IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014
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Partial Tracking Error Constrained Fuzzy Dynamic Surface Control for a Strict Feedback Nonlinear Dynamic System Seong I. Han, Member, IEEE, and Jang M. Lee, Senior Member, IEEE
Abstract—As an extension of the conventional output constraint problem, this paper proposes the use of nonlinear dynamic surface control (DSC) combined with adaptive fuzzy logic control to constrain the partial tracking errors of strict feedback nonlinear dynamic systems with uncertainties. Transformation of the tracking errors into new virtual error variables is used to recursively design a DSC controller, and a careful selection of the controller design parameters ensures that partial state tracking errors are confined at all times within the prescribed bounds. The stability and boundedness of all closed-loop signals were confirmed using the Lyapunov theorem. The simulation results highlight the efficacy and utility of the proposed control scheme. Index Terms—Adaptive fuzzy control, dynamic surface control (DSC), partial state tracking error constraint, prescribed performance function, strict feedback nonlinear dynamic system.
I. INTRODUCTION VER the past two decades, adaptive nonlinear control for a variety of nonlinear dynamic systems has attracted considerable attention as a means of coping with the rapid technical advances in sophisticated industrial equipment. As a recursive controller design, adaptive backstepping control [1] provides a systematic procedure for designing the stabilizing controllers used in nonlinear systems with uncertainties by guaranteeing global stability and the tracking properties. Such control also removes the restriction of the matching conditions and the useful cancellations of nonlinearities as compared with feedback linearization techniques. Fuzzy logic control (FLC) and neural network (NN)-based robust adaptive backstepping [2]–[9] have been investigated as a means of dealing with various unknown parameters and modeling uncertainties. Various studies have contributed to the popularity and rapid application of backstepping control. On the other hand, certain coupling issues between the design steps have been encountered owing to the repeated differentiation of the intermediate variables, which
O
Manuscript received February 15, 2013; revised June 14, 2013; accepted July 28, 2013. Date of publication August 23, 2013; date of current version October 2, 2014. This work was supported by the Ministry of Trade, Industry and Energy, Korea, under the Human Resources Development Program for Special Environment Navigation/Localization National Robotics Research Center Support Program supervised by the National IT Industry Promotion Agency (H1502-13-1001). The authors are with the Department of Electronic Engineering, Pusan National University, Busan 609-735, Korea (e-mail:
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TFUZZ.2013.2279543
greatly increases the complexity of the controller. Dynamic surface control (DSC) [10], [11] was, therefore, developed as an alternative to backstepping control to overcome the problem of an explosion of terms encountered when using a first-order filter for the synthetic input at each step of the backstepping design procedure. Because their controller has a simpler structure than the backstepping-based controller, several adaptive DSCs combined with FLC and NNs have been developed [12]–[14]. In most uncertain models, however, although the approximations are utilized through FLC or NNs, undesirable transient behaviors can appear in the early stages of parameter adaptation owing to the large initial parameter estimation errors. This problem can be typically averted through an appropriate control gain selection and by providing sufficiently small initial estimation errors. Bounding the transient performance and guaranteeing the convergence of steady-state tracking errors to a residual set by adjusting only the control gains are fundamentally difficult to accomplish using traditional backstepping and DSC-based adaptive controls without depending on a trial-and-error process, even for cases involving known nonlinearities. In this paper, we propose a prescribed performance constraint (PPC) method that overcomes the aforementioned difficulties encountered in the regulation of the time-domain performances. This proposed method guarantees that the maximum overshoot, convergence rate, and steady-state size of the tracking errors are consistently constrained within certain prescribed values provided that the stability of the transformed error system is assured. Using a devised transformed error variable derived from several specific conditions for a prescribed performance function, we designed a robust adaptive controller based on the DSC scheme combined with adaptive FLC [12], [24] for the purpose of compensating for unknown uncertainties. The tracking errors for the states and intermediate stabilizing functions were then ensured to fall within the prescribed bounds, and all other signals of the closed-loop systems were continuously bounded into a small residual set. Similar error constraint methods include the PPC method combined with backstepping control, as proposed by Rovithakis et al. [15]–[17]. Their methods used a tangent hyperbolic function as the transformation function, which is combined with a prescribed smooth function to transform any existing tracking errors. The controller was designed using the inverse of the transformation function used to obtain the transformed error. However, when a fast transient convergence rate is selected under a constraint condition, the PPC may be violated and an actuator saturation problem attributed to a singularity may arise.
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014
Ilchman et al. [18], [19] proposed the used of funnel control as a PPC method. This method was subsequently extended for use in more general systems by Hackl et al. [20], [21]. However, the application of funnel control is limited to linear or nonlinear class S systems with relative degrees of one or two, stable zero-dynamics, and a known high-frequency gain sign. A state constraint method adopting a barrier Lyapunov function (BLF) also was developed [22]–[25]. In a BLF constraint method, however, the BLF controller was redesigned to accommodate the change in Lyapunov function and establishment of the stability of the closed-loop system and bound condition parameters. In addition, because a piecewise smooth BLF was also adopted in an asymmetric BLF design, extra effort is required to ensure the continuity and differentiability of the piecewise smooth stabilizing functions. The main advantages of our proposed constraint method over conventional methods are that the proposed prescribing performance technique does not use a complex transformation procedure, thereby avoiding singularity problems, and is not limited by the system class found in previous techniques [15]–[21]. Furthermore, an output tracking constrained problem using the proposed constraint method in this paper can be extended to partial tracking error constrained controller similar to that of Rovithakis et al. [17]. As a special case, the proposed partial tracking error constraint can, therefore, incorporate both full-state constraint and an output constraint. The selection of the control parameter, which guarantees that partial tracking errors remain within the prescribed tracking error bounds, is considerably difficult to achieve compared with the selection of the output constraint because the parameters of the prescribed constraint functions must be defined to accommodate the physical relationships between the states and intermediate stabilizing functions. This issue has not been addressed in the partial state feedback control [17], and therefore, remains a challenging problem in the determination of the optimal control parameter values for the partial tracking error constraint problem. For this paper, the control and adaptive gains were carefully selected to ensure that they do not violate the constrained conditions or closed-loop stability of the given control system. As an application example, an interconnected inverted pendulum was simulated to verify the efficacy of the proposed control scheme.
where n1 + · · · + nm = n; xi,j i represent the states of the ith subsystem; x = [xT1,n 1 , . . . , xTm ,n m ]T ∈ Rn are the state vectors; x ¯i,j i = [xi,1 , . . . , xi,j i ]T ∈ Rj i ; ui ∈ R and yi ∈ R denote the input and output of the jth subsystem, respectively; fi,j i (·) and gi,j i (·) are smooth linear or nonlinear functions that depend only upon x ¯i,j i (and are strictly feedback); and Fu i,j i are the bounded uncertainties including the nonlinear friction, external disturbances, and coupled dynamic terms. The strict feedback system formation of (1) should be constructed to develop a recursive control design for backstepping and DSC systems. Most mechanical dynamic systems, such as robotic manipulators and servo systems can be built into this type of system. The states of the ith subsystem are partitioned into the states x ¯i,j l = [xi,1 , xi,2 , . . . , xi,n c ]T ∈ Rn c for the constrained errors, and free states x ¯i,j k = [xi,n c +1 , xi,n c +2 , . . . , xi,n i ]T ∈ Rn r for the unconstrained errors, where nc + nr = ni and the number sequences are ascending. The constrained errors for the states, x ¯i,j l (t), must remain in the set |ei,j l (t)| < ρi,j l (t), jl = 1, . . . , nc ∀t ≥ 0. The partial constraint problem is a generalized ¯i,j i and x ¯i,j k = 0 provide a full error conone because x ¯i,j l = x ¯i,j k = [xi,2 , . . . , xi,n i ]T straint problem, and x ¯i,j l = xi,1 and x correspond to the output tracking error constraint problem. The control objective for a nonlinear dynamic system is to determine a state feedback control system such that the system states, xi,j l , can track the desired trajectory, zi,j l , while ensuring that all closed-loop signals are bounded and that the partial error constraints are not violated. Assumption 1: The signs of the control gain functions gi,j i (·), ji = 1, . . . , ni , are known, and positive constants 0 < gi,j i m in ≤ gi,j i m ax exist such that gi,j i m in ≤ |gi,j i | ≤ gi,j i m ax . If the sign of the control gain function, gi,j i (·), is unknown, the Nussbaum function method can be considered [26]. Such a case was excluded from this paper. Assumption 2: |Fu i,j i | are bounded such that unknown positive constants, |Fu i,j i | ≤ σu i,j i , ji = 1, . . . , ni , and i = 1, . . . , m exist. Assumption 3: The desired trajectory vectors are continuous and available, and with a known compact set Ωz i,j i = {[zi,j i , 2 2 2 + z˙i,j + z¨i,j ≤ δz i,j i }, where δz i,j i > 0 is z˙i,j i , z¨i,j i ]T : zi,j i i i a constant. B. Performance Function and Error Transformation
II. PROBLEM FORMULATION
A smooth decreasing performance function ρi,j l (t) : R+ → R+ /{0} with limt→∞ ρi,j l (t) = ρ∞i,j l can be defined to satisfy the partial error constraints as follows:
A. Description of the Nonlinear Multiple-Input and Multiple-Output Dynamic System Consider a multiple-input and multiple-output (MIMO) strict feedback nonlinear system in the presence of uncertainty, whose dynamic equation can be described as xi,j i ) + gi,j i (¯ xi,j i )xi,j i +1 + Fu i,j i x˙ i,j i = fi,j i (¯ ji = 1, . . . , ni − 1
ρi,j l (t) = (ρ0i,j l − ρ∞i,j l )e−a i , j l t + ρ∞i,j l
where ρ0i,j l , ρ∞i,j l , and ai,j l are appropriately defined positive constants. The prescribed transient and steady-state bounds are then guaranteed using the following constraint conditions:
.. .
i = 1, . . . , m
−δi,j l ρi,j l (t) < ei,j l (t) < ρi,j l (t)
if ei,j l (0) ≥ 0
(3)
−ρi,j l (t) < ei,j l (t) < δi,j l ρi,j l (t)
if ei,j l (0) < 0
(4)
or
x˙ i,n i = fi,n i (¯ xi,n i ) + gi,n i (¯ xi,n i )ui + Fu i,n i yi = xi,1 ,
(2)
(1)
HAN AND LEE: PARTIAL TRACKING ERROR CONSTRAINED FUZZY DYNAMIC SURFACE CONTROL
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Proof: First, the prescribed constraint conditions (3) and (4) ⇒ 0 < ξj,1 < 1 are proven from the initial conditions ei,j l (0) and the preceding statements for several conditions. Case I (ei,j l (0) ≥ 0): The constraint can be written as −δi,j l ρi,j l (t) < ei,j l (t) < ρi,j l (t) from (3). Then, η¯i,j l = ρi,j l (t) and η i,j (t)= − δi,j l ρi,j l (t). l 1) If ei,j l (t) ≥ 0, then ηi,j l = η¯i,j l = ρi,j l (t). Given that ei,j l (t)/ρi,j l (t) > 0, the following condition is obtained from (3) and (5): 0 < ξi,j l < 1.
(10)
2) If ei,j l (t) < 0, then ηi,j l = η i,j = −δi,j l ρi,j l (t). Because l ei,j l (t)/ − δi,j l ρi,j l (t) > 0, the following condition is obtained from (3) and (5): 0 < ξi,j l < 1.
(11)
Case II (ei,j l (0) < 0): The constraint can be selected as −ρi,j l (t) < ei,j l (t) < δi,j l ρi,j l (t) from (4). 1) If ei,j l (t) > 0, then ηi,j l = η¯i,j l = δi,j l ρi,j l (t). Because ei,j l (t)/δi,j l ρi,j l (t) > 0, the following condition is obtained from (4) and (5): 0 < ξi,j l < 1. Fig. 1. Behavior of the prescribed performance function. (a) ei , j l (0) ≥ 0. (b) ei , j l (0) < 0.
where 0 < δi,j l ≤ 1 are the prescribed scale parameters. Fig. 1 illustrates the transient performance function. The constant ρ∞i,j l confines the size of the tracking errors under a steady state. The decreasing rate of ρi,j l (t) regulates the required speed of convergence of the tracking errors, and the maximum overshoot and undershoot can be controlled by selecting δi,j l ρi,j l (0) and ρi,j l (0). Next, the transformed constraint error, ξi,j l , can then be defined as follows: (5)
ηi,j l = q η¯i,j l + (1 − q)η i,j
l
(6)
where q = 1 if ei,j l (t) ≥ 0, and q = 0 if ei,j l (t) < 0. The parameters η¯i,j l and η i,j can be defined as follows: l
η¯i,j l = ρi,j l (t) η i,j (t) = −δi,j l ρi,j l (t)
if ei,j l (0) ≥ 0
(7)
l
η¯i,j l = δi,j l ρi,j l (t) η i,j (t) = −ρi,j l (t)
2) If ei,j l (t) < 0, then ηi,j l = η i,j = −ρi,j l (t). Because l ei,j l (t)/ − ρi,j l (t) > 0, the following condition is obtained from (4) and (5): 0 < ξi,j l < 1.
(13)
Proving that 0 < ξi,j l < 1 ⇒ the prescribed constraint conditions, (3) and (4), are straightforward using the reverse of the aforementioned procedure. Thus, from (10)–(13), (9) is proven. C. Function Approximation Using a Fuzzy System
ei,j l (t) ξi,j l (t) = ηi,j l (t)
(12)
if ei,j l (0) < 0.
R(l) : IF x1 is F1l and . . . and xn is Fnl then y is y¯l , l = 1, 2, . . . , M
(8)
l
Theorem 1: The following condition of the transformed error that is defined in (5): 0 < ξi,j l < 1 ∀t > 0
The basic configuration of a fuzzy system is comprised of a fuzzifier, fuzzy rule base, fuzzy inference engine, and a defuzzifier. The fuzzy inference engine maps an input linguistic vector x = [x1 , . . . , xn ]T ∈ Rn onto an output linguistic scalar variable y ∈ R. The fuzzy rule base consists of a collection of fuzzy IF–THEN rules. The lth IF–THEN rules can be described as follows:
(9)
holds if and only if ρ0i,j l , ρ∞i,j l , ai,j l , and δi,j l are selected such that they satisfy (3) and (4).
(14)
where Fil , i = 1, . . . , n are the fuzzy sets, y¯l is the fuzzy singleton for the output of the ith rule, and M is the number of rules in the fuzzy rule base. The output of a fuzzy system with a center-average defuzzifier, product inference and singleton fuzzifier can be expressed as M l n ¯ l=1 y i=1 μF il (xi ) y(x) = (15) M n l=1 i=1 μF il (xi )
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014
where μF il (xi ) is the degree of membership of xi in Fil . Equation (15) can be rewritten as follows: y(x) = WyT Ξ(x) 1
(16)
where Γi,j i = diag(γw i,j i ) > 0, i = 1, . . . , m, ji = 1, . . . , ni are a γ(·) > 0, and ψi,j i > 0 are constants. By setting the tracking command, ydi,1 = zi,1 , the surfaces of the error and virtual errors can be defined as follows:
M T
y , . . . , y¯ ] is an adjustable parameter vecwhere Wy = [¯ tor grouping all consequence parameters, and Ξ(x) = [Ξ1 , . . . , ΞM ]T is a set of fuzzy basis function defined as follows: n i=1 μF l (xi ) l . Ξ (x) = i (17) M n l=1 i=1 μF il (xi ) A fuzzy logic system has been proven to uniformly approximate any nonlinear continuous function to an arbitrary degree of accuracy if sufficient rules are provided. Therefore, a fuzzy logic system performs a universal approximation in the sense that, given any real continuous functions, f (·) : Rn → R on a sufficiently large compact set, Ωy ⊂ R, and arbitrary εm > 0, there is a fuzzy logic system y(x) in the form of (16) such that sup |f (x) − y(x)| ≤ εm .
(18)
x∈Ω
The functionf (x)can be expressed as (19)
∗
where |ε | ≤ εm , ε is the error of the fuzzy approximation and Wy∗ is chosen as the value of Wy that minimizes the fuzzy approximation error, ε∗ , i.e. Wy∗ = arg min (20) sup |f (x) − WyT Ξ(x)| . W o ∈R M
x∈Ω
ˆ y , which Because Wy∗ is unknown, it will be replaced by W ∗ is an estimation of Wy . The adaptation law is necessary to ˆ y , online to asymptotically minimize update the parameter, W the reference tracking error. The optimal fuzzy output function can be rewritten as follows: ˆ T Ξ(x) + W ˜ T Ξ(x) Wy∗T Ξ(x) = W y y
(21)
ˆy. ˜y = W∗ − W where W y III. CONTROLLER AND COMPENSATOR DESIGN In this section, the virtual control functions, adaptive laws, and control laws are derived based on the recursive DSC design procedures. Note that the compact set Ωsi,j i can be defined as follows: nc 1 T ˜ oi,j , σ ˜ ] : S2 Ωsi,j i = [Si,j i , βi,j i +1 , W i,j i i 2 j =1 i,j l l
ni n i −1 1 1 2 + Si,j + β2 k 2 j =n +1 2 j =1 i,j i +1 c
k
ni
1 + 2j +
j i =1
2γσ i,j i
≤ ψi,j i
(24) (25)
Si,j k = xi,j k − zi,j k , jk = nc + 1, . . . , ni − 1
(26)
Si,n i = xi,n i − zi,n i
(27)
where zi,j i is a virtual filtering control. The time derivative of the error surfaces can be given as follows: S˙ i,j l =
1 (1 − ξi,j l )2 ηi,j l
(fi,j l (¯ xi,j l ) + gi,j l (¯ xi,j l )xi,j l +1
+ Fu i,j l − η˙ i,j l ξi,j l − z˙i,j l ) 1 (1 − ξi,j l )2 ηi,j l
(Wy∗Ti,j l Ξi,j l (¯ xi,j l )
xi,j l )xi,j l +1 − η˙ i,j l ξi,j l − z˙i,j l ) + gi,j l (¯ +
1 (ε∗ + Fu i,j l ), jl = 1, . . . , nc (28) (1 − ξi,j l )2 ηi,j l i,j l
S˙ i,j k = fi,j k (¯ xi,j k ) + gi,j k (¯ xi,j k )xi,j k +1 + Fu i,j k − z˙i,j k = Wy∗Ti,j k Ξi,j k (¯ xi,j k ) + gi,j k (¯ xi,j k )xi,j k +1 + ε∗j,i j + Fdj,i j − z˙i,j k , jk = nc + 1, . . . , ni − 1
(29)
S˙ i,n i = fi,n i (¯ xi,n i ) + gi,n i (¯ xi,n i )ui + Fu i,n i − z˙i,n i = Wy∗Ti,n i Ξi,n i (¯ xi,n i ) + gi,n i (¯ xi,n i )ui + ε∗i,n i + Fu i,n i − z˙i,n i
(30)
where it is assumed that |ε∗i,j i | ≤ εm i,j i , |εm i,j i + Fu i,j i | ≤ ∗ ∗ ∗ , σi,j are positive constants, and σ ˆi,j i are estimates of σi,j . σi,j i i i The filtering virtual controls, zi,j i , are introduced and αi,j i is passed through first-order filters with time constants, τi,j i , as follows: τi,j i z˙i,j i +1 + zi,j i +1 = αi,j i , zi,j i +1 (0) = αi,j i (0) ji = 1, . . . , ni − 1.
(31)
The output errors of these filters are set as follows: βi,j i +1 = zi,j i +1 − αi,j i βi,j i +1 τi,j i +1
ji = 1, . . . , ni − 1.
(32) (33)
βi,j i +1 β˙ i,j i +1 = z˙i,j i +1 − α˙ i,j i = − τi,j i +1
i =1
2 σ ˜i,j i
(23)
The time derivative of the output errors of the filter becomes
˜ yTi,j Γ−1 W ˜ y i,j W i i,j i i 1
ξi,j l , jl = 1, . . . , nc 1 − ξi,j l
z˙i,j i +1 = −
i
ni
Si,j l =
=
f (x) = Wy∗T Ξ(x) + ε∗ ∀x ∈ Ω ⊂ Rn ∗
ei,j l = xi,j l − zi,j l , jl = 1, . . . , nc ei,j l ξi,j l = , jl = 1, . . . , nc ηi,j l
(22)
ˆ y i,1 , . . . W ˆ y i,j + Φi,j i +1 (Si,1 , . . . , Si,j i , W i Ξi,1 , . . . , Ξi,j i , σ ˆi,1 , . . . , σ ˆi,j i , zi,j l , z˙i,j l , z¨i,j l )
(34)
HAN AND LEE: PARTIAL TRACKING ERROR CONSTRAINED FUZZY DYNAMIC SURFACE CONTROL
where
Taking the time derivative of (39) nc m Si,j l ˆ yTi,j Ξi,j (¯ ˙ (W xi,j l ) V1 = l l 2η (1 − ξ ) i,j i,j l l i=1 j =1
Φi,j i +1 (Si,1 , . . . , Si,j i , Wy i,1 , . . . , Wy i,j i Ξi,1 , . . . , Ξi,j i , ρˆi,1 , . . . , ρˆi,j i , ydi , y˙ di , y¨di )
l
∂αi,j i ∂αi,j i ˙ ∂αi,j i ˙ ∂αi,j i ˙ Si,j i + Wy i,j i + Ξi,j i = x ¯˙ i,j i + ∂x ¯i,j i ∂Si,j i ∂Wy i,j i ∂Ξi,j i +
+ gi,j l (¯ xi,j l )xi,j l +1 − η˙ i,j l ξi,j l −z˙i,j l ) + βi,j l β˙ i,j l
Si,j l T ∗ ˜ + (Wy i,j l Ξi,j l (¯ xi,j l ) + εi,j l + Fu i,j l ) (1 − ξi,j l )ηi,j l nc m Si,j l ˆ yTi,j Ξi,j (¯ ≤ (W xi,j l ) l l 2η (1 − ξ ) i,j i,j l l i=1 j =1
∂αi,j i ˙ ∂αi,j i ∂αi,j i z˙i,j i + z¨i,j i σ ˆ i,j i + ∂σ ˆi,j i ∂zi,j i ∂ z˙i,j i ji = 1, . . . , ni − 1
is a continuous function and has a maximum Mi,j i +1 , on the compact set Ωz i,j i × Ωsi,j i defined in Assumption 3 and (22). From the definitions in (23), (24), and (32), the states variable can be represented as follows: xi,2 = ξi,2 ηi,2 + βi,2 + αi,1
(35)
xi,j l = ξi,j l ηi,j l + βi,j l + αi,j l −1 , jl = 3, . . . , nc
(36)
l
+ gi,j l (¯ xi,j l)xi,j l +1 − η˙ i,j l ξi,j l − z˙i,j l + σ ˆi,j l)+βi,j l β˙ i,j l
Si,j l T ˜ (W + Ξi,j l (¯ xi,j l ) + σ ˜i,j l ) (46) (1 − ξi,j l )2 |ηi,j l | y i,j l
xi,n i = Si,n i + zi,n i . 3
[Si,j k (Wy∗Ti,j k Ξi,j k (¯ xi,j k ) + gi,j k (¯ xi,j k )xi,j k +1
i=1 j k =1
− z˙i,j k ) + Si,j k (ε∗i,j k + Fu i,j k ) + βi,j k +1 β˙ i,j k +1 ]
(38)
The Lyapunov function candidate can be defined as follows:
ni m
V˙ 2 =
xi,j k = Si,j k + βi,j k + αi,j k −1 , jk = nc + 1, . . . , ni − 1 (37)
V =
1053
n i −1
m
≤
ˆ T Ξi,j (¯ [Si,j k (W xi,j k )+gi,j k (¯ xi,j k )xi,j k +1 y i,j k k
i=1 j k =n c +1
Vj + U
(39)
ˆ T Ξi,j (¯ +σ ˆi,j k − z˙i,j k ) + Si,j k (W xi,j k ) + σ ˜i,j k ) y i,j k k
j =1
+ βi,j k +1 β˙ i,j k +1 ]
where V1 =
nc
m 1
1 2 2 Si,j + βi,j l l +1 2 2
i=1 j l =1
V2 =
n i −1
m
i=1 j k =n c +1
V3 =
m 1 i=1
U=
2
(40)
i=1 j i =1
1 2 1 2 Si,j k + βi,j k +1 2 2
(41)
˜ T Ξi,n (¯ − z˙i,n i ) + Sj,i j W xi,n i ) + Si,n i σ ˜i,n i ] y i,n i i
ni m 1 ˙ T −1 ˜ ˙ ˜ ˙ U= Wy i,j i Γi,j i W y i,j i + σ ˜i,j i σ ˜ i,j i . γσ i,j i i=1 j =1
(48) (49)
(42) Using (45) 1 ˜T ˜ y i,j + 1 σ W Γ−1 W ˜2 i 2 y i,j i i,j i 2γi,j i i,j i
ˆ y i,j , σ ˜ y i,j = −W ˜i,j i = and W i i following equation:
≤
ˆ T Ξi,n (¯ [Si,n i (gi,n i (¯ xi,n i )ui + W xi,n i ) + σ ˆi,n i y i,n i i
i
Wy∗i,j i
|βi,j i +1 |Φi,j i +1 ≤
m i=1
2 Si,n i
ni m
V˙ 3 ≤
(47)
1 2ζi,j i +1 1 2ζi,j i +1
∗ σi,j i
(43)
2 2 βi,j Mi,j + i +1 i +1
βi,j i +1 β˙ i,j i +1 ≤ − ≤− +
τi,j i +1
+
ˆ T Ξi,j (¯ Si,j k (W xi,j k ) y i,j k k
m
ˆ T Ξi,n (¯ Si,n i (gi,n i (¯ xi,n i )ui + W xi,n i ) y i,n i i
i=1
+σ ˆi,n i − z˙i,n i )
+ |βi,j i +1 |Φi,j i +1 ,
ji = 1, . . . , ni − 1.
n i −1
+ gi,j k (¯ xi,j k )xi,j k +1 + σ ˆi,j k − z˙i,j k ) (44)
+
2 βi,j 1 i +1 + β2 M2 τi,j i +1 2ζi,j i +1 i,j i +1 i,j i +1
ζi,j i +1 2
m
i=1 j k =n c +1
ζi,j i +1 > 0, ji = 1, . . . , ni − 1, and from (34) 2 βi,j i +1
Si,j l ˆ T Ξi,j (¯ (W xi,j l ) y i,j l l (1 − ξi,j l )2 ηi,j l
+ gi,j l (¯ xi,j l )xi,j l +1 − η˙ i,j l ξi,j l + σ ˆi,j l − z˙i,j l ) +
ζi,j i +1 2 ζi,j i +1 2
nc m i=1 j l =1
−σ ˆi,j i . Using the
2 βi,j Φ2i,j i +1 + i +1
V˙ ≤
nc m
˜ yTi,j W l
i=1 j l =1
(45)
+
nc m i=1 j l =1
σ ˜i,j l
Si,j l Ξi,j l (¯ xi,j l ) ˆ˙ − Γ−1 i,j l W y i,j l 2 (1 − ξi,j l ) |ηi,j l |
Si,j l Ξi,j l (¯ xi,j l ) − γσ i,j l σ ˆ˙ i,j l (1 − ξi,j l )2 |ηi,j l |
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014
+
m
ni
are the design constants. The following where k(·) > 0 and γ(·) inequalities are then satisfied: ˆi,j l |Si,j l | σ 0≤ (1 − ξi,j l )2 |ηi,j l | ⎛ ⎞
˜ yTi,j (Si,j − Γ−1 W ˆ˙ y i,j ) W k k i,j k k
i=1 j k =n c +1
+
m
ni
σ ˜i,j k (Si,j k − γσ i,j k σ ˆ˙ i,j k )
Si , j l σ ˆi , jl
i=1 j k =n c +1
−
nc
m
τi,j l +1
i=1 j l =1
−
1
n i −1
m
i=1 j k =n c +1
+
−
1 2ζi,j l +1
1 τi,j k +1
−
ˆi,j l Si,j l σ ⎜ (1−ξ i , j l ) 2 |η i , j l | ⎟ tanh ⎝ − ⎠ 2 (1 − ξi,j l ) |ηi,j l | κi,j l
2 2 Mi,j βi,j l +1 l +1
1 2ζi,j k +1
2 Mi,j k +1
⎤ 2 ⎦ βi,j k +1
m n i −1 ζi,j i +1 . 2 i=1 j =1
(50)
i
The virtual control laws, control law, and adaptive laws can be expressed as follows:
αi,j l =
1 gi,j l
(58)
≤ 0.2785κi,j k = κ i,j k , jk = nc + 1, . . . , ni
(59)
1 2 2 + βi,j , ji = 1, . . . , ni (60) Si,j i βi,j i +1 ≤ Si,j i i +1 4
1 2 2 S gi,j i Si,j i Si,j i +1 ≤ gi,j i m ax Si,j + , ji = 1, . . . , ni i 4 i,j i +1 (61) ˜T W ˆ y i,j ≤ − γw i,j i W y i,j i i
ˆ T χi,j 2 (¯ xi,j l ) −ki,j l ηi,j l Si,j l − W oi,j l l
i,jl
˜T W ˜ y i,j γw i,j i W i y i,j i 2
+
∗T ∗ Woi,j γw i,j i Woi,j i i
2
, ji = 1, . . . , ni (62)
+ η˙ i,j l ξi,j l − gi,j l −1 ηi,j l +1 ξi,j l +1 ⎞ ⎞ ⎛ S ρˆ −ˆ ρi,j l
≤ 0.2785κi,j l = κ i,j l , jl = 1, . . . , nc
Si,j k σ ˆi,j k ˆi,j k − Si,j k σ ˆi,j k tanh 0 ≤ |Si,j k | σ κi,j k
i,jl 2
(1−ξ i , j l ) |η i , j l | ⎠+ z˙i,j l⎠, jl = 1, . . . , nc tanh⎝ κi,j l
˜i,j i σ ˆi,j i ≤ − γσ i,j i σ
2 ˜i,j γσ i,j i σ i
2
+
∗2 γσ i,j i σi,j i
2
, ji = 1, . . . , ni (63)
where κi,j i , ji = 1, . . . , ni , are positive constants.
(51)
IV. STABILITY ANALYSIS 1 ˆ yTi,j Ξi,j (¯ αi,j k = xi,j k ) −ki,j k Si,j k −gi,j k −1 Si,j k −1 − W k k gi,j k Theorem 2: Under Assumptions 1–3, consider the following.
The closed-loop partial states of a constrained strict feedback Si,j k σ ˆi,j k −ˆ σi,j k tanh + z˙i,j k , jk = nc + 1, . . . , ni system consist of a plant (1), virtual stabilizing functions (51) κi,j k and (52), control law (53), and the adaption laws (54)–(57). A (52) compact set can be defined as 1 ui = [−ki,n i Si,n i − gi,n i −1 m a x Si,n i −1 Ωi,j l := {S¯i,j l ∈ Rn c , ϕ¯ˆi ∈ Rn i : |Si,j l | ≤ Dsi,j l , ϕ¯ˆi,j l gi,n i m a x ≤ Dϕi,j (64) ¯ˆ l , jl = 1, . . . , nc } ˆ yTi,n Ξi,n (¯ −W xi,n i ) − σ ˆi,n i + z˙i,n i ] (53) i i m
μ Si,j l Ξi,j l (¯ xi,j l ) ˙ + ˆ ˆ D , jl = 1, . . . , nc (65) := S + 2 V W y i,j l = Γi,j l W − γ s y i,j ϕ i , j w i,j l i,j l l l C (1 − ξi,j l )2 |ηi,j l | i=1 σ ˆ˙ i,j l
jl = 1, . . . , nc
Si,j l Ξi,j l (¯ xi,j l ) − γσ i,j l σ = γσ i,j l ˆi,j l (1 − ξi,j l )2 |ηi,j l | jl = 1, . . . , nc
(54)
μ 2 V (0) + (66) , jk = nc + 1, . . . , ni C 2μV , ji = 1, . . . , ni (67) := ϕ¯∗i,j i m ax + λm in (¯ γi,j i )C
Ds i , j k := (55)
Dϕ¯ˆ i ,j i
ˆ˙ y i,j = Γ−1 (Si,j − γ ˆ W w i,j k Wy i,j k ), jk = nc + 1, . . . , ni k k i,j k (56)
Vϕ
ni 1 = S 2 2 j =n +1 i,j k c
k
σ ˆ˙ i,j k = γσ i,j k (Si,j k − γσ i,j k σ ˆi,j k ), jk = nc + 1, . . . , ni (57)
+
ni
1 2j
i =1
λm ax (¯ γi,j i ) ϕ¯∗i,j i m ax − ϕ¯ˆi,j i (0) 2
(68)
HAN AND LEE: PARTIAL TRACKING ERROR CONSTRAINED FUZZY DYNAMIC SURFACE CONTROL
nc 1 S 2 + Vϕ 2 j =1 i,j l
V =
(69)
l
ˆ y i,j , σ where ϕ¯ˆi,j i = [W ˆi,j i ]T , γ¯i,j i = [Γ−1 i i,j i , diag(1/γσ i,j i )]. The upper bounds for Si,j i in the compact set Ωi,j i can be expressed as sup Si,j i (xi,j i , zi,j i (0), ϕ¯ˆi,j i (0)) ≤ Si,j i
1055
Assumption 4: Sufficient small time-constants of the low-pass filter are chosen such that 1 1 1 2 + Ci,j l (75) ≥ Mi,j + l +1 τi,j l +1 2ζi,j l +1 4(1 − ξi,j l )2 |ηi,j l | 1 1 1 ≥ M2 + + Ci,j k , τi,j k +1 2ζi,j k +1 i,j k +1 4 jk = nc + 1, . . . , ni − 1. (76)
xi , j i
i = 1, . . . , m, ji = 1, . . . , ni . (70) The following properties then hold. i) The error surfaces Si,j i (t) and ϕ¯ˆi,j i , i = 1, . . . , m, ji = 1, . . . , ni remain in the sets Ωi,j i and Ωϕ¯ˆ i , j ∀t > 0, defined as i
Ωi,j i := {Si,n i ∈ R
: |Si,j l | ≤ Ds i , j l , jl = 1, . . . , nc
ni
|Si,j k |≤ Dsi,j k , jk = nc + 1, . . . , ni } Ωϕ¯ˆ i , j := {ϕ¯ˆi,j i ∈ R
(71)
ˆ¯i,j i ≤ Dϕ¯ˆ i , j }. : ϕ
ni
i
(72)
i
ii) The partial tracking errors ei,j l (t) remain in the set Ωei,j l ∀t > 0, which is defined as i = 1, . . . , m, jl = 1, . . . , nc }.
(73)
The partial tracking error constraints are thus never violated. iii) All closed-loop signals are bounded. iv) The partial tracking errors are smaller than the prescribed error bounds and the size of the tracking errors can be decreased arbitrarily based on an appropriate selection of the design parameters. Proof: i) Considering (51)–(63), (50) can be written as V˙ ≤
i=1
⎩
nc
−
j l =1
n i −1
−
ki,j l − gi,j l m ax (1 − ξi,j l )2
(ki,j k − gi,j k
2 m ax )Si,j k
ki,j l = gi,j l m ax (1 − ξi,j l )2 + Ci,j l , jl = 1, . . . , nc ki,j k = gi,j k m ax + Ci,j k , jk = nc + 1, . . . , ni − 1 ki,n i = Ci,n i γw i,j i = 2Ci,j i , ji = 1, . . . , ni γσ i,j i = 2Ci,j i , ji = 1, . . . , ni .
2 Si,j l
−
ni ˜T W ˜ y i,j γw i,j i W i y i,j i
2
j i =1 nc
−
j l =1
1
τi,j l +1
n i −1
−
j k =n c +1
−
−
2ζi,j l +1 1
τi,j k +1
−
1 2ζi,j k +1
2 Mi,j k +1
μ=
i=1 j i =1
2
γw i,j i Wy∗i,j i 2
i
c μ 1 S 2 + Vϕ + . 2 i=1 j =1 i,j l C
m
≤
)2 |η
i,j l |
2 βi,j l +1
2 ⎦+μ βi,j k +1
!
n
(81)
l
Thus m
⎤
∗2 γσ i,j i σi,j i + 2
ni m n m i −1 ζi,j i +1 + + κ i,j i . 2 i=1 j =1 i=1 j =1 i
n
l
2 Si,j ≤ l
i=1
μ 2 , jl = 1, . . . , nc Si,j + 2 V + ϕ l C i=1
m
(82)
which provides m
ni m
(79)
Integrating (79) over[0, t] leads to μ −C t μ μ 0 ≤ V ≤ V (0) − e ≤ V (0) + . (80) + C C C From (40) and (80), the constrained virtual error surfaces can further be represented as m
(74)
where
(78)
c 1 μ μ 2 ≤V + Si,j ≤ V (0) + l 2 i=1 j =1 C C
1
1 − 4
k =1
2 Ci,k βi,k +1 + μ = −CV + μ
2
4(1 − ξi,j l
k =1
d (V eC t ) ≤ μeC t . dt
ni 2 γσ i,j i σ ˜ i,j i
2 Mi,j − l +1
n i −1
#
where C = min[2Ci,1 , . . . , 2Ci,j i ] for i = 1, . . . , m, ji = 1, . . . , ni . Furthermore, multiplying (78) by eC t yields
2 − ki,n i Si,n i
j i =1
1
k =1
k =1
j k =n c +1
−
(77)
Therefore, by considering Assumption 4 and (77) " n ni ni m i 2 2 ˜ yTi,k W ˜ y i,k − V˙ ≤ Ci,k Si,k − Ci,k W Ci,k σ ˜i,k − i=1
Ωei,j l := {ei,j l (t) ∈ Rn c : |ei,j l (t)| ≤ |ηi,j l (t)|
⎧ m ⎨
The following control gains are selected:
|Si,j l
m |≤ S
i,j l
i=1
i=1
μ , jl = 1, . . . , nc . + 2 Vϕ + C (83)
Next, from (41) and (42) m
ni
i=1 j k =n c +1
1 2 μ S ≤ V (0) + 2 i,j k C
(84)
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014
and
μ |Si,j k | ≤ 2 V (0) + , jk = nc + 1, . . . , ni . (85) C i=1
transformation can be expressed as
m
Therefore, Si,j i remains in the compact set Ωi,j i ∀t > 0. Next, because ϕ¯∗i,j i ≤ ϕ¯∗i,j i m ax ⎛ nc ni m 1 2 μ 2 ⎝ = Si,j i (0) + V (0) + Si,j (0) i C 2 i=1 j =1 j =n +1 i
i
c
ni 1 γ¯i,j ϕ¯∗i,j i − ϕ¯ˆi,j i (0) + 2 i j =1
⎞ 2
⎠
i
+
μ μ ≤V + C C
(86)
−1 1 2 where nj ii=1 2 βi,j i +1 (0) = 0 owing to Si,j i +1 (0) = αi,j i (0). It is clear that V ≤ V (0) + μ/C ≤ V + μ/C. Hence m 1 i=1
2
λm in (¯ γi,j i ) ϕ¯∗i,j i
− ϕ¯ˆi,j i 2 ≤ V +
μ C
ji = 1, . . . , ni .
(87) $ γi,j i )C. ThereThus, ϕ¯ˆi,j i ≤ ϕ¯∗i,j i m ax + 2μV /λm in (¯ fore, ϕ¯ˆi,j i remains in the compact set Ωϕ¯ˆ i , j ∀t. i ii) Given that 0 < ξi,j l (0) < 1, and ξi,j l (t) is continuous, it can also be concluded from Theorem 1 that 0 < ξi,j l < 1 ∀t > 0. From the definition of the transformed errors, 0 < ξi,j l < 1 gives |ei,j l (t)| < |ηi,j l (t)|, and vice versa. Hence, the partial tracking errors ei,j l (t) remain in the set Ωei,j l . This guarantees that the transient and steady-state performances of the partial tracking errors remain within the prescribed constraint bounds. iii) From the results of i and ii, it is easy to show that |Si,j l | ≤ Dsi,j l , |Si,j k | ≤ Dsi,j k , |ei,j l | ≤ |ηi,j l |, and ϕ¯ˆi,j i ≤ Dϕ¯ˆ i , j for jl = 1, . . . , nc , jk = nc + 1, . . . , ni ,ji = 1, . . . , ni . i Thus, it has been shown progressively that the control input ui (S¯i,j i , z¯i,j i , ϕ¯ˆi,j i ), and the remaining signals are all bounded. iv) From (40) and (81) 1 2 μ −C t μ e Si,j l ≤ V (0) − + , jl = 1, . . . , nc . 2 i=1 C C (88) Some manipulations lead to the inequality m
m
|ei,j l (t)| ≤
i=1
m √ 2 |ηi,j l |(1 − ξi,j l )
i=1
μ −C t μ e + , jl = 1, . . . , nc . (89) C C √ m m $As t → ∞, i=1 |ei,j l (t)| ≤ 2 i=1 |ηi,j l |(1 − ξi,j l ) μ/C for jl = 1, . . . , nc . Therefore, the partial tracking errors are smaller than the prescribed error bounds, and the sizes of the errors can be arbitrarily minimized as small values from the definitions of C and μ, as well as parameter through the selection of the prescribed performance function given in (2). Remark 1: If the constrained method proposed by Rovithakis et al. [15]–[17] is used instead of the proposed method, the error ×
V (0) −
ei,j l (t) = ρi,j l (t)Ti (Sj,j l (t))
ei,j l (t) ei,j l (t) −1 Si,j l (t) = Ti,j = R i,j l l ρi,j l (t) ρi,j l (t)
(90) (91)
where Ti,j 1 (·) and its inverse Ri,j l (·) are given as Ti,j l (Si,j l (t)) ⎧ exp(Si,j l (t)) − δi,j exp(−Si,j l (t)) ⎪ ⎪ , ⎪ ⎨ exp(Si,j l (t)) + exp(−Si,j l (t)) = ⎪ ⎪ δ exp(Si,j l (t)) − exp(−Si,j l (t)) ⎪ ⎩ i,j , exp(Si,j l (t)) + exp(−Si,j l (t)) Ri,j l (ei,j l (t)/ρi,j l (t)) ⎧ 1 ei,j l (t)/ρi,j l (t) + δi,j l ⎪ ⎪ ln , ⎪ ⎨2 1 − ei,j l (t)/ρi,j l (t) = ⎪ ⎪ ⎪ 1 ln ei,j l (t)/ρi,j l (t) + 1 , ⎩ 2 δi,j l − ei,j l (t)/ρi,j l (t)
if ei,j l ≥ 0 (92) if ei,j l < 0
if ei,j l (0) ≥ 0 (93) if ei,j l (0) < 0.
The time derivatives of Si,j l (t) can be expressed as
∂Ri,j l ˙ Si,j l = xi,j l ) + gi,j l (¯ xi,j l )xi,j l +1 + Fu i,j l fi,j l (¯ ρi,j l ρ˙ i,j l − ei,j l (t) − z˙i,j l ρi,j l
∂Ri,j l = xi,j l ) + gi,j l (¯ xi,j l )xi,j l +1 Wy∗Ti,j l Ξi,j l (¯ ρi,j l ρ˙ i,j l ∂Ri,j l ∗ − ei,j l (t) − z˙i,j l + (εi,j l + Fu i,j l ) ρi,j l ρi,j l jl = 1, . . . , nc
(94)
where ∂Ri,j l = ⎧ 1 + δi,j l 1 ⎪ ⎪ if ei,j l (0) ≥ 0 ⎪ ⎨ 2 (ei,j l /ρi,j l + δi,j l )(1 − ei,j l /ρi,j l ) ⎪ 1 + δi,j l 1 ⎪ ⎪ ⎩ if ei,j l (0) < 0. 2 (ei,j l /ρi,j l + 1)(δi,j l − ei,j l /ρi,j l ) (95) A virtual control laws can be then specified as follows:
1 ki,1 ρi,1 ˆ y i,1 Ξi,1 (¯ αi,1 = Si,1 − W xi,1 ) − gi,1 (¯ xi,1 ) ∂Ri,1
Si,1 σ ˆi,1 ρ˙ i,1 ei,1 (t) + z˙i,1 −σ ˆi,1 tanh + (96) κi,1 ρi,1
1 gi,j l −1 ∂Ri,j l −1 ki,j ρi,j l Si,j l − Si,j l −1 αi,j l = − l gi,j l (¯ xi,j l ) ∂Ri,j l ρi,j l −1
Si,j l σ ˆi,j l ˆ − Wy i,j l Ξi,j l (¯ xi,j l ) − σ ˆi,j l tanh κi,j l ρ˙ i,j l + ei,j l (t) + z˙i,j l , jl = 2, . . . , nc . (97) ρi,j l
HAN AND LEE: PARTIAL TRACKING ERROR CONSTRAINED FUZZY DYNAMIC SURFACE CONTROL
1057
Tf 1 and Tf 2 are assumed to be a LuGre friction model [27] Tf i = s0 εi + s1 ε˙i + s2 θ˙i , ε˙i = θ˙i −
s0 |θ˙i | , ˙ θ˙2 |2 ) Tc + (Ts − Tc ) exp(−|θ/
i = 1, 2 (102)
where s0 = 1 Nm, s1 = 1 Nms, s1 = 1 Nms, θ˙s = 0.1 rad/s, Ts = 2 Nm, and Tc = 1 Nm. The state-space equations of (98) and (99) that define x1,1 = θ1 , x1,2 = θ˙1 , x2,1 = θ2 , and x2,2 = θ˙2 can be represented as Fig. 2.
Two inverted pendulums connected by a spring and a damper.
x˙ 1,1 = x1,2
Remark 2: In a prescribed constraint performance function, such as (2), which is also the constraint function used by Rovithakis et al. [15]–[17], a larger exponential decreasing rate, ai,i l , causes a fast approach of ρi,j l (t) to ρ∞i,j l . This makes the values of ρ∞i,j l and error ei,j l (t) closer or equal to each other. Therefore, the possibility that |ρ∞j,j l | = |ei,j l (t)| at an earlier time increases substantively. In particular, in (97), if the values of the constraint parameter, ρi,j l (t) and δi,j l , are selected as ei,j l (t)/ρi,j l (t) → ±1 or ei,j l (t)/ρi,j l (t) → ±δi,j l , then ∂Ri,j l → ±∞, and the resulting error increases, which can give rise to the closed-loop system stability problem and to a violation of the prescribed constraint conditions. V. APPLICATION EXAMPLE In this section, we present a simulation conducted using a nonlinear MIMO system to evaluate the performance of our prescribed partial error constraints against uncertainties. Two inverted pendulums composed of spring and damper connections, and nonlinear friction, as shown in Fig. 2, were used as a control system for the constraint problem of the full-state tracking errors [17]. The pendulum angle and angular velocity were controlled using the torque inputs generated by a servomotor at each base. The dynamic equation of the inverted pendulum can be described as J1 θ¨1 = m1 gr sin θ1 − 0.5F r cos(θ1 − θ) − Tf 1 + u1
(98)
J2 θ¨2 = m2 gr sin θ2 + 0.5F r cos(θ2 − θ) − Tf 2 + u2
(99)
where θ1 and θ2 are angular positions, J1 = 0.5 kgm2 and J2 = 0.625 kgm2 are the moments of inertia, m1 = 2 kg and m2 = 2.5 kg are the masses, r = 0.5 m, F = k(p − l) + bp˙ denotes the force applied by the spring and damper at the connection points, and p is the distance between the connection points as follows: r2 p = d2 + dr(sin θ1 − sin θ2 ) + [1 − cos(θ2 − θ1 )] 2 (100) where d = 0.5 m, k = 150 N/m, and b = 1 Ns/m. The relative angular position θ can be defined as
r 2 (cos θ2 − cos θ1 ) θ = tan−1 . (101) d + 2r (sin θ1 − sin θ2 )
x˙ 1,2 = f1,2 (¯ x2,2 ) + g1,2 u1 x˙ 2,1 = x2,2 x˙ 2,2 = f2,2 (¯ x2,2 ) + g2,2 u2
(103)
x2,2 ) = g1,2 [m1 gr sin x1 − 0.5F r cos(x1 − x) − where f1,2 (¯ Tf 1 ], g1,1 = 1/J1 , f2,2 (¯ x2,2 ) = g2,2 [m2 gr sin x2,1 + 0.5F r cos(x2,2 − x) − Tf 2 ], g2,2 = 1/J2 , and x = θ. A. Output Tracking Error Constraint We first considered the output error constraint problem. The control objective was for x1,1 (t) and z2,1 (t) to follow the desired commands, z1,1 (t) = 0.5 sin(2πt) and z2,1 (t) = 0.4 sin(πt), subject to the error performance functions as follows: ρ1,1 (t) = (ρ01,1 − ρ∞1,1 )e−a 1 , 1 t + ρ∞1,1 , −a 2 , 1 t
ρ2,1 (t) = (ρ02,1 − ρ∞2,1 )e
+ ρ∞2,1 ,
if e1,1 (0) ≥ 0 if e2,1 (0) < 0.
The initial points selected for each state were x ¯1,2 (0) = (π/3.6, 0) and x ¯2,2 (0) = (π/4, 0). The parameters of the controller were selected as k1,1 = 1.2, k1,2 = 3, γw 1,2 = 20, γw 1,2 = 0.5, γσ 1,2 = 0.5, γσ 1,2 = 0.01, τ1,2 = 0.01, κ1,2 = 0.1, k2,1 = 1.2, k2,2 = 3, γw 2,2 = 15, γw 2,2 = 0.5, γσ 2,2 = 1, γσ 2,2 = 0.01, κ2,2 = 0.1, and τ2,2 = 0.01. The parameters of the prescribed performance functions were selected for the following two cases. Case I: ρ01,1 = π/3, ρ∞1,1 = π/360, δ1,1 = 0.8, a1,1 = 1, (e1,1 (0) ≥ 0) ρ02,1 = π/2, ρ∞2,1 = π/360, δ2,1 = 0.5, a2,1 = 1, (e2,1 (0) < 0) Case II: ρ01,1 = π/3, ρ∞1,1 = π/360, δ1,1 = 0.8, a1,1 = 80, (e1,1 (0) ≥ 0) ρ02,1 = π/2, ρ∞2,1 = π/360, δ2,1 = 0.5, a2,1 = 30, (e2,1 (0) < 0). The fuzzy membership functions were selected as follows: μF i12 =
1 1 + exp[(Si,2 − 10−2 )2 ]
2 μF i22 = exp[−0.5(Si,2 − 0.75 × 10−2 )2 /ςi,2 ] 2 μF i22 = exp[−0.5(Si,2 − 0.5 × 10−2 )2 /ςi,2 ] 2 μF i22 = exp[−0.5(Si,2 − 0.25 × 10−2 )2 /ςi,2 ] 2 μF i22 = exp[−0.5(Si,2 + 10−2 )2 /ςi,2 ]
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2 μF i22 = exp[−0.5(Si,2 + 0.25 × 10−2 )2 /ςi,2 ] 2 μF i22 = exp[−0.5(Si,2 + 0.5 × 10−2 )2 /ςi,2 ] 2 μF i22 = exp[−0.5(Si,2 + 0.75 × 10−2 )2 /ςi,2 ]
μF i52 =
1 , i = 1, 2 1 + exp[(Si,2 + 10−2 )2 ]
where ςi,2 denotes the standard deviation of the Gaussian function. For a comparative simulation, three control systems were designed: 1) the proposed PPC combined with DSC and adaptive FLC (PPDSC); 2) a conventional DSC without a PPC combined with adaptive FLC (DSC); and 3) Rovisthakis’ PPC combined with DSC and adaptive FLC (RPDSC). Fig. 3 shows the simulated responses for Case I. Both the RPDSC and PPDSC systems satisfied the prescribed output error performance constraints because a relatively small convergence rate for Case I was selected along with small control gains, whereas the conventional DSC violated the prescribed constrained bounds. Therefore, the output performance of the proposed PPPC system is less sensitive to variations of the control gain than a conventional DSC scheme. For Case II, Fig. 4(a) and (b) shows the position tracking errors and given position error constraints for the PPDSC and RPDSC systems, where the position tracking errors of the RPPC systems violated the prescribed position error constraint boundaries and approached infinity because the large value of the convergence rate in Case II resulted in a high value for ∂Ri,j l . Therefore, this gave rise to the aforementioned singularity and stability problem in Remark 2 in the RPDSC system, as shown in Fig. 4(c) and (d), where the values of ∂Ri,1 were increased significantly; these excessive inputs led to prescribed tracking error violations in the RPDSC systems. On the other hand, the PPDSC system satisfied the prescribed performance conditions, and the tracking errors remained in the prescribed error performance bound despite the high exponential convergence rate, as shown in Fig. 4(a) and (b). B. Full-State Tracking Error Constraint In this section, the full-state tracking error problem was dealt with as a special case of the partial-state tracking error problem. All parameters of the controller and the adaptive laws were selected to be equal to the output error constraint. The following parameters of the prescribed performance functions were selected: ρ01,1 = π/3, ρ∞1,1 = π/360, δ1,1 = 0.8, a1,1 = 1, (e1,1 (0) ≥ 0) ρ02,1 = π/2, ρ∞2,1 = π/360, δ2,1 = 0.5, a2,1 = 1, (e2,1 (0) < 0) ρ01,2 = 0.5, ρ∞1,2 = 0.05, δ1,2 = 1, a1,1 = 1, (e1,2 (0) ≥ 0) ρ02,2 = 0.5, ρ∞2,2 = 0.05, δ2,2 = 1, a2,2 = 1, (e2,1 (0) < 0). In this control, only Case I was considered because it was shown that the RPDSC is weak under the fast convergence rate condition described in the previous section. Fig. 5 shows the state tracking responses for the given sine input position commands and the intermediate virtual velocity control commands, where the outputs tracked the given commands well
Fig. 3. Position tracking responses for the output tracking error constraint (Case I). (a) Position tracking output of pendulum 1 of PPDSC. (b) Position tracking output of pendulum 2 of PPDSC. (c) Position tracking error of pendulum 1 for DSC (dash-dotted line), RPDSC (dashed line), and PPDSC (solid line). (d) Position tracking error of pendulum 2 for DSC (dash-dotted line), RPDSC (dashed line), and PPDSC (solid line).
without an excessive overshoot. In Fig. 6, all tracking errors for the position and velocity of the full-state constraint system fell within the prescribed performance bounds, whereas the velocity tracking errors of the output error constraint system violated the prescribed velocity performance conditions despite the
HAN AND LEE: PARTIAL TRACKING ERROR CONSTRAINED FUZZY DYNAMIC SURFACE CONTROL
Fig. 4. Position tracking errors for the output tracking error constraint for Case II. (a) Position tracking error of pendulum 1 for RPDSC (dashed line) and PPDSC (solid line). (b) Position tracking error of pendulum 2 for RPDSC (dashed line) and PPDSC (solid line). (c) Position tracking error (dashed line) and ∂R 1 , 1 (solid line) of RPDSC in pendulum 1. (d) Position tracking error (dashed line) and ∂R 2 , 1 (solid line) of RPDSC in pendulum 2.
position tracking error remaining in the prescribed position tracking error bounds because the constraints on the velocity tracking error were not imposed on the output constraint control system. Fig. 7(a)–(d) shows that Theorem 1 is satisfied such that |ei,j l | ≤ |ηi,j l | ⇔ 0 < ξi,j l < 1, i.e., the absolute values of the
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Fig. 5. State tracking responses for the full-state tracking error constraint. (a) Position and (b) velocity tracking responses of pendulum 1. (c) Position and (d) velocity tracking responses of pendulum 2.
tracking errors were smaller than those of the prescribed constraints. In addition, −ki,j l ηi,j l Si,j l prevented the generation of a large envelope of tracking errors by pressing them in opposite the direction, as shown in Figs. 7(e)–(f), which affected the control inputs shown in Fig. 7(i).
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Fig. 6. State tracking errors for the full-state tracking error constraint in output (dashed line) and full-state (solid line) constraint systems. (a) Output and (b) velocity tracking errors of pendulum 1. (c) Output and (d) velocity tracking errors of pendulum 2.
VI. CONCLUSION In this paper, we examined the partial prescribed tracking error performance constraint for a MIMO strict feedback nonlinear dynamic system as an extension of the previously studied output error constraint problem. By transforming the state tracking errors into new virtual variables the state tracking errors were confined within the prescribed constraint bounds in the direction opposite to that of the error evolvement. Adaptive FLC was used to approximate unknown nonlinear functions. Using the
Fig. 7. State tracking responses for the full-state tracking error constraint system. (a)–(d) |ei , j l | (dashed line) and |η i , j l |(solid line) for i = 1, 2, j = 1, 2. (e)–(h) ei , j l (dashed line) and −k i , j l η i , j l S i , j l (solid line) for i = 1, 2, j = 1, 2. (i) Control inputs.
Lyapunov stability theorem, we confirmed that partial tracking error constraints and the boundedness of all signals in the closed loop were guaranteed. The results of a simulation conducted using two inverted connecting pendulums confirmed the utility and efficacy of our proposed control scheme.
HAN AND LEE: PARTIAL TRACKING ERROR CONSTRAINED FUZZY DYNAMIC SURFACE CONTROL
REFERENCES [1] M. Kristic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York, NY, USA: Wiley, 1995. [2] S. C. Tong, X. L. Li, and H. G. Zhang, “A combined backstepping and small-gain approach to robust adaptive fuzzy output feedback control,” IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1059–1069, Oct. 2009. [3] S. C. Tong and Y. Li, “Observer-based fuzzy adaptive control for strictfeedback nonlinear systems,” Fuzzy Sets Syst., vol. 160, pp. 1749–1764, 2009. [4] Q. Zhou, P. Shi, J. Lu, and S. Xu, “Adaptive output-feedback fuzzy tracking control for a class of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 972–982, Oct. 2011. [5] S. C. Tong and Y. Li, “Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown deadzones,” IEEE Trans. Fuzzy Syst., vol. 20, no. 1, pp. 168–180, Feb. 2012. [6] S. C. Tong, Y. Li, and P. Shi, “Observer-based adaptive fuzzy backstepping output feedback control of uncertain MIMO pure-feedback nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 20, no. 4, pp. 771–785, Aug. 2012. [7] S. C. Tong and Y. Li, “Adaptive fuzzy output feedback control of MIMO nonlinear systems with unknown dead-zone inputs,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 134–146, Feb. 2013. [8] S. C. Tong, T. Wang, Y. Li, and B. Chen, “A combined backstepping and stochastic small-gain approach to robust adaptive fuzzy output feedback control,” IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 314–327, Apr. 2013. [9] C. Kwan and F. L. Lewis, “Robust backstepping control of nonlinear systems using neural networks,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 30, no. 6, pp. 753–766, Nov. 2000. [10] S. Swaroop, J. K. Hedrick, P. P. Yip, and J. C. Gerdes, “Dynamic surface control for a class of nonlinear systems,” IEEE Trans. Autom. Control, vol. 45, no. 10, pp. 1893–1899, Oct. 2000. [11] J. K. Hedrick and P. P. Yip, “Multiple sliding surface control: Theory and application,” Trans. Amer. Soc. Mech. Eng., vol. 122, pp. 586–593, 2000. [12] T. S. Li, S. C. Tong, and G. Feng, “A novel adaptive-fuzzy-tracking control for a class of nonlinear multi-input/multi-output systems,” IEEE Trans. Fuzzy Syst., vol. 18, no. 1, pp. 150–160, Feb. 2010. [13] S. C. Tong, Y. Li, G. Feng, and T. S. Li, “A DSC approach to robust adaptive NN tracking control for strict-feedback nonlinear system,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 915–927, Jun. 2010. [14] W. S. Chen, “Adaptive backstepping dynamic surface control for systems with periodic disturbances using neural networks,” IET Control Theory and Appl., vol. 3, no. 10, pp. 1383–1394, 2009. [15] C. P. Benchlioulis and G. A. Rovithakis, “Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance,” IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 2090–2099, Oct. 2008. [16] C. P. Benchlioulis and G. A. Rovithakis, “Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems,” Automatica, vol. 45, pp. 532–538, 2009. [17] C. P. Benchlioulis and G. A. Rovithakis, “Robust partial-state feedback prescribed performance control of cascade systems with unknown nonlinearities,” IEEE Trans. Autom. Control, vol. 56, no. 9, pp. 2224–2230, Sep. 2011. [18] A. Ilchman and H. Schuster, “Tracking control with prescribed transient behavior degree,” Syst. Control Lett., vol. 55, pp. 396–406, 2006. [19] A. Ilchman, E. P. Ryan, and S. Trenn, “PI-funnel control for two mass systems,” IEEE Trans. Autom. Control, vol. 54, no. 4, pp. 981–923, Apr. 2009.
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[20] C. M. Hackl, C. Endisch, and D. Schroder, “Contribution to non-identifier based adaptive control in mechatronics,” Robot. Auton. Syst., vol. 57, pp. 996–1005, 2009. [21] C. M. Hackl, “High-gain position control,” Int. J. Control, vol. 84, no. 10, pp. 1695–1716, 2011. [22] K. P. Tee, S. S. Ge, and E. H. Tay, “Barrier Lyapunov functions for the output-constrained nonlinear systems,” Automatica, vol. 45, no. 4, pp. 918–927, 2009. [23] B. B. Ren, S. S. Ge, K. P. Tee, and T. H. Lee, “Adaptive neural control for output feedback nonlinear systems using a barrier Lyapunov function,” IEEE Trans. Neural Netw., vol. 21, no. 8, pp. 1339–1345, Aug. 2010. [24] S. I. Han and J. M. Lee, “Adaptive fuzzy backstepping dynamic surface control for output-constrained non-smooth nonlinear dynamic system,” Int. J. Control Autom. Syst., vol. 10, no. 4, pp. 684–696, 2012. [25] Y. M. Li, T. S. Li, and X. J. Jing, “Indirect adaptive fuzzy control for input and output constrained nonlinear systems using barrier Lyapunov function,” Int. J. Adaptive Control Signal Process., May 2013. [26] S. S. Ge and J. Wang, “Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients,” IEEE Trans. Autom. Control, vol. 48, no. 8, pp. 1463–1469, Aug. 2003. [27] C. Canudas de Wit, H. Olsson, and K. J. Astrom, “A new model for control of systems with friction,” IEEE Trans. Autom. Control, vol. 40, no. 3, pp. 419–425, Mar. 1995.
Seong I. Han (M’12) received the B.S. and M.S. degrees in mechanical engineering, and the Ph. D. degree in mechanical design engineering from Pusan National University, Busan, Korea, in 1987, 1989, and 1995, respectively. From 1995 to 2009, he was an Associate Professor of electrical automation with Suncheon First College, Suncheon-si, Korea. He is currently with the Department of Electronic Engineering, Pusan National University. His research interests include intelligent control, nonlinear control, robotic control, vehicle system control, and steel process control.
Jang M. Lee (SM’03) received the B.S. and M.S. degree in electronic engineering from Seoul National University, Seoul, Korea, in 1980 and 1982, respectively, and the Ph. D. degree in computer engineering from the University of Southern California, Los Angeles, CA, USA, in 1990. Since 1992, he has been a Professor with Pusan National University, Busan, Korea, where he was the Leader of the “Brain Korea 21 Project.” His research interests include intelligent robotics, advanced control algorithm, and specialized environment navigation/localization. Prof. Lee is a past President of the Korean Robotics Society.
