UDE-based Robust Control for Nonlinear Systems with Mismatched Uncertainties and Input Saturation

Transcription

1 UDEbased Robust Control for Nonlinear Systems with Mismatched Uncertainties and Input Saturation DAI Jiguo, REN Beibei, ZHONG QingChang. Department of Mechanical Engineering, Texas Tech University, Lubbock, TX, 795, USA Department of of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, IL, 666, USA Abstract: In this paper, the uncertainty and disturbance estimator (UDE)based control is adopted for a class of nonlinear systems with mismatched uncertainties and input saturation. By integrating with the backstepping approach, the robustness for the mismatched uncertainties is guaranteed by the recursive structure of backstepping. The disturbancelike terms which consist of the unknown dynamics, disturbances and the system intermediate tracking errors are estimated by lowpass filters with the knowledge of spectrum information. An auxiliary system is constructed to deal with the input saturation while only the system order information is known. Stability analysis of the closedloop system shows that the uniformly ultimate boundedness is guaranteed. A numerical example is provided to illustrate the effectiveness of the proposed method. Key Words: Uncertainty and disturbance estimator, Mismatched uncertainties, Input saturation Introduction With the increasing demand of precise control in industry, alleviating the effect of uncertainties in the nonlinear systems has drawn a lot of attention in both theoretic and practical aspects among control community. A number of techniques to handle the uncertainties have been proposed in the past decades [, ]. Adaptive control which adjusts the control parameters is capable of handling the parametric uncertainties []. With the basis of approximation theorem in a compact set, nonparametric uncertainties like the unknown dynamics, can be approximated by universal approximators e.g, neural networks (NNs) or fuzzy logic (FL) []. By using the high gain feedback to force the system to slide along a sliding mode surface, sliding mode control is another effective way to handle the uncertainties [3]. Besides above methods, it is well known that once the estimation of uncertainties is available, the uncertainties can be compensated directly. This idea results in a large class of uncertainties compensation methods, e.g, the active disturbance rejection control (ADRC) [], the disturbance observer (DOB) [5], the equivalent input disturbance (EID) [6], the extended state observer (ESO) [7], the uncertainty and disturbance estimator (UDE)based control [8]. Both unknown dynamics and external disturbances are regarded as the uncertainties in the system. If the uncertainties are estimated in real time then they can be compensated effectively. Hence, the focal point is how the uncertainties can be estimated. In the framework of ADRC and ESO, the uncertainties are formulated as an augmented state and estimated by a state observer. DOB is another way which is based on the inverse of the nominal transfer function of the plant. EID regards the uncertainties as an unknown disturbance in the input, then a state observer is used to estimate it. Based on the observation of uncertainties, UDEbased control adopts a lowpass filter to reconstruct an estimation of the uncertainties, then the control action is modified by this estimation. However, most above methods require the uncertainties satisfy the socalled matching condition which implies that the uncertainties act via the same channel as the control input, as discussed in [9, ] for ESO and DOB. In addition, the UDEbased control requires a structure constraint [8]. Unfortunately, in many applications, the uncertainties will affect the states directly rather than through the input channel. Therefore, solving the mismatched uncertainties has more general and wide applicability. To tackle with the mismatched uncertainties, some extension results for above methods have been presented. In [9], a generalized version of ESO is proposed for general systems with mismatched uncertainties by appropriately developing a disturbance compensation gain. The similar technique is also utilized with DOB to handle the mismatched uncertainties in []. In [], DOB combining with sliding mode control solves the mismatched uncertainties. Another effective technique to handle the mismatched uncertainties is the adaptive backstepping approach []. With the recursive structure, adaptive backstepping is capable for both matched and mismatched uncertainties. Though UDEbased control is very suitable for handling uncertainties and has received more and more attention [ ], all the existing UDE related works only handle the matched uncertainties due to the structural constraint [8]. This paper will utilize the recursive structure of backstepping approach to extend the UDEbased control to systems with mismatched uncertainties. Another issue that should be considered in the real application is the input saturation since actuators are usually limited by the physical characteristics or under some safety considerations [5]. When a controller is developed without considering the input saturation, it will degrade the performance of the system and even result in the instability [6]. The antiwindup control is a common technique to deal with the input saturation for linear systems [7]. A saturation compensator is designed to remove the effect of input saturation by using the difference between the actual input and controller output. For nonlinear systems, auxiliary system [8, 9], command filter [, ], approximationbased methods [, 3] are proposed to deal with the input saturation. To cope with the input saturation as well as the uncertainties, ADRC and antiwindup are combined in []; DOB

2 and NNs are integrated in [3]; ESO and sliding mode are merged in [5]. To the best of our knowledge, there are few works related to the UDEbased control considering the input saturation. This paper will integrate an auxiliary system with the UDE to tackle the input saturation. The objective of this paper is to develop the UDEbased control for systems with mismatched uncertainties and input saturation. The main contributions of this paper are summarized as follows: ) The backstepping technique is employed to extend the UDEbased method to cope with the mismatched uncertainties; ) The reference model is utilized in the design process which relaxes the derivation of virtual controls and avoids the complexity explosion in the conventional backstepping approach; 3) The disturbancelike terms, which consist of the unknown dynamics, disturbances and the system intermediate tracking errors, are estimated and compensated by the UDEbased control only with the knowledge of their spectrum information; ) An auxiliary system is constructed to handle the input saturation and only the system order is required to be known. The remainder of this paper is organized as follows: In Section, the problem formulation is presented. In Section 3, the UDEbased backstepping design with an auxiliary system is developed. Then the stability of the closedloop system is investigated in Section. Finally, a numerical example and some concluding remarks are provided in Section 5 and Section 6, respectively. Problem Formulation and Preliminaries Consider the following nthorder singleinputsingleoutput (SISO) nonlinear system ẋ i = f i ( x i ) + x i+ + d i, i =,, n ẋ n = f n ( x n ) + sat(u) + d n () y = x, x n () = x where x i R, i =,, n, x i = [x,, x i ] T are the system states and state vectors, y R is the output, u R is the controller output and sat(u) R is the actual input, f i ( x i ) are the system dynamics which are continuous and d i R are the external disturbances. x is the initial value and sat( ) is the saturation function, sat(u) = u, ρ max, ρ min, ρ min u ρ max u > ρ max u < ρ min () where ρ max and ρ min are upper and lower bounds of u. The control objective is to make the system output y to asymptotically track the desired trajectory y d in the presence of unknown dynamics, disturbance and input saturation. Define e = y y d be the tracking error then the objective is lim t e. Assumption. [] The system () is inputtostate stable (ISS). Remark. [] The Assumption is reasonable, because an unstable system can not be stabilized in the presence of input saturation. Assumption 3. The system dynamics f i (), i =,, n are continuous and can be completely or partially unknown. In this paper, they are all regarded as unknown dynamics. Assumption. The external disturbances d i, i =,, n are bounded, but their bounds are not necessary to be known. 3 Controller Design The basic idea is integrating the UDEbased control with the recursive structure of backstepping method to handle the system uncertainties and disturbances at each step. In order to suppress the effect of the input saturation, the following auxiliary system will be applied into the controller design [9] λ i = c i λ i + λ i+, i =,, n λ n = c n λ n + u (3) λ i () = where λ i, i =,, n are auxiliary states and constants c i, i =,, n are chosen to make the polynomial c s n + c s n ++c n s+c n be Hurwitz. Let u = sat(u) u denotes the difference between the actual input and the controller output. Then the auxiliary signals λ i can be represented as filtered versions of u and remain zero while no saturation occurs, and nonzero while saturation occurs. The following change of coordinates are defined, z = x y d λ z i = x i β i λ i, i =,, n () = x i y i α i λ i where z,, z n are intermediate system errors and β,, β n are reference signals generated by the following firstorder reference models through the virtual controls α,, α n, β i = a i β i + b i α i, i =,, n (5) β i () = α i () where a i, b i are design parameters. Furthermore, β i can be regarded as a filtered version of the virtual control α i and y i are defined as y i = β i α i, i =,, n. Usually, taking the derivative of the virtual controls in the backstepping approach will result in the complexity explosion, but with the help of the reference models inherited from UDE, the complexity explosion can be avoided. The reference dynamics (5) are also in the same spirit of dynamic surface control [6]. The control design consists of n steps: Step Taking the derivative of the first equation in () results in ż = ẋ ẏ d λ = f + d + x ẏ d + c λ λ = f + d + z + β + λ ẏ d + c λ λ = f + d + z + y + α ẏ d + c λ = + α ẏ d + c λ (6) where = f + d + z + y is regarded as the lumped disturbancelike term. If the virtual control is chosen as α = κ z + ẏ d c λ, κ > then ż = κ z which is asymptotically stable. However, is totally unknown and should be estimated. According to the idea of the UDEbased method [8], the virtual control consists of two parts, α = α l + α d (7)

3 where the uncertainty compensation term α d will be designed later and the linear feedback term α l is chosen as such that From (9), it has α l = κ z + ẏ d c λ (8) ż = κ z + α d + (9) = ż + κ z α d () The information of the disturbancelike term can be estimated by using a proper stable lowpass filtering with a suitable bandwidth, ˆ = g f [ż + κ z α d ] () where g f is a lowpass filter with the unity gain and zero phase shift in the spectrum of and is the convolution operator. Hence, the uncertainty compensation term α d can be designed as α d = ˆ () Substituting () into (), solving for α d yields α d = L Gf [ż + κ z ] (3) G f where G f is the Laplace transform of g f and L is the inverse Laplace operator. Hence, the virtual control has the form of α = κ z + ẏ d c λ L Gf [ż + κ z ] () G f Substituting (7), (8), () into (6), the error dynamics is obtained as ż = κ z + where = ˆ. Step i (i =,, n ) Taking the derivative of the ith equation in () and considering (3), (5), ż i = ẋ i β i λ i = f i + d i + x i+ β i + c i λ i + λ i+ = f i + d i + z i+ + β i+ + λ i+ β i + c i λ i λ i+ = f i + d i + z i+ + y i+ + α i+ β i + c i λ i = i + α i+ + a i β i b i α i + c i λ i (5) where i = f i + d i + z i+ + y i+ is regarded as the lumped disturbancelike term. The virtual control consists of α i+ = α (i+)l + α (i+)d with the linear feedback term α (i+)l = κ i z i a i β i + b i α i c i λ i. (6) Similar to Step, the uncertainty compensation term is designed as α (i+)d = ˆ i = L Gfi [ż i + κ i z i ] G fi Then the virtual control is obtained as α i+ = κ i z i a i β i + b i α i c i λ i L Gfi [ż i + κ i z i ] (7) G fi The error dynamics can be obtained as ż i = κ i z i + i (8) where i = i ˆ i. Step n Taking the derivative of the nth equation in () and considering (3), (5), there is ż n = ẋ n β n λ n = ẋ n β n + c n λ n (sat(u) u) = f n + d n + sat(u) β n + c n λ n (sat(u) u) = n + u + a n β n b n α n + c n λ n (9) where n = f n +d n is regarded as the lumped disturbancelike term. Thus the control input consists of u = u l + u d with the linear feedback term u l = κ n z n a n β n + b n α n c n λ n () and the uncertainty compensation term u d = ˆ n = L Gfn [ż n + κ n z n ] () G fn Then the final control input is obtained as u = κ n z n a n β n + b n α n c n λ n L Gfn [ż n + κ n z n ] () G fn Furthermore, the error dynamics is obtained as where n = n ˆ n. Stability Analysis ż n = κ n z n + n (3) The proposed control structure is shown in Fig.. The following theorem shows the stability and control performance of the closedloop system. Theorem 5. Consider the closedloop system which consists of the nthorder system (), the auxiliary system (3), the reference dynamics (5), the virtual controls () (7) and the control input (). Define the compact sets as x = (x,, x n ) R n, Ω x = { x : x p }, z = (z,, z n ) R n,ω z := { z : z p } and ȳ = (y,, y n ) R n, Ω y := {ȳ : ȳ p 3 }, λ = (λ,, λ n ) R n, Ω λ = { λ } : λ p, p, p, p 3, p are predefined arbitrary positive constants. If the following conditions are satisfied, C) (y d, ẏ d ) N y, N y is a compact set in R. C) (d,, d n ) N d, N d is a compact set in R n. C3) c >, c i >, i =,, n, and c n > 3. C) κ i >, a i >, i =,, n. then i) All signals in the closedloop system are bounded. ii) The system tracking error z can be adjusted within an arbitrary small compact set by changing the control parameters κ,, κ n, c,, c n, a, b,, a n, b n appropriately.

4 l yd y d z + x UDEbased controller a l b z UDEbased b = ab+ ba b a + 3 controller x a n l b = a b + b a n n n n n l ln b n b n + x n l = c l + l l = c l +Du z n n n n UDEbased controller Du u + sat() sat( u) d d x n dn x = f + x + d x = f + sat( u) + d n n n x x y Fig. : The proposed control structure. Proof. By substituting the virtual controls (), (7) and the control input () into the original system () results in the following closedloop system ż i = κ i z i + i, i =,, n ż n = κ n z n + n () where i = i ˆ i and i = f i + d i + z i+ + y i+, i =,, n, n = f n + d n. Using the Young s inequality, there is z i ż i = κ i z i + z i i κ i zi + z i + i = κ i zi + i (5) where κ i = κ i, and condition C guarantees that κ i >. Furthermore, consider the following Lyapunov function candidate V z = zi. (6) Taking the derivative of both sides of the above equation leads to V z = z i ż i κ i zi + n= c z V z + i i (7) where c z = min{ κ i, i =,, n}, and the functions i = (δ g f ) i = (δ g f ) (f i +d i +z i+ +y i+ ) are continuous functions of x,, x i, z,, z i, y,, y i, d,, d i, where δ is the Dirac delta function. According to (5), there is thus ẏ i = a i y i + (b i a i )α i α i (8) ẏ i + a i y i ξ i (y d, ẏ d, ÿ d, z, z i, y,, y i ) (9) where ξ i (y d, ẏ d, ÿ d, z, z i, y,, y i, λ,, λ i ) = (b i a i )α i α i is a continuous function of variables y d, ẏ d, ÿ d, z,, z i, y,, y i, λ,, λ i. Furthermore, ẏ i y i a i y i + y i ξ i a i y i + y i + ξ i = ā i y i + ξ i (3) where ā i = a i, and condition C guarantees that ā i >. Consider the Lyapunov function candidate thus, V y = V y = y i ẏ i yi (3) ā i yi + ξi c y V y + ξi (3) where c y = min{ā i, i =,, n}. From (3), while i =,, n, there is While i = n, λ i λi = c i λ i + λ i λ i+ c i λ i + λ i + λ i+ λ n λn = c n λ n + λ n u Consider the Lyapunov function candidate (33) c n λ n + λ n + u (3) V λ = λ i (35)

5 Taking its derivative results in V λ = n c i λ i + λ i λ i+ + λ n u c i λ i + λ n + λ i + 3 λ n + u c i λ i + u c λ V λ + u (36) where c = c, c i = c i, i =,, n, c n = c n 3 and c λ = min{ c i, i =,, n}. The condition C3 guarantees that c i >. Let V = V z +V y +V λ be the composite Lyapunov function candidate, thus V = V z + V y + V λ c z V z c y V y c λ V λ + i + ξi + u (37) Under the conditions C and C, firstly, consider the compact set Ω = N d Ω x Ω z Ω y. The functions i are continuous functions inside Ω, thus there exists a maximum value for n i, say M. Next, consider the compact set Ω = N y Ω z Ω y Ω λ. Since the functions ξ i are continuous functions inside Ω, there exists a maximum value for n ξ i, say M. Similarly, the function u is also a continuous function in Ω, and it must have a maximum value say M 3. Therefore, (37) can be rewritten as V µv + ζ (38) where µ = min{ c z, c y,, c λ } > and ζ = M +M +M >. Solving (38) gives V (t) V ()e µt + ζ µ ( e µt ) (39) Let p = p + p 3 + p, if the control parameters, κ,, κ n, c,, c n, a, b,, a n, b n, are chosen to satisfy that p > ζ µ and from (38) there is V < on V = p. Furthermore, if the initial condition satisfies that V () p ζ µ, there is V (t) p for all t. As shown in Fig., the compact set Ω = Ω x Ω z Ω y is an invariant set. It can be concluded W V mv + z z p m V () V < Fig. : The invariant set p that: (i) the signals x,, x n, z,.., z n, y,, y n, λ,, λ n are bounded. As t, V (t) < ζ µ, then z = x y d λ < ζ/µ; (ii) by adjusting the control parameters κ,, κ n, c,, c n, a, b,, a n, b n, we can increase µ and decrease ζ to make ζ/µ, such that z can be achieved and z can be bounded in an arbitrarily small compact set. Furthermore, from (36), it is obvious that λ can be guaranteed by tuning the control parameters appropriately. Thus lim t e = lim t (x y d ) can be achieved. 5 Simulation Results To illustrate the proposed method, the following example is considered, ẋ = x x + x + d ẋ = sat(u) x e x + d y = x () where d =. sin(x ) and d =. cos(x ). The reference signal is given as y d = sin t. The input saturation is described by.8, u >.8 sat(u) = u, u.8 (), u < The sampling time in the simulation is.s and the initial conditions of the states are chosen as x () =.5, x () =.. The lowpass filters G f = G f =.5s+ are used in the UDEbased method. Other parameters are chosen as c = c = 5 and a = b =. The results are shown in Fig. 3. As expected, the tracking performance is shown in Figs. 3 (a) and (d). It can be seen that the output signal can successfully track the reference signal very well. Figs. 3 (b) and (e) demonstrate the controller output u and the actual input sat(u) respectively, where the actual input sat(u) is saturated when the controller output u exceeds the limitation. Figs. 3 (c) and (f) illustrate that the uncertainties, are estimated accurately which results in the good performance of the UDEbased control in handling the mismatched uncertainties as well as the input saturation. 6 Conclusions In this paper, an UDEbased backstepping control method was developed for nonlinear systems with mismatched uncertainties and input saturation. The backstepping approach was integrated with the UDE to deal with the mismatched uncertainties. An auxiliary system was constructed to handle the input saturation. In the development of the UDEbased control, only the spectrum of uncertainties and the system order information are required. Stability analysis has shown that all signals in the closedloop system are bounded. Furthermore, a simulation example verified the effectiveness of the proposed method. References [] M. Krstic, P. V. Kokotovic, and I. Kanellakopoulos, Nonlinear and adaptive control design. John Wiley & Sons, Inc., 995.

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