The guaranteed cost controller for nonlinear systems with time-varying parameters and input saturation

Abstract

This paper investigates the uniformly asymptotically/exponentially stable guaranteed cost control problem for a class of nonlinear systems with time-varying parameters and input saturation. The nonlinearities and the time-varying nature of these systems render it quite challenging to solve the guaranteed cost control problem. This paper presents a new Lyapunov functional and a novel controller structure as nonlinear and parameter-varying forms incorporating the state, the time-varying parameters and its derivative. Based on Lyapunov stability theory, the solvable conditions for the existence of guaranteed cost controllers are obtained in terms of state-and-parameter-dependent linear matrix inequalities, which can be efficiently solved via sum-of-squares convex programming. Finally, the effectiveness of the proposed method is demonstrated by the conversion flight control of a tilt rotor aircraft.

Keywords

Nonlinear time-varying systems input saturation sum of squares guaranteed cost control

Introduction

The guaranteed cost control (GCC) problem has been extensively investigated by many scholars due to its importance in applications to practical systems, such as turbine generator, robot, unmanned aerial vehicles, the piezoelectrically actuated clamped-clamped micro-beam and so on (e.g. Chang and Peng, 1972; Coutinho et al., 2002; Huang, 2018; Ma et al., 2017; Petersen and Mcfarlane, 1994; Vafamand et al., 2017; Xu et al., 2017; Yu, 2000). Among those results, Petersen and McFarlane (1994) studied the GCC problem of linear systems, where a state-feedback control law can be obtained via solving the parameterized Riccati equation. A less conservative linear matrix inequality (LMI) condition has been provided in Coutinho et al. (2002) by polynomial Lyapunov functional for the GCC problems of nonlinear systems. Such framework has been introduced in Ma et al. (2017) where the desired GCC strategy has been designed by solving Hamilton-Jacobi equation. However, the solution of the Hamilton-Jacobi equation is usually difficult to address because of its nonlinearity. Specially, when it comes to the time-varying feature, the GCC problems will be more complicated and challenging, and little related studies can be seen from the literature (e.g. Zhao et al., 2014). In Zhao et al. (2014), the GCC problem has been addressed for a class of polytopic linear parameter-varying (LPV) systems, where the infinite-dimensional optimization problem had been transformed into a finite-dimensional convex optimization problem by basis functions.

In practical applications, it is generally acknowledged that the nonlinearity and the time-varying nature are two inherent characteristics (e.g. Li and Li, 2018; Wang et al., 2010). The analysis and synthesis problems for nonlinear time-varying (NTV) systems have been extensively investigated by many scholars (e.g. He et al., 2016; Mazenc and Malisoff, 2017; Fu et al., 2018a,b; Tee et al., 2011). Essentially, numerous NTV systems can be modeled in a state-and-parameter-dependent (SDP) linear-like form as $\overset{\cdot}{x} = A (x, θ (t)) x + B (x, θ (t)) u$ (e.g. Fu et al., 2018; Rugh and Shamma, 2000). Also, it allows us to maintain the time-varying feature and the nonlinearity of the system simultaneously. As a result, the exponential stabilization problem of nonlinear systems with time-varying parameters (hereinafter, it is shortened to NPV systems) has been solved, and solvable conditions are provided by applying sum-of-squares (SOS) convex programming (e.g. Fu et al., 2018a). Also, the mixed stability of NPV systems was considered in Fu et al. (2018b).

Note that many available results for the GCC are related to linear time-invariant system. Unfortunately, these results cannot be directly extended to NPV systems. This is because, for NPV systems, SDP matrices would make the systems’ analysis and synthesis a fairly challenging task, and hence the control problems have not been adequately investigated and deserved further study. The mechanical systems (which are nonlinear systems) are ultimately driven by an electromotor that can only offer a limited amount of torque in numerous practical control applications. Therefore, the magnitude of the control effort is always constrained. In particular, the input saturation is suspicious of deteriorating the control performance and causing instability in systems (e.g. Du et al., 2016; Guo et al., 2019; Hu et al., 2008; Xia and Huo, 2016; Vafamand et al., 2017; Zhao et al., 2017). To the best of our knowledge, there are no published results devoted to studying the GCC for NPV systems with input saturation. It is, therefore, our objective to solve this problem.

In this research, our main aim is to design a class of guaranteed cost controllers for NPV systems with/without input saturation, which takes the nonlinear time-invariant systems in Vafamand et al. (2017) as special cases. Some distinguishing features of this paper can be summarized as follows. (1) An effective method is provided to solve the GCC problem for NPV systems. To the best of our knowledge, this is the first time to propose such a method that the GCC problem of NPV systems can be solved. (2) A novel SPD structure can be used for the choice of the Lyapunov functional as well as the control laws, which potentially brings less conservatism than that in the published work (e.g. Coutinho et al., 2002; Huang, 2018; Ma et al., 2017; Petersen and Mcfarlane, 1994; Xu et al., 2017; Yu, 2000). (3) Compared with the stabilisation problem in Fu et al. (2018), GCC problem is investigated and the condition for the parameter derivative is relaxed. It is worth noting that the parameter derivative has been regarded as an independent SOS variable in the solvable condition.

The notations are standard in this paper. For the sake of brevity and readability, the notations are displayed in the form of Table 1, as follows.

Table 1.

Notations and abbreviations of this paper.

$R^{n}$ / $R^{n \times m}$	the set of real vectors/ matrices of dimension $n$ / $n \times m$
$Φ_{sos}$	the set of SOS polynomials
$I$	the identity matrix
$He (A)$	$A + A^{T}, A \in R^{n \times n}$
$‖ \cdot ‖_{2}$	the Euclidean norm
*	the symmetric block of a symmetric matrix

Preliminaries and problem description

Definition 1: (Papachristodoulou et al., 2016) A polynomial $h (x)$ in $x \in R^{n}$ is an SOS, if there exist polynomials $h_{1} (x)$ , $h_{2} (x)$ , ⋯, $h_{k} (x)$ such that

h (x) = \sum_{i = 1}^{k} h_{i}^{2} (x) .

It is obvious that $h (x) \in Φ_{SOS}$ implies that $h (x) \geq 0$ for all $x \in R^{n}$ , but the converseness is not all true. Although the SOS property for a polynomial is only sufficient for its non-negativity, a large amount of numerical experiments imply that the gap between them is insignificant (Reznick, 2000). In some cases, they are even equivalent.

Consider a class of NPV systems as follows

\overset{\cdot}{x} = A (x, θ (t)) x + B (x, θ (t)) u

(1)

where $x \in R^{n}$ and $u \in R^{m}$ are the state and the control input, respectively. $θ (t) \in R^{s}$ is the time-varying parameter vector. $A (x, θ (t))$ and $B (x, θ (t))$ are polynomial matrices in $x$ and $θ (t)$ .

Note that the input saturation is ubiquitous in practical systems. Thus, input saturation is considered in this study. NPV systems in (1) can be modified as

\overset{\cdot}{x} = A (x, θ (t)) x + B (x, θ (t)) sat (u)

(2)

where $sat (\cdot) : R^{m} \to R^{m}$ represents the saturation function, and it is supposed that the control input $u \in R^{m}$ satisfies the following constraints

sat (u_{i}) = {\begin{matrix} {\bar{u}}_{i}, {\bar{u}}_{i} < u_{i}, \\ u_{i}, - {\bar{u}}_{i} \leq u_{i} \leq {\bar{u}}_{i}, \\ - {\bar{u}}_{i}, - u_{i} < {\bar{u}}_{i} . \end{matrix}

Let $A_{i} (x, θ (t))$ denote the $i$ th row of the matrix $A (x, θ (t))$ , $H = {i_{1}, i_{2}, \dots, i_{m}}$ denote the row indices of $B (x, θ (t))$ whose corresponding row is zero, and define $\tilde{x} = {[x_{i_{i}}, \dots, x_{i_{m}}]}^{T}$ .

Suppose the time-varying parameter vector $θ (t) = {[\begin{matrix} θ_{1} (t) & θ_{2} (t) & \dots & θ_{s} (t) \end{matrix}]}^{T}$ is continuous over time, and $θ (t) \in A_{θ} \subset R^{s}$ , $\overset{\cdot}{θ} (t) \in A_{v} \subset R^{s}$

A_{θ} = {θ (t) \in R^{s} : f_{i} (θ (t)) \geq 0, i = 1, 2, \dots, r}

(3a)

and

A_{v} = {{\overset{\cdot}{θ}}_{i} \in R^{s} : {\underline{v}}_{i} \leq {\overset{\cdot}{θ}}_{i} (t) \leq {\bar{v}}_{i}, i = 1, 2, \dots, s}

(3b)

where $f_{i}$ are polynomial functions of $θ (t)$ that specify the boundaries of parameter set $A_{θ}$ .

Consider the following state-feedback controller

u = K (x, θ (t), \overset{\cdot}{θ} (t)) x

(4)

where $K (x, θ (t), \overset{\cdot}{θ} (t))$ is $m \times n$ dimensional matrix in $x$ , $θ (t)$ and $\overset{\cdot}{θ} (t)$ .

From (1) and (4), the corresponding closed-loop system can be obtained as follows

\overset{\cdot}{x} = (A (x, θ (t)) + B (x, θ (t)) K (x, θ (t), \overset{\cdot}{θ} (t))) x

(5)

The cost function of system (1) is

J = \int_{0}^{\infty} (x^{T} Qx + u^{T} Ru) dt

(6)

where $Q$ and $R$ are given positive-definite weighting matrices. For simplicity, the brief form $θ$ and $\overset{\cdot}{θ}$ will be used instead of $θ (t)$ and $\overset{\cdot}{θ} (t)$ in the rest of this paper.

Definition 2: (Chen et al., 2004) If there exists a controller $u$ and a positive scalar $J^{*}$ , such that the closed-loop system (5) is stable and the cost function (6) of the system satisfies $J \leq J^{*}$ , then $J^{*}$ is said to be a guaranteed cost and $u$ is a guaranteed cost controller for the system (1).

The control goal of this paper is given as follows:

Problem 1: Design a guaranteed controller in the form of (4) for the system given by (1) without input saturation such that the corresponding closed-loop system is globally uniformly asymptotically stable (GUAS) and the cost function has an upper bound $J^{*}$ .

Problem 2: Design a guaranteed controller in the form of (4) for the system given by (2) with input saturation such that the corresponding closed-loop system is uniformly asymptotically stable (UAS) and the cost function has an upper bound $J^{*}$ .

Problem 3: Design a guaranteed controller in the form of (4) for the system given by (2) with input saturation such that the corresponding closed-loop system is UAS and the upper bound is minimized.

Remark 1: It is worth mentioning that Problem 2 cannot be viewed as a special case of Problem 1. This is because, Problem 2 is more difficult to deal with than Problem 1 because of the existence of the input saturation constraint. The reason of considering both Problems 1 and 2 in this paper is to enlarge the potential application field of the proposed algorithms.

Before moving on, the following assumption and lemmas are introduced, which are crucial for the further development of our results.

Assumption 1: The parameter-varying vector $θ$ and its derivative $\overset{\cdot}{θ}$ are measurable in real-time.

Lemma 1: (Prajna et al., 2004) For a symmetric polynomial matrix $P (x)$ that is nonsingular for all $x$ , we have

\frac{\partial P}{\partial x_{i}} (x) = - P (x) \frac{\partial P^{- 1}}{\partial x_{i}} (x) P (x), i = 1, 2, \dots, n .

Lemma 2: (Jarvis-Wloszek, 2003) Given ${g_{i} (x)}_{i = 1}^{m} \in R [x]$ , if there exist ${s_{i} (x)}_{i = 1}^{m} \in Φ_{sos}$ such that

g_{0} (x) - \sum_{i = 1}^{m} s_{i} (x) g_{i} (x) \in Φ_{sos}

then

{x \in R^{n} | g_{1} (x) \geq 0, \dots, g_{m} (x) \geq 0} \subset {x \in R^{n} | g_{0} (x) \geq 0}

Lemma 3: (Du and Zhang, 2009) For saturation function $sat (u_{i})$ , as long as $| u_{i} | < \frac{{\bar{u}}_{i}}{α}$ , then

{(sat (u) - \frac{1 + α}{2} u)}^{T} (sat (u) - \frac{1 + α}{2} u) \leq {(\frac{1 - α}{2})}^{2} u^{T} u .

GCC via state feedback

This section provides a new approach to solving Problems 1–3. Firstly, we will present a method of designing a controller without input saturation. Then, we give the main results by taking input saturation into consideration.

State-feedback control without input saturation

Without considering the input saturation, this subsection is to get the solvable conditions for Problem 1.

Theorem 1: Consider system (1) satisfying Assumption 1. Given positive-definite symmetric matrices $Q$ , $R$ , and polynomial scalars $ε_{i} (\tilde{x}, θ) > 0 (i = 1, 2)$ , $ε_{3} (x, θ, \overset{\cdot}{θ}) > 0$ with $ε_{2} (\tilde{x}, θ) > ε_{1} (\tilde{x}, θ)$ , if there exists a symmetric polynomial matrix $P (\tilde{x}, θ) \in R^{n \times n}$ , a polynomial matrix $L (x, θ, \overset{\cdot}{θ}) \in R^{m \times n}$ , such that

z_{1}^{T} (P (\tilde{x}, θ) - ε_{1} (\tilde{x}, θ) I) z_{1} \in Φ_{sos}

(7)

z_{2}^{T} (ε_{2} (\tilde{x}, θ) I - P (\tilde{x}, θ)) z_{2} \in Φ_{sos}

(8)

\begin{matrix} - z_{3}^{T} (Ξ_{1} + ε_{3} (x, θ, \overset{\cdot}{θ}) I) z_{3} - \sum_{i = 1}^{r} σ_{i} (z_{3}, θ, \overset{\cdot}{θ}) f_{i} (θ) \\ - \sum_{i = 1}^{s} μ_{i} (z_{3}, θ) ({\overset{\cdot}{θ}}_{i} - {\underline{v}}_{i}) ({\bar{v}}_{i} - {\overset{\cdot}{θ}}_{i}) \in Φ_{sos} \end{matrix}

(9)

where $z_{i} (i = 1, 2, 3)$ are column vectors of proper dimensions, $σ_{i} (z_{3}, θ, \overset{\cdot}{θ})$ and $μ_{i} (z_{3}, θ)$ are SOS multipliers, and

Ξ_{1} = [\begin{matrix} Π_{1} & * & * \\ P (\tilde{x}, θ) & - Q^{- 1} & * \\ L (x, θ, \overset{\cdot}{θ}) & 0 & - R^{- 1} \end{matrix}],

\begin{matrix} Π_{1} = He (A (x, θ) P (\tilde{x}, θ) + B (x, θ) L (x, θ, \overset{\cdot}{θ})) \\ - \sum_{i \in H} \frac{\partial P}{\partial x_{i}} (\tilde{x}, θ) A_{i} (x, θ) x - \sum_{i = 1}^{s} \frac{\partial P}{\partial θ_{i}} (\tilde{x}, θ) {\overset{\cdot}{θ}}_{i} \end{matrix}

then Problem 1 is solvable, and the corresponding controller and an upper bound of the cost function in (6) being, respectively, given by

K (x, θ, \overset{\cdot}{θ}) = L (x, θ, \overset{\cdot}{θ}) P^{- 1} (\tilde{x}, θ)

(10)

and

J^{*} = x {(0)}^{T} P^{- 1} (\tilde{x} (0), θ (0)) x (0)

(11)

Proof: Define the following polynomial Lyapunov functional candidate

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

Next, we will demonstrate that the corresponding closed-loop system under the state feedback controller (4) is GUAS at the zero equilibrium and the cost function (6) has an upper bound $J^{*}$ .

According to the conditions in (7) and (8), we have

0 < ε_{1} (\tilde{x}, θ) I \leq P (\tilde{x}, θ) \leq ε_{2} (\tilde{x}, θ) I,

which means $P (\tilde{x}, θ)$ is invertible for $\tilde{x}$ and $θ$ , and

0 < ε_{2}^{- 1} (\tilde{x}, θ) I \leq P^{- 1} (\tilde{x}, θ) \leq ε_{1}^{- 1} (\tilde{x}, θ) I .

Therefore

0 < ε_{2}^{- 1} (\tilde{x}, θ) ‖ x ‖_{2}^{2} \leq V (x, θ) \leq ε_{1}^{- 1} (\tilde{x}, θ) ‖ x ‖_{2}^{2},

that is, $V (x, t)$ is positive definite and bounded, and $V (x, t) \to \infty$ as $‖ x ‖_{2} \to \infty$ .

By employing Lemma 2, the condition in (9) is equivalent to the following statement

Ξ_{1} + ε_{3} (x, θ, \overset{\cdot}{θ}) I \leq 0

(12)

for all $θ \in A_{θ}$ , $\overset{\cdot}{θ} \in A_{v}$ .

From (12), one has

Ξ_{1} = [\begin{matrix} Π_{1} & * & * \\ P (\tilde{x}, θ) & - Q^{- 1} & * \\ L (x, θ, \overset{\cdot}{θ}) & 0 & - R^{- 1} \end{matrix}] < 0

(13)

By applying the Schur Complement Lemma to (13), the matrix inequality (13) holds if and only if

\begin{matrix} Π_{1} < - {[\begin{matrix} P (\tilde{x}, θ) \\ L (x, θ, \overset{\cdot}{θ}) \end{matrix}]}^{T} [\begin{matrix} Q & 0 \\ 0 & R \end{matrix}] [\begin{matrix} P (\tilde{x}, θ) \\ L (x, θ, \overset{\cdot}{θ}) \end{matrix}] \\ = - (P (\tilde{x}, θ) QP (\tilde{x}, θ) + L^{T} (x, θ, \overset{\cdot}{θ}) RL (x, θ, \overset{\cdot}{θ})) \end{matrix}

(14)

Pre- and post- multiplying (14) by $P^{- 1} (\tilde{x}, θ)$ , based on Lemma 1 and denoting $K (x, θ, \overset{\cdot}{θ}) = L (x, θ, \overset{\cdot}{θ}) P^{- 1} (\tilde{x}, θ)$ , it is easy to conclude that

\begin{matrix} He (P^{- 1} (\tilde{x}, θ) A (x, θ) + P^{- 1} (\tilde{x}, θ) B (x, θ) K (x, θ, \overset{\cdot}{θ})) \\ + \sum_{i \in H} \frac{\partial P^{- 1}}{\partial x_{i}} (\tilde{x}, θ) A_{i} (x, θ) x \\ + \sum_{i = 1}^{s} \frac{\partial P^{- 1}}{\partial θ_{i}} (\tilde{x}, θ) {\overset{\cdot}{θ}}_{i} + Q + K^{T} (x, θ, \overset{\cdot}{θ}) RK (x, θ, \overset{\cdot}{θ}) \leq 0 . \end{matrix}

The above inequality implies

\begin{matrix} \overset{\cdot}{V} (x, t) = {\overset{\cdot}{x}}^{T} P^{- 1} (\tilde{x}, θ) x + x^{T} {\overset{\cdot}{P}}^{- 1} (\tilde{x}, θ) x + x^{T} P^{- 1} (\tilde{x}, θ) \overset{\cdot}{x} \\ = x^{T} {He (P^{- 1} (\tilde{x}, θ) A (x, θ) + P^{- 1} (\tilde{x}, θ) B (x, θ) K (x, θ, \overset{\cdot}{θ})) \\ + \sum_{i \in H} \frac{\partial P^{- 1}}{\partial x_{i}} (\tilde{x}, θ) A_{i} (x, θ) x + \sum_{i = 1}^{s} \frac{\partial P^{- 1}}{\partial θ_{i}} (\tilde{x}, θ) {\overset{\cdot}{θ}}_{i}} x \\ \leq - x^{T} (Q + K^{T} (x, θ, \overset{\cdot}{θ}) RK (x, θ, \overset{\cdot}{θ})) x < 0 . \end{matrix}

(15)

By (15), it is clear that (9) implies $\overset{\cdot}{V} (x, t) < 0$ . Thus, the corresponding closed-loop system is GUAS at the zero equilibrium.

Furthermore, integrating inequality (15) from 0 to infinity yields

\begin{matrix} J = \int_{0}^{\infty} (x^{T} Qx + u^{T} Ru) dt & \leq - \int_{0}^{\infty} \overset{\cdot}{V} (x, t) dt = V (x (0), θ (0)) \\ = x {(0)}^{T} P^{- 1} (\tilde{x} (0), θ (0)) x (0) = J^{*} . \end{matrix}

(16)

Therefore, the cost function has an upper bound $J^{*}$ . This completes the proof.

Remark 2: Observe that the matrix $P (\tilde{x}, θ)$ is polynomial in $\tilde{x}$ and $θ$ , which brings flexibility in constructing the Lyapunov functional and the controller. $P (\tilde{x}, θ)$ can reinforce the success probability than that of $P (θ)$ in dealing with the synthesis problem. Moreover, it has shown beneficial effects in improving the system performance while $P (\tilde{x}, θ)$ also can lead to a higher-order Lyapunov functional. It is worth noting that the form of $P (\tilde{x}, θ)$ is employed, rather than $P (θ)$ , is to avoid causing the non-convexity of the solvable conditions. In fact, a similar structure can be found in Prajna et al. (2004) and Fu et al. (2018a).

Remark 3: It is noteworthy that the polynomial positive scalars $ε_{1} (\tilde{x}, θ)$ , $ε_{2} (\tilde{x}, θ)$ and $ε_{3} (x, θ, \overset{\cdot}{θ})$ are given from an intuitive sense. In other words, if the controlled system and the controller are time-varying and nonlinear, the scalars should be nonlinear and time-varying as well. Thus, $ε_{1} (\tilde{x}, θ)$ , $ε_{2} (\tilde{x}, θ)$ and $ε_{3} (x, θ, \overset{\cdot}{θ})$ are designed as polynomials in $x$ , $\tilde{x}$ , $θ$ and $\overset{\cdot}{θ}$ , which can add useful flexibility in solving SPDLMIs. Furthermore, when $ε_{1} (\tilde{x}, θ)$ , $ε_{2} (\tilde{x}, θ)$ and $ε_{3} (x, θ, \overset{\cdot}{θ})$ are reduced to positive constants, we can easily draw the conclusion that the closed-loop system in (5) is globally uniformly exponentially stable at the origin. In fact, the analogous proof method for exponentially stable of NPV system can be found in Fu et al. (2018a).

Remark 4: The researches concerning LPV (e.g. Hu et al., 2018; Nguyen et al., 2018), often need to verify the parameter-dependent solvable condition over the entire space of parameter or use a gridding method. On the other hand, the convex covering method can also be adopted to address the parameter-dependent condition. However, the solution obtained from the above methods are conservative. With the aid of SOS technique, we can incorporate the parameter and its derivative into the solvable conditions. Our method faithfully reflects the time-varying nature and the nonlinear dynamics of NPV systems, and leads to less conservative results.

State-feedback control with input saturation

Taking input saturation into consideration, (2) is equivalent to

\begin{matrix} \overset{\cdot}{x} = A (x, θ) x + B (x, θ) sat (u) \\ = A (x, θ) x + \frac{1 + α}{2} B (x, θ) u + B (x, θ) (sat (u) - \frac{1 + α}{2} u), \end{matrix}

(17)

where $0 < α < 1$ .

Substituting the state feedback controller in (4) into (17) yields

\begin{matrix} \overset{\cdot}{x} & = (A (x, θ) + \frac{1 + α}{2} B (x, θ) K (x, θ, \overset{\cdot}{θ})) x \\ + B (x, θ) (sat (u) - \frac{1 + α}{2} u) . \end{matrix}

(18)

The following theorem presents a sufficient condition to solve Problem 2.

Theorem 2: Consider system (2) and assume Assumption 1 holds. Given positive-definite symmetric matrices $Q$ , $R$ , and scalars $λ > 0$ and $0 < α < 1$ , polynomial scalars $ε_{i} (\tilde{x}, θ) > 0 (i = 1, 2)$ , $ε_{3} (x, θ, \overset{\cdot}{θ}) > 0$ with $ε_{2} (\tilde{x}, θ) > ε_{1} (\tilde{x}, θ)$ , if there exists a symmetric polynomial matrix $P (\tilde{x}, θ) \in R^{n \times n}$ , a polynomial matrix $L (x, θ, \overset{\cdot}{θ}) \in R^{m \times n}$ , such that the conditions in (7), (8) and the following conditions

\begin{matrix} - z_{3}^{T} (Ξ_{2} + ε_{3} (x, θ, \overset{\cdot}{θ}) I) z_{3} - \sum_{i = 1}^{r} σ_{i} (z_{3}, θ, \overset{\cdot}{θ}) f_{i} (θ) \\ - \sum_{i = 1}^{s} μ_{i} (z_{3}, θ) ({\overset{\cdot}{θ}}_{i} - {\underline{v}}_{i}) ({\bar{v}}_{i} - {\overset{\cdot}{θ}}_{i}) \in Φ_{sos} \end{matrix}

(19)

and

z_{4}^{T} [\begin{matrix} {(α^{- 1} {\bar{u}}_{k})}^{2} I & * \\ λ L_{k}^{T} (x, θ, \overset{\cdot}{θ}) & λ P (\tilde{x}, θ) \end{matrix}] z_{4} \in Φ_{sos}

(20)

hold, where $z_{i} (i = 1, 2, 3, 4)$ are column vectors of proper dimensions, $σ_{i} (z_{3}, θ, \overset{\cdot}{θ})$ and $μ_{i} (z_{3}, θ)$ are SOS multipliers, and

\begin{matrix} Ξ_{2} = [\begin{matrix} Π_{2} & * & * & * & * \\ B^{T} (x, θ) & - τ I & * & * & * \\ P (\tilde{x}, θ) & 0 & - Q^{- 1} & * & * \\ L (x, θ, \overset{\cdot}{θ}) & 0 & 0 & - R^{- 1} & * \\ τ L (x, θ, \overset{\cdot}{θ}) & 0 & 0 & 0 & - τ {(2 {(1 - α)}^{- 1})}^{2} I \end{matrix}], \\ Π_{2} = He (A (x, θ) P (\tilde{x}, θ) + \frac{1 + α}{2} B (x, θ) L (x, θ, \overset{\cdot}{θ})) \\ - \sum_{i \in H} \frac{\partial P}{\partial x_{i}} (\tilde{x}, θ) A_{i} (x, θ) x - \sum_{i = 1}^{s} \frac{\partial P}{\partial θ_{i}} (\tilde{x}, θ) {\overset{\cdot}{θ}}_{i}, \end{matrix}

then Problem 2 is solvable with the corresponding controller and an upper bound being given by (10) and (11), and the controller in (4) is designed for the domain $Ω_{1} = {x | x^{T} P^{- 1} (\tilde{x}, θ) x < λ}$ .

Proof: Define the following polynomial Lyapunov functional candidate

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

On the basis of Theorem 1, the conditions in (7) and (8) can ensure that $V (x, t)$ is positive definite and bounded, and $V (x, t) \to \infty$ as $‖ x ‖_{2} \to \infty$ .

The time derivative of $V (x, t)$ along the trajectory of the closed-loop system in (18) is given by

\begin{matrix} \overset{\cdot}{V} (x, t) + \overset{\cdot}{J} & = {\overset{\cdot}{x}}^{T} P^{- 1} (\tilde{x}, θ) x + x^{T} {\overset{\cdot}{P}}^{- 1} (\tilde{x}, θ) x \\ + x^{T} P^{- 1} (\tilde{x}, θ) \overset{\cdot}{x} + x^{T} Qx + u^{T} Ru \\ = x^{T} P^{- 1} (\tilde{x}, θ) Π_{3} P^{- 1} (\tilde{x}, θ) x \\ + x^{T} K^{T} (x, θ, \overset{\cdot}{θ}) RK (x, θ, \overset{\cdot}{θ}) x + x^{T} Qx \\ + x^{T} P^{- 1} (\tilde{x}, θ) B (x, θ) (sat (u) - \frac{1 + α}{2} u) \\ + {(sat (u) - \frac{1 + α}{2} u)}^{T} B^{T} (x, θ) P^{- 1} (\tilde{x}, θ) x \\ = {[\begin{matrix} P^{- 1} (\tilde{x}, θ) x \\ sat (u) - \frac{1 + α}{2} u \end{matrix}]}^{T} Ξ_{3} [\begin{matrix} P^{- 1} (\tilde{x}, θ) x \\ sat (u) - \frac{1 + α}{2} u \end{matrix}] \end{matrix}

(21)

where

\begin{matrix} Ξ_{3} = & [\begin{matrix} Π_{4} & * \\ B^{T} (x, θ) & 0 \end{matrix}], Π_{4} = Π_{3} \\ + P (\tilde{x}, θ) (K^{T} (x, θ, \overset{\cdot}{θ}) RK (x, θ, \overset{\cdot}{θ}) + Q) P (\tilde{x}, θ), \end{matrix}

\begin{matrix} Π_{3} = & He ((A (x, θ) + \frac{1 + α}{2} B (x, θ) K (x, θ, \overset{\cdot}{θ})) P (\tilde{x}, θ)) \\ - \sum_{i \in H} \frac{\partial P}{\partial x_{i}} (\tilde{x}, θ) A_{i} (x, θ) x - \sum_{i = 1}^{s} \frac{\partial P}{\partial θ_{i}} (\tilde{x}, θ) {\overset{\cdot}{θ}}_{i} . \end{matrix}

The matrix $Ξ_{3}$ in (21) has a zero diagonal element. Thus, the negative definiteness of $Ξ_{3}$ cannot be guaranteed. To solve this problem, we apply Lemma 3. Substituting (3) into Lemma 3 results in

\begin{matrix} {(sat (u) - \frac{1 + α}{2} u)}^{T} (sat (u) - \frac{1 + α}{2} u) \\ \leq {(\frac{1 - α}{2})}^{2} x^{T} K^{T} (x, θ, \overset{\cdot}{θ}) K (x, θ, \overset{\cdot}{θ}) x \end{matrix}

The above inequality is equivalent to

{[\begin{matrix} P^{- 1} (\tilde{x}, θ) x \\ sat (u) - \frac{1 + α}{2} u \end{matrix}]}^{T} Ξ_{4} [\begin{matrix} P^{- 1} (\tilde{x}, θ) x \\ sat (u) - \frac{1 + α}{2} u \end{matrix}] \geq 0

(22)

where

\begin{matrix} Ξ_{4} = [\begin{matrix} Π_{5} & 0 \\ 0 & - I \end{matrix}], \\ Π_{5} = {(\frac{1 - α}{2})}^{2} P (\tilde{x}, θ) K^{T} (x, θ, \overset{\cdot}{θ}) K (x, θ, \overset{\cdot}{θ}) P (\tilde{x}, θ) \end{matrix}

Applying the S-procedure, there exists positive scalar $τ$ , a sufficient condition in (21) and (22) is given by

[\begin{matrix} Π_{4} + τ Π_{5} & B (x, θ) \\ B^{T} (x, θ) & - τ I \end{matrix}] < 0

(23)

By letting $K (x, θ, \overset{\cdot}{θ}) = L (x, θ, \overset{\cdot}{θ}) P^{- 1} (\tilde{x}, θ)$ in (23) and then employing the Schur Complement Lemma, one has

Ξ_{2} = [\begin{matrix} Π_{2} & * & * & * & * \\ B^{T} (x, θ) & - τ I & * & * & * \\ P (\tilde{x}, θ) & 0 & - Q^{- 1} & * & * \\ L (x, θ, \overset{\cdot}{θ}) & 0 & 0 & - R^{- 1} & * \\ τ L (x, θ, \overset{\cdot}{θ}) & 0 & 0 & 0 & - τ {(2 {(1 - α)}^{- 1})}^{2} I \end{matrix}] < 0

(24)

It can be easily concluded that the negative definiteness of (24) is enforced if condition (19) is satisfied. Thus

\overset{\cdot}{V} (x, t) \leq - (x^{T} Qx + u^{T} Ru) < 0

Furthermore, integrating the above inequality from 0 to infinity, we can obtain that the cost function has an upper bound.

It is worth noting that Lemma 3 always holds in the following domain

Ω_{2} = {x | | u_{k} | < \frac{{\bar{u}}_{k}}{α}, k = 1, \dots, m}

(25)

Hence, sufficient conditions for satisfying the domain must be derived. Accounting for the set inverse analysis put forwarded by Vafamand et al. (2016), the condition that ensures the ellipsoid $Ω_{1} = {x | x^{T} P^{- 1} (\tilde{x}, θ) x < λ}$ being inside the domain $Ω_{2}$ , i.e. $Ω_{1} \subset Ω_{2}$ , is given by

\begin{matrix} L_{k} (x, θ, \overset{\cdot}{θ}) P^{- 1} (\tilde{x}, θ) {(\frac{P^{- 1} (\tilde{x}, θ)}{λ})}^{- 1} P^{- 1} (\tilde{x}, θ) L_{k}^{T} (x, θ, \overset{\cdot}{θ}) \\ \leq {(\frac{{\bar{u}}_{k}}{α})}^{2}, k = 1, \dots, m \end{matrix}

(26)

where $L_{k} (x, θ, \overset{\cdot}{θ})$ is the $k$ th row of $L (x, θ, \overset{\cdot}{θ})$ . By using the Schur complement Lemma on (26), the SOS decomposition (20) can be obtained. Thus, the Problem 2 is solvable. This completes the proof.□

Remark 5: It is worth mentioning that, compared with the form of controller in the existing literatures (e.g. Muksidani, 2005; Vafamand et al., 2017; Zhao et al., 2014) (which are constant matrix or state-dependent matrix), the form of the controller in Theorems 1 and 2 is a SDP matrix along with its derivative $\overset{\cdot}{θ}$ . It is obtained from an intuitive idea, if the controlled system is time-varying, the corresponding controller should also be time-varying. Fortunately, the SOS technique has a property that avails the construction of matrices to combine the state, the time-varying parameter and its derivative, without ignoring the nonlinearities or hiding the time-varying nature. The state and time-varying parameter can be treated as free variables, for which SOS technique is introduced to provide a validity method. Therefore, a gap between the time-varying parameter and state of the NPV system has been eliminated from the point of computation. The analogous method can be found in Fu et al. (2018a).

Remark 6: The derivative of parameter $\overset{\cdot}{θ}$ has been taken for a SOS variable in conditions (9) and (19). This will avoid calculating vertex points of $A_{v}$ . Different from the existing conditions (e.g. Hu et al., 2018; Nguyen et al., 2018) in terms of LMIs that depend on parameter, for a given NPV system, the computational cost of SOS-based method is fixed. Less conservatism relaxation and computationally tractable conditions are given in Theorem 1 and Theorem 2.

To address Problem 3, we will transform the SOS feasibility problem into the following optimization problem.

Theorem 3: Consider system (2) and assume Assumption 1 holds. If the following optimization problem

min_{L (x, θ, \overset{\cdot}{θ}), P (\tilde{x}, θ)} J_{1}

s. t. (7), (8), (19), (20)

z_{5}^{T} [\begin{matrix} J_{1} & * \\ x (0) & P (\tilde{x}, θ) \end{matrix}] z_{5} \in Φ_{sos}

(27)

and

λ - J_{1} \in Φ_{sos}

(28)

has a solution, then Problem 3 is solvable and the form of controller is given in (10).

Proof: Similarly, the conditions (7)–(8) and (19)–(20) hold by Theorem 2.

In order to minimize the upper bound of cost function, consider that

x {(0)}^{T} P^{- 1} (\tilde{x} (0), θ (0)) x (0) \leq J_{1}

(29)

Therefore, condition (29) is equivalent to

[\begin{matrix} J_{1} & x {(0)}^{T} \\ x (0) & P (\tilde{x} (0), θ (0)) \end{matrix}] \geq 0

(30)

Since the unknown SOS variable $P (\tilde{x}, θ)$ is not obviously appeared in (30), the condition (30) is not a modality of the common SOS decomposition. Nevertheless, if the condition (27) has a feasible solution, it is true for all $t$ , especially when $t = 0$ . Consequently, (30) is indicated by the SOS condition (27). To guarantee that the system’s initial condition $x (0)$ comprises the region $Ω_{1}$ . Substituting $x (0)$ in $Ω_{1}$ , we have

x^{T} (0) P^{- 1} (\tilde{x} (0), θ (0)) x (0) < λ

(31)

By comparing (29) with (31), we infer that if (31) has a feasible solution for $J_{1} \leq λ$ . Hence, the condition (28) is satisfied. This completes the proof.

Interestingly, when $A (x, θ)$ and $B (x, θ)$ in (1) become nonlinear time-invariant matrices, the NPV system in (1) is reduced to the state-dependent nonlinear system (e.g. Prajna et al., 2004)

\overset{\cdot}{x} = A (x) x + B (x) u

(32)

and when we remove the condition given by (8) and the dependence of $P (\tilde{x}, θ)$ and $L (x, θ, \overset{\cdot}{θ})$ on $θ$ and $\overset{\cdot}{θ}$ , Theorem 3 is reduced to Theorem 1 in Vafamand et al. (2017). For clarity, the above statement can be summarized in the following Corollary.

Corollary 1: Consider system in (32) with input saturation, given positive-definite symmetric matrices $Q$ , $R$ , and scalars $λ > 0$ and $0 < α < 1$ , polynomial scalars $ε_{1} (\tilde{x}) > 0$ and $ε_{2} (x) > 0$ , if there exists a symmetric polynomial matrix $P (\tilde{x}) \in R^{n \times n}$ , a polynomial matrix $L (x) \in R^{m \times n}$ , if the following optimization problem

min_{L (x), P (\tilde{x})} J_{1}

s . t . z_{1}^{T} (P (\tilde{x}) - ε_{1} (\tilde{x}) I) z_{1} \in Φ_{sos}

z_{2}^{T} [\begin{matrix} J_{1} & * \\ x (0) & P (\tilde{x}) \end{matrix}] z_{2} \in Φ_{sos}

- z_{3}^{T} (Ξ_{5} + ε_{3} (x) I) z_{3} \in Φ_{sos}

z_{4}^{T} [\begin{matrix} {(α^{- 1} {\bar{u}}_{k})}^{2} I & * \\ λ L_{k}^{T} (x) & λ P (\tilde{x}) \end{matrix}] z_{4} \in Φ_{sos}

λ - J_{1} \in Φ_{sos}

has a solution, then corresponding optimization problem is solvable. The controller is designed in the domain $Ω_{3} = {x | x^{T} P^{- 1} (\tilde{x}) x < λ}$ and the form of controller is given by

$K (x) = L (x) P^{- 1} (\tilde{x})$ ,

where $z_{i} (i = 1, 2, 3, 4)$ are column vectors of proper dimensions, and

\begin{matrix} Ξ_{5} = [\begin{matrix} Π_{6} & * & * & * & * \\ B^{T} (x) & - τ I & * & * & * \\ P (\tilde{x}) & 0 & - Q^{- 1} & * & * \\ L (x) & 0 & 0 & - R^{- 1} & * \\ τ L (x) & 0 & 0 & 0 & - τ {(2 {(1 - α)}^{- 1})}^{2} I \end{matrix}], \\ Π_{6} = He (A (x) P (\tilde{x}) + \frac{1 + α}{2} B (x) L (x)) - \sum_{i \in H} \frac{\partial P}{\partial x_{i}} (\tilde{x}) A_{i} (x) x_{i} . \end{matrix}

Remark 7: Corollary 1 includes the results of Theorem 1 in Vafamand et al. (2017) as special cases. Different from the condition in Vafamand et al. (2017), the SDPLMIs are related to the time-varying parameter and its derivative. Thus, the solvable condition in Theorem 3 can be treated as the time-varying version of the method proposed in Vafamand et al. (2017) and also can be regarded as an extension of the polynomial nonlinear time-invariant system to the NPV system. In particular, the NPV system is very common because it can represent the dynamics of a large number of real-world plants (e.g. Fu et al., 2018a,b). Thus, it is meaningful to investigate the control problems for the NPV system.

Analogously, when the state gain matrices are independent of the state, the system in (1) can be reduced into an LPV system

\overset{\cdot}{x} = A (θ) x + B (θ) u

(33)

Next, we will consider the GCC of LPV system subject to input saturation. The following corollary will be given.

Corollary 2: When we remove the dependence of $P (\tilde{x}, θ)$ , $L (x, θ, \overset{\cdot}{θ})$ , $ε_{i} (\tilde{x}, θ)$ , $i = 1, 2$ and $ε_{3} (x, θ, \overset{\cdot}{θ})$ on $x$ and $\tilde{x}$ in Theorem 3, if the corresponding optimization problem has a solution, then the LPV system in (33) with input saturation is asymptotically stable, and the controller is given by

K (θ, \overset{\cdot}{θ}) = L (θ, \overset{\cdot}{θ}) P^{- 1} (θ)

Remark 8: A systematic method for designing a guaranteed cost controller for LPV system with input saturation based on the SOS technique is proposed in Corollary 2. The algorithm can be easily applied to the polynomial system and the controller is obtained straightforwardly by the semi-definite programming. Different from Zhao et al. (2014), the condition for LPV system with input saturation is formulated in terms of parameterized LMIs, and they are converted into solvability LMIs by applying an efficient parameter relaxation method. Thus, the controller gain $K (θ, \overset{\cdot}{θ})$ and $P (θ)$ become constant matrices that may bring conservatism. It further illustrates that the forms of $K (x, θ, \overset{\cdot}{θ})$ and $P (\tilde{x}, θ)$ have shown beneficial effects in adding flexibility.

Simulations

Consider the error model of tilt rotor aircraft, whose longitudinal dynamics equations can be characterized as (e.g. Fu et al., 2018a)

\overset{\cdot}{x} = [\begin{matrix} 0 & - m^{- 1} F_{yt *} & 0 & 0 & 0 \\ A_{21} & A_{22} & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ A_{51} & - V_{*} & V_{*} & 0 & 0 \end{matrix}] x + [\begin{matrix} 1 / m & B_{12} & 0 \\ B_{21} & B_{22} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 / I_{z} \\ 0 & 0 & 0 \end{matrix}] u

(34)

A_{21} = - m^{- 1} (F_{xt}^{*} α^{*} + F_{yt}^{*}) (α \cdot Δ V + 2 α V^{*} + b), A_{51} = Δ ϑ - Δ α,

\begin{matrix} A_{22} = - m^{- 1} F_{xt}^{*} (a {(V^{*} + Δ V)}^{2} + b (V^{*} + Δ V) + c), \\ B_{12} = - (α_{*} + Δ α) / m, \end{matrix}

\begin{matrix} B_{21} = - m^{- 1} (α^{*} + Δ α) (a {(V^{*} + Δ V)}^{2} + b (V^{*} + Δ V) + c), \\ B_{22} = - m^{- 1} (a {(V^{*} + Δ V)}^{2} + b (V^{*} + Δ V) + c) . \end{matrix}

a = 8.791 \times 10^{- 4}, b = - 0.03274, c = 0.3491

where $x = {[\begin{matrix} Δ V & Δ α & Δ ϑ & Δ q & Δ H \end{matrix}]}^{T}$ is the state, $Δ V$ , $Δ α$ , $Δ ϑ$ , $Δ q$ and $Δ H$ are the error of velocity, the angle of attack, the pitch angle, the pitch rate and the altitude, respectively. $u = {[\begin{matrix} Δ F_{xt} & Δ F_{yt} & Δ M_{z} \end{matrix}]}^{T}$ is the control input, $Δ F_{xt}$ , $Δ F_{yt}$ and $Δ M_{z}$ are components of the error of force along the body $x$ and $y$ axes and the error of pitching moment, respectively. $V^{*}$ , $α^{*}$ , $ϑ^{*}$ , $q^{*}$ and $H^{*}$ are the desired reference trajectory. $F_{xt}^{*}$ , $F_{yt}^{*}$ and $M_{z}^{*}$ are the reference control input. $m$ and $I_{z}$ are the mass of the aircraft and the moment of inertia about the pitch axis.

The initial state in simulation is given by $x_{0} = {[\begin{matrix} 1 & 0.01 & 0.01 & 0.01 & 0.1 \end{matrix}]}^{T}$ . We can derive the guaranteed cost controller (it is represented by “Our approach”) by the application of Theorem 1 and Theorem 3. To show the advantages of the proposed approach based on the NPV systems, we also linearize the model in (34) along the same generalized conversion corridor and obtain an LPV system by polynomial fitting. And based on the LPV system, we also can obtain the guaranteed cost controller (it is represented by “LPV approach”) with the same parameters.

The performance analysis for NPV systems without input saturation

The parameters of simulation are chosen as $ε_{1} = 1 \times 10^{- 4}$ , $ε_{2} = 1 \times 10^{3}$ , $ε_{3} = 1 \times 10^{- 3}$ , $α = 0.3$ , $Q = 1 \times 10^{3}$ , $R = 1 \times 10^{- 1}$ , $m = 2 kg$ , $I_{z} = 0.0514 kg \cdot m^{2}$ . The guaranteed cost controller for NPV system and LPV system without saturation can be obtained by the application of Theorem 1.

The performance analysis for NPV systems with input saturation

The parameters of simulation are chosen as $ε_{1} = 1 \times 10^{- 3}$ , $ε_{2} = 1 \times 10^{3}$ , $ε_{3} = 1 \times 10^{- 3}$ , $λ = 1 \times 10^{4}$ , $α = 0.1$ , ${\bar{u}}_{1} = 100 N$ , ${\bar{u}}_{2} = 50 N$ , ${\bar{u}}_{3} = 20 N . m$ , $Q = 1 \times 10^{3}$ , $R = 1 \times 10^{- 1}$ , $m = 2 kg$ , $I_{z} = 0.0514 kg \cdot m^{2}$ . We can minimize the guaranteed cost function and derive the SOS-based state feedback controller by the application of Theorem 3. Furthermore, the cost function upper bound of the guaranteed cost controller in our approach and LPV approach is 0.13937 and 3.6531, which illustrates that our approach achieves a less conservation cost function bound than that acquired in the LPV system.

It can be observed from Figures 1 –5 and Figures 9 –13 that the corresponding closed-loop system is stable and all states converge to the desived values under both controllers (with and without input saturation). However, there are some extent tracking errors for the LPV controller during the conversion process. Therefore, the proposed approach has been shown to be superior to the LPV controller. Furthmore, Figures 6 –8 and Figures 14 –16 provide the control input profile for the guaranteed cost controller (with and without input saturation). As shown by Figures 6 –8 and Figures 14 –16, we can see that the control effort required by the LPV controller is much bigger than that by the proposed approach. Clearly, the proposed approach outperforms the LPV controller in the terms of convergence and control effort.

Conclusions

The GCC problem of nonlinear systems with time-varying parameters and input saturation has been addressed in this paper. Based on the Lyapunov stability theory, the sufficient SPDLMIs conditions are given in such way that they can be reformulated as SOS programming problems to get a feasible solution by the SOS technique. Simulation results imply that the proposed method is more effective than the published methods in the literature by the example of a tilt rotor aircraft. Interesting future work include, but are not limited to, (i) extending this work to a class of time-delay nonlinear systems and (ii) designing an output feedback controller to stabilize the NPV system.

Footnotes

Author’s note

Jingyao Wang is also affiliated with the Shenzhen Research Institute of Xiamen University, China.

Declaration of conflicting interests

The author(s) declared no potential conflict of interests 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 work is supported by the National Natural Science Foundation of China (grant numbers U1713223 and 61803319), the Natural Science Foundation of Fujian Province of China (grant numbers 2019J05021), Shenzhen Science and Technology Projects (JCYJ20180306172720364) and the Fundamental Research Funds for the Central Universities of China (20720190015).

ORCID iD

Pingfang Zhu

References

Chang

Peng

(1972) Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Transactions on Automatic Control 17(4): 474–483.

Chen

Guan

(2004) Delay-dependent output feedback guaranteed cost control for uncertain time-delay systems. Automatica 40(7): 1263–1268.

Coutinho

Trofino

(2002) Guaranteed cost control of uncertain nonlinear systems via polynomial Lyapunov functions. IEEE Transactions on Automatic Control 47(9): 1575–1580.

Zhang

(2009) Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint. IEEE Transactions on Fuzzy Systems 17(2): 343–356.

Krstic

, et al. (2016) Robust dynamic positioning of ships with disturbances under input saturation. Automatica 73: 207–214.

Guo

Luo

Wang

, et al. (2019) An adaptive cascade trajectory tracking control for over-actuated autonomous electric vehicles with input saturation. Science China Technological Science 62: 2153–2160.

Sun

Zeng

(2018a) Exponential stabilisation of nonlinear parameter-varying systems with applications to conversion flight control of a tilt rotor aircraft. International Journal of Control 92(11): 2473–2483.

Zeng

Duan

(2018b)

H_{\infty}

mixed stabilization of nonlinear parameter-varying systems. International Journal of Robust and Nonlinear Control 28: 5232–5246.

(2016) Vibration control of flexible marine riser systems with input saturation. IEEE/ASME Transactions on Mechatronics 21(1): 254–265.

10.

Teel

Zaccarian

(2008) Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance. Automatica 44(2): 512–519.

11.

Duan

Tan

(2018). Control of LPV systems subject to state constraints and input saturation. Transactions of the Institute of Measurement and Control 40(14): 3985–3993.

12.

Huang

(2018) Optimal guaranteed cost control of uncertain non-linear systems using adaptive dynamic programming with concurrent learning. IET Control Theory & Applications 12(8): 1025–1035.

13.

Jarvis-Wloszek

(2003) Lyapunov based analysis and controller synthesis for polynomial systems using sum-of-squares optimization. PhD Thesis, University of California, Berkeley, CA.

14.

(2018) Adaptive tracking control for nonlinear time-varying delay systems with full state constraints and unknown control coefficients. Automatica 93: 444–453.

15.

Wang

Liu

, et al. (2017) A note on guaranteed cost control for nonlinear stochastic systems with input saturation and mixed time-delay. International Journal of Robust & Nonlinear Control 27(3): 4443–4456.

16.

Mazenc

Malisoff

(2017) Stabilization of nonlinear time-varying systems through a new prediction based approach. IEEE Transactions on Automatic Control 62(6): 2908–2915.

17.

Muksidani

(2005) A new approach to robust guaranteed cost control for uncertain multimodeling systems. Automatica 41: 1055–1062.

18.

Nguyen

Chevrel

Claveau

(2018) Gain-scheduled static output feedback control for saturated LPV systems with bounded parameter variations. Automatica 89: 420–424.

19.

Papachristodoulou

Anderson

Valmorbida

, et al. (2016) SOSTOOLS: Sum of squares optimisation toolbox for MATLAB (version 3.01).

20.

Petersen

Mcfarlane

(1994) Optimal guaranteed cost control and filtering for uncertain linear systems. IEEE Transactions on Automatic Control 39(9): 1971–1977.

21.

Prajna

Papachristodoulou

(2004) Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach. In: The 5th Asian Control Conference, Melbourne, Australia, 20–23 July 2004, pp. 157–165.

22.

Reznick

(2000) Some concrete aspects of Hilbert’s 17th problem. Contemporary Mathematics, 253: 251–272.

23.

Rugh

Shamma

(2000) Research on gain scheduling. Automatica 36(10): 1401–1425.

24.

Tee

K P

Ren

(2011) Control of nonlinear systems with time-varying output constraints. Automatica 47(11): 2511–2516.

25.

Vafamand

Asemani

Khayatiyan

(2016) A robust L1 controller design for continuous-time TS systems with persistent bounded disturbance and actuator saturation. Engineering Applications of Artificial Intelligence 56: 212–221.

26.

Vafamand

Mardani

Khayatian

, et al. (2017) Non-iterative SOS-based approach for guaranteed cost control design of polynomial systems with input saturation. IET Control Theory & Applications 11(16): 2724–2730.

27.

Wang

DWC

Dong

, et al. (2010) Robust

H_{\infty}

finite-horizon control for a class of stochastic nonlinear time-varying systems subject to sensor and actuator saturations. IEEE Transactions on Automatic Control 55(7): 1716–1722.

28.

Xia

Huo

(2016) Robust adaptive backstepping neural networks control for spacecraft rendezvous and docking with input saturation. ISA Transactions 62(3): 249–257.

29.

Nian

Wang

, et al. (2017) Robust guaranteed cost tracking control of quadrotor UAV with uncertainties. ISA Transactions 69: 157–165.

30.

(2000) Optimal guaranteed cost control of linear uncertain system: an LMI approach. Control Theory & Applications 17(3): 423–428.

31.

Zhao

Yang

Lei

(2014)

H_{\infty}

guaranteed cost control for LPV systems subject to the gain constraint. Asian Journal of Control 16(2): 609–616.

32.

Zhao

Liu

Luo

(2017) Output feedback boundary control of an axially moving system with input saturation constraint. ISA Transactions 68: 22–32.