H ∞ Dynamic output feedback control of LPV time-delay systems via dilated linear matrix inequalities

Abstract

This paper uses the linear matrix inequality dilation approach to deal with robust stability and H_∞ dynamic output feedback controller synthesis for linear parameter varying delayed systems with variable delay. This approach can express the original non-convex problem in terms of convex linear matrix inequalities and consequently reduces the conservatism of linear matrix inequality synthesis without dilation. Both delay-dependent stability and H_∞ performance are studied in a quadratic context. Furthermore, a Lyapunov–Krasovskii functional is used to derive a delay-dependent criterion formulated in terms of a linear matrix inequality that will be used to search for an H_∞ linear parameter varying delayed dynamic output feedback controller. To achieve this aim we use an integral inequality which plays a key role in the derivation of this criterion and enables the reduction of the H_∞ cost in comparison to other results.

Keywords

Linear parameter-varying delayed systems dilated linear matrix inequalities Lyapunov–Krasovskii functional

Introduction

In recent years, a considerable effort has been made to deal with the analysis and control of systems in which a time delay occurs (Choi et al., 2016; Cheng and Yi, 2014; Hongmin and Xinyong, 2016; Kao and Rantzer, 2007; Li et al., 2014; Suplin et al., 2006). In this context, the robust stability analysis of linear uncertain delayed systems has received attention in particular (Beibei et al., 2017, 2009; He et al., 2004; Park et al., 2015; Zhong, 2006; Xia and Jia, 2003).

The stability criteria can be classified into two classes according to the dependence or non-dependence of the criteria on the size of the delay. The delay-independent criteria are more conservative than the delay dependent ones.

In other fields of research, linear parameter varying (LPV) systems have been studied intensively (Onat et al., 2009; Seron and De Doná, 2015; Xia and Eisaka, 2004; Yingbo et al., 2016) because they can be regarded as a particular type of nonlinear system. Therefore, the first control strategy for these systems was based on a gain scheduling adaptive methodology (Rugh and Shamma, 2000; Shamma and Athans, 1991). This strategy provides a reasonable compromise between performance and robustness at the expense of higher complexity and stability problems in the switching zones. In order to obtain guarantees of stability and performance against dependent parameters, another approach based on the concept of parameter-dependent controllers has been investigated (Daafouz et al., 2008; Wu, 2001; Xie, 2008). The sought controller has the same dependence on the time-varying parameters as the plant. Thus, the performances are higher than those obtained with a classical Linear Time Invariant controller because conservatism is reduced. At this level, significant progress has been made in this area, notably in the development of linear matrix inequality (LMI) techniques which allowed for the formulation of the analysis problem and the synthesis of the controller in terms of LMI problems.

Most LMI characterizations in the quadratic theory involve the product of the Lyapunov variables with the controller’s. This enforces a particular (and restrictive) choice of Lyapunov variables in order to get around bilinearities. To reduce the conservatism generated by this method (Ebihara and Hagiwara, 2005; Oliveira and Peres, 2005) showed that dilation of the LMI characterizations helps to avoid the restriction of using a common Lyapunov variable.

Motivated by the LPV control theory, many reports (He et al., 2004; Zhang et al., 2002, 2014) deal with the analysis and control of LPV time-delay systems using the Lyapunov–Krasovskii functional. To the best of the authors’ knowledge, only the analysis and the state feedback problem have been investigated (Briat et al., 2009; Wu and Grigoriadis, 2001). However, the output feedback control problem in LPV time-delay systems remains open and major improvements are still expected to be made. This paper examines the delay-dependent stability and H_∞ output feedback control of LPV time-delay systems.

The main contributions of this paper are as follows.

We provide a new type H_∞ output feedback control and develop a less conservative delay-dependent stability for LPV time-delay systems based on the approach of Briat et al. (2013).

We propose a general approach for the dilated LMI characterizations.

Additionally, to analyse the stability of LPV delay systems and to provide a delay-dependent criterion, we choose a Lyapunov–Krasovskii functional so that its derivative has an explicit dependence on the maximal value of the system delay considered to be variable. The derived condition is expressed in the LMI framework and is broadened to take into account an H_∞ performance. This is the purpose of section ‘Analysis of LPV time-delay systems’. In section ‘H_∞ dynamic output feedback’ we deal with the problem of the H_∞ dynamic output feedback control of LPV delayed systems. The sought controller is itself LPV delayed as it has the same dependence on the varying parameters as the system to be controlled. We use the dilated LMI approach to reduce the conservatism of the method as much as possible. The section named ‘Numerical example’ provides a numerical example to illustrate the results. Finally, we conclude the paper in the final section.

Analysis of LPV time-delay systems

This section focuses on the stability analysis of LPV time-delay systems with an H_∞ constraint.

We consider the following state-space model of an LPV time-delay system

{\begin{matrix} \overset{\cdot}{x} (t) = A (θ) x (t) + A_{d} (θ) x (t - τ (t)) + B_{1} (θ) w (t) \\ z (t) = C_{1} (θ) x (t) + D_{11} (θ) w (t) \\ x (t) = η (t), \forall t \in [- τ, 0] \end{matrix}

(1)

where $x (t) \in ℜ^{n}$ , $x (t - τ (t)) \in ℜ^{n}$ , $w (t) \in ℜ^{p}$ , $z (t) \in ℜ^{m}$ and $η (t)$ are the system state, the delayed state, the external disturbance, the controlled output and the functional initial condition, respectively.

The system matrices $A (θ)$ , $A_{d} (θ)$ , $B_{1} (θ)$ , $C_{1} (θ)$ and $D_{11} (θ)$ are parameter dependent matrices of compatible dimensions of a time-varying parameter $θ (t) = [θ_{1} (t), θ_{2} (t), . . ., θ_{r} (t)]^{T} \in ℜ^{r}$ such that $\underline{θ_{i}} \leq θ_{i} (t) \leq \bar{θ_{i}}$ .

Moreover, we assume that the following is true.

The state space matrices $A (θ)$ , $A_{d} (θ)$ , $B_{1} (θ)$ , $C_{1} (θ)$ and $D_{11} (θ)$ are continuous and bounded functions and depend linearly on $θ (t)$ .

The real parameter vector $θ (t)$ that can be identified by online measurement values varies in a polytope $Θ$ as

θ \in Θ (t) = {\sum_{i = 1}^{N} α_{i} θ_{i} (t), α_{i} \geq 0, \sum_{i = 1}^{N} α_{i} = 1}

(2)

with $N = 2^{r}$ being the number of the polytope vertices.

We have $N = 2^{r}$ because each parameter $θ_{i} (t)$ varies between two values $\underline{θ_{i}}$ and $\bar{θ_{i}}$ for $i = 1, 2, . . ., r$ .

For example, if $r = 2$ then we have $N = 2^{r} = 4$ vertices

(\begin{matrix} \underline{θ_{1}} \\ \underline{θ_{2}} \end{matrix}), (\begin{matrix} \bar{θ_{1}} \\ \underline{θ_{2}} \end{matrix}), (\begin{matrix} \underline{θ_{1}} \\ \bar{θ_{2}} \end{matrix}), (\begin{matrix} \bar{θ_{1}} \\ \bar{θ_{2}} \end{matrix})

The matrix inputs are dependent on a time-varying parameter $θ (t)$ which belongs to the operating domain of the system $Θ$ , so that the LPV system is described through a polytopic model. Let

\begin{matrix} (\begin{matrix} A (θ (t)) & A_{d} (θ (t)) & B_{1} (θ (t)) \\ C_{1} (θ (t)) & 0 & D_{11} (θ (t)) \end{matrix}) = \sum_{i = 1}^{N} α_{i} (\begin{matrix} A_{i} & A_{di} & B_{1 i} \\ C_{1 i} & 0 & D_{11 i} \end{matrix}) \end{matrix}

(3)

where $α_{i}$ are time invariant and belong to unitary simplex $Λ$ defined by

Λ : = {α_{i} \geq 0, \sum_{i = 1}^{N} α_{i} = 1, i = 1, . . ., N}

We provide here a result of the stability for LPV time-delay systems with H_∞ constraint using the method defined by Briat et al. (2013).

Theorem 1: System (1) is robustly asymptotically stable for any dependent time delay $τ (t)$ satisfying both $0 < τ (t) \leq τ_{\max}$ and $\frac{∥ z ∥_{2}}{∥ w ∥_{2}} < γ$ if there exist positive-definite matrices $P (θ) \in ℜ^{n \times n}$ , $Q (θ) \in ℜ^{n \times n}$ , $R (θ) \in ℜ^{n \times n}$ and matrices $M 1 (θ)$ , $M 2 (θ) \in ℜ^{n \times n}$ such that for all $θ (t) \in Θ$

(\begin{matrix} ϕ_{11} (θ) & ϕ_{12} (θ) & P (θ) B_{1} (θ) \\ (*) & ϕ_{22} (θ) & 0 \\ (*) & (*) & - γ I \\ (*) & (*) & (*) \\ (*) & (*) & (*) \\ (*) & (*) & (*) \end{matrix} \begin{matrix} C_{1}^{T} (θ) & τ_{max} A^{T} (θ) R (θ) & τ_{max} M 1^{T} (θ) \\ 0 & τ_{max} A_{d}^{T} (θ) R (θ) & τ_{max} M 2^{T} (θ) \\ D_{11}^{T} (θ) & τ_{max} B_{1}^{T} (θ) R (θ) & 0 \\ - γ I & 0 & 0 \\ (*) & - τ_{max} R (θ) & 0 \\ (*) & (*) & - τ_{max} R (θ) \end{matrix}) < 0

(4)

with

\begin{matrix} ϕ_{11} (θ) = A^{T} (θ) P (θ) + P (θ) A (θ) + Q (θ) + M 1 (θ)^{T} + M 1 (θ), \\ ϕ_{12} (θ) = P (θ) A_{d} (θ) - M 1^{T} (θ) + M 2 (θ), and \\ ϕ_{22} (θ) = - (1 - μ) Q (θ) - M 2^{T} (θ) - M 2 (θ), \end{matrix}

where $μ$ is an upper bound of the delay derivative $| \overset{\cdot}{τ} | \leq μ$ , $\forall t \geq 0$ .

To prove this theorem, we use the following Lemma 1 that gives an integral inequality, helping in the derivation of both the analysis and synthesis criteria.

Lemma 1 (Zhang et al., 2005): Let $x (t) \in ℜ^{n}$ be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices $M 1 (θ$ ), $M 2 (θ) \in ℜ^{n \times n}$ and $R (θ) = R (θ)^{T} > 0$ , and a scalar function $τ : = τ (t) \geq 0$

- \int_{t - τ}^{t} {\overset{\cdot}{x}}^{T} (ω) R (θ) \overset{\cdot}{x} (ω) d ω \leq ξ^{T} (t) ϒ ξ (t) + τ ξ^{T} (t) Γ^{T} R^{- 1} (θ) Γ ξ (t)

(5)

where

\begin{matrix} ϒ = (\begin{matrix} M 1^{T} (θ) + M 1 (θ) & - M 1^{T} (θ) + M 2 (θ) \\ (*) & - M 2^{T} (θ) - M 2 (θ) \end{matrix}) \end{matrix}

Γ^{T} : = (\begin{matrix} M 1^{T} (θ) \\ M 2^{T} (θ) \end{matrix}), ξ (t) : = (\begin{matrix} x (t) \\ x (t - τ) \end{matrix})

Proof (of Theorem 1): The main result of the stability analysis of LPV time-delay systems is based on the use of the following parameter-dependent Lyapunov–Krasovskii functional

\begin{matrix} V (t) = & x^{T} (t) P (θ) x (t) + \int_{t - τ (t)}^{t} x^{T} (ω) Q (θ) x (ω) d ω \\ + \int_{- τ_{\max}}^{0} \int_{t + ω}^{t} {\overset{\cdot}{x}}^{T} (s) R (θ) \overset{\cdot}{x} (s) dsd ω \end{matrix}

(6)

where $P (θ) = \sum_{i = 1}^{N} α_{i} P_{i}$ , $Q (θ) = \sum_{i = 1}^{N} α_{i} Q_{i}$ and $R (θ) = \sum_{i = 1}^{N} α_{i} R_{i}$ are positive-definite matrices that need to be determined.

Calculating the derivative of $V (t)$ along the trajectories of the solutions of system (1) leads to

\begin{matrix} \overset{\cdot}{V} (t) = & x^{T} (t) [A^{T} (θ) P (θ) + P (θ) A^{T} (θ) + Q (θ)] x (t) \\ + 2 x^{T} (t - τ (t)) A_{d}^{T} (θ) P (θ) x (t) + 2 w^{T} (t) B_{1} (θ) P (θ) x (t) \\ - (1 - \overset{\cdot}{τ}) x^{T} (t - τ (t)) Q (θ) x (t - τ (t)) + τ_{max} {\overset{\cdot}{x}}^{T} (t) R (θ) \overset{\cdot}{x} (t) + I \end{matrix}

(7)

with

I = - \int_{t - τ_{max}}^{t} {\overset{\cdot}{x}}^{T} (ω) R (θ) \overset{\cdot}{x} (ω) d ω

Note that $- (1 - \overset{\cdot}{τ}) \leq - (1 - μ)$ .

Using Lemma 1 we have

\overset{\cdot}{V} (t) \leq X^{T} (t) Φ (θ) X (t) \leq 0

(8)

with

\begin{matrix} Φ (θ) = & (\begin{matrix} ϕ_{11} (θ) & ϕ_{12} (θ) & P (θ) B_{1} (θ) \\ (*) & ϕ_{22} (θ) & 0 \\ (*) & (*) & 0 \end{matrix}) \\ + τ_{max} Z^{T} (θ) R (θ) Z (θ) + Γ^{T} R^{- 1} (θ) Γ^{T} \\ X (t) = & (\begin{matrix} x (t) \\ x (t - τ (t)) \\ w (t) \end{matrix}), Z (θ) = (\begin{matrix} A (θ) & A_{d} (θ) & B_{1} (θ) \end{matrix}) \end{matrix}

The criterion $J_{zw}$ , which characterizes the $L_{2}$ -gain from $w$ to $z$ to be minimized, is considered as

J_{zw} (t) = γ^{2} w^{T} (t) w (t) - z^{T} (t) z (t)

(9)

$L_{2}$ performances are introduced in the criteria through the Hamiltonian function $H$ defined by

H (t) = V (t) - \int_{0}^{t} [γ^{2} w^{T} (θ) w (θ) - z^{T} (θ) z (θ)] dt

(10)

The derivative of the function can be bounded from above by

\overset{\cdot}{H} \leq \overset{\cdot}{V} + z^{T} (t) z (t) - γ^{2} w^{T} (t) w (t) dt

(11)

Next, expanding the expression of $z (t)$ into the expression of $\overset{\cdot}{H}$ and performing a Schur complement, we obtain LMI (4).

Obviously (4) is not linear in the terms $P (θ) A^{T} (θ)$ , $P (θ) A_{d} (θ$ ), $A^{T} (θ) R (θ)$ and $A_{d}^{T} (θ) R (θ)$ . Usually, this problem is bypassed by using a constant Lyapunov matrix function. However, this method is very conservative. That is why the LMI dilation approach can be used to reduce this conservatism.

This technique applied to the result of Theorem 2 leads to a less conservative criterion.

Theorem 2: System (1)is robustly asymptotically stable for any dependent time delay $τ$ satisfying both $0 < τ (t) \leq τ_{\max}$ and $\frac{∥ z ∥_{2}}{∥ w ∥_{2}} < γ$ if there exist positive-definite matrices $P (θ) \in ℜ^{n \times n}$ , $Q (θ) \in ℜ^{n \times n}$ , $R (θ) \in ℜ^{n \times n}$ , matrices $M 1 (θ) \in ℜ^{n \times n}$ , $M 2 (θ) \in ℜ^{n \times n}$ and $X \in ℜ^{n \times n}$ such that for all $θ (t) \in Θ$

\begin{matrix} (\begin{matrix} - X - X^{T} & Φ_{12} (θ) & Φ_{13} (θ) & Φ_{14 (θ)} \\ (*) & Φ_{22} (θ) & - M 1 (θ)^{T} + M 2 (θ)^{T} & 0 \\ (*) & (*) & Φ_{33} (θ) & 0 \\ (*) & (*) & (*) & - γ I \\ (*) & (*) & (*) & (*) \\ (*) & (*) & (*) & (*) \\ (*) & (*) & (*) & (*) \\ (*) & (*) & (*) & (*) \end{matrix} \begin{matrix} 0 & X^{T} & τ_{max} R (θ) & 0 \\ C_{1} {(θ)}^{T} & 0 & 0 & τ_{max} M 1 (θ)^{T} \\ 0 & 0 & 0 & τ_{max} M 2 (θ)^{T} \\ D_{11 (θ)}^{T} & 0 & 0 & 0 \\ - γ I & 0 & 0 & 0 \\ (*) & - P (θ) & - τ_{max} R (θ) & 0 \\ (*) & (*) & - τ_{max} R (θ) & 0 \\ (*) & (*) & (*) & - τ_{max} R (θ) \end{matrix}) < 0 \end{matrix}

(12)

with

\begin{matrix} Φ_{22} (θ) = - P (θ) + Q (θ) + M 1 (θ)^{T} + M 1 (θ), \\ Φ_{12} (θ) = P (θ) + X^{T} A (θ), \\ Φ_{13} (θ) = X^{T} A_{d} (θ), Φ_{14} (θ) = X^{T} B_{1} (θ), and \\ Φ_{33} (θ) = - (1 - μ) Q (θ) - M 2 (θ)^{T} - M 2 (θ) \end{matrix}

Proof (of Theorem 2) : The proof is inspired by Tuan et al. (2001) and Tuan et al. (2003). Let us rewrite (12) as

\begin{matrix} Ψ (θ) + P^{T} (θ) X^{T} Q + Q^{T} X P (θ) < 0 \end{matrix}

(13)

\begin{matrix} Ψ (θ) = (\begin{matrix} 0 & P (θ) & 0 & 0 \\ (*) & ψ_{22} (θ) & - M 1 (θ)^{T} + M 2 (θ)^{T} & 0 \\ (*) & (*) & ψ_{33} (θ) & 0 \\ (*) & (*) & (*) & - γ I \\ (*) & (*) & (*) & (*) \\ (*) & (*) & (*) & (*) \\ (*) & (*) & (*) & (*) \\ (*) & (*) & (*) & (*) \end{matrix} \begin{matrix} 0 & 0 & τ_{max} R (θ) & 0 \\ C_{1} {(θ)}^{T} & 0 & 0 & τ_{max} M 1 (θ)^{T} \\ 0 & 0 & 0 & τ_{max} M 2 (θ)^{T} \\ D_{11 (θ)}^{T} & 0 & 0 & 0 \\ - γ I & 0 & 0 & 0 \\ (*) & - P (θ) & - τ_{max} R (θ) & 0 \\ (*) & (*) & - τ_{max} R (θ) & 0 \\ (*) & (*) & (*) & - τ_{max} R (θ) \end{matrix}) \end{matrix}

\begin{matrix} ψ_{22} (θ) = - P (θ) + Q (θ) + M 1^{T} (θ) + M 1 (θ, \\ ψ_{33} (θ) = - (1 - μ) Q (θ) - M 2^{T} (θ) - M 2 (θ), \\ P (θ) = (\begin{matrix} - I & A (θ) & A_{d} (θ) & B_{1} (θ) & 0 & I & 0 & 0 \end{matrix}) \end{matrix}

Q = (\begin{matrix} I & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 & 0 & 0 \end{matrix}), X = {(\begin{matrix} X^{T} & 0 & 0 & 0 \end{matrix})}^{T}

Noting that the explicit basis of the null-space of $P$ and $Q$ is given by

\begin{array}{l} \ker (P) = (\begin{matrix} A (θ) & A_{d} (θ) & B_{1} (θ) & I & 0 & 0 & 0 \\ I & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & I \end{matrix}), \\ \ker (Q) = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ I & 0 & 0 \\ 0 & 0 & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix}) \end{array}

and applying the projection Lemma (Gahinet and Apkarian, 1994), the feasibility of LMI(13) implies the feasibility of the two underlying LMIs

\begin{matrix} \begin{matrix} {\begin{matrix} \ker {(P (θ))}^{T} Ψ (θ) \ker (P (θ)) < 0 \\ \ker {(Q (θ))}^{T} Ψ (θ) \ker (Q (θ)) < 0 \end{matrix} \end{matrix} \end{matrix}

(14)

The first LMI in expression (14) is equivalent to LMI (4) modulo a Schur complement whereas the second LMI one is the LMI given by

(\begin{matrix} - γ I & 0 & 0 \\ 0 & - τ_{max} R (θ) & 0 \\ 0 & 0 & - τ_{max} R (θ) \end{matrix}) < 0

This LMI is a relaxed form of the right bottom $3 \times 3$ block of the inequality (4) and it is satisfied for all positive definite matrices $R (θ)$ .

In this sense, we can confirm that the feasibility of (12), written in the form of (13), implies the feasibility of (4).

H_∞ dynamic output feedback

This section is devoted to the design of an H_∞ exact memory dynamic output feedback controller. $τ (t)$ is assumed to be known exactly. The change-of-variables technique used in this section is based on the results from Ebihara and Hagiwara (2004) and the references therein (Apkarian et al., 2001; Chilali and Gahinet, 1996; Scherer et al., 1997).

Problem formulation

Let the LPV time-delay system $S (θ)$ with state-space realization

S (θ) : {\begin{matrix} \overset{\cdot}{x} (t) = A (θ) x (t) + A_{d} (θ) x (t - τ (t)) + B_{1} (θ) w (t) + B_{2} (θ) u (t) \\ z (t) = C_{1} (θ) x (t) + D_{12} (θ) u (t) + D_{11} (θ) w (t) \\ y (t) = C_{2} (θ) x (t) + D_{21} (θ) w (t) \\ θ (t) \in Θ \forall t \geq 0 \end{matrix}

(15)

where $x (t) \in ℜ^{n}$ , $w (t) \in ℜ^{m_{1}}$ , $u (t) \in ℜ^{m_{2}}$ , $z (t) \in ℜ^{p_{1}}$ , $y (t) \in ℜ^{p_{2}}$ are the state, the exogenous noise input, the control, the controlled output and the measured output vectors, respectively. The matrix entries are dependent on a time-varying parameter vector $θ (t)$ which belongs to the operating domain of the system $Θ$ . The system can be defined via a matrix polytope with summits $S_{i}$

S_{i} = (\begin{matrix} A_{i} & A_{di} & B_{1 i} & B_{2 i} \\ C_{1 i} & 0 & D_{11 i} & D_{12 i} \\ C_{2 i} & 0 & D_{21 i} & 0 \end{matrix})

and $S (θ)$ is determined by its barycentric coordinates

S (θ) = \sum_{i = 1}^{N} α_{i} (θ) S_{i}; α_{i} (θ) \geq 0; \sum_{i = 1}^{N} α_{i} (θ) = 1

(16)

The problem addressed in this section is to find, over all trajectories $θ (t) \in Θ$ , an LPV dynamic output feedback controller that:

asymptotically stabilizes the system (15);

provides a guaranteed $L_{2}$ performance attenuation gain from $w$ to $z$ satisfying $∥ z ∥_{2} < γ ∥ w ∥_{2}$ .

H _∞ dynamic output feedback synthesis

In this part, our goal is to look for an output-feedback controller of the form

K_{c} (θ) : {\begin{matrix} {\overset{\cdot}{x}}_{K} (t) = A_{K} (θ) x_{K} (t) + A_{Kd} (θ) x_{K} (t - τ (t)) + B_{K} (θ) y (t) \\ u (t) = C_{K} (θ) x_{K} (t) + D_{K} (θ) y (t) \end{matrix}

(17)

We choose the same polytopic structure for the controller as the system; that is

K_{c} (θ) = \sum_{i = 1}^{N} α_{i} K_{ci}, α_{i} \geq 0, \sum_{i = 1}^{N} α_{i} = 1

(18)

with $K_{ci} = (\begin{matrix} A_{Ki} & A_{Kdi} & B_{Ki} \\ C_{Ki} & 0 & D_{Ki} \end{matrix})$ .

We recall that each online measure of $θ (t)$ corresponds to a combination of $α_{i}$ as its barycentric coordinates in the polytope $Θ$ . These coordinates will be the same as those used to determine the controller $K_{c} (θ) = \sum_{i = 1}^{N} α_{i} K_{ci}$ as shown by the control schema of Figure 1.

Figure 1.

LPV control.

The state space representation of the closed-loop system is

{\begin{matrix} {\overset{\cdot}{x}}_{cl} (t) = A_{cl} (θ) x_{cl} (t) + A_{cld} (θ) x_{cl} (t - τ (t)) + B_{cl} (θ) w (t) \\ z (t) = C_{cl} (θ) x_{cl} (t) + D_{cl} (θ) w (t) \end{matrix}

(19)

with

\begin{matrix} A_{cl} (θ) = (\begin{matrix} A (θ) + B_{2} (θ) D_{K} (θ) C_{2} (θ) & B_{2} (θ) C_{K} (θ) \\ B_{K} (θ) C_{2} (θ) & A_{K} (θ) \end{matrix}) \\ A_{cld} (θ) = (\begin{matrix} A_{d} (θ) & 0 \\ 0 & A_{Kd} (θ) \end{matrix}) \\ B_{cl} (θ) = (\begin{matrix} B_{1} (θ) + B_{2} (θ) D_{K} (θ) D_{21} (θ) \\ B_{K} (θ) D_{21} (θ) \end{matrix}) \\ C_{cl} (θ) = (\begin{matrix} C_{1} (θ) + D_{12} (θ) D_{K} (θ) C_{2} (θ) & D_{12} (θ) C_{K} (θ) \end{matrix}) \\ D_{cl} (θ) = D_{11} (θ) + D_{12} (θ) D_{K} (θ) D_{21} (θ) \\ x_{cl} (t) = (\begin{matrix} x (t) \\ x_{K} (t) \end{matrix}), x_{cl} (t - τ (t)) = (\begin{matrix} x (t - τ (t)) \\ x_{K} (t - τ (t)) \end{matrix}) \end{matrix}

(20)

Clearly, these matrices are bilinear with respect to the parameters of the controller $K_{c} (θ)$ .

To avoid this bilinearity, we can assume that the input and output matrices are invariant (Apkarian et al., 1995).That is: $B_{2} (θ) = B_{2}, C_{2} (θ) = C_{2}, D_{12} (θ) = D_{12}$ and $D_{21} (θ) = D_{21}$ .

The desired characterization for dynamic output feedback synthesis can be derived in the following steps:

1) We introduce a partition of $X$ in (12) and its inverse $X^{- 1} = Y$ in the form

X = (\begin{matrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix}), Y = (\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix})

with $X_{11} \in ℜ^{n \times n}$ and $Y_{11} \in ℜ^{n \times n}$ . Assuming that $X_{21}$ and $Y_{21}$ are invertible, we then introduce the notation

Π_{X} = (\begin{matrix} X_{11} & I \\ X_{21} & 0 \end{matrix}), Π_{Y} = (\begin{matrix} I & Y_{11} \\ 0 & Y_{21} \end{matrix})

$Π_{X}$ and $Π_{Y}$ are invertible matrices which readily verify the identity

Y Π_{X} = Π_{Y}, X Π_{Y} = Π_{X}

(21)

2) Now we define the change in the controller variables as follows

\begin{matrix} {\hat{A}}_{K} (θ) = X_{11}^{T} A (θ) Y_{11} + X_{21}^{T} A_{K} (θ) Y_{21} + X_{11}^{T} B_{2} C_{K} (θ) Y_{21} \\ + X_{21}^{T} B_{K} (θ) C_{2} Y_{11} + X_{11}^{T} B_{2} D_{K} (θ) C_{2} Y_{11} \\ {\hat{A}}_{Kd} (θ) = X_{11}^{T} A (θ) Y_{11} + X_{21}^{T} A_{Kd} (θ) Y_{21} \\ {\hat{B}}_{K} (θ) = X_{21}^{T} B_{K} (θ) + X_{11}^{T} B_{2} D_{K} (θ) \\ {\hat{C}}_{K} (θ) = C_{K} (θ) Y_{21} + D_{K} (θ) C_{2} Y_{11} \\ {\hat{D}}_{K} (θ) = D_{K} (θ) \\ the variable U = X_{11}^{T} Y_{11} + X_{21}^{T} Y_{21} \end{matrix}

(22)

3) Then we apply the LMI (12) of Theorem 2 for the closed-loop system (19) and we made the congruence by $Diag {Π_{Y}, Π_{Y}, Π_{Y}, I, I, Π_{Y}, Π_{Y}, Π_{Y}}$

This congruence leads to a matrix inequality which involves the terms

\begin{matrix} \begin{matrix} Π_{Y}^{T} X^{T} A_{cl} (θ) Π_{Y} = Π_{X}^{T} A_{cl} (θ) Π_{Y} \\ = (\begin{matrix} X_{11}^{T} A (θ) + {\hat{B}}_{K} (θ) C_{2} & {\hat{A}}_{K} (θ) \\ A (θ) + B_{2} {\hat{D}}_{K} (θ) C_{2} & A (θ) Y_{11} + B_{2} {\hat{C}}_{K} (θ) \end{matrix}) \end{matrix} \\ Π_{Y}^{T} X^{T} A_{cld} (θ) Π_{Y} = Π_{X}^{T} A_{cld} (θ) Π_{Y} = (\begin{matrix} X_{11}^{T} A_{d} (θ) & {\hat{A}}_{Kd} (θ) \\ A_{d} (θ) & A_{d} (θ) Y_{11} \end{matrix}) \\ Π_{Y}^{T} X^{T} B_{cl} (θ) = Π_{X}^{T} B_{cl} (θ) = (\begin{matrix} X_{11} B_{1} (θ) + {\hat{B}}_{K} (θ) D_{21} \\ B_{1} (θ) + B_{2} {\hat{D}}_{K} (θ) D_{21} \end{matrix}) \\ Π_{Y}^{T} C_{cl} {(θ)}^{T} = {(\begin{matrix} C_{1} (θ) + D_{12} {\hat{D}}_{K} (θ) C_{2} & C_{1} (θ) Y_{11} + D_{12} {\hat{C}}_{K} (θ) \end{matrix})}^{T} \\ Π_{Y}^{T} P (θ) Π_{Y} = \hat{P} (θ) = (\begin{matrix} {\hat{P}}_{11} (θ) & {\hat{P}}_{12} (θ) \\ {\hat{P}}_{12} (θ) & {\hat{P}}_{22} (θ) \end{matrix}) \\ Π_{Y}^{T} Q (θ) Π_{Y} = \hat{Q} (θ) = (\begin{matrix} {\hat{Q}}_{11} (θ) & {\hat{Q}}_{12} (θ) \\ {\hat{Q}}_{12} (θ) & {\hat{Q}}_{22} (θ) \end{matrix}) \\ Π_{Y}^{T} R (θ) Π_{Y} = \hat{R} (θ) = (\begin{matrix} {\hat{R}}_{11} (θ) & {\hat{R}}_{12} (θ) \\ {\hat{R}}_{12} (θ) & {\hat{R}}_{22} (θ) \end{matrix}) \\ Π_{Y}^{T} M 1 (θ) Π_{Y} = \hat{M} 1 (θ) = (\begin{matrix} \hat{M} 1_{11} (θ) & \hat{M} 1_{12} (θ) \\ \hat{M} 1_{21} (θ) & \hat{M} 1_{22} (θ) \end{matrix}) \\ Π_{Y}^{T} M 2 (θ) Π_{Y} = \hat{M} 2 (θ) = (\begin{matrix} \hat{M} 2_{11} (θ) & \hat{M} 2_{12} (θ) \\ \hat{M} 2_{21} (θ) & \hat{M} 2_{22} (θ) \end{matrix}) \\ Π_{Y}^{T} X Π_{Y} = Π_{Y}^{T} Π_{X} = (\begin{matrix} X_{11}^{T} & U \\ I & Y_{11}^{T} \end{matrix}) \end{matrix}

(23)

This leads to the following convex problem with linear constraints

(\begin{matrix} Ξ_{11} (θ) & Ξ_{12} (θ) & Ξ_{13} (θ) & Ξ_{14} (θ) \\ (*) & Ξ_{22} (θ) & Ξ_{23} (θ) & 0 \\ (*) & (*) & Ξ_{33} (θ) & 0 \\ (*) & (*) & (*) & - γ I \\ (*) & (*) & (*) & (*) \\ (*) & (*) & (*) & (*) \\ (*) & (*) & (*) & (*) \\ (*) & (*) & (*) & (*) \end{matrix} \begin{matrix} 0 & Ξ_{16} (θ) & Ξ_{17} (θ) & 0 \\ Ξ_{25} (θ) & 0 & 0 & Ξ_{28} (θ) \\ 0 & 0 & 0 & Ξ_{38} (θ) \\ Ξ_{45} (θ) & 0 & 0 & 0 \\ - γ I & 0 & 0 & 0 \\ (*) & Ξ_{66} (θ) & Ξ_{67} (θ) & 0 \\ (*) & (*) & Ξ_{77} (θ) & 0 \\ (*) & (*) & (*) & Ξ_{88} (θ) \end{matrix}) < 0

(24)

with

Ξ_{11} (θ) = (\begin{matrix} - X_{11} - X_{11}^{T} & - U - I \\ - U^{T} - I & - Y_{11} - Y_{11}^{T} \end{matrix})

Ξ_{12} (θ) = (\begin{matrix} {\hat{P}}_{11} (θ) + X_{11}^{T} A (θ) + {\hat{B}}_{K} (θ) C_{2} & {\hat{P}}_{12} (θ) + {\hat{A}}_{K} (θ) \\ {\hat{P}}_{21} (θ) + A (θ) + B_{2} (θ) {\hat{D}}_{K} (θ) C_{2} & {\hat{P}}_{22} (θ) + A (θ) Y_{11} + B_{2} {\hat{C}}_{K} (θ) \end{matrix})

\begin{matrix} Ξ_{13} (θ) = (\begin{matrix} X_{11}^{T} A_{d} (θ) & {\hat{A}}_{Kd} (θ) \\ A_{d} (θ) & A_{d} (θ) Y_{11} \end{matrix}) \end{matrix} Ξ_{14} (θ) = (\begin{matrix} X_{11}^{T} B_{1} (θ) + {\hat{B}}_{K} (θ) D_{21} \\ B_{1} (θ) + B_{2} {\hat{D}}_{K} (θ) D_{21} \end{matrix})

\begin{matrix} Ξ_{16} (θ) = (\begin{matrix} X_{11}^{T} & U \\ I & Y_{11} \end{matrix}) \\ Ξ_{17} (θ) = (\begin{matrix} τ_{max} {\hat{R}}_{11} (θ) & τ_{max} {\hat{R}}_{12} (θ) \\ τ_{max} {\hat{R}}_{12} (θ) & τ_{max} {\hat{R}}_{22} (θ) \end{matrix}) \\ \begin{matrix} Ξ_{22} (θ) = (\begin{matrix} - {\hat{P}}_{11} (θ) + {\hat{Q}}_{11} (θ) + \hat{M} 1_{11} {(θ)}^{T} + \hat{M} 1_{11} (θ) \\ - {\hat{P}}_{12} (θ) + {\hat{Q}}_{12} (θ) + \hat{M} 1_{21} {(θ)}^{T} + \hat{M} 1_{21} (θ) \end{matrix} \begin{matrix} - {\hat{P}}_{12} (θ) + {\hat{Q}}_{12} (θ) + \hat{M} 1_{12} {(θ)}^{T} + \hat{M} 1_{12} (θ) \\ - {\hat{P}}_{22} (θ) + {\hat{Q}}_{22} (θ) + \hat{M} 1_{22} {(θ)}^{T} + \hat{M} 1_{22} (θ) \end{matrix}) \end{matrix} \end{matrix}

\begin{matrix} Ξ_{23} (θ) = (\begin{matrix} - \hat{M} 1_{11} {(θ)}^{T} + \hat{M} 2_{11} (θ) & - \hat{M} 1_{12} {(θ)}^{T} + \hat{M} 2_{12} (θ) \\ - \hat{M} 1_{21} {(θ)}^{T} + \hat{M} 2_{21} (θ) & - \hat{M} 1_{22} {(θ)}^{T} + \hat{M} 2_{22} (θ) \end{matrix}) \\ Ξ_{25} (θ) = (\begin{matrix} C_{1} {(θ)}^{T} + C_{2}^{T} {\hat{D}}_{K} {(θ)}^{T} D_{12}^{T} \\ Y_{11}^{T} C_{1} {(θ)}^{T} + {\hat{C}}_{K} {(θ)}^{T} D_{12}^{T} \end{matrix}) \\ Ξ_{28} (θ) = {(\begin{matrix} τ_{max} \hat{M} 1_{11} (θ) & τ_{max} \hat{M} 1_{12} (θ) \\ τ_{max} \hat{M} 1_{21} (θ) & τ_{max} \hat{M} 1_{22} (θ) \end{matrix})}^{T} \end{matrix}

\begin{matrix} \begin{matrix} Ξ_{33} (θ) = (\begin{matrix} - (1 - μ) {\hat{Q}}_{11} (θ) - \hat{M} 2_{11} {(θ)}^{T} - \hat{M} 2_{12} (θ) \\ - (1 - μ) {\hat{Q}}_{12} (θ) - \hat{M} 2_{21} {(θ)}^{T} - \hat{M} 2_{21} (θ) \end{matrix} \begin{matrix} - (1 - μ) {\hat{Q}}_{12} (θ) - \hat{M} 2_{12} {(θ)}^{T} - \hat{M} 2_{12} (θ) \\ - (1 - μ) {\hat{Q}}_{22} (θ) - \hat{M} 2_{22} {(θ)}^{T} - \hat{M} 2_{22} (θ) \end{matrix}) \end{matrix} \\ Ξ_{38} (θ) = {(\begin{matrix} τ_{max} \hat{M} 2_{11} (θ) & τ_{max} \hat{M} 2_{12} (θ) \\ τ_{max} \hat{M} 2_{21} (θ) & τ_{max} \hat{M} 2_{22} (θ) \end{matrix})}^{T} \\ Ξ_{45} (θ) = D_{11} (θ) + D_{12} D_{k} (θ) D_{21} \\ Ξ_{66} (θ) = (\begin{matrix} - {\hat{P}}_{11} (θ) & - {\hat{P}}_{12} (θ) \\ - {\hat{P}}_{12} (θ) & - {\hat{P}}_{22} (θ) \end{matrix}) \\ Ξ_{67} (θ) = Ξ_{77} (θ) = Ξ_{88} (θ) = (\begin{matrix} - τ_{max} {\hat{R}}_{11} (θ) & - τ_{max} {\hat{R}}_{12} (θ) \\ - τ_{max} {\hat{R}}_{12} (θ) & - τ_{max} {\hat{R}}_{22} (θ) \end{matrix}) \end{matrix}

4) Finally, we can determine the variables ${\hat{A}}_{Ki}$ , ${\hat{A}}_{Kdi}$ , ${\hat{B}}_{Ki}$ , ${\hat{C}}_{Ki}$ , ${\hat{D}}_{Ki}$ , $X_{11}$ , $Y_{11}$ , ${\hat{P}}_{i}$ , ${\hat{Q}}_{i}$ , ${\hat{R}}_{i}$ , $\hat{M} 1_{i}$ and $\hat{M} 2_{i}$ by solving LMI (24) with

\begin{matrix} {\hat{Q}}_{i} = (\begin{matrix} {\hat{Q}}_{11 i} & {\hat{Q}}_{12 i} \\ {\hat{Q}}_{12 i}^{T} & {\hat{Q}}_{22 i} \end{matrix}), {\hat{R}}_{i} = (\begin{matrix} {\hat{R}}_{11 i} & {\hat{R}}_{12 i} \\ {\hat{R}}_{12 i}^{T} & {\hat{R}}_{22 i} \end{matrix}), \\ \hat{M} 1_{i} = (\begin{matrix} \hat{M} 1_{11 i} & \hat{M} 1_{12 i} \\ \hat{M} 1_{21 i}^{T} & \hat{M} 1_{22 i} \end{matrix}), \hat{M} 2_{i} = (\begin{matrix} \hat{M} 2_{11 i} & \hat{M} 2_{12 i} \\ \hat{M} 2_{21 i}^{T} & \hat{M} 2_{22 i} \end{matrix}) \end{matrix}

Once these variables are determined, we can reconstruct the $K_{ci}$ controller as follows:

compute the two matrices $X_{21}$ and $Y_{21}$ by the following factorization

$X_{21}^{T} Y_{21} = U - X_{11}^{T} Y_{11}$

compute the controller data $A_{Ki}$ , $A_{Kdi}$ , $B_{Ki}$ , $C_{Ki}$ and $D_{Ki}$ by reversing the formulas in (22).

Numerical example

Here, we consider the LPV time-delay system (25) defined by (Briat et al., 2009)

\begin{matrix} \overset{\cdot}{x} (t) = & (\begin{matrix} 0 & 1 + ϕ \sin (t) \\ - 2 & - 3 + δ \sin (t) \end{matrix}) x (t) \\ + (\begin{matrix} ϕ \sin (t) & 0.1 \\ - 0.2 + δ \sin (t) & - 0.3 \end{matrix}) \\ x (t - τ (t)) + (\begin{matrix} 0.2 \\ 0.2 \end{matrix}) w (t) + (\begin{matrix} 0.2 \\ 0.2 \end{matrix}) u (t) \\ z (t) = & (\begin{matrix} 0 & 10 \\ 0 & 0 \end{matrix}) x (t) + (\begin{matrix} 0 \\ 0.1 \end{matrix}) u (t) \\ y (t) = & (\begin{matrix} 0 & 1 \\ 0.5 & 0 \end{matrix}) x (t) \end{matrix}

(25)

where $ϕ = 0.2$ and $δ = 0.1$ . Define $θ (t) = \sin (t)$ as a varying parameter which varies in the following uncertainty range: $θ (t) \in [- 11]$ .

This leads to a polytopic system with two vertices

S_{1} = (\begin{matrix} A_{1} & A_{d 1} & B_{1} & B_{2} \\ C_{1} & 0 & 0 & D_{12} \\ C_{2} & 0 & 0 & 0 \end{matrix}) and S_{2} = (\begin{matrix} A_{2} & A_{d 2} & B_{1} & B_{2} \\ C_{1} & 0 & 0 & D_{12} \\ C_{2} & 0 & 0 & 0 \end{matrix})

With

\begin{matrix} A_{1} = (\begin{matrix} 0 & 0.8 \\ - 2 & - 3.1 \end{matrix}), A_{d 1} = (\begin{matrix} - 0.2 & 0.1 \\ - 0.3 & - 0.3 \end{matrix}), \\ A_{2} = (\begin{matrix} 0 & 1.2 \\ - 2 & - 2.9 \end{matrix}), A_{d 2} = (\begin{matrix} 0.2 & 0.1 \\ - 0.1 & - 0.3 \end{matrix}) \end{matrix}

We follow the procedure detailed in subsection ‘H_∞ dynamic output feedback synthesis’ to derive an H_∞ dynamic output feedback problem.

The solution of the LMIs (24) leads to a performance level $γ = 0.466$ for the time delay $τ_{\max}$ =0.5 and $μ = 0.5$ . The same system is state feedback controlled with H_∞ performance in Briat et al. (2013) and Zhang and Grigoriadis (2005). Their methods lead to a performance level $γ = 4.164$ and $γ = 3.09$ , respectively. Hence, for this example, the control method presented in this paper provides a lower and better H_∞ performance level.

According to our design method, the maximum obtained delay is $1.7$ and the maximum delay variation rate is $μ = 0.9$ , while based on the method in Zhang and Grigoriadis (2005) the maximum obtained delay is 1.65 and the maximum delay variation rate is $μ = 0.5$ . Thus, our results can provide the controller with a wider delay range than those in Zhang and Grigoriadis (2005).

In the simulation, we choose $τ (t) = 0.5 | \sin (t) |$ and the maximum delay variation rate is $μ = 0.5$ . Figure 2 describes the time-delay evolution.

Figure 2.

Delay $τ (t) = 0.5 | \sin (t) |$ .

The external disturbance is a rectangular signal defined as follows

w (t) = {\begin{matrix} 0, 0 < t < 1 \\ 1, 1 \leq t < 4 \\ 0, t \geq 4 \end{matrix}

For this disturbance $w (t)$ , we simulate the closed-loop behaviour of the system using the LPV dynamic output feedback controller. The system responses and control input profiles are shown in Figures 3 and 4. Note that both responses $z_{1}$ , $z_{2}$ converge to zero very rapidly.

Figure 3.

The system responses evolution $z_{1}$ , $z_{1}$ ( $τ (t) = 0.5 | \sin (t) |$ ).

Figure 4.

Control input evolution $u (t)$ ( $τ (t) = 0.5 | \sin (t) |$ ).

Conclusion

This paper examined the robust stability and H_∞ dynamic output feedback control synthesis for LPV delayed systems. A new LMI representation to H_∞ performance analysis and dynamic output feedback control synthesis has been presented for the LPV delayed system using a parameter-dependent Lyapunov–Krasovskii functional and Jensen’s inequality; an approach which has proven its efficiency. Moreover, this approach involves an additional variable and avoids the product of the Lyapunov matrices and the system matrices. This allows us to find a linearising change of variable for the stabilization problem. The results in this paper are less conservative and can provide a controller with a wider delay range and a better quadratic performance level for the LPV system with rate bounded time-varying state delays. In future work, we will consider the nonlinear delayed system and design an output-feedback controller for this type of system.

Footnotes

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) received no financial support for the research, authorship, and/or publication of this article.

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H ∞ Dynamic output feedback control of LPV time-delay systems via dilated linear matrix inequalities

Abstract

Keywords

Introduction

Analysis of LPV time-delay systems

H∞ dynamic output feedback

Problem formulation

H ∞ dynamic output feedback synthesis

Numerical example

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

H_∞ dynamic output feedback

H _∞ dynamic output feedback synthesis