Finite-time control for polynomial parameter-varying systems and its application in mode transition of turbine-based combined cycle engines

Abstract

This article investigates the problem of finite-time control for the mode transition of turbine-based combined cycle (TBCC) engines described in the polynomial parameter-varying (PPV) framework. In order to realize small-deviation mode transition, a PPV model that can be applied to different opening rates of an inlet splitter is established for a TBCC project put forward by Xiamen University named Xiamen Turbine-based Ejector Ramjet (XTER). The concepts of $H_{\infty}$ uniformly finite-time stability and $H_{\infty}$ uniformly finite-time output tracking performance are proposed, together with the corresponding state-and-parameter-dependent controller design conditions which can be efficiently solved by using sum-of-squares programming. Meanwhile, based on the PPV system of XTER, the controllers are designed in the designed flight conditions and the non-designed flight conditions, respectively. Finally, simulation results on the mode transition of the XTER are given to verify the feasibility and effectiveness of the proposed control scheme.

Keywords

Nonlinear parameter-varying systems finite-time control turbine-based combined cycle engine mode transition sum of squares

Introduction

The turbine-based combined cycle (TBCC) engine is a promising hypersonic air-breathing propulsion system. It has been widely concerned because of its great application prospects in space transportation, space development, and so on (Chen and Tang, 2009; Huang et al., 2014; Lv et al., 2022; Ma et al., 2018; Zhou and Molder, 2019). The parallel TBCC usually consists of a gas turbine engine (TE) and a ramjet/scramjet engine in parallel, in which the TE and the ramjet/scramjet engine mainly provide thrust at low speed and high speed, respectively (Zheng et al., 2020). The process of replacing the main thrust provider between the TE and ramjet/scramjet engine is called mode transition (MT), and the MT control is a critical issue in TBCC control (Gao et al., 2023; Huang et al., 2014; Zheng et al., 2019, 2020). In addition, during the MT, it is generally necessary to adjust the variable-geometry inlet to improve efficiency and safety (Liu et al., 2018, 2021; Xiang et al., 2015). It should be particularly pointed out that the inlet splitter adjustment process will have an impact on the thrust–drag relationship of aero-engine and aircraft (Xiang et al., 2015; Zheng et al., 2020), and the aerodynamic/propulsive couplings should not be neglected, especially for MT of the TBCC engine (Zheng et al., 2019). Therefore, the influence of an inlet splitter on the required thrust of aircraft cannot be ignored. However, the detailed research progress on MT in TBCC remains limited in the open literature, and there is a gap in the study of MT when the splitter cannot run according to the given trajectory (such as in the presence of disturbances or temporary task changes).

Typical models in aero-engine control include single-point linearization model (Otto et al., 1951), linear parameter-varying (LPV) model (Jia et al., 2017; Wang et al., 2019), switched model (Yang and Zhao, 2017; Zong et al., 2022), etc. Obviously, the nonlinearity and the time-varying properties of aero-engine cannot be fully reflected by these models (Jia et al., 2017; Otto et al., 1951; Wang et al., 2019; Yang and Zhao, 2017; Zong et al., 2022), because modern aero-engines usually have a large and varying working range and an effective model should be able to describe the aero-engine over a wide range of parameter variation. However, these features are inherent to aero-engine, and even more noteworthy for TBCC, since TBCC consists of multiple engines working together during MT. Based on the LPV model, Previdi and Lovera propose the nonlinear parameter-varying (NLPV) model, which intensifies the characterization of the nonlinearity and time-varying properties of plant (Previdi and Lovera, 1999), but NLPV models generally assume that the nonlinear part lies in a bounded convex region (Morato et al., 2021; Zhou et al., 2021). Therefore, Fu et al. proposed a class of NLPV model with the general from of $\overset{\cdot}{x} = A (x, σ (t)) x + B (x, σ (t)) u$ (we refer to it as polynomial parameter-varying (PPV) systems), which avoid the aforementioned restriction on the nonlinear part and and can be analyzed/synthesized by sum-of-squares (SOS) techniques (Fu et al., 2018, 2019). In addition, this PPV system has been tested and validated on tilt-rotor aircraft (Fu et al., 2019; Zhu et al., 2020) and turbofan engine (Zhu et al., 2023). However, papers (Fu et al., 2018, 2019; Zhu et al., 2020, 2023) mainly focus on asymptotic stability and exponential stability.

On the other hand, finite-time stability (FTS) is a special kind of asymptotic stability, meaning that an asymptotically stable system reaches an equilibrium point in finite time. It should be pointed out that, in order to avoid confusion, the FTS mentioned in this article is distinguished from finite-time boundedness (Amato et al., 2010, 2019; Fu et al., 2018). Compared with asymptotically stable systems, the FTS systems have the advantages of faster convergence, higher accuracy, stronger robustness, and so on (Bhat and Bernstein, 2000; Du et al., 2011; Li et al., 2019). Therefore, it has been widely concerned by the control community (Moulay and Perruquetti, 2003, 2008; Shen and Huang, 2012; Sun et al., 2019; Yu et al., 2005, 2018; Zhu et al., 2011). Since Bhat and Bernstein proposed the Lyapunov-like theory of FTS (Bhat and Bernstein, 2000), the Lyapunov condition of FTS has been extensively developed. In addition, Moulay and Perruquetti (2003) gives the Lyapunov condition of uniformly finite-time stable (UFTS) for nonlinear time-varying systems. References (Shen and Huang, 2012; Yu et al., 2005) combine FTS and exponential stable to put forward a new definition of FTS. This effectively alleviates the problem that the convergence rate of FTS is slower than the exponential one when the initial state is far away from the origin (Bhat and Bernstein, 2000; Sun et al., 2019). However, constructing a suitable Lyapunov function to declare FTS is inherently not easy, especially in complicated environments (Moulay and Perruquetti, 2008), and the research on FTS of the PPV system is still lacking.

Motivated by the above observations. In this article, the problem of finite-time control for PPV systems and its application in MT thrust control of TBCC engines are investigated. First, the PPV model of the Xiamen Turbine-based Ejector Ramjet (XTER), which is a TBCC project put forward by Xiamen University (Guo et al., 2019; He et al., 2023) is established. In particular, the opening rate of a splitter is regarded as one of the flight conditions, and it is incorporated into the system matrices as a time-varying parameter during modeling. Then, in the sense of finite time, the stabilization and tracking control synthesis conditions are established based on state-and-parameter-dependent matrix inequalities. Finally, simulation results about the MT thrust control problem between the TE and the ejector-ramjet (ER) engine for XTER are provided to demonstrate the effectiveness of the proposed scheme.

The main contributions of the present study can be summarized as follows. (i) To the authors’ knowledge. The MT control of TBCC is investigated in the framework of PPV and finite time for the first time, and the resulting finite-time stable system shows better performance compared with the exponential stable system (Fu et al., 2018, 2019). (ii) The concept of $H_{\infty}$ output tracking performance has been extended from linear systems (Zhao and Zhao, 2021) and LPV systems (Yang and Zhao, 2017) to PPV systems and finite time. The stabilization and tracking control synthesis conditions are established based on state-and-parameter-dependent matrix inequalities, which can be solved by SOS programming effectively. (iii) The PPV model of XTER is established for the first time, which can adapt to different opening rates of the splitter. And compared with He et al. (2023), the nonlinearity and time-varying properties of XTER are all preserved.

The rest of the article is organized as follows. In section “Preliminaries and problem description,” model and problem description are given. The main results and control system structure diagram are presented in section “Controller design.” In section “MT thrust control simulations,” the application to the MT thrust control of XTER is given. Finally, a conclusion is drawn in Section “Conclusions and outlooks.”

Notations: $R^{n}$ denotes a series of $n \times 1$ real column vectors, $R^{n \times m}$ a series of $n \times m$ real matrices, 0 and $I$ the zero matrix and the identity matrix, $Φ_{SOS}$ a series of SOS polynomials. $He (E) = E + E^{T}$ . $diag {c_{1}, c_{2}, \dots, c_{n}}$ denotes block diagonal matrix consisting of $c_{1}, c_{2}, \dots, c_{n}$ . * denotes the symmetric term.

Preliminaries and problem description

Preliminaries

Definition 1. (Moulay and Perruquetti, 2003). Consider a nonlinear time-varying system

\overset{\cdot}{ζ} (t) = f (t, ζ (t)), f (t, 0) = 0, ζ (t_{0}) = ζ_{0}, t \geq t_{0} \geq 0,

(1)

where $f : [0, \infty) \times D_{0} \to R^{n}$ is continuous on an open neighborhood $D_{0} \subseteq R^{n}$ of the origin $ζ = 0$ for all $t \geq 0$ , then $ζ (t, t_{0}, ζ_{0})$ denotes a solution starting from $(t_{0}, ζ_{0})$ . The equilibrium $ζ = 0$ of system (1) is UFTS if:

(1) the origin $ζ = 0$ is uniformly Lyapunov stable,

(2) there exists $δ > 0$ , for all $t_{0} \geq 0$ , if $∥ ζ_{0} ∥< δ$ , then there exists $0 \leq T (ζ_{0}) < + \infty$ such that $ζ (t, t_{0}, ζ_{0}) = 0$ for all $t \geq t_{0} + T (ζ_{0})$ .

Let $T_{0} (ζ_{0}) = \inf {T (ζ_{0}) \geq 0$ ; $∥ ζ (t, t_{0}$ , $ζ_{0}) ∥ = 0, \forall t \geq t_{0} + T (ζ_{0})}$ , we say that $ζ (t, t_{0}$ , $ζ_{0})$ tends uniformly to the equilibrium $ζ = 0$ with the settling time $T_{0} (ζ_{0})$ .

Lemma 1. (Shen and Huang, 2012). Consider the nonlinear system

\overset{\cdot}{ζ} (t) = f (ζ (t)), f (0) = 0, ζ (0) = ζ_{0},

(2)

where $f : D_{0} \subseteq R^{n} \to R^{n}$ is continuous in an open neighborhood $D_{0}$ of the origin $ζ = 0$ . The origin of system (2) is FTS, if there exist a Lyapunov function $V : D_{0} \to R$ , real numbers $c, l > 0$ , $α \in (0, 1)$ and open neighborhood $D \subseteq D_{0}$ of the origin such that

\overset{\cdot}{V} (ζ) \leq - c V^{α} (ζ) - lV (ζ), \forall x \in D \ {0}, \forall t \in [0, \infty) .

(3)

Lemma 1 provide a general Lyapunov condition for FTS of the nonlinear system, which can be easily extended to the nonlinear time-varying system by replacing $V (ζ)$ with $V (ζ, t)$ . Therefore, the following corollary is succinctly given without proof.

Corollary 1. Consider a nonlinear time-varying system with the form of (1), if there exist a Lyapunov function $V (ζ, t)$ , continuous positive definite functions $H_{1} (ζ)$ and $H_{2} (ζ)$ , such that $H_{1} (ζ) \leq V (ζ, t) \leq H_{2} (ζ)$ and $\overset{\cdot}{V} (ζ, t) \leq - c V^{α} (ζ, t) - lV (ζ, t)$ . Then, the equilibrium $ζ = 0$ is UFTS.

Finally, the definition of SOS and a formula used in the theoretical analysis are given.

Definition 2. (Papachristodoulou et al., 2016). If a polynomial function $p (k)$ in $k \in R^{n}$ is called SOS, there exist polynomials $p_{1} (k)$ , $p_{2} (k), \dots, p_{d} (k)$ , such that

p (k) = \sum_{i = 1}^{d} p_{i}^{2} (k) .

(4)

Obviously, $p (k) \in Φ_{SOS}$ implies $p (k) \geq 0$ , $\forall k \in R^{n}$ . However, the contrary cannot be guaranteed. That is, the above property is not necessary for a polynomial to be non-negative, but most numerical studies show that the gap between them is negligible (Reznick, 2000).

Lemma 2. (Prajna et al., 2004). For a symmetric polynomial matrix $M (υ)$ , which is nonsingular for all $υ$ , we have

\frac{\partial M (υ)}{\partial υ_{i}} = - M (υ) \frac{\partial M^{- 1}}{\partial υ_{i}} (υ) M (υ), i = 1, \dots, n .

(5)

XTER model and problem description

The data of XTER comes from He et al. (2023). As shown in Figure 1, the XTER consists of four channels and three powers. The upper channel is a series structure composed of Ejector and Ramjet, the lower channel and the left/right channels are Scramjet and TEs, respectively, and the four channels are connected by an adjustable inlet and nozzle.

We focus on the MT between TE and ER. The specific control objectives of this article are as follows:

Design controllers to make each engine achieve the reference thrust in the designed flight condition (DFC) with exogenous disturbance, so that the total thrust is within the acceptable range during MT.

Design controllers to make each engine achieve the reference thrust in the DFC and non-DFC (NDFC) with exogenous disturbance, so that the total thrust is within the acceptable range during MT.

Remark 1. The emphasis of this article is not thrust distribution, but to give an MT control scheme and controller design method to adapt to different flight conditions, such as different opening rates of the splitter. Therefore, it is assumed that different intermediate changes do not affect the expected total thrust when the starting and ending states of the splitter are the same, and the total thrust is the sum of the thrust of each engine. Then, when each engine realizes its own reference thrust, the XTER can achieve the expected total thrust.

For the purpose of controller design, a PPV model with exogenous disturbance is obtained by means of the system identification technique (Ling et al., 2021). The PPV models of TE and ER are given as follows (see Appendix 2 for details).

{\begin{matrix} [\begin{matrix} Δ \overset{\cdot}{n} \\ Δ \overset{\cdot}{π} \end{matrix}] = [\begin{matrix} a_{11} (Δ n, β) a_{12} (Δ n, β) \\ a_{21} (Δ n, β) a_{21} (Δ n, β) \end{matrix}] [\begin{matrix} Δ n \\ Δ π \end{matrix}] \\ + [\begin{matrix} b_{11} (Δ n, β) \\ b_{21} (Δ n, β) \end{matrix}] Δ W f_{TE} + [\begin{matrix} e_{11} (Δ n, β) \\ e_{21} (Δ n, β) \end{matrix}] ω, \\ Δ F_{TE} = [\begin{matrix} c_{11} (Δ n, β) c_{12} (Δ n, β) \end{matrix}] [\begin{matrix} Δ n \\ Δ π \end{matrix}], \end{matrix}

(6)

{\begin{matrix} [\begin{matrix} Δ {\overset{\cdot}{P}}_{t} \\ Δ {\overset{\cdot}{T}}_{t} \end{matrix}] = [\begin{matrix} a_{11} (Δ P_{t}, β) a_{12} (Δ P_{t}, β) \\ a_{21} (Δ P_{t}, β) a_{21} (Δ P_{t}, β) \end{matrix}] [\begin{matrix} Δ P_{t} \\ Δ T_{t} \end{matrix}] \\ + [\begin{matrix} b_{11} (Δ P_{t}, β) \\ b_{21} (Δ P_{t}, β) \end{matrix}] Δ W f_{ER} + [\begin{matrix} e_{11} (Δ P_{t}, β) \\ e_{21} (Δ P_{t}, β) \end{matrix}] ω, \\ Δ F_{ER} = [\begin{matrix} c_{11} (Δ P_{t}, β) c_{12} (Δ P_{t}, β) \end{matrix}] [\begin{matrix} Δ P_{t} \\ Δ T_{t} \end{matrix}], \end{matrix}

(7)

where system (6) is the TE and the system (7) is the ER, $Δ n = n - n_{ref},$ $Δ π = π - π_{ref},$ $Δ W f_{TE} = W f_{TE} - W f_{TEref},$ $Δ F_{TE} = F_{TE} - F_{TEref},$ $Δ P_{t} = P_{t} - P_{tref},$ $Δ T_{t} = T_{t} - T_{tref}$ , and $Δ W f_{ER} = W f_{ER} - W f_{ERref}$ , respectively. In addition, $ω$ represents the exogenous disturbance and the total thrust is defined as $F_{XTER} = F_{TE} + F_{ER}$ .

Remark 2. The model coupling between the TE and the ER is realized through the splitter opening. The splitter opening is regarded as one of the flight conditions and is modeled as a time-varying parameter for the following reasons. (a) Similar to the $H$ and $Ma$ , which are generally modeled as time-varying parameters in control-oriented models of aero-engines, the change of the splitter will affect the inlet flow conditions of each engine. (b) The change of the splitter will affect the thrust–drag relationship of the aircraft. As a result, to maintain the same $H$ and $Ma$ , the thrust required for different splitter variation laws may be different. The loop-matching problem of the splitter opening rate with the expected thrust may arise when the splitter opening is treated as a control input and can be avoided by taking it as a time-varying parameter. (c) For the deviation systems such as (6) and (7), the splitter opening rate in DFC and NDFC corresponds to different reference thrust, respectively. The thrust to be compensated can be obtained in advance by comparing the reference thrust corresponding to the splitter opening rate with the expected total.

Without loss of generality, the above XTER engine model can be extended to the following general PPV system:

{\begin{matrix} \overset{\cdot}{ζ} = A (ζ, θ (t)) ζ + B (ζ, θ (t)) u + E (ζ, θ (t)) ω, \\ y = C (ζ, θ (t)) ζ + F (ζ, θ (t)) ω, \end{matrix}

(8)

where $ζ \in R^{n}$ , $θ (t) \in R^{s}$ , $u \in R^{m}$ , $ω \in L_{2}^{d} [0, + \infty),$ and $y \in R^{q}$ are the state, time-varying parameter, control input, exogenous disturbance, and controlled input, respectively. $A (ζ, θ (t))$ , $B (ζ, θ (t))$ , $C (ζ, θ (t))$ , $E (ζ, θ (t)),$ and $F (ζ, θ (t))$ are polynomial matrices about $ζ$ and $θ (t)$ .

Remark 3. The PPV system retains the nonlinearity and the time-varying properties of the controlled plant via $ζ$ and $θ (t)$ in the system matrices. In addition, after removing $ζ$ or $θ (t)$ , the PPV system can be “degraded” into an LPV or a polynomial nonlinear system, respectively.

In order to facilitate stability analysis and controller design, the next assumptions are made throughout this article.

Assumption 1. The $θ (t)$ and its time derivative $\overset{\cdot}{θ} (t)$ are both measurable in real-time.

Assumption 2. The matrix $A (ζ, θ (t))$ is nonsingular for all $ζ \in R^{n}$ and $θ (t) \in R^{s}$ .

The problem of thrust control in DFC can be solved by designing a stabilization controller for the deviation model established based on the flight conditions, and the problem of thrust control in NDFC can be solved by designing a tracking controller that compensates for thrust. In order to design the tracking controller, the concept of $H_{\infty}$ output tracking performance has been extended from linear systems (Zhao and Zhao, 2021) and LPV systems (Yang and Zhao, 2017) to PPV systems and finite time. The tracking of the reference signal can be realized by stabilizing the error-augmented system. For the purpose of describing the problem more precisely, we set up the following augmented system first.

The reference signal is given as $r$ , then, the output tracking error is

e = y - r .

(9)

Let $z = \int_{t_{0}}^{t} e d s$ , and combining with (8) and (9), we have

\begin{matrix} \overset{\cdot}{z} = e = y - r \\ = C (ζ, θ (t)) ζ + F (ζ, θ (t)) ω - r . \end{matrix}

(10)

Let $μ = [\begin{matrix} ζ^{T} z^{T} \end{matrix}]^{T}$ , $\bar{ω} = [\begin{matrix} ω^{T} r^{T} \end{matrix}]^{T}$ , we get the following augmented system according to (8) and (10):

{\begin{matrix} \overset{\cdot}{μ} = \hat{A} (μ, θ (t)) μ + \hat{B} (μ, θ (t)) u + \hat{E} (μ, θ (t)) \bar{ω}, \\ e = \hat{C} (μ, θ (t)) μ + \hat{F} (μ, θ (t)) \bar{ω}, \end{matrix}

where

\begin{matrix} \hat{A} (μ, θ (t)) = [\begin{matrix} A (μ, θ (t)) & 0 \\ C (μ, θ (t)) & 0 \end{matrix}], \hat{B} (μ, θ (t)) = [\begin{matrix} B (μ, θ (t)) \\ 0 \end{matrix}], \\ \hat{C} (μ, θ (t)) = [\begin{matrix} C (μ, θ (t)) 0 \end{matrix}], \\ \hat{E} (μ, θ (t)) = [\begin{matrix} E (μ, θ (t)) & 0 \\ F (μ, θ (t)) & - I \end{matrix}], \hat{F} (μ, θ (t)) = [\begin{matrix} F (μ, θ (t)) - I \end{matrix}] . \end{matrix}

(11)

Then, we introduce some definitions and redescribe the control objectives in the framework of $H_{\infty}$ and finite time.

Definition 3. Given a scalar $γ > 0$ , system (8) is said to be $H_{\infty}$ UFTS under the effect of control input $u$ , if the following statements are satisfied:

(i) When $ω \equiv 0$ , the system (8) is UFTS at zero equilibrium point.

(ii) When $ω \in L_{2}^{d} [0, + \infty)$ , with zero initial condition, the output $y$ satisfies the following:

\int_{0}^{\infty} y^{T} y d t \leq γ^{2} \int_{0}^{\infty} ω^{T} ω d t .

(12)

Definition 4. System (8) is said to have $H_{\infty}$ uniformly finite-time output tracking performance under the effect of control input $u$ , if the augmented system (11) is been $H_{\infty}$ UFTS.

Combining the above definitions, the control objectives of this article are restated as follows, corresponding to the control objectives at the beginning of this article, respectively.

Problem 1. Design a state-feedback controller for system (8) such that the corresponding closed-loop system is $H_{\infty}$ UFTS.

Problem 2. Design a state-feedback controller for system (11) such that system (8) has $H_{\infty}$ uniformly finite-time output tracking performance.

Controller design

In this section, we will give the controller design approaches to solve Problems 1 and 2. Some important theories are developed by SOS techniques and are presented in the form of solvability conditions, and the control system of MT control is given.

Stability conditions

In order to facilitate the subsequent derivation, making $θ_{i} (t)$ represents the $i$ th row of $θ (t)$ , $A_{j} (ζ, θ (t))$ represents the $j$ th row of $A (ζ, θ (t))$ , $J = {j_{1}, \dots, j_{k}}$ represents the row indices of $B (ζ, θ (t))$ whose associated row is equal to zero, and let $\tilde{ζ} = [ζ_{j 1}, \dots, ζ_{jk}]^{T}$ . Without loss of generality, assuming the initial time $t_{0} = 0$ . Solving Problem 1, we come to the following conclusion and some matrix-dependent terms have been omitted for simplification.

Theorem 1. For the PPV system (8), given a positive scalar $γ$ , suppose there exist a symmetric polynomial matrix $Q (\tilde{ζ}, θ (t))$ , polynomial matrices $W (ζ, θ (t), \overset{\cdot}{θ} (t))$ , $G (ζ, θ (t), \overset{\cdot}{θ} (t))$ , two real numbers $c_{2} > 0$ , $α_{2} \in (0, 1)$ , and some constants $ε_{p} > 0 (p = 1, 2, 3, 4)$ with $ε_{2} > ε_{1}$ , such that

υ_{1}^{T} (Q (\tilde{ζ}, θ (t)) - ε_{1} I) υ_{1} \in Φ_{SOS},

(13)

υ_{2}^{T} (ε_{2} I - Q (\tilde{ζ}, θ (t))) υ_{2} \in Φ_{SOS},

(14)

\begin{matrix} - υ_{3}^{T} ([\begin{matrix} Φ & E (ζ, θ (t)) & Q (\tilde{ζ}, θ (t)) C^{T} (ζ, θ (t)) \\ * & - γ^{2} I & F^{T} (ζ, θ (t)) \\ * & * & - I \end{matrix}] + ε_{3} I) \\ υ_{3} \in Φ_{SOS}, \end{matrix}

(15)

\begin{matrix} - υ_{4}^{T} (He (BG) + ε_{4} I) υ_{4} \in Φ_{SOS}, \end{matrix}

(16)

\begin{matrix} Φ = He (A (ζ, θ (t)) Q (\tilde{ζ}, θ (t)) + B (ζ, θ (t)) W (ζ, θ (t), \overset{\cdot}{θ} (t))) \\ - \sum_{j = 1}^{j = k} \frac{\partial Q}{\partial ζ_{j}} (\tilde{ζ}, θ (t)) \cdot A_{j} (ζ, θ (t)) ζ - \sum_{i = 1}^{i = s} \frac{\partial Q}{\partial θ_{i}} (\tilde{ζ}, θ (t)) \cdot \frac{d θ_{i}}{dt}, \end{matrix}

(16)

where $υ_{p} (p = 1, 2, 3, 4)$ are column vectors with suitable dimensions. Then, Problem 1 is solvable with the controller given by

u = W Q^{- 1} ζ + c_{2} ∥ ζ ∥^{α_{2} - 1} G Q^{- 1} ζ .

(17)

Proof. Given a Lyapunov candidate

V (ζ, t) = ζ^{T} Q^{- 1} (\tilde{ζ}, θ (t)) ζ .

(18)

From (13) and (14), we have

0 < \frac{1}{ε_{2}} I \leq Q^{- 1} (\tilde{ζ}, θ (t)) \leq \frac{1}{ε_{1}} I .

(19)

Consequently,

0 < \frac{1}{ε_{2}} ∥ ζ ∥^{2} \leq V (ζ, t) \leq \frac{1}{ε_{1}} ∥ ζ ∥^{2} .

(20)

Thus, $V (ζ, t)$ is positive definite and bounded, and $V (ζ, t) \to \infty$ as $∥ ζ ∥ \to \infty$ .

When $\bar{ω} \equiv 0$ , the time derivative of $V (ζ, t)$ is given by

\begin{matrix} \overset{\cdot}{V} (ζ, t) = {\overset{\cdot}{ζ}}^{T} Q^{- 1} (\tilde{ζ}, θ (t)) ζ + ζ^{T} \frac{d Q^{- 1}}{dt} (\tilde{ζ}, θ (t)) ζ \\ = {(K_{A, W} ζ)}^{T} Q^{- 1} ζ + ζ^{T} Q^{- 1} (K_{A, B} ζ) \\ + ζ^{T} Q^{- 1} (\tilde{ζ}, θ (t)) \overset{\cdot}{ζ} \\ + ζ^{T} (\sum_{j = 1}^{j = k} \frac{\partial Q^{- 1}}{\partial ζ_{j}} \cdot A_{j} (ζ, θ (t)) ζ + \sum_{i = 1}^{i = s} \frac{\partial Q^{- 1}}{\partial θ_{i}} \cdot \frac{d θ_{i}}{dt}) ζ \\ + {(BG Q^{- 1} c_{2} ∥ ζ ∥^{α_{2} - 1} ζ)}^{T} Q^{- 1} ζ \\ + ζ^{T} Q^{- 1} (BG Q^{- 1} c_{2} ∥ ζ ∥^{α_{2} - 1} ζ) \\ = ζ^{T} (He (Q^{- 1} K_{A, W}) + \sum_{j = 1}^{j = k} \frac{\partial Q^{- 1}}{\partial ζ_{j}} (\tilde{ζ}, θ (t)) \\ \cdot A_{j} (ζ, θ (t)) ζ + \sum_{i = 1}^{i = s} \frac{\partial Q^{- 1}}{\partial θ_{i}} (\tilde{ζ}, θ (t)) \cdot \frac{d θ_{i}}{dt}) ζ \\ + c_{2} ∥ ζ ∥^{α_{2} - 1} ζ^{T} Q^{- 1} He (BG) Q^{- 1} ζ, \\ K_{A, W} \overset{Δ}{=} A (ζ, θ (t)) + B (ζ, θ (t)) W (ζ, θ (t), \overset{\cdot}{θ} (t)) Q^{- 1} (\tilde{ζ}, θ (t)) . \end{matrix}

(21)

The conditions (15) and (16) imply that $Φ \leq - ε_{3} I$ and $He (BG) \leq - ε_{4} I$ , respectively. Then, pre- and post-multiplying of $Φ \leq - ε_{3} I$ and $He (BG) \leq - ε_{4} I$ with $Q^{- 1} (\tilde{ζ}, θ (t))$ , and utilizing Lemma 2, we can obtain

\begin{matrix} He (Q^{- 1} K_{A, W}) + \sum_{j = 1}^{j = k} \frac{\partial Q^{- 1}}{\partial ζ_{j}} (\tilde{ζ}, θ (t)) \cdot A_{j} (ζ, θ (t)) ζ \\ + \sum_{i = 1}^{i = s} \frac{\partial Q^{- 1}}{\partial θ_{i}} (\tilde{ζ}, θ (t)) \cdot \frac{d θ_{i}}{dt} + ε_{3} Q^{- 2} (\tilde{ζ}, θ (t)) \leq 0 . \end{matrix}

(22)

Q^{- 1} He (BG) Q^{- 1} \leq - ε_{4} Q^{- 2} .

(23)

Pre- and post-multiplying of (22) and (23) with $ζ^{T}$ and $ζ$ , respectively, and combining with (19), we have

\begin{matrix} ζ^{T} (He (Q^{- 1} K_{A, W}) + \sum_{j = 1}^{j = k} \frac{\partial Q^{- 1}}{\partial ζ_{j}} (\tilde{ζ}, θ (t)) \cdot A_{j} (ζ, θ (t)) ζ \\ + \sum_{i = 1}^{i = s} \frac{\partial Q^{- 1}}{\partial θ_{i}} (\tilde{ζ}, θ (t)) \cdot \frac{d θ_{i}}{dt}) ζ \\ \leq - \frac{ε_{3}}{ε_{2}} V (ζ, t) < 0, \forall t \geq 0, \forall ζ \neq 0 . \end{matrix}

(24)

ζ^{T} Q^{- 1} He (BG) Q^{- 1} ζ \leq - \frac{ε_{4}}{ε_{2}} ζ^{T} Q^{- 1} ζ .

(25)

Multiplying $c_{2} ∥ ζ ∥^{α_{2} - 1}$ on the left-hand of (25), and combining with (19), one obtains

\begin{matrix} c_{2} (t) ∥ ζ ∥^{α_{2} - 1} (ζ^{T} Q^{- 1} He (BG) Q^{- 1} ζ) \\ \leq - \frac{ε_{4}}{ε_{2}} c_{2} ∥ ζ ∥^{α_{2} - 1} (ζ^{T} Q^{- 1} ζ) \\ \leq - c_{2} \frac{ε_{4} ε_{1}^{(α_{2} - 1) / 2}}{ε_{2}} {(ζ^{T} \frac{1}{ε_{1}} ζ)}^{(α_{2} - 1) / 2} ζ^{T} Q^{- 1} ζ \\ \leq - c_{2} \frac{ε_{4}}{ε_{2} ε_{1}^{(1 - α_{2}) / 2}} V^{(α_{2} + 1) / 2} (ζ, t) < 0, \forall t \geq 0, \forall ζ \neq 0 . \end{matrix}

(26)

Combining with (24) and (26), we have

\begin{matrix} \overset{\cdot}{V} (ζ, t) \leq - \frac{ε_{3}}{ε_{2}} V (ζ, t) - c_{2} \frac{ε_{4}}{ε_{2} ε_{1}^{(1 - α_{2}) / 2}} V^{(α_{2} + 1) / 2} (ζ, t) \\ \overset{Δ}{=} - lV (ζ, t) - c V^{α} (ζ, t) < 0, \forall t \geq 0, \forall ζ \neq 0, \end{matrix}

(27)

where $α \in (0.5, 1)$ .

This concludes that the system (8) is UFTS at zero equilibrium point.

When $ζ (0) = 0$ , $ω \in L_{2}^{d} [0, + \infty)$ , for any $T_{0} > 0$ , define $J = \int_{0}^{T_{0}} (y^{T} y - γ^{2} ω^{T} ω) d t$ , then

\begin{array}{l} J = \int_{0}^{T_{0}} (y^{T} y - γ^{2} ω^{T} ω + \dot{V} (ζ, t)) d t - V (ζ (T_{0}), T_{0}) \\ \leq \int_{0}^{T_{0}} (\begin{array}{l} {[\begin{matrix} ζ \\ ω \end{matrix}]}^{T} [\begin{matrix} C^{T} C & C^{T} F \\ F^{T} C & F^{T} F \end{matrix}] [\begin{matrix} ζ \\ ω \end{matrix}] + {[\begin{matrix} ζ \\ ω \end{matrix}]}^{T} [\begin{matrix} 0 & 0 \\ * & - γ^{2} I \end{matrix}] [\begin{matrix} ζ \\ ω \end{matrix}] \\ + {[\begin{matrix} ζ \\ ω \end{matrix}]}^{T} [\begin{matrix} Ξ & Q^{- 1} (\tilde{ζ}, θ (t)) E (ζ, θ (t)) \\ * & 0 \end{matrix}] [\begin{matrix} ζ \\ ω \end{matrix}] - c V^{α_{2}} \end{array}) \\ d t - V (ζ (T_{0}), T_{0}) \\ \leq \int_{0}^{T_{0}} ({[\begin{matrix} ζ \\ ω \end{matrix}]}^{T} [\begin{matrix} Ξ + C^{T} C & Q^{- 1} (\tilde{ζ}, θ (t)) E (ζ, θ (t)) + C^{T} F \\ * & - γ^{2} I + F^{T} F \end{matrix}] [\begin{matrix} ζ \\ ω \end{matrix}] \\ - c V^{α_{2}}) d t - V (ζ (T_{0}), T_{0}) \\ \leq \int_{0}^{T_{0}} ({[\begin{matrix} ζ \\ ω \end{matrix}]}^{T} [\begin{matrix} Ξ + C^{T} C & Q^{- 1} (\tilde{ζ}, θ (t)) E (ζ, σ (t)) + C^{T} F \\ * & - γ^{2} I + F^{T} F \end{matrix}] \\ [\begin{matrix} ζ \\ ω \end{matrix}]) d t . \end{array}

where

\begin{matrix} Ξ = He (Q^{- 1} (\tilde{ζ}, θ (t)) K_{A, W}) + \sum_{j = 1}^{j = k} \frac{\partial Q^{- 1}}{\partial ζ_{j}} (\tilde{ζ}, θ (t)) \cdot A_{j} (ζ, θ (t)) ζ \\ + \sum_{i = 1}^{i = s} \frac{\partial Q^{- 1}}{\partial θ_{i}} (\tilde{ζ}, θ (t)) \cdot \frac{d θ_{i}}{dt} . \end{matrix}

(27)

Note that condition (15) implies

[\begin{matrix} Φ & E (ζ, θ (t)) & Q (\tilde{ζ}, θ (t)) C^{T} (ζ, θ (t)) \\ * & - γ^{2} I & F^{T} (ζ, θ (t)) \\ * & * & - I \end{matrix}] < 0 .

(28)

Multiplying (28) from the both sides by $diag {Q^{- 1} (\tilde{ζ}, θ (t)), I, I}$ and using Lemma 2, we have

[\begin{matrix} Ξ & Q^{- 1} (\tilde{ζ}, θ (t)) E (ζ, θ (t)) & C^{T} \\ * & - γ^{2} I & F^{T} (ζ, θ (t)) \\ * & * & - I \end{matrix}] < 0 .

By Schur complements, we can easily conclude that when $T_{0} \to \infty$ ,

[\begin{matrix} Ξ + C^{T} C & Q^{- 1} (\tilde{ζ}, θ (t)) E (ζ, θ (t)) + C^{T} F \\ * & - γ^{2} I + F^{T} F \end{matrix}] < 0 .

Thus, $\int_{0}^{\infty} (y^{T} y - γ^{2} ω^{T} ω) d t < 0$ .

This completes the proof of Theorem 1. □

Remark 4. The control law given by (17) is bounded and converges to zero when the system state converges to the equilibrium $ζ = 0$ . Because, when $α_{2} \in (0, 1)$ , $\lim_{∥ ζ ∥ \to 0} ∥ ζ ∥^{α_{2} - 1} ζ = 0$ , for $ζ \in R^{n}$ (Bohn and Sanyal, 2016). In addition, the controller in Theorem 1 achieves a split design with great flexibility, that is, after the matrix $W (ζ, θ (t), \overset{\cdot}{θ} (t))$ and $Q (\tilde{ζ}, θ (t))$ in the controller (17) are determined, the matrix $G (ζ, θ (t), \overset{\cdot}{θ} (t))$ can be independently designed since condition (16) can be independent of conditions (13) to (15). However, it is obvious that condition (16) has no solution in some cases, such as when $B (ζ, θ (t))$ has zero rows. This means that augmented system (11) cannot be stabilized by using Theorem 1.

For Problem 2, we come to the following conclusion.

Theorem 2. For the augmented PPV system (11), given a positive scalar $γ$ , suppose there exist a symmetric polynomial matrix $Q (\tilde{μ}, θ (t))$ , polynomial matrices $W (μ, θ (t), \overset{\cdot}{θ} (t))$ , $G (μ, θ (t), \overset{\cdot}{θ} (t))$ , a real number $α \in (0, 1)$ , and some constants $ε_{p} > 0 (p = 1, 2, 3, 4)$ with $ε_{3} > ε_{4}$ for equation (32b), and $ε_{2} > ε_{1}$ , such that

υ_{1}^{T} (Q (\tilde{μ}, θ (t)) - ε_{1} I) υ_{1} \in Φ_{SOS},

(29)

υ_{2}^{T} (ε_{2} I - Q (\tilde{μ}, θ (t))) υ_{2} \in Φ_{SOS},

(30)

\begin{matrix} - υ_{3}^{T} ([\begin{matrix} \bar{Φ} & E (μ, θ (t)) & Q (\tilde{μ}, θ (t)) C^{T} (μ, θ (t)) \\ * & - γ^{2} I & F^{T} (μ, θ (t)) \\ * & * & - I \end{matrix}] + ε_{3} I) \end{matrix}

(31)

υ_{3} \in Φ_{SOS},

(31)

υ_{4}^{T} ({\bar{Φ}}_{G} - ε_{4} I) υ_{4} \in Φ_{SOS},

(32a)

or {\begin{matrix} - υ_{4}^{T} {\bar{Φ}}_{G} υ_{4} \in Φ_{SOS}, \\ υ_{5}^{T} ({\bar{Φ}}_{G} + ε_{4} I) υ_{5} \in Φ_{SOS}, \end{matrix}

(32b)

\begin{matrix} \bar{Φ} = He (A (μ, θ (t)) Q (\tilde{μ}, θ (t)) + B (μ, θ (t)) W (μ, θ (t), \overset{\cdot}{θ} (t))) \\ - \sum_{j = 1}^{j = k} \frac{\partial Q}{\partial μ_{j}} (\tilde{μ}, θ (t)) \cdot A_{j} (μ, θ (t)) μ - \sum_{i = 1}^{i = s} \frac{\partial Q}{\partial θ_{i}} (\tilde{μ}, θ (t)) \cdot \frac{d θ_{i}}{dt}, \\ {\bar{Φ}}_{G} = He (A (μ, θ (t)) Q (\tilde{μ}, θ (t)) + B (μ, θ (t)) G (μ, θ (t), \overset{\cdot}{θ} (t))) \\ - \sum_{j = 1}^{j = k} \frac{\partial Q}{\partial μ_{j}} (\tilde{μ}, θ (t)) \cdot A_{j} (μ, θ (t)) μ - \sum_{i = 1}^{i = s} \frac{\partial Q}{\partial θ_{i}} (\tilde{μ}, θ (t)) \cdot \frac{d θ_{i}}{dt}, \end{matrix}

where $υ_{p} (p = 1, 2, 3, 4, 5)$ are column vectors with suitable dimensions. Then, Problem 2 is solvable with the controller given by

u = W Q^{- 1} μ + c_{2} ∥ μ ∥^{α_{2} - 1} W Q^{- 1} μ - c_{2} ∥ μ ∥^{α_{2} - 1} G Q^{- 1} μ .

(33)

Proof. The proof of Theorem 2 is similar to the proof of Theorem 1.

Given a Lyapunov candidate

V (μ, t) = μ^{T} Q^{- 1} (\tilde{μ}, θ (t)) μ .

(34)

From (29) and (30), we have $V (μ, t)$ is positive definite and bounded, and $V (μ, t) \to \infty$ as $∥ μ ∥ \to \infty$ .

When $\bar{ω} \equiv 0$ , the time derivative of $V (μ, t)$ is given by

\begin{matrix} \overset{\cdot}{V} (μ, t) = {\overset{\cdot}{μ}}^{T} Q^{- 1} (\tilde{μ}, θ (t)) μ \\ + μ^{T} \frac{d Q^{- 1}}{dt} (\tilde{μ}, θ (t)) μ + μ^{T} Q^{- 1} (\tilde{μ}, θ (t)) \overset{\cdot}{μ} \\ = μ^{T} (He (Q^{- 1} {\bar{K}}_{A, B}) + \sum_{j = 1}^{j = k} \frac{\partial Q^{- 1}}{\partial μ_{j}} (\tilde{μ}, θ (t)) \cdot A_{j} (μ, θ (t)) μ \\ + \sum_{i = 1}^{i = s} \frac{\partial Q^{- 1}}{\partial θ_{i}} (\tilde{μ}, θ (t)) \cdot \frac{d θ_{i}}{dt}) μ + c_{2} ∥ μ ∥^{α_{2} - 1} μ^{T} \\ Q^{- 1} He (BW) Q^{- 1} μ - c_{2} ∥ μ ∥^{α_{2} - 1} μ^{T} Q^{- 1} He (BG) Q^{- 1} μ \\ = μ^{T} \bar{Ξ} μ + c_{2} ∥ μ ∥^{α_{2} - 1} μ^{T} (\bar{Ξ}) μ - c_{2} ∥ μ ∥^{α_{2} - 1} μ^{T} ({\bar{Ξ}}_{G}) μ . \end{matrix}

(35)

where

\begin{matrix} {\bar{K}}_{A, W} = A (μ, θ (t)) + B (μ, θ (t)) W (μ, θ (t), \overset{\cdot}{θ} (t)) Q^{- 1} (\tilde{μ}, θ (t)), \\ {\bar{K}}_{A, G} = A (μ, θ (t)) + B (μ, θ (t)) G (μ, θ (t), \overset{\cdot}{θ} (t)) Q^{- 1} (\tilde{μ}, θ (t)), \\ \bar{Ξ} = He (Q^{- 1} {\bar{K}}_{A, W}) + \sum_{j = 1}^{j = k} \frac{\partial Q^{- 1}}{\partial μ_{j}} (\tilde{μ}, θ (t)) \cdot A_{j} (μ, θ (t)) μ \\ + \sum_{i = 1}^{i = s} \frac{\partial Q^{- 1}}{\partial θ_{i}} (\tilde{μ}, θ (t)) \cdot \frac{d θ_{i}}{dt}, \\ {\bar{Ξ}}_{G} = He (Q^{- 1} {\bar{K}}_{A, G}) + \sum_{j = 1}^{j = k} \frac{\partial Q^{- 1}}{\partial μ_{j}} (\tilde{μ}, θ (t)) \cdot A_{j} (μ, θ (t)) μ \\ + \sum_{i = 1}^{i = s} \frac{\partial Q^{- 1}}{\partial θ_{i}} (\tilde{μ}, θ (t)) \cdot \frac{d θ_{i}}{dt} . \end{matrix}

(35)

From condition (31) and utilizing Lemma 2, we can obtain

μ^{T} \bar{Ξ} μ \leq - \frac{ε_{3}}{ε_{2}} V (μ, t), \forall t \geq 0, \forall μ \neq 0 .

(36)

\begin{matrix} c_{2} (t) ∥ μ ∥^{α_{2} - 1} μ^{T} (\bar{Ξ}) μ \leq - c_{2} \frac{ε_{3}}{ε_{2} ε_{1}^{(1 - α_{2}) / 2}} V^{(α_{2} + 1) / 2} (μ, t) < 0, \\ \forall t \geq 0, \forall μ \neq 0 . \end{matrix}

(37)

From condition equation (32) and utilizing Lemma 2, we can obtain

When condition equation (32a) holds, we have

\begin{matrix} c_{2} ∥ μ ∥^{α_{2} - 1} μ^{T} ({\bar{Ξ}}_{G}) μ \leq - c_{2} \frac{ε_{4}}{ε_{2} ε_{1}^{(1 - α_{2}) / 2}} V^{(α_{2} + 1) / 2} (μ, t), \\ \forall t \geq 0, \forall μ \neq 0 . \end{matrix}

(38)

When condition equation (32b) holds, we have

\begin{matrix} 0 \leq - c_{2} ∥ μ ∥^{α_{2} - 1} μ^{T} ({\bar{Ξ}}_{G}) μ \\ \leq c_{2} \frac{ε_{4}}{ε_{2} ε_{1}^{(1 - α_{2}) / 2}} V^{(α_{2} + 1) / 2} (μ, t) < 0, \forall t \geq 0, \forall μ \neq 0 . \end{matrix}

(39)

Combining (34)–(39), we have

When condition equation (32a) holds, we have

\begin{matrix} \overset{\cdot}{V} (μ, t) \leq - \frac{ε_{3}}{ε_{2}} V (μ, t) - c_{2} \frac{ε_{3} + ε_{4}}{ε_{2} ε_{1}^{(1 - α_{2}) / 2}} V^{(α_{2} + 1) / 2} (μ, t) \\ \overset{Δ}{=} - lV (μ, t) - c V^{α} (μ, t) < 0, \forall t \geq 0, \forall μ \neq 0 . \end{matrix}

(40)

When condition equation (32b) holds, then

\begin{matrix} \overset{\cdot}{V} (μ, t) \leq - \frac{ε_{3}}{ε_{2}} V (μ, t) - c_{2} \frac{ε_{3} - ε_{4}}{ε_{2} ε_{1}^{(1 - α_{2}) / 2}} V^{(α_{2} + 1) / 2} (μ, t) \\ \overset{Δ}{=} - lV (μ, t) - c V^{α} (μ, t) < 0, \forall t \geq 0, \forall μ \neq 0 . \end{matrix}

(41)

where $α \in (0.5, 1)$ .

This conclude that the system (11) is UFTS.

When $μ (0) = 0$ , $\bar{ω} \in L_{2}^{d} [0, + \infty)$ , for any $T_{0} > 0$ , define $J = \int_{0}^{T_{0}} (e^{T} e - γ^{2} {\bar{ω}}^{T} \bar{ω}) d t$ , then

\begin{array}{l} J = \int_{0}^{T_{0}} (e^{T} e - γ^{2} {\bar{ω}}^{T} \bar{ω} + \dot{V} (μ, t)) d t - V (μ (T_{0}), T_{0}) \\ \leq \int_{0}^{T_{0}} (\begin{array}{l} {[\begin{matrix} μ \\ \bar{ω} \end{matrix}]}^{T} [\begin{matrix} C^{T} C & C^{T} F \\ F^{T} C & F^{T} F \end{matrix}] [\begin{matrix} μ \\ \bar{ω} \end{matrix}] + {[\begin{matrix} μ \\ \bar{ω} \end{matrix}]}^{T} \\ [\begin{matrix} \bar{Ξ} & Q^{- 1} (\tilde{μ}, θ (t)) E (μ, θ (t)) \\ * & - γ^{2} I \end{matrix}] [\begin{matrix} μ \\ \bar{ω} \end{matrix}] \\ + c_{2} ∥ μ ∥^{α_{2} - 1} μ^{T} (\bar{Ξ}) μ - c_{2} ∥ μ ∥^{α_{2} - 1} μ^{T} ({\bar{Ξ}}_{G}) μ \end{array}) \\ d t - V (μ (T_{0}), T_{0}) \\ \leq \int_{0}^{T_{0}} ({[\begin{matrix} μ \\ \bar{ω} \end{matrix}]}^{T} [\begin{matrix} α \bar{Ξ} + C^{T} C & Q^{- 1} (\tilde{μ}, θ (t)) E (μ, θ (t)) + C^{T} F \\ * & - γ^{2} I + F^{T} F \end{matrix}] \\ [\begin{matrix} μ \\ \bar{ω} \end{matrix}]) d t . \end{array}

Note that (31) implies

[\begin{matrix} \bar{Φ} & E (μ, θ (t)) & Q (\tilde{μ}, θ (t)) C^{T} (μ, θ (t)) \\ * & - γ^{2} I & F^{T} (μ, θ (t)) \\ * & * & - I \end{matrix}] < 0 .

(42)

Similar to the proof of Theorem 1, we can obtain, when $T_{0} \to \infty$ ,

[\begin{matrix} \bar{Ξ} + C^{T} C & Q^{- 1} (\tilde{μ}, θ (t)) E (μ, θ (t)) + C^{T} F \\ * & - γ^{2} I + F^{T} F \end{matrix}] < 0 .

This completes the proof of Theorem 2. □

Remark 5. The conservativeness of the proposed theory is mainly due to four aspects. (a) SOS is only a sufficient condition for the non-negativity of polynomials, as explained after Definition 2. (b) The value of the normal number $ε_{i} (i = 1, 2, 3, 4)$ in the solvable condition. For example, the smaller the $ε_{3}$ , the smaller the gap between (15) and (28). (c) The requirement is that the $\overset{\cdot}{σ} (t)$ is measurable and bounded. (d) The condition (16) in Theorem 1 has no solution in some cases, as mention in Remark 4. And the matrices $Q (\tilde{μ}, θ (t))$ , $W (μ, θ (t), \overset{\cdot}{θ} (t))$ , and $G (μ, θ (t), \overset{\cdot}{θ} (t))$ in Theorem 2 are coupled to each other.

PPV control system

In this section, Figure 2 is given to illustrate the MT control system of XTER in the framework of PPV. The conditions in Theorems 1 and 2 are solved in SOSTOOLS to obtain the parameters of the controller.

MT thrust control simulations

In this section, we apply Theorems 1 and 2 to the MT thrust control between the TE and the ER of XTER in DFC and NDFC, respectively. The PPV model of XTER is established by employing a system identification technique, assuming that the thrust is measurable, as shown in (6) and (7). Then, in order to test the effectiveness of our results, the proposed control design scheme (the results denoted as “Proposed”) is compared to the exponential stable $H_{\infty}$ control method (the results denoted as “Compared”) comes from Fu et al. (2018, 2019).

The MT is carried out at $H = 15.5 km, Ma = 2.5$ . The reference values of thrust and fuel flow, the opening rate of the splitter, and the compensated thrust in NDFC are shown in Figure 3. As shown in Figure 3, during MT, the reference thrust of the turbofan engine gradually decreases to zero, while the reference thrust of the ER gradually increases from zero and finally stabilizes at a constant value, and the opening of the splitter gradually changes from fully open to fully closed. The reference thrust is different between DFC and NDFC, and there is a thrust that needs to be compensated. All the simulations in the subsequent section are carried out with the initial total thrust deviation of nearly 10%.

Case I (MT in DFC). Consider the problem of MT thrust control in DFC with exogenous disturbance $ω = e^{- 2 t}$ given by the health parameter input (Yang and Zhao, 2017). The state feedback stabilizing controllers are designed for the systems (6) and (7) by using Theorem 1. The parameters used in controller design are shown in Table 1, and the simulation results are shown in Figure 4 as follows.

Table 1.

Parameters used in simulation.

Case	Engine	Method	$ε_{1}$	$ε_{2}$	$ε_{3}$	$ε_{4}$	$γ$	$α_{2}$	$c_{2}$
I	TE	Proposed	Free variable	$5.05 \times 10^{4}$	$1 \times 10^{- 5}$	$1 \times 10^{- 5}$	1.5	0.8	1
		Compared		$5.05 \times 10^{4}$	$1 \times 10^{- 5}$	–	1.5	–	–
	ER	Proposed	Free variable	$3.1 \times 10^{4}$	$1 \times 10^{- 1}$	1	1.5	0.8	10
		Compared		$3.1 \times 10^{4}$	$1 \times 10^{- 1}$	–	1.5	–	–
II	TE	Proposed	Free variable	$3.28 \times 10^{5}$	$2 \times 10^{- 3}$	$2 \times 10^{- 3}$	1.8	0.8	1
		Compared		$3.8 \times 10^{5}$	$1 \times 10^{- 4}$	–	1.5	–	–
	ER	Proposed	Free variable	$2.23 \times 10^{4}$	$3 \times 10^{- 1}$	$1 \times 10^{- 1}$	1.8	0.8	1
		Compared		$7.1 \times 10^{4}$	$1 \times 10^{- 3}$	–	1.5	–	–

TE: turbine engine; ER: ejector-ramjet.

The parameters are obtained by tuning, and for the detailed usage of SOSTOOLS and detailed definition of parameters refer to Papachristodoulou et al. (2016).

As shown in Figure 4, the total thrust error of the proposed method converges to <2% in 0.94 seconds and eventually converges to zero within 2 seconds. But the compared method is >2 seconds. It can be seen that, compared with the compared method, the thrust of the proposed method can approach the target more quickly and with less vibration.

Case II (MT in DFC and NDFC). Consider the problem of MT thrust control in DFC and NDFC with exogenous disturbance $ω = e^{- 2 t}$ . The state feedback tracking controllers are designed for the systems (6) and (7) by using Theorem 2 with equations (32a) and (32b), respectively. The parameters of controller design are shown in Table 1. The simulation results of DFC are shown in Figure 5, and the simulation results of NDFC are shown in Figure 6.

As shown in Figure 5, the total thrust error of the proposed method converges to <2% in 0.21 seconds and eventually converges to zero within 2 seconds. But the compared method is <2 seconds. It can be seen that, compared with the compared method, the proposed method has better performance when tracking the constant value compensated thrust. As shown in Figure 6, the total thrust error of the proposed method quickly converges to <2% in 0.22 seconds and no longer exceeds. And the error fluctuation of the compared method is larger. It can be seen that, compared with the compared method, the proposed method has better performance when tracking the time-varying compensated thrust.

In summary, the simulation results show that even in the case of a large initial error, the designed stabilizing controllers can guarantee a satisfactory performance in DFC, and the designed tracking controllers can guarantee a satisfactory performance in both DFC and NDFC.

Conclusions and outlooks

In this article, the problem of $H_{\infty}$ uniformly finite-time stabilization control and $H_{\infty}$ uniformly finite-time output tracking control for PPV systems are addressed. The sufficient conditions are given in the format of state-and-parameter-dependent matrix inequalities. In order to adapt different opening rates of the splitter, a new model design scheme is presented of XTER and the established PPV model can retain the nonlinearity and time-varying properties. Furthermore, the controllers are designed for MT thrust control in DFC and NDFC, respectively. In DFC, the stabilizing controllers make the total thrust error <2% in 0.94 seconds and eventually converge to zero. In NDFC, the tracking controllers make the total thrust error <2% in 0.21 seconds and no longer exceeds. Moreover, the tracking controllers also realize the MT thrust control in DFC with a total thrust error <2% in 0.22 seconds and eventually converges to zero. The simulation results show the effectiveness of the proposed control design scheme.

However, as noted in Remark 5, the proposed theories in this article can be further optimized. The challenge works for further research are to make Theorem 11 avoid the requirement that $B (ζ, θ (t))$ has no zero rows, and decouple the $Q (\tilde{μ}, θ (t))$ , $W (μ, θ (t), \overset{\cdot}{θ} (t))$ , and $G (μ, θ (t), \overset{\cdot}{θ} (t))$ in Theorem 2 from each other. In addition, considering that the local controller may have better performance than the global controller in practice and reduce the difficulty of the design, the local theory is also our interest in future research.

Footnotes

Appendix 1

Appendix 2

The PPV system of the TE is shown as follows:

{\begin{matrix} [\begin{matrix} Δ \overset{\cdot}{n} \\ Δ \overset{\cdot}{π} \end{matrix}] = [\begin{matrix} a_{11} (Δ n, β) & a_{12} (Δ n, β) \\ a_{21} (Δ n, β) & a_{21} (Δ n, β) \end{matrix}] [\begin{matrix} Δ n \\ Δ π \end{matrix}] \\ + [\begin{matrix} b_{11} (Δ n, β) \\ b_{21} (Δ n, β) \end{matrix}] Δ W f_{TE} + [\begin{matrix} e_{11} (Δ n, β) \\ e_{21} (Δ n, β) \end{matrix}] ω, \\ Δ F_{TE} = [\begin{matrix} c_{11} (Δ n, β) c_{12} (Δ n, β) \end{matrix}] [\begin{matrix} Δ n \\ Δ π \end{matrix}], \end{matrix}

where

(42)

\begin{matrix} a_{11} (Δ n, β) = - 0.4586 β^{2} - 0.0748 β - 3.8861 Δ n^{2} \\ - 1.6470 Δ n - 0.0064, \\ a_{12} (Δ n, β) = 5.1789 β^{2} - 2.4228 β + 6.6081 Δ n^{2} \\ + 1.9143 Δ n - 6.0903, \\ a_{21} (Δ n, β) = 1.2786 β^{2} + 0.1805 β - 1.3167 Δ n^{2} \\ + 2.8286 Δ n + 0.0984, \\ a_{22} (Δ n, β) = - 25.1867 β^{2} + 11.3071 β - 3.5310 Δ n^{2} \\ - 3.0092 Δ n - 6.2332, \\ b_{11} (Δ n, β) = - 5.1468 β^{2} + 18.3988 β - 3.9981 Δ n^{2} \\ + 1.0024 Δ n + 28.1826, \\ b_{21} (Δ n, β) = - 1.5679 β^{2} + 6.7450 β + 0.8096 Δ n^{2} \\ - 0.8669 Δ n - 0.0482, \\ c_{11} (Δ n, β) = 0.0139 β^{2} + 0.2266 β + 2.1431 Δ n^{2} \\ - 0.2192 Δ n + 0.0375, \\ c_{12} (Δ n, β) = 0.6803 β^{2} - 2.3124 β + 2.0005 Δ n^{2} \\ + 2.0979 Δ n + 8.2296, \\ e_{11} (Δ n, β) = 0.1, e_{11} (Δ n, β) = 0.2 . \end{matrix}

The PPV system of the ER is shown as follows:

{\begin{matrix} [\begin{matrix} Δ {\overset{\cdot}{P}}_{t} \\ Δ {\overset{\cdot}{T}}_{t} \end{matrix}] = [\begin{matrix} a_{11} (Δ P_{t}, β) & a_{12} (Δ P_{t}, β) \\ a_{21} (Δ P_{t}, β) & a_{21} (Δ P_{t}, β) \end{matrix}] [\begin{matrix} Δ P_{t} \\ Δ T_{t} \end{matrix}] \\ + [\begin{matrix} b_{11} (Δ P_{t}, β) \\ b_{21} (Δ P_{t}, β) \end{matrix}] Δ W f_{ER} + [\begin{matrix} e_{11} (Δ P_{t}, β) \\ e_{21} (Δ P_{t}, β) \end{matrix}] ω, \\ Δ F_{ER} = [\begin{matrix} c_{11} (Δ P_{t}, β) c_{12} (Δ P_{t}, β) \end{matrix}] [\begin{matrix} Δ P_{t} \\ Δ T_{t} \end{matrix}], \end{matrix}

where

\begin{matrix} a_{11} (Δ P_{t}, β) = 4.0690 β^{2} + 3.9058 β - 58.7863 Δ {P_{t}}^{2} \\ + 16.8611 Δ P_{t} - 26.9154, \\ a_{12} (Δ P_{t}, β) = - 3.8269 β^{2} - 3.4098 β - 142.0510 Δ P^{2} \\ - 39.8867 Δ P + 65.7770, \\ a_{21} (Δ P_{t}, β) = 1.8500 β^{2} - 7.0879 β + 11.0749 Δ {P_{t}}^{2} \\ - 3.0581 Δ P_{t} + 0.9059, \\ a_{22} (Δ P_{t}, β) = - 33.6542 β^{2} + 64.4108 β - 157.8792 Δ {P_{t}}^{2} \\ - 7.5324 Δ P_{t} - 27.4723, \\ b_{11} (Δ P_{t}, β) = - 214.1510 β^{2} - 126.5153 β + 3456.9527 Δ P^{2} \\ + 135.9355 Δ P - 95.4461, \\ b_{21} (Δ P_{t}, β) = 97.9049 β^{2} - 10.7585 β - 976.9404 Δ {P_{t}}^{2} \\ - 167.9824 Δ P_{t} + 89.5602, \\ c_{11} (Δ P_{t}, β) = - 0.6514 β^{2} + 0.0476 β + 1.3266 Δ {P_{t}}^{2} \\ - 0.3415 Δ P_{t} + 1.1822, \\ c_{12} (Δ P_{t}, β) = - 0.0823 β^{2} + 0.0888 β + 13.6094 Δ {P_{t}}^{2} \\ - 1.8466 Δ P_{t} + 0.0384, \\ e_{11} (Δ P_{t}, β) = 0.3, e_{11} (Δ P_{t}, β) = 0.5 . \end{matrix}

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Chinese Aviation Science Foundation (20220058068001) and the 1912 Project.

ORCID iDs

Lianchen Yu

Jianping Zeng

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