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Procedia Computer Science 00 (2018) 000–000 Procedia Computer Science 125 (2018) 25–33
www.elsevier.com/locate/procedia
6th International Conference on Smart Computing and Communications, ICSCC 2017, 7-8 December 2017, Kurukshetra, India
RBF Neural Control Design for SISO Nonaffine Nonlinear Systems Pramendra Kumara , Naveen Kumarb,∗, Vikas Panwara a Department
b Department
of Applied Mathematics, Gautam Buddha University, Greater Noida-201308, India of Mathematics, National Institute of Technology (NIT), Kurukshetra, Kurukshetra-136119, Haryana, India
Abstract In the present paper, an RBF neural control scheme is designed for regulatory control of SISO nonaffine systems facing unknown nonlinearities. Using Taylor series expansion, the nonaffine part of the system is converted into affine form. RBF network is utilized to estimate the equivalent affine system. The parameters of RBF network are updated online based on Lyapunov stability theory. To avoid the requirement of measurement of the states of the system, an observer is designed, which provides the estimated values of the system’s states. Using Lyapunov theory, the signals of the system are shown to be asymptotically stable. To validate the effectiveness of the presented scheme, numerical simulation study has been performed. c 2018 The Authors. Published by Elsevier B.V. ⃝ Peer-review under responsibility of the scientific committee of the 6th International Conference on Smart Computing and Communications. Keywords: Nonaffine Nonlinear System, RBF Neural Network, Lyapunov stability theory.
1. Introduction During last decade, a large number of control schemes are developed for uncertain systems with the assumption that the the system is affine (linear) in the control input. However in many industrial applications such as flight control[1], chemical systems [2], wind turbines [3], tank reactors [4] etc., the system to be controlled is nonaffine. Therefore controller design for nonaffine systems has evolved as a challenging problem. Many remarkable controllers are designed for nonaffine systems. Previously control schemes incorporated backstepping techniques for control of nonaffine systems[5, 6, 7, 8, 9, 10]. In [5] an adaptive scheme was discussed for non-affine systems using neural networks. By utilizing time-scale seperation to estimate the inversion of nonaffine system, a novel method was presented by singular perturbation method in [6] and [7]. In [8] a neural control based synthesis approach was used for nonaffine systems and the assumption of fixed-point was eliminated. A disturbance observer was designed for nonaffine system in [9] to estimate the compounded disturbances of the system. In [10] a control scheme was designed for hysteretic systems by combining the merits of hysteresis model with backstepping technique with no requirement of hysteresis inverse. ∗
Corresponding author. Tel.: +91-1744-233508 E-mail address:
[email protected] (Naveen Kumar). c 2018 The Authors. Published by Elsevier B.V. 1877-0509 ⃝ Peer-review under responsibility of the scientific committee of the 6th International Conference on Smart Computing and Communications.
1877-0509 © 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 6th International Conference on Smart Computing and Communications 10.1016/j.procs.2017.12.006
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However, in backstepping design the complexity of the control design increases with the system order as a result of the recurring differentiation of the control input. To overcome this issue, another technique named as dynamic surface control (DSC), was used for control of nonaffine system[11, 12, 13]. In [11] novel DSC scheme was presented using multilayer neural networks and a filter was utilized to eliminate the aforementioned issue of the repeated differentiation. In [12] by incorporating DSC approach with NN based approach, backstepping based controller was designed for nonlinear systems. However, it is a recursive method while requiring O(n) number of NNs and auxiliary low-pass filters for an nth order system. Adaptive backstepping-free controllers were also studied [14, 15, 16, 17, 18, 19, 20]. In [14], an adaptive backstepping-free controller was presented. To approximate the states and unknown nonlinearities, NN observer was proposed. In [15] an adaptive scheme was proposed for nonaffine systems. A fuzzy logic based recurrent scheme was presented for non-affine system by utilizing observer for the states. in [16]. In [17] an adaptive controller, based on observer, was presented for uncertain SISO non-affine systems. In [18] neural network based control scheme was proposed for non-affine systems by considering the delay problem of the system. In [19], different optimal control schemes were presented for systems having nonlinearities. For a nonaffine system with nonlinearities of dead-zone and facing external disturbances, an asymptotically stable control scheme was presented in [20]. In the present paper, we have designed an RBF neural network based control scheme for nonaffine systems. Firstly nonaffine part of the system is transformed into affine form by utilizing taylor series approach. Then RBF neural network is utilized to estimate the equivalent affine system. In many industrial applications, the exact knowledge of the states of the system are not available. Therefore the states of the system are estimated by using the designed observer. Lyapunov approach is utilized for the purpose of stability analysis and based on the analysis, all signals of the system are shown to be bounded. Finally the scheme is validated through numerical simulation studies. The paper is organized as follows. In Section 2, the problem is formulated together with the description of RBF neural network. In Section 3, the design of control system are described in details. In Section 4, the detailed stability analysis is presented. Numerical simulation studies are performed in Section 5. Finally conclusive remark are described in Section 6.
2. Problem Formulation We Consider the following form of SISO nonaffine nonlinear system x˙(t) = Mx(t) + NG(x(t), u(t)) y(t) = Dx(t)
(1)
where M ∈ Rn×n , N ∈ Rn×n and D ∈ Rn×n are system matrices (known). Also u(t) ∈ R is the input of the system and y(t) ∈ R is the output of the system. The the measurement of the state vector x(t) ∈ Rn , is not available and G(x, u) is smooth function (unknown). The following commonly found assumptions are used: Assumption 1: G(x, u) ∈ C 1 ∀ (x, u) ∈ Rn+1 . n+1 Assumption 2: ∂G . ∂u ̸= 0 ∀ (x, u) ∈ R The objective of the present study is to outline a state feedback control scheme for SISO nonaffine system so that the output of the system follows a reference signal, keeping all signals of system to be bounded. In the presented work, RBFNN is utilized to design the control scheme. In the area of control engineering, NN is being used to compensate the unknown function. It has been shown in the literature that a linear combination of Gaussian functions can approximate any continuous function[21]. In RBFNN, the input space is mapped into an intermediate space by a nonlinear transformation performed by hidden layer. Then the output of the network is obtained by linearly combining the outputs of the intermediate layer. We can described RBFNN as h(x) = Wht (x)θh† + ϵh (x) ϵh (x) is the NN reconstruction errors.
(2)
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3. Control System Design To obtain the desired goal, firstly the non-affine part of the system is transformed into affine form utilizing Taylor series expansion method as G(x, u) = d(x) + h(x)u + q(x, u)
(3)
where d(x) = G(x, u0 ) − ( h(x) = (
∂G )u u0 (x) ∂u 0
(4)
∂G )u=u0 (x) ∂u
(5)
with q(x, u) containing the higher degree terms and u0 (x) is utilized to minimize the function |q(x, u)|. As a result, we obtained the affine system equivalent to (1) x˙ = Mx + N{d(x) + h(x)u + q(x, u)} y = Dx
(6)
Now RBFNN is used to estimate the unknown d(x) and h(x). ˆ and h(x) ˆ Let θd and θh be the estimate of θd† and θh† , then we define the RBF neural network approximation d(x) of d(x) and h(x) as follows: d(x) = Wdt (x)θd† + ϵd (x) ˆ = W t (x)θd d(x)
(7)
h(x) = ˆ h(x) =
(8)
d Wht (x)θh† Wht (x)θh
+ ϵh (x)
Using (7) and (8) in (6), we get x˙ = Mx + N{Wdt (x)θd† + ϵd (x) + (Wht (x)θh† + ϵh (x))u + q(x, u)} y = Dx
(9) (10)
Since the state of the system is unmeasurable, we need to have an observer to approximate the state vector x(t). An appropriate observer for x(t) is given by x˙ˆ = M xˆ + N{Wdt ( xˆ)θd + (Wht ( xˆ)θh )u + S (y − yˆ )} yˆ = D xˆ
(11) (12)
where xˆ is the estimate of x whereas yˆ (t) is output of the observer and S ∈ Rn×1 . Now, the error vector of the estimation can be defined by e = x − xˆ, then we have e˙ = x˙ − x˙ˆ
e˙ = Mx + N[Wdt (x)θd† + ϵd (x) + (Wht (x)θh† + ϵh (x))u + q(x, u)] − [M xˆ + N{Wdt ( xˆ)θd + (Wht ( xˆ)θh )u + S (y − yˆ )}]
e˙ = (M − S D)e + N[Wdt (x)θd† − Wdt ( xˆ)θd + (Wht (x)θh† − Wht ( xˆ)θh )] + N[ϵd (x) + uϵh (x) + q(x, u)]
(13)
In order to get the error equation of the system (1), we expand W(x) = W( xˆ + e) in terms of e as a Taylor series: Wd (x) = Wd ( xˆ + e) = Wd ( xˆ) + Jd e + HOT
(14)
Wh (x) = Wh ( xˆ + e) = Wh ( xˆ) + Jh e + HOT
(15)
Where e = [e1 , e2 , e3 , ..., en ]t and Jd can be defined as:
4 28
Kumar et al. / Procedia Computer Science 00 (2018) 000–000 Pramendra Kumar et al. / Procedia Computer Science 125 (2018) 25–33
∂Wd1 ∂x1 ∂Wd2 ∂x Jd = . 1 .. ∂Wdn ∂x1
∂Wd1 ∂x2 ∂Wd2 ∂x2
.. .
∂Wdn ∂x2
··· ··· .. . ···
.. . ∂Wdn ∂Wd1 ∂xn ∂Wd2 ∂xn
∂xn
Similarly we can define Jh and HOT is higher order term. Then using (14) and (15), (13) is rewritten as e˙ = (M − S D)e + N[Wdt ( xˆ)θd† − Wdt ( xˆ)θd + (Wht ( xˆ)θh† − Wht ( xˆ)θh )] + N[ϵd (x) + et Jd θd† +uϵh (x) + et Jh θh† u + q(x, u)]
e˙ = (M − S D)e + N[Wdt ( xˆ)(θd† − θd ) + (Wht ( xˆ)(θh† − θh )] + N[ϵd (x) + et Jd θd† + uϵh (x) + et Jh θh† u + q(x, u)] e˙ = (M − S D)e + N[Wdt ( xˆ)θ˜d + Wht ( xˆ)θ˜h u] + N[ϵd (x) + et Jd θd† + uϵh (x) + et Jh θh† u + q(x, u)]
(16)
e˙ = (M − S D)e + N[Wdt ( xˆ)θ˜d + Wht ( xˆ)θ˜h u + w]
(17)
where w = [ϵd (x) + et Jd θd† + uϵh (x) + et Jh θh† u + q(x, u)], θ˜d = θd† − θd and θ˜h = θh† − θh . It is assumed that w satisfies |w(t)| ≤ ϵN with ϵN as some positive constant. The desired controller is given as: u=
1 ˆ ϕ(h)
ˆ xˆ) − (−E xˆ − d(
1 t¯ N P xˆ + rm ) 2δ2
(18)
ˆ is selected so that ϕ(h) ˆ = h0 , when hˆ → 0 and ϕ(h) ˆ = h, ˆ when hˆ ̸= 0, where h0 is a small positive design where ϕ(h) parameter. E ∈ Rn is the gain vector, δ > 0 is a constant, rm is the reference signal of the input and P¯ = P¯ T is a symmetric matrix satisfying the Ricatti equation as follows ¯ E + µ2 P¯ 2 = −Q2 ϕtE P¯ + Pϕ
(19)
where µ > 0 and Q2 is symmetric positive definite matrix. Using (18) in (11), we get 1 x˙ˆ = (M − NE) xˆ + N(rm − 2 N t P¯ xˆ) + S De 2δ
(20)
Where E and S are chosen so that ϕ = (M − NE) and ϕS = (M − S D) are Hurwitz matrices whereas P satisfies the following Ricatti equation as ϕtS P + PϕS +
1 1 PNN t P + 2 Dt S t S D = −Q1 δ2 µ
(21)
where Q1 is symmetric positive definite matrix. 4. Stability Analysis If the nonaffine system is given as (1) and the control scheme be chosen as (18), then for t ≥ 0, error and the weight estimates are UUB whereas H ∞ tracking performance is contented as ∫ T [et (t)Q1 e(t) + xˆ(t)Q2 xˆ(t)]dt J= 0 ∫ T 1 ˜t 1 ˜t θd (0)θ˜d (0) + θh (0)θ˜h (0) + δ2 [rmt (t)rm (t) + wt (t)w(t)]dt} ≤ {et (0)Pe(0) + xˆt (0)P¯ xˆ(0) + 2γd 2γh 0
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Proof. For the stability analysis, the Lyapunov function is chosen as 1 ˜t ˜ 1 ˜t ˜ V(t) = et Pe + xˆ P¯ xˆ + θd θd + θ θh 2γd 2γh h
(22)
The time derivative of (22) gives ˙ = e˙ t Pe + et P˙e + x˙ˆ P¯ xˆ + xˆ P¯ x˙ˆ + 1 θ˜t θ˙˜d + 1 θ˜t θ˙˜h V(t) γd d γh h
(23)
Substituting (9), (11), (13) and (20) in (23), we get ˙ = et (ϕtS P + PϕS )e + xˆ(ϕtE P¯ + Pϕ ¯ E ) xˆ + (ϕtS P + PϕS ) + 2θ˜t Wd N t Pe + 1 θ˜t θ˙˜d V(t) d γd d 1 1 ¯ t¯ ¯ De) ¯ m ) + (2et PNw + 2 xˆ PS + 2θ˜ht Wh N t Peu + θ˜ht θ˙˜h − ( 2 xˆt PNN P xˆ − 2 xˆt PNr γh δ ¯ E ) xˆ + (ϕtS P + PϕS ) + 2θ˜t Wd N t Pe + 1 θ˜t θ˙˜d ˙ = et (ϕtS P + PϕS )e + xˆ(ϕtE P¯ + Pϕ V(t) d γd d 1 1 1 1 1 + 2θ˜ht Wh N t Peu + θ˜ht θ˙˜h − ( N t P¯ xˆ − δrm )t ( N t P¯ xˆ − δrm ) − ( N t Pe − δw)t ( N t Pe − δw) γh δ δ δ δ 1 1 1 1 t 2 t t 2 t t t t − ( S De − µP¯ xˆ) ( S De − µP¯ xˆ) + δ (rm rm + w w) + µ xˆ P¯2 xˆ + 2 e D S S De + 2 et PNN t Pe µ µ µ δ 1 1 ¯ E + µ2 P¯2 ) xˆ + 2θ˜t Wd N t Pe ≤ et (ϕtS P + PϕS + 2 TC t S t S D + 2 PNN t P)e + xˆ(ϕtK P¯ + Pϕ d µ δ 1 1 + θ˜dt θ˙˜d + 2θ˜ht Wh N t Peu + θ˜ht θ˙˜h + δ2 (rmt rm + wt w) γd γh
(24)
(25)
If the neural network weights are updated as: θ˙˜d = 2γd Wd N t Pe θ˙˜h = 2γh Wh N t Peu
(26)
Then using (19), (21), (25) can be rewritten as ˙ ≤ −et Q1 e − xˆt Q2 xˆ + δ2 (rmt rm + wt w) V(t) ˙ ≤ −λmin (Q1 )∥e∥2 − λmin (Q2 )∥ xˆ(t)∥2 + δ2 (∥rm ∥2 + ∥w∥2 ) V(t)
(27)
where λmin (Q1 ) and λmin (Q2 ) denote the singular value of matrix Q1 and Q2 respectively. Whenever ∥e(t)∥2 + ∥ xˆ(t)∥2 ≥
δ2 (∥rm ∥2 + ∥w∥2 ) λmin (Q)
˙ ≤ 0, We get from (27) V(t) where λmin (Q) = min{λmin (Q1 ), λmin (Q2 )}. Then integrating (28) between the limits t = 0 to t = T , we get ∫ T [et Q1 e + xˆt Q2 xˆ]dt J= 0 ∫ T 2 (rmt rm + wt w)dt ≤ V(0) + δ 0 ∫ T 1 ˜t 1 ˜t 2 t t ˆ ˜ ˜ ¯ θ (0)θd (0) + θ (0)θh (0) + δ [rmt (t)rm (t) + wt (t)w(t)]dt} ≤ {e (0)Pe(0) + x (0)P xˆ(0) + 2γd d 2γh h 0 This completes the proof.
(28)
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5. Simulation Results In this section, simulation study has been performed for the proposed controller. For the simulation purpose, we consider the following nonaffine system x˙1 = x2 x˙2 = x12 + 0.15u3 + .1(1 + x22 )u + sin(0.1u) y = x1
(29)
where A = [0 1; 0 0], B = [0; 1], C = [1 0]. The gain vectors are chosen as K = [1 3] and L = [3 1]′ so that ϕ and ϕS are Hurwitz matrices. The matrix P is obtained as P = [201 − 100; −100 500]. The design parameters are taken as µ = 0.1, δ = 100, g0 = 0.1, rm (t) = 0.5 for case 1 and rm (t) = 1.0 for case 2. For the simulation purpose we consider 10 nodes of RBFNN structure. The matrices γ f and γg are chosen as γ f = 50I5 , γg = 50I5 . The controller is simulated with MATLAB environment. The effectiveness of the proposed controller is shown with the Figs. 1-3 for case 1 and with the Figs. 4–6 for case 2. From the figures, it is clear that the error are quickly converging to zero levels and the proposed observer based RBF neural controller can achieve the desired performance. 1
State x1 State x2
0.5
0.5
0
0
Trajectories
Trajectories
1
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
0
5
10
time (sec)
15
20
−2
25
Observer Estimate for State x1 Observer Estimate for State x2
0
5
10
time (sec)
15
20
25
Fig. 1. (a) Trajectories of x1 (t) and x2 (t) (Case 1) ; (b) Trajectories of Observer estimate xˆ1 (t) and xˆ2 (t) (Case 1)
−5
0.5
3
y(t) rm(t)
Error Estimate for State x1
2.5
0
2
Estimation Error
Trajectories
x 10
−0.5
−1
1.5 1 0.5 0
−1.5
−0.5 −2
0
5
10
time (sec)
15
20
25
−1
0
5
10
time (sec)
15
Fig. 2. (a) Error estimate for x1 (t) and xˆ1 (t) (Case 1) ; (b) Error estimate for x2 (t) and xˆ2 (t) (Case 1)
20
25
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−3
8
x 10
Error Estimate for State x2
6
Estimation Error
4 2 0 −2 −4 −6 −8
0
5
10
time (sec)
15
20
25
Fig. 3. The output of the closed loop system (Case 1)
1
State x1 State x2
0.5
0.5
0
0
Trajectories
Trajectories
1
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
0
5
10
time (sec)
15
20
−2
25
Observer Estimate for State x1 Observer Estimate for State x2
0
5
10
time (sec)
15
20
25
Fig. 4. (a) Trajectories of x1 (t) and x2 (t) (Case 2) ; (b) Trajectories of Observer estimate xˆ1 (t) and xˆ2 (t) (Case 2) −5
1
3
y(t) r (t) m
2
Estimation Error
0
Trajectories
Error Estimate for State x1
2.5
0.5
−0.5
−1
1.5 1 0.5
−1.5
−2
x 10
0
0
5
10
time (sec)
15
20
25
−0.5
0
5
10
time (sec)
15
Fig. 5. (a) Error estimate for x1 (t) and xˆ1 (t) (Case 2) ; (b) Error estimate for x2 (t) and xˆ2 (t) (Case 2)
20
25
32 8
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8
x 10
Error Estimate for State x2
6
Estimation Error
4 2 0 −2 −4 −6 −8
0
5
10
time (sec)
15
20
25
Fig. 6. The output of the closed loop system (Case 2)
6. Conclusion In the present paper, an RBF neural network based scheme is presented for the regulatory control of nonaffine nonlinear systems with unknown structure of nonlinearities. Taylor series expansion is utilized to convert the nonaffine part of the system affine form. The RBF neural network is utilized to compensate the unknown dynamics of the system without the requirement of offline learning. In many industrial applications, the exact knowledge of the states of the system are not available. Therefore, an observer is designed, which provides the estimated states of the system. From the stability analysis it is shown that all signals of the system are uniformly ultimately bounded and satisfy the given H∞ criteria. It can ce concluded from the computer simulation studies that the presented controller achieves the desired performance adequately. In future the investigation and design of varying trajectory tracking and study of MIMO system will be interesting. References [1] A. Young, C. Cao, N. Hovakimyan, An adaptive approach to nonaffine control design for aircraft applications, AIAA Guidance, Navigation, and Control Conference (2006) 1–28. [2] Y. Akahane, M. Kato, Y. Miyake, A study on adaptation mechanism of physarum based on chemo-mechanical system, 39 th SICE Annual Conference (2000) 191–196. [3] W. Meng, Q. Yang, Y. Ying, Y. Sun, Z. Yang, Y. Sun, Adaptive power capture control of variable-speed wind energy conversion systems with guaranteed transient and steady-state performance, IEEE Transactions on Energy Conversion 28 (3) (2013) 716–725. [4] T. Zhang, M. Guay, Adaptive control of uncertain continuously stirred tank reactors with unknown actuator nonlinearities, ISA Transactions 44 (2005) 55–68. [5] S. S. Ge, T. Zhang, Neural-network control of nonaffine nonlinear system with zero dynamics by state and output feedback, IEEE Transactions on Neural Networks 14 (2003) 900–918. [6] N. Hovakimyam, E. Lavretsky, A. Sasane, Dynamic inversion for nonaffine-in-control systems via time-scale separation: part i, Journal of Dynamical and Control Systems, 13 (4) (2007) 451–465. [7] E. Lavretsky, N. Hovakimyam, Dynamic inversion for nonaffine-in-control systems via time-scale separation: part ii, In Proc. American control conference (2005) 3548–3553. [8] B. Yang, A. J. Calise, Adaptive control of a class of nonaffine systems using neural networks, IEEE Transactions on Neural Networks 18 (4) (2007) 1149–1159. [9] M. Chen, S. S. Ge, Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer, IEEE Transactions on Cybernetics 43 (4) (2013) 1213–1255. [10] Y. H. Liu, L. Huang, D. Xiao, Y. Guo, Global adaptive control for uncertain nonaffine nonlinear hysteretic systems, ISA Transactions 58 (2015) 255–261. [11] T. P. Zhang, S. S. Ge, Adaptive control of a class of nonaffine systems using neural networks, IEEE Transactions on Neural Networks 44 (7) (2008) 1895–1903. [12] D. Wang, Neural network-based adaptive dynamic surface control of uncertain nonlinear pure-feedback systems, International Journal of Robust and Nonlinear Control 21 (5) (2011) 527–541.
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