Global robust stabilization for nonholonomic systems with dynamic uncertainties

Global Robust Stabilization for Nonholonomic Systems with Dynamic Uncertainties

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Global Robust Stabilization for Nonholonomic Systems with Dynamic Uncertainties Jiangbo Yu, Yan Zhao PII: DOI: Reference:

S0016-0032(19)30756-2 https://doi.org/10.1016/j.jfranklin.2019.10.024 FI 4222

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Journal of the Franklin Institute

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Please cite this article as: Jiangbo Yu, Yan Zhao, Global Robust Stabilization for Nonholonomic Systems with Dynamic Uncertainties, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.10.024

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Global Robust Stabilization for Nonholonomic Systems with Dynamic Uncertainties Jiangbo Yu∗ , Yan Zhao School of Science, Shandong Jianzhu University, Jinan, 250101, China

Abstract In this paper, we investigate the global robust stabilization problem for a class of uncertain chained-form nonholonomic systems. Remarkably, the studied system allows the nonlinear dynamic uncertainties. Characterizing the system of dynamic uncertainties via Sontag’s input-to-state stability (ISS) and ISS-Lyapunov function, we present a novel robust stabilization scheme via output feedback. The input-state scaling transformation is employed in this procedure to design a discontinuous time-invariant control law. A technical lemma is proposed in this paper to handle the dynamic uncertainties combining with the changing supply rates technique. Moreover, a switching control strategy based on the state magnitude is applied to get around the smooth stabilization burden associated with nonholonomic systems. The simulation results illustrate the efficacy of the presented algorithm. Keywords: Nonholonomic systems, dynamic uncertainty, input-to-state stable (ISS), switching strategy, stabilization control 1. Introduction It is well known that the nonholonomic control systems represent a wide class of practical systems with nonholonomic (nonintegrable) constraints, such as the mobile robots, under-actuated ships, n-level trailer systems [1]. Because of its high significance both in theory and application, there has been an increasing attention devoted to the control of nonholonomic systems in last two decades, see, [2]-[9], etc. However, as is known from the Brockett’s ∗

Corresponding author. Tel.:+86 531 86367051. E-mails addresses: [email protected](J.Yu), [email protected](Y.Zhao).

Preprint submitted to Journal of the Franklin Institute

November 4, 2019

condition [10], a nonholonomic system is not stabilizable in the Lyapunov sense by any stationary continuous control laws although it is open-loop controllable. This makes the stabilization control for such class of nonlinear systems become a challenging issue in the control field. In order to overcome this obstruction, many approaches have been proposed for the problem, such as discontinuous feedback, time-varying feedback, and hybrid stabilization, see [1]. The discontinuous feedback stabilization proposed by [3] is an effective tool to yield the asymptotic stabilization with rapid convergence. In past years, the stabilization problem is solvable through this method for several classes of nonholonomic systems, see, for example, [11]-[27], etc. In almost all engineering control systems, the presence of disturbances, model uncertainties and nonlinear model parts are inevitable, which could influence system performance and lead to instability [28]. As a result, the robustness issue for the uncertain nonlinear systems has always been a hot topic, and recently active research on this problem has generated many fruitful results [29]-[34]. As a special class of nonlinear systems, the nonlinear uncertainties could destroy the exponential convergence property or even cause instability phenomenon achieved by previous stabilization algorithms for systems in ideal chained-form nonholonomic systems. In [11], Jiang proposes a robust control scheme for uncertain chained-form nonholonomic systems with bounded time-varying disturbances. Subsequently, the adaptive global stabilization is studied for nonholonomic systems with parameter uncertainty in [12] and [13]. The output feedback adaptive stabilization control design is proposed for nonholonomic chained systems with strong nonlinear drifts in [14]-[16]. The work [17] and [18] give the stabilization schemes for the highorder nonholonomic systems with time-delay. The stabilization results on nonholonomic systems with stochastic noise are reported in [19]-[22]. Gao et al proposes the finite-time stabilization results in [23] and [24]. Recently, the stabilization and tracking control problems for the nonholonomic systems in the presence of nonvanishing disturbances are investigated in [25] and [26]. However, the work mentioned above does not involve the dynamic uncertainties or unmodelled dynamics. It is known that some unmodelled dynamics or external disturbances that are neither controllable nor observable can be modelled by the unknown dynamic uncertainties. For example, the bounded time-varying disturbances may be generated by some neutrally stable dynamic systems [35][36]. Additionally, as shown in Chap.7 in [37], the dynamic uncertainties may cause the parameter drift instability, and hence result in a degradation of system performance. Therefore, it is of interest 2

to design the robust controller such that the nonholonomic systems with dynamic uncertainties are feedback stable. In [38], a global stabilization control strategy is developed for the uncertain chained-form nonholonomic systems in the presence of ISS dynamic uncertainties. The nonlinear small gain argument is invoked to deal with the unmeasured dynamic states for the interconnected systems between the considered nonholonomic system and the ISS dynamic subsystem. This control scheme depends on the system output as well as other measured states. In practice, it is a more challenging problem to design an output feedback controller using only the output information for the uncertain nonholonomic systems under consideration. As our continuing research work, in this paper, we are concerned with the global robust stabilization control via output feedback for a class of uncertain chained-form nonholonomic systems with dynamic uncertainties and nonlinear drifts. Different from the recently reported results in [23][25], the unknown states are reconstructed with the help of a novel high-gain observer whose gain comes from a Riccati matrix differential equation. In summary, the main contributions in our manuscript are highlighted as follows: (i) The robustification tools for nonholonomic system design in literatures fail to be applied into the stabilization control for nonholonomic systems with dynamic disturbances. Using the notion of ISS in [39], for the first time, we put forward a novel robust output feedback stabilization control scheme, and improve the existing results related to robust stabilization control of uncertain chained-form nonholonomic systems. (ii) The unmeasured dynamic disturbances appearing in nonlinear drifts brings about the technical difficulty for the output feedback design and stability analysis. Sufficient local small gain type conditions are explored in this paper to identify for the existence of the nonlinear output feedback stabilization control laws. Moreover, a modified version of changing supply rates technique is applied to handle the unmeasured dynamic states for the interconnected systems between the coupled nonholonomic system and the ISS external disturbance subsystem. The remainder of this paper is organized as follows. In Section 2, we clarify the research motivation and formulate the investigated problem as well as some preliminaries. Section 3 presents the controller design procedure using the backstepping recursive method and a switching control scheme for the singular case. In Section 4, we illustrate our robust control scheme via a nonholonomic system with dynamic uncertainty. Finally, some concluding remarks are contained in Section 5. Notations The following notations are adopted in the paper. N rep3

resents the natural number set. If x(t) is a possibly time-varying vector, then kx(t)k is the Euclidean norm of x at time t, kxk∞ = sup0≤t |x(t)|, and x ∈ L∞ when kxk∞ exists. For a n-dimension vector x = (x1 , · · · , xn ) ∈ Rn , we denote x[i] = (x1 , · · · , xi ) when i = 2, · · · , n − 1. A continuous function α: [0, ∞) → [0, ∞) is said to be a class-K∞ function if it vanishes at the origin, strictly increases, and has no upper bounds. 2. Problem statement and preliminaries In this paper,  η˙      x˙ 0 x˙ i   x˙ n    y

we consider the following uncertain nonholonomic systems = = = = =

q(η, x), u0 + x0 ϕ0 (x0 ), xi+1 u0 + φdi (u0 , x0 , x, η), 1 ≤ i ≤ n − 1, u + φdn (u0 , x0 , x, η), (x0 , x1 )T ,

(1)

where (x0 , xT ) = (x0 , x1 , · · · , xn )T ∈ Rn+1 , u = (u0 , u) ∈ R2 and y ∈ R2 are the system state, control input and system measurable output, respectively; η ∈ Rr means the unmeasurable dynamic uncertainty; ϕ0 (x0 ) is a known smooth nonnegative function, and φdi (·) ∈ R (i = 1, · · · , n) are the (unknown) nonlinear drifts. If the η subsystem and the nonlinear drifts do not exist, i.e., φdi (·) = 0, i = 1, · · · , n, then the system (1) degenerates to the standard chained-form nonholonomic systems in [2]. Remark 1: In the absence of unmeasured η subsystem, the chainedform nonholonomic system (1) has been commonly studied in literatures. For example, Jiang proposes a robust design scheme in [11] for chained-form nonholonomic systems in perturbed form. Then, the other extensions such as parameter uncertainty [12][13], nonlinear drifts [14]-[16], the delayed case [17][18], and stochastic noise [19]-[22], etc, are also discussed for such class of uncertain chained-form nonholonomic systems. Suppose that the system (1) satisfies the following Assumptions 1-3. Assumption 1: For the η-subsystem, there exists a positive definite smooth ISS-Lyapunov function U0 (η), such that αη (kηk) ≤ U0 (η) ≤ αη (kηk),

∂U0 ∂η

q(η, x) ≤ −α0 (kηk) + γ0 (|x1 |),

where αη (·), αη (·), α0 (·), and γ0 (·) are class-K∞ functions. 4

(2) (3)

Assumption 2: For each i = 1, · · · , n, there exist known nonnegative smooth functions ϕi (u0 , x0 , x1 ), and unknown nonnegative smooth functions ψi (η), such that   d φi (u0 , x0 , x, η) ≤ |x1 | ϕi (u0 , x0 , x1 ) + ψi (η) . (4) Assumption 3: For the functions γ0 (·) and ψi (·)(i = 1, · · · , n) in Assumptions 1 and 2, the following local small-gain type condition holds: < +∞, lim sup γ0s(s) 2 lim sup s→0+

s→0+ ψi2 (s) < α0 (s)

+∞, i = 1, · · · , n.

(5) (6)

Remark 2: It is known that a very useful application of ISS is in investigation of stability of cascaded systems [40]. Here, we use the notion of ISS and ISS-Lyapunov functions in Assumption 1 to characterize the system of dynamic uncertainty. Assumption 2 shows that the growth of the nonlinear drifts could allow the system output as well as the unmeasured η. It is a more general growth condition than that in [11] in the case of output feedback. The local small gain type conditions (5) (6) in Assumption 3 are used to deal with the dynamic uncertainty and identify for the existence of the nonlinear output feedback stabilization control laws. The following lemmas are used in the output feedback control design. Lemma 1[41]: For any x, y ∈ Rn , any scalar  > 0, any positive definite matrix Q ∈ Rn×n , the following inequality holds 2 xT y ≤ −1 xT Q x +  y T Q−1 y.

Lemma 2: If α e, α ∈ K∞ satisfies lim sup s→0+

then lim sup s→0+

α e(s) < ∞, α(s)

α em (s) < ∞, m ∈ N. α(s)

(7)

(8)

(9)

e(s) Proof: In view of lim sup αα(s) < ∞, there exist positive constants c1 and

s1 , such that

s→0+

α e(s) ≤ c1 α(s), s ∈ [0, s1 ]. 5

(10)

Considering α e(s) ≥ 0 a continuous function on [0, s1 ], there exists a maximum value α e(¯ s), s¯ ∈[0, s1 ]. As a result, i.e.,

which shows

α e2 (s) = α e(s) · α e(s) ≤ α e(¯ s)·c1 α(s), s ∈ [0, s1 ],

(11)

α e2 (s) ≤ c1 α e(¯ s)·α(s), s ∈ [0, s1 ],

(12)

lim sup s→0+

α e2 (s) < ∞. α(s)

(13)

Similarly, the case of m > 2 also holds in the same way. Remark 3: It is shown that the high-order gain functions related to the unmeasured dynamic disturbances will appear in the case of output feedback. Assuming the gain functions satisfy some kind of local small gain type condition (30) in [42], the work in [42] investigates the decentralized adaptive output feedback stabilization for large scale stochastic nonlinear systems. Motivated by this assumption in [42], we propose a more general lemma 2 in this paper to handle these high-order gain functions and derive an input-output dissipation inequality suitable for the stability analysis. 3. Output feedback control design The design process is divided into two separate stages. First, we design the control law u0 to stabilize the x0 -subsystem. Then the control u is chosen such that the rest states converge to zero. 3.1. State scaling We first consider the case of x0 (t0 ) 6= 0, and the case for x0 (t0 ) = 0 will be discussed later. Here, take u0 as follows u0 = −λ0 x0 − x0 ϕ0 (x0 ),

(14)

where λ0 > 0 is a design parameter. Using the first control law u0 in (14), we establish the following lemma, whose proof can be found in [11]. Lemma 3: Applying the control signal (14) into the x0 -subsystem in (1), if x0 (t0 ) 6= 0, then the state x0 (t) asymptotically converges to zero as t tends to infinity but doesn’t cross zero at any time instant. 6

Next, we will design the control law u for the x-subsystem. Thanks to x0 (t) 6= 0 as t ≥ t0 , we introduce the input-state scaling transformation: xi zi = n−i , i = 1, · · · , n, (15) u0 then we get  z˙i = zi+1 − (n − i) uu˙ 00 zi + φ¯di (u0 , x0 , x, η), 1 ≤ i ≤ n − 1, z˙n = u + φ¯dn (u0 , x0 , x, η),

(16)

φdi (u0 ,x0 ,x,η) , i = 1, · · · , n. u0n−i drifts φ¯di (u0 , x0 , x, η) (i = 1, · · · , n),

with φ¯di (u0 , x0 , x, η) =

For the nonlinear the following lemma holds true. Lemma 4: For each φ¯di (u0 , x0 , x, η) (i = 1, · · · , n), we have the following upper bounds:   d i−1 ¯ φi (u0 , x0 , x, η) ≤ |u0 ||z1 | ϕi (u0 , x0 , x1 ) + ψi (η) . (17) Proof: According to Assumption 2, it can be verified that φd (u , x , x, η) 0 0 φ¯di (u0 , x0 , x, η) ≤ i n−i u0  |x1 |  ϕ (u , x , x ) + ψ (η) ≤ i 0 0 1 i |un−i 0 |   = |u0i−1 ||z1 | ϕi (u0 , x0 , x1 ) + ψi (η) .

(18)

The proof is completed.

  φ¯d1 (·)  0 In−1 .  ¯ d (u0 , x0 , x, η) =  Take the notations of A = , Φ  .. , 0 0 φ¯dn (·) b = [0, · · · , 1]T , and L = diag{n − 1, · · · , 1, 0}, then the system (16) can be rewritten into the compact form:  u˙ 0  ¯ d (u0 , x0 , x, η). z˙ = A − L z + b u + Φ (19) u0 

Construct the following full-order observer:  u˙ 0  zˆ˙ = A − L zˆ + b u − P CC T zˆ, u0 7

(20)

with C = [1, 0, · · · , 0]T and the dynamic gain P = (pij )n×n with pij = pji (i, j = 1, · · · , n) coming from ( T    P˙ = P A − uu˙ 00 L + A − uu˙ 00 L P − P CC T P + In , (21) P (0) = P0 > 0. It is noted that the following lemma ensures that the observer (20) and (21) makes sense, whose proof can be found in [15]. Lemma 5[15]: For any continuous function µ0 (t), there exist two strictly positive real numbers pmin and pmax such that the unique solution P (t) of the following matrix differential equation:  P˙ = P (A − µ0 (t)L)T + (A − µ0 (t)L)P − P CC T P + In , (22) P (0) = P0 > 0, satisfies pmin In ≤ P (t) ≤ pmax In , t ≥ 0. Define the error ε = z − zˆ, and it follows from (19) and (20) that   u˙ 0 ¯ d (u0 , x0 , x, η). ε˙ = A − L − P CC T ε + z1 P C + Φ u0

(23)

Lemma 6: For the error system (23), choose Vε (ε, P ) = εT P −1 (t)ε, then we have the following conclusion: 1 V˙ ε (ε, P ) ≤ − εT P −2 ε + z12 ϕz1 (x0 , x1 ) + 2p2max 2

n X

ψi4 (η),

(24)

i=1

where ϕz1 (x0 , x1 ) is a known smooth function depending on (x0 , x1 ). z }| ˙ { Proof: In light of P −1 (t) = −P −1 (t)P˙ (t)P −1 (t), the time derivative of Vε (ε, P ) along (21) and (23) satisfies ¯ d (u0 , x0 , x, η) − εT P −2 ε. V˙ ε = −ε21 + 2ε1 z1 + 2εT P −1 Φ

(25)

With the help of the completion of squares, the following holds 2ε1 z1 ≤ ε21 + z12 ,

(26)

and by means of Lemma 1, we have ¯ d (u0 , x0 , x, η) ≤ 1 εT P −2 ε + 2p2 kΦ ¯ d (u0 , x0 , x, η)k2 . 2εT P −1 Φ max 2 8

(27)

Using Lemma 4 and the completion of squares again, it can be verified that   d 2 2 2 2n−2 2 ¯ kΦ (·)k ≤ 2z1 ϕ1 (·) + · · · + u0 ϕn (·) + z14 + · · · + z14 u4n−4 0 +

n X

ψi4 (η).

(28)

i=1

As a result, we have  1 T −2 2 ˙ ϕ2n (·)) Vε ≤ − ε P ε + z1 1 + 4p2max (ϕ21 (·) + · · · + u2n−2 0 2 n  X +2p2max z12 + · · · + 2p2max z12 u04n−4 + 2p2max ψi4 (η).

(29)

i=1

ϕ2n (·)) +2p2max z12 + Denote ϕz1 (x0 , x1 ) = 1 + 4p2max (ϕ21 (·) + u20 ϕ22 (·) + · · · + u2n−2 0 , and the proof is completed. · · · + 2p2max z12 u4n−4 0 3.2. Backstepping design We now use the backstepping technique to design the control law. For the subsequent feedback design, the system for control design is rewritten as follows:  u˙ 0 d  z˙1 = zˆ2 + ε2 − (n − 1) u0 z1 + φ¯1 (u0 , x0 , x, η), zˆ˙ i = zˆi+1 − (n − i) uu˙ 00 zˆi − CiT P CC T zˆ, 2 ≤ i ≤ n − 1, (30)  ˙ T T zˆn = u − Cn P CC zˆ.

Step 1: Define ξ1 = z1 and ξ2 = zˆ2 − α1 , where α1 is the first virtual control law, and ξ2 is an error variable. Choose the Lyapunov function: 1 V1 = Vε + ξ12 , 2

(31)

and in view of Lemma 6 and (30), the time derivative of V1 verifies n X 1 V˙ 1 ≤ − εT P −2 ε + z12 ϕz1 (x0 , x1 ) + 2p2max ψi4 (η) 2 i=1   u˙ 0 + ξ1 zˆ2 + ε2 − (n − 1) z1 + φ¯d1 (u0 , x0 , x, η) . u0

9

(32)

It follows from Lemma 1 and Lemma 4 that 1 2 2 p ξ , ν > 0, 4ν max 1 1 ξ1 φ¯d1 (u0 , x0 , x, η) ≤ ξ12 ϕ1 (u0 , x0 , x1 ) + ξ14 + ψ12 (η). 4 ξ1 ε2 ≤ νεT P −2 ε +

Let ϕ¯1 (x0 , x1 ) =

1 2 p 4ν max

(33) (34)

+ ϕz1 (·) + ϕ1 (·) + 41 ξ12 , and then, we have

 1 u˙ 0  T −2 ˙ V1 ≤ −( − ν)ε P ε + ξ1 α1 + z1 ϕ¯1 (x0 , x1 ) − (n − 1) z1 2 u0 n X ψi4 (η) + ψ12 (η) + ξ1 ξ2 . (35) +2p2max i=1

In view of ξ1 = z1 , select the virtual control law α1 in the form of α1 (x0 , z1 ) = −l1 (x0 , ξ1 )ξ1 − z1 ϕ¯1 (x0 , x1 ) + (n − 1)

u˙ 0 z1 , u0

(36)

where l(x0 , ξ1 ) dependent on (x0 , ξ1 ) is a smooth positive function to be designed later. Then, a direct substitution into (35) leads to 1 V˙ 1 ≤ −( − ν)εT P −2 ε − l1 (x0 , ξ1 )ξ12 + ξ1 ξ2 + 2p2max 2

n X

ψi4 (η) + ψ12 (η). (37)

i=1

Step i (2 ≤ i ≤ n): Assume that, in Step i − 1, we have designed the virtual control αj (x0 , z1 , zˆ[j] , P ) such that with ξj = zˆj − αj−1 (2 ≤ j ≤ i), the derivative of 1 2 1 Vi−1 = Vε + ξ12 + · · · + ξi−1 2 2

(38)

satisfies i−1 1    X T −2 2 ˙ Vi−1 ≤ − − (i − 1)ν ε P ε − l1 (x0 , ξ1 ) − (i − 1) ξ1 − lj ξj2 2 j=2

+ξi−1 ξi +

2p2max

n X i=1

ψi4 (η) + (i − 1)ψ12 (η),

with some design constants lj > 0 (2 ≤ j ≤ i − 1). 10

(39)

In the sequel, one shows that a similar property of (39) holds in Step i. Let ξi+1 = zˆi+1 − αi , and we consider the function 1 Vi = Vi−1 + ξi2 . 2

(40)

To be first, according to (30), we have the following calculations n X u˙ 0 ∂αi−1 ∂αi−1 ξ˙i = ξi+1 + αi − (n − i) zˆi − CiT P C zˆ1 − p˙kl − x˙ 0 u0 ∂p ∂x kl 0 k,l=1

− −

i−1 X ∂αi−1 j=2

∂αi−1 ∂αi−1 ∂αi−1 u˙ 0 zˆ˙ j − zˆ2 − ε2 + (n − 1) z1 ∂ zˆj ∂z1 ∂z1 ∂z1 u0

∂αi−1 ¯d φ (u0 , x0 , x, η). ∂z1 1

(41)

Using Lemma 1 and Lemma 4 again, we have −ξi

 ∂α 2 1 ∂αi−1 i−1 ε2 ≤ νεT P −2 ε + p2max ξi2 , ∂z1 4ν ∂z1

(42)

and 1 2  ∂αi−1 2 2 1 2  ∂αi−1 2 2 ∂αi−1 ¯d 2 φi (·) ≤ ξ1 + ξi ϕ1 (·) + ξi z1 + ψ12 (η). (43) −ξi ∂z1 4 ∂z1 4 ∂z1   2  1 2 1 2 1 2 + (·) + , Let the smooth function ϕ¯i (x0 , z1 , zˆ[i−1] ) = ∂α∂zi−1 p ϕ z 4ν max 4 1 4 1 1 and a direct substitution leads to i−1 1    X V˙ i ≤ − − iν εT P −2 ε − l1 (x0 , ξ1 ) − i ξ12 − lj ξj2 + ξi ξi+1 2 j=2

n  X ∂αi−1 ∂αi−1 u˙ 0 p˙kl − x˙ 0 +ξi αi + ξi−1 − (n − i) zˆi − CiT P C zˆ1 − u0 ∂p ∂x kl 0 k,l=1

−

 ∂αi−1 ∂αi−1 u˙ 0 zˆ˙ j − zˆ2 + (n − 1) z1 + ξi ϕ¯i (x0 , z1 , zˆ[i−1] ) ∂ zˆj ∂z1 ∂z1 u0

i−1 X ∂αi−1 j=2

+2p2max

n X

ψi4 (η) + iψ12 (η).

(44)

i=1

11

Choose the virtual control law ∂αi−1 u˙ 0 zˆi + CiT P C zˆ1 + x˙ 0 u0 ∂x0 n i−1 X X ∂αi−1 ∂αi−1 ˙ ∂αi−1 u˙ 0 + p˙kl + zˆj − (n − 1) z1 ∂p ∂ z ˆ ∂z u kl j 1 0 j=2 k,l=1

αi (x0 , z1 , zˆ[i] , P ) = −li ξi − ξi−1 + (n − i)

+

∂αi−1 zˆ2 − ξi ϕ¯i (x0 , z1 , zˆ[i−1] ), ∂z1

(45)

and one get i    1 X T −2 2 ˙ − iν ε P ε − l1 (x0 , ξ1 ) − i ξ1 − lj ξj2 + ξi ξi+1 Vi ≤ − 2 j=2

+2p2max

n X

ψi4 (η) + iψ12 (η).

(46)

i=1

Specially, when i = n, we obtain the control law in the form of u = αn (x0 , z1 , zˆ, P ) = −ln ξn − ξn−1 + +

CnT P C zˆ1

n n−1 X X ∂αn−1 ∂αn−1 ∂αn−1 ˙ + p˙kl + x˙ 0 + zˆj ∂pkl ∂x0 ∂ zˆj j=2 k,l=1

∂αn−1 ∂αn−1 u˙ 0 zˆ2 − (n − 1) z1 − ξn ϕ¯n (x0 , z1 , zˆ[n−1] ), ∂z1 ∂z1 u0

(47)

which guarantees the Lyapunov function n

1X 2 Vn = Vε + ξ 2 i=1 i

(48)

verify n 1    X T −2 2 ˙ Vn ≤ − − nν ε P ε − l1 (x0 , ξ1 ) − n ξ1 − li ξi2 2 i=2

+2p2max

n X

ψi4 (η) + nψ12 (η).

i=1

12

(49)

Next, we will deal with the dynamic uncertainties ψi (η) in (49) by means of the changing supply rates technique [43]. To this end, we construct another ISS-Lyapunov function in the form of Z

U 0 (η) =

U0 (η)

ρ(s) ds,

(50)

0

where ρ(·) is a positive continuous function. In accordance with Assumption 1, the time derivative of U 0 (η) satisfies 1 U˙ 0 (η) ≤ ρ ◦ αη ◦ α0−1 ◦ 2γ0 (|x1 |)γ0 (|x1 |) − ρ ◦ αη (kηk)α0 (kηk). (51) 2 Considering the local small-gain type condition in Assumption 3: lim sup s→0+

ψi2 (s) < +∞, i = 1, · · · , n, α0 (s)

(52)

according to Lemma 2, we have lim sup s→0+

ψi4 (s) < +∞, i = 1, · · · , n. α0 (s)

(53)

Together with (52) and (53), there is a desired function ρ such that n n o X 1 2 ρ ◦ αη (kηk)α0 (kηk) ≥ max 2pmax ψi4 (η), nψ12 (η) . 8 i=1

(54)

Take the Lyapunov function n

V = Vε +

1X 2 ξ + U 0 (η), 2 i=1 i

(55)

then V˙

   1 − nν εT P −2 ε − ρ ◦ αη (kηk)α0 (kηk) − l1 (x0 , ξ1 ) − n ξ12 2 4 n X − li ξi2 + ρ ◦ αη ◦ α0−1 ◦ 2γ0 (|x1 |) γ0 (|x1 |). (56)

≤ −

1

i=2

13

< +∞ in Assumption 3 and (14), there exists a In view of lim sup γ0s(s) 2 s→0+

smooth nondecreasing function γ b0 (s), such that

b0 (x0 , ξ1 ). γ0 (|x1 |) = γ0 (|u0n−1 ξ1 |) ≤ ξ12 γ

Therefore,

(57)

ρ ◦ αη ◦ α0−1 ◦ 2γ0 (|x1 |)γ0 (|x1 |) ≤ ρ ◦ αη ◦ α0−1 ◦ 2γ0 (|ξ1 |) γ b0 (x0 , ξ1 ) ξ12 . (58)

Furthermore, the following calculations hold true: V˙

1



n

X 1 − nν ε P ε − ρ ◦ αη (kηk)α0 (kηk) − ≤ − li ξi2 2 4 i=2   −1 b0 (x0 , ξ1 ) ξ12 . − l1 (x0 , ξ1 ) − n − ρ ◦ αη ◦ α0 ◦ 2γ0 (|x1 |) γ T

Take the constants 0 < ν ≤

−2

1 , 4n

(59)

li ≥ 1(i = 2, · · · , n), and the smooth function

b0 (x0 , ξ1 ), l1 (x0 , ξ1 ) ≥ n + 1 + ρ ◦ αη ◦ α0−1 ◦ 2γ0 (|x1 |) γ

(60)

and we have

n

V˙

X 1 1 ≤ − εT P −2 ε − ρ ◦ αη (kηk)α0 (kηk) − ξi2 . 4 4 i=1

(61)

This together with (55) and Lemma 3 guarantees that the signals of (x0 (t), ε(t), η(t), ξ(t)) in closed-loop are bounded. Consequently, it is known from ε1 ∈ L∞ , ξ1 ∈ L∞ , and ε1 = z1 − zˆ1 that zˆ1 ∈ L∞ . Furthermore, in view of (36), we conclude that α1 is bounded. This together with ξ2 = zˆ2 − α1 and ξ2 ∈ L∞ implies zˆ2 ∈ L∞ . Using a recursive manner, it follows that zˆi (i = 3, · · · , n) are bounded. In view of xi = zi u0n−i (i = 1, · · · , n) and u0 ∈ L∞ , we know xi ∈ L∞ , i = 1, · · · , n. This shows that the finite time escape will not happen. Therefore, it is natural that [0, ∞) is its maximal interval of the existence and uniqueness. Additionally, from LaSalle’s Invariant Theorem [37], it further concludes that (ε(t), η(t), ξ(t)) converge to the origin as t tends to infinity. As a result, we know from ξ1 = z1 and lim ξ1 (t) = 0 that lim z1 (t) = 0, and then t→∞

t→∞

lim zˆ1 (t) = 0. In view of (36), it is concluded that α1 is convergent as

t→∞

t → ∞, which in turn implies lim zˆ2 (t) = 0. In the similar way, we have t→∞

14

lim zˆi (t) = 0(i = 3, · · · , n). This together with ε = z − zˆ and lim εi (t) =

t→∞

t→∞

0(i = 1, · · · , n) implies lim zi (t) = 0(i = 1, · · · , n). Finally, in view of xi = t→∞

zi un−i (i = 1, · · · , n) in (15) and u0 ∈ L∞ , it is known that 0 lim xi (t) = 0, i = 1, · · · , n.

t→∞

(62)

The above analysis is summarized into the following theorem: Theorem 1: Under Assumptions 1-3, if the designed control laws (14) and (47) are applied to (1) with suitable function l1 (x0 , ξ1 ) in (60) and design 1 , li ≥ 1(i = 2, · · · , n), the global uniform parameters λ0 ≥ 1, 0 < ν ≤ 4ν asymptotic stability of the closed-loop system is achieved for x0 (t0 ) 6= 0, and moreover,   lim |η(t)| + |x0 (t)| + |x(t)| = 0. (63) t→∞

Remark 4: The input-state scaling transformation is used in this paper to achieve the robust stabilization control for system (1) in the presence of dynamic uncertainty. As a consequence, the η subsystem is driven by the coupled ξ1 and u0 , which makes the classical changing supply rates technique in [43] fail to directly apply here. The key issue is how to compensate the unknown dynamic disturbances η using the gain function γ0 in supply pair (α0 , γ0 ). We remove this burden by means of variable separation principle and the established Lemma 2. This permits the chained-form uncertain nonholonomic systems to reject a more broader class of nonlinear disturbances than the existing results such as [11]-[22], etc. In the next subsection, a switching strategy based on the magnitude of the state of the x0 -subsystem is derived to prevent the singular phenomenon of uncontrollability when x0 (t0 ) = 0. It is also shown that a finite time escape is avoided before the given switching time. 3.3. Switching control strategy In case of x0 (t0 ) = 0, we consider the constant control u0 = c, c > 0

(64)

before switching to drive the state x0 away from zero. Since x0 (t0 ) = 0, then x˙ 0 (t0 ) = c + x0 (t0 )ϕ0 (x0 (t0 )) = c. 15

(65)

Considering x0 (t0 )ϕ0 (x0 (t0 )) = 0 < c and the smoothness of ϕ0 (x0 ), there exists a small neighborhood Ω of x0 (t0 ) = 0 such that x0 ϕ0 (x0 ) < c. Suppose x∗0 satisfies x∗0 ϕ0 (x∗0 ) = c. That is, x0 (t) increases in Ω until x0 ϕ0 (x0 ) = c. Then, 0 < x0 (t) < x∗0 . During the time period satisfying x0 (t) ≤ x∗0 , applying the similar procedure in Subsection 3.2, we design a backstepping-based output feedback control u(x0 , x) such that the states x0 , x and η are bounded with u0 defined as (64). When x0 (t) = x∗0 , the constant control u0 and the output feedback law u(x0 , x) are switched into (14) and (47), respectively. Using the switching control strategy when x0 (t0 ) = 0, we have the Theorem 2. Theorem 2: Under Assumptions 1-3, if the above switching control strategy and the control laws (14) and (47) are applied to system (1), with an appropriate choice of the design function l1 (x0 , ξ1 ) in (60) and the parameters 1 λ0 ≥ 1, 0 < ν ≤ 4ν , li ≥ 1(i = 2, · · · , n), then, the uncertain system (1) is uniformly globally asymptotically regulated at the origin whenever x0 (t0 ) = 0. 4. Simulation study 4.1. An illustrative example In this section, we will verify our proposed control strategies using an illustrative example. We consider the following three-dimensional nonholonomic system with dynamic uncertainties:    η˙ = −k η + x1 , k > 0,  x˙ 0 = u0 + x0 , (66) x˙ 1 = x2 u0 + d(t) x1 + η x21 ,    x˙ 2 = u,

where η is a dynamic disturbance satisfying ISS with state η and input x1 , d(t) is some bounded external disturbance satisfying |d(t)| ≤ dmax for all t ≥ 0 where dmax > 0 is a known positive constant. Remark 5: It is noted that using the coordinates transformation, the kinematics of the unicycle-type mobile robot can be transformed into the

16

chained-form nonholonomic system such as   x˙ 0 = u0 , x˙ 1 = x2 u0 ,  x˙ 2 = u.

(67)

In [11], Jiang proposes a robust exponential regulation controller for such class of nonholonomic systems in perturbed chained form with input- and state-driven disturbances. Subsequently, for such class of chained-form nonholonomic systems, there are some improvements in several directions, such as linear parameter uncertainty [12][13], strongly nonlinear disturbances and drift terms [14]-[16], the inputs saturation nonlinearity [23], nonvanishing disturbances [25][26], etc. To the authors’ knowledge, there are no results reported in literatures related to such class of nonholonomic systems with ISS dynamic uncertainties. Here, we use a disturbed version of the ideal chained-form nonholonomic systems (66) to illustrate the proposed robust control scheme for the nonholonomic systems in the presence of ISS dynamic uncertainties. Assuming t0 = 0. As in Section 3, we first consider the case of x0 (0) 6= 0. Take u0 = −λ0 x0 − x0 with λ0 > 0, then, x˙ 0 = −λ0 x0 , and x0 (t) = x0 (0) e−λ0 t , u0 (t) = −(λ0 + 1)x0 (0)e−λ0 t . Let z1 = ux10 , z2 = x2 , and we get  z˙1 = z2 + d(t)z1 + η z12 u0 − uu˙ 00 z1 , (68) z˙2 = u,

which can be rewritten into the following compact form:  u˙ 0  ¯ d (u0 , x0 , x, η) + b u z˙ = A − L z + Φ (69) u0       0 1 d(t)z1 + η z12 u0 0 d ¯ ,b= , and with A = , Φ (u0 , x0 , x, η) = 0 1 0 0 L = diag{1, 0}. Design the observer like (20) and (21). Let the estimation error ε = z − zˆ, and we have   u˙ 0 T ¯ d (u0 , x0 , x, η). ε˙ = A − L − P CC ε + P CC T z + Φ (70) u0 Similar to the calculations in Lemma 6, the time derivative of the Lyapunov function Vε (ε) = εT P −1 (t)ε verifies 1 V˙ ε ≤ − εT P −2 (t)ε + z12 (5 + z16 u40 ) + η 4 . 2 17

(71)

Using the proposed control design scheme in Section 3, we obtain the following control law ∂α1  u˙ 0  ∂α1 u = −l2 ξ2 − ξ1 + C2T P CC T zˆ + zˆ2 − z1 + u˙ 0 ∂ξ1 u0 ∂u0  2  2 ∂α1 ∂α1 1 (72) − ξ2 z14 u40 − ξ2 4ν ∂ξ1 ∂ξ1 1 2 pmax ξ1 + uu˙ 00 z1 − 14 ξ15 u20 −5z1 −z17 u40 , C T = [1 0], with α1 = −l1 (x0 , x1 )ξ1 −ξ1 − 4ν C2T = [0 1]. When x(0) = 0, we use the switching strategy in Subsection 3.3 to design the control laws u0 and u. For simulation purposes, the design function l1 (x0 , x1 ) and the external disturbance d(t) are chosen as l1 (x0 , x1 ) = 21 u20 (x0 ) + 14 x21 u20 (x0 ) + 1.5, d(t) = sin(0.5 t). The design parameters and initial conditions are taken as follows: c = 1, λ0 = 1, l2 = 1, k = 2.5, ν = 14 , dmax = 1, and x0 (0) = 0, x1 (0) = 0, x2 (0) = 1, η(0) = 1, z1 (0) = −1, z2 (0) = 1, zˆ1 (0) = 2, zˆ2 (0) = 1, ε1 (0) = 1, ε2 (0) = 0, and P (0) = I2 . The simulation results are shown in Figs.1-3. It is clear that all of the system states as well as the control laws converge to zero. It can be seen that the proposed discontinuous switching control strategy gets around the smooth stabilization burden associated with the nonholonomic systems, which shows the proposed output feedback control scheme works well for the nonholonomic systems with ISS dynamic uncertainties. Remark 6: As one reviewer points out, the design parameters may effect the control performance. Using the state x0 as an example, the convergence speed of state x0 depends on the design parameters λ0 and c. Specifically, the switching time changes with different parameter c’s, and the larger the constant c, the smaller the switching time ts is (see Table 1 and Fig.4). Furthermore, according to switching control strategy, a larger λ0 may accelerate the convergence rate after the time instant ts .

Table 1: The switching time ts at different constant c

the value of c 0.50 1.50 2.50

switching time ts 1.1433 0.5257 0.3485

18

x0

1

x1

0.05

switching time ts

0

0.5

-0.05 -0.1

0 -0.15 0

5

10

0

5

time/s

10

time/s

x2

1

1

0.5 0.5 0 0

-0.5 0

5

10

15

0

5

time/s

10

15

time/s

Fig.1 The profiles of states in closed-loop (66)-(72) using our method. 3

1.5

2

1

1

0.5

switching time ts

0

0

-1

-0.5

-2

-1 0

5

10

15

0

5

time/s

10

15

10

15

time

1.5 3 1 2

0.5

1

0

0

-0.5

-1

-1 0

5

10

15

0

time/s

5

time/s

Fig.2 The profiles of estimates, errors, and gains in closed-loop (66)-(72) using our method. 19

1 0 -1

1

0

0.5

time/s

1

0

-1 0

5

time/s

10

15

10

15

5 0 -5

0

-10

-10

-15

-20 0

0.5

-20 0

5

time/s time/s

1

Fig.3 The profiles of control laws in closed-loop (66)-(72) using our method.

20

1.2 state x0 at different c's using switching scheme

1 c=0.5 0.8

0.6

c=1.5

0.4

0.2

0

c=2.5

0

2

4

6

8

10

time/s

Fig.4 The profiles of state x0 at different c’s using our switching scheme.

21

4.2. Comparison In this subsection, the output feedback control approach developed in [25] is applied into the same example (66) for comparison. By means of the design scheme in [25], the control u0 is taken as u0 = −λ0 x0 − x0 when x(0) 6= 0. Correspondingly, it is verified that uu˙ 00 = −λ0 , β = −λ0 and φ¯0 (t, x0 ) = 0. Let z1 = ux10 and z2 = x2 , and one get 

z˙1 = z2 + d(t)z1 + η z12 u0 − z˙2 = u,

u˙ 0 z, u0 1

(73)

which is rewritten into the following compact form: z˙ = Az + Φd (u0 , x0 , z, η) + b u (74)       2 u 0 λ0 1 d(t)z + η z 0 1 1 d ¯ (u0 , x0 , z, η) = , and b = . with A = ,Φ 0 1 0 0 Like the estimator design in [25], we design the following estimator:  ξ˙0 = A0 ξ0 + Kz1 , (75) υ˙ = A0 υ + bu,       ξ01 υ1 λ0 − k1 1 T where ξ0 = , υ = , A0 = A − KC = , K = υ2 − k2 0   ξ02  k1 1 , C = , and k1 , k2 are design parameters chosen to make A0 a k2 0 Hurwitz matrix. In view of the control direction known a priori in (66), we define zˆ = ξ0 +υ, and then the following holds zˆ˙ = A0 zˆ + Kz1 + bu.

(76)

Let ε = z − zˆ, and one obtains the error dynamic system ¯ d (u0 , x0 , z, η). ε˙ = A0 ε + Φ

(77)

Now, we obtain the augmented system for feedback design as follows:  z˙1 = zˆ2 + 2 − uu˙ 00 z1 + d(t)z1 + ηz12 u0 , (78) zˆ˙ 2 = −k2 zˆ1 + u + k2 z1 . 22

Using the design scheme in [25], the real control law is designed as follows: ∂α1 ∂α1 ∂α1 u = −l2 ξ2 − ξ1 + k2 zˆ1 − k2 ξ1 − λ0 x0 + zˆ2 + λ0 ξ1 ∂x0 ∂ξ1 ∂ξ1 ρ2  ∂α1 2 1  ∂α1 2 2 1  ∂α1 2 4 2 − ξ2 − ξ2 dmax − ξ2 ξ1 u0 4 ∂ξ1 2 ∂ξ1 2 ∂ξ1

(79)

with α1 = −l1 (x0 , ξ1 )ξ1 + uu˙ 00 z1 −2ρ1 λ2max (Q)d2max ξ1 −ρ21 λ4max (Q)u40 ξ17 −dmax ξ1 − 1 ξ − 14 ξ15 u20 , and Q = QT > 0 satisfying AT 0 Q + QA0 = −2I. 2 1 When x(0) = 0, the switching strategy based on the magnitude of the state x0 in [25] is invoked to design the control laws u0 and u. Using the control scheme in [25], the simulation results plotted in the Figs.5-6. For comparison, we choose the same initial conditions x0 (0) = 0, x1 (0) = 0, x2 (0) = 1, η(0) = 1, zˆ1 (0) = 2, zˆ2 (0) = 1, ε1 (0) = 1, ε2 (0) = 0 with the same design function l1 (x0 , x1 ) = 21 u20 (x0 ) + 14 x21 u20 (x0 ) + 1.5, the external disturbance d(t) = sin(0.5 t), and the design parameters c = 1, λ0 = 1, l2 = 1, k = 2.5 with k1 = 2, k2 = 1, λmax (Q) = 3.62. From the profiles shown in Figs.5-6 we can see that the two kind of discontinuous feedback controllers have satisfactory control performance. It can also be seen that the control strategy in [25] have faster convergence rate for the observer states as well as the estimate errors. However, it does require more control efforts than our control scheme. In fact, the estimate states in [25] have faster convergence rate by increasing the static gain K in (76), which in turn yields the larger control gains. 5. Conclusions This paper investigated the global stabilizing control for a class of uncertain nonholonomic systems. The considered system permits much complicated nonlinear drifts which contain the unmeasured states from the unmodeled dynamics. Using the notions of ISS and ISS-Lyapunov function, we propose a novel output feedback robust control methodology. A switching control strategy is employed to handle the singular case. The efficacy of the proposed robust control scheme is demonstrated by a simulation example. The proposed approach can be extended to a larger class of chained-form nonholonomic systems with more general disturbances and uncertainties, such as the parameter uncertainties, and nonvanishing disturbances, etc. Some more complex models, such as the nonholonomic systems with the L´ evy noise or Markov switching are also interesting research directions. 23

1.5 x0

1

0.5

6

0

4

-0.5

2

-1

0.5

0 x1

-1.5 0

x2

-2

-2 0

5

10

0

time/s

5

10

0

time/s

1.5

10

15

time/s 1.5

6

1

4

1

5

0.5 2 0

0.5 0 0

-0.5

-2 0

5

10

-1 0

time/s

5

10

0

time/s

5

10

time/s

Fig.5 The profiles of states, estimates, and errors in closed-loop (73)-(79) using the control method in [25]. 2 1 0 -1 -2 0

5

10

15

10

15

time/s

20 0 -20 -40 -60 0

5

time/s

Fig.6 The profiles of control laws in closed-loop (73)-(79) using the control method in [25]. 24

Acknowledgment The authors would like to express their highly appreciation to the associate editor and the reviewers for their insightful and constructive comments in improving the quality of this paper. This work was supported by the National Natural Science Foundation of China under Grant 61803228 and 61803229, and the China Postdoctoral Science Foundation under Grant 2017M612271. The first author also thanks the School of Control Science and Engineering when he is currently doing his postdoctoral research in Shandong University. References References [1] I. Kolmanovsky, N.H. Mcclamroch, Development in nonholonomic control problems, IEEE Control Systems 15(6)(1996) 20-36, 1996. [2] R.M. Murray, S. Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Transactions on Automatic Control 38(5)(1993) 700716. [3] A. Astolfi, Discontinuous control of nonholonomic systems, Systems & Control Letters 27(1)(1996) 37-45. [4] S. Mobayen, S. Javadi, Disturbance observer and finite-time tracker design of disturbed third-order nonholonomic systems using terminal sliding mode, Journal of Vibration and Control 23(2)(2017) 181-189. [5] O. Mofid, S. Mobayen, Adaptive sliding mode control for finite-time stability of quad-rotor UAVs with parametric uncertainties, ISA Transactions 72(1)(2018) 1-14. [6] Y.Q. Wu, B. Wang, G.D. Zong, Finite-time tracking controller design for nonholonomic systems with extended chained form, IEEE Transactions on Circuits and Systems-II: Express Briefs 52(11)(2005) 798-802. [7] J. Fu, T.Y. Chai, C.Y. Su, Y. Jin, Motion/force tracking control of nonholonomic mechanical systems via combining cascaded design and backstepping, Automatica 49(12)(2013) 3682-3686. 25

[8] H.B. Du, G.H. Wen, X.H. Yu, S.H. Li, M.Z.Q. Chen, Finite-time consensus of multiple nonholonomic chained-form systems based on recursive distributed observer, Automatica 62(2)(2015) 236-242. [9] S. Shi, S.Y. Xu, X. Yu, Y.M. Li, Z.Q. Zhang, Finite-time tracking control of uncertain nonholonomic systems by state and output feedback, International Journal of Robust and Nonlinear Control 28(2)(2018) 19421959. [10] R.W. Brockett, Asymptotic stability and feedback stabilization, Differential Geometric Control Theory 27(1)(1983) 181-191. [11] Z.P. Jiang, Robust exponential regulation of nonholonomic systems with uncertainties, Automatica 36(2)(2000) 189-209. [12] K.D. Do, J. Pan, Adaptive global stabilization of nonholonomic systems with strong nonlinear drifts, Systems & Control Letters 46(3)(2002) 195205. [13] S.S. Ge, Z.P. Wang, T.H. Lee, Adaptive stabilization of uncertain nonholonomic systems by state and output feedback, Automatica 39(8)(2003) 1451-1460. [14] Y.G. Liu, J.F. Zhang, Output feedback adaptive stabilization control design for nonholonomic systems with strong nonlinear drifts, International Journal of Control 78(7)(2005) 474-490. [15] Z.R. Xi, G.Feng, Z.P.Jiang, D.Z.Cheng, Output feedback exponential stabilization of uncertain chained systems, Journal of the Franklin Institute 344(1)(2007) 36-57. [16] Y.Q. Wu, Y. Zhao, J.B. Yu, Global asymptotic stability controller of uncertain nonholonomic systems, Journal of the Franklin Institute 350(5)(2013) 1248-1263. [17] Y.Y. Wu, Y.Q. Wu, Robust stabilization of delayed nonholonomic systems with strong nonlinear drifts, Nonlinear Analysis: Real World Applications 11(5)(2010) 3620-3627. [18] Z.G. Liu, Y.Q. Wu, Z.Y. Sun, Output feedback control for a class of highorder nonholonomic systems with complicated nonlinearity and timevarying delay, Journal of the Franklin Institute 354(11)(2017) 4289-4310. 26

[19] Y. Zhao, J.B. Yu, Y.Q. Wu, State-feedback stabilization for a class of more general high order stochastic nonholonomic systems, International Journal of Adaptive Control and Signal Processing 25(8)(2011) 687-706. [20] F.Z. Gao, F.S. Yuan, Adaptive stabilization of stochastic nonholonomic systems with nonlinear parameterization, Applied Mathematics and Computation 219(16)(2013) 8676-8686. [21] K.D. Do, Global inverse optimal stabilization of stochastic nonholonomic systems, Systems & Control Letters 75(1)(2015) 41-55. [22] H. Wang, Q.X. Zhu, Adaptive output feedback control of stochastic nonholonomic systems with nonlinear parameterization, Automatica 98(12)(2018) 247-255. [23] F.Z. Gao, Y.Q. Wu, Finite-time output feedback stabilisation of chainedform systems with inputs saturation, International Journal of Control 90(7)(2017) 1466-1477. [24] F.Z. Gao, Y.Q. Wu, Finite-time stabilisation for a class of outputconstrained nonholonomic systems with its application, International Journal of Systems Science 49(10)(2018) 2155-2169. [25] K. Wu, C.Y. Sun, Output feedback control for nonholonomic systems with non-vanishing disturbances, International Journal of Control, https://doi.org/10.1080/00207179.2019.1566633, published online, 2019. [26] F. Bayat, S. Mobayen, S. Javadi, Finite-time tracking control of nthorder chained-form nonholonomic systems in the presence of disturbances, ISA Transactions 63(6)(2016) 78-83. [27] S. Mobayen, F. Tchier, Nonsingular fast terminal sliding-mode stabilizer for a class of uncertain nonlinear systems based on disturbance observer, Scientia Iranica D 24(3)(2017) 1410-1418. [28] A. Ajorlou, M.M. Asadi, A.G. Aghdam, S. Blouin, Distributed consensus control of unicycle agents in the presence of external disturbances, Systems & Control Letters 82(8)(2015) 86-90.

27

[29] J.S. Huang, C.Y. Wen, W. Wang, Z.P. Jiang, Adaptive stabilization and tracking control of a nonholonomic mobile robot with input saturation and disturbance, Systems & Control Letters 62(3)(2013) 234-241. [30] H. Wang, Q.X. Zhu, Finite-time stabilization of high-order stochastic nonlinear systems in strict-feedback form, Automatica 54(4)(2015) 284291. [31] B. Wang, Q.X. Zhu, Stability analysis of semi-Markov switched stochastic systems, Automatica 94(8)(2018) 72-80. [32] Q.X. Zhu, Stability analysis of stochastic delay differential equations with L´ evy noise, Systems & Control Letters 118(8)(2018) 62-68. [33] W. Sun, J.W. Xia, G.M. Zhuang, X. Huang, H. Shen, Adaptive fuzzy asymptotically tracking control of full state constrained nonlinear system based on a novel Nussbaum-type function, Journal of the Franklin Institute 356(4)(2019) 1810-1827. [34] Y.C. Man, Y.G. Liu, Global output-feedback stabilization for a class of uncertain time-varying nonlinear systems, Systems & Control Letters 90(4)(2016) 20-30. [35] W. Liu, J. Huang, Cooperative global robust output regulation for a class of nonlinear multi-agent systems with switching network, IEEE Transactions on Automatic Control 60(7)(2015) 1963-1968. [36] C.J. Qian, C.B. Schrader, W. Lin, Global regulation of a class of uncertain nonlinear systems using output feedback, Proceedings of the American Control Conference, Denver, Colorado, June, 2003, 1542-1547. [37] M. Krstic, I. Kanellakopoulos, P.V. Kokotovic. Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [38] Y. Zhao, C.X. Wang, J.B. Yu, Partial-state feedback stabilization for a class of generalized nonholonomic systems with ISS dynamic uncertainties, International Journal of Control, Automation, and Systems 16(1)(2018) 79-86. [39] E.D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control 34(4)(1989) 435-443. 28

[40] J.B. Yu, Y. Zhao, Y.Q. Wu, Global robust output tracking control for a class of uncertain cascaded nonlinear systems, Automatica 93(7)(2018) 274-281. [41] K. Liu, A. Seuret, Comparison of bounding methods for stability analysis of systems with time-varying delays, Journal of the Franklin Institute 354(7)(2017) 2979-2993. [42] S.J. Liu, J.F. Zhang, Z.P. Jiang, Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems, Automatica 43(12)(2007) 238-251. [43] E.D. Sontag, A. Teel, Changing supply functions in input-state stable systems, IEEE Transactions on Automatic Control 40(8)(1995) 14761478.

29