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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2020.2996985, IEEETransactions on Automatic Control

Observer-Based Controllers for IncrementallyQuadratic Nonlinear Systems with Disturbances

Xiangru Xu, Behcet Acıkmese, Martin J. Corless

Abstract—Robust global stabilization of nonlinear systems byobserver-based feedback controllers is a challenging task. Thispaper investigates the problem of designing observer-based stabi-lizing controllers for incrementally quadratic nonlinear systemswith external disturbances. The nonlinearities considered in thesystem model satisfy the incremental quadratic constraints, whichare characterized by incremental multiplier matrices and en-compass many common nonlinearities. The simultaneous searchfor the observer and the controller gain matrices is formulatedas a feasibility problem of linear matrix inequalities, for twoparameterizations (i.e., the block diagonal parameterization andthe block anti-triangular parameterization) of the incrementalmultiplier matrices, respectively. The closed-loop system imple-menting the observer-based feedback controller is proven to beinput-to-state stable with respect to external disturbances. Usingthe proposed continuous-time observer-based controllers, event-triggered controllers with time regularization are constructed forglobally Lipschitz systems, such that the closed-loop system isZeno-free and input-to-state practically stable.

Index Terms—Incrementally quadratic nonlinearity, Observer-based control, Stabilization of nonlinear systems, Event-triggeredcontrol

I. INTRODUCTION

As the state variables of a system are difficult or expen-sive to measure in practice, output feedback control designhas received a lot of attention (see, e.g., [1], [2], [3], [4])and found applications in biological systems [5], [6], [7],mechanical systems [8], [9], [10], power systems [11], [12],and networked control systems [13], [14], among others.For linear systems, the output feedback stabilizing controldesign problem can be solved by designing the state-feedbackcontroller and the state observer independently, which isknown as the controller-observer separation principle. Fornonlinear systems, a certainty-equivalence implementation ofa globally stabilizing state-feedback controller with an asymp-totic observer can lead to finite escape time (e.g., see thecounter-examples in [15], [16]), which makes observer-basedstabilizing controller design a challenging problem [17]. Byusing a high-gain observer [18], [19], separation principlesfor input-output linearizable systems were studied in [20],[21], [22], [23] and semiglobal asymptotic stability of theresulting closed-loop systems was proven in these papers.

This research was funded in part by ONR grants N00014-16-1-3144 andN00014-17-1-2623. (Corresponding author: Xiangru Xu.)

X. Xu is with the Department of Mechanical Engineering,University of Wisconsin-Madison, Madison, WI, USA (email:[email protected]).

B. Acıkmese is with the Department of Aeronautics & Astronautics,University of Washington, Seattle, WA, USA (email: [email protected]).

M. J. Corless is with the School of Aeronautics & Astronautics, PurdueUniversity, W. Lafayette, IN, USA (email: [email protected]).

Separation principles for some other special class of nonlinearsystems were also investigated, such as bilinear systems [24],[25], non-affine nonlinear systems [26], systems with non-decreasing or slope-restricted nonlinearities [27], [28], [29],and cascaded systems [30], [31]. Apart from the certainty-equivalence approach, interdependent design of the controllerand the observer was investigated in [32], [33], [34].

Linear matrix inequalities (LMIs) provide a computationallyefficient approach for the synthesis of observer-based outputfeedback controllers [35], where the main difficulty lies in thecoupling between the unknown matrices of the observer andthe controller and the Lyapunov matrices. For linear systems,LMI-based conditions were proposed for the robust observer-based stabilization of linear systems with state perturbations[36] or with parametric uncertainties [37], [38]. For nonlinearsystems, the synthesis problem is often formulated as the fea-sibility of bilinear matrix inequalities (BMIs), which is knownto be an NP-hard problem [39]. Different approaches thataim to transform the non-convex BMI conditions to convexLMI conditions have been proposed: [38] studied observer-based controller design for Lipschitz nonlinear systems withuncertain parameters, and developed an LMI-based designtechnique that relies on the linearization of the correspondingBMIs; [40] investigated observer-based control design for theinterconnection of a linear system and an uncertain nonlinearoperator satisfying the integral quadratic constraint, and pro-posed a sequential LMI algorithm to solve BMIs; [41] studiedoutput feedback control of discrete-time parametric uncertainLure systems, and developed an LMI-based iterative algorithmto solve BMIs. Moreover, [42] investigated H∞ stabilizationof discrete-time globally Lipschitz nonlinear systems, andprovided LMI-based conditions that compute simultaneouslythe observer and controller gains; [43] considered asymptoticstabilization of continuous-time Lipschitz nonlinear systemsand developed LMI-based conditions that synthesize the gainmatrices of the observer-based controller.

This paper considers observer-based output feedback globalstabilization of a class of nonlinear systems whose nonlin-earities satisfy incremental quadratic constraints. The incre-mental quadratic constraint is characterized by an incremen-tal quadratic inequality with incremental multiplier matrices[44], [45], [46], [47], [48], [49]. This characterization withincremental multiplier matrices provides a general frameworkto represent many common classes of nonlinearities (e.g.,globally Lipschitz nonlinearities, incrementally sector boundednonlinearities, non-decreasing nonlinearities and the polytopicJacobian nonlinearities), implying a wide range of applicabil-ity for the proposed theoretical results. Observer design for

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systems with nonlinearities satisfying incremental quadraticconstraints was studied in [48], which was later generalizedto the systems with bounded exogenous disturbances in [50].Observer-based control design for some special classes ofincrementally quadratic nonlinear systems have been inves-tigated in [28], [43], [51], [52], [53].

Motivated by the development of networked control sys-tems, event-triggered control (ETC) has recently received a lotof attention as it provides a new control paradigm to reducethe resource consumption of networked control systems whosecommunication bandwidth and computational power are usu-ally limited [54], [55], [56]. Most of the ETC results assumethat full-state information is available, but this assumptionis restrictive since many systems only have information ontheir measured outputs. Extending results on observer-based,event-triggered control design from linear systems (e.g., see[57], [58], [59]) to nonlinear systems is difficult [60]. Existingresults on ETC design for nonlinear systems mostly assumethat the continuous-time observer-based controllers are alreadygiven, but the observers and controllers themselves can behard to construct. When external disturbances or measurementnoise are present, the triggering rules also need to be carefullydesigned to rule out the Zeno phenomenon [60], [61], [62],e.g., using time regularization to enforce a built-in lower boundfor inter-execution times.

The main contributions of the paper are summarized asfollows. For incrementally quadratic nonlinear systems af-fected by external disturbances and measurement noise, LMI-based sufficient conditions are developed for the design ofrobust stabilizing observer-based controllers. The simultaneoussearch for the observer and the controller gain matrices is for-mulated as a feasibility problem of LMIs when the incrementalmultiplier matrices are parameterized as the block diagonalmatrices or the block anti-triangular matrices. The resultingclosed-loop system is proven to be input-to-state stable withrespect to disturbances. Using the proposed continuous-timeobserver-based controller, event-triggered controllers are con-structed for globally Lipschitz systems affected by externaldisturbances and measurement noise where the triggering ruleis designed with an enforced positive lower-bound on inter-execution times. The resulting closed-loop system is Zeno-freeand input-to-state practically stable with respect to externaldisturbances. A preliminary version of this work appeared in[63]. The present paper is different from [63] in the followingimportant ways: the system model considered is subject to ex-ternal disturbances and measurement noise; the event-triggeredmechanism is considered; complete proofs are included, andmore discussion is added. The remainder of the paper isorganized as follows: Section II introduces preliminaries onincremental quadratic constraints and input-to-state practicalstability, Section III develops LMI-based conditions for thedesign of robust stabilizing observer-based controllers for twoparameterizations of the incremental multiplier matrices, Sec-tion IV presents the event-triggered controller design, SectionV provides a simulation example, and Section VI provides theconclusions.

Notation. R+0 denotes the set of non-negative real numbers;

‖x‖ denotes the 2-norm of a vector x; ‖P‖ denotes the maxi-

mum singular value of a matrix P ; λm(P ) and λM (P ) denotethe minimum and maximum eigenvalues of a symmmetricmatrix P , respectively; In denotes an identity matrix of sizen; 0n1×n2

and 0n denote the zero matrix of size n1×n2 andthe zero vector of size n, respectively, where the subscript willbe omitted when clear from context. For symmetric matrices,∗ denotes entries whose values follow from symmetry. Fora matrix M , M � 0, M � 0, M ≺ 0, M � 0 mean Mis positive definite, positive semi-definite, negative definite,and negative semi-definite, respectively. A continuous functionf : R+

0 → R+0 belongs to class K (denoted as f ∈ K) if it

is strictly increasing and f(0) = 0; f belongs to class K∞(denoted as f ∈ K∞) if f ∈ K and f(r) → ∞ as r → ∞.A continuous function f : R+

0 × R+0 → R+

0 belongs to classKL (denoted as f ∈ KL) if for each fixed s, the functionf(·, s) ∈ K∞ and for each each fixed r, the function f(r, ·)is decreasing and f(r, s)→ 0 as s→ 0.

II. PRELIMINARIES

Consider the following nonlinear systemx = Ax+Bu+ Ep(q) + Eww,

y = Cx+Du+ Fww,

q = Cqx,

(1)

where x ∈ Rnx is the state, u ∈ Rnu is the control input,y ∈ Rny is the measured output, p : Rnq → Rnp is the knownnonlinearity of the system, w ∈ Rnw is the unknown externaldisturbance or measurement noise, and A ∈ Rnx×nx , B ∈Rnx×nu , C ∈ Rny×nx , D ∈ Rny×nu , Cq ∈ Rnq×nx , E ∈Rnx×np , Ew ∈ Rnx×nw , Fw ∈ Rny×nw are constant matricesof appropriate dimensions.

The characterization of the nonlinearity p is based onincremental multiplier matrices [48], [49].

Definition 1. Given a function p : Rnq → Rnp , a symmetricmatrix M ∈ R(nq+np)×(nq+np) is called an incrementalmultiplier matrix (δ-MM) for p if it satisfies the followingincremental quadratic constraint for all q1, q2 ∈ Rnq :(

δqδp

)>M

(δqδp

)≥ 0 (2)

where δq = q2 − q1, δp = p(q2)− p(q1).

For a given nonlinearity p, its δ-MM is not unique. DenoteM as the set of incremental multiplier matrices for p. If M ∈M, then λM ∈M for any λ ≥ 0.

Remark 1. The global Lipschitz condition ‖p(q2)−p(q1)‖ ≤γ‖q2 − q1‖ where γ > 0 can be expressed in the form of (2)with

M =

(γ2I 00 −I

). (3)

The incrementally sector bounded nonlinearity (δp −K1δq)

>S(δp −K2δq) ≤ 0 where S = S> can be expressedin the form of (2) with

M =

(−K>1 SK2 −K>2 SK1 ∗

S(K1 +K2) −2S

). (4)




The nondecreasing nonlinearity, which satisfies δp>δq ≥ 0,can be expressed in the form of (2) with

M =

(0 II 0

). (5)

Refer to [48], [49] for some other nonlinearities that can beexpressed using the incremental quadratic constraint.

Next, we introduce input-to-state practical stability andits characterization using Lyapunov functions. Consider thesystem

x = f(x, u) (6)

where f : Rnx × Rnu → Rnx is a locally Lipschitz functionand u : R → Rnu is a measurable essentially bounded input.Define x(t, x0, u) as the solution of (6) with initial state x0and input u, which satisfies x(0, x0, u) = x0.

Definition 2. (Def. 2.1 of [64]) The system (6) is called input-to-state practically stable (ISpS) w.r.t. u, if there exist functionsβ1 ∈ KL, β2 ∈ K and a non-negative constant d such thatfor every initial state x0 and every measurable essentiallybounded u defined on [0,∞), the solution x(t, x0, u) existson [0,∞) and satisfies

‖x(t, x0, u)‖ ≤ β1(‖x0‖, t) + β2(‖u‖∞) + d, ∀t ≥ 0 (7)

where ‖u‖∞ := ess supt≥0‖u(t)‖.

When (7) is satisfied with d = 0, the system is said to beinput-to-state stable (ISS) w.r.t. u [65].

Definition 3. (Remark 2.2 of [64]) A smooth function V :Rn → R is said to be an ISpS-Lyapunov function for thesystem (6) if V is radially unbounded, positive definite andthere exist functions γ ∈ K∞, χ ∈ K and a non-negativeconstant d such that the following condition holds:

∇V (x)T f(x, u) ≤ −γ(‖x‖) + χ(‖u‖) + d. (8)

Instead of requiring inequality (8), the ISpS-Lyapunov func-tion can be also defined equivalently as follows: a smooth,positive definite, radially unbounded function V is an ISpS-Lyapunov function for the system (6) if there exist a positive-definite function γ, a class K function χ and a non-negativeconstant d such that the following condition holds (Def. 2.2of [64]):

‖x‖ ≥ χ(‖u‖) + d ⇒ ∇V (x)T f(x, u) ≤ −γ(‖x‖). (9)

The existence of an ISpS-Lyapunov function is a necessaryand sufficient condition for the ISpS property.

Proposition 1. [66] The system (6) is ISpS (resp. ISS) if andonly if it has an ISpS- (resp. ISS-) Lyapunov function.

In particular, if there exist a symmetric and positive definitematrix P = P> � 0, two constants α > 0, d ≥ 0 anda function χ ∈ K∞ such that the positive definite functionV (x) = x>Px satisfies

∇V (x)T f(x, u) ≤ −αV (x) + χ(‖u‖) + d, (10)

then V is an ISpS-Lyapunov function satisfying (8) with

γ(‖x‖) = αλm(P )‖x‖2, implying that (6) is ISpS w.r.t. u.

III. LMI-BASED CONDITIONS FOR ROBUST GLOBALSTABILIZATION OF INCREMENTALLY QUADRATIC

NONLINEAR SYSTEMS

Consider a system described by (1) where the nonlinearterm p satisfies the incremental quadratic constraint (2) forsome M ∈ M. In this section, a continuous-time observerand a feedback controller will be designed for (1), such thatthe closed-loop system is ISS w.r.t. w. LMI-based sufficientconditions will be given for the simultaneous design of theobserver and controller gain matrices.

The following observer is proposed:˙x=Ax+Bu+ Ep(q + L1(y − y)) + L2(y − y),

y=Cx+Du,

q=Cqx,

(11)

where L1, L2 are gain matrices to be designed. This observercontains a copy of the plant and two correction terms, thenonlinear injection term L1(y − y) and the Luenberger-typecorrection term L2(y−y). Based on observer (11), we designthe feedback controller u as

u = k(x) (12)

where k : Rnx → Rnu is a function that has the form of

k(x) = K1x+K2p(Cqx) (13)

with gain matrices K1 ∈ Rnu×nx , K2 ∈ Rnu×np to bedesigned. Defining the estimation error by

e(t) = x(t)− x(t)

the input (12) can be rewritten as

u = k(x)−∆k(x, x)

where ∆k(x, x) = k(x) − k(x). Recalling (13), ∆k can beexpressed as ∆k = K1e−K2∆p where

∆p = p(q)− p(q). (14)

The closed-loop system resulting from the observer-basedcontroller (12) can now be expressed as{

x=(A+BK1)x+(E+BK2)p−B∆k+Eww,

e=(A+L2C)e−Eδp+(Ew+L2Fw)w,(15)

where {δp = p(q + δq)− p(q),δq = −(Cq + L1C)e− L1Fww.

(16)

Defining z =

(xe

), dynamics (15) are expressed compactly

as

z = Acz +H1p+H2δp+H3∆p+H4w (17)

where ∆p is given in (14), δp is given in (16), and

Ac =

(A+BK1 −BK1

0 A+ L2C

), (18)




H1 =

(E +BK2

0

), H2 =

(0

−E

),

H3 =

(BK2

0

), H4 =

(Ew

Ew + L2Fw

).

(19)

The following proposition provides a sufficient conditionfor the closed-loop system (17) to be ISS w.r.t. w.

Proposition 2. Consider the system described by (1)-(2)with p(0) = 0. Suppose that there exist matrices L1 ∈Rnq×ny , L2 ∈ Rnx×ny ,K1 ∈ Rnu×nx ,K2 ∈ Rnu×np , P ∈R2nx×2nx with P � 0, and real numbers α0 > 0, µ > 0, σ1 ≥0, σ2 ≥ 0, σ3 ≥ 0 such that(

S0 S1

∗ 0

)+ σ1S

>2 MS2 + σ2S

>3 MS3 + σ3S

>4 MS4

− µS>5 S5 � 0 (20)

where

S0 = PAc +A>c P + α0P,

S1 = (PH1 PH2 PH3 PH4) ,

S2 =

(Cq 0nq×(nx+3np+nw)

0np×2nxInp

0np×(2np+nw)

),

S3 =

(0nq×nx

− (Cq + L1C) 0nq×3np− L1Fw

0np×(2nx+np) Inp0np×(np+nw)

),

S4 =

(0nq×nx

− Cq 0nq×(3np+nw)

0np×(2nx+2np) Inp 0np×nw

),

S5 = (0nw×(2nx+3np) Inw).

Then the closed-loop system (17) is ISS w.r.t. w and satisfiesV ≤ −α0V + µ‖w‖2 where V (z) = z>Pz.

Proof. Since M is a δ-MM for p and p(0) = 0, it holds that(qp

)>M

(qp

)≥0,

(δqδp

)>M

(δqδp

)≥0,

(∆q∆p

)>M

(∆q∆p

)≥0,

where δp, δq are given in (16), ∆p is given in (14), and

∆q = Cqx− Cqx = −Cqe. (21)

With ξ = (x> e> p> δp> ∆p> w>)>,(qp

)= S2ξ,

(δqδp

)= S3ξ,

(∆q∆p

)= S4ξ.

Hence, ξ>S>2 MS2ξ ≥ 0, ξ>S>3 MS3ξ ≥ 0, ξ>S>4 MS4ξ ≥ 0.Pre- and post-multiply (20) by ξ> and ξ, respectively. Sinceσ1, σ2, σ3 are non-negative, we obtain that

ξ>(S0 S1

∗ 0

)ξ − µξ>S>5 S5ξ ≤ 0. (22)

Consider the positive definite function defined by V (z) =z>Pz. Then, it is easy to check that V + α0V − µw>w isequal to the left hand side of (22) where V is the derivativeof V along the trajectories of (17). Therefore, V is an ISS-Lyapunov function since V ≤ −α0V +µ‖w‖2. The conclusionfollows from Proposition 1.

Clearly, matrix inequality (20) is not a LMI. In the nexttwo subsections, we will consider two parameterizations of the

δ-MM M and provide LMI conditions which can be used tosolve for M and gain matrices L1, L2,K1,K2 simultaneously.

A. Block Diagonal ParameterizationThis subsection considers a block diagonal parameterization

of the δ-MM for p. We first make the following two assump-tions on the parameterizations of M .

Assumption 1. There exist a set N1 of matrix pairs (X1, Y1)with X1 ∈ Rnq×nq , Y1 ∈ Rnp×np symmetric, and an invertiblematrix T1 with

T1 =

(T11 T12T13 T14

)(23)

and T14 ∈ Rnp×np invertible, such that M1 given below is aδ-MM of p for all (X1, Y1) ∈ N1:

M1 = T>1 M1T1 where M1 =

(X1 00 −Y1

). (24)

Assumption 2. There exist a set N2 of matrix pairs (X2, Y2)with X2 ∈ Rnq×nq , Y2 ∈ Rnp×np symmetric and invertible,and an invertible matrix T2 with

T2 =

(T21 T22T23 T24

)(25)

and T24 ∈ Rnp×np invertible, such that M2 given below is aδ-MM of p for all (X2, Y2) ∈ N2:

M2 = T>2 M2T2 where M2 =

(X−12 00 −Y −12

). (26)

Remark 2. For the globally Lipschitz nonlinearity ‖p(q2) −p(q1)‖ ≤ γ‖q2−q1‖, the matrix M in (3) satisfies Assumption1 and 2 if we choose

T1 =T2 =

(γI 00 I

), N1 =N2 ={(λI, λI)|λ > 0}.

For the incrementally sector bounded nonlinearity (δp −K1δq)

>S(δp − K2δq) ≤ 0 where S is symmetric and in-vertible, the matrix M in (4) satisfies Assumption 1 and 2 ifwe choose

T1 =T2 =

(K2 −K1 0K2 +K1 −2I

), N1 =N2 ={(λS, λS)|λ > 0}.

For the nondecreasing nonlinearity δp>δq ≥ 0, the matrix Min (5) satisfies Assumption 1 and 2 if we choose

T1 =T2 =

(I II −I

), N1 =N2 ={(λI, λI)|λ > 0}.

N1 and N2 do not have to be the set of scalings of amatrix pair as in the examples above. For instance, for thenonlinearity whose Jacobian is confined within a polytope ora cone, N1 that satisfies Assumption 1 (or N2 that satisfiesAssumption 2) is characterized via matrix inequalities (seeSection 5 in [48] for more details). Furthermore, T1 does notnecessarily has to be chosen to be equal to T2.

Because T1 in Assumption 1 and T2 in Assumption 2 areinvertible, the matrix Γi1(i = 1, 2) defined as

Γi1 = Ti1 − Ti2T−1i4 T13 (27)




is also invertible by the matrix inversion lemma. Furthermore,we define the matrix Γi2(i = 1, 2) as

Γi2 = Ti2T−1i4 . (28)

The following theorem provides sufficient conditions for thedesign of matrices L1, L2 in the observer (11) and matricesK1,K2 in the controller (12), when the δ-MM can be param-eterized in a block diagonal manner.

Theorem 1. Consider the system described by (1)-(2) withp(0) = 0. Suppose that1) Assumption 1 holds;

2) Assumption 2 holds with M2 =

(M21 M22

M23 M24

)where

M24 ∈ Rnp×np and M24 ≺ 0;3) there exist positive numbers α1, α2, µ1, µ2, matricesR1, R2, R3, R4, symmetric and positive definite matricesP1, P2, X1, X2, Y2 and a symmetric matrix Y1, such that(X1, Y1) ∈ N1, (X2, Y2) ∈ N2 and

(observer ineq.)(

Φ− ϕ>Y1ϕ φ>

φ −X1

)� 0, (29)

(controller ineq.)(

Ψ− ϕ>Y2ϕ ψ>

ψ −X2

)� 0, (30)

where

Φ =

Φ0 −P1E1 P1Ew +R1

∗ 0 0∗ ∗ −µ1I

, (31)

Φ0 = A>1 P1 + P1A1 + C>R>1 +R1C + α1P1, (32)

Ψ =

Ψ0 E2Y2 +BR4 Ew∗ 0 0∗ ∗ −µ2I

, (33)

Ψ0 = A2P2 + P2A>2 +BR3 +R>3 B

> + α2P2, (34)φ = (−(X1Γ11Cq +R2C), X1Γ12,−R2Fw), (35)ϕ = (0np×nx , Inp ,0np×nw), (36)ψ = (Γ21CqP2,Γ22Y2,0nq×nw), (37)

Ai = A− ET−1i4 Ti3Cq, i = 1, 2, (38)

Ei = ET−1i4 , i = 1, 2, (39)

with Γi1(i = 1, 2) given in (27) and Γi2(i = 1, 2) given in(28). Then, the closed-loop system (15) is ISS w.r.t. w with

L1 = Γ−111 X−11 R2,

L2 = P−11 R1 + ET−114 T13L1,

K1 = R3P−12 +K2T

−124 T23Cq,

K2 = R4Y−12 T24.

(40)

Proof. The proof proceeds in five steps.1) Firstly, we derive dynamics of the system under trans-

formations of variables q and p via T1 and T2. Since M1

(resp. M2) satisfies Assumption 1 (resp. Assumption 2) withan invertible matrix T1 (resp. T2), we introduce variabletransformations from (q, p) to (qi, pi) as follows:(

qipi

)= Ti

(qp

). (41)

Since pi = Ti3q + Ti4p and Ti4 is invertible, we have p =T−1i4 pi − T−1i4 Ti3q and qi = Γi1q + Γi2pi for i = 1, 2, whereΓi1,Γi2 are given in (27),(28). Recall that Γi1 is invertiblesince Ti is invertible.

Substituting p = T−124 p2 − T−124 T23q into (15), we have

x = (A2 +BK1)x+ (E2 +BK2)p2 −B∆k + Eww, (42)

where p2 = p2(Cqx), A2 is given in (38), E2 is given in (39),

K1 = K1 −K2T−124 T23Cq, K2 = K2T

−124 , (43)

and∆k = K1e− K2∆p, (44)

where

∆p = p2(Cqx)− p2(Cqx). (45)

Define(δq1δp1

)= T1

(δqδp

). Then, δp1 = T13δq+T14δp, which

implies that δp = T−114 δp1−T−114 T13δq. Substituting this form

of δp and (16) into (15), we have

e = (A1 + L2C)e− E1δp1 + (Ew + L2Fw)w, (46)

where A1 is given in (38), E1 is given in (39), and L2 isdefined as

L2 = L2 − ET−114 T13L1. (47)

Equations (42) and (46) are the dynamics of the closed-loopsystem after transformations of variables via T1 and T2.

2) We now consider the performance of the observer. From(40) we have R1 = P1L2 where L2 is given in (47), andR2 = X1Γ11L1. Plugging R1 into Φ in (31), we haveΦ0 = P1(A1 + L2C) + (A1 + L2C)>P1 + α1P1, and the(1, 3) entry of Φ to be P1Ew + R1Fw = P1(Ew + L2Fw);plugging R2 into φ in (35) we have φ = X1φ0 whereφ0 := (−Γ11(Cq+L1C),Γ12,−Γ11L1Fw). Recalling ϕ givenin (36) and applying Schur’s complement to (29), we have

Φ +

(φ0ϕ

)>M1

(φ0ϕ

)� 0. (48)

Define ξ1 = (e>, δp>1 , w>)>. Pre- and post-multiplying the

inequality (48) by ξ>1 and ξ1, respectively, we have

ξ>1 Φξ1 + ξ>1

(φ0ϕ

)>M1

(φ0ϕ

)ξ1 ≤ 0. (49)

Note that δq1 = T11δq + T12δp = T11δq + T12T−114 δp1 −

T12T−114 T13δq = Γ11δq + Γ12δp1 = −Γ11(Cq + L1C)e +

Γ12δp1 − Γ11L1Fww. Therefore,(δq1δp1

)=

(φ0ϕ

)ξ1. Since(

δqδp

)>M

(δqδp

)≥ 0, we have

(δq1δp1

)>M1

(δq1δp1

)≥ 0, and

therefore, ξ>1

(φ0ϕ

)>M1

(φ0ϕ

)ξ1 ≥ 0. Thus, ξ>1 Φξ1 ≤ 0

from (49), which is equivalent to 2e>P1[(A+L2C)e−E1δp1+(Ew + L2Fw)w] + α1e

>P1e− µ1‖w‖2 ≤ 0.

Define V1(e) = e>P1e. Then the derivative of V1 along the




trajectory of (46) satisfies

V1 = 2e>P1[(A+ L2C)e− E1δp1 + (Ew + L2Fw)w]

≤ −α1e>P1e+ µ1‖w‖2. (50)

3) We now prove that ‖∆k‖/‖e‖ is bounded where ∆k isgiven in (44). Since M24 = T>22X

−12 T22 − T>24Y −12 T24 ≺ 0

and T24 is invertible, we have

Γ>22X−12 Γ22 − Y −12 = T−>24 M24T

−124 ≺ 0. (51)

Recall that ∆q = −Cqe in (21) and define ∆q := q2(Cqx)−q2(Cqx). Then, ∆q = −Γ21Cqe+ Γ22∆p where ∆p is givenin (45). Define ζ = (e>,∆p>)>. Therefore,

ζ>(−Γ21Cq Γ22

0 I

)>M2

(−Γ21Cq Γ22

0 I

)ζ

=

(−Γ21Cqe+ Γ22∆p

∆p

)>M2

(−Γ21Cqe+ Γ22∆p

∆p

)=

(∆q∆p

)>T>2 M2T2

(∆q∆p

)≥ 0,

where the last equality is from (26) in Assumption 2.Hence, e>C>q Γ>21X

−12 Γ21Cqe − 2e>C>q Γ>21X

−12 Γ22∆p +

∆p>(Γ>22X−12 Γ22 − Y −12 )∆p ≥ 0. From (51), the inequality

above implies that κ1‖e‖2+κ2‖e‖‖∆p‖−κ3‖∆p‖2 ≥ 0 whereκ1 = λmax(C>q Γ>21X

−12 Γ21Cq), κ2 = 2‖C>q Γ>21X

−12 Γ22‖,

κ3 = λmin(Y −12 − Γ>22X−12 Γ22). Clearly, κ1, κ3 > 0, κ2 ≥ 0.

Therefore, we have ‖∆p‖ ≤ κ‖e‖ where κ := (κ2 +√κ22 + 4κ1κ3)/2κ3 > 0.

Since ∆k = K1e+ K2∆p by (44), we have

‖∆k‖ ≤ κ‖e‖ (52)

for all x, e, where κ = ‖K1‖ + ‖K2‖κ > 0, which bounds‖∆k‖/‖e‖.

4) Next, we analyse controller performance. From (40) wehave R3 = K1P2 and R4 = K2Y2 where K1, K2 are givenin (43). Plugging R3, R4 into (33), we have Ψ0 = (A2 +BK1)P2 +P2(A2 +BK1)>+α2P2, and the (1, 2) entry of Ψto be (E2 +BK2)Y2. Pre- and post-multiplying the inequality(30) by the matrix diag(In, Y

−12 , Inw

, Inq), and then applying

Schur’s complement, we have

Ψ +

(ψ1

ϕ

)>M2

(ψ1

ϕ

)� 0, (53)

where

Ψ =

Ψ0 E2 +BK2 Ew∗ 0 0∗ ∗ −µ2I

,

and ψ1 = (Γ21CqP2,Γ22,0nq×nw), Ψ0 is shown above, ϕis given in (36). Let P3 = P−12 and pre- and post-multiplythe inequality (53) by diag(P3, Inp

, Inw) and its transpose,

respectively. This results in

Ψ +

(ψ0

ϕ

)>M2

(ψ0

ϕ

)� 0, (54)

where ψ0 = (Γ21Cq,Γ22,0nq×nw) and

Ψ =

Ψ0 P3(E2 +BK2) P3Ew∗ 0 0∗ ∗ −µ2I

, (55)

Ψ0 = P3(A2 +BK1) + (A2 +BK1)>P3 + α2P3. (56)

Define ξ2 = (x>, p>2 , w>)>. Pre- and post-multiplying the

inequality (54) by ξ>2 and ξ2, respectively, we have

ξ>2 Ψξ2 + ξ>2

(ψ0

ϕ

)>M2

(ψ0

ϕ

)ξ2 ≤ 0. (57)

By (26) and (41), we have(q2p2

)>M2

(q2p2

)≥ 0. Since

q2 = Γ21q + Γ22p2 = Γ21Cqx + Γ22p2,(q2p2

)=

(ψ0

ϕ

)ξ2,

ξ>2

(ψ0

ϕ

)>M2

(ψ0

ϕ

)ξ2 ≥ 0. Thus, ξ>2 Ψξ2 ≤ 0 from (57),

which is equivalent to 2x>P3[(A2+BK1)x+(E2+BK2)p2+Eww] + α2x

>P3x− µ2‖w‖2 ≤ 0.

Let V2(x) = x>P3x. Then the derivative of V2 along thetrajectory of (42) satisfies

V2 = 2x>P3[(A2 +BK1)x+ (E2 +BK2)p2 −B∆k + Eww]

≤ −α2x>P3x+ µ2‖w‖2 + 2‖P3B‖‖x‖‖∆k‖.

Recalling (52), we have

V2 ≤ −α2x>P3x+ µ2‖w‖2 + θ‖x‖‖e‖. (58)

where θ = 2‖P3B‖κ.5) Finally, we prove that the closed-loop system ex-

pressed by (42) and (46) is ISS with respect to w. Choosetwo constants c1, c2 as c1 = α1λm(P1)/λM (P1), c2 =α1λm(P3)/λM (P3). Since c1 > 0, c2 > 0, we can choose twoconstants α0 > 0, β0 > 0 such that α0 <min{c1, c2}, β0 ≥

θ2

4λM (P1)λM (P3)(c1−α0)(c2−α0). Then, it is easy to check that

the matrix P0 :=

(P0 θ/2

θ/2 P0

)is negative semi-definite where

P0 = −α2λm(P3) + α0λM (P3) and P0 = β0(−α1λm(P1) +

α0λM (P1)). Define a matrix P as P =

(P3 00 β0P1

).

Clearly, P is positive definite. We can verify that the candidateLyapunov function V (x, e) := z>Pz satisfies V (x, e) =β0V1(e) + V2(x), and its derivative along the trajectory of(42) and (46) satisfies

V + α0V ≤− α1β0e>P1e− α2x

>P3x+ θ‖x‖‖e‖+ α0V

≤(‖x‖, ‖e‖)P0(‖x‖, ‖e‖)> + (µ1β0 + µ2)‖w‖2

≤(µ1β0 + µ2)‖w‖2. (59)

Therefore, the closed-loop system (42) and (46), or equiv-alently (15), satisfies (8) with K∞ functions γ(‖(x, e)‖) =−α0λm(P )‖(x, e)‖2 and χ(‖w‖) = (µ1β0 + µ2)‖w‖2. Thiscompletes the proof.

Remark 3. In the proof of Theorem 1, we first prove thatthe observer error e is ISS w.r.t. w (see (50) in step 2),then the state x is ISS w.r.t. w and e (see (58) in step 4),




and finally (x, e) is ISS w.r.t. w (see (59) in step 5). Thisprocedure of proving global stabilization is similar to thecertainty equivalence proof used in [25].

Remark 4. If α1 is fixed, then (29) is an LMI in decisionvariables µ1, P1, R1, R2, X1, Y1 that are used to determineobserver gains L1, L2; if α2 is fixed, then (30) is an LMIin decision variables µ2, P2, R3, R4, X2, Y2 that are usedto determine controller gains K1,K2. There is no couplingin decision variables in LMIs (29) and (30), implying aseparation of the controller and observer designs.

Remark 5. The proof of Theorem 1 indicates that largerα1, α2 result in a larger function γ(·) ∈ K∞ in (8), which inturn indicates a faster convergence rate for the system (17).The convergence rate guarantee given in the proof of Theorem1, α0, can be improved by finding a new ISS-Lyapunov functionV = z>Pz via Proposition 2.

Remark 6. The condition M24 ≺ 0 in Theorem 1 is used toprove that ‖∆k‖/‖e‖ is bounded, which holds automaticallywhen p is globally Lipschitz. The condition M24 ≺ 0 canbe replaced by a growth condition on p similar to Theorem2 of [28]. Specifically, for a system described by (1)-(2)where Ew = Fw = 0, if all the conditions of Theorem 1but M24 ≺ 0 hold, and there exist a K function g1 anda non-decreasing function g2 : [0,∞) → [0,∞) such that‖p(Cq(x+ ∆x))− p(Cqx)‖ ≤ g1(‖∆x‖)‖Cqx‖ for all x,∆xthat satisfy ‖Cqx‖ ≥ g2(‖∆x‖), then the feedback controller(12) with L1, L2,K1,K2 given by (40) renders the closed-loopsystem (15) globally exponentially stable.

The condition M24 ≺ 0 can be eliminated by using asimpler form of u. Specifically, suppose that the observer-based controller u has the form u(t) = K1x(t), all theconditions of Theorem 1 but M24 ≺ 0 hold with R4 = 0,and L1, L2,K1 are given by (40), then the closed-loop system(15) is ISS w.r.t. w. In this case, the LMI (30) is less likely tobe satisfied by fixing R4 = 0.

B. Block Anti-Triangular Parameterization

In this subsection, we consider a block anti-triangular pa-rameterization of the δ-MM for p. The following assumptionon the parameterization of M is given first.

Assumption 3. There exist a set N of matrix pairs (X,Y )with X ∈ Rnq×np , Y ∈ Rnp×np , and an invertible matrixT ∈ R(np+nq)×(np+nq), such that M given below is a δ-MMof p for all (X,Y ) ∈ N :

M = T>MT where M =

(0 XX> Y

). (60)

The following theorem provides sufficient conditions for thedesign of matrices L1, L2,K1,K2 when the δ-MM M can beparameterized in a block anti-triangular manner.

Theorem 2. Consider the system described by (1)-(2) withp(0) = 0. Suppose that1) Assumption 3 holds for some T1,N1 and T2,N2, respec-tively, where T1 and T2 are partitioned as in (23) and in (25),respectively, with T14, T24 invertible;

2) there exist positive constants α1, α2, µ1, µ2, matricesR1, R2, R3, R4, X1, Y1.X2, Y2, and symmetric and positivedefinite matrices P1, P2, such that (X1, Y1) ∈ N1, (X2, Y2) ∈N2, and

Φ + Υ>1 M1Υ1 + Υ>2 Υ1 + Υ>1 Υ2 � 0, (61)

Ψ + Υ>3 M2Υ3 + Υ>4 Υ3 + Υ>3 Υ4 � 0, (62)

Γ>12X1 +X>1 Γ12 + Y1 ≺ 0, (63)

where

M1 =

(0 X1

X>1 Y1

), M2 =

(0 X2

X>2 Y2

), (64)

Ψ =

Ψ0 E2 +BR4 Ew∗ 0 0∗ ∗ −µ2I

, (65)

Υ1 =

(−Γ11Cq Γ12 0nq×nw

0np×nx Inp 0np×nw

),

Υ2 =

(0nq×nx 0nq×np 0nq×nw

−R2C 0np−R2Fw

),

Υ3 =

(0nq×nx

Γ22 0nq×nw

0np×nxInp

0np×nw

),

Υ4 =

(0nq×nx

0nq×np0nq×nw

X2Γ21CqP2 0np 0np×nw

),

(66)

with Γi1(i = 1, 2) given in (27), Γi2(i = 1, 2) given in (28),Φ given in (31), Φ0 given in (32), Ψ0 given in (34), ϕ givenin (36), Ai(i = 1, 2) given in (38) and Ei(i = 1, 2) given in(39). If X1 has full row rank, then the closed-loop system (15)is ISS w.r.t. w with L2, K1 given by (40), and L1,K2 givenby

L1 = Γ−111 X†1R2, K2 = R4T24, (67)

where X†1 is the right inverse of X1.

Proof. As shown in (42) and (46), dynamics of the closed-loopsystem under transformations can be described as

x = (A2 +BK1)x+ (E2 +BK2)p2 −B∆k + Eww,

e = (A1 + L2C)e− E1δp1 + (Ew + L2Fw)w,

where K1, K2 are given in (43), ∆k is given in (44), p2 isgiven in (41), and L2 is given in (47). From (40) and (67), wehave R1 = P1L2 and R2 = X1Γ11L1. We claim that (61) isequivalent to

Φ +Q>1 M1Q1 � 0 (68)

and (62) is equivalent to

Ψ +Q>2 M2Q2 � 0 (69)

where

Q1 =

(−Γ11(Cq + L1C) Γ12 −Γ11L1Fw

0np×nx Inp 0np×nw

),

Q2 =

(Γ21CqP2 Γ22 0nq×nw

0np×nxInp

0np×nw

).




Indeed, Q1 can be written as Q1 = Υ1 + Υ2 where Υ1 isgiven in (66) and

Υ2 =

(−Γ11L1C 0nq×np −Γ11L1Fw0np×nx

0np×np0np×nw

).

It is easy to verify that Υ2 = M1Υ2 and Υ>2 M1Υ2 = 0.Therefore,

Q>1 M1Q1 = (Υ1 + Υ2)>M1(Υ1 + Υ2)

= Υ>1 M1Υ1 + Υ>2 M1Υ1 + Υ>1 M1Υ2 + Υ>2 M1Υ2

= Υ>1 M1Υ1 + Υ>2 Υ1 + Υ>1 Υ2.

Similarly, Q2 can be written as Q2 = Υ3 + Υ4 where Υ3 isgiven in (66) and

Υ4 =

(Γ21CqP2 0nq×np 0nq×nw

0np×nx 0np×np 0np×nw

).

It is easy to verify that Υ4 = M2Υ4 and Υ>4 M2Υ4 =0. Therefore, Q>2 M2Q2 = (Υ3 + Υ4)>M2(Υ3 + Υ4) =Υ>3 M2Υ3 + Υ>3 Υ4 + Υ>4 Υ3. Hence, our claim is proved.

Plugging R1 into Φ0 and Φ, we have Φ0 = P1(A1 +L2C) + (A1 + L2C)>P1 +α1P1, and the (1, 3) entry of Φ isP1(Ew+ L2Fw). Define ξ1 = (e>, δp>1 , w

>)>. Pre- and post-multiplying (68) by ξ>1 and ξ1, respectively, we have ξ>1 Φξ1+

ξ>1 Q>1 M1Q1ξ1 ≤ 0. Since Q1ξ1 =

(δq1δp1

)= T1

(δqδp

)and

M1 satisfies Assumption 3, we have ξ>1 Q>1 M1Q1ξ1 ≥ 0,

which implies that ξ>1 Φξ1 ≤ 0. Hence, 2e>P1[(A+ L2C)e−E1δp1 + (Ew + L2Fw)w] + α1e

>P1e− µ1‖w‖2 ≤ 0. DefineV1(e) = e>P1e. Then, we have

V1 ≤ −α1e>P1e+ µ1‖w‖2.

Define ∆q = Cqx − Cqx and ∆q := q1(Cqx) − q1(Cqx).Then, ∆q = −Cqe and ∆q = −Γ11Cqe+Γ12∆p where ∆p =p1(Cqx)− p1(Cqx). Define ζ = (e>,∆p>)>. Therefore,

ζ>(−Γ11Cq Γ12

0 I

)>M1

(−Γ11Cq Γ12

0 I

)ζ

=

(∆q∆p

)>T>1 M1T1

(∆q∆p

)≥ 0,

where the last equality is from Assumption 3. Hence,−2e>C>q Γ>11X1∆p + ∆p>(Γ>12X1 + X>1 Γ12 + Y1)∆p ≥ 0.From (63), the inequality above implies that κ1‖e‖‖∆p‖ −κ2‖∆p‖2 ≥ 0, where κ1 = 2‖C>q Γ>11X1‖ and κ2 =−λmin(Γ>12X1 +X>1 Γ12 +Y1). Noticing that κ1 ≥ 0, κ2 > 0,we have ‖∆p‖ ≤ κ1

κ2‖e‖. Noting that ∆k = K1e + K2∆p

with ∆p = p1(Cqx) − p1(Cqx), K1 = K1 − K2T−114 T13Cq ,

K2 = K2T−114 , we have ‖∆k‖ ≤ κ‖e‖ for all x, e, where

κ = ‖K1‖+ ‖K2‖κ1/κ2 ≥ 0.From (40) and (67) we have R3 = K1P2 and R4 = K2

where K1, K2 are defined in (43). Plugging R3, R4 into Ψ0

and Ψ, we have Ψ0 = (A2 +BK1)P2 + P2(A2 +BK1)> +α2P2, and the (1, 2) entry of Ψ is E2 +BK2. Let P3 = P−12

and pre- and post-multiply (69) by diag(P3, Inp, Inw

) and itstranspose, respectively. This results in

Ψ +Q>3 M2Q3 � 0, (70)

where Q3 =

(Γ21Cq Γ22 0nq×nw

0np×nxInp

0np×nw

)and Ψ is given in

(55) with Ψ0 given in (56). Define ξ2 = (x>, p>2 , w>)>.

Pre- and post-multiplying (70) by ξ>2 and ξ2, respectively,we have ξ>2 Ψξ2 + ξ>2 Q

>3 M2Q3ξ2 ≤ 0. Since Q3ξ2 =(

q2p2

)= T2

(qp

)and M2 satisfies Assumption 3, it follows

that ξ>2 Q>3 M2Q3ξ2 ≥ 0. Hence, we have ξ>2 Ψξ2 ≤ 0, which

is equivalent to 2x>P3[(A2 + BK1)x + (E2 + BK2)p2 +Eww]+α2x

>P3x−µ2‖w‖2 ≤ 0. Let V2(x) = x>P3x. Then,we have V2 = 2x>P3[(A2+BK1)x+(E2+BK2)p2−B∆k+Eww] ≤ −α2x

>P3x+µ2‖w‖2+2‖P3B‖‖x‖‖∆k‖. Recalling‖∆k‖ ≤ κ‖e‖, we have

V2 ≤ −α2x>P3x+ µ2‖w‖2 + θ‖x‖‖e‖

where θ = 2‖P3B‖κ. The rest of the proof proceeds as thatgiven in part 5) of the proof of Theorem 1.

Remark 7. Inequality (61) is an LMI in decision variablesµ1, P1, R1, R2 when α1 is fixed, (62) is an LMI in decisionvariables µ2, P2, R3, R4 when α2 and X2 are fixed, and (63)is an LMI in decision variables X1, Y1. Hence, we can fixα1, α2, X2 and solve for (61)-(63). When L1, L2,K1,K2 areobtained, a re-computation for P, α0, µ using Proposition 2may result in a better convergence rate guarantee.

Remark 8. The LMIs (29) and (30) both have dimensions(nx+np+nq+nw)×(nx+np+nq+nw), the LMIs (61) and(62) both have dimensions (nx+np+nw)×(nx+np+nw), andthe LMI (63) has dimension np×np. These LMIs can be solvedreliably and efficiently by the interior point method algorithmsof convex optimization with a polynomial-time complexity.Exploring for what class of systems these LMIs are guaranteedto be feasible (i.e., analytical verification of feasibility) is stillunder our investigation. Furthermore, these LMI conditionsmight be conservative compared with specific results that focuson certain special nonlinearities such as the globally Lipschitznonlinearity.

IV. EVENT-TRIGGERED CONTROL DESIGN

In this section, we discuss event-triggering mechanisms(ETMs) within the observer-based controller designed in thepreceding section for the system described by (1)-(2) wherethe nonlinearity p is assumed to be globally Lipschitz. Forcertain incrementally quadratic nonlinearities that imply theglobal Lipschitzness (such as the incremental sector boundednonlinearity and the nonlinearities with Jacobians in polytopes[48]), using their corresponding incremental matrix character-izations, instead of the matrix characterizations for global Lip-schitzness, makes the associated LMIs in the design procedureless conservative, while benefiting from having the Lipschitzproperty needed for the upcoming ETM-related results to hold.

A. Configuration I: The Controller Channel Is ImplementedBy ETM

In this subsection, we discuss the configuration shown inFigure 1 where the plant is described by (1)-(2), the observeris given in (11), the continuous-time feedback controller is




w

PlantController

ObserverZOH

u y

xs

ETM

x

w

PlantZOHu y

ObserverControllerx

ETM

1

Fig. 1. Configuration where the ETM is implemented in the controllerchannel.

given in (12), and the ETM only has the information of x,the state of the observer. We will assume that ‖w‖∞ ≤ ω0

where ω0 is a positive constant indicating the bound of thedisturbance in this subsection and the next subsection.

The feedback controller u(t) is implemented by an ETMsuch that it is only updated at certain triggering time instancest1, t2, ... where tk < tk+1 for any k ≥ 0 and kept constantduring consecutive time instances. Define t0 = 0 and thepiecewise constant signal xs as

xs(t) = x(tk), ∀t ∈ [tk, tk+1).

Then the control input u(t) is given by

u(t) = K1xs(t) +K2p(Cqxs(t)) (71)

where K1,K2 are matrices to be designed. The input u(t)has the same form as that in (12), but it is updated attriggering time instances t1, t2, . . . , which are determined bythe following type of triggering rule:

tk+1 = inf{t | t ≥ tk + τ, ‖xe(t)‖ > σ‖x(t)‖+ ε} (72)

where xe is defined as xe(t) = xs(t)− x(t) and τ, σ, ε are allpositive numbers to be specified. The time-updating rule (72)guarantees that the inter-execution times {tk+1−tk} are lowerbounded by the built-in positive constant τ , which means thatZeno phenomenon (i.e., infinite executions happen in a finiteamount of time) will not occur [55].

Remark 9. The triggering rule (72) only depends on the in-formation of x and xe, which are available from the proposedobserver. This triggering rule is a combination of a mixed ETMand a time regularization technique. There are several moti-vations for choosing this type of rule. It was known that evenin the absence of disturbances, inter-execution times of manyETMs converge to zero for output-based control configurations[59]. To exclude the Zeno phenomenon, time regularizationor periodic event-triggered control, which enforces a built-in lower bound for inter-execution times, has been utilizedin recent works on observer-based ETMs [62], [57], [61].Furthermore, mixed ETM is known to be robust to externaldisturbances or measurement noise, while relative ETM andabsolute ETM have zero robustness to disturbance/noise [60].Additionally, an event-triggering rule with time regularizationcan benefit from using mixed ETMs in terms of the number ofevents that are generated (e.g., see Example 3 in [60]).

The closed-loop system that combines system (1)-(2), ob-

server (11) and event-triggered controller (71) is expressedcompactly as

z = Acz +H1p+H2δp+H3δp+H4w +H5xe (73)

where δp, δq are given in (16), Ac is given in (18),H1, H2, H3, H4 are given in (19), and

δp = p(Cqxs)− p(Cqx), (74)

H5 =

(BK1

0

). (75)

Theorem 3. Consider the configuration shown in Figure 1where the plant is described by (1)-(2) with p(0) = 0 and‖w‖∞ ≤ ω0 with ω0 a positive number. Suppose that thereexists ` > 0 such that ‖p(r) − p(s)‖ ≤ `‖r − s‖ for anyr, s. Suppose that there exist positive numbers α0 > 0, µ > 0,and matrices P � 0,K1,K2, L1, L2 such that the closed-loopsystem (17) with controller (12) and observer (11) satisfiesV ≤ −α0V + µ‖w‖2 where V = z>Pz. Choose any ε > 0and

σ =%α0λm(P )

2√

2s> 0 (76)

where 0 < % < 1 and s = ‖PH5‖ + `‖PH3‖‖Cq‖. Chooseτ > 0 as the solution to the equation φ(τ) = 1 where φ is thesolution of the following ODE:

φ =√

2(η4 + η2φ)(1 + σφ), φ(0) = 0,

with

η1 = ‖Ac‖+ `√b21 + b22,

η2 = ‖H5‖+ `‖H3‖‖Cq‖,η3 = `‖H2‖‖L1Fw‖+ ‖H4‖,η4 = η1√

2σ+ η3ω0

ε ,

b1 = ‖H1‖‖Cq‖,b2 = ‖H2‖‖Cq + L1C‖+ ‖H3‖‖Cq‖.

(77)

Then, the closed-loop system (73) that implements the trigger-ing rule (72) is ISpS w.r.t. w.

Proof. Since the derivative of V along the trajectory of theclosed-loop system (17) satisfies V ≤ −α0V + µ‖w‖2, thederivative of V along the trajectory of the closed-loop system(73) satisfies V ≤ −α0V + µ‖w‖2 + 2z>P [H5xe +H3(δp−∆p)] ≤ −α0λm(P )‖z‖2 + µ‖w‖2 + 2‖z‖(‖PH5‖‖xe‖ +‖PH3‖‖δp − ∆p‖). Clearly, ‖δp − ∆p‖ = ‖p(Cqxs) −p(Cqx)‖ ≤ `‖Cq(xs − x)‖ ≤ `‖Cq‖‖xe‖. Then, we have

V ≤ −α0λm(P )‖z‖2 + µ‖w‖2 + 2s‖z‖‖xe‖≤ −(1− %)α0λm(P )‖z‖2 + µ‖w‖2

+ ‖z‖[2s‖xe‖ − %α0λm(P )‖z‖

]. (78)

For any x, e, we have ‖z‖ =√‖x‖2 + ‖x− x‖2 =√

‖x‖2 + 2‖x‖2 − 2x>x ≥ ‖x‖/√

2, meaning that ‖x‖ ≤√2‖z‖. Therefore, the condition

‖xe‖ ≤ σ‖x‖+ ε (79)




implies

‖xe‖ ≤√

2σ‖z‖+ ε, (80)

which is equivalent to the inequality 2s‖xe‖ −%α0λm(P )‖z‖ ≤ 2sε.

Choose a constant c such that 0 < c < (1 − %)α0λm(P ).Then, as long as (80) holds, from (78) we have

V ≤ −[(1− %)α0λm(P )− c]‖z‖2 + µ‖w‖2 +s2ε2

c. (81)

Recalling that p(0) = 0 and ` is the Lipschitz constantof p, we have ‖p‖ ≤ `‖Cq‖‖x‖, ‖δp‖ ≤ `‖δq‖ ≤ `(‖Cq +L1C‖‖e‖ + ‖L1Fw‖‖w‖), and ‖δp‖ ≤ `‖Cq‖(‖xe‖ + ‖e‖).Therefore, from (73) we have

‖z‖ ≤ ‖Ac‖‖z‖+ ‖H5‖‖xe‖+ `‖H1‖‖Cq‖‖x‖+ `‖H2‖(‖Cq + L1C‖‖e‖+ ‖L1Fw‖‖w‖)+ `‖H3‖‖Cq‖(‖xe‖+ ‖e‖) + ‖H4‖‖w‖

≤ η1‖z‖+ η2‖xe‖+ η3‖w‖

where the second inequality follows from Cauchy’s inequalityb1‖x‖+ b2‖e‖ ≤

√b21 + b22‖z‖.

Because ‖z‖ =√‖x‖2 + ‖e‖2 =

√‖x‖2 + ‖x− ˙x‖2 =√

‖ ˙x‖2 + 2‖x‖2 − 2x> ˙x ≥ ‖ ˙x‖/√

2 and ‖ ˙xe‖ = ‖ ˙x‖, wehave ‖ ˙xe‖ ≤

√2‖z‖.

Let v(t) = ‖xe(t)‖√2σ‖z(t)‖+ε . Then for any h > 0,

v(t+ h)− v(t) =‖xe(t+ h)‖√

2σ‖z(t+ h)‖+ ε− ‖xe(t)‖√

2σ‖z(t)‖+ ε

=‖xe(t+h)‖(

√2σ‖z(t)‖+ε)−‖xe(t)‖(

√2σ‖z(t+h)‖+ε)

(√

2σ‖z(t+h)‖+ε)(√

2σ‖z(t)‖+ε)

=(‖xe(t+ h)− ‖xe(t)‖)(

√2σ‖z(t)‖+ ε)

(√

2σ‖z(t+ h)‖+ ε)(√

2σ‖z(t)‖+ ε)

−√

2σ‖xe(t)‖(‖z(t+ h)‖ − ‖z(t)‖)(√

2σ‖z(t+ h)‖+ ε)(√

2σ‖z(t)‖+ ε)

and hence

D+v(t) = lim suph→0+

v(t+ h)− v(t)

h

=D+‖xe(t)‖√2σ‖z(t)‖+ ε

−√

2σ‖xe(t)‖D+‖z(t)‖(√

2σ‖z(t)‖+ ε)2. (82)

When z(t) 6= 0, D+‖z(t)‖ = z(t)T z(t)‖z(t)‖ and therefore

|D+‖z(t)‖| ≤ ‖z(t)‖. When z(t) = 0, D+‖z(t)‖ =

lim suph→0+‖z(t+h)‖−‖z(t)‖

h = lim suph→0+ ‖z(t+h)h ‖ =

‖z(t)‖. Thus, in all cases |D+‖z(t)‖| ≤ ‖z(t)‖. Similarly,|D+‖xe(t)‖| ≤ ‖ ˙xe(t)‖. Dropping the argument t, it nowfollows from (82) that

D+v ≤ ‖ ˙xe‖√2σ‖z‖+ ε

+

√2σ‖xe‖‖z‖

(√

2σ‖z‖+ ε)2

≤√

2‖z‖√2σ‖z‖+ ε

+

√2σ‖xe‖‖z‖

(√

2σ‖z‖+ ε)2

=

√2‖z‖√

2σ‖z‖+ ε(1 +

σ‖xe‖√2σ‖z‖+ ε

)

≤√

2(η4 + η2‖xe‖√

2σ‖z‖+ ε)(1 + σ

‖xe‖√2σ‖z‖+ ε

)

=√

2 (η4 + η2v) (1 + σv)

where the following facts are used to derive the last inequality:

η1‖z‖√2σ‖z‖+ ε

≤ η1√2σ,

η3‖w‖√2σ‖z‖+ ε

≤ η3ω0

ε.

Since v(tk) = 0, it now follows from the comparison lemmathat v(t) ≤ φ(t − tk). Since the time it takes for v to evolvefrom 0 to 1 is lower bounded by τ , (80) holds during thetime interval [tk, tk + τ ]. For any k ≥ 0, if tk+1 = tk + τ ,then (80) holds during the interval [tk.tk+1) as shown above; iftk+1 > tk+τ , then, during the interval [tk+τ, tk+1), condition(79) holds, which implies that (80) holds. Therefore, (80) holdsduring any interval [tk, tk+1) for any k ≥ 0, i.e., it holds forany t ≥ 0. Since satisfaction of (80) implies the inequality(81), we conclude that the function V is an ISpS-Lyapunovfunction since it satisfies (8) for any t ≥ 0 with γ(‖z‖) =[(1 − %)α0λm(P ) − c]‖z‖2 ∈ K∞, χ(‖w‖) = µ‖w‖2 ∈ Kand d = s2ε2/c > 0. The conclusion follows by Proposition1.

Remark 10. In the proof of Theorem 3, the equation for τis given explicitly, and any τ ′ ∈ (0, τ ] also makes the proofvalid. The parameter ε can be chosen arbitrarily, but thereare trade-offs in choosing ε: on one hand, the value of d inthe inequality (8) or (10) increases as ε increases, meaningthat the ultimate bound for x increases as ε increases; on theother hand, the explicit equation of τ depends on ε, with τdecreasing to 0 when ε approaches 0. Hence, parameters inthe triggering rule should be chosen appropriately to balancethe execution times and the performance. Finding the maximallower-bound of the inter-execution times is an interesting andchallenging problem that will be investigated in our futurework.

B. Configuration II: The Controller and Observer ChannelsAre Both Implemented By ETMs

ETM

ZOHy

w

PlantController

ObserverZOH

u y

xs

ETM

x

1

Fig. 2. Configuration where ETMs are implemented in both the observer andcontroller channels asynchronously

In this subsection, we discuss the configuration shown inFigure 2 where the ETM for the output is triggered by theinformation of y and the ETM for the input is triggered bythe information of x, in an asynchronous manner.

Consider a system described by (1)-(2). The observer in theconfiguration of Figure 2 only has sampled information ys(t)




of the output y(t) where ys(t) is updated at time instancesty1, t

y2, ... by

ys(t) = y(tyk), ∀t ∈ [tyk, tyk+1).

Here, ty0 = 0 and the triggering times ty1, ty2, . . . are determined

by the following triggering rule:

tyk+1 = inf{t | t ≥ tyk + τy, ‖ye(t)‖ > σy‖y(t)‖+ εy} (83)

where ye(t) = ys(t) − y(t) and τy, σy, εy are all positivenumbers to be specified.

With the sampled information ys(t), the observer nowbecomes

˙x=Ax+Bu+Epp(q+L1(y−ys))+L2(y−ys),y=Cx+Du,

q=Cqx,

(84)

where L1, L2 are matrices to be designed.

The observer-based controller u(t) has the form shown in(71) where xs(t) is updated at time instances tu1 , t

u2 , ... by

xs(t) = x(tuk), ∀t ∈ [tuk , tuk+1).

Here, tu0 = 0 and the triggering times tu1 , tu2 , . . . are deter-

mined by the following triggering rule:

tuk+1 = inf{t | t ≥ tuk + τu, ‖xe(t)‖ > σu‖x(t)‖+ εu} (85)

where xe(t) = x(tk) − x(t) and τu, σu, εu are all positivenumbers to be specified. Note that the information of x andxe are available from the proposed observer.

The time-updating rule (83) (or (85)) provides a built-inpositive lower bound τy (or τu) for inter-execution times{tyk+1 − tyk} (or {tuk+1 − tuk}), implying that the Zeno phe-nomenon will not occur. Although there is no bound guaranteeon the inter-execution times between tyk and tuk , this willnot cause a problem since these two ETMs are implementedseparately.

Since ye(t) = ys(t) − y(t), the closed-loop system thatcombines system (1)-(2), observer (84) and event-triggeredcontroller (71) is expressed compactly as

z=Acz+H1p+H2δp+H3δp+H4w+H5xe+H6ye (86)

where Ac, H1, H2, H3, H4, H5 are given in (18), (19), (75),respectively, δp is given in (74), and

δp = p(q + δq)− p(q),δq = −(Cq + L1C)e− L1Fww − L1ye,

H6 =

(0L2

).

Theorem 4. Consider the configuration shown in Figure 2where the plant is described by (1)-(2) with D = 0, p(0) = 0,and ‖w‖∞ ≤ ω0 with ω0 a positive number. Suppose thatthere exists ` > 0 such that ‖p(r) − p(s)‖ ≤ `‖r − s‖ forany r, s. Suppose that there exist constants α0 > 0, µ > 0,and matrices P � 0,K1,K2, L1, L2 such that the closed-loopsystem (17) with controller (12) and observer (11) satisfiesV ≤ −α0V +µ‖w‖2 where V = z>Pz. Choose any εy, εu >

0, and

σy =a2%α0λm(P )

2‖C‖s2, σu =

a1%α0λm(P )

2√

2s1, (87)

where 0 < % < 1, s1 = ‖PH5‖ + `‖PH3‖‖Cq‖, s2 =‖PH6‖ + `‖PH2‖‖L1‖, and a1, a2 are two constants sat-isfying 0 < a1, a2 < 1 and a1 + a2 = 1. Choose τu > 0as the solution to the equation φ1(τu) = 1 where φ1 is thesolution of the following ODE

φ1 =√

2(1 + σuφ1)(η5 + η2φ1 + d1η7), φ1(0) = 0

and choose τy > 0 as the solution to the equation φ2(τy) = 1where φ2 is the solution of the following ODE

φ2 = ‖C‖(1 + σyφ2)(η6 + η7φ2 + d2η2), φ2(0) = 0

where

η5 = η1√2σu

+ η2ω0

εu,

η6 = η1σy‖C‖ + η3ω0

εy,

η7 = `‖H2‖‖L1‖+ ‖H6‖,d1 = max{ εyεu ,

σy‖C‖√2σu},

d2 = max{ εuεy ,√2σu

σy‖C‖},

(88)

and η1, η2, η3 are given in (77). Then, the closed-loop system(86) that impllements triggering rules (83) and (85) is ISpSw.r.t. w.

Proof. If the derivative of V along the trajectory of (17)satisfies V ≤ −α0V + µ‖w‖2, then the derivative of V alongthe trajectory of the closed-loop system (86) satisfies

V ≤ −α0V + µ‖w‖2 + 2z>P[H2(δp− δp)

+H3(δp−∆p) +H5xe +H6ye

]≤ −(1− %)α0λm(P )‖z‖2 + µ‖w‖2 + ‖z‖

(2s1‖xe‖

+ 2s2‖ye‖ − %α0λm(P )‖z‖)

(89)

where the following facts are used: ‖δp−δp‖ ≤ `‖δq−δq‖ ≤`‖L1‖‖ye‖, ‖δp−∆p‖ ≤ `‖Cq‖‖xe‖. As ‖z‖ ≥ ‖x‖/

√2 and

‖z‖ ≥ ‖x‖ ≥ ‖y‖/‖C‖, we have

‖z‖ ≥ a1‖x‖√2

+a2‖y‖‖C‖

. (90)

From (89) and (90) we have

V ≤ −(1− %)α0λm(P )‖z‖2 + µ‖w‖2 + ‖z‖[2s1‖xe‖

− a1%α0λm(P )√2

‖x‖]

+ ‖z‖[2s2‖ye‖ −

a2%α0λm(P )

‖C‖‖y‖].

The condition ‖xe‖ ≤ σu‖x‖+ εu implies

‖xe‖ ≤√

2σu‖z‖+ εu, (91)

and the condition ‖ye‖ ≤ σy‖y‖+ εy implies

‖ye‖ ≤ σy‖C‖‖z‖+ εy. (92)




As long as (91) and (92) hold, we have

V ≤ −[(1− %)α0λm(P )− c]‖z‖2 + µ‖w‖2 +ε204c

(93)

where ε0 = 2(s1εu + s2εy), and c is a constant satisfying0 < c < (1− %)α0λm(P ).

Since ‖p‖ ≤ `‖Cq‖‖x‖, ‖δp‖ ≤ `‖Cq‖(‖xe‖ + ‖e‖), and‖δp‖ ≤ `(‖Cq +L1C‖‖e‖+ ‖L1Fw‖‖w‖+ ‖L1‖‖ye‖), from(86) we have ‖z‖ ≤ η1‖z‖+ η2‖xe‖+ η3‖w‖+ η7‖ye‖.

Similar to the argument in the proof of Theorem 3, wecan show the following inequality holds when ‖xe‖ 6= 0 and‖z‖ 6= 0:

d

dt(

‖xe‖√2σu‖z‖+ εu

) ≤√

2(1 +σu‖xe‖√

2σu‖z‖+ εu)×

(η5 +η2‖xe‖√

2σu‖z‖+ εu+

η7‖ye‖√2σu‖z‖+ εu

). (94)

It is easy to verify that η7‖ye‖√2σu‖z‖+εu

≤ d1 η7‖ye‖σy‖C‖‖z‖+εy . Hence,

from (94) we have

d

dt(

‖xe‖√2σu‖z‖+ εu

) ≤√

2(1 +σu‖xe‖√

2σu‖z‖+ εu)×

(η5 +η2‖xe‖√

2σu‖z‖+ εu+ d1

η7‖ye‖σy‖C‖‖z‖+ εy

).

When ‖xe‖ = 0 or ‖z‖ = 0, the upper right-hand derivative of‖xe‖√

2σu‖z‖+εucan be calculated similar to the proof of Theorem

3, which can still be captured by the inequality above.Since ‖ye‖ = ‖y‖ ≤ ‖C‖‖x‖ ≤ ‖C‖‖z‖, we can show

that the following inequality holds using arguments similar tothose used above:

d

dt(

‖ye‖σy‖C‖‖z‖+ εy

) ≤ ‖C‖(1 +σy‖ye‖

σy‖C‖‖z‖+ εy)×

(η6 +η7‖ye‖

σy‖C‖‖z‖+ εy+ d2

η2‖xe‖√2σu‖z‖+ εu

)

where the discussion on using the upper right-hand derivativeis omitted since it is similar to that used in the proof ofTheorem 3.

It is not hard to show that the time it takes for ‖xe‖ (resp.‖ye‖) to evolve from 0 to

√2σu‖z‖+εu (resp. σy‖C‖‖z‖+εy)

is lower bounded by τu (resp. τy), which implies that (91)holds during [tuk , t

uk + τu), and (92) holds during [tyk, t

yk + τy),

for any k ≥ 0. Recalling that ‖xe‖ ≤ σu‖x‖+ εu implies (91)and ‖ye‖ ≤ σy‖y‖+ εy implies (92), the triggering rules (83)and (85) guarantee that (91) holds during the interval [tuk , t

uk+1)

for any k ≥ 0, and (92) holds during the interval [tyk, tyk+1) for

any k ≥ 0. Hence, (93) holds for any t ≥ 0, implying that thefunction V is an ISpS-Lyapunov function since it satisfies (8)with γ(‖z‖) = [(1− %)α0λm(P )− c]‖z‖2 ∈ K∞, χ(‖w‖) =µ‖w‖2 ∈ K and d = ε20/4c > 0. The conclusion follows byProposition 1.

Remark 11. The assumption V ≤ −α0V +µ‖w‖2 in Theorem3 and 4 can be verified by using Theorem 1. Therefore,Theorem 1, 3 and 4 altogether provide a systematic and con-structive approach to design observer-based event-triggeredcontrollers. One limitation of this ETC design, however, is that

it relies on the global Lipschitz constant which is normally veryconservatively computed.

Remark 12. Similar to Remark 10, there are trade-offsin choosing parameters in triggering rules (83) and (85);for example, smaller εu, εy reduces the ultimate bounds butdecreases the inter-execution times.

V. SIMULATION EXAMPLE

In this section, we use a single-link robot arm example givenin [62] and the configuration of Figure 2 to illustrate Theorem4. Dynamics of the single-link robot arm are expressed as:

x1 = x2,

x2 = − sin(x1) + u+ w,

y = x1,

where x = (x1, x2)> is the state representing the angleand the rotational velocity, u is the input representing thetorque, and w is the external disturbance. The system can be

written in the form of (1) with A =

(0 10 0

), B =

(01

),

C = (1, 0), D = 0, E =

(0−1

), Ew =

(01

), Fw = 0,

Cq = (1, 0) and p(q) = sin(q). The nonlinearity p satisfies

(2) with M =

(1 00 −1

). Recalling Remark 2, p satis-

fies Assumption 1 and 2 with T1 = T2 =

(I 00 I

)and

N1 = {(λ1I, λ1I)|λ1 > 0}, N2 = {( 1λ2I, 1

λ2I)|λ2 > 0},

which means that X1 = Y1 = λ1I , X2 = Y2 = λ2I .Additionally, the corresponding M24 = −1 < 0. By lettingα1 = α2 = 1, µ1 = µ2 = 0.1, the LMIs (29)-(30)with variables λ1, λ2, P1, P2 are feasible, from which we

can obtain matrix gains L1 = −1, L2 =

(−5.1294−18.0352

),

K1 = (−7.3936,−3.9937), K2 = 1. The observer is givenin (84) with L1, L2 above, and the controller is given in (71)with K1,K2 above. We then let α0 = 0.25, w0 = 0.02 andrecompute P via (20) with the objective to be minimizingthe condition number of P . With % = 0.8, a1 = a2 = 0.5,εu = εy = 0.005, we can calculate that σy = 0.0017,σu = 0.0023, and τu ≥ 1.07 × 10−4 s, τu ≥ 7.68 × 10−5

s. In the simulations, we suppose that the random disturbancew is uniformly generated from [−w0, w0], and the initialconditions of the plant and the observer are (0.1,−0.15) and(−0.1, 0.05), respectively. The simulation results are shown inFigure 3 through Figure 7.

Figure 3 and Figure 4 show trajectories of the state x andthe estimation error e, respectively. Both x and e eventuallyenter a small neighborhood of the origin as expected. Figure 5shows inter-execution times {tyk+1− t

yk} in the observer ETM

(83), and Figure 6 shows inter-execution times {tuk+1 − tuk}in the controller ETM (85). Figure 7 shows the trajectory ofthe piecewise constant input u(t) that is fed into the plant. Itis readily seen that the control input u(t) updates its valuesat each sampling time t = tuk , which is determined by thetriggering rule (85).




Denote τmin[T1,T2]and τavg[T1,T2]

as the minimal and averageinter-execution times during the time interval [T1, T2], respec-tively. The values of τ [0,20]min , τ [0,20]avg , τ [3,20]min , τ [3,20]avg for theobserver ETM and the controller ETM are summarized inTable I. We notice that after 3 seconds, the controller inputis updated about every 0.36 seconds on average, and the plantoutput is updated about every 1.09 seconds on average, whichshows the effectiveness of our control design.

TABLE IMINIMAL AND AVERAGE INTER-EXECUTION TIMES FOR OBSERVER AND

CONTROLLER ETMS

τmin[0,20] τavg[0,20] τmin[3,20] τavg[3,20]

Observer ETM 0.0106 s 0.1945 s 0.2104 s 1.0977 s

Controller ETM 0.0013 s 0.0663 s 0.0903 s 0.3665 s

0 2 4 6 8 10 12 14 16 18 20-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 3. Trajectory of the plant state x.

0 2 4 6 8 10 12 14 16 18 20-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 4. Trajectory of the estimation error e = x− x.

VI. CONCLUSION

In this paper, we studied observer-based, global stabilizingcontrol design for incrementally quadratic nonlinear systemsaffected by external disturbances and measurement noise. Weproposed LMI-based sufficient conditions for the simultaneousdesign of the observer and the controller in the continuous-time domain for two parameterizations of the incrementalmultiplier matrices. Based on that, we investigated ETMdesign within the observer-based controller setting for globally

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

Fig. 5. Inter-execution times {tyk+1 − tyk} in the observer ETM (83).

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Fig. 6. Inter-execution times {tuk+1 − tuk} in the controller ETM (85).

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 7. Trajectory of the input u(t).

Lipschitz systems. The simulation example showed the effec-tiveness of the controller design and the proposed triggeringrule.

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Xiangru Xu is currently an Assistant Professor inthe Department of Mechanical Engineering at theUniversity of Wisconsin-Madison, Madison, Wis-consin, USA. He received his B.S. degree fromBeijing Normal University, Beijing, and his Ph.D.degree from Chinese Academy of Sciences, Bei-jing. Before joining UW-Madison, he served as apostdoctoral scholar in the Department of ElectricalEngineering and Computer Science at the Universityof Michigan, Ann Arbor, and the Department ofAeronautics & Astronautics at the University of

Washington, Seattle. His research interests include safety-critical control,nonlinear control, and autonomy.

Behcet Acıkmese is a Professor in the Departmentof Aeronautics and Astronautics at the Universityof Washington, Seattle. He received his Ph.D. fromPurdue University and was a Visiting Assistant Pro-fessor at Purdue University before joining NASAJet Propulsion Laboratory (JPL) in 2003. He was asenior technologist at JPL and a lecturer at Caltech,where he developed GN&C algorithms for planetarylanding, formation flying spacecraft, and asteroidand comet sample return missions. He is the devel-oper of the “flyaway” GN&C algorithms in Mars

Science Laboratory, which landed on Mars in August, 2012. He was a facultyat the University of Texas at Austin from 2012 to 2015.

Martin J. Corless is currently a Professor in theSchool of Aeronautics and Astronautics at PurdueUniversity, West Lafayette, Indiana, USA. He isalso an Adjunct Honorary Professor in the Hamil-ton Institute at The National University of Ireland,Maynooth, Ireland. He received a B.E. from Univer-sity College Dublin, Ireland and a Ph.D. from theUniversity of California at Berkeley; both degreesare in mechanical engineering. He is the recipientof a National Science Foundation Presidential YoungInvestigator Award. His research is concerned with

obtaining tools which are useful in the robust analysis and control of systemscontaining significant uncertainty and in applying these results to aerospaceand mechanical systems and to sensor and communication networks.


xu.me.wisc.edu...0018-9286 (c) 2020 ieee. personal use is permitted, but...

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