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Output-Tracking-Error-Constrained Robust Positioning Control for a Nonsmooth Nonlinear Dynamic System Seong I. Han, Member, IEEE, and Jang M. Lee, Senior Member, IEEE
Abstract—An output-tracking-error-constrained dynamic surface control (DSC) is proposed for the robust output positioning of a multiple-input–multiple-output nonlinear dynamic system in the presence of both friction and deadzone nonsmooth nonlinearities. An error transformation method and simple barrier Lyapunov function are also proposed to ensure the prescribed output tracking performance and stability without requiring specific observations of the friction and deadzone parameters. In addition, a new adaptive cerebellar model articulation controller–echo state neural networks system is proposed to deal with an unknown nonlinear function to improve the positioning performance. The boundedness of the overall closed-loop signals and the prescribed performance constraints were guaranteed, and precise positioning performance was also ensured regardless of the effects of friction and deadzone. The proposed control scheme was evaluated by simulation and experiment. Index Terms—Barrier Lyapunov function (BLF), cerebellar model articulation controller (CMAC)–echo state network (ESN), friction and deadzone, multiple-input–multiple-output (MIMO) dynamic surface control (DSC), prescribed tracking error constraint.
I. I NTRODUCTION
I
N MANY mechanical control systems, nonsmooth nonlinearities, such as deadzone and friction, are the main obstacles to maintaining high control performance. Friction between a moving part and guide surface causes problems, such as stick slip, limit cycle, frictional lag, presliding hysteresis, and steadystate error. As the classical static friction model, the Gaussian friction model [1] was developed because of its ease of implementation. Dynamic friction models, such as LuGre [2] and Leuven [3], were developed to provide better descriptions of the friction phenomena than the classical models. On the other hand, a precise friction model cannot be constructed because the friction characteristics often vary according to the operating conditions and contact surface circumstances. Furthermore, Manuscript received April 4, 2013; revised September 5, 2013; accepted February 22, 2014. Date of publication April 9, 2014; date of current version September 12, 2014. This work was supported by the Ministry of Knowledge Economy, Korea, through the Human Resources Development Program for Specialized Environment Navigation/Localization Technology Research Center support program supervised by the National IT Industry Promotion Agency under NIPA-2012-H1502-12-1002. The authors are with the Department of Electronic Engineering, Pusan National University, Busan 609-735, Korea (e-mail:
[email protected]; jmlee@ pusan.ac.kr). 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/TIE.2014.2316263
when an accurate friction model is required for model-based control, considerable offline time-consuming identification work is required [4]. On the other hand, elaborate robust control should be designed because most of the friction phenomena have time-varying properties. In addition, the deadzone nonlinearity in actuator and transmission devices causes inaccuracies in the control system. Therefore, although many deadzone compensation methods have been developed to reduce the deadzone nonlinear effect [5], complete deadzone compensation is not expected because laborious deadzone identification is also required similar to friction model identification. Furthermore, when both nonsmooth nonlinearities exist in an actuator simultaneously, a controller designed to compensate for the friction or deadzone alone might result in poor performance due to the overlap of friction and deadzone. This paper proposes an improved friction and deadzone compensation method using the prescribed tracking error constraint and barrier Lyapunov function (BLF) without introducing a friction and deadzone estimate. This method guarantees the prescribed tracking performance, such as the transient and steady-state responses as well as the compensating friction and deadzone nonlinear effects. Recently, a constraint problem in the control design has attracted attention because any violation of the constraints can lead to performance degradation, hazards, or system damage. The design of BLF in the Lyapunov theorem was proposed to handle the issue of the system state constraint in the output constraint [6], [7] and time-varying output feedback system [8]. In this constraint method, the tracking errors were indirectly constrained by constraining the behavior of the state variables. On the other hand, the designed controllers have complex structures because these control methods adopt complex BLF for guaranteeing the asymmetric and time-varying state variable constraints, as well as an adaptive backstepping control scheme [9] with extensive complexity problems due to the repeated differentiation of the virtual control functions. Furthermore, in these BLF-based control schemes, nonsmooth nonlinearities, such as friction and deadzone, were not considered, and no experimental verification was performed. The cerebellar model articulation controller (CMAC) has the advantages of fast learning, good generalization, and simple calculation [10] in the approximation of unknown nonlinear functions. Therefore, the CMAC has been adopted for the closed-loop control of complex dynamic systems [11]. To improve the drawbacks of static neural networks (NNs), recurrent NNs (RNNs) [12], [13] have
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been developed to involve dynamic structures in the form of feedback connections used as internal memories. Most RNNs have large computational cost, and these algorithms can only be applied to small nets. Echo state networks (ESNs) have been proposed by Jaeger as advanced RNNs [15]. This paper proposes a new CESN technique that utilizes the advantages of CMAC and ESN systems to improve the approximation performance of unknown nonlinear functions. The main contributions of this paper are as follows. 1) The performance constraint variable is proposed to constrain the prescribed error bound by using a virtual control of the dynamic surface control (DSC) [15] and BLF for a multiple-input–multiple-output (MIMO) strict feedback nonlinear system. 2) The proposed control scheme can compensate for the friction and deadzone nonsmooth nonlinear effects without depending on the specific compensator. 3) A new algorithm that synthesizes CMAC and ESNs to enhance the approximating performance of nonlinear function is proposed. 4) The experimental verification for the BLF-based constraint problem was executed, whereas the conventional BLF-based controls [6]–[8] were verified only numerically. 5) The proposed constraint method can be more easily applied to constant and time-varying constraints than those in [6]–[8].
Assumption 1: gi,j (·), j = 1, . . . , ni , are positive without loss of generality, and there are positive constants, g0i,j , such that 0 < g0i,j ≤ |gi,j (xi,j )|. Assumption 2: The desired trajectory vectors are continuous and available, and [ydi , y˙ di , y¨di ]T ∈ Ωdi , and Ωdi is a known 2 2 + y˙ di,1 + compact set, i.e., Ωdi = {[ydi,1 , y˙ di1, , y¨di,1 ]T : ydi,1 2 y¨di,1 ≤ δyi }, where δy > 0 is a constant. The mathematical models of the deadzone nonlinearity and the deadzone inverse are represented in [5] and [18]. The Gaussian friction as a classical static friction model and the estimation of it are well explained in [17] and [18]. The detail expressions for the deadzone and Gaussian friction nonlinearities are omitted. The control objectives for a nonlinear dynamic system are as follows. 1) Determine a state feedback control system, such that the system output xi,1 can track a desired trajectory ydi,1 while ensuring that all closed-loop signals are bounded. 2) The prescribed output tracking error bounds for ei,1 (t) = xi,1 (t) − ydi,1 (t) are always satisfied without depending on compensation for the nonlinear effects using friction and deadzone observers.
II. P ROBLEM F ORMULATION
ψi,1 (t) = (ψ0i,1 − ψssi,1 )e−ai,1 t + ψssi,1
A. Description of the Nonlinear MIMO Dynamic System Consider a MIMO strict feedback nonsmooth nonlinear system in the presence of friction and deadzone whose dynamic equation can be expressed as x˙ i,j = fi,j (xi,j ) + gi,j (xi,j )xi,j+1 − Ff i,j + Fui,j , .. . x˙ i,ni = fi,ni (xi,ni ) + gi,ni (xi,ni )Di (ui ) − Ff i,ni + Fui,ni i = 1, . . . , m
A smooth decreasing performance function ψi,1 (t) : R+ → R+ \ {0} with limt→∞ ψi,1 (t) = ψssi,1 was selected to satisfy the second control objective as follows: (2)
where ψ0i,1 , ψssi,1 and ai,1 are the appropriately defined positive constants. The transient performance is then guaranteed by the following prescribed constraint condition: −Mli,1 ψi,1 (t) < ei,1 (t) < ψi,1 (t) if ei,1 (0) ≥ 0
(3)
−ψi,1 (t) < ei,1 (t) < Mhi,1 ψi,1 (t) if ei,1 (0) < 0
(4)
or
j = 1, . . . , ni − 1,
yi,1 = xi,1 ,
B. Performance Function and Error Transformation
(1)
where n1 + · · · + nm = n, xi,j represents the states of the ith subsystem, x = [xT1,n1 , . . . , xTm,nm ]T ∈ Rn are the state vectors; xi,j = [xi,1 , . . . , xi,j ]T ∈ Rj , ui ∈ R and yi,1 ∈ R denote the control input and output of the ith subsystem, respectively; fi,j (·) and gi,j (·) are smooth linear or nonlinear functions; Ff i,j are nonlinear friction, and Fui,j are the bounded un∗ , including the external disturbances certainties |Fui,j | ≤ Fui,j and unmodelled and coupled dynamic terms. The signs of the control gain function, i.e., Di (ui ), were assumed to be the same as control inputs containing deadzone nonlinearity. Subscript i, which denotes the ith subsystem, is assumed to be repeated as i = 1, . . . , m, and this notation is omitted through this paper for convenience as long as a special notation is not required.
where 0 < Mhi 1 , Mli,1 ≤ 1 are the prescribed scale parameters. This performance function is illustrated in Figs. 2 and 3. The constant, i.e., ψssi,1 , confines the size of the tracking errors under a steady state. The decreasing rate, i.e., ai,1 , of ψi,1 (t) regulates the required speed of convergence of the tracking errors, and the maximum overshoot and undershoot can be controlled by selecting Mhi,1 ψi,1 (0), Mli,1 ψi,1 (0), and ψi,1 (0). Next, a transformed error, i.e., ξi,1 , is defined as follows: ξi,1 (t) =
ei,1 (t) ηi,1 (t)
(5)
ηi,1 = pηpi,1 + (1 − p)ηni,1
(6)
where p = 1 if ei,1 (t) ≥ 0 and p = 0 if ei,1 (t) < 0. ηpi,1 and ηni,1 are defined as follows: ηpi,1 = ψi,1 (t) and ηni,1 (t) = −Mli,1 ψi,1 (t), if ei,1 (0) ≥ 0
(7)
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Fig. 1. Proposed CESNs architecture.
ηpi,1 = Mhi,1 ψi,1 (t) and ηni,1 (t) = −ψi,1 (t), if ei,1 (0) < 0.
(8)
where Ωζi = [ζ i , W i , ρi ]T , ςli,1, and ςhi,1 are class K∞ functions. Let Vi (θi ) := Vb (ξ i ) + Ui (Ωζi ) and ξ i (0) ∈ Σi . If the following inequality holds:
Given that |ei,1 (0)| ≤ |ηi,1 (0)|, the following condition was obtained by checking the given equations (3)–(8):
∂Vi V˙ i = ϕi ≤ −Ci Vi + μi ∂θi
0 ≤ ξi,1 < 1.
(9)
in the set θi ∈ Ni and Ci , μi are constants, then Ωζi remains bounded, and ξ i (t) remains in the open set Σi ∀ t ∈ [0, ∞). Remark 1: A BLF given in (10) is much simpler than the asymmetric BLF used in [6]–[8] as follows:
A BLF similar to that used in [6]–[8] could be defined to ensure the tracking error constraint condition given in (9). Definition 1: A BLF is defined as the following form:
2p 2p kbi,1 (t) kai,1 (t) q(ei,1 ) 1 − q(ei,1 ) log 2p log 2p Vb = 2p + 2p 2p kbi,1 (t)−ei,1 kai,1 (t)−e2p i,1 (15)
C. BLF
Vb =
2 1 ξi,1 2 . 2 1 − ξi,1
(10)
This BLF approaches infinity at ξi,1 = 1 and is a valid Lyapunov function because Vb is positive definite and C 1 in the set Σi := {ξi,1 ∈ R : 0 ≤ ξi,1 < 1} ⊂ R. Similar to [6], the following lemma 1 could be presented and proven easily using the procedure reported in [8]. Lemma 1: Let Σi := {ξi,1 ∈ R : ξi,1 < 1} ⊂ Rm and Ni := 3m R × Σi ⊂ R4m be open sets. Consider the following system: θ˙ i = ϕi (t, θi )
(11)
where θi := [ξ i , ζ i , W oi , ρi ]T ∈ Ni is the state, ξ i = [ξ1,1 , . . . , ˆ o1,1 , . . . , W ˆ om,1 ]T , ξm,1 ]T , ζ i = [ζi,2 , . . . , ζm,2 ]T , W oi = [W T ρi = [ˆ ρ1,1 , . . . , ρˆm,1 ] , and the function ϕi : R+ × Ni → R4m satisfies the conditions that ϕi is locally Lipschitz in θi , and ϕi is locally integrable on t. Suppose that there are functions Vb : Σi → R+ , and Ui : R3m → R+ that are continuously differentiable and positive definite in their respective domains, such that the following statements are true: Vb (ξ i ) → ∞ as ξi,1 → 1 ςli,1 Ωζi ≤ Ui (Ωζi ) ≤ ςhi,1 Ωζi ,
(12)
i = 1, . . . , m
(13)
(14)
where p is a positive integer satisfying 2p ≥ ni . Therefore, the designed controller based on the BLF given in (15) also becomes too complex for the designed controller from (10). D. Function Approximation Using CESNs System Fig. 1 presents the CESNs system. The signal propagation and basic function in each space of the CMAC are introduced as follows. 1) Input space X: For a given x = [x1 , x2 , . . . , xna ]T ∈ Rna , each input state variable is quantized into discrete regions (called elements) according to the given control space. 2) Association memory space A: Several elements can be accumulated as a block, and the number of blocks, i.e., nb , normally exceeds two. A denotes an association memory space with nc (nc = na × nb ) components. In this space, each block performs a receptive-field basis function, i.e., (xi − mjk )2 Φjk = exp − , for k = 1, 2, . . . , nb (16) 2 vjk where Φjk is the output of the kth block receptive-field basis function for the jth input xj with mean mjk and variance vjk .
HAN AND LEE: ROBUST POSITIONING CONTROL FOR A NONSMOOTH NONLINEAR DYNAMIC SYSTEM
3) Receptive-field space T : In this paper, the number of receptive fields, i.e., nc , is equal to nb . The following kth multidimensional receptive-field function is defined:
Step i, 1: The error surfaces are defined by the following equations:
bk (xi , mk , v k ) =
na j=1
= exp ⎣−
na (xi − mjk )2 j=1
2 vjk
⎤ ⎦ , k = 1, 2, . . . , nc (17)
where mk = [mk1 , . . . , mkna ]T ∈ Rna , v k = [vk1 , . . . , vkna ]T ∈ Rna . The multidimensional receptive-field function can be expressed in a vector notation as follows: Θ(xi , m, v) = [b1 , . . . , bnc ]T
(23)
Si,1 ηi,1
(24)
Si,2 = xi,2 − zi,2
(25)
X˙ = (1 − β)X + F (Win u + Wd X + Wf b y + υ)
(20)
where Wo ∈ RN ×1 is the output weight matrix. The initial value of the output weight matrix is selected as Wo = χT ydi . The function f (x) can be expressed as follows: f (x) = Wo∗T χ(x) + ε∗ ,
∀ x ∈ Ω ⊂ Rn
Tsi,2 z˙i,2 + zi,2 = αi,1 ,
(21)
where |ε∗ | ≤ εm , ε∗ is the error of the CESNs approximation, and Wo∗ is chosen as
∗ T ˆ (22) Wo = arg min sup f (x) − Wo χ(x) . x∈Ω
Wo∗
ˆ o , which is Because is unknown, it will be replaced with W an estimate of Wo . III. D ESIGN OF C ONTROLLER AND C OMPENSATOR Here, the virtual control functions, adaptive laws, and control laws are derived using recursive DSC design procedures [15].
zi,2 (0) = αi,1 (0)
(26)
the output errors of these filters were obtained as follows: ζi,2 = zi,2 − αi,1 .
(27)
From (26) and (27), we have z˙i,2 = −
ζi,2 . Tsi,2
(28)
Therefore, the time derivative of the filter output error can be expressed as
(19)
where X is the N -dimensional reservoir activation state, β > 0 is the leaking rate of the reservoir neuron, and F is a threshold function (tanh(·) function in this study). Win ∈ RN ×M , Wd ∈ RN ×N , and Wf b ∈ RN ×1 are the input, internal, and feedback connection weight matrices, respectively, and υ is a suitably normalized noise vector. u denotes the M -dimensional external input and corresponds to the output of the receptive-field space. 5) Output space O: The output equation is given as follows: y = WoT χ
where ydi,1 is a command output, and zi,2 is a filtering virtual control. Let a virtual control αi,1 pass through the first-order filters with time constants Tsi,2 by introducing filtering virtual controls zi,2 , as follows:
(18)
where m = [mT1 , . . . , mTk , . . . , mTnc ]T ∈ Rna nc , and v = [v T1 , . . . , v Tk , . . . , v Tnc ]T ∈ Rna nc . 4) Internal hidden layer space H: This space consists of a dynamic reservoir of leaky integrator ESNs [14], whose continuous time dynamic can be described as
Wo ∈RN ×1
Si,1 = xi,1 − ydi,1 ξi,1 =
Φjk ⎡
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ζi,2 − α˙ i,1 . ζ˙i,2 = − Tsi,2 The Lyapunov function defined as m 2 1 ξi,1 V1 = 2 + 2 1 − ξi,1 i=1
(29)
candidate, including a BLF, can be 1 2 ζ 2 i,2
1 ˜ T −1 ˜ 1 2 + Woi,1 Γi,1 Woi,1 + ρ˜ . 2 2γρi,1 i,1
(30)
The time derivative of (30) is given as m ξi,1 ˙ ˆ T χi,1 (xi,1 ) V1 ≤ gi,1 (xi,1 )αi,1 + W oi,1 2 2η 1 − ξ i,1 i=1 i,1 + ρˆi,1 − ξi,1 η˙ i,1 − y˙ di,1 gi,1 (xi,1 )ξi,1 Si,2 gi,1 (xi,1 )ξi,1 ζi,2 + + 2 2 2 2η 1 − ξi,1 ηi,1 1 − ξi,1 i,1 ξ i,1 ˙ T −1 ˆ oi,1 ˜ oi,1 χi,1 (xi,1 ) +W − Γi,1i W 2 2η 1−ξi,1 i,1 ξi,1 ρ˜i,1 1 − ρ˜i,1 ρˆ˙ i,1 2 2 γ ρi,1 1 − ξi,1 ηi,1 2 ζi,2 − + |ζi,2 ||α˙ i,1 | Tsi,2 +
(31)
˜ o1 = W ˆ o1 − where Γi,1 > 0 is a diagonal constant matrix, W ∗ , γρi,1 > 0 is a constant, ρˆi,1 is an estimate of ρ∗i,1 , Wo1
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∗ ρi,1 = ρˆi,1 − ρ∗i,1 , and |Fui,1 + ε∗i,1 | ≤ ρ∗i,1 . The virtual control and estimation law can be expressed as 1 ˆ T χi,1 (xi,1 ) αi,1 = − ci,1 ξi,1 ηi,1 − W oi,1 gi,1 (xi,1 ) ρˆi,1 ξi,1 − + ξi,1 η˙ i,1 + y˙ di,1 (32) ξi,1 + κi,1 ξi,1 ˙ˆ ˆ oi,1 W oi,1 = Γi,1 − κwi,1 W (33) 2 2η 1 − ξi,1 i,1 2 ξi,1 ˙ρˆi,1 = 1 − kρi,1 ρˆi,1 2 2η γρi,1 1 − ξi,1 i,1 (|Si,1 | + κi,1 ) (34)
where ci,1 > 0 and κi,1 > 0 are the design constants. The virˆ T , ρˆi,1 , ξi,1 , and tual control αi,1 is bounded because Si,1 , W oi,1 ydi,1 are bounded, and α˙ i,1 is then bounded such that |α˙ i,1 | ≤ ∗ ∗ and α˙ mi,1 > 0 is a bounded constant. Considering the α˙ mi,1 following relations: gi,1 (xi,1 )ξi,1 ζi,2 ≤ 2 2η 1 − ξi,1 i,1
2 gi,1 (xi,1 )ξi,1 2 2η 1 − ξi,1 i,1
+
2 gi,1 (xi,1 )ζi,2 2 2η 4 1 − ξi,1 i,1
1 ∗2 2 |ζi,2 ||α˙ i,1 | ≤ ζi,2 + α˙ mi,1 . 4
(35)
The Lyapunov function candidate is defined as V2 =
m 1 i=1
(42) j = 2, . . . , ni − 1 m 1 ˜T 1 −1 ˜ V3 = ρ˜2i,ni . (43) Si,ni + W oi,ni Γi,ni Woi,ni + 2 2γ ρi,n i i=1 The time derivatives of (42) and (43) are given as m T ˆ oi,j V˙ 2 ≤ χi,j (xi,j )+ ρˆi,j − z˙i,j Si,j gi,j (xi,j )xi,j+1 + W i=1
V˙ 3 ≤
m i=1
(36)
The following relation can be easily obtained: −
2 ξi,1 ≤ −ξi,1 + κi,1 . ξi,1 + κi,1
1 2 1 ˜ T −1 ˜ 1 2 W Si,j + ζi,j+1 + W Γ + ρ ˜ , oi,j 2 2 2 oi,j i,j 2γρi,j i,j
T ˜ oi,j ˆ˙ +W χi,j (xi,j ) Si,j − Γ−1 ˜i,j i,j W oi,j + Si,j ρ 2 ζi,j+1 1 − ρ˜i,j ρˆ˙ i,j − − |ζi,j+1 ||α˙ i,j | , γρi,j Tsi,j+1
j = 2, . . . , ni − 1
(44)
ˆ T χi,n (xi,n ) Si,ni gi,ji (xi,ji )udi + W oi,ni i i ˜ T χi,n (xi,n ) + ρˆi,ni − z˙i,ni + W oi,ni i i ˆ˙ × Si,ni − Γ−1 ˜i,ni i,ni W oi,ni + Si,ni ρ 1 − ρ˜i,ni ρˆ˙ i,ni (45) γρi,ni
(37)
where Γi,j > 0 are diagonal constant matrices, γρi,j > 0 are constants, ρˆi,j are estimates of ρ∗i,j , ρi,j = ρˆi,j − ∗ ∗ + ε∗i,j | ≤ ρ∗i,j , |Fui,n + ε∗i,ni + ΔDi | ≤ ρ∗i,ni , and ρ∗i,j , |Fui,j i ∗ |α˙ i,j | ≤ α˙ mi,j . The virtual controls, control input, and estimation laws are expressed as follows: gi,1 (xi,1 )ξi,1 Si,2 T ˆ oi,1 ×W χi,1 (xi,1 ) + kρi,1 ρ˜i,1 ρˆi,1 + 1 ρˆi,2 |Si,2 | 2 T ˆ oi,2 2 −ci,2 Si,2 − W χi,2 (xi,2 )− αi,2 = 1 − ξi,1 ηi,1 gi,2 (xi,2 ) |Si,2 |+κi,2 1 gi,1 (xi,1 ) 2 − − − 1 ζi,2 gi,1 (xi,1 )ξi,1 2 2η Tsi,2 − + z˙i,2 (46) 4 1 − ξi,1 i,1 2 2η 1−ξi,1 i,1 ∗ κi,1 ρi,1 1 ∗2 + α˙ mi,1 + . (38) 1 ρˆi,j |Si,j | T 2 ˆ oi,j 2 4 χi,j (xi,j )− αi,j = −ci,j Si,j − W 1 − ξi,1 ηi,1 gi,j (xi,j ) |Si,j |+κi,j Step i, 2: Similar procedures were repeated to ni step. The −gi,j−1 (xi,j−1 )Si,j−1 + z˙i,j , error surfaces were defined as
Substituting (32)–(37) into (31) gives m 2 − (ci,1 − gi,1 (xi,1 )) ξi,1 ˜ T χi,1 (xi,1 ) V˙ 1 ≤ + kwi,1 W 2 oi,1 2 1 − ξi,1 i=1
Si,j = xi,j − zi,j ,
j = 2, . . . , ni
(39)
where zi,j is filtering virtual control. Introducing filtering virtual controls zi,j and allowing virtual control αi,j−1 to pass through the first-order filters with time constants Tsi,j , the time derivatives of the error surfaces are expressed as S˙ i,j = xi,j (xi,j )xi,j + fi,j (xi,j ) + Fui,j − z˙j T T ˆ oi,j ˜ oi,j = gi,j xi,j+1 + W χi,j (xi,j ) − z˙j + W χi,j (xi,j ) + ε∗i,j + Fui,j ,
j = 2, . . . , ni − 1
S˙ i,ni = gi,ni (xi,ni )Di (ui ) + fi,ni (xi,ni ) + Fui,ni .
(40) (41)
udi
j = 3, . . . , ni − 1 (47) 1 ˆ T χi,n (xi,n ) −ci,ni Si,ni − W = oi,ni i i gi,ni (xi,ni ) ρˆi,ni |Si,ni | − gi,ni −1 (xi,ni −1 ) |Si,ni | + κi,ni × Si,ni −1 + z˙i,ni (48) ˆ oi,j , j = 1, . . . , ni = Γi,j Si,j −κwi,j W (49) |Si,j |2 1 −kρi,j ρˆi,j , j = 2, . . . , ni (50) = γρi,j (|Si,j |+κi,j ) −
ˆ˙ oi,j W ρˆ˙ i,j
HAN AND LEE: ROBUST POSITIONING CONTROL FOR A NONSMOOTH NONLINEAR DYNAMIC SYSTEM
where ci,j > 0 and κi,j > 0 are the design constants. From the results in [19], the output tracking errors ei,1 (t) remain in the set Ωei,1 := {ei,1 (t) ∈ R : |ei,1 (t)| ≤ |ηi,1 (t)|, i = 1, . . . , m} ∀ t > 0, and the output tracking error constraints are never violated if the initial condition |ei,1 (0)| < |ηi,1 (0)| is satisfied. Considering the previous results, the following expression is obtained: V˙ =
3
V˙ i ≤
i=1
×
2 2 × Si,k − ci,ni Si,n − i ni kρi,k ρ˜2i,k
×
2 ζi,2
2 −
−
ni kρi,k ρ∗2 i,k
−
1 Tsi,2
1 Tsi,k
n i −1
(ci,k − gi,k (xi,k ))
k=2
k=1
ni kρi,k ρ∗i,k
gi,1 (xi,1 ) − −1 2 2η 4 1 − ξi,1 i,1
m
k=1
k=1
⎞ +
1
(51)
k=2
The control and adaptive gains are selected as follows: 2 Ξi,1 /2 ci,1 = gi,1 + 1 − ξi,1 ci,k = gi,k + Ξi,k /2, k = 2, . . . , ni − 1 ci,ni = Ξi,ni /2 kwi,k = Ξi,k , k = 1, . . . , ni kρi,k = Ξi,k , k = 1, . . . , ni .
(52)
Assumption 3: The time constant of the low-pass filter is chosen to be sufficiently small such that
κi,1 ρ∗i,1 2 2η − ξi,1 i,1
4
∗2 α˙ mi,k +
k=1
ni
⎟ κi,k ρ∗i,k⎠ .
k=2
Therefore, all error signals are semiglobally uniformly and ultimately bounded provided that CESNs are chosen to cover a bounded compact set of sufficiently large size. From (56), if V (0) = μ/Ξ, the following inequality holds: μ |Sj,k | ≤ = μ∗i,k , i = 1, . . . , m, k = 2, . . . , ni . (57) Ξ If V (0) = μ/Ξ, from (56), given any μi,k > μ∗i,k , there exist Ts,k , such that for any t > Ts,k , |Si,k | ≤ μi,k . In particular, given any μi,k μ μ −Ξt μ e μi,k = 2 V (0) − + V (0) = , Ξ Ξ Ξ i = 1, . . . , m
(53) Ts,k
1
1 ≥ gi,j max (xi,j )+2Ξi,j +1, j = 2, . . . , ni −1. (54) Tsi,j+1 4
+
ni 1
Multiplying (55) by eΞt and integrating it over [0, t] leads to the following inequality: μ −Ξt μ e (56) + , ∀ t ≥ 0. 0 ≤ V (t) ≤ V (0) − Ξ Ξ
then
gi,1 max (xi,1 ) ≥ +2Ξi,1 +1 Tsi,2 4 1−ξ 2 2 ηi,1 i,1 1
k=2
⎛ ∗ 2 ni k ni m W wi,k kρi,k ρ∗i,k oi,k ⎜ + μi = ⎝ 2 2 i=1
1 2 − gi,k (xi,k ) − 1 ζi,k 4
+
k=1
+
i=1
2
κi,1 ρ∗i,1 2 2η 2 1 − ξi,1 i,1 k=1 ni ni 1 ∗2 + α˙ mi,k + κi,k ρ∗i,k . 4
2
k=1
where Ξ = min[Ξi,1 , . . . , Ξi,k ] for i = 1, . . . , m, k = 1, . . . , ni , and μ=
ni ˜T W ˜ kwi,k W oi,k oi,k k=1
n i −1 k=3
+
2 Ξi,k+1 ζi,k+1 +
k=1
2 − (ci,1 − gi,1 (xi,1 )) ξi,1 2 2 1 − ξi,1
k=1
1 2
∗ 2 ni k W wi,k oi,k
κi,1 ρ∗i,1 2 2η 2 1 − ξi,1 i,1 k=1 n n i i 1 ∗2 + α˙ mi,k + κi,k ρ∗i,k = −ΞL + μ (55) 4
+
m i=1
−
−
n i −1
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μ μ2i,k − Ξ 1 = − ln μ Ξ 2 V (0) − Ξ
lim |Si,k | → μ∗i,k , i = 1, . . . , m.
t→∞
(58)
(59) (60)
Therefore, (51) can be expressed as V˙ ≤
m i=1
ni 2 Ξi,1 ξi,1 1 2 Ξi,k Si,k − − 2 2 2 1 − ξi,1 k=2 i i 1 ˜T W ˜ oi,k − 1 − Ξi,k W Ξi,k ρ˜2i,k oi,k 2 2
n
n
k=1
k=1
IV. A PPLICATION E XAMPLES A. Simulated Example As a control system for the constraint problem of the output tracking errors, two inverted pendulums composed of a spring and a damper connection and nonlinear friction were used in [20].
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The dynamic equation of the inverted pendulum can be described as follows: J1 θ¨1 = m1 gr sin θ1 − 0.5F r cos(θ1 − θ) − Tf1 + D1 (u1 ) (61) ¨ J2 θ2 = m2 gr sin θ2 + 0.5F r cos(θ2 − θ) − Tf2 + D2 (u2 ) (62) where θ1 and θ2 are the angular positions, J1 = 0.5 kg · m2 and J2 = 0.625 kg · m2 are the moments of inertia, m1 = 2 kg and m2 = 2.5 kg are the masses, r = 0.5 m, Di (ui ) are the deadzone inputs, Tf i are the friction torques, F = k(p−l)+bp˙ denotes the force applied by the spring and damper at the connection points, and p = d2 + dr(sin θ1 −sin θ2 ) + (r2 /2)[1−cos(θ2 −θ1 )]. d = 0.5 m, k = 150 N/m, and b = 1 N sec/m. The deadzone widths were chosen as dr1 = 20 Nm, dl1 = −20 Nm dr2 = 10 Nm, dl2 = −10 Nm, the slope of the deadzone mri = mli = 1, and θ = tan−1 (((r/2)(cos θ2 − cos θ1 )/d + (r/2)(sin θ1 − sin θ2 ))). Tf 1 and Tf 2 are assumed to be the LuGre friction model [2], i.e., Tf i = σ0 zi + σ1 z˙i + Tv θ˙i z˙i = θ˙i −
σ0 θ˙i
, i = 1, 2 (63)
˙ ˙ 2 θs ) Tc + (Ts − Tc ) exp(− θ/
where σ0 = 100 Nm, σ1 = 1 Nm sec/rad, Tv = 1 Nm/sec, θ˙s = 0.1 rad/sec, Ts = 5 Nm, and Tc = 4.8 Nm. The control objective was for x1,1 (t) and x2,1 (t) to follow the desired commands yd1,1 (t) = 0.5 sin(2πt), yd2,1 (t) = 0.4 sin(πt), and subject to the error performance functions as follows: ψ1,1 (t) = (ψ01,1 −ψss1,1 )e−a1,1 t + ψss1,1 ,
if e1,1 (0) ≥ 0
ψ2,1 (t) = (ψ02,1 −ψss2,1 )e−a2,1 t + ψss2,1 ,
if e2,1 (0) < 0.
The selected initial points of each state were selected as x1,2 (0) = (π/3.6, 0) and x2,2 (0) = (π/4, 0). The parameters of the controller were selected as c1,1 = 5, c1,2 = 10, c2,1 = 5, c2,2 = 10, γw1,2 = 1, Ts1,2 = 0.01, κ1,2 = 0.1, γw2,2 = 1, κ2,2 = 0.1, and Ts2,2 = 0.01. The following parameters of the prescribed performance functions were selected: ψ01,1 = π/3, ψss1,1 = π/360 Ml1,1 = 0.2, a1,1 = 1, (e1,1 (0) ≥ 0) ψ02,1 = π/2, ψss2,1 = π/360 Mh2,1 = 0.2, a2,1 = 1, (e2,1 (0) < 0) . nr = 5 and nt = 4 in CMAC (nt is the number of elements in a complete block) were used in simulation. In the ESN system, N = 20, M = 8, and β = 0.7. The weight matrices Win ∈ R20×8 , Wd ∈ R20×20 , and Wf b ∈ R20×1 are randomly selected, respectively, and υ = 0.001 × WfTb . For a comparative simulation, six control systems were designed as follows: 1) QLF-CMAC-WO: the conventional quadratic Lyapunov function (QLF) based on DSC with CMAC and without friction and deadzone compensation;
2) QLF-CESN-WO: the conventional QLF based on DSC with CESNs and without friction and deadzone compensation; 3) QLF-CESN-WF: the conventional QLF based on DSC with CESNs and only friction compensation; 4) QLF-CESN-WDF: the conventional QLF based on DSC with CESNs and with friction and deadzone compensation; 5) BLF-CESN-WDF: the proposed BLF-based prescribed performance constraint combined with CESNs and with friction and deadzone compensation; 6) BLF-CESN-WO: the proposed BLF-based prescribed performance constraint combined with CESNs without friction and deadzone compensation. Fig. 2 shows the simulated responses. The QLF-based control system violated the prescribed constrained bounds, whereas the BLF-based control systems satisfied the prescribed output error performance constraints. In Fig. 2(a) and (b), the QLF-CESNWDF system violates the prescribed tracking error constraint despite compensation for both friction and deadzone using CESNs and inverse deadzone method given in (3). On the other hand, the proposed BLF-CESN systems did not transgress the prescribed tracking error constraint, although deadzones were not compensated for due to the BLF-based controller. Finally, the sizes of the tracking error of the proposed CESN-based control were lower than those of the CMAC-based control, as shown in Fig. 2(a) and (b) and Table I. B. Experimental Example This section describes the experimental application of the proposed control scheme to the Scorbot robot system. A photograph and manipulator of the Scorbot robot is represented in [18]. The dynamic equation for two degrees of freedom links of the Scorbot robot manipulator can be shown in [18]. A sinewave motion of the end-effector was chosen as the desired trajectory. The following tracking commands were chosen: link1 : qd1 (t) = 0.1 sin(2.5132t)(rad) link2 : qd2 (t) = 0.1 sin(2.5132t)(rad). Initial conditions were x1,1 (0) = −0.08 rad and x2,1 (0) = 0.15. The preceding transient and steady-state tracking error bounds are prescribed as the following performance functions: ψ2,1 (t) = (ψ02,1 −ψss2,1 )e−a2,1 t +ψss2,1 ,
if e2,1 (0) < 0.
ψ2,1 (t) = (ψ02,1 −ψss2,1 )e−a2,1t +ψss2,1 ,
if e2,1 (0) < 0.
The CMAC parameters are selected to be nr = 5 and nt = 4 (nt is the number of elements in a complete block). The input ranges of the CMAC system were selected as xi,1 = [−0.035 −0.021 −0.007 0.021 0.035], i = 1, 2. The parameters of the ESN system were selected similar to simulation case. The experiment system is the same as that given in [18]. The parameters of the controller of the BLF-based controller were selected as c1,1 = 1, c1,2 = 0.1, c2,1 = 0.5, c2,2 = 0.1, γw1,2 = 0.2, κ1,2 = 0.001, γw2,2 = 0.1, κ2,2 = 0.001, Ts1,2 = 2, Ts2,2 = 2, ψ01,1 = 0.2, ψss1,1 = 0.0075,
HAN AND LEE: ROBUST POSITIONING CONTROL FOR A NONSMOOTH NONLINEAR DYNAMIC SYSTEM
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Fig. 2. Simulated position tracking responses for the output tracking error constraint. (a) Position tracking errors of pendulum 1 of QLF-CMAC and CESN. (b) Position tracking errors of pendulum 2 of QLF-CMAC and CESN. (c) Position tracking errors of pendulum 1: BLF-CESN-WDF and BLF-CESN-WO. (d) Position tracking error of pendulum 2: BLF-CESN-WDF and BLF-CESN-WO. TABLE I ROOT M EAN S QUARE (RMS) T RACKING E RROR VALUES OF E ACH S YSTEM IN S IMULATION
Ml1,1 = 0.5, a1,1 = 1, ψ02,1 = 0.2, ψss2,1 = 0.005, Mh2,1 = 0.5, and a2,1 = 1. For comparative experiments, QLF-CMACWO, QLF-CESN-WO, QLF-CESN-WF, QLF-CESN-WDF, BLF-CESN-WF, BLF-CESN-WDF, and BLF-CESN-WO were designed to be similar to the simulation case. The output tracking errors of the QLF-based control were sensitive to the use of friction and deadzone compensations and the control gain values, and a higher gain made the error size smaller as shown in Table III. In Table II, the tracking performance of the QLF control system using CESNs were improved over the CMAC-based one like simulated result. Fig. 3(a) and (b) shows the output tracking errors when the control gains of the QLF-based control systems were selected as c1,1 = 100 and c2,1 = 50, respectively, under fixed gain values, c1,2 = c2,2 = 5. However, in these figures, QLF-based control systems violated the prescribed error constraint, although the high control gains (100 times larger than the BLF case) were selected, and the
TABLE II RMS T RACKING E RROR VALUES OF THE QLF S YSTEM IN E XPERIMENT (c1,1 = 100, c2,1 = 50)
friction and deadzone were compensated by the friction and deazone observers given in [18]. Fig. 3(c) and (d) and Table III present the comparative experiments of BLF-CESN-WF, BLFCESN-WDF, and BLF-CESN-WO, along with the prescribed performance functions. The approach results in an output tracking performance that falls within the prescribed transient and steady-state bounds, and the error size is similar to friction and deadzone compensated system, regardless of the friction and deadzone compensation. V. C ONCLUSION A robust positioning scheme was developed with the prescribed performance constraints for a strict feedback nonlinear MIMO dynamic system in the presence of both friction and deadzone. A more convenient prescribed error transformation method based on the specified plausible conditions and a simple sign function and BLF, which is in contrast to the conventional approaches that require a complex BLF
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Fig. 3. Experimental responses. (a) The tracking errors of link 1 of QLF-CESN systems: QLF-CESN-WF and QLF-CESN-WDF. (b) The tracking error of link 2 of QLF-CESN systems: QLF-CESN-WF and QLF-CESN-WDF. (c) The tracking errors of link 1 of BLF-CESN-WF, BLF-CESN-WDF, and BLF-CESN-WO. (d) The tracking errors of link 2 of BLF-CESN-WF, BLF-CESN-WDF, and BLF-CESN-WO.
TABLE III RMS T RACKING E RROR VALUES OF THE BLF S YSTEM IN E XPERIMENT
and parameter estimation for nonsmooth nonlinearities. The adaptive CESNs were developed to enhance the approximation performance compared with that of the conventional CMAC. To demonstrate the application of the method, the tracking performance of a double inverted pendulum and Scorbot robot manipulator were evaluated numerically and experimentally. The proposed control scheme did not violate the prescribed performance bound and provided favorable position tracking performance without the need for friction and deadzone parameter estimations. R EFERENCES [1] H. Chaoui and P. Sicard, “Adaptive fuzzy logic control of permanent magnet synchronous machines with nonlinear friction,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1123–1133, Feb. 2012. [2] 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. [3] J. Swevers, F. Al-Bender, C. Ganseman, and T. Prajogo, “An integrated friction model structure with improved structure with improved presliding behavior for accurate friction model structure,” IEEE Trans. Autom. Control, vol. 45, no. 4, pp. 675–686, Apr. 2000.
[4] W. S. Huang, C. W. Liu, P. L. Hsu, and S. S. Yeh, “Precision control and compensation of servomotors and machine tools via disturbance observer,” IEEE Trans. Ind. Electron., vol. 57, no. 1, pp. 420–429, Jan. 2010. [5] G. Tao and P. V. Kokotovic, “Adaptive control of plants with unknown dead-zones,” IEEE Trans. Autom. Control, vol. 39, no. 1, pp. 59–68, Jan. 1994. [6] 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, Apr. 2009. [7] 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. [8] P. Tee, B. Ren, and S. S. Ge, “Control of nonlinear systems with timevarying output constraints,” Automatica, vol. 47, no. 11, pp. 2511–2516, Nov. 2011. [9] M. Kristic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. Hoboken, NJ, USA: Wiley, 1995. [10] J. S. Albus, “A new approach to manipulator control: The Cerebellar Model Articulation Controller (CMAC),” Trans. ASME, J. Dyn. Syst. Meas. Control, vol. 97, no. 3, pp. 220–227, Sep. 1975. [11] C. H. Chiu, “The design and implementation of a wheeled inverted pendulum using an adaptive output recurrent cerebellar model articulation controller,” IEEE Trans. Ind. Electron., vol. 57, no. 5, pp. 1814–1822, May 2010. [12] Y. Pan and J. Wang, “Model predictive control of unknown nonlinear dynamical systems based on recurrent neural networks,” IEEE Trans. Ind. Electron., vol. 59, no. 8, pp. 3089–3101, Aug. 2012. [13] X. Gao, D. You, and S. Katayama, “Seam tracking monitoring based on adaptive Kalman filter embedded Elman neural network during highpower fiber laser welding,” IEEE Trans. Ind. Electron., vol. 59, no. 11, pp. 4315–4325, Nov. 2012. [14] H. Jaeger, L. Mantas, P. Dan, and S. Udo, “Optimization and application of echo state networks with leaky integrator neurons,” Neural Netw., vol. 20, no. 3, pp. 335–352, Apr. 2007. [15] 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.
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[16] T. S. Li, S. C. Tong, and G. Geng, “A novel robust 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. [17] S. N. Huang, K. K. Tan, and T. H. Lee, “Sliding-mode monitoring and control of linear driver,” IEEE Trans. Ind. Electron., vol. 56, no. 9, pp. 3532–3540, Sep. 2009. [18] S. I. Han and J. M. Lee, “Precise positioning of nonsmooth dynamic systems using fuzzy wavelet echo state networks and dynamic surface sliding mode control,” IEEE Trans. Ind. Electron., vol. 60, no. 11, pp. 5124–5136, Nov. 2013. [19] S. I. Han and J. M. Lee, “Improved prescribed performance constraint control for a strict feedback non-linear dynamic system,” IET Control Theory Appl., vol. 7, no. 14, pp. 1818–1827, Sep. 2013. [20] 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. 2224–2239, Sep. 2011.
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, 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.
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Jang M. Lee (SM’03) received the B.S. and M.S. degrees 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. Dr. Lee was the former President of the Korean Robotics Society. He is a Member of the Industrial Electronics Society.
