Delay-Dependent $H_\infty$ Control for LPV Time-Delay Systems via Dynamic Output Feedback

Abstract

In this paper, we present a new systematic method to design a dynamic output feedback controller (DOF) for linear parameter-varying (LPV) systems with time delay and exogenous disturbances. An LPV controller is proposed due to the presence of LPV terms in the structure of the system. The purpose is the design of an LPV DOF controller such that the resulting closed-loop system is robustly asymptotically stable while satisfying a prescribed $H_\infty$ performance level. This problem is originally non-convex because of the coupling between system matrices and decision variables. The design conditions are transformed into LMI problems by using some technical lemmas for straightforward computation of the controller matrices. Finally, two examples are presented to demonstrate the validity and effectiveness of the theoretical results.

1 Introduction

Time delay is frequently encountered in dynamic systems such as underwater vehicles, rolling mills, communication systems, and nuclear reactors [16, 18, 24, 29]. Ignoring the effect of time delays in the controller design may lead to deterioration of the closed-loop performance and even destabilization of the overall control system [25]. Thus, the control of time-delay systems (TDSs) has become a topic of great practical and theoretical importance [10, 17, 28, 38, 40, 44, 45, 49, 51, 52, 54, 56, 57, 61].

During the past decades, linear parameter-varying (LPV) systems have attracted much attention because of their wide applications in various practical systems. LPV systems can be considered as a representation of nonlinear systems that is well suited for controlling dynamical systems with parameter variations [3]. Such systems depend on a priori unknown but online measurable parameters.

The control problem of LPV TDSs has been widely studied by the control community [5, 8, 23, 35, 46, 60]. LPV TDSs belong to the intersection of LPV systems and TDSs and hence inherit from the difficulties of each one [7]. The control problem of such systems is expressed by nonlinear matrix inequalities (NMIs), and there is no effective algorithm to solve it. Therefore, the controller design for LPV TDSs is quite challenging. To obtain computationally tractable criteria, there are many techniques presented in the literature [27, 30, 36, 37, 39, 62].

Besides, external disturbances may be appeared in many practical applications, which are a source of instability and poor performance of systems [9, 13, 22]. The $H_\infty$ control method is used to minimize the external disturbances effects [19]. In recent years, many researchers have focused on $H_\infty$ controller design for LPV TDSs.

Delay-dependent $H_{\infty }$ control results for LPV TDSs were first presented in [53]. However, the delay variation rate was not used in this work. Although the rate information of delay variations was used in [60], the results were still conservative since this approach employs a model transformation introducing additional dynamics in the system. To overcome this shortcoming, a descriptor model transformation and an improved bounding technique were employed in [46]. In [6], a parameter-dependent state feedback controller was designed based on Jensen’s inequality and projection approach. This method led to an NMI, which was solved using an iterative LMI algorithm. In [7], a ${\delta }$-memory-resilient state feedback controller was presented, where the delay implementation error was considered in the design. By using slack variables, a relaxation method was proposed to derive less conservative results in [67, 68]. In many industrial systems, only system outputs are available and internal states cannot be directly measured for technological and/or economic reasons. In [20], under the assumption that the state vector is not available for feedback, an observer-based finite-time $H_\infty$ controller was designed such that the resulting closed-loop system is uniformly finite-time bounded and a prescribed $H_\infty$ performance level in a finite-time interval is satisfied. The design problem of dynamic output feedback (DOF) controller for LPV TDSs was investigated in [21, 23, 68].

In the study of TDSs, there exist two key indexes to evaluate the conservatism of the derived conditions: maximum allowable delay bound (MADB) and $H_\infty$ performance level. For a prescribed $H_\infty$ performance level, the larger the MADB, the less conservative the conditions; for a given time delay, the smaller the $H_\infty$ performance level, the better the conditions.

It is well known that the selection of appropriate Lyapunov–Krasovskii functional (LKF) and employing improved bounding techniques to estimate time derivative of LKF result in improvement in stability region [15, 32]. In recent years, several approaches have been proposed to reduce conservatism, for instance, reciprocally convex combination [26, 31, 65], free matrix-based integral inequality [50], delay partitioning approach [12], Wirtinger-based inequality [42, 63], Bessel–Legendre inequality [43], and the augmented LKF approach [34, 48]. Although much research has been done on TDSs, the existing results have some conservatism since they are only sufficient conditions. This motivates us to propose a new method to reduce conservatism in designing the controller for LPV TDSs.

In this paper, the $H_\infty$ control problem for LPV TDSs subject to $L_2$-norm disturbances is addressed. The delay-dependent sufficient conditions are derived for the existence of an $H_\infty$ DOF controller based on the Lyapunov–Krasovskii theorem. The control design problem for LPV TDSs is originally non-convex. The main strategy used to derive the design conditions is to introduce degrees of freedom allowing one to decouple the decision variables from the system matrices. The key idea lies in the use of the Young’s relation in a judicious manner. The design conditions are linearized using the Young’s relation and some technical lemmas resulting in LMI conditions providing the controller matrices. The additional decision variables enable improvements in the feasibility of the derived LMIs. Finally, two examples are provided to illustrate the effectiveness and less conservatism of the proposed criteria.

In general, the main advantages of the proposed method over recent existing results can be summarized as follows:

• Larger MADB for stability of the closed-loop system

• Better disturbance attenuation effect

The rest of the current paper is organized as follows: The problem formulation is given in Sect. 2. The main results are given in Sect. 3. The simulation results are presented in Sect. 4. Finally, the conclusion is presented in Sect. 5.

2 Problem Formulation

Consider the following state-space representation for an LPV TDS:

$$\begin{aligned} S(\rho ):\left\{ \begin{array}{l} \dot{x}(t)=A(\rho ) x(t)+A_{d}(\rho ) x(t-\tau (t))+B_{1}(\rho ) w(t)+B_{2}(\rho ) u(t), \\ z(t)=C_{1}(\rho ) x(t)+D_{12}(\rho ) u(t)+D_{11}(\rho ) w(t), \\ y(t)=C_{2}(\rho ) x(t)+D_{21}(\rho ) w(t),\\ x(\nu )=\varphi (\nu ), \forall \varphi \in [-h, 0], \end{array}\right. \end{aligned}$$

(1)

where $x(t)\in \mathbb {R}^{n_x}$ is the state vector, $u(t)\in \mathbb {R}^{n_u}$ is the control input vector, $z(t)\in \mathbb {R}^{n_z}$ is the controlled output vector, $y(t)\in \mathbb {R}^{n_y}$ is the output measurement vector, $w(t)\in \mathbb {R}^{n_w}$ is the disturbance vector with finite energy in the space $L_2[0,\infty )$, and $\varphi (\nu )$ denotes the functional initial condition. The system matrices are parameter-dependent matrices of compatible dimensions. $\rho (t)=\left[ \rho _{1}(t), \rho _{2}(t), \ldots , \rho _{r}(t)\right] ^\mathrm {T} \in \mathbb {R}^r$ is the time-varying parameter vector. $\tau (t)$ is a time-varying delay satisfying

$$\begin{aligned} 0< \tau (t) \le h,\quad 0\le \dot{\tau }(t) \le \mu < 1. \end{aligned}$$

(2)

Assumption 1

The state-space matrices are continuous and bounded functions and depend affinely on $\rho (t)$ [55].

Assumption 2

The real parameters $\rho (t)$, which can be known in advance or online measurement values, vary in a polytope $\mathcal {L}$ of vertices $\varrho _1, \varrho _2, \ldots , \varrho _N$ [55]. That is:

$$\begin{aligned} \begin{aligned} \rho (t) \in \mathcal {L}:=&\mathrm {Co}\{\varrho _1, \varrho _2, \ldots , \varrho _N\} \\=&\left\{ \sum _{i=1}^{N} \alpha _{i}(t) \varrho _{i}, \alpha _{i}(t) \ge 0, \sum _{i=1}^{N} \alpha _{i}(t)=1, N=2^{r}\right\} . \end{aligned} \end{aligned}$$

(3)

According to the above assumptions, the LPV system (1) can be expressed as the following polytopic model:

$$\begin{aligned}&\left( \begin{array}{cccc} A(\rho ) &{} A_{d}(\rho ) &{} B_{1}(\rho ) &{} B_{2}(\rho ) \\ C_{1}(\rho ) &{} 0 &{} D_{11}(\rho ) &{} D_{12}(\rho ) \\ C_{2}(\rho ) &{} 0 &{} D_{21}(\rho ) &{} 0 \end{array}\right) \nonumber \\&\quad =\sum \limits _{i=1}^{N} \alpha _{i}(t)\left( \begin{array}{cccc} A(\varrho _i) &{} A_{d}(\varrho _i) &{} B_{1}(\varrho _i) &{} B_{2}(\varrho _i) \\ C_{1}(\varrho _i) &{} 0 &{} D_{11}(\varrho _i) &{} D_{12}(\varrho _i) \\ C_{2}(\varrho _i) &{} 0 &{} D_{21}(\varrho _i) &{} 0 \end{array}\right) . \end{aligned}$$

(4)

We construct the following DOF controller:

$$\begin{aligned} K_c(\rho ): \left\{ \begin{array}{l} \dot{x}_{c}(t)=A_{K}(\rho ) x_{c}(t)+A_{K d}(\rho ) x_{c}(t-\tau (t))+B_{K}(\rho ) y(t), \\ u(t)=C_{K}(\rho ) x_{c}(t)+D_{K}(\rho ) y(t), \end{array}\right. \end{aligned}$$

(5)

where $x_c \in \mathbb {R}^{n_x}$ is the controller state vector.

Based on the polytopic approach, the applied controller $K_c(\rho )$ is a convex combination of the controllers synthesized at the vertices of the polytope [1]:

$$\begin{aligned} K_{c}(\rho )=\sum _{i=1}^{N} \alpha _{i}(t) K_{c}(\varrho _i), \alpha _{i}(t) \ge 0, \sum _{i=1}^{N} \alpha _{i}(t)=1, \end{aligned}$$

(6)

where

$$\begin{aligned} K_{c}(\varrho _i)=\left( \begin{array}{ccc} A_{K}(\varrho _i) &{} A_{Kd}(\varrho _i) &{} B_{K}(\varrho _i) \\ C_{K}(\varrho _i) &{} 0 &{} D_{K}(\varrho _i) \end{array}\right) . \end{aligned}$$

Applying the DOF controller (5) to the system (1) leads to

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x}_{c l}(t)=A_{c l}(\rho ) x_{c l}(t)+A_{c l d}(\rho ) x_{c l}(t-\tau (t))+B_{c l}(\rho ) w(t), \\ z(t)=C_{c l}(\rho ) x_{c l}(t)+D_{c l}(\rho ) w(t), \end{array}\right. \end{aligned}$$

(7)

where

$$\begin{aligned} \begin{array}{l} A_{c l}(\rho )=\left( \begin{array}{cc} A(\rho )+B_{2}(\rho ) D_{K}(\rho ) C_{2}(\rho ) &{} B_{2}(\rho ) C_{K}(\rho ) \\ B_{K}(\rho ) C_{2}(\rho ) &{} A_{K}(\rho ) \end{array}\right) , \\ A_{c l d}(\rho )=\left( \begin{array}{cc} A_{d}(\rho ) &{} 0 \\ 0 &{} A_{K d}(\rho ) \end{array}\right) , \\ B_{c l}(\rho )=\left( \begin{array}{c} B_{1}(\rho )+B_{2}(\rho ) D_{K}(\rho ) D_{21}(\rho ) \\ B_{K}(\rho ) D_{21}(\rho ) \end{array}\right) , \\ C_{c l}(\rho )=\left( C_{1}(\rho )+D_{12}(\rho ) D_{K}(\rho ) C_{2}(\rho ) \quad D_{12}(\rho ) C_{K}(\rho )\right) , \\ D_{c l}(\rho )=D_{11}(\rho )+D_{12}(\rho ) D_{K}(\rho ) D_{21}(\rho ), \\ x_{c l}(t)=\left( \begin{array}{c} x(t) \\ x_{c}(t) \end{array}\right) , \quad x_{c l}(t-\tau (t))=\left( \begin{array}{c} x(t-\tau (t)) \\ x_{c}(t-\tau (t)) \end{array}\right) . \end{array}\end{aligned}$$

The objective of this study is to design an LPV DOF controller of the form (5) such that the following conditions are satisfied:

• The closed-loop system (7) is asymptotically stable when $w(t)=0$.

• Under the zero-initial condition, the controlled output z(t) satisfies

  $$\begin{aligned} \Vert z(t)\Vert _{2} \le \gamma \Vert w(t)\Vert _{2} \end{aligned}$$

  (8)

  for all nonzero $w(t) \in L_2[0,+\infty )$ and a prescribed attenuation level $\gamma >0$.

Now, we present some mathematical preliminaries that are used to prove our main results.

Lemma 1

[66] Let $x(t) \in \mathbb {R}^n$ be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices $M_1, M_2 \in \mathbb {R}^{n \times n}$ and $R = R^\mathrm {T}>0$, and a scalar function $\tau (t) \ge 0$:

$$\begin{aligned} \begin{aligned} -\int _{t-\tau (t)}^{t} \dot{x}^\mathrm {T}(\omega ) R \dot{x}(\omega ) \mathrm {d} \omega \le \xi ^\mathrm {T}(t) \varUpsilon \xi (t)+\tau (t) \xi ^\mathrm {T}(t) \varTheta ^\mathrm {T} R^{-1} \varTheta \xi (t), \end{aligned} \end{aligned}$$

(9)

where

$$\begin{aligned} \begin{array}{c} \varUpsilon =\left( \begin{array}{cc} M_1^\mathrm {T}+M_1 &{} -M_1^\mathrm {T}+M_2 \\ (*) &{} -M_2^\mathrm {T}-M_2 \end{array}\right) , \ \varTheta ^\mathrm {T}:=\left( \begin{array}{cc} M_1^\mathrm {T} \\ M_2^\mathrm {T} \end{array}\right) , \ \xi (t):=\left( \begin{array}{cc} x(t) \\ x(t-\tau (t)) \end{array}\right) . \end{array} \end{aligned}$$

Lemma 2

(Young’s relation) [58] For given matrices $\mathcal {X}$ and $\mathcal {Y}$ with appropriate dimensions, the following property holds for any symmetric positive definite matrix $\mathcal {Q}$ and scalar $\varepsilon >0$:

$$\begin{aligned} \mathcal {X}^\mathrm {T} \mathcal {Y}+\mathcal {Y}^\mathrm {T} \mathcal {X} \le \varepsilon ^{-1} \mathcal {X}^\mathrm {T} \mathcal {Q} \mathcal {X}+\varepsilon \mathcal {Y}^\mathrm {T} \mathcal {Q}^{-1} \mathcal {Y}. \end{aligned}$$

(10)

Lemma 3

(Schur complement lemma) [4] Let $S_{11} = S_{11}^\mathrm {T}$ and $S_{22} = S_{22}^\mathrm {T}$, the following conditions are equivalent:

$$\begin{aligned} \begin{aligned} \bullet \quad&\left[ \begin{array}{ll} S_{11} &{} S_{12} \\ S_{12}^\mathrm {T} &{} S_{22} \end{array}\right]<0,\\ \bullet \quad&S_{11}<0, \text {and} \, \, S_{22}-S_{12}^\mathrm {T} S_{11}^{-1} S_{12}<0, \\ \bullet \quad&S_{22}<0, \text {and} \, \, S_{11}-S_{12} S_{22}^{-1} S_{12}^\mathrm {T}<0. \end{aligned} \end{aligned}$$

(11)

According to the Schur complement lemma, the following result is obtained.

Corollary 1

Consider

$$\begin{aligned} A=\left[ \begin{array}{cccc} A_{11} &{} A_{12} &{} \cdots &{} A_{1 r} \\ A_{12}^{\mathrm {T}} &{} A_{22} &{} \cdots &{} A_{2 r} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ A_{1 r}^{\mathrm {T}} &{} A_{2 r}^{\mathrm {T}} &{} \cdots &{} A_{r r} \end{array}\right] . \end{aligned}$$

Then, $A<0$ implies $A_{ii}<0, i=1, \ldots , r$ [14].

3 Main Results

In this section, we develop the sufficient conditions for the $H_\infty$ control of LPV TDSs.

Theorem 1

For some given positive scalars h and $\mu$, the system (1) is asymptotically stable and satisfies $\Vert z(t)\Vert _{2} \le \gamma \Vert w(t)\Vert _{2}$ for any time-varying delay $\tau (t)$ satisfying (2), if there exist symmetric positive-definite matrices $P, Q, R \in \mathbb {R}^{2n_{x} \times 2n_{x}}$, matrices $M_{1}, M_{2} \in \mathbb {R}^{2n_{x} \times 2n_{x}}$, $A_{K}(\rho ), A_{K d}(\rho ) \in \mathbb {R}^{n_{x} \times n_{x}}$, $B_{K}(\rho ) \in \mathbb {R}^{n_{x} \times n_{y}}$, $C_{K}(\rho ) \in \mathbb {R}^{n_{u} \times n_{x}}$, $D_{K}(\rho ) \in \mathbb {R}^{n_{u} \times n_{y}}$ and a positive scalar $\gamma$ such that the following inequality holds:

$$\begin{aligned} \begin{aligned}&\varSigma (\rho )=\left( \begin{array}{cccccc} \varSigma _{11}(\rho ) &{} \varSigma _{12}(\rho ) &{} \varSigma _{13}(\rho ) &{} C_{cl}^\mathrm {T}(\rho ) &{} {\varSigma }_{15}(\rho ) &{} h M_1^\mathrm {T} \\ (*) &{} \varSigma _{22} &{} 0 &{} 0 &{} {\varSigma }_{25}(\rho ) &{} h M_2^\mathrm {T} \\ (*) &{} (*) &{} -\gamma I &{} D_{cl}^\mathrm {T}(\rho ) &{} {\varSigma }_{35}(\rho ) &{} 0 \\ (*) &{} (*) &{} (*) &{} -\gamma I &{} 0 &{} 0 \\ (*) &{} (*) &{} (*) &{} (*) &{} -h R &{} 0 \\ (*) &{} (*) &{} (*) &{} (*) &{} (*) &{} -h R \end{array}\right) <0,\\&\varSigma _{11}(\rho )=A_{cl}^\mathrm {T}(\rho ) P+PA_{cl}(\rho )+Q+M_1^\mathrm {T}+M_1, \\&\varSigma _{12}(\rho )=P A_{cld}(\rho )-M_1^\mathrm {T}+M_2, \ \ \varSigma _{13}(\rho )=P B_{cl}(\rho ), \\&{\varSigma }_{15}(\rho )=h A_{cl}^\mathrm {T}(\rho ) R,\ \ \varSigma _{22}=-(1-\mu ) Q-M_2^\mathrm {T}-M_2, \\&{\varSigma }_{25}(\rho )=h A_{cld}^\mathrm {T}(\rho ) R,\ \ {\varSigma }_{35}(\rho )=h B_{cl}^\mathrm {T}(\rho ) R. \end{aligned} \end{aligned}$$

(12)

Proof

Consider the following candidate LKF:

$$\begin{aligned} \begin{aligned} V(t)&=x^\mathrm {T}(t) P x(t) +\int _{t-\tau (t)}^{t} x^\mathrm {T}(\omega ) Q x({\omega }) \mathrm {d} \omega +\int _{-h}^{0} \int _{t+{\omega }}^{t} \dot{x}^\mathrm {T}(s) R \dot{x}(s) \mathrm {d} s \mathrm {d}{\omega }. \end{aligned} \end{aligned}$$

(13)

Then, the time derivative of (13) becomes

$$\begin{aligned} \begin{aligned} \dot{V}(t)&= x^\mathrm {T}(t)\left( P A_{cl}(\rho )+A_{cl}^\mathrm {T}(\rho ) P+Q\right) x(t) \\&\quad +2 x^\mathrm {T}(t-\tau (t)) A_{cld}^\mathrm {T}(\rho ) P x(t)+2 w^\mathrm {T}(t) B_{cl}^\mathrm {T}(\rho ) P x(t) \\&\quad -(1-\dot{\tau }(t)) x^\mathrm {T}(t-\tau (t)) Q x(t-\tau (t)) +h \dot{x}^\mathrm {T}(t) R \dot{x}(t) \\ {}&\quad -\int _{t-h}^{t} \dot{x}^\mathrm {T}(\omega ) R \dot{x}(\omega ) \mathrm {d} \omega . \end{aligned} \end{aligned}$$

(14)

Note that

$$\begin{aligned} \begin{aligned}&-\int _{t-h}^{t} \dot{x}^\mathrm {T}(\omega ) R \dot{x}(\omega ) \mathrm {d} \omega \\&\quad =-\int _{t-\tau (t)}^{t} \dot{x}^\mathrm {T}(\omega ) R \dot{x}(\omega ) \mathrm {d} \omega -\int _{t-h}^{t-\tau (t)} \dot{x}^\mathrm {T}(\omega ) R \dot{x}(\omega ) \mathrm {d} \omega \\&\quad \le -\int _{t-\tau (t)}^{t} \dot{x}^\mathrm {T}(\omega ) R \dot{x}(\omega ) \mathrm {d} \omega . \end{aligned} \end{aligned}$$

(15)

From Lemma 1 and inequality (15), we obtain

$$\begin{aligned} \begin{aligned} \dot{V}(t)&\le x^\mathrm {T}(t)\left( P A_{cl}(\rho )+A_{cl}^\mathrm {T}(\rho ) P+Q \right) x(t) \\&\quad +2 x^\mathrm {T}(t-\tau (t)) A_{cld}^\mathrm {T}(\rho )P x(t)+2 w^\mathrm {T}(t) B_{cl}^\mathrm {T}(\rho )P x(t) \\&\quad -(1-\mu ) x^\mathrm {T}(t-\tau (t)) Q x(t-\tau (t))+h \dot{x}^\mathrm {T}(t) R \dot{x}(t)\\&\quad +\xi ^\mathrm {T}(t)( \varUpsilon +h \varTheta ^\mathrm {T} R^{-1} \varTheta ) \xi (t), \end{aligned} \end{aligned}$$

(16)

where $\xi (t)={[x(t)^\mathrm {T} \ \ x(t-\tau (t))^\mathrm {T}]}^\mathrm {T}$.

Consider the following performance index:

$$\begin{aligned} \mathcal {J}=\int _{0}^{\infty }\left( \gamma ^{-1}z^\mathrm {T}(t)z(t)- \gamma w^\mathrm {T}(t)w(t)\right) \mathrm {d} t. \end{aligned}$$

(17)

Under zero-initial conditions, the performance index $\mathcal {J}$ is transformed into the following inequality:

$$\begin{aligned} \begin{aligned} \mathcal {J}&=\int _{0}^{\infty }\left( \gamma ^{-1}z^\mathrm {T}(t) z(t)-\gamma w^\mathrm {T}(t) w(t)+\dot{V}(t)\right) \mathrm {d} t+\left. V(t)\right| _{t=0}-\left. V(t)\right| _{t=\infty }\\ {}&\le \int _{0}^{\infty }\left( \gamma ^{-1}z^\mathrm {T}(t) z(t)-\gamma w^\mathrm {T}(t) w(t)+\dot{V}(t)\right) \mathrm {d} t. \end{aligned} \end{aligned}$$

(18)

Thus, if

$$\begin{aligned} \gamma ^{-1}z^\mathrm {T}(t) z(t)-\gamma w^\mathrm {T}(t) w(t)+\dot{V}(t)<0, \end{aligned}$$

(19)

then $\mathcal {J}<0,$ which implies that $\Vert z(t)\Vert _2 \le \gamma \Vert w(t)\Vert _2$ for all nonzero $w(t) \in L_2[0,+\infty )$.

Substituting the expression of z(t) and (16) into (19), we obtain

$$\begin{aligned} \begin{aligned} \eta ^\mathrm {T}(t)\varXi \eta (t)<0, \end{aligned} \end{aligned}$$

(20)

where

$$\begin{aligned} \begin{aligned} \varXi =&\varLambda (\rho )+\gamma ^{-1}[C_{cl}(\rho )\, \ 0 \, \ D_{cl}(\rho )]^\mathrm {T} [C_{cl}(\rho )\, \ 0 \, \ D_{cl}(\rho )]\\ {}&+h[A_{cl}(\rho )\, \ A_{cld}(\rho ) \, \ B_{cl}(\rho )]^\mathrm {T} R [A_{cl}(\rho )\, \ A_{cld}(\rho ) \, \ B_{cl}(\rho )]\\ {}&+ h[M_1\, \ M_2 \, \ 0]^\mathrm {T} R^{-1}[M_1\, \ M_2 \, \ 0]. \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \varLambda (\rho )=\left( \begin{array}{ccc} \varSigma _{11}(\rho ) &{} \varSigma _{12}(\rho ) &{} \varSigma _{13}(\rho )\\ (*) &{} \varSigma _{22} &{} 0\\ (*) &{} (*) &{} -\gamma I \end{array}\right) , \ \ \eta (t)=\left( \begin{array}{c} x(t) \\ x(t-\tau (t)) \\ w(t) \end{array}\right) . \end{aligned}$$

From the inequalities (19) and (20), we have

$$\begin{aligned} \gamma ^{-1}z^\mathrm {T}(t) z(t)-\gamma w^\mathrm {T}(t) w(t)+\dot{V}(t)\le \eta ^\mathrm {T}(t)\varXi \eta (t)<0. \end{aligned}$$

(21)

By applying the Schur complement lemma to $\varXi <0$, one can get the inequality (12). Therefore, if the inequality (12) is satisfied, the inequality $\varXi <0$ is fulfilled. In this case, for $w(t)=0$, we have

$$\begin{aligned} \left. \left[ \gamma ^{-1} z^{\mathrm {T}}(t) z(t)-\gamma w^{\mathrm {T}}(t) w(t)+\dot{V}(t)\right] \right| _{w(t)=0}\le \eta ^\mathrm {T}(t)\varXi \eta (t)|_{w(t)=0}< 0. \end{aligned}$$

(22)

From the above inequality, we can deduce that

$$\begin{aligned} \dot{V}(t)|_{w(t)=0}<-\gamma ^{-1} z^{\mathrm {T}}(t) z(t)|_{w(t)=0}. \end{aligned}$$

(23)

Hence, since the right-hand side of the above inequality is always negative, the closed-loop system (7) with $w(t)=0$ is asymptotically stable. This completes the proof. $\square$

Remark 1

The inequality (12) is still an NMI because of the presence of the coupling between the closed-loop system matrices and Lyapunov–Krasovskii matrices. Therefore, it is very hard to compute controller gains by Theorem 1.

Next, based on Theorem 1, delay-dependent sufficient conditions for the existence of an $H_\infty$ DOF controller are derived in terms of LMIs.

Theorem 2

For some given positive scalars h, $\mu$, and $\varepsilon$, the closed-loop system (7) is asymptotically stable and satisfies $\Vert z(t)\Vert _{2} \le \gamma \Vert w(t)\Vert _{2}$ for any time-varying delay $\tau (t)$ satisfying (2), if there exist symmetric positive-definite matrices $\bar{Q}, \bar{R} \in \mathbb {R}^{2 n_{x} \times 2 n_{x}}, \mathcal {S}, \mathcal {R} \in \mathbb {R}^{n_{x} \times n_{x}}$, matrices $G(\rho ) \in \mathbb {R}^{n_{u} \times n_{y}}, J(\rho ), F(\rho ) \in \mathbb {R}^{n_{x} \times n_{x}}, E(\rho ) \in \mathbb {R}^{n_{x} \times n_{y}}, H(\rho ) \in \mathbb {R}^{n_{u} \times n_{x}}, \bar{M_{1}}, \bar{M_{2}} \in \mathbb {R}^{2n_{x} \times 2n_{x}}$ and a positive scalar $\gamma$ such that the following inequality holds:

$$\begin{aligned}&\left( \begin{array}{cccccccc}{\varOmega _{11}(\rho )} &{} {\varOmega _{12}(\rho )} &{} {\varOmega _{13}(\rho )} &{} {\varOmega _{14}(\rho )} &{} {0} &{} {\varOmega _{16}} &{} {\varOmega _{17}(\rho )} &{} {0} \\ {(*)} &{} {\varOmega _{22}} &{} {0} &{} {0} &{} {0} &{} {\varOmega _{26}} &{} {\varOmega _{27}(\rho )} &{} {0}\\ {(*)} &{} {(*)} &{} {-\gamma I} &{} {\varOmega _{34}(\rho )} &{} {0} &{} {0} &{} {\varOmega _{37}(\rho )} &{} {0}\\ {(*)} &{} {(*)} &{} {(*)} &{} {-\gamma I} &{} {0} &{} {0}&{} {0} &{} {0}\\ {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {\varOmega _{55}} &{} {0} &{} {0} &{} {\varOmega _{58}}\\ {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {\varOmega _{66}}&{} {0} &{} {0}\\ {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {\varOmega _{77}} &{} {0}\\ {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {(*)} &{} {\varOmega _{88}} \end{array}\right) <0, \nonumber \\&{\varPi _{11}(\rho )= \begin{pmatrix} A(\rho )\mathcal {R}+B_2H(\rho ) &{} A(\rho )+B_2G(\rho )C_2 \\ F(\rho ) &{} \mathcal {S}A(\rho )+E(\rho )C_2 \end{pmatrix}}, \nonumber \\&{\varPi _{12}(\rho )=\begin{pmatrix} A_{d}(\rho )\mathcal {R} &{} A_{d}(\rho ) \\ J(\rho ) &{} \mathcal {S}A_{d}(\rho ) \end{pmatrix}},\nonumber \\&{\varPi _{13}(\rho )=\begin{pmatrix} B_{1}(\rho )+B_{2}G(\rho )D_{21} \\ \mathcal {S}B_{1}(\rho )+E(\rho )D_{21} \end{pmatrix}},\nonumber \\&{\varPi _{14}(\rho )=\begin{pmatrix} \mathcal {R}C_{1}(\rho )^\mathrm {T}+H(\rho )^\mathrm {T}D_{12}^\mathrm {T}\\ C_{1}(\rho )^\mathrm {T}+C_{2}^\mathrm {T}G(\rho )^\mathrm {T}D_{12}^\mathrm {T} \end{pmatrix}}, \nonumber \\&{\varPi _{34}(\rho )=\begin{pmatrix} D_{11}(\rho )^\mathrm {T}+D_{21}^\mathrm {T}G(\rho )^\mathrm {T}D_{12}^\mathrm {T} \end{pmatrix}}, \nonumber \\&{\varPi _P=\begin{pmatrix} \mathcal {R} &{} I \\ I &{} \mathcal {S} \end{pmatrix}},\nonumber \\&{\varOmega _{11}(\rho )=\varPi _{11}(\rho )+{\varPi _{11}(\rho )}^\mathrm {T}+\bar{Q}+\bar{M}_{1}+{\bar{M}_{1}}^\mathrm {T}},\nonumber \\&{\varOmega _{12}(\rho )=\varPi _{12}(\rho )-\bar{M_{1}}^\mathrm {T}+\bar{M_2}},\quad {\varOmega _{13}(\rho )=\varPi _{13}(\rho )},\quad {\varOmega _{14}(\rho )=\varPi _{14}(\rho )},\nonumber \\&{\varOmega _{16}=h \bar{M}_1^\mathrm {T}}, \quad {\varOmega _{17}(\rho )=h \varPi _{11}^\mathrm {T}(\rho )},\quad {\varOmega _{22}=-(1-\mu ) \bar{Q}-\bar{M}_2^\mathrm {T}-\bar{M}_2},\nonumber \\&{\varOmega _{26}=h \bar{M}_2^\mathrm {T}},\quad {\varOmega _{27}(\rho )=h \varPi _{12}^\mathrm {T}(\rho )}, \quad {\varOmega _{34}(\rho )=\varPi _{34}(\rho )},\quad {\varOmega _{37}(\rho )=h \varPi _{13}^\mathrm {T}(\rho )},\nonumber \\&{\varOmega _{55}=-h \bar{R}},\quad {\varOmega _{58}=\varepsilon \bar{R}}, \quad {\varOmega _{66}=-h \bar{R}},\quad {\varOmega _{77}=-\varepsilon \varPi _P}, \quad {\varOmega _{88}=-\varepsilon \varPi _P}. \end{aligned}$$

(24)

Then, the controller gain matrices are calculated as

$$\begin{aligned} \left\{ \begin{aligned}&A_{K}(\rho )=\mathcal {N}^{-1}(F(\rho )-\mathcal {S}A(\rho )\mathcal {R}-\mathcal {S}B_2D_K(\rho )C_2\mathcal {R}-NB_K(\rho )C_2\mathcal {R}\\&\qquad \quad \quad -\mathcal {S}B_2C_K(\rho )\mathcal {M}^\mathrm {T})\mathcal {M}^\mathrm {-T}, \\ {}&A_{Kd}(\rho )=\mathcal {N}^{-1}(J(\rho )-\mathcal {S}A_d(\rho )\mathcal {R})\mathcal {M}^\mathrm {-T}, \\ {}&B_K(\rho )=\mathcal {N}^{-1}(E(\rho )-\mathcal {S}B_2D_K(\rho )), \\ {}&C_K(\rho )=(H(\rho )-D_K(\rho )C_2\mathcal {R})\mathcal {M}^\mathrm {-T}, \\ {}&D_K(\rho )=G(\rho ), \end{aligned}\right. \end{aligned}$$

(25)

where matrices $\mathcal {M}$ and $\mathcal {N}$ can be any nonsingular matrices satisfying the following condition:

$$\begin{aligned} \mathcal {M N}^\mathrm {T}=I-\mathcal {R S}. \end{aligned}$$

(26)

Proof

The left-hand side of inequality (12) can be decomposed as

$$\begin{aligned} \varSigma (\rho )=\tilde{\varSigma }(\rho )+\mathcal {X}^\mathrm {T}(\rho )\mathcal {Y}+\mathcal {Y}^\mathrm {T} \mathcal {X}(\rho ), \end{aligned}$$

(27)

where

$$\begin{aligned} \tilde{\varSigma }(\rho )=\left( \begin{array}{ccccccc} \varSigma _{11}(\rho ) &{} \varSigma _{12}(\rho ) &{} \varSigma _{13}(\rho ) &{} C_{cl}^\mathrm {T}(\rho ) &{} 0 &{} hM_{1}^\mathrm {T}\\ (*) &{} \varSigma _{22} &{} 0 &{} 0 &{} 0 &{} hM_{2}^\mathrm {T} \\ (*) &{} (*) &{} -\gamma I &{} D_{cl}^\mathrm {T}(\rho ) &{} 0&{}0 \\ (*) &{} (*) &{} (*) &{} -\gamma I &{}0 &{} 0\\ (*) &{} (*) &{} (*) &{} (*) &{} -hR &{} 0\\ (*) &{} (*) &{} (*) &{} (*) &{} (*) &{} -hR \end{array}\right) , \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l} \mathcal {X}(\rho )=\left( \begin{array}{llllll} hA_{cl}(\rho ) &{} hA_{cld}(\rho ) &{} hB_{cl}(\rho ) &{} 0 &{}0 &{}0 \end{array}\right) , \\ \mathcal {Y}=\left( \begin{array}{llllll} 0 &{}0 &{}0 &{}0 &{} R &{} 0 \end{array}\right) . \end{array} \end{aligned}$$

From Lemma 2 and relation (27), we deduce that

$$\begin{aligned} {\varSigma }(\rho ) \le \tilde{\varSigma }(\rho )+\varepsilon ^{-1} \mathcal {X}^\mathrm {T}(\rho ) \mathcal {Q} \mathcal {X}(\rho )+\varepsilon \mathcal {Y}^\mathrm {T} \mathcal {Q}^{-1} \mathcal {Y}. \end{aligned}$$

(28)

Choosing $\mathcal {Q}=P$ provides a solution to remove some of the cross terms in (27). Hence, the inequality ${\varSigma }(\rho )<0$ holds if the following one is fulfilled:

$$\begin{aligned} \begin{aligned} \tilde{\varSigma }(\rho )+\varepsilon ^{-1} \mathcal {X}^\mathrm {T}(\rho ) P \mathcal {X}(\rho )+\varepsilon \mathcal {Y}^\mathrm {T} P^{-1} \mathcal {Y}<0. \end{aligned} \end{aligned}$$

(29)

By the Schur complement lemma, the inequality (29) is equivalent to

$$\begin{aligned}&\left( \begin{array}{cccccccc} \varSigma _{11}(\rho ) &{} \varSigma _{12}(\rho ) &{} \varSigma _{13}(\rho )&{} C_{cl}^\mathrm {T}(\rho ) &{} 0&{} h M_{1}^\mathrm {T} &{} \varSigma _{17}(\rho ) &{}0 \\ (*) &{} \varSigma _{22} &{} 0 &{} 0 &{} 0 &{} h M_{2}^\mathrm {T} &{} \varSigma _{27}(\rho ) &{} 0 \\ (*) &{} (*) &{} -\gamma I &{} D_{cl}^\mathrm {T}(\rho ) &{} 0 &{} 0 &{} \varSigma _{37}(\rho ) &{} 0 \\ (*) &{} (*) &{} (*) &{} -\gamma I &{} 0 &{} 0 &{} 0 &{} 0 \\ (*) &{} (*) &{} (*) &{} (*) &{} -h R &{} 0 &{} 0 &{} \varepsilon R \\ (*) &{} (*) &{} (*) &{} (*) &{} (*) &{} -h R &{} 0 &{} 0 \\ (*) &{} (*) &{} (*) &{} (*) &{} (*) &{} (*) &{} -\varepsilon P &{} 0 \\ (*) &{} (*) &{} (*) &{} (*) &{} (*) &{} (*) &{} (*) &{} -\varepsilon P \end{array}\right) <0,\nonumber \\&\varSigma _{17}(\rho )=h A_{cl}^\mathrm {T}(\rho )P, \ \ \varSigma _{27}(\rho )=h A_{cld}^\mathrm {T}(\rho ) P, \ \ \varSigma _{37}(\rho )=h B_{cl}^\mathrm {T}(\rho ) P. \end{aligned}$$

(30)

According to the method in [41], we partition P and $P^{-1}$ as follows:

$$\begin{aligned} P=\left( \begin{array}{cc} \mathcal {S} &{} \mathcal {N} \\ \mathcal {N}^\mathrm {T} &{} \otimes \end{array}\right) , \quad P^{-1}=\left( \begin{array}{cc} \mathcal {R} &{} \mathcal {M} \\ \mathcal {M}^\mathrm {T} &{} \otimes \end{array}\right) , \end{aligned}$$

where $\otimes$ represents elements that will not be used in the sequel.

Note that $PP^{-1} = I$ leads to Eq. (26).

We define the following matrices:

$$\begin{aligned} \varGamma _{1}=\left( \begin{array}{cc} \mathcal {R} &{} I \\ \mathcal {M}^\mathrm {T} &{} 0 \end{array}\right) ,\quad \varGamma _{2}=\left( \begin{array}{cc} I &{} \mathcal {S} \\ 0 &{} \mathcal {N}^\mathrm {T} \end{array}\right) , \end{aligned}$$

such that $P\varGamma _{1}=\varGamma _{2}.$

Pre- and post-multiplying (30) by $\mathrm {diag}(\varGamma _1,\varGamma _1,I,I,\varGamma _1,\varGamma _1,\varGamma _1,\varGamma _1)^\mathrm {T}$ and its transpose leads to a matrix inequality which involves the following terms:

$$\begin{aligned}&\begin{aligned}&\varGamma _1^\mathrm {T}PA_{cl}(\rho ) \varGamma _1=\varGamma _2^\mathrm {T}A_{cl}(\rho ) \varGamma _1=\left( \begin{array}{cc} \varPsi _{11}(\rho ) &{} \varPsi _{12}(\rho ) \\ \varPsi _{21}(\rho ) &{}\varPsi _{22}(\rho ) \end{array}\right) ,\\&\varPsi _{11}(\rho )=A(\rho )\mathcal {R}+B_2(\rho )D_K(\rho )C_2(\rho )\mathcal {R}+B_2(\rho )C_K(\rho )\mathcal {M}^\mathrm {T},\\&\varPsi _{12}(\rho )=A(\rho )+B_2(\rho )D_K(\rho )C_2(\rho ),\\&\varPsi _{21}(\rho )=\mathcal {S}A(\rho )\mathcal {R}+\mathcal {S}B_2(\rho )D_K(\rho )C_2(\rho )\mathcal {R}+\mathcal {N}B_K(\rho )C_2(\rho )\mathcal {R}\\&\qquad \quad \qquad +\mathcal {S}B_2(\rho )C_K(\rho )\mathcal {M}^\mathrm {T}+\mathcal {N}A_K(\rho )\mathcal {M}^\mathrm {T},\\&\varPsi _{22}(\rho )=\mathcal {S}A(\rho )+\mathcal {S}B_2(\rho )D_K(\rho )C_2(\rho )+\mathcal {N}B_K(\rho )C_2(\rho ). \end{aligned} \end{aligned}$$

(31)

$$\begin{aligned}&\begin{aligned}&\varGamma _1^\mathrm {T}PA_{cld}(\rho ) \varGamma _1=\varGamma _2^\mathrm {T}A_{cld}(\rho ) \varGamma _1=\left( \begin{array}{cc} A_d(\rho )\mathcal {R}&{} A_d(\rho )\\ \hat{\varPsi }_{21}(\rho )&{}\mathcal {S}A_d(\rho ) \end{array}\right) ,\\&\hat{\varPsi }_{21}(\rho )=\mathcal {S}A_d(\rho )\mathcal {R}+\mathcal {N}A_{Kd}(\rho )\mathcal {M}^\mathrm {T}. \end{aligned} \end{aligned}$$

(32)

$$\begin{aligned}&\begin{aligned}&\varGamma _1^\mathrm {T}PB_{cl}(\rho )=\varGamma _2^\mathrm {T}A_{cld}(\rho )=\left( \begin{array}{c} \bar{\varPsi }_{11}(\rho )\\ \bar{\varPsi }_{21}(\rho ) \end{array}\right) ,\\&\bar{\varPsi }_{11}(\rho )=B_1(\rho )+B_2(\rho )D_K(\rho )D_{21}(\rho ),\\&\bar{\varPsi }_{21}(\rho )=\mathcal {S}B_1(\rho )+\mathcal {S}B_2(\rho )D_K(\rho )D_{21}(\rho )+\mathcal {N}B_K(\rho )D_{21}(\rho ). \end{aligned} \end{aligned}$$

(33)

$$\begin{aligned}&\begin{aligned}&\varGamma _1^\mathrm {T}C_{cl}(\rho )^\mathrm {T}=\left( \begin{array}{c} \tilde{\varPsi }_{11}(\rho )\\ \tilde{\varPsi }_{21}(\rho ) \end{array}\right) ,\\&\tilde{\varPsi }_{11}(\rho )=\mathcal {R}C_1^\mathrm {T}(\rho )+\mathcal {R}C_2^\mathrm {T}(\rho )D_K^\mathrm {T}(\rho )D_{12}^\mathrm {T}(\rho )+ \mathcal {M}C_K^\mathrm {T}(\rho )D_{12}^\mathrm {T}(\rho ),\\&\tilde{\varPsi }_{21}(\rho )=C_1^\mathrm {T}(\rho )+C_2^\mathrm {T}(\rho )D_K^\mathrm {T}(\rho )D_{12}^\mathrm {T}(\rho ). \end{aligned} \end{aligned}$$

(34)

$$\begin{aligned}&\varGamma _{1}^\mathrm {T} R \varGamma _{1}=\bar{R}, \end{aligned}$$

(35)

$$\begin{aligned}&\varGamma _{1}^\mathrm {T} Q \varGamma _{1}=\bar{Q}, \end{aligned}$$

(36)

$$\begin{aligned}&\varGamma _{1}^\mathrm {T} M_{1} \varGamma _{1}=\bar{M}_{1}, \end{aligned}$$

(37)

$$\begin{aligned}&\varGamma _{1}^\mathrm {T} M_{2} \varGamma _{1}=\bar{M}_{2}, \end{aligned}$$

(38)

$$\begin{aligned}&\varGamma _{1}^\mathrm {T} P \varGamma _{1}=\varGamma _{1}^\mathrm {T} \varGamma _{2}=\left( \begin{array}{cc} \mathcal {R} &{} I \\ I &{} \mathcal {S} \end{array}\right) =\varPi _{P}. \end{aligned}$$

(39)

Note that we assume that the input and output matrices are parameter-independent to formulate the design problem as a convex problem with LMI constraints [1, 35]. That is:

$$\begin{aligned}C_{2}(\rho )=C_{2}, B_{2}(\rho )=B_{2}, D_{21}(\rho )=D_{21},\, \text {and} \ D_{12}(\rho )=D_{12}.\end{aligned}$$

After some simple calculations and using the following change of variables:

$$\begin{aligned} \left\{ \begin{aligned}&E(\rho )=\mathcal {S}B_2D_K(\rho )+\mathcal {N}B_K(\rho ), \\ {}&F(\rho )=\mathcal {S}A(\rho )\mathcal {R}+\mathcal {S}B_2D_K(\rho )C_2\mathcal {R} +\mathcal {N}B_K(\rho )C_2\mathcal {R}\\&\quad \quad \quad \quad +\mathcal {S}B_2C_K(\rho )\mathcal {M}^\mathrm {T}+\mathcal {N}A_K(\rho )\mathcal {M}^\mathrm {T}, \\ {}&G(\rho )=D_K(\rho ), \\ {}&H(\rho )=D_K(\rho )C_2\mathcal {R}+C_K(\rho )\mathcal {M}^\mathrm {T}, \\ {}&J(\rho )=\mathcal {S}A_d(\rho )\mathcal {R}+\mathcal {N}A_{Kd}(\rho )\mathcal {M}^\mathrm {T}, \end{aligned}\right. \end{aligned}$$

(40)

LMI (24) can be obtained. This completes the proof. $\square$

Assumption 3

It is assumed that the delay is time-differentiable and bounded with bounded derivatives. In addition, since the conditions are dependent on the upper bound of time delay and its derivative (i.e., $\mu$ and h), the values of these parameters should be known in advance.

Remark 2

According to Corollary 1, if the LMI (24) holds, then $\varOmega _{88}<0$, which implies that $\varGamma _P>0$. In addition, based on the relation (39), $\varGamma _P>0$ means that $P>0$. Therefore, we do not need to add the constraint $P>0$ to the problem.

Remark 3

Using the proposed method, the NMI (12) is successfully transformed into the LMI (24), which can be easily solved via convex optimization algorithms.

Remark 4

In practical systems, the system output signals are easy to access, while the full state information is difficult to obtain [59]. In this paper, it is assumed that system states are not available for technological and/or economical reasons. To this end, a method for designing a DOF controller has been developed.

Remark 5

The constraint (24) is a parameter-dependent LMI for a constant $\varepsilon$. The optimal solution of this problem can be readily approached by solving the LMI-based problem on a grid in $\varepsilon$. The appropriate selection of the parameter $\varepsilon$ is definitely effective in the reduction in conservatism.

Remark 6

The main novelty of this paper is that the results are obtained with less conservatism. Therefore, the designed controller can provide a better disturbance attenuation effect and a larger delay range than previous methods in the literature.

4 Simulation Results

In this section, two examples are provided to illustrate the advantages of the proposed design methodology. The LMI problem is solved by using the YALMIP interface [33] and MOSEK solver [2]. In the examples, the initial conditions are chosen to be zero.

Example 1

This example is motivated by control of chatter during the milling process, whose simplified mechanical model is shown in Fig. 1. The dynamic model of this system can be formulated as an LPV time-delay system [37, 47].

Milling is the machining process using rotary cutters to remove material by advancing a cutter into a workpiece. This system consists of a two-blade cutter of mass $m_1$ and a spindle of mass $m_2$. Moreover, two springs with stiffness $k_1$ and $k_2$ and a damping with the coefficient c are lumped in the model. The motion equations of the system can be expressed as

Fig. 1

Simplified geometry of a milling process [11]

$$\begin{aligned} \begin{aligned}&m_1\ddot{x}_1(t)+k_1(x_1(t)-x_2(t))=f(t),\\&m_2\ddot{x}_2(t)+c\dot{x}_2(t)+k_1(x_2(t)-x_1(t))+k_2x_2(t)=u(t), \end{aligned} \end{aligned}$$

(41)

with

$$\begin{aligned} \begin{aligned} f(t)&=k \sin (\phi (t)+\beta ) l(t)-w(t), \\ l(t)&=\sin (\phi (t))\left[ x_{1}(t-\tau (t))-x_{1}(t)\right] , \end{aligned} \end{aligned}$$

where $x_1(t)$ and $x_2(t)$ are the displacements of the cutter and spindle, respectively. The angle $\beta$ depends on the particular material and the spindle used. The angle $\phi (t)$ denotes the angular position of the blade, k is the cutting force coefficient, u(t) denotes the control input, w(t) denotes the disturbance, and $\tau (t)$ denotes the time-varying delay that is approximated to be $\pi /\omega (t)$, where $\omega (t)$ is the rotation speed of the blade [11].

The system equations (41) can be rewritten as

$$\begin{aligned} \begin{aligned} \ddot{x}_{1}(t)&=\frac{1}{m_{1}}\left[ -k_{1} x_{1}(t)+k_{1} x_{2}(t)-k \sin (\phi (t)+\beta ) \sin (\phi (t)) x_{1}(t)\right. \\&\quad \left. +k \sin (\phi (t)+\beta ) \sin (\phi (t)) x_{1}(t-\tau (t))-w(t)\right] , \\ \ddot{x}_{2}(t)&=\frac{1}{m_{2}}\left[ k_{1} x_{1}(t)-k_{1} x_{2}(t)-k_{2} x_{2}(t)-c \dot{x}_{2}(t)+u(t)\right] . \end{aligned} \end{aligned}$$

(42)

Consider the parameters: $m_1 = 1 \mathrm {kg}$, $m_2 = 2 \mathrm {kg}$, $k_1 = 10 \mathrm {N}/{\mathrm {m}}$, $k_2 = 20 \mathrm {N}/{\mathrm {m}}$, $k = 2 \mathrm {N}/{\mathrm {m}}$, $c=0.5 \mathrm {N}/{\mathrm {m.s}}$, and $\beta = 70^\circ$.

The derived model depends on two measurable parameters $\phi (t)$ and $\omega (t)$. Notice that

$$\begin{aligned} \sin (\phi (t)+\beta ) \sin (\phi (t))=0.5[\cos (\beta )-\cos (2 \phi (t)+\beta )]=0.1710-0.5 \cos (2 \phi (t)+\beta ). \end{aligned}$$

(43)

We define the scheduling parameter vector as $\rho (t)=[\rho _1(t) \ \rho _2(t)]^\mathrm {T}$ with $\rho _1(t)=\cos (2\phi (t)+1.22)$ and $\rho _2(t)=\omega (t)$.

It is assumed that $\omega (t)$ varies between 200 $\mathrm {rpm}$ and 2000 $\mathrm {rpm}$, and the maximum rate of variation is 1000 $\mathrm {rpm}/\mathrm {sec}$. Therefore, we have

$$\begin{aligned} \begin{array}{l} \rho _{1}(t) \in [-1 \ \ 1], \quad \left| \dot{\rho }_{1}(t)\right| =|-2 \omega (t) \sin (2 \phi (t)+1.22)| \le 418.9(\mathrm {rad} / \mathrm {s}), \\ \rho _{2}(t) \in [20.94 \ \ 209.4](\mathrm {rad} / \mathrm {s}), \quad \left| \dot{\rho }_{2}(t)\right| =52.35\left( \mathrm {rad} / \mathrm {s}^{2}\right) . \end{array} \end{aligned}$$

For the time delay ${\tau }(t)={\pi }/{\omega (t)}$, we have

$$\begin{aligned} 0.015< {\tau }(t)< 0.15, \ \ |{\dot{\tau }(t)}| \le 0.75<1. \end{aligned}$$

(44)

Considering the state vector $x(t)=[x_1(t),x_2(t),\dot{x}_1(t),\dot{x}_2(t)]^\mathrm {T}$, the state-space LPV representation of the system can be described as follows [64]:

$$\begin{aligned} \begin{aligned} \dot{x}(t)&=\begin{pmatrix} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ -10.34+\rho _{1}(t) &{} 10 &{} 0 &{} 0 \\ 5 &{} -15 &{} 0 &{} -0.25 \end{pmatrix} x(t)\\&\quad +\begin{pmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 \\ 0.34-\rho _{1}(t) &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \end{pmatrix} x\left( t-\pi / \rho _{2}(t)\right) + {\begin{pmatrix} 0 \\ 0 \\ -1 \\ 0 \end{pmatrix} w(t)+\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0.5 \end{pmatrix} u(t)},\\ z(t)&=\begin{pmatrix} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \end{pmatrix} x(t)+\begin{pmatrix} 0 \\ 0 \\ 0.1 \end{pmatrix} u(t),\\ y(t)&=\begin{pmatrix} 0&1&0&0 \end{pmatrix} x(t). \end{aligned} \end{aligned}$$

In the simulation, the profile of $\omega (t)$ in $\mathrm {rpm}$ unit is chosen as

$$\begin{aligned} \omega (t)=\left\{ \begin{array}{ll} 200 &{} 0 \le t<2 \\ 800(t-2)+200 &{} 2 \le t<3 \\ 1000&{} 3\le t<7 \\ -800(t-7)+1000 &{} 7 \le t<8 \\ 200 &{} 8 \le t \end{array}\right. \end{aligned}$$

and the exogenous disturbance is set to

$$\begin{aligned} w(t)=\left\{ \begin{array}{l} 1,\quad 0 \le t \le 4 \\ 0,\quad 4<t \end{array}\right. \end{aligned}$$

If no control signal is applied to the spindle, the system exhibits chattering behavior. Figure 2 illustrates that it takes a long time for the displacements of the two masses to vanish when no active control is applied. This justifies the use of a controller. To reduce chattering during the milling process, a force u(t) is to be applied to the spindle.

The measurement vector y(t) implies that only the position of the spindle is available. Moreover, the controlled output vector z(t) consists of the displacements of the cutter and the spindle, and the control effort. Note that the penalty on the control effort is 0.1. Therefore, we seek to design an $H_\infty$ controller to reduce the cutter and spindle displacements and penalize large control signal.

By solving the LMIs in Theorem 2 with $\varepsilon =1.5$, the $H_\infty$ performance level is obtained as $\gamma _{\min }=0.268$. The comparative results are listed in Table 1. From the table, the proposed control scheme leads to a better disturbance attenuation performance than the previous control methods [23, 60, 67, 68]. Therefore, it is clear that our result is less conservative than those results reported in the literature.

Table 1 Comparison of minimum $H_\infty$ performance level

Fig. 2

Displacements of the cutter and spindle without control

Fig. 3

Displacements of the cutter and spindle with control

Fig. 4

Control effort for the milling process

Fig. 5

Time response of the ratio ${\sqrt{\int _0^\infty z^\mathrm {T}(t)z(t)\mathrm {d}t}}\Bigm /{\sqrt{\int _0^\infty w^\mathrm {T}(t)w(t)\mathrm {d}t}}$ in Example 1

The obtained simulation results are compared with those in [68]. The displacements of the two masses and the control signal are plotted in Figs. 3 and 4. As can be seen in Fig. 3, by using the proposed method, the closed-loop system has better disturbance attenuation performance and faster convergence rate than the method in [68]. Figure 4 shows that the amplitude of the control signal is more reduced. Moreover, Fig. 5 illustrates the time response of the ratio ${\sqrt{\int _0^\infty z^\mathrm {T}(t)z(t)\mathrm {d}t}}/{\sqrt{\int _0^\infty w^\mathrm {T}(t)w(t)\mathrm {d}t}}$ under w(t) and zero-initial conditions. We can discover that the ratio is smaller than the minimum $H_\infty$ performance level $\gamma _{\min }=0.268$. Therefore, simulations performed validate the better disturbance attenuation effect of the proposed control scheme.

Example 2

Let us consider the following system [35]:

$$\begin{aligned} \dot{x}(t)= & {} \left( \begin{array}{cc} 0 &{} 1+\phi \sin (t) \\ -2 &{} -3+\delta \sin (t) \end{array}\right) x(t)+\left( \begin{array}{cc} \phi \sin (t) &{} 0.1 \\ -0.2+\delta \sin (t) &{} -0.3 \end{array}\right) x(t-\tau (t)) \\&+\left( \begin{array}{c} 0.2 \\ 0.2 \end{array}\right) w(t)+\left( \begin{array}{c} 0.2 \\ 0.2 \end{array}\right) u(t), \\ z(t)= & {} \left( \begin{array}{cc} 0 &{} 10 \\ 0 &{} 0 \end{array}\right) x(t)+\left( \begin{array}{c} 0 \\ 0.1 \end{array}\right) u(t), \\ y(t)= & {} \left( \begin{array}{cc} 0 &{} 1 \\ 0.5 &{} 0 \end{array}\right) x(t), \end{aligned}$$

where $\delta =0.1$ and $\phi =0.2$. Define $\rho (t)=\sin (t)$ as a varying parameter which varies in the uncertainty range $\rho (t)\in [-1 \ 1].$

The LPV system can be described in a polytopic form with the following vertices:

$$\begin{aligned} S(\overline{\rho })=\left( \begin{array}{cccc} A_(\overline{\rho }) &{} A_{d}(\overline{\rho }) &{} B_{1} &{} B_{2} \\ C_{1} &{} 0 &{} 0 &{} D_{12} \\ C_{2} &{} 0 &{} 0 &{} 0 \end{array}\right) \text{ and } S_(\underline{\rho })=\left( \begin{array}{cccc} A_(\underline{\rho }) &{} A_d(\underline{\rho }) &{} B_{1} &{} B_{2} \\ C_{1} &{} 0 &{} 0 &{} D_{12} \\ C_{2} &{} 0 &{} 0 &{} 0 \end{array}\right) , \end{aligned}$$

where

$$\begin{aligned} \begin{array}{l} A(\underline{\rho })=\left( \begin{array}{cc} 0 &{} 0.8 \\ -2 &{} -3.1 \end{array}\right) , A_{d}(\underline{\rho })=\left( \begin{array}{cc} -0.2 &{} 0.1 \\ -0.3 &{} -0.3 \end{array}\right) , \\ A(\overline{\rho })=\left( \begin{array}{cc} 0 &{} 1.2 \\ -2 &{} -2.9 \end{array}\right) , A_{d}(\overline{\rho })=\left( \begin{array}{cc} 0.2 &{} 0.1 \\ -0.1 &{} -0.3 \end{array}\right) . \end{array} \end{aligned}$$

It is observed from Theorem 2 that the $H_\infty$ performance index $\gamma$ indicates the attenuation disturbance level. Therefore, a better disturbance attenuation performance can be achieved by minimizing the value of $\gamma$.

As can be seen in Table 2, by Theorem 2 with $\mu =0.5, h=0.5$, and $\varepsilon =25$, the minimum $H_\infty$ performance level is found as $\gamma _{\min }=0.1021$, which is smaller than the one in [35].

Comparing the proposed method with [35], the MADB for $\mu =0.9$ is presented in Table 3, which shows that our method provides a controller for a larger delay range.

As mentioned before, the $H_\infty$ performance level and MADB are two key indexes to judge the conservatism of the derived criteria. These comparisons show that the proposed method leads to less conservative results than the method in [35].

Table 2 Minimum $H_\infty$ performance level for $h=0.5$ and $\mu =0.5$

Table 3 MADB for $\mu =0.9$

Figure 6 demonstrates the minimized $H_\infty$ performance level under different combinations of the MADB h and delay derivative upper bound $\mu$ by solving the LMIs in Theorem 2. It is evident from the figure that either increasing h or $\mu$ leads to larger values of $\gamma _{\min }$, which indicates a degraded performance level.

For the simulation, the disturbance is set as

$$\begin{aligned} w(t)=\left\{ \begin{array}{l} 0, \quad 0<t<1 \\ 1,\quad 1 \le t \le 4 \\ 0,\quad 4<t \end{array}\right. \end{aligned}$$

The controller’s coefficient matrices are computed as follows:

$$\begin{aligned}&A_{K}(\underline{\rho })=\begin{pmatrix} -74.64 &{} -60.54\\ -30.79 &{} -42.84 \end{pmatrix},\ A_{Kd}(\underline{\rho })=\begin{pmatrix} 10.16 &{} 10.48\\ -19.06 &{} -19.01 \end{pmatrix},\\&B_{K}(\underline{\rho })=\begin{pmatrix} 834.71 &{} -2916.86\\ -876.63 &{} 2668.48 \end{pmatrix},\ C_{K}(\underline{\rho })=\begin{pmatrix} 0.09&0.05 \end{pmatrix},\\&D_{K}(\underline{\rho })=\begin{pmatrix} -148.9&-685.1 \end{pmatrix},\ D_{K}(\overline{\rho })=\begin{pmatrix} -149.53&-685.48 \end{pmatrix},\\&A_{K}(\overline{\rho })=\begin{pmatrix} -74.33 &{} -59.66\\ -27.23 &{} -39.80 \end{pmatrix},\ A_{Kd}(\overline{\rho })=\begin{pmatrix} 29.74 &{} 29.23\\ -26.18 &{} -25.87 \end{pmatrix},\\&B_{K}(\overline{\rho })=\begin{pmatrix} 846.82 &{} -3105.6\\ -883.12 &{} 2794.12 \end{pmatrix},\ C_{K}(\overline{\rho })=\begin{pmatrix} 0&-0.04 \end{pmatrix}. \end{aligned}$$

Fig. 6

Relation between minimum $H_\infty$ performance level $\gamma _{\min }$, MADB h, and delay variation $\mu$

Therefore, the applied controller is calculated as

$$\begin{aligned} K_c(\rho )=\alpha _1(t) K_{c}(\underline{\rho })+\alpha _2(t) K_{c}(\overline{\rho }), \end{aligned}$$

(45)

where

$$\begin{aligned} K_{c}(\underline{\rho })=\left( \begin{array}{ccc} A_{K}(\underline{\rho }) &{} A_{K d}(\underline{\rho }) &{} B_{K}(\underline{\rho }) \\ C_{K}(\underline{\rho }) &{} 0 &{} D_{K}(\underline{\rho }) \end{array}\right) , K_{c}(\overline{\rho })=\left( \begin{array}{ccc} A_{K}(\overline{\rho }) &{} A_{K d}(\overline{\rho }) &{} B_{K}(\overline{\rho }) \\ C_{K}(\overline{\rho }) &{} 0 &{} D_{K}(\overline{\rho }) \end{array}\right) , \end{aligned}$$

and

$$\begin{aligned} \alpha _1(t)=\frac{1-\rho (t)}{2}, \quad \alpha _2(t)=\frac{\rho (t)+1}{2}. \end{aligned}$$

Fig. 7

System states of Example 2

Fig. 8

Controlled outputs of Example 2

Fig. 9

Time response of the ratio ${\sqrt{\int _0^\infty z^\mathrm {T}(t)z(t)\mathrm {d}t}}\Bigm /{\sqrt{\int _0^\infty w^\mathrm {T}(t)w(t)\mathrm {d}t}}$ in Example 2

The system states and the controlled outputs are depicted in Figs. 7 and 8, respectively. Figure 7 shows that the system responses converge to zero asymptotically. Figure 8 demonstrates that the external disturbance is well attenuated in the controlled outputs. Figure 9 shows the time response of the ratio ${\sqrt{\int _0^\infty z^\mathrm {T}(t)z(t)\mathrm {d}t}}/{\sqrt{\int _0^\infty w^\mathrm {T}(t)w(t)\mathrm {d}t}}$ under w(t) and zero-initial conditions. As can be seen from Fig. 9, the ratio is evidently less than the minimum disturbance attenuation level $\gamma _{\min }=0.1021$, and thus the designed controller achieves satisfactory performance.

5 Conclusion

In this paper, a new design methodology is proposed to solve the $H_\infty$ control problem for LPV TDSs. Due to the inherent non-convexity of these problems, they are difficult to solve. This non-convexity is due to the coupling between the decision variables and the system matrices. To handle the non-convex terms, the Young’s relation is used in a judicious manner. This leads to LMI conditions that are numerically tractable with any convex optimization algorithm. The proposed method allows to have additional degrees of freedom in the controller design. These additional decision variables improve the feasibility of the LMI conditions. By solving the LMIs presented, an $H_\infty$ DOF controller is designed. The obtained results demonstrate that the disturbance attenuation level is smaller, and the controller is provided for a larger delay range than those in the existing methods. In future work, we will design an $H_\infty$ finite-time controller for LPV systems with time-varying delay.