Stabilization of non-minimum phase switched ... - Semantic Scholar
Stabilization of non-minimum phase switched nonlinear systems with application to multi-agent systems ∗ Hao Yang
a †
, Bin Jiang a , and Huaguang Zhang
b
a
College of Automation Engineering Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China. b
School of Information Science and Engineering Northeastern University, Shenyang, 110004, China.
Abstract This paper addresses the stabilization issue of a class of switched nonlinear systems where each mode may be non-minimum phase, and the states of linearized dynamics of all modes compose the whole state space. Time dependent and state dependent stabilization switching laws are provided with considering both common and multiple Lyapunov functions. The new results are applied to aggregation problem of nonlinear multi-agent systems with designable switching connection topology. Finally, an aircraft team example illustrates the efficiency of the proposed approaches.
Keywords: Switched systems; Non-minimum Phase; Stabilization; Multi-agent systems.
1
Introduction
Many engineering applications can be modeled by switched systems due to the existence of various jumping parameters [1]. Fruitful results have been reported on stability and stabilization of switched nonlinear systems, e.g., [2, 3, 4, 5] to name a few. Stabilization of non-minimum phase nonlinear systems is a quite challenging problem, several fundamental methods have been proposed including state feedback control [6], output feedback contro [7, 8, 9]. The main idea behind these methods is to compensate for the unstable zero dynamics by means of output synthesis or auxiliary systems such that the system becomes stable. The controllability of the zero dynamics is the basic requirement, otherwise, the stabilization can not be achieved. Some contributions have also been devoted to non-minimum phase switched nonlinear systems where each nonlinear mode may be non-minimum phase. In [10], H∞ control goal is achieved for a class of non-minimum phase cascade switched nonlinear systems where the internal dynamics of ∗
This work is supported by National Natural Science Foundation of China (61034005, 61010121, 61104116) and Doctoral Fund of Ministry of Education of China (20113218110011) and NUAA Research Funding (NS2011016). † Corresponding author: Tel: +86 25 84892301-6060, Fax: +86 25 84892300. Email: [email protected] (H. Yang), [email protected] (B. Jiang), [email protected] (H. Zhang).
1
each mode is assumed to be asymptotically stabilizable. Output tracking of non-minimum phase switched nonlinear systems has been considered in [11], where an approximated minimum phase model is utilized. The same problem is also investigated in [12] by means of an inversion-based control strategy. The main idea of the above results consists of two steps : 1. Design the individual controller respectively in each mode to compensate for its own unstable internal dynamics such that all modes become stable individually. This can be done by using the existing techniques for non-minimum phase nonlinear systems. 2. Apply the standard stability condition of switched systems, e.g. common/multiple Lyapunov functions methods to achieve the stability of the whole switched system. This idea is natural and extends the approaches of non-switching systems to the switched one. However, it is well known that the stabilization for non-minimum phase nonlinear systems is quite difficult, and is even impossible to be achieved if the unstable zero dynamics is uncontrollable. In this work, we focus on the stabilization of non-minimum phase switched nonlinear systems where the internal dynamics of each mode may be unstable and uncontrollable. Different from [10, 11, 12], we do not try to compensate for the unstable internal dynamics respectively in each mode. Instead, we achieve the stabilization from the overall system point of view. It will show that under some conditions, the negative effects of internal dynamics in some modes may be compensated by other modes, and the overall switching process can still be stable. The similar idea can be seen in [13] for switched linear systems and [14] for switched nonlinear systems. This special property of switched system provides a new clue of stabilizing the non-minimum phase systems. Two benefits follow : 1. It relaxes significantly the condition on the internal dynamics (it is allowed to be unstable and uncontrollable simultaneously). 2. It makes easier the design of each mode’s individual controller. We consider a class of switched nonlinear systems where each mode may be non-minimum phase, and the states of linearized dynamics of all modes compose the whole state space. Consequently, we provide novel time dependent and state dependent stabilization switching laws with considering both common and multiple Lyapunov functions. The proposed results are more general and flexible than that in [14] where only a time-dependent switching law is provided. As an important application of the new results, we consider the target aggregation problem of nonlinear multi-agent systems which can be divided into several groups of subsystems with each group being in the leader-following structure, and only one group is allowed to be interconnected at one time. Inspired by linearized feedback control idea [19], we design a novel “feedback control topology” for each group, under which the multi-agent system with switching topologies can be regarded as a non-minimum phase switched system and the proposed switched system results are applied. 2
In the rest of the paper, some preliminaries are given in Section 2. Sections 3 and 4 propose time dependent and state dependent stabilization switching laws respectively, which are applied to multi-agent systems in Section 5, followed by some concluding remarks in Section 6.
2
Preliminaries
Consider the following switched nonlinear control systems x˙ = fσ (x) + gσ (x)uσ
(1)
where x ∈ 0 is called “dwell-time” [1]. We also assume that the states do not jump at the switching instants. > > Suppose that for each mode i ∈ M, we can find a function yi and a partition x = [¯ x> i , xi ] where xi ∈ x = [x> ♦ 1 , ..., xm ] . Assumption 2.1 implies that the states of unstable internal dynamics in each mode can be controlled and linearized in other modes. The states of all linearized dynamics compose the whole state space as illustrated in Fig. 1. This allows us to achieve the stabilization by fully utilizing the tradeoff among different modes of the switched system. Assumption 2.1 can be relaxed to a more moderate case where more than one mode may share some states in their linearized dynamics, the proposed methods can be straightly extended to this case. 3
Remark 2.1 : We would like to show how to fully utilize the switching properties to achieve the stabilization goal. Therefore, an idea of “overall system point of view” will be followed through out the paper. For more general switched systems where Assumption 2.1 is not satisfied, the combination between the existing control techniques for non-switching systems and the switching properties can be developed. ♦
x1
linearized dynamics
internal dynamics linearized dynamics
x2
internal dynamics
... ... ... ...
internal dynamics internal dynamics linearized dynamics
xm mode 1
mode m
mode 2
Figure 1: The structure of switched systems
3
Time-dependent switching law
In this section, we provide time-dependent stabilizing switching laws for the switched system satisfying Assumption 2.1.
3.1
Common Lyapunov function
Define a special function that will be used in the following sections: Definition 3.1 : A class GKL function γ : [0, ∞) × [0, ∞) → [0, ∞) if γ(·, t) is of class K1 for each fixed t ≥ 0 and γ(s, t) increases to infinity as t → ∞ for each fixed s ≥ 0. 2 Assumption 3.1 : For the switched system (1) satisfying Assumption 2.1, we can design ui for each mode i under which there exists a continuous non-negative function V : 0, ∆t ≥ τ such that when mode i is activated and Vi (tik ) > ², xj dynamics (j 6= i) satisfies © ª φj (Vj (tik ), ∆t) + maxk∈M−{j} γjk (η m−1 δ0 ) ≤η (9) Vj (tik ) ♦ 5
It follows from (8) that the left side of (9) is an upper bound of the gain of Vj in the period ∆t, if kVk k[tik ,tik +∆t) ≤ η m−1 δ0 . Inequality (9) implies that the diverging speed of each mode’s internal dynamics is bounded during a interval that is not less than the dwell time τ . Since the switching sequence and instants are designed a priori, on-line informations can not be used, such fixed upper bound has to be imposed. η is not hard to be found if γjk is small. Theorem 3.1 : Consider a switched system (1) satisfying assumptions 2.1, 3.1 and 3.2. Under switching law S1 , there exist ² > 0 such that for any given δ0 > 0, limt→∞ sup V (t) = m², ∀Vi (0) ≤ δ0 , i ∈ M. Proof : We first suppose that Vj (t) > ², ∀t ≥ 0. Consider xi dynamics in the first period [0, T ). Since the switching sequence is mode 1 → mode 2 → · · · mode i · · · → mode m, the solution of Vi (i 6= 1) at t = ∆t can be represented as © ª φj (Vi (0), ∆t) + maxk∈M−{i} γik (kVk k[0,∆t) ) Vi (0) (10) Vi (∆t) ≤ Vi (0) Assumption 3.2 ensures that Vi (∆t) ≤ ηVi (0) if kVk k[0,∆t) ≤ η m−1 δ0 for k ∈ M − {i}. Note that the initial Vj (0) ≤ δ0 , ∀j ∈ M. Therefore at [0, ∆t), V1 (t) would decrease, other Vi (t) increase until t = ∆t at which Vi (∆t) ≤ ηVi (0) ≤ ηδ0 . By induction, it is not hard to find that under S1 , Vi would decrease when mode i is activated, and increase when other modes are working. It follows from Assumption 3.2 that kVk k[0,T ) ≤ η m−1 δ0 for k ∈ M − {i}. Since λi can be assigned arbitrarily by ui , we choose λi such that ? m∆t
e−λi ∆t η m−1 ≤ e−λ
, for λ? > 0
(11)
One further has that ?T
Vi (T ) ≤ η m−i · e−λi ∆t · η i−1 · Vi (0) ≤ e−λ
Vi (0)
(12)
Thus Vi is bounded in [0, T ) and its value decreases once a period of switching is completed as shown in Fig.2. Combining all Vi yields ?T
V (T ) ≤ e−λ
V (0)
(13)
In the second interval [T, 2T ), one has kVk k[T,2T ) ≤ δ, for δ < δ0 . Assumption 3.2 still holds, It ? follows that V (2T ) ≤ e−λ T V (T ). ?
Following the similar way, we can obtain that V ((k + 1)T ) ≤ e−λ T V (kT ), k = 0, 1, 2.... Once at some time instants t = t? , Vj (t? ) ≤ ², Vi may not decrease. Therefore limt→∞ sup Vi (t) = ², and limt→∞ sup V (t) = m². This completes the proof. 2 Remark 3.1 : The number ² is designed according to system structure. A small gain γjk allows a small ² as shown in Assumption 3.2. This is consistent with the small gain idea in [22] for large scale interconnected systems. If there is no interconnection among all xi dynamics (as in Section 5), γjk disappears. In this case, we could let ² = 0, the origin is rendered asymptotically 6
stable. An alternative stabilization approach follows [14] that makes xi input-to-state stable in each [kT, (k + 1)T ], i.e. |xi ((k + 1)T )| ≤ β(|xi (kT )|, (k + 1)T ) +
©
max
k∈M−{i}
ª γ¯ik (kxk k[kT,(k+1)T ) ) , β ∈ KL, γ¯ik ∈ K∞ (14)
and then applies the small gain conditions among γ¯ik , ∀i ∈ M. However, such input-to-state stability is not easy to verify for general switched nonlinear systems. ♦
3.2
Multiple Lyapunov functions
This section extends the result in Section 3.1 to the case of multiple Lyapunov functions. Assumption 3.3 : For each mode i ∈ M of system (1) satisfying Assumption 2.1, we can design ui under which there exists a continuous non-negative function there exists a continuous non-negative function (i) (i) V (i) : ², xj dynamics (j 6= i) satisfies © ª (i) φj (Vj (tik ), ∆t) + maxk∈M−{j} γjk (ζ m−1 (δ0 )) (i)
Vj (tik )
≤η
(17) ♦
Theorem 3.2 : Consider a switched system (1) satisfying assumptions 2.1, 3.3 and 3.4. Under switching law S1 , there exists ² > 0 such that for any given δ0 > 0, limt→∞ sup V (1) (t) = m², (1) ∀Vi (0) ≤ δ0 , i ∈ M. (j)
Proof : Consider xi dynamics in the first period [0, T ). Suppose that Vi (t) > ², ∀j ∈ M, t ≥ 0. Similar to the proof of Theorem 3.1, we have (m)
Vi
(T − ) ≤ ηζ m−1−i (e−λi ∆t ζ i−1 (δ0 )) (1)
(18) (m)
Note that ηζ m−1−i (e−λi ∆t ζ i−1 (δ0 )) ≤ ηe−λi ∆t ζ m−2 (δ0 ), and Vi (T ) ≤ ζ(Vi such that ? (1) e−λi ∆t ζ m−1 (δ0 ) ≤ e−λ T Vi (0), for λ? > 0 7
(T − )). Assign λi (19)
we further obtain that (1)
?T
Vi (T ) ≤ e−λ (1)
Combining all Vi
(1)
Vi (0)
(20)
V (1) (0)
(21)
yields ?T
V (1) (T ) ≤ e−λ
Thus all V (i) , i ∈ M are bounded in [0, T ], and V (1) decreases once a period of switching is completed. ?
Following the similar way, we can obtain that V (1) ((k + 1)T ) ≤ e−λ T V (1) (kT ), k = 0, 1, 2... (j) (j) (1) Once at some time instants t = t? , Vi (t? ) ≤ ², Vi may not decrease, consequently, Vi may also not decrease. Therefore limt→∞ sup V (1) (t) = m². This completes the proof. 2
4
State-dependent switching law
In this section, we propose a more flexible state-dependent switching law where the dwell period of each mode is not designed and fixed a priori but relies on real-time state values. We will focus on the decreasing behavior of the whole system in each mode respectively. This is quite different from the idea in Section 3.
4.1
Common Lyapunov function
Similar to Section 3.1, consider a series of functions V1 (x1 ), V2 (x2 ), ..., Vm (xm ) with Vk (xk ) ∈ C 1 : P ².
(22) ♦
Assumption 4.1 means that the diverging speed of each mode’s internal dynamics is bounded. Moreover, if Vi at each tik is maximal among all Vi? , i? ∈ M, then during each activating period τ of mode i, the increasing amount of V¯i is less than the decreasing amount of Vi . The decay rate of Vi can be designed arbitrarily. Inequality (22) is satisfied if the increasing rate of V¯i is small enough, and can be checked conveniently when V¯i follows the exponential diverging form as shown in Section 5. A state-dependent switching law is designed : Switching law S2 1. Let k = 0, t0 = 0, choose ² > 0, κ > 0. 2. At t = tk , choose i = min arg maxi? ∈M Vi? (tk ). 8
3. If Vi (tk ) > ², go to 4; Else, go to 5. Until t = t? such that ∆V¯i (t? )tk = −∆Vi (t? )tik − υ, let k = k + 1,
4. Choose σ(tk ) = i. go to 2.
5. If k = 0, choose σ(t0 ) = i, let k = k + 1, go to 6; Else, go to 6. 6. Let mode σ(tk−1 ) work until t = t?? such that ∃j ∈ M, Vj (t?? ) > ² and V (t?? ) < m² + κ. let tk = t?? , go to 2. ¥ Vi
∋
... t b t b+ 1
t a t a+ 1
0
t c t c+ 1
td
t
V
m +κ m ∋
...
∋
0
t1
t2
t3
t4
t5
t6
t
Figure 3: Behaviors of Vi and V under S2 The main idea behind S2 is that once Vi is maximal among all functions (Step 2), mode i is activated, both Vi and V are decrease (Step 4). The decreasing performance of V is expected to be maintained at each switching instant until V ≤ m² as illustrated in Fig. 3. Theorem 4.1 : Consider a switched system (1) satisfying assumptions 2.1 and 4.1. Under switching law S2 , there exists ² > 0 such that for any given δ0 > 0, limt→∞ sup V (t) = m² + κ, where κ > 0 can be arbitrarily small, ∀Vi (0) ≤ δ0 , i ∈ M. Proof : Consider the interval [0, t1 ) during which mode i is supposed to be activated, t1 ≥ τ . We consider two cases: Case 1: Vi (0) > ². We have V (t1 ) = Vi (t1 ) + V¯i (t1 ) = Vi (0) + ∆Vi (t1 )0 + V¯i (0) + ∆V¯i (t1 )0
(23)
Assumption 4.1 guarantees that ∆V¯i (τ )0 ≤ −∆Vi (τ )0 − υ. there always exists t1 ≥ τ such that ∆V¯i (t1 )0 = −∆Vi (t1 )0 − υ
(24)
V (t1 ) < V (0)
(25)
Substituting (24) into (23) yields Case 2: Vi (0) ≤ ². It follows that V (0) ≤ m². Once Vi (t?? ) > ² and V (t?? ) < m² + κ, at t = t?? , for i ∈ M, the switching would occur at t = t?? , thus t1 = t?? . We have V (t1 ) < m² + κ 9
(26)
Combining (25) and (26) leads to V (t1 ) < max[V (0), m² + κ]
(27)
By induction, we further have V (t2 ) < max[V (t1 ), m² + κ] ... V (tk ) < max[V (tk−1 ), m² + κ], k = 1, 2, ...
(28)
Finally we obtain V (t) < max[V (0), m² + κ], for t > tk , k = 0, 1, ... One can find that V (t) would decrease if it is larger than m² + κ. Once V (t) ≤ m², it may decrease or increase and is always not larger than m² + κ. Therefore limt→∞ sup V (t) = m² + κ. This completes the proof. 2 Remark 4.1 : The decreasing speed of V depends on υ, the smaller is υ, the slower V decreases. ² could also be small if −∆Vi (tik + τ )tik is large enough compared with ∆V¯i (tik + τ )tik . If (22) holds for all Vi (tik ), we can choose an arbitrarily small ², in this case, the states converges to an arbitrarily small region. ♦
4.2
Multiple Lyapunov functions
Now we extend the result in Section 4.1 to the case of multiple Lyapunov functions. Assumption 4.2 : For each mode i ∈ M of system (1) satisfying Assumption 2.1, we can design ui under which there exists a continuous non-negative function there exists a continuous non-negative (i) function that satisfies (15) with Vk (xk ) satisfies (6)-(7), (16) and (i)
Vi
(i)
≥ Vj , ∀j ∈ M − {i}
(29)
(i) (i) Moreover, ∆V¯i (t)tik < β(V¯i (tik ), t − tik ) for β ∈ GKL. There exists positive ², υ > 0 such that (i)
(i)
∆V¯i (tik + τ )tik ≤ −∆Vi (tik + τ )tik − υ − ² (i)
for Vi (tik ) > 2².
♦
We can design ui and choose appropriate find a K∞ function denoted as ζ¯ such that
(i) Vi
to satisfy (29). Under condition (16), we can also
¯ (q) (x)) V (p) (x) ≤ ζ(V
(30)
Assumption 4.3 : When mode i ∈ M is activated at t = tik , there exists a positive ² > 0 such (i) that V (i) (tik ) − ζ¯−1 ◦ ζ¯−1 (V (i) (tik )) < ², ∀i ∈ M, for Vi (tik ) > 2². ♦ A state-dependent switching law is designed : Switching law S3 1. Let k = 0, t0 = 0.
Choose ² > 0, κ > 0, 0 < υ < ². (i? )
2. At t = tk , choose i = min arg maxi? ∈M Vi? (tk ). 10
(i)
3. If Vi (tk ) > 2², go to 4; Else, go to 5. (i)
(i)
Until t = t? such that ∆V¯i (t? )tk = ∆Vi (t? )tk − υ − ², let k =
4. Choose σ(tk ) = i. k + 1, go to 3.
5. If k = 0, choose σ(t0 ) = i, let k = k + 1, go to 6; Else, go to 6. 6. Let mode σ(tk−1 ) work until t = t?? such that V (σ(tk−1 )) (t?? ) > 2m² and ζ(V (σ(tk−1 )) (t?? )) < 2m² + κ, for i ∈ M. let tk = t?? , go to 2.
¥
Theorem 4.2 : Consider a switched system (1) satisfying assumptions 2.1, 4.2 and 4.3. Under S3 , there exists ² > 0 such that for any given δ0 > 0, limt→∞ sup V (i) (t) = 2m² + κ, ∀i ∈ M where κ (i) can be arbitrarily small, ∀Vi (0) ≤ δ0 , i ∈ M. Proof : Consider the interval [0, t1 ) during which mode i is supposed to be activated. We still consider two cases: (i)
Case 1: Vi (0) > 2². It follows from Assumption 4.2 that V (i) (t1 ) < V (i) (0) − ²
(31)
We also have ∀j ∈ M − {i} ³ ´ V (j) (t1 ) ≤ ζ¯ V (i) (t1 ) ³ ´ ζ¯−1 V (i) (0) ≤ V (j) (0)
(32) (33)
Assumption 4.3 ensures that V (i) (0) − ζ¯−1 ◦ ζ¯−1 (V (i) (0)) < ²
(34)
V (i) (t1 ) < ζ¯−1 ◦ ζ¯−1 (V (i) (0))
(35)
Combining (31) and (34) yields
One further has ³ ´ ³ ´ ζ¯ V (i) (t1 ) < ζ¯−1 V (i) (0)
(36)
This together with (32)-(33) leads to V (j) (t1 ) < V (j) (0)
(37)
Inequality (37) implies that at t1 , V (j) , ∀j ∈ M − {i} also decreases. (i)
Case 2: Vi (0) ≤ 2². It follows from condition (29) that V (j) (0) ≤ 2m², ∀j ∈ M. The switching would occurs at t = t1 if V (i) (t1 ) > 2m². Moreover, whatever mode j is activated at t = t1 , it holds that V (j) (t1 ) < 2m² + κ. To this end we have V (i) (t1 ) < max[V (i) (0) − ², 2m² + κ] V
(j)
(t1 ) < max[V
(j)
(0), 2m² + κ], ∀j ∈ M − {i} 11
(38) (39)
By induction, we further have V (σ(tk )) (tk+1 ) < max[V (σ(tk )) (tk ) − ², 2m² + κ] V (j) (tk+1 ) < max[V (j) (tk ), 2m² + κ], ∀j ∈ M − {σ(tk )} k = 1, 2, ...
(40) (41)
Similar to the proof of Theorem 4.1, we finally obtain that limt→∞ sup V (i) (t) ≤ 2m² + κ. This completes the proof. 2 Remark 4.2 : Each V (i) decreases at each switching instants of the whole process until V (i) ≤ 2m² whatever which mode is activated. A key condition to achieve this goal is Assumption 4.3, which ¯ among V (i) can be compensated if the decreasing amount of V (i) is implies that the effect of ζ(·) (i) large enough when mode i is activated. This requires that both the percent rate of Vi in V (i) and (i) the decreasing amount of Vi are large enough. ♦
5
Application to multi-agent systems
This section applies the results in Sections 3-4 to the target aggregation problem of a class of multi-agent systems. Previous assumptions will be further verified.
5.1
Problem formulation
The system consists of n agents modeled by a directed graph G = (N , E), where N = {1, 2, ..., n} is the set of agents and E is the set of arcs, (j, i) ∈ E denotes an arc from agent j to agent i, such that agent i can receive information directly from agent j. Agent j is the neighbor of agent i if (j, i) exists. The dynamics of agents are given as : x˙ i = gi (xi ) + uci (x1 , ..., xn ), i ∈ N
(42)
where xi ∈ l [xi1 , ..., xiri ] . Suppose that iri ∈ N , i.e., in each group, there is a unique leader who knows the target information. The problem to be solved is to let the states of all agents (42) satisfying Assumption 5.1 reach χ in the case that only one group Ni , i ∈ M is allowed to be interconnected at one time.
5.2
Feedback control topology
Motivated by linearized feedback control idea [19], we propose a “feedback control topology” with its corresponding cooperative controllers. Since there are m groups of subsystems, we build m connection topologies respectively by using the following rule. Rule R of building topology i (i ∈ M) 1. Pick agents in Ni . 2. Set arcs (is+1 , is ), for s = 1, 2, ..., ri − 1. 3. Set arcs (iι , iri ), for ι = 1, ..., ri − 1.
¥
In the topology resulting from R, there is a chain from the leader to followers, and each follower also feedback its information to the leader. The cooperative law of agent is is designed as ∂V ( is )> ¯∂xis ¯2 (−Ki Φi ) s = ri ¯ ∂Vi ¯ s¯ ¯ Under topology i =⇒ ucis = ¯ ∂xis ¯ a(x s = 1, ..., ri − 1 is+1 − xis )
13
(45)
where Φi , [Vi1 , Vi2 , ..., Viri ]> . Ki = [ki1 ki2 ... kiri ] is the feedback gain vector with each element ¯ ¯ ¯ ∂V ¯2 kis > 0. a ≥ λ1 is a positive number. Due to the structure of Vis , the term ¯ ∂xiis ¯ is impossible to s become zero unless xis − χ = 0. Theorem 5.1 : Consider a group Ni of multi-agent system (42) satisfying Assumption 5.1. The cooperative law (45) under R guarantees that |xi (t) − χ| exponentially tends to zero with the arbitrary decay rate. Proof : According to Assumption 5.1, the time derivative of Vis , s = 1, ..., ri −1 along (42) satisfies V˙ is ≤ λ1 Vis + 2a(xis − χ)> P (xis+1 − xis ) ³ ´ ≤ λ1 Vis + 2a −(xis − χ)> P (xis − χ) + (xis − χ)> P (xis+1 − χ)
(46)
There exists a constant matrix η ∈ η. We further have 2(xis − χ)> P (xis+1 − χ) ≤ (xis − χ)> η > η(xis − χ) + (xis+1 − χ)> η > η(xis+1 − χ) ≤ Vis + Vis+1
(47)
Substituting (47) into (46) yields V˙ is ≤ (λ1 − a)Vis + aVis+1 ≤ aVis+1
(48)
One can also get from Assumption 5.1 that the time derivative of Viri along (42) satisfies V˙ iri ≤ −λ0 Viri − Ki Φi ≤ −Ki Φi
(49)
Combining (48)-(49) leads to the compact form Φ˙ i ≤ A¯i Φi where
A¯i =
0 .. .
a .. .
0 −λ0 − ki1
0 −ki2
(50) ··· 0 .. .. . . ··· a · · · −kiri
Since the eigenvalues of A¯i can be assigned arbitrarily according to Ki , there exists a nonnegative function ¯ Ωi (Φi ) , Φ> (51) i P Φi with P¯ ∈ 0. This completes the proof. uci
Remark 5.1 : The similar target aggregation problem is also addressed in [16] with = P j6=i aij (xj − xi ), where aij is the weight of arc between agent i and agent j. aij > 0 if (j, i) exists, otherwise aij = 0. The strategy provided in [16] requires each agent to calculate weights 14
of its own arcs that depend on the weights of other arcs. Comparatively, a benefit of the cooperative law (45) under the feedback control topology is that all calculations are done only by the leader, which significantly reduces the real-time computational burden of the following agents, and is simpler to be implemented. ♦ Now let us turn to other groups in N − Ni under topology i. Since there is no interconnection in these groups, all cooperative laws are set zero, i.e. Under topology i =⇒ ucj = 0, ∀j ∈ N − Ni
(53)
According to Assumption 5.1, we further obtain that Ω˙ k (Φk ) ≤ λ?? Ωk (Φk ) ∀k ∈ M − {i}
(54)
for λ?? > 0, where Ωk is defined in (58). We can find that under topology i, the whole multi-agent system can be regarded as a nonminimum phase system with states Φ , [Φ1 Φ2 · · · Φm ]>
(55)
where the group Ni is controllable and linearizable, while other groups of agents compose the “uncontrollable internal dynamics” which may be unstable.
5.3
Target aggregation via switching topology
As the topology switch among m candidate ones, the system process naturally become a switched system with m modes where mode i corresponds to the topology i. The dynamics of group Ni can be regarded as xi dynamics in Section 2. Assumption 2.1 is naturally satisfied. Moreover, each Φi (i ∈ M) is continuous, all modes share a common Lyapunov function Ω,
m X
Ωi
i=1
Next we provide two kinds of switching topologies. To avoid arbitrary fast switching, the “dwell time” τ is also involved. Theorem 5.2 (time-dependent switching topology) : Consider a multi-agent system (42) satisfying Assumption 5.1. limt→∞ Ω(t) = 0, ∀Ωi (Φi (0)) ≤ δ0 , i ∈ M with any given δ0 > 0 if 1) Each group takes the cooperative law (45) and (53) under R. ?? ∆t
2) There exists positive numbers η > 0, ∆t ≥ τ such that eλ ¯ > 0. for λ
? ∆t
≤ η, and e−λ
¯
η m−1 ≤ e−λm∆t
3) m candidate topologies are switched following S1 . Proof : Condition 1) guarantees that under each topology, Assumption 3.1 is satisfied. Since there is no interconnection among groups, Assumption 3.2 is also satisfied under condition 2), we can choose ² = 0. Therefore, it follows from Theorem 3.1 that limt→∞ Ω(t) = 0 under S1 . 2 15
Theorem 5.3 (state-dependent switching topology) : Consider a multi-agent system (42) satisfying P Assumption 5.1. limt→∞ sup m i=1 Ω(t) = ς, with ς > 0 an arbitrarily small number, ∀Ωi (Φi (0)) ≤ δ0 , i ∈ M with any given δ0 > 0 if 1) Each group has the cooperative law (45) and (53) under R. ?? τ
2) (m − 1)(eλ
?
− 1) < 1 − eλ τ .
3) m candidate topologies are switched following S2 with Ωi replacing Vi and Ω replacing V . Proof : The proof is based on Theorem 4.1. It follows from (54) that ?? τ
∆Ωj (Φj (tik + τ ))tik ≤ (eλ
− 1)Ωj (Φj (tik ))
(56)
Since mode i is activated at t = tik if Ω(Φi (tik )) is maximal among all functions, we further have X
?? τ
∆Ωj (Φj (tik + τ ))tik ≤ (m − 1)(eλ
− 1)Ωj (Φi (tik ))
(57)
j∈M−{i}
On the other hand, one has ?
∆Ωi (Φi (tik + τ ))tik ≥ (1 − eλ τ )Ωi (Φi (tik ))
(58)
Combining (57), (58) and Condition 2) yields X
∆Ωj (Φj (tik + τ ))tik ≤ Ωi (Φi (tik )) − υ
(59)
j∈M−{i}
for υ > 0 that can be arbitrarily small. Therefore, under conditions 1)-2), Assumption 4.1 is satisfied. It follows from Theorem 4.1 that limt→∞ sup Ω(t) = ς under S2 . 2
5.4
Unmanned aerial vehicles team example
We are now taking an unmanned aerial vehicles (UAVs) team example to illustrate the above results. Consider a team of 4 UAVs divided into two groups. UAV 1 and UAV 3 are leaders that determines the flying behavior of the group, while UAVs 2 and 4 have no behavior information by themselves. Such system can be naturally modeled by a multi-agent system with each UAV being an agent. The UAV’s longitudinal differential equations under small attack angle can be expressed as [24]: ½ ϑ˙ i = ωi ω˙ i = Miω ωi + Miϑ cos(ϑi − αi ) + Mi ui i = 1, 2, ..., 4 For UAV i, the two states ϑi and ωi denote respectively the pitch angle and the pitch rate, the input ui is the elevator deflection angle, αi denotes the attack angle. Miω , Miϑ and Mi are the longitudinal dynamics parameters.
16
Consider the “climbing” process which requires that all UAVs in the team have the same pitch rates ω ? = 5 (deg/s). The initial values ωi (0) = 2 (deg/s), for i = 1, 2, 3, 4. The dynamic equations of pitch rates under appropriate controllers are : UAV i :
ω˙ 1 = −(ωi − ω ? ) + uci , i = 1, 3
(60)
UAV i :
ω˙ i = 0.5(ωi − ω ? ) + uci , i = 2, 4
(61)
It can be seen that UAVs 1 and 3 knows the prescribed pitch rate. However, without cooperation, UAVs 2 and 4 may track their own pitch rates, ω2 and ω4 run far away from ω ? . Define Vi = (ωi − ω ? )2 , i = 1, 2, 3, 4, Assumption 5.1 is satisfied with λ0 = 2, λ1 = 1. The topology 1 for group 1 is built based on R: (1, 2) and (2, 1). Let ω1 − ω ? (−3V1 − 5V2 ), 2(ω1 − ω ? )2 uc2 = ω1 − ω2 uc1 =
for k1 , k2 > 0
Similarly, the topology 2 for group 2 is arcs (3, 4) and (4, 3). Let ω3 − ω ? (−3V3 − 5V4 ) 2(ω3 − ω ? )2 uc4 = (ω3 − ω4 ) uc3 =
Theorem 5.1 guarantees that under topologies 1 and 2, groups 1 and 2 approach ω ? respectively. Suppose that only one group can be connected at one time due to the communication cost limitation. In this case, the switching among two topologies has to be applied. ¸ · 7.5 1 > > . Simple Define Ξ1 , [V1 V2 ]P [V1 V2 ] and Ξ2 , [V3 V4 ]P [V3 V4 ] with P = 1 0.5 calculation yields ½ Ξ˙ 1 ≤ −1.2Ξ1 Under topology 1 : Ξ˙ 2 ≤ 0.4Ξ2 ½ Ξ˙ 2 ≤ −1.2Ξ2 Under topology 2 : Ξ˙ 1 ≤ 0.4Ξ1 One can find that group 1 (resp. 2) is uncontrollable under topology 2 (resp. 1), which can be regarded as an unstable zero dynamics. Now we illustrate switching laws S1 and S2 . Choose dwell time τ = 0.5 (s), η = 1.5, ∆t = 1 (s), all conditions of Theorem 5.2 hold. Fig. 4(a) shows the switching function and trajectories of 4 pitch rates under S1 , from which we can see that all pitch rates reach ω ? . All conditions of Theorem 5.3 also hold. Choose ² = υ = 0.2, Fig. 4(b) shows the switching law and trajectories of 4 pitch rates under S2 , all pitch rates still reach ω ? . The performance under S1 is better than that under S2 , this is because υ is chosen very small, which makes the decay speed of states under S2 slower.
6
Conclusion
This paper discusses the stabilization problem for a class of non-minimum phase switched nonlinear systems as well as its application. Some further remarks are as follows: 17
switching functions
2
1
0
trajectories of pitch rates
3
0
1
2
3 t/s
4
5
6 4 ω1 ω2 ω3 ω4
2 0 0
1
2
3 t/s
4
5
2
1
0
6
trajectories of pitch rates
switching function
3
6
(a) under S1
0
1
2
3 t/s
4
5
6
6 4 ω1 ω2 ω3 ω4
2 0 −2
0
1
2
3 t/s
4
5
6
(b) under S2
Figure 4: Trajectories of pitch rates 1. The developed results rely on a series of Lyapunov-like functions that are established systematically in the target aggregation problem of multi-agent systems in Section 5. The existence and constructions of these functions for general switched systems deserve further investigation. 2. In this work, all states are available, output-feedback control together with observer design would be considered in the absence of full state measurements. 3. The dynamics of each mode is supposed to be known and fixed. Robust issue would be addressed in the presence of disturbance or unknown parameters, etc.
References [1] D. Liberzon, Switching in Systems and Control, Boston, MA: Birkhauser, 2003. [2] J. P. Hespanha, D. Liberzon, D. Angeli, and E. D. Sontag, Nonlinear norm-observability notions and stability of switched systems, IEEE Trans. on Automatic Control, 50(2), 154-168, 2005. [3] J. Zhao, and D. J. Hill, On stability, L2 gain and H∞ control for switched systems, Automatica, 44(5), 1220-1232, 2008. [4] P. Colaneri, J. C. Geromel, and A. Astolfi, Stabilization of continuous-time switched nonlinear systems, Systems & Control Letters, 57(1), 95-103, 2008. [5] T-T Han, S. S. Ge, and T. H. Lee, Persistent dwell-time switched nonlinear systems: variation paradigm and gauge design, IEEE Trans. on Automatic Control, 55(2), 321-337, 2010. [6] M. Cannon, M. Bacic, and B. Kouvaritakis, Dynamic non-minimum phase compensation for SISO nonlinear, affine in the input systems, Automatica, 42(11), 1969-1975, 2006.
18
[7] A. Isidori, A tool for semiglobal stabilization of uncertain non-minimum-phase nonlinear systems via output feedback, IEEE Trans. on Automatic Control, 45(10), 1817-1827, 2000. [8] S. Nazrulla, and H. K. Khalil, Robust stabilization of non-minimum phase nonlinear systems using extended high-gain observers, IEEE Trans. on Automatic Control, 56(4), 802-813, 2011. [9] S. M. Hoseini, M. Farrokhi, A. J. Koshkouei, Robust adaptive control of nonlinear nonminimum phase systems with uncertainties, Automatica, 47(2), 348-357, 2011. [10] M. Wang, G. M. Dimirovski, and J. Zhao, H∞ control for a class of non-minimum-phase cascade switched nonlinear Systems, Proc. of 2008 American Control Conference, Seattle, USA, 5080-5085, 2008. [11] M. Oishi and C. Tomlin, Switching in nonminimum phase systems: Applications to a vstol aircraft, Proc. of 2000 American Control Conference, Chicago, USA, 487-491, 2000. [12] M. Benosman, and K.Y. Lum, Output trajectory tracking for a switched nonlinear nonminimum phase system, Proc. of 16th IEEE International Conference on Control Applications, Singapore, 262-269, 2007. [13] G. M. Xie, and L. Wang, Periodic stabilizability of switched linear control systems, Automatica, 45(9), 2141-2148, 2009. [14] H. Yang, B. Jiang, V. Cocquempot, and H. Zhang, Stabilization of switched nonlinear systems with all unstable modes: applications to multi-agent systems, IEEE Transactions on Automatic Control, 56(9), 2230-2235, 2011. [15] H. Yang, V. Cocquempot, and B. Jiang, On stabilization of switched nonlinear systems with unstable modes, Systems & Control Letters, 58(10-11), 703-708, 2009. [16] H. Yang, M. Staroswiecki, B. Jiang, and J. Liu, Fault tolerant cooperative control for a class of nonlinear multi-agent systems, Systems & Control Letters, 60(4), 271-277, 2011. [17] J. Zhao, D. Hill, and T. Liu, Synchronization of complex dynamical networks with switching topology: A switched system point of view, Automatica, 45(11), 2502-2511, 2009. [18] G. Shi, and Y. Hong, Global target aggregation and state agreement of nonlinear multi-agent systems with switching topologies, Automatica, 45(5), 1165-1175, 2009. [19] A. Isidori, Nonlinear Control Systems, 3rd edition, Springer Verlag, London, 1995. [20] J. Yu, and L. Wang, Group consensus in multi-agent systems with switching topologies and communication delays, Systems & Control Letters, 59(6), 340-348, 2010. [21] R. O. Saber, and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. on Automatic Control, 49(9), 1520-1533, 2004.
19
[22] Z. P. Jiang, and Y. Wang, A generalization of the nolinear small-gain theorem for large-scale complex systems, Proc. of the 7th World Congress on Intelligent Control and Automation, Chongqing, China, 1188-1193, 2008. [23] F. Giulietti, L. Pollini, and M. Innocenti, Autonomous formation flight, IEEE Control Systems Magazine, 20(6), 34-44, 2000. [24] X. Mu, W. Zhang, and W. Zhang, An adaptive backstepping design for longitudinal flight path control, Proc. of 7th World Congress on Intelligent Control and Automation, Chongqing, China, 5249-5251, 2008.
20
a †
, Bin Jiang a , and Huaguang Zhang
b
a
College of Automation Engineering Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China. b
School of Information Science and Engineering Northeastern University, Shenyang, 110004, China.
Abstract This paper addresses the stabilization issue of a class of switched nonlinear systems where each mode may be non-minimum phase, and the states of linearized dynamics of all modes compose the whole state space. Time dependent and state dependent stabilization switching laws are provided with considering both common and multiple Lyapunov functions. The new results are applied to aggregation problem of nonlinear multi-agent systems with designable switching connection topology. Finally, an aircraft team example illustrates the efficiency of the proposed approaches.
Keywords: Switched systems; Non-minimum Phase; Stabilization; Multi-agent systems.
1
Introduction
Many engineering applications can be modeled by switched systems due to the existence of various jumping parameters [1]. Fruitful results have been reported on stability and stabilization of switched nonlinear systems, e.g., [2, 3, 4, 5] to name a few. Stabilization of non-minimum phase nonlinear systems is a quite challenging problem, several fundamental methods have been proposed including state feedback control [6], output feedback contro [7, 8, 9]. The main idea behind these methods is to compensate for the unstable zero dynamics by means of output synthesis or auxiliary systems such that the system becomes stable. The controllability of the zero dynamics is the basic requirement, otherwise, the stabilization can not be achieved. Some contributions have also been devoted to non-minimum phase switched nonlinear systems where each nonlinear mode may be non-minimum phase. In [10], H∞ control goal is achieved for a class of non-minimum phase cascade switched nonlinear systems where the internal dynamics of ∗
This work is supported by National Natural Science Foundation of China (61034005, 61010121, 61104116) and Doctoral Fund of Ministry of Education of China (20113218110011) and NUAA Research Funding (NS2011016). † Corresponding author: Tel: +86 25 84892301-6060, Fax: +86 25 84892300. Email: [email protected] (H. Yang), [email protected] (B. Jiang), [email protected] (H. Zhang).
1
each mode is assumed to be asymptotically stabilizable. Output tracking of non-minimum phase switched nonlinear systems has been considered in [11], where an approximated minimum phase model is utilized. The same problem is also investigated in [12] by means of an inversion-based control strategy. The main idea of the above results consists of two steps : 1. Design the individual controller respectively in each mode to compensate for its own unstable internal dynamics such that all modes become stable individually. This can be done by using the existing techniques for non-minimum phase nonlinear systems. 2. Apply the standard stability condition of switched systems, e.g. common/multiple Lyapunov functions methods to achieve the stability of the whole switched system. This idea is natural and extends the approaches of non-switching systems to the switched one. However, it is well known that the stabilization for non-minimum phase nonlinear systems is quite difficult, and is even impossible to be achieved if the unstable zero dynamics is uncontrollable. In this work, we focus on the stabilization of non-minimum phase switched nonlinear systems where the internal dynamics of each mode may be unstable and uncontrollable. Different from [10, 11, 12], we do not try to compensate for the unstable internal dynamics respectively in each mode. Instead, we achieve the stabilization from the overall system point of view. It will show that under some conditions, the negative effects of internal dynamics in some modes may be compensated by other modes, and the overall switching process can still be stable. The similar idea can be seen in [13] for switched linear systems and [14] for switched nonlinear systems. This special property of switched system provides a new clue of stabilizing the non-minimum phase systems. Two benefits follow : 1. It relaxes significantly the condition on the internal dynamics (it is allowed to be unstable and uncontrollable simultaneously). 2. It makes easier the design of each mode’s individual controller. We consider a class of switched nonlinear systems where each mode may be non-minimum phase, and the states of linearized dynamics of all modes compose the whole state space. Consequently, we provide novel time dependent and state dependent stabilization switching laws with considering both common and multiple Lyapunov functions. The proposed results are more general and flexible than that in [14] where only a time-dependent switching law is provided. As an important application of the new results, we consider the target aggregation problem of nonlinear multi-agent systems which can be divided into several groups of subsystems with each group being in the leader-following structure, and only one group is allowed to be interconnected at one time. Inspired by linearized feedback control idea [19], we design a novel “feedback control topology” for each group, under which the multi-agent system with switching topologies can be regarded as a non-minimum phase switched system and the proposed switched system results are applied. 2
In the rest of the paper, some preliminaries are given in Section 2. Sections 3 and 4 propose time dependent and state dependent stabilization switching laws respectively, which are applied to multi-agent systems in Section 5, followed by some concluding remarks in Section 6.
2
Preliminaries
Consider the following switched nonlinear control systems x˙ = fσ (x) + gσ (x)uσ
(1)
where x ∈ 0 is called “dwell-time” [1]. We also assume that the states do not jump at the switching instants. > > Suppose that for each mode i ∈ M, we can find a function yi and a partition x = [¯ x> i , xi ] where xi ∈ x = [x> ♦ 1 , ..., xm ] . Assumption 2.1 implies that the states of unstable internal dynamics in each mode can be controlled and linearized in other modes. The states of all linearized dynamics compose the whole state space as illustrated in Fig. 1. This allows us to achieve the stabilization by fully utilizing the tradeoff among different modes of the switched system. Assumption 2.1 can be relaxed to a more moderate case where more than one mode may share some states in their linearized dynamics, the proposed methods can be straightly extended to this case. 3
Remark 2.1 : We would like to show how to fully utilize the switching properties to achieve the stabilization goal. Therefore, an idea of “overall system point of view” will be followed through out the paper. For more general switched systems where Assumption 2.1 is not satisfied, the combination between the existing control techniques for non-switching systems and the switching properties can be developed. ♦
x1
linearized dynamics
internal dynamics linearized dynamics
x2
internal dynamics
... ... ... ...
internal dynamics internal dynamics linearized dynamics
xm mode 1
mode m
mode 2
Figure 1: The structure of switched systems
3
Time-dependent switching law
In this section, we provide time-dependent stabilizing switching laws for the switched system satisfying Assumption 2.1.
3.1
Common Lyapunov function
Define a special function that will be used in the following sections: Definition 3.1 : A class GKL function γ : [0, ∞) × [0, ∞) → [0, ∞) if γ(·, t) is of class K1 for each fixed t ≥ 0 and γ(s, t) increases to infinity as t → ∞ for each fixed s ≥ 0. 2 Assumption 3.1 : For the switched system (1) satisfying Assumption 2.1, we can design ui for each mode i under which there exists a continuous non-negative function V : 0, ∆t ≥ τ such that when mode i is activated and Vi (tik ) > ², xj dynamics (j 6= i) satisfies © ª φj (Vj (tik ), ∆t) + maxk∈M−{j} γjk (η m−1 δ0 ) ≤η (9) Vj (tik ) ♦ 5
It follows from (8) that the left side of (9) is an upper bound of the gain of Vj in the period ∆t, if kVk k[tik ,tik +∆t) ≤ η m−1 δ0 . Inequality (9) implies that the diverging speed of each mode’s internal dynamics is bounded during a interval that is not less than the dwell time τ . Since the switching sequence and instants are designed a priori, on-line informations can not be used, such fixed upper bound has to be imposed. η is not hard to be found if γjk is small. Theorem 3.1 : Consider a switched system (1) satisfying assumptions 2.1, 3.1 and 3.2. Under switching law S1 , there exist ² > 0 such that for any given δ0 > 0, limt→∞ sup V (t) = m², ∀Vi (0) ≤ δ0 , i ∈ M. Proof : We first suppose that Vj (t) > ², ∀t ≥ 0. Consider xi dynamics in the first period [0, T ). Since the switching sequence is mode 1 → mode 2 → · · · mode i · · · → mode m, the solution of Vi (i 6= 1) at t = ∆t can be represented as © ª φj (Vi (0), ∆t) + maxk∈M−{i} γik (kVk k[0,∆t) ) Vi (0) (10) Vi (∆t) ≤ Vi (0) Assumption 3.2 ensures that Vi (∆t) ≤ ηVi (0) if kVk k[0,∆t) ≤ η m−1 δ0 for k ∈ M − {i}. Note that the initial Vj (0) ≤ δ0 , ∀j ∈ M. Therefore at [0, ∆t), V1 (t) would decrease, other Vi (t) increase until t = ∆t at which Vi (∆t) ≤ ηVi (0) ≤ ηδ0 . By induction, it is not hard to find that under S1 , Vi would decrease when mode i is activated, and increase when other modes are working. It follows from Assumption 3.2 that kVk k[0,T ) ≤ η m−1 δ0 for k ∈ M − {i}. Since λi can be assigned arbitrarily by ui , we choose λi such that ? m∆t
e−λi ∆t η m−1 ≤ e−λ
, for λ? > 0
(11)
One further has that ?T
Vi (T ) ≤ η m−i · e−λi ∆t · η i−1 · Vi (0) ≤ e−λ
Vi (0)
(12)
Thus Vi is bounded in [0, T ) and its value decreases once a period of switching is completed as shown in Fig.2. Combining all Vi yields ?T
V (T ) ≤ e−λ
V (0)
(13)
In the second interval [T, 2T ), one has kVk k[T,2T ) ≤ δ, for δ < δ0 . Assumption 3.2 still holds, It ? follows that V (2T ) ≤ e−λ T V (T ). ?
Following the similar way, we can obtain that V ((k + 1)T ) ≤ e−λ T V (kT ), k = 0, 1, 2.... Once at some time instants t = t? , Vj (t? ) ≤ ², Vi may not decrease. Therefore limt→∞ sup Vi (t) = ², and limt→∞ sup V (t) = m². This completes the proof. 2 Remark 3.1 : The number ² is designed according to system structure. A small gain γjk allows a small ² as shown in Assumption 3.2. This is consistent with the small gain idea in [22] for large scale interconnected systems. If there is no interconnection among all xi dynamics (as in Section 5), γjk disappears. In this case, we could let ² = 0, the origin is rendered asymptotically 6
stable. An alternative stabilization approach follows [14] that makes xi input-to-state stable in each [kT, (k + 1)T ], i.e. |xi ((k + 1)T )| ≤ β(|xi (kT )|, (k + 1)T ) +
©
max
k∈M−{i}
ª γ¯ik (kxk k[kT,(k+1)T ) ) , β ∈ KL, γ¯ik ∈ K∞ (14)
and then applies the small gain conditions among γ¯ik , ∀i ∈ M. However, such input-to-state stability is not easy to verify for general switched nonlinear systems. ♦
3.2
Multiple Lyapunov functions
This section extends the result in Section 3.1 to the case of multiple Lyapunov functions. Assumption 3.3 : For each mode i ∈ M of system (1) satisfying Assumption 2.1, we can design ui under which there exists a continuous non-negative function there exists a continuous non-negative function (i) (i) V (i) : ², xj dynamics (j 6= i) satisfies © ª (i) φj (Vj (tik ), ∆t) + maxk∈M−{j} γjk (ζ m−1 (δ0 )) (i)
Vj (tik )
≤η
(17) ♦
Theorem 3.2 : Consider a switched system (1) satisfying assumptions 2.1, 3.3 and 3.4. Under switching law S1 , there exists ² > 0 such that for any given δ0 > 0, limt→∞ sup V (1) (t) = m², (1) ∀Vi (0) ≤ δ0 , i ∈ M. (j)
Proof : Consider xi dynamics in the first period [0, T ). Suppose that Vi (t) > ², ∀j ∈ M, t ≥ 0. Similar to the proof of Theorem 3.1, we have (m)
Vi
(T − ) ≤ ηζ m−1−i (e−λi ∆t ζ i−1 (δ0 )) (1)
(18) (m)
Note that ηζ m−1−i (e−λi ∆t ζ i−1 (δ0 )) ≤ ηe−λi ∆t ζ m−2 (δ0 ), and Vi (T ) ≤ ζ(Vi such that ? (1) e−λi ∆t ζ m−1 (δ0 ) ≤ e−λ T Vi (0), for λ? > 0 7
(T − )). Assign λi (19)
we further obtain that (1)
?T
Vi (T ) ≤ e−λ (1)
Combining all Vi
(1)
Vi (0)
(20)
V (1) (0)
(21)
yields ?T
V (1) (T ) ≤ e−λ
Thus all V (i) , i ∈ M are bounded in [0, T ], and V (1) decreases once a period of switching is completed. ?
Following the similar way, we can obtain that V (1) ((k + 1)T ) ≤ e−λ T V (1) (kT ), k = 0, 1, 2... (j) (j) (1) Once at some time instants t = t? , Vi (t? ) ≤ ², Vi may not decrease, consequently, Vi may also not decrease. Therefore limt→∞ sup V (1) (t) = m². This completes the proof. 2
4
State-dependent switching law
In this section, we propose a more flexible state-dependent switching law where the dwell period of each mode is not designed and fixed a priori but relies on real-time state values. We will focus on the decreasing behavior of the whole system in each mode respectively. This is quite different from the idea in Section 3.
4.1
Common Lyapunov function
Similar to Section 3.1, consider a series of functions V1 (x1 ), V2 (x2 ), ..., Vm (xm ) with Vk (xk ) ∈ C 1 : P ².
(22) ♦
Assumption 4.1 means that the diverging speed of each mode’s internal dynamics is bounded. Moreover, if Vi at each tik is maximal among all Vi? , i? ∈ M, then during each activating period τ of mode i, the increasing amount of V¯i is less than the decreasing amount of Vi . The decay rate of Vi can be designed arbitrarily. Inequality (22) is satisfied if the increasing rate of V¯i is small enough, and can be checked conveniently when V¯i follows the exponential diverging form as shown in Section 5. A state-dependent switching law is designed : Switching law S2 1. Let k = 0, t0 = 0, choose ² > 0, κ > 0. 2. At t = tk , choose i = min arg maxi? ∈M Vi? (tk ). 8
3. If Vi (tk ) > ², go to 4; Else, go to 5. Until t = t? such that ∆V¯i (t? )tk = −∆Vi (t? )tik − υ, let k = k + 1,
4. Choose σ(tk ) = i. go to 2.
5. If k = 0, choose σ(t0 ) = i, let k = k + 1, go to 6; Else, go to 6. 6. Let mode σ(tk−1 ) work until t = t?? such that ∃j ∈ M, Vj (t?? ) > ² and V (t?? ) < m² + κ. let tk = t?? , go to 2. ¥ Vi
∋
... t b t b+ 1
t a t a+ 1
0
t c t c+ 1
td
t
V
m +κ m ∋
...
∋
0
t1
t2
t3
t4
t5
t6
t
Figure 3: Behaviors of Vi and V under S2 The main idea behind S2 is that once Vi is maximal among all functions (Step 2), mode i is activated, both Vi and V are decrease (Step 4). The decreasing performance of V is expected to be maintained at each switching instant until V ≤ m² as illustrated in Fig. 3. Theorem 4.1 : Consider a switched system (1) satisfying assumptions 2.1 and 4.1. Under switching law S2 , there exists ² > 0 such that for any given δ0 > 0, limt→∞ sup V (t) = m² + κ, where κ > 0 can be arbitrarily small, ∀Vi (0) ≤ δ0 , i ∈ M. Proof : Consider the interval [0, t1 ) during which mode i is supposed to be activated, t1 ≥ τ . We consider two cases: Case 1: Vi (0) > ². We have V (t1 ) = Vi (t1 ) + V¯i (t1 ) = Vi (0) + ∆Vi (t1 )0 + V¯i (0) + ∆V¯i (t1 )0
(23)
Assumption 4.1 guarantees that ∆V¯i (τ )0 ≤ −∆Vi (τ )0 − υ. there always exists t1 ≥ τ such that ∆V¯i (t1 )0 = −∆Vi (t1 )0 − υ
(24)
V (t1 ) < V (0)
(25)
Substituting (24) into (23) yields Case 2: Vi (0) ≤ ². It follows that V (0) ≤ m². Once Vi (t?? ) > ² and V (t?? ) < m² + κ, at t = t?? , for i ∈ M, the switching would occur at t = t?? , thus t1 = t?? . We have V (t1 ) < m² + κ 9
(26)
Combining (25) and (26) leads to V (t1 ) < max[V (0), m² + κ]
(27)
By induction, we further have V (t2 ) < max[V (t1 ), m² + κ] ... V (tk ) < max[V (tk−1 ), m² + κ], k = 1, 2, ...
(28)
Finally we obtain V (t) < max[V (0), m² + κ], for t > tk , k = 0, 1, ... One can find that V (t) would decrease if it is larger than m² + κ. Once V (t) ≤ m², it may decrease or increase and is always not larger than m² + κ. Therefore limt→∞ sup V (t) = m² + κ. This completes the proof. 2 Remark 4.1 : The decreasing speed of V depends on υ, the smaller is υ, the slower V decreases. ² could also be small if −∆Vi (tik + τ )tik is large enough compared with ∆V¯i (tik + τ )tik . If (22) holds for all Vi (tik ), we can choose an arbitrarily small ², in this case, the states converges to an arbitrarily small region. ♦
4.2
Multiple Lyapunov functions
Now we extend the result in Section 4.1 to the case of multiple Lyapunov functions. Assumption 4.2 : For each mode i ∈ M of system (1) satisfying Assumption 2.1, we can design ui under which there exists a continuous non-negative function there exists a continuous non-negative (i) function that satisfies (15) with Vk (xk ) satisfies (6)-(7), (16) and (i)
Vi
(i)
≥ Vj , ∀j ∈ M − {i}
(29)
(i) (i) Moreover, ∆V¯i (t)tik < β(V¯i (tik ), t − tik ) for β ∈ GKL. There exists positive ², υ > 0 such that (i)
(i)
∆V¯i (tik + τ )tik ≤ −∆Vi (tik + τ )tik − υ − ² (i)
for Vi (tik ) > 2².
♦
We can design ui and choose appropriate find a K∞ function denoted as ζ¯ such that
(i) Vi
to satisfy (29). Under condition (16), we can also
¯ (q) (x)) V (p) (x) ≤ ζ(V
(30)
Assumption 4.3 : When mode i ∈ M is activated at t = tik , there exists a positive ² > 0 such (i) that V (i) (tik ) − ζ¯−1 ◦ ζ¯−1 (V (i) (tik )) < ², ∀i ∈ M, for Vi (tik ) > 2². ♦ A state-dependent switching law is designed : Switching law S3 1. Let k = 0, t0 = 0.
Choose ² > 0, κ > 0, 0 < υ < ². (i? )
2. At t = tk , choose i = min arg maxi? ∈M Vi? (tk ). 10
(i)
3. If Vi (tk ) > 2², go to 4; Else, go to 5. (i)
(i)
Until t = t? such that ∆V¯i (t? )tk = ∆Vi (t? )tk − υ − ², let k =
4. Choose σ(tk ) = i. k + 1, go to 3.
5. If k = 0, choose σ(t0 ) = i, let k = k + 1, go to 6; Else, go to 6. 6. Let mode σ(tk−1 ) work until t = t?? such that V (σ(tk−1 )) (t?? ) > 2m² and ζ(V (σ(tk−1 )) (t?? )) < 2m² + κ, for i ∈ M. let tk = t?? , go to 2.
¥
Theorem 4.2 : Consider a switched system (1) satisfying assumptions 2.1, 4.2 and 4.3. Under S3 , there exists ² > 0 such that for any given δ0 > 0, limt→∞ sup V (i) (t) = 2m² + κ, ∀i ∈ M where κ (i) can be arbitrarily small, ∀Vi (0) ≤ δ0 , i ∈ M. Proof : Consider the interval [0, t1 ) during which mode i is supposed to be activated. We still consider two cases: (i)
Case 1: Vi (0) > 2². It follows from Assumption 4.2 that V (i) (t1 ) < V (i) (0) − ²
(31)
We also have ∀j ∈ M − {i} ³ ´ V (j) (t1 ) ≤ ζ¯ V (i) (t1 ) ³ ´ ζ¯−1 V (i) (0) ≤ V (j) (0)
(32) (33)
Assumption 4.3 ensures that V (i) (0) − ζ¯−1 ◦ ζ¯−1 (V (i) (0)) < ²
(34)
V (i) (t1 ) < ζ¯−1 ◦ ζ¯−1 (V (i) (0))
(35)
Combining (31) and (34) yields
One further has ³ ´ ³ ´ ζ¯ V (i) (t1 ) < ζ¯−1 V (i) (0)
(36)
This together with (32)-(33) leads to V (j) (t1 ) < V (j) (0)
(37)
Inequality (37) implies that at t1 , V (j) , ∀j ∈ M − {i} also decreases. (i)
Case 2: Vi (0) ≤ 2². It follows from condition (29) that V (j) (0) ≤ 2m², ∀j ∈ M. The switching would occurs at t = t1 if V (i) (t1 ) > 2m². Moreover, whatever mode j is activated at t = t1 , it holds that V (j) (t1 ) < 2m² + κ. To this end we have V (i) (t1 ) < max[V (i) (0) − ², 2m² + κ] V
(j)
(t1 ) < max[V
(j)
(0), 2m² + κ], ∀j ∈ M − {i} 11
(38) (39)
By induction, we further have V (σ(tk )) (tk+1 ) < max[V (σ(tk )) (tk ) − ², 2m² + κ] V (j) (tk+1 ) < max[V (j) (tk ), 2m² + κ], ∀j ∈ M − {σ(tk )} k = 1, 2, ...
(40) (41)
Similar to the proof of Theorem 4.1, we finally obtain that limt→∞ sup V (i) (t) ≤ 2m² + κ. This completes the proof. 2 Remark 4.2 : Each V (i) decreases at each switching instants of the whole process until V (i) ≤ 2m² whatever which mode is activated. A key condition to achieve this goal is Assumption 4.3, which ¯ among V (i) can be compensated if the decreasing amount of V (i) is implies that the effect of ζ(·) (i) large enough when mode i is activated. This requires that both the percent rate of Vi in V (i) and (i) the decreasing amount of Vi are large enough. ♦
5
Application to multi-agent systems
This section applies the results in Sections 3-4 to the target aggregation problem of a class of multi-agent systems. Previous assumptions will be further verified.
5.1
Problem formulation
The system consists of n agents modeled by a directed graph G = (N , E), where N = {1, 2, ..., n} is the set of agents and E is the set of arcs, (j, i) ∈ E denotes an arc from agent j to agent i, such that agent i can receive information directly from agent j. Agent j is the neighbor of agent i if (j, i) exists. The dynamics of agents are given as : x˙ i = gi (xi ) + uci (x1 , ..., xn ), i ∈ N
(42)
where xi ∈ l [xi1 , ..., xiri ] . Suppose that iri ∈ N , i.e., in each group, there is a unique leader who knows the target information. The problem to be solved is to let the states of all agents (42) satisfying Assumption 5.1 reach χ in the case that only one group Ni , i ∈ M is allowed to be interconnected at one time.
5.2
Feedback control topology
Motivated by linearized feedback control idea [19], we propose a “feedback control topology” with its corresponding cooperative controllers. Since there are m groups of subsystems, we build m connection topologies respectively by using the following rule. Rule R of building topology i (i ∈ M) 1. Pick agents in Ni . 2. Set arcs (is+1 , is ), for s = 1, 2, ..., ri − 1. 3. Set arcs (iι , iri ), for ι = 1, ..., ri − 1.
¥
In the topology resulting from R, there is a chain from the leader to followers, and each follower also feedback its information to the leader. The cooperative law of agent is is designed as ∂V ( is )> ¯∂xis ¯2 (−Ki Φi ) s = ri ¯ ∂Vi ¯ s¯ ¯ Under topology i =⇒ ucis = ¯ ∂xis ¯ a(x s = 1, ..., ri − 1 is+1 − xis )
13
(45)
where Φi , [Vi1 , Vi2 , ..., Viri ]> . Ki = [ki1 ki2 ... kiri ] is the feedback gain vector with each element ¯ ¯ ¯ ∂V ¯2 kis > 0. a ≥ λ1 is a positive number. Due to the structure of Vis , the term ¯ ∂xiis ¯ is impossible to s become zero unless xis − χ = 0. Theorem 5.1 : Consider a group Ni of multi-agent system (42) satisfying Assumption 5.1. The cooperative law (45) under R guarantees that |xi (t) − χ| exponentially tends to zero with the arbitrary decay rate. Proof : According to Assumption 5.1, the time derivative of Vis , s = 1, ..., ri −1 along (42) satisfies V˙ is ≤ λ1 Vis + 2a(xis − χ)> P (xis+1 − xis ) ³ ´ ≤ λ1 Vis + 2a −(xis − χ)> P (xis − χ) + (xis − χ)> P (xis+1 − χ)
(46)
There exists a constant matrix η ∈ η. We further have 2(xis − χ)> P (xis+1 − χ) ≤ (xis − χ)> η > η(xis − χ) + (xis+1 − χ)> η > η(xis+1 − χ) ≤ Vis + Vis+1
(47)
Substituting (47) into (46) yields V˙ is ≤ (λ1 − a)Vis + aVis+1 ≤ aVis+1
(48)
One can also get from Assumption 5.1 that the time derivative of Viri along (42) satisfies V˙ iri ≤ −λ0 Viri − Ki Φi ≤ −Ki Φi
(49)
Combining (48)-(49) leads to the compact form Φ˙ i ≤ A¯i Φi where
A¯i =
0 .. .
a .. .
0 −λ0 − ki1
0 −ki2
(50) ··· 0 .. .. . . ··· a · · · −kiri
Since the eigenvalues of A¯i can be assigned arbitrarily according to Ki , there exists a nonnegative function ¯ Ωi (Φi ) , Φ> (51) i P Φi with P¯ ∈ 0. This completes the proof. uci
Remark 5.1 : The similar target aggregation problem is also addressed in [16] with = P j6=i aij (xj − xi ), where aij is the weight of arc between agent i and agent j. aij > 0 if (j, i) exists, otherwise aij = 0. The strategy provided in [16] requires each agent to calculate weights 14
of its own arcs that depend on the weights of other arcs. Comparatively, a benefit of the cooperative law (45) under the feedback control topology is that all calculations are done only by the leader, which significantly reduces the real-time computational burden of the following agents, and is simpler to be implemented. ♦ Now let us turn to other groups in N − Ni under topology i. Since there is no interconnection in these groups, all cooperative laws are set zero, i.e. Under topology i =⇒ ucj = 0, ∀j ∈ N − Ni
(53)
According to Assumption 5.1, we further obtain that Ω˙ k (Φk ) ≤ λ?? Ωk (Φk ) ∀k ∈ M − {i}
(54)
for λ?? > 0, where Ωk is defined in (58). We can find that under topology i, the whole multi-agent system can be regarded as a nonminimum phase system with states Φ , [Φ1 Φ2 · · · Φm ]>
(55)
where the group Ni is controllable and linearizable, while other groups of agents compose the “uncontrollable internal dynamics” which may be unstable.
5.3
Target aggregation via switching topology
As the topology switch among m candidate ones, the system process naturally become a switched system with m modes where mode i corresponds to the topology i. The dynamics of group Ni can be regarded as xi dynamics in Section 2. Assumption 2.1 is naturally satisfied. Moreover, each Φi (i ∈ M) is continuous, all modes share a common Lyapunov function Ω,
m X
Ωi
i=1
Next we provide two kinds of switching topologies. To avoid arbitrary fast switching, the “dwell time” τ is also involved. Theorem 5.2 (time-dependent switching topology) : Consider a multi-agent system (42) satisfying Assumption 5.1. limt→∞ Ω(t) = 0, ∀Ωi (Φi (0)) ≤ δ0 , i ∈ M with any given δ0 > 0 if 1) Each group takes the cooperative law (45) and (53) under R. ?? ∆t
2) There exists positive numbers η > 0, ∆t ≥ τ such that eλ ¯ > 0. for λ
? ∆t
≤ η, and e−λ
¯
η m−1 ≤ e−λm∆t
3) m candidate topologies are switched following S1 . Proof : Condition 1) guarantees that under each topology, Assumption 3.1 is satisfied. Since there is no interconnection among groups, Assumption 3.2 is also satisfied under condition 2), we can choose ² = 0. Therefore, it follows from Theorem 3.1 that limt→∞ Ω(t) = 0 under S1 . 2 15
Theorem 5.3 (state-dependent switching topology) : Consider a multi-agent system (42) satisfying P Assumption 5.1. limt→∞ sup m i=1 Ω(t) = ς, with ς > 0 an arbitrarily small number, ∀Ωi (Φi (0)) ≤ δ0 , i ∈ M with any given δ0 > 0 if 1) Each group has the cooperative law (45) and (53) under R. ?? τ
2) (m − 1)(eλ
?
− 1) < 1 − eλ τ .
3) m candidate topologies are switched following S2 with Ωi replacing Vi and Ω replacing V . Proof : The proof is based on Theorem 4.1. It follows from (54) that ?? τ
∆Ωj (Φj (tik + τ ))tik ≤ (eλ
− 1)Ωj (Φj (tik ))
(56)
Since mode i is activated at t = tik if Ω(Φi (tik )) is maximal among all functions, we further have X
?? τ
∆Ωj (Φj (tik + τ ))tik ≤ (m − 1)(eλ
− 1)Ωj (Φi (tik ))
(57)
j∈M−{i}
On the other hand, one has ?
∆Ωi (Φi (tik + τ ))tik ≥ (1 − eλ τ )Ωi (Φi (tik ))
(58)
Combining (57), (58) and Condition 2) yields X
∆Ωj (Φj (tik + τ ))tik ≤ Ωi (Φi (tik )) − υ
(59)
j∈M−{i}
for υ > 0 that can be arbitrarily small. Therefore, under conditions 1)-2), Assumption 4.1 is satisfied. It follows from Theorem 4.1 that limt→∞ sup Ω(t) = ς under S2 . 2
5.4
Unmanned aerial vehicles team example
We are now taking an unmanned aerial vehicles (UAVs) team example to illustrate the above results. Consider a team of 4 UAVs divided into two groups. UAV 1 and UAV 3 are leaders that determines the flying behavior of the group, while UAVs 2 and 4 have no behavior information by themselves. Such system can be naturally modeled by a multi-agent system with each UAV being an agent. The UAV’s longitudinal differential equations under small attack angle can be expressed as [24]: ½ ϑ˙ i = ωi ω˙ i = Miω ωi + Miϑ cos(ϑi − αi ) + Mi ui i = 1, 2, ..., 4 For UAV i, the two states ϑi and ωi denote respectively the pitch angle and the pitch rate, the input ui is the elevator deflection angle, αi denotes the attack angle. Miω , Miϑ and Mi are the longitudinal dynamics parameters.
16
Consider the “climbing” process which requires that all UAVs in the team have the same pitch rates ω ? = 5 (deg/s). The initial values ωi (0) = 2 (deg/s), for i = 1, 2, 3, 4. The dynamic equations of pitch rates under appropriate controllers are : UAV i :
ω˙ 1 = −(ωi − ω ? ) + uci , i = 1, 3
(60)
UAV i :
ω˙ i = 0.5(ωi − ω ? ) + uci , i = 2, 4
(61)
It can be seen that UAVs 1 and 3 knows the prescribed pitch rate. However, without cooperation, UAVs 2 and 4 may track their own pitch rates, ω2 and ω4 run far away from ω ? . Define Vi = (ωi − ω ? )2 , i = 1, 2, 3, 4, Assumption 5.1 is satisfied with λ0 = 2, λ1 = 1. The topology 1 for group 1 is built based on R: (1, 2) and (2, 1). Let ω1 − ω ? (−3V1 − 5V2 ), 2(ω1 − ω ? )2 uc2 = ω1 − ω2 uc1 =
for k1 , k2 > 0
Similarly, the topology 2 for group 2 is arcs (3, 4) and (4, 3). Let ω3 − ω ? (−3V3 − 5V4 ) 2(ω3 − ω ? )2 uc4 = (ω3 − ω4 ) uc3 =
Theorem 5.1 guarantees that under topologies 1 and 2, groups 1 and 2 approach ω ? respectively. Suppose that only one group can be connected at one time due to the communication cost limitation. In this case, the switching among two topologies has to be applied. ¸ · 7.5 1 > > . Simple Define Ξ1 , [V1 V2 ]P [V1 V2 ] and Ξ2 , [V3 V4 ]P [V3 V4 ] with P = 1 0.5 calculation yields ½ Ξ˙ 1 ≤ −1.2Ξ1 Under topology 1 : Ξ˙ 2 ≤ 0.4Ξ2 ½ Ξ˙ 2 ≤ −1.2Ξ2 Under topology 2 : Ξ˙ 1 ≤ 0.4Ξ1 One can find that group 1 (resp. 2) is uncontrollable under topology 2 (resp. 1), which can be regarded as an unstable zero dynamics. Now we illustrate switching laws S1 and S2 . Choose dwell time τ = 0.5 (s), η = 1.5, ∆t = 1 (s), all conditions of Theorem 5.2 hold. Fig. 4(a) shows the switching function and trajectories of 4 pitch rates under S1 , from which we can see that all pitch rates reach ω ? . All conditions of Theorem 5.3 also hold. Choose ² = υ = 0.2, Fig. 4(b) shows the switching law and trajectories of 4 pitch rates under S2 , all pitch rates still reach ω ? . The performance under S1 is better than that under S2 , this is because υ is chosen very small, which makes the decay speed of states under S2 slower.
6
Conclusion
This paper discusses the stabilization problem for a class of non-minimum phase switched nonlinear systems as well as its application. Some further remarks are as follows: 17
switching functions
2
1
0
trajectories of pitch rates
3
0
1
2
3 t/s
4
5
6 4 ω1 ω2 ω3 ω4
2 0 0
1
2
3 t/s
4
5
2
1
0
6
trajectories of pitch rates
switching function
3
6
(a) under S1
0
1
2
3 t/s
4
5
6
6 4 ω1 ω2 ω3 ω4
2 0 −2
0
1
2
3 t/s
4
5
6
(b) under S2
Figure 4: Trajectories of pitch rates 1. The developed results rely on a series of Lyapunov-like functions that are established systematically in the target aggregation problem of multi-agent systems in Section 5. The existence and constructions of these functions for general switched systems deserve further investigation. 2. In this work, all states are available, output-feedback control together with observer design would be considered in the absence of full state measurements. 3. The dynamics of each mode is supposed to be known and fixed. Robust issue would be addressed in the presence of disturbance or unknown parameters, etc.
References [1] D. Liberzon, Switching in Systems and Control, Boston, MA: Birkhauser, 2003. [2] J. P. Hespanha, D. Liberzon, D. Angeli, and E. D. Sontag, Nonlinear norm-observability notions and stability of switched systems, IEEE Trans. on Automatic Control, 50(2), 154-168, 2005. [3] J. Zhao, and D. J. Hill, On stability, L2 gain and H∞ control for switched systems, Automatica, 44(5), 1220-1232, 2008. [4] P. Colaneri, J. C. Geromel, and A. Astolfi, Stabilization of continuous-time switched nonlinear systems, Systems & Control Letters, 57(1), 95-103, 2008. [5] T-T Han, S. S. Ge, and T. H. Lee, Persistent dwell-time switched nonlinear systems: variation paradigm and gauge design, IEEE Trans. on Automatic Control, 55(2), 321-337, 2010. [6] M. Cannon, M. Bacic, and B. Kouvaritakis, Dynamic non-minimum phase compensation for SISO nonlinear, affine in the input systems, Automatica, 42(11), 1969-1975, 2006.
18
[7] A. Isidori, A tool for semiglobal stabilization of uncertain non-minimum-phase nonlinear systems via output feedback, IEEE Trans. on Automatic Control, 45(10), 1817-1827, 2000. [8] S. Nazrulla, and H. K. Khalil, Robust stabilization of non-minimum phase nonlinear systems using extended high-gain observers, IEEE Trans. on Automatic Control, 56(4), 802-813, 2011. [9] S. M. Hoseini, M. Farrokhi, A. J. Koshkouei, Robust adaptive control of nonlinear nonminimum phase systems with uncertainties, Automatica, 47(2), 348-357, 2011. [10] M. Wang, G. M. Dimirovski, and J. Zhao, H∞ control for a class of non-minimum-phase cascade switched nonlinear Systems, Proc. of 2008 American Control Conference, Seattle, USA, 5080-5085, 2008. [11] M. Oishi and C. Tomlin, Switching in nonminimum phase systems: Applications to a vstol aircraft, Proc. of 2000 American Control Conference, Chicago, USA, 487-491, 2000. [12] M. Benosman, and K.Y. Lum, Output trajectory tracking for a switched nonlinear nonminimum phase system, Proc. of 16th IEEE International Conference on Control Applications, Singapore, 262-269, 2007. [13] G. M. Xie, and L. Wang, Periodic stabilizability of switched linear control systems, Automatica, 45(9), 2141-2148, 2009. [14] H. Yang, B. Jiang, V. Cocquempot, and H. Zhang, Stabilization of switched nonlinear systems with all unstable modes: applications to multi-agent systems, IEEE Transactions on Automatic Control, 56(9), 2230-2235, 2011. [15] H. Yang, V. Cocquempot, and B. Jiang, On stabilization of switched nonlinear systems with unstable modes, Systems & Control Letters, 58(10-11), 703-708, 2009. [16] H. Yang, M. Staroswiecki, B. Jiang, and J. Liu, Fault tolerant cooperative control for a class of nonlinear multi-agent systems, Systems & Control Letters, 60(4), 271-277, 2011. [17] J. Zhao, D. Hill, and T. Liu, Synchronization of complex dynamical networks with switching topology: A switched system point of view, Automatica, 45(11), 2502-2511, 2009. [18] G. Shi, and Y. Hong, Global target aggregation and state agreement of nonlinear multi-agent systems with switching topologies, Automatica, 45(5), 1165-1175, 2009. [19] A. Isidori, Nonlinear Control Systems, 3rd edition, Springer Verlag, London, 1995. [20] J. Yu, and L. Wang, Group consensus in multi-agent systems with switching topologies and communication delays, Systems & Control Letters, 59(6), 340-348, 2010. [21] R. O. Saber, and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. on Automatic Control, 49(9), 1520-1533, 2004.
19
[22] Z. P. Jiang, and Y. Wang, A generalization of the nolinear small-gain theorem for large-scale complex systems, Proc. of the 7th World Congress on Intelligent Control and Automation, Chongqing, China, 1188-1193, 2008. [23] F. Giulietti, L. Pollini, and M. Innocenti, Autonomous formation flight, IEEE Control Systems Magazine, 20(6), 34-44, 2000. [24] X. Mu, W. Zhang, and W. Zhang, An adaptive backstepping design for longitudinal flight path control, Proc. of 7th World Congress on Intelligent Control and Automation, Chongqing, China, 5249-5251, 2008.
20
