UDE-based Dynamic Surface Control for Strict-feedback Systems with Mismatched Disturbances

Transcription

1 16 American Control Conference ACC) Boston Marriott Copley Place July 6-8, 16. Boston, MA, USA UDE-based Dynamic Surface Control for Strict-feedback Systems with Mismatched Disturbances Jiguo Dai, Beibei Ren, Qing-Chang Zhong Abstract This paper proposes a new method for the tracking control problem of strict-feedback systems with mismatched disturbances. The method, called the uncertainty disturbance estimator UDE), is introduced into the conventional backstepping scheme to hle the mismatched disturbances due to its excellent performance in hling uncertainties disturbances yet a simple control scheme. The dynamic surface control DSC) is adopted to overcome the explosion of complexity problem in the backstepping scheme. Compared to the existing work in the literature, the backstepping deviation term between the synthetic virtual control system state, is lumped into the disturbance-like term, together with the DSC filtering error mismatched disturbances. The lumped term is then hled by the UDE method at each step, which simplifies the control design analysis. In addition, the marriage of UDE with backstepping also relaxes the structural constraint of the UDE design for mismatched uncertainties. The uniformly ultimate boundedness of the closed-loop system is proved. Both numerical simulation experimental validation are carried out to demonstrate the effectiveness of the proposed approach. I. INTRODUCTION Most of modern control systems suffer various uncertainties like parameter perturbation, unmodeled dynamics external disturbances. In order to attenuate the undesired influence of uncertainties, numerous adaptive robust control strategies have been proposed by the control community, such as the sliding mode control integrator backstepping control. Unlike the sliding mode control requires the matched conditions, the integrator backstepping control is an effective popular strategy for a large class of nonlinear systems which are in the strict-feedback form with mismatched uncertainties [1], []. But an obvious problem comes from the explosion of complexity, since it requires obtaining the derivatives of virtual controls. To overcome the above complexity, a methodology named dynamic surface control DSC) was proposed in [3], [4] by introducing the first-order filtering of the synthetic controls at each step of the traditional backstepping approach. Jiguo Dai Beibei Ren are with the Department of Mechanical Engineering, Texas Tech University, Lubbock, , USA. jiguo.dai@ttu.edu, beibei.ren@ttu.edu). Qing-Chang Zhong is with the Department of Electrical Computer Engineering, Illinois Institute of Technology, Chicago, IL 6616, USA. zhongqc@ieee.org). Since the DSC method avoids the calculation of derivatives of virtual control signals, it has drawn a lot of attention has been applied to many applications in the recent decade. In [5], the DSC method was adopted into an adaptive neural network controller for a class of uncertain nonlinear systems in strict-feedback form. Furthermore, the DSC method has been adopted for hling different uncertainties nonlinearities such as the unknown dead zone [6], stochastic systems [7], unknown time-delay [8], hysteresis systems [9], unknown direction of control gain input saturation [1]. And in the practical engineering applications, the DSC method has been applied to the formation control trajectory tracking of autonomous vehicles [11], [1], [13], flexible-joint robot control [14], permanent magnet synchronous motors control [15], etc. However, the common feature of above works is the utilization of neural networks NNs) to approximate complex nonlinear functions uncertainties. But it will cause another problem that the complexity of nonlinear functions approximated by NNs will grow dramatically as the order of the system increases. Same problems exist in fuzzy control based DSC control [6], [16], [17], etc. Recently, a robust control methodology named uncertainty disturbance estimator UDE) has demonstrated its excellent performance in hling uncertainties disturbances has been successfully applied in various systems [18], [19], [], [1], []. As an easily implemented real time scheme, the UDE method uses a strictly-proper stable filter to reconstruct the uncertainty disturbance signals it does not need any knowledge of these unknown signals [18], [3]. All the uncertainties including parameter perturbation, unmodeled dynamics, external disturbances even the coupled states can be regarded as a disturbancelike term. Then the effects of the disturbance-like term on the system can be observed by the measured states input signals a compensator is constructed based on the filtering of this observation. However, the UDE method has a structural constraint which limits its application range [18], e.g., mismatched disturbances. Therefore, introducing the backstepping scheme into the conventional UDE is a promising solution to relax this constraint. Motivated by the above results, this paper synthesizes both the UDE method DSC method into the traditional backstepping approach to solve the tracking problem for strict /$ AACC 4743

2 feedback systems with mismatched disturbances. Compared to the existing works in the literature, the backstepping deviation term between the synthetic virtual control system state, is lumped into the disturbance-like term, together with the DSC filtering error mismatched disturbances. The lumped term is then hled by the UDE method at each step, which simplifies the control design analysis. In addition, the marriage of UDE with backstepping also relaxes the structural constraint in the conventional UDE design for mismatched uncertainties. The remainder of this paper is organized as follows: In Section II, the UDE-based dynamic surface control via backstepping is developed for a third-order system, a numerical example is provided for the validation. Then, Section III generalizes the strategy to the nth-order systems gives a stability analysis. Finally, the experiment validation concluding remarks are presented in Section IV Section V, respectively. II. AN ILLUSTRATIVE EXAMPLE For the simplicity, consider the following third-order strictfeedback system with mismatched disturbances: ẋ 1 = f 1 x 1 ) + x + d 1 ẋ = f x 1, x ) + x 3 + d 1) ẋ 3 = f 3 x 1, x, x 3 ) + u + d 3 y = x 1 where x 1, x, x 3 are the system states, f 1 ), f ), f 3 ) are known continuous smooth functions, d 1, d, d 3 are bounded smooth external disturbances u is the control input. Without loss of generality, it is assumed that f i,..., ) =. The control objective is to force the system output y = x 1 to track a smooth reference signal x 1r, i.e., x 1 x 1r as t. In order to facilitate the control design, first define the variables z 1, z, z 3 as the system tracking errors, x r, x 3r as the virtual controls, ω r, ω 3r as the filtered virtual controls y, y 3 as the virtual control errors. The UDE-based DSC control design is based on the following coordinates transformation [1], [4], z 1 = x 1 x 1r z = x ω r = x y x r ) z 3 = x 3 ω 3r = x 3 y 3 x 3r where y i = ω ir x ir, i =, 3 the recursive design procedure contains three steps. The design procedure begins at the first equation which is progressively stabilized by virtual control that appears in the outer equations. The procedure terminates when the final external control is reached. Step 1: Consider the first equation in 1). Since z 1 = x 1 x 1r = ż 1 = f 1 + x + d 1 ẋ 1r 3) applying z = x y x r results in ż 1 = f 1 + z + y + x r + d 1 ẋ 1r 4) where the virtual control x r will be derived to stabilize the system 4). Following the UDE-based control design [18], the virtual control x r consists of the feedback linearization part uncertainty compensation part as x r = x rl + x rd 5) where x rl is for the feedback linearization x rd is for the uncertainty compensation. Since z y are unknown for system 4), let u d1 = z +y +d 1 as the lumped disturbancelike term in 4). This is different from the conventional backstepping method where z remains in this step will be eliminated in the next step. Select where k 1 > x rl = k 1 z 1 f 1 + ẋ 1r 6) x rd = z + y + d 1 ) = u d1 7) the closed-loop system becomes ż 1 = k 1 z 1 8) where k 1 >. Therefore, the system state z 1 is exponentially stable as z 1 = e k1t z 1 ), with t. However, u d1 is unknown, can not be used directly. The disturbance compensation should be redesigned. Substituting 5), 6) into the system 4) yields Solving u d1 there is ż 1 = k 1 z 1 + x rd + u d1. 9) u d1 = ż 1 + k 1 z 1 x rd 1) which indicates that the uncertainty term can be observed by the known signals z 1, x rd. Using a proper low-pass filter g f1 which has the unity gain zero phase shift in the spectrum of u d1, the uncertainty term can be estimated by û d1 = g f1 ż 1 + k 1 z 1 x rd ) 11) where is the convolution operator the disturbance compensation is written as x rd = û d1 = g f1 ż 1 + k 1 z 1 x rd ). 1) Solving x rd yields x rd = L 1 Gf1 1 G f1 ) ż 1 + k 1 z 1 ) 13) where G f1 is the Laplace transfer of g f1, L 1 is the inverse Laplace operator. Hence, the virtual control 5) for the system becomes x r = [ k 1 z 1 f 1 + ẋ 1r ] ) L 1 Gf1 ż 1 + k 1 z 1 ) 14) 1 G f1 To obtained a filtered virtual control, x r is passed through a first-order filter τ ω r + ω r = x r ω r ) = x r ) 15) 4744

3 with τ > is a time constant. Consequently, substituting 5), 6), 11) 1) into 4) results in the closed-loop system ż 1 = k 1 z 1 + u d1 û d1 ) 16) Step Consider the second equation in 1) z = x ω r ż = f + x 3 + d ω r 17) Applying z 3 = x 3 y 3 x 3r results in ż = f + z 3 + y 3 + x 3r + d ω r 18) Similarly, as z 3 y 3 are unknown for system 18), let u d = z 3 +y 3 +d as the lumped disturbance-like term follow the same procedure with Step 1. The virtual control x 3r is obtained as where the feedback linearization part is x 3r = x 3rl + x 3rd 19) x 3rl = k z f + ω r ) with k > the uncertainty compensation part is ) x 3rd = L 1 Gf ż + k z ). 1) 1 G f Hence, the virtual control is x 3r = [ k z f + ω r ] ) L 1 Gf ż + k z ) ) 1 G [ f = k z f + x ] r ω r τ ) L 1 Gf ż + k z ) 3) 1 G f Then, we pass x 3r though a first-order filter to obtain the filtered virtual control τ 3 ω 3r + ω 3r = x 3r ω 3r ) = x 3r ) 4) with τ 3 > is a time constant the resulting closed-loop system is ż = k z + u d û d ) 5) Step 3 Consider the last equation in 1), Actual/desired output Tracking error Control effort x 1r x a) b) c) Figure 1. Simulation results of Example 1 where k 3 >, ẋ 3r is obtained from the previous step. Then the resulting closed-loop system is ż 3 = k 3 z 3 + u d3 û d3 ) 9) Example 1. Consider the following third-order system ẋ 1 = x 1 + x + d 1 3) ẋ = x 1 x + x 3 + d 31) ẋ 3 = u + d 3 3) z 3 = x 3 ω 3r ż 3 = f 3 + u + d 3 ω 3r 6) Let the lumped disturbance-like term be u d3 = d 3 follow the same procedure with Steps 1, the control input is obtained as u = [ k 3 z 3 f 3 + ω 3r ] ) L 1 Gf3 ż 3 + k 3 z 3 ) 7) 1 G [ f3 = k 3 z 3 f 3 + x ] 3r ω 3r τ ) 3 L 1 Gf3 ż 3 + k 3 z 3 ) 8) 1 G f The disturbances are d 1 =.5 sin πt, d =.15 sin πt, d 3 =.5 cos πt the initial value of the system is chosen as x 1 = x = x 3 =. The design parameters of the controller are taken as k 1 = 3, k = k 3 = 5, the πs first-order filters are G f1 = G f = G f3 = s +πs+π. The time constants for DSC are set as τ = τ 3 =.1s. The output signal x 1 is required to track the reference signal x 1r = sin πt. Fig. 1 shows the performance of the proposed method. The system output can successfully track the reference signal the tracking error is remained below.5%.

4 III. CONTROL DESIGN AND STABILITY ANALYSIS FOR A nth-order SYSTEM More generally, the results in Section II can be generalized to the nth-order case: ẋ 1 = f 1 x 1 ) + x + d 1 ẋ = f x 1, x ) + x 3 + d... 33) ẋ = f x 1,..., x ) + x n + d ẋ n = f n x 1,..., x n ) + u + d n y = x 1 where x i, i = 1,,..., n, are the system states, f i ) are known continuous smooth functions, d i are bounded smooth external disturbances, y is the system output, u is the control input. Using the following coordinate transformation z 1 = x 1 x 1r 34) z i = x i ω ir = x i y i x ir 35) y i = ω ir x ir 36) τ i ω ir + ω ir = x ir, ω ir ) = x ir ) 37) where i =,..., n, τ i > are time constants, x ir are virtual controls, ω ir are filtered virtual controls, y i are deviations of virtual controls. The UDE-based DSC algorithm is summarized as the following Algorithm 1. Algorithm 1 UDE-based DSC for strict-feedback systems 33) 1: z 1 = x 1 x 1r : x r = [ k 1 z 1 f 1 + ẋ 1r ] L 1 G f1 1 G f1 ) ż 1 + k 1 z 1 ) 3: τ ω r + ω r = x r, ω r ) = x r ) 4: z = x ω r 5: for i = 3 to n do 6: x ir = [ k i 1 z i 1 f i 1 + x i 1)r ω i 1)r τ i 1 ] L 1 G fi 1 G fi ) ż i 1 + k i 1 z i 1 ) 7: τ i ω ir + ω ir = x ir, ω ir ) = x ir ) 8: z i = x i ω ir 9: i = i + 1 1: end for 11: u = [ k n z n f n + xnr ωnr τ n ] L 1 G fn 1 G fn ) ż n +k n z n ) The following theorem shows the stability control performance of the closed-loop system. Theorem Stability). Consider the nth-order strictfeedback system 33) the Algorithm 1, with a series of virtual controls x r, x 3r,..., x nr the control input u. Assume d i, i = 1,..., n are bounded the control parameters satisfy that k i > 1, k n > 1, < 1, i =,..., n 1, then the closed-loop system is stable, i.e., the system tracking error x 1 x 1r is bounded the bound is adjustable by control parameters. Proof: By using the Algorithm 1, the resulting closedloop system is as follows ż i = k i z i + u di û di ) 38) ż n = k n z n + u dn û dn ) 39) where k i, k n >, u di = z i+1 + y i+1 + d i, i = 1,,..., n 1 u dn = d n. The filtered errors y i = ω ir x ir from 37) will result in the following filtered error dynamics ẏ i+1 = 1 y i+1 ẋ i+1)r i = 1,..., n 1 4) By differentiating the virtual controls 4), there is ẏ + 1 y ξ 41) τ Furthermore, ẏi y i+1 ξ i+1 4) where ξ z 1, y, x 1r, ẋ 1r, ẍ 1r ) ) = k 1 ż 1 f 1 x 1 ẋ 1 + ẍ 1r t L 1 Gfi 1 G fi ż 1 + k 1 z 1 ) ξ i+1 z 1,..., z i+1, y,..., y i+1, x 1r, ẋ 1r, ẍ 1r ) = k i ż i i f i k=1 x k ẋ k + 1 τ i ẋ ir ω ir ) t L 1 Gfi 1 G fi ) ż i + k i z i ) i = 1,..., n 1 are continuous functions. Therefore, ẏ i+1 y i+1 y i+1 + y i+1 ξ i+1 y i+1 + y i ξ i+1 43) Consider the Lyapunov function cidates V zi = 1 z i i = 1,..., n 44) V yi+1) = 1 y i+1 i = 1,..., n 1 45) Using the Young s inequality 43) results in V zi = z i ż i = k i zi + z i u di û di ) k i zi + z i + u di û di ) 46) V yi+1) = y i+1 ẏ i+1 y i+1 + y i ξ i+1 47) Define the composite Lyapunov function cidate V = n V zi + V yi+1) 48) 4746

5 By differentiating V respect to t results in n V = V zi + V yi+1) [ ] n k i zi + z i + u di û di ) + y i+1 + yi ) τ i+1 4 ξ i+1 n k i + 1 ) zi + 1 ) + 1 yi+1 + n 1 g f ) u di ) + Consider the compact sets Ω pi = Ω r = { x 1r, ẋ 1r, ẍ 1r ) R 3} { z 1,..., z i, y,..., y i ) R i 1 } i i 1 zk + yk+1 p k=1 k=1 1 4 ξ i+1 49) where p >. Therefore, ξ i+1 z 1,..., z i+1, y,..., y i+1, x 1r, ẋ 1r, ẍ 1r ) has a maximum, say M i+1 on Ω r Ω pi+1), i = 1,..., n 1. Furthermore, since the external disturbances, d i, are assumed to be bounded u di = z i+1 + y i+1 + d i, i = 1,,..., n 1 u dn = d n. Thus, for each 1 g f ) u di there exists a bound say N i, i = 1,,..., n. If the control parameters are fixed to satisfy k i > 1, k n > 1 < 1, i = 1,...,, V + 1 n k i 1 ) zi n Ni M i ) y i+1 ρv + c 5) { } where ρ = min k i 1, k n 1, 1 1, i = 1,..., n 1 > c = 1 n N i M i+1 >. Let ρ > c p, then V < on V = p. Thus the set Ω p = { z 1,..., z n, y,..., y n ) R V p } is an invariant set, i.e., if V ) + c ρ p then V t) p for all t. Solving the inequality 5) gives V t) V ) e ρt + c 1 e ρt ) t 51) ρ thus, V t) is bounded by c ρ while t all signals of the closed-loop system are bounded, i.e., z 1,..., z n, y,..., y n are uniformly ultimately bounded. By increasing parameters k 1,..., k n, or decreasing τ,..., τ n, the quantity c ρ can be adjusted to be arbitrarily small. Then the tracking error z 1 = x 1 x 1r can be made arbitrarily small. This completes the proof IV. EXPERIMENTAL VALIDATION In order to illustrate the performance of the proposed method, an experimental study is carried out on a rotary servo system, as shown in Fig.. The servo system can be modeled Figure. DC servo motor by the following second-order system ẋ 1 = x + d 1 5) ẋ = 3x + 8u + d 53) where x 1, x represent the position velocity, respectively; u is the control input d 1, d are unmodeled dynamics or disturbances appearing in the system. It is reasonable to assume these uncertainties to be bounded in the real physical system. The control objective is to make the position x 1 track a step change. The set-point is chosen as 4 degree, which is associated to the sensed voltage 1.14V. To overcome the jump point at t =, a reference model is selected as ẋ 1r = 5x 1r , which makes the reference signal x 1r become a smooth continuous function. To facilitate the UDE design, the time constant in the first-order G f1 s) = G f s) = 1 τs+1 is chosen as τ =.5s the DSC time constant is chosen as τ =.1s. In addition, the control parameters are chosen as k 1 = 75 k = 5. The performance is shown in Fig. 3 a) - c). It can be seen that the servo arm successfully tracks the set point after 1 second the steady state error is achieved at.5%. V. CONCLUSION The UDE-based dynamic surface control has been developed for strict-feedback systems with mismatched external disturbances. Taking advantages of the backstepping approach which could hle the mismatched disturbances), the DSC method which could avoid the calculation of derivatives of the virtual controls), this paper introduced the UDE method for the uncertainty compensation. Inheriting from the conventional UDE method, the developed control approach could achieve the very good tracking performance in real-time is easily to be implemented. The stability analysis of the closed-loop system was also presented. Finally, the performance of the proposed method was validated by the numerical simulation experimental studies. REFERENCES [1] M. Krstić, I. Kanellakopoulos, P. V. Kokotović, Nonlinear Adaptive Control Design. New York: Wiley, 1995.

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