Optimal design of robust control for positive fuzzy dynamic systems with one-sided control constraint

Abstract

The paper considers an optimal design problem for a class of uncertain systems. The systems are nonlinear and the state is constrained to be positive. The uncertainty of the system is time-varying and bounded, with the bound lies within a prescribed fuzzy set. The control input of the system may also be constrained to be one-sided (i.e., either positive or negative). A transformation of the state is proposed to release the state constraint. Based on a partial sign-definiteness knowledge of the uncertainty, a one-sided robust control is presented. The control structure is deterministic and is not fuzzy if-then rule-based. By using the fuzzy description of uncertainty, the paper proposes an optimal design problem of the one-sided robust control. It is proven that the global solution to this optimal design problem always exists and is unique. The performance of the resulting controlled system is deterministically guaranteed as well as fuzzily optimized. The control design is illustrated by applying to a drug administration problem.

Keywords

Fuzzy dynamic system state constraint control constraint fuzzy set theory optimal design

1 Introduction

Positive systems [1] stand for the systems whose state variables are restricted to be positive due to their physical meanings. There are examples of such systems in engineering, ecology, economics, etc. For instance, in ecological and biological systems, the populations of the species are state variables, which can never be negative [2 –5].

One-sided control stands for that the control input can only be either positive or negative. This is mainly due to the physical nature of the control/actuator [6]. For example, in a braking system, one may control the clamp force of the brake calipers, which is unidirectional; that is, the clamp force can never be negative [7]. Other examples may include the control being the harvest rate in ecological systems [8] and the control being levels of government taxation, government expenditure and open-market purchases of bonds in macroeconomic systems [9].

The dynamic systems with state and control constraints are considered in the paper. Due to the imperfect knowledge of the system parameters and/or disturbance, uncertainty is inevitable in system models. By incorporating fuzzy descriptions of the uncertainties, the system is called the fuzzy dynamic systems [10]. The uncertainty in the fuzzy dynamic systems is (possibly fast) time-varying and bounded. The bound of the uncertainty is prescribed by a fuzzy number. In this fuzzy description, the exact bound of uncertainty is unknown, but is within a threshold. It adds more insight to the system modeling and is another way to combine fuzzy set theory and control theory [11]. In this approach, no Takagi-Sugeno fuzzy modeling [12] or fuzzy Mamdani if-then rule-based control [13] is invoked.

The desired system performance with the resulting control has two features: deterministic and fuzzy. From the deterministic perspective, the system is guaranteed to meet certain criteria regardless of the uncertainty. From the fuzzy perspective, the minimization of a fuzzy performance index is considered. As a result, the resulting controlled system is both performance guaranteeing and performanceoptimal.

The main contributions of this paper are fourfold. First, we propose a novel state transformation. The transformed system is then free from the state constraint.

Second, based on the information of the uncertainty bound, a robust control for the transformed system is designed. By using the robust control, both uniform boundedness and uniform ultimate boundedness are guaranteed. The control assures the state is always constrained regardless of the uncertainty.

Third, we propose to further explore the “sign” of the effect of uncertainty, which means some “worst case” uncertainty may never occur in certain domain. As a result, with the extra property of the uncertainty, the control input can be “on” in certain uncertainty situations and “off” in other situations. This results in a one-sided robust control to meet the control constraint. This one-sided robust control still guarantees uniform boundedness and uniform ultimate boundedness.

Fourth, an optimal design associated with the one-sided robust control is formulated. A fuzzy based performance index for the fuzzy dynamic system is constructed. Through minimizing this performance index, a design parameter can be selected. It is proven that the global solution to this optimal design problem exists and is unique. The solution can be found by solving a closed-form (i.e., analytic) algebraic equation. The control design is illustrated by applying to a drug administration problem.

2 Mathematical preliminaries

There are several mathematical preliminaries shown in the following for the later control and optimization design [14].

nFuzzy number: Let G be the fuzzy set in R; G is called a fuzzy number if: (1) G is normal, (2) G is convex, (3) the support of G is bounded, and (4) all α-cuts are closed intervals in R.

nFuzzy arithmetic: Let G and H be two fuzzy numbers and $G_{α} = [g_{α}^{-}, g_{α}^{+}]$ , $H_{α} = [h_{α}^{-}, h_{α}^{+}]$ be their α-cuts, α ∈ [0, 1]. The addition, subtraction, multiplication and division of G and H are given by, respectively $(G + H)_{α} = [g_{α}^{-} + h_{α}^{-}, g_{α}^{+} + h_{α}^{+}],$ (1) $\begin{matrix} (G - H)_{α} & = & [min (g_{α}^{-} - h_{α}^{-}, g_{α}^{+} - h_{α}^{+}), \\ max (g_{α}^{-} - h_{α}^{-}, g_{α}^{+} - h_{α}^{+})], \end{matrix}$ (2) $\begin{matrix} (G \cdot H)_{α} & = & [min (g_{α}^{-} h_{α}^{-}, g_{α}^{-} h_{α}^{+}, g_{α}^{+} h_{α}^{-}, g_{α}^{+} h_{α}^{+}), \\ max (g_{α}^{-} h_{α}^{-}, g_{α}^{-} h_{α}^{+}, g_{α}^{+} h_{α}^{-}, g_{α}^{+} h_{α}^{+})], \end{matrix}$ (3) $\begin{matrix} (G / H)_{α} & = & [min (\frac{g_{α}^{-}}{h_{α}^{-}}, \frac{g_{α}^{-}}{h_{α}^{+}}, \frac{g_{α}^{+}}{h_{α}^{-}}, \frac{g_{α}^{+}}{h_{α}^{+}}), \\ max (\frac{g_{α}^{-}}{h_{α}^{-}}, \frac{g_{α}^{-}}{h_{α}^{+}}, \frac{g_{α}^{+}}{h_{α}^{-}}, \frac{g_{α}^{+}}{h_{α}^{+}})] . \end{matrix}$ (4)

nDecomposition theorem: Define a fuzzy set ${\tilde{D}}_{α}$ in U with the membership function $μ_{{\tilde{D}}_{α}} = α I_{D_{α}} (x)$ where I_{nD_nα} (x) =1 if x ∈ D_nα and I_{nD_nα} (x) =0 if x ∈ U - D_nα. Then the fuzzy set D is obtained as $D = ⋃_{α \in [0, 1]} {\tilde{D}}_{α},$ (5) where ∪ is the union of the fuzzy sets (i.e., sup over α ∈ [0, 1]).

Fuzzy mathematics allows fuzzy algebraic computations of fuzzy numbers. To add two fuzzy numbers, for example, we first obtain their α-cuts. Then, we apply (1) to add them. The decomposition theorem is, then, applied to build the membership function of the resulting fuzzy number. Proceeding similarly, we are able to perform algebraic operations on fuzzy numbers and construct the resulting membershipfunction.

3 Positive uncertain system

Consider the following class of uncertain dynamical systems:

$\begin{matrix} \dot{x} (t) & = & f (x (t), σ (t), t) + (B (x (t), t) + Δ B (x (t), \\ σ (t), t)) τ (t) + Δ f (x (t), σ (t), t), \end{matrix}$ (6)

where t ∈ ℛ is the independent variable, $x (t) \in ℛ_{+}^{n}$ is the state and only assumes positive value, τ (t) ∈ ℛ^m is the control, and σ (t) ∈ Σ ⊂ ℛ^p is the uncertain parameter. Here the Σ ⊂ ℛ^p is known and compact, which stands for the possible bound of σ. The functions f (·): $ℛ_{+}^{n} \times Σ \times ℛ \to ℛ^{n}$ , B (·): $ℛ_{+}^{n} \times ℛ \to ℛ^{n \times m}$ , ΔB (·): $ℛ_{+}^{n} \times Σ \times ℛ \to ℛ^{n \times m}$ , and Δf (·): $ℛ_{+}^{n} \times Σ \times ℛ \to ℛ^{n}$ are continuous. In the following development, only B (·) needs to be known. Other functions may be unknown.

Assumption 1. (1) For each entry of x₀ namely, x_0i, i = 1, 2,. . . , n, there exists a fuzzy set X_0i in an universe of discourse Ξ_ni ∈ R characterized by a membership function μ_{nΞ_ni}: Ξ_ni → [0, 1]. That is $X_{0 i} = {(x_{0 i}, μ_{Ξ_{i}} (x_{0 i})) | x_{0 i} \in Ξ_{i}} .$ (7) Here Ξ_ni is known and compact. (2) For each entry of σ, namely σ_ni, i = 1, 2,. . . , p, the function σ_ni (·) is Lebesgue measurable. (3) For each σ_ni, there exists a fuzzy set S_ni in an universe of discourse Σ_ni ∈ R characterized by a membership function μ_ni: Σ_ni → [0, 1] . That is $S_{i} = {(σ_{i}, μ_{Σ_{i}} (σ_{i})) | σ_{i} \in Σ_{i}} .$ (8) Here, Σ_ni ∈ R is known and compact, Σ = Σ₁ × Σ₂ × ⋯ × Σ_np.

Remark. This assumption imposes fuzzy restrictions on the uncertainty x₀ and σ (t).

Assumption 2. There exists a constant vector $x^{*} = [x_{1}^{*}, x_{2}^{*}, . . ., x_{n}^{*}]^{T} > 0$ (this is interpreted as each component being positive; that is, $x_{i}^{*} > 0$ , i = 1, 2,. . . , n) such that f (x^*, σ, t) =0 for all (σ, t) ∈ Σ × ℛ.

Assumption 3. There exists a matrix E (x, σ, t) such that $Δ B (x, σ, t) = BE (x, σ, t) .$ (9) In addition, there exists $\underline{λ} > - 1$ , such that for all $(x, t) \in ℛ_{+}^{n} \times ℛ$ , $min_{σ \in Σ} \frac{1}{2} λ_{min} (E (x, σ, t) + E^{T} (x, σ, t)) \geq \underline{λ} > - 1 .$ (10)

Consider that x (t₀) = x₀ > 0 where t₀ is the initial time. The objective of the control is to drive the state x (t) of (6) to be close to x^* subject to the constraint that x (t) >0 for all t ≥ t₀.

4 State transformation and control design

As the constraint of positive state, we propose a bijective transformation to transform the constrained x to a new state without constraint. Let $y_{i} = ln (\frac{x_{i}}{x_{i}^{*}}), i = 1, 2, . . ., n,$ (11) where for all 0< x_ni < + ∞, the corresponding y_ni is -∞ < y_ni < + ∞. Furthermore, $y_{i} = 0 \Leftrightarrow x_{i} = x_{i}^{*}$ . Therefore $x_{i} = x_{i}^{*} e^{y_{i}}, i = 1, 2, . . ., n .$ (12)

By (12), we have

$\begin{matrix} \dot{y} & = & [\begin{matrix} x_{1}^{* - 1} e^{- y_{1}} {\dot{x}}_{1} \\ x_{2}^{* - 1} e^{- y_{2}} {\dot{x}}_{2} \\ ⋮ \\ x_{n}^{* - 1} e^{- y_{n}} {\dot{x}}_{n} \end{matrix}] \\ = & Ψ (y) [f (x, σ, t) + (B (x, t) + Δ B (x, σ, t)) τ \\ + Δ f (x, σ, t)], \end{matrix}$ (13) where Ψ : = diag $[x_{i}^{* - 1} e^{- y_{i}}]_{n \times n}$ . Let $\hat{f} (y, σ, t) : = f (x^{*} e^{y}, σ, t),$ (14) $\hat{B} (y, t) : = B (x^{*} e^{y}, t),$ (15) $Δ \hat{B} (y, σ, t) : = Δ B (x^{*} e^{y}, σ, t),$ (16) $Δ \hat{f} (y, σ, t) : = Δ f (x^{*} e^{y}, σ, t),$ (17) where $x^{*} e^{y} : = {[x_{1}^{*} e^{y_{1}}, x_{2}^{*} e^{y_{2}}, \dots, x_{n}^{*} e^{y_{n}}]}^{T} .$ (18)

We can rewrite the system in (13) as

$\begin{matrix} \dot{y} & = & \tilde{f} (y, σ, t) + (\tilde{B} (y, t) + Δ \tilde{B} (y, σ, t)) τ (t) \\ + Δ \tilde{f} (y, σ, t), \end{matrix}$ (19) where $\tilde{f} (y, σ, t) : = Ψ (y) \hat{f} (y, σ, t),$ (20) $\tilde{B} (y, t) : = Ψ (y) \hat{B} (y, t),$ (21) $Δ \tilde{B} (y, σ, t) : = Ψ (y) Δ \hat{B} (y, σ, t),$ (22) $Δ \tilde{f} (y, σ, t) : = Ψ (y) Δ \hat{f} (y, σ, t) .$ (23)

Remark. Under the provision of Assumption 3, we also have for all (y, t) ∈ ℛⁿ × ℛ, $min_{σ \in Σ} \frac{1}{2} λ_{min} (\hat{E} (y, σ, t) + {\hat{E}}^{T} (y, σ, t)) \geq \underline{λ} > - 1,$ (24) where $\hat{E} (y, σ, t) : = E (x^{*} e^{y}, σ, t)$ .

Assumption 4. There is a Lyapunov function V (·): ℛⁿ × ℛ → ℛ₊, KR functions γ_ni (·): ℛ₊ → ℛ₊, i = 1, 2, 3, and a scalar constant γ₄ ≥ 0, such that for all (y, σ, t) ∈ ℛⁿ × Σ × ℛ, $γ_{1} (‖ y ‖) \leq V (y, t) \leq γ_{2} (‖ y ‖),$ (25) $\frac{\partial V (y, t)}{\partial t} + \frac{\partial V (y, t)}{\partial y} \tilde{f} (y, σ, t) \leq - γ_{3} (‖ y ‖) + γ_{4} .$ (26)

Remark. If γ₄ = 0, then Assumption 2 (which implies $\tilde{f} (0, σ, t) = 0$ for all (σ, t) ∈ Σ × ℛ) and Assumption 4 assure that y = 0 of the system $\dot{y} = \tilde{f} (y, σ, t)$ (27) is globally asymptotically stable. If γ₄ > 0, then y (t) of (19) is (globally) uniformly ultimately bounded. The size of the uniformly ultimately bounded region is dependent on γ₄.

Remark. Both $\tilde{f} (y, σ, t)$ and $Δ \tilde{f} (y, σ, t)$ are dependent on the uncertain parameter σ. The reason to distinguish them in (19) is that $\tilde{f} (y, σ, t)$ provides a “stable” portion for the system regardless of the appearance/effect of σ (per Assumption 4). The other portion $Δ \tilde{f} (y, σ, t)$ may not provide the same advantage for the system. It will be compensated by the controller, as will be shown later.

Assumption 5. Let $α (y, t) : = {\tilde{B}}^{T} (y, t) \nabla_{y} V (y, t)$ . (1) There exist continuous functions β (·): ℛⁿ × Σ × ℛ → ℛ^m, with β (y, σ, t) : = [β₁ (y, σ, t) , β₂ (y, σ, t) ,. . . , β_nm (y, σ, t)] ^nT, such that for all (y, σ, t) ∈ ℛⁿ × Σ × ℛ, $\nabla_{y}^{T} V (y, t) Δ \tilde{f} (y, σ, t) = α (y, t) β (y, σ, t) .$ (28) (2) There are scalars ζ_nij (σ, t) ′s and scalar functions ρ_nij (y, t) ′s, j = 1, 2,. . . , l, such that ‖β_ni (y, σ, t) ‖ can be upper-bounded as

$\begin{matrix} ‖ β_{i} (y, σ, t) ‖ & \leq & [ζ_{i 1} (σ, t) ζ_{i 2} (σ, t) \dots ζ_{il} (σ, t)] \\ \times [\begin{matrix} ρ_{i 1} (y, t) \\ ρ_{i 2} (y, t) \\ ⋮ \\ ρ_{il} (y, t) \end{matrix}] \\ = : & ζ_{i} (σ, t) ρ_{i} (y, t) . \end{matrix}$ (29)

From (29), we have $‖ β_{i} (y, σ, t) ‖ \leq {\hat{ζ}}_{i} (σ, t) {\hat{ρ}}_{i} (y, t),$ (30) where ${\hat{ζ}}_{i} (σ, t) : = ‖ ζ_{i} (σ, t) ‖$ , ${\hat{ρ}}_{i} (y, t) : = ‖ ρ_{i} (y, t) ‖$ .

Remark. The scalar ${\hat{ζ}}_{i}$ depends on the possible bound of uncertainty. The associated membership function can be determined via the individual membership functions for ζ_nij’s by using the fuzzy arithmetic and decomposition theorem.

We propose the following robust control as $τ_{i} (t) = - κ α_{i} (y (t), t) (1 + \underline{λ})^{- 1} {\hat{ρ}}_{i}^{2} (y (t), t),$ (31) where i = 1, 2,. . . , m, the scalar κ is a constant design parameter and κ > 0.

Remark. The control τ_ni (t) is based on the nominal system, the transformed state y (t), the known function ${\hat{ρ}}_{i} (y, t) (\cdot)$ which relates the bound of uncertainty and the design parameter. This proposed control is deterministic and is not if-then rules-based.

Theorem 1. Consider the system (6) and (19). Suppose the system subject to the Assumptions 1–5, then the control (31) renders the system state y (t) of the system (19) to the following performance:

Uniform boundedness: For any r > 0 with ‖y (t) ‖ ≤ d (r) for all t ≥ t₀;

Uniform ultimate boundedness: For any r > 0 with ‖y (t₀) ‖ ≤ r, there exists a $\underline{d} > 0$ such that $‖ y (t) ‖ \leq \bar{d}$ for any $\underline{d} > \bar{d}$ as $t \geq t_{0} + T (\underline{d}, r)$ , where $T (\underline{d}, r) < \infty$ .

Proof. Consider the Lyapunov function candidate V (y, t). For any given admissible uncertainty σ (·), the time derivative of V is

Based on the Assumption 4, we have

$\begin{matrix} \dot{V} (y, t) & \leq & - γ_{3} (‖ y ‖) + \nabla_{y}^{T} V (y, t) (\tilde{B} (y, t) \\ + Δ \tilde{B} (y, σ, t)) τ (t) + \nabla_{y}^{T} V (y, t) \\ \times Δ \tilde{f} (y, σ, t) . \end{matrix}$ (33) By Assumption 5, we can show that

$\begin{matrix} \dot{V} & \leq & - γ_{3} (‖ y ‖) + \nabla_{y}^{T} V (y, t) \tilde{B} (y, t) (I \\ + \hat{E} (y, σ, t)) τ (t) + \nabla_{y}^{T} V (y, t) Δ \tilde{f} (y, σ, t) \\ \leq & - γ_{3} (‖ y ‖) + α^{T} (y, t) (I + \hat{E} (y, σ, t)) τ (t) \\ + \sum_{i = 1}^{m} α_{i} β_{i} . \end{matrix}$ (34)

According to the Assumption 3 and the control (31), we have

$\begin{matrix} α^{T} (I + \hat{E}) τ \\ = [α_{1} α_{2} \dots α_{m}] (I + \hat{E}) {(1 + \underline{λ})}^{- 1} κ \\ \times [\begin{matrix} - α_{1} {\hat{ρ}}_{1}^{2} \\ - α_{2} {\hat{ρ}}_{2}^{2} \\ ⋮ \\ - α_{m} {\hat{ρ}}_{m}^{2} \end{matrix}] \\ \leq - \sum_{i = 1}^{m} κ ‖ α_{i} ‖^{2} {\hat{ρ}}_{i}^{2} . \end{matrix}$ (35)

By Assumption 5, we can get

$\begin{matrix} \sum_{i = 1}^{m} α_{i} β_{i} & \leq & \sum_{i = 1}^{m} ‖ α_{i} ‖ ‖ β_{i} ‖ \\ \leq \sum_{i = 1}^{m} ‖ α_{i} ‖ {\hat{ζ}}_{i} {\hat{ρ}}_{i} . \end{matrix}$ (36)

Substituting (35) and (36) into (34), then

$\begin{matrix} \dot{V} & \leq & - γ_{3} (‖ y ‖) - \sum_{i = 1}^{m} κ ‖ α_{i} ‖^{2} {\hat{ρ}}_{i}^{2} + \sum_{i = 1}^{m} ‖ α_{i} ‖ {\hat{ζ}}_{i} {\hat{ρ}}_{i} \\ \leq & - γ_{3} (‖ y ‖) + \sum_{i = 1}^{m} \frac{{\hat{ζ}}_{i}^{2}}{4 κ} . \end{matrix}$ (37)

Invoking the standard arguments as in [15, 16], we can conclude the uniform boundedness with $d (r) = {\begin{matrix} (γ_{1}^{- 1} \circ γ_{2}) (R), if r \leq R, \\ (γ_{1}^{- 1} \circ γ_{2}) (r), if r > R, \end{matrix}$ (38) $R = γ_{3}^{- 1} (\sum_{i = 1}^{m} \frac{ζ}{4 κ}),$ (39) where $ζ = max_{σ \in Σ} {\hat{ζ}}_{i}^{2} .$ (40) Furthermore, uniform ultimate boundedness also follows with $\underline{d} = (γ_{1}^{- 1} \circ γ_{2}) (R), \bar{d} > \underline{d},$ (41) $T (\bar{d}, r) = {\begin{matrix} 0, if r \leq \bar{R}, \\ \frac{γ_{2} (r) - γ_{1} (\bar{R})}{γ_{3} (\bar{R}) - \sum_{i = 1}^{m} \frac{{\hat{ζ}}_{i}^{2}}{4 κ}}, otherwise, \end{matrix}$ (42) $\bar{R} = (γ_{2}^{- 1} \circ γ_{1}) (\bar{d}) .$ (43) Q.E.D.

Remark. Since all universes of discourse are compact (hence closed and bounded), ${\hat{ζ}}_{i}$ (i = 1, 2,. . . , m) is bounded and R is bounded. From (39), we can conclude that the larger the control magnitude κ, the smaller the size of the uniform ultimate boundedness region.

5 One-sided robust control

Many practical designs require that the control components τ_ni (t) can only assume one-sided value (i.e., either τ_ni (t) ≤0 or τ_ni (t) ≥0). We now take this control constraint into consideration.

Assumption 6. Let α_ni (y, t) denote the i-th component of α (y, t). (1) There exist continuous functions $\tilde{ω} (\cdot)$ : ℛⁿ × Σ × ℛ → ℛ^m, for all (y, σ, t) ∈ ℛⁿ × Σ × ℛ, $\frac{\partial V (y, t)}{\partial y} Δ \tilde{f} (y, σ, t) = α^{T} (y, t) \tilde{ω} (y, σ, t),$ (44) $\begin{matrix} \tilde{ω} (y, σ, t) \\ : = {[{\tilde{ω}}_{1} (y, σ, t), {\tilde{ω}}_{2} (y, σ, t), . . ., {\tilde{ω}}_{m} (y, σ, t)]}^{T} . \end{matrix}$ (45) (2) There is a set J ⊆ I : = {1, 2,. . . , m} such that for all i ∈ J and (y, t) with α_ni (y, t) ≥0,

${\tilde{ω}}_{i} (y, σ, t) \leq 0, σ \in Σ .$ (46) (3) There are scalars ${\tilde{ζ}}_{ij} (σ, t)^{'} s$ and scalar functions ${\tilde{ρ}}_{ij} (y, t)^{'} s$ , i = 1, 2,. . , m, j = 1, 2,. . . , l, such that $‖ {\tilde{ω}}_{i} (y, σ, t) ‖$ can be upper-bounded as

$\begin{matrix} ‖ {\tilde{ω}}_{i} (y, σ, t) ‖ & \leq & [{\tilde{ζ}}_{i 1} (σ, t) {\tilde{ζ}}_{i 2} (σ, t) \dots {\tilde{ζ}}_{il} (σ, t)] \\ \times [\begin{matrix} {\tilde{ρ}}_{i 1} (y, t) \\ {\tilde{ρ}}_{i 2} (y, t) \\ ⋮ \\ {\tilde{ρ}}_{il} (y, t) \end{matrix}] \\ = : & {\tilde{ζ}}_{i} (σ, t) {\tilde{ρ}}_{i} (y, t) . \end{matrix}$ (47)

Remark. Assumption 6 (2) imposes a partial sign-definiteness condition on the uncertainty ${\tilde{ω}}_{i} (y, σ, t)$ , which means ${\tilde{ω}}_{i} (y, σ, t)$ to be opposite in sign to α_ni (y, t) when α_ni (y, t) ≥0. However, when α_ni (y, t) <0, there is no constraint on ${\tilde{ω}}_{i} (y, σ, t)$ , hence ${\tilde{ω}}_{i} (y, σ, t)$ can be both positive and negative. This, in general, can be verified without knowing the exact value of ${\tilde{ω}}_{i} (y, σ, t)$ , which is unknown since σ is unknown.

Remark. Condition (46) can be met in a number of situations. These include, for example, when ${\tilde{ω}}_{i} (y, σ, t) = - α_{i} (y, t) φ_{i} (y, σ, t),$ (48) where φ_ni (y, σ, t) ≥0 for all (y, σ, t) ∈ ℛⁿ × Σ × ℛ.

Another example is ${\tilde{ω}}_{i} (y, σ, t) \leq 0 \forall (y, σ, t) \in ℛ^{n} \times Σ \times ℛ .$ (49) Based on (49), the condition (46) can be met. For i ∈ J, a heuristic chart in Fig. 1 will illustrates a possible choice of ${\tilde{ω}}_{i}$ .

From (47), we can have $‖ {\tilde{ω}}_{i} (y, σ, t) ‖ \leq {\underline{\tilde{ζ}}}_{i} (σ, t) {\underline{\tilde{ρ}}}_{i} (y, t),$ (50) where

$\begin{array}{l} {\underline{\tilde{ζ}}}_{i} (σ, t) : = ‖ {\tilde{ζ}}_{i} (σ, t) ‖, \\ {\underline{\tilde{ρ}}}_{i} (y, t) : = ‖ {\tilde{ρ}}_{i} (y, t) ‖ . \end{array}$ (51)

Remark. Based on the Assumption 6 (2), we can describe the bound of the uncertainty ${\tilde{ω}}_{i}$ by scalars ${\tilde{ζ}}_{i}$ ’s and scalar functions ${\tilde{ρ}}_{i} (\cdot)$ ’s.

The following one-sided robust control is proposed: $τ_{i} (t) = {\begin{matrix} - κ α_{i} (y (t), t) (1 + \underline{λ})^{- 1} {\underline{\tilde{ρ}}}_{i}^{2} (y (t), t), \\ if α_{i} (y (t), t) < 0, (52) \\ 0, \\ if α_{i} (y (t), t) \geq 0, (53) \end{matrix}$ where i = 1, 2,. . . , m, κ ∈ ℛ is a constant design parameter with κ > 0.

Remark. The control τ_ni is only to compensate the uncertainty when α_ni (y, t) <0. The design sets the control to be zero when α_ni (y, t) ≥0. By the design in (52) and (53), we can show that τ_ni (t) ≥0 for any time t. This control is deterministic and not fuzzy if-then rule-based.

Theorem 2. Consider the system (6) and (19). Suppose the system is subject to Assumptions 1-4 and 6, then the one-sided control in (52) and (53) renders the system (19) the following performance:

Uniform boundedness: For any r > 0 with ‖y (t) ‖ ≤ d (r) for all t ≥ t₀;

Uniform ultimate boundedness: For any r > 0 with ‖y (t₀) ‖ ≤ γ, there exists a $\underline{d} > 0$ such that $‖ y (t) ‖ \leq \bar{d}$ for any $\underline{d} > \bar{d}$ as $t \geq t_{0} + T (\underline{d}, r)$ , where $T (\underline{d}, r) < \infty$ .

Proof. Suppose the Lyapunov function candidate be V (y, t) in (32). By using Assumptions 4 and 6, for any admissible uncertainty σ (·), the derivative of V (y, t) is

$\begin{matrix} \dot{V} (y, t) & = & \frac{\partial V (y, t)}{\partial t} + \nabla_{y} V (y, t) [\tilde{f} (y, σ, t) \\ + (\tilde{B} (y, t) + Δ \tilde{B} (y, σ, t)) τ (t) \\ + Δ \tilde{f} (y, σ, t)] \\ \leq & - γ_{3} (‖ y ‖) + \nabla_{y}^{T} V (y, t) \tilde{B} (y, t) (I \\ + \hat{E} (y, σ, t)) τ (t) + \nabla_{y}^{T} V (y, t) Δ \tilde{f} (y, σ, t) \\ \leq & - γ_{3} (‖ y ‖) + α^{T} (y, t) (I + \hat{E} (y, σ, t)) \\ \times τ (t) + \sum_{i = 1}^{m} α_{i} ω_{i} . \end{matrix}$ (54)

We can rewrite the second term of (55) as

$\begin{matrix} α^{T} (y, t) (I + \hat{E} (y, σ, t)) τ (t) \\ = [α_{1} α_{2} \dots α_{m}] (I + \hat{E}) (1 + \underline{λ})^{- 1} κ [\begin{matrix} - α_{1} π_{1}^{2} \\ - α_{2} π_{2}^{2} \\ ⋮ \\ - α_{m} π_{m}^{2} \end{matrix}] \\ \leq - \sum_{i = 1}^{m} κ ‖ α_{i} ‖^{2} π_{i}^{2}, \end{matrix}$ (55) where π_ni ≥ 0. If α_ni < 0, $π_{i} = {\underline{\tilde{ρ}}}_{i}^{2}$ , otherwise, π_ni = 0. Then we can substitute the (55) into (54) and have $\dot{V} (y, t) \leq - γ_{3} (‖ y ‖) - \sum_{i = 1}^{m} κ ‖ α_{i} ‖^{2} π_{i}^{2} + \sum_{i = 1}^{m} α_{i} ω_{i} .$ (56)

Considering the condition of the control design, we can develop the (63) under two possible situations of the inequalities about α_ni (y, t).

(1) For any j ∉ J and j ∈ K ⊆ I : = {1, 2,. . . , m}, which means α_nj (y, t) <0, we have

$\begin{matrix} - κ ‖ α_{j} ‖^{2} π_{j}^{2} + α_{j} ω_{j} \\ \leq - κ ‖ α_{j} ‖^{2} {\underline{\tilde{ρ}}}_{j}^{2} + ‖ α_{j} ‖ ‖ ω_{j} ‖ \\ \leq - κ ‖ α_{j} ‖^{2} {\underline{\tilde{ρ}}}_{j}^{2} + ‖ α_{j} ‖ {\underline{\tilde{ζ}}}_{j} {\underline{\tilde{ρ}}}_{j} \\ \leq \frac{{\underline{\tilde{ζ}}}_{j}^{2}}{4 κ} . \end{matrix}$ (57)

(2) For any i ∈ J, which means α_ni (y, t) ≥0, we have

$- κ ‖ α_{i} ‖^{2} π_{i}^{2} + α_{i} ω_{i} \leq 0 + α_{i} ω_{i} \leq α_{i} ω_{i} .$ (58) By Assumption 6 (2), we have ω_ni ≤ 0, when α_ni ≥ 0, then α_ninω_ni ≤ 0.

From (58) and (59), we can conclude $\sum_{i = 1}^{m} (- κ ‖ α_{i} ‖^{2} π_{i}^{2}) + \sum_{i = 1}^{m} α_{i} ω_{i} \leq \sum_{j = 1}^{k} \frac{{\underline{\tilde{ζ}}}_{j}^{2}}{4 κ},$ (59) where k is the number of the set K and k ≤ m. Then $\dot{V} (y, t) \leq - γ_{3} (‖ y ‖) + \sum_{j = 1}^{k} \frac{{\underline{\tilde{ζ}}}_{j}^{2}}{4 κ} .$ (60) By (60), We can describe the system performance with uniform boundedness and uniform ultimate boundedness: $d (r) = {\begin{matrix} (γ_{1}^{- 1} \circ γ_{2}) (R), if r \leq R, \\ (γ_{1}^{- 1} \circ γ_{2}) (r), if r > R, \end{matrix}$ (61) $R = γ_{3}^{- 1} (\sum_{j = 1}^{k} \frac{\underline{\tilde{ζ}}}{4 κ}),$ (62) where $\underline{\tilde{ζ}} = max_{σ \in Σ} {\underline{\tilde{ζ}}}_{j}^{2} .$ (63) Furthermore, uniform ultimate boundedness also follows with $\underline{d} = (γ_{1}^{- 1} \circ γ_{2}) (R), \bar{d} > \underline{d},$ (64) $T (\bar{d}, r) = {\begin{matrix} 0, if r \leq \bar{R}, \\ \frac{γ_{2} (r) - γ_{1} (\bar{R})}{γ_{3} (\bar{R}) - \sum_{j = 1}^{k} \frac{{\underline{\tilde{ζ}}}_{j}^{2}}{4 κ}}, otherwise, \end{matrix}$ (65) $\bar{R} = (γ_{2}^{- 1} \circ γ_{1}) (\bar{d}) .$ (66) Q.E.D.

Remark. From (58), for any j ∈ K and α_nj < 0, the uncertainty term α_njnω_nj can be dealt with the control design in (52). From (57), for any i ∈ J and α_ni ≥ 0, the uncertainty term α_ninω_ni is sign-definiteness (α_ninω_ni ≤ 0). As more knowledge of the uncertainty is utilized, we can achieve the system performance in two kinds. The first kind performance is uniform boundedness and uniform ultimate boundedness. The second kind performance is (globally) asymptotically stable. If the set J ⊂ I, we can get the system performance in the first kind. If the set J = I, we can achieve the system performance in the second kind.

Remark. The size of the uniform ultimate boundedness can be determined by the choice of control gain κ. We may obtain a smaller size of the uniform ultimate boundedness by a lager κ. Based on the trade-off between system performance and control cost, an optimal design of the one-sided robust control is proposed.

6 Optimal design for one-sided robust control

In previous analysis, the size of the uniform ultimate boundedness decreases as the magnitude of the robust control gain κ increases. If κ approaches to infinity, the size of the uniform ultimate boundedness approaches to 0. This strong system performance requires a large control effort, which can be decided by choosing the control gain κ. From the practical control design point of view, the designer may be interested in seeking an optimal choice of κ for a compromise among various conflicting criteria.

From the uniform boundedness and uniform ultimate boundedness expressed in (61–66), we can view $(γ_{1}^{- 1} \circ γ_{2} \circ γ_{3}^{- 1}) (\sum_{i = 1}^{k} ({\underline{\tilde{ζ}}}_{i}^{2} / 4 κ))$ as the upper bound of the steady state. For the given $\bar{R} > 0$ and $\bar{d} > 0$ , if the initial state $‖ y (t_{0}) ‖ > \bar{R}$ , $T (\bar{d}, r)$ stands for the upper bound of the time interval it will take for the state y (t) to enter the set $Ω_{\infty} = {‖ y ‖ \leq \bar{d}}$ . Let $η (t_{0}) : = \frac{γ_{2} (‖ y (t_{0}) ‖) - γ_{1} (\bar{R})}{γ_{3} (\bar{R}) - \frac{δ}{κ}},$ (67) $η_{\infty} (δ, κ) : = (γ_{1}^{- 1} \circ γ_{2} \circ γ_{3}^{- 1}) (\frac{δ}{κ}),$ (68) where $δ : = \sum_{i = 1}^{k} ({\underline{\tilde{ζ}}}_{i}^{2} / 4 κ)$ .

In (67, 68), η (t₀) and η_∞ (δ, γ) depend on the initial state y₀ and δ which are in fuzzy description. There is no exact value of initial state y₀ and the uncertainty δ, while the values are characterized by the membership functions.

Remark. When the initial state y₀ meets the condition that $‖ y (t_{0}) ‖ > \bar{R}$ , the time interval of the transient state is calculated by η (t₀) in (67). Otherwise, the time interval of transient state is 0 shown in (65). Thus we may relate η (t₀) to the transient state performance. As the η_∞ (δ, κ) regarded as the upper bound of the uniform ultimate boundedness, we can relate η_∞ (δ, κ) to the steady state performance.

Definition. Consider a fuzzy set $N = {(ν, μ_{N}) | ν \in N}$ . For any function f (·): N → R, the D-operation D [f (ν)] is given by $D [f (ν)] = \frac{\int_{N} f (ν) μ_{N} (ν) d ν}{\int_{N} μ_{N} (ν) d ν} .$ (69)

Remark. In a sense, the D-operation D [f (ν)] takes an average value of f (ν) over μ_nN (ν). In the special case that f (ν) = ν, this is reduced to the well-known center-of-gravity defuzzification method (see, e.g., [14]). If $N$ is crisp (i.e., μ_nN = 1 for all ν ∈ N), then D [f (ν)] = f (ν).

Lemma 1. For any crisp constant a ∈ R $D [af (ν)] = aD [f (ν)] .$ (70)

We now propose the following quadratic performance index: For any t₀ and $r > \bar{R}$ , let

$\begin{matrix} J (κ, t_{0}) & : = & λ_{1} D [η^{2} (t_{0})] + λ_{2} D [η_{\infty}^{2} (δ, κ)] + λ_{3} κ^{2} \\ = : & λ_{1} J_{1} (t_{0}) + λ_{2} J_{2} (κ) + λ_{3} J_{3} (κ), \end{matrix}$ (71) where λ₁, λ₂, λ₃ > 0 are the weighting factors. This performance index consists three parts. The first part is J₁ (t₀) which can be interpreted as the average (via D-operation) of the transient performance related with initial state y₀. The second part J₂ (κ) is viewed as the average (via D-operation) of the steady-state performance. The third part J₃ (κ) is due to the control cost. The objective of our optimal design is to choose an appropriate κ > 0, so that the performance index J (κ, t₀) is minimized.

Remark. A standard LQG (Linear Quadratic Gaussian) problem in stochastic control is to minimize a performance index which is the average (via the expectation value operation in probability) of the overall state and control accumulation. In this paper, the proposed new optimal design may be viewed as, loosely speaking, a parallel problem in the fuzzy setting. However, we need to note the differences in the essence of these two problems. The Gaussian probability distribution implies that the uncertainty is unbounded (although a higher bound is predicted by a lower probability). In the current consideration, the uncertainty bound is always finite. In addition, LQG does not take the parameter uncertainty into account.

We assume that

$\begin{matrix} D [η^{2} (t_{0})] & = & h (\frac{1}{κ}), \\ D [η_{\infty}^{2} (δ, κ)] & = & g (\frac{1}{κ}), \end{matrix}$ (72) where the function h (·): ℛ₊ → ℛ₊ is continuous and strictly increasing and

$\begin{matrix} h_{1} (x) : = \frac{\partial h (x)}{\partial x}, h_{1} (\cdot) : ℛ_{+} \to ℛ_{+}; \\ h_{2} (x) : = \frac{\partial^{2} h (x)}{\partial x^{2}}, h_{2} (\cdot) \geq 0 . \end{matrix}$ (73) and the function g (·): ℛ₊ → ℛ₊ is also continuous and strictly increasing and

$\begin{matrix} g_{1} (x) : = \frac{\partial g (x)}{\partial x}, g_{1} (\cdot) : ℛ_{+} \to ℛ_{+}; \\ g_{2} (x) : = \frac{\partial^{2} g (x)}{\partial x^{2}}, g_{2} (\cdot) \geq 0 . \end{matrix}$ (74) Therefore, we have $J (κ, t_{0}) : = λ_{1} h (\frac{1}{κ}) + λ_{2} g (\frac{1}{κ}) + λ_{3} κ^{2} .$ (75)

The new proposed optimal design problem can be described as the following constrained optimization problem: For any t₀ and given $\bar{R}$ , let $κ_{opt} \in {min_{κ} J (κ, t_{0}) | κ > \frac{δ}{γ_{3} (\bar{R})}} .$ (76)

By using the performance index expressed in (75), we can take the first order derivative of J (κ, t₀) with respect to κ as $\frac{\partial J}{\partial κ} = - λ_{1} \frac{\partial h (\frac{1}{κ})}{\partial (\frac{1}{κ})} \frac{1}{κ^{2}} - λ_{2} \frac{\partial g (\frac{1}{κ})}{\partial (\frac{1}{κ})} \frac{1}{κ^{2}} + 2 λ_{3} κ .$ (77) Then $\frac{\partial J}{\partial κ} = 0$ (78) leads to $- λ_{1} \frac{\partial h (\frac{1}{κ})}{\partial (\frac{1}{κ})} \frac{1}{κ^{2}} - λ_{2} \frac{\partial g (\frac{1}{κ})}{\partial (\frac{1}{κ})} \frac{1}{κ^{2}} + 2 λ_{3} κ = 0,$ (79) or $λ_{1} h_{1} (\frac{1}{κ}) + λ_{2} g_{1} (\frac{1}{κ}) - 2 λ_{3} κ^{3} = 0 .$ (80) This a well-formulated closed-form scalar algebraic equation. We shall show the solution to this equation is the optimal solution to the constrained optimization problem (670).

Since $κ > δ / γ_{3} (\bar{R})$ , we have

$\begin{matrix} lim_{κ \to {\frac{δ}{γ_{3} (\bar{R})}}^{+}} λ_{1} h_{1} (\frac{1}{κ}) + λ_{2} g_{1} (\frac{1}{κ}) - 2 λ_{3} κ^{3} \\ = lim_{κ \to λ_{1} {\frac{δ}{γ_{3} (\bar{R})}}^{+}} \frac{δ (γ_{2} (r) - γ_{1} (\bar{R}))}{(γ_{3} (\bar{R}) - \frac{δ}{κ})^{2}} + λ_{2} g_{1} (\frac{γ_{3} (\bar{R})}{δ}) \\ - 2 λ_{3} {(\frac{γ_{3} (\bar{R})}{δ})}^{3} \\ = + \infty, \end{matrix}$ (81) and

$\begin{matrix} lim_{κ \to + \infty} λ_{1} h_{1} (\frac{1}{κ}) + λ_{2} g_{1} (\frac{1}{κ}) - 2 λ_{3} κ^{3} \\ = lim_{κ \to + \infty} λ_{1} \frac{δ (γ_{2} (r) - γ_{1} (\bar{R}))}{γ_{3}^{2} (\bar{R})} + λ_{2} g_{1} (\frac{1}{κ}) - 2 λ_{3} κ^{3} \\ = - \infty . \end{matrix}$ (82) By (675) and (676), ∂J/∂κ ∈ (- ∞ , + ∞). The value of κ, which lets ∂J/∂κ = 0, exists.

Taking the second order derivative of J with respect to κ and considering λ₁, λ₂, λ₃, h₁ (·) >0, h₂ (·) >0, g₁ (·) >0 and g₂ (·) >0, for $κ > δ / γ_{3} (\bar{R})$ , we have

$\begin{matrix} \frac{\partial^{2} J}{\partial κ^{2}} = 2 λ_{1} h_{1} (\frac{1}{κ}) \frac{1}{κ^{3}} + λ_{1} h_{2} (\frac{1}{κ}) \frac{1}{κ^{4}} + 2 λ_{2} \\ \times g_{1} (\frac{1}{κ}) \frac{1}{κ^{3}} + λ_{2} g_{2} (\frac{1}{κ}) \frac{1}{κ^{4}} + 2 λ_{3} > 0 . \end{matrix}$ (83)

Theorem 3. For any t₀ and given $\bar{R} > γ_{3}^{- 1} (δ / κ)$ , the solution $κ > δ / γ_{3} (\bar{R})$ of (674) always exists and is unique, which can globally minimize the performance index J (κ, t₀).

Proof. Since $lim_{κ \to (δ / γ_{3} (\bar{R}))^{+}} (λ_{1} h_{1} (1 / κ) + λ_{2} g_{1} (1 / κ) - 2 λ_{3} κ^{3}) = + \infty$ , $lim_{κ \to + \infty} (λ_{1} h_{1} (1 / κ) + λ_{2} g_{1} (1 / κ) - 2 λ_{3} κ^{3}) = - \infty$ and the function f (κ) is continuous for $κ > δ / γ_{3} (\bar{R})$ , the solution κ for f (κ) =0 exists. Thus we can conclude the solution κ for ∂J/∂κ = 0 exists.

From (677), for any t₀, $\bar{R}$ and $κ > δ / γ_{3} (\bar{R})$ , we can calculate ∂²nJ/∂κ² > 0. Then ∂J/∂κ is continuous and non-increasing in κ. Therefore, the positive solution $κ_{opt} > δ / γ_{3} (\bar{R})$ for (672) always exists and is unique, which solves the constrained minimization problem in (670). Q.E.D.

Remark. The one-sided robust control using the optimal design of κ renders the state y of the closed-loop nonlinear system uniformly bounded and uniformly ultimately bounded. The performance index J is globally minimized. The solution can be achieved by a close form algebraic equation. As the complex nonlinear form of the functions h₁ (·) and g₁ (·), we may not obtain the explicit solution κ in (673). In the practical application, we can achieve an approximate solution κ by simulation results. The solution κ from simulation is close to the exact value of κ_opt. From a pragmatic viewpoint, the engineer can take this approximate solution in this optimal design.

7 An illustrative example

Consider a drug administration system, which can be described as [18] $\dot{ξ} (t) = - a (t) ξ (t) + b (t) u (t),$ (84) $ξ (0) = ξ_{0},$ (85) where ξ (t) is the drug concentration level of a body compartment, u (t) is the drug infusion rate, a (t) is the decrease rate coefficient of the drug concentration, and b (t) is the drug effectiveness coefficient. We suppose that the parameters a (t) and b (t) are unknown, which can be decomposed into $a (t) = \bar{a} + Δ a (t),$ (86) $b (t) = \bar{b} + Δ b (t),$ (87) where $\bar{a}$ and $\bar{b}$ are the known nominal values, and Δa (·) and Δb (·) are bounded unknown functions of t.

The physical meanings of the drug concentration ξ and the drug infusion rate u imply that they both need to be positive. The objective is to design the control which drives the drug concentration to a desired level even in the presence of uncertainty while observing the state and control constraint. This is a very practical problem which is important in biomedical engineering. A standard control scheme, which does not take the constraint into account, would not apply. Let $y = ln \frac{ξ}{ξ^{*}} .$ (88) The system (84) can be represented in terms of y:

$\begin{matrix} \dot{y} & = & (\bar{a} + Δ a) (e^{- y} - 1) \\ + \frac{(\bar{b} + Δ b)}{ξ^{*}} e^{- y} u - (\bar{a} + Δ a) e^{- y} . \end{matrix}$ (89) The system (89) is in the form of (19) by taking σ = [Δa, Δb] ^nT, $\tilde{f} (y, σ) = (\bar{a} + Δ a) (e^{- y} - 1),$ (90) $\tilde{B} (y) = \frac{\bar{b}}{ξ^{*}} e^{- y},$ (91) $Δ \tilde{B} (y, σ) = \frac{Δ b}{ξ^{*}} e^{- y},$ (92) $Δ \tilde{f} (y, σ) = - (\bar{a} + Δ a) e^{- y} .$ (93)

We can formulate the Lyapunov function of the system (89) as $V (y) = (\hat{a})^{- 1} (e^{y} - 1 - y),$ (94) where $\hat{a}$ is a known strictly positive constant such that for all t ∈ ℛ, $a (t) \geq \hat{a}$ .

To satisfy (25) of Assumption 4, we choose $γ_{1} (ψ) = min {V (y) | ψ = ∥ y ∥},$ (95) $γ_{2} (ψ) = max {V (y) | ψ = ∥ y ∥} .$ (96)

In order to show that γ₁ (ψ) and γ₂ (ψ) satisfy Assumption 4, we will deduce their explicit forms. Let $φ (y) = y^{2} - ψ^{2}, for ψ \leq 0 .$ (97) By using the Lagrange multiplier method, we can extremize V (y) subject to the constraint φ (y) =0 as $γ_{1} (ψ) = (\hat{a})^{- 1} (e^{- ψ} - 1 + ψ),$ (98) $γ_{2} (ψ) = (\hat{a})^{- 1} (e^{ψ} - 1 - ψ) .$ (99) For (845) and (846), it is seen that γ_ni (0) =0 and $lim_{ψ \to \infty} γ_{i} = \infty$ , i = 1, 2, and $\frac{d γ_{1} (ψ)}{d ψ} = (\hat{a})^{- 1} (- e^{- ψ} + 1) > 0,$ (100) $\frac{d γ_{2} (ψ)}{d ψ} = (\hat{a})^{- 1} (e^{ψ} - 1) > 0,$ (101) for all ψ > 0. As d (γ₂)/dψ > d (γ₁)/dψ ∀ψ > 0, we could get γ₂ (ψ) > γ₁ (ψ) ∀ψ > 0.

From (90), we can get

$\begin{matrix} \frac{\partial V}{\partial y} \tilde{f} (y, σ) \\ = (\hat{a})^{- 1} (e^{y} - 1) (\bar{a} + Δ a) (e^{- y} - 1) \\ = \frac{\bar{a} + Δ a}{\hat{a}} e^{- y} (e^{y} - 1) (1 - e^{y}) \\ = - \frac{\bar{a} + Δ a}{\hat{a}} e^{- y} {(e^{y} - 1)}^{2} \leq - e^{- y} {(e^{y} - 1)}^{2} . \end{matrix}$ (102) By (102), let $γ_{3} (ψ) = e^{ψ} {(e^{- ψ} - 1)}^{2},$ (103) $γ_{4} = 0 .$ (104) Note γ₃ (0) =0, $lim_{ψ \to \infty} γ_{3} (ψ) = \infty$ , and

$\begin{matrix} \frac{d γ_{3} (ψ)}{d ψ} & = & e^{ψ} {(e^{- ψ} - 1)}^{2} - 2 (e^{- ψ} - 1) > 0, \\ \forall ψ > 0 . \end{matrix}$ (105) Thus the conditions of Assumption 4 are indeed satisfied.

We have $α (y) = \frac{\bar{b}}{ξ^{*}} (\hat{a})^{- 1} (e^{y} - 1) e^{- y} .$ (106) Then

$\begin{matrix} \frac{\partial V}{\partial y} Δ \tilde{f} (y, σ) \\ = - (\hat{a})^{- 1} (e^{y} - 1) (\bar{a} + Δ a) e^{- y} \\ = - \frac{ξ^{*}}{\bar{b}} (\bar{a} + Δ a) \frac{\bar{b}}{ξ^{*}} (\hat{a})^{- 1} (e^{y} - 1) e^{- y} \\ = : α (y) \tilde{ω} (y, σ) . \end{matrix}$ (107) Let the positive constant a_max > a (t), ∀t ∈ ℛ. Then $‖ \tilde{ω} (y, σ) ‖ \leq \frac{ξ^{*}}{\bar{b}} a_{max} = : \underline{\tilde{ρ}} .$ (108) As $\tilde{ω} (y, σ) < 0$ for all (y, σ) ∈ ℛ × Σ and (108), the conditions of Assumption 6 are satisfied.

By (52, 53), the control is proposed to be $u (t) = {\begin{matrix} - κ α (y (t)) (1 + \underline{λ})^{- 1} {\underline{\tilde{ρ}}}^{2}, \\ if α_{i} (y (t)) < 0, (109) \\ 0, \\ if α_{i} (y (t)) \geq 0 . (110) \end{matrix}$

For numerical simulations, we choose ξ^* = 0.8, $\bar{a} = 1$ , $\bar{b} = 4$ . Suppose the uncertainty exist in initial condition ξ₀ and σ. Let the fuzzy descriptions of ξ₀ and σ be “ξ₀ is close to 1.5” and “Δa is close to 0.4”, respectively. The corresponding membership functions of the uncertain terms are (both triangular): $μ_{ξ_{0}} (ν) = {\begin{matrix} 2 ν - 2, 1.0 \leq ν \leq 1.5, \\ - 2 ν + 4, 1.5 \leq ν \leq 2.0, \end{matrix}$ (111) $μ_{Δ a} (ν) = {\begin{matrix} \frac{5}{2} ν, 0 \leq ν \leq 0.4, \\ - \frac{5}{2} ν + 2, 0.4 \leq ν \leq 0.8 . \end{matrix}$ (112) By (71), the performance index J (κ, t₀) of the drug administration system is $J (κ, t_{0}) = λ_{1} J_{1} (t_{0}) + λ_{2} J_{2} (κ) + λ_{3} J_{3} (κ) .$ (113) By using the fuzzy arithmetic on the α-cuts, the decomposition theorem, and D-operation, we can calculate J₁, J₂ and J₃ with a known κ. According to Theorem 3, for the given weighting factors λ₁, λ₂ and λ₃, there always exists an unique optimal κ_opt which globally minimizes the performance index J (κ, t₀). With different choices of weighting factors λ₁, λ₂ and λ₃, the corresponding κ_opt and J_min are shown in Table 1.

For simulations, the uncertainties are selected to be $Δ a (t) = 0.4 \bar{a} sin (π t),$ (114) $Δ b (t) = 0.4 \bar{b} cos (π t) .$ (115) Figure 2 shows the drug concentration ξ results with two different initial conditions ξ¹ (0) =1.5 and ξ² (0) =0.3 (under κ_opt = 26, i.e., when (λ₁, λ₂, λ₃)=(100, 100, 1)). The corresponding drug concentrations ξ¹ and ξ² stay in a very small zone around the desired concentration level ξ^* after a finite time by employing the robust control. Figures 3–4 demonstrate the drug concentrations ξ¹ and ξ² with or without the control. With the control, ξ¹ and ξ² converge to the desired concentration level ξ^* in less than 1 second. Without the control, ξ¹ and ξ² can not stay close to the desired concentration level ξ^*. The corresponding drug infusion rates u¹ and u² (for ξ¹ and ξ², respectively) are shown in Fig. 5, which are strictly positive. For comparisons, Figures 6–7 show the drug concentrations and the drug infusion rates under different κ_opt.

The relationship between the optimal parameter κ_opt, initial value ξ₀, and the uncertainty bound of Δa is shown in Fig. 8. It demonstrates the effect of initial condition and uncertainty bound on the variation of κ_opt.

8 Conclusion

We consider a one-sided robust control design problem for positive uncertain systems. The uncertainty, including the unknown parameters and external disturbance, is (possibly fast) time-varying and unknown. It lies within a prescribed fuzzy set. There are four major results of the paper. First, a transformation of the state is provided. The system is converted to be state constraint free. Second, a robust control is designed for this system. It guarantees uniform boundedness and uniform ultimate boundedness as well as state constraints regardless of the uncertainty. Third, the sign definiteness of the uncertainty is proposed to be an extra insight of the uncertainty property. By this extra information of the uncertainty, the paper constructs a one-sided robust control. The control is deterministic and not fuzzy if-then rule-based. Fourth, a fuzzy based optimal design is formulated as a constrained optimization problem. We show that the global solution to this problem always exists and is unique. Furthermore, the solution can be obtained by solving a well-formulated closed-form scalar algebraic equation. The one-sided robust control, with optimal control design parameter, guarantees the deterministic performance regardless of the uncertainty. Furthermore, it renders an optimal system performance. The design is illustrated by applying to a drug administration problem, in which both the drug concentration level and infusion rate are to remain positive at all time regardless of the uncertainty.

Footnotes

Acknowledgments

Ruiying Zhao is supported by the China Scholarship Council (No. 201206560002) and the Fundamental Research Funds for the Central Universities (No. 310825161009). Shengjie Jiao is supported by the National Science and Technology Pillar Program (No. 2015BAF07B08).

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