Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006
FrB12.6
Adaptive Output Control of Nonlinear Time-Delayed Systems with Uncertain Dead-Zone Input Jing Zhou∗ , Meng Joo Er and Kalyana C. Veluvolu
Abstract— In this paper, we present a new scheme to design adaptive controllers for uncertain time-delayed systems preceded by dead-zone nonlinearity. The unknown time delays have been compensated by using appropriate Lyapunov-Krasovskii functionals. The applied control effort of the backstepping design is to eliminate the effects arisen from the dead-zone nonlinearity in the input by introducing the dead-zone inverse. It is shown that the proposed controller not only can guarantee stability, but also transient performance.
Keywords: Adaptive control, backstepping, dead-zone, timedelay, nonlinear systems I. INTRODUCTION It is well known that dead-zone and time-delayed characteristics are frequently encountered in various engineering systems and can be cause of instability. Adaptive control is popular in engineering and science. However, it still faces many important challenges, such as the handling of deadzone nonlinearity or time-delayed phenomenon. Dead-zone is common in mechanical connection, hydraulic servo valves, piezoelectric translators, and electric servomotors. On the other hand, time-delay phenomenon is commonly found in chemical processes, biological reactors, rolling mills, communication networks, etc. In fact, the existence of dead-zone nonlinearity and time delay phenomenon usually deteriorates the system performance. Several adaptive control schemes have been proposed to handle dead-zone nonlinearity, see for examples [1], [2], [3], [4]. In these papers, an adaptive inverse approach was presented to deal with dead-zone nonlinearity in the design of continuous time model reference adaptive controllers. In the controller design, the uncertain parameters of the system must be within known bounded intervals. Dead-zone precompensation using fuzzy logic or neural network have also been used in feedback control systems [5], [6]. But in controller design, it is required that all system states are within a known compact set to handle approximation errors caused by fuzzy logic or NN approximation and the weights must be bounded with known bounds. However, the states of the system cannot be guaranteed in a compact set before stability of the closed-loop system is established. In [7], [8], [9], the state feedback control was considered for nonlinear uncertain systems, where the dead-zone was treated in a similar way to disturbance. The characteristics of the dead-zone nonlinearity were not considered in the Intelligent Systems Centre, Research TechnoPlaza, Nanyang Technological University, Singapore 637533 ∗ Corresponding Author. E-mail:
[email protected]; Tel: +656790-6962; Fax: +65-6790-6961
1-4244-0210-7/06/$20.00 ©2006 IEEE
controller design, so the performance, especially steady state performance, may not be good enough. It was also assumed that the dead-zone slopes in both positive and negative sides must be the same. In [10], a new smooth dead-zone inverse was developed to compensate the effect of dead-zone nonlinearity. Stabilization problem of control systems with time-delay have received much attention, see for examples, [11], [12], [13]. A state delay-dependent feedback linearization algorithm was proposed for nonlinear systems with delay on the state and the input in [13]. In [14], the robust stability problem of linear discrete singular time-delay systems with structured parameter uncertainties is investigated. An iterative design procedure has been proposed in [15] to construct a robust controller to stabilize a state-delay nonlinear system with a triangular structure. In [16], methods for systematic construction of control Lyapunov-Razumikhin functions for time-delay systems have been proposed for several classes of nonlinear time-delay systems. However, the problem on unknown dead-zone nonlinearity and parameters uncertainties are not discussed in above works. In this paper, we will address the output feedback control of a class of time-delayed nonlinear systems, in the presence of dead-zone actuator nonlinearity. We take the deadzone into account in our controller design. An inverse of the dead-zone will be introduced to compensate the effect of the dead-zone in controller design with backstepping approach as in [2]. The specific treatment of the deadzone may bring performance improvement. The unknown time delays have been compensated by using appropriate Lyapunov-Krasovskii functionals. As system output feedback is employed, a state observer is required. To obtain such an observer, a new parametrization of the state observer for the plant is proposed to include two sets of parameters: one from the dead-zone nonlinearity and the other from the plant, similar to [3]. Besides showing stability of the system, the transient performance in terms of L2 norm of the tracking error is derived to be an explicit function of design parameters and thus our scheme allows designers to obtain the closed loop behavior by tuning design parameters in an explicit way. This paper is organized as follows: The problem considered is formulated in Section 2. In Section 3, a state observer is parameterized. In Section 4, we present the adaptive control design approach based on the backstepping technique and analyze the stability and performance of the closed loop system. Simulation results are presented to illustrate the effectiveness of our scheme in Section 5. Finally, the paper
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is concluded in Section 6.
σ(X) =
II. P ROBLEM S TATEMENT
σr (t) σl (t)
A. System Model Consider the following class of single-input-single-output (SISO) nonlinear time-delay systems in the feedback form x˙ i
= xi+1 + aT ψi (y) + φi (y) + fi (¯ xi (t − τi ))
x˙ n
i = 1, . . . , n − 1 = u + aT ψn (y) + φn (y) + fn (x(t − τn ))
(1)
= eT1 x, u = DZ(v)
(2)
y
if X is true otherwise
(8)
= σ u(t) > 0 or v(t) > br = σ u(t) < 0 or v(t) < bl
(9) (10)
As θ is unknown, the actual control input to the plant ud (t) is designed as ud (t)
where x ¯ = [x1 , · · · , xi ]T ∈ Ri , x = [x1 , · · · , xn ]T ∈ n R , u ∈ R and y ∈ R are system states, input and output respectively, a ∈ Rr is uncertain parameter vector, ψi ∈ Rr and φi are known smooth nonlinear functions, fi (¯ xi (t − τi )), i = 1, . . . , n denote unknown smooth functions and τi are unknown time delays of the states, v(t) is the output from the controller, u(t) is the input to the system and y(t) is the system output. The actuator nonlinearity DZ(v) is described as a dead-zone characteristic.
1 0
= −θˆT ω(t)
(11)
where θˆ is an estimate of θ, i.e. θˆ = [m r , m l ]T
(12)
Then corresponding control output v(t) is given by d (t)) DI(u ud (t) + br )σr (t) = ( m r ud (t) + bl )σl (t) +( m l
v(t) =
(13)
The resulting error between u and ud is u(t) − ud (t) = (θˆ − θ)T ω(t)
(14)
The control objective is to design an output feedback control law for v(t) to ensure that all closed-loop signals are bounded and the plant output y(t) converges to zero asymptotically.
III. S TATE O BSERVER
For the development of control laws, the following assumptions are made. Assumption 1: The reference signal yr (t) and its first n-th derivatives are known and bounded. Assumption 2: The dead-zone parameters mr and ml satisfy mr ≥ mr0 and ml ≥ ml0 , where mr0 and ml0 are two small positive constants. xi (t − τi )) Assumption 3: The time-delayed functions fi (¯ satisfy the following inequality
As we consider output feedback, a state observer is required. To design such an observer, we re-write plant equation (1) as
fi (¯ xi (t − τi )) 2 ≤ y(t − τi )f¯i (y(t − τi ))
x˙ = Ax + Ψa + Φ + F + uen y = cx, u = DZ(v) where
⎤ ⎤ ⎡ T 0 ψ1 (y) ⎥ ⎥ ⎢ ⎢ .. A = ⎣ ... In−1 ⎦ , Ψ = ⎣ ⎦, . T 0 ...0 ψn (y) ⎤ ⎤ ⎡ ⎡ φ1 (y) f1 (x1 (t − τ1 )) ⎥ ⎥ ⎢ ⎢ .. .. Φ = ⎣ ⎦,F = ⎣ ⎦, . . φn (y) fn (x(t − τn )) ⎡ ⎤ ⎡ ⎤T 0 1 ⎢ .. ⎥ ⎢ .. ⎥ c = ⎣ . ⎦ , en = ⎣ . ⎦ . (16)
(3)
where f¯i are known smooth functions. B. Dead-zone Characteristic The dead-zone characteristic DZ(.) can be represented as u(t) = DZ(v(t)) ⎧ ⎨ mr (v(t) − br ) 0 = ⎩ ml (v(t) − bl )
v(t) ≥ br bl < v(t) < br (4) v(t) ≤ bl
where br ≥ 0, bl ≤ 0 are known constants and mr > 0, ml > 0 are uncertain constants. The essence of compensating deadzone effect is to employ a dead-zone inverse. To design adaptive controller for the system, we parameterize the dead-zone as in [2] u(t) = −θ T ω where θ = [mr , ml ]T ω(t) = [−σr (t)(v(t) − br ), −σl (t)(v(t) − bl )]T
(15)
⎡
1
0
To construct an observer for (15), we choose k = [k1 , . . . , kn ]T such that all eigenvalues of A0 = A − kc are at some desired stable locations. If the signal u(t) were available then we would implement the following filters as in [10], [17]:
(5) (6) (7)
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x ˆ(t) = ξ0 +
r
ai ξi + η
(17)
i=1
η˙ ˙ξ0 ξ˙i
= A 0 η + en u
(18)
= A0 ξ0 + ky + Φ + χ = A0 ξi + Ψei , i = 1, . . . , r
(19) (20)
where χ is a design signal specified later. It can be shown that the state estimation error = x(t) − x ˆ(t) satisfies ˙ = A0 + F − χ
z˙1
(21)
Then the derivative of lyapunov function of estimation error V = 2l11 T P is given as V˙
• Step 1: We start with the equation for the tracking error z1 obtained from (15) and (27) to obtain
1 T 1 1 (P A0 + AT0 P ) − T P χ + T P F 2l1 l1 l1 1 1 ≤ − T − T P χ 4l1 l1 n 1 + P 2 fk2 (¯ xk (t − τk )) l1
=
(22)
k=1
where P = P T > 0 satisfies the equation P A0 + AT0 P = −I. Note that the signal u(t) is not available. Thus the signal η d . With in (20) needs to be re-parameterized. Let p denote dt Δ(p) = det(pI − A0 ), we express η(t) as η(t) = [η1 (t), η2 (t), . . . , ηn (t)]T 1 u(t) = [q1 (p), q2 (p), . . . , qn (p)] Δ(p) T
+ 2 + f1 (y(t − τ1 )) ˆ where θ˜ = θ − θ. Now select the first virtual control law α1 as 1 α1 = −(c1 + )z1 − ξ02 − a ˆT ξ2 4 n P 2 − − nf¯1 (y(t)) f¯k (y(t)) l1
z˙1
qi (p)I2 ω(t) Δ(p)
We define a positive definite function V1 as V1
(24)
(25) (26)
where I2 is a 2 × 2 identity matrix. Based on (24), ωi is available for controller design in place of u. Denoting the second component of ξi as ξi2 , i = 0, . . . , r, we have x ˆ2
= ξ02 −
r
ai ξi2 − θT ω2 (t)
(27)
i=1
ω2 (t) =
Different from the usual step in backstepping approach, the following change of coordinates is made. zi
(29)
(i−2) = −θˆT ω2 − αi−1 , i = 2, 3, . . . , ρ
(30)
1 2 1 ˜T −1 ˜ 1 T −1 z + θ Γθ θ + a ˜ Γa a ˜ + V 2 1 2 2 n P 2 t + y(ξ)f¯k (y(ξ))dξ l1 t−τ k k=1 t y(ξ)f¯1 (y(ξ))dξ +n
(34)
where a ˜ =a−a ˆ, Γθ , Γa are positive definite matrices. Remark 2: The existence of time delays will render the control problem more difficult and different from that of controlling the pure nonlinear systems. The time-delay parts should be considered. Different from normal Lyapunov function used in backstepping design, the idea of Lyapunovt Krasovskii function t−τk y(ξ)f¯k (y(ξ))dξ is used to handle the unknown time-delay terms. Then from (22), (33) to (34) and using Assumption 2, we obtain the time derivative of V1 as
(p + k1 )I2 ω(t) (28) pn + k1 pn−1 + . . . + kn−1 p + kn
= y
=
t−τ1
IV. D ESIGN OF A DAPTIVE C ONTROLLERS
z1
1 = −(c1 + )z12 + a ˜T ξ2 + z2 − θ˜T ω2 (t) − nf¯1 (y(t)) 4 n P 2 + 2 (33) f¯k (y(t)) + f1 (y(t − τ1 )) − l1 k=1
(23)
where ωi (t) =
(32)
where a ˆ is am estimate of a. n ¯ P 2 and Remark 1: Note that the term k=1 fk (y(t)) l1 nf¯1 (y(t)) are used here to deal with the unknown time-delay xi (t − τi )). function fi (¯ Then the choice results in the following system
for some known polynomials qi (p), i = 1, . . . , n. Using (23) and u(t) = −θ T ω(t), we obtain ηi (t) = −θ T ωi (t)
(31)
k=1
k=1
1 1 ≤ − T − T P χ 4l1 l1 n 1 + P 2 y(t − τk )f¯k (y(t − τk )) l1
= ξ02 + aT ξ2 + z2 + α1 − θ˜T ω2 (t)
where αi−1 is the virtual control at the ith step and will be determined in later discussion. The first and the last steps of the design are elaborated in details. The results of other steps, i.e. step i, i = 2, . . . , n−1 are only presented without elaboration.
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P ˙ ˆ˙ ˜T Γ−1 a ˙ = z1 z˙1 − θ˜T Γ−1 · a ˆ + V + θ θ−a l1 k=1 [y(t)f¯k (y(t)) − y(t − τk )f¯k (y(t − τk ))] n
V˙ 1
2
+ny(t)f¯1 (y(t)) − nf¯1 (y(t − τ1 ))y(t − τ1 ) ˆ˙ ≤ −c1 z12 + θ˜T (ω2 z1 − Γ−1 ˆ˙ ) ˜T (ξ2 z1 − Γ−1 a a θ θ) + a 1 1 T 1 + z1 z2 − z12 + T [e2 z1 − P χ] − l1 4l1 4 +z1 f1 (y(t − τ1 )) − y(t − τ1 )f¯1 (y(t − τ1 )) −(n − 1)f¯1 (y(t − τ1 ))y(t − τ1 ) ˆ˙ ≤ −c1 z12 + θ˜T Γ−1 ˜T (τa1 − Γ−1 ˆ˙ ) a a θ (τθ1 − θ) + a 1 1 T + z 1 z2 + T [τχ1 − P χ] − l1 4l1
−(n − 1)f¯1 (y(t − τ1 ))y(t − τ1 ) τa1 τθ1
= ξ2 z 1 = −Γθ ω2 (t)z1
(35) (36)
τχ1
= e2 z 1
(37)
1 Vn = Vn−1 + zn2 2
1 ∂αi−1 2 )zi − zi−1 − βi 4 ∂y ∂αi−1 T ∂αi−1 ˆT ∂αi−1 a ˆ ξ2 − θ ω2 (t) + Γa τai + ∂y ∂y ∂ˆ a ∂αi−1 ∂αi−1 Γθ τΘi + l1 P −1 τχi + ∂ξ0 ∂ θˆ i−1 ∂αk−1 ∂αi−1 ξ2 + zk − ∂ˆ a ∂y k=2 ∂αk−1 ∂αi−1 l1 P −1 e2 − ∂ξ0 ∂y i−1 ∂αk−1 ∂αi−1 ω2 − zk (38) ∂y ∂ θˆ k=3 ∂αi−1 ξ2 zi τai−1 − (39) ∂y ∂αi−1 zi e2 τχi−1 − (40) ∂y ∂αi−1 zi ω 2 τθi−1 − (41) ∂y 1 2 z + Vi−1 (42) 2 i i ˆ˙ − ck zk2 + θ˜T Γ−1 ˜T (τai − Γ−1 ˆ˙ ) a a θ (τθi − θ) + a
a ˆ˙ = Γa τan ˙ θˆ = P roj(−Γθ τΘn )
= −(ci +
τai
=
τχi
=
τθi
=
Vi
=
V˙ i
≤
k=1
1 1 T P χ] − + zi zi+1 l1 4l1 −(n − i)f¯1 (y(t − τ1 ))y(t − τ1 )
+ T [τχi −
(43)
where βi contains all known terms. Step n: Using (13) and (28), we have (n−1) θˆT ω2
(pn + k1 pn−1 )I2 ω(t) pn + k1 pn−1 + . . . + kn−1 p + kn (44) = ud (t) + ω0
= −
χ = l1 P −1 τχn
(k2 pn−2 + . . . + kn−1 p + kn )I2 ω(t) (45) pn + k1 pn−1 + . . . + kn−1 p + kn
(n−2) −αn−1 With this equation, the derivative of zn = −θˆT ω2 is ∂αn−1 T ∂αn−1 T a ξ2 + θ ω2 (t) z˙n = ud + βn − ∂y ∂y ∂αn−1 ˙ ∂αn−1 ˆ˙ ∂αn−1 ∂αn−1 2 − χ− a ˆ− θ− ∂ˆ a ∂ξ0 ∂y ∂ θˆ ∂αn−1 f1 (y(t − τ1 )) (46) − ∂y
(49)
(50)
Finally the control law is given by v(t) = ( ud
ud (t) ud (t) + br )σ(ud > 0) + ( + bl )σ(ud < 0) m r m l (51)
= αn
(52)
With this choice and using the property −θ˜T Γ−1 θ P roj(τθ ) ≤ −θ˜T Γ−1 τ , the derivative of V becomes θ n θ V˙ n
≤ −
n
ci zi2 −
i=1
1 T 4l1
(53)
From (53), we get the following theorem.
Theorem 1: Consider the system consisting of the parameter estimators given by (48) and (49), adaptive controllers designed using (52) with virtual control laws (32) and (38), and time-delayed plant (1) with a dead-zone nonlinearity (4). The system is stable in the sense that all signals in the closed loop are bounded. Furthermore • The system output approaches 0 asymptotically, i.e., lim y(t) = 0
= θˆT
(48)
where P roj(.) is a smooth projection operation to ensure the estimates m ˆ r (t) ≥ mr0 and m ˆ l (t) ≥ ml0 . Such an operation can be found in [17]. The design signal χ as
t→∞
where ω0 is given by ω0
(47)
We choose the update laws for a ˆ and θˆ
• Step i, i = 2, . . . , n: As detailed in [18], we choose αi
where βn contains all known terms. Define a positive definite Lyapunov function Vn as
(54)
• The transient performance of output y is given by 1 1 1 ˜ 2 a ˜(0)T Γ−1 ˜(0) + Γ−1 θ(0) y(t) 2 ≤ √ a a c1 2 2 θ 1/2n 1 + (0)2 (55) 2l1 with zi (0) = 0, i = 1, . . . , n, ˆa ˆ, are Proof: From (53), we have that z1 , . . . , zn , θ, bounded. The tracking error performance can be obtained from (53) following similar approaches to those in [18]. What we need to prove is the boundedness of state x, controller output v and plant input u. From state observers ξi in (20), we have that ξ0 , . . . , ξr are bounded. Re-writing plant (1) as pn y +
r i=1
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ai Yi y, py, . . . , pn−1 y = bu
(56)
and using (23), we have η2
pn q2 (p) q2 (p) q2 (p) u= y+ Δ(p) bΔ(p) bΔ(p)
=
r
ai Yi (y) (57)
i=1
q2 (p) Since Δ(p) = pn + k1 pn−1 + . . . + kn is Hurwitz, so bΔ(p) is stable. We have that η2 is bounded because y is bounded. From (24), we have
= −θ T ω2 (t)
η2
where u represents the output of the dead-zone nonlinearity, the dead-zone br = 0.5, bl = −0.5, parameter a is unknown and dead-zone parameters mr , ml are unknown, but mr ≥ 0.1, ml ≥ 0.1, x(t−τ ) is a time-delayed function. The actual parameter values are chosen as a = 1,mr = 1, ml = 1.5 and time-delayed term τ = 2. The objective is to control the system state x to zero. Firstly we choose the dead-zone ˆ d (t)) as in (11) and the filters inverse v(t) = DI(u
(58)
ξ˙0 ξ˙1
∞
So θ ω2 ∈ L . Express (28) as T
ω2 (t)
θT ω2 (t)
q2 (p) (v(t) − br )σr (t), Δ(p) q2 (p) − (v(t) − bl )σl (t)]T Δ(p) q2 (p) (v(t) − br )σr (t) = −mr Δ(p) q2 (p) −ml (v(t) − bl )σl (t) Δ(p)
=
η˙
−
ω2 (59)
k
(60)
A0
(66)
1 − e−x(t) 1 + e−x(t) = [k1 , k2 ]T = [1, 3]T −k1 1 −1 1 = = −3 0 −12 0 =
(67) (68) (69)
VI. C ONCLUSION This paper presents an output feedback backstepping adaptive controller design scheme for a class of uncertain time-delayed nonlinear system preceded by dead-zone actuator nonlinearity. We propose a new adaptive inverse to compensate the effect of the unknown dead-zone. The inverse function is employed in the backstepping controller design. The unknown time delays have been compensated by using appropriate Lyapunov-Krasovskii functionals. Besides showing stability, we also give an explicit bound on the L2 performance of the system output in terms of design parameters. Simulation results illustrate the effectiveness of our proposed scheme.
In this section, we illustrate the above methodology on the following example. Consider a nonlinear time-delayed system as x ¨ = a u y
(64) (65)
Then we apply our control design to the plant. In the simulations, taking c1 = c2 = 2, Γa = 0.1, Γθ = [0.1, 0.1]T ˆ and the initial parameters a ˆ(0) = 1.5, θ(0) = [1.1, 1.2]T . The initial state is chosen as x(0) = 1.0. The system output y and the controller output v(t) are shown in Figure 1 and 2. Clearly, the simulation results verify our theoretical findings and show the effectiveness of our control scheme.
V. S IMULATION S TUDIES
1 − e−x(t) + x(t − τ ) + u 1 + e−x(t) = DZ(v) = x
(63)
= A0 ξ1 + Y1 e2 = A0 η + e2 u, p + k1 = I2 [ω] p2 + k1 p + k 2
where Y1
2 (p) Because σr ∈ L∞ , σl ∈ L∞ and qΔ(p) is stable, we obtain that ω2 is bounded. Since θˆT ω2 and z2 are bounded, from z2 = −θˆT ω2 − α1 we can obtain the boundedness of α1 . From (38), α2 , . . . , αn are bounded, and so is χ. From (52) we have that ud (t) is bounded, and so are v = DI(u d ) and u = DI(v). It following from (25) that ωi ∈ L∞ , i = 1, . . . , n. From (20), we have that η is bounded. Then x ˆ is bounded from (17) and finally x(t) = x ˆ(t) + (t) is bounded from (17-20). From (54) and (55), we can discuss how the initial estimate errors and the choices of the adaptation gains affect the transient performance in terms of L2 norm of the system output. Remark 3: The closer the initial estimates to the true values, the better the transient performance. Remark 4: We can decrease the effects of the initial error estimates on the transient performance by increasing the adaptation gains Γa , Γθ . However, increasing these gains may influence other performance such as x, ˙ following similar discussion in [18]. Remark 5: In fact, our proposed scheme can also be applied to deal with the tracking problem by doing some modification as in [10]. The tracking error will converge to a small neighborhood of zero.
= A0 ξ0 + ky + χ
(61) (62)
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R EFERENCES [1] H. Cho and E. W. Bai, “Convergence results for an adaptive dead zone inverse,” International Journal of Adaptive Control and Signal Process, vol. 12, pp. 451–466, 1998. [2] G. Tao and P. V. Kokotovic, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. New York: John Willey & Sons, 1996. [3] G. Tao, Adaptive control design and analysis. New York: John Willey & Sons, 2003. [4] A. Taware, G. Tao, and C. Teolis, “Design and analysis of a hybrid control scheme for sandwich nonsmooth nonlinear systems,” Automatica, vol. 40, pp. 145–150, 2002. [5] F. L. Lewis, W. K. Tim, L. Z. Wang, and Z. X. Li, “Dead-zone compensation in motion control systems using adaptive fuzzy logic control,” IEEE Transactions on Control System Technology, vol. 7, pp. 731–741, 1999. [6] R. R. Selmis and F. L. Lewis, “Dead-zone compensation in motion control systems using neural networks,” IEEE Transactions on Automatic Control, vol. 45, pp. 602–613, 2000.
1.2
[16] M. Jankovic, “Control lyapunov-razumikhin functions and robust stabilization of time delay systems,” IEEE Transactions on Automatic Control, vol. 46, pp. 1048–1060, 2001. [17] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [18] J. Zhou, C. Wen, and Y. Zhang, “Adaptive backstepping control of a class of uncertain nonlinear systems with unknown backlash-like hysteresis,” IEEE Transactions on Automatic Control, vol. 49, pp. 1751–1757, 2004.
1
Output y(t)
0.8
0.6
0.4
0.2
0
−0.2
0
1
2
3
4
5 t(sec)
6
7
8
9
10
Fig. 1. System output with the controller designed using proposed scheme. 1
0
Input signal v(t)
−1
−2
−3
−4
−5
−6
0
1
2
3
4
5 t(sec)
6
7
8
9
10
Fig. 2. Control signal with the controller designed using proposed scheme.
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