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Output feedback tracking control for nonlinear time-delay systems with tracking errors and input constraints
Author's Accepted Manuscript
Output Feedback Tracking Control for Nonlinear Time-delay Systems with Tracking Errors and Input Constrains Changchun Hua, Guopin Liu, Liuliu Zhang, Xinping Guan
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)01179-0 http://dx.doi.org/10.1016/j.neucom.2015.08.026 NEUCOM15955
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Neurocomputing
Received date: 12 January 2015 Revised date: 24 June 2015 Accepted date: 9 August 2015 Cite this article as: Changchun Hua, Guopin Liu, Liuliu Zhang, Xinping Guan, Output Feedback Tracking Control for Nonlinear Time-delay Systems with Tracking Errors and Input Constrains, Neurocomputing, http://dx.doi.org/10.1016/j. neucom.2015.08.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Output Feedback Tracking Control for Nonlinear Time-delay Systems with Tracking Errors and Input Constrains Changchun Hua, Guopin Liu, Liuliu Zhang, Xinping Guan Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004, China
Email: [email protected]
Abstract The adaptive tracking control problem is considered for a class of nonlinear time-delay systems in the presence of input and tracking error constraint. An reduced-order observer is designed to estimate the unmeasured state variables at first. Then, a constraint variable is utilized to ensure that the tracking error is within the prescribed boundaries. An auxiliary state is introduced to deal with the input saturation constraint. With the time-delay functions unavailable, we employ adaptive RBF neural network systems to approximate unknown functions. It is proved that the resulting closed-loop system is stable in the sense of semiglobal uniformly ultimately boundedness. The simulations are performed and the results demonstrate the effectiveness of the proposed approach. Keywords: Tracking control; Time-delay systems; Input constraint; Error constraint; RBF neural network 1. Introduction Time-delay exists in a variety of practical systems, for example, teleoperation systems, electrical networks and so on. The existence of time-delay is an unavoidable factor affecting the stability of the closed-loop system. It also makes the controller design more difficult([1, 2]). Hence, the research for time-delay systems is one of the most meaningful and challenging problems. Meanwhile, rapid progress has been made on the problem of stability analysis and control for time-delay systems, see [2, 3, 4, 5] and the references therein. Preprint submitted to Neurocomputing
August 13, 2015
Unknown nonlinearity is inherent in many practical systems. In order to deal with this challenging problem, adaptive control method is widely used for nonlinear systems that exhibit unknown nonlinearities. Typically, RBF neural network is usually used as a tool to approximate the unknown nonlinear functions. Under the assumption that the growth conditions for the unknown nonlinear functions are partially known, [6] investigated the adaptive finite-time stabilization of a class of switched nonlinear systems using RBFNN. In [7], by using a RBFNN based compensator to approximate the uncertain nonlinear function, a neural network based robust finite-time control strategy was proposed for the trajectory tracking of robotic manipulators with structured and unstructured uncertainties. The requirement of performance constraint arises naturally in many industrial control systems. For example, physical stoppage, saturation performance and other measures are taken to prevent system damage. Three methods are commonly employed for output/error constraint problem, (i) the barrier Lyapunov function (BLF) method ([8]), (ii) the performance transformation function method ([9]), and (iii) the funnel control method ([10]). BLF method takes a logarithmic function which yields a value that approaches infinity whenever its arguments approach some limits. The state variable can be constrained directly. [8] proposed a control design method for nonlinear systems with time-varying output constraints. The performance transformation function method is to construct a prescribed performance function, then the inverse of a transformation function is obtained. We study the inverse function in the transformed system. Thus, the performance of system output or tracking error can be characterized by a prescribed constraint function. [9] investigated the robust adaptive control for strict feedback nonlinear system. [11] considered the problem of output feedback control for interconnected time-delay systems with prescribed performance. On the other hand, the funnel control proposed by Illchaman et al. and Hackl et al. also guarantees the prescribed transient behavior([12]). Many practical dynamic systems have constraints on their inputs, such as saturation, dead-zone and so on. Saturation is a potential problem for actuators of control systems and often severely limits system performance, giving rise to undesirable inaccuracy or leading instability. Control design for nonlinear systems with actuator constraints presents a tremendous challenge. During the past decades, several schemes of adaptive control design have been proposed for systems with input saturation([13, 14, 15, 16] and the references therein). [13] investigated adaptive fuzzy output-feedback control for a class 2
of output constrained uncertain nonlinear system with input saturation. In [14], adaptive tracking control was proposed for uncertain MIMO nonlinear systems with input saturation. [15] considered adaptive control of single input uncertain nonlinear systems in the presence of input saturation and unknown external disturbance. However, to our best knowledge, there is no result on tracking control for nonlinear time-delay systems with input and tracking error constraint. Motivated by the above observation and inspired by the work of [11, 12, 13], we investigate the adaptive tracking control for a class of time-delay systems with input and tracking error constraint in this paper. The reduced-order observer is designed at first. Then an error-constraint variable is introduced which is simpler than the inverse function transformation in [9]. Unknown time-varying delay is approximated by neural networks and controller is constructed by backstepping method. Compared with the existing works on nonlinear systems, the main contributions of this paper are as follows: (i) A new error constraint variable is introduced, which leaves out the procedure of solving the inverse function and makes the method simpler than [9]. Furthermore, the designed parameters make the boundaries tunable. (ii) For the input and output constrained problem, an adpative method is proposed to deal with the bounded saturation approximation error function. The rest of this paper is organized as follows. System formulation and preliminaries are given in section 2. The observer design is performed in section 3. Our main result is given in section 4, which consists of two subsections. The first subsection introduces a tracking error constraint variable. The controller design is given in the second subsection. Then a simulation example is given section 5. Section 6 concludes the work of this paper. Notations: Rn denotes the real n-dimensional space. |a| denotes the absolute value of scalar a. For any vector x = [x1 , x2 , · · · , xn ]T ∈ Rn , xi = [x1 , · · · , xi ]T ∈ Ri denotes the vector of partial variables, x denotes state n 2 x for Euclidean norm of vector x, i.e.,x = i=1 i . For x ∈ R, sign(x) denotes the signum function, that is, sign(x) = 1 when x > 0, sign(x) = 0 when x = 0, and sign(x) = −1 when x < 0. I represents the identity matrix with appropriate dimension. The argument of functions will be omitted or simplified whenever no confusion can arise from the context. For example, we may denote f (x(t)) by f (x), f (·), or f .
3
2. System Formulation and Preliminaries In this paper, we consider the nonlinear system described by the following equations ⎧ i = 1, 2..n − 1 ⎨ x˙ i (t) = xi+1 (t) + gi (xi (t)) + fi (t, y(t), y(t − di (t))) x˙ n (t) = sat (u (t)) + gn (xn (t)) + fn (t, y(t), y(t − dn (t))) ⎩ y (t) = x1 (t) (1) T n where x(t) = [x1 (t), x2 (t), · · · , xn (t)] ∈ R , u ∈ R and y ∈ R denote state variable, control input and output of the system, respectively. gi (·) represent known smooth nonlinear functions. fi (·) are unknown time-delay functions. di (t) is the time-varying delay of state xi (t) satisfying d˙i (t) ≤ di ≤ 1. di are positive scalars. State variables are unmeasurable except x1 . Input sat(u(t)) is described as: sign(u(t))uM , |u(t)| ≥ uM sat(u(t)) = (2) u(t), |u(t)| < uM where uM is known bound of sat(u(t)). The applied control sat(u) has a sharp corner when |u(t)| = uM , so we can not apply backstepping method directly. As in [13, 15], we use a smooth function to approximate the saturation. Then, system input is transformed into sat(u) = g(u) + δ(u)
(3)
where the smooth function g(u(t)) = uM × tanh(u/uM ). On the other hand, δ(u) is a bounded saturation approximation error function, i.e. |δ(u)| = |sat(u) − g(u)| ≤ δ1 . δ1 is a constant satisfying δ1 ≥ uM × (1 − tanh(1)). Remark 1: It should be pointed out that low-triangular nonlinear system (1) can cover many state-space models of delay systems and can be used to represent many important physical systems. For example, cold rolling mills, wind tunnel, and water resources systems. See [17] and the references therein. Considering that input saturation exists in many practical systems, we highlight the influence of input saturation. Thus, (1) focuses on a more general condition in practical control systems. The output feedback tracking control problem will be investigated in this paper with prescribed performance. We define the tracking error as e1 (t) = y(t) − yd (t), where yd (t) is known reference signal. A positive decreasing smooth function Fb (t) = ξ0 e−at + ξ∞ is selected to address the transient and 4
s 1Fb e1 (t )
- s 2 Fb
Figure 1: Prescribed performance requirement
steady-state performance. With a, ξ0 , ξ∞ are positive constans, we know that Fb (0) = ξ0 + ξ∞ and Fb (t) = ξ∞ at t = ∞. The objective of this paper is to design the output feedback controller such that: (i) The trajectory of the tracking error e1 satisfies −s2 Fb < e1 (t) < s1 Fb , where s1 and s2 are tunable papameters about the prescribed performance requirement, as shown in Fig. 1. (ii) All the state variables of the closed-loop system are bounded in the presence of input saturation. To complete the description of system (1), we make the following assumptions. Assumption 1: Smooth functions gi (·) satisfy the following condition xi )| ≤ ρi xi − x i |gi (xi ) − gi ( where ρi are known positive constants. Assumption 2: Nonlinear functions fi (·) satisfy |fi (t, y(t), y(t − di (t)))|2 ≤ fi12 (y(t)) + fi22 (y(t − di (t)))
(4)
where fi1 (·) and fi2 (·) are unknown smooth functions with fi1 (0) = fi2 (0) = 0. Remark 2: Assumption 2 is frequently imposed on time-delay functions, see [11] and the references therein. The uncertain functions are required to be bounded by the functions of output variables. We define e1 (t) = y(t) − yd(t). For the functions fi1 and fi2 , we have fi12 (y(t)) ≤ e1 (t)Hi1 (e1 (t)) + i1 (yd (t)) 5
(5)
fi22 (y(t − di (t))) ≤ e1 (t − di (t))Hi2 (e1 (t − di (t))) + i2 (yd (t − di (t)))
(6)
where Hi1 (·), Hi2 (·) are unknown functions, and i1 (·), i2 (·) are bounded functions with respect to their arguments. With inequalities (5), (6) and assumption 2, we have |fi (t, y(t), y(t − di(t)))|2 ≤ e1 (t)Hi1 (e1 (t)) + e1 (t − di (t))Hi2 (e1 (t − di (t))) + i (7) where i ≥ i1 (yd (t)) + i2 (yd (t − di (t))) are positive constants. RBF neural network is an efficient approximator for unknown function. The combination of neural networks and adaptive control technique has been proposed by a number of researchers ([18]). In this paper, we approximate an unknown continuous function via the following RBF neural network f (x) = θ∗T ξ(x) + ε(x),
∀x ∈ Ω
(8)
where the input vector x ∈ Ω ⊂ Rq . θ∗ = [θ1∗ , θ2∗ , · · · , θl∗ ]T ∈ Rl is the ideal weight vector, ξ(·) = [ξ1 (·), ξ2 (·), · · · , ξl (·)]T is the radial basis function vector with the NN node number l > 1. ε(x) is the function approximation error. The basis function ξi (x) is chosen as the commonly used Gaussian function of the following form
(x − μi )T (x − μi ) , i = 1, 2, · · · , l ξi (x) = exp − ηi2 where μi = [μi1 , μi2, · · · , μiq ]T is the center of the receptive field and ηi is the width of the Gaussian function. The optimal weight vector θ∗ is chosen as the value of θ that minimizes ε for all x ∈ Ω, i.e. ∗ T θ := arg min sup|f (x) − θ ξ(x)| θ∈Rl
y∈Ω
With the help of RBF neural network and adaptive control technique, a lot of controller design methods are presented for various classes of nonlinear systems, see [19, 20, 21] and the references therein. 3. Observer Design To estimate the unmeasurable states in system (1), we design the delayindependent reduced-order observer (9) as follows ⎧ ⎨ λ˙ i (t) = λi+1 (t) + ki+1 x1 (t) + gˆi (x1 , λi ) − ki (g1 + λ2 + k2 x1 ) (9) λ˙ (t) = sat(u) + gˆn (x1 , λn ) − kn (g1 + λ2 + k2 x1 ) ⎩ n xˆi = λi + ki x1 (t) i = 2, 3, · · · , n − 1 6
where gˆi (x1 , λi ) = gi (ˆ xi ). Parameters ki are the elements of matrix A, which will be introduced later. ki should be selected properly to satisfy (48). Since x1 are measurable, we could apply the signal directly. Consequently, observer states start from xˆ2 . We define the estimation errors ei = xi − xˆi for i = 2, 3...n, then the estimation errors can be expressed as follows: e˙ = Ae + G(x, xˆ) + F where
⎡
−k2 −k3 .. .
⎢ ⎢ ⎢ A=⎢ ⎢ ⎣ −kn−1 −kn
0 ··· 0 ··· .. . ··· 0 0 0 0 ··· 0 0 0 0 ··· 1 0 .. .
0 1 .. .
0 0 .. .
(10) ⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 1 ⎦ 0
e = [e2 , e3 , e4 , · · · , en ]T
T G(x, xˆ) = g2 (x2 ) − g2 (ˆ x2 ), g3 (x3 ) − g3 (ˆ x3 ), · · · , gn (xn ) − gn (ˆ xn ) F = [f2 − k2 f1 , f3 − k3 f1 , · · · , fn − kn f1 ]T Remark 3: To reconstruct the state of nonlinear time-delay systems, the observer design for such systems has been attracting the attention for researchers in recent years ([22]). The full-order observer is frequently proposed for output feedback control, see [3, 13] and the references therein. While in this paper, we introduce the reduced-order observers instead of full-order ones. Consequently, the structure of observer and computation complexity are simplified. 4. Tracking Error Constraint Design To meet the requirement of output performance constraint, firstly, we select a proper performance boundary function as Fb = ξ0 e−at + ξ∞
(11)
where ξ0 > 0, ξ∞ > 0 are proper parameters, and |e1 (0)| < Fb (0). We define the following state transformation as the error constraint variable. e1 (t) e1 (t) q(e1 (t)) + (1 − q(e1 (t))) (12) z1 (t) = σ1 Fb − e1 σ2 Fb + e1 7
where σ1 and σ2 are proper parameters to modulate the tracking error boundaries. Fb satisfies the condition given in (11) and q(e1 (t)) satisfies 1 e1 (t) ≥ 0 q(e1 (t)) = (13) 0 e1 (t) < 0 The time derivative of z1 is: 1 [e˙ 1 (σ1 Fb − e1 ) − e1 (σ1 F˙ b − e˙ 1 )]q (σ1 Fb − e1 )2 1 [e˙ 1 (σ2 Fb + e1 ) − e1 (σ2 F˙ b + e˙ 1 )](1 − q) + (σ2 Fb + e1 )2
z˙1 =
(14)
Through some computation, we obtain
where ΦF =
z˙1 = Fb ΦF e˙ 1 − F˙ b ΦF e1
(15)
σ1 q σ2 (1 − q) + (σ1 Fb − e1 )2 (σ2 Fb + e1 )2
(16)
Remark 4: From the definition of z1 and q(e1 (t)), we can know that when tracking error approaches boundary Fb , z1 increases consequently. We design controller such that the resulting closed-loop system is stable in the sense of semiglobal uniform ultimate boundedness. Then, the tracking error will be forced within the prescribed boundaries, i.e. −σ2 Fb < e1 (t) < σ1 Fb . In addition, there is one further point to make, it means that the introduced state transformation will result in z1 = 0 only at e1 = 0. Compared with the method proposed in [9], we leave out the procedure of seeking the inverse function, which simplifies the computation complexity. Furthermore, compared with [12], parameters σ1 and σ2 make the boundary tunable. 5. Controller Design and Stability Analysis In this section, an adaptive RBF neural network controller is developed in the framework of backstepping method. System stability will be proved at the same time. We choose the Lyapunov function as V = Ve + V1 + V2 8
(17)
with Ve = e(t)T P e(t)
(18)
V1 = V11 + V1i + V1n ⎧ 1 2 ⎪ z + 12 θT Λ−1 θ ⎨V11 = 2 1 1 2 V1i = n−1 i=2 2 zi ⎪ ⎩ V1n = 12 zn2 + 12 δ2
(19) (20)
t q1 V2 = e1 (ξ)H12 (e1 (ξ))dξ 1 − d1 t−d1 (t) t n q2 e1 (ξ)Hi2 (e1 (ξ))dξ + 1 − di t−di (t) i=2
(21)
where q1 , q2 are positive scalars to be defined later. The time derivative of Ve along (18) is V˙ e = 2eT P (Ae + G(x, xˆ) + F ) x)) + 2eT P F = eT (P A + AT P ) + 2eT P (g(x) − g(ˆ with Assumption 1 and Young’s inequality, we get V˙ e ≤ eT (P A + AT P + c1 P P + 2c2 P P + c−1 1 + c−1 2
n i=2
fi2 + c−1 2
n
n
ρ2i I)e
i=2
ki2 f12
(22)
i=2
where c1 and c2 are positive design parameters. Then we define the states transformation as follows zi = λi (t) − αi−1 i = 2, 3, ...n − 1 zn = λn − αn−1 −
(23)
where αi are virtual control and is the auxiliary state to be designed to deal with the saturation input. Next, we use the backstepping method to design the controller. step 1: Taking the time derivation of V11 yields ˙ V˙ 11 = z1 z˙1 − θT Λ−1 θˆ 9
(24)
where θˆ = θ∗ − θ is the estimation of θ∗ . Substuting (15) into (24), we obtain ˙ V˙ 11 = z1 Fb ΦF (x2 + g1 + f1 − y˙ d ) − z1 F˙ b ΦF e1 − θT Λ−1 θˆ = z1 Fb ΦF (z2 + α1 + k2 x1 + e2 + g1 + f1 − y˙ d ) ˙ − z1 F˙ b ΦF e1 − θT Λ−1 θˆ + z1 ω(e1 ) − z1 ω(e1 )
(25)
where ω(e1 ) are unknown nonlinear functions to be specialized later. We use RBF neural network to approximate the functions, i.e z1 ω(e1 ) = z1 (θ∗T ϕ(e1 ) + ε)
(26)
taking some computation, one has 1 1 z1 ω(e1 ) ≤ z1 θT ϕ(e1 ) + z1 θˆT ϕ(e1 ) + z12 + ε2 2 2 select the virtual control and adaptive law as F˙ b e1 c11 1 α1 = −k2 x1 − g1 + y˙ d + − + z1 Fb ΦF Fb 2 4 ˆ 1) c1 z1 + 12 θϕ(e − − Fb ΦF Fb ΦF ˙ θˆ = z1 Λϕ(e1 ) − Λθˆ
(27)
(28) (29)
Substituting (27), (28), (29) into (25), we have 1 2 2 T ˆ 1 2 V˙ 11 ≤ z1 z2 Fb ΦF − c1 z12 + c−1 11 e2 + f1 + θ θ + ε − z1 ω(e1 ) 2 2
(30)
It has been proved that the virtual control α1 is piecewise continuously differentiable with respect to z1 over the condition e|(0)| < Fb (0) in [12]. On the other hand, dynamic surface control(DSC) technique may be employed to get the time derivative of α1 via an first-order filter. step 2: The derivative of V12 is V12 = z2 (λ˙ 2 − α˙ 1 )
∂α1 ∂α1 = z2 z3 + α2 + k3 x1 + gˆ2 − k2 (g1 + λ2 + k2 x1 ) − − y˙ ∂t ∂y 10
(31)
1 through some computation on the term of − ∂α y, ˙ we obtain ∂y
∂α1 ∂α1 y˙ ≤ −z2 (ˆ x2 + g1 ) + −z2 ∂y ∂y
c11 1 + 2 4
2 ∂α1 1 + c−1 e2 + f12 (32) z2 ∂y 2 11 2
substituting (32) into (31) yields
∂α 1 V˙ 12 ≤ z2 z3 + α2 + k3 x1 + gˆ2 − k2 (g1 + λ2 + k2 x1 ) − ∂t 2 c11 1 1 ∂α1 ∂α1 + c−1 e2 + f12 − z2 (ˆ x2 + g1 ) + + z2 ∂y 2 4 ∂y 2 11 2
(33)
select the virtual control as ∂α1 α2 = −c2 z2 − k3 x1 − gˆ2 + k2 (g1 + λ2 + k2 x1 ) + ∂t 2 c11 1 ∂α1 ∂α1 (ˆ x2 + g1 ) − + z2 − z1 Fb ΦF + ∂y 2 4 ∂y
(34)
Substituting (32), (34) into (33), we have 1 V˙ 12 ≤ z2 z3 − z1 z2 Fb ΦF + c−1 e2 + f12 − c2 z22 2 11 2
(35)
step i: Similar with step 2, the time derivative of V1i is V˙ 1i = zi zi+1 + αi + ki+1 x1 + gˆi − ki (g1 + λ2 + k2 x1 ) i−1 ∂αi−1 ∂αi−1 ∂αi−1 ˙ − y˙ λj − − ∂t ∂λj ∂y j=2 ≤ zi zi+1 + αi + ki+1 x1 + gˆi − ki (g1 + λ2 + k2 x1 ) i−1 ∂αi−1 ∂αi−1 ∂αi−1 ˙ λj − zi − (ˆ x2 + g1 ) − ∂t ∂λj ∂y j=2 2 c11 1 ∂αi−1 1 2 2 + + + c−1 zi 11 e2 + f1 2 4 ∂y 2
11
(36)
select the virtual control as αi = −ci zi − ki+1 x1 − gˆi + ki (g1 + λ2 + k2 x1 ) i−1 ∂αi−1 ∂αi−1 ∂αi−1 ˙ + (ˆ x2 + g1 ) λj + ∂t ∂λ ∂y j j=2 2 ∂αi−1 c11 1 zi − zi−1 + − 2 4 ∂y
+
(37)
Substituting (37) into (36), we have 1 2 2 V˙ 1i ≤ zi zi+1 − ci zi2 − zi zi−1 + c−1 11 e2 + f1 2
(38)
step n: The time derivative of V1n is ˙ V˙ 1n = zn (λ˙ n − α˙ n−1 − ) ˙ − δδˆ
(39)
we define the dynamic of as ˙ = − + g(u) − u
(40)
substituting (40) into (39), one obtains V˙ 1n = zn [δ(u) + + u + gˆn − kn (g1 + λ2 + k2 x1 ) ∂αn−1 ∂αn−1 ˙ ∂αn−1 ˙ − − y] ˙ − δδˆ λj − ∂t ∂λj ∂y j=2 n−1
≤ zn [δ(u) + + u + gˆn − kn (g1 + λ2 + k2 x1 ) 1 ∂αn−1 ∂αn−1 ˙ 2 2 − λj ] + c−1 − 11 e2 + f1 ∂t ∂λ 2 j j=2 c11 1 ∂αn−1 − zn (g1 + λ2 + k2 x1 ) + + ∂y 2 4 2 ∂αn−1 ˙ − δδˆ × zn ∂y n−1
(41)
We know that δ(u) is a bounded saturation approximation error function, then zn δ(u) ≤ |zn |δ(u) ≤ |zn |δ1 is bounded. We design the controller and 12
adaptive law as follows u = − − gˆn + kn (g1 + λ2 + k2 x1 ) +
∂αn−1 ∂t
n−1 ∂αn−1
∂αn−1 (g1 + λ2 + k2 x1 ) λ˙ j + ∂λj ∂y j=2 2 ∂αn−1 c11 1 ˆ zn − cn zn − δsign(z + − n) 2 4 ∂y +
˙ δˆ = |zn | − δˆ
(42)
(43)
where δˆ = δ − δ is the estimation of δ(u). Substituting (42), (43) into (41), we have 1 e2 + f12 + δδˆ (44) V˙ 1n ≤ −cn zn2 + c−1 2 11 2 Remark 5: From the description of sat(u), we know that the backstepping technique cannot be applied directly. Inspired by the work [13, 15], we approximate the saturation input by a smooth function g(u) with a bounded approximation error function. [13] did not deal with the approximation error. However, as we know the bound of input, namely, the uM , we can deal with the approximation error via adaptive method, as we did in step n. Then, we are ready to present our main result in this note. Theorem 1: For nonlinear systems (1), under Assumption 1-2 and initial condition |e1 (0)| < Fb (0), the observer-based controller (28), (34), (37), (42) with adaptive law (29), (43) and auxiliary state (40), guarantee that all signals in the closed loop systems are bounded in the presence of input saturation, and the tracking error remains in the prescribed boundaries for all t > 0. Proof: From the definition of V2 , we have q1 e12 (t)H12 (e1 (t)) − q1 e1 (t)H12 (e1 (t − d1 (t))) 1 − d1 n n q2 e1 (t)Hi2 (e1 (t)) − q2 e1 (t)Hi2 (e1 (t − di (t))) + 1 − di i=2 i=2
V˙ 2 ≤
13
(45)
With the observer (9), virtual control (28), (34), (37), controller (42) and adaptive laws (29), (43), we obtain n n T T −1 ˙ ˙ Ve + V1 ≤ e ρ2i I + c−1 e P A + A P + (c01 + 2c02 )P P + c01 11 I 2 i=2 −
n
ci zi2
+ q2
i=1
where q1 = c−1 02 q1 f12
+ q2
n i=2
n i=2
n
1 fi2 + q1 f12 − zi ω(e1 ) + θT θˆ + ε2 + δδˆ 2
(46)
ki2 + n, q2 = c−1 02 . With remark 1, we have
fi2 ≤ q1 [e1 H11 (e1 ) + e1 (t − d1 (t))H12 (e1 (t − d1 (t)))]
i=2
n [e1 Hi1 (e1 ) + e1 (t − di (t))Hi2 (e1 (t − di (t)))] + q2 i=2
where = q1 1 + q2
+ n
(47)
i=2 i ,
we let
T
P A + A P + (c01 + 2c02 )P P +
c−1 01
n
ρ2i I +
i=2
n −1 c I ≤ −Q 2 11
q1 e1 (t)H11 (e1 (t)) + q1 e1 (t)H11 (e1 (t)) 1 − d1 n n q2 e1 (t)Hi1 (e1 (t)) + q2 e1 (t)Hi1 (e1 ) + 1 − di i=2 i=2
(48)
z1 ω(e1 ) =
(49)
then, from (46), (47), (48), (49), we come to the result V˙ ≤ −eT Qe −
n i=1
we know that
1 ci zi2 + θT θˆ + ε2 + δδˆ + 2
1 2 1 ∗ 2 θT θˆ ≤ − θ + θ 2 2 1 1 δδˆ ≤ − δ2 + δ 2 2 2 14
(50)
(51) (52)
We define η = 12 θ∗ 2 + 12 δ 2 + 12 ε2 + , substituting (51), (52) into (50), one has n 1 2 1 2 T ˙ V ≤ −e Qe − ci zi2 − θ − δ +η (53) 2 2 i=1 Thus completes the proof. Remark 6: In this paper, we introduce a error constraint variable z1 (t) to address error constraint σ1 Fb < e1 < σ2 Fb . σ1 and σ2 are parameters to modulate the prescribed boundaries. [12] omits them and the time-delay free systems it considered do not contain the input constraint. Its error could only fall between symmetrical boundaries. However, this paper considers a more complex condition, so that the controller design and the stability of the closed-loop system are more challenging. Remark 7: It should be pointed out that we design a dynamic output feedback controller with the constructed observer, adaptive laws and the auxiliary states. While, from an implementation perspective, static outputfeedback control law is more welcome. There are many results on static output-feedback control, such as [23, 24] and the references therein. It is meaningful and challenging to extend the proposed method to the static output-feedback case, which can be set as a future direction. 6. Simulation Example In this section, an example will be used to illustrate the validity of the control approach proposed in this paper. Consider a nonlinear time-delay system as follows ⎧ ⎪ ⎨x˙ 1 = x2 + 0.3x1 + cos(t)x1 (t − d1 (t)) (54) x˙ 2 = sat(u) + 0.5x2 + 0.5x2 sin(x1 ) + sin(t)x1 (t − d2 (t)) ⎪ ⎩ y(t) = x1 (t) where the time varying delay d1 (t) = d2 (t) = 0.4(1 + sin(t)). We select k2 = 3.5 to satisfy inequality (48), consequently, we have the reduced-order observer as follows λ˙ 2 = sat(u) − 11.55x1 − 3λ2 + 0.5λ2 sin(x1 ) + 1.75x1 sin(x1 )
(55)
Then, the performance boundary function is selected as Fb = ξ0 e−at + ξ∞ 15
(56)
with ξ0 , a, ξ∞ are chosen as 0.9, 7, 0.17, respectively. Parameters σ1 and σ2 in z1 are specified σ1 = 1, σ2 = 0.3. According to the selected parameters, the virtual control is constructed as α1 = −3.8x1 + cos(t) +
F˙ b e1 10.5z1 θˆT ξ(e1 ) − 0.75z1 Fb ΦF − − Fb Fb ΦF Fb ΦF
(57)
˙ and adaptive laws θˆ are ˙ θˆ = z1 ξ T (e1 ) − θˆ
(58)
where ⎡ ⎢ ⎢ ξ (e1 ) = ⎢ ⎢ ⎣ T
2
e−0.5∗(e1 −8) 2 e−0.5∗(e1 −4) 2 e−0.5∗(e1 ) 2 e−0.5∗(e1 +4) 2 e−0.5∗(e1 +8)
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(59)
the constructed controller is u = − + 3λ2 + 11.55x1 − 0.5(λ2 + 3.5x1 )sin(x1 ) ∂α1 ∂α1 ∂α1 2 + (λ2 + 3.8x1 ) − 0.75( ) z2 + ∂t ∂t ∂t
(60)
The initial values are chosen as x1 (0) = 0.45, x2 (0) = 0.25, λ2(0) = −0.1.
(61)
The simulation results of the closed loop control system are illustrated in Fig.2-Fig.6. The response of system output and tracking signal is shown in Fig.2, while Fig.3 shows the tracking error and the symmetrical boundaries. Fig.4 and Fig.5 provide the response of adaptive laws of the system. Fig.4 ˙ ˙ demonstrates the adaptive laws of θˆi and Fig.5 the δˆ0 . The control signal u and applied control input sat(u) are shown in Fig.6. Here we set the uM = 5. We design parameters uM , Fb to tune the input saturation boundary and the transient and steady performance of output. Fig.7 and Fig.8 show a rapid response of the system by decreasing the parameter −a of Fb . We can see that the system tends to be stable rapidly, and the output can track the reference signal using less time than Fig.2. It should be pointed out 16
1.5 Y Yd 1
0.5
0
−0.5
−1 0
2
4
6
8
10
Figure 2: system output and tracking signal
1.2 upper bound lower bound tracking error
1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 0
2
4
6
8
10
Figure 3: tracking errors with prescribed boundaries
0.35 ˆ1 θ ˆ2 θ ˆ3 θ ˆ4 θ ˆ5 θ
0.3 0.25 0.2 0.15 0.1 0.05 0 0
2
4
6
Figure 4: response of adaptive laws θˆi
17
8
10
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
2
4
6
8
10
Figure 5: response of adaptive law δˆ
30 u sat(u)
20 10 0 −10 −20 −30 −40 −50 0
0.2
0.4
0.6
0.8
Figure 6: control signal u and control input sat(u)
18
1
1.5 Y Yd 1
0.5
0
−0.5
−1 0
2
4
6
8
10
Figure 7: rapid output response and tracking signal
upper bound lower bound tracking error
0.8
0.6
0.4
0.2
0
−0.2 0
2
4
6
8
10
Figure 8: tracking errors with prescribed boundaries
that the constraint of input and output could be arbitrarily small in theory. However, when the constraint is too restrictive, it may lead the failure of the controller. From this point, the controller is weak. In practical engineering, we can choose proper parameters to reach the desired result. 7. Conclusion In this paper, we investigate the adaptive control for a class of time-delay systems with tracking error and input constraint. The delay independent reduced order observer is designed at first. Then, a novel error constraint variable is introduced to ensure a prescribed tracking performance. Next, a output feedback controller is constructed by the well known backstepping 19
method step by step, during which, RBF neural networks are used to approximate the unknown nonlinear functions. Finally, a simulation example is performed to verify the effectiveness of the proposed method.
References [1] K. Gu, V.L. Kharitonov, J. Chen, ‘Stability of time-delay systems’, Berlin, Germany Birkhauser (2003) [2] M.S. Mahmoud, ‘Robust control and filtering for time-delay systems’, CRC Press, New York (2000) [3] C.C. Hua, Q.G. Wang, X.P. Guan, ‘Adaptive fuzzy output-feedback controller disign for nonlinear time-delay systems with unknown control direction’, IEEE Transactions on Systems,Man, and Cybernetics-PART B: Cybernetics 39(2)(2009) 363-374 [4] Z.X. Yu, S.G. Li, ‘Neural-network-based output-feedback adaptive dynamic surface control for a class of stochastic nonlinear time-delay systems with unknown control directions’, Neurocomputing, 129(2014) 540547 [5] S.C. Tong, Y.M. Li, ‘Observer-based fuzzy adaptive robust control of nonlinear systems with time delays and unmodeled dynamics’, Neurocomputing, 74(2010) 369-378 [6] M.J. Cai, Z.R. Xiang, ‘Adaptive neural finite-time control for a class of switched nonlinear systems’, Neurocomputing, 155(2015) 177-185 [7] H.T. Liu, T. Zhang, ‘Neural network-based robust finite-time control for robotic manipulators considering actuator dynamics’, Robotics and Computer-Integrated Manufacturing, 29(2)(2013) 301-308 [8] K.P. Tee, B.B. Ren, S.S. Ge, ‘Control of nonlinear systems with timevarying output constraints’, Automatica, 47(2011) 2511-2516 [9] C.P. Bechlioulis, G.A. Rovithakis, ‘Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems’, Automatica, 45(2009) 532-538 20
[10] A. Ilchman, E.P. Ryan, S. Trenn, ‘PI-funnel control for two mass systems’, IEEE Transactions on Affective Computing, 54(4)(2009) 918-923 [11] C.C. Hua, L.L. Zhang and X.P. Guan, ‘Output feedback control for interconnected time-delay systems with prescribed performance’, neurocomputing, 129(10)(2014) 208-215 [12] S. Han, J. Lee, ‘Recurrent fuzzy neural network backstepping control for the prescribed output tracking performance of nonlinear dynamic systems’, ISA Transactions, 53(2014) 33-43 [13] Y.M. Li, S.C. Tong, T.S. Li, ‘Adaptive fuzzy output-feedback control for output constrained nonlinear systems in the presence of input saturation’, FUZZY Sets and Systems, 248(1)(2014) 138-155 [14] M. Chen, S.S. Ge, B.B. Ren, ‘Adaptive tracking control of uncertain MIMO nonlinear systems with input constrains’, Automatica, 47(2011) 452-465 [15] C.Y. Wen, J. Zhou, Z.T. Liu, ‘Robust adaptive control of uncertain nonlinear systems in the presence of input stauration and external disturbance’, IEEE Transactions on Automatic Control, 56(7)(2011) 1672-1678 [16] W. Wu, S. Jayasuriya, ‘A robust synthesis methodology for neutrally stable uncertain SISO plants under input amplitude saturation’, International Journal of Systems Science, 44(3)(2013) 461-470 [17] C.C. Hua, X.P. Guan, P. Shi, ‘Robust backstepping control for a class of time delay systems’, IEEE Transactions on Automatic Control, 50(6)(2005) 894-899 [18] C.P. Bechlioulis, G.A. Rovithakis, ‘Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance’, IEEE Transactions on Automatic Control, 53(9)(2008) 2090-2099 [19] Y. Liu, S. Tong, D. Wang, ‘adaptive neural output feedback control using neural networks’, IEEE Transactions on Neural Networks, 22(8)(2011) 1328-1334 [20] B.B. Ren, S.S. Ge, K.P. Tee, T.H. Lee, ‘Adaptive neural control for output feedcack nonlinear systems using a Barrier Lyapunov Function’, IEEE Transactions on Neural Networks, 21(8)(2010) 1339-1345 21
[21] Q. Zhu, S. Fei, T. Zhang, T. Li, ‘Adaptive RBF neural-networks control for a class of time-delay nonlinear systems’, Neurocomputing, 71(2008) 3617-3624 [22] G. Malek, D. Jesus, L. Barbot, ‘Observer design for nonlinear systens under unknown time-varing delays’, IEEE Transactions on Automatic Control, 58(6)(2013) 1529-1534 [23] X.W. Li, H.J. Gao, ‘A heuristic approach to static output-feedback controller synthesis with restricted frequency-domain specifications’, IEEE Transactions on Automatic Control, 59(4)(2014) 1008-1014 [24] C. Agulhari, R. Oliveira and P. Peres, ‘LMI relaxations for reducedorder robust H∞ control of continuous-time uncertain linear systems’, IEEE Transactions on Automatic Control, 57(6)(2012) 1532-1537
22
Output Feedback Tracking Control for Nonlinear Time-delay Systems with Tracking Errors and Input Constrains Changchun Hua, Guopin Liu, Liuliu Zhang, Xinping Guan
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)01179-0 http://dx.doi.org/10.1016/j.neucom.2015.08.026 NEUCOM15955
To appear in:
Neurocomputing
Received date: 12 January 2015 Revised date: 24 June 2015 Accepted date: 9 August 2015 Cite this article as: Changchun Hua, Guopin Liu, Liuliu Zhang, Xinping Guan, Output Feedback Tracking Control for Nonlinear Time-delay Systems with Tracking Errors and Input Constrains, Neurocomputing, http://dx.doi.org/10.1016/j. neucom.2015.08.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Output Feedback Tracking Control for Nonlinear Time-delay Systems with Tracking Errors and Input Constrains Changchun Hua, Guopin Liu, Liuliu Zhang, Xinping Guan Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004, China
Email: [email protected]
Abstract The adaptive tracking control problem is considered for a class of nonlinear time-delay systems in the presence of input and tracking error constraint. An reduced-order observer is designed to estimate the unmeasured state variables at first. Then, a constraint variable is utilized to ensure that the tracking error is within the prescribed boundaries. An auxiliary state is introduced to deal with the input saturation constraint. With the time-delay functions unavailable, we employ adaptive RBF neural network systems to approximate unknown functions. It is proved that the resulting closed-loop system is stable in the sense of semiglobal uniformly ultimately boundedness. The simulations are performed and the results demonstrate the effectiveness of the proposed approach. Keywords: Tracking control; Time-delay systems; Input constraint; Error constraint; RBF neural network 1. Introduction Time-delay exists in a variety of practical systems, for example, teleoperation systems, electrical networks and so on. The existence of time-delay is an unavoidable factor affecting the stability of the closed-loop system. It also makes the controller design more difficult([1, 2]). Hence, the research for time-delay systems is one of the most meaningful and challenging problems. Meanwhile, rapid progress has been made on the problem of stability analysis and control for time-delay systems, see [2, 3, 4, 5] and the references therein. Preprint submitted to Neurocomputing
August 13, 2015
Unknown nonlinearity is inherent in many practical systems. In order to deal with this challenging problem, adaptive control method is widely used for nonlinear systems that exhibit unknown nonlinearities. Typically, RBF neural network is usually used as a tool to approximate the unknown nonlinear functions. Under the assumption that the growth conditions for the unknown nonlinear functions are partially known, [6] investigated the adaptive finite-time stabilization of a class of switched nonlinear systems using RBFNN. In [7], by using a RBFNN based compensator to approximate the uncertain nonlinear function, a neural network based robust finite-time control strategy was proposed for the trajectory tracking of robotic manipulators with structured and unstructured uncertainties. The requirement of performance constraint arises naturally in many industrial control systems. For example, physical stoppage, saturation performance and other measures are taken to prevent system damage. Three methods are commonly employed for output/error constraint problem, (i) the barrier Lyapunov function (BLF) method ([8]), (ii) the performance transformation function method ([9]), and (iii) the funnel control method ([10]). BLF method takes a logarithmic function which yields a value that approaches infinity whenever its arguments approach some limits. The state variable can be constrained directly. [8] proposed a control design method for nonlinear systems with time-varying output constraints. The performance transformation function method is to construct a prescribed performance function, then the inverse of a transformation function is obtained. We study the inverse function in the transformed system. Thus, the performance of system output or tracking error can be characterized by a prescribed constraint function. [9] investigated the robust adaptive control for strict feedback nonlinear system. [11] considered the problem of output feedback control for interconnected time-delay systems with prescribed performance. On the other hand, the funnel control proposed by Illchaman et al. and Hackl et al. also guarantees the prescribed transient behavior([12]). Many practical dynamic systems have constraints on their inputs, such as saturation, dead-zone and so on. Saturation is a potential problem for actuators of control systems and often severely limits system performance, giving rise to undesirable inaccuracy or leading instability. Control design for nonlinear systems with actuator constraints presents a tremendous challenge. During the past decades, several schemes of adaptive control design have been proposed for systems with input saturation([13, 14, 15, 16] and the references therein). [13] investigated adaptive fuzzy output-feedback control for a class 2
of output constrained uncertain nonlinear system with input saturation. In [14], adaptive tracking control was proposed for uncertain MIMO nonlinear systems with input saturation. [15] considered adaptive control of single input uncertain nonlinear systems in the presence of input saturation and unknown external disturbance. However, to our best knowledge, there is no result on tracking control for nonlinear time-delay systems with input and tracking error constraint. Motivated by the above observation and inspired by the work of [11, 12, 13], we investigate the adaptive tracking control for a class of time-delay systems with input and tracking error constraint in this paper. The reduced-order observer is designed at first. Then an error-constraint variable is introduced which is simpler than the inverse function transformation in [9]. Unknown time-varying delay is approximated by neural networks and controller is constructed by backstepping method. Compared with the existing works on nonlinear systems, the main contributions of this paper are as follows: (i) A new error constraint variable is introduced, which leaves out the procedure of solving the inverse function and makes the method simpler than [9]. Furthermore, the designed parameters make the boundaries tunable. (ii) For the input and output constrained problem, an adpative method is proposed to deal with the bounded saturation approximation error function. The rest of this paper is organized as follows. System formulation and preliminaries are given in section 2. The observer design is performed in section 3. Our main result is given in section 4, which consists of two subsections. The first subsection introduces a tracking error constraint variable. The controller design is given in the second subsection. Then a simulation example is given section 5. Section 6 concludes the work of this paper. Notations: Rn denotes the real n-dimensional space. |a| denotes the absolute value of scalar a. For any vector x = [x1 , x2 , · · · , xn ]T ∈ Rn , xi = [x1 , · · · , xi ]T ∈ Ri denotes the vector of partial variables, x denotes state n 2 x for Euclidean norm of vector x, i.e.,x = i=1 i . For x ∈ R, sign(x) denotes the signum function, that is, sign(x) = 1 when x > 0, sign(x) = 0 when x = 0, and sign(x) = −1 when x < 0. I represents the identity matrix with appropriate dimension. The argument of functions will be omitted or simplified whenever no confusion can arise from the context. For example, we may denote f (x(t)) by f (x), f (·), or f .
3
2. System Formulation and Preliminaries In this paper, we consider the nonlinear system described by the following equations ⎧ i = 1, 2..n − 1 ⎨ x˙ i (t) = xi+1 (t) + gi (xi (t)) + fi (t, y(t), y(t − di (t))) x˙ n (t) = sat (u (t)) + gn (xn (t)) + fn (t, y(t), y(t − dn (t))) ⎩ y (t) = x1 (t) (1) T n where x(t) = [x1 (t), x2 (t), · · · , xn (t)] ∈ R , u ∈ R and y ∈ R denote state variable, control input and output of the system, respectively. gi (·) represent known smooth nonlinear functions. fi (·) are unknown time-delay functions. di (t) is the time-varying delay of state xi (t) satisfying d˙i (t) ≤ di ≤ 1. di are positive scalars. State variables are unmeasurable except x1 . Input sat(u(t)) is described as: sign(u(t))uM , |u(t)| ≥ uM sat(u(t)) = (2) u(t), |u(t)| < uM where uM is known bound of sat(u(t)). The applied control sat(u) has a sharp corner when |u(t)| = uM , so we can not apply backstepping method directly. As in [13, 15], we use a smooth function to approximate the saturation. Then, system input is transformed into sat(u) = g(u) + δ(u)
(3)
where the smooth function g(u(t)) = uM × tanh(u/uM ). On the other hand, δ(u) is a bounded saturation approximation error function, i.e. |δ(u)| = |sat(u) − g(u)| ≤ δ1 . δ1 is a constant satisfying δ1 ≥ uM × (1 − tanh(1)). Remark 1: It should be pointed out that low-triangular nonlinear system (1) can cover many state-space models of delay systems and can be used to represent many important physical systems. For example, cold rolling mills, wind tunnel, and water resources systems. See [17] and the references therein. Considering that input saturation exists in many practical systems, we highlight the influence of input saturation. Thus, (1) focuses on a more general condition in practical control systems. The output feedback tracking control problem will be investigated in this paper with prescribed performance. We define the tracking error as e1 (t) = y(t) − yd (t), where yd (t) is known reference signal. A positive decreasing smooth function Fb (t) = ξ0 e−at + ξ∞ is selected to address the transient and 4
s 1Fb e1 (t )
- s 2 Fb
Figure 1: Prescribed performance requirement
steady-state performance. With a, ξ0 , ξ∞ are positive constans, we know that Fb (0) = ξ0 + ξ∞ and Fb (t) = ξ∞ at t = ∞. The objective of this paper is to design the output feedback controller such that: (i) The trajectory of the tracking error e1 satisfies −s2 Fb < e1 (t) < s1 Fb , where s1 and s2 are tunable papameters about the prescribed performance requirement, as shown in Fig. 1. (ii) All the state variables of the closed-loop system are bounded in the presence of input saturation. To complete the description of system (1), we make the following assumptions. Assumption 1: Smooth functions gi (·) satisfy the following condition xi )| ≤ ρi xi − x i |gi (xi ) − gi ( where ρi are known positive constants. Assumption 2: Nonlinear functions fi (·) satisfy |fi (t, y(t), y(t − di (t)))|2 ≤ fi12 (y(t)) + fi22 (y(t − di (t)))
(4)
where fi1 (·) and fi2 (·) are unknown smooth functions with fi1 (0) = fi2 (0) = 0. Remark 2: Assumption 2 is frequently imposed on time-delay functions, see [11] and the references therein. The uncertain functions are required to be bounded by the functions of output variables. We define e1 (t) = y(t) − yd(t). For the functions fi1 and fi2 , we have fi12 (y(t)) ≤ e1 (t)Hi1 (e1 (t)) + i1 (yd (t)) 5
(5)
fi22 (y(t − di (t))) ≤ e1 (t − di (t))Hi2 (e1 (t − di (t))) + i2 (yd (t − di (t)))
(6)
where Hi1 (·), Hi2 (·) are unknown functions, and i1 (·), i2 (·) are bounded functions with respect to their arguments. With inequalities (5), (6) and assumption 2, we have |fi (t, y(t), y(t − di(t)))|2 ≤ e1 (t)Hi1 (e1 (t)) + e1 (t − di (t))Hi2 (e1 (t − di (t))) + i (7) where i ≥ i1 (yd (t)) + i2 (yd (t − di (t))) are positive constants. RBF neural network is an efficient approximator for unknown function. The combination of neural networks and adaptive control technique has been proposed by a number of researchers ([18]). In this paper, we approximate an unknown continuous function via the following RBF neural network f (x) = θ∗T ξ(x) + ε(x),
∀x ∈ Ω
(8)
where the input vector x ∈ Ω ⊂ Rq . θ∗ = [θ1∗ , θ2∗ , · · · , θl∗ ]T ∈ Rl is the ideal weight vector, ξ(·) = [ξ1 (·), ξ2 (·), · · · , ξl (·)]T is the radial basis function vector with the NN node number l > 1. ε(x) is the function approximation error. The basis function ξi (x) is chosen as the commonly used Gaussian function of the following form
(x − μi )T (x − μi ) , i = 1, 2, · · · , l ξi (x) = exp − ηi2 where μi = [μi1 , μi2, · · · , μiq ]T is the center of the receptive field and ηi is the width of the Gaussian function. The optimal weight vector θ∗ is chosen as the value of θ that minimizes ε for all x ∈ Ω, i.e. ∗ T θ := arg min sup|f (x) − θ ξ(x)| θ∈Rl
y∈Ω
With the help of RBF neural network and adaptive control technique, a lot of controller design methods are presented for various classes of nonlinear systems, see [19, 20, 21] and the references therein. 3. Observer Design To estimate the unmeasurable states in system (1), we design the delayindependent reduced-order observer (9) as follows ⎧ ⎨ λ˙ i (t) = λi+1 (t) + ki+1 x1 (t) + gˆi (x1 , λi ) − ki (g1 + λ2 + k2 x1 ) (9) λ˙ (t) = sat(u) + gˆn (x1 , λn ) − kn (g1 + λ2 + k2 x1 ) ⎩ n xˆi = λi + ki x1 (t) i = 2, 3, · · · , n − 1 6
where gˆi (x1 , λi ) = gi (ˆ xi ). Parameters ki are the elements of matrix A, which will be introduced later. ki should be selected properly to satisfy (48). Since x1 are measurable, we could apply the signal directly. Consequently, observer states start from xˆ2 . We define the estimation errors ei = xi − xˆi for i = 2, 3...n, then the estimation errors can be expressed as follows: e˙ = Ae + G(x, xˆ) + F where
⎡
−k2 −k3 .. .
⎢ ⎢ ⎢ A=⎢ ⎢ ⎣ −kn−1 −kn
0 ··· 0 ··· .. . ··· 0 0 0 0 ··· 0 0 0 0 ··· 1 0 .. .
0 1 .. .
0 0 .. .
(10) ⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 1 ⎦ 0
e = [e2 , e3 , e4 , · · · , en ]T
T G(x, xˆ) = g2 (x2 ) − g2 (ˆ x2 ), g3 (x3 ) − g3 (ˆ x3 ), · · · , gn (xn ) − gn (ˆ xn ) F = [f2 − k2 f1 , f3 − k3 f1 , · · · , fn − kn f1 ]T Remark 3: To reconstruct the state of nonlinear time-delay systems, the observer design for such systems has been attracting the attention for researchers in recent years ([22]). The full-order observer is frequently proposed for output feedback control, see [3, 13] and the references therein. While in this paper, we introduce the reduced-order observers instead of full-order ones. Consequently, the structure of observer and computation complexity are simplified. 4. Tracking Error Constraint Design To meet the requirement of output performance constraint, firstly, we select a proper performance boundary function as Fb = ξ0 e−at + ξ∞
(11)
where ξ0 > 0, ξ∞ > 0 are proper parameters, and |e1 (0)| < Fb (0). We define the following state transformation as the error constraint variable. e1 (t) e1 (t) q(e1 (t)) + (1 − q(e1 (t))) (12) z1 (t) = σ1 Fb − e1 σ2 Fb + e1 7
where σ1 and σ2 are proper parameters to modulate the tracking error boundaries. Fb satisfies the condition given in (11) and q(e1 (t)) satisfies 1 e1 (t) ≥ 0 q(e1 (t)) = (13) 0 e1 (t) < 0 The time derivative of z1 is: 1 [e˙ 1 (σ1 Fb − e1 ) − e1 (σ1 F˙ b − e˙ 1 )]q (σ1 Fb − e1 )2 1 [e˙ 1 (σ2 Fb + e1 ) − e1 (σ2 F˙ b + e˙ 1 )](1 − q) + (σ2 Fb + e1 )2
z˙1 =
(14)
Through some computation, we obtain
where ΦF =
z˙1 = Fb ΦF e˙ 1 − F˙ b ΦF e1
(15)
σ1 q σ2 (1 − q) + (σ1 Fb − e1 )2 (σ2 Fb + e1 )2
(16)
Remark 4: From the definition of z1 and q(e1 (t)), we can know that when tracking error approaches boundary Fb , z1 increases consequently. We design controller such that the resulting closed-loop system is stable in the sense of semiglobal uniform ultimate boundedness. Then, the tracking error will be forced within the prescribed boundaries, i.e. −σ2 Fb < e1 (t) < σ1 Fb . In addition, there is one further point to make, it means that the introduced state transformation will result in z1 = 0 only at e1 = 0. Compared with the method proposed in [9], we leave out the procedure of seeking the inverse function, which simplifies the computation complexity. Furthermore, compared with [12], parameters σ1 and σ2 make the boundary tunable. 5. Controller Design and Stability Analysis In this section, an adaptive RBF neural network controller is developed in the framework of backstepping method. System stability will be proved at the same time. We choose the Lyapunov function as V = Ve + V1 + V2 8
(17)
with Ve = e(t)T P e(t)
(18)
V1 = V11 + V1i + V1n ⎧ 1 2 ⎪ z + 12 θT Λ−1 θ ⎨V11 = 2 1 1 2 V1i = n−1 i=2 2 zi ⎪ ⎩ V1n = 12 zn2 + 12 δ2
(19) (20)
t q1 V2 = e1 (ξ)H12 (e1 (ξ))dξ 1 − d1 t−d1 (t) t n q2 e1 (ξ)Hi2 (e1 (ξ))dξ + 1 − di t−di (t) i=2
(21)
where q1 , q2 are positive scalars to be defined later. The time derivative of Ve along (18) is V˙ e = 2eT P (Ae + G(x, xˆ) + F ) x)) + 2eT P F = eT (P A + AT P ) + 2eT P (g(x) − g(ˆ with Assumption 1 and Young’s inequality, we get V˙ e ≤ eT (P A + AT P + c1 P P + 2c2 P P + c−1 1 + c−1 2
n i=2
fi2 + c−1 2
n
n
ρ2i I)e
i=2
ki2 f12
(22)
i=2
where c1 and c2 are positive design parameters. Then we define the states transformation as follows zi = λi (t) − αi−1 i = 2, 3, ...n − 1 zn = λn − αn−1 −
(23)
where αi are virtual control and is the auxiliary state to be designed to deal with the saturation input. Next, we use the backstepping method to design the controller. step 1: Taking the time derivation of V11 yields ˙ V˙ 11 = z1 z˙1 − θT Λ−1 θˆ 9
(24)
where θˆ = θ∗ − θ is the estimation of θ∗ . Substuting (15) into (24), we obtain ˙ V˙ 11 = z1 Fb ΦF (x2 + g1 + f1 − y˙ d ) − z1 F˙ b ΦF e1 − θT Λ−1 θˆ = z1 Fb ΦF (z2 + α1 + k2 x1 + e2 + g1 + f1 − y˙ d ) ˙ − z1 F˙ b ΦF e1 − θT Λ−1 θˆ + z1 ω(e1 ) − z1 ω(e1 )
(25)
where ω(e1 ) are unknown nonlinear functions to be specialized later. We use RBF neural network to approximate the functions, i.e z1 ω(e1 ) = z1 (θ∗T ϕ(e1 ) + ε)
(26)
taking some computation, one has 1 1 z1 ω(e1 ) ≤ z1 θT ϕ(e1 ) + z1 θˆT ϕ(e1 ) + z12 + ε2 2 2 select the virtual control and adaptive law as F˙ b e1 c11 1 α1 = −k2 x1 − g1 + y˙ d + − + z1 Fb ΦF Fb 2 4 ˆ 1) c1 z1 + 12 θϕ(e − − Fb ΦF Fb ΦF ˙ θˆ = z1 Λϕ(e1 ) − Λθˆ
(27)
(28) (29)
Substituting (27), (28), (29) into (25), we have 1 2 2 T ˆ 1 2 V˙ 11 ≤ z1 z2 Fb ΦF − c1 z12 + c−1 11 e2 + f1 + θ θ + ε − z1 ω(e1 ) 2 2
(30)
It has been proved that the virtual control α1 is piecewise continuously differentiable with respect to z1 over the condition e|(0)| < Fb (0) in [12]. On the other hand, dynamic surface control(DSC) technique may be employed to get the time derivative of α1 via an first-order filter. step 2: The derivative of V12 is V12 = z2 (λ˙ 2 − α˙ 1 )
∂α1 ∂α1 = z2 z3 + α2 + k3 x1 + gˆ2 − k2 (g1 + λ2 + k2 x1 ) − − y˙ ∂t ∂y 10
(31)
1 through some computation on the term of − ∂α y, ˙ we obtain ∂y
∂α1 ∂α1 y˙ ≤ −z2 (ˆ x2 + g1 ) + −z2 ∂y ∂y
c11 1 + 2 4
2 ∂α1 1 + c−1 e2 + f12 (32) z2 ∂y 2 11 2
substituting (32) into (31) yields
∂α 1 V˙ 12 ≤ z2 z3 + α2 + k3 x1 + gˆ2 − k2 (g1 + λ2 + k2 x1 ) − ∂t 2 c11 1 1 ∂α1 ∂α1 + c−1 e2 + f12 − z2 (ˆ x2 + g1 ) + + z2 ∂y 2 4 ∂y 2 11 2
(33)
select the virtual control as ∂α1 α2 = −c2 z2 − k3 x1 − gˆ2 + k2 (g1 + λ2 + k2 x1 ) + ∂t 2 c11 1 ∂α1 ∂α1 (ˆ x2 + g1 ) − + z2 − z1 Fb ΦF + ∂y 2 4 ∂y
(34)
Substituting (32), (34) into (33), we have 1 V˙ 12 ≤ z2 z3 − z1 z2 Fb ΦF + c−1 e2 + f12 − c2 z22 2 11 2
(35)
step i: Similar with step 2, the time derivative of V1i is V˙ 1i = zi zi+1 + αi + ki+1 x1 + gˆi − ki (g1 + λ2 + k2 x1 ) i−1 ∂αi−1 ∂αi−1 ∂αi−1 ˙ − y˙ λj − − ∂t ∂λj ∂y j=2 ≤ zi zi+1 + αi + ki+1 x1 + gˆi − ki (g1 + λ2 + k2 x1 ) i−1 ∂αi−1 ∂αi−1 ∂αi−1 ˙ λj − zi − (ˆ x2 + g1 ) − ∂t ∂λj ∂y j=2 2 c11 1 ∂αi−1 1 2 2 + + + c−1 zi 11 e2 + f1 2 4 ∂y 2
11
(36)
select the virtual control as αi = −ci zi − ki+1 x1 − gˆi + ki (g1 + λ2 + k2 x1 ) i−1 ∂αi−1 ∂αi−1 ∂αi−1 ˙ + (ˆ x2 + g1 ) λj + ∂t ∂λ ∂y j j=2 2 ∂αi−1 c11 1 zi − zi−1 + − 2 4 ∂y
+
(37)
Substituting (37) into (36), we have 1 2 2 V˙ 1i ≤ zi zi+1 − ci zi2 − zi zi−1 + c−1 11 e2 + f1 2
(38)
step n: The time derivative of V1n is ˙ V˙ 1n = zn (λ˙ n − α˙ n−1 − ) ˙ − δδˆ
(39)
we define the dynamic of as ˙ = − + g(u) − u
(40)
substituting (40) into (39), one obtains V˙ 1n = zn [δ(u) + + u + gˆn − kn (g1 + λ2 + k2 x1 ) ∂αn−1 ∂αn−1 ˙ ∂αn−1 ˙ − − y] ˙ − δδˆ λj − ∂t ∂λj ∂y j=2 n−1
≤ zn [δ(u) + + u + gˆn − kn (g1 + λ2 + k2 x1 ) 1 ∂αn−1 ∂αn−1 ˙ 2 2 − λj ] + c−1 − 11 e2 + f1 ∂t ∂λ 2 j j=2 c11 1 ∂αn−1 − zn (g1 + λ2 + k2 x1 ) + + ∂y 2 4 2 ∂αn−1 ˙ − δδˆ × zn ∂y n−1
(41)
We know that δ(u) is a bounded saturation approximation error function, then zn δ(u) ≤ |zn |δ(u) ≤ |zn |δ1 is bounded. We design the controller and 12
adaptive law as follows u = − − gˆn + kn (g1 + λ2 + k2 x1 ) +
∂αn−1 ∂t
n−1 ∂αn−1
∂αn−1 (g1 + λ2 + k2 x1 ) λ˙ j + ∂λj ∂y j=2 2 ∂αn−1 c11 1 ˆ zn − cn zn − δsign(z + − n) 2 4 ∂y +
˙ δˆ = |zn | − δˆ
(42)
(43)
where δˆ = δ − δ is the estimation of δ(u). Substituting (42), (43) into (41), we have 1 e2 + f12 + δδˆ (44) V˙ 1n ≤ −cn zn2 + c−1 2 11 2 Remark 5: From the description of sat(u), we know that the backstepping technique cannot be applied directly. Inspired by the work [13, 15], we approximate the saturation input by a smooth function g(u) with a bounded approximation error function. [13] did not deal with the approximation error. However, as we know the bound of input, namely, the uM , we can deal with the approximation error via adaptive method, as we did in step n. Then, we are ready to present our main result in this note. Theorem 1: For nonlinear systems (1), under Assumption 1-2 and initial condition |e1 (0)| < Fb (0), the observer-based controller (28), (34), (37), (42) with adaptive law (29), (43) and auxiliary state (40), guarantee that all signals in the closed loop systems are bounded in the presence of input saturation, and the tracking error remains in the prescribed boundaries for all t > 0. Proof: From the definition of V2 , we have q1 e12 (t)H12 (e1 (t)) − q1 e1 (t)H12 (e1 (t − d1 (t))) 1 − d1 n n q2 e1 (t)Hi2 (e1 (t)) − q2 e1 (t)Hi2 (e1 (t − di (t))) + 1 − di i=2 i=2
V˙ 2 ≤
13
(45)
With the observer (9), virtual control (28), (34), (37), controller (42) and adaptive laws (29), (43), we obtain n n T T −1 ˙ ˙ Ve + V1 ≤ e ρ2i I + c−1 e P A + A P + (c01 + 2c02 )P P + c01 11 I 2 i=2 −
n
ci zi2
+ q2
i=1
where q1 = c−1 02 q1 f12
+ q2
n i=2
n i=2
n
1 fi2 + q1 f12 − zi ω(e1 ) + θT θˆ + ε2 + δδˆ 2
(46)
ki2 + n, q2 = c−1 02 . With remark 1, we have
fi2 ≤ q1 [e1 H11 (e1 ) + e1 (t − d1 (t))H12 (e1 (t − d1 (t)))]
i=2
n [e1 Hi1 (e1 ) + e1 (t − di (t))Hi2 (e1 (t − di (t)))] + q2 i=2
where = q1 1 + q2
+ n
(47)
i=2 i ,
we let
T
P A + A P + (c01 + 2c02 )P P +
c−1 01
n
ρ2i I +
i=2
n −1 c I ≤ −Q 2 11
q1 e1 (t)H11 (e1 (t)) + q1 e1 (t)H11 (e1 (t)) 1 − d1 n n q2 e1 (t)Hi1 (e1 (t)) + q2 e1 (t)Hi1 (e1 ) + 1 − di i=2 i=2
(48)
z1 ω(e1 ) =
(49)
then, from (46), (47), (48), (49), we come to the result V˙ ≤ −eT Qe −
n i=1
we know that
1 ci zi2 + θT θˆ + ε2 + δδˆ + 2
1 2 1 ∗ 2 θT θˆ ≤ − θ + θ 2 2 1 1 δδˆ ≤ − δ2 + δ 2 2 2 14
(50)
(51) (52)
We define η = 12 θ∗ 2 + 12 δ 2 + 12 ε2 + , substituting (51), (52) into (50), one has n 1 2 1 2 T ˙ V ≤ −e Qe − ci zi2 − θ − δ +η (53) 2 2 i=1 Thus completes the proof. Remark 6: In this paper, we introduce a error constraint variable z1 (t) to address error constraint σ1 Fb < e1 < σ2 Fb . σ1 and σ2 are parameters to modulate the prescribed boundaries. [12] omits them and the time-delay free systems it considered do not contain the input constraint. Its error could only fall between symmetrical boundaries. However, this paper considers a more complex condition, so that the controller design and the stability of the closed-loop system are more challenging. Remark 7: It should be pointed out that we design a dynamic output feedback controller with the constructed observer, adaptive laws and the auxiliary states. While, from an implementation perspective, static outputfeedback control law is more welcome. There are many results on static output-feedback control, such as [23, 24] and the references therein. It is meaningful and challenging to extend the proposed method to the static output-feedback case, which can be set as a future direction. 6. Simulation Example In this section, an example will be used to illustrate the validity of the control approach proposed in this paper. Consider a nonlinear time-delay system as follows ⎧ ⎪ ⎨x˙ 1 = x2 + 0.3x1 + cos(t)x1 (t − d1 (t)) (54) x˙ 2 = sat(u) + 0.5x2 + 0.5x2 sin(x1 ) + sin(t)x1 (t − d2 (t)) ⎪ ⎩ y(t) = x1 (t) where the time varying delay d1 (t) = d2 (t) = 0.4(1 + sin(t)). We select k2 = 3.5 to satisfy inequality (48), consequently, we have the reduced-order observer as follows λ˙ 2 = sat(u) − 11.55x1 − 3λ2 + 0.5λ2 sin(x1 ) + 1.75x1 sin(x1 )
(55)
Then, the performance boundary function is selected as Fb = ξ0 e−at + ξ∞ 15
(56)
with ξ0 , a, ξ∞ are chosen as 0.9, 7, 0.17, respectively. Parameters σ1 and σ2 in z1 are specified σ1 = 1, σ2 = 0.3. According to the selected parameters, the virtual control is constructed as α1 = −3.8x1 + cos(t) +
F˙ b e1 10.5z1 θˆT ξ(e1 ) − 0.75z1 Fb ΦF − − Fb Fb ΦF Fb ΦF
(57)
˙ and adaptive laws θˆ are ˙ θˆ = z1 ξ T (e1 ) − θˆ
(58)
where ⎡ ⎢ ⎢ ξ (e1 ) = ⎢ ⎢ ⎣ T
2
e−0.5∗(e1 −8) 2 e−0.5∗(e1 −4) 2 e−0.5∗(e1 ) 2 e−0.5∗(e1 +4) 2 e−0.5∗(e1 +8)
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(59)
the constructed controller is u = − + 3λ2 + 11.55x1 − 0.5(λ2 + 3.5x1 )sin(x1 ) ∂α1 ∂α1 ∂α1 2 + (λ2 + 3.8x1 ) − 0.75( ) z2 + ∂t ∂t ∂t
(60)
The initial values are chosen as x1 (0) = 0.45, x2 (0) = 0.25, λ2(0) = −0.1.
(61)
The simulation results of the closed loop control system are illustrated in Fig.2-Fig.6. The response of system output and tracking signal is shown in Fig.2, while Fig.3 shows the tracking error and the symmetrical boundaries. Fig.4 and Fig.5 provide the response of adaptive laws of the system. Fig.4 ˙ ˙ demonstrates the adaptive laws of θˆi and Fig.5 the δˆ0 . The control signal u and applied control input sat(u) are shown in Fig.6. Here we set the uM = 5. We design parameters uM , Fb to tune the input saturation boundary and the transient and steady performance of output. Fig.7 and Fig.8 show a rapid response of the system by decreasing the parameter −a of Fb . We can see that the system tends to be stable rapidly, and the output can track the reference signal using less time than Fig.2. It should be pointed out 16
1.5 Y Yd 1
0.5
0
−0.5
−1 0
2
4
6
8
10
Figure 2: system output and tracking signal
1.2 upper bound lower bound tracking error
1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 0
2
4
6
8
10
Figure 3: tracking errors with prescribed boundaries
0.35 ˆ1 θ ˆ2 θ ˆ3 θ ˆ4 θ ˆ5 θ
0.3 0.25 0.2 0.15 0.1 0.05 0 0
2
4
6
Figure 4: response of adaptive laws θˆi
17
8
10
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
2
4
6
8
10
Figure 5: response of adaptive law δˆ
30 u sat(u)
20 10 0 −10 −20 −30 −40 −50 0
0.2
0.4
0.6
0.8
Figure 6: control signal u and control input sat(u)
18
1
1.5 Y Yd 1
0.5
0
−0.5
−1 0
2
4
6
8
10
Figure 7: rapid output response and tracking signal
upper bound lower bound tracking error
0.8
0.6
0.4
0.2
0
−0.2 0
2
4
6
8
10
Figure 8: tracking errors with prescribed boundaries
that the constraint of input and output could be arbitrarily small in theory. However, when the constraint is too restrictive, it may lead the failure of the controller. From this point, the controller is weak. In practical engineering, we can choose proper parameters to reach the desired result. 7. Conclusion In this paper, we investigate the adaptive control for a class of time-delay systems with tracking error and input constraint. The delay independent reduced order observer is designed at first. Then, a novel error constraint variable is introduced to ensure a prescribed tracking performance. Next, a output feedback controller is constructed by the well known backstepping 19
method step by step, during which, RBF neural networks are used to approximate the unknown nonlinear functions. Finally, a simulation example is performed to verify the effectiveness of the proposed method.
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