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Sampled-Data Output-Feedback Fuzzy Controller for Nonlinear Systems Based on Polynomial Fuzzy Model-Based Control Approach

  • Hak-Keung LamEmail author
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Part of the Studies in Systems, Decision and Control book series (SSDC, volume 64)

Abstract

This chapter considers a sampled-data output-feedback polynomial fuzzy model-based control system which is formed by a nonlinear plant represented by the polynomial fuzzy model and a sampled-data output-feedback polynomial fuzzy controller connected in a closed loop. SOS-based stability analysis considering the effect due to sampling and zero-order-hold activities is performed using the input-delay method. SOS-based stability conditions are obtained to determine the system stability and synthesize the controller. A simulation example is presented to demonstrate the design procedure and the results show that the sampled-data output-feedback polynomial fuzzy controller can be designed to stabilize a nonlinear system using the obtained SOS-based stability conditions.

Keywords

Polynomial Fuzzy Model Fuzzy Controller Sampled-data Output Feedback Input Delay Approach Stability Analysis Results 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

8.1 Introduction

Sampled-data control is welcome by a lot of control applications due to the control strategy can be implemented efficiently and flexibly by microcontrollers or digital computers, which are available at low cost nowadays.

A sampled-data fuzzy control system is shown in Fig. 8.1 which consists of a nonlinear plant, a sampler, a discrete-time fuzzy controller and a zero-order-hold (ZOH). The nonlinear plant is the system to be controlled which will be represented by a polynomial fuzzy model. A discrete-time fuzzy controller which can be implemented by microcontrollers or digital computers is employed to control the nonlinear plant. The system state vector \(\mathbf {x}(t)\) (when full-state feedback is considered) or output vector \(\mathbf {y}(t)\) (when output feedback is considered) will be captured by the sampler with sampling period of T seconds to produce the sampled system state vector \(\mathbf {x}(KT)\) or output vector \(\mathbf {y}(KT)\), \(K = 1, 2, \ldots , \infty \). The discrete-time controller then processes on the sampled system state vector \(\mathbf {x}(KT)\) or output vector \(\mathbf {y}(KT)\) to produce the control signal \(\mathbf {u}(KT)\) at time instant KT. A ZOH is then employed to hold the control signal \(\mathbf {u}(KT)\) during the sampling period to produce the control signal \(\mathbf {u}(t)\) for the control of the nonlinear plant, i.e., \(\mathbf {u}(t) = \mathbf {u}(KT)\) during the sampling period.
Fig. 8.1

A block diagram of sampled-data polynomial fuzzy model-based control system

Because of the existence of the sampler and ZOH, the sample-and-hold activity introduces discontinuity to the closed-loop system which complicates the system dynamics and makes the stability analysis difficult. Consequently, the existing stability analysis for either continuous/discrete-time FMB control systems cannot be applied.

Linear [1] and nonlinear [2, 3] sampled-data control systems have been investigated for decades. Emulation is a commonly used method to design a controller based on the continuous-time plant and then followed by a discretization process to obtain the discrete-time controller. Various stability properties were developed in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] to guarantee the stability of the sampled-data control system. As pointed out by some work [8] that this approach may not be practical as the sampling rate is required to be sufficiently high which may exceed the hardware limitations. Recently, input delay approach [17, 18] has been proposed for the investigation of linear sampled-data control systems. By representing the sampled control signal as an input delay form, the analysis of time-delay linear or nonlinear systems can be applied. The input delay approach was extended to sampled-data polynomial systems [19], sampled-data FMB control systems [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], sampled-data FMB control systems with analogue-to-digital converter and digital-to-analogue converter [38], time-delay sampled-data FMB control systems [20] and sampled-data PFMB control systems [39].

In this chapter, the stability of sampled-data output-feedback (SDOF) PFMB control system [39], which is formed by a nonlinear plant represented by a polynomial fuzzy model and a SDOF fuzzy controller, is investigated. Instead of using full-state feedback compensation, output-feedback [39, 40] is suitable for the systems that only the system output is available. As shown in Fig.  5.1, there are three categories of PFMB control systems, namely perfectly, partially and imperfectly matched premises. Perfectly matched premises require that the polynomial fuzzy model and polynomial fuzzy controller share the same premise membership functions. However, even the membership functions of both polynomial fuzzy model and sampled-data fuzzy controller are the same in the sampled-data PFMB control system, the membership grades of the sampled-data fuzzy controller will be in general different from those of the polynomial fuzzy model except at the sampling instant. Thus, there is no point in considering the category of perfectly matched premises. Consequently, we shall only consider the partially and imperfectly matched premises.

The organization of this chapter is as follows. In Sect. 8.2, the SDOF fuzzy controller and SDOF PFMB control system are presented. In Sect. 8.3, the stability analysis of SDOF PFMB control system is investigated using the Lyapunov stability theory. SOS-based stability conditions are obtained to guarantee the system stability and synthesize the SDOF fuzzy controller. In Sect. 8.4, simulation examples are given to demonstrate the design process and the merits of the proposed SDOF fuzzy control scheme. In Sect. 8.5, a conclusion is drawn.

8.2 Preliminaries

An SDOF fuzzy controller is introduced to control the nonlinear plant represented by a polynomial fuzzy model using the system output. By connecting the polynomial fuzzy model and SDOF fuzzy controller in a closed-loop as shown in Fig. 8.1, an SDOF PFMB control system is obtained. Its system stability will be investigated based on Lyapunov stability theory through SOS-based analysis.

8.2.1 Polynomial Fuzzy Model

Let p be the number of fuzzy rules describing the behavior of a nonlinear plant. The ith rule is of the following format:
$$\begin{aligned} \text {Rule }i\text {: }&\text {IF } f_1(\mathbf {x}(t)) \text { is } M_1^i \text { AND } \cdots \text { AND } f_{ \Psi }(\mathbf {x}(t)) \text { is } M_{ \Psi }^i \nonumber \\&\text {THEN } \dot{\mathbf {x}}(t) = \mathbf {A}_i(\mathbf {x}(t)) \hat{\mathbf {x}}(\mathbf {x}(t)) + \mathbf {B}_i(\mathbf {x}(t))\mathbf {u}(t), \nonumber \\&\qquad \quad \mathbf {y}(t) = \mathbf {C}\hat{\mathbf {x}}(t), i = 1, \ldots , p. \end{aligned}$$
(8.1)
The system dynamics is described as follows:
$$\begin{aligned} \displaystyle \dot{\mathbf {x}}(t)&= \sum _{i=1}^p w_i(\mathbf {x}(t))(\mathbf {A}_i(\mathbf {x}(t)) \hat{\mathbf {x}}(\mathbf {x}(t)) + \mathbf {B}_i(\mathbf {x}(t))\mathbf {u}(t)), \end{aligned}$$
(8.2)
$$\begin{aligned} \mathbf {y}(t)&= \mathbf {C}\hat{\mathbf {x}}(t), \end{aligned}$$
(8.3)
where the variables are defined and details are given in Sect.  7.2.1.

Remark 8.1

The polynomial model (8.2) and (8.3) are different from ( 7.2) and ( 7.3) in Chap. 7 that the membership function \(w_i(\mathbf {x}(t))\) is in \(\mathbf {x}(t)\) in this chapter while the one in Chap.  7 is in \(\mathbf {y}(t)\), i.e., \(w_i(\mathbf {y}(t))\). In Chap.  7, as the category of perfectly matched premises is considered, the output-feedback polynomial fuzzy controller needs to consider \(w_i(\mathbf {y}(t))\) in terms of \(\mathbf {y}(t)\) for implementation purposes. However, when the category of imperfectly matched premises is considered, the output-feedback polynomial fuzzy controller can have freedom to choose its own membership functions. Consequently, the output-feedback polynomial fuzzy controller are allowed to have membership functions \(w_i(\mathbf {x}(t))\) in terms of \(\mathbf {x}(t)\), which is not required for the implementation of output-feedback polynomial fuzzy controller.

8.2.2 Sampled-Data Output-Feedback Fuzzy Controller

An SDOF fuzzy controller described by c rules of the following format is proposed to control the nonlinear plant represented by the polynomial fuzzy model (8.2) and (8.3).
$$\begin{aligned} \text {Rule }j\text {: }&\text {IF } g_1(\mathbf {y}(t_\gamma )) \text { is } N_1^j \text { AND } \cdots \text { AND } g_{\varOmega }(\mathbf {y}(t_\gamma )) \text { is } N_{\varOmega }^j \nonumber \\&\text {THEN } \mathbf {u}(t) = \mathbf {G}_j\mathbf {y}(t_\gamma ), j = 1, \ldots , c, \end{aligned}$$
(8.4)
where \(N_{\beta }^j\) is the fuzzy term of rule j corresponding to the function \(g_{\beta }(\mathbf {y}(t_\gamma ))\), \(\beta = 1, \ldots , {\varOmega }\); \(j = 1, \ldots , c; {\varOmega }\) is a positive integer; \(\mathbf {G}_j \in \mathfrak {R}^{m \times N}\) are constant feedback gains to be determined; \(t_\gamma = \gamma h_s\), \(\gamma = 1, 2, \ldots , \infty \), where \(h_s > 0\), denotes the constant sampling time. The SDOF fuzzy controller is defined as,
$$\begin{aligned} \displaystyle \mathbf {u}(t)&= \sum _{j=1}^c m_j(\mathbf {y}(t_\gamma ))\mathbf {G}_j\mathbf {y}(t_\gamma ), \end{aligned}$$
(8.5)
where
$$\begin{aligned} \displaystyle \sum _{j=1}^c m_j(\mathbf {y}(t_\gamma )) = 1, \end{aligned}$$
(8.6)
$$\begin{aligned} m_j(\mathbf {y}(t_\gamma )) \ge 0 \; \forall \; j, \end{aligned}$$
(8.7)
$$\begin{aligned} m_j(\mathbf {y}(t_\gamma )) = \frac{\displaystyle \prod _{l=1}^{\varOmega }\mu _{N_l^j}(g_l(\mathbf {y}(t_\gamma )))}{\displaystyle \sum _{k=1}^c \prod _{l=1}^{\varOmega }\mu _{N_l^k}(g_l(\mathbf {y}(t_\gamma )))} \; \forall \; j, \end{aligned}$$
(8.8)
\(m_j(\mathbf {y}(t_\gamma ))\) are the normalized grades of membership; \(\mu _{N_{\beta }^{j}}(g_{\beta }(\mathbf {y}(t_\gamma )))\) is the grade of membership corresponding to the fuzzy term of \(N_{\beta }^{j}\).
An input delay approach [17] is employed to represent the SDOF fuzzy controller (8.5). Denoting \(\tau _s(t) = t - t_\gamma < h_s\) for \(t_\gamma \le t < t_{\gamma +1}\), the SDOF fuzzy controller (8.5) can be rewritten as the following input delay from.
$$\begin{aligned} \displaystyle \mathbf {u}(t)&= \sum _{j=1}^c m_j(\mathbf {y}(t_\gamma ))\mathbf {G}_j\mathbf {y}(t_\gamma - t + t) \nonumber \\&= \sum _{j=1}^c m_j(\mathbf {y}(t_\gamma ))\mathbf {G}_j\mathbf {C} \hat{\mathbf {x}}(\mathbf {x}(t-\tau _s(t))), t_\gamma \le t < t_{\gamma +1}. \end{aligned}$$
(8.9)

Remark 8.2

The control signal \(\mathbf {u}(t) = \mathbf {u}(t_\gamma )\) holds constant using the ZOH during the period of \(t_\gamma \le t < t_{\gamma +1}\).

Remark 8.3

The SDOF fuzzy controller becomes a sampled-data full-state feedback fuzzy controller when \(\mathbf {C}\) is a full rank matrix, for example, \(\mathbf {C} = \mathbf {I}\), where \(\mathbf {I}\) is the identity matrix of compatible dimensions.

Remark 8.4

The output vector \(\mathbf {C}\) is considered to be a constant vector to ease the stability analysis and control design using the properties of ( 7.17) and ( 7.18). When \(\mathbf {C}\) is system state dependent, the analysis will become more complicated.

8.2.3 Sampled-Data Output-Feedback Polynomial Fuzzy Model-Based Control System

Connecting the polynomial fuzzy model (8.2) and the SDOF fuzzy controller (8.9) in a closed loop and using the fact that \(\sum _{i=1}^p \sum _{j=1}^c w_i(\mathbf {x}(t)) m_j(\mathbf {y}(t_\gamma )) = 1\), we obtain the following SDOF PFMB control system.
$$\begin{aligned} \displaystyle \dot{\mathbf {x}}(t)&= \sum _{i=1}^p \sum _{j=1}^c w_i(\mathbf {x}(t)) m_j(\mathbf {y}(t_\gamma )) \big ( \mathbf {A}_i(\mathbf {x}(t))\hat{\mathbf {x}}(\mathbf {x}(t)) + \mathbf {B}_i(\mathbf {x}(t)) \mathbf {G}_j \mathbf {C}\hat{\mathbf {x}}_s(t) \big ), \end{aligned}$$
(8.10)
where \(\hat{\mathbf {x}}_s(t) \equiv \hat{\mathbf {x}}(\mathbf {x}(t-\tau _s(t)))\).

The control objective is to obtain the feedback gains \(\mathbf {G}_j\) and the sampling period \(h_s\) such that the SDOF PFMB control system (8.10) is asymptotically stable, i.e., \(\mathbf {x}(t) \rightarrow \mathbf {0}\) as time \(t \rightarrow \infty \).

8.3 Stability Analysis

We shall investigate the stability of SDOF PFMB control system (8.10) based on the Lyapunov stability theory. SOS-based stability conditions will be obtained to check the system stability and synthesize the SDOF fuzzy controller. For brevity, in the following analysis, \(w_i(\mathbf {x}(t))\), \(m_j(\mathbf {y}(t_\gamma ))\), \(\mathbf {x}(t)\), \(\mathbf {x}(t-\tau _s(t))\), \(\hat{\mathbf {x}}(\mathbf {x}(t))\) and \(\hat{\mathbf {x}}_s(t)\) are denoted as \(w_i\), \(m_j\), \(\mathbf {x}\), \(\mathbf {x}_s\), \(\hat{\mathbf {x}}\) and \(\hat{\mathbf {x}}_s\), respectively.

To proceed with the stability analysis, the constant transformation matrix \(\varvec{\Gamma } \in \mathfrak {R}^{N \times N}\) [41] demonstrating the properties in Lemma  7.1 is considered.

From the definition of \(\hat{\mathbf {x}}\), we have,
$$\begin{aligned} \dot{\hat{\mathbf {x}}}&= \frac{\partial \hat{\mathbf {x}}}{\partial {\mathbf {x}}} \frac{d\mathbf {x}}{dt} = \mathbf {H}(\mathbf {x})\dot{\mathbf {x}}, \end{aligned}$$
(8.11)
where \(\mathbf {x} = [x_1, \ldots , x_n]^T\), \(\hat{\mathbf {x}} = [\hat{x}_1, \ldots , \hat{x}_N]^T\) and
$$\begin{aligned} \mathbf {H}(\mathbf {x}) = \frac{\partial \hat{\mathbf {x}}}{\partial {\mathbf {x}}} = \left[ \begin{array}{ccc} \frac{\partial \hat{x}_{1}(\mathbf {x})}{\partial x_{1}} &{} \cdots &{} \frac{\partial \hat{x}_{1}(\mathbf {x})}{\partial x_{n}} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial \hat{x}_{N}(\mathbf {x})}{\partial x_{1}} &{} \cdots &{} \frac{\partial \hat{x}_{N}(\mathbf {x})}{\partial x_{n}}\end{array} \right] . \end{aligned}$$
(8.12)
From (8.10) and (8.11), we have,
$$\begin{aligned} \dot{\mathbf {z}}&= \varvec{\Gamma }^{-1}\dot{\hat{\mathbf {x}}} \nonumber \\&= \sum _{i=1}^p \sum _{j=1}^c w_{i}m_{j}(\tilde{\mathbf {A}}_i(\mathbf {x})\mathbf {z} + \tilde{\mathbf {B}}_i(\mathbf {x})\mathbf {G}_j\mathbf {C}\varvec{\Gamma }\mathbf {z}_s), \end{aligned}$$
(8.13)
where \(\varvec{\Gamma }^{-1}\) is defined in Lemma  7.1, \(\mathbf {z} = \varvec{\Gamma }^{-1}\hat{\mathbf {x}}\), \(\mathbf {z}_s = \varvec{\Gamma }^{-1}\hat{\mathbf {x}}_s\), \(\tilde{\mathbf {A}}_i(\mathbf {x}) = \varvec{\Gamma }^{-1}\mathbf {H}(\mathbf {x})\mathbf {A}_i(\mathbf {x})\varvec{\Gamma }\) and \(\tilde{\mathbf {B}}_i(\mathbf {x}) = \varvec{\Gamma }^{-1}\mathbf {H}(\mathbf {x})\mathbf {B}_i(\mathbf {x})\).

Remark 8.5

It should be noted that \(\varvec{\Gamma }\) is non-singular. \(\mathbf {z} = \mathbf {0}\) implies \(\mathbf {x} = \mathbf {0}\). Consequently, the system stability of the transformed SDOF PFMB control system (8.13) implies that of the SDOF PFMB control system (8.10).

We consider the following Lyapunov functional to investigate the stability of (8.13).
$$\begin{aligned} \displaystyle V(t)&= \mathbf {z}^{T}\mathbf {P}_1\mathbf {z} + \int _{-h_s}^{0}\int _{t+\sigma }^{t} \dot{\mathbf {z}}(\varphi )^{T}\mathbf {R}\dot{\mathbf {z}}(\varphi )d{\varphi }d{\sigma }, \end{aligned}$$
(8.14)
where \(0 < \mathbf {P}_1 = \mathbf {P}_1^T \in \mathfrak {R}^{N \times N}\), \(0 < \mathbf {R} = \mathbf {R}^T \in \mathfrak {R}^{N \times N}\).

Remark 8.6

The feedback gains \(\mathbf {G}_j\) in the SDOF fuzzy controller (8.9) are chosen to be constant matrices to ease the stability analysis. As output-feedback sampled-data fuzzy control is considered, if the feedback gains are considered to be polynomial matrices, they must be a function of \(\mathbf {y}(t_\gamma )\), i.e., \(\mathbf {G}_j(\mathbf {y}(t_\gamma ))\). By using such a feedback gain, it can be deduced from the following analysis that the stability conditions depend on system state vector \(\mathbf {x}\) and sampled-output vector \(\mathbf {y}(t_\gamma )\) where the feedback gains are the only decision variables depending on \(\mathbf {y}(t_\gamma )\) causing difficulty in solving the solution. Introducing \(\mathbf {y}(t_\gamma )\) to the Lyapunov functional will take other decision variables depending on \(\mathbf {y}(t_\gamma )\) to the stability. However, the Lyapunov functional is not differentiable at the sampling instant \(t_\gamma \). Consequently, the work around is to use constant feedback gains \(\mathbf {G}_j\) in the SDOF fuzzy controller (8.9), and constant \(\mathbf {P}_{1}\) and \(\mathbf {R}\) in the Lyapunov functional (8.14).

In the following, we consider two cases to investigate the stability of the SDOF PFMB control system (8.13), namely, \(c = p\) and \(c \ne p\).

8.3.1 Stability Analysis with \(c = p\)

From (8.13) and (8.14), with \(c = p\), we have,
$$\begin{aligned} \displaystyle \dot{V}(t)&= \mathbf {z}^{T}\mathbf {P}_{1}\dot{\mathbf {z}} + \dot{\mathbf {z}}^{T}\mathbf {P}_{1}\mathbf {z} + h_s\dot{\mathbf {z}}^{T}\mathbf {R}\dot{\mathbf {z}} - \int _{t-h_s}^{t} \dot{\mathbf {z}}(\varphi )^{T}\mathbf {R}\dot{\mathbf {z}}(\varphi )d{\varphi } \nonumber \\&= \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}m_{j}\mathbf {h}^T (\mathbf {P}^T\mathbf {Q}_{ij}(\mathbf {x}) + \mathbf {Q}_{ij}(\mathbf {x})^T\mathbf {P})\mathbf {h} \nonumber \\&\quad + h_s\dot{\mathbf {z}}^{T}\mathbf {R}\dot{\mathbf {z}} - \int _{t-h_s}^{t} \dot{\mathbf {z}}(\varphi )^{T}\mathbf {R}\dot{\mathbf {z}}(\varphi )d{\varphi }, \end{aligned}$$
(8.15)
where \(\mathbf {h} = \left[ \begin{array}{c} \mathbf {z} \\ \mathbf {z}_s\end{array}\right] \), \(\mathbf {P} = \left[ \begin{array}{cc}\mathbf {P}_1 &{} \mathbf {0} \\ \mathbf {P}_2 &{} \mathbf {P}_3\end{array}\right] \), both \(\mathbf {P}_2 \in \mathfrak {R}^{N \times N}\) and \(\mathbf {P}_3 \in \mathfrak {R}^{N \times N}\) are arbitrary matrices, and \(\mathbf {Q}_{ij}(\mathbf {x}) = \left[ \begin{array}{cc} \tilde{\mathbf {A}}_i(\mathbf {x}) &{} \tilde{\mathbf {B}}_i(\mathbf {x})\mathbf {G}_j\mathbf {C}\mathbf {\Gamma } \\ \mathbf {0} &{} \mathbf {0}\end{array}\right] \).

To deal with the term \(\int _{t-h_s}^{t} \dot{\mathbf {z}}(\varphi )^{T}\mathbf {R}\dot{\mathbf {z}}(\varphi )d{\varphi }\) in (8.15), the following Lemma is introduced.

Lemma 8.1

Considering \(\mathbf {h} \in \mathfrak {R}^{2N}\), \(\dot{\mathbf {z}}(\varphi ) \in \mathfrak {R}^{2N}\), \(0 < \mathbf {R} = \mathbf {R}^T \in \mathfrak {R}^{N \times N}\), arbitrary polynomial matrices \(\mathbf {T}_{ij}(\mathbf {x}) \in \mathfrak {R}^{N \times N}\), \(\mathbf {V}_{ij}(\mathbf {x}) \in \mathfrak {R}^{N \times N}\) and the fact that \(0 \le \tau _s(t) = t - t_{\gamma } < h_s\), the following inequality holds.
$$\begin{aligned}&h_s \sum _{i=1}^{p}\sum _{j=1}^{p}\sum _{k=1}^{p}\sum _{l=1}^{p}w_{i}m_{j}w_{k}m_{l} \mathbf {h}^T\left[ \begin{array}{c} \mathbf {T}_{ij}(\mathbf {x}) \\ \mathbf {V}_{ij}(\mathbf {x}) \end{array}\right] \mathbf {R}^{-1} \left[ \begin{array}{c} \mathbf {T}_{kl}(\mathbf {x}) \\ \mathbf {V}_{kl}(\mathbf {x}) \end{array}\right] ^T\mathbf {h} \nonumber \\&\qquad + \int _{t-\tau _s(t)}^{t}\mathbf {h}^T\sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}m_{j}\left[ \begin{array}{c} \mathbf {T}_{ij}(\mathbf {x}) \\ \mathbf {V}_{ij}(\mathbf {x}) \end{array}\right] \dot{\mathbf {z}}(\varphi )d\varphi \nonumber \\&\qquad + \int _{t-\tau _s(t)}^{t}\dot{\mathbf {z}}(\varphi )^T\sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}m_{j}\left[ \begin{array}{c} \mathbf {T}_{ij}(\mathbf {x}) \\ \mathbf {V}_{ij}(\mathbf {x}) \end{array}\right] ^T\mathbf {h}d\varphi \nonumber \\&\quad \ge - \int _{t-h_s}^{t}\dot{\mathbf {z}}(\varphi )^T\mathbf {R}\dot{\mathbf {z}}(\varphi )d\varphi \end{aligned}$$
(8.16)

Proof

Using the fact that \(- \int _{t-\tau _s(t)}^{t}\dot{\mathbf {z}}(\varphi )^T\mathbf {R}\dot{\mathbf {z}}(\varphi )d\varphi \ge - \int _{t-h_s}^{t}\dot{\mathbf {z}}(\varphi )^T\mathbf {R}\dot{\mathbf {z}}(\varphi )d\varphi \), the proof follows immediately by expanding the following inequality:
$$\begin{aligned}&\int _{t-\tau _s(t)}^{t}\Bigg (\Bigg (\sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}m_{j}\left[ \begin{array}{c} \mathbf {T}_{ij}(\mathbf {x}) \\ \mathbf {V}_{ij}(\mathbf {x}) \end{array}\right] ^T\mathbf {h} + \mathbf {R}\dot{\mathbf {z}}(\varphi )\Bigg )^T \nonumber \\&\quad \times \mathbf {R}^{-1}\Bigg (\sum _{k=1}^{p}\sum _{l=1}^{p}w_{k}m_{l}\left[ \begin{array}{c} \mathbf {T}_{kl}(\mathbf {x}) \\ \mathbf {V}_{kl}(\mathbf {x}) \end{array}\right] ^T\mathbf {h} + \mathbf {R}\dot{\mathbf {z}}(\varphi )\Bigg ) \Bigg )d\varphi \ge 0. \end{aligned}$$
(8.17)
    \(\blacksquare \)
Applying Lemma 8.1, the upper bound of the term \(- \int _{t-h_s}^{t} \dot{\mathbf {z}}(\varphi )^{T}\mathbf {R}\dot{\mathbf {z}}(\varphi )d{\varphi }\) can be obtained. Applying the Newton–Leibniz rule to the term \(\int _{t-\tau _s(t)}^{t}\dot{\mathbf {z}}(\varphi )d\varphi \), we have \(\int _{t-\tau _s(t)}^{t}\dot{\mathbf {z}}(\varphi )d\varphi \) = \(\mathbf {z}(t) - \mathbf {z}(t - \tau _s(t))\) = \(\left[ \begin{array}{c} \mathbf {I} \\ -\mathbf {I}\end{array}\right] ^T\mathbf {h}\). Consequently, (8.15) becomes
$$\begin{aligned} \displaystyle \dot{V}(t)&\le \sum _{i=1}^{p}\sum _{j=1}^{p}\sum _{k=1}^{p}\sum _{l=1}^{p}w_{i}m_{j}w_{k}m_{l}\mathbf {h}^T \Bigg (\mathbf {P}^T\mathbf {Q}_{ij}(\mathbf {x}) + \mathbf {Q}_{ij}(\mathbf {x})^T\mathbf {P} \nonumber \\&\quad + h_s\left[ \begin{array}{c} \mathbf {T}_{ij}(\mathbf {x}) \\ \mathbf {V}_{ij}(\mathbf {x}) \end{array}\right] \mathbf {R}^{-1}\left[ \begin{array}{c} \mathbf {T}_{kl}(\mathbf {x}) \\ \mathbf {V}_{kl}(\mathbf {x}) \end{array}\right] ^T + \left[ \begin{array}{c} \mathbf {T}_{ij}(\mathbf {x}) \\ \mathbf {V}_{ij}(\mathbf {x}) \end{array}\right] \left[ \begin{array}{c} \mathbf {I} \\ -\mathbf {I}\end{array}\right] ^T \nonumber \\&\quad + \left[ \begin{array}{c} \mathbf {I} \\ -\mathbf {I}\end{array}\right] \left[ \begin{array}{c} \mathbf {T}_{ij}(\mathbf {x}) \\ \mathbf {V}_{ij}(\mathbf {x}) \end{array}\right] ^T\Bigg )\mathbf {h} + h_s\dot{\mathbf {z}}^{T}\mathbf {R}\dot{\mathbf {z}}, \end{aligned}$$
(8.18)
where \(\mathbf {T}_{ij}(\mathbf {x}) \in \mathfrak {R}^{N \times N}\) and \(\mathbf {V}_{ij}(\mathbf {x}) \in \mathfrak {R}^{N \times N}\) are arbitrary polynomial matrices.

In the following, we shall develop SOS-based stability conditions to achieve \(\dot{V}(t) < 0\) for system stability. We define \(\mathbf {X} = \mathbf {P}^{-1} = \left[ \begin{array}{cc}\mathbf {X}_1 &{} \mathbf {0} \\ \mathbf {X}_2 &{} \mathbf {X}_3\end{array}\right] \), where \(0 < \mathbf {X}_1 = \mathbf {X}_1^T \in \mathfrak {R}^{N \times N}\), \(\mathbf {X}_2 = \mathbf {X}_2^T \in \mathfrak {R}^{N \times N}\) and \(\mathbf {X}_3 = \mathbf {X}_3^T \in \mathfrak {R}^{N \times N}\).

As proposed in [41], we choose
$$\begin{aligned} \mathbf {X}_1 = \left[ \begin{array}{cc} \mathbf {X}_{11} &{} \mathbf {0} \\ \mathbf {0} &{} \mathbf {X}_{22}\end{array}\right] , \end{aligned}$$
(8.19)
where \(\mathbf {X}_{11} = \mathbf {X}_{11}^T \in \mathfrak {R}^{l \times l}\) and \(\mathbf {X}_{22} = \mathbf {X}_{22}^T \in \mathfrak {R}^{(N-l) \times (N-l)}\).

Remark 8.7

The off-diagonal elements of \(\mathbf {X}_1\) is chosen to be zero matrices to ease the stability analysis for obtaining convex stability conditions.

From ( 7.18), we have
$$\begin{aligned} \mathbf {C}\varvec{\Gamma } = \left[ \begin{array}{cc} \mathbf {I}_l&\mathbf {0} \end{array}\right] , \end{aligned}$$
(8.20)
where \(\mathbf {I}_l \in \mathfrak {R}^{l \times l}\) is the identify matrix.
Considering the term \(\tilde{\mathbf {B}}_i(\mathbf {x})\mathbf {G}_j\mathbf {C}\varvec{\Gamma }\) in \(\mathbf {Q}_{ij}(\mathbf {x})\), it makes the terms \(\mathbf {P}^T\mathbf {Q}_{ij}(\mathbf {x})\) and \(\mathbf {Q}_{ij}(\mathbf {x})^T\mathbf {P}\) not convex in \(\mathbf {P}\) and \(\mathbf {G}_j\). To circumvent this problem, we rewrite the term \(\tilde{\mathbf {B}}_i(\mathbf {x})\mathbf {G}_j\mathbf {C}\varvec{\Gamma }\) in \(\mathbf {Q}_{ij}(\mathbf {x})\) as \(\tilde{\mathbf {B}}_i(\mathbf {x})\mathbf {G}_j\mathbf {C}\varvec{\Gamma }\mathbf {X}_1\mathbf {X}_1^{-1}\). Let the feedback gains be \(\mathbf {G}_j = \mathbf {N}_j\mathbf {X}_{11}^{-1}\), \(j = 1, \ldots , p\), where \(\mathbf {N}_j \in \mathfrak {R}^{m \times l}\). From (8.20), we have,
$$\begin{aligned} \mathbf {G}_j\mathbf {C}\varvec{\Gamma }\mathbf {X}_1&= \mathbf {N}_j\mathbf {X}_{11}^{-1}\mathbf {C}\varvec{\Gamma }\mathbf {X}_1 \nonumber \\&= \mathbf {N}_j\mathbf {X}_{11}^{-1}\left[ \begin{array}{cc} \mathbf {I}_l&\mathbf {0} \end{array}\right] \mathbf {X}_1 \nonumber \\&= \left[ \begin{array}{cc} \mathbf {N}_j&\mathbf {0} \end{array}\right] . \end{aligned}$$
(8.21)
Choose \(\mathbf {X}_2 = \varepsilon _2\mathbf {X}_1\in \mathfrak {R}^{N \times N}\), \(\mathbf {X}_3 = \varepsilon _3\mathbf {X}_1 \in \mathfrak {R}^{N \times N}\), where \(\varepsilon _2\) and \(\varepsilon _3 \ne 0\) are constant scalars to be determined. The term \(\tilde{\mathbf {B}}_i(\mathbf {x})\mathbf {G}_j\mathbf {C}\varvec{\Gamma }\) will become \(\tilde{\mathbf {B}}_i(\mathbf {x}) \left[ \begin{array}{cc} \mathbf {N}_j&\mathbf {0} \end{array}\right] \mathbf {X}_1^{-1}\). Premultiplying \(\mathbf {X}\) to the left and post-multiplying \(\mathbf {X}^T\) to the right of \(\mathbf {P}^T\mathbf {Q}_{ij}(\mathbf {x})\), we can obtainAs it can be seen from (8.22) that each element has only one decision variable matrix (\(\mathbf {X}_1\) or \(\mathbf {N}_j\)), it will help the development of convex SOS stability conditions in the following analysis.

Denote \(\mathbf {M} = \mathbf {M}^T = \mathbf {R}^{-1} \in \mathfrak {R}^{N \times N}\), \(\mathbf {U}_{ij}(\mathbf {x}) = \mathbf {X}_1\mathbf {T}_{ij}(\mathbf {x})\mathbf {X}_1 \in \mathfrak {R}^{N \times N}\), \(\mathbf {W}_{ij}(\mathbf {x}) = \mathbf {X}_1\mathbf {V}_{ij}(\mathbf {x})\mathbf {X}_1 \in \mathfrak {R}^{N \times N}\), \(\mathbf {Z}(t) = \left[ \begin{array}{c}\mathbf {Z}_1(t) \\ \mathbf {Z}_2(t)\end{array}\right] = \mathbf {X}^{-1}\left[ \begin{array}{c} \mathbf {z} \\ \mathbf {z}_s\end{array}\right] \) and \(\dot{\mathbf {Z}}_1(t) = \mathbf {X}_1^{-1}\dot{\mathbf {z}}\).

From (8.18) and (8.21), we have
$$\begin{aligned} \dot{V}(t) \le \mathbf {Z}(t)^T\mathbf {\Xi }(\mathbf {x})\mathbf {Z}(t), \end{aligned}$$
(8.23)
where
$$\begin{aligned} \displaystyle \mathbf {\Xi }(\mathbf {x})&= \sum _{i=1}^{p}\sum _{j=1}^{p}\sum _{k=1}^{p}\sum _{l=1}^{p}w_{i}m_{j}w_{k}m_{l} \Big ( \varvec{\Theta }_{ij}(\mathbf {x}) + \varvec{\Theta }_{ij}(\mathbf {x})^T \nonumber \\&\quad + h_s\varvec{\Phi }_{ij}(\mathbf {x})\mathbf {X}_1^{-1}\mathbf {M}\mathbf {X}_1^{-1}\varvec{\Phi }_{kl}(\mathbf {x})^T + h_s\varvec{\Psi }_{ij}(\mathbf {x})\mathbf {M}^{-1}\varvec{\Psi }_{kl}(\mathbf {x})^T \Big ), \end{aligned}$$
(8.24)
$$\begin{aligned} \varvec{\Theta }_{ij}^{(11)}(\mathbf {x}) = \tilde{\mathbf {A}}(\mathbf {x})\mathbf {X}_1 + \varepsilon _2\varvec{\Upsilon }_{ij}(\mathbf {x}) + (1-\varepsilon _2)\mathbf {U}_{ij}(\mathbf {x})^T + (1-\varepsilon _2)\varepsilon _2\mathbf {W}_{ij}(\mathbf {x})^T, \end{aligned}$$
$$\varvec{\Theta }_{ij}^{(12)}(\mathbf {x}) = \varepsilon _3\varvec{\Upsilon }_{ij}(\mathbf {x}) + (1-\varepsilon _2)\varepsilon _3 \mathbf {W}_{ij}(\mathbf {x})^T,$$
$$\varvec{\Theta }_{ij}^{(21)}(\mathbf {x}) = -\varepsilon _3\mathbf {U}_{ij}(\mathbf {x})^T - \varepsilon _2\varepsilon _3\mathbf {W}_{ij}(\mathbf {x})^T,$$
$$\varvec{\Theta }_{ij}^{(22)}(\mathbf {x}) = -\varepsilon _3^2\mathbf {W}_{ij}(\mathbf {x})^T,$$
$$\varvec{\Upsilon }_{ij}(\mathbf {x}) = \tilde{\mathbf {B}}_i(\mathbf {x})\mathbf {G}_j\mathbf {C}\varvec{\Gamma }\mathbf {X}_1 = \tilde{\mathbf {B}}_i(\mathbf {x})\left[ \begin{array}{cc} \mathbf {N}_j&\mathbf {0} \end{array}\right] ,$$
$$\varvec{\Phi }_{ij}(\mathbf {x}) = \left[ \begin{array}{c} \mathbf {U}_{ij}(\mathbf {x}) + \varepsilon _2\mathbf {W}_{ij}(\mathbf {x}) \\ \varepsilon _3\mathbf {W}_{ij}(\mathbf {x}) \end{array}\right] ,$$
$$\varvec{\Psi }_{ij}(\mathbf {x}) = \left[ \begin{array}{c} \varvec{\Psi }_{ij}^{(1)}(\mathbf {x}) \\ \varvec{\Psi }_{ij}^{(2)}(\mathbf {x}) \end{array}\right] ,$$
$$\varvec{\Psi }_{ij}^{(1)}(\mathbf {x}) = \mathbf {X}_1\tilde{\mathbf {A}}_i(\mathbf {x})^T + \varepsilon _2\varvec{\Upsilon }_{ij}(\mathbf {x})^T,$$
$$\varvec{\Psi }_{ij}^{(2)}(\mathbf {x}) = \varepsilon _3\varvec{\Upsilon }_{ij}(\mathbf {x})^T.$$
It is required according to the Lyapunov stability theory that \(V(t) > 0 \) and \(\dot{V}(t) < 0\) for all \(\mathbf {x}\) (excluding for \(\mathbf {x} = \mathbf {0}\)) have to be achieved to guarantee the stability of the SDOF PFMB control system (8.10). As seen from (8.24), \(\dot{V}(t) < 0\) can be achieved by satisfying \(\varvec{\Xi }(\mathbf {x}) < 0\). However, taking a closer look to (8.24), \(\varvec{\Xi }(\mathbf {x}) < 0\) is not a convex stability condition. In what follows, Schur complement and the following lemma will be applied to circumvent the difficulty to obtain convex stability conditions.

Lemma 8.2

Considering an arbitrary scalar \(\xi \), an arbitrary square symmetric matrix \(\mathbf {X}_1\) and an arbitrary positive-definite matrix \(\mathbf {M}\), the following inequality holds.
$$\begin{aligned} \mathbf {X}_1\mathbf {M}^{-1}\mathbf {X}_1 \ge 2\xi \mathbf {X}_1 - \xi ^2\mathbf {M} \end{aligned}$$
(8.25)

Proof

The proof follows immediately by expanding the inequality of \((\mathbf {X}_1 - \xi \mathbf {M})^T\mathbf {M}^{-1}(\mathbf {X}_1 - \xi \mathbf {M}) \ge 0\).    \(\blacksquare \)

By applying Schur complement and Lemma 8.2, the satisfaction of \(\varvec{\Xi }(\mathbf {x}) < 0\) can be achieved by the satisfaction of the following inequality.
$$\begin{aligned} \displaystyle \hat{\varvec{\Xi }}(\mathbf {x}) = \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}m_{j}\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) < 0, \end{aligned}$$
(8.26)
where
$$\begin{aligned} \hat{\varvec{\Xi }}_{ij}(\mathbf {x}) = \left[ \begin{array}{ccc} \varvec{\Theta }_{ij}(\mathbf {x}) + \varvec{\Theta }_{ij}(\mathbf {x})^T &{} * &{} * \\ h_s\varvec{\Psi }_{ij}(\mathbf {x})^T &{} -h_s\mathbf {M} &{} *\\ h_s\varvec{\Phi }_{ij}(\mathbf {x})^T &{} \mathbf {0} &{} -h_s(2\xi \mathbf {X}_1 - \xi ^2\mathbf {M}) \end{array}\right] , \end{aligned}$$
and “*” denotes the transposed element at the corresponding position.

Collecting the analysis result obtained above, the SDOF PFMB control system (8.10) is asymptotically stable by the satisfaction of \(\mathbf {X}_1 > 0\), \(\mathbf {M} > 0\) and \(\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) < 0\) for all i and j. However, as explained in the previous chapters, MFI stability analysis always leads to conservative stability analysis result as the membership functions are not considered.

In the following, we employ the MFD stability analysis approach to relax the stability conditions by considering the information of membership functions below:
$$\begin{aligned} m_j(\mathbf {y}(t_\gamma )) - w_j(\mathbf {x}) + \rho _j \ge 0 \quad \forall \; j, \mathbf {x}(t), \mathbf {y}(t_\gamma ), \end{aligned}$$
(8.27)
where \(\rho _j\) are constant scalars to be determined.

Remark 8.8

The value of \(\rho _j\) is the upper bound of \(w_j(\mathbf {x}) - m_j(\mathbf {y}(t_\gamma ))\). It can be found numerically by considering the form of \(w_j(\mathbf {x})\) and \(m_j(\mathbf {y}(t_\gamma ))\). In general, the trivial value of \(\rho _j\) is 1 as \(0 \le w_j(\mathbf {x}) \le 1\) and \(0 \le m_j(\mathbf {y}(t_\gamma )) \le 1\) are independent of each other (one depends on \(\mathbf {x}\) and another depends on \(\mathbf {y}(t_\gamma )\)). Unlike the previous chapters, both membership functions of polynomial fuzzy model and polynomial fuzzy controller depend on the same state vector \(\mathbf {x}\), which will lead to non-trivial lower and upper bounds. The trivial upper bounds \(\rho _j = 1\) for all j will not help relax the stability conditions. A work around is to consider the situation that the change between \(\mathbf {x}(t)\) and \(\mathbf {y}(t_\gamma )\) (note that \(\mathbf {y}(t_\gamma ) = \mathbf {C}\hat{\mathbf {x}}(t_\gamma )\)) is sufficiently slow during the sampling period which will lead to non-trivial \(\rho _j\). However, verification has to be done to make sure that the assumption on small change is valid. More details are discussed in the simulation examples in Sect. 8.4.

It follows from (8.26) that we have
$$\begin{aligned} \displaystyle \hat{\varvec{\Xi }}(\mathbf {x})&= \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}(m_{j}+w_{j}-w_{j}+\rho _j-\rho _j)\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) \nonumber \\&= \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}w_{j}\Big (\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) - \sum _{k=1}^{p}\rho _{k}\hat{\varvec{\Xi }}_{ik}(\mathbf {x})\Big ) \nonumber \\&\quad + \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}(m_{j}-w_{j}+\rho _j)\hat{\varvec{\Xi }}_{ij}(\mathbf {x}). \end{aligned}$$
(8.28)
Using the fact that \(\sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}(m_{j}-w_{j}) = 0\) resulting from the property of membership functions and the introduction of the arbitrary polynomial matrices \(\varvec{\Lambda }_i(\mathbf {x}) \in \mathfrak {R}^{4N \times 4N}\), \(i = 1, \ldots , p\), leading to \(\sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}(m_{j}-w_{j})\varvec{\Lambda }_i(\mathbf {x}) = \mathbf {0}\), it follows from (8.28) that we have
$$\begin{aligned} \displaystyle \hat{\varvec{\Xi }}(\mathbf {x})&= \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}w_{j}\Big (\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) - \sum _{k=1}^{p}\rho _{k}\hat{\varvec{\Xi }}_{ik}(\mathbf {x})\Big ) \nonumber \\&\quad + \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}(m_{j}-w_{j}+\rho _j)\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) \nonumber \\&\quad + \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}(m_{j}-w_{j}+\rho _j-\rho _j)\varvec{\Lambda }_i(\mathbf {x}) \nonumber \\&= \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}w_{j}\Big (\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) - \sum _{k=1}^{p}\rho _{k}\big (\hat{\varvec{\Xi }}_{ik}(\mathbf {x}) + \varvec{\Lambda }_i(\mathbf {x})\big )\Big ) \nonumber \\&\quad + \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}(m_{j}-w_{j}+\rho _j)(\hat{\varvec{\Xi }}_{ij}(\mathbf {x})+\varvec{\Lambda }_i(\mathbf {x})) \nonumber \\&= \frac{1}{2}\sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}w_{j}\Big (\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) - \sum _{k=1}^{p}\rho _{k}\big (\hat{\varvec{\Xi }}_{ik}(\mathbf {x}) + \varvec{\Lambda }_i(\mathbf {x})\big ) \nonumber \\&\quad + \hat{\varvec{\Xi }}_{ji}(\mathbf {x}) - \sum _{k=1}^{p}\rho _{k}\big (\hat{\varvec{\Xi }}_{jk}(\mathbf {x}) + \varvec{\Lambda }_j(\mathbf {x})\big )\Big ) \nonumber \\&\quad + \sum _{i=1}^{p}\sum _{j=1}^{p}w_{i}(m_{j}-w_{j}+\rho _j)(\hat{\varvec{\Xi }}_{ij}(\mathbf {x})+\varvec{\Lambda }_i(\mathbf {x})). \end{aligned}$$
(8.29)
It is required that \(\dot{V}(t) < 0\) excluding for \(\mathbf {x} = \mathbf {0}\) to guarantee the system stability which can be achieved by satisfying \(\hat{\varvec{\Xi }}(\mathbf {x}) < 0\). It can be seen from (8.29) that \(\hat{\varvec{\Xi }}(\mathbf {x}) < 0\) can be achieved when \(\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) - \sum _{k=1}^{p}\rho _{k}\big (\hat{\varvec{\Xi }}_{ik}(\mathbf {x}) + \varvec{\Lambda }_i(\mathbf {x})\big ) + \hat{\varvec{\Xi }}_{ji}(\mathbf {x}) - \sum _{k=1}^{p}\rho _{k}\big (\hat{\varvec{\Xi }}_{jk}(\mathbf {x}) + \varvec{\Lambda }_j(\mathbf {x})\big ) < 0\) and \(\hat{\varvec{\Xi }}_{ij}(\mathbf {x})+\varvec{\Lambda }_i(\mathbf {x}) < 0\) for all i and j. The stability analysis result is summarized in the following theorem.

Theorem 8.1

The SDOF PFMB control system (8.10), formed by a nonlinear plant represented by the polynomial fuzzy model (8.2) and the SDOF fuzzy controller (8.5) connected in a closed loop, is guaranteed to be asymptotically stable if there exist matrices \(\mathbf {M} = \mathbf {M}^T \in \mathfrak {R}^{N \times N}\), \(\mathbf {N}_j \in \mathfrak {R}^{m \times l}\) and \(\mathbf {X}_1 = \mathbf {X}_1^T = \left[ \begin{array}{cc} \mathbf {X}_{11} &{} \mathbf {0} \\ \mathbf {0} &{} \mathbf {X}_{22}\end{array} \right] \in \mathfrak {R}^{N \times N}\), and polynomial matrices \(\varvec{\Lambda }_i(\mathbf {x}) \in \mathfrak {R}^{4N \times 4N}\), \(i = 1, \ldots , p\) such that the following SOS conditions are satisfied.
$$\mathbf {r}^T(\mathbf {X}_1 - \upsilon _1\mathbf {I})\mathbf {r} \mathrm {\;is \;SOS},$$
$$\mathbf {r}^T(\mathbf {M} - \upsilon _2\mathbf {I})\mathbf {r} \mathrm {\;is \;SOS},$$
$$\begin{aligned}&-\mathbf {s}^T \Big (\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) - \sum _{k=1}^{p}\rho _{k}\big (\hat{\varvec{\Xi }}_{ik}(\mathbf {x}) + \varvec{\Lambda }_i(\mathbf {x})\big ) + \hat{\varvec{\Xi }}_{ji}(\mathbf {x}) \\&- \sum _{k=1}^{p}\rho _{k}\big (\hat{\varvec{\Xi }}_{jk}(\mathbf {x}) + \varvec{\Lambda }_j(\mathbf {x})\big ) + \upsilon _3(\mathbf {x})\mathbf {I}\Big )\mathbf {s} \mathrm {\;is \;SOS \; \forall \;} i, j, \end{aligned}$$
$$-\mathbf {s}^T (\hat{\varvec{\Xi }}_{ij}(\mathbf {x})+\varvec{\Lambda }_i(\mathbf {x}) + \upsilon _4(\mathbf {x})\mathbf {I}) \mathbf {s} \mathrm {\;is \;SOS \; \forall } \; i\;, j,$$
where \(\mathbf {r} \in \mathfrak {R}^{N}\) and \(\mathbf {s} \in \mathfrak {R}^{4N}\) are arbitrary vectors independent of \(\mathbf {x}\); \(\rho _j\), \(i = 1, \ldots , p\), are predefined scalars satisfying \(m_j - w_j + \rho _j \ge 0\) for all j, \(\mathbf {x}\) and \(\mathbf {y}(t_\gamma )\); \(h_s > 0\) is the predefined sampling period; \(\xi \), \(\varepsilon _2\), \(\varepsilon _3 \ne 0\), \(\upsilon _1 > 0\) and \(\upsilon _2 > 0\) are predefined scalars; \(\upsilon _3(\mathbf {x}) > 0\) and \(\upsilon _4(\mathbf {x}) > 0\) are predefined scalar polynomials and the feedback gains are chosen as \(\mathbf {G}_j = \mathbf {N}_j\mathbf {X}_{11}^{-1}\), \(j = 1, \ldots , p\).

Remark 8.9

The matrix inequality in the form of double fuzzy summations \(\sum _{i=1}^p \sum _{j=1}^p w_i w_j \mathbf {H}_{ij}\), where \(\mathbf {H}_{ij}\) is an arbitrary symmetric matrix, is usually found in the stability analysis of FMB control systems. In this chapter, the method in [42] is employed to handle the matrix inequality (Lemma  6.2). More relaxed result can be obtained by using the MFI methods introduced in [43, 44, 45, 46, 47, 48] and MFD methods introduced in Chaps.  3 5.

Remark 8.10

The introduction of the slack matrix variables \(\mathbf {U}_{ij}(\mathbf {x})\), \(\mathbf {W}_{ij}(\mathbf {x})\) and \(\varvec{\Lambda }_j(\mathbf {x})\) will increase the computational demand on solving the solution to the stability conditions. Using common slack matrix variables \(\mathbf {U}_{ij}(\mathbf {x}) = \mathbf {U}(\mathbf {x})\), \(\mathbf {W}_{ij}(\mathbf {x}) = \mathbf {W}(\mathbf {x})\) and \(\varvec{\Lambda }_j(\mathbf {x}) = \varvec{\Lambda }(\mathbf {x})\) for all i and j can reduce the computational demand, however, conservativeness may be introduced.

8.3.2 Stability Analysis with \(c \ne p\)

In this section, we consider the case that the number of rules and the membership functions of the polynomial fuzzy model and SDOF fuzzy controller are different. As a result, as discussed in other chapters, e.g. Chap.  5, it offers a greater design flexibility and lower complexity to the controller when less number of rules and less complicated membership functions are used for the controller. However, the stability analysis will be more complicated due to the mismatched premised membership functions.

We consider the SDOF fuzzy controller with c fuzzy rules and denote the membership functions as \(m_j\), \(j = 1, \ldots , c\). To deal with the mismatched premised membership functions \(w_i\) and \(m_j\), we use the upper bound \(\gamma _{ij}\) of \(w_im_j\), which is a scalar to be determined, such that the following inequality is satisfied:
$$\begin{aligned} \gamma _{ij} - w_im_j \ge 0 \quad \forall \; i, j, \mathbf {x}, \mathbf {y}(t_\gamma ). \end{aligned}$$
(8.30)
Considering the slack matrices \(0 \le \mathbf {H}_{ij}(\mathbf {x}) = \mathbf {H}_{ij}(\mathbf {x}) ^T \in \mathfrak {R}^{4N \times 4N}\) for all i and j, it obtains \(\sum _{i=1}^p \sum _{j=1}^c ( \gamma _{ij} - w_im_j ) \mathbf {H}_{ij}(\mathbf {x}) \ge 0\). From (8.26), we have
$$\begin{aligned} \displaystyle \hat{\varvec{\Xi }}(\mathbf {x})&= \sum _{i=1}^p\sum _{j=1}^c w_{i} m_{j}\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) \nonumber \\&\le \sum _{i=1}^p\sum _{j=1}^c w_{i} m_{j}\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) + \sum _{i=1}^p \sum _{j=1}^c ( \gamma _{ij} - w_im_j ) \mathbf {H}_{ij}(\mathbf {x}) \nonumber \\&= \sum _{i=1}^p\sum _{j=1}^c w_{i} m_{j} \big ( \hat{\varvec{\Xi }}_{ij}(\mathbf {x}) - \mathbf {H}_{ij}(\mathbf {x}) + \sum _{r=1}^p\sum _{s=1}^c \gamma _{rs} \mathbf {H}_{rs}(\mathbf {x}) \big ). \end{aligned}$$
(8.31)
It can be seen from (8.31) that \(\hat{\varvec{\Xi }}(\mathbf {x}) < 0\) leading to \(\dot{V}(t) < 0\) excluding for \(\mathbf {x} = \mathbf {0}\) can be achieved by \(\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) - \mathbf {H}_{ij}(\mathbf {x}) + \sum _{r=1}^p\sum _{s=1}^c \gamma _{rs} \mathbf {H}_{rs}(\mathbf {x}) < 0\) for all i and j. The stability analysis result for the SDOF PFMB control system (8.10) with the polynomial fuzzy model and SDOF fuzzy controller not sharing the same number of fuzzy rules is summarized in the following theorem.

Theorem 8.2

The SDOF PFMB control system (8.10), formed by a nonlinear plant represented by the polynomial fuzzy model (8.2) with p rules and the SDOF fuzzy controller (8.5) with c rules connected in a closed loop, is guaranteed to be asymptotically stable if there exist matrices \(\mathbf {M} = \mathbf {M}^T \in \mathfrak {R}^{N \times N}\), \(\mathbf {N}_j \in \mathfrak {R}^{m \times l}\) and \(\mathbf {X}_1 = \mathbf {X}_1^T = \left[ \begin{array}{cc} \mathbf {X}_{11} &{} \mathbf {0} \\ \mathbf {0} &{} \mathbf {X}_{22}\end{array} \right] \in \mathfrak {R}^{N \times N}\), and polynomial matrices \(\mathbf {H}_{ij}(\mathbf {x}) = \mathbf {H}_{ij}(\mathbf {x})^T \in \mathfrak {R}^{4N \times 4N}\), \(i = 1, 2, \ldots , p\), \(j = 1, 2, \ldots , c\) such that the following SOS conditions are satisfied.
$$\mathbf {r}^T(\mathbf {X}_1 - \upsilon _1\mathbf {I})\mathbf {r} \mathrm {\;is \;SOS},$$
$$\mathbf {r}^T(\mathbf {M} - \upsilon _2\mathbf {I})\mathbf {r} \mathrm {\;is \;SOS},$$
$$\mathbf {s}^T \mathbf {H}_{ij} \mathbf {s} \mathrm {\;is \;SOS} \; \forall \; i, j,$$
$$\begin{aligned}&-\mathbf {s}^T (\hat{\varvec{\Xi }}_{ij}(\mathbf {x}) - \mathbf {H}_{ij}(\mathbf {x}) + \sum _{r=1}^p\sum _{s=1}^c \gamma _{rs} \mathbf {H}_{rs}(\mathbf {x}) \\ {}&+ \upsilon _3(\mathbf {x})\mathbf {I} ) \mathbf {s} \mathrm {\;is \;SOS \; \forall } \; i\;, j, \end{aligned}$$
where \(\mathbf {r} \in \mathfrak {R}^{N}\) and \(\mathbf {s} \in \mathfrak {R}^{4N}\) are arbitrary vectors independent of \(\mathbf {x}\); \(\gamma _{ij}\), \(i = 1, \ldots , p\), \(j = 1, \ldots , c\) are predefined scalars satisfying \(\gamma _{ij} - w_im_j \ge 0\) for all i, j, \(\mathbf {x}\) and \(\mathbf {y}(t_\gamma )\); \(h_s > 0\) is the predefined sampling period; \(\xi \), \(\varepsilon _2\), \(\varepsilon _3 \ne 0\), \(\upsilon _1 > 0\) and \(\upsilon _2 > 0\) are predefined scalars; \(\upsilon _3(\mathbf {x}) > 0\) is a scalar polynomial and the feedback gains are chosen as \(\mathbf {G}_j = \mathbf {N}_j\mathbf {X}_{11}^{-1}\), \(j = 1, \ldots , c\).

Remark 8.11

In the sampled-data fuzzy control paradigm, employing the membership functions of the fuzzy model for the sampled-data fuzzy controller cannot help relax the stability analysis. Because of the sampling activity, only system states or outputs at the sampling instant can be used for the sampled-data fuzzy controller. Consequently, \(m_j\) will depend on \(\mathbf {x}(t_\gamma )\) (full-state feedback case) or \(\mathbf {y}(t_\gamma )\) (output-feedback case) causing mismatch in the membership grades of the fuzzy model and sampled-data fuzzy controller even \(m_i = w_i\) for all i is considered.

Remark 8.12

Because of the mismatched membership grades between polynomial fuzzy model and sampled-data polynomial fuzzy controller, and \(m_j\) is independent of \(w_i\), it is difficult to find the non-trivial upper bounds \(\rho _i\) satisfying (8.27) for the case of \(c = p\) and \(\gamma _{ij}\) satisfying (8.30) for the case of \(c \ne p\). The trivial upper bounds \(\rho _i = 1\) and \(\gamma _{ij} = 1\) can be obtained, which cannot help relax the stability conditions. However, non-trivial upper bounds \(\rho _i\) or \(\gamma _{ij}\) can be obtained by assuming that the closed-loop system is slow enough such that the deviation between \(w_i\) and \(m_j\) is small. Although non-trivial upper bounds can be achieved, verification is required to show that the assumption of slow closed-loop system is valid. Further elaboration can be found in Sect. 8.4.

Remark 8.13

The largest sampling period can be obtained by gradually increasing \(h_s\) during solving the solution of the stability conditions until no feasible solution is found.

8.4 Simulation Examples

Two simulation examples are given in this section to demonstrate the design procedure and effectiveness of the proposed SOS-based SDOF fuzzy control approach using Theorems 8.1 and 8.2. The first examples employing output state-feedback control consider the SDOF fuzzy controller sharing the same number of fuzzy rules as the polynomial fuzzy model. The polynomial fuzzy model and the SDOF fuzzy controller have two fuzzy rules for each. For demonstration purposes, we choose \(w_1(x_1(t)) = \mu _{M_1^1}(x_1(t)) = e^{\frac{{-x_1(t)}^{2}}{2^2}}\) and \(w_2(x_1(t)) = \mu _{M_1^2}(x_1(t)) = 1 - w_1(x_1(t))\) as the membership functions for the polynomial fuzzy model while \(m_1(x_1(t_\gamma )) = \mu _{N_1^1}(x_1(t_\gamma )) = e^{\frac{{-x_1(t_\gamma )}^{2}}{2^2}}\) and \(m_2(x_1(t_\gamma )) = \mu _{N_1^2}(x_1(t_\gamma )) = 1 - m_1(x_1(t_\gamma ))\) are the membership functions for the SDOF fuzzy controller.

In the second example, we consider the polynomial fuzzy model and the SDOF fuzzy controller not sharing the same number of fuzzy rules. The polynomial fuzzy model is considered to have three fuzzy rules with the membership functions of \(w_1(x_1(t)) = 1 - \frac{1}{1 + e^\frac{-(x_1(t)+2)}{2}}\), \(w_2(x_1(t)) = 1 - w_1(x_1(t)) - w_3(x_1(t))\) and \(w_3(x_1(t)) = \frac{1}{1 + e^\frac{-(x_1(t)-2)}{2}}\). The SDOF fuzzy controller has two fuzzy rules with the membership functions of \(m_1(x_1(t_\gamma )) = 1 - \frac{1}{1 + e^\frac{-x_1(t_\gamma )}{0.8}}\) and \(m_2(x_1(t_\gamma )) = 1-m_1(x_1(t_\gamma ))\).

Example 8.1

(\(c = p\)) Consider a nonlinear plant represented by a 2-rule polynomial fuzzy model in the form of (8.2) and (8.3), where \(\hat{\mathbf {x}} = \mathbf {x}\) and the system, input and output matrices are chosen as follows:
$$\mathbf {A}_1(x_1) = \left[ \begin{array}{cc} 3.85-0.68x_1 &{} 1 \\ 1 &{} -10.38-0.85x_1^2 \end{array}\right] $$
$$\mathbf {A}_2(x_1) = \left[ \begin{array}{cc} 0.21-4.16x_1-0.21x_1^2 &{} 0.02 \\ 1-0.25x_1 &{} -0.25-0.72x_1^2 \end{array}\right] ,$$
$$\mathbf {B}_1(x_1) = \left[ \begin{array}{c} 1+0.32x_1^2 \\ 0 \end{array}\right] ,$$
$$\mathbf {B}_2(x_1) = \left[ \begin{array}{c} 8+0.1x_1+0.02x_1^2 \\ 0 \end{array}\right] $$
and
$$\mathbf {C} = [1 \quad 0].$$
The chosen output matrix \(\mathbf {C}\), according to Lemma  7.1, offers \(\varvec{\Gamma } = \left[ \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \end{array}\right] \). The membership functions of the polynomial fuzzy model are chosen as follows:
$$w_1(x_1(t)) = \mu _{M_1^1}(x_1(t)) = e^{\frac{{-x_1(t)}^{2}}{2^2}}$$
and
$$w_2(x_1(t)) = \mu _{M_1^2}(x_1(t)) = 1 - w_1(x_1(t)).$$
We employ a 2-rule SDOF fuzzy controller in the form of (8.2) to stabilize the nonlinear plant, where the membership functions are chosen as follows:
$$m_1(x_1(t_\gamma )) = \mu _{N_1^1}(x_1(t_\gamma )) = e^{\frac{{-x_1(t_\gamma )}^{2}}{2^2}}$$
and
$$m_2(x_1(t_\gamma )) = \mu _{N_1^2}(x_1(t_\gamma )) = 1 - m_1(x_1(t_\gamma )).$$
It is assumed that \(\dot{x}_1(t) \in [-20, \quad 20]\). Considering the time period of \(t_\gamma \) to t, where \(t_\gamma \le t \le t_\gamma + h_s\), we have \(x_1(t) - x_1(t_\gamma ) = \int _{t_\gamma }^{t}\dot{x}_1(t) dt\) leading to \(x_1(t_\gamma ) - 20h_s \le x_1(t) \le x_1(t_\gamma ) + 20h_s\). By choosing the sampling period \(h_s = 0.02\) (sampling frequency of 50 Hz), it is found numerically that \(\rho _1 = \rho _2 = 0.2394\) satisfies \(m_j(x_1(t_\gamma )) -w_j(x_1(t)) + \rho _j \ge 0\) for j = 1, 2.
As the number of rules of the polynomial fuzzy model and the SDOF fuzzy controller are the same, the SOS-based stability conditions in Theorem 8.1 are employed to determine the system stability and obtain the feedback gains. Choosing \(\varepsilon _2 = 0\), \(\varepsilon _3 = 60\), \(\xi = 1\), using the third-party Matlab toolbox SOSTOOLS [49], the feedback gains are found as follows:
$$G_1 = -5.4832$$
and
$$G_2 = -4.0533.$$
The phase plots of the SDOF PFMB control system with various initial conditions are shown in Fig. 8.2 which demonstrates that the nonlinear plant can be stabilized successfully by the proposed SDOF fuzzy controller. It can be verified from the simulation result that \(\dot{x}_1\) is within the predefined operating range showing the values of \(\rho _1\) and \(\rho _2\) are valid.
For demonstration purposes, the time response of the system states \(x_1\) and \(x_2\), and the control signal with \(\mathbf {x}(0) = [1 \quad 0]^T\) are shown in Figs. 8.3 and 8.4, respectively. It can be seen that the control signal is a staircase signal with value being kept constant during the sampling period.
Fig. 8.2

Phase plot of \(x_1(t)\) and \(x_2(t)\) for Example 8.1, where ‘\(\circ \)’ indicates the initial condition

Fig. 8.3

Time responses of \(x_1(t)\) and \(x_2(t)\) for Example 8.1

Fig. 8.4

Control signal u(t) for Example 8.1

Example 8.2

(\(c \ne p\)) A 3-rule polynomial fuzzy model is considered, where \(\hat{\mathbf {x}} = \mathbf {x}\), and the system, input and output matrices are chosen as follows:
$$\mathbf {A}_1(x_1) = \left[ \begin{array}{cc} 0.55+0.68x_1 &{} -0.6 \\ 1 &{} -10.38-0.85x_1^2 \end{array}\right] ,$$
$$\mathbf {A}_2(x_1) = \left[ \begin{array}{cc} 3.65-0.02x_1-0.33x_1^2 &{} -0.02x_1 \\ 1-0.25x_1 &{} -0.25 \end{array}\right] ,$$
$$\mathbf {A}_3(x_1) = \left[ \begin{array}{cc} 1.31-0.15x_1 &{} 0.5 \\ 1 &{} -5.16-0.17x_1^2 \end{array}\right] ,$$
$$\mathbf {B}_1(x_1) = \left[ \begin{array}{c} 1+0.12x_1^2 \\ 0 \end{array}\right] ,$$
$$\mathbf {B}_2(x_1) = \left[ \begin{array}{c} 5-0.06x_1+0.41x_1^2 \\ 1 \end{array}\right] $$
,
$$\mathbf {B}_3(x_1) = \left[ \begin{array}{c} 10+0.02x_1^2 \\ -1 \end{array}\right] $$
and
$$\mathbf {C} = [1 \quad 0].$$
With the chosen output matrix, according to Lemma  7.1, \(\varvec{\Gamma } = \left[ \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \end{array}\right] \) is obtained. The membership functions of the polynomial fuzzy model are chosen as follows:
$$w_1(x_1(t)) = 1 - \frac{1}{1 + e^\frac{-(x_1(t)+2)}{2}},$$
$$w_2(x_1(t)) = 1 - w_1(x_1(t)) - w_3(x_1(t))$$
and
$$w_3(x_1(t)) = \frac{1}{1 + e^\frac{-(x_1(t)-2)}{2}}.$$
We employ a 2-rule SDOF fuzzy controller to stabilize the nonlinear plant where the membership functions are chosen as follows:
$$m_1(x_1(t_\gamma )) = 1 - \frac{1}{1 + e^\frac{-x_1(t_\gamma )}{0.8}}$$
and
$$m_2(x_1(t_\gamma )) = 1-m_1(x_1(t_\gamma )).$$
Similar to the first example, it is assumed that \(\dot{x}_1(t) \in [-20, \quad 20]\). By choosing the sampling period \(h_s = 0.02\) (sampling frequency of 50 Hz), it is found numerically that \(\gamma _{11} = \gamma _{32} = 0.9960\), \(\gamma _{12} = \gamma _{31} = 0.0480\) and \(\gamma _{21} = \gamma _{22} = 0.7181\) satisfy \(\gamma _{ij} - w_i(x_1(t))m_j(x_1(t_\gamma )) \ge 0\) for \(i = 1, 2, 3\) and \(j = 1, 2\). As the number of rules between the polynomial fuzzy model and the SDOF fuzzy controller are different, the SOS-based stability conditions in Theorem 8.2 are employed to determine the system stability and obtain the feedback gains. Choosing \(\varepsilon _2 = 0\), \(\varepsilon _3 = 60\), \(\xi = 10\), the feedback gains are found as follows:
$$G_1 = -1.8480$$
and
$$G_2 = -0.9913.$$
The SDOF fuzzy controller with the obtained feedback gains is employed to stabilize the nonlinear plant. The phase plots of \(x_1\) and \(x_2\) for the SDOF PFMB control system with various initial conditions are shown in Fig. 8.5. The state response and the control signal corresponding to \(\mathbf {x}(0) = [1 \quad 0]^T\) are shown in Figs. 8.6 and 8.7, respectively.
Fig. 8.5

Phase plot of \(x_1(t)\) and \(x_2(t)\) for Example 8.2, where ‘\(\circ \)’ indicates the initial condition

Fig. 8.6

Time responses of \(x_1(t)\) and \(x_2(t)\) for Example 8.2

Fig. 8.7

Control signal u(t) for Example 8.2

8.5 Conclusion

This chapter has investigated the stability of sampled-data output feedback polynomial fuzzy model-based control systems consisting of a nonlinear plant represented by a polynomial fuzzy model and a sampled-data output-feedback fuzzy controller connected in a closed-loop. Two cases of sampled-data output feedback polynomial fuzzy model-based control systems, namely \(c = p\) and \(c \ne p\), have been considered. Membership function-dependent analysis approach has been proposed to investigate the system stability by considering the upper-bound information of the membership functions. SOS-based stability conditions have been obtained to guarantee the system stability and facilitate the synthesis of the sampled-data output-feedback fuzzy controller. Simulation examples have been given to illustrate the effectiveness of the proposed approach.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of InformaticsKing’s College LondonLondonUK

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