具輸出約束之高階非線性系統有限時間穩定化控制：一種階層齊次占優技術

科技部補助專題研究計畫報告

具輸出約束之高階非線性系統有限時間穩定化控制：一種階層

齊次占優技術

報 告 類 別 ： 成果報告

計 畫 類 別 ： 個別型計畫

計 畫 編 號 ： MOST

108-2221-E-006-211-執 行 期 間 ： 108年08月01日至109年07月31日

執 行 單 位 ： 國立成功大學系統及船舶機電工程學系（所）

計 畫 主 持 人 ： 陳智強

計畫參與人員： 碩士班研究生-兼任助理：陳冠勳

碩士班研究生-兼任助理：丁齊萱

碩士班研究生-兼任助理：王聲瑞

碩士班研究生-兼任助理：劉桓維

大專生-兼任助理：莊榕珊

大專生-兼任助理：張心瑜

報 告 附 件 ： 出席國際學術會議心得報告

本研究具有政策應用參考價值：■否　□是，建議提供機關

（勾選「是」者，請列舉建議可提供施政參考之業務主管機關）

本研究具影響公共利益之重大發現：□否　□是　

中　華　民　國　109　年　08　月　24　日

中 文 摘 要 ： 有鑑於諸多優點，如狀態收斂速率較快、控制精確度較高與抗干擾

能力較優異等，探討控制系統的有限時間穩定化控制問題是至關重

要的。此外，在有限時間穩定化控制問題中，基於系統性能表現與

安全性考量，經常需要針對系統之輸出進行限制；意即，受控系統

不僅要能完成有限時間穩定化任務，同時也須滿足某些特定的輸出

約束條件。由於高階非線性系統涵蓋了許多常見的實際物理系統之

模型，針對高階非線性系統進行具輸出約束之有限時間穩定化控制

設計不僅具有理論重要性，同時也深具應用價值。據申請人所知

，目前文獻上仍未有關於高階非線性系統的具輸出約束之有限時間

穩定化控制設計成果被提出，且許多適用於嚴格回授系統的現存方

法也因高階非線性系統所具有的高度非線性特性（控制設計拓撲阻

礙），而無法用來設計其具輸出約束之有限時間穩定化控制器。因

此，在本計畫中，我們將考慮一類含有不確定參數與模型不確定性

的高階非線性系統；透過提出一種『階層齊次系統』的新概念，我

們將發展一套稱為『階層齊次占優技術』的新方法，並同時給出用

以履行輸出約束之barrier Lyapunov函數的新設計，藉此有效地處

理高階非線性系統的具輸出約束之有限時間穩定化控制設計問題。

中 文 關 鍵 詞 ： 非線性系統控制、有限時間穩定化問題、輸出約束、局部不可控性

英 文 摘 要 ： In view of benefits including faster state convergence

rate, higher control accuracy, and better robustness, the

study of finite-time stabilization problem for control

systems has become of paramount importance. Besides, for

reasons of system performance specifications and/or safety,

systems to be considered in the finite-time stabilization

problem may usually be subjected to an output constraint

and/or partial state constraints. That is, the system to be

controlled should not only perform the finite-time

stabilization task but also prevent violation of the output

constraint (partial states constraints). Because many

real-world systems can be modeled as high-order nonlinear

systems, the design of finite-time stabilizing controllers

with an output constraint for high-order nonlinear systems

is of practical and theoretical importance. To the best of

the applicant’s knowledge, there is no research result

available yet concerning the design of finite-time

stabilizing controllers with an output constraint for

high-order nonlinear systems; moreover, most of existing results

for strict-feedback systems and/or triangular systems are

inapplicable to dealing with high-order nonlinear systems

due to the high-order nonlinearity and the potential

topological obstruction in controller design. Therefore,

how to design suitable finite-time stabilizing controllers

with an output constraint for high-order nonlinear systems

is a critical and challenging problem. In this project, we

will consider the problem of designing finite-time

class of high-order nonlinear systems, which suffer from

uncertain parameters and model uncertainties. By

introducing a new concept “hierarchical homogeneous

system”, we shall develop a new approach called

“hierarchical homogeneous domination technique” and a new

type of barrier Lyapunov functions for preventing violation

of the output constraint to designing finite-time

stabilizing controllers with an output constraint. With the

developed approach, the problem of designing finite-time

stabilizing controllers with an output constraint for

high-order nonlinear systems can be resolved successfully and

delicately.

英 文 關 鍵 詞 ： Nonlinear system control, finite-time stabilization

problem, output constraint, local uncontrollability

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由於諸多專家學者與前輩們的努力，線性系統分析與控制設計理論在過去的數十年內 已趨近於完整。然而，由於大多數的物理或控制系統都隸屬於非線性系統，透過線性系統 的分析與設計所獲得之結果不可避免地只適用於特定操作點（operation point）附近的局部範 圍（local region）。為了能夠使得控制系統具有更大的有效操作範圍並獲得更理想的控制性 能，從1970年代開始，許多控制領域的專家學者相繼地投入其研究能量於非線性系統分析 與設計之研究議題，這些專家包含有 Petar Kokotovic、Mathukumalli Vidyasagar、Alberto Isidori、Shankar Sastry、Arjan van der Schaft、Eduardo Sontag、Miroslav Krstic、Andrea Bacciotti、Henk Nijmeijer、Jean-Jacques Slotine、Wassim Haddad 與 Hassan Khalil 等 人

（詳見文獻[1–12]）。在這些專家學者的投入之下，許多經典的非線性分析與控制理論相繼

地被提出，包含有中心流型理論（center manifold theory）、步階遞歸設計（recursive

back-stepping design）、奇異擾動理論（singular perturbation theory）、回授線性化方法（feedback

linearization approach）、順滑模控制（sliding mode control）_{、非線性H∞控制（nonlinear H∞}

control）、模糊控制（fuzzy control）等（見文獻[13–28]及其參考文獻）。伴隨著這些經典非線

性控制技術的發展、運用與延伸，對於過去難以使用線性控制方法來處理的複雜系統與高性 能控制問題，大多數都已獲得有效之解決。建立於前人的知識之上，如何進一步改善或拓展 非線性系統之控制與分析技術，目前仍是系統理論中的一大研究熱點。

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近年來，基於實際系統控制與性能分析之需求，一類被稱為“高階非線性系統（high-order nonlinear system）”的動態系統被廣泛地討論[1, 2, 8, 10, 29–53]，該系統的結構（ structure）可 由下列非線性微分方程式（nonlinear differential equation）所描述：

˙xi = di(x, t)xpi+1i + φi(x, t), i = 1, . . . , n− 1

˙xn= dn(x, t)u + φn(x, t)

y = x1 (1)

其中x = (x1, . . . , xn)T ∈ Rn, u∈ R分別為系統狀態（state）與控制輸入（control input），對所

有i = 1, . . . , n，pi ∈ R+odd:= {r1/r2 | r1與r2皆為奇整數（odd number）}，φi :Rn× R+→ R為

非線性項（nonlinear term），di :Rn× R+ → R為不確定參數（uncertain parameter）。有別於

非線性項φi(x, t)，xpi+1i 又被特別稱之為系統(1)的非線性高階項（nonlinear high-order term）

[8, 10, 29–53]。明顯地，系統(1)可被視為嚴格回授系統（strict-feedback system）[1–3]或三角

系統（triangular system）[54, 55]的推廣（generalization）。此外，系統(1)也經常用來描述許多

實際的物理系統，例如質量非線性彈簧系統（mass nonlinear spring system）[8]、鍋爐渦輪系 統（boiler turbine system）[30]與欠驅動弱耦合倒立單擺系統（underactuated weakly coupled inverted pendulum system）[37, 40, 41, 50]等。

有鑑於系統(1)可用來描述許多實際的物理系統，針對系統(1)進行控制設計與分析不僅 具有理論研究之重要性，同時也深具實際應用價值。對於系統(1)來說，許多經典的控制 方法（如回授線性化方法、步階遞歸設計、順滑模控制等）都無法有效地用來設計對應之 控制器（controller）並使其成功地完成所需之控制任務。而造成系統(1)之控制設計問題十 分困難的主要理由來自於系統(1)所具有的特殊結構及高度非線性（high-order nonlinearity） 特性[53, 56–67]。事實上，觀察系統(1)可發現，當pi < 1時，系統(1)在原點的線性化系統 （linearization system）並不存在；當pi > 1時，系統(1)在原點的線性化系統則為不可控系 統，換句話說，即便在原點附近的局部區域（local region），想要有效地控制系統(1)的系統 狀態都是非常不容易的。因此，對於系統(1)來說，其控制器或穩定器（stabilizer）設計是極 端困難的任務。 另外一方面，雖然在大多數的情況下受控系統（controlled system）具有一致漸進穩定之

特性已足夠，但由於一致漸進穩定之系統的系統狀態（理論上）需要歷經無窮時間（infinite-time horizon）才能夠到達控制目標（完成控制任務），往往會造成累積誤差（accumulation

error）或控制精確度（accuracy）下降之缺點。對於探討如何設計一個適當的控制器（穩定 器）使得閉迴路系統不僅為一致漸進穩定，同時更具有有限時間收斂之特性的問題，一 般稱之為有限時間穩定化控制問題（finite-time stabilization control problem）[46, 68–89]。有 限時間穩定化控制（finite-time stabilization control）不僅能夠使得閉迴路系統擁有有限時 間（狀態）收斂之特性，還能使得閉迴路系統具更好的穩健性（robustness）[46, 68–89]。截 至目前為止，對於系統(1)的有限時間穩定化控制問題而言，仍僅有少數的研究成果（例

如[47–49, 51, 52, 59, 90–97]）對其提出適當的解決方案（solution），且往往只有在非常嚴苛的系

統結構（假設）條件下，對應的有限時間穩定化控制器（finite-time stabilizing controller）才 能夠順利地被設計。

除了有限時間穩定化控制任務外，人們經常不僅希望受控系統能夠成功地完成有限時 間穩定化任務，同時更要求其輸出（部分系統狀態）能夠滿足某些特定的輸出約束（output

constraint），以便達成對應的性能表現、硬體限制或安全性需求。針對一非線性系統，如 何有效地解決其有限時間穩定化問題並同時使得閉迴路系統之輸出（部分狀態）滿足對應 的輸出約束，便是具輸出約束之有限時間穩定化控制問題（finite-time stabilization control problem with output constraint）所要探討的主要議題。對於非線性嚴格回授系統[1–3]或三角 系統[54, 55]來說，其具輸出約束之（漸進）穩定化控制問題與具輸出約束之有限時間穩定化 控制問題，在過去的二十幾年內已被諸多專家學者成功地解決，許多研究成果也相繼地被提

出（例如[98–114]）。然而，對於高階非線性系統(1)來說，截至目前為止，文獻上仍未有任何

研究成果或報告提出有效或系統化的方法，來設計其具輸出約束之有限時間穩定化控制器。 由此，本計畫之主要研究目的如下：

“考考考慮慮慮如如如系系系統統統(1)所所所描描描述述述之之之高高高階階階非非非線線線性性性系系系統統統（（（high-order nonlinear system））），，，我我我們們們該該該 如

如如何何何有有有效效效地地地克克克服服服其其其所所所具具具有有有的的的高高高度度度非非非線線線性性性（（（high-order nonlinearity）））特特特性性性，，，並並並發發發展展展一一一套套套系系系 統

統統化化化的的的（（（systematical）））方方方法法法來來來設設設計計計其其其具具具輸輸輸出出出約約約束束束之之之有有有限限限時時時間間間穩穩穩定定定化化化控控控制制制器器器（（（finite-time

stabilizing controller with output constraint）））？？？”

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有限時間穩定化控制問題而言，文獻上已有許多嚴謹且極具應用潛力的研究成果也相繼 地被提出（例如[16, 46, 59, 63, 68–89, 115, 116]）。在這些現存的結果中，大部分的設計方法都 是採用非連續控制（discontinuous control）的技術來達成有限時間穩定化任務。需要特別強 調的是，上述之非連續有限時間穩定化控制設計結果[69, 72–76, 78–81, 83, 115]並無法直接地 用來設計系統(1)的有限時間穩定器，其主要原因歸咎於系統(1)中具有不確定參數di(x, t)及 可能潛藏於非線性項φi(x, t)中的模型不確定性。這些不確定性直接地造成以最佳控制為基礎 的bang-bang控制[69, 74, 115]失效。此外，由於系統(1)的特殊結構，其非線性項φi(x, t)中的 模型不確定性對於終端順滑模控制設計[16,72,73,75,76,78–81,83,88,116–118]來說，便是所謂 的非匹配不確定性而無法被有效補償補償[8, 11, 119]，同時亦無法克服系統(1)所造成的控制 設計『拓撲阻礙』。 另一方面，目前文獻上已有許多研究成果專注於解決其具輸出約束之（漸進或有限時 間）穩定化控制問題，包含有不變集合（invariant set）方法[108, 109]、模型預測控制（model predictive control）方法[106,107,111]、參考調節（reference governors）方法[105]與barrier punov函數（barrier Lyapunov function）設計方法[98–104, 110, 112, 113]等。由於barrier Lya-punov函數設計方法其主要思想是建立在傳統的步階遞歸設計框架下，且系統(1)不為嚴 格回授型式，其所具有高度非線性特性及潛在的模型或參數不確定性，將會造成barrier Lyapunov函數設計方法無法有效地處理系統(1)的具輸出約束之穩定化控制問題（故亦無法用

來處理系統(1)的具輸出約束之有限時間穩定化控制問題）。 現存之有限時間穩定化控制設計結果[47–49, 51, 52, 59, 90–97]與具輸出約束（漸進）穩定 化控制設計結果[98–113]，不論將其單獨應用或互相結合產生綜合性技術，都無法用來有效 地解決/處理系統(1)的具輸出約束之有限時間穩定化控制問題。由此再次說明了本計畫欲探 討並解決的問題仍是一個未有解答的開放性問題，而此計畫之研究成果將可提供控制工程師 一套系統化的設計方法來幫助其解決具輸出約束之有限時間穩定化控制問題。

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首先，針對系統(1)我們提出兩項假設，其中假設1主要涉及系統之顯然不確定性，而假 設2則針對系統之非線性項進行增長條件（growth condition）之規範。 假 假假設設設 1 對所有i = 1, . . . , n，存在正常數d_i與di使得 d_{i ≤ d}i(x, t)≤ di 對所有(x, t) ∈ Rn_{× R}+_。 假 假假設設設 2 對所有i = 1, . . . , n存在常數σ < 0及平滑（smooth）函數φ_i(xi)≥ 0使得 |φi(x, t)| ≤ φi(x1, . . . , xi)  |x1| ri+σ r1 ₊|x₂|ri+σr2 ₊∙ ∙ ∙ + |x_i|ri+σri  (2) 對所有(x, t) ∈ Rn_{× R}+_其中r 1, . . . , rn由下列遞迴式所定義 r1 = 1 且 rj+1 = rj + σ pj > 0 對所有 j = 1, . . . , n。 (3) 此外，我們亦需要下列引理來幫助完成相關的分析，關於這些引理的證明，已清楚地描 述於我們所發表的論文[120]中。 引 引引理理理 1 _{假設p ≥ 1為常數，則函數f(s) = dse}p _{= |s|}p_{sign(p)為連續可微分（continuously} differentiable），且f0(s) = p|s|p−1_。 引 引引理理理 2 假設m1 ≥ 1, m2 > 0, m3 > 0, m4 > 0為常數，則下列不等式成立： m1|x|m3|y|m4 ≤ m2|x|m3+m4 + m4 (m3+ m4)  m3 (m3+ m4)m2 m3 m4 m (m3+m4) m4 1 |y|m3+m4 對所有x ∈ R與y ∈ R。 引 引引理理理 3 _{假設p ≥ 1與q > 0為常數，則下列兩不等式成立：}  

 dxep− dxep ≤ p 2p−2+ 2
|x − y|p_{+|x − y| ∙ |y|}p−1

   dxepq − dxe q p    ≤ 21−p1    dxeq− dxeq 1 p

對所有x ∈ R與y ∈ R。 引 引引理理理 4 假設p > 0為常數，則下列不等式成立： (|x| + |y|)p ≤ 2p−1+ 1
(|x|p_+|y|p₎ 對所有x ∈ R與y ∈ R。 另外，由於系統(1)僅是一連續系統（continuous system）（不一定滿足Lipschitz條件）， 傳統針對Lipschitz系統進行barrier Lyapunov函數定義之結果並不適用於系統(1)。下列我們 給出對於系統(1)之barrier Lyapunov函數的精確定義。 定

定定義義義 考慮一時變非線性系統（time-varying nonlinear system）如下

˙x = f (x, t), x(t0)∈ Rn, t0 ∈ R+ (4)

其中f : _Rn _{× R}+ _{→ R}n_{為連續函數（continuous function）。假設B ⊂ R}n_{為一開連通（open}

and connected）集合且0 ∈ B，V : B → R為一正定（positive definite）且連續可微分函數。

若存在L ∈ R+_{使得對任意從x(t} 0) ∈ Rn出發之系統(4)的解x(t)都有V (x(t)) → ∞當x(t) → ∂B及V (x(t)) ≤ L對所有t ≥ t0，則V (x)稱為系統(4)的一個barrier Lyapunov函數。 引 引引理理理 5 考慮系統(4)及兩個正實數εl與εu。定義集合 Si(εl, εu) =  (x1, . . . , xi) | (x1, . . . , xi)∈ Ri 且 − εl < x1 < εu ⊂ Ri

若 存 在 兩 連 續 可 微 分 函 數U1 : S1(εl, εu) → R與U2 : Sn(εl, εu) → R， 其 中U1(x1)為 正

定 且U2(x)為 非 負（nonnegative）， 及 一 連 續 正 定 函 數U3 : Sn(εl, εu) → R使 得U1(x1) →

∞當x1 → −εl或x2 → εu，且U(x) = U1(x1) + U2(x)滿足 ∂U (x) ∂x f (x, t)≤ −U3(x) 對所有(x, t) ∈ Sn(εl, εu)×R+，則對任意從x(t0)∈ Rn出發之系統(4)的解x(t)都在區間[t0,∞)有 定義並滿足x(t) ∈ Sn(εl, εu)對所有t≥ t0。 證 證證明明明: 詳見[120]。 下面我們給出此計畫的主要結果。 定 定定理理理 _{當假設1與2成立，則存在一連續之狀態回授控制律使得對任一個由初始狀態x(0) ∈} Sn(εl, εu)所出發之解x(t)在[0,∞)有定義並滿足有限時間內x(t) → 0及−εl < y(t) = x1(t) < εu對所有t ≥ 0。 證

證證明明明: 整體證明過程可分成三大步驟：(i) barrier Lyapunov函數設計；(ii) 連續控制律設

計；(iii) 分析狀態有限時間收斂性，我們分述如下。 5

步 步步驟驟驟一一一 –《《《barrier Lyapunov函函數函數數設設設計計計》》》 我們設計VB :S1(εl, εu)→ R為 VB(x1) = ε2μ−σ_l ε2μ−σ_u |x1|2μ−σ (2μ_{− σ) (ε}u− x1)2μ−σ(εl+ x1)2μ−σ (5) 其中 μ_{≥ ω ≥ max} 1≤i≤n{ri} (6) 且ri定義在假設2，ω為輔助參數。明顯地，VB(x1)為正定（positive definite）並滿足VB(x1)→ ∞當x1 → −εl或x2 → εu。此外，我們亦有 ∂VB(x1) ∂x1 = %(x1)dx1e2μ−1−σ 對所有 x1 ∈ R (7) 其中 %(x1) = ε2μ_l −σε2μ−σ u (x21+ εuεl) (εu− x1)2μ+1−σ(εl+ x1)2μ+1−σ 由此可見，VB(x1)為連續可微分函數。 步 步步驟驟驟二二二 –《《《連連連續續續控控控制制制律律律設設設計計計》》》 在此我們透過數學歸納法來證明控制器之型式與存在性，並決定其控制增益。首先，定 義ξ1 =dx1eω並令V1 :S1(εl, εu)→ R為V1(x1) = VB(x1)且設計 x∗₂(x1) = −β1(x1)dξ1e r2 ω 且 β₁(x₁) =  n + %(x1)φ1(x1) %(x1)d1  1 p1 (8) 其中β1 :S1(εl, εu)→ (0, ∞)為平滑函數，則根據假設1與2我們有 ˙ V1(x1)≤ −n|ξ1| 2μ ω + d₁(t)%(x₁)dξ₁e 2μ−1−σ ω (xp1 2 − x∗p2 1) (9) 對所有(t, x, u) ∈ R+_{× S} n(εl, εu)× R。 接著，我們定義ξ2 := dx2eω/r2 − dx2∗eω/r2並令V2 :Sε2 → R為 V2(x1, x2) = V1(x1) + W2(x1, x2) (10) 其中 W2(x1, x2) = Z x2 x∗ 2 l dser2ω _{− dx}∗ 2(x1)e ω r2 m(2μ−r2−σ) ω ds 根據我們的論文[120]可知，V2(x1, x2)為連續可微分函數。根據假設1與2我們有 ˙ V2(x1, x2)≤ −n|ξ1| 2μ ω + d 1(t)%(x1)dξ1e 2μ−1−σ ω (xp1 2 − x∗p2 1) +dξ2e 2μ_−r2−σ ω φ 2(t, x, u) +∂W2(x1, x2) ∂x1 ˙x1+ d2(t)dξ2e 2μ_−r2−σ ω xp2 3 (11)

對所有(t, x, u) ∈ R+_{× S} n(εl, εu)× R。透過引理1–4可得下列不等式： d1(t)%(x1)dξ1e 2μ−1−σ ω (xp1 2 − x∗p2 1)≤ 1 3|ξ2| 2μ ω + %k(x₁)g₂(x₁, x₂)|ξ₂| 2μ ω dξ2e 2μ_−r2−σ ω φ₂(t, x, u)≤ 1 3|ξ2| 2μ ω + ˆg₂(x₁, x₂)|ξ₂| 2μ ω ∂W2(x1, x2) ∂x1 ˙x1 ≤ 1 3|ξ2| 2μ ω + ˜g₂(x₁, x₂)|ξ₂| 2μ ω 由上述不等式我們進一步可獲得 ˙ V2(x1, x2)≤ −(n − 1)  |ξ1| 2μ ω +|ξ 2| 2μ ω  + d2(t)dξ2e 2μ_−r2−σ ω (xp2 3 − x∗p3 2) (12) 對所有(t, x, u) ∈ R+_{× S} n(εl, εu)× R，其中 x∗₃ =_−β2(x1, x2)dξ2e r3 σ 且 β₂(x₁, x₂) =  n_{− 1 + %}k_(x 1)g2(x1, x2) + ˆg2(x1, x2) + ˜g2(x1, x2) d₂ 1 p2 (13) 且g2 :S2(εl, εu) → (0, ∞)為平滑函數。延續著相同的方法我們可證明，存在一連續可微分函 數Vn :Sεn → R如下 Vn(x) = V1(x1, . . . , xn−1) + n X i=2 Wi(x) Wi(x) = Z xi x∗ i(x1,...,xi−1) l dseriω _{− dx}∗ i(x1, . . . , xi−1)e ω ri m2μ_−ri−σ ω ds 及連續控制律 u =_−βn(x)dξne rn+σ ω 其中ξj(x1, . . . , xj) =dxje ω rj _{− dx}∗ j(x1, . . . , xj−1)e ω rj_{對所有j = 2, . . . , n，使得} ˙ Vn(x)≤ − n X j=1 |ξj| 2μ ω (14) 對 所 有(t, x, u) ∈ R+ _{× S} n(εl, εu) × R。 因 此 ， 根 據 引 理5， 對 任 一 個 由 初 始 狀 態x(0) ∈ Sn(εl, εu)所出發之解x(t)在[0,∞)有定義並滿足−εl< y(t) = x1(t) < εu對所有t ≥ 0。 步 步步驟驟驟三三三 –《《《分分分析析析狀狀狀態態態有有有限限限時時時間間間收收收斂斂斂性性性》》》 考慮x(0) ∈ Sn(εl, εu)，從上述不等式我們可知 0_{≤ V}1(x1(t)) + n X i=2 L1|xi− x∗i| 2μ−σ ri _{≤ Vn(x(t0))} 其 中L1 > 0。 由 此 可 直 接 地 推 得|ξi(x1(t), . . . , xi−1(t))|2μ/ω為 一 致 連 續（uniformly continu-ous）及 lim t→∞ Z t t0 |ξi(x1(s), . . . , xi−1(s))| 2 ω ds <∞ 對所有 i = 1, . . . , n 7

因此，根據Barbalat引理可知，當x(0) ∈ Sn(εl, εu)，對任一個由初始狀態x(0) ∈ Sn(εl, εu)所 出發之解x(t)將有x(t) → 0當t → ∞。基於此結果，不難證明存在集合 H = {x ∈ Sn(εl, εu) | Vn(x)≤ L2} ⊆ Sn(εl, εu) 使得_{H為不變集合（invariant set）且} ˙ Vn(x) + 4 −4μ−1 2 V 2μ 2μ−σ n (x)≤ − 1 2 n X i=1 |ξi| 2μ ω ≤ 0 對所有(t, x, u) ∈ R+_{× H × R。根據上列之不等式，我們可得對任一個由初始狀態x(0) ∈} Sn(εl, εu)所出發之解x(t)將滿足有限時間內收斂至零。

五

五

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、

、 模

模

模擬

擬

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驗

驗證

證

證

為了驗證定理之有效性，在此我們考慮二皆非線性系統如下： ˙x1 = x32 ˙x2 = u + ln(1 + x21) y = x1. (15) 透過均值定理（mean-value theorem）可知 ln 1 + x2₁
= ln  1 +_|x1| 1 15 30 ≤ 29|x1| 1 15 故假設2中的參數可被取為 r1 = 1, r2 = 4 15, σ =− 1 5 再根據定理之證明，我們取 r3 = 1 15, μ = ω = 1 則可得到對應之控制器。 在 模 擬 中 ， 我 們 選 擇 初 始 條 件 為 (x1(0), x2(0))T = (1, 5)T及 兩 組 不 同 的 約 束 條 件(εl, εu) = (1, 2)與(εl, εu) = (400, 400)。模擬結果如圖1–2所示。明顯地，所設計之控制 律為一個有效的有限時間穩定器，同時使得系統之輸出滿足對應之輸出約束。

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-1

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圖 圖圖 1. 閉迴路系統狀態x1(t)之時間響應

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圖 圖圖 2. 閉迴路系統狀態x2(t)之時間響應 9

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結

結論

論

論

在本計畫中，我們成功地透過發展一套稱系統化的設計方法，來處理高階非線性系統的 具輸出約束之有限時間穩定化控制問題。根據我們所知，本計畫所提出之方法為文獻上首次 成功地解決該問題的研究成果。有鑑於理論之突破性，我們亦專注於研究後續諸多相關之 議題，如非線性切換系統（nonlinear switching systems）的具輸出約束之有限時間穩定化控 制問題，以及高階非線性系統的具輸出約束之輸出回授（output feedback）穩定化控制問題 等，目前相關結果正於研究收尾階段，後續亦會將相關成果整理並投稿至國際控制期刊。在 此，我們想再次感謝科技部對我們團隊的補助；同時，我們也希望未來能夠持續地獲得國內 控制領域中專家學者與先進前輩們的支持，讓我們可以在控制理論的領域中，持續地傾心研 究，增加台灣在國際控制圈的能見度，並努力培育台灣控制理論圈的下一代。

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感

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技

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及

及參

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究

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、 參

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參考

考

考文

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文獻

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獻

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1

科技部補助專題研究計畫出席國際學術會議心得報告

日期：108 年 9 月 3 日

一、 參加會議經過

此次參加的會議名稱為 2019 第 15 屆國際智慧無人機系統會議（The 15th

In-ternational Conference on Intelligent Unmanned Systems）

。該會議對於控制理論與應

用學者來說，是場必須參加的國際重要盛會之一。整體會議所涉及之議題包含了

探討無人機系統之最新研究成果，及世界各地等先進無人機技術及其相關應用，

並特別著重於探討無人機系統的未來發展方向和趨勢。該會議今年是北京科技大

學（University of Science and Technology Beijing）及淡江大學（Tamkang University）

主辦，並由 International Society of Intelligent Unmanned Systems、International

So-ciety of Mechatronic Engineering、Office of Naval Research Global、中國自動化學會

青年工作委員會（Youth Academic Committee of Chinese Association of Automation）

、

IEEE SMC Beijing Capital Region Chapter、IEEE SMC TC on Autonomous Bionic

計畫編號

MOST 108－2221－E－006－211－

計畫名稱

具輸出約束之高階非線性系統有限時間穩定化控制：一種階層齊次

占優技術

出國人員姓名

陳智強

服務機構及

職稱

國立成功大學/助理教授

會議時間

108 年 8 月 27 日至

108 年 8 月 29 日

會議地點

中國

/北京-北京科技大學

會議名稱

(中文) 2019 第 15 屆國際智慧無人機系統會議

(英文) 2019 The 15

th

_{International Conference on Intelligent}

Unmanned Systems

發表題目

(中文) 非線性不確定系統之固定時間穩定化控制

(英文) Fixed-time stabilization for a class of uncertain nonlinear

sys-tems

Robotic Aircraft、北京市高校高精尖學科（Beijing Top Discipline for Artificial

Intel-ligent Science and Engineering, University Science and Technology Beijing）

、School of

Automation and Electrical Engineering, University of Science and Technology Beijing、

Institute of Artificial Intelligence, University of Science and Technology Beijing 等單位

協辦。會議期間為西元 2019 年 8 月 27 日至 8 月 29 日，地點為北京科技大學、天

宮大廈 B 座樓（The Third Floor Corridor of the Techart Plaza）

。本人在 8 月 27 日先

由台南出發至桃園國際機場，搭乘飛機（中國國際航空）直達飛抵中國北京首都

國際機場。本人的論文報告日期被安排在 8 月 28 日下午 1 點 30 分開始得分組報

告（由本人之學生：丁齊萱進行報告）

，主題為 Space Robotic Systems Modelling and

Autonomous Control。在發表論文前，本人提前至會場參觀世界各地先進的研究，

對於世界各國目前對於無人機技術與其應用之研究成果及發展留下深刻之印象。

此行開拓了個人的視野，自覺收穫豐碩。會議期間本人也遊覽北京科技大學校園，

對於北京科技大學校園及其各大研究中心的磅礡建築，感到無比的讚嘆。對於此

次前往參加會議，我們不僅能夠學習到國際頂尖無人機設計、分析、控制技術與

應用，還能夠受到諸多國際學者對於研究的認真與努力態度的薰陶，收穫甚多。

結束會議與參訪活動後我們於 9 月 1 日下午搭飛機按原路線飛返台灣。

二、 與會心得

參加此次國際智慧無人機系統會議的各國專家學者甚多，包含有中國大陸、

印度、美國、越南、韓國等國家，探討的主題從無人機系統設計技術之開發，到

其整體系統之控制分析與其在各類實際問題上之應用等等應有盡有，內容涵蓋相

當廣泛。在這次會議中，我們被安排在 28 日的下午進行口頭。在報告完畢後，許

多學者向我們（由本人之學生：丁齊萱，做報告）提出了許多有價值的問題，並

在會後我們做了更進一步的討論，在討論後我們也獲得了共同的解答。此外，在

此次會議中，我們與多位在無人機控制技術發展等領域中，極其頂尖的專家學者

們見面，並向其自我介紹，並在交談中獲得了許多寶貴的研究想法與觀念，自覺

收穫良多。在會議期間我們也前往聆聽與自己研究領域相關的主題，同時觀摩各

3

國學者專家呈現研究成果的方式，並從中學習做為往後若再次獲得發表論文機會

時的最好準備。參加此次會議除了能夠親自目睹及體會著名學者的風範及執著、

認真的研究精神，萌生見賢思齊與自我期許的成長動力，同時能夠增進自己在國

際會議發表論文的膽識並獲得國際學術界最新的研究動態。在會議期間許多研究

無人機系統的先進前輩所提出的問題及給予之意見不但具體，且往往能明確地點

出問題之所在，進而提供進一步的研究方向與可能解答之輪廓，個人自覺收穫甚

多。在未來，希望自己能有擁有更多的機會參與類似的國際會議。在這次的會議

後，我們帶回大會給予的論文資料，使我們可以盡覽此次會議的所有成果。在此

願再一次的感謝科技部的經費補助，使我們能順利的參與這次的研究成果發表饗

宴。

三、 發表論文之摘要

This paper investigates the problem of fixed-time stabilization for a class of

multivaria-ble uncertain nonlinear systems. A new approach is proposed by skillfully revamping

the technique of adding a power integrator whereby a state feedback controller and a

suitable Lyapunov function for verifying fixed-time convergence can be explicitly

con-structed to render the closed-loop system fixed-time stable. The novelty of this paper

owes to the develop-ment of a subtle strategy that provides a new solution to the

prob-lem of fixed-time stabilization for multivariable nonlinear systems. Finally, the

devel-oped approach is applied to the attitude stabilization of a spacecraft to show the

effec-tiveness of the resultant controller.

四、 建議

此行前往大陸北京科技大學參與本次會議後，我們深感學術交流與國際視野

開拓的重要性。此外，由於世界各國，特別是中國大陸，等諸多大學資金充裕，

研究資源豐富，在各專業領域中不乏有知名專家學者。在與其交流的過程中，不

僅能夠享受到高手交流之樂，更能了解每位專家所關注的焦點，使本人對整體研

究趨勢有更深入的了解，有助於本人掌握新的研究方向。有鑑於此，本人亦誠心

地建議科技部或相關單位，往後能多鼓勵並盡可能地補助國內年輕學者或博士研

究生，使其能夠多與國際學者們進行交流訪問或參與國際學術會議，藉此開拓其

國際視野並邁向國際，同時提高臺灣之國際學術能見度。

五、 攜回資料名稱及內容

大會論文集隨身碟一枚及紙本一本。

六、 發表論文之全文

詳見最末頁。

附件一：

（議程）

ICIUS 2019, Beijing, China Paper ID 26



Abstract—This paper investigates the problem of fixed-time stabilization for a class of multivariable uncertain nonlinear sys-tems. A new approach is proposed by skillfully revamping the technique of adding a power integrator whereby a state feedback controller and a suitable Lyapunov function for verifying fixed-time convergence can be explicitly constructed to render the closed-loop system fixed-time stable. The novelty of this paper owes to the development of a subtle strategy that provides a new solution to the problem of fixed-time stabilization for multivariable nonlinear systems. Finally, the developed approach is applied to the attitude stabilization of a spacecraft to show the effectiveness of the resultant controller.

Keywords—Uncertain nonlinear systems, adding a power inte-grator technique, fixed-time stabilization.

I. INTRODUCTION

HE stabilization control of nonlinear system has always been crucial in performing additional control tasks, such as output tracking, disturbance attenuation and/or decoupling. Global asymptotic stabilization of nonlinear systems has gained tremendous progress due to the development of mathematical tools, including backstepping design [1], feedback linearization [2], sliding mode control [3, 4], fuzzy control [5, 6], has tremendous progress by mathematical tools.

As is well-known, finite-time stabilization is more attrac-tive compared with asymptotic stabilization [7] because the systems with finite-time convergence usually exhibit superi-or properties [7-10], which are rather impsuperi-ortant fsuperi-or demand-ing applications. Bedemand-ing aware of these features, the fi-nite-time stabilization problem has been intensively studied, and numerous interesting results have been proposed in the past decades [11-15]. For instance, owing to the benefits including fast response and ease of implementation, terminal sliding mode control design [14] is one of most important techniques for finite-time stabilization of nonlinear system. By constructing a discontinuous controller while design a suitable nonlinear sliding surface, the phase of terminal sliding mode can be achieved in finite-time, thereby guaran-teeing finite-time stabilization of the closed-loop system [14-16].

It should be mentioned that the information of initial states is critical for the settling-time estimates of finite-time stabilization schemes; however, the availability of initial states will prevent us from applying finite-time schemes [17,

1_{C.-C. Chen is an Assistant Professor with Department of Systems and}

Naval Mechatronic Engineering, National Cheng Kung University, Tainan 70101, Taiwan (e-mail: ccchenevan@mail.ncku.edu.tw).

2_{C.-H. Ding and G.-S. Chen are master students with Department of}

Systems and Naval Mechatronic Engineering, National Cheng Kung Uni-versity, Tainan 70101, Taiwan.

18]. Fortunately, the notion of fixed-time stability together with its Lyapunov-like criteria has been recently presented in the seminal work [17] in which the potential obstruction of finite-time schemes was resolved effectively. To be more specific, as stated in [17], by fixed-time controller design, it implies global uniform finite-time stability while providing a settling time function to be uniformly bounded by a tunable constant, which independent of initial states [17, 18-24].

Due to the complexity of multivariable nonlinear systems and the lack of systematic strategies for ensuring the fixed-time convergence, the problem on how to design a fixed-time stabilizing controller for multivariable nonlinear systems remains unclear and largely open. In this paper, by introducing extra manipulations in the feedback domination to delicately revamp the technique of adding a power inte-grator [18], a new approach is developed to the synthesis the fixed-time stabilizer together with the Lyapunov function for multivariable uncertain nonlinear systems.

II. PRELIMINARIES A. Problem Formulation

Consider a class of nonlinear systems described by

𝐱 𝐱𝟐

𝐱2 𝐟 𝑡,𝐱 𝐺 𝑡,𝐱 𝐮 𝐝 𝑡,x (1)

where 𝐱 𝐱 , 𝐱 ∈ ℝ denotes the system states with 𝐱 𝑥 , … , 𝑥 ∈ ℝ and 𝐱 𝑥 , … , 𝑥 ∈ ℝ , 𝐝 𝑡, 𝐱 𝑑 𝑡, 𝐱 , . . . , 𝑑 𝑡, 𝐱 ∈ ℝ describes the mod-el uncertainties and/or external disturbances, 𝐮 ∈ ℝ rep-resents the control input, and 𝐟 𝑡, 𝐱 and 𝐺 𝑡, 𝐱 are smooth functions with the uniform rank 𝐺 𝑡, 𝐱 𝑛) for all 𝑡, 𝐱 ∈ ℝ ℝ which in turn ensures the controlla-bility of system (1) (see, e.g., [25]). The initial time de-scribed by 𝑡 is set to be zero, i.e., 𝑡 0, and the initial state of system (1) is denoted by 𝐱 0 𝟎 ∈ ℝ . It is worth mentioning that a very large class of physical systems can be represented by system (1). Besides, the solutions of system (1) are understood in the sense of Filippov [26] since the control input 𝐮 𝐮 𝑡, 𝐱 is admitted to be discontinu-ous (piecewise continudiscontinu-ous) and 𝐝 𝑡, 𝐱 is assumed to be piecewise continuous and bounded as follows.

Assumption 1. There exists a constant 𝜌 0 such that |𝑑 𝑡, 𝐱 | 𝜌

for all 𝑡, 𝐱 ∈ ℝ ℝ and 𝑖 1, . . . , 𝑛.

Under Assumption 1, the main objective of this paper is to design a controller 𝐮 𝐮 𝑡, 𝐱 that renders the origin of system (1) fixed-time stable in the sense of the following definition.

Definition 1 ([17]). Consider the following nonlinear system

𝐱 𝐠 𝑡, 𝐱

Fixed-Time Stabilization for a Class of Uncertain Nonlinear Systems

Chih-Chiang Chen1_{, Chi-Hsuan Ding}2_{, and Guan-Shiun Chen}2

ICIUS 2019, Beijing, China Paper ID 26 where 𝐱 ∈ ℝ ,𝑡 ∈ ℝ , and 𝐠 ∶ ℝ ℝ → ℝ is

discon-tinuous (piecewise condiscon-tinuous). The initial time is 𝑡 0 and the initial state is 𝐱 0 𝐱 . The solutions of system (2) are understood in the sense of Filippov [26]. Then, the origin of system (2) is said to be fixed-time stable if it is globally uniformly finite-time stable (see, e.g., [27]) and the set-tling-time function 𝑇 𝐱 is globally uniformly bounded by a positive constant; i.e., there exists a positive stant 𝑇 0 such that 𝑇 𝐱 𝑇 for all 𝐱 ∈ ℝ .

Remark 1. Compared to global uniform finite-time stability,

the key feature of fixed-time stability is the uniformity of its settling time. To see this point more clearly, the following two examples are considered. First, the origin of the system 𝑥 3 2⁄ 𝑥 ⁄ _{is globally uniformly finite-time stable}

with the settling-time function 𝑇 𝑥 0 𝑥 / _{0 because}

its solutions take the form of

𝑥 𝑡 𝑠𝑖𝑔𝑛 𝑥 0 𝑥 0 𝑡 , 0 𝑡 𝑥 0

0, 𝑡 𝑥 0

. However, with an additional drift term, the solutions of the system 𝑥 3 2⁄ 𝑥 ⁄ _{3 2⁄ 𝑥} ⁄ _{can be found directly}

as follows 𝑥 𝑡 𝑠𝑖𝑔𝑛 𝑥 𝑡𝑎𝑛 𝑡𝑎𝑛 |𝑥 | 𝑡 , 0 𝑡 𝜋 2 0, 𝑡 𝜋 2 . Therefore, the origin of system (3) is fixed-time stable with the settling-time function 𝑇 𝑥 0 satisfying 𝑇 𝑥 0 𝜋 2⁄ uniformly in 𝑥 0 .

B. Technical Lemmas

Lemma 1. Let 𝑚 1 is a ratio of two odd integers. For any 𝑥, 𝑦 ∈ ℝ one has

|𝑥 𝑦| 2 |𝑥 𝑦 |.

Lemma 2 ([28]). Let 𝑚 , 𝑚 , 𝛾 0. For any 𝑥, 𝑦 ∈ ℝ, one has

|𝑥| |𝑦| 𝛾𝑚

𝑚 𝑚 |𝑥|

𝛾 𝑚

𝑚 𝑚 |𝑦| .

Lemma 3. Let 𝑚 0. For any 𝑦 ∈ ℝ, 𝑖 1, . . . , 𝑛, one has

|𝑦 | ⋯ |𝑦 | 𝑐 |𝑦 | ⋯ |𝑦 | where 𝑐 𝑛 if 𝑚 1 and 𝑐 1 if 𝑚 1.

III. FIXED-TIME STABILIZING CONTROLLER DESIGN

We first summarize our approach to the construction of a fixed stabilizing controller for system (1) as follows.

A. Theorem 1.

Under Assumption 1, the origin of system (1) is fixed-time stable with the settling-time estimate

𝑇 𝐱 𝑇 2𝑛

2 𝜏

1

2 𝜏

If the controller 𝐮 𝑡, 𝐱 𝐮 is designed as 𝐮 G 𝑡, 𝐱 𝔏 𝐱 ξ 𝔏 𝐱 ξ

𝐟 𝑡, 𝐱 𝜌sign 𝛏𝟐 4

with 𝛏𝟐 𝐱 ⁄ 2𝐱 2𝐱 ⁄ where

τ 𝜏 ⁄𝜏 ∈ 1 2⁄ , 0 and 𝜏 𝜏 ⁄𝜏 ∈ 0,1 are parameters satisfying 𝜏 and 𝜏 , for 𝑖 1,2, being posi-tive even integers and posiposi-tive odd integers, respecposi-tively, and 𝔏 x ∈ ℝ and 𝔏 x ∈ ℝ are square matrices defined as

𝔏 𝐱 𝑔 Φ 𝐱 𝑔 𝐼 𝐼 1 𝜏

2 Φ 𝐱 𝔏 𝐱 𝑔 Φ 𝐱 𝐼 Φ 𝐱 diag ϕ 𝐱 , . . . , ϕ 𝐱 with ϕ 𝐱 , 𝑔 , 𝑔 and 𝑔 of the following form

Proof:

Part I— Design the controller

A two-step design approach is developed to construct the controller.

Step 1: Choose V 𝐱 𝐱 𝐱 𝑎𝑠 𝑡ℎ𝑒 scalar function, which is obviously positive definite, proper and continuous-ly differentiable. Select the virtual control 𝐱∗ _𝐱

2 𝑥 𝑥 . It follows that

V 𝐱 2 𝑥 𝑥 𝑥 𝑥 𝑥∗ ₅

for all x ∈ ℝ . By Lemmas 1 and 2, (5) becomes

V 𝐱 3

2𝑥 2𝑥

𝑔 𝑥 𝑥∗ 6

for all x ∈ ℝ .

Step 2: With 𝐱∗ _{𝐱 2 𝑥 𝑥 , we define} 𝛏𝟏 𝜉 , . . . , 𝜉 𝑥

𝛏_𝟐 𝜉 , . . . , 𝜉 𝑥 𝑥∗ 7 Since 1 𝜏 is a ratio of two positive odd integers, there is a one-to-one correspondence between 𝛏𝟏, 𝛏𝟐 and

𝐱 , 𝐱 . Then, we choose V 𝐱 V 𝐱 V 𝐱 with

V 𝐱 𝑠 𝑥∗ 𝑥 𝑑𝑠.

∗

Note that, V 𝐱 is positive definite, continuously differen-tiable and proper (i.e., radially unbounded), besides, it is easy to see from (7), Lemma 1, Lemma 2 and Lemma 3. V 𝐱 V 𝐱 V 𝐱

𝑥 𝑥 𝑔 𝜙 x |𝜉 |

1 𝜏

2 𝜙 𝐱 𝑔 𝜙 𝐱 𝑔 |𝜉 |

𝜉 𝐟 𝑡,𝐱 G 𝑡,𝐱 𝐮 𝐝 𝑡,𝐱 9 for all 𝑡, x ∈ ℝ ℝ ∖ 𝒩 . Substituting the controller (4) into (9) yields

V 𝐱 𝑥 𝑥 𝜉 𝜉