On-line T-S fuzzy control using Riccati differential equation

Abstract

In order to assure the stability of Takagi-Sugeno (T-S) fuzzy systems, the linear matrix inequality (LMI) should be applied. However, the LMI method cannot be applied online, because its computational complexity. In this paper, the T-S fuzzy control is transformed into a time-varying system. By using Riccati differential equation (RDE) and a special optimal-like controller, the T-S fuzzy control can be applied online. We prove that the T-S fuzzy control is stable and the trajectory tracking error converges to a bounded zone. Since RDE can be solved online, the novel T-S fuzzy control is adaptive and more simple and effective than LMI-based methods. We apply successfully this online fuzzy control to an autonomous underwater vehicle.

Keywords

T-S fuzzy system on-line control Riccati differential equation optimal-like control

1 Introduction

The Takagi-Sugeno (T-S) fuzzy model [1] uses several fuzzy rules, where the conclusion parts are linear time-invariant plants, to represent a nonlinear system. Each rule utilizes the local dynamics by a linear system model. The T-S fuzzy model uses the fuzzy set theory, which has been applied to fuzzy fixed point theory [2] and fuzzy topology [3]. T-S fuzzy model can approximate a large class of nonlinear systems, while keeps the simplicity of the linear models [4]. It has been applied for nonlinear stabilization control. In [5], a parallel distributed compensation (PDC) technique is proposed to design fuzzy logic controller. The T-S fuzzy control is first analyzed by Lyapunov stability theorem in [6]. A nonlinear system can be approximated by several piecewise linear systems, which are T-S fuzzy models. However, the stability of all these locally linear systems cannot guarantee the nonlinear system to be stable. Many optimization problems in control theory and system identification can be formulated into linear matrix inequality (LMI) problems [7]. In order to find common stability conditions for all subsystems, the solutions of LMIs are needed.

Most of T-S fuzzy systems use LMI to analyze the stability, because LMI is feasible for the convex problems, even Lyapunov and Riccati inequalities can also be written as LMIs. LMIs are a useful tool for solving the stability problem of T-S fuzzy systems. In [8], a T-S fuzzy system is used to model an unknown nonlinear system. LMI is applied to prove the stability of the observer-based fuzzy controller. In [9], quadratic stability is proven by LMIs, which also guarantees the existence of bounded control. In [10], the digital redesign problem of T-S fuzzy system is solved by the convex minimization problem with LMIs. For large-scale systems, they can be controlled by T-S fuzzy models with PDC [11], or by the switched quadratic Lyapunov function method [12]. Both of them need LMI to give the common matrix. In [13], PDC is applied to guarantee robust stability by transforming the problem to standard LMIs. In [14], modified PDC with multi-indexed matrix approach is presented. The premise variable spaces is splited in [12], a convex condition to design the fuzzy control is proposed. However, they still need LMIs. The stabilization problem of T-S fuzzy systems with time-varying state delay is studies in [15, 16]. The stability conditions are formulated into LMIs, which are solved by using existing LMI optimization techniques [17]. T-S fuzzy H_∞ output tracking control is discussed in [18]. Sufficient conditions for the stability are obtained by the two-step LMI method. The stability conditions represented by LMI are relaxed in [4]. They are suitable for designing fuzzy state feedback controllers. In [24], the networked control systems is represented by a T-S fuzzy model. The robust controller gains are obtained by solving LMIs. The paper [19] uses T-S fuzzy system to control a solar power generation system. LMI is used to prove the stability.

There are several problems of using LMI for T-S fuzzy control: 1) The stability condition of LMI is sufficient. Only after the controller is designed, we can check if the common positive definite matrix exists, and if the stability is guaranteed. 2) The common quadratic Lyapunov function tends to be conservative for highly nonlinear complex systems. 3) The computational complexity increases with fuzzy rule number. Studies on less conservative methods have been widely developed. In [20], the linear rule consequent method [21] is used to avoid the LMI problem. However, it still needs to find a common positive definite matrix [22]. The main disadvantage of LMI for T-S fuzzy model is it cannot be applied on-line.

The piecewise Lyapunov function [23] and the weighting dependent Lyapunov function [6] need to check several inequalities. The piecewise Lyapunov functions require certain boundary conditions [25]. In [26], the weighting dependent Lyapunov function method is simplified by estimating the distance between two successive states in T-S fuzzy discrete system. The time optimal problem of T-S fuzzy model is discussed via Lie algebra in [27]. The existence of a solution reveals the controllability of T-S fuzzy system and a rank condition. It does not guarantee stability. By using sliding mode technique, the solving LMI for T-S fuzzy models is avoided in [28]. However, the controller is not longer smooth, and in order to assure the fuzzy model to be stable, LMIs are still needed [29]. In [24], polynomial form Lyapunov functions is applied to obtain the membership-dependent conditions, which do not need LMIs.

Algebraic Riccati equations are applied in the optimal control design. The control values at any time can be found using the solution of the Riccati equation with the current observations state variables. This method is also applied for fuzzy controller design. In [30], an uncertain fuzzy system is stabilized if an algebraic Riccati equation or a set of algebraic Riccati equations have solutions. They use piecewise Lyapunov functions. In order to find algebraic Riccati equations, the solutions of LMIs are required. In [31], the fuzzy controller design is directly transformed into optimal control form. Riccati-like equation and Riccati-like differential equation are used to obtain the optimal control. The stability of the closed-loop system is bounded, not in the sense of Lyapunov, so LMIs are not need. In this paper, we use one Riccati differential equation without LMIs. The new controller is an on-line version of T-S fuzzymodel.

In this paper, the locally linear time-invariant systems are transformed into a time-varying system. By using matrix Riccati differential equation and a local optimal-like control, we prove that the tracking error of the T-S fuzzy control is bounded. The main contributions of the paper are

The normal off-line LMI design for T-S fuzzy control is modified into on-line process. The key technique is we use Riccati differential equation.

The stability of the on-line T-S fuzzy control and existence of the controller are proven.

The on-line T-S fuzzy control is applied to a underwater vehicle.

Experimental results show that this novel T-S fuzzy control has many advantages over the popular LMI method.

2 Takagi-Sugeno fuzzy control for unknown nonlinear system

It is not easy to obtain exact nonlinear models for many complex physical systems. The T-S model expresses the physical systems with several local models. The nonlinear uncertain system to be controlled is given by the following fuzzy dynamic model [18 , 30]

$\begin{matrix} R^{i} : IF x_{1} is F_{1}^{i} and x_{2} is F_{2}^{i} and \dots x_{n} is F_{n}^{i} \\ THEN \dot{x} (t) = A^{i} x (t) + B^{i} u (t) \end{matrix}$ (1) where $x (t) \in R^{n}$ is the state vector of the system, $u (t) \in R^{l}$ is a given control action, l is the dimension of control input, Rⁱ is the i-th fuzzy rule, i = 1, 2 ⋯ m, m is the number of inference rules. $F_{j}^{i}$ (i = 1 ⋯ m, j = 1 … n) are the fuzzy sets for the system states x_j. The matrices $A^{i} \in R^{n \times n}$ and $B^{i} \in R^{n \times l}$ define the i-th local T-S model.

Considering uncertainties, (1) becomes

$\begin{matrix} R^{i} : IF x_{1} is F_{1}^{i} and x_{2} is F_{2}^{i} and \dots x_{n} is F_{n}^{i} \\ THEN \dot{x} (t) = A^{i} x (t) + B^{i} u (t) + ξ^{i} + {\tilde{f}}^{i} \end{matrix}$ (2) where ${\tilde{f}}^{i} \in R^{n}$ is the modeling error for each sub-model, $ξ^{i} \in R^{n}$ denotes bounded external disturbance, x = [x₁, ⋯ , x_n] ^T.

Using a fuzzy standard inference method, i.e., product inference, center-average and singleton fuzzifier, the m T-S models (2) can be rewritten as $\begin{matrix} \dot{x} (t) & = & \sum_{i = 1}^{m} α_{i} (x) [A^{i} x (t) + B^{i} u (t)] \\ + \sum_{i = 1}^{m} α_{i} (x) {\tilde{f}}^{i} + \sum_{i = 1}^{m} α_{i} (x) ξ^{i} \end{matrix}$ (3) where α_i (x) is defined as, $α_{i} (x) = \prod_{j = 1}^{m} μ_{j}^{i} / \sum_{i = 1}^{m} \prod_{j = 1}^{m} μ_{j}^{i},$ $μ_{j}^{i}$ is the membership functions of the fuzzy sets $F_{j}^{i} .$ Obviously, 0 ≤ α_i (x) ≤ 1, $\sum_{i = 1}^{m} α_{i} (x) = 1$ .

If we define $A (t) = \sum_{i = 1}^{m} α_{i} (x) A^{i},$ $B (t) = \sum_{i = 1}^{m} α_{i} (x) B^{i}$ , $η (t) = \sum_{i = 1}^{m} α_{i} (x) {\tilde{f}}^{i},$ and $ξ (t) = \sum_{i = 1}^{m} α_{i} (x) ξ^{i},$ (3) can be simplified as $\dot{x} (t) = A (t) x (t) + B (t) u (t) + η (t) + ξ (t)$ (4) where η (t) is the modeling error, ξ (t) is defined in (2).

We design Aⁱ and Bⁱ such that the following control goal is reached. It is to force the system states to track desired signals, generated by a nonlinear reference model given by ${\dot{x}}_{ref} (t) = s [x_{ref} (t)]$ (5) where $x_{ref} (t) \in R^{n},$ x_ref (0) = x₀, ∀t ≥ 0 . The function $s (\cdot) : R^{n} \to R^{n}$ is nonlinear and Lipchitz, it satisfies ${‖ s (x_{r e f}) - s (0) ‖}^{2} \leq L_{s} {‖ x_{r e f} ‖}^{2}$ (6) where L_s > 0 . Based on the orthogonal decomposition [33], (2) can be rewritten as $s [x_{ref} (t)] = Bv (t) + B^{⊥} r (t)$ (7) where B^⊥ is the orthogonal function of B, v (t) and $r (t) \in R^{n \times 1},$ are given by

$\begin{matrix} v (t) & = & {[B^{T} B]}^{- 1} B^{T} s [x_{ref} (t)] \\ r (t) & = & {[{(B^{⊥})}^{T} B^{⊥}]}^{- 1} {(B^{⊥})}^{T} s [x_{ref} (t)] \end{matrix}$ (8)

Define B⁺ as the pseudo-inverse of B in the sense of Moore-Penrose [33], then $B^{+} = {(B^{T} B)}^{- 1} B^{T}, B^{⊥} = I - {BB}^{+}$ (9)

In this paper the following assumptions are considered to be satisfied:

A1: The reference x_ref (t) is bounded by ${‖ x_{r e f} (t) ‖}^{2} \leq γ_{r e f} %, γ_{r e f} > 0$ (10)

A2: The perturbations are bounded by ${‖ ξ [x (t), t] ‖}^{2} \leq γ_{ξ}, γ_{ξ} > 0$ (11)

To establish a feasible control problem, we define an admissible control set as $U_{a d m} = {u : {‖ u (t) ‖}^{2} \leq γ_{u 1} + γ_{u 2} {‖ x (t) ‖}^{2}}$ (12) where γ_u1 and γ_u2 are positive constants. U_adm guarantees the right hand side of (3) is locally Lipchitz. Obviously, the normal state feedback control satisfies γ_u1 = 0 and γ_u2 > 0.

The controller design process is to find u (t) ∈ U_adm, such that the trajectory tracking error is bounded as $\lim_{t \to \infty} \sup ‖ x (t) - x_{r e f} (t) ‖ \leq β_{r e f}$ (13) where β_ref > o, which defines the quality of the trajectory tracking control.

A3: The pairs (Aⁱ, Bⁱ), i = 1, 2 ⋯ m, are controllable.

The modelling error $η (t) = \sum_{i = 1}^{m} α_{i} {\tilde{f}}^{i}$ is assumed to satisfy the following sector condition

A4: The modeling error η (t) in (4) satisfies ${‖ η (t) ‖}^{2} \leq f_{0} + f_{1} {‖ x (t) ‖}^{2}$ (14) where f₀ and f₁ are known positive constants.

The sector condition (14) is more feasible than the bounded condition as ||η (t) ||² ≤ f₀, here f₀ is the upper bound of the uncertainty.

3 On-line fuzzy control with differential Riccati equation

This paper does not discuss the fuzzy modeling problem, so the minimization of the modeling error is out of the scope of this paper. We will design a fuzzy control such that the tracking error (13) is minimized. We use the model (4) and A4.

Now we define the trajectory tracking error by $Δ = x - x_{ref}$ (15)

The following theorem gives the main result of the paper. It guarantees the tracking error (15) is bounded. It also provides an explicit and easy design method for the T-S fuzzy control.

Theorem 1.If there exists a matrix K (t) and a positive scalar α such that the following matrix Riccati differential equation have positive definite solution

$\begin{matrix} \dot{P} (t) + P (t) {\bar{A}}_{1} + {\bar{A}}_{1}^{T} P (t) P (t) \\ - 3 P (t) R_{1} (t) + Q_{1} (t) = 0 \end{matrix}$ (16) where P (t) = P^T (t) > 0, ${\bar{A}}_{1} (t) = A (t) + \frac{α}{2} I,$

$\begin{matrix} R_{1} (t) & = & A (t) Λ_{1} A^{T} (t) + S_{1} (t) Λ_{4} S_{1}^{T} (t) \\ + S_{2} (t) Λ_{4} S_{2}^{T} (t) + Λ_{2} + Λ_{3} \\ Q_{1} (t) & = & 2 λ_{max} (Λ_{2}^{- 1}) f_{1} I_{n \times n} \\ S_{1} (t) & = & B (t) {[B^{T} (t) B (t)]}^{- 1} B^{T} (t) \end{matrix}$ (17)S₂ (t) = B^⊥ (t) [(B^⊥ (t)) ^TB^⊥ (t)] ^-1 (B^⊥ (t)) ^T, Λ_i (i = 1 ⋯4) are known positive definite matrices, f₁ is defined in (14), B^⊥ is defined in (9), A (t) and B (t) are defined in (4), I_n×n is the identity matrix of n × n, and the feedback control is in the following form

$\begin{matrix} u (t) & = & - K (t) Δ (t) + v (t), K (t) \in R^{m \times n} \\ K^{T} (t) & = & 2 {[B^{T} (t) B (t)]}^{- 1} B^{T} (t) R_{1} (t) P (t) \end{matrix}$ (18) where v (t) is defined in (8). Then the trajectory tracking error Δ satisfies $\lim_{t \to \infty} \sup ‖ x (t) - x_{r e f} (t) ‖ \leq \frac{β}{α λ_{\min} [P (t)]}$ (19) where $\begin{matrix} β & = & λ_{max} (Λ_{1}^{- 1}) γ_{ref} + λ_{max} (Λ_{2}^{- 1}) f_{0} \\ + 2 λ_{max} (Λ_{2}^{- 1}) f_{1} γ_{ref} + λ_{max} (Λ_{3}^{- 1}) γ_{ξ} \\ + λ_{max} (Λ_{4}^{- 1}) L_{s} γ_{ref} + λ_{max} (Λ_{4}^{- 1}) L_{s} γ_{ref} \end{matrix}$ f₀ and f₁ are defined in (14), γ_ref is defined in (10), γ_ξ is defined in (11), L_s is defined in (6), λ_max (A) and λ_min [A] are maximum and minimum eigenvalues of matrix A .

Proof 1. In order to prove the tracking error is bounded, we use the following Lyapunov function $V = Δ^{T} P (t) Δ$ (20) where P (t) is a time-variant positive definite matrix, P (t) = P^T (t) > 0. From (7), (4) and (15), the dynamic of the tracking error Δ is given by

$\begin{matrix} \dot{Δ} (t) & = & A (t) x (t) + B (t) [u (t) - v] \\ - B^{⊥} (t) r (t) + η + ξ \end{matrix}$ (21)

Using (21), the derivative of the Lyapunov function (20) is

$\begin{matrix} \dot{V} & = & Δ^{T} \dot{P} (t) Δ + 2 Δ^{T} (t) P (t) A (t) x (t) \\ + 2 Δ^{T} (t) P (t) B (t) u (t) + 2 Δ^{T} (t) P (t) η \\ + 2 Δ^{T} (t) P (t) ξ - 2 Δ^{T} (t) P (t) B (t) v \\ - 2 Δ^{T} (t) P (t) B^{⊥} (t) r (t) \end{matrix}$ (22)

Using x = Δ + x_ref and (18), (22) becomes

$\begin{matrix} \dot{V} & = & Δ^{T} \dot{P} (t) Δ + Δ^{T} (t) {P (t) [A (t) - B (t) K (t)] \\ + [A (t) - B (t) K (t)]^{T} P (t)} Δ (t) \\ + 2 Δ^{T} (t) P (t) A (t) x_{ref} + 2 Δ^{T} (t) P (t) η \\ + 2 Δ^{T} (t) P (t) ξ - 2 Δ^{T} (t) P (t) B (t) v \\ - 2 Δ^{T} (t) P (t) B^{⊥} (t) r (t) \end{matrix}$ (23)

For any matrices $X, Y \in R^{m \times n},$ they satisfy the following inequality, $X^{T} Y + Y^{T} X \leq X^{T} Λ X + Y^{T} Λ^{- 1} Y$ (24) where Λ is a positive definite matrix, Λ = Λ^T > 0 . Now we apply this inequality to $\begin{matrix} 2 Δ^{T} (t) P (t) A (t) x_{ref} \\ \leq Δ^{T} (t) P (t) A (t) Λ_{1} A^{T} (t) P (t) Δ (t) \\ + λ_{max} (Λ_{1}^{- 1}) γ_{ref} \end{matrix}$ (25)

The other cross terms have similar relations. The derivative of the Lyapunov function (23) becomes

$\begin{matrix} \dot{V} & \leq & Δ^{T} (t) {\dot{P} (t) + P (t) [A (t) - B (t) K (t)] \\ + [A (t) - B (t) K (t)]^{T} P (t)} Δ (t) \\ + P (t) [A (t) Λ_{1} A^{T} (t) + Λ_{2} + Λ_{3}] P (t) \\ + 2 λ_{max} (Λ_{2}^{- 1}) f_{1} * I_{n \times n}} Δ (t) \\ + λ_{max} (Λ_{1}^{- 1}) γ_{ref} + λ_{max} (Λ_{2}^{- 1}) f_{0} \\ + 2 λ_{max} (Λ_{2}^{- 1}) f_{1} γ_{ref} + λ_{max} (Λ_{3}^{- 1}) γ_{ξ} \\ - 2 Δ^{T} (t) P (t) B (t) v - 2 Δ^{T} (t) P (t) B^{⊥} (t) r (t) \end{matrix}$ (26)

Adding and subtracting the term αV = αΔ^TP (t) Δ to (26), and using (8), (26) becomes $\begin{matrix} \dot{V} & \leq & Δ^{T} (t) {\dot{P} (t) + P (t) [A (t) - B (t) K (t) + \frac{α}{2} I] \\ + [A (t) - B (t) K (t) + \frac{α}{2} I]^{T} P (t)} Δ (t) \\ + P (t) [A (t) Λ_{1} A^{T} (t) + Λ_{2} + Λ_{3}] P (t) \\ + 2 λ_{max} (Λ_{2}^{- 1}) f_{1} * I_{n \times n}} Δ (t) \\ - α V (t) + λ_{max} (Λ_{1}^{- 1}) γ_{ref} + λ_{max} (Λ_{2}^{- 1}) f_{0} \\ + 2 λ_{max} (Λ_{2}^{- 1}) f_{1} γ_{ref} + λ_{max} (Λ_{3}^{- 1}) γ_{ξ} \\ - 2 Δ^{T} (t) P (t) B (t) {[B^{T} (t) B (t)]}^{- 1} B^{T} (t) s (x_{ref}) \\ - 2 Δ^{T} (t) P (t) B^{⊥} (t) {[{(B^{⊥} (t))}^{T} B^{⊥} (t)]}^{- 1} \\ * {(B^{⊥} (t))}^{T} s (x_{ref}) \end{matrix}$ (27)

Applying (24) to $- 2 Δ^{T} (t) P (t) B {[B^{T} B]}^{- 1} B^{T} s (x_{ref})$ then

$\begin{matrix} - 2 Δ^{T} (t) P (t) B (t) {[B^{T} (t) B (t)]}^{- 1} B^{T} (t) s (x_{r e f}) \\ \leq Δ^{T} (t) P (t) S_{1} (t) Λ_{4} S_{1}^{T} (t) P (t) Δ (t) \\ + λ_{\max} (Λ_{4}^{- 1}) {‖ s (x_{r e f}) ‖}^{2} \end{matrix}$ (28) where $S_{1} (t) = \bar{B} (t) {[B^{T} (t) B (t)]}^{- 1} B^{T} (t)$ . Under the assumption on s (x_ref) in (5), and s (0) = 0, ${‖ s (x_{r e f}) ‖}^{2} \leq L_{s} γ_{r e f}$ (29)

So (27) becomes $\dot{V} \leq Δ^{T} (t) L_{t} Δ (t) - α V + β$ (30) where α > 0, β > 0,

$\begin{matrix} L_{t} & = & \dot{P} (t) + P (t) A_{1} (t) + A_{1}^{T} (t) P (t) \\ + P (t) R_{1} (t) P (t) + Q_{1} (t) \end{matrix}$ (31) where $A_{1} (t) = A (t) - B (t) K (t) + \frac{α}{2} I,$ R₁ (t) and Q₁ (t) are defined in (17). In order to obtain feedback control (18), we define ${\bar{A}}_{1} (t) = A_{1} (t) - B (t) K (t),$ $\begin{matrix} L_{t} & = & \dot{P} (t) + P (t) {\bar{A}}_{1} (t) + {\bar{A}}_{1}^{T} (t) P (t) \\ - P (t) R_{1} (t) P (t) + Q_{1} (t) - P (t) B (t) K (t) \\ - K^{T} (t) B^{T} (t) P (t) + 2 P (t) R_{1} (t) P (t) \end{matrix}$ (32) (32) can be formed into the following form $\begin{matrix} L_{t} & = & {\bar{L}}_{t} + (\sqrt{2} P (t) B (t) {\bar{R}}_{1}^{1 / 2} (t) - G (t)) \\ * {(\sqrt{2} P (t) B (t) {\bar{R}}_{1}^{1 / 2} (t) - G (t))}^{T} \end{matrix}$ (33) where $G (t) = \frac{K^{T} (t) {\bar{R}}_{1}^{- 1 / 2}}{\sqrt{2}}$ and

$\begin{matrix} {\bar{L}}_{t} & = & \dot{P} (t) + P (t) {\bar{A}}_{1} + {\bar{A}}_{1}^{T} P (t) - P (t) R_{1} (t) P (t) \\ + Q_{1} (t) - \frac{K^{T} (t) {\bar{R}}_{1}^{- 1} (t) K (t)}{2} \end{matrix}$ (34) where $R_{1} (t) = B (t) {\bar{R}}_{1} B^{T} (t) .$ To assure L_t = 0, we need ${\bar{L}}_{t} = 0$ and $K (t) = 2 {\bar{R}}_{1} (t) B^{T} (t) P (t)$ (35)

Substituting (35) into (34) and using $R_{1} (t) = B (t) {\bar{R}}_{1} B^{T} (t),$

$\begin{matrix} {\bar{L}}_{t} & = & \dot{P} (t) + P (t) {\bar{A}}_{1} + {\bar{A}}_{1}^{T} P (t) \\ - 3 P (t) R_{1} (t) P (t) + Q_{1} (t) \end{matrix}$ (36)

The conditions L_t = 0 and (35) are (16) and (18). So L_t = 0, (30) becomes $\dot{V} \leq - α V + β$ (37)

Using the comparison principle to solve the differential inequality (37), $V (t) \leq e^{- α t} V (0) + \frac{β}{α} (1 - e^{- α t}) .$ (38) (38) means $lim_{t \to \infty} sup V (t) \leq \frac{β}{α}$

According to the concept of ultimate boundedness theory [21], after finite time t, the tracking error Δ (t) remains in a bounded region, since $lim_{t \to \infty} sup {‖ Δ (t) ‖}^{2} \leq \frac{β}{α λ_{min} [P (t)]}$

It is (19).

Remark 1. In order to prove the stability of several linear systems (1) and (3) for T-S fuzzy control, there are two popular methods: common Lyapunov function and LMI method. Both of them try to find common stability conditions for all linear systems. This paper combines all these linear system as in (4). However, it becomes a time-varying linear system. We use Riccati differential equation to avoid the complexity of the common Lyapunov function method and LMI method.

Remark 2. The control action u (t) in (18) minimizes the energetic function Ψ_t (u) at each time t,

$\begin{matrix} Ψ_{t} (u) & = & (\sqrt{2} P (t) B (t) {\bar{R}}_{1}^{1 / 2} (t) - G (t)) \\ * {(\sqrt{2} P (t) B (t) {\bar{R}}_{1}^{1 / 2} (t) - G (t))}^{T} \end{matrix}$ (39)

To calculate the control action u (t), which minimizes Ψ_t (u), we have to fulfill Ψ_t (u) = 0. So u (t) can be regarded as a local control to minimize the energy of (30), because it is calculated based only on “local” information available at time t.

The T-S fuzzy controller (18) needs the solution P (t) of the Riccati differential equation (16). We do not calculate the analytical solution. (16) can be written as

$\begin{matrix} \dot{P} (t) & = & 3 P (t) R_{1} (t) P (t) \\ - P (t) {\bar{A}}_{1} - {\bar{A}}_{1}^{T} P (t) - Q_{1} (t) \end{matrix}$ (40)

The solution P (t) of (40) can be simulated provided an initial condition P (0) . Since (40) has time-varying parameters, it is not easy to discuss the existence conditions for P (t) . The following lemma shows how to use a RDE with time invariant parameters to decide the solution of a RDE with time-varying parameters.

Lemma 1. Consider the following two matrix differential Riccati equations,

$\begin{matrix} P_{1}^{\cdot} (t) + A^{T} (t) P_{1} (t) + P_{1} (t) A (t) \\ - P_{1} (t) R (t) P_{1} (t) + Q (t) = 0 \\ A_{2}^{T} P_{2} + P_{2} A_{2} - P_{2} R_{2} P_{2} + Q_{2} = 0 \end{matrix}$ (41) one is Riccati differential equation with time-varying parameter, the other one is algebraic Riccati equation. If the initial condition satisfies $P_{1} (0) > P_{2},$ (42) then the corresponding Hamiltonians are given by $H_{1, t} = [\begin{matrix} Q (t) & A (t)^{T} \\ A (t) & - R (t) \end{matrix}] H_{2} = [\begin{matrix} Q_{2} & A_{2}^{T} \\ A_{2} & - R_{2} \end{matrix}]$ and $H_{2} \geq H_{1, t} \geq 0$ (43) the pair (A₂, R₂) is stable, i.e., Reλ_i (A₂ - kR₂) < 0, ∃k, then $P_{1} (t) \geq P_{2} \geq 0, \forall t > 0$ (44)

Proof 2. Let us define e_t = P₁ (t) - P₂ . Using condition (43) and the definition $H = [\begin{matrix} H_{11} & H_{12} \\ H_{21} & H_{22} \end{matrix}]$ , the Riccati equation (41) in the Hamiltonian form is $\begin{matrix} - P_{1}^{\cdot} (t) = [\begin{matrix} I & P_{1} (t) \end{matrix}] H_{1, t} [\begin{matrix} I \\ P_{1} (t) \end{matrix}] \\ [\begin{matrix} I & P_{2} \end{matrix}] H_{2} [\begin{matrix} I \\ P_{2} \end{matrix}] = 0 \end{matrix}$ we derive e_t, $\begin{matrix} - {\dot{e}}_{t} & = & [\begin{matrix} [c] cc I & e_{t} + P_{2} \end{matrix}] H_{1, t} [\begin{matrix} [c] c I \\ e_{t} + P_{2} \end{matrix}] \\ \leq & [\begin{matrix} [c] cc I & e_{t} \end{matrix} + \begin{matrix} [c] cc 0 & P_{2} (t) \end{matrix}] \\ * H_{2} [\begin{matrix} [c] c I \\ e_{t} \end{matrix} + \begin{matrix} [c] c I \\ P_{2} (t) \end{matrix}] \\ = & (A_{2}^{T} + P_{2} R_{2}) e_{t} + e_{t} (A_{2} - R_{2} P_{2}) \\ - e_{t} R_{2} e_{t} + Q_{t} - Q_{t} = L_{t} - Q_{t} \end{matrix}$ where $\begin{matrix} L_{t} & = & (A_{2}^{T} + P_{2} R_{2}) e_{t} + e_{t} (A_{2} - R_{2} P_{2}) \\ - e_{t} R_{2} e_{t} + Q_{t} \end{matrix}$

Based on Theorem 3 of [37], the term $(A_{2}^{T} - P_{2} R_{2})$ is stable when (A₂, R₂) is stable. From (42), e_t=0 > 0 . By Lemma 1 of [37], there exit Q_t=0 > 0 such that L_t=0 = 0 . This leads to ${\dot{e}}_{t} \geq Q_{t} > 0 .$ (45)

The solution of the Riccati differential equation is a continuous function, if its time-varying parameters are continuous. So we conclude that for time t > 0, there exists a ɛ > 0 such that $Q_{τ} > 0 \forall τ \in [t, t + ɛ] .$

As a result, $\begin{matrix} e_{t + ɛ} & = & e_{t} + \int_{t}^{t + ɛ} e_{τ}^{\cdot} d τ \\ \geq & e_{t} + \int_{t}^{t + ɛ} Q_{τ} d τ \geq e_{t} + inf_{τ} [λ_{min} (Q_{t})] I ɛ > 0 \end{matrix}$ that leads to $P_{1} (τ) > P_{2} (τ) \forall τ \in [0, 0 + ɛ]$

Iterating this procedure for the next time interval [ɛ, 2ɛ] , we obtain the final result (44).

P₁ (t) should be the function of t and x (t) , $P_{1}^{\cdot} (t) = \frac{\partial}{\partial x (t)} P_{1} (t) \dot{x} (t) .$ However, in this paper we do not need the exact form of $P_{1}^{\cdot} (t)$ ,i.e., $\frac{\partial}{\partial x (t)} P_{1} (t) \dot{x} (t)$ also satisfies (41). In Appendix, we can also see that we do not calculate $P_{1}^{\cdot} (t) .$

Lemma 1 shows that the existence condition, or the matrix DRE (41) has solution, of the on-line fuzzy control is $[\begin{matrix} [c] cc Q_{1} (t) & {\bar{A}}_{1}^{T} (t) \\ {\bar{A}}_{1} (t) & - 3 R_{1} (t) \end{matrix}] > [\begin{matrix} [c] cc Q & A^{T} \\ A & - R \end{matrix}]$ (46) where A, Q, and R are constant matrices which are chosen such that they satisfy the following conditions

the pair (A, R^1/2) is controllable,

the pair (Q^1/2, A) is observable.

These two conditions are equivalent to the following local frequency condition [33] $A^{T} R^{- 1} A - Q \geq 0$ (47)

The detail proof of (47) can be found in [32].

From Lemma 1, the solution of (16) P (t) is not less than the solution of $A^{T} P + PA - PRP + Q = 0$ (48) providing that the initial condition of (16) is bigger than that of (48). So if we design a T-S fuzzy control such that (46) is satisfied, and the initial condition of (16) is big enough, then Theorem 1 is established. (46) requires to calculate all membership functions. It is the design limit of this method.

For an unknown nonlinear system, to obtain A (t) and B (t) of the T-S fuzzy system (4), some learning methods are needed, one may refer to [34]. This paper does not focus on this aspect.

4 Simulation results

In order to validate the performances of our Riccati differential equation based T-S fuzzy control, we design an autonomous underwater vehicle (AUV), see Fig. 1. It is a small-size AUV designed for real-world applications. It is equipped two propellers to control the vehicle in X-Y positions. In order to obtain a model of this AUV, we define two frames: the inertial reference frame X_I - Y_I, and the body frame X_B - Y_B. The origin of the body frame is chosen as the center of gravity of the AUV. In the inertial reference frame, X_B can be regarded as longitudinal axis (form aft to fore), Y_B is regarded as transversal axis (starboard direction).

Fig.1

An autonomous underwater vehicle (AUV).

A body-frame model for the under-actuated AUV with two independent propellers is [36] $M_{ine} \dot{χ} (t) + C (χ (t)) χ (t) + D (χ (t)) χ (t) = τ (t)$ (49) where M_ine is the inertia matrix. C (χ) is the Coriolis and centripetal matrix. D (χ) is the damping matrix, $χ = {[\begin{matrix} [c] ccc p, & q, & w \end{matrix}]}^{T}$ is the velocity vector. p, q are linear velocities in X_B-axis and Y_B-axis, respectively, w is the angular velocity with respect to the vertical axis Z_B, φ is the yaw angle in the inertial reference frame X_I - Y_I, τ_p = F_p + F_s is the total force along the X_B-axis, τ_w = (F_p - F_s) l is the torque with respect to Z_B-axis, F_p is the port thrust, F_s is the starboard thrust, and ξ (t) is perturbation. If we neglect hydrodynamic damping, consider the AUV as a disc, and the propellers are located at the center of mass, then a simplified dynamic model is $\begin{matrix} \dot{p} (t) & = & q (t) w (t) + τ_{p} (t) + ξ (t) \\ \dot{q} (t) & = & - p (t) w (t) \\ \dot{w} (t) & = & τ_{w} (t) \end{matrix}$ (50)

We use 27 T-S fuzzy rules to control this AUV, the i-th rule is

$\begin{matrix} If p \in F_{p}^{i} and q \in F_{q}^{i} and w \in F_{w}^{i}, \\ then \frac{d}{dt} χ (t) = A^{i} χ (t) + B^{i} τ (t) + {\tilde{f}}^{i} (χ (t)) + ζ^{i} (t) \end{matrix}$ (51) where i = 1 ⋯27, $A^{i} \in R^{3 \times 3}$ and $B^{i} \in R^{3 \times 2}$ . The membership functions of $F_{p}^{i},$ $F_{q}^{i}$ and $F_{w}^{i}$ are selected as Fig. 2. Aⁱ and Bⁱ are chosen as (43) such that the Riccati differential equation (16) has solution. The first four elements are $A^{1} = [\begin{matrix} [c] ccc 0 & 1 & 1 \\ - 1 & 0 & - 2 \\ 0 & 0 & 0 \end{matrix}],$ $A^{2} = [\begin{matrix} [c] ccc 0 & 1 & 0.5 \\ - 1 & 0 & - 1 \\ 0 & 0 & 0 \end{matrix}],$ $A^{3} = [\begin{matrix} [c] ccc 0 & 2 & 0.8 \\ - 2 & 0 & - 0.5 \\ 0 & 0 & 0 \end{matrix}],$ $A^{4} = [\begin{matrix} [c] ccc 0 & 5.3 & 1.2 \\ - 5.3 & 0 & - 3.5 \\ 0 & 0 & 0 \end{matrix}],$ $B^{1} = B^{2} = B^{3} = B^{4} = [\begin{matrix} [c] cc 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{matrix}],$ $F_{x}^{1} = N$ , $F_{x}^{2} = Z$ , $F_{x}^{3} = P .$ For (43),we choose A₂ = diag [- 1, - 1, - 1] , Q₂ = diag [0.2, 0.2, 0.2] , R₂ = diag [1.5, 1.5, 1.5] . Λ_i (i = 1 ⋯4) in (17) are chosen as identity matrices.

Fig.2

Membership functions of $F_{p}^{i},$ $F_{q}^{i}$ and $F_{w}^{i}$ .

Based on the planning motion method, the desired reference has the same structure as the AUV, it satisfies $s (x_{ref}, t) = [\begin{matrix} [c] c 2 + q_{ref} (t) w_{ref} (t) \\ - p_{ref} (t) w_{ref} (t) \\ 2 \end{matrix}] .$ With the initial conditions $χ_{0} = {[\begin{matrix} [c] ccc 2 & - 2 & 4 \end{matrix}]}^{T},$ $s (x_{ref}, 0) = {[\begin{matrix} [c] ccc 0 & 0 & 0 \end{matrix}]}^{T},$ the final feedback control is (18), here P (t) is the numerical solution of (40). The control gain K (t) is shown in Fig. 3. This controller is applied to the original system (50).

Fig.3

One element of the control gain K (t) based on RDE.

There are several results on fuzzy control for AUV. However, they do not care about the stability issue, so they do use LMIs or Lyapunov function [35]. Now we compare our method (RDE) with the popular LMI based T-S fuzzy control [7]. LMI method has PDC form $τ (t) = K_{LMI} (t) Δ (t), K_{LMI} (t) = \sum_{i = 1}^{27} α_{i} (t) K_{i}$ (52)

The first four K_i are $\begin{matrix} K_{1} & = & [\begin{matrix} [c] ccc 17.7187 & - 33.6441 & 5.6486 \\ 3.7922 & - 58.4148 & 22.7813 \end{matrix}] \\ K_{2} & = & [\begin{matrix} [c] ccc 20.0000 & - 74.0000 & 5.5000 \\ 5.0000 & - 75.0000 & 20.0000 \end{matrix}] \\ K_{3} & = & [\begin{matrix} [c] ccc - 4.3751 & 59.6067 & - 6.6742 \\ - 6.5349 & 14.6499 & 12.5751 \end{matrix}] \\ K_{4} & = & [\begin{matrix} [c] ccc 13.6403 & - 3.2136 & 4.9247 \\ 3.7247 & - 5.6222 & 10.4597 \end{matrix}] \end{matrix}$

The simulation is in Windows 7. We use Matlab 2012a and Simulink as the simulation environment. LMI toolbox in Matlab is applied to solve (52) off-line. Our on-line controller needs the dynamic system (40), which runs in Simulink. The control period is 10 ms. Within this time (10 ms), both fixed gain of LMI and time-varying gain of our controller work well. So the computation time of our on-line algorithm is less than 10 ms .

The experimental platform includes an internal computer system connected to an external computer, this configuration is shown in Fig. 4. The external computer is a laptop with Microsoft Windows operating system. The internal computer system consists of an electronic (controller) board that integrates: power stages, an electronic speed controller (ESC) for each thruster, and an ATmega2560 AVR microcontroller.

Fig.4

Configuration of the AUV control platform.

The AUV is equipped with three T100 Blue Robotics brushless thrusters: port and starboard thrusters located at the bottom of the vehicle and a vertical thruster located at the center of gravity of the vehicle. The AUV is equipped with two sensors: An inertial measurement unit (IMU) Invesense MPU-9150 to measure the orientation (heading) of the AUV and a pressure sensor Measurement Specialities MS5803-14B to measure the depth the vehicle is located underwater.

The experimental results are shown in Figs. 5 and 6. The references in the top figures are constant, while the references in the bottom figures are sine-waves. We can see that our Riccati differential equation (RDE) based method need 4 seconds to stabilize the AUV, while the LMI based method needs 10 seconds. Also LMI method cannot track the trajectory at the beginning. The fast speed of our RDE method may come from the dynamic system (40) is more simple and faster than the LMI method.

Fig.5

Trajecory tracking with T-S fuzzy control in X-axis.

Fig.6

Trajecory tracking with T-S fuzzy control in Y-axis.

The stability analysis of T-S fuzzy system based on LMI is complex. However, it is only an off-line and analytic task. There are also many relaxing condition for LMI technique. In on-line case, we need to run the dynamic system (40). The control gain is time-varying, while the gains of the LMI method are constant. For the computation speed, LMI is better than our RDE method. However, both of them are very fast. Simulation results show our RDE has faster convergence speed than LMI, because the feedback control gain of this paper is time-varying, which depends on the state P (t) of the dynamic system (40). P (t) needs both fuzzy model A (t) and B (t) , and the state x (t) . While the feedback control gains of LMIs only use the fuzzy model A (t) and B (t) .

The T-S fuzzy control is worse than the model-based control, such as feedback linearization. This is because the fuzzy model (51) is not exact model of the controlled system (50). We hope some adaptive techniques, such as ANFIS, can improve the fuzzy modeling accuracy in our future work.

Another behavior limit of this T-S fuzzy control is the adaptive gain depends on P (t) of RDE. The control amplitude always varies, even in the steady state, see Fig. 3. This may lead chattering problem for mechanical systems.

5 Conclusions

Normal T-S fuzzy control cannot be applied on-line, because it needs LMIs or the other complex techniques to guarantee stability. In this paper, a Riccati differential equation and an optimal-like feedback controller are used to guarantee the stability of the T-S fuzzy closed-loop system. By using these techniques, the T-S fuzzy controller becomes on-line version. We also propose an existence condition for the Riccati differential equation. The novel on-line adaptive fuzzy control is applied to an autonomous underwater vehicle. Experimental results show that our design method is more simple and effective than LMI-based methods. The further work is to extend the results to robust T-S fuzzy control.

References

Takagi

and Sugeno

, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans Syst, Man, Cybern15(1) (1985), 116–132.

Das

P.C.

, Statistically convergent fuzzy sequence spaces by fuzzy metric, Kyungpook Math Journal54(3) (2014), 413–423.

Tripathy

B.C.

and Ray

G.C.

, Weakly continuous functions on mixed fuzzy topological spaces, Acta Scientiarum Technology36(2) (2014), 331–335.

Feng

, A survey on analysis and design of model-based fuzzy control systems, IEEE Trans Fuzzy Syst14(5) (2006), 676–697.

Tanaka

, Hori

and Wang

H.O.

, A multiple Lyapunov function approach to stabilization of fuzzy control systems, IEEE Trans Fuzzy Syst11(4) (2003), 582–589.

Tanaka

and Wang

H.O.

, Fuzzy control systems design and analysis: A linear matrix inequality approach, John Wiley & Sons Inc., New York, USA, 2004.

Tuan

H.D.

, Apkarian

, Narikiyo

and Yamamoto

, Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Trans, Fuzzy Syst9 (2001), 324–332.

Tseng

C.-S.

, Chen

B.-S.

and Uang

H.-J.

, Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model, IEEE Trans Fuzzy Syst9(3) (2001), 381–392.

Liu

and Zhang

, Approaches to quadratic stability conditions and H_∞ control designs for T-S fuzzy systems, IEEE Trans Fuzzy Syst11(6) (2003), 830–839.

10.

Lee

H.-J.

, Kim

, Joo

Y.-H.

, Chang

and Park

J.-B.

, A new intelligent digital redesign for T-S fuzzy systems: Global approach, IEEE Trans Fuzzy Syst12(2) (2004), 274–284.

11.

Wang

W.-J.

and Lin

W.-W.

, Decentralized PDC for large-scale T-S fuzzy systems, IEEE Trans Fuzzy Syst13(6) (2005), 381–392.

12.

Dong

and Yang

G.-H.

, H_∞ controller synthesis via switched PDC scheme for discrete-time T-S fuzzy systems, IEEE Trans Fuzzy Syst17(3) (2009), 544–555.

13.

Zhao

, Zhang

and Gao

, New approach to guaranteed cost control of T-S fuzzy dynamic systemsWith interval parameter uncertainties, IEEE Trans Syst, Man, Cybern, B, Cybern39(6) (2009), 1516–1527.

14.

Xie

, Yue

, Ma

and Zhu

, Further studies on control synthesis of discrete-time T-S fuzzy systems via augmented multi-indexed matrix approach, IEEE Trans Cybern44(12) (2014), 2784–2791.

15.

, Su

, Shi

and Qiu

, A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time varying delay systems, IEEE Trans Syst, Man, Cybern, B, Cybern41(1) (2011), 273–286.

16.

, Shi

, Wu

and Song

Y.-D.

, A novel control design on discrete-time takagi–sugeno fuzzy systems with time-varying delays, IEEE Transactions on Fuzzy Systems21(4) (2013), 655–671.

17.

, Wu

, Shi

and Song

Y.-D.

, A novel approach to output feedback control of fuzzy stochastic systems, Automatica50(12) (2014), 3268–3275.

18.

Lin

, Wang

Q.-G.

and Lee

T.-H.

, Output tracking control for nonlinear systems via T-S fuzzy model approach, IEEE Trans Syst, Man, Cybern, B, Cybern36(2) (2004), 450–457.

19.

Chiu

C.-S.

, T-S fuzzy maximum power point tracking control of solar power generation systems,–, IEEE Trans Energy Convers25(4) (2010), 1123–1132. 7.

20.

and Li

S.-Y.

, Stability analysis and design of T-S fuzzy control system with simplified linear rule consequent, IEEE Trans Syst, Man, Cybern, B, Cybern34(1) (2004), 799–795.

21.

Narendra

K.S.

and Balakrishnan

, The changing face of adaptive control: The use of multiple models, Annual Reviews in Control35(1) (2011), 1–12.

22.

Joh

, Chen

Y.-H.

and Langari

, On the stability issues of linear Takagi-Sugeno fuzzy models, IEEE Trans Fuzzy Syst6 (1998), 402–410.

23.

Johansson

, Rantzer

and Arzen

K.E.

, Piecewise quadratic stability of fuzzy systems, IEEE Trans Fuzzy Syst7 (1999), 713–722.

24.

Zhao

, Zhang

, Shi

and Karimi

H.R.

, Novel stability criteria for T–S fuzzy systems, IEEE Trans Fuzzy Syst22(2) (2014), 313–323.

25.

Qiu

, Feng

and Gao

, Static-output-feedback H_∞ control of continuous-time T-S fuzzy affine systems via piecewise lyapunov functions, IEEE Trans Fuzzy Syst21(2) (2013), 245–3261.

26.

Wang

W.-J.

and Sun

C.-H.

, A relaxed stability criterion for T-S fuzzy discrete systems, IEEE Trans Syst, Man, Cybern, B, Cybern34(5) (2004), 2155–2158.

27.

Lin

P.-T.

, Wang

C.-H.

and Lee

T.-T.

, Time-optimal control of TS fuzzy models via lie algebra, IEEE Trans Fuzzy Syst17(4) (2009), 737–749.

28.

Chang

Y.-H.

, Chan

W.-S.

and Chang

C.-W.

, T-S fuzzy model-based adaptive dynamic surface control for ball and beam system, IEEE Trans Ind Electron60(6) (2013), 2251–2263.

29.

Gao

, Feng

and Xi

, A new design of robust H_∞ sliding mode control for uncertain stochastic T-S fuzzy time-delay systems, IEEE Trans Cybern44(9) (2014), 1556–1566.

30.

Feng

, Analysis and Synthesis of Fuzzy Control Systems: A Model-Based Approach, CRC Press, Boca Roton, USA, 2010.

31.

S.-J.

and Lin

C.-T.

, Optimal fuzzy controller design in continuous fuzzy system: Global concept approach, IEEE Transactions on Fuzzy Systems8(6) (2000), 713–729.

32.

Poznyak

A.S.

, Sanchez

E.N.

and Yu

, Differential Neural Networks for Robust Nonlinear Control, World Scientific Publishing Co., Singapore, 2001.

33.

Willems

J.C.

, Least squares optimal control and algebraic riccati equations, IEEE Transactions on Automatic Control16(2) (1971), 621–634.

34.

and Li

, Fuzzy identification using fuzzy neural networks with stable learning algorithms, IEEE Transactions on Fuzzy Systems12(3) (2004), 411–420.

35.

Chang

W.-J.

, Chang

and Liu

H.-H.

, Model-based fuzzy modeling and control for autonomous underwater vechicles in the horizonal plane, Journal of Marine Science and Technology11(3) (2003), 155–163.

36.

Lefeber

, Pettersen

K.Y.

and Nijmeijer

, Tracking control of an underactuated ship, IEEE Transactions on Control Systems Technology11(1) (2003), 52–61.

37.

Wimmer

H.K.

, Monotonicity of maximal solutions of algebraic riccati equations, System and Control Letters5(4) (1985), 317–319.