Research Article
Adaptive Fuzzy Backstepping Control against Actuator Faults
College of Information Science and Engineering, Huaqiao University, China
Fujiang Jin
College of Information Science and Engineering, Huaqiao University, China
The design and development of Fault-Tolerant Control (FTC) have attracted more and more attention in recent years. Since, the active FTC can usually obtain better control performance, many researchers are interested in this branch of FTC. Adaptive method is widely used for FTC design because of its ability of on-line updating. For linear systems, Tao et al. (2004) presented adaptive FTC against lock-in-place actuator faults, Ye and Yang (2006) developed an adaptive FTC method and Ye and Yang (2006) and Yang and Ye (2006) deal with loss of effectiveness of actuator. Tang et al. (2005,2007) presented some adaptive control methods for nonlinear systems which are subject to lock-in-place actuator faults. However, they are applicable to the systems with only parameter uncertainties, when there are structural uncertainties in the controlled systems, they are failed for control purpose.
Since, Wang (1993) proved that fuzzy logic systems are universal approximators and proposed stable adaptive fuzzy control for unknown nonlinear systems in (Wang, 1993), some researchers began to develop FTC scheme based on fuzzy approximation theory, with which structural uncertainties were considered. Polycarpou and Helmicki (1995) proposed a general framework for FTC design using learning approximation Polycarpou et al. (2004), Mao et al. (2006) and Xue and Jiang (2006) developed FTC methods with neural networks and Zhang et al. (2004, 2006) did with fuzzy logic systems. These results were obtained on the basis of FDD mechanism. Diao and Passino (2001) and Li and Yang (2008a) presented adaptive fuzzy FTC without FDD but the controlled nonlinear systems are required to satisfy the matching conditions. As the control design, Yang and Zhou (2005), Tong (2006), Zhang et al. (2005) and Chen et al. (2009) removed the above restriction by embedding adaptive fuzzy approximators in backstepping procedures. Li and Yang (2008b, 2009b, c) developed FTC against actuator faults with backstepping adaptive fuzzy approaches. Then in order to further release the design conditions, Li and Yang (2009b, c) proposed adaptive FTC scheme with nonlinearly parameterized fuzzy approximators where the fuzzy basis functions of the approximators need not to be completely known. However, the on-line updating of parameters brings much computation burden to the controllers.
In this study, a novel adaptive fuzzy FTC scheme is proposed to reduce the computation burden. By employing nonlinearly parameterized fuzzy systems to approximate the unknown functions in a backstepping procedure, not only the controlled system does not require to satisfy the matching conditions but also the fuzzy basis functions need not be completely known. Then the adaptive laws are synthesized to update the 2-norm of on-line parameter vectors. Thus only a few parameters need to be updated which reduces the computation burden considerably. The FTC scheme can guarantee the closed-loop stability and desired output tracking performance of the controlled system against both lock-in-place and loss of effectiveness faults of the actuators. A simulation example is also included to show its effectiveness.
PROBLEM STATEMENT AND PRELIMINARIES
Consider the following nonlinear system with m actuators:
(1) |
where, xi and di are the state variable and external disturbance, respectively, , with Ωi being a compact set in Ri, u = [u1, u2,..., um]T εRm is the input vector whose components may fail, yεR is the output, and gnj for i = 1, 2,..., n-1, j = 1, 2,..., m are unknown continuous nonlinear functions, gi and gnj are smooth. The state xi is measurable.
The actuator faults considered are lock-in-place and loss of effectiveness whose models are given as follows.
• | Lock-in-place fault: |
(2) |
where, is a constant standing for the place at which the jth actuator is locked, 1≤p≤m-1 is the number of the locked actuator, tj is the time instant when the fault occurs.
• | Loss of effectiveness fault: |
(3) |
where, vi (t) is the applied control input, ti is the instant of fault-occurrence. is an indicator showing the effectiveness of the ith actuator, is the lower bound of ρi. When , the ith actuator is normal.
The control objective of this study is to design a FTC control law for system (1) to ensure that all signals in the closed-loop system remain bounded and the output y(t) tracks the given reference signal ym(t) as closely as possible though there are unknown actuator faults (5) and (6). The reference signal ym(t) meets the following assumption.
Assumption 1: ym (t) and , i = 1, 2,.., n, are continuous and bounded, where, is the ith order derivative of ym (t).
The system functions gi and gnj (i = 1, 2,.., n-1, j = 1, 2,.., m) and the external disturbances di (i = 1, 2,.., n) satisfy the following assumptions, respectively.
Assumption 2: There exist some unknown constants gi0 and gnj0 such that
Assumption 3: There exist positive smooth functions such that , however is not required to be known.
Remark 1: Without loss of generality, we further assume that for i = 1, 2,.., n-1, j = 1, 2,.., m.
Taking the actuator faults (2) and (3) into account, one has:
(4) |
with v(t) = [v1 (t)vm (t)]T, and ρ = diag {ρ1, ρ2,..., ρm}, β = diag {β1, β2,..., βm}, where, βj = 1 if and βj = 0 otherwise.
Assumption 4: System (1) is so constructed that it can be driven to track the reference signal ym (t) as long as Rank (β)<n and for .
Fuzzy logic system can be expressed in the following form:
(5) |
where, is the membership function of which is generally chosen as:
(6) |
with and denoting the width and center, respectively and pi = maxyεR μBi (y). Denote p = [p1, p2,..., pN]T and ξ (x) with:
Then the fuzzy logic system (5) can be rewritten as:
(7) |
Lemma 1: Wang (1993): For any given real continuous function F (x) on a compact set Ω⊆Rn, there exists a fuzzy logic system in the form of (10) such that ∀ε>0:
(8) |
Remark 2: If the parameters and given in Eq. 6 are completely known, the fuzzy logic system (5) is linearly parameterized because only the weights pi need to be updated on-line. When and are unknown, the fuzzy logic system is nonlinearly parameterized.
Define parameter vectors and . Then the optimal parameter vectors p*, c* and σ* are defined as:
CONTROL DESIGN AND ANALYSIS
With the solution to the nominal plant u0, the controller can be constructed as:
(9) |
with being continuous functions which satisfy , are the lower-bound values of the functions . Suppose there are p control signals uj1, uj2,..., ujp cannot be available during (tk, tk+1), k = 0, 1,..., q, 0≤q≤m-1. Meanwhile the other actuators may lose their effectiveness partly, that is ui (t) = ρivi, I≠j1, j2,..., jp, as long as ρi ε[ρi, 1]. From Eq. 4 and 9, one can get that:
(10) |
with u0 being designed by the following backstepping procedure.
Step 1: Define z1 = x1-ym and
Take the derivative of , one can get:
(11) |
From Assumption 3 and triangular inequality, it can be seen that:
(12) |
Where,
α1 is the virtual control designed later, z2 = x2-α1, η11 is a positive constant defined by designer and with Ωz1 being a compact set in R3. Denote that pi, ci, σi and ξi (Zi), i = 1, 2,..., n, are the corresponding parameters and the basis functions of the fuzzy approximator in the ith step of the backstepping procedure and p*, c*, σ* are their optimal values, respectively. From Lemma 1, we have:
(13) |
with ε1 being an arbitrarily small positive constant. Define:
and:
(14) |
where, Mi is the total number of the rules for the fuzzy approximator in step 1. For convenience, denotes the short form of throughout the following statement. Then:
(15) |
where, the fact is used, a1 and b1 are positive constants.
By Taylor series expansion of at (Z1, 0, 0), one gets that:
(16) |
where, is the error term. Define:
and:
for i = 1, 2, ,..., n. Then can be formulated as:
(17) |
with l1 and r1 being positive constants determined by designer. Take Eq. 17 into Eq. 16, one gets:
(18) |
Then Eq. 14 can be rewritten as:
(19) |
Design the virtual control α1 and on-line updating laws for θ1, φ1 and δ1 as:
(20) |
(21) |
where, and are positive constants, θ1, φ1 and δ1 are the estimation of and , respectively. By choosing Lyapunov function V1 as:
(22) |
with , one can get:
(23) |
Where,
will be dealt with in the next step.
Step i (2≤i≤n-1): Likewise, the virtual control αi and the updating laws are designed as:
(24) |
(25) |
Choose Lyapunov function Vi as:
(26) |
Take the derivative of Vi, one can obtain that:
(27) |
Step n: Under the control structure (9) and following the above procedure, we design:
(28) |
(29) |
Theorem 1: The proposed FTC scheme with the structure (9), the control law (28) and the adaptive laws (29), together with the virtual control laws (20) and (24), as well as the adaptive laws (21) and (25), can guarantee the following properties for system (1), though there are structural uncertainties and actuator faults.
• | All signals in the closed-loop system remain bounded only if the initial conditions are bounded |
• | The tracking error z1 = x1-ym = y1-ym converges to an arbitrarily small neighborhood of the origin. |
Proof: Similar to the design of Step 1, choose Lyapunov function as:
(30) |
(31) |
Where,
By selecting:
and
one can rewrite as follows:
(32) |
Fig. 1: | The curves of y and ym |
Fig. 2: | The curves of u1 and u2 |
From Eq. 32, one can deduce that all signals of the closed-loop system remain bounded only if the initial conditions are bounded and the tracking error satisfies:
Since μ and γ are determined by designer and can be chosen arbitrarily small, thus z1 converges to an arbitrarily small neighborhood of the origin. The results of theorem 1 holds in (0, t1) and in turn, in (t1, t2),..., (tq, t∞), thus the results of Theorem 1 can be obtained in (o, +∞).
SIMULATION EXAMPLE
An unknown nonlinear system is considered here:
(33) |
The actuator faults introduced for simulation are u1 (t) = -5 when t≥4 and u2 (t) = 0.4v1 for t≥11. For simulation study, we choose
Choose ym = 3+sin (0.5t)+0.5sin (1.5t), k1 = 5, k2 = 15, M1 = 7, M2 = 49, a1 = a2 = 1000, b1 = b2 = 3000, l1 = l1 = 500, r1 = r2 = 100, . The initial conditions are given as x (0) = (3.05, 1.25)T, (θ1 (0), φ1 (0), δ1 (0))T = (θ2 (0), φ2 (0), δ2 (0))T = (0, 0, 1)T. The simulation results are shown in Fig. 1 and 2.
In present study, a novel FTC scheme is proposed to deal with both lock-in-place and loss of effectiveness actuator faults for a class of unknown nonlinear systems. With adaptive fuzzy approximators, the FTC scheme is designed without resort to FDD mechanism. So the undesired results caused by false alarms, miss alarms and the time-delays from detection to alarm are avoided. The fuzzy approximators employed are nonlinearly parameterized which removes the restriction that the fuzzy basis functions must be completely known before control design. Besides, the computation burden is not large by appropriately design the parameter updating laws. Therefore the proposed scheme is more practical for FTC. It is proved in theory and shown in simulation that the FTC scheme can guarantee the closed-loop stability and good tracking performance of the controlled system though there are structural uncertainties and unknown actuator faults.