Output Feedback Control for a Class of Nonlinear Systems

International Journal of Automation and Computing 3 2006 25-22 Output Feedback Control for a Class of Nonlinear Systems Keylan Alimhan, Hiroshi Inaba Department of Information Sciences, Tokyo Denki University, Hatoyama-machi, Hiki-gun, Saitama 350-0394, Japan Abstract: This paper studies the global stabilization problem by an output controller for a family of uncertain nonlinear systems satisfying some relaxed triangular-type conditions and with dynamics which may not be exactly known Using a feedback domination design method, we explicitly construct a dynamic output compensator which globally stabilizes such an uncertain nonlinear system The usefulness of our result is illustrated with an example Keywords: Nonlinear system, global stabilization, output feedback Introduction The problem of controlling nonlinear systems by output feedback is one of the most important problems in the field of nonlinear control Unlike in the case of linear systems, the separation principle generally does not hold for general nonlinear systems [] Due to this, the problem has been more difficult and challenging for the systems and control community In recent years, many important results for the problem have been obtained However, as investigated in [], some extra growth conditions on the immeasurable states of a system are usually necessary for the global stabilization of nonlinear systems via output feedback Since this work, a great deal of subsequent research work has focused on the output feedback stabilization of nonlinear systems under various structural or growth conditions [2 8] For example, it was assumed that the nonlinear terms of a given system satisfy triangular conditions in [2] and [8], or some global Lipschitz-like condition in [7], etc In this paper, we consider essentially the same class of nonlinear systems as treated in [2,5 8] So far, it seems that one of most relaxed conditions imposed on the nonlinear terms of a given system, is a triangulartype condition as far as output feedback control is concerned, as shown in [2,5,6] Most recently, as a new way of understanding observers, a backstepping-like design procedure for observers was introduced in [2] and [8], in which global stabilization was achieved by a linear output feedback Manuscript received December 8, 2005; revised February 5, 2006 This work was supported in part by the Japanese Ministry of Education, Science, Sports and Culture under both the Grant- Aid of General Scientific Research No C-5560387, and the 2st Century Center of Excellence COE Program Corresponding author E-mail address: keylan@jdendaiacjp controller under a triangular condition The main purpose of this paper is to develop a global stabilizer for a class of uncertain nonlinear systems by linear output feedback under a far more relaxed condition on the nonlinear terms of a given system than the triangular-type condition In fact, we consider the following class of uncertain nonlinear systems ẋ = x 2 + δ t, x, u ẋ 2 = x 3 + δ 2 t, x, u ẋ n = u + δ n t, x, u y = x x = [x, x 2,,x n ] T R n is the state, and u R and y R are input and output of the system, respectively One feature of this paper is that our design method for global stabilizing controllers does not require a detailed structure for the nonlinear terms δ i : R R n R R, i =,,n, including a triangular-type condition see 3 below, except that they are Lipschitz continuous and satisfy the following assumptions Assumption For system, there exist some constants c > 0 and 0 < α such that for any s 0, α, the inequality s i δ i t, x, u c s i x i 2 is satisfied It is not difficult to see that if a triangular condition is satisfied for δ i t, x, u as in [,2,4,5,6,8], ie, δ i t, x, u c i x j 3 j=

26 International Journal of Automation and Computing 3 2006 25-22 then Assumption is always satisfied, but not vice versa In fact, if condition 3 is satisfied, then for any s 0, α s i δ i t, x, u c x + cs x + x 2 + + cs n x + + x n c + s + + s n x + cs + s + + s n 2 x 2 + + cs n x n n n c s i s i x i and hence Assumption is satisfied However, it is clear that the converse may not always hold true 2 Global stabilization by output feedback In this section, we prove that there exists a dynamic output compensator of the form ξ = fξ, y, u = hξ, y 4 such that closed-loop system with dynamic output compensator 4 satisfies lim xt, ξt = 0, 0 t That is to say that the system is stabilized by the dynamic output compensator 4 The dynamic output compensator we propose is made of a linear high gain observer and a linear high gain controller as follows: Theorem Under Assumption, there is a dynamic output compensator of the form 4, which solves the global stabilization problem for an uncertain nonlinear system of the form Proof We begin by introducing the following dynamic system ˆx = ˆx 2 + ra x ˆx ˆx 2 = ˆx 3 + r 2 a 2 x ˆx 5 ˆx n = u + r n a n x ˆx r is a gain parameter to be determined later, and a i i =,,n are the coefficients of any Hurwitz polynomial ρ n + a ρ n + + a n ρ + a n Next, treat 5 as an observer for system, and consider the estimation error e i = x i ˆx i, i n 6 Then it follows from and 5 that ė = e 2 ra e + δ t, x, u ė 2 = e 3 r 2 a 2 e + δ 2 t, x, u 7 ė n = r n a n e + δ n t, x, u Further, introduce the scaled estimation error ε by and set Then one obtains or equivalently ε i = r i e i, i n 8 ε = [ε ε 2 ε n ] T R n 9 ε = rε 2 a ε + δ t, x, u ε 2 = rε 3 a 2 ε + r δ 2t, x, u 0 ε n = ra n ε + r n δ nt, x, u ε = raε + Φ a 0 0 a 2 0 0 A = a n 0 0 a n 0 0 0 Φ = [δ t, x, u, r δ 2t, x, u,, r n δ nt, x, u] T 2 Now consider the quadratic function given as V := ε T Pε 3 P is a positive definite symmetric matrix satisfying A T P + PA = I 4 Then it follows from 0 and 4 that the time derivative of V along the solution of satisfies V =rε T A T P + PAε + 2ε T PΦ r ε 2 + 2ε T PΦ 5 From Assumption and the fact that r, one can obtain Φ δ + r δ 2 + + r n δ n j n i c x i r r nc j= r i x i

K Alimhan and H Inaba/Output Feedback Control for a Class of Nonlinear Systems 27 Further, a simple computation with 6 and 8 gives 2ε T PΦ 2 ε P nc 2 ε P nc 2 ε P nc n r i x i r i ˆx i + ε i r i ˆx i + = ε i n 2 ε P nc r i ˆx i + n ε n 2 P nc r i ˆx i ε + n ε 2 n 2 P nc 2 n2 n + nc P r 2i ˆx i 2 + ε 2 n2 n + nc P ε 2 n k ε 2 + k r 2i ˆx2 i r 2i ˆx i 2 + + n ε 2 k = cn2 n+n P is a constant, independent of r Then from 5, one obtains V r k ε 2 + k n Next introduce ξ i = ˆx i, i n ri r 2i ˆx2 i 6 and set ξ = [ξ ξ 2 ξ n ] T R n Then, ξ = rξ 2 + ra ε ξ 2 = rξ 3 + ra 2 ε 7 ξ n = r r n u + ra n ε and hence inequality 7 can be written as V r k ε 2 + k ξ 2 8 Now we design a compensator of the form u = r n b n ξ + b n ξ 2 + + b ξ n 9 verify that ξ-subsystem 7 with controller 9 can be expressed as B = ξ = rbξ + rε [a, a 2,,a n ] T 20 0 0 0 0 b n b n b Further, choose a quadratic function of the form V 2 := ξ T Qξ 2 Q is a positive definite symmetric matrix satisfying B T Q + QB = 2I 22 Then one can easily obtain the inequality And similarly V 2 = ξ T Qξ + ξ T Q ξ = rξ T B T Q + QBξ + 2rξ T Qaε 23 2r ξ 2 + 2rξ T Qaε 2rξ T Qaε 2r ξ Q [a,,a n ] T ε 2r 2 ξ 2 + 2 Q [a,,a n ] T ε 2 r ξ 2 + r Q [a,, a n ] T 2 ε 2 r ξ 2 + rk 2 ε 2 k 2 Q [a,, a n ] T 2 is a constant, independent of r Therefore, inequality 23 can be written as V 2 r ξ 2 + rk 2 ε 2 24 Next, we observe that the closed-loop system with 5 and 9 can be treated as an interconnection of a ε-subsystem and ξ-subsystem Now, consider the function V := k 2 + V + V 2 = k 2 + ε T Pε + ξ T Qξ 25 It easily follows from 8 and 24 that V = k 2 + V + V 2 r k k 2 + ε 2 + k k 2 + ξ 2 r ξ 2 + rk 2 ε 2 r k k 2 + ε 2 r k k 2 + ξ 2 = r k k 2 + ε 2 + ξ 2 26 Clearly, if we choose the gain parameter r to be r + k k 2 + b i are the coefficients of any Hurwitz polynomial ρ n + b ρ n + + b n ρ + b n Then, it is easy to then V 2 ε 2 + ξ 2

28 International Journal of Automation and Computing 3 2006 25-22 This implies εt 0, ξt 0 as t u, y R are input and output respectively, and z, x R m R n is the state, as long as the following assumption is satisfied and hence that closed-loop system with 5 and 9 is globally asymptotically stable This completes the proof The new approach proposed does not need to go through a recursive design procedure as in [8] It can determine all the observer and controller parameters in one step, rather than n-steps [2,8] Example To check the effectiveness of the result, consider the following simple system x ẋ = x 2 + c x 2 2 + x 2 2 ẋ 2 = u + ln + x 2 2 c2 27 y = x c and c 2 are constant It is easy to check that system 27 satisfies Assumption Therefore, by Theorem, a globally stabilizing output dynamic compensator can be constructed To construct such a compensator by following the proof of Theorem, choose the coefficients of the two Hurwitz polynomials to be a = a 2 = and b = /4, b 2 = 20 Then, the compensator given by 9 can now be described as a ˆx = ˆx 2 + ry ˆx ˆx 2 = u + r 2 y ˆx 28 u = rb 2 rˆx + b ˆx 2 For our numerical simulation, we choose r 8339, and the initial states to be x 0, x 2 0, ˆx 0, ˆx 2 0 =, 5, 3, 5 Simulation results are shown in Fig They show the effectiveness of output dynamic compensator 28 From the design procedure of Theorem, it is clear that there is a linear output feedback controller 5 9 which makes the entire family of nonlinear systems simultaneously asymptotically stable, as long as they satisfy Assumption The global stabilization idea above can be extended to a family of nonlinear systems of the following form ż = fz + gt, z, x, u ẋ = x 2 + δ t, z, x, u ẋ 2 = x 3 + δ 2 t, z, x, u 29 ẋ n = u + δ n t, z, x, u y = x b Fig State trajectories: a shows the results by the proposed method and b the results by the method in [8] Assumption 2 For system 29, suppose that i ż = fz is globally exponentially stable at z = 0 ii There exist some constants ĉ > 0, c > 0, and 0 < α, such that for any s 0, α, the inequalities gt, z, x, u ĉ x

K Alimhan and H Inaba/Output Feedback Control for a Class of Nonlinear Systems 29 s i δ i t, z, x, u c s i z + x i 30 are satisfied Theorem 2 Under Assumption 2, there is a dynamic output compensator of the form 4, which solves the global stabilization problem for an uncertain nonlinear system of the form 29 Proof Since system 29 satisfies Assumption 2, by the converse theorem of globally exponential stability [9], there is a positive and radically unbounded function V z such that This in turn implies V z fz z 2 z V z z c z with c > 0 V z fz+gz, x, u z z 2 + V z z gz, x, u z 2 + cĉ z x 3 4 z 2 + cĉ 2 x 2 3 Now, one can construct a dynamic system 5 with a gain parameter r to be determined later Next, assume that 5 is an observer for system 29, and consider the estimation error e i = x i ˆx i, i n 32 Then, it follows from 29 and 5 that ė = e 2 ra e + δ t, z, x, u ė 2 = e 3 r 2 a 2 e + δ 2 t, z, x, u 33 ė n = r n a n e + δ n t, z, x, u Further, introduce the scaled estimation error ε by and Then one obtains ε i = r i e i, i n 34 ε = [ε ε 2 ε n ] T R n 35 ε = raε + Φ 36 Φ =[δ t, z, x, u, r δ 2t, z, x, u,, r n δ nt, z, x, u] T 37 a 0 0 a 2 0 0 A = a n 0 0 a n 0 0 0 Now, consider the function V := V z + ε T Pε 38 P is a positive definite symmetric matrix satisfying A T P + PA = I 39 Then, it follows from 3, 36, and 39 that the time derivative of V along the solution of 36 satisfies V = V z ż + rε T A T P + PAε + 2ε T PΦ z 3 4 z 2 + cĉ 2 x 2 r ε 2 + 2ε T PΦ 40 From Assumption 2 ii and the fact that r, one can obtain Φ δ + r δ 2 + + r n δ n c j= r n 2 c z + n c j n i z + x i r r i x i Further, a simple computation with 32 and 34 gives 2ε T PΦ 2 ε P n 2 c z + n c r i x i 2 ε P n 2 c z + 2 ε P n c n 2 ε P n c r i ˆx i + ε i n 2 ε P n c r i ˆx i + n ε n 2 P n c r i ˆx i ε + n ε 2 r i ˆx i + ε i

220 International Journal of Automation and Computing 3 2006 25-22 2 P n c n 2 r ˆx i 2 + ε 2 + n ε 2 2i n2 n + n c P n2 n + n c P ε 2 4 z 2 + k ε 2 + k r 2i ˆx2 i r 2i ˆx i 2 + k = c 2 + cn2 n + n P is a constant, independent of r Then, from 40 one obtains V n 2 z 2 r k ε 2 + cĉ 2 x 2 +k r 2i ˆx2 i 4 Next, introduce ξ = [ξ ξ 2 ξ n ] T R n by Then, ξ i = ˆx i, i n ri Then, one can easily obtain the inequality and similarly V 2 = ξ T Qξ + ξ T Q ξ = rξ T B T Q + QBξ + 2rξ T Qaε 48 2r ξ 2 + 2rξ T Qaε 2rξ T Qaε 2r ξ Q [a,,a n ] T ε r ξ 2 + r Q [a,, a n ] T 2 ε 2 r ξ 2 + rk 2 ε 2 k 2 = Q [a,, a n ] T 2 is a constant, independent of r Therefore, inequality 48 can be written as V 2 r ξ 2 + rk 2 ε 2 49 Next, we observe that closed-loop system 29 with 5 and 44 can be treated as an interconnection of a ε-subsystem and ξ-subsystem Now, consider the function ξ = rξ 2 + ra ε ξ 2 = rξ 3 + ra 2 ε 42 ξ n = r r n u + ra n ε and hence inequality 4 can be written as V 2 z 2 r k ε 2 + k ξ 2 43 Now, we design a compensator of the form u = r n b n ξ + b n ξ 2 + + b ξ n 44 b i are the coefficients of any Hurwitz polynomial ρ n + b ρ n + + b n ρ + b n Then, it is easy to verify that ξ-subsystem 42 with controller 44 can be expressed as B = ξ = rbξ + rε [a, a 2,, a n ] T 45 0 0 0 0 b n b n b Further, choose a quadratic function of the form V 2 := ξ T Qξ 46 Q is a positive definite symmetric matrix satisfying B T Q + QB = 2I 47 V := k 2 + V +V 2 = k 2 +V z+ε T Pε+ξ T Qξ 50 It easily follows from 43 and 49 that V =k 2 + V + V 2 2 z 2 r k k 2 + ε 2 + k k 2 + ξ 2 r ξ 2 + rk 2 ε 2 2 z 2 r k k 2 + ε 2 r k k 2 + ξ 2 = 2 z 2 r k k 2 + ε 2 + ξ 2 5 Clearly, if we choose the gain parameter r to be then This implies r + k k 2 + V 2 2 z 2 + ε 2 + ξ 2 εt 0, ξt 0 as t and hence closed-loop system 29 with 5 and 44 is globally asymptotically stable This completes the proof 3 Conclusions We presented a new result for the global stabilization of a class of uncertain nonlinear systems using a dynamic output compensator By integrating the idea

K Alimhan and H Inaba/Output Feedback Control for a Class of Nonlinear Systems 22 of the use of an output feedback domination design method [2], we gave an explicit method for constructing a globally stabilizing output dynamic compensator for a family of uncertain nonlinear systems References [] F Mazenc, L Praly, W Dayawansa Global Stabilization by Output Feedback: Examples & Counterexamples Systems and Control Letters, vol 23, no 2, pp 9 25, 994 [2] C Qian, W Lin Output Feedback Control of a Class of Nonlinear Systems: A Nonseparation Principle Paradigm IEEE Transactions on Automatic Control, vol 47, no 0, pp 70 75, 2002 [3] J Tsinias Backstepping Design for Time-varying Nonlinear Systems with Unknown Parameters Systems and Control Letters, vol 39, no 4, pp 29 227, 2000 [4] R Marino, P Tomei Robust Stabilization of Feedback Linearizable Time-Varying Uncertain Nonlinear Systems Automatica, vol 29, no, pp 8 89, 993 [5] R Rajamani Observers for Lipschitz Nonlinear Systems IEEE Transactions on Automatic Control, vol 43, no 3, pp 397 40, 998 [6] J Tsinias A Theorem on Global Stabilization of Nonlinear Systems by Linear Feedback Systems and Control Letters, vol 7, no 5, pp 357 362, 99 [7] L Praly Asymptotic Stabilization via Output Feedback for Lower Triangular Systems with Output Dependent Incremental Rate IEEE Transactions on Automatic Control, vol 48, no 6, pp 03 08, 2003 [8] C Qian, C B Schrader, W Lin Global Regulation of a Class of Uncertain Nonlinear Systems Using Output Feedback In Proceedings of American Control Conference, Denver, Colorado, USA, pp 542 547, 2003 [9] H K Khalil Nonlinear Systems, 3rd ed, Prentice Hall, New Jersey, 996 Keylan Alimhan received the BS degree in mathematics in 985 from Xinjiang University, China, and MS degree in Information Sciences in 998 from Tokyo Denki University, Japan and he finished his doctoral course work in 2003 at Tokyo Denki University From 985 to 996, he served as a lecturer in the Department of Mathematics, Xinjiang University From 2003 to 2004, he was a research associate in the Department of Information Sciences, Tokyo Denki University Since Spring 2004, he has been Instructor in the Department of Information Sciences,Tokyo Denki University His research interests include nonlinear control theory, in particular, output feedback control of nonlinear systems and nonlinear robust control Mr Alimhan is a member of the Society of Instrument and Control Engineers of Japan Hiroshi Inaba received the PhD degree from the University of Texas at Austin in 972 From 974 to 976 he served as an Assistant Professor in the Department of Aerospace Engineering, the University of Texas at Austin In 977 he joined as Professor in the Department of Mathematical Sciences, Tokyo Denki University TDU, and since 986 he has been with the Department of Information Sciences of TDU From 993 to 997 he served as the Dean of the Graduate School of Science and Engineering, and from 2000 to 2004 as the Director of the Research Institute for Science and Technology, both at TDU His main research interests include mathematical systems theory, particularly emphasizing the theories for infinitedimensional linear systems, linear systems over rings, neural networks and dynamical machine vision Dr Inaba is a member of the Institute of Systems, Control and Information Engineers of Japan, the Society of Instrument and Control Engineers of Japan, the Institute of Electrical and Electronics Engineers IEEE, the Society for Industrial and Applied Mathematics SIAM, and the International Linear Algebra Society ILAS, respectively