LMI-Based Composite Nonlinear Feedback Tracker for Uncertain Nonlinear Systems With Time Delay and Input Saturation

This study proposes a composite nonlinear feedback approach for the robust tracking control problem of uncertain nonlinear systems with input saturation, Lipschitz nonlinear functions, multivariable time-delays, and disturbances. The composite nonlinear feedback technique includes two components: a linear feedback portion constructed in such a way that it changes the damping ratio so as to speed up the system’s response. A nonlinear feedback controller is designed to further increment the damping ratio in a way that it ensures the tracking whilst reducing the overshoot created by the linear portion. By creating a suitable Lyapunov functional and by using the linear matrix inequality (LMI) approach, the LMI conditions are determined to guarantee system stability and obtain the required design parameters. The performance of the proposed approach is assessed using a simulation study of a two-dimensional system along with a Chua’s circuit system. The advantages of the proposed approach are its less-restrictive assumptions, improved transient performance, and steady-state precision.

of particular importance, the ability of the controller to track 23 the input command and remove the effects of perturbations 24 from the controlled system is critical [10]. Input saturation 25 The associate editor coordinating the review of this manuscript and approving it for publication was Haibin Sun . is a constraint commonly found in practical systems that do 26 not allow the control input to exceed the specified control 27 bounds. Ignoring the input saturation constraint in the control 28 design stage can lead to undesirable behavior and potential 29 instability of the closed-loop system. Input saturation and 30 time delay frequently appear simultaneously in control sys-31 tems [11], [12]. Various approaches have been devised in the 32 literature to mitigate these problem, such as adaptive neural 33 tracking control [13], robust tracking control [14], [15], H ∞ 34 output tracking control [16], observer-based adaptive fuzzy 35 tracking control [17], quantized state feedback [18], and 36 robust H ∞ control [19]. 37 Stabilization and tracking control are critical for systems 38 requiring robust and optimal performance such electron-39 ics, chemical processes, helicopter flight control systems. 40  In [42], the CNF procedure with a nonlinear term has been 92 studied in terms of smooth and quick regulation with the 93 uncertainty, external disturbances and input saturation ren-94 dezvous for the spacecrafts regardless of the nonlinearities 95 and time delays. The CNF control procedure was studied for a 96 category of linear/nonlinear systems with parallel distributed 97 recovery via sliding mode control method in [43]. In [44], 98 a quick and precise robust path-following control approxi-99 mate has been conducted for a fully-actuated marine surface 100 vessel with external disturbances. In [45], the quick and 101 precise chaos synchronization of uncertain chaotic systems 102 with Lipschitz nonlinear terms and disturbances have been 103 investigated. 104 An active front-steering control that combines CNF with 105 a disturbance observer was proposed in [46], to obtain a fast 106 damping rate tracking response and yield robustness to exter-107 nal disturbances. An adaptive nonlinear gain-based CNF con-108 troller was proposed in [47] to optimize the system dynamic 109 efficiency. However, disturbances, uncertainties, and time 110 delays were not considered in the design. A CNF technique 111 was proposed in [48] for the tracking problem of a class 112 of single-input single-output nonlinear systems subject to 113 input saturation. The CNF control problem based on the 114 event-triggered strategy was investigated in [49], for saturated 115 systems. In [50], the adaptive tracking control problem of 116 uncertain large-scale nonlinear time-delayed systems in the 117 presence of input saturations is studied. In [51], the bound-118 edness property of the robust tracking CNF controller has 119 been studied for time-delay uncertain systems with input 120 saturation. In [52], a hybrid controller design for a quar-121 ter car is developed. A design that combines active distur-122 bance rejection control with a fuzzy control approach was 123 proposed in [53].

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For the intermittent control, its control signal is updated 125 in a continuous manner on control time intervals. To over-126 come the limitation, time-triggered intermittent control is 127 considered in [54]. In [55], the time-triggered intermittent 128 control is proposed to examine the exponential synchro-129 nization issue of chaotic Lur'e systems. In [56], a control 130 approach is suggested for the robust stabilization of the iner-131 tial wheel inverted pendulum, with norm-bounded parametric 132 uncertainties and both motion constraints and actuator satu-133 ration. The robust stabilization of a class of continuous-time 134 nonlinear systems via an affine state-feedback control law 135 using the linear matrix inequality approach is studied in [57]. 136 In [58], an enhanced composite nonlinear feedback technique 137 using adaptive control is developed for a nonlinear delayed 138 system subject to input saturation and exogenous distur-139 bances. In [59], the precise tracking problem for electrostatic 140 micromirror systems with disturbances and input saturation is 141 studied. A composite nonlinear feedback-based adaptive inte-142 gral sliding mode controller with a reaching law for fast and 143 accurate control of a servo position system subject to external 144 disturbance is presented in [60]. In [61], an L 2 −L ∞ /H ∞ 145 optimization control issue is studied for a family of non-146 linear plants by Takagi-Sugeno (T-S) fuzzy approach with 147 actuator failure. In [62], the problem of resilient event-148 trigger-based security controller design is investigated for 149 nonlinear networked control systems described by interval 150 type-2 fuzzy models subject to non-periodic denial of service 151 attacks. In [63], an adaptive performance guaranteed tracking 152

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Consider the following nonlinear system with time delay, 176 input saturation, uncertainties and disturbances: where u i (t) is the maximum value of the i th control input.

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The main control purpose is to synthesize a CNF law so 192 that the output y(t) can follow the reference output y m (t), 193 as quickly and smoothly as possible. The reference model is where A m and C m are constant matrices, x m (t)∈R n m represents 199 the state vector of the reference model with x m (t) ≤M , 200 where M is a positive scalar. The reference model is selected 201 so that there exist two matrices G∈R n×n m and H ∈R m×n m 202 satisfying: Assumption 1: There are matrices with continuous bound-205 ary conditions N (.), which the bounds of the uncertainties are provided by Remark 1: Assumption 1 illustrates that the uncertain 215 nonlinear system with time-delay and input saturation (1) 216 contains a special structure which is commonly denominated 217 by a matching condition for parametric uncertainties; a stan-218 dard assumption in robust control problems. According to 219 the matching conditions, all external disturbances, and uncer-220 tainties must be controlled in the control vector space, thus 221 limiting the structure of the system. The boundary conditions 222 are continuous and bounded matrix functions, meaning that 223 the arguments of the matrix are bounded and continuous 224 functions.

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Assumption 2: The nonlinear function f is uncertain and 226 Lipschitz for all x(t)∈R n and x m (t)∈R n so that [64], [65]: In other words, it is stated as The tracking error e(t) and auxiliary vector tildex(t) are

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Since C is bounded, then x(t) → 0 implies 256 e(t) → 0. Hence, to prove the perfect tracking one has 257 to show that x(t) is convergent.

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The CNF control design method is suggested for the tracking

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where G and H are obtained from (4), and K denotes a gain 266 matrix that is specified in the LMI form. Then define the 267 nonlinear function as:

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where P denotes a positive-symmetric matrix, and ψ(x(t)) is a  (10), and (14), it can be obtained By applying Assumption 1, Eq. (16) is stated as Then, from (6) and (19), and determining the bounds 303 ρ = sup F(r, s, ν, q, x m ) and x m (t) ≤ M , the following 304 inequality is obtained The nonlinear function ψ(x(t)) in (13) is defined as in (21), 307 shown at the bottom of the page, where σ (x(t))∈R + is any 308 positive uniform continuous bounded function with where σ 0 is a bounded constant.

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The liberty in choosing a nonlinear expression σ (x(t)) 312 causes the control law to be adjusted and the perfor-313 mance is improved as the system output converges to the 314 reference input. The nonlinear expression σ (x(t)) applies to 315 the following features 316 (1) Because σ (x(t)) a function of x(t) , the following 317 equation is established: .
put, the function σ (x(t)) becomes larger, as a result, 321 the function shrinks and the efficacy of the nonlinear 322 portion of the CNF law reduces.

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(3) If the system output converges the reference output, σ (x(t)) function becomes very small and reaches its 325 lowest value, thus increasing the |ψ(x(t))| value and 326 therefore, the efficacy of the nonlinear controller will 327 be evident.

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Since the choice of the nonlinear function σ (x(t)) is free so 329 it can be expressed in different ways. To compatible alteration 330 of the tracking aim, a nonlinear function is defined as follows: The nonlinear expression ψ(x(t)) converges from the pri-  Then, the control feedback gain K is obtained by K = Y Q −1 .

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Proof: To prove the sustainability of the system, we con-356 struct the Lyapunov candidate functional as where P and R i , i = 1, · · ·, N are the weighting matrices that 359 is characterized by LMI. By deriving (29) along the directions 360 of the system in (15) outcomes According to Assumption 1, it can obtain that: The following equation holds for all In the following, we investigate four states of the saturation 409 function.

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Case 1: In this case, all input channels are higher than the 411 upper bound, that is, u > u i for this situation, we have

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Therefore, all paths from the inside of X δ will approach the 420 reference, and hence 423 From (33), we obtain 425 since ψ(x(t)) is a nonpositive expression, it results: Case 2: In this case, all input channels are assumed to 428 be smaller than their lower bound, that is, u < −u i then, 429 we have: Similarly, we obtain:   Case 3: In this case, we will have the combination of 441 modes 1 and 2 which means that some of the control inputs 442 are saturated and some are not. In order to represent the 443 unsaturated factors, we have: and, (48), as shown at the bottom of the next page, where 461 (47), we achieve From (49) and (50), we have Then, it follows that since σ (x(t)) ≤σ , and overline (σ is a bounded constant.

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Based on the equation (52), all of the responses of the system 473 are bounded, and also according to inequality (22), we con- so, the auxiliary vector tilde (x(t)) of the system (15) tends 482 uniformly to zero and thereby it follows from (11) that the 483 tracking error e(t) reduces asymptotically to zero.

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The condition dotV (x, t) < 0 holds if there exists a 485 scalar γ so that Q 1 < 0. In order to satisfy the inequality 486 (48) with the form of LMIs assuming Q = P −1 , K = 487 Y Q −1 , S i = QR i Q, and pre-and post-multiplying (48) By applying the Schur complement, the following inequality 492 is obtained as in (57)  Note that, the proposed robust tracking controller by CNF 497 improves the transient efficiency and steady-state precision 498 at the same time. Obviously, the CNF control law leads 499 to a linear controller when the nonlinear portion tends to 500 zero. As a result, the added nonlinear expression allows 501 modifying the linear control law to recover system tran-502 sient performance and the error converges to zero. Selec-503 tion of the nonlinear feedback part is important because the 504 delays and disturbance are considered as:

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A di (ν(t)) , i = 1, 2 are the uncertain parameters and q(t) 530 is the disturbance. The disturbance and uncertain bounds are 531 as follows:

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(|r 1 (t)| ≤ 0.5, |r 2 (t)| ≤ 1, |q(t)| ≤ 0.5, | v(t |≤ 1.5) and Then, from (5) and (58), we haveW (q(t)) = q(t), N (r(t)) =   Figure 1 shows the state vector of the reference model. 561 Figure 2 displays a diagram of state vector paths. The trajec-562 tories of the tracking error are illustrated in Figure 3 and the 563 nonlinear state-feedback controller is depicted in Figure 4. 564 Note that all the state paths approach the origin. So, the 565 simulation results show that the system is resistant to time 566

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where f 1 (x(t)) = −5x 2 1 (t) + 11.42 x 1 (t) + 2.14 (| x 1 (t)+ 579 1| − |x 1 (t) − 1 |) and A di , i = 1, 2 are fixed parameters, 580 r 1 (t), r 2 (t) and A di (v(t)), i = 1, 2 are the uncertain param-      conditions for the tracking controllers and also proved the 623 stability of the system and convergence of the tracking errors 624 to the origin. Implementation of the proposed approach to a 625 two-dimensional system and the Chua's circuit system con-626 firmed its superior performance and robustness to external 627 disturbances and parametric uncertainties. Addressing the 628 design problem of disturbance observer for nonlinear systems 629 under input saturation using the CNF method can be the topic 630 for future investigations. 631 back tracking of uncertain input-saturated MIMO nonlinear systems with 694 application to 2-DOF helicopter,'' Mathematics, vol. 9, no. 9, p. 1062,