# LMI-Based Analysis and Stabilization of Nonlinear Descriptors with Multiple Delays via Delayed Nonlinear Controller Schemes

Authors: Javier Adrián Romero-Vega; Raúl Villafuerte-Segura; Víctor Estrada-Manzo
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper presents a convex approach for nonlinear descriptor systems with multiple delays; it allows designing delayed nonlinear controllers such that the closed-loop system holds exponential estimates for convergence. The proposal takes advantage of an equivalent convex representation of the given descriptor model together with Lyapunov-Krasovskii functionals; thus, the conditions are in the form of linear matrix inequalities, which can be efficiently solved by commercially available software. To avoid possible saturation in the actuators, conditions for bounding the control input are also given. Numerical and academic examples illustrate the performance of the proposal.


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## Body

## 1. Introduction

In the last decades, a large number of results concerning the analysis and stabilization of systems by means of the direct Lyapunov method [1], since the publication of the book [2], linear matrix inequalities (LMIs) have become a preferred solution to many control problems [3, 4], as they can be effectively solved by means of convex optimization techniques already implemented in commercially available software [5, 6]. These ideas have been extended to the analysis of time-delay systems (TDSs) via LyapunovKrasovskii (L-K) functionals [7] or LyapunovRazumikhin (L-R) functions [8]. In this context, there are several results that provide sufficient stability conditions using LMI-based approaches for different classes of TDS, such as linear time-delay systems [9–13], uncertain linear time-delay systems [14–16], neutral linear systems [17–20], systems with uncertain time-invariant delays [21], descriptor system approach for TDS [22], linear parameter-varying (LPV) time-delay systems [23], systems with time-varying delays [24–29], exponential estimates for TDS [30, 31], systems with polytopic-type uncertainties [32], singular systems [33], neural networks with time delay [34, 35], and genetic regulatory networks with probabilistic time delays [36]. Recently, in [37] convex approaches are employed to provide robust stability conditions based on quasi-polynomials.In general, delays are undesirable phenomena, because they can destabilize or produce a poor performance in the system response. However, in recent years, it has been shown that delays can also stabilize and improve the close-loop performance of a system. Moreover, the deliberate induction of delays by means of the control law is an efficient alternative to stabilize systems [38, 39]; these types of controllers are known as delayed ones. For example, in [40–42], a proportional control with an appropriate delay replaces a traditional proportional-derivative one; thus, the system response is fast and insensitive to high-frequency noise. In [43], a scheme called time-delayed feedback control (TDFC) is proposed, originating different investigations [44–55].As mentioned above, LMI-based approaches have become important in the control community; however, in the context of TDS, there is an inherent conservatism for stability and stabilization conditions even for linear setups [37]; this leaves room for improvements. Moreover, a problem little explored by the community is obtaining stability conditions for nonlinear TDS. Although, originally, LMI-based stability conditions were given for linear time-invariant (LTI) systems, these have also been used on LPV [3] and nonlinear setups via exact Takagi-Sugeno (TS) [56]. The latter case employs the sector of no linearity approach [57] which allows rewriting the original nonlinear model as a convex one by means of scalar convex functions that capture uncertainties and nonlinearities. This technique has also been applied to a class of nonlinear TDS; for instance, in [58, 59], sufficient LMI conditions are proposed; in [60], uncertain TS systems are considered; in [61], sufficient LMI conditions have been given for a class of nonlinear systems. A larger family of functionals is explored in [62]. Nonetheless, none of these previous works deal with nonlinear descriptor systems; they appear when using the EulerLagrange formalism for modeling plants [63]. In the context of convex descriptor models without delays, there are some recent works [64, 65]; time-delay nonlinear descriptor systems are a few works in the literature; for instance, in [66], LMI stability and stabilization conditions have been developed for systems with only one time-delay.Contribution: this paper proposes an LMI methodology for analysis and stability of nonlinear descriptor systems with multiple delays, thus overcoming recent results in the literature. For example, the work [4] only considers linear in standard form systems with multiple delays, [52] only studies nonlinear systems in standard form, and [66] treats nonlinear systems in descriptor with one delay. Additionally, to avoid possible saturation in the actuators, LMI conditions for bounding the control signal are established. Numerical and academic examples illustrate that including delays in the controller can reduce noise in the control signal, which increases the useful life of the actuators.The paper is organized as follows: the problem statement and preliminary results are shown in Section2. LMI-based stability analysis and delayed nonlinear controller design conditions for a class of nonlinear descriptors systems with multiple delays are given in Section 3, and additionally conditions for input constraints are also given. In Section 4, the implementation and numerical validation of the previous theoretical results are provided. Concluding remarks are stated in Section 5.


## 2. Problem Statement and Preliminary Results

### 2.1. Problem Statement

Let us consider a nonlinear descriptor system under multiple delays of the following form:(1)Exx˙t=Axxt+∑h=1dAτhxxt−τh+Bxu,xθ=ϕθ,θ∈−τ,0,where x∈ℝn is the state vector, u∈ℝm is the input vector, Ex, Ax, Bx, and Aτhx are matrix functions assumed to be smooth and bounded, 0<τ1<τ2<⋯<τd=τ are time delays, and ϕ∈C−τ,0,ℝn are the initial functions, where C−τ,0,ℝn is the Banach space of real continuous functions on the intervals −τ,0 with the following norm:(2)ϕτ=maxθ∈−τ,0ϕθ,where ⋅ stands for the Euclidean norm in ℝn. It is assumed that for each initial condition ϕ∈C−τ,0,ℝn, t>0 there exists a unique solution xt,ϕ of the system; moreover, xtϕ=xt+θ,ϕ:θ∈−τ,0. In this work, matrix Ex is assumed to be invertible, at least in a region including the origin; this is a common assumption when studying systems (1) derived from the EulerLagrange formalism [63, 65].In order to obtain LMI conditions, the sector nonlinearity approach [57] is employed to compute an exact convex representation of (1). This methodology begins by defining a premise vector zx∈ℝp whose entries are different nonconstant terms in Ax, Aτhx, Bx, h∈1,2,…,d; similarly, ζx∈ℝq is the premise vector with nonconstant terms in Ex. It is assumed that each entry of the vectors zx and ζx is bounded in the compact set Ωx that includes the origin, that is, zix∈zi0,zi1 and ζkx∈ζk0,ζk1. Thus, each of them can be expressed as convex sums of their bounds:(3)zix=zi0w0ix+zi1w1ix,i∈1,2,…,p,ζkx=ζk0ω0kx+ζk1ω1kx,k∈1,2,…,q,where(4)w0iz=zi1−zixzi1−zi0,w1ix=1−w0iz,ω0kζ=ζk1−ζkxζk1−ζk0,ω1kζ=1−ω0kζ,are scalar convex functions holding the convex sum property for all x∈Ωx, i.e., 0≤wiz≤1, w1i+w0i=1, 0≤ωkζ≤1, and ω1k+ω0k=1. Then, the so-called scheduling (membership) functions can be computed:(5)wiz=wi11xwi22x⋯wirpx,ij∈0,1,ωkζ=ωk11xωk22x⋯ωkρqx,kj∈0,1,where i∈1,2,…,r, r=2p, and indexes i1i2⋯ip are chosen as a p-digit binary representation of i−1; similarly, k∈1,2,…,ρ, ij∈0,1, ρ=2q; and the set k1k2⋯kq is a q-digit binary representation of k−1. The scheduling functions also hold the convex sum property in Ωx. Finally, an equivalent convex representation of (1) is [67](6)∑k=1ρωkζEkx˙t=∑i=1rwizAixt+∑h=1dAτhixt−τh+Biu,where Ek=Ex|ωkζ=1, Ai=Ax|wiz=1, Aτhi=Aτhx|wiz=1, h∈1,2,…,d, and Bi=Bx|wiz=1 are constants matrices; r=2p and ρ=2q are the number of vertices for the right and left side of (6), respectively. It is important to notice that (6) is a convex rewriting of (1); thus, all the conclusions derived from the former directly apply to the latter.


### 2.2. Notation and Properties

In the following, convex sums of matrices will be shortly represented by(7)ϒw=∑i=1rwizϒi,ϒω=∑k=1ρωkζϒk,ϒw−1=∑i=1rwizϒi−1,ϒwwω=∑i=1r∑j=1r∑k=1ρwizwjzωkζϒijk.Thus, (6) is expressed as Eωx˙t=Awxt+∑h=1mAτhwxt−τh+Bwut. Additionally, an asterisk ∗ will be employed in matrix expressions to denote the transpose of the symmetric element; for in-line ones, it indicates the transpose of the terms in its left-hand side, that is, A+B+AT+BT+C=A+B+∗+C.Usually, when deriving LMI conditions for convex descriptor models, the designer is faced to inequalities of the formϒwwω<0; the scheduling functions are dropped off by means of the following relaxation lemma:Lemma 1 (see [68]).
Letϒijk=ϒijkT, i,j=1,2,…,r2, and k∈1,2,…,ρ be matrices of adequate sizes. Then,(8)∑i=1r∑j=1r∑k=1ρwizwjzωkζϒijk<0,holds if the following LMIs,(9)2r−1ϒiik+ϒijk+ϒjik<0,are satisfied for all i,j=1,2,…,r2, k∈1,2,…,ρ.The following results establish the exponential estimates for time-delay nonlinear systems:Lemma 2 (see [30]).
Consider system (1). If there exists a functional V⋅ and positive constants c1, c2, and α, such that(1)
c1x2≤Vxt≤c2xtτ2,(2)
V˙xt+2αVxt<0,then, the solutions xt,ϕ of the system (1) satisfy the exponential estimates:(10)xt,ϕ≤c2c1e−αtϕτ.As customary, for the analysis and design of convex descriptor models, the so-called descriptor redundancy is employed [69]; in our case, the augmented vectors x¯t=xTtx˙TtT and x¯t−τh=xTt−τhx˙Tt−τhT, h∈1,2,…,d, are employed to rewrite (6) as follows:(11)E¯x¯˙t=A¯wωx¯t+∑h=1dA¯τhwx¯t−τh+B¯wu,with(12)E¯=I000,A¯wω=0IAw−Eω,A¯τhw=00Aτhw0,B¯w=0Bw,h∈1,2,…,d.In what follows, the stability and stabilization conditions are derived from the augmented system (11); nevertheless, it is important to stress that the system under study has the form (1).Let us recall previous works on the subject. The work [66] studies the stability and stabilization of a nonlinear descriptor system with only one delay, that is (1) with d=1. For stabilization purposes, the following control law is proposed:(13)u=Kwωxt+Fwωxt−τ,with Kwωxt=∑j=1r∑k=1ρwjzxωkζxKjk and Fwω=∑j=1r∑k=1ρwjzxωkζxFjk; it is a nonlinear control law with nonlinearities of both sides of the nonlinear descriptor model. In the Section 3, a generalization of this controller will be presented.


## 2.1. Problem Statement

Let us consider a nonlinear descriptor system under multiple delays of the following form:(1)Exx˙t=Axxt+∑h=1dAτhxxt−τh+Bxu,xθ=ϕθ,θ∈−τ,0,where x∈ℝn is the state vector, u∈ℝm is the input vector, Ex, Ax, Bx, and Aτhx are matrix functions assumed to be smooth and bounded, 0<τ1<τ2<⋯<τd=τ are time delays, and ϕ∈C−τ,0,ℝn are the initial functions, where C−τ,0,ℝn is the Banach space of real continuous functions on the intervals −τ,0 with the following norm:(2)ϕτ=maxθ∈−τ,0ϕθ,where ⋅ stands for the Euclidean norm in ℝn. It is assumed that for each initial condition ϕ∈C−τ,0,ℝn, t>0 there exists a unique solution xt,ϕ of the system; moreover, xtϕ=xt+θ,ϕ:θ∈−τ,0. In this work, matrix Ex is assumed to be invertible, at least in a region including the origin; this is a common assumption when studying systems (1) derived from the EulerLagrange formalism [63, 65].In order to obtain LMI conditions, the sector nonlinearity approach [57] is employed to compute an exact convex representation of (1). This methodology begins by defining a premise vector zx∈ℝp whose entries are different nonconstant terms in Ax, Aτhx, Bx, h∈1,2,…,d; similarly, ζx∈ℝq is the premise vector with nonconstant terms in Ex. It is assumed that each entry of the vectors zx and ζx is bounded in the compact set Ωx that includes the origin, that is, zix∈zi0,zi1 and ζkx∈ζk0,ζk1. Thus, each of them can be expressed as convex sums of their bounds:(3)zix=zi0w0ix+zi1w1ix,i∈1,2,…,p,ζkx=ζk0ω0kx+ζk1ω1kx,k∈1,2,…,q,where(4)w0iz=zi1−zixzi1−zi0,w1ix=1−w0iz,ω0kζ=ζk1−ζkxζk1−ζk0,ω1kζ=1−ω0kζ,are scalar convex functions holding the convex sum property for all x∈Ωx, i.e., 0≤wiz≤1, w1i+w0i=1, 0≤ωkζ≤1, and ω1k+ω0k=1. Then, the so-called scheduling (membership) functions can be computed:(5)wiz=wi11xwi22x⋯wirpx,ij∈0,1,ωkζ=ωk11xωk22x⋯ωkρqx,kj∈0,1,where i∈1,2,…,r, r=2p, and indexes i1i2⋯ip are chosen as a p-digit binary representation of i−1; similarly, k∈1,2,…,ρ, ij∈0,1, ρ=2q; and the set k1k2⋯kq is a q-digit binary representation of k−1. The scheduling functions also hold the convex sum property in Ωx. Finally, an equivalent convex representation of (1) is [67](6)∑k=1ρωkζEkx˙t=∑i=1rwizAixt+∑h=1dAτhixt−τh+Biu,where Ek=Ex|ωkζ=1, Ai=Ax|wiz=1, Aτhi=Aτhx|wiz=1, h∈1,2,…,d, and Bi=Bx|wiz=1 are constants matrices; r=2p and ρ=2q are the number of vertices for the right and left side of (6), respectively. It is important to notice that (6) is a convex rewriting of (1); thus, all the conclusions derived from the former directly apply to the latter.


## 2.2. Notation and Properties

In the following, convex sums of matrices will be shortly represented by(7)ϒw=∑i=1rwizϒi,ϒω=∑k=1ρωkζϒk,ϒw−1=∑i=1rwizϒi−1,ϒwwω=∑i=1r∑j=1r∑k=1ρwizwjzωkζϒijk.Thus, (6) is expressed as Eωx˙t=Awxt+∑h=1mAτhwxt−τh+Bwut. Additionally, an asterisk ∗ will be employed in matrix expressions to denote the transpose of the symmetric element; for in-line ones, it indicates the transpose of the terms in its left-hand side, that is, A+B+AT+BT+C=A+B+∗+C.Usually, when deriving LMI conditions for convex descriptor models, the designer is faced to inequalities of the formϒwwω<0; the scheduling functions are dropped off by means of the following relaxation lemma:Lemma 1 (see [68]).
Letϒijk=ϒijkT, i,j=1,2,…,r2, and k∈1,2,…,ρ be matrices of adequate sizes. Then,(8)∑i=1r∑j=1r∑k=1ρwizwjzωkζϒijk<0,holds if the following LMIs,(9)2r−1ϒiik+ϒijk+ϒjik<0,are satisfied for all i,j=1,2,…,r2, k∈1,2,…,ρ.The following results establish the exponential estimates for time-delay nonlinear systems:Lemma 2 (see [30]).
Consider system (1). If there exists a functional V⋅ and positive constants c1, c2, and α, such that(1)
c1x2≤Vxt≤c2xtτ2,(2)
V˙xt+2αVxt<0,then, the solutions xt,ϕ of the system (1) satisfy the exponential estimates:(10)xt,ϕ≤c2c1e−αtϕτ.As customary, for the analysis and design of convex descriptor models, the so-called descriptor redundancy is employed [69]; in our case, the augmented vectors x¯t=xTtx˙TtT and x¯t−τh=xTt−τhx˙Tt−τhT, h∈1,2,…,d, are employed to rewrite (6) as follows:(11)E¯x¯˙t=A¯wωx¯t+∑h=1dA¯τhwx¯t−τh+B¯wu,with(12)E¯=I000,A¯wω=0IAw−Eω,A¯τhw=00Aτhw0,B¯w=0Bw,h∈1,2,…,d.In what follows, the stability and stabilization conditions are derived from the augmented system (11); nevertheless, it is important to stress that the system under study has the form (1).Let us recall previous works on the subject. The work [66] studies the stability and stabilization of a nonlinear descriptor system with only one delay, that is (1) with d=1. For stabilization purposes, the following control law is proposed:(13)u=Kwωxt+Fwωxt−τ,with Kwωxt=∑j=1r∑k=1ρwjzxωkζxKjk and Fwω=∑j=1r∑k=1ρwjzxωkζxFjk; it is a nonlinear control law with nonlinearities of both sides of the nonlinear descriptor model. In the Section 3, a generalization of this controller will be presented.


## 3. LMI Conditions for Descriptor Systems with Multiple Delays

In this section, the developments are based on the following LyapunovKrasovskii functional candidate:(14)Vxτ=x¯TtE¯TP¯wx¯t+∑h=1d∫−τh0x¯Tt+θE¯TQ¯he2αθx¯t+θdθ,with(15)P¯w=P10P2wP3w,Q¯h=Qh000,E¯TP¯w=P¯wTE¯≥0,α>0,P1>0,Qh>0,h∈1,2,…,d.Note that, the functional (14) is a valid L-K functional candidate as it reduces to(16)Vxτ=xTtP1xt+∑h=1d∫−τh0xTt+θQhe2αθxt+θdθ,which is clearly positive definite and bounded by c1x2≤Vxt≤c2xtτ2, with c1=λminP1 and c2=λmaxP1+∑h=1dτhλmaxQh, thus fulfilling conditions (1) in Lemma 2.For the analysis of system (1), i.e., when u=0, we have the following result.Theorem 1.
The origin of system (1), with u=0 and an exact convex representation (6), is exponentially stable if the exist matrices P1>0, P2j, P3j, Qh>0 with h∈1,2,…,d, j∈1,2,…,r, and a scalar α>0 such that LMIs (9) hold with(17)ϒijk≔P2jTAi+AiTP2j+2αP1+∑h=1dQh∗∗∗∗P1−EkTP2j+P3jTAi−EkTP3j−P3jTEk⋯∗∗P2jTAτ1iAτ1iTP3j−e−2ατ1Q1⋯∗⋮⋮⋯⋱∗P2jTAτdiAτdiTP3j0⋯−e−2ατdQd.wherei,j=1,2,…,r2 and k∈1,2,…,ρ. Additionally, the solution xt,ϕ of (1) satisfies the exponential estimates:(18)xt,ϕ≤c2c1e−αtϕτ,with c1=λminP1 and c2=λmaxP1+∑h=1dτhλmaxQh,h∈1,2,…,d.Proof.
The time derivative of (14) along the trajectories of (11) is V˙xτ=V1+V2 with(19)V1=x¯TtP¯wTE¯x¯˙t+x¯˙TtE¯TP¯wx¯t,V2=∑h=1mddt∫−τh0x¯Tt+θE¯TQ¯he2αθx¯t+θdθ,which once the dynamics of (11) are substituted and using Leibniz’s rule yield(20)V1=x¯TtP¯wTA¯wωx¯t+∑h=1dA¯τhwx¯t−τh+A¯wωx¯t+∑h=1dA¯τhwx¯t−τhTP¯wx¯t,V2=∑h=1dx¯TtE¯TQ¯hx¯t−x¯Tt−τhe−2ατhE¯TQ¯hx¯t−τh−2α∫−τh0x¯Tt+θE¯TQ¯he2αθx¯t+θdθ.
From the latter, we have thatV˙xt+2αVxt is equal to(21)x¯tx¯t−τ1⋮x¯t−τdTP¯wTA¯wω+A¯wωTP¯w+2αE¯TP¯w+∑h=1dE¯TQ¯h∗⋯∗A¯τ1wTP¯w−e−2ατ1E¯Q¯1⋯∗⋮⋮⋱∗A¯τdwTP¯w0⋯−e−2ατdE¯Q¯dx¯tx¯t−τ1⋮x¯t−τd.
Thus, in order to fulfill condition (2) in Lemma2, V˙xt+2αVxt<0 can be established after some algebraic manipulations when developing vectors x¯t and x¯t−τh,h∈1,2,…,d, by the following inequality:(22)P2wTAw+AwTP2w+2αP1+∑h=1dQh∗∗∗∗P1−EωTP2w+P3wTAw−EωTP3w−P3wTEω⋯∗∗P2wTAτ1wAτ1wTP3w−e−2ατ1Q1⋯∗⋮⋮⋯⋱∗P2wTAτdwAτdwTP3w0⋯−e−2ατdQd<0.
Finally, in order to drop off the scheduling functionsw and ω via the relaxation scheme in Lemma 1, the proof is concluded.
Let us consider system (1). Now, the task is to design a multiple delayed PDC control law of the following form:(23)u=Kwωxt+∑h=1dFhwωxt−τh,where Kwω=∑j=1r∑k=1ρwjzxωkζxKjk and Fhwω=∑j=1r∑k=1ρwjzxωkζxFhjk,h∈1,2,…,d are nonlinear gains to be designed via the augmented system (11); thus, (23) can be expressed as ut=K¯wωx¯t+∑h=1dF¯hwωx¯t−τh with K¯wω=Kwω0 and F¯hwω=Fhwω0. The following result provides LMI conditions for the design of the control law (23). It is based on a slightly modification of the L-K functional candidate (14), that is,(24)Vxτ=x¯TtE¯TP¯w−1x¯t+∑h=1d∫−τh0x¯Tt+θE¯TQ¯he2αθx¯t+θdθ,with(25)E¯TP¯w−1=P¯w−TE¯≥0,P¯w−1=P10P2wP3w−1,Q¯h=Qh000,P1>0,Qh>0,α>0,h∈1,2,…,d.Theorem 2.
The origin of system (1) with an exact convex representation (6), under the law of control (23), is exponentially stable if existing matrices P1>0, P2j, P3j, Rh>0, Mjk, Nhjk,h∈1,2,…,d,j∈1,2,…,r, and a scalar α>0 if the LMIs (9) hold with the following:(26)ϒijk≔P2jT+P2j+2αP1+∑h=1dRh∗∗∗∗AiP1+BiMjk−EkP2j+P3jT−EkP3j−P3jTEkT⋯∗∗0P1Aτ1iT+N1jkTBiT−e−2ατ1R1⋯∗⋮⋮⋯⋱∗0P1AτdiT+NdjkTBiT0⋯−e−2ατdRd.Then, the vertex control gains are computed asKjk=MjkP1−1 and Fhjk=NhjkP1−1,j∈1,2…,r,k∈1,2,…,ρ,andh∈1,2,…,d. Moreover, Qh=P1−1RhP1−1,h∈1,2,…,d and the solution satisfies the following exponential estimates:(27)xt,ϕ≤c2c1e−αtϕτ,where c1=λminP1−1 and c2=λmaxP1−1+∑h=1dτhλmaxQh.Proof.
Using the augmented system (11) and its corresponding control law, the closed-loop system is(28)E¯x¯˙t=A¯wω+B¯wK¯wωx¯t+∑h=1dA¯wτh+B¯wF¯hwωx¯t−τh.
Similar to the proof of Theorem1, consider the functional (24) and its time derivative V˙xt=V1+V2 with(29)V1=x¯˙TtE¯TP¯w−1x¯t+x¯TtP¯w−TE¯x¯˙t,V2=∑h=1dddt∫−τh0x¯Tt+θE¯TQ¯he2αθx¯t+θdθ.
Substituting the dynamics of (28) in V1 while using Leibniz’s rule in V2, we have(30)V˙1=A¯wω+B¯wK¯wωx¯t+∑h=1dA¯wτh+B¯wF¯hwωx¯t−τhTP¯w−1x¯t+x¯TtP¯w−TA¯wω+B¯wK¯wωx¯t+∑h=1dA¯wτh+B¯wF¯hwωx¯t−τh,V2=∑h=1dx¯TtE¯TQ¯hx¯t−x¯Tt−τhe−2ατhE¯TQ¯hx¯t−τh−2α∫−τh0x¯Tt+θE¯TQ¯he2αθx¯t+θdθ.
Therefore,V˙xt+2αVxt is equivalent to(31)x¯tx¯t−τ1⋮x¯t−τdTP¯w−TA¯wω+B¯wK¯wω+∗+2αE¯TP¯w−1+∑h=1dE¯TQ¯h∗…∗A¯τ1w+B¯wF¯1wωTP¯w−1−e−2ατ1E¯TQ¯1…∗⋮…⋱∗A¯τdw+B¯wFdwωTP¯w−10…−e−2ατdE¯TQ¯dx¯tx¯t−τ1⋮x¯t−τd.
After some simplifications,V˙xt+2αVxt<0 (condition (2) in Lemma 2 holds if(32)−P1−TP2wP3w−TAw+BwKwω+∗+2αP1−1+∑h=1dQh∗∗∗∗P3w−TAw+BwKwω+Pi−1+EωTP3w−1P1−1−EωTP3w−1−P3w−TEω…∗∗−Aτ1wT+F1wωTBWTP3w−1P2wP1−1Aτ1wT+F1wωTP3w−1−e−2ατ1Q1…∗⋮⋮…⋱∗−AτdwT+FdwωTBWTP3w−1P2wP1−1AτdwT+FdwωTBWTP3w−10…−e−2ατdQd<0,holds too; however, from the previous inequality, one cannot directly obtain LMI conditions. Thus, by means of the congruence property, that is, pre- and postmultiplying by the matrix block−diagP10P2wP3w,P1,…,P1︸d gives(33)ϒwwω≔P2wT+P2w+2αP1+∑h=1dRh∗∗∗∗AwP1+BwMwω−EωP2w+P3wT−EωP3w−P3wTEωT…∗∗0P1Aτ1wT+N1wωTBwT−e−2ατ1R1…∗⋮⋮…⋱∗0P1AτdwT+NdwωTBwT0…−e−2ατdRd<0,with the definitions Mwω=KwωP1 and Nhwω=FhwωP1,h∈1,2,…,d. Now, the previous inequality can be translated into the LMI conditions in the theorem once the relaxation scheme in Lemma 1 is applied.
Recall that, by hypothesis matrix,Ex in (1) is invertible in a region around the origin; thus, it is always possible to calculate a standard state-space form as follows:(34)x˙t=A˜xxt+∑h=1dA˜τhxxt−τh+B˜xu,where A˜x=E−1xAx, B˜x=E−1xBx, and A˜τhx=E−1xAτhx,h∈1,2,…,d. Naturally, it is possible to obtain a convex representation of system (34), that is,(35)x˙t=A˜wxxt+∑h=1dA˜τhwxxt−τh+B˜wxu,with A˜w=∑i=1r˜wizA˜i, A˜τhw=∑i=1r˜wizA˜τhi, and B˜w=∑i=1r˜wizB˜i, where r˜ is the number of vertex models, h∈1,2,…,d. Even though nonlinear systems (34) and (1) and their convex forms are equivalent, establishing exponential stability of them via LMIs may lead to different feasibility set solution. Keeping the original descriptor form (1) results in a convex representation with less vertex matrices; this, in general, yields less conservative results [64].
Thus, the following result provides stability and stabilization conditions for systems of the form (34) by means of the L-K functional (14) (for stability) and (24) (for stabilization), respectively.Corollary 1.
Stability: the origin of system (1), with u=0 and an exact convex representation (35), is exponentially stable if the exist matrices P1>0, P2j, P3j, and Qh>0 with h∈1,2,…,d, j∈1,2,…,r˜, and a scalar α>0 such that LMIs,(36)2r˜−1ϒii+ϒij+ϒji<0,∀i,j=1,2,…,r˜2,hold with(37)ϒij≔P2jTA˜i+A˜iTP2j+2αP1+∑h=1dQh∗∗∗∗P1−P2j+P3jTA˜i−P3j−P3jT⋯∗∗P2jTA˜τ1iA˜τ1iTP3j−e−2ατ1Q1⋯∗⋮⋮⋯⋱∗P2jTA˜τdiA˜τdiTP3j0⋯−e−2ατdQd.Moreover, the solutionxt,ϕ satisfies the following exponential estimates:(38)xt,ϕ≤c2c1e−αtϕτ.Stabilization: the origin of system (1) with an exact convex representation (35), under the law of control u=K˜wωxt+∑h=1dF˜hwωxt−τh, is exponentially stable if existing matrices P1>0, P2j, P3j, Rh>0, M˜j, N˜hj,h∈1,2,…,d,j∈1,2,…,r˜, and a scalar α>0 if the LMIs (36) hold with(39)ϒij≔P2jT+P2j+2αP1+∑h=1dRh∗∗∗∗A˜iP1+B˜iM˜j−P2j+P3jT−P3j−P3jT…∗∗0P1A˜τ1iT+N˜1jTB˜iT−e−2ατ1R1…∗⋮⋮…⋱∗0P1A˜τdiT+N˜djTB˜iT0…−e−2ατdRd.Then, the vertex control gains are computed asK˜j=M˜jP1−1 and F˜hj=N˜hjP1−1, F˜hj=N˜hjP1−1,j∈1,2…,r˜,h∈1,2,…,d. Then, the solution satisfies the following exponential estimates:(40)xt,ϕ≤c2c1e−αtϕτ.Proof.
It follows a similar path than results in Theorems1 and 2, respectively.
The result above employs the same L-K functional of the descriptor approach, and thus the slack matricesP2w and P3w are also considered into the LMI conditions. Another set of LMI conditions for establishing the exponential estimates of systems in standard form (34) and its convex representation (35) can be done via a L-K functional without slack matrices, namely,(41)Vxτ=xTtPxt+∑h=1d∫−τh0xTt+θQhe2αθxt+θdθ,whose positiveness is inferred by P>0, Qh>0,h∈1,2,…,d and its boundedness c1x2≤Vxt≤c2xtτ2, with c1=λminP and c2=λmaxP+∑h=1dτhλmaxQh. This is summarized in the following result.Corollary 2.
The origin of the system (1) with an exact convex representation (35) and ut=0 is exponentially stable if there exists matrices P>0, Qh>0 with h∈1,2,…,d and a scalar α>0 such that,(42)PA˜i+A˜iTP+2αP+∑h=1dQh∗⋯∗A˜τ1iTP−e−2ατ1Q1⋯∗⋮⋮⋱∗A˜τdiTP⋯⋯−e−2ατdQd<0,holds for i∈1,2,…,r˜. Then, the solution satisfies the following exponential estimates:(43)xt,ϕ≤c2c1e−αtϕτ,with c1=λminP and c2=λmaxP+∑h=1dτhQh.Proof.
It follows a similar path than previous results.Remark 1. Theorem1.
establishes LMI conditions for the exponential estimates for the origin of system (1), these conditions include results in [66, Theorem 1] are always included when d=1 (system (1) with only one delay). Moreover, conditions in Corollary 2 always include those for linear systems in [4, Theorem 2], to see this set r=1.Remark 2.
The numerical complexity of the LMI problems in the above results can be approximated bylog10=nd3nl, where nl is the number of total LMI rows and nd is the number of scalar decision variables [70]. For Theorem 1 we have nl=nd+2r2ρ+nd+1 and nd=0.5nn+11+d+2rn2; as for Theorem 2, nl=nd+2r2ρ+nd+1 and nd=0.5nn+11+d+2rn2+nmrρ1+d; as for standard systems, Corollary 1 (stability) is nl=nd+2r˜2+nd+1 and nd=0.5nn+11+d+2r˜n2; Corollary 1 (stabilization) is nl=nd+2r˜2+nd+1 and nd=0.5nn+11+d+2r˜n2+nmr˜1+d; while Corollary 2 is nl=nd+1r˜+nd+1 and nd=0.5nn+11+d.
Results in Theorem2 can be directly applied for real-world setups; nevertheless, the LMIs might render controller gains whose magnitude cannot be applied in practice. To alleviate this issue as well as to avoid damages in the actuators, the following result provides conditions for bounding the control input (23); they can be combined with those of Theorem 2.Theorem 3.
Consider the delayed nonlinear controller given in (23); then, this controller satisfies that u<μ, for any μ>0, if the following inequalities hold:(44)P1MwωTMwωμ22I>0,(45)e−2ατhRhNhwωTNhwωμ22I>0,h∈1,2,…,d,(46)τ−1ϕTθϕTθ⋯ϕTθϕθP10⋯0ϕθ0R1⋯0⋮⋮⋮⋱0ϕθ00⋯Rd>0,∀θ∈−τ,0.Proof.
Observe thatu2=uTu with the delayed PDC (23) yields(47)u2=Kwωxt+∑h=1dFhwωxt−τhTKwωxt+∑h=1dFhwωxt−τh=xTtKwωTKwωxt+2∑h=1dxTtKwωTFhwωxt−τh+∑h=1dxTt−τhFhwωTFhwωxt−τh≤2xTtKwωTKwωxt+∑h=1dxTt−τhFhwωTFhwωxt−τh≤2xTtKwωTKwωxt+∑h=1d∫−τh0xTt+θFhwωTFhwωxt+θdθ≤μ2,which is satisfied if the following holds:(48)xTtKwωT2μ−2Kwωxt+∑h=1d∫−τh0xTt+θFhwωT2μ−2Fhwωxt+θdθ≤1.
On the other hand, let us consider the following inequality on functional (14):(49)Vxt=xTtP1−1xt+∑h=1d∫−τh0xTt+θe2αθQhxt+θdθ≤ϕT0P1−1ϕ0+∑h=1d∫−τh0ϕTθe2αθQhϕθdθ≤ϕT0P1−1ϕ0+∑h=1d∫−τ0ϕTθe2αθQhϕθdθ≤1.
Now, combining (48) and (49), it follows that(50)0<xTtKwωT2μ−2Kwωxt+∑h=1d∫−τh0xTt+θFhwωT2μ−2Fhwωxt+θdθ≤xTtP1−1xt+∑h=1d∫−τh0xTt+θe2αθQhxt+θdθ≤ϕT0P1−1ϕ0+∑h=1d∫−τh0ϕTθe2αθQhϕθdθ<ϕT0P1−1ϕ0+∑h=1d∫−τ0ϕTθP1−1Qh−1QhQh−1P1−1ϕθdθ≤1,or equivalently(51)xTtP1−1−KwωT2μ−2Kwωxt+∑h=1d∫−τh0xTt+θe2αθQh−FhwωT2μ−2Fhwωxt+θdθ>0,(52)∫−τ0τ−1−ϕTθP1−1ϕθ−∑h=1dϕTθX1Qh−1X1ϕθdθ>0.
Thus, (51) is satisfied if(53)P1−1−KwωT2μ−2Kwω>0,e−2ατhQh−FhwωT2μ−2Fhwω>0,h∈1,…,dhold too. From the latter inequalities and using the Schur complement together with congruence property with block−diagP1,I, conditions (44) and (45) follow with Mwω=KwωP1−1, Nhwω=FhwωP1−1, and Rh=P1QhP1,h∈1,2,…,d. Once again, employing the Schur complement on (52) gives(54)τ−1ϕTθϕTθ⋯ϕTθϕθP10⋯0ϕθ0P1Q1P1⋯0⋮⋮⋮⋱0ϕθ00⋯P1QdP1>0,∀θ∈−τ,0,which yields (46) after the substitution of Rh=P1QhP1,h∈1,2,…,d.


## 4. Examples

Next, a numerical example as well as the well-known inverted pendulum on a car is employed in order to illustrate the effectiveness of the proposed results. The LMI conditions have been checked with the LMIToolbox [6] within MATLAB 2109a.


### 4.1. Systems in Standard Form versus Descriptor Form

The following numerical example illustrates the advantages of the descriptor structure over standard state-space representations. Firstly, it compares stability at the origin via the LMIs in Theorem1 and those in Corollary 2 by means of their feasibility sets. Secondly, a delayed nonlinear control law is designed via Theorem 2.Consider a nonlinear system with two delaysd=2 in the descriptor form (1):(55)Exx˙t=Axxt+Aτ1xxt−τ1+Aτ2xxt−τ2+Bu,xθ=ϕθ,θ∈−τ,0,where the time delays are τ1=0.1yτ2=0.3=τ, and matrices are as follows:(56)Ex=0.80.1−35x22+50.080.97,Ax=−0.5cosx2−7.5−10.0837x1−7sinx1x1+a−5.5,B=01,Aτ1x=9.5+b14−6.6−0.067155x1−10sinx1x1,Aτ2x=cosx2−4.2−6.23.65.1.Descriptor form: following the sector nonlinearity approach, the following nonlinear terms have been identified:ζ1=x22+1−1∈0,1, z1=cosx2∈−1,1, and z2=sinx1/x1∈−0.2,1; their bounds have been calculated in the region Ωx=ℝ2. Thus the vertex matrices are(57)A1=−7−10.5+a−5.5,A2=−7−1−0.2+a−5.5,A3=−8−10.5+a−5.5,A4=−8−1−0.2+a−5.5,Aτ11=9.4+b14.2−6.6−10.2,Aτ12=9.4+b14.2−6.6−9.4,Aτ13=9.4+b14.2−6.6−10.2,Aτ14=9.4+b14.2−6.6−9.4,Aτ21=−5.2−6.23.65.1,Aτ22=−5.2−6.23.65.1,Aτ23=−3.2−6.23.65.1,Aτ24=−3.2−6.23.65.1,B1=B2=B3=B4=01,E1=0.80.10.080.97,E2=0.8−0.50.080.97.Commonly, systems of the form (55) are analyzed in the following standard form:(58)x˙=A˜xxt+A˜τ1xxt−τ1+A˜τ2xxt−τ2+B˜xu,where(59)A˜x=E−1xAx=x22+10.768x22+0.8160.97−0.1+35x22+5−0.080.8−0.5cosx2−7.5−10.0837x1−7sinx1x1+a−5.5,A˜τ1x=E−1xAτ1x=x22+10.768x22+0.8160.97−0.1+35x22+5−0.080.89.5+b14−6.6−0.067155x1−10sinx1x1,A˜τ2x=E−1xAτ2x=x22+10.768x22+0.8160.97−0.1+35x22+5−0.080.8cosx2−4.2−6.23.65.1,B˜x=E−1xB=x22+10.768x22+0.8160.97−0.1+35x22+5−0.080.801.In this case, four nonlinear terms are defined as follows:z1=cosx2∈−1,1, z2=sinx1/x1∈−0.2,1, z3=x22+1−1∈0,1, and z4=x22+1/0.768x22+0.816∈1.2255,1.3021; their bounds hold within Ωx. Some of the vertex matrices are given below:(60)A˜1=−8.3824−0.51471.1765−5.2942,A˜10=−10.1694−0.54691.3542−5.6251,A˜15=−9.6324−4.55890.5882−5.2942,A˜τ11=11.982918.13−7.3922−11.3922,A˜τ110=12.73219.263−7.8543−12.104,A˜τ115=7.1311.12−7.3922−10.608,A˜τ21=−6.6226−7.99524.03925.6079,A˜τ210=−4.5105−8.49494.08345.9584,A˜τ215=−1.5981−4.24513.84325.6079,B˜1=−0.122550.9804,B˜10=−0.130211.0417,B˜15=0.612750.9804.Note that for this example, we have the following:(i)
With respect to its exact convex representation, the descriptor one hasEk,Ai,Aτi,B,k=1,2,i=1,2,3,4, i.e., 8 vertexes, while the standard has A˜i,A˜τi,B˜i,i=1,2,…,24, i.e., 16 vertexes. Additionally, the descriptor keeps a constant matrix B; thus, descriptor form requires less computational resources [64]. Indeed, in this case with n=2, m=1, r=4, ρ=2, r˜=16, and d=2, the computational complexity using Theorem 1 is 7.2567 with 35 LMIs, using Corollary 1 is 9.7228 with 35 LMIs, and using Corollary 2 is 4.8713 with 19 LMIs.(ii)
With respect to feasibility sets for Theorem1, and Corollaries 1 and 2, when u=0 and parameter values as a∈0,0.8, b∈−0.2,0.5 for convex representations of systems (55) and (58), respectively. In Figure 1, the regions marked with a circle ∘ correspond to the feasibility sets using Theorem 1, while the regions marked by × and + are the feasibility sets obtained using Corollaries 1 and 2, respectively. It can be seen that by using the descriptor form the feasibility set is larger, i.e., the descriptor approach provides more relaxed results than the standard approach.Thus, results given in Theorem1 improve the previously classic results found in the literature.Figure 1
Feasibility sets for Theorem1, Corollary 1, and Corollary 2 are applied to the example given in Subsection 4.1.On the other hand, Figure2 shows that the system response (55) does not converge to the trivial equilibrium point when a=−2.5, b=6.2, u=0, τ1=0.1, τ2=0.3, and ϕθ=−55T, θ∈−0.3,0.Figure 2
System response of the example given in Subsection4.1 (descriptor form) when a = − 2.5, b = 6.2, and u = 0.Next, a delayed nonlinear controller of the form (23) is given by(61)u=Kwωxt+∑h=12Fhwωxt−τh,which is employed to stabilize this system. To this end, LMI conditions in Theorem 2 with an exponential decay α=0.5 together with those from in Theorem 3 for u<48=μ render feasible solution providing the following values:(62)P1−1=0.206030.194580.194580.21717,Q1=3.58663.32473.32473.1135,Q2=1.09721.2151.2151.4539,K11=−11.0453−13.9169,F111=−11.7060−6.5843,F211=2.56062.2870,K12=−7.3722−12.1755,F112=−13.9524−8.5677,F212=3.28943.1315,K21=−10.4266−13.6762,F121=−10.9789−6.6378,F221=2.27351.9269,K22=−6.7172−11.7399,F122=−13.2816−8.7126,F222=3.04482.8193,K31=−9.8536−13.3457,F131=−9.7933−4.8426,F231=−0.20401.5048,K32=−6.7529−11.8054,F132=−13.8964−8.4844,F232=0.62963.0749,K41=−9.0894−12.9706,F141=−9.2846−5.0947,F241=−0.35991.2441,K42=−5.7661−11.1366,F142=−13.6875−9.0667,F242=0.56192.9637.Effectively, system (55) is stabilized at the origin as shown in Figure 3. The evolution in time of the system state is shown in Figure 3(a), while the guaranteed exponential decay in the system response is depicted in Figure 3(b).Figure 3
System response of the example given in Section4.1 (descriptor form) using delayed controller (61). (a) Time evolution of the closed-loop system. (b) Guaranteed exponential decay α = 0.5 for the closed-loop system.
(a)(b)


### 4.2. Nonlinear Controller versus Delayed Nonlinear Controller

The following example is to illustrate the advantages of the use of artificial delays in controllers when there is the presence of noise, as mentioned in the introduction and its corroboration by various results found in the literature.Consider the system known as the car-pendulum, whose scheme is shown in Figure4, a mathematical model is given by(63)M1+M2−M2lcosθ−M2lcosθJ+M2l2x˜¨tθ¨t+cx˜˙+M2lθ˙2sinθγθ˙−M2glsinθ=F0.Figure 4
Schematic of the inverted pendulum system.For illustrative purposes, a delayed measurementτ>0 in the positions of the car and pendulum are intentionally added. Also, we define x1=x˜, x2=x˜˙, x3=θ, x4=θ˙, and u=F. Thus, for g=9.81m/s2, l=0.304m, M2=0.2kg, M1=1.3282kg, J=M2l2/3, c=0.001, γ=0.001, and τ=0.05, the system (63) can be rewritten as(64)Exx˙t=Axxt+Aτxxt−τ+But.where(65)Ex=100001.54820−0.0669cosx300100−0.0669cosx300.0271,Aτx=00000.10000000000.20,Ax=01000−0.0010−0.0669x4sinx30001000.6561sinx3x3−0.001,B=0100.Considering the regionΩx=x:x1≤2m,x2≤3m/s,x3≤π/3rad,x4≤4rad/s, the nonconstant terms and their bounds are ζ1=cosx3∈0.5,1, z1=x4sinx3∈−0.4238,0.4238, and z2=sinx3/x3∈0.827,1; the vertex matrices are(66)A1=01.0000−0.0010−0.02580001.000−0.4933−0.001,A2=01000−0.0010−0.025000100−0.596−0.001,A3=01000−0.00100.0258000100−0.4933−0.001,A4=01000−0.00100.0258000100−0.5964−0.001,Aτ1=Aτ2=Aτ3=Aτ4=00000.10000000000.20,B1=B2=B3=B4=0100,E1=100001.52820−0.030400100−0.030400.0370,E2=1.000001.54820−0.060800100−0.060800.0370.As mentioned above, with the purpose of showing the advantages of using a controller with delayed action in the presence of noise; for this example, two controllers are used: a nonlinear controller of the following form is(67)u=Kwωxt,and a delayed nonlinear controller of the form (23) is given by(68)u=Kwωxt+Fwωxt−τ.To illustrate the effectiveness of Theorem3, the controllers (68) are conditioned to satisfy that ut<12. For the controller (67), the same condition is requested, for which Theorem 3 can be used, after some simple adjustments when considering free-delays controller. For the controller (67), the corresponding LMI conditions in Theorem 2 with an exponential decay α=0.003 are found feasible with the following values:(69)P1=0.020240.044110.1428−0.0059290.044110.18740.6166−0.026980.14280.61666.0360.1574−0.005929−0.026980.15740.2537,Q1=0.00770.02250.14550.000050.022580.096030.48360.00420.14550.48367.8790.07780.000050.00420.07780.1026,K11=−0.3685−1.71−8.965−3.468,K12=−0.4382−1.979−8.867−2.879,K21=−0.3567−1.626−5.913−3.452,K22=−0.4368−1.95−6.884−2.852,K31=−0.3619−1.668−8.823−3.516,K32=−0.435−1.955−8.748−2.915,K41=−0.3483−1.573−5.728−3.505,K42=−0.4333−1.924−6.765−2.889.On the other hand, for the delayed controller (68), LMI conditions in Theorem 2 with an exponential decay α=0.003 are also feasible with the following values:(70)P1−1=0.00200.00620.02261−0.00050.00620.04220.1531−0.0030.02260.15312.3660.07454−0.0005−0.00300.07450.1213,Q1=0.00030.00160.01340.00020.00160.01140.07890.0020.01340.07892.2840.0310.00020.0020.0310.0418,K11=−0.1389−0.9876−4.926−2.323,F11=−0.092−0.0006−5.6910.0014,K12=−0.1457−0.99−4.341−2.052,F12=−0.0867−0.0001−3.540.0004,K21=−0.1368−0.9425−2.012−2.263,F21=−0.0932−0.001−5.9560.0011,K22=−0.1479−0.9792−2.502−2.001,F22=−0.0870−0.0001−3.6340.0004,K31=−0.1394−0.9793−4.867−2.345,F31=−0.0928−0.0008−5.8290.0017,K32=−0.1459−0.9801−4.278−2.078,F32=−0.0870−0.0001−3.6280.0004,K41=−0.1373−0.9317−1.924−2.284,F41=−0.0944−0.00135−6.1360.0019,K42=−0.1485−0.9693−2.432−2.025,F42=−0.0873−0.0002−3.740.0005.To simulate the noise present in the sensors, a random signal with variance0.001 and a step 0.001 is introduced at the system input. In Figure 5, the applied control signals are plotted, and it can be seen that the control signal presents less noisy when using the delayed controller than the other.Figure 5
Time evolution of control laws (67) and (68).In Figure6, the evolution in time of the state of the system (64), with ϕθ=0.50π/30T, θ∈−0.05,0, under the control laws (67) and (68), is shown. It can be seen that the overshoot is greater when using a controller without a delayed action.Figure 6
System response of the example given in Section4.2. Time evolution of the closed-loop system with (a) (28) and (b) (68).
(a)(b)Remark 3.
Systems of the form (63) can be stabilized by a free-delay controller of the form (67). However, using delayed controllers of the form (68) or (23) to stabilize this system class may be a better option when systems have inherent noise.


## 4.1. Systems in Standard Form versus Descriptor Form

The following numerical example illustrates the advantages of the descriptor structure over standard state-space representations. Firstly, it compares stability at the origin via the LMIs in Theorem1 and those in Corollary 2 by means of their feasibility sets. Secondly, a delayed nonlinear control law is designed via Theorem 2.Consider a nonlinear system with two delaysd=2 in the descriptor form (1):(55)Exx˙t=Axxt+Aτ1xxt−τ1+Aτ2xxt−τ2+Bu,xθ=ϕθ,θ∈−τ,0,where the time delays are τ1=0.1yτ2=0.3=τ, and matrices are as follows:(56)Ex=0.80.1−35x22+50.080.97,Ax=−0.5cosx2−7.5−10.0837x1−7sinx1x1+a−5.5,B=01,Aτ1x=9.5+b14−6.6−0.067155x1−10sinx1x1,Aτ2x=cosx2−4.2−6.23.65.1.Descriptor form: following the sector nonlinearity approach, the following nonlinear terms have been identified:ζ1=x22+1−1∈0,1, z1=cosx2∈−1,1, and z2=sinx1/x1∈−0.2,1; their bounds have been calculated in the region Ωx=ℝ2. Thus the vertex matrices are(57)A1=−7−10.5+a−5.5,A2=−7−1−0.2+a−5.5,A3=−8−10.5+a−5.5,A4=−8−1−0.2+a−5.5,Aτ11=9.4+b14.2−6.6−10.2,Aτ12=9.4+b14.2−6.6−9.4,Aτ13=9.4+b14.2−6.6−10.2,Aτ14=9.4+b14.2−6.6−9.4,Aτ21=−5.2−6.23.65.1,Aτ22=−5.2−6.23.65.1,Aτ23=−3.2−6.23.65.1,Aτ24=−3.2−6.23.65.1,B1=B2=B3=B4=01,E1=0.80.10.080.97,E2=0.8−0.50.080.97.Commonly, systems of the form (55) are analyzed in the following standard form:(58)x˙=A˜xxt+A˜τ1xxt−τ1+A˜τ2xxt−τ2+B˜xu,where(59)A˜x=E−1xAx=x22+10.768x22+0.8160.97−0.1+35x22+5−0.080.8−0.5cosx2−7.5−10.0837x1−7sinx1x1+a−5.5,A˜τ1x=E−1xAτ1x=x22+10.768x22+0.8160.97−0.1+35x22+5−0.080.89.5+b14−6.6−0.067155x1−10sinx1x1,A˜τ2x=E−1xAτ2x=x22+10.768x22+0.8160.97−0.1+35x22+5−0.080.8cosx2−4.2−6.23.65.1,B˜x=E−1xB=x22+10.768x22+0.8160.97−0.1+35x22+5−0.080.801.In this case, four nonlinear terms are defined as follows:z1=cosx2∈−1,1, z2=sinx1/x1∈−0.2,1, z3=x22+1−1∈0,1, and z4=x22+1/0.768x22+0.816∈1.2255,1.3021; their bounds hold within Ωx. Some of the vertex matrices are given below:(60)A˜1=−8.3824−0.51471.1765−5.2942,A˜10=−10.1694−0.54691.3542−5.6251,A˜15=−9.6324−4.55890.5882−5.2942,A˜τ11=11.982918.13−7.3922−11.3922,A˜τ110=12.73219.263−7.8543−12.104,A˜τ115=7.1311.12−7.3922−10.608,A˜τ21=−6.6226−7.99524.03925.6079,A˜τ210=−4.5105−8.49494.08345.9584,A˜τ215=−1.5981−4.24513.84325.6079,B˜1=−0.122550.9804,B˜10=−0.130211.0417,B˜15=0.612750.9804.Note that for this example, we have the following:(i)
With respect to its exact convex representation, the descriptor one hasEk,Ai,Aτi,B,k=1,2,i=1,2,3,4, i.e., 8 vertexes, while the standard has A˜i,A˜τi,B˜i,i=1,2,…,24, i.e., 16 vertexes. Additionally, the descriptor keeps a constant matrix B; thus, descriptor form requires less computational resources [64]. Indeed, in this case with n=2, m=1, r=4, ρ=2, r˜=16, and d=2, the computational complexity using Theorem 1 is 7.2567 with 35 LMIs, using Corollary 1 is 9.7228 with 35 LMIs, and using Corollary 2 is 4.8713 with 19 LMIs.(ii)
With respect to feasibility sets for Theorem1, and Corollaries 1 and 2, when u=0 and parameter values as a∈0,0.8, b∈−0.2,0.5 for convex representations of systems (55) and (58), respectively. In Figure 1, the regions marked with a circle ∘ correspond to the feasibility sets using Theorem 1, while the regions marked by × and + are the feasibility sets obtained using Corollaries 1 and 2, respectively. It can be seen that by using the descriptor form the feasibility set is larger, i.e., the descriptor approach provides more relaxed results than the standard approach.Thus, results given in Theorem1 improve the previously classic results found in the literature.Figure 1
Feasibility sets for Theorem1, Corollary 1, and Corollary 2 are applied to the example given in Subsection 4.1.On the other hand, Figure2 shows that the system response (55) does not converge to the trivial equilibrium point when a=−2.5, b=6.2, u=0, τ1=0.1, τ2=0.3, and ϕθ=−55T, θ∈−0.3,0.Figure 2
System response of the example given in Subsection4.1 (descriptor form) when a = − 2.5, b = 6.2, and u = 0.Next, a delayed nonlinear controller of the form (23) is given by(61)u=Kwωxt+∑h=12Fhwωxt−τh,which is employed to stabilize this system. To this end, LMI conditions in Theorem 2 with an exponential decay α=0.5 together with those from in Theorem 3 for u<48=μ render feasible solution providing the following values:(62)P1−1=0.206030.194580.194580.21717,Q1=3.58663.32473.32473.1135,Q2=1.09721.2151.2151.4539,K11=−11.0453−13.9169,F111=−11.7060−6.5843,F211=2.56062.2870,K12=−7.3722−12.1755,F112=−13.9524−8.5677,F212=3.28943.1315,K21=−10.4266−13.6762,F121=−10.9789−6.6378,F221=2.27351.9269,K22=−6.7172−11.7399,F122=−13.2816−8.7126,F222=3.04482.8193,K31=−9.8536−13.3457,F131=−9.7933−4.8426,F231=−0.20401.5048,K32=−6.7529−11.8054,F132=−13.8964−8.4844,F232=0.62963.0749,K41=−9.0894−12.9706,F141=−9.2846−5.0947,F241=−0.35991.2441,K42=−5.7661−11.1366,F142=−13.6875−9.0667,F242=0.56192.9637.Effectively, system (55) is stabilized at the origin as shown in Figure 3. The evolution in time of the system state is shown in Figure 3(a), while the guaranteed exponential decay in the system response is depicted in Figure 3(b).Figure 3
System response of the example given in Section4.1 (descriptor form) using delayed controller (61). (a) Time evolution of the closed-loop system. (b) Guaranteed exponential decay α = 0.5 for the closed-loop system.
(a)(b)


## 4.2. Nonlinear Controller versus Delayed Nonlinear Controller

The following example is to illustrate the advantages of the use of artificial delays in controllers when there is the presence of noise, as mentioned in the introduction and its corroboration by various results found in the literature.Consider the system known as the car-pendulum, whose scheme is shown in Figure4, a mathematical model is given by(63)M1+M2−M2lcosθ−M2lcosθJ+M2l2x˜¨tθ¨t+cx˜˙+M2lθ˙2sinθγθ˙−M2glsinθ=F0.Figure 4
Schematic of the inverted pendulum system.For illustrative purposes, a delayed measurementτ>0 in the positions of the car and pendulum are intentionally added. Also, we define x1=x˜, x2=x˜˙, x3=θ, x4=θ˙, and u=F. Thus, for g=9.81m/s2, l=0.304m, M2=0.2kg, M1=1.3282kg, J=M2l2/3, c=0.001, γ=0.001, and τ=0.05, the system (63) can be rewritten as(64)Exx˙t=Axxt+Aτxxt−τ+But.where(65)Ex=100001.54820−0.0669cosx300100−0.0669cosx300.0271,Aτx=00000.10000000000.20,Ax=01000−0.0010−0.0669x4sinx30001000.6561sinx3x3−0.001,B=0100.Considering the regionΩx=x:x1≤2m,x2≤3m/s,x3≤π/3rad,x4≤4rad/s, the nonconstant terms and their bounds are ζ1=cosx3∈0.5,1, z1=x4sinx3∈−0.4238,0.4238, and z2=sinx3/x3∈0.827,1; the vertex matrices are(66)A1=01.0000−0.0010−0.02580001.000−0.4933−0.001,A2=01000−0.0010−0.025000100−0.596−0.001,A3=01000−0.00100.0258000100−0.4933−0.001,A4=01000−0.00100.0258000100−0.5964−0.001,Aτ1=Aτ2=Aτ3=Aτ4=00000.10000000000.20,B1=B2=B3=B4=0100,E1=100001.52820−0.030400100−0.030400.0370,E2=1.000001.54820−0.060800100−0.060800.0370.As mentioned above, with the purpose of showing the advantages of using a controller with delayed action in the presence of noise; for this example, two controllers are used: a nonlinear controller of the following form is(67)u=Kwωxt,and a delayed nonlinear controller of the form (23) is given by(68)u=Kwωxt+Fwωxt−τ.To illustrate the effectiveness of Theorem3, the controllers (68) are conditioned to satisfy that ut<12. For the controller (67), the same condition is requested, for which Theorem 3 can be used, after some simple adjustments when considering free-delays controller. For the controller (67), the corresponding LMI conditions in Theorem 2 with an exponential decay α=0.003 are found feasible with the following values:(69)P1=0.020240.044110.1428−0.0059290.044110.18740.6166−0.026980.14280.61666.0360.1574−0.005929−0.026980.15740.2537,Q1=0.00770.02250.14550.000050.022580.096030.48360.00420.14550.48367.8790.07780.000050.00420.07780.1026,K11=−0.3685−1.71−8.965−3.468,K12=−0.4382−1.979−8.867−2.879,K21=−0.3567−1.626−5.913−3.452,K22=−0.4368−1.95−6.884−2.852,K31=−0.3619−1.668−8.823−3.516,K32=−0.435−1.955−8.748−2.915,K41=−0.3483−1.573−5.728−3.505,K42=−0.4333−1.924−6.765−2.889.On the other hand, for the delayed controller (68), LMI conditions in Theorem 2 with an exponential decay α=0.003 are also feasible with the following values:(70)P1−1=0.00200.00620.02261−0.00050.00620.04220.1531−0.0030.02260.15312.3660.07454−0.0005−0.00300.07450.1213,Q1=0.00030.00160.01340.00020.00160.01140.07890.0020.01340.07892.2840.0310.00020.0020.0310.0418,K11=−0.1389−0.9876−4.926−2.323,F11=−0.092−0.0006−5.6910.0014,K12=−0.1457−0.99−4.341−2.052,F12=−0.0867−0.0001−3.540.0004,K21=−0.1368−0.9425−2.012−2.263,F21=−0.0932−0.001−5.9560.0011,K22=−0.1479−0.9792−2.502−2.001,F22=−0.0870−0.0001−3.6340.0004,K31=−0.1394−0.9793−4.867−2.345,F31=−0.0928−0.0008−5.8290.0017,K32=−0.1459−0.9801−4.278−2.078,F32=−0.0870−0.0001−3.6280.0004,K41=−0.1373−0.9317−1.924−2.284,F41=−0.0944−0.00135−6.1360.0019,K42=−0.1485−0.9693−2.432−2.025,F42=−0.0873−0.0002−3.740.0005.To simulate the noise present in the sensors, a random signal with variance0.001 and a step 0.001 is introduced at the system input. In Figure 5, the applied control signals are plotted, and it can be seen that the control signal presents less noisy when using the delayed controller than the other.Figure 5
Time evolution of control laws (67) and (68).In Figure6, the evolution in time of the state of the system (64), with ϕθ=0.50π/30T, θ∈−0.05,0, under the control laws (67) and (68), is shown. It can be seen that the overshoot is greater when using a controller without a delayed action.Figure 6
System response of the example given in Section4.2. Time evolution of the closed-loop system with (a) (28) and (b) (68).
(a)(b)Remark 3.
Systems of the form (63) can be stabilized by a free-delay controller of the form (67). However, using delayed controllers of the form (68) or (23) to stabilize this system class may be a better option when systems have inherent noise.


## 5. Conclusions

In this paper, analysis and design using a convex approach for nonlinear descriptor systems with multiple delays have been presented. This analysis allows synthesizing delayed nonlinear controllers to ensure convergence of the system trajectories with a guaranteed exponential decay; moreover, conditions for bounding the control input avoid possible saturation in the actuators have been provided. It also has been shown that keeping the descriptor form increases the possibility of obtaining feasibility in the LMI conditions, unlike the use of standard forms. Also, it is observed that including deliberately delays in the controller can reduce noise in the control signal, thus avoiding mechanical wear of the actuators. As future work, an extension of the proposed results in nonlinear descriptor systems with multiple time-varying delays is in course, since it will allow the synthesis of controllers for a larger class of systems.



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