of solutions. Moreover, we propose two approaches for analyzing (input-to-state) stability of the resulting nonlinear closed-loop systems: (i) circle-criterion-like conditions based on (measured) frequency response data, and (ii) LMI-based conditions exploiting a new construction of piecewise quadratic Lyapunov functions. A motion control example is used to illustrate the results.


Introduction
For the control of linear motion systems, linear time invariant (LTI) control theory is appealing due to its well-understood and straightforward controller design with guaranteed stability and performance properties. However, LTI control designs suffer from fundamental performance limitations such as Bode's gain-phase relationship and the waterbed effect due to Bode's sensitivity integral (Freudenberg, Middleton, & Stefanpoulou, 2000;Seron, Braslavsky, & Goodwin, 1997). In the context of closed-loop performance, this typically results in design trade-offs.
Inspired by the advantages of reset control, in this paper we are interested in formalizing, as an alternative to reset control, a new nonlinear integrator referred to as the hybrid integrator-gain system (HIGS), which offers the same phase advantages, but without the need for hard resets of the (integrator) state.
The HIGS is designed to keep its input-output relation bounded in the sector [0, k h ], where k h ∈ R >0 denotes the gain parameter, thereby inheriting the hinted phase advantage of reset control (as the input and output of the HIGS have the same sign). However, the HIGS avoids resetting the integrator state, and exploits projection of the (controller) dynamics in a manner resulting in continuous control signals. In particular, a HIGS element acts as a linear integrator as long as its input-output pair lies inside the mentioned sector (called the 'integrator mode'). At moments when the sector condition tends to be violated, the vector field of the HIGS element is altered via projection in such a way that the resulting trajectories stay on the boundary of the sector. This second mode of operation is referred to as the 'gain mode' of the HIGS, explaining the terminology of hybrid integrator-gain systems. Interestingly, upon switching from integrator to gain mode, the integrator buffer is preserved as much as possible while respecting the sector condition, instead of being completely depleted by resets. This leads to increased potential for improving closed-loop performance for this hybrid integrator in comparison to, for instance, the Clegg integrator.
In this paper, we formalize the above idea based on extensions of so-called projected dynamical systems (PDS) (Dupuis & Nagurney, 1993;Henry, 1973;Nagurney & Zhang, 2012), a class of discontinuous dynamical systems introduced in the early 1990s. PDS are described by differential equations of which the solutions are restricted to a constraint set. At moments when the solutions tend to leave this set, the vector field of the system is changed by means of projection so that the solutions remain inside the constraint set. Although the PDS philosophy resembles that of the HIGS, there are essential differences that prevent direct description of the HIGS as a PDS. First of all, the constraint set of the HIGS (the sector), does not satisfy the regularity requirements that the PDS framework commonly requires, see, e.g., Henry (1973). Secondly, in the case of PDSs, the complete vector field is projected on (the tangent cone of) the constraint set. In the context of control, however, when considering a HIGS element in feedback interconnection with a physical plant, it is only possible to project the dynamics of the controller (HIGS) and not the full dynamics (including the plant dynamics). This calls for important generalizations of PDS, as provided in this paper, for which we coin the term extended projected dynamical systems (ePDS). Based on this new ePDS framework, which naturally captures the design philosophy of HIGS, we provide a formal mathematical description of HIGS-controlled systems. Interestingly, the representation used in our preliminary work (Deenen, Heertjes, Heemels, & Nijmeijer, 2017) follows from the ePDS-based formulation of HIGS-controlled systems. Furthermore, we establish the well-posedness of HIGS-controlled systems in the sense of existence and forward completeness of solutions, under mild assumptions as generally satisfied for linear motion systems.
In our preliminary work (Deenen et al., 2017), a circlecriterion-like argument is presented for stability analysis of HIGS-controlled systems (however, without a proof of the stability and without a well-posedness proof). Clearly, this circle-criterion approach offers great advantages in terms of easyto-check graphical conditions based on accurate and quickly measurable frequency response functions. A potential drawback, however, is that it may yield conservative bounds on closed-loop stability due to (i) the underlying use of a common quadratic Lyapunov function for a piecewise linear closed-loop system (Sontag, 1981), and (ii) solely using the hybrid integrator's sector-boundedness instead of its complete nonlinear dynamic behavior. Furthermore, its application is limited to control configurations where a stable LTI system is used in feedback connection with a HIGS element. One of the key contributions of this paper, next to providing a full proof of the circle-criterion-like stability condition for the first time, is to also provide less conservative stability conditions in terms of linear matrix inequalities (LMIs). This approach is inspired by Aangenent et al. (2009) and Zaccarian et al. (2005Zaccarian et al. ( , 2011 for reset control systems, where LMIs guarantee stability based on a piecewise quadratic (PWQ) Lyapunov function by partitioning the two-dimensional inputoutput plane of the reset element into double cones with the apices at the origin. The HIGS differs from the previously considered reset integrators in the sense that its switching dynamics are determined in a three-dimensional space. Therefore, an extension of the planar approach is proposed that uses volumetric partitions in a spherical coordinate system leading to LMI-based (input-to-state) stability conditions for HIGS-controlled systems. Both methods will be used to verify stability of an LTI motion system in feedback with a HIGS to illustrate, on the one hand, the practical convenience of the circle-criterion-like approach, and, on the other hand, the increased potential of the LMI-based conditions in terms of reduced conservativeness regarding parameter bounds.
The remainder of this paper is organized as follows. Section 2 contains preliminaries and notation. The HIGS is described in Section 3 and proven to be well-posed in Section 4. In Section 5, the LMI-based closed-loop stability conditions and their derivation are discussed, followed by the circle-criterion-like frequency-domain stability conditions in Section 6. In Section 7, these stability conditions are compared using an illustrative motion control example. Section 8 states the conclusions.
A function w : I → R nw , with I ⊆ R is called a Bohl function, denoted by w ∈ B I , if there exist matrices H ∈ R nw ×n F , F ∈ R n F ×n F , and a vector v ∈ R n F such that w(t) = He Ft v for all t ∈ I.
Note that piecewise Bohl functions can be discontinuous, but they are continuous from the right in the sense that for each T ∈ R ≥0 it holds that w(T ) = lim t↓T w(t).

An absolutely continuous (AC) function
Definition 2.4 (Rockafellar & Wets, 1998). The tangent cone to a set S ⊂ R n at a point ξ ∈ S, denoted by T S (ξ ), is the set of all vectors v ∈ R n for which there exist sequences {ξ i } i∈N ∈ S and

System description
In this section we consider the closed-loop system setup in Fig. 1, consisting of a linear time-invariant (LTI), single-input single-output (SISO) plant G interconnected with a (SISO) HIGS element H . The plant G contains the linear part of the closedloop system including the plant to be controlled and possibly an LTI controller, given by the state-space representation G : with states x g taking values in R ng , performance output e in R, control input v in R and exogenous disturbances and references denoted by w taking values in R nw . Moreover, the realization (A g , B gv , C g ) is assumed to be minimal. As our key area of application involves motion systems containing floating masses, the following assumption is typically satisfied.
Assumption 3.1. The LTI system G as in Fig. 1 is such that C g B gw = 0 and C g B gv = 0.
The HIGS element H has as its preferred mode of operation the linear integrator dynamicṡ where the state x h takes values in R, the (HIGS) input e and the (HIGS) output u both take values in R and ω h ∈ [0, ∞) denotes the integrator frequency. This mode of operation of the HIGS is referred to as the integrator mode. The integrator mode (2) can only be used as long as the input-output pair (e, u) of H remains inside where k h ∈ (0, ∞) denotes the gain parameter of H . Note that (e, u) ∈ F implies equal sign of the input e and the output u of the HIGS as eu ≥ 0, see Fig. 2. At moments when the input-output pair (e, u) of H tends to leave the sector F we will ''project'' the integrator dynamics in (2) such that (e, u) ∈ F remains true along the trajectories of the system. We will formalize this operation of the HIGS in the upcoming subsections.

Projection-based representation
To mathematically introduce the operation of the HIGS, we directly use the interconnection of the HIGS element H and the linear system G described by (1), resulting in a closed-loop system as in Fig. 1 where x g and x h are the states of G and H , respectively and thus n = n g + 1. Note that the constraint (e, u) ∈ F translates to x ∈ S with S = K ∪ −K, where K is a polyhedral cone given by . In fact, F 1 x = k h e − u and F 2 x = u such that (e, u) ∈ F if and only if x ∈ S. When H operates in the integrator mode, by combining (1) and (2) we obtain the state space representation for the HIGS-controlled system in Fig. 1, given bẏ where y = [e u] ⊤ , and As indicated above, when the state trajectory tends to leave the set S, which in terms of Definition 2.4 happens when for x(t) ∈ S, the vector field of (6), is altered by partial projection such that the resulting trajectory remains inside S. Using this perspective, we can formally introduce the HIGS-controlled system where for x ∈ S and f ∈ R n Π S,E (x, f ) := arg min with Π S,E : S × R n → R n an operator, which projects the dynamics f onto the tangent cone of the set S at point x, in the direction imE. In the case of (9), E = [0 ⊤ np×1 1] ⊤ such that the correction of the dynamics (6) is only possible for the dynamics of the HIGS and not for the (physical) plant dynamics (1), which can clearly not be modified (we cannot directly modifyẋ g ). Note that the projection operator Π S,E , is well-defined in the sense that it provides a unique outcome for every x ∈ S and each f ∈ R n , in the setting considered here (see . The model (10) resembles so-called projected dynamical systems (PDS) (Henry, 1973;Nagurney & Zhang, 2012) given bẏ where f : R n → R n is a general vector field and S ⊆ R n is a constraint set. Our representation (10) differs from (11) in two essential ways. First of all, we have partial projection of dynamics as a result of using the matrix E, the image of which specifies the direction of projection. It should be noted that the matrix E is not limited to the choice made in (9) and should be chosen depending on the specific case under consideration. Secondly, in the PDS literature, the PDS (11) is shown to be well-defined for constraint sets that satisfy certain regularity conditions. In particular, Henry (1973) and Nagurney and Zhang (2012) restrict the constraint sets to be convex, while in Hauswirth, Bolognani, and Dorfler (2021) (and some references therein) convexity is relaxed to Clarke regularity and prox-regularity of the constraint set for existence and uniqueness of Carathéodory solutions, respectively. However, the constraint set S considered here does not satisfy any of the above mentioned regularity requirements, cf. (4). In addition, note that (10) is a generalization of (11) since by taking imE = R n and restricting S to satisfy the required conditions, one recovers the classical PDS as in (11). In fact, for these reasons we refer to the class of systems (10) as extended projected dynamical systems (ePDS).
Remark 3.1. Note that we could extend the dynamics (9) that are currently defined for initial states x(0) ∈ S, such that they are also defined for x(0) / ∈ S. In case x(0) ̸ ∈ S, we can use to reset the state to a state inside S. Note that this reset only occurs at the initial time and not afterwards, as the state never leaves S for time t ∈ R >0 .

Remark 3.2.
It is easy to see that (9) satisfies This symmetry property will prove to be useful in Section 4, in showing well-posedness of the system.

Discontinuous PWL model
In this subsection we reformulate (9) as an equivalent piecewise linear (PWL) model. To explicitly compute (9), we first note that where (7), (12), and Assumption 3.1, we obtain that The proof of the statement above can be established by comparing the algebraic expressions of (12) and (13) for states lying in the interior of S where F 1 x > (<)0 and F 2 x > (<)0, and its boundaries where F 1 x = 0 or F 2 x = 0. Due to space limitations a complete proof is omitted here. As a result of the discussion above, S 1 is the region where the integrator mode of H is active. Moreover, when (8) holds. Based on (9) and (12) (with f = A 1 x + Bw and S as in (4)), when x ∈ S 2 , by solving (9), through manipulating (10) (see equation (17) in Sharif et al., 2019) and resorting to the Karush-Kuhn-Tucker (KKT) optimality conditions for constrained optimization (Boyd & Vandenberghe, 2004), we obtaiṅ We refer to (15) as the gain mode dynamics. By considering both modes of operation (given by (6) and (15)) and their corresponding regions, we obtain the explicit discontinuous PWL model for (9). Note that S 1 has a non-empty interior while S 2 does not (it is part of the lower-dimensional sub-space kerF 1 ). The matrices A 1 , B and, C have been explicitly computed in (7). We can also compute A 2 from (15) aṡ As a result of Assumption 3.1, this simplifies tȯ and thus for (16) we have Hence, (16) with (7) and (18) is an explicit PWL formulation of the HIGS controlled system in Fig. 1.

Remark 3.3.
As observed from the expressions of S 1 and S 2 the switching in (16) is based on where F 2 x = x h = u is the output of the HIGS element (input u to the linear system G in Fig. 1) and which is a function of e (output of the linear plant) and u (output of H ). Lastly, F 1 (A 1 x) = k hė − ω h e is a function ofė, the first derivative of the plant output, and the plant output e. Hence, the regions S 1 and S 2 can be fully described in terms of e,ė and u.
Indeed, one has x ∈ S 2 when (ė, e, u) ∈ F 2 , where where F is as defined in (3).
A graphical illustration of the regions F 1 and F 2 is provided in Fig. 3. As a result, an (internally) equivalent representation of (16) is given by Σ : We will use (9), (16), and (21) interchangeably.
From (16), we also see that we are dealing with a discontinuous differential equation, which makes proving (global) existence of solutions, given an initial state x 0 and external signal w, a difficult problem, since typical continuity properties used for studying differential equations/inclusions (such as upper-semicontinuity of the right-hand side cf. Aubin & Cellina, 1984) are not fulfilled, see also Cortes (2008).

Well-posedness analysis
In this section we show that the HIGS-controlled system (16) is well-posed in the sense of global existence of solutions. To this end, we first prove in Section 4.1 that (16) is locally well-posed, i.e., for each initial state x(0) ∈ S and exogenous signal of interest w, the system admits a solution on [0, ϵ] for some ϵ > 0. We select here the class of exogenous signals (disturbances, references, etc.) to be of piecewise Bohl (PB) nature (see Definition 2.2). Note that sines, cosines, exponentials, polynomials, and their sums are all Bohl functions, thereby showing that the class of PB functions is sufficiently rich to accurately describe (deterministic) disturbances frequently encountered in practice. In particular, any piecewise constant signal is PB, and thus this class of functions can approximate any measurable function arbitrarily closely. 1 Building on the local existence results of Section 4.1, in Section 4.2 we prove that all (maximal) solutions are forward complete, i.e., are defined for all times t ∈ R ≥0 .
To make this discussion precise, we will formalize the solution concept. (16) holds almost everywhere in T.
The solutions in Definition 4.1 are Carathéodory-type solutions, see also Cortes (2008) for more details regarding solution concepts for discontinuous dynamical systems.

Local well-posedness
Definition 4.2. We call the HIGS-controlled system (16) locally well-posed if for all x 0 ∈ S and w ∈ PB, there exists an ϵ > 0 such that the system admits a solution on [0, ϵ] with initial state x 0 and input w. Proof. Take x 0 ∈ S and w ∈PB. Without loss of generality we can take w ∈ B [0,ε] by selectingε > 0 sufficiently small. Hence, w can be represented as for some matrices H w ∈ R nw ×n Fw , F w ∈ R n Fw ×n Fw and a vector v w ∈ R n Fw . In other words, w is generated (on [0,ε]) by the exo-systeṁ Combining this exo-system with (16) yieldṡ as an equivalent description of (16) with w as in (22) 1 An interesting future research direction is establishing the existence of solutions for larger classes of input signals.
For proving local well-posedness of (24) (and thus of (16) with w as in (22)), we define the set In fact, sinceŜ int ⊆Ŝ 1 , we conclude that eÂ 1 tx 0 is also a solution to (24) on a non-trivial time window [0, ε] for some 0 < ε ≤ε. Next, we will also show that for eachx 0 ∈Ŝ \Ŝ int , a local solution to (24) exists, so that it is established that for all x 0 ∈Ŝ a local solution exists. In order to do so, we first rewrite (25) in a more algebraic form. Using the definition ofŜ, one can rewrite (25) aŝ By using the Taylor series expansion of eÂ 1 t together with the Cayley-Hamilton theorem, the characterizationŜ int which can be verified based on the expressions ofF 2 ,Â 1 , andF 1 .

Forward completeness
Hence, a maximal solution is a solution that cannot be prolonged (is not a strict prefix of another ''larger'' solution for the same input). (16) for w ∈ PB are forward complete.

Theorem 4.2. All maximal solutions to HIGS-controlled system
Proof. Consider a maximal solution x : T → R n of (16) for initial state x 0 ∈ S and w ∈ PB. We will show that if T is equal to [0, T ] or [0, T ) with T ∈ R ≥0 a finite number, the left-limit x(T ) := lim t↑T x(t) ∈ S exists, and we can exploit the local existence result to prolong x to a solution on [0, T + ε).
This would contradict the maximality of the solution and thus T = R ≥0 , hence x has to be forward complete.

To show the existence of lim t↑T x(t), let us remark that if
T is equal to [0, T ], the solution is AC on [0, T ] and thus the left-limit trivially exists. So, the exciting case to handle is [0, T ). By Definition 2.2, w can be represented on [t i , T ] as in (22) for some t i < T (in fact, t i is the largest value in the set {t k } k∈N , which is strictly smaller than T ). Thus, (16) can be equivalently written as (24) on [t i , T ]. This implies the existence of a constant M ∈ R such that the vector field of (24) satisfies the linear growth condition ∥ΠŜ ,Ê (x,Â 1x )∥ ≤ M∥x∥, for allx ∈ R n+n Fw , because ΠŜ ,Ê (x,Â 1x ) ∈ {Â 1x ,Â 2x }. As a result of (32), By applying Gronwall's Lemma (Khalil, 2002), one concludes that Hence, the solution x(t) is Lipschitz continuous on t ∈ [t i , T ], and thus also absolutely continuous and uniformly continuous. Thereby, the limitx(T ) := lim t↑Tx (t) exists, as required. □ Since we proved local existence of solutions and forward completeness of all maximal solutions, it is concluded that for each initial state x 0 ∈ S and w ∈ PB a global solution exists on [0, ∞) and all solutions can be extended to be defined on [0, ∞).

Time-domain stability analysis
In this section, we present a Lyapunov-based stability analysis for HIGS-controlled systems. In particular, an input-to-state stability (ISS) condition in terms of LMIs is proposed that guarantees the existence of a PWQ Lyapunov function (Johansson & Rantzer, 1998), in which a novel partitioning is used to reduce conservativeness.
Definition 5.1 ( 2 Khalil, 2002). The closed-loop system (16) is said to be input-to-state stable (ISS), if there exist a KL-function β and a K-function γ such that for any initial state x(0) ∈ S and any bounded w ∈ PB, any corresponding solution x : .

Three-dimensional partitioning
For the system given by (16) (or equivalently (21)), numerically tractable stability conditions can be formulated using LMIs. A piecewise quadratic Lyapunov function (Johansson & Rantzer, 1998) is pursued inspired by Aangenent et al. (2009) and Zaccarian et al. (2005Zaccarian et al. ( , 2011 for reset control systems. In the cited works, however, the flow set is partitioned only in the input-output plane of the reset element, as its nonlinear behavior is captured within this two-dimensional space. As explained in Remark 3.3, the HIGS' switching dynamics, by contrast, are determined in the (ė, e, u)-space, requiring a novel three-dimensional partitioning to reduce conservativeness. To ensure a partitioning of the (ė, e, u)-space such that the edges of (some of) the resulting regions exactly coincide with the boundaries of regions where different modes are active, first note that all boundaries of the regions F 1 and F 2 pass through the origin (see Remark 3.3 for analytic expressions of F 1 and F 2 ). Hence, a spherical coordinate system in the (ė, e, u)-space can be used to realize such a partitioning. That is, the azimuthal angle θ and polar angle φ can be used to divide this space into polyhedral (double) cones, see C ij in Fig. 4(a), which will be used to partition F 1 , the region where the integrator mode is active. For F 2 (the region where the gain mode is active), which is a subset of the plane u = k h e, a partitioning using the spherical coordinate system is possible using φ (and fixed θ), yielding regions such as depicted by T j in Fig. 4(b).
The construction of the N × M polyhedral cells is as follows. Define the N + 1 azimuthal angles and the M + 1 polar angles where φ M 1 = arctan(k h /(ω h cos(θ N ))). The angle θ N is chosen specifically such that it describes the sector boundary u = k h e. Similarly, the angle φ M 1 is defined such that the vector at angular coordinates (θ N , φ M 1 ) (red vector in Fig. 4(b)) coincides with the dynamics switching boundary at the intersection of the planes ω h e = k hė and the plane u = k h e, at which the HIGS switches back from gain mode to integrator mode.
Let the subsets C ij and T j , depicted in Fig. 4, that partition the regions where HIGS' integrator mode and gain mode are active, respectively, be given by where the inequalities in (35a) hold element-wise, and These matrices follow from the patching hyperplanes described where the latter is obtained by computing the cross product of the unit vectors with angular coordinates (θ i−1 , φ j ) and (θ i , φ j ), i.e., the two unit vectors that individually span the two region corners (successive in θ-direction) shared by the regions C ij and C i(j+1) , and together spanning the common boundary plane between these regions. Note that as a result of the symmetry of the system with respect to the origin of the (ė, e, u) space, the subsets C ij as in (35a) are defined such that if z ∈ C ij then −z ∈ C ij .
For the actual partitioning, let the index sets be defined as N = {1, 2, . . . , N} and M = M 1 ∪ M 2 , where M 1 = {1, . . . , M 1 } and M 2 = {M 1 + 1, . . . , M}. The closure of the integrator mode flow set F 1 = F can then be partitioned by C ij for i ∈ N , j ∈ M, and the gain mode flow set closure F 2 ⊂ F 1 by T j for j ∈ M 2 , i.e., Note that T j captures the boundary plane of C Nj where u = k h e, hence T j ⊂ C Nj .

LMI-based stability condition
In addition to N and M, defineÑ = N \{N} andM = M\{M}, and let S n and S n ≥0 denote the sets of n × n symmetric matrices, the latter consisting of nonnegative elements only. The following result then states a sufficient condition for ISS of a closed-loop system with HIGS as described by (16) (or (21)).
Theorem 5.1. If there exist matrices U ij , W ij ∈ S 4 ≥0 for i ∈ N , j ∈ M, and V j ∈ S 2 ≥0 for j ∈ M 2 such that P ij ∈ S n for i ∈ N , j ∈ M satisfy the LMIs wherê with C ij and T j from (36), arbitrary scalars v j,1 , v j,2 , v j,3 ∈ R,C as in (21a(i)) and whereΘ i,⊥ ,Φ ij,⊥ ∈ R n×(n−1) andΦ M,⊥ ∈ R n×(n−2) are matrices of full column rank such that im( respectively, then the system in (16) (or equivalently (21)) is ISS.
Proof. Since z =Cx (21a(i)), we can define the setŝ to describe the polytopic partitions in the state space of the closed-loop system. For this system, consider the radially un- To prove ISS, we will show that V is an appropriate ISS Lyapunov function. By (35a) and (46a) together with the nonnegativity of the elements in W ij ∈ S 4 ≥0 , it holds that As a result, using the S-procedure (41) implies that thereby ensuring positive definiteness of V . Next, using Finsler's lemma continuity of V is imposed over the boundary between two cells connected in azimuthal direction by (42) and in polar direction by (43). Since φ 0 = 0 and φ M = π, we also require (44) to ensure continuity of V over the boundary between the first and last regions in polar direction. Note that the former two constraints ensure continuity over hyperplanes of dimension n − 1, whereas the latter does so in only n − 2 dimensions. This is due to the fact that in (ė, e, u)-space, all region boundaries are double-conical subsets of planes, except for the boundary line between the first and Mth region in φ-direction, i.e., on the intersection C i1 ∩ C iM . In fact, since φ 0 = 0 and φ M = π this boundary coincides with theė-axis, i.e., at e = u = 0, from which it is easily seen that this leaves only n − 2 directions in the state space of the closed-loop system in which continuity over this boundary must be ensured. Hence, V is a locally Lipschitz continuous function.
Inspecting the time derivative of V in integrator mode, note that due to (35a), (46a), and U ij ∈ S 4 ≥0 it holds that which via the S-procedure ensures that (39) yields if x ∈Ĉ ij , for some ε ij,1 > 0 and (i, j) ∈ N × M by strictness of the matrix inequalities. For the gain mode, let us first observe that (35b), (46b), and V j ∈ S 2 ≥0 , imply forV j as in (45) with arbitrary v j,1 , v j,2 , v j,3 ∈ R. Simultaneously employing the S-procedure and Finsler's lemma results in (40) to ensure if x ∈T j , with ε Nj,2 > 0 and j ∈ M 2 by strictness of the LMIs in (40). Combining (50) and (52) and applying Young's inequality yields almost everywhere the upper bound on the time derivative of V over both modeṡ with constants ε = min (i,j,q) ε ij,q − 1 δ > 0 and ρ = δ max (i,j) ∥P ij B∥, where (i, j, q) for sufficiently large δ > 0. Thus, V is an appropriate PWQ ISS Lyapunov function by which the closed-loop system with HIGS is ISS. □ Remark 5.1. The constraints (42) and (43) are sufficient conditions for continuity of V , as they in fact demand continuity over the entire (n − 1)-dimensional hyperplanes rather than only over the angularly bounded subset of such a hyperplane shared by two neighboring partitions. By contrast, (44) is a necessary and sufficient condition, requiring continuity of V only where it is truly needed. Moreover, for (44) one may remark that it would suffice to only impose this condition for any single i ∈ N , provided that (42) is satisfied, since theė-axis is a common boundary for all regionsĈ i1 andĈ iM , i ∈ N . In fact, the HIGS' dynamics are such that every crossing of this boundary, i.e., every zero crossing of e, leads to trajectories traveling fromT M intô C 11 (except for trajectories traveling through z = 0, for which continuity of V is already guaranteed by both (42) and (43)), and hence it would be sensible to require (44) only for i = N.

Discussion
The main strength of the proposed LMI-based conditions is that the discontinuous PWL dynamics are explicitly incorporated, and that the flexibility of a PWQ Lyapunov function is used to reduce conservativeness. Moreover, since the conditions pose a convex optimization problem, they can efficiently be solved by numerical algorithms. Furthermore, this LMI-based approach is general in nature in the sense that it makes no restrictive demands on G (only Assumption 3.1 is required), as opposed to the approach presented in Section 6, which requires stability of the linear system G . Hence, the LMI-based approach is in principle applicable to any HIGS-controlled (motion) system that can be written in the form (16). However, being an LMI-based stability analysis, it has two aspects that might be experienced as less desirable. First, it requires an accurate parametric state-space model, which for high-precision industrial (motion) systems may not be straightforward to obtain. Second, if infeasible, the evaluated conditions provide no direction to the control engineer on how to (re)design the controller or how to guarantee robustness margins.

Frequency-domain conditions
In this section we discuss circle-criterion-like conditions used in Deenen et al. (2017), which similarly to the circle criterion (Khalil, 2002) exploits the sector-boundedness of the HIGS' input-output behavior to enable nonparametric stability analysis in the frequency domain.

Circle-criterion-like condition
Similar to the classical circle criterion, we inspect the frequency response function of G that connects the HIGS' input and output in the loop of Fig. 1, given by in relation to the HIGS' input-output sector. The following theorem states the sufficient condition for ISS.
Theorem 6.1. Consider the system in Fig. 1 described by (16) with fixed ω h ∈ R >0 and k h ∈ R >0 . This system is ISS in the sense of Definition 5.1 if the following conditions are satisfied: (I) The system matrix A g of (1) is Hurwitz; (II) The transfer function G ev (s) as in (54) Proof. The proof is based on modifications of the circle criterion as proposed in van Loon et al. (2017). Different from van Loon et al. (2017), however, is the absence of an explicit additional detectability condition when considering the scalar-state HIGS, and the fact that, besides for the integrator dynamics, the Lyapunov function must be proven to decrease for an additional set of flow dynamics resulting from the gain mode. The proof is divided into the following steps: (1) Initially, the internal dynamics of H are disregarded. Using the circle criterion, the sector-boundedness of its input-output pair (e, u) is exploited to prove ISS of G with respect to w by construction of a quadratic ISS Lyapunov function (Sontag, 1995) V g via the Kalman-Yakubovich-Popov (KYP) lemma (Khalil, 2002). (2) A quadratic Lyapunov-like function V h is constructed for the HIGS in isolation, and an upper bound on its time derivative is found through explicit use of the sector condition and mode constraints, showing that the hybrid element is a (state) strictly passive system. (3) The functions V g and V h constructed in the previous two steps are combined into a (common) quadratic ISS Lyapunov function V c for the closed-loop system including the HIGS to prove the theorem.
Step 1: The KYP Lemma (Khalil, 2002), shows that the conditions (I) and (II) and minimality of (A g , B gv , C g ) imply the existence of a positive definite matrix P g ∈ S ng , a matrix L, and a positive constant ε g that satisfy Hence, the Lyapunov function V g (x g ) = x ⊤ g P g x g satisfies λ(P g )∥x g ∥ 2 ≤ V g (x g ) ≤ λ(P g )∥x g ∥ 2 , where λ(P g ) and λ(P g ) denote the minimum and maximum eigenvalues of the matrix P g ≻ 0, respectively. Following the derivation in Step 1 of the proof of Theorem 6 in van Loon et al. (2017), in which the sector condition eu ≥ 1 k h u 2 from (3) is explicitly used twice, we find that the time derivative of V g along solutions of (1) satisfies almost everywherė where c 1 = ε g λ(P g ) − 1 δ 1 > 0 for sufficiently large δ 1 , and c 2 = δ 1 (λ(P g )∥B gw ∥) 2 > 0, from which we conclude that V g is indeed an ISS Lyapunov function for G with respect to w.
Step 2: Consider the quadratic Lyapunov function where c 3 = 1−δ 2 ω h > 0 with 0 < δ 2 < 1, for the isolated HIGS. In integrator mode (2), the corresponding time derivative is given bẏ which using u = x h and the sector condition u 2 ≤ k h eu from (3) can be rewritten aṡ where c 4 = δ 2 k h > 0 such that the last equality holds. Note that (60) shows that the integrator mode of the isolated HIGS is (state) strictly passive (Khalil, 2002). For the gain mode, let us first note that substitution of u = k h e into the quadratic condition ω h e 2 > k hė e in (19) yields We consecutively use (2), (61), and the gain mode constraint u = k h e from (19) to rewrite the time derivative of V h in gain mode aṡ which is equal to (60), and thereby denotes the uniform upper bound onV h (almost everywhere) over both the integrator and gain mode, showing strict passivity of H . Employing Young's inequality and ∥u∥ ≤ k h ∥e∥ from (3), we find almost everywherė where (1) gives almost everywherė where c 6 = c 5 ∥C g ∥ 2 > 0.
Step 3: For the closed-loop system consisting of the interconnection of (16) (or equivalently (21)) as depicted in Fig. 1 with state where P c = , and 0 < µ < c 1 c 6 . V c is positive definite and radially unbounded according to An upper bound for the time derivative is given bẏ almost everywhere, with ε c = min(c 1 − µc 6 , µc 4 ) > 0 and ρ c = c 2 > 0.
Consequently, we conclude that the system is ISS in the sense of Definition 5.1. □

Discussion
Condition (I) of Theorem 6.1 restricts the theorem's applicability to stable systems G . For motion systems with floating masses, i.e., where the transfer function of the open-loop system contains poles at s = 0, this means that G ev (s) may only describe an already stabilized closed-loop system. Another weakness of this approach with respect to Theorem 5.1 is the fact that this approach potentially yields a conservative estimate on stability for two main reasons. First, the Lyapunov function that underlies the circle criterion is a common quadratic one. Second, the actual nonlinear behavior of the HIGS is not taken into account, but instead condition (II) only considers the sector following from the HIGS gain k h . In fact, from the Nyquist diagram of the linear system k h G ev we find that condition (II) directly leads to k h,cc ≤ k h,gain (which becomes an equality in case the largest negative real part of G ev is on the real axis), where k h,cc denotes the smallest upper bound on the gain k h that satisfies condition (II) of Theorem 6.1, and k h,gain denotes the supremum of the gain value by which the closed loop G ev /(1 + k h G ev ) is stable. In other words, using Theorem 6.1 one can only guarantee stability for gain values k h that also render the individual gain mode subsystem stable.
Another indication of the circle-criterion-like approach being more conservative than the LMI-based conditions is found by the inequality in (67), which implies that any Lyapunov function V c (x) = x ⊤ P c x that follows from satisfying the conditions in Theorem 6.1 also satisfies the conditions (39)-(41) of Theorem 5.1 with N = M = 1 and P 11 = P c (rendering the continuity constraints trivial), and with U 11 , V 1 , and W 11 being zero matrices. Conversely, satisfying the conditions in Theorem 5.1 need not imply that the conditions in Theorem 6.1 are satisfied, hinting that the former may possibly guarantee stability in cases where the latter cannot.
The main strength of the circle-criterion-like approach lies in its convenience of application and compatibility with current industrial practice for nonparametric frequency-domain stability evaluation during the controller design process. That is, classical linear loopshaping techniques can be used to design a linear controller contained in L that stabilizes G ev , thereby satisfying condition (I). Also, for typical motion systems and as formalized by Assumption 3.1, it holds that G ev (jω) → 0 for ω → ∞, implying (55) of condition (II) is satisfied. Thus, only (56) of condition (II) remains to be verified, which boils down to a graphical evaluation of (measured) frequency response data of G ev with respect to the design parameter k h in a Nyquist diagram. As such, this provides the control engineer with frequency-specific information regarding violations and robustness margins toward satisfying condition (II), which can directly be used for controller redesign.
Remark 6.1. Let us briefly elaborate on robustness issues of a discontinuous differential equatioṅ with x(t) ∈ R n , w(t) ∈ R nw , and X ⊆ R n , just as our HIGS-controlled system (9). As discussed in Goebel, Sanfelice, and Teel (2012), in order to obtain robust (with respect to arbitrary small state perturbations) stability guarantees, it is important to consider the Krasovskii regularization of (68), defined aṡ where B(x(t), δ) is the open ball of radius δ around x(t) and for a set Ω ⊆ R n , co(Ω) denotes its closed convex hull and X denotes the closure of X . In case of (16), its Krasovskii regularization is given aṡ It can be shown that Theorems 5.1 and 6.1 hold for (69) as well, and thus stronger stability guarantees for (16) can be obtained including state perturbations (see Goebel et al., 2012 for more details).

Illustrative example
In this section, we demonstrate how the two stability analysis approaches of Sections 5 and 6 can be used to evaluate stability of a typical control system including a HIGS. The reader interested in time and frequency domain simulations/experiments of successful applications of HIGS-based control, is referred to Deenen et al. (2017), van den Eijnden, Heertjes, and Nijmeijer (2019) and Heertjes, van den Eijnden, Sharif, Heemels, and Nijmeijer (2019). Moreover note that in the recent work (van den Eijnden, Heertjes, Heemels, & Nijmeijer, 2020) it is shown how HIGS-based controllers overcome fundamental limitations of LTI control.

System description
Consider the SISO motion control tracking problem depicted in Fig. 5, where the LTI open-loop system L represents the series interconnection of a single-mass plant P and the stabilizing (nominal) linear controller C consisting of a PD-controller and first-order lowpass filter. The corresponding Laplace transforms are given by with mass m = 1 kg, PD-controller parameters k p = 1 N/m and k d = 0.2 Ns/m, and lowpass corner frequency ω lp = 7 rad/s, resulting in a bandwith of 1 rad/s. The plant output y l ∈ R is the position of the mass, which must track the reference r ∈ R. Using an appropriate (non-causal) feedforward signal u f the effects of r are assumed to be fully canceled in the closed loop. As a result, the tracking error is given by e = −y l , and the exogenous input d ∈ R represents only the disturbance effects. Furthermore, corresponding to Fig. 1, the linear part G (red dashed box) representing the baseline linear control system is in feedback with the HIGS H , the latter generating the control signal u = −v ∈ R. In particular, the transfer function G ev (s) is given by in which we recognize the complementary sensitivity function, which similarly to L (s) has relative degree two, thereby satisfying Assumption 3.1.

LMI-based stability analysis
Evaluation of the conditions in Theorem 5.1 requires a state-space representation of G as in (1). To this end, consider first the state-space model of the open-loop system L given by L : with states denote the controller and plant states, respectively, and corresponding matrices To obtain a state-space description of G in the form of (1), we close the loop using e = −y l and u = −v, resulting in G : where additionally x g = x l is substituted as no pole-zero cancellation is found to occur by closing the loop. An extended closed-loop state-space representationG follows from augmenting the output in (74) toẽ = [ė e] ⊤ , which using the fact that C l B l = 0 (by Assumption 3.1) results in the matrices with which we can construct closed-loop system (21). To demonstrate the added benefit of higher-dimensional partitioning of the state space, we evaluate stability using three different sets of LMI conditions: (1) Without partitioning, which results in a common quadratic Lyapunov function. The corresponding LMIs are given by (39)-(41) for N = M = 1.
(3) Volumetric partitioning using all conditions stated in Theorem 5.1, implying the existence of a three-dimensionally PWQ Lyapunov function. For fair comparison, we choose the same number of regions N = 10 in the azimuthal direction as in the previous case, but now also use M 1 = 3 and M 2 = 2 to partition the state space in polar direction.
The LMI-based conditions are solved using the YALMIP toolbox (Lofberg, 2004) with SDPT3 (Tutuncu, Toh, & Todd, 2003) in MATLAB. In the implementation, the right-hand sides of (39) and (40) are tightened to −ϵ 1 I, and the right-hand side of (41) is set to ϵ 1 I, where ϵ 1 = 10 −3 is sufficiently large with respect to the machine precision such that the resulting LMIs may be solved in a non-strict manner for solver compatibility. Furthermore, similar to Remark 4 in Zaccarian et al. (2011), the equality constraints (42)-(44) are replaced by auxiliary inequality constraints with a small tolerance of ϵ 2 = 10 −8 to reduce numerical problems. Moreover, a balancing state transformationx g = T g x g with T g ∈ R ng ×ng is applied to the linear system with matrices (75) to improve numerical conditioning. Stability is evaluated using Theorem 5.1 for a grid of HIGS parameter values (k h , ω h ), the results of which are shown in Fig. 7 and will be discussed in more detail in Section 7.4.

Frequency-domain stability analysis
To evaluate stability of the closed-loop system using the circle-criterion-like approach, we simply inspect the complementary sensitivity function (71). In verifying the conditions of Theorem 6.1, we first observe that condition (I) is satisfied by design of the nominal controller C , as can also be seen from Fig. 6(a) using the Nyquist criterion. Next, since G ev (s) has a relative degree of two, it holds that G ev (jω) → 0 as ω → ∞, meaning (55) in condition (II) is satisfied for any k h > 0. For (56) in condition (II), we inspect the Nyquist diagram of (71) shown in Fig. 6(b), from which it follows that the closed-loop system including HIGS is guaranteed to be ISS by Theorem 6.1 for any ω h ∈ (0, ∞) and k h < k h,cc = 0.12, as indicated in Fig. 7 by the region to the left of dashed red line.

Comparison
For a grid of HIGS parameters (k h , ω h ), Fig. 7 visualizes the range of parameter values for which the different approaches are able to guarantee closed-loop stability of the HIGS-controlled example system (70). In addition, the figure shows the parameter range for which the system has been concluded to be stable on the basis of time-series simulation studies. Fig. 7 illustrates the conservativeness associated with this frequency-based approach, which can only guarantee stability for HIGS gains up to k h,cc = 0.12 (dashed red line). This conservativeness is thought to stem partly from inadequately accounting for the true nonlinear dynamics determined by the parameter pair (k h , ω h ), and instead using an upper bound based only on k h , which is evident in Fig. 7 as the stable parameter region according to Theorem 6.1 does not depend on ω h .
The LMI-based approach without partitioning, on the contrary, does explicitly include the nonlinear closed-loop dynamics. Consequently, the corresponding range of verifiably stable parameter values (black dots) increases with respect to the circle criterion, see Fig. 7(a). Comparing to the simulation-based stable region (gray), however, a considerable degree of conservativeness remains due to the LMIs demanding the existence of a common Lyapunov function. In Fig. 7(b), it is shown that the planar partitioning as described by Theorem 5.1 for M = 1 is able to partly alleviate this problem, resulting in a significantly larger set of parameters for which stability can be guaranteed. For parameter pairs close to the edge of the stable region, also this approach is consistently unable to guarantee stability. Finally, Fig. 7(c) illustrates the potential of the conditions in Theorem 5.1 in terms of reducing the conservativeness by extending the partitioning to three dimensions. The resulting range of (k h , ω h )-values for which closed-loop stability can be concluded on the basis of LMI conditions approximately coincides with the stable parameter region found by time-series simulation . Clearly, the LMI-based analysis outperforms the circle-criterion-like approach in terms of conservativeness. Nevertheless, the latter may sometimes be the preferable option due to its conditions being easier to verify. This is especially true in practice, in case accurate state-space models such as (72) are not available. The other potential drawbacks of the LMI-based approach are caused by the conditions being evaluated qualitatively, numerically, and for only a single parameter pair (k h , ω h ) at a time. Moreover, using numerical solvers may cause sensitivity to numerical inaccuracies, especially those related to the continuity constraints (42)-(44). In particular, if ϵ 2 is chosen too small, the solver may be unable to find a feasible solution for some values (k h , ω h ) within the stable region, while for ϵ 2 too large, increasingly many false positive conclusions on stability occur outside the stable region. Moreover, the suitable tolerance values may depend on the number of partitions N, M 1 , and M 2 , which in turn affects the number of LMIs and thereby the required solver time. The process of finding a suitable partitioning and corresponding tolerance values, combined with the fact that each value of the pair (k h , ω h ) must be evaluated individually, renders this approach more time-consuming compared to the circle-criterion-like analysis.
To reduce the numerical sensitivity of the LMI-based approach in this example, we tuned the PD-controller with reduced robustness margins, allowing for a fair comparison of the conservativeness of the different approaches (i.e., independent of numerical issues).

Conclusion
In this paper, we have introduced the formalization of the hybrid integrator-gain system (HIGS). The HIGS is a nonlinear integrator that projects its dynamics onto a sector, thereby keeping the sign of its input and output the same while maintaining a continuous control signal. We have presented an appropriate mathematical framework for the formal description of HIGS-controlled systems based on generalizations of projected dynamical systems, called extended projected dynamical systems (ePDS), which naturally describes the main design philosophy behind the HIGS. The ePDS framework was used in showing the fundamental property of well-posedness of HIGS-controlled systems in the sense of existence and forward completeness of solutions, thereby laying down a mathematical framework for formal studies of HIGS-based controllers. Moreover, two approaches for analyzing closed-loop stability of a motion system including a HIGS have been presented. The first involves LMI-based conditions that guarantee ISS of the closed-loop system via a PWQ Lyapunov function. Its main strength lies in a novel three-dimensional partitioning of the state space specifically tailored to the HIGS' dynamics, which reduces conservativeness of the conditions to a degree similar to what would be expected from a necessary condition on closed-loop stability. The second approach involves a circle-criterion-like analysis. Although potentially more conservative and only applicable to certain (common) feedback configurations, this approach allows for a nonparametric frequency-domain evaluation of input-to-state stability of the closed-loop system including HIGS. Both methods have been demonstrated on a motion system. Since their strengths and weaknesses are largely complementary, together they form a powerful set of tools for the stability analysis of HIGS-controlled systems.