Nonlinear Dyn DOI 10.1007/s11071-012-0439-6
O R I G I N A L PA P E R
Robust tracking control method based on composite nonlinear feedback technique for linear systems with time-varying uncertain parameters and disturbances Saleh Mobayen · Vahid Johari Majd
Received: 22 November 2011 / Accepted: 24 April 2012 © Springer Science+Business Media B.V. 2012
Abstract In this paper, the composite nonlinear feedback control method is considered for robust tracking and model following of uncertain linear systems. The control law guarantees that the tracking error decreases asymptotically to zero in the presence of time varying uncertain parameters and disturbances. For performance improvement of the dynamical system, the proposed robust tracking controller consists of linear and nonlinear feedback parts without any switching element. The linear feedback law is designed to allow the closed loop system have a small damping ratio and a quick response while the nonlinear feedback law increases the damping ratio of the system as the system output approaches the output of the reference model. A new collection of different nonlinear functions used in the control law are offered to improve the reference tracking performance of the system. The proposed robust tracking controller improves the transient performance and steady state accuracy simultaneously. Finally, the simulations are provided to verify the theoretical results. Keywords Composite nonlinear feedback · Robust tracking · Nonlinear functions · Uncertain dynamical system S. Mobayen · V.J. Majd () Intelligent Control Systems Laboratory, School of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran e-mail:
[email protected] url: http://www.modares.ac.ir/~majd
1 Introduction During the past decades, the robust tracking and model following problem for uncertain systems has been widely investigated [1]. Nonlinear state feedback controllers [2, 3] as well as linear state feedback controllers [4–6] are employed for robust tracking of dynamical systems. In [5, 6], a continuous robust tracking controller is proposed that only guarantees ultimate boundedness of the tracking error. The robust tracking controllers proposed in [1–6] cannot yield asymptotic tracking due to the existence of disturbances. In [7, 8], another class of continuous state feedback controllers is proposed for robust tracking of linear dynamical systems with time-varying uncertainties and disturbances, which guarantees the asymptotic tracking. To the best of our knowledge, no attempts have been made in the performance improvement of the robust model following problem. Quick response and small overshoot are usually the main requirements in the tracking problems [9, 10]. However, quick response provides high overshoot which is undesirable in many applications. This problem can be solved by using the composite nonlinear feedback (CNF) method [11]. The CNF control law consists of linear and nonlinear feedback laws without any switching element [12, 13]. The CNF design includes three steps [14]. In the first step, a linear feedback control law is designed to yield small damping ratio to achieve a quick response in the closed-loop system. In the second step, the nonlinear feedback control is designed to gradually change the damping
S. Mobayen, V.J. Majd
ratio of the closed-loop system as the system output approaches the output of the reference model, and consequently reduce the overshoot caused by the linear portion [15]. In the final step, the linear and nonlinear feedback laws are combined to form the CNF control law. In [16], an enhanced CNF technique with an integral term is introduced to remove the steady-state bias caused by unknown constant disturbances. However, the integral control is not robust against the variations of the disturbance level and may lead to the integralwindup phenomenon which is not desired in practical systems. In [17], a robust CNF control method is proposed to achieve fast and accurate set-point tracking for disturbed linear systems. The drawbacks of [16] and [17] are that the disturbance input is assumed constant and the parametric uncertainties for the system are not considered. Moreover, to the best of our knowledge, no attempts have been made in performance improvement of the robust model following problem using CNF approach. Besides the theoretical advances, the CNF method have been applied to design various servo systems, such as HDD servo systems [15, 18], helicopter flight control systems [19, 20], and position servo systems [17]. In this paper, similar to [7], we consider the robust tracking and model following problem for linear systems with time varying uncertain parameters and disturbances. We would like to develop a design procedure of the CNF control law for robust tracking and to improve the tracking performance of the closed-loop system. We offer some new forms of the nonlinear function for CNF and compare their results as well as those of [7, 8]. The paper is organized as follows. In Sect. 2, the problem of robust tracking and model following is stated and the required assumptions are discussed. The main theoretical results are developed in Sect. 3. The selection of the nonlinear function in CNF for performance improvement of the overall system is discussed in Sect. 4. The simulation results are illustrated in Sect. 5, and conclusions are given in Sect. 6. 2 System description and assumptions Consider the uncertain dynamical system described by [21]: x(t) ˙ = A + A r(t) x(t) + B + B s(t) u(t) (1) + W q(t) , y(t) = Cx(t),
where t ∈ [t0 , ∞), x(t) ∈ R n is the measurable state vector, u(t) ∈ R m is the control input, y(t) ∈ R p is the output vector which is to track the reference output ym (t), and matrices A(.) and B(.) represent the system uncertainties which are continuous in their arguments. Moreover, the vector W (q(t)) ∈ R n is the external disturbance. The uncertain parameters ((r(t), s(t), q(t)) ∈ Ω are Lebesgue measurable, where Ω is a compact bounding set. The objective is to design a feedback control law such that the controlled output y(t) tracks the reference output ym (t) in the presence of uncertainties and without experiencing large overshoot. The reference model is described by the differential equations of the form [8]: x˙m (t) = Am xm (t), ym (t) = Cm xm (t),
(2)
where Am and Cm are constant matrices of appropriate dimensions, xm (t) ∈ R nm is the state vector of the reference model, and ym (t) has the same dimension as y(t). The state vector of the reference model is considered to be bounded, i.e., there exists a positive constant M such that xm (t) ≤ M for all t ≥ t0 . Moreover, the reference model is chosen such that there exist matrices G ∈ R n×nm and H ∈ R m×nm satisfying [7, 8]: A B G GAm . (3) = Cm C 0 H If the solution of this algebraic equation cannot be found, a different reference model must be chosen [2]. On the other hand, the following assumptions on the original system are required. Assumption 1 Pair (A, B) is completely controllable. It follows from this assumption that for any given symmetric positive definite matrix Q ∈ R n×n , there exists an unique symmetric positive definite matrix P ∈ R n×n as the solution of the following algebraic Riccati equation [7, 22]: AT P + P A − ηP BB T P = −Q,
(4)
where η is a positive scalar. Assumption 2 There exist continuous and bounded matrix functions N (.), E(.), and W˜ (.) of appropriate dimensions such that [23]:
Robust tracking control method based on composite nonlinear feedback technique for linear systems
A r(t) = BN r(t) , B s(t) = BE s(t) , W q(t) = B W˜ q(t) .
(5)
This assumption defines the matching condition on the uncertainties and is rather a standard assumption in robust control problems. For convenience, the following notations which represent the bounds of the uncertainties are introduced: ρr = maxN r(t) , r∈Ω μ = maxE s(t) , (6) s∈Ω ρq = maxW˜ q(t) . q∈Ω
Assumption 3 For some δ > 0, E(s(t)) + E T (s(t)) + 2I ≥ δI , where I is identity matrix.
3 Main results The output tracking error e(t) and a new auxiliary state vector x(t) ˜ are defined as [5, 24]: e(t) = y(t) − ym (t),
(7)
x(t) ˜ = x(t) − Gxm (t),
(8)
where G is assumed to satisfy (3). From (2), (3), (7), and (8), one can obtain e(t) = C x(t) − Gxm (t) = C x(t). ˜ (9) It follows from (9) that e(t) ≤ Cx(t) ˜ .
(10)
Since C ≤ ∞, one can conclude that x(t) ˜ →0 yields e(t) → 0. Therefore, the proof of stability of x(t) ˜ is sufficient for the tracking goal. The linear feedback control law for system (1) is designed as [6]: uL (t) = Kx(t) + (H − KG)xm (t),
(11)
u(t) = uL (t) + uN (t) = Kx(t) + (H − KG)xm (t) T ˜ (14) + ψ x(t) ˜ B P x(t). Equations (1), (2), (8), and (14) yield ˙˜ = A + BK + A r(t) x(t) T + B + B s(t) ψ x(t) ˜ B P + B s(t) K x(t) ˜ + g(r, s, q, xm ),
(12)
where γ is a positive scalar and the real symmetric matrix P > 0 is the solution of the algebraic Riccati equation (4). The nonlinear feedback control law uN (t) is given by T uN (t) = ψ x(t) ˜ B P x(t), ˜ (13)
(15)
where
g(r, s, q, xm ) = A r(t) G + B s(t) H xm (t) + W q(t) . (16) Then, using the matching condition, Eq. (16) can be written as g(r, s, q, xm ) = BF (r, s, q, xm ),
(17)
where:
F (r, s, q, xm ) = N r(t) G + E s(t) H xm (t) + W˜ q(t) .
(18)
Furthermore, defining the bound: ρ = max F (r, s, q, xm ) : (r, s, q, xm ) ∈ Ω,
xm (t) ≤ M, t ∈ R ,
(19)
and using (6), one can write that ρ ≤ [ρr G + μH ]M + ρq .
(20)
In this paper, the function ψ(x(t)) ˜ in (13) is defined as ψ x(t) ˜ =−
with K = −γ B T P ,
where ψ(x(t)) ˜ is a choice of nonpositive locally Lipschitz function in x(t), ˜ which is used to change the damping ratio of the closed-loop system as the controlled output approaches the reference output to reduce the overshoot caused by the linear part. The linear and nonlinear feedback control laws obtained in (11) and (13) are now combined to form a CNF controller as follows:
2 ˜ (ρ + ρr x(t)) , (1 + μ)[B T P x(t)(ρ ˜ + ρr x(t)) ˜ + σ (x(t))] ˜ (21)
where σ (x(t)) ˜ ∈ R + is any positive uniform continuous bounded function that satisfies: t σ x(τ ˜ ) dτ ≤ σ¯ < ∞, (22) lim
t→∞ t 0
where σ¯ is any positive constant. The selection procedure of σ (x(t)) ˜ will be discussed later.
S. Mobayen, V.J. Majd
Remark 1 It is obvious that since σ > 0, the following holds for any x(t) ˜ ∈ Rn : uN (t) (ρ + ρr (1 + μ)[B T P x(t)(ρ ˜ + ρr x(t)) ˜ + σ (x(t))] ˜ ρ + ρr x(t) ˜ = , 1+μ
˜ +ρ ˜ + 2 ρr x(t) ≤ −x˜ T (t)Qx(t) × B T P x(t) ˜ −2
2 B T P x(t) x(t)) ˜ ˜
=
which shows that the nonlinear control law 13 is uniformly bounded. Theorem 1 Consider the closed loop system (15) and the CNF control law of (14). Suppose that Assumptions 1–3 are satisfied. Then the CNF control law in (14) drives the system output to asymptotically track the reference output. Proof Constructing the Lyapunov function as ˜ V (x, ˜ t) = x˜ T (t)P x(t),
(23)
and taking its derivative along the trajectories of the closed loop system in (15) yields T ˙˜ ˜ + x˜ T (t)P x(t) V˙ (x, ˜ t) = x˙˜ (t)P x(t) T T = x˜ (t) A P + P A + K T B T P + P BK
+ N T B T P + P BN + K T E T B T P + P BEK + P Bψ x(t) ˜ I + ET B T P T + P B(I + E)ψ x(t) ˜ B P x(t) ˜ + F T (r, s, q, xm )B T P x(t) ˜ + x˜ T (t)P BF (r, s, q, xm ).
(24)
In the light of (4), (12), (21), and Assumption 3, one can obtain that V˙ (x, ˜ t) ≤ x˜ T (t) AT P + P A + N T B T P + P BN − γ P B E + E T + 2I B T P T + 2P B(I + E)ψ x(t) ˜ B P x(t) ˜ + 2x˜ (t)P BF (r, s, q, xm ) ˜ ≤ x˜ T (t) AT P + P A − γ δP BB T P x(t) T ˜ N r(t) x(t) + 2 B P x(t) ˜ T T + 2(1 + μ)x˜ (t)P Bψ x(t) ˜ B P x(t) ˜ T + 2B P x(t) ˜ F (r, s, q, xm ) T ˜ ≤ −x˜ (t)Qx(t) ˜ + 2ρr B T P x(t) ˜ x(t) T + 2(1 + μ)x˜ T (t)P Bψ x(t) ˜ B P x(t) ˜ T + 2ρ B P x(t) ˜ T
2 (ρ + ρr x(t)) ˜ + ρr x(t)) ˜ + σ (x(t)) ˜
B T P x(t)(ρ ˜
2 × B T P x(t) ˜
≤ −x˜ T (t)Qx(t) ˜ T P x(t)σ (ρ + ρr x(t))B ˜ ˜ (x(t)) ˜ . +2 T B P x(t)(ρ ˜ + ρr x(t)) ˜ + σ (x(t)) ˜ (25) Considering the fact that σ 0≤ ≤ σ, ∀σ, > 0, (26) +σ the last term in (25) becomes less than 2σ , and thus it follows that: 2 ˜ + 2σ x(t) V˙ (x, ˜ t) ≤ −λmin (Q)x(t) ˜ . (27) On the other hand, there always exist two positive constants c1 and c2 such that for t ≥ t0 : 2 2 ˜ ≤ V x(t) ˜ . c1 x(t) ˜ ≤ c2 x(t) (28) From (27) and (28), the following derived: 2 ˜ 0 ≤ c1 x(t) t ≤ V x(t) ˜ = V x(t ˜ 0) + V˙ x(τ ˜ ) dτ 2 ˜ 0 ) − ≤ c2 x(t +2
t0
t
2 ˜ ) dτ λmin (Q)x(τ
t0 t
σ x(τ ˜ ) dτ,
(29)
t0
and thus t 2 2 ˜ ≤ c2 x(t ˜ 0 ) + 2 σ x(τ ˜ ) dτ. (30) 0 ≤ c1 x(t) t0
Notice that for t ≥ t0 , we have t sup σ x(τ ˜ ) dτ ≤ σ¯ .
(31)
t∈[t0 ,∞) t0
Using (30) and (31) yields: 2 2 ˜ ≤ c2 x(t ˜ 0 ) + 2σ¯ . 0 ≤ c1 x(t)
(32)
Moreover, if taking the limit of (29) as t approaches infinity follows that: t 2 2 ˜ ) dτ ˜ 0 ) − lim 0 ≤ c2 x(t λmin (Q)x(τ + 2 lim
t→∞ t 0
t
t→∞ t 0
σ x(τ ˜ ) dτ.
(33)
Robust tracking control method based on composite nonlinear feedback technique for linear systems
From (22) and (33), one can obtain t 2 2 ˜ ) dτ ≤ c2 x(t ˜ 0 ) + 2σ¯ . (34) lim λmin (Q)x(τ
t→∞ t 0
It follows from (32) that x(t) ˜ is uniformly bounded. Since x(t) ˜ is continuous, it is uniformly continuous, and consequently, the term λmin (Q)x(τ ˜ )2 in (34) is also uniformly continuous. Applying the Barbalat lemma to (34) results in 2 ˜ = 0. lim λmin (Q)x(t) (35) t→∞
Because λmin (Q) > 0, one can conclude that (36) ˜ = 0. lim x(t)
t→∞
Therefore, the closed loop auxiliary system described by (15) is uniformly bounded and its auxiliary state x(t) ˜ converges uniformly asymptotically to zero. Then it can be obtained from (10) that the tracking error e(t) decreases asymptotically towards zero. This completes the proof.
4 The selection of the nonlinear function σ (x(t)) ˜ The selection procedure of nonlinear function σ (x(t)) ˜ is discussed in this section. The freedom of choosing the nonlinear function σ (x(t)) ˜ enables one to tune the control law so as to improve the closed loop system performance as the output approaches the command input. The main purpose of adding the nonlinear term to the CNF control law is to speed up the settling time, or equivalently to contribute an important value to the linear control input when the tracking error, e(t), is small. The selection of an appropriate nonlinear function σ (x(t)) ˜ is the main difficulty of CNF controller design. This function needs to be chosen such that it satisfies the following properties: (1) Since σ acts over the norm x(t), ˜ and should satisfy (31), then it follows that σ x(t) ˜ = σ −x(t) ˜ ≥ 0. (2) When the system output y(t) is far away from the reference model output, σ (x(t)) ˜ must have its high value so that |ψ(x(t))| ˜ becomes small, and therefore the effect of the nonlinear part of CNF control law becomes negligible.
Fig. 1 Interpretation of the nonlinear function ψ(x(t)) ˜
(3) When the system output y(t) approaches the refer˜ must converge ence model output ym (t), σ (x(t)) to its lowest value so that |ψ(x(t))| ˜ becomes large, and thus the nonlinear part of the CNF control law becomes effective. The closed loop system (15) can be expressed as T ˙˜ = A + BK + Bψ x(t) ˜ + Bω, (37) x(t) ˜ B P x(t) where ω = [A(r(t)) + B(s(t))K + B(s(t)) × T P ]x(t) ˜ + [A(r(t))G + B(s(t))H ] × ψ(x(t))B ˜ xm (t) + W (q(t)). The eigenvalues of the closed loop system (37) change by the variation of the nonlinear function σ (x(t)) ˜ over time. Such a mechanism can be interpreted using the classical feedback control concept as shown in Fig. 1, where the auxiliary system (Gaux (s)) is defined as Gaux (s) = Caux (sI − Aaux )−1 Baux ,
(38)
where Aaux = A + BK, Baux = B, Caux
(39)
= BT P .
Therefore, one can write Gaux (s) = B T P (sI − A − BK)−1 B.
(40)
Similar to the classical root-locus theory, the poles of the closed loop system (37), which are the functions of the tuning parameter ψ(x(t)), ˜ start from the open loop poles of the auxiliary system, i.e., the eigenvalues of A + BK, when ψ(x(t)) ˜ = 0, and end up at the open loop zeros, as |ψ(x(t))| ˜ becomes larger. The designer should select the invariant zeros of Gaux (s) such that the dominated poles have a large damping ratio, yielding smaller overshoot. Remark 2 It is shown in [12] that the auxiliary system (40) is stable and is minimum phase with n − 1 stable invariant zeros. The invariant zeros of Gaux (s) play a
S. Mobayen, V.J. Majd
significant role in the pole selection of the closed loop system (37). There is freedom in preselecting the locations of the invariant zeros by selecting a suitable Q in (4). The nonlinear function σ (x(t)) ˜ is nonunique and is defined in various forms as follows: Form 1 The nonlinear function can be chosen in the form of an exponential function as − ˜α + ε, (41) σ x(t) ˜ = βe x(t) ε = ε0 e−τ0 t ,
(42)
where α, β, ε0 , τ0 > 0 are the tuning parameters. Function σ (x(t)) ˜ reaches its maximum value β + ε0 when x(t) ˜ tends to infinity and reaches its minimum value 0 as x(t) ˜ tends to zero. Form 2 A modified version of (41) is given as follows:
(1−(y(t)−y0 )/(ym (t)−y0 ))2 −1 2 e + ε, σ x(t) ˜ =β e1 − 1 (43) where β is a positive tuning parameter and y(t) = ˜ The function σ (x(t)) ˜ changes from ym (t) + C x(t). β + ε0 to 0 as the tracking error approaches zero. Form 3 To avoid the problem of division by zero when ym (t) = y0 for some t, (43) can be modified to form a smooth and positive function as (44) σ x(t) ˜ = βeα(y(t)−ym (t)−y0 −ym (t)) + ε, where α, β are appropriate positive scalars which can be chosen to yield a desired performance, i.e., fast settling time and small overshoot. This function changes from β + ε0 to 0 as the tracking error approaches zero. Form 4 To adapt the variation of the tracking target, the nonlinear function can be chosen as −α0 α (45) σ x(t) ˜ = βe y(t)−ym (t) + ε, where α0 =
y0 − ym (t) 1
y0 = ym (t) y0 = ym (t).
(46)
The scaling parameter α0 changes with different tracking target values ym (t), and then closed loop system has stronger robustness to the variations of tracking targets than the nonscaled functions.
5 Simulation results Consider an uncertain nominal system which is proposed in [2, 6] with disturbances and without timedelay as ⎛⎡ ⎤ 0 1 0 x(t) ˙ = ⎝⎣ 0 1 2⎦ −1 −2 0 ⎤⎞ ⎡ 0 0 0 + ⎣ 0.1r1 0 0.1r2 ⎦⎠ x(t) (47) r1 0 r2 ⎛⎡ ⎤ ⎡ ⎤⎞ ⎡ ⎤ 0 0 0 + ⎝⎣ 0.1 ⎦ + ⎣ 0.1s ⎦⎠ u(t) + ⎣ 0.1 ⎦ q, 1 s 1 y = 1 0 0 x(t), where the uncertainties and disturbances have the following bounds: |r1 | ≤ 0.15, |r2 | ≤ 0.2, |s| ≤ 0.5 and |q| ≤ 0.2. The reference model is specified by 0 1 x˙m (t) = x (t), −1 0 m (48) ym (t) = 1 0 xm (t). The eigenvalues of the nominal system given in (47) are: [−0.433, 0.7164 ± j 2.026]. From (3), the solution of the matrices G and H can be obtained as follows: ⎡ ⎤ 1 0 ⎦, G=⎣ 0 1 −0.5786 −0.5711 H = 1.5711 1.4214 . From (5) and (47), we obtain: N (r(t)) = [r1 (t) 0 r2 (t)], E(s(t)) = s(t), W˜ (q(t)) = q(t). Then one can obtain from (6), (19), and (48) that: ρr = 0.25, μ = 0.5, ρ = 1.26 and xm (t) ≤ M = 1. For the given constant γ = 1 and the positive-definite matrix Q = diag{20, 6, 1}, the solution of the Lyapunov equation (4) is obtained as: ⎡ ⎤ 22.218 10.32 2.77 P = ⎣ 10.32 13.216 4.575 ⎦ . 2.77 4.575 3.936 The gain matrix K is determined from (12) as: K = [−5.704 − 8.845 − 6.59]. The eigenvalues of A + BK are located at [−1.593 ± j 1.157, −3.289]. The nonlin-
Robust tracking control method based on composite nonlinear feedback technique for linear systems Fig. 2 Variations of nonlinear functions. (a) σ (x(t)) ˜ and (b) ψ(x(t)) ˜
ear function ψ(x(t)) ˜ is defined in the form of (21). The constant parameters are selected as: ε0 = 0.5, τ0 = 1, β = 100, α = 1. Only for the simulation purposes, we set the uncertain parameters r1 (t), r2 (t), s(t), q(t), and initial conditions as follows: r1 (t) = 0.15 sin(t), s(t) = 0.5 sin(2t), T xm (0) = 1 0 ,
r2 (t) = 0.2 sin(t), q(t) = 0.2 sin(3t). T x(0) = 0 0 0 .
Variations of the four forms of nonlinear functions σ (x(t)) ˜ and ψ(x(t)) ˜ are demonstrated in Fig. 2. We can find from Fig. 2 that the convergence tendency of the CNF-Form 4 controller is much faster than other ones. In the following, the simulation results of the proposed controller with Form 4 nonlinear function are illustrated in comparison with the works of [7] and also [8] but without time-delay. Figure 3 shows the trajectories of the system states. This figure significantly demonstrates the importance of adding the nonlinear function ψ(x(t)) ˜ for improving the system response. The trajectory of the tracking errors and control input are illustrated in Fig. 4. As it can be seen from the simulation results, our proposed controller makes fine responses, indicating its superiority over the robust tracking controller of [7] and the model reference adaptive controller of [8]. Simultaneously, we can also find from simulations that the responses ob-
tained by CNF-Form 4 controller have better control performance, in terms of tracking accuracy and quick response compared to the controllers of [7, 8]. Also, Fig. 5 demonstrates the tracking trajectory of the system and the error between the desired and actual outputs. This figures confirm that our proposed controller has better control performance, representing its advantage over other controllers obtained in [7, 8]. Consequently, from the above simulation results, it is obvious that the proposed method provides faster response and better transient response over the robust tracking controller of [7] and the model reference adaptive controller of [8]. Since we consider a tracking-control problem, a direct and simple performance index is the integral of absolute-value of error (IAE) with the following formula: t I1 (χ) = |χ| dt, (49) 0
where χ is a signal. Also, to address the settling time and overshoot of the transient response, the integral of time multiplied by absolute-value of error (ITAE) is presented: t I2 (χ) = t|χ| dt. (50) 0
The IAE and ITAE values of the tracking errors, control inputs, and the auxiliary state vectors using CNF controllers in comparison with [7, 8] are shown
S. Mobayen, V.J. Majd Fig. 3 Trajectory of the system states. (a) plot of x1 (t), (b) plot of x2 (t), (c) plot of x3 (t)
Fig. 4 Control performance: (a) control input, (b) trajectory of the tracking errors
in Table 1. As it can be seen, the IAE and ITAE performance indices values are decreased when using the CNF-Form 4 controller compared with other ones. For example, the tracking error and control input IAE improvement using CNF-Form 4 controller is 36.26 % and 17.22 % than those of [7] and 27.62 % and 19.06 % than those of [8], respectively.
6 Conclusions In this paper, the algebraic Riccati equation and CNF technique were used for robust tracking and model following of a class of linear systems with time varying uncertainties and disturbances. Different forms of the nonlinear function σ (x(t)) ˜ were proposed to im-
Robust tracking control method based on composite nonlinear feedback technique for linear systems Fig. 5 System output y(t) and model output ym (t)
Table 1 Performance indices values using CNF controllers with different forms of nonlinear functions compared with [7] and [8] (without time-delay) IAE I1 (x˜1 )
ITAE I1 (x˜2 )
I1 (e)
I1 (u)
I2 (x˜1 )
I2 (x˜2 )
I2 (e)
I2 (u)
CNF-Form 1
1.221
1.134
1.221
12.624
1.413
1.781
1.413
182.61
CNF-Form 2
1.252
1.172
1.252
12.491
1.574
1.939
1.574
178.28
CNF-Form 3
1.46
1.349
1.46
12.518
3.077
3.433
3.077
178.28
CNF-Form 4
1.2504
1.166
1.2504
12.503
1.539
1.911
1.539
183.62
Wu, H. [7]
1.962
1.695
1.962
15.104
6.106
5.7201
6.1064
285.59
Wu, H. [8] (without delay)
1.7276
2.564
1.7276
15.448
4.881
7.548
4.881
503.25
prove the reference tracking performance of the system. Among them, the CNF-Form 4 controller was shown to have better control performance. Also, the simulation results demonstrate that the new technique has better IAE and ITAE performance indices values in comparison with other related works. It is shown from the simulation results that the proposed method is effective and feasible and can be considered as a promising way for applying on practical robust tracking problems of uncertain dynamical systems.
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