Parallel distributed compensation controller design for Markovian jump system with time-varying delays using Bessel-Legendre inequality method and improved positive definite rule

Abstract

In this paper, based on time-varying delay Markovian jump system (MJS), Bessel-Legendre inequality method and improved positive definite condition of Lyapunov function are used, and parallel distributed compensation (PDC) state feedback controller is also introduced. The conservatism of the system is reduced by using this inequality method and positive definite conditions, and further decreases with the increase of the Legendre parameter N. The PDC controller considers two control parameters simultaneously, which can represent the actual system more truthfully. Finally, Example 1 proves the effectiveness of the improved method in this paper. Example 2 considers time-varying probability transition perturbations. Example 3 obtains the parameters of PDC controller with different parameters N. Example 4 illustrates the practical significance of methods in this paper by introducing a time-delay inverted pendulum system with Markovian parameters.

Keywords

Markovian jump system (MJS)Bessel-Legendre inequality PDC controller time-varying delay linear matrix inequalities (LMIs)inverted pendulum system

Introduction

In practical engineering, many systems have undergone sudden structural changes due to changes in external environment, internal interconnected systems, failure of internal components and changes in working points. Such systems can be described as Markovian jump system (MJS), such as solar receiver control (Sworder and Rogers, 1983), manufacturing system (Shen and Buscher, 2012), networked control system (Song et al., 2013), fault detection system (Ge and Han, 2014), flight target tracking (Li and Bar-Shalom, 1993), and so forth. MJS was proposed by Krasovskii and Lidskii in 1961 (Krasovskii and Lidskii, 1961). It consists of several subsystems described by differential or difference equations and random switching rules among these subsystems.

Because MJS can describe many practical engineering systems better and more accurately, the research on MJS has achieved many valuable results (Li, Shi et al., 2017; Luan et al., 2018; Sun et al., 2018; Wu et al., 2018; Zhang et al., 2019). A robust $H_{2}$ state feedback controller for MJS is designed in Xiong and Lam (2009) and Oliveira et al. (2009) to satisfy the upper and lower bounds of transition probability matrix elements. In Geromel and Gabriel (2015), the optimal thermal control problem for a class of MJS based on sampled data is studied. For the multiplicative noise in the system, the hybrid $H_{2} / H_{\infty}$ control problem of MJS is studied in Sheng et al. (2014). Based on the neural network method, the passivity problem of a class of MJS with interval time-varying delays is analyzed in Chen et al. (2015). In addition, many studies have extended Markov chains to nonlinear systems, such as adaptive event-triggered fault detection for semi-MJSs with output quantization (Yan et al., 2019), prescribed performance cooperative control for multiagent systems with input quantization (Yang et al., 2019).

On the other hand, in the industrial process, data transmission delay, measurement delay and aging of system components are the main reasons for the time-delay phenomenon. The existence of time-delay makes the analysis and synthesis of control systems more complex and difficult, and is often the main reason for the instability and performance deterioration of the system. Therefore, the study of time-delay MJS is a hot topic in control theory and engineering (Li, Du et al., 2017; Li et al., 2018; Sun and Wu, 2014 ; Xu et al., 2017). In Li et al. (2014), the problem of robust exponential $H_{\infty}$ controller design for a class of MJS with time-varying delays and saturated inputs is studied. In Wang et al. (2009), the exponential mean square stability of a class of stochastic MJS with mixed time-varying delays is studied by using the neural network method.

However, in the design of controller for MJS, most literatures only consider the internal time-delay phenomena in the system, but ignore the possible time-delay phenomena in the controller, which may lead to instability or even collapse of the control system. Parallel distributed compensation (PDC) control law considers both memoryless and delayed state feedback. So the MJS using PDC controller is closer to the real system. In addition, in order to ensure that the Lyapunov function is positive definite in the stability analysis of the system, such as

V (t) = x^{T} (t) H x (t) + \int_{t - h}^{t} x^{T} (t) G x (t) + \cdot \cdot \cdot

Lyapunov matrices H, G are always greater than 0. This method needs further improvement.

To sum up, this paper improves the above mentioned problems in three aspects. Firstly, the convergence conditions of inequalities are improved by using Bessel-Legendre inequality. With the increase of N, the parameters in Bessel-Legendre inequality can better reduce the conservatism of the system. Then, the positive definite conditions are improved by using the improved interactive convex method and Bessel-Legendre inequality. Finally, the traditional controller is replaced by the PDC controller. In the final numerical simulation, the effectiveness of the improved method in this paper is verified by Example 1, Example 2 is a comparative experiment under time-varying probability transfer perturbation, and different PDC controller parameters are obtained by setting different parameters N. In order to further illustrate the effectiveness of this method, the actual inverted pendulum system is introduced to obtain specific controller parameters.

In this paper, $R^{n \times n}$ represents arbitrary real matrix of $n \times n$ dimension. * represents the symmetric transpose term of diagonal line in linear matrix inequalities (LMIs). $r (t)$ represents the probability transfer factor of MJS. $I_{n}$ represents the unit matrix of $n \times n$ dimensions.

Problem formulation

Consider the following MJS with time-varying-delayed

{\begin{matrix} \overset{\cdot}{x} (t) = A (r (t)) x (t) + A_{d} (r (t)) x (t - h) + B (r (t)) u (t) \\ x (t) = φ (t), \forall t \in [- h_{2}, 0] \end{matrix}

(1)

where $x (t) \in R^{n \times n}$ stands for the state vector. $A (\cdot)$ , $A_{d} (\cdot)$ , $B (\cdot)$ are suitable dimension system matrices with state probability transition factor. $h (t)$ stands for time-varying delays, and satisfy $h_{1} ⩽ h (t) ⩽ h_{2}$ and $d_{1} ⩽ \overset{\cdot}{h} (t) ⩽ d_{2}$ . The state transition rate matrix $Ξ = (π_{ij})_{N \times N}$ is represented by

P {r (t + Δ) = j | r (t) = i} = {\begin{matrix} π_{ij} Δ + o (Δ); if j \neq i, \\ 1 + π_{ii} Δ + o (Δ); if j = i \end{matrix}

where $π_{ij} ⩾ 0$ , if $j \neq i$ , $π_{ii} = - \sum_{j = 1, j \neq i}^{N} π_{ij}$ .

In this paper, parallel distributed compensation (PDC) state feedback controllers are shown below

u (t) = u_{m} (t) + u_{d} (t)

(2)

with

u_{m} (t) = K (r (t)) x (t)

and

u_{d} (t) = K_{d} (r (t)) x (t - h (t))

In summary, combined with MJS (1) and PDC controller (2), the following closed-loop system is obtained

\begin{matrix} \overset{\cdot}{x} (t) = (A (r (t)) + B (r (t)) K (r (t))) x (t) \\ + (A_{d} (r (t)) + B (r (t)) K_{d} (r (t))) x (t - h (t)) \end{matrix}

(3)

Lemma 1: (Park et al., 2011) Let n be a positive integer and symmetric matrices $R_{1} > 0$ , $R_{2} > 0$ be in $R^{n \times n}$ . For all $α = 0, 1$ , if there exist symmetric matrices $X_{1}$ , $X_{2}$ in $R^{n \times n}$ , and matrices $Y_{1}$ , $Y_{2}$ in $R^{n \times n}$ such that the following hold

[\begin{matrix} R_{1} & 0 \\ * & R_{2} \end{matrix}] ⩾ [\begin{matrix} α X_{1} & α Y_{1} + (1 - α) Y_{2} \\ * & (1 - α) X_{2} \end{matrix}]

then, the following inequality is obtained

[\begin{matrix} \frac{1}{α} R_{1} & 0 \\ * & \frac{1}{1 - α} R_{2} \end{matrix}] - [\begin{matrix} R_{1} & 0 \\ * & R_{2} \end{matrix}] ⩾ [\begin{matrix} (1 - α) X_{1} & (1 - α) Y_{2} + α Y_{1} \\ * & α X_{2} \end{matrix}]

Lemma 2: (Bessel-Legendre inequality, Seuret and Frdric, 2017) If a, b are arbitrary constants and $a < b$ , for any positive integer $N ⩾ 0$ and any symmetric matrix $R > 0 \in R^{n \times n}$ , the following inequality is obtained

\frac{1}{b - a} \int_{a}^{b} x^{T} (s) Rx (s) ds ⩾ ϕ_{N}^{T} {\tilde{R}}_{N} ϕ_{N}

where

ϕ_{N} = \frac{1}{b - a} \int_{a}^{b} L_{N} (\frac{s - a}{b - a}) x (s) ds

{\tilde{R}}_{N} = diag {R, 3 R, \dots, (2 N - 1) R}

and

L_{N} (u) = [L_{0} (u) I_{n} L_{1} (u) I_{n} \dots L_{N} (u) I_{n}]^{T}

1_{N} = L_{N} (1) = [\begin{matrix} I_{n} \\ I_{n} \\ ⋮ \\ I_{n} \end{matrix}] {\bar{1}}_{N} = L_{N} (0) = [\begin{matrix} I_{n} \\ - I_{n} \\ ⋮ \\ {(- 1)}^{N} I_{n} \end{matrix}]

Remark 1: Legendre polynomial $L_{N}$ satisfies the following two differential operations, which will be used in the proof of next part

\frac{d}{du} L_{N} (u) = {\bar{Γ}}_{N} L_{N} (u) = Γ_{N} L_{N - 1} (u)

\frac{d}{du} (u L_{N} (u)) = L_{N} (u) + Θ_{N} L_{N} (u)

where ${\bar{Γ}}_{N} = [Γ_{N} 0_{n (N + 1), n}]$ , $Γ_{N} = γ_{N} \otimes I_{n}$ , $Θ_{N} = θ_{N} \otimes I_{n}$ . Matrices $γ_{N} \in R^{(N + 1) \times N}$ and $θ_{N} \in R^{(N + 1) \times (N + 1)}$ are defined by

γ_{N} (k, i) = {\begin{matrix} 0 if k ⩾ i, \\ (2 k - 1) (1 - {(- 1)}^{i + k}) if k < i \end{matrix}

θ_{N} (k, i) = {\begin{matrix} 0 if k > i, \\ k if k = i, \\ (2 k - 1) if k < i \end{matrix}

Remark 2: Explanation on parameter N: The value of N directly affects the augmented terms in Legendre polynomials. With the increase of N, the more complete the system information contained in Legendre polynomials is. Jensen’s inequality (Gu et al., 2003) is a special case of $N = 0$ , the Wirtinger-based integral inequality (Seuret and Gouaisbaut, 2013) is a special case of $N = 1$ , the auxiliary-function based integral inequality (Park et al., 2015) is a special case of $N = 2$ . In Theorem 1 and Corollary 1, the value of N will also affect the dimension of matrix in Lyapunov functional. Although the increase of N will improve the convergence effect of inequalities, it also increases the complexity of function matrix, and the waiting time is too long in simulation. So, in the example part of this paper, only small parameters N are selected to do the comparative experiment.

Design of PDC controller and stabilization analysis

In this part, firstly, Theorem 1 discusses the sufficient conditions for the stability of the system under the positive definite condition of the traditional standard. Secondly, Corollary 1 discusses how to improve the traditional positive definite condition by using Bessel-Legendre inequality.

Theorem 1: For a given positive integer $N ⩾ 0$ , and arbitrary time-varying-delayed $h (t)$ , MJS (1) with PDC state feedback controller (2) is stochastic stable, if there exist positive definite symmetric matrices $P_{iN} = [\begin{matrix} P_{i 11 N} & P_{i 12 N} & P_{i 13 N} \\ * & P_{i 22 N} & P_{i 23 N} \\ * & * & P_{i 33 N} \end{matrix}] \in R^{(2 N + 3) n \times (2 N + 3) n}$ , S, Q, $R \in R^{n \times n}$ , and matrices $Y_{1}$ , $Y_{2} \in R^{(N + 2) n \times (N + 2) n}$ , $F_{i}$ , $F_{di}$ , X with appropriate dimensions, and two scalars $ε_{1}$ and $ε_{2}$ , the following is held

{\overset{̆}{Ψ}}_{N} (h (t), \overset{\cdot}{h} (t)) = [\begin{matrix} ψ_{11} & ψ_{12} & ψ_{13} & ψ_{14} & ψ_{15} & ψ_{16} & \frac{h_{2} - h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{1} & 0 \\ * & ψ_{22} & ψ_{23} & ψ_{24} & ψ_{25} & ψ_{26} & - \frac{h_{2} - h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{1} & \frac{h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{2}^{T} \\ * & * & ψ_{33} & ψ_{34} & ψ_{35} & 0 & 0 & - \frac{h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{2}^{T} \\ * & * & * & ψ_{44} & ψ_{45} & ψ_{46} & - \frac{h_{2} - h (t)}{h_{2}} Γ_{N + 1}^{T} {\bar{Y}}_{1} & 0 \\ * & * & * & * & ψ_{55} & ψ_{56} & 0 & - \frac{h (t)}{h_{2}} Γ_{N + 1}^{T} {\bar{Y}}_{2}^{T} \\ * & * & * & * & * & ψ_{66} & 0 & 0 \\ * & * & * & * & * & * & - \frac{h_{2} - h (t)}{h_{2}} {\bar{\tilde{R}}}_{N + 1} & 0 \\ * & * & * & * & * & * & * & - \frac{h (t)}{h_{2}} {\bar{\tilde{R}}}_{N + 1} \end{matrix}] ⩽ 0

(4)

where

ψ_{11} = He ({\bar{P}}_{i 12 N} 1_{N}) + \bar{S} - 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} 1_{N + 1} - \frac{h_{2} - h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} 1_{N + 1} + \sum_{j = 1}^{N} π_{ij} {\bar{P}}_{j 11 N} + A_{i} X + X^{T} A_{i}^{T} + B_{i} F_{i} + F_{i}^{T} B_{i}^{T}

\begin{matrix} ψ_{12} = (\overset{\cdot}{h} (t) - 1) {\bar{P}}_{i 12 N} {\bar{1}}_{N} + (1 - \overset{\cdot}{h} (t)) {\bar{P}}_{i 13 N} 1_{N} + 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} {\bar{1}}_{N + 1} + \frac{h_{2} - h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} {\bar{1}}_{N + 1} - \\ \frac{h_{2} - h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{2} 1_{N + 1} - \frac{h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{1} 1_{N + 1} + A_{di} X + B_{i} F_{di} + ε_{1} (X^{T} A_{i}^{T} + F_{i}^{T} B_{i}^{T}) \end{matrix}

ψ_{13} = - {\bar{P}}_{i 13 N} {\bar{1}}_{N} + \frac{h_{2} - h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{2} {\bar{1}}_{N + 1} + \frac{h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{1} {\bar{1}}_{N + 1}

ψ_{14} = {\bar{P}}_{i 12 N} ((\overset{\cdot}{h} (t) - 1) {\bar{Γ}}_{N} - \overset{\cdot}{h} (t) Θ_{N}) + h (t) 1_{N}^{T} {\bar{P}}_{i 22 N}^{T} + 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} + \frac{h_{2} - h (t)}{h_{2}} 1_{N + 1} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} + h (t) \sum_{j = 1}^{N} π_{ij} {\bar{P}}_{j 12 N} I_{nN}

ψ_{15} = {\bar{P}}_{i 13 N} (- {\bar{Γ}}_{N} + \overset{\cdot}{h} (t) Θ_{N}) + (h_{2} - h (t)) 1_{N}^{T} {\bar{P}}_{i 23 N} + \frac{h_{2} - h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{2} Γ_{N + 1} + \frac{h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{1} Γ_{N + 1} + (h_{2} - h (t)) \sum_{j = 1}^{N} π_{ij} {\bar{P}}_{j 13 N} I_{nN}

ψ_{16} = {\bar{P}}_{i 11 N} - X + ε_{2} (X^{T} A_{i}^{T} + F_{i}^{T} B_{i}^{T})

\begin{matrix} ψ_{22} = - (1 - \overset{\cdot}{h} (t)) (\bar{S} - \bar{Q}) - {\bar{1}}_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} {\bar{1}}_{N + 1} - \frac{h_{2} - h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} {\bar{1}}_{N + 1} + \frac{h_{2} - h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{2}^{T} {\bar{1}}_{N + 1} + \frac{h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{1}^{T} {\bar{1}}_{N + 1} + \\ \frac{h_{2} - h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{2} 1_{N + 1} + \frac{h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{1} 1_{N + 1} - 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} 1_{N + 1} - \frac{h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} 1_{N + 1} + \\ ε_{1} (A_{di} X + X^{T} A_{di}^{T} + B_{i} F_{di} + F_{di}^{T} B_{i}^{T}) \end{matrix}

ψ_{23} = - \frac{h_{2} - h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{2} {\bar{1}}_{N + 1} - \frac{h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{1} {\bar{1}}_{N + 1} + 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} {\bar{1}}_{N + 1} + \frac{h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} {\bar{1}}_{N + 1}

\begin{matrix} ψ_{24} = h (t) (\overset{\cdot}{h} (t) - 1) {\bar{1}}_{N}^{T} {\bar{P}}_{i 22 N}^{T} + h (t) (1 - \overset{\cdot}{h} (t)) 1_{N}^{T} {\bar{P}}_{i 23 N}^{T} - {\bar{1}}_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} - \frac{h_{2} - h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} + \\ \frac{h_{2} - h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{Y}}_{2}^{T} Γ_{N + 1} + \frac{h (t)}{h_{2}} 1_{N + 1} {\bar{Y}}_{1}^{T} Γ_{N + 1} \end{matrix}

\begin{matrix} ψ_{25} = (h_{2} - h (t)) (\overset{\cdot}{h} (t) - 1) {\bar{1}}_{N}^{T} {\bar{P}}_{i 23 N} + (h_{2} - h (t)) (1 - \overset{\cdot}{h} (t)) 1_{N}^{T} {\bar{P}}_{i 33 N}^{T} - \frac{h_{2} - h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{2} Γ_{N + 1} - \frac{h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{1} Γ_{N + 1} + \\ 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} + \frac{h (t)}{h_{2}} 1_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} \end{matrix}

ψ_{26} = - ε_{1} X + ε_{2} (X^{T} A_{di}^{T} + F_{di}^{T} B_{i}^{T})

ψ_{33} = - \bar{Q} - {\bar{1}}_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} {\bar{1}}_{N + 1} - \frac{h (t)}{h_{2}} {\bar{1}}_{N + 1} {\bar{\tilde{R}}}_{N + 1} {\bar{1}}_{N + 1}

ψ_{34} = - h (t) {\bar{1}}_{N}^{T} {\bar{P}}_{i 23 N}^{T} - \frac{h_{2} - h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{2}^{T} Γ_{N + 1} + \frac{h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{Y}}_{1}^{T} Γ_{N + 1}

ψ_{35} = - (h_{2} - h (t)) {\bar{1}}_{N}^{T} {\bar{P}}_{i 33}^{T} - {\bar{1}}_{N + 1} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} - \frac{h (t)}{h_{2}} {\bar{1}}_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1}

ψ_{44} = He (h (t) {\bar{P}}_{i 22 N} ((\overset{\cdot}{h} (t) - 1) {\bar{Γ}}_{N} - \overset{\cdot}{h} (t) Θ_{N})) - Γ_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} - \frac{h_{2} - h (t)}{h_{2}} Γ_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} + h^{2} (t) I_{nN}^{T} \sum_{j = 1}^{N} π_{ij} {\bar{P}}_{j 22 N} I_{nN}

\begin{matrix} ψ_{45} = h (t) {\bar{P}}_{i 23 N} (- {\bar{Γ}}_{N} + \overset{\cdot}{h} (t) Θ_{N}) + (h_{2} - h (t)) ((\overset{\cdot}{h} (t) - 1) {\bar{Γ}}_{N}^{T} - \overset{\cdot}{h} (t) Θ_{N}^{T}) {\bar{P}}_{i 23 N} - \frac{h_{2} - h (t)}{h_{2}} Γ_{N + 1}^{T} {\bar{Y}}_{2} Γ_{N + 1} - \\ \frac{h (t)}{h_{2}} Γ_{N + 1}^{T} {\bar{Y}}_{1} Γ_{N + 1} + h (t) (h_{2} - h (t)) I_{nN}^{T} \sum_{j = 1}^{N} π_{ij} {\bar{P}}_{j 23 N} I_{nN} \end{matrix}

ψ_{46} = h (t) {\bar{P}}_{i 12 N}^{T}

ψ_{55} = He ((h_{2} - h (t)) {\bar{P}}_{i 33 N} (- {\bar{Γ}}_{N} + \overset{\cdot}{h} (t) Θ_{N})) - Γ_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} - \frac{h (t)}{h_{2}} Γ_{N + 1}^{T} {\bar{\tilde{R}}}_{N + 1} Γ_{N + 1} + (h_{2} - h (t))^{2} I_{nN}^{T} \sum_{j = 1}^{N} π_{ij} {\bar{P}}_{j 33 N} I_{nN}

ψ_{56} = (h_{2} - h (t)) {\bar{P}}_{i 13 N}^{T}

ψ_{66} = h_{2}^{2} \bar{R} - ε_{2} (X + X^{T})

{\bar{\tilde{R}}}_{N + 1} = diag {\bar{R}, 3 \bar{R}, \cdot \cdot \cdot, (2 N + 1) \bar{R}, (2 N + 3) \bar{R}}

PDC control gains are obtained: $K_{i} = F_{i} X^{- 1}$ , $K_{di} = F_{di} X^{- 1}$ .

Proof:

Consider the following Lyapunov-Krasovskii functional (LKF) $V_{N} (t) = V_{1} (t) + V_{2} (t)$

\begin{matrix} V_{1} (t) = x^{T} (t) P_{11 N} (r (t)) x (t) + ϕ_{1, N}^{T} (t) P_{12 N}^{T} (r (t)) x (t) + ϕ_{2, N}^{T} (t) P_{13 N}^{T} (r (t)) x (t) + \\ x^{T} (t) P_{12 N} (r (t)) ϕ_{1, N} (t) + ϕ_{1, N}^{T} (t) P_{22 N} (r (t)) ϕ_{1, N} (t) + ϕ_{2, N}^{T} (t) P_{23 N}^{T} (r (t)) ϕ_{1, N} (t) + \\ x^{T} (t) P_{13 N} (r (t)) ϕ_{2, N} (t) + ϕ_{1, N}^{T} (t) P_{23 N} (r (t)) ϕ_{2, N} (t) + ϕ_{2, N}^{T} (t) P_{33 N} (r (t)) ϕ_{2, N} (t) \end{matrix}

(5)

V_{2} (t) = \int_{t - h (t)}^{t} x^{T} (s) Sx (s) ds + \int_{t - h_{2}}^{t - h (t)} x^{T} (s) Qx (s) ds + h_{2} \int_{t - h_{2}}^{t} \int_{θ}^{t} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) dsd θ

(6)

where

ϕ_{1, N} (t) = \int_{- h (t)}^{0} L_{N} (\frac{s + h (t)}{h (t)}) x_{t} (s) ds = h (t) \frac{1}{h (t)} \int_{- h (t)}^{0} L_{N} (\frac{s + h (t)}{h (t)}) x_{t} (s) ds = h (t) Ψ_{1, N} (t)

ϕ_{2, N} (t) = \int_{- h_{2}}^{- h (t)} L_{N} (\frac{s + h_{2}}{h_{2} - h (t)}) x_{t} (s) ds = (h_{2} - h (t)) \frac{1}{h_{2} - h (t)} \int_{- h_{2}}^{- h (t)} L_{N} (\frac{s + h_{2}}{h_{2} - h (t)}) x_{t} (s) ds = (h_{2} - h (t)) Ψ_{2, N} (t)

Derivative of LKF $V_{1} (t)$ and $V_{2} (t)$

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) = He (x^{T} (t) P_{i 11 N} \overset{\cdot}{x} (t)) + He (ϕ_{1, N}^{T} (t) P_{i 12 N}^{T} \overset{\cdot}{x} (t)) + He (ϕ_{2, N}^{T} (t) P_{i 13 N}^{T} \overset{\cdot}{x} (t)) + He (x^{T} (t) P_{i 12 N} {\overset{\cdot}{ϕ}}_{1, N} (t)) + \\ He (ϕ_{1, N}^{T} (t) P_{i 22 N} {\overset{\cdot}{ϕ}}_{1, N} (t)) + He (ϕ_{2, N}^{T} (t) P_{i 23 N}^{T} {\overset{\cdot}{ϕ}}_{1, N} (t)) + He (x^{T} (t) P_{i 13 N} {\overset{\cdot}{ϕ}}_{2, N} (t)) + \\ He (ϕ_{1, N}^{T} (t) P_{i 23 N} {\overset{\cdot}{ϕ}}_{2, N} (t)) + He (ϕ_{2, N}^{T} (t) P_{i 33 N} {\overset{\cdot}{ϕ}}_{2, N} (t)) + x^{T} (t) \sum_{j = 1}^{N} π_{ij} P_{j 11 N} x (t) + ϕ_{1, N}^{T} (t) \sum_{j = 1}^{N} π_{ij} P_{j 12 N}^{T} x (t) + \\ ϕ_{2, N}^{T} (t) \sum_{j = 1}^{N} π_{ij} P_{j 13 N}^{T} x (t) + x^{T} (t) \sum_{j = 1}^{N} π_{ij} P_{j 12 N} ϕ_{1, N} (t) + ϕ_{1, N}^{T} (t) \sum_{j = 1}^{N} π_{ij} P_{j 22 N} ϕ_{1, N} (t) + ϕ_{2, N}^{T} (t) \sum_{j = 1}^{N} π_{ij} P_{j 23 N}^{T} ϕ_{1, N} (t) + \\ x^{T} (t) \sum_{j = 1}^{N} π_{ij} P_{j 13 N} ϕ_{2, N} (t) + ϕ_{1, N}^{T} (t) \sum_{j = 1}^{N} π_{ij} P_{j 23 N} ϕ_{2, N} (t) + ϕ_{2, N}^{T} (t) \sum_{j = 1}^{N} π_{ij} P_{j 33 N} ϕ_{2, N} (t) \end{matrix}

{\overset{\cdot}{V}}_{2} (t) = x^{T} (t) Sx (t) - (1 - \overset{\cdot}{h} (t)) x^{T} (t - h (t)) (S - Q) x (t - h (t)) - x^{T} (t - h_{2}) Qx (t - h_{2}) + h_{2}^{2} {\overset{\cdot}{x}}^{T} (t) R \overset{\cdot}{x} (t) - h_{2} \int_{- h_{2}}^{0} {\overset{\cdot}{x}}_{t}^{T} (s) R {\overset{\cdot}{x}}_{t} (s) ds

(7)

Using Lemma 1 and Lemma 2 for the last term in ${\overset{\cdot}{V}}_{2} (t)$

h_{2} \int_{- h_{2}}^{0} {\overset{\cdot}{x}}_{t}^{T} (s) R {\overset{\cdot}{x}}_{t} (s) ds ⩾ ξ_{N}^{T} (t) ϖ_{N}^{T} [ϑ_{N} (h) - ϑ_{N 0} (h)] ϖ_{N} ξ_{N} (t)

(8)

where

ϑ_{N} (h (t)) = [\begin{matrix} \frac{2 h_{2} - h (t)}{h_{2}} {\tilde{R}}_{N + 1} & \frac{h_{2} - h (t)}{h_{2}} Y_{2} + \frac{h (t)}{h_{2}} Y_{1} \\ * & \frac{h_{2} + h (t)}{h_{2}} {\tilde{R}}_{N + 1} \end{matrix}] ϑ_{N 0} (h (t)) = [\begin{matrix} \frac{h_{2} - h (t)}{h_{2}} Y_{1} {\tilde{R}}_{N + 1}^{- 1} Y_{1}^{T} & 0 \\ 0 & \frac{h (t)}{h_{2}} Y_{2}^{T} {\tilde{R}}_{N + 1}^{- 1} Y_{2} \end{matrix}]

and

ϖ_{N} = [\begin{matrix} 1_{N + 1} & - {\bar{1}}_{N + 1} & 0 & - Γ_{N + 1} & 0 & 0 \\ 0 & 1_{N + 1} & - {\bar{1}}_{N + 1} & 0 & - Γ_{N + 1} & 0 \end{matrix}]

ξ_{N} (t) = col {x (t) x (t - h (t)) x (t - h_{2}) Ψ_{1, N}^{T} (t) Ψ_{2, N}^{T} (t) \overset{\cdot}{x} (t)}

Next, consider the following null term with the slack variables $Z_{1}$ , $Z_{2}$ and $Z_{3}$

2 (x^{T} (t) Z_{1} + x^{T} (t - h (t)) Z_{2} + {\overset{\cdot}{x}}^{T} (t) Z_{3}) \times ((A_{i} + B_{i} K_{i}) x (t) + (A_{di} + B_{i} K_{di}) x (t - h (t)) - \overset{\cdot}{x} (t)) = 0

which is equivalent to

ξ_{N}^{T} (t) ϒ_{N} ξ_{N} (t) = 0

(9)

where

ϒ_{N} = [\begin{matrix} ϒ_{N}^{11} & ϒ_{N}^{12} & 0 & 0 & 0 & - Z_{1} + {(A_{i} + B_{i} K_{i})}^{T} Z_{3}^{T} \\ * & ϒ_{N}^{22} & 0 & 0 & 0 & - Z_{2} + {(A_{di} + B_{i} K_{di})}^{T} Z_{3}^{T} \\ * & * & 0 & 0 & 0 & 0 \\ * & * & * & 0 & 0 & 0 \\ * & * & * & * & 0 & 0 \\ * & * & * & * & * & - Z_{3} - Z_{3}^{T} \end{matrix}]

and

\begin{matrix} ϒ_{N}^{11} = Z_{1} (A_{i} + B_{i} K_{i}) + (A_{i} + B_{i} K_{i})^{T} Z_{1}^{T} \\ ϒ_{N}^{12} = Z_{1} (A_{di} + B_{i} K_{di}) + (A_{i} + B_{i} K_{i})^{T} Z_{2}^{T} \\ ϒ_{N}^{22} = Z_{2} (A_{di} + B_{i} K_{di}) + (A_{di} + B_{i} K_{di})^{T} Z_{2}^{T} \end{matrix}

Considering the coupling between parameters memoryless $K_{i}$ , delayed state feedback $K_{di}$ in PDC controller and LKF, let $Z_{1} = X^{- T}$ , $Z_{2} = ε_{1} X^{- T}$ , $Z_{3} = ε_{2} X^{- T}$ , $ε_{1}$ and $ε_{2}$ are arbitrary constants. The coefficient symbols in matrix (9) are expressed concretely, then left-multiplied matrix $diag {X, X, X, X, X, X, X_{N + 1}, X_{N + 1}}^{T}$ and right-multiplied matrix $diag {X, X, X, X, X, X, X_{N + 1}, X_{N + 1}}$ , where the dimension of $X_{N + 1}$ matches the diagonal matrix ${\tilde{R}}_{N + 1}$ . Then the following LMIs hold

{\overset{̆}{ϒ}}_{N} = [\begin{matrix} {\overset{̆}{ϒ}}_{N}^{11} & {\overset{̆}{ϒ}}_{N}^{12} & 0 & 0 & 0 & - X + ε_{2} (X^{T} A_{i}^{T} + F_{i}^{T} B_{i}^{T}) \\ * & {\overset{̆}{ϒ}}_{N}^{22} & 0 & 0 & 0 & - ε_{1} X + ε_{2} (X^{T} A_{di}^{T} + F_{di}^{T} B_{i}^{T}) \\ * & * & 0 & 0 & 0 & 0 \\ * & * & * & 0 & 0 & 0 \\ * & * & * & * & 0 & 0 \\ * & * & * & * & * & - ε_{2} (X + X^{T}) \end{matrix}]

(10)

and

\begin{matrix} {\overset{̆}{ϒ}}_{N}^{11} = A_{i} X + X^{T} A_{i}^{T} + B_{i} F_{i} + F_{i}^{T} B_{i}^{T} \\ {\overset{̆}{ϒ}}_{N}^{12} = A_{di} X + B_{i} F_{di} + ε_{1} (X^{T} A_{i}^{T} + F_{i}^{T} B_{i}^{T}) \\ {\overset{̆}{ϒ}}_{N}^{22} = ε_{1} (A_{di} X + X^{T} A_{di}^{T} + B_{i} F_{di} + F_{di}^{T} B_{i}^{T}) \end{matrix}

The parameters are replaced equivalently as follows: $F_{i} = K_{i} X$ , $F_{di} = K_{di} X$ , ${\bar{P}}_{iN} = X^{T} P_{iN} X$ , ${\bar{Y}}_{1} = X^{T} Y_{1} X$ , ${\bar{Y}}_{2} = X^{T} Y_{2} X$ , ${\bar{\tilde{R}}}_{N + 1} = X^{T} {\tilde{R}}_{N + 1} X$ , $\bar{S} = X^{T} SX$ , $\bar{Q} = X^{T} QX$ , $\bar{R} = X^{T} RX$ .

Combining (7), (8) and (10), we can get the following

{\overset{\cdot}{V}}_{N} (t) ⩽ {\overset{̆}{Ψ}}_{N} (h (t), \overset{\cdot}{h} (t))

where ${\overset{̆}{Ψ}}_{N} (h (t), \overset{\cdot}{h} (t))$ is shown in (4). As long as condition (4) is satisfied, ${\overset{\cdot}{V}}_{N} (t) ⩽ 0$ is satisfied, which means that the MJS is asymptotically stable. Also, the PDC controller is obtained $K_{i} = F_{i} X^{- 1}$ , $K_{di} = F_{di} X^{- 1}$

This completes the proof.

Remark 3: When constructing a PDC controller, we confine the parameters of the controller to an arbitrary matrix separately, which can avoid too many coupling nonlinear terms between the controller matrix and the system matrix, reduce the complexity of the system, and this arbitrary matrix will not affect the stability of the system itself. Finally, the coupling terms are decoupled by similarity matrix transformation.

Next, we will improve the positive definite condition in $V_{N} (t)$ .

Corollary 1: For a given positive integer $N ⩾ 0$ , the LKF $V_{N} (t)$ is positive definite, if there exist symmetric matrices $P_{iN} = [\begin{matrix} P_{i 11 N} & P_{i 12 N} & P_{i 13 N} \\ * & P_{i 22 N} & P_{i 23 N} \\ * & * & P_{i 33 N} \end{matrix}] \in R^{(2 N + 3) n \times (2 N + 3) n}$ , $S > 0$ , $Q > 0$ , $R > 0 \in R^{n \times n}$ and a matrix $U \in R^{n (N + 1) \times n (N + 1)}$ , the following

Φ_{N} (h_{2}) = \frac{1}{h_{2}} [\begin{matrix} h_{2} P_{i 11 N} & h_{2} P_{i 12 N} & h_{2} P_{i 13 N} \\ * & {\tilde{S}}_{N} + h_{2} P_{i 22 N} & U + h_{2} P_{i 23 N} \\ * & * & {\tilde{Q}}_{N} + h_{2} P_{i 33 N} \end{matrix}] > 0

(11)

hold, where ${\tilde{S}}_{N} = diag {S, 3 S, \dots, (2 N + 1) S}$ and ${\tilde{Q}}_{N} = diag {Q, 3 Q, \dots, (2 N + 1) Q}$ .

Proof:

The LKF $V_{N} (t)$ are shown in (5) and (6). Then, applying Lemma 2 to the order N yields

\int_{t - h (t)}^{t} x^{T} (s) Sx (s) ds ⩾ h (t) Ψ_{1, N}^{T} (t) {\tilde{S}}_{N} Ψ_{1, N} (t)

\int_{t - h_{2}}^{t - h (t)} x^{T} (s) Qx (s) ds ⩾ (h_{2} - h (t)) Ψ_{2, N}^{T} (t) {\tilde{Q}}_{N} Ψ_{2, N} (t)

where $Ψ_{1, N} (t)$ and $Ψ_{2, N} (t)$ are shown in Theorem 1, then we can get

V_{N} (t) ⩾ {[\begin{matrix} x (t) \\ Ψ_{1, N} (t) \\ Ψ_{2, N} (t) \end{matrix}]}^{T} [\begin{matrix} P_{i 11 N} & P_{i 12 N} & P_{i 13 N} \\ * & \frac{1}{h (t)} {\tilde{S}}_{N} + P_{i 22 N} & P_{i 23 N} \\ * & * & \frac{1}{h_{2} - h (t)} {\tilde{Q}}_{N} + P_{i 33 N} \end{matrix}] [\begin{matrix} x (t) \\ Ψ_{1, N} (t) \\ Ψ_{2, N} (t) \end{matrix}] + h_{2} \int_{t - h_{2}}^{t} \int_{θ}^{t} {\overset{\cdot}{x}}^{T} (s) R \overset{\cdot}{x} (s) dsd θ

Following Lemma 1, consider $X_{1} = X_{2} = 0$ and $Y_{1} = Y_{2} = U \in R^{n (N + 1) \times n (N + 1)}$ , then, if $Φ_{N} (h_{2}) > 0$ , $S > 0$ , $Q > 0$ and $R > 0$ , the LKF $V_{N} (t)$ is positive definite. This completes the proof.

Remark 4: Corollary 1 removes the condition of positive definite $P_{iN}$ in Theorem 1 and constructs (11) by using Bessel-Legendre inequality. That is to say, positive definite $P_{iN}$ is a special case of (11). As long as Corollary 1 holds, Theorem 1 must hold, so Corollary 1 improves the condition of positive definite $P_{iN}$ in Theorem 1.

Numerical examples

In this part, four examples are given to illustrate the effectiveness of the proposed method in this paper. Firstly, the validity of Bessel-Legendre inequality method and improved positive definite condition are illustrated by analyzing the time-varying delay stability of MJS without considering the PDC state feedback controller.

Example 1:

Consider MJS (1) with two modes and the matrix parameters (Nuo and Liankun, 2018)

A_{1} = [\begin{matrix} 0.5 & - 1 \\ 0 & - 3 \end{matrix}] A_{2} = [\begin{matrix} - 5 & 1 \\ 1 & 0.2 \end{matrix}] A_{d 1} = [\begin{matrix} 0.5 & - 0.2 \\ 0.2 & 0.3 \end{matrix}] A_{d 2} = [\begin{matrix} - 0.3 & 0.5 \\ 0.4 & - 0.5 \end{matrix}]

with transition rates matrix

Ξ = [\begin{matrix} - 7 & 7 \\ 3 & - 3 \end{matrix}]

Let $d_{2} = 0.1$ , $h_{1} = 0.5$ , $ε_{1} = 10^{- 1}$ , $ε_{2} = 10^{- 2}$ . By choosing different Legendre parameter N, different upper bounds of delay $h_{2}$ can be obtained. The final comparison results are obtained by Table 1. It is shown that the method presented in this paper is less conservative than improved Wirtinger-based integral inequality in Nuo and Liankun (2018) when $N ⩾ 1$ .

Table 1.

Upper bounds of delay $h_{2}$ for Example 1.

	By Nuo and Liankun (2018)	N=1	N=2	N=3
$h_{2}$	2.842	3.023	3.115	3.195

When $N = 1$ , set state vector $x (t) = [- 0.2 0.2]$ . The response curves obtained are shown in following. Figure 1 is the state transition response curve of MJS (1), Figure 2 is a curve that responds according to transition probability under given initial state vector. The results of Figure 2 show that the curve is convergent, which shows that the method presented in this paper is effective.

Example 2:

This example takes into account the time-varying probability transfer disturbance. In order to make a comparative experiment, this disturbance is added to Theorem 1, which is transformed as follows

{\overset{̆}{Ψ}}_{N} (h, \overset{\cdot}{h}) = \sum_{l = 1}^{M} r_{l}^{2} (t) {\overset{̆}{Ψ}}_{N}^{(ll)} + \sum_{l = 1}^{M - 1} \sum_{s = l + 1}^{M} r_{l} (t) r_{s} (t) ({\overset{̆}{Ψ}}_{N}^{(ls)} + {\overset{̆}{Ψ}}_{N}^{(sl)}) ⩽ 0

The deformations of $P_{N} (r (t))$ with Markov probability transfer factor is as follows

\begin{matrix} {\overset{\cdot}{P}}_{N} (r (t)) = P_{iN} (r (t)) + \sum_{j = 1}^{N} π_{ij} (r (t)) P_{jN} (r (t)) + \frac{d (P_{iN} (r (t)))}{dt} \\ = P_{iN}^{l} + \sum_{j = 1}^{N} π_{ij}^{l} P_{jN}^{s} + \sum_{n = 1}^{M - 1} \pm (P_{iN}^{n} - P_{iN}^{M}) \end{matrix}

So, the LMI (4) of Theorem 1 is ultimately transformed into

{\overset{̆}{Ψ}}_{N} (h (t), \overset{\cdot}{h} (t)) ⩽ 0 \Rightarrow {\overset{̆}{Ψ}}_{N}^{ls} + {\overset{̆}{Ψ}}_{N}^{sl} ⩽ 0, 1 ⩽ l ⩽ s ⩽ M

Consider MJS (1) with two modes and the matrix parameters (Ding and Liu, 2016)

A_{1} = [\begin{matrix} - 4.5 & 2 \\ - 0.8 & - 4.3 \end{matrix}] A_{2} = [\begin{matrix} - 3.5 & 1.2 \\ 1.0 & - 1.9 \end{matrix}] A_{d 1} = [\begin{matrix} 0.6 & 1.6 \\ 1.8 & - 1.1 \end{matrix}] A_{d 2} = [\begin{matrix} 1.6 & 0.5 \\ - 0.7 & 1.0 \end{matrix}]

The time-varying probability transfer matrix follows the following rules $Ξ = si n^{2} (t) Ξ^{(1)} + co s^{2} (t) Ξ^{(2)}$ and

Ξ^{(1)} = [\begin{matrix} - 0.8 & 0.8 \\ 0.6 & - 0.6 \end{matrix}] Ξ^{(2)} = [\begin{matrix} - 0.6 & 0.6 \\ 0.8 & - 0.8 \end{matrix}]

Let $d_{2} = 0.6014$ , $h_{1} = 0.1$ , $ε_{1} = 10^{- 1}$ , $ε_{2} = 10^{- 2}$ . As shown in Table 2, we can see that the results in this paper have a higher upper bound of delay when considering time-varying probability transition perturbations.

Table 2.

Upper bounds of delay $h_{2}$ for Example 2.

	By Ding and Liu (2016)	N=1	N=2	N=3
$h_{2}$	1	1.3586	1.3640	1.3701

Next, we discuss MJS with PDC controller (3). According to different Legendre parameter N, different controller parameters are obtained.

Example 3:

Consider MJS (1) with the following two modes and the matrix parameters

A_{1} = [\begin{matrix} 0 & 0.6 \\ 0 & 1 \end{matrix}] A_{d 1} = [\begin{matrix} 0.5 & 0.9 \\ 0 & 2 \end{matrix}] B_{1} = [\begin{matrix} 1 \\ 1 \end{matrix}]

A_{2} = [\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}] A_{d 2} = [\begin{matrix} 0.9 & 0 \\ 1 & 1.6 \end{matrix}] B_{1} = [\begin{matrix} 1 \\ 1 \end{matrix}]

with transition rates matrix

Ξ = [\begin{matrix} - 0.5 & 0.5 \\ 0.3 & - 0.3 \end{matrix}]

Let $d_{2} = 0.1$ , $h_{1} = 0.01$ , $ε_{1} = 10^{- 1}$ , $ε_{2} = 10^{- 2}$ . By choosing different N, different upper bounds $h_{2}$ , memoryless $K_{i}$ and delayed state feedback $K_{di}$ parameters can be obtained. The results are shown in Table 3. It can be seen that with the increase of N, the upper bound of time delay $h_{2}$ is higher, and Bessel-Legendre inequality is more effective and less conservative.

Table 3.

$h_{2}$ and controller parameters for Example 3.

	N=0	N=1	N=2	N=3	N=4
$h_{2}$	3.67	3.72	3.85	3.92	4.01
$K_{1}$	[0.1561 -0.7999]	[0.3659 -0.6817]	[0.2298 -0.6110]	[0.2825 -0.7219]	[0.2505 -0.7159]
$K_{2}$	[-0.9236 0.0727]	[-0.7435 0.2241]	[-0.6900 0.1044]	[-0.8186 0.1712]	[-0.8201 0.1461]
$K_{d 1}$	[0.8094 -0.9012]	[4.4110 2.9467]	[4.2742 2.6872]	[4.2902 2.7658]	[4.3695 2.8906]
$K_{d 2}$	[0.0312 -0.1922]	[3.6931 3.6082]	[3.5356 3.3861]	[3.5927 3.4360]	[3.6643 3.5638]

Finally, the validity of Bessel-Legendre inequality and improved positive definite conditions are illustrated by the actual inverted pendulum system, and the controller parameters of the time-delay inverted pendulum system are obtained.

Example 4:

Consider the following PDC control inverted pendulum system with markovian parameters (Zhang et al., 2011)

{\begin{matrix} \overset{\cdot}{x} (t) & = & A (r (t)) x (t) + A_{d} (r (t)) x (t - h (t)) + B_{1} (r (t)) u (t - h (t)) + B_{2} (r (t)) ω (t) \\ u (t - h (t)) & = & K_{d} (r (t)) x (t - h (t)) \\ x (t) & = & φ (t), t \in [- h_{2}, 0] \end{matrix}

(12)

where $x (t)$ represents the state vector, $h (t)$ represents the time-varying delay, $u (\cdot)$ represents the PDC controller, $ω (t)$ represents the external disturbance, $A (\cdot)$ , $A_{d} (\cdot)$ , $B_{1} (\cdot)$ and $B_{2} (\cdot)$ represent the constant matrices of the appropriate dimension, $K_{d} (\cdot)$ represents the controller parameter, $r (t)$ represents the markovian parameter.

The actual system matrix parameters are as follows

A (r (t)) = [\begin{matrix} 1 & 0 & T & 0 \\ 0 & 1 & 0 & T \\ 0 & - \frac{Tmg}{M} & 1 - \frac{T c_{r}}{M} & 0 \\ 0 & \frac{T (M + m) g}{Ml} & \frac{T c_{r}}{Ml} & 1 \end{matrix}] A_{d} (r (t)) = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & \frac{T g_{ω}}{M} & 0 & 0 \\ 0 & \frac{- T g_{ω}}{Ml} \end{matrix}] B_{1} (r (t)) = [\begin{matrix} 0 \\ 0 \\ \frac{T}{M} \\ \frac{- T}{Ml} \end{matrix}]

where M is the mass of the slider on the bracket, m is the mass of a damped pendulum, l is the length of the pendulum, g is gravitational acceleration, T is sampling time, $g_{ω}$ is an external disturbance parameter. Select parameters: $M = 1.378 kg$ , $m = 0.051 kg$ , $l = 0.325 m$ , $g = 9.8 m / s^{2}$ , $c_{r} = 5.98 kg / s$ , $g_{ω} = 1.2 kg / (ms)$ , $T = 0.025 s$ , $θ = 15^{°}$ . Then, we have

A_{1} = [\begin{matrix} 1 & 0 & 0.025 & 0 \\ 0 & 1 & 0 & 0.025 \\ 0 & - 0.0091 & 0.8915 & 0 \\ 0 & 0.7817 & 0.3338 & 1 \end{matrix}] A_{d 1} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0.0218 & 0 & 0 \\ 0 & - 0.067 & 0 & 0 \end{matrix}] B_{11} = [\begin{matrix} 0 \\ 0 \\ 0.0181 \\ - 0.056 \end{matrix}]

A_{2} = [\begin{matrix} 1 & 0 & 0.025 & 0 \\ 0 & 1 & 0 & 0.025 \\ 0 & - 0.01 & 0.8915 & 0 \\ 0 & 0.9954 & 0.3855 & 1 \end{matrix}] A_{d 2} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0.0218 & 0 & 0 \\ 0 & - 0.0773 & 0 & 0 \end{matrix}] B_{12} = [\begin{matrix} 0 \\ 0 \\ 0.0181 \\ - 0.065 \end{matrix}]

and transition rates matrix

Ξ = [\begin{matrix} - 7 & 7 \\ 3 & - 3 \end{matrix}]

Let $d_{2} = 0.1$ , $h_{1} = 0.1$ , $ε_{1} = 10^{- 1}$ , $ε_{2} = 10^{- 2}$ , $ω (t) = 0$ . When $N = 0$ , the upper bound of time delay $h_{2} = 4.67$ , and delayed state feedback control parameter of inverted pendulum with time-varying-delayed

K_{d 1} = [100.8355 - 328.9065 128.8979 - 22.5886]

K_{d 2} = [85.4179 - 325.2413 - 129.0472 - 93.9115]

Conclusions

In this paper, the MJS based on PDC state feedback controller is proposed, and Bessel-Legendre inequality and improved positive definite rules are introduced. The final examples are given to illustrate the effectiveness of the proposed method under general or time-varying probability transition perturbations, and the parameters of PDC controller with different parameters N are obtained. Moreover, the time-delay inverted pendulum system with Markovian parameters is introduced to obtain the controller parameters of the actual system.

Footnotes

Acknowledgements

The authors are very indebted to the Editor and the anonymous reviewers for their insightful comments and valuable suggestions that have helped improve the academic research.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project is supported by the National Natural Science Foundation of China (Grant Nos. 61403278,61503280).

ORCID iDs

Nuo Xu

Liankun Sun

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