Simplified robust fuzzy output regulator design for discrete-time nonlinear systems

Abstract

This paper proposes the simplified robust T-S fuzzy output regulator for discrete-time affine nonlinear systems in the presence of parametric uncertainties and external disturbance. First, a virtual desired variable (VDV) based fuzzy output regulator is developed by involving integral compensation and virtual desired state profile. Meanwhile, the sufficient condition of asymptotic output regulation is formulated in LMIs. Since the VDV-based fuzzy regulator is unavailable for the system corrupted by parametric uncertainty and external disturbance, we further modify the above one to a simplified non-VDV fuzzy output regulator which only consists of an integral-type PDC controller without VDV calculation. The asymptotic output regulation is achieved for the uncertain system without disturbance. Afterward, the robustness is enhanced to satisfy an H^∞ criterion for attenuating the effect of disturbance and modelling error via robust LMI-based gain design. Compared with traditional methods, the proposed fuzzy controller is more simplified and robust without using error coordinate transformation and solving regulation equations. Finally, a truck-trailer system is taken as applications to show expected control performance.

Keywords

T-S fuzzy contol discrete-time nonlinear system output regulation linear matrix inequality (LMI)

1 Introduction

Over the past decades, intelligent control method like neural networks [1, 2] and fuzzy logic control [3 –6] have been successfully applied to solve control problems of complex or poorly modelled nonlinear systems. Among them, the Takagi-Sugeno (T-S) fuzzy model based control is the most popular approach for controlling nonlinear systems because of its ability to approximate complex nonlinear systems by a set of linear subsystems [7] and to systematically design parallel distributed compensation (PDC) [8]. Moreover, the stability can be strictly proven by Lyapunov direct method, while the control gains are obtained via the powerful linear matrix inequality (LMI) technique [9]. For example, T-S fuzzy model-based stabilization control has been proposed in [10 –13]. Discrete-time nonlinear systems also receive much attention by T-S fuzzy model control, e.g., [14 –16]. Based on advantaged LMI techniques, some papers further consider the robustness to uncertainty [17 –19]. However, these works require complex stability analysis, such as using multiple fuzzy Lyapunov function or polynomial Lyapunov function. In other words, the robust control of discrete-time T-S fuzzy systems is still required to be improved.

In contrast to continuous-time fuzzy systems, only a few of papers develop the output regulation control for discrete-time T-S fuzzy systems, e.g., [20 –22]. Generally speaking, complex coordinate transformation is required to construct an error system for the regulator design. To avoid this drawback, the regulation theory [23] is extended to solve the output regulation control problem of discrete-time T-S fuzzy systems, e.g., [24, 25]. Nevertheless, these papers have to solve some regulation equations for exact output regulation and are failed when considering uncertainty/disturbance or bias terms, cf. [26]. This means that the regulation theory based approaches are difficultly applied on more general discrete-time systems. To remove the regulation equations, a virtual fuzzy reference model design concept is proposed in [22]. However, the controller order of [22] becomes higher due to adding a virtual dynamic reference model to track. Moreover, the compensation of uncertainty and disturbance is complicated. In light of the above discussion, we need a new design method for the output regulation of discrete-time T-S fuzzy systems in the simultaneous presence of uncertainty, disturbance, and bias term. This problem is still a challenging open research.

The above motivates us to further develop a simplified T-S fuzzy regulator for discrete-time affine nonlinear systems subject to parametric uncertainties and external disturbance. First, the integral-type compensation is introduced to cope with the bias terms in the control system. Meanwhile, a set of virtual desired variables (VDVs) is involved for the augmented control system. Different to the virtual fuzzy reference model based design [22], the VDV of this paper is a static solution so that the controller order is smaller. By representing the error system in a T-S fuzzy model, the controller design is reduced to a stabilization problem. Furthermore, the simplified T-S fuzzy controller is derived to avoid solving VDVs, where the operational point value is compensated by the error integrator. Then, the asymptotic output regulation control is guaranteed for non-perturbed systems via proper controller gain compensation even if considering system uncertainty. In the presence of external disturbance, an H^∞ robust performance is also achieved from the LMI-based robustness design. Since the coordinate transformation, VDVs, fuzzy reference model, and regulation equations are removed in the simplified fuzzy controller, the implementation of the proposed controller is easier and more straightforward than traditional methods. In addition, compared with [24, 25], the bias terms and uncertainty are allowed in this paper.

The rest of this paper is organized as follows. First, a VDV-based T-S fuzzy regulator design is introduced for discrete affine nonlinear system in Section 2. In Section 3, the synthesis of the simplified non-VDV fuzzy regulator is presented to remove the VDV design and to enhance robustness. Numerical simulations are carried out for a truck-trailer system in Section 4. Finally, the conclusions and future works are made in Section 5.

Notations:Rⁿ and R^n×m denote the n-dimensional Euclidean space and the set of all n × m real matrices, respectively. The superscript T stands for matrix and vector transpose. The notation P = P^T > 0 (≥0) means that the matrix P is symmetric and positive (semi-) definite. I and 0 represent the identity matrix and a zero matrix with appropriate dimensions, respectively. Furthermore, “∗” denotes the symmetric terms in a block matrix and diag{⋯} denotes the block-diagonal matrix.

Preliminary: The following Lemmas are needed in this paper.

Lemma 1.LetM_ibe a matrix andμ_ibe a normalized weighting. For 0 ≤ μ_i ≤ 1, the following property is satisfied $\sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} μ_{j} M_{i}^{T} M_{j} \leq \sum_{i = 1}^{r} μ_{i} M_{i}^{T} M_{i}$ (1)

2 T-S fuzzy regulator

First of all, a VDV-based T-S fuzzy regulator is introduced for the output regulation control. Consider a discrete-time affine nonlinear system described by the following dynamic equation: $\begin{matrix} x_{p} (k + 1) & = & f (x_{p} (k)) + g (x_{p} (k)) u (k) \\ + η + J ω (k) \end{matrix}$ (2) $y (k) = h (x_{p} (k))$ (3) where x_p (k) ∈ Rⁿ, u (k) ∈ R^q, y (k) ∈ R^m (q ≤ m) are the system state, the input, and the output, respectively; f (·), g (·) and h (·) are nonlinear function vectors with appropriate dimensions; η is a bias term; ω (k) denotes an external disturbance; J is a known matrix; and k is the time index. For this system, the control objective is to drive the output y (k) to a desired constant value y_d. To achieve zero steady-state regulation error, the integral-type fuzzy compensation is introduced to cope with the bias term and uncertainty. A new state variable is added to account for the accumulation of output regulation error. In light of this, the new state variable is defined as follows: $x_{e} (k) = \sum_{i = 1}^{k} (y_{d} - y (i - 1))$ (4) This means that the dynamics of the error integrator is $x_{e} (k + 1) = x_{e} (k) + y_{d} - y (k)$ (5) By combining (2), (3) and (5), the overall augmented dynamics becomes $\begin{matrix} x_{p} (k + 1) & = & f (x_{p} (k)) + g (x_{p} (k)) u (k) + η + J ω (k) \\ x_{e} (k + 1) & = & x_{e} (k) + y_{d} - h (x_{p} (k)) \end{matrix}$ (6) In this augmented dynamic form, the uncertainty and disturbance can be compensated in an easy and straightforward form discussed later. Let the new state vector $x (k) = {[x_{p}^{T} (k) x_{e}^{T} (k)]}^{T}$ , the state-space form of the above dynamics can be rewritten below:

$\begin{matrix} x (k + 1) & = & \tilde{A} (x_{p} (k)) x (k) + \tilde{B} (x_{p} (k)) u (k) \\ + \tilde{η} + \tilde{J} ω (k) \end{matrix}$ (7) where $\tilde{A} (x_{p} (k))$ , $\tilde{B} (x_{p} (k))$ , $\tilde{J}$ , and $\tilde{η}$ are given as follows: $\begin{matrix} \tilde{A} (x_{p} (k)) & = & [\begin{matrix} A (x_{p} (k)) & 0 \\ - C (x_{p} (k)) & I \end{matrix}], \\ \tilde{B} (x_{p} (k)) & = & [\begin{matrix} g (x_{p} (k)) \\ 0 \end{matrix}], \\ \tilde{J} & = & [\begin{matrix} J \\ 0 \end{matrix}], \tilde{η} = [\begin{matrix} η \\ y_{d} \end{matrix}] . \end{matrix}$ Note that the matrices A (x_p (k)), C (x_p (k)) and g (x_p (k)) are well defined in the discussed region Ω. The terms satisfy A (x_p (k)) x_p (k) = f (x_p (k)) and C (x_p (k)) x_p (k) = h (x_p (k)). If the output regulation control is achieved, then there is an operational point $x_{d} = [x_{pd}^{T} x_{ed}^{T}]$ for y (k) = y_d, where x_pd is an operational point for the state x_p, and x_ed is the operational point for the integral state x_e. Since we only need to control the output, the operational point x_d can be designed later. As a result, x_d is called the virtual desired variable (VDV) for the system state. In addition, the operational point x_d will be compensated by the error integrator, such that the controller can be simplified later.

Let us define the error states ${\tilde{x}}_{p} (k) = x_{p} (k) - x_{pd}$ , ${\tilde{x}}_{e} (k) = x_{e} (k) - x_{ed}$ , and $\tilde{x} (k) = x (k) - x_{d}$ . The nonlinear regulation error system is expressed in the form: $\begin{matrix} \tilde{x} (k + 1) & = & \tilde{A} (x_{p} (k)) \tilde{x} (k) + \tilde{B} (x_{p} (k)) u (k) \\ + \tilde{A} (x_{p} (k)) x_{d} + \tilde{η} - x_{d} + \tilde{J} ω (k) \\ = & \tilde{A} (x_{p} (k)) \tilde{x} (k) + \tilde{B} (x_{p} (k)) τ (k) + φ (k) \\ + \tilde{J} ω (k) \end{matrix}$ (8) where the terms $\tilde{B} (x_{p} (k)) τ (k)$ and φ (k) satisfy

$\begin{matrix} φ (k) & = & \tilde{B} (x_{p} (k)) u (k) + \tilde{A} (x_{p} (k)) x_{d} + \tilde{η} - x_{d} \\ - \tilde{B} (x_{p} (k)) τ (k) \end{matrix}$ (9) If letting φ (k) =0, the term τ (k) becomes a virtual control input to (8). Next, according to the T-S fuzzy modelling method [27] and letting φ (k) =0, the nominal part of the nonlinear regulation error system (8) can be represented by the following T-S fuzzy rules:

$\begin{matrix} Plant Rule i : \\ IF z_{1} (k) is F_{1 i} and \dots and z_{ℓ} (k) is F_{ℓ i} \\ THEN \tilde{x} (k + 1) = {\tilde{A}}_{i} \tilde{x} (k) + {\tilde{B}}_{i} τ (k), \\ i = 1, 2, . . ., r \end{matrix}$ (10) where z₁ (k) ∼ z_ℓ (k) are premise variables which consist of proper state variables of the system; F_ji (j = 1, 2, …, ℓ) are the fuzzy sets; r is the number of fuzzy rules; ${\tilde{A}}_{i}$ and ${\tilde{B}}_{i}$ are proper subsystem matrices. Using singleton fuzzifier, product inference, and weighted defuzzifier, the inferred output of the fuzzy system (10) is $\tilde{x} (k + 1) = \sum_{i = 1}^{r} μ_{i} (z (k)) {\tilde{A}}_{i} \tilde{x} (k) + \sum_{i = 1}^{r} μ_{i} (z (k)) {\tilde{B}}_{i} τ (k)$ (11) where z (k) = [z₁ (k) z₂ (k) ⋯ z_ℓ (k)] ^T and $μ_{i} (z (k)) = \frac{\prod_{j = 1}^{ℓ} F_{ji} (z_{j} (k))}{\sum_{i = 1}^{r} \prod_{j = 1}^{ℓ} F_{ji} (z_{j} (k))} \geq 0$ for all k in the regarded discussion region Ω; and $\sum_{i = 1}^{r} μ_{i} (z (k)) = 1$ . In the fuzzy model (11), the membership function F_ji (z_j (k)) and the subsystem matrices ${\tilde{A}}_{i}$ , ${\tilde{B}}_{i}$ will be properly chosen such that $\sum_{i = 1}^{r} μ_{i} (z (k)) {\tilde{A}}_{i} = \tilde{A} (x_{p} (k))$ and $\sum_{i = 1}^{r} μ_{i} (z (k))$ ${\tilde{B}}_{i} = \tilde{B} (x_{p} (k))$ . If considering fuzzy modelling error, the control system (8) with φ (k) =0 can be expressed in terms of the fuzzy inferred output (11) as follows: $\begin{matrix} \tilde{x} (k + 1) & = & \sum_{i = 1}^{r} μ_{i} (z (k)) {\tilde{A}}_{i} \tilde{x} (k) + \sum_{i = 1}^{r} μ_{i} (z (k)) {\tilde{B}}_{i} τ (k) \\ + {\tilde{J}}_{d} ϑ (k) \end{matrix}$ (12) where ${\tilde{J}}_{d} = [\begin{matrix} \tilde{J} & J_{m} \end{matrix}]$ is a known matrix; and $ϑ (k) = [\begin{matrix} ω^{T} (k) & ω_{m}^{T} (k) \end{matrix}]^{T}$ with the fuzzy modelling error ω_m (k).

At this step, if the perturbed term ${\tilde{J}}_{d} ϑ (k)$ is assumed to be zero, then the state x (k) is driven to the VDV x_d by designing the virtual control input τ (k). The output regulation control is thus transformed to a stabilization problem, while the VDV x_d and the control input u (k) is determined from the constraints (9) and C (x_pd) x_pd = y_d. According to the IF-THEN fuzzy rules in (10), the T-S fuzzy stabilization law is set to: $\begin{matrix} Controller Rule i : \\ IF z_{1} (k) is F_{1 i} and \dots and z_{ℓ} (k) is F_{ℓ i} \\ THEN τ (k) = - K_{i} \tilde{x} (k), i = 1, 2, \dots, r \end{matrix}$ where the $K_{i}^{'} s$ are feedback gains. The inferred output of the fuzzy controller is $τ (k) = - \sum_{i = 1}^{r} μ_{i} (z (k)) K_{i} \tilde{x} (k)$ (13) By sub stituting (13) into the regulation control system (12), the closed-loop system is thus obtained below: $\tilde{x} (k + 1) = \sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (z (k)) μ_{j} (z (k)) G_{ij} \tilde{x} (k) + {\tilde{J}}_{d} ϑ (k)$ (14) where $G_{ij} = {\tilde{A}}_{i} - {\tilde{B}}_{i} K_{j}$ . The remaining design for the output regulator is to determine x_d, u (k) and feedback gain K_j. For simplification, the perturbed term ${\tilde{J}}_{d} ϑ (k)$ is firstly assumed to be zero for the gain design here.

Consider a quadratic Lyapunov function candidate as $V (\tilde{x} (k)) = {\tilde{x}}^{T} (k) P \tilde{x} (k)$ , where P is a symmetric positive-definite matrix. Taking the time difference of $V (\tilde{x} (k))$ along the dynamics (14) with ${\tilde{J}}_{d} ϑ (k) = 0$ leads to

$\begin{matrix} Δ V (\tilde{x} (k)) & = & \sum_{i = 1}^{r} \sum_{j = 1}^{r} \sum_{s = 1}^{r} \sum_{t = 1}^{r} μ_{i} (x_{p} (k)) \\ \times μ_{j} (x_{p} (k)) μ_{s} (x_{p} (k)) μ_{t} (x_{p} (k)) {\tilde{x}}^{T} (k) \\ {G_{ij}^{T} {PG}_{st} - P} \tilde{x} (k) \end{matrix}$ (15) According to Lemma 1, $Δ V (\tilde{x} (k))$ is further reduced to $\begin{matrix} Δ V (\tilde{x} (k)) \\ \leq \sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (z (k)) μ_{j} (z (k)) {\tilde{x}}^{T} (k) {G_{ij}^{T} {PG}_{ij} \\ - P + DPD} \tilde{x} (k) - {\tilde{x}}^{T} (k) DPD \tilde{x} (k) \end{matrix}$ where D is a diagonal positive-definite matrix chosen by designer for adjusting the control performance. Therefore, $Δ V (\tilde{x} (k)) < - {\tilde{x}}^{T} (k) DPD \tilde{x} (k)$ under the stability condition: $G_{ij}^{T} {PG}_{ij} - P + DPD < 0,$ (16) for all i, j. This implies that the error trajectory $\tilde{x} (k)$ asymptotically converges to zero as k→ ∞ once the inequality (16) is satisfied and ${\tilde{J}}_{d} ϑ (k) = 0$ . Afterward, the stability condition (16) is converted to LMIs to determine the control gain K_j. After pre-multiplying and post-multiplying (16) by X = P⁻¹ and using Schur’s complement, we obtain the following LMIs: $[\begin{matrix} - X & X {\tilde{A}}_{i}^{T} - M_{j}^{T} {\tilde{B}}_{i}^{T} & XD \\ {\tilde{A}}_{i} X - {\tilde{B}}_{i} M_{j} & - X & 0 \\ DX & 0 & - X \end{matrix}] < 0,$ (17) for i, j = 1, 2, …, r, where M_j = K_jX. In other word, if the above LMIs have a feasible solution, then the control gain K_j is obtained from M_jX⁻¹ = K_j.

Next, the design procedure for the virtual desired variable x_d and the practical controller input u (k) is performed. According to the constraint φ (k) =0 in (9), we partition the matrices as follows: $\begin{matrix} x_{d} = [\begin{matrix} \bar{x_{d}} \\ - - \\ \underline{x_{d}} \end{matrix}], \tilde{B} (x_{p} (k)) = [\begin{matrix} \bar{B} (x_{p} (k)) \\ - - - - - \\ 0 \end{matrix}], \\ \tilde{A} (x_{p} (k)) = [\begin{matrix} \bar{A} (x_{p} (k)) \\ - - - - - \\ \underline{A} (x_{p} (k)) \end{matrix}], \tilde{η} = [\begin{matrix} \bar{η} \\ - - \\ \underline{η} \end{matrix}], \end{matrix}$ where $\bar{x_{d}} \in R^{q}$ , $\underline{x_{d}} \in R^{(n + m - q)}$ , $\bar{B} (x_{p} (k)) \in R^{q \times q}$ is the nonzero term of g (x_p (k)) ∈ R^n×q and 0^{inR (n +m − q) ×q} is a zero matrix; $\bar{A} (x_{p} (k)) \in R^{q \times (n + m)}$ ; $\underline{A} (x_{p} (k)) \in R^{(n + m - q) \times (n + m)}$ ; $\bar{η} \in R^{q \times 1}$ ; $\underline{η} R^{(n + m - q) \times 1}$ . The condition φ (k) =0 in (9) can be rewritten in the form:

$\begin{matrix} [\begin{matrix} \bar{B} (x_{p} (k)) (u (k) - τ (k)) \\ - - - - - - - - - - \\ 0 \end{matrix}] \\ = [\begin{matrix} - \bar{A} (x_{p} (k)) x_{d} - \bar{η} + \bar{x_{d}} \\ - - - - - - - - - - \\ - \underline{A} (x_{p} (k)) x_{d} - \underline{η} + \underline{x_{d}} \end{matrix}] \end{matrix}$ (18) From the upper part of (18), the VDV-based T-S fuzzy regulation law is formed as follows: $u (k) = τ (k) + {\bar{B}}^{- 1} (x_{p} (k)) {{\bar{x}}_{d} - \bar{η} - \bar{A} (x_{p} (k)) x_{d}}$ (19) where τ (k) has been given in (13); x_d is solved from the lower part of (18), i.e., $\underline{A} (x_{p} (k)) x_{d} + \underline{η} - \underline{x_{d}} = 0$ (20) Since the number of the above constraint is lower than the dimension of x_d, there exist m redundant freedoms for solutions of x_d, i.e., x_ed is unlimited for the controller under the output regulation objective y_d = C (x_pd) x_pd. Overall, the result is summarized below.

Proposition 1. By using the VDV-based fuzzy controller (13), (19) and solving LMIs (17) and VDVs of (20), asymptotically stable output regulation is assured for the closed-loop regulation system (14) without disturbance and fuzzy modelling error. □

In order to demonstrate the implementation procedure of the T-S fuzzy regulator, the following example is given.

Example. Consider a nonlinear truck-trailer system [28] described by the dynamic model: $\begin{matrix} x_{1} (k + 1) & = & (1 - \frac{ν T_{s}}{l_{2}}) x_{1} (k) + (\frac{ν T_{s}}{l_{1}}) u (k) \\ x_{2} (k + 1) & = & x_{2} (k) + \frac{ν T_{s}}{l_{2}} x_{1} (k) + γ \\ x_{3} (k + 1) & = & x_{3} (k) + ν T_{s} sin (x_{2} (k) + \frac{ν T_{s}}{2 l_{2}} x_{1} (k)) \end{matrix}$ where x₁ (k) is the angle difference between truck and trailer; x₂ (k) is the angle between the trailer and x-axis; x₃ (k) is the vertical position of the rear end of trailer; u (k) is the steering angle of the front wheels of the truck, which is the control input of this system; ν, T_s, l₁, l₂ are system parameters; and γ is a system bias, i.e., η = [0 γ 0] ^T in this example.

The control objective is to drive the output y (k) = x₃ (k) to a desired value y_d. For the regulation control purpose, the integral error is set to $x_{e} (k + 1) = x_{e} (k) + y_{d} - y (k)$ (21) By letting x (k) = [x₁ (k) x₂ (k) x₃ (k) x_e (k)] ^T, the overall system dynamics is written in the form: $\begin{matrix} x (k + 1) & = & [\begin{matrix} 1 - \frac{ν T_{s}}{l_{2}} & 0 & 0 & 0 \\ \frac{ν T_{s}}{l_{2}} & 1 & 0 & 0 \\ \frac{(ν T_{s})^{2}}{2 l_{2}} σ (z (k)) & ν T_{s} σ (z (k)) & 1 & 0 \\ 0 & 0 & - 1 & 1 \end{matrix}] x (k) \\ + [\begin{matrix} \frac{ν T_{s}}{l_{1}} \\ 0 \\ 0 \\ 0 \end{matrix}] u (k) + [\begin{matrix} 0 \\ γ \\ 0 \\ y_{d} \end{matrix}] \\ \equiv & \tilde{A} (x_{p} (k)) x (k) + \tilde{B} (x_{p} (k)) u (k) + \tilde{η} \end{matrix}$ (22) where $σ (z (k)) = \frac{sin (z (k))}{z (k)}$ with $z (k) = x_{2} (k) + \frac{ν T_{s}}{2 l_{2}} x_{1} (k)$ is well-defined in the discussion region Ω ≡ (− π π); and the system matrices $\tilde{A} (x_{p}), \tilde{B} (x_{p}), \tilde{η}$ are thus defined with appropriate dimensions. From the state-space model form (22), the term z (k) is taken as the premise variable. According to the fuzzy modelling method [27], the membership functions are chosen such that $\tilde{A} (x_{p} (k)) = \sum_{i = 1}^{r} μ_{i} (z (k)) {\tilde{A}}_{i}$ , and $\tilde{B} (x_{p} (k)) = \sum_{i = 1}^{r} μ_{i} (z (k)) {\tilde{B}}_{i}$ . Then, (22) can be exactly represented by the following T-S fuzzy model:

$\begin{matrix} Plant Rule i : \\ IF z (k) is F_{i} \\ THEN x (k + 1) = {\tilde{A}}_{i} x (k) + {\tilde{B}}_{i} x (k) + \tilde{η}, \\ i = 1, 2 \end{matrix}$ (23) where the fuzzy sets are F₁ = {about 0 rad} and F₂ = {about ± π rad} with corresponding membership functions: $F_{1} (z (k)) = \frac{σ (z (k)) - β}{α - β}, F_{2} (z (k)) = \frac{α - σ (z (k))}{α - β}$ where $α \equiv \max_{z (k) \in Ω} σ (z (k))$ , $β \equiv \min_{z (k) \in Ω} σ (z (k))$ . From the above, F₁ + F₂ = 1, σ (z (k)) = F₁α + F₂β. The subsystem matrices of fuzzy rules are given by: $\begin{matrix} {\tilde{A}}_{1} & = & [\begin{matrix} 1 - \frac{ν T_{s}}{l_{2}} & 0 & 0 & 0 \\ \frac{ν T_{s}}{l_{2}} & 1 & 0 & 0 \\ α \frac{(ν T_{s})^{2}}{2 l_{2}} & α ν T_{s} & 1 & 0 \\ 0 & 0 & - 1 & 1 \end{matrix}], {\tilde{B}}_{1} = [\begin{matrix} \frac{ν T_{s}}{l_{1}} \\ 0 \\ 0 \\ 0 \end{matrix}], \\ {\tilde{A}}_{2} & = & [\begin{matrix} 1 - \frac{ν T_{s}}{l_{2}} & 0 & 0 & 0 \\ \frac{ν T_{s}}{l_{2}} & 1 & 0 & 0 \\ β \frac{(ν T_{s})^{2}}{2 l_{2}} & β ν T_{s} & 1 & 0 \\ 0 & 0 & - 1 & 1 \end{matrix}], {\tilde{B}}_{2} = [\begin{matrix} \frac{ν T_{s}}{l_{1}} \\ 0 \\ 0 \\ 0 \end{matrix}], \\ \tilde{η} & = & {[0 γ 0 y_{d}]}^{T} . \end{matrix}$ Notice that no fuzzy modelling error exists in the discussion region Ω based on the above chosen fuzzy membership functions and subsystem matrices. On the other hand, the constraint φ (k) =0 in (9) becomes $\begin{matrix} [\begin{matrix} \frac{ν T_{s}}{l_{1}} \\ 0 \\ 0 \\ 0 \end{matrix}] (τ (k) - u (k)) \\ = [\begin{matrix} 1 - \frac{ν T_{s}}{l_{2}} & 0 & 0 & 0 \\ \frac{ν T_{s}}{l_{2}} & 1 & 0 & 0 \\ \frac{(ν T_{s})^{2}}{2 l_{2}} σ (z (k)) & ν T_{s} σ (z (k)) & 1 & 0 \\ 0 & 0 & - 1 & 1 \end{matrix}] [\begin{matrix} x_{1 d} \\ x_{2 d} \\ x_{3 d} \\ x_{ed} \end{matrix}] \\ + [\begin{matrix} 0 \\ γ \\ 0 \\ y_{d} \end{matrix}] - [\begin{matrix} x_{1 d} \\ x_{2 d} \\ x_{3 d} \\ x_{ed} \end{matrix}] \end{matrix}$ (24) where x_d = [x_1d x_2d x_3d x_ed] ^T. From the lower three rows of (24), we obtain $x_{1 d} = \frac{- γ l_{2}}{ν T_{s}}$ , $x_{2 d} = \frac{γ}{2}$ , x_3d = y_d, and x_ed is any value. Meanwhile, the control input from the first row of (24) is $\begin{matrix} u (k) & = & τ (k) + \frac{l_{1}}{l_{2}} x_{1 d} \\ = & - \sum_{i = 1}^{r} μ_{i} (x_{p} (k)) K_{i} \tilde{x} (k) - \frac{γ l_{1}}{ν T_{s}} \end{matrix}$ (25)Remark 1. From the aforementioned analysis, the VDV-based control law is a powerful scheme for output regulation. However, the solution of VDVs of (20) is unavailable when uncertainty is considered. Moreover, the disturbance and fuzzy modelling error should be omitted. □

3 Controller simplification and robustness enhancement

3.1 Controller simplification

Since the above VDV-based regulator is complex due to requiring to solve the virtual desired variables which are dependent on the system model, the controller simplification is proposed in the following procedures.

Step 1. Remove the VDV design

To avoid solving the VDVs and calculating the complex control law, the term φ (k) in (9) with the virtual control input (13) is rewritten in the following form:

$\begin{matrix} φ (k) & = & \tilde{B} (x_{p} (k)) (u (k) + \sum_{i = 1}^{r} μ_{i} (z (k)) K_{i} \tilde{x} (k)) + \tilde{η} \\ + \tilde{A} (x_{p} (k)) x_{d} - x_{d} \\ = & \tilde{B} (x_{p} (k)) (u (k) + \sum_{i = 1}^{r} μ_{i} (z (k)) K_{i} x (k)) \\ + \bar{φ} (x_{d}) + Δ φ (k) \end{matrix}$ (26) where

$\begin{matrix} \bar{φ} (x_{d}) & = & - \tilde{B} (x_{pd}) \sum_{i = 1}^{r} μ_{i} (z_{d}) K_{i} x_{d} \\ + \tilde{η} + \tilde{A} (x_{pd}) x_{d} - x_{d} \\ Δ φ (k) & = & \tilde{A} (x_{p} (k)) x_{d} - \tilde{B} (x_{p} (k) \sum_{i = 1}^{r} μ_{i} (z (k)) K_{i} x_{d} \\ - [\tilde{A} (x_{pd})) x_{d} - \tilde{B} (x_{pd}) \sum_{i = 1}^{r} μ_{i} (z_{d}) K_{i} x_{d}] \end{matrix}$ (27) and z_d corresponding to z is the premise variable composed of x_d. According to (26), a simplified non-VDV fuzzy output regulator is set to $u (k) = - \sum_{i = 1}^{r} μ_{i} (z (k)) K_{i} x (k)$ (28) The controller constraint is further changed to $\bar{φ} (x_{d}) = 0$ (29) This implies φ (k) = Δφ (k) in (26) under (28) and (29), i.e., the term Δφ (k) becomes a control error to the closed-loop system. As a result, by applying the new design (28), (29) under the designed virtual control input (13), the dynamics of the closed-loop system is obtained as follows:

$\begin{matrix} \tilde{x} (k + 1) & = & \sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (x_{p} (k)) μ_{j} (x_{p} (k)) G_{ij} \tilde{x} (k) \\ + Δ φ (k) + {\tilde{J}}_{d} ϑ (k) \end{matrix}$ (30) On the other hand, by taking the same partition as (18), the new controller constraint $\bar{φ} (x_{d}) = 0$ from (27) can be separated into two equations: $\begin{matrix} \bar{B} (x_{pd}) \sum_{i = 1}^{r} μ_{i} (z_{d}) [K_{pi} x_{pd} + K_{ei} x_{ed}] \\ = \bar{A} (x_{pd}) x_{d} + \bar{η} - {\bar{x}}_{d} \end{matrix}$ (31) $\begin{matrix} \underline{A} (x_{pd}) x_{d} + \underline{η} - \underline{x_{d}} = 0 \end{matrix}$ (32) Here, x_pd can be firstly solved from (32), then x_ed is found from (31). The number of the constraint equals to the dimension of x_d, so that there exists a feasible solution x_d satisfying the constraint $\bar{φ} (x_{d}) = 0$ . Notice that the VDV x_d is not required to solve for the controller (28). This is because the error integrator compensates the operational point value, so that the VDVs are removed from the controller.

Step 2. Redesign controller gain

Based on the simplified fuzzy output regulator (28), the error term Δφ (k) should be compensated by proper gain design. Since the error term Δφ (k) in (27) is independent on disturbance and modelling error, the error term Δφ (k) has a benefited property from local Lipschitz-like dynamic functions $\tilde{A} (x_{p} (k))$ , $\tilde{B} (x_{p} (k))$ and μ_i (z (k)). The local Lipschitz-like condition implies the error term Δφ (k) can be upper bounded by the control error $\tilde{x} (k)$ from the mean value theorem. It yields that there is a positive-definite matrix ΔE such that $Δ φ (k)^{T} Δ φ (k) \leq \tilde{x} (k)^{T} Δ E^{T} Δ E \tilde{x} (k)$ (33) for the fuzzy discussion region Ω .

Remark 2. The local Lipschitz-like condition means that a dynamic function s (x_p) has well defined and upper bounded ∂s (x_p)/∂x_p in the considered control region Ω. Since $\tilde{A} (x_{p} (k))$ , $\tilde{B} (x_{p} (k))$ and μ_i (z (k)) are functional matrices of the system dynamic functions f (x_p) and g (x_p), the local Lipschitz-like condition of Δφ (k) can be assured from local Lipschitz-continuous dynamic functions. □

Afterward, the gain design is given in the following theorem.

Theorem 1. By using the simplified fuzzy output regulator (28), the closed-loop system (30) in an ideal case with ${\tilde{J}}_{d} ϑ (k) = 0$ is asymptotically stable for the output regulation objective if there exist a symmetric positive-definite matrix X > 0 and control gain K_j satisfying the following LMIs: $\begin{matrix} [\begin{matrix} - X & X {\tilde{A}}_{i}^{T} - M_{j}^{T} {\tilde{B}}_{i}^{T} + X Δ E^{T} & XD \\ {\tilde{A}}_{i} X - {\tilde{B}}_{i} M_{j} + Δ EX & - X & 0 \\ DX & 0 & - X \end{matrix}] \\ < 0, \end{matrix}$ (34)

where M_j = K_jX and i, j = 1, 2, … , r ; and D = D^T > 0.

Proof. First, the ideal case without disturbance and fuzzy modelling error is considered. By choosing the Lyapunov function candidate $V (\tilde{x} (k)) = {\tilde{x}}^{T} (k) P \tilde{x} (k)$ with P = P^T > 0 and taking the time difference of $V (\tilde{x} (k))$ along the error dynamics (30) with ${\tilde{J}}_{d} ϑ (k) = 0$ , we have

$\begin{matrix} Δ V (\tilde{x} (k)) \\ = {\sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (x_{p} (k)) μ_{j} (x_{p} (k)) G_{ij} \tilde{x} (k) + Δ φ (k)}^{T} \\ \times P {\sum_{s = 1}^{r} \sum_{t = 1}^{r} μ_{s} (x_{p} (k)) μ_{t} (x_{p} (k)) G_{st} \tilde{x} (k) + Δ φ (k)} \\ - {\tilde{x}}^{T} (k) P \tilde{x} (k) \end{matrix}$ (35) Due to the property (33), $Δ V (\tilde{x} (k))$ further satisfies the following inequality:

$\begin{matrix} Δ V (\tilde{x} (k)) \\ \leq {\sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (x_{p} (k)) μ_{j} (x_{p} (k)) (G_{ij} + Δ E) \tilde{x} (k)}^{T} \\ \times P {\sum_{s = 1}^{r} \sum_{t = 1}^{r} μ_{s} (x_{p} (k)) μ_{t} (x_{p} (k)) (G_{st} + Δ E) \tilde{x} (k)} \\ - {\tilde{x}}^{T} (k) P \tilde{x} (k) \\ = \sum_{i = 1}^{r} \sum_{j = 1}^{r} \sum_{s = 1}^{r} \sum_{t = 1}^{r} μ_{i} (x_{p} (k)) μ_{j} (x_{p} (k)) \\ \times μ_{s} (x_{p} (k)) μ_{t} (x_{p} (k)) {\tilde{x}}^{T} (k) {(G_{ij} + Δ E)^{T} P \\ \times (G_{st} + Δ E) - P} \tilde{x} (k) \end{matrix}$ (36) According to Lemma 1, the above can be rewritten as follows:

$\begin{matrix} Δ V (\tilde{x} (k)) & \leq & \sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (x_{p} (k)) μ_{j} (x_{p} (k)) {\tilde{x}}^{T} (k) \\ \times {(G_{ij} + Δ E)^{T} P (G_{ij} + Δ E) - P \\ + DPD} \tilde{x} (k) - {\tilde{x}}^{T} (k) DPD \tilde{x} (k) \end{matrix}$ (37) As a result, $Δ V (\tilde{x} (k)) < - {\tilde{x}}^{T} (k) DPD \tilde{x} (k)$ once the following inequality is satisfied $(G_{ij} + Δ E)^{T} P (G_{ij} + Δ E) - P + DPD < 0$ (38) for all i, j. Since $V (\tilde{x} (k)) > 0$ and $Δ V (\tilde{x} (k)) < 0$ , the error $\tilde{x} (k)$ asymptotically converges to zero as k→ ∞. This implies that the simplified fuzzy regulator assures the asymptotic output regulation. Furthermore, the stability condition can be transformed to LMI conditions. After pre-multiplying and post-multiplying (49) by X = P⁻¹, it leads to

$\begin{matrix} ({\tilde{A}}_{i} X - {\tilde{B}}_{i} M_{j} + Δ EX)^{T} X^{- 1} \\ \times ({\tilde{A}}_{i} X - {\tilde{B}}_{i} M_{j} + Δ EX) - X + {XDX}^{- 1} DX < 0 \end{matrix}$ (39) where M_j = K_jX. Then, the LMIs (34) are obtained by applying Schur’s complement technique on (39). □

Remark 3. Based on the simplified fuzzy output regulator (28), the output regulation is achieved without needing coordinate transformation and solving the virtual desired variables. The control performance can be adjusted by the LMI design. □

3.2 Robustness to uncertainties and disturbance

Furthermore, system uncertainty, fuzzy modelling error, and disturbance are taken consideration into the closed-loop system in the following form (cf. (30)): $\begin{matrix} \tilde{x} (k + 1) \\ = \sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (x_{p} (k)) μ_{j} (x_{p} (k)) (G_{ij} + Δ G_{ij}) \tilde{x} (k) \\ + Δ φ (k) + {\tilde{J}}_{d} ϑ (k) \end{matrix}$ (40) where ϑ (k) is composed of disturbance and fuzzy modeling error defined in (12); $Δ G_{ij} = Δ {\tilde{A}}_{i} - Δ {\tilde{B}}_{i} K_{j}$ denotes the fuzzy model uncertainties; and $Δ {\tilde{A}}_{i}$ , and $Δ {\tilde{B}}_{i}$ are unknown time-varying parametric uncertainties, which hold the norm-bounded condition: $[Δ {\tilde{A}}_{i} Δ {\tilde{B}}_{i}] = U_{i} Φ_{i} (k) [H_{1 i} H_{2 i}]$ (41) where U_i, H_1i, and H_2i are known real constant matrices; and Φ_i (k) is an unknown matrix function with Lebesgue-measurable elements and satisfies $Φ_{i} (k)^{T} Φ_{i} (k) \leq I$ for all k, in which I is an identity matrix with appropriate dimension. According to (41), the uncertain term ΔG_ij can be expressed in the form: $\begin{matrix} Δ G_{ij} & = & Δ {\tilde{A}}_{i} - Δ {\tilde{B}}_{i} K_{j} \\ \equiv & U_{i} Φ_{i} (k) H_{ij} \end{matrix}$ where H_ij = H_1i − H_2iK_j. The closed-loop system (40) can be rewritten as follows: $\begin{matrix} \tilde{x} (k + 1) \\ = \sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (x_{p} (k)) μ_{j} (x_{p} (k)) {G_{ij} + U_{i} Φ_{i} (k) \\ \times H_{ij}} \tilde{x} (k) + Δ φ (k) + {\tilde{J}}_{d} ϑ (k) \end{matrix}$ (42) To cope with the perturbed term ϑ (k), the control problem is to let the controlled system having the following H^∞ criterion:

$\begin{matrix} \sum_{k = 0}^{t_{f}} {\tilde{x}}^{T} (k) D^{T} PD \tilde{x} (k) \\ < {\tilde{x}}^{T} (0) P \tilde{x} (0) + \frac{1}{1 + ρ^{2}} \sum_{k = 0}^{t_{f}} ϑ^{T} (k) {\tilde{J}}_{d}^{T} {\tilde{J}}_{d} ϑ (k) \end{matrix}$ (43) where $\tilde{x} (0)$ is the initial state; P is a symmetric positive-definite matrix; and ρ > 0 is an attenuation factor. For the above objective, the robust gain design is given in the following theorem.

Theorem 2.Considering the closed-loop system (42) endowed with the simplified fuzzy control law (28), theH^∞regulation performance (43) is guaranteed for a desired attenuated level $\frac{1}{\sqrt{1 + ρ^{2}}}$ if there exist a symmetric positive-definite matrixX > 0 andM_j = K_jXsatisfying the LMIs $\begin{matrix} [\begin{matrix} - X & * & * & * & * & * \\ 0 & \frac{- 1}{1 + ρ^{2}} {\tilde{J}}_{d}^{T} {\tilde{J}}_{d} & * & * & * & * \\ (\begin{matrix} {\tilde{A}}_{i} X - {\tilde{B}}_{i} M_{j} \\ + Δ EX \end{matrix}) & {\tilde{J}}_{d} & - X & * & * & * \\ H_{1 i} X - H_{2 i} M_{j} & 0 & 0 & - ɛ_{ij} I & * & * \\ 0 & 0 & U_{i}^{T} & 0 & - ɛ_{ij}^{- 1} I & * \\ DX & 0 & 0 & 0 & 0 & - X \end{matrix}] \\ \begin{matrix} < \end{matrix} 0, \end{matrix}$ (44)

fori, j = 1, 2, …, rand someɛ_ij > 0.

Proof. Choosing the Lyapunov function candidate $V (\tilde{x} (k))$ = ${\tilde{x}}^{T} (k) P \tilde{x} (k)$ with P = P^T > 0 and taking the time difference of $V (\tilde{x} (k))$ along the error dynamics (42) yields $\begin{matrix} Δ V (\tilde{x} (k)) \\ = {\sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (x_{p} (k)) μ_{j} (x_{p} (k)) \\ \times (G_{ij} + U_{i} Φ_{i} (k) H_{ij}) \tilde{x} (k) \\ + Δ φ (k) + {\tilde{J}}_{d} ϑ (k)}^{T} P {\sum_{s = 1}^{r} \sum_{t = 1}^{r} μ_{s} (x_{p} (k)) μ_{t} (x_{p} (k)) \\ \times (G_{st} + U_{s} Φ_{s} (k) H_{st}) \tilde{x} (k) + Δ φ (k) + {\tilde{J}}_{d} ϑ (k)} \\ - {\tilde{x}}^{T} (k) P \tilde{x} (k) \end{matrix}$ According to the Lemma 1 and the property (33), $Δ V (\tilde{x} (k))$ satisfies

$\begin{matrix} Δ V (\tilde{x} (k)) \\ \leq \sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} (x_{p} (k)) μ_{j} (x_{p} (k)) {[\begin{matrix} \tilde{x} (k) \\ ϑ (k) \end{matrix}]}^{T} \\ \times {{[\begin{matrix} G_{ij} + U_{i} Φ_{i} (k) H_{ij} + Δ E & {\tilde{J}}_{d} \end{matrix}]}^{T} P \\ \times [\begin{matrix} G_{ij} + U_{i} Φ_{i} (k) H_{ij} + Δ E & {\tilde{J}}_{d} \end{matrix}] \\ + [\begin{matrix} - P + DPD & 0 \\ 0 & - \frac{1}{1 + ρ^{2}} {\tilde{J}}_{d}^{T} {\tilde{J}}_{d} \end{matrix}]} [\begin{matrix} \tilde{x} (k) \\ ϑ (k) \end{matrix}] \\ - {\tilde{x}}^{T} (k) DPD \tilde{x} (k) + \frac{1}{1 + ρ^{2}} ϑ^{T} (k) {\tilde{J}}_{d}^{T} {\tilde{J}}_{d} ϑ (k) \end{matrix}$ (45) If the following matrix inequality $\begin{matrix} {[\begin{matrix} {\tilde{A}}_{i} - {\tilde{B}}_{i} K_{j} + U_{i} Φ_{i} (k) H_{ij} + Δ E & {\tilde{J}}_{d} \end{matrix}]}^{T} P \\ \times [\begin{matrix} {\tilde{A}}_{i} - {\tilde{B}}_{i} K_{j} + U_{i} Φ_{i} (k) H_{ij} + Δ E & {\tilde{J}}_{d} \end{matrix}] \\ + [\begin{matrix} - P + DPD & 0 \\ 0 & - \frac{1}{1 + ρ^{2}} {\tilde{J}}_{d}^{T} {\tilde{J}}_{d} \end{matrix}] < 0 \end{matrix}$ (46) is satisfied, then

$\begin{matrix} Δ V (\tilde{x} (k)) & < & - {\tilde{x}}^{T} (k) DPD \tilde{x} (k) \\ + \frac{1}{1 + ρ^{2}} ϑ^{T} (k) {\tilde{J}}_{d}^{T} {\tilde{J}}_{d} ϑ (k) \end{matrix}$ (47) Therefore, by summing both sides of the inequality (47) from k = 0 up to k = t_f, the H^∞ criterion (43) is assured under the condition (46).

Next, the robust gain design is performed. By using Schur complement, the stability condition (46) is equivalent to $\begin{matrix} [\begin{matrix} - P + DPD & * & * \\ 0 & - \frac{1}{1 + ρ^{2}} {\tilde{J}}_{d}^{T} {\tilde{J}}_{d} & * \\ {\tilde{A}}_{i} - {\tilde{B}}_{i} K_{j} + U_{i} Φ_{i} (k) H_{ij} + Δ E & {\tilde{J}}_{d} & - P^{- 1} \end{matrix}] \\ < 0 \end{matrix}$ (48)

Due to Φ_i (k) ^TΦ_i (k) ≤ I, we further have $\begin{matrix} [\begin{matrix} - P + DPD & * & * & * & * \\ 0 & - \frac{1}{1 + ρ^{2}} {\tilde{J}}_{d}^{T} {\tilde{J}}_{d} & * & * & * \\ {\tilde{A}}_{i} - {\tilde{B}}_{i} K_{j} + Δ E & {\tilde{J}}_{d} & - P^{- 1} & * & * \\ H_{1 i} - H_{2 i} K_{j} & 0 & 0 & - ɛ_{ij} I & * \\ 0 & 0 & U_{i}^{T} & 0 & - ɛ_{ij}^{- 1} I \end{matrix}] \\ < 0, \end{matrix}$ (49)

for some positive constants ɛ_ij > 0. After pre-multiplying and post-multiplying (49) by diag {P⁻¹, I, I, I, I}, the inequality (49) is transformed into $\begin{matrix} [\begin{matrix} - P^{- 1} {PP}^{- 1} + P^{- 1} {DPDP}^{- 1} & * \\ 0 & - \frac{1}{1 + ρ^{2}} {\tilde{J}}_{d}^{T} {\tilde{J}}_{d} \\ {\tilde{A}}_{i} P^{- 1} - {\tilde{B}}_{i} K_{j} P^{- 1} + Δ {EP}^{- 1} & {\tilde{J}}_{d} \\ H_{1 i} P^{- 1} - H_{2 i} K_{j} P^{- 1} & 0 \\ 0 & 0 \end{matrix} \\ \begin{matrix} \end{matrix} \begin{matrix} * & * & * \\ * * & * & * \\ - P^{- 1} & * & * \\ 0 & - ɛ_{ij} I & * \\ U_{i}^{T} & 0 & - ɛ_{ij}^{- 1} I \end{matrix}] < 0 \end{matrix}$ (50) Finally applying Schur complement to (50) again with denoting X = P⁻¹ and M_j = K_jX, the LMI stability condition (40) is obtained.

Therefore, if there exists a feasible solution satisfying the LMIs (44), the simplified fuzzy output regulator (28) can drive the system to the desired operational point with y (k) = y_d even if the system uncertainty and disturbance are considered. The effect of disturbance and fuzzy modeling error is attenuated to $\frac{1}{\sqrt{1 + ρ^{2}}}$ level. □

Remark 4. Generally speaking, the above common Lyapunov function method may be conservative for the robust gain design. To relax the stability limitation, the piecewise-fuzzy Lyapunov function method [13, 16] can be further applied on the gain design. The benefit is that the simplified fuzzy regulator can achieve higher robustness and apply on more complex systems. However, since the piecewise-fuzzy Lyapunov function method is more complex than the above analysis method, we take it as a future work of this paper. □

Remark 5. The proposed fuzzy regulator is capable of extending to underlying systems with Markovian jumping parameters. To deal with the regulation problem of Markovian jump systems, the gain design of the regulator should be performed by properly choosing a Markovian Lyapunov functional (e.g., [30]). In the issue of Markovian systems with time delay, a Markovian Lyapunov-Krasovskii functional (MLKF) (cf. [31 –33]) would be considered to modified the gain design. The delay-dependent stability analysis is more relax than delay-independent methods. The final results will be also converted to LMI based stability for auxiliary design. □

4 Numerical simulations

In this section, numerical simulations are given to verify the validity of the proposed simplified fuzzy regulator.

Example (Continued). Continuing the Example in Section 2, steering the truck-trailer system is taken as the control application. The system parameters are set to l₁ = 2.8m, ν = -1m/s, l₂ = 5.5m, and T_s = 2s, while system parameters are uncertain but bounded within 5% of their nominal values. For the T-S fuzzy model (23), the fuzzy parameters are chosen as $α = 1, β = \frac{10^{- 2}}{π}$ . In other words, the norm-bounded condition of uncertainties is regarded with $\begin{matrix} U_{1} & = & U_{2} = 0.05 I_{4 \times 4} \\ H_{11} & = & [\begin{matrix} 1 - \frac{ν T_{s}}{l_{2}} & 0 & 0 & 0 \\ \frac{ν T_{s}}{l_{2}} & 0 & 0 & 0 \\ α \frac{(ν T_{s})^{2}}{2 l_{2}} & α ν T_{s} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}], \\ H_{12} & = & [\begin{matrix} 1 - \frac{ν T_{s}}{l_{2}} & 0 & 0 & 0 \\ \frac{ν T_{s}}{l_{2}} & 0 & 0 & 0 \\ β \frac{(ν T_{s})^{2}}{2 l_{2}} & β ν T_{s} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}], H_{21} = H_{22} [\begin{matrix} \frac{ν T_{s}}{l_{1}} \\ 0 \\ 0 \\ 0 \end{matrix}] . \end{matrix}$ Moreover, we assume that the external disturbance is ω (k) =0.05 sin(0.01k) with $\tilde{J} = [0 1 0 0]^{T}$ , and the fuzzy modelling error is ω_m (k) =0 with J_m = [0 0 0 0] ^T, and the unknown bias is γ = 0.2 (see (22)). For the simplified fuzzy regulator, the control parameters are set to D = diag {0.5, 0.3, 0.05, 0.5} , ΔE = diag {0.005, 0.003, 0.001, 0.01}, ρ = 0.46, ɛ₁₁ = 26, ɛ₁₂ = 26, ɛ₂₁ = 16, and ɛ₂₂ = 16. After solving the LMIs (44) in Theorem 2, the feasible controller gains are obtained as follows: $\begin{matrix} K_{1} & = & [\begin{matrix} - 3.9067 & 6.2654 & - 1.816 & 0.252 \end{matrix}], \\ K_{2} & = & [\begin{matrix} - 3.9067 & 6.2654 & - 1.816 & 0.252 \end{matrix}] . \end{matrix}$ First, the fixed regulation is considered to steer the truck-trailer with the desired output y_d = 2. By letting the initial states of the truck-trailer system to be x (0) = [0.3 − 0.1 0.4] ^T, the output regulation results of simplified fuzzy regulator (28) are obtained as shown in Figs. 1 and 2. The output quickly tends to the desired value by the simplified T-S fuzzy controller without solving the VDVs and performing coordinate transformation.

Next, a piecewise constant signal is also taken as the desired output command. Based on the above controller setting, the control responses are shown in Figs. 3 and 4. To demonstrate the benefit of the simplified non-VDV fuzzy regulator, the VDV-based fuzzy controller (25) is also applied on the truck-trailer system. The VDVs are solved from the nominal system part as Example, i.e., the VDVs are unavailable for practical uncertain system. The controller gains of the VDV-based fuzzy regulator are solved according to LMIs (17) with the control parameter D = diag {0.05, 0.03, 0.005, 0.05} below: $\begin{matrix} K_{1} & = & [\begin{matrix} - 5.9941 & 14.9025 & - 7.3467 & 0.0205 \end{matrix}], \\ K_{2} & = & [\begin{matrix} - 5.9941 & 14.9026 & - 7.3471 & 0.0205 \end{matrix}] . \end{matrix}$ Under the same initial condition, the VDV-based output regulation control results for the piecewise-constant regulation are shown in Figs. 5 and 6. In comparison with the simplified non-VDV fuzzy regulator, the VDV-based fuzzy regulator results in large oscillation and non-zero steady-state error due to weak gain design and inexact solution of the VDVs for uncertain model. Furthermore, the VDV-based fuzzy regulator cannot be easily implemented because the error coordinate transformation and VDV calculation are required. In addition, the VDV-based regulator has to be redesigned when changing the desired command (e.g., piecewise-constant output command).

5 Conclusion

In this paper, the synthesis of the simplified robust T-S fuzzy output regulator has been proposed for discrete affine nonlinear systems subject to parametric uncertainties and external disturbance. Although the VDV-based fuzzy regulator can be designed in a straightforward manner by involving integral compensation, the system uncertainty and disturbance are non-trivial for the VDV design. In contrast, the simplified non-VDV fuzzy output regulator has higher capacity of coping with system uncertainty, disturbances, and bias terms due to removing the VDV calculation and enhancing robust gain design. Since no VDVs are used, the proposed fuzzy regulator is easily implemented without requiring any coordinator transformation and solving equilibrium point, i.e., the drawbacks in traditional method are removed. This benefit implies that the simplified fuzzy regulator does not need redesign for a multi-objective regulation with varying operational points which cannot be achieved by traditional methods. Furthermore, the LMI robust gain design assures robust H^∞ output regulation performance to disturbance and modelling error. Finally, the numerical simulations show satisfactory results and better performance than the VDV-based fuzzy regulator.

Motivated by [13, 16], the stability conditions of the regulator could be relaxed by using the piecewise-fuzzy Lyapuvov function method in the future work. If the limitation of the regulator is reduced, then the applications become more general. In addition, the proposed method can be further extended to cope with Markovian jump systems [30 –33] or time delay systems in the future.

Acknowledgments

This work was supported by Ministry of Science and Technology, R.O.C., under grants MOST 102-2221-E-033-053 and MOST 103-2632-E-033-001.

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