Bounded UDE-based Controller for Systems with Input Constraints

Transcription

1 Bonded UDE-ased Controller for Systems with Inpt Constraints Yeqin Wang, Beiei Ren, and Qing-Chang Zhong Astract The ncertainty and distrance estimator (UDE- ased control is a rost control method, which is proposed as a replacement of the time-delay controller (TDC. With the filter design in the UDE-ased controller, the challenging prolem of designing a rost controller is converted into designing a filter. In this paper, a onded UDE-ased control is developed for systems with inpt constraints. The onded controller otpt is achieved throgh nonlinear Lyapnov analysis, and an additional time-varying variale is introdced into the error dynamics to natrally avoid the integrator windp. The ondedness design is emedded into the existing UDE framework to form a onded UDE-ased controller withot integrator windp via a simple strctre and clear gidelines of parameter selections. Both theoretical analysis and simlation stdies are provided to validate the proposed design. Index Terms Uncertainty and distrance estimator (UDE- ased controller, inpt constraint, anti-windp. I. INTRODUCTION A rost control method, the ncertainty and distrance estimator (UDE-ased controller, was proposed in [] to handle ncertainties and distrances for linear timeinvariant (LTI systems as a replacement of the time-delay controller (TDC []. Compared to the TDC [], the UDEased controller does not need to measre the derivative of the states, and no oscillations in control signal []. In the UDE design, a filter is adopted to estimate and compensate ncertainties and distrances, then the challenging prolem of designing a rost controller is converted into the design of a filter. In recent years, the UDE-ased control demonstrates excellent performances in road practical applications, e.g., solar system control [3], crrent control for permanentmagnet synchronos motor (PMSM drives [4], power electronics control [5] [8], qadrotors [9], and root maniplator tracking []. The idea of UDE is also extended to the sliding-mode control [], []. Some frther theoretical work aot UDE are condcted in [3] [6]. A two-degreeof-freedom natre of the UDE-ased controller is disclosed in [3]. A rogh first-order pls time delay (FOPDT model is introdced into the UDE-ased controller in [4] to handle apparent lag and time delay. The asymptotic reference tracking and distrance rejection of the UDE-ased controller is achieved in [5] ased on the internal model principle. The tradeoff etween the tracking and distrance rejection nder Yeqin Wang is with the National Wind Institte, Texas Tech University, Lock, TX 7949-, USA. ( yeqin.wang@tt.ed. Beiei Ren is with the Department of Mechanical Engineering, Texas Tech University, Lock, TX 7949-, USA. ( eiei.ren@tt.ed. Qing-Chang Zhong is with the Department of Electrical and Compter Engineering, Illinois Institte of Technology, Chicago, IL 666, USA. ( zhongqc@ieee.org. finite andwidth constraints in the UDE-ased controller is investigated in [6]. The constraint of the system inpt or controller otpt is a common prolem in practice, de to the physical limitation of the actators or the staility of the system reqirement. The UDE-ased controller faces the same challenges. However, it is not an easy task to design the ondedness for the UDE-ased controller, as it incldes the integral terms to eliminate tracking error. If a satration nit is simply applied to the UDE-ased controller, this often leads to the instaility de to the prolem of the integrator windp [7], [8]. To overcome this isse, the anti-windp designs [8] sally are adopted for the controllers with integral property, t this still can no longer garantee system staility in the original form or reqire additional knowledge of the system strctre and parameters [7], [9]. Recently, a onded integral controller (BIC is proposed in [7] to handle inpt constraint with integral control, which atonomosly provides the ondedness of the controller otpt withot any switches. The BIC is frther applied to many applications, e.g. the onded voltage and freqency of virtal synchronos machines [9], onded droop controller [], and solar systems [3]. Thogh the BIC can handle oth inpt constraint and integrator windp simltaneosly, its rost performance is still limited y the single integral control. Moreover, the ondedinpt onded-otpt (BIBO staility of original system is reqired for the BIC. Motivated y the BIC in [7], a ondedness design is proposed in this paper for the conventional UDE-ased controller to deal with inpt constraints. The ondedness of the final controller otpt is developed throgh nonlinear Lyapnov analysis. With an additional time-varying variale introdced into the error dynamics, the dynamic integral gain is achieved. The integral property will converge to zero, when the controller otpt converges to its limits, which natrally avoids the integrator windp sject to the inpt constraint. The staility analysis of closed-loop system is investigated throgh Lyapnov method. Compared to the existing anti-windp designs with the axiliary systems [], [], the proposed ondedness design is emedded into the existing UDE strctre, and the whole controller ecomes a onded UDE-ased controller. Compared to the existing UDE-ased controller, only two additional design parameters are introdced in the ondedness design. This onded UDE-ased controller inherits the merits of the existing UDE method with a simple strctre and clear gidelines of parameter selections sject to inpt constraints, and the BIBO assmption in [7] is relaxed.

2 G f3 (s = TABLE I FILTER DESIGN FOR THE UDE-BASED CONTROLLER G f G f (s sg f (s G f (s G f (s = a + a s+a s a a G f (s = s + +a s+a s s(s +ω (s+a (s +a s+ω + a s + a (s+a s +ω a s+a a s+a a + a + a (a s ω s +ω II. PROBLEM FORMULATION A. Overview of the UDE-ased controller Consider a class of LTI systems ẋ = Ax + f(x + B(t + d(t ( where x(t = [x, x,..., x n ] R n is the system state, (t R is the system inpt, A R n n is the known system matrix, B R n is known control vector, f(x R n is nonlinear part or nknown dynamics, and d(t R n is the onded external distrance. f(x is assmed with f(x F x where denotes the Eclidean norm, F is a positive constant. Then, a stale reference model is selected as ẋ m (t = A m x m (t + B m c(t ( where x m (t R n is the reference state vector, c(t = [c, c,..., c r ] R r is a piecewise continos and niformly onded command for the reference model, A m R n n and B m R n r are selected to meet the desired specification. The ojective is to design the control law (t sch that the state x(t can asymptotically track its reference x m (t, where the tracking error satisfies the following error dynamics e x (t = x m (t x(t (3 ė x = (A m + Ke x (4 where K R n n is a constant error feedack gain matrix and (A m + K shold e Hrwitz. Comining (-(4, there is A m x m + B m c Ax B f(x d = (A m + Ke x. The control law (t is designed as B = A m x m + B m c Ax d (A m + Ke x (5 where the lmped term d = f(x + d denotes the nknown term f(x and distrance d(t. According to the system dynamics (, d can e written as d = f(x + d(t = ẋ Ax B. Following the UDE procedres in [], d can e otained as û d = L {G f (s} d = L {G f (s} (ẋ Ax B where is the convoltion operator and G f (s is the UDE filter with oth strictly proper stale manner and appropriate andwidth to cover the spectrm of d. L means inverse Laplace transformation. Replacing d with û d in (5, there is B = A m x m + B m c Ax (A m + Ke x L {G f (s} (ẋ Ax B. Then, the final UDE-ased control law is formlated as [ { } = B + Ax + L G f (s [A m x m + B m c (A m + Ke x ] { } ] L sgf (s x G f (s where B + = (B T B B T. The scheme of the conventional UDE-ased controller is shown in Fig.. B. Integral property in the UDE-ased controller In the UDE-ased controller (6, there are two terms involving the filter, G f (s and sg f G f (s. As pointed ot in [3], [5], the filter design plays a very important role to achieve the good performances of the UDE-ased controller, e.g., asymptotic reference tracking and distrance rejection. Based on the internal model principle [5], the low-pass filter with G f ( = is reqired to handle step distrance and step reference. Tale I lists three freqently-sed UDE filters, G f (s in [], [3], [3], G f (s in [5], and G f3 in [6], [5]. It is interesting to notice that the integral property is inclded in the G f (s term to provide good steady-state performances. However, the sg f (s G f (s term does not inclde the integral property de to the s term in the nmerator. Thogh the integral property in G f (s can handle step distrance and step reference, it might case the integrator windp, if the system inpt is sject to a constraint. The anti-windp designs [8] are commonly adopted to deal with this isse, however, system strctre and parameters are sally reqired to garantee the closed-loop system staility [7], [9]. Frthermore, anti-windp designs with axiliary systems [], [] will ecome very complex with the nmer of design parameters increasing, if the system order increases. Is it possile to have a simple onded design for the UDE-ased controller to handle the integrator windp and to garantee the closed-loop system staility? III. BOUNDED UDE-BASED CONTROLLER Inspired y the BIC in [7], a ondedness design is proposed for the UDE-ased controller to handle inpt (6

3 Reference Model ( Plant A Fig.. The scheme of the existing UDE-ased controller []. Reference Model Bondedness design ( (9 and ( Plant A Fig.. The scheme of the proposed onded UDE-ased controller. constraint withot integrator windp. Unlike the axiliary system design, this ondedness design is emedded into the existing UDE strctre to reslt in a onded UDEased controller. The staility of the closed-loop system is investigated. A. Design of the onded UDE-ased controller Instead of the error dynamics in eqation (4, a new error dynamics is designed as ė x = k (t(a m + Ke x (7 where < k (t is an additional time-varying variale to e determined y the following ondedness design. Following the same UDE design procedres in Section II, the new modified UDE-ased control law can e otained as [ { } n = B + Ax + L G f (s [A m x m + B m c k (A m + Ke x ] { } ] L sgf (s x. (8 G f (s Inspired y the BIC in [7], a ondedness design is added to (8 to reglate the final controller otpt (t inside a given range of (,, ( = k + k ( k = k k + k k k( n (9 + k k ( n ( where k > and k > are positive constants. k (t is frther introdced into new error dynamics (7. Lemma. Throgh the ondedness design in (9 and (, the final controller otpt (t is reglated within a given range, i.e. (,. Proof: Consider the following Lyapnov fnction candidate V (t = + k. Taking the time derivative of V (t along with ondedness design (9 and (, it yields V (t = + k k ( + k k k ( n ( k k + k + k k ( n = k ( ( = k + k + k = k V + k V. ( Then, solving ( gives V (t = = e kt V ( e kt V ( V ( + e kt ( ( V (. Throgh the initial design with k ( =, and ( =, i.e., V ( =, then V ( = V (t =, t. Conseqently, it always holds that + k =. So, (t is kept within a given range of (,. The scheme of the proposed onded UDE-ased controller is shown in Fig.. Compared to the existing UDEased controller [], shown in Fig., it can e seen that the ondedness design is well emedded into the existing UDE framework with a simple strctre. Note that if there are any

4 Fig. 3. An illstration of the onded controller otpt and the additional variale k with the proposed onded UDE-ased controller (9 and (. nmerical errors or parameter drifts in V (t, V (t will still converge to with t, according to eqation (, and the rate of convergence can e adjsted y the parameter k. In this way, with the introdction of the of the additional controller state k (t, the controller states, (t and k (t will start and always remain on the ellipse + k = (3 which indicates that (t is onded in the given range (,, and k (, ], no matter how n (t in (8 changes. In the steady-state, (t and k (t will e reglated to with zero tracking error. When = and k =, oth controller states (t and k (t will converge to an eqilirim point ( e, k e, as shown in Fig. 3. Remark. With V (t =, the ondedness design (9 and ( are redced to = k k ( n k = k k ( n. When the final controller otpt (t is not close to the imm vale and k (t does not converge to, (t will converge to n (t, and the rate of convergence can e adjsted y the parameter k. So, the parameter k shold e well designed and ig enogh to cover all the effective andwidths of the modified controller otpt n. When (tis close to the imm vale, i.e., (t ± {, the variale } k (t will converge to. Then, the term L G f (s [k (A m + Ke x ] with integral property in (8 will converge to zero as k (t. This means that the integral property in (8 slows down and converges to zero, when (t goes to the limits, which can natrally prevent the integrator windp prolem. Compared to the conventional UDE-ased controller [], this onded UDE-ased controller (8, (9 and ( only introdce two additional design parameters, k, and k, and oth of them have the clear selection gidelines. B. Staility analysis Theorem 3. Consider the closed-loop system { shown in Fig.. Given} any compact set Ω x = x(t x(t < q, q >, if the initial state x( is within this set Ω x, and the system inpt within the range (, is capale to stailize the system ( within the set Ω x, i.e., k (t does not converge to, the closed-loop system is stale in the sense of ondedness. Proof: As shown in Fig., the ondedness design (9 and ( and the plant can e comined together as a new plant (inside the dashed ox with the system inpt n (t, where oth x(t and k (t are system states. Then, the ondedness design can e treated as a distrance R n for the new plant as ẋ = Ax + f(x + B n (t + d(t + (4 where = B( n. And, the lmped term ecomes d = f(x + d(t +. Consider the following Lyapnov fnction candidate V x (t = x T x. Taking the derivative of V x (t along with the new plant (4 and the modified UDE-ased control law (8, there is V x (t = ẋ T x + x T ẋ = [ x T A T + f T (x + T n B T + d T + T ] x +x T [Ax + f(x + B n + d + ] = k x T (A T m + K T + A m + Kx + ( k ( x T ma T mx + x T A m x m k ( x T m K T x + x T Kx m + c T B T mx + x T B m c +L { G f (s} [ f T (xx + x T f(x ] +L { G f (s} [( d T + T x + x T (d + ] k λ (Q x + ( k A m x m x +k K x m x + B m c x +L { G f (s} (F x +L { G f (s} [ x ( d + ] [k λ (Q + F ] x + ζ x (5 where Q = A T m + K T + A m + K is negative semi-definite with the Hrwitz matrix (A m + K, λ (Q < is the imm eigenvale of Q, and ζ = ( k A m x m + k K x m + B m c + d +. Within the set Ω x, is onded, according to Remark. Therefore, ζ has an pper ond p >, which is a fnction of q. By applying the Yong s ineqality to (5, there is V x (t [ k λ (Q + F + ε ] x + ζ ε λ V x (t + p ε (6 where λ = [ min t (k λ (Q + F + ε ], and ε > is a tning coefficient to determine the size of λ. The proper error feedack gain matrix (A m + K always can e fond to flfill λ > p qε, if k (t does not converge to. Then, V x (t < when V x (t q. In other word, the set Ω x is an invariant set. Therefore, x(t < q for all t > if x( q. The closed-loop system is stale in the sense of ondedness.

5 It is worth noting that the closed-loop system still cold e stale when k (t converges to, if the plant ( is BIBO stale, according to [7]. C. Performance analysis For the new plant (4 and the modified UDE-ased control law (8, as illstrated in Fig., the performances of the tracking error will e analyzed. When the UDE filter is sed to estimate the ncertain term d, and d is replaced with û d for the modified UDE-ased control law (8, the actal error dynamics (7 ecomes ė x = k (t(a m + Ke x ũ d (7 where ũ d d û d is the estimation error of the lmped term d, and d = f(x + d(t +. The estimation error is ũ d = L { G f (s} d. (8 Consider the following Lyapnov fnction candidate Then, V e (t = e T x e x. V e (t = k e T x Qe x ũ T d e x e T x ũ d k λ (Q e x + ũ d e x (9 [k λ (Q + ] e x + ũ d V e (t + λ 4 ( where = [min t (k λ (Q + ], λ 4 = t ( ũ d >. Then, solving ( gives Therefore, V e (t V e (e λ3t + λ 4 ( e λ3t. ( e x e x ( e λ3t + λ 4 ( e λ3t. If k (t does not converge to zero, the error feedack gain matrix (A m +K can e well designed to flfill >. The term e x ( e λ3t will gradally decay and converge to zero, and the term λ4 ( e λ3t will gradally increase and converge to λ4, when t. Therefore, e x will finally λ converge to 4 λ. The vale of 4 can e adjsted throgh the design of the error feedack gain matrix (A m +K. If the filter G f (s is well designed [5] as a strictly proper stale filter with nity gain and appropriate andwidth to cover the spectrm of the lmped term d (t, the estimation error (8 will e close to zero, so as λ 4. Then, the tracking error e x will converge to zero. Note that the steady-state performances cannot e garanteed if k (t converges to, even thogh the closed-loop system is stale given the BIBO staility of the plant (. System state Tracking error E TABLE II CONTROL PARAMETERS Parameters Vales Parameters Vales a m, m k a m + k k G f (s s+ 4.5 Bonded UDE E E 3 E 4 x m x Time (s Time (s Controller otpt Time (s n System state 3 (a System states Tracking error ( Tracking error 5 Controller otpt UDE+Sat x m x Time (s Time (s (c Controller otpt Time (s Fig. 4. Simlation reslts with the onded UDE-ased controller (left colmn and with the conventional UDE-ased controller [] pls a satration nit (UDE+Sat (right colmn. IV. SIMULATION STUDIES In order to verify the proposed ondedness design for the UDE-ased controller, a nmerical simlation is provided. Consider the following system: ẋ = ax + F x + + d where a =, F =, and d is a step distrance with d =.5 applied at t = 6 s. The onded system inpt is reqired within the range of ( 4.5, 4.5. The reference model is chosen as ẋ m = a m + m c, where c(t is a mltiple step command with c = at t = s, c =.4 at t = s, and c = at t = 4 s. And the error dynamics is designed as ė = k (t(a m + ke. The control parameters for the proposed onded UDE-ased controller (8-( are shown in Tale II. How to choose parameters for the UDE filter and the error dynamics in the UDE-ased controller can refer to [3], [5]. In order to show the advantages of the proposed ondedness design, the conventional UDE-ased controller (6 [] y adding a satration nit is also provided for the comparison. The system responses are shown Fig.. Before t = s, oth the proposed onded UDE and the conventional UDE pls a satration nit have the same performances in reference tracking when the controller otpt is not constrained. The tracking errors are almost zero at steady-state. When the reference is changed to.4 at t = s, oth controllers cannot reach the reference target ecase of the inpt constraint. sat

6 (a Within time-domain Fig. 5. The illstration of the controller states and k with the proposed ondedness design. ( Within the k ellipse The tracking errors of oth controllers are aot.5 at steady-state in Fig. 4(a and (. From Fig. 4(c, it shows that the proposed onded UDE can garantee the inpt constraint withot any integrator windp, and n in eqation (8 stops increasing and converges to steady-state, ecase of the effects from k (t. However, the conventional UDE pls a satration nit sffers from the integrator windp, ecase of the integral property in the UDE-ased controller. After t = 4s, the onded UDE still can achieve reference tracking qickly, whereas the conventional UDE pls a satration nit cannot achieve reference tracking till aot t = 5 s de to the effects of the integrator windp. After t = 6 s, oth controllers can reject the distrance with similar performances. Therefore, the proposed onded UDE control design can handle the inpt constraint withot the integrator windp. The controller states (t and k (t always remain on the ellipse, which is clearly illstrated in Fig. 5, where the eqilirim points E, E, E 3, and E 4 represent the for steady-states of the system, at 5 s s, 5 s 4 s, 45 s 6 s, and 65 s 8 s respectively. V. CONCLUSION In this paper, a ondedness design has een emedded into the conventional UDE-ased controller to form a onded UDE-ased controller. An additional time-varying variale has een introdced into the design of error dynamics to handle the integrator windp isse. The proposed onded UDE-ased control has a simple strctre with the clear gidelines of parameter selections. Both theoretical analysis and simlation stdies have demonstrated the effectiveness of the proposed approach. REFERENCES [] Q.-C. Zhong and D. Rees, Control of ncertain LTI systems ased on an ncertainty and distrance estimator, ASME Trans. J. Dyn. Sys. Meas. Control, vol. 6, no. 4, pp. 95 9, Dec. 4. [] K. Yocef-Tomi and O. Ito, A time delay controller for systems with nknown dynamics, ASME Trans. J. Dyn. Sys. Meas. Control, vol., no., pp. 33 4, Mar. 99. [3] Y. Wang and B. Ren, Falt ride-throgh enhancement for grid-tied PV systems with rost control, IEEE Trans. Ind. Electron., vol. 65, no. 3, pp. 3 3, Mar 8. [4] J. Ren, Y. Ye, G. X, Q. Zhao, and M. Zh, Uncertainty-anddistrance-estimator-ased crrent control scheme for PMSM drives with a simple parameter tning algorithm, IEEE Trans. Power Electron., vol. 3, no. 7, pp , Jl. 7. [5] Y. Wang, B. Ren, and Q.-C. Zhong, Rost power flow control of grid-connected inverters, IEEE Trans. Ind. Electron., vol. 63, no., pp , Nov. 6. [6] Q.-C. Zhong, Y. Wang, and B. Ren, UDE-ased rost droop control of inverters in parallel operation, IEEE Trans. Ind. Electron., vol. 64, no. 9, pp , Sept. 7. [7] S. Y. Gadelovits, Q. C. Zhong, V. Kadirkamanathan, and A. Kperman, UDE-ased controller eqipped with a mlti-and-stop filter to improve the voltage qality of inverters, IEEE Trans. Ind. Electron., vol. 64, no. 9, pp , Sept. 7. [8] Y. Ye and Y. Xiong, UDE-ased crrent control strategy for LCCLtype grid-tied inverters, IEEE Trans. Ind. Electron., vol. 65, no. 5, pp , May 8. [9] R. Sanz, P. Garcia, Q.-C. Zhong, and P. Alertos, Rost control of qadrotors ased on an ncertainty and distrance estimator, ASME Trans. Jornal of Dyn. Sys. Meas. Con., vol. 38, no. 7, p. 76, 6. [] J. P. Kolhe, M. Shaheed, T. S. Chandar, and S. E. Taloe, Rost control of root maniplators ased on ncertainty and ditrance estimation, International Jornal of Rost and Nonlinear Control, vol. 3, no. Jan., pp. 4, 3. [] S. Talole and S. Phadke, Model following sliding mode control ased on ncertainty and distrance estimator, ASME Trans. J. Dyn. Sys. Meas. Control, vol. 3, pp. 5, 8. [] D. D. Dhadekar and B. M. Patre, UDE-ased decopled fll-order s- liding mode control for a class of ncertain nonlinear MIMO systems, Nonlinear Dynamics, vol. 88, no., pp , Apr. 7. [3] Q.-C. Zhong, A. Kperman, and R. Stoart, Design of UDE-ased controllers from their two-degree-of-freedom natre, International Jornal of Rost and Nonlinear Control, vol., pp , Nov.. [4] L. Sn, D. Li, Q. C. Zhong, and K. Y. Lee, Control of a class of indstrial processes with time delay ased on a modified ncertainty and distrance estimator, IEEE Trans. Ind. Electron., vol. 63, no., pp , Nov. 6. [5] B. Ren, Q.-C. Zhong, and J. Dai, Asymptotic reference tracking and distrance rejection of UDE-ased rost control, IEEE Trans. Ind. Electron., vol. 64, no. 4, pp , Apr. 7. [6] I. Aharon, D. Shmilovitz, and A. Kperman, Uncertainty and distrance estimator-ased controllers design nder finite control andwidth constraint, IEEE Trans. Ind. Electron., vol. 65, no., pp , Fe. 8. [7] G. C. Konstantopolos, Q.-C. Zhong, B. Ren, and M. Krstic, Bonded integral control of inpt-to-state practically stale non-linear systems to garantee closed-loop staility, IEEE Trans. Atom. Control, vol. 6, no., pp , Dec. 6. [8] S. Taroriech and M. Trner, Anti-windp design: an overview of some recent advances and open prolems, IET Contr. Theory Appl., vol. 3, no., pp. 9, Jan. 9. [9] Q.-C. Zhong, G. C. Konstantopolos, B. Ren, and M. Krstic, Improved synchronverters with onded freqency and voltage for smart grid integration, IEEE Trans. Smart Grid, vol. 9, no., pp , Mar. 8. [] G. C. Konstantopolos, Q.-C. Zhong, B. Ren, and M. Krstic, Bonded droop controller for parallel operation of inverters, Atomatica, vol. 53, pp. 3 38, Mar. 5. [] K. Esfandiari, F. Adollahi, and H. A. Talei, Adaptive control of ncertain nonaffine nonlinear systems with inpt satration sing neral networks, IEEE Trans. Neral Netw. Learn. Syst., vol. 6, no., pp. 3 3, Oct. 5. [] J. Zho and C. Wen, Adaptive Backstepping Control of Uncertain Systems: Nonsmooth Nonlinearities, Interactions or Time-Variations. Springer, 8. [3] B. Ren, Y. Wang, and Q.-C. Zhong, UDE-ased control of varialespeed wind trine systems, International Jornal of Control, vol. 9, no., pp. 36, Jan. 7.