Composite Nonlinear Feedback Control for Output Regulation Problem of Linear Systems with Input Saturation

Composite Nonlinear Feedback Control for Output Regulation Problem of Linear Systems with Input Saturation Bo Zhang Weiyao Lan Dan Wang Department of Automation, Xiamen University, Fujian, China 361005 (e-mail:xmu zb@126com) Department of Automation, Xiamen University, Fujian, China 361005 (e-mail: wylan@xmueducn) Marine Engineering College, Dalian Maritime University, Dalian, China 116026 (e-mail: dwangdl@gmailcom) Abstract: This paper addresses output regulation problem for general linear systems subject to input saturation by composite nonlinear feedback (CNF) control The CNF control law is constructed based on a switch control law which solves the output regulation problem by introducing a nonlinear feedback part The purpose of the nonlinear feedback part is to improve the transient performance of the output response of the closed-loop system It is shown that the nonlinear part of CNF control law does not change the solvability conditions of the output regulation problem The effectiveness of the CNF control law is illustrated with an example by comparing the CNF control law and the original switch control law Keywords: output regulation; input saturation; composite nonlinear feedback; transient performance 1 INTRODUCTION The output regulation problem for linear systems subject to input saturation was extensively studied in the literature Lin et al (1996) proposed the low-gain feedback technique to solve the output regulation problem for the linear systems with all eigenvalues of system matrixaare in the closed left half-plane In Hu and Lin (2000), a switch control law was constructed to solve the output regulation problem for general linear system with input saturation A bounded continuous feedback control law is designed for the output regulation problem in Hu and Lin (2004) These works on the output regulation problem are focused on the solvability of the problem However, the transient performance of the output response is seldom addressed in the literature In this paper, we are trying to improve the transient performance of the output regulation problem by using composite nonlinear feedback (CNF) control technique The CNF control was firstly proposed by Lin et al (1998) to improve the transient performance for the tracking problem of second order system with input saturation The essence of the CNF control is manipulating the damping ratio by nonlinear feedback, ie when the tracking error is large, the controller decrease the damping ratio to get quick response and when the tracking error is small the controller increase the damping ratio to inhabit the overshoot In the following years, Turner et al (2000) extended the CNF control to the multi-variable system Chen et al (2003) developed a CNF control to a more general class of systems with measurement feedback, and applied the CNF control technique to design an HDD servo system The The work is partially supported by National Nature Science Foundation of China (61074004,61074017), Natural Science Foundation of Fujian Province of China (2008J0033), Program for New Century Excellent Talents in University (NCET-09-0674), Program for Liaoning Excellent Talents in Universities under Grant 2009R06, and Program of 211 Innovation Engineering on Information in Xiamen University (2009-2011) extension to multi-variable systems is reported in He et al (2005) He et al (2007) and Lan et al (2006) extended the CNF control technique to a class of nonlinear systems Moreover, the CNF control is applied to design various servo systems, such as HDD servo systems, helicopter flight control systems, and position servo systems, please see Lan et al (2010a) Chen et al (2006) Li et al (2007), Cai et al (2008) and Cheng and Peng (2007) In this paper, we will introduce the CNF control technique into the output regulation problem for the linear systems with input saturation to improve the transient performance of the closed-loop system In Lan et al (2010b), a CNF control law is designed to solve the semi-global output regulation problem for the linear systems without anti-stable eigenvalues Based on the regulatable region and the switch control law proposed by Hu and Lin (2000), we will design a CNF control law to solve the output regulation problem for general linear systems The improvement of the transient performance of the CNF control law is shown by comparing the closed-loop performance under the CNF control law and the original switch control law The remainder of this paper is organized as follows Section 2 formulates the output regulation problem and presents some preliminaries of the regulatable regions of the output regulation problem In Section 3, a CNF control law is constructed to solve the output regulation problem Section 4 illustrates the effectiveness of the CNF control law by an example Finally, Section 5 concludes the paper with some remarks 2 PROBLEM STATEMENT AND PRELIMINARIES Consider a linear system subject to input saturation: Copyright by the International Federation of Automatic Control (IFAC) 1398

ẋ=ax+bsat(u)+pw e=cx+qw (1) Where x R n is the system state, u R m is the control input,w R q is the exogenous signal consisting of disturbance and/or reference, Pw is the disturbance of the system, Qw is the signal required to track So e R p is the error output The mappingsat( ) : R m R m is defined as: sat(u) = sat(u 1 ) sat(u 2 ) sat(u m ) (2) where sat(u i ) = sgn(u i )min{u imax, u i } with u imax being the saturation level of the ith input channel Without loss of generality, we assume that u imax = 1 in this paper LetY 0 R n R q be a compact set containing the origin, and define X 0 = {x 0 R n : (x 0,0) Y 0 } The output regulation problem for the system (1) is described as follows Output Regulation Problem by State Feedback: Find a state feedback control law u = φ(x,w) (3) with φ(0,0) = 0 such that the following requirements are satisfied, R1: (Internal Stability) Whenw = 0, the equilibrium point of ẋ = Ax+Bsat(φ(x,0)) is asymptotically stable with X 0 contained in its domain of attraction R2: (Output Regulation) For all (x(0),w(0)) Y 0, the state feedback control lawφ(x,w) results in lim t e(t) = 0 To solve the above described problem, the following assumptions are quite standard: A1: The pair (A,B) is stabilizable; A2: The eigenvalues of S are on the imaginary axis and are simple; A3: There exist matrixes Π and Γ that solve the regulator equations ΠS =AΠ+BΓ+P (4) 0=CΠ+Q (5) Because of the actuator saturation, the output regulation problem of the system (1) cannot be solved globally In fact, it is shown in Lin et al (1996) that the output regulation requirement R2 is satisfied only if sup Γe St w 0 < 1 t 0 Thus, we will consider the exosystem with the initial conditions in the following compact set W 0 = {w 0 R r : Γw(t) = Γe St w 0 γ} for some γ [0,1) The regulatable region, ie the maximal set of initial conditions(x 0,w 0 ) on which the output regulation problem is solvable, is discussed in Hu and Lin (2000) and Hu and Lin (2004) Specifically, let z = x Πw (6) the system (1) can be rewritten as: ż=az +Bsat(u) BΓw e=cz (7) It is clear that e(t) goes to zero asymptotically if z(t) goes to zero asymptotically The precise definition of the regulatable region is described as follows Definition 1(Hu and Lin (2004)): Given T > 0, a pair (z 0,w 0 ) R n W 0 is regulatable in time T if there exists an admissible controlu( ) 1, such that the response of(1) satisfies z(t) = 0; A pair (z 0,w 0 ) is regulatable if there exist a finite T > 0 and an admissible control u( ) such that z(t) = 0 The set of all (z 0,w 0 ) regulatable in time T is denoted as R g (T) And the set of all regulatable (z 0,w 0 ) is referred to as the regulatable region and is denoted as R g The set of all (z 0,w 0 ) for which there exists an admissible control u( ) such that the response of (1) satisfies lim t e(t) = 0 is referred to as the asymptotically regulatable region and is denoted asr α g The regulatable region R α g closely relate to the asymptotically null controllable region of the system v = Av +Bu, u 1 which is defined as: Definition 2(Hu and Lin (2004)): The asymptotically null controllable region region, denoted as C α, is the set of all v 0 that can be driven to the origin asymptotically by a admissible control The null controllable region at thet, denoted asc(t), is { T } C(T) = e Aτ Bu(τ)dτ : u 1 0 and the null controllable region, denoted as C is C = T (0, ) C(T) Remark 1 Without loss of generality, we assume that A = [ ] [ ] A1 0 B1, B = 0 A 2 B 2 where A 1 has all its eigenvalues in the closed left half-plane, and A 2 is anti-stable Let C 2 be the null controllable region for the unstable system v = A 2 v +B 2 u, u 1 V 2 is the solution of A 2 V 2 +V 2 S = B 2 Γ It is shown in Hu and Lin (2000) and Hu and Lin (2004) that the asymptotically regulatable region is given by R α g = R n1 R α 2g 1 u is referred to as an admissible control if it is measurable and u 1 1399

where R α 2g = {(z 2,w) R n2 W 0 : z 2 V 2 w C 2 } Please refer to Hu et al (2002) and Hu and Lin (2004) for the details on null controllable region and regulatable region for the linear systems with input saturation Remark 2 The solvability of the output regulation problem for the linear systems with input saturation has been investigated in the literature For example, the feedback control laws that solve the output regulation problem are explicitly constructed in Hu and Lin (2000) and Hu and Lin (2004) However, the transient performance of the regulation system has not been fully addressed In this paper, we will present a composite nonlinear feedback control law to improve the transient performance of the closed-loop system The CNF control law is constructed based on the state feedback control law given in Hu and Lin (2000) 3 CNF CONTROL LAW In this section, we will construct a CNF control law to solve the output regulation for linear systems subject to input saturation The CNF control law is designed based on the feedback control law proposed in Hu and Lin (2000) Let F R m R n be such that A + BF is stable The equilibrium point v = 0 of the closed-loop system v = Av +Bsat(Fv) has a domain of attraction S C a Let X > 0 be the positive define solution of the following Lyapunov inequality (A+BF) T X +X(A+BF) < 0 and c δ > 0 be such that Fv 1 for all v in the following set E := {v R n : v T Xv c δ } It is obvious that E S Assume that A4: There exists a matrix V satisfying AV +VS = BΓ (8) Then, for a given α (0,1), there exists a positive integer N such that for all w W 0 Define α N X 1 2 Vw 2 < 1 γ =: δ (9) D zw ={(z,w) R n W 0 : z Vw E}, D 1 zw ={(z,w) R n W 0 : z αvw αe} D k zw ={(z,w) R n W 0 : z α k Vw α k E} D N+1 zw ={(z,w) R n W 0 : z α N+1 Vw δe} The CNF control law is given by: u=g(z,w,α,n) = where f N+1 (z,w) if(z,w) Ω N+1 D N+1 zw f k (z,w,α) if(z,w) Ω k D k zw \ N+1 j=k+1 Dj zw f(z Vw) if(z,w) Ω S W 0 \ N+1 j=0 Dj zw (10) f N+1 (z,w)=γw+δsat((f +ρb T X)z/δ); f k (z,w,α)=(1 α k )Γw +α k sat((f +ρb T X)(z α k Vw)/α k ); f(z Vw)=(F +ρb T X)(z Vw); whereρis a non-positive function Remark 3 If ρ = 0, the CNF control law (10) is reduced to the feedback control law proposed in Hu and Lin (2000) The extra nonlinear term is introduced to improve the transient performance of the closed-loop system The effectiveness of the CNF control will be illustrated in next section It is clear that the selection of the nonlinear function ρ is a key step in the design of the CNF control law In this paper, we will let β 1 ρ = 1 exp( 1) (exp( e ) exp( 1)) e +β 2 for some β 1 > 0 and β 2 > 0 Please refer to Lan et al (2010a) for the details on the selection of the nonlinear functionρin the CNF control law Theorem 4 Let x 1/2 Vw 2 α 0 = max w W x 1/2 Vw 2 +1 Choose any α (α 0,1) and N is specified by (9), then the CNF control law (10) solves the output regulation problem for all(z 0,w 0 ) D zw Proof Follow the lines of Hu and Lin (2000), we just need to show the following to two facts Fact 1: For the system ż=az +Bsat(f k (z,w,α)) BΓw Dzw k is an invariant set Moreover, if k = 0,1,,N, for all (z 0,w 0 ) Dzw,lim k t (z(t) α k Vw(t)) = 0; ifk = N+1, for all (z 0,w 0 ) Dzw N+1,lim t (z(t)) = 0 Fact 2: For the system ż=az +Bsat(f(z Vw)) BΓw D zw is an invariant set and for all(z 0,w 0 ) D zw, lim t (z(t) Vw(t)) = 0 Consider the following system ż = Az +Bsat(f k (z,w,α)) BΓw; for k = 0,,N Let v k = z α k Vw the system can be rewritten as follows: v k = Av k +α k Bsat((Fv k +u n )/α k ) (11) 1400

whereu n = ρb T Xv k Denote f w = α k sat((fv k +u n )/α k ) Fv k we have: v k = (A+BF)v k +Bf w Consider the Lyapunov function of the system V(v k ) = v T kxv k ; wherex > 0 is the positive definite solution of (A+BF) T X +X(A+BF) = Q < 0 we have Denote B T Xv k = V(v k )= v T k Xv k +vkx T v k =((A+BF)v k +Bf w ) T Xv k +v T kx((a+bf)v k +Bf w ) =v T k(a+bf) T Xv k +f T wb T Xv k +v T kx(a+bf)v k +v k XBf w = v T kqv k +2v T kxbf w q 1 q 2 ;Fv k = q m p 1 p 2 ;f w = p m θ 1 θ 2 ; Without loss of generality, we assume that the first ith input channels are unsaturated; and the(i+1)th tojth input channels are saturated from upper bound; and the(j +1)th tomth input channels are saturated from lower bound, that is, Then we have, θ 1 θ 2 θ m = ρq 1 ρq i α k p i+1 α k p j α k p j+1 α k p m θ m V(v k ) = vkqv T k +2[q 1,q 2,q m ] θ m θ 1 θ 2 = v T kqv k +2(q 1 θ 1 +q 2 θ 2 + +q m θ m ) For(z,w) D k zw, Fv k α k, that is, p l α k (12) for l = 1,,m Forl = 1,,i, q l θ l = ρq 2 l 0 becauseρis a non-positive function Forl = i+1,,j, In this case, q l θ l = q l (α k p l ) (p l +ρq l )/α k > 1 Using (12), we have ρq l α k p l 0 which implies q l 0 We also can conclude from (12) that 0 θ l = α k p l 2α k So q l θ l < 0 Forl = j +1,,m, q l θ l = q l ( α k p l ) In this case, (p l +ρq l )/α k < 1 Using (12), we have ρq l α k p l 0 thus,q l 0 Using (12) again, we have It means Thus, Now consider the system 2α k θ l = α k p l 0 q l θ l < 0 V(v k ) v T kqv k < 0 (13) ż=az +Bsat(f N+1 (z,w)) BΓw =Az +Bsat(Γw+δsat((Fz +u n )/δ)) BΓw whereu n = ρb T Xz Because f N+1 (z,w) 1, ż = Az +δbsat((f z +u n )/δ) Obviously, it has the same form of equation (11) Let V(z) = z T Xz it is not difficult to show that V(z) z T Qz < 0 (14) The Fact 1 is concluded from (13) and (14) Finally, consider the closed-loop system ż=az +Bsat(f(z Vw)) BΓw Definev = z Vw, the system can be transformed into: v = Av +Bsat(Fv +u n ) where u n = ρb T Xv The equation above has same form of equation (11), the proof of Fact 2 can be completed by the same reasoning in the proof of Fact 1 Remark 5 As described in Hu and Lin (2000), Assumption A4 is not necessary for the solvability of the output regulation problem Since the introducing of the nonlinear feedback in the CNF control does not change the solvability conditions of the output regulation problem, Assumption A4 can also be removed here Suppose that the system (7) has the following form, ] [ ][ ] [ ] [ż1 A1 0 z1 B1 = + (sat(u) Γw) (15) ż2 0 A 2 z 2 B 2 where A 1 has all its eigenvalues in the closed left half-plane, and A 2 is anti-stable We know that there exists a function 1401

f 2 (z 2 ) 1 for all z 2 R n 2 such that the equilibrium point of the system z 2 = A 2 z 2 +B 2 f 2 (z 2 ) has a domain of attractione 2 Then by the Lemma 2 of Hu and Lin (2000), the system z 2 = A 2 z 2 +δb 2 f 2 (z 2 /δ) has a domain of attraction δe 2 By Teel (1996), there exists a control lawu = δsat(h(z)) that can stabilize the system ż = Az +δbsat(h(z)) in a domain of attractions δ = R n 1 E 2 For the subsystem: z 2 = A 2 z 2 +B 2 sat(u) B 2 Γw (16) there exitsv 2 such that A 2 V 2 +V 2 S = B 2 Γ By Theorem 4, we can construct a controller such that for any (z 20,w 0 ) satisfying z 20 V 2 w 0 E 2, u = g(z 2,w,α,N) can stabilize (16), ie lim t z 2 = 0 Hence z(t) will enter into the regions δ = R n 1 E 2 Final, the controller { g(z2,w,α,n) if z 2 E 2 \δe 2 u = Γw+δsat(h(z)) if z 2 E 2 can stabilize the system (15) 4 AN ILLUSTRATIVE EXAMPLE To demonstrate the improvement of the transient performance of the CNF control law, a tracking control problem is considered by comparing the CNF control law and its original switch control law Consider the following system with A= ẋ=ax+bsat(u)+pw e=cx+qw [ ] 0 1 2 3 C =[0 1];S = [ 1 ;B = ;P = 1] [ 0 1 1 0 [ ] 0 0 4 1 ] ;Q = [ 1 0] and u max = 1 Here we select a sine signal as the reference input Qw(t) = Ampsin(wt) with Amp = 045, w = 1 and Qw(0) = 02 The system matrixahas two unstable eigenvalues1and2 It is not difficult to verify that the above system satisfies assumptions A1-A4 Thus, we can design a CNF controller as described in Section 3 Firstly, solving the regulation equations yields: ΠS =AΠ+BΓ+P Π = 0=CΠ+Q [ ] [ 0 0 1 ;Γ = 1 0 0] Then, the equation AV +VS = BΓ can be solved immediately, which yields, System output and referance input 05 04 03 02 01 0 01 02 03 04 Switch controller CNF controller Reference signal 05 0 2 4 6 8 10 Time(sec) Fig 1 The profile of output responses tracking error e(t) 015 01 005 0 005 01 015 CNF controller Switch controller 02 0 2 4 6 8 10 Time(sec) Fig 2 The profile of tracking error [ ] 07 11 V = 01 07 Let F =[30000 210000]; such that the poles ofa+bf have a small damping ratio The CNF control law is given by (10), where ρ = β 1 exp( 1) (exp( e e +0005 ) exp( 1)) e When e is far from0, e +0005 closes to 1,ρis small and the effect of nonlinear part is very weak When e approaches e 0, e +0005 closes to0, the system matrix becomesa+bf βbb T X The parameter β is designed such that the poles of A+BF βbb T X are in the desired location to increase the damping ratio of the closed-loop system The simulation results with x 0 = [0;0] are shown in Fig 1 and Fig 2 Fig 1 shows comparison of the output response of the closed-loop system under the CNF control law and the switch control law The switch control law is obtained from the CNF control law by setting the nonlinear part of the CNF control to zero, ie, ρ = 0 The tracking error which we are most interested in is shown in Fig 2 From Fig 2 and Fig 1, we can see that the CNF control law can improve the transient performance significantly 1402

5 CONCLUSION In this paper, we presented a methodology to improve the transient performance of the output regulation problem for general linear system subject to input saturation The control law is designed based on a stabilizing switch controller We embedded a nonlinear feedback part in the stabilizing controller to improve the transient performance of the closed-loop system The embedded nonlinear control part does not change the solvability conditions of the original output regulation problem The effectiveness of the CNF control is illustrated by a tracking control example In the paper, only the state feedback control law is considered It is interesting to extend the our result to output feedback case REFERENCES Cai, G, Chen, BM, Peng, K, Lee, TH, and Dong, M (2008) Comprehensive modeling and control of the yaw channel of a uav helicopter IEEE Transactions on Industrial Electronics, 55(9), 3426 3434 Chen, BM, Lee, TH, Peng, K, and Venkataramanan, V (2003) Composite nonlinear feedback control for linear systems with input saturation: theory and an application IEEE Transactions On Automatic Control, 48(3), 427 439 Chen, BM, Lee, TH, Peng, K, and Venkataramanan, V (2006) Hard Disk Drive Servo Systems, 2nd Edn Springer, London Cheng, G and Peng, K (2007) Robust composite nonlinear feedback control with application to a servo positioning system IEEE Transactions on Industrial Electronics, 54(2), 11325 1140 He, Y, Chen, BM, and Lan, W (2007) On improving transient performance in tracking control for a class of nonlinear discrete-time systems with input saturation IEEE Transactions on Automatic Control, 52(7), 1307 1313 He, Y, Chen, BM, and Wu, C (2005) Composite nonlinear control with state and measurement feedback for general multivariable systems with input saturation Systems and Control Letters, 64, 455 569 Hu, T and Lin, Z (2000) Output regulation of general linear systems with saturating actuators In Proc39th IEEE Conf on Decision and Control, volume 4, 3242 3247 Sydney, Australia Hu, T and Lin, Z (2004) Output regulation of linear system with bounded continuous feedback IEEE Transactions On Automatic Control, 49(11), 1941 1953 Hu, T, Lin, Z, and Qiu, L (2002) An explicit discription of the null controllable regions of linear systems with saturating actuators Systems and Control Letters, 47(1), 65 78 Lan, W, Chen, BM, and He, Y (2006) On imporvement of transient performance in tracking control for a class of nonlinear systems with input saturation Systems and Control Letters, 55, 132 138 Lan, W, Thum, CK, and Chen, BM (2010a) A hard disk drive servo system design using composite nonlinear feedback control with optimal nonlinear gain tuning methods IEEE Transactions on Industrial Electronics, 57(5), 1735 1745 Lan, W, Zhou, XM, and Wang, D (2010b) Output regulation for linear systems with saturation by composite nonlinear feedback control In Proc 29th Chinese Control Conference, volume 1, 660 664 Beijing, China Li, Y, Venkataramanan, V, Guo, G, and Wang, Y (2007) Dynamic nonlinear control for fast seek-settling performance in hard disk drives IEEE Transactions on Industrial Electronics, 54(2), 951 962 Lin, Z, Pachter, M, and Banda, S (1998) Toward improvement of tracking performance-nonlinear feedback for linear systems Int J Control, 70(1), 1 11 Lin, Z, Stoorvogel, AA, and Saberi, A (1996) Output regulation for linear systems subject to input saturation Automatica, 32(1), 29 47 Teel, AR (1996) A nonlinear small gain theorem for the analysis of control systems IEEE Transactions On Automatic Control, 42, 1256 1270 Turner, MC, Postlethwaite, I, and Walker, DJ (2000) Nonlinear tracking control for multivariable constrained input linear systems Int J Control, 73, 1160 1172 1403