Robust Predictive Control of Input Constraints and Interference Suppression for Semi-Trailer System

Transcription

1 Vol.7, No.7 (4), htt://dx.doi.org/.457/ica Robust Predictive Control of Inut Constraints and Interference Suression for Semi-Trailer System Zhao, Yang Electronic and Information Technology Deartment, Jiangmen Polytechnic, Jiangmen 599, China Abstract In this aer, an online receding robust redictive control scheme is roosed for inutconstrained semi-trailer system with delay and disturbance. The controller method meets the requirements of control constraint and, based on dual-mode control, the method is obtained by online otimization of erformance index. The erformance index is the cumulative sum of quadratic weighted value of minimal states. Controller outut is calculated by means of linear matrix inequality (LMI), and the controller itself can ensure the asymtotic stability and disturbance attenuation of semi-trailer closed-loo system. Finally, simulation results confirm the effectiveness of the method. Keywords: semi-trailer system, time-delay and disturbance system, robust redictive control, dual-mode control, LMI. Introduction With the develoment of transortation technology, driving safety and driving comfort have become an increasingly imortant indicator for vehicles []. A semi-trailer is an efficient means for long-distance transortation. Its fuel consumtion er ton kilometers is 4% lower than that of a bus, and its maintenance cost is lower, too. Therefore, vigorously develoing semi-trailer transortation has become an imortant aroach to imrove transortation efficiency. Because of factors such as the large mass and high center of gravity of semi-trailer, its driving stability and safety are great concerns. There are frequent accidents as rollover, folding and shimmy, not only damaging vehicles and drivers, but causing significant harm to other vehicles on road as well. Therefore ride erformance of semi-trailer has become one of the imortant issues that need to be addressed. Ride erformance is the ability of the vehicle system to ease and attenuate road shock so as to kee a certain comfort for drivers in a vibration environment when the vehicle is on the road, and also the ability of trucks to maintain the intactness of goods. However, there are conditions of longitudinal and transverse rams or uneven road surface, such as sandstone road, mountainous road, forest and lateau. Under such conditions, once a vibration roblem occurs, the longitudinal and lateral adhesion erformance of semi-trailer s tires will be reduced. For loaded goods, vibration can cause a negative imact on the reliability of materials [, ]. Semi-trailer s vibration comes mainly from the ground roughness, followed by change of force from the engine, transmission system, steering system, and brake system as well as the imbalance and deflection of wheels, etc. These are the maor factors affecting semi-trailer s ride erformance. The vibrations caused by the imact can also reduce the life of the semi-trailer and its steering stability. Therefore, to imrove the semi-trailer s ride erformance, vibration must be reduced and semi-trailer s anti-vibration erformance must be imroved. ISSN: IJCA Coyright c 4 SERSC

2 Vol.7, No.7 (4) In the driving rocess, because of time-delay [3] of the semi-trailer as well as external disturbance and inut constraint, conventional control methods such as LQG/LTR [4, 5] and PD [6] cannot obtain good effect. Some engineers have turned their eyes to modern control theory. Model redictive control (MPC), based on redictive model [7], is an efficient control method to deal with control and state constraints [8~]. Comared with single-state feedback control, redictive control relaces global one-time otimization with receding otimization, which means that the otimization rocess is not an offline one-time rocess but rather reeated online otimizations and receding imlementations, so that uncertainties caused by model mismatch, time-varying and interference can get a timely remedy. This receding otimization strategy has taken into account both the ideal otimization for a sufficient long eriod of time in the future and the imact of actual uncertainties. Therefore, the establishment of a receding otimization strategy in a finite time domain will be more effective.literature [3] gives a detailed summary of research on robust redictive control. In view of bounded disturbances of linear time-invariant system, H controller with receding horizon is designed in literature [4], which can guarantee the stability of a closed-loo system and can meet the H index. Since H control directly designs controller in state sace and it has strengths such as recise calculation and maximum otimization, it rovides the uncertain MIMO system of model erturbation with a controller design method which can ensure robust stability of the control system and can otimize certain erformance indicators [5]. Therefore, many scholars have combined the method with redictive control algorithm. In literature [6], an MPC algorithm is roosed for linear systems with inut constraints and unknown time-delay. This algorithm is given under the assumtion that the terminal matrix of time-delay is a constant matrix. Under this assumtion, even though the otimal cost function should be transformed into a simle otimization roblem of two equivalents, it can be relatively easy to obtain a controller with asymtotic stability. In literature [7], a new Lyaunov function is constructed for the same system to imrove the raidity of erformance indexes. But the system has not taken into account disturbance factor. Literature [7-9] alies redictive control to network control, in which literature [8] considers a nonlinear system; literature [9] designs an outut feedback redictive control method for a class of constrained linear systems; literature [] designs a minimum-maximum redictive controller for a class of nonlinear network with delay and acket loss. It is known that as the calculation amount of redictive control algorithm is large, the imlementation of redictive control algorithm usually requires the use of high-erformance comuting devices. However, with the develoment of field control devices and embedded system as well as the continued imrovements in the frequency of related microcontrollers, redictive control algorithm has also begun to enetrate underlying control and has been alied in PLC, FPGA and other devices []. Based on the above analysis, a redictive controller design method is ut forward for semi-trailer system s inut constraints, uncertain time-delay and vibration. In redictive control, the dual mode control is used, and aroriate Lyaunov function is chosen to imrove the erformance of the system. A H controller with receding horizon is designed based on that, which can guarantee the stability of the system and disturbance attenuation. 37 Coyright c 4 SERSC

3 Vol.7, No.7 (4). Descrition of Problems Consider a semi-trailer system []: Consider as Figure : U[+] x3 L x[+] 3 l x 3[-] U[-] x[+] x [+] x [-] x [-] Figure. Semi-trailer Structure Chart where x ( k ), x ( k ), x ( k ), u ( k ) resectively reresent the angle difference between truck and 3 trailer, trailer angle, vertical osition of trailer tail and swinging angle of truck; truck length is l, trailer length is L, reverse seed is v, a [,) refers to lagged variable coefficient and T refers to samling eriod. The swinging angle of truck is bounded by the condition u( k). When x ( k ) and u( k ) change slightly, the mathematical model is as follows: x ( k ) x ( k ) v T L x ( k ) v T L u ( k ) x ( k ) v T L x ( k ) x ( k ) x ( k ) v T sin 3 v T L x ( k ) x ( k ) x ( k ) 3 As it is a non-rigid chain link from the truck to the trailer, let a [,) be a lag coefficient, and add formula () to lags: x ( k ) x ( k ) a v T L x ( k ) a v T L x ( k d ) v T L u ( k ) x ( k ) a v T L x ( k ) a v T L x ( k d ) x ( k ) x ( k ) v T s in a 3 v T L x ( k ) a v T L x ( k d ) x ( k ) x ( k ) 3 3 In formula (), x ( k) is with a linear form. Let it be an aroximation linear and consider vibration of truck and trailer in the rocess of driving, and then formula () can be converted into formula (3) which is shown below: () () Coyright c 4 SERSC 373

4 Vol.7, No.7 (4) x ( k ) x ( k ) a v T L x ( k ) a v T L x ( k d ) v T L u ( k ) b w ( k ) x ( k ) a v T L x ( k ) a v T L x ( k d ) x ( k ) b w ( k ) x ( k ) a v T L x ( k ) a v T L 3 x ( k d ) x ( k ) x ( k ) b w ( k ) 3 3 Convert formula (3) into a matrix form: x ( k ) A ( k ) x ( k ) A ( k ) x ( k d ) where x ( k ) B ( k ) u ( k ) B ( k ) w ( k ) n R, u ( k ) (4) m R and w ( k ) R P (3) resectively refer to the system state, inut and disturbance signal. d (,, ) is a unknown time-delay constant. Suose d d, where d is a known constant. Meanwhile, for the convenience of calculation, d d ( i ) i is set. A( k ), A ( k ) and (,, ), B ( k ), B ( k ) refer to uncertain time-varying matrix with corresonding dimensionality: A( k ) a a a a a a a a a n n n n n n a a a a a a A ( k ) a a a B ( k ) n n n n n n b ( k ) b b ( k ) b, B ( k ) b ( k ) m b ( k ) ( k ) ( k ) System (4) also satisfies the assumed condition: ersistent disturbance w( k) uer bound, i.e., w ( k ) r, r An evaluation signal is defined: has a known z ( k ) [ C x ( k ), C x ( k d ), C x ( k d ) / ( ), ( )] T C x k d R u k (5) 374 Coyright c 4 SERSC

5 Vol.7, No.7 (4) q where C, C (,, ) R n resectively refer to weighting matrix for quantity of state and m m quantity of time-delay state; R R refers to a ositive definite symmetric weighting matrix for inut quantity. Quadratic weighted accumulative value of minimized state is taken as the erformance index. Meanwhile disturbance factor is introduced into the erformance index. A controller is designed for system (4), so that the corresonding closed-loo system can meet the following erformance: ) The controller can satisfy the constraint condition, namely, u ( k ) U k,, m (6), m ax and the following erformance index can reach the minimum value. m in m ax J ( k) u [ A ( k ) A ( k ) B ( k ) B ( k ) ] J ( k ) J ( k ) J ( k ) N i ( z ( k i k ) w ( k i k ) ) i N ( z ( k i k ) w ( k i k ) ) where J ( k ) reresents the finite time domain, and J ( k ) reresents the infinite time domain, which can be converted into terminal cost function. x ( k i / k ) indicates a state redictive value at k i moment based on model (4) at k moment, and u ( k i / k ) indicates the redictive control at the i th ste of k moment. ) When w( k ), the system is asymtotically stable; when w( k ), L norm from (7) disturbing w( k ) signal to evaluation signal z( k ) is no more than, namely, z w 3. Main Results 3.. Controller Design This controller is designed by the thinking of dual-mode control []. Assume that the arameters in system (4) are accurate, and it is a nominal model, i.e., the "closest" model to the actual system. When system constraints are not considered, closed-loo linear feedback form is used: u ( k i / k ) F ( k i / k ) x ( k i / k ) (8) The above state feedback control law allows unconstrained nominal system to have a stable closed-loo and to be otimal in a certain sense. When system constraints are considered, a set of auxiliary variables e ( k i / k ) are added to control law (8), and then (8) becomes: u ( k i / k ) F ( k i / k ) x ( k i / k ) e ( k i / k ) (9) in which: F ( k i / k ) F ( k i / k ), i, N Coyright c 4 SERSC 375

6 Vol.7, No.7 (4) F ( k N / k ) F ( k ) As e ( k i / k ) is a comensation term added to control law to meet the constraint condition, e ( k i / k ) when the system constraints do not work. Let the rediction horizon be N. Combine formula (8) and (9) together: u ( k i / k ) F ( k i / k ) x ( k i / k ) e ( k i / k ) F ( / ), i {, N } u ( k i / k ) F ( k ) x ( k i / k ) i N () Formula () and () are the dual-mode control strategy: in the rediction horizon, control law () with auxiliary variables is used to ensure that the system terminal state enters the robust invariant set structured by feedback control law and constraint; outside of the rediction horizon, control law () is used to ensure the stability of the closed-loo system. It should be noted that F ( k i / k ), F ( k ) in formula () and formula () are called feedback gain in conventional control theory. There are used here to reflect the gain scheduling thinking of redictive control. Theorem : Suose F ( k ) YQ in formula () and () for the system described in the equation (4). If there are variables Y, Q, Q Q, r, F ( k i / k ), e ( k i / k ), r, ( i,,,..., ) i () N, then the control law () and () is used to minimize erformance index online under the condition of satisfying control constraints, in which these variables are the otimal solutions for the following LMI roblems. m in r, ri, Q, Q,... Q, Y, e ( k i / k ) N r r () i m ( k N / k ) l N, l, l i H Q N K Y r R (3) (4) 376 Coyright c 4 SERSC

7 Vol.7, No.7 (4) r r i li... l, l C x ( k i d / k ) I li... l, l C x ( k i d / k ) li... l, l F ( k i / k ) x ( k i / k ) e ( k i / k ) (5) U Y, m a x T Q I R F ( k i / k ) x ( k i / k ) e ( k i / k ) U (7), m ax where d ia g { ( Q, Q,, Q ), r }, l l l l l H A Q B Y A Q A Q B N d ia g { C Q, C Q, C Q,, }, d ia g { r, r,, r }, K d ia g { Q, Q, Q,,, },, l [,, L ] d ia g { ( d Q, ( d d ) Q ( d d ) Q ), I, I, I }, l N, l, l m ( k N / k ); l L, N, includes all ossible m ( k N / k ) comact sets, and m ( k i / k ) [ x ( k i / k ), x ( k i / k ), T T x ( k i d / k ) ], i,,, N d ia g [ Q, d Q, ( d ) Q Q, d d d ( d d ) Q, ( d d ) Q Q,, ( d d ) Q, ( d d ) Q Q ] P P T T (6) d d The symbol indicates symmetric items in the symmetric matrix, and the following relationshis exist: Q rp, Q rp,... Q rp, Y F ( k ) Q See aendix for the roof of Theorem. Therefore, the following algorithm stes can be obtained by using Theorem : Coyright c 4 SERSC 377

8 Vol.7, No.7 (4) Ste Let x ( k / k ) x ( k ) at k moment. Ste By solving LMI (8) -(3), the exressions of F ( k ), F ( k i / k ), ( i,, N ) can be obtained. e ( k i / k ) Ste 3 u ( k / k ) F ( k / k ) x ( k / k ) e ( k / k ) is alied to system (4), and x( k ) is calculated. Ste 4 Let k k, and reeat from Ste to Ste Analysis of Stability and Disturbance Attenuation Theorem : If the algorithm obtained by Theorem has a feasible solution at the initial moment k, the system is feasible for any k, and the closed-loo system is with asymtotical stability and H disturbance attenuation. Proof: Assume that Theorem has a feasible solution at the initial moment k (exressed by the symbol *): F ( k ), e ( k i / k ), ( k ), at this oint, the following solution is taken as a feasible solution at k moment: u ( k i / k ) F ( k i / k ) x ( k i / k ) e ( k i / k ) i,,... N u ( k i N / k ) F ( k ) x ( k i N / k ) ( i ) ( k ) ( k ) J ( k ) N i i [ z ( k i / k ) w ( k i / k ) ] T m ( k N / k ) ( k ) m ( k N / k ) N [ z ( k i / k ) w ( k i / k ) ] T m ( k N / k ) ( k ) m ( k N / k ) At this oint, Equation (A) is set u in the aendix, so the following can be obtained: m ( k N / k ) m ( k N / k ) ( k ) ( k ) z ( k N / k ) w ( k N / k ) Add Equation (9) to equation (8): N i J ( k ) [ z ( k i / k ) w ( k i / k ) ] T m ( k N / k ) ( k ) m ( k N / k ) J ( k ) J ( k ) J ( k ) z ( k / k ) w ( k / k ) (8) (9) At this oint, J ( k ) J ( k ) J ( k ) J ( k ) z ( k / k ) w ( k / k ) 378 Coyright c 4 SERSC

9 Vol.7, No.7 (4) Herein J ( k ) is obtained by otimizing Theorem at moment k. At this oint, let J ( k) serve as Lyaunov function in system (4). If w( k ), the system is asymtotically stable; if w( k ), the system has disturbance attenuation. 4. Simulation Research Consider semi-trailer system (). The arameters come from literature []. Take l.8 m, L 5.5 m, v. m / s and t.. [, ], a.7, T., and then the system arameters are: A A. 5 9 B B. 8 In the simulation, the initial state is selected as x ( ) [ ] T, where the mean value of disturbance is taken as, and the variance is.5 white noise signal. The weighting matrix in evaluation signal is selected as C, C and R, time delay arameter d d 5. Take d 3 in the simulation, L gain.8. Several exeriments show that the values of arameter C, C and L gain have no significant effect on the control result. The simulation results have shown the following: based on the robust redictive control algorithm roosed in the aer, Figure indicates that the system is asymtotically stable and the system resonse seed is rather fast in situations without disturbance. Figure 3 shows that the control inut switches back and forth largely at the beginning, and then tends to be stable gradually. Figure 4 shows that controller solved like this imroves the erformance index of the system, so that the uer bound value of erformance index can quickly aroach to the minimum value. When a disturbance occurs, Figure 5 shows the system can be raidly stabilized. Figure 6 shows that control inut fluctuates on a small scale and satisfies the constraint conditions. From the viewoint of actual hysical alication, the controller suits well. Figure 7 shows that by using the method in this aer, there is no great change between disturbance erformance index and non-disturbance erformance index, and it can aroach to the minimum value quickly. Coyright c 4 SERSC 379

10 Vol.7, No.7 (4) 4 x x x3 x(t) k Figure. State Curve of the System Without Disturbance u(t) k Figure 3. Control Inut Curve without Disturbance J-inf k Figure 4. Uer Bound Value of Performance Index without Disturbance 38 Coyright c 4 SERSC

11 Vol.7, No.7 (4) 6 4 x x x3 x(t) k Figure 5. State Curve with Disturbance k Figure 6. Control Inut Curve with Disturbance J-inf u(t) k Figure 7. Uer Bound Value of Performance Index with Disturbance Comared with literature [], algorithm in the aer does not require a comlex fuzzy model of semi-trail system and the controller s design is simle with better control effects. Coyright c 4 SERSC 38

12 Vol.7, No.7 (4) 5. Conclusions In this aer, uncertain time-delay and disturbance was considered and a robust redictive control method was roosed for the inut-constrained semi-trailer system, which can reresent a class of linear time-varying and uncertain time-delay system. Through the selection of dual mode control and Lyaunov function, it was ensured that the semi-trailer system has asymtotic stability and disturbance attenuation, and the disturbance is raidly restrained by the quick online minimum erformance index. These characteristics are very imortant for semi-trailer system traveling under different road conditions and with severe disturbance. Ultimately, the simulation verified the effectiveness of the method roosed. Furthermore, the robust redictive control method is also alicable for n-order system with time-varying uncertain and time-delay disturbance. Acknowledgements This work was suorted by technical service roect(3h36). References [] Hongwu and F. Huang, China Journal of Highway and Transort, vol., no. 9, (996). [] J. Kong and Z. Liu, Journal of Highway and Transortation Research and Develoment, vol. 3, no. 5, (6). [3] G. Rachakit, Alied Mathematics & Information Sciences, vol. 3, no. 6, (). [4] T. Zhu, C. Zong and H. Zheng, Journal of System Simulation, vol., no., (8). [5] C. Zong, T. Zhu and M. Li, Journal of Mechanical Engineering, vol., no. 44, (8). [6] Z. Zhu, J. Chen and C. Niao, Journal of Agricultural Machine, vol. 7, no. 37, (6). [7] S. Jun and D. Uhm, Mathematics & Information Sciences, vol. 5, no. 7, (3). [8] M. V. Kothare and V. B. Morari, Automatica, vol., no. 3, (996). [9] B. Kouvaritakis, J. A. Rossiter and J. Schuurmans, IEEE Trans. Auto. Cont, vol. 8, no. 45, (). [] H. H. J. Bloemen, Automatica, vol. 6, no. 38, (). [] Y. Lu and Arkun Y, Automatica, vol. 4, no. 36, (). [] Ding B C and Xi Y G, Automatica, vol., no. 4, (7). [3] D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. Scokaert, Automatica, vol. 6, no. 36, (3). [4] K. B. Kim, IEEE Transactions on Automatic Control, vol. 5, no. 49, (4). [5] J. B. Rawlings, IEEE Trans. Automatic Control, vol., no., (). [6] S. C. Jeong and P. G. Park, IEEE Transactions on Automatica Control, vol., no. 5, (5). [7] X.-B. Hu and W.-H. Chen, Int. J. Robust Nonlinear Control, vol. 3,no. 4, (4). [8] H. Li and Y. Shi, IEEE Transactions on Industrial Electronics, vol. 3, no. 6, (3). [9] H. Li and Y. Shi, Systems & Control Letters, vol. 4, no. 6, (3). [] H. Li and Y. Shi, International Journal of Control, vol. 4, no. 86, (3). [] X. Yugeng, L.Dewei and L.Shu, Acta Automatica Sinica, vol.3, no. 39, (3). [] K. Tanaka and M. Sano, IEEE Transactions on Fussy systems, vol., no., (994). [3] Z. Min and L. Shaoyuan, System Science and Mathematics, vol. 3, no.7, (7). Author Zhao, Yang received the master degree in control engineering from Changchun University of Technology of China in 8. Currently, he is a teacher in at Jiangmen Polytechnic, China. His maor research interests include intelligent control and Predictive Control. 38 Coyright c 4 SERSC