On Direct Adaptive Control for Uncertain Dynamical Systems - Synthesis and… 891 4.4 Discrete-time Active Suspension System We use the quarter car model as the mathematical description of the suspen- sion system, given by (Laila, 2003) )),(( )1( 0 0 0 )( 0 0 0 )())( 1 1 0 100 0 )1( 1 1 001 ()1( 2 2 2 22 kxu T kd T kxk T T T TT T kx ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + − ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ −Δ+ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + − ++ − =+ ρ ρ ρω ω ρ ρω ρ ω (128) where .)3.0sin(01.0)( ., 800k , ,0 800k ),( 1115 0000 0000 00010 )( , .100 ,0 1000 ),20sin(10 0 ,0 )( kk k k k kk k kd =Θ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ≠ =Θ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − =Δ > ≤< ≤ = ππ [] T kxkxkxkxkx )()()()()( 4321 = , and 1 x is tire defection, 2 x is unsprung mass velocity, 3 x is suspension deflection, 4 x is sprung mass velocity, sec 20 rad πω = and 10= ρ are unknown parameters, 001.0=T is sampling time, )(kd is disturbance modeling the isolate bump with the bump height mA 01.0= , and )(kΔ is the perturbation on system dynamics. Next, let c A is asymptotically stable , 1 0 0 0 , 1.0001.0 5.01.000 1.013.075.0 175.011 0 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − −− −− − = BA c We apply the framework from Corollary 4.2 and choosing the design matrices Manufacturing the Future: Concepts, Technologies & Visions 892 ,05.0 , 82.0000 0900 0040 0001 , 1000 0400 0010 0002 03.0 = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅= qRY 0 0.5 1 1.5 2 −0.02 0 0.02 0.04 0.06 Tire Deflection Time(sec) X 1 0 0.5 1 1.5 2 −2 −1 0 1 2 Unsprung Mass Velocity Time(sec) X 2 Figure 9 Tire defection and unsprung mass Velocity 0 0.5 1 1.5 2 −0.4 −0.2 0 0.2 0.4 Suspension Deflection Time(sec) X 3 0 0.5 1 1.5 2 −20 −10 0 10 20 Sprung Mass Velocity Time(sec) X 4 Figure 10 Suspension deflection and mass velocity On Direct Adaptive Control for Uncertain Dynamical Systems - Synthesis and… 893 P satisfies the Lyapunov equation (121). The simulation start with [] T x 001.0005.0)0( = . To demonstrate the efficacy of the controller, the states are perturbed to [] T x 5.002.000)800( = at 800=k , and the system parameters are changed to 4= ρ . The controller stabilizes the system in sec 2 under no information of the system changes, either the perturbation of the states. Figure 9 depicts tire defection and unsprung mass velocity versus the time steps, Figure 10 shows the suspension deflection and sprung mass veloc- ity versus the time step, Figure 11 and Figure 12 illustrate the control inputs and adaptive gains at each time step. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 −3 −25 −20 −15 −10 −5 0 5 10 15 20 25 Control Input U Time(sec) Figure 11 Control Input Manufacturing the Future: Concepts, Technologies & Visions 894 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 −3 − 40 − 30 − 20 − 10 0 10 20 30 40 Feedback Gain Time(sec) K(1,1) K(1,2) K(1,3) K(1,4) Figure 12 Adaptive Gains 4.4 Nonlinear Discrete-time Uncertain System We consider the uncertain nonlinear discrete-time system in normal form given by (Fu & Cheng, 2004); (Fu & Cheng, 2005) )),(( 10 01 00 )()( ))(cos()()( )( ()1( 3 13 22 2 1 2 kxu kdxkcx kxkbxkax kx kx ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + +=+ (129) where a , b , c, and d are unknown parameters. Next, let ))(( kxf c to be ))),(( ˆ ))(())((( 0 )()( ))(cos()()( 0 )())(( 1 3 13 22 2 10 kxfkxfkxfB B kdxkcx kxkbxkaxkxAkxf unuuns s c Φ+Θ−Θ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ++= − (130) On Direct Adaptive Control for Uncertain Dynamical Systems - Synthesis and… 895 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = )( )( )( )( ))(cos()( )( ))(( , )( )( ))(( ˆ , )( )( ))(cos()( ))(( 2 1 3 1 3 22 2 1 2 1 3 1 3 22 2 1 kx kx kx kx kxkx kx kxF kx kx Kxf kx kx kxkx x kxf uu and n Θ and n Φ are chosen such that ).( ˆ ))(( ˆ ))(( kxAkxfkxf unun =Φ+Θ (131) where 32 ˆ × ∈ R A is arbitrary, such that ),()( ˆ ~ ))(( 0 kxAkx A A kxf cc = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = (132) and c A is asymptotically stable, specifically, chose , 10 01 00 , 9.05.03.0 1.04.05.0 010 0 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − −= BA c First, we apply the update law (113) and choosing the design matrices 6 1.0 IY = , 3 2.0 IR = , and 005.0=q , where P satisfies the Lyapunov condition RPAAP c T c += . The simulation start with [] T x 15.01)0( −= , and let 5.0=a , 1.0=b , 3.0=c , and 5.0=d . At time 19=k , the states are perturbed [] T x 5.05.01)19( −= , and the system parameters are changed to 65.0=a , 25.0=b , 45.0=c , and 55.0=d . The controller does not have the information of the system parameters, either the perturbation of the states. Figure 13 – Fig- ure 15 show the states versus the time step, Figures 16 shows the control in- puts at each time step, and Figure 17 shows the update gains. The results indi- cate that the proposed controller can stabilize the system with uncertainty in Manufacturing the Future: Concepts, Technologies & Visions 896 the system parameters and input matrix. In addition, re-adapt system while perturbation occurs. The only assumption required is sign definiteness of the input matrix and disturbance weighting matrix. 0 10 20 30 40 5 0 − 0.5 0 0.5 1 x 1 Time step Figure 13 1 x 0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x 2 Time step Figure 14. 2 x On Direct Adaptive Control for Uncertain Dynamical Systems - Synthesis and… 897 0 10 20 30 40 5 0 −1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 x 3 Time step Figure 15. 3 x 0 5 10 15 20 25 30 35 40 45 50 −2 −1.5 −1 −0.5 0 x 10 −11 Control U 1 Time step 0 5 10 15 20 25 30 35 40 45 50 −2.5 −2 −1.5 −1 −0.5 0 x 10 −8 Control U 2 Time step Figure 16. Control Signal Manufacturing the Future: Concepts, Technologies & Visions 898 0 5 10 15 20 25 30 35 40 45 5 0 − 1.8 − 1.6 − 1.4 − 1.2 −1 − 0.8 − 0.6 − 0.4 − 0.2 0 x 10 −11 Gain Time step K(1,1) K(1,2) K(1,3) K(1,4) K(1,5) K(1,6) 0 5 10 15 20 25 30 35 40 45 50 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 x 10 −8 Gain Time step K(2,1) K(2,2) K(2,3) K(2,4) K(2,5) K(2,6) Figure 17. Update Gains On Direct Adaptive Control for Uncertain Dynamical Systems - Synthesis and… 899 5. Conclusion In this Chapter, both discrete-time and continuous-time uncertain systems are investigated for the problem of direct adaptive control. Noted that our work were all Lyapunov-based schemes, which not only on-line adaptive the feed- back gains without the knowledge of system dynamics, but also achieve stabil- ity of the closed-loop systems. We found that these approaches have following advantages and contributions: 1. We have successfully introduced proper Lyapunov candidates for both dis- crete-time and continuous-time systems, and to prove the stability of the resulting adaptive controllers. 2. A series of simple direct adaptive controllers were introduced to handle uncertain systems, and readapt to achieve stable when system states and parameters were perturbed. 3. Based on our research, we claim that a discrete-time counterpart of con- tinuous-time direct adaptive control is made possible. However, there are draw backs and require further investigation: 1. The nonlinear system is confined to normal form, which restrict the results of the proposed frameworks. 2. The assumptions of (63), (64), and (72) still limit our results. Our future research directions along this field are as following: 1. Further investigate the optimal control application, i.e. to seek the adaptive control input 2 Lu ∈ or 2 lu ∈ , minimize certain cost function )(uf , such that not only a constraint is satisfied, but also satisfies Lyapunov hypothesis. 2. Stochastic control application, which require observer design under the ex- tension of direct adaptive scheme. 3. Investigate alternative Lyapunov candidates such that the assumptions of (63), (64), and (72) could be released. 4. Application to ship dynamic control problems. 5. Direct adaptive control for output feedback problems, such as Manufacturing the Future: Concepts, Technologies & Visions 900 )()()( )),(())(()())(()( ),())(())(())(())(()1( kykKku kxukxIkxkxHky kwkxJkxukxGkxfkx = += ++=+ or )()()( )),(())(()())(( ),())(())(())(())(( tytKtu txutxItxtxHy twtxJtxutxGtxfx = += ++= & 6. References Bar-Kana, I. (1989), Absolute Stability and Robust Discrete Adaptive Control of Multivariable Systems, Control and Dynamic Systems, pp. 157-183, Vol. 31. Chantranuwathana, S. & Peng, H. (1999), Adaptive Robust Control for Active Suspension, Proceedings of the American Control Conference, pp. 1702-1706, San Diego, California, June, 1999. de Leòn-Morales, J.; Alvarez-Leal, J. G.; Castro-Linares, R. & Alvarez-Gallego, J. A. (2001), Control of a Flexible Joint Robot Manipulator via a Nonlinear Control-Observer Scheme, Int. J. Control, vol. 74, pp. 290—302 . Fu, S. & Cheng, C. (2003, a), Direct Adaptive Control Design for Reachable Linear Discrete-time Uncertain Systems, in Proceedings of IEEE Interna- tional Symposium on Computational Intelligence in Robotics and Automation, pp. 1306-1310, Kobe, Japan, July, 2003. Fu, S. & Cheng, C. (2003, b), Direct Adaptive Control Design for Linear Dis- crete-time Uncertain Systems with Exogenous Disturbances and 2 l Dis- turbances, in Proceedings of IEEE International Symposium on Computa- tional Intelligence in Robotics and Automation, pp. 1306-1310, Kobe, Japan, July, 2003. Fu, S. & Cheng, C. (2004, a), Adaptive Stabilization for Normal Nonlinear Dis- crete-time Uncertain Systems, in Proceedings of Fifth ASCC, pp. 2042— 2048, Melbourne, Australia, July, 2004. Fu, S. & Cheng, C. (2004, b), Direct Adaptive Control for a Class of Linear Dis- crete-time Systems, in Proceedings of Fifth ASCC, pp. 172—176, Melbourne, Australia, July, 2004. [...]... Narendra, K (199 6), Control of Nonlinear Dynamical Systems using Neural Networks -Part II: Observability, Identification, and Control, IEEE Trans Neural networks, Vol 7, pp 30-42 Loria, A.; Panteley, E.; Nijmeijer, H & Fossen, T (199 8), Robust Adaptive Control of Passive Systems with Unknown Disturbances, IFAC NOLCOS, pp 866-872, Enschede, The Netherlands, 199 8 Mareels, I & Polderman, J (199 6), Adaptive... J (199 6), Adaptive Systems An Introduction, Birkhauser 902 Manufacturing the Future: Concepts, Technologies & Visions Venugopal, R & Bernstein, D (199 9), Adaptive Disturbance Rejection Using ARMARKOV System Representations, Proc of the 36th IEEE CDC, pp 1654-1658, San Diego, CA, December 199 9 Shibata, H.; Li, D.; Fujinaka, T & Maruoka, G (199 6), Discrete-time Simplified Adaptive Control Algorithm and... International Conference on Mechatronics, pp 881-886, Taipei, Taiwan, July 2005 Fukaom, T.; Yamawaki, A & Adachi, N (199 9), Nonlinear and H ∞ Control of Active Syspension Systems with Hydraulic Actuators, Proceedings of the 38th IEEE CDC, pp 5125—5128, Phoenix, Arizona, December, 199 9 Guo, L (199 7), On Critical Stability of Discrete-time Adaptive Nonlinear Control, IEEE Transactions on Automatic Control,... & Anderson, B D O (196 9), Discrete Positive-real Functions and Their Application to System Stability, Proc IEE, pp 153-155, Vol 116 Johansson, R (198 9), Global Lyapunov Stability and Exponential Convergence of Direct Adaptive Control, Int J Control, pp 859-869, Vol 50 Laila, D S (2003), Integrated Design of Discrete-time Controller for an Active Suspension System, Proceedings of the 42th IEEE CDC,... 947-954 903 Corresponding Author List Kazem Abhary School of Advanced Manufacturing and Mechanical Engineering University of South Australia Australia Jose Barata Universidade Nova de Lisboa – DEE Portugal Thierry Berger LAMIH, University Valenciennes France Felix T S Chan Department of Industrial and Manufacturing Systems Engineering The University of Hong Kong P.R China Che-Wei Chang Graduate Institute... Engineering, Sultan Qaboos University Sultanate of Oman Mehmet Savsar Department of Industrial & Management Systems Engineering College of Engineering & Petroleum Kuwait University Cem Sinanoglu Erciyes University Engineering Faculty Department of Mechanical Engineering Turkey 905 Bill Tseng Department of Mechanical and Industrial Engineering The University of Texas at El Paso USA Joze Tavcar Faculty of Mechanical... Application to a Motor Control, IEEE Industrial Electronics, pp 248-253, Vol 1 Zhao, J & Kanellakopoulos I (199 7), Discrete-Time Adative Control of Output-Feedback Nonlinear Systems, in IEEE Conference on Decision and Control, pp 4326-4331, San Diego, CA, December 199 7 Zhihong, M.; Wu, H R & Palaniswami, M (199 8), An Adaptive Tracking Controller Using Neural Network for a Class of Nonlinear Systems, IEEE Transactions... Paso USA Joze Tavcar Faculty of Mechanical Engineering University of Ljubljana Slovenia A.M.M Sharif Ullah Department of Mechanical Engineering College of Engineering United Arab Emirates University Yong Yin Department of Public Policy & Social Studies Yamagata University Japan Chao Zhang Department of Mechanical & Materials Engineering University of Western Ontario Canada ... of Electronic Technology PR China Tatsushi Nishi Department of Systems Innovation Graduate School of Engineering Science Osaka University Japan Che Ruhana Isa Faculty of Business and Accountancy University of Malaya Malaysia Mario Pena-Cabrera Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas IIMAS-UNAM, Mexico Tritos Laosirihongthong Department of Industrial Engineering, Faculty of Engineering... Institute of Manufacturing Engineering, National Cheng Kung University Taiwan, ROC Cheng Siong Chin Nanyang Technological University Singapore Jorge Corona-Castuera CIATEQ A.C Advanced Technology Centre, Queretaro Mexico Alexandre Dolgui Division for Industrial Engineering and Computer Sciences Ecole des Mines de Saint Etienne France Ming Dong Shanghai JiaoTong University P.R China Jerry Fuh Ying Hsi Department . at 800=k , and the system parameters are changed to 4= ρ . The controller stabilizes the system in sec 2 under no information of the system changes, either the perturbation of the states. Figure. time 19= k , the states are perturbed [] T x 5.05.01 )19( −= , and the system parameters are changed to 65.0=a , 25.0=b , 45.0=c , and 55.0=d . The controller does not have the information of the. parameters, either the perturbation of the states. Figure 13 – Fig- ure 15 show the states versus the time step, Figures 16 shows the control in- puts at each time step, and Figure 17 shows the update