AbstractThis paper presents a method of enhancing tracking in repetitive processes which can be approximated by first order plus dead-time (FOPDT) models. Enhancement is achieved through the iterative learning control (ILC) scheme known as filter-based ILC. Using the approximate model, the design of the ILC parameters is done in the frequency domain. In a water heating plant, this ILC scheme is effortlessly added onto a PI controller, with minimal implementation requirements. The results show good improvements in tracking. For even better performance, the trajectory is segmented into piecewise smooth sections with ILC applied separately to each. For constant level sections, the feed-forward signal is initialised to the desired control estimated from previous iterations. This gives the best empirical results.1.IntroductionTrajectory tracking is very important in many industrial applications. The control objective is for system outputs to track a specified profile as tightly as possible over a given finite period. This task, known as an o ptimal t racking c ontrol p roblem (OTCP) over a finite interval, is very difficult to achieve in practice. One frequently encountered problem is the lack of exact system information for accurate derivation of the precise control effort required for tracking.Usually, perfect tracking can only be achieved asymptotically - the initial tracking will be conspicuously poor within the finite interval. Thus, standard controllers, like the widely used PI or PID, are not able to achieve satisfactory tracking performance for the whole operation period. Moreover, most advance control schemes only guarantee asymptotic convergence along the time horizon, and thus are unable to improve transient response within the finite tracking duration.On the other hand, most OTCP industrial processes are batch operations, which by virtue are repeated many times with the same desired tracking profile. The same tracking performance will thus be observed, albeit with hindsight from previous operations. Clearly, these continual repetitions make it conceivable to improve tracking, potentially over the entire task duration, by using information from past operations.For enhancing tracking in repeated operations, ILC schemes developed hitherto well cater to the needs [1-7]. Basically, ILC uses repetitions as experience to improve tracking without exact system knowledge [8]. Hence, ILC improves tracking by using previous control and error signals in the current operation. Numeric processing on these data yields a feed-forward signal added to the current feedback control. Clearly, ILC requirements are minimal - a memory store for past data plus some simple data operations to derive the feed-forward signal. With its utmost simplicity, ILC can very easily be added on top of existing (predominantly PID batch) facilities without any hassle at all.This paper illustrates the application of ILC to a class of processes which can be well approximated by a FOPDT model. The control scheme known as filter-based ILC is applied to a water heating plant. Using non-causal zero-phase filtering, this ILC scheme simply involves 2 parameters - the filter length and the learning gain, both easily tuned using the approximate model. Also, this scheme is practically robust to random system noise.The remainder parts of this paper are organised as follows. Section 2 describes the water heating plant, its modelling work and the control objective. Section 3 gives an overview of filter-based ILC with its convergence analysis. Section 4 shows the controller design work and the experimental results. From these results, an improved ILC scheme, with profile segmentation and feed-forward initialisation, is used to improve tracking performance even further. Finally, Section 5 concludes the paper.2.Modelling the Water Heating Plant andProblem StatementA.Modelling the Water Heating PlantFigure 1 - Schematic Diagram of the Water Heating Plant Fig. 1 is a schematic diagram showing the relevant portions of the water heating plant. Water from Tank A is pumped (N1) through a heat exchanger as the cooling stream. The heating stream on the other side of the exchanger is supplied from a heated reservoir. ThisEnhancing Trajectory Tracking for a Class of Process Control Problems usingIterative LearningJian Xin Xu, Tong Heng Lee, Yang Quan Chen and Hou TanNational University of SingaporeDepartment of Electrical Engineering10 Kent Ridge CrescentSingapore 119260Singapore(E-mail: elexujx@.sg)heated stream is pumped (N2) through the exchanger before returning to the reservoir where it is heated by aheating rod (PWR).Figure 2 - Model and Plant Response to 200W StepFig. 2 shows the actual response of the plant and the response of its distributed PDE model to a step input of 200W at the heater. (The heater is switched to 200W at t=0s and then switched off at t=12.5Ks .) Both the simulation and the experimental responses show that the plant can be effectively approximated by the FOPDT system:ses s PWR s T s G 50128651390.0)()(2)(-+==(1)B. Problem StatementIn general, the control problem is to enhance trajectory tracking in repeated batch operations where the process can be approximated with the FOPDT model:asT τs Ke G(s)+-=1(2)where K : Plant gainτ : Apparent dead-time a T :Apparent time constantThe desired trajectory is a piecewise continuous profile commonly found in process control eg. desired temperature profile, desired concentration profile, etc. For the water heating plant, the control objective is for T2 (water temperature in the heated reservoir) to track the desired profile shown in Fig. 3 as closely as possible over its entire duration, by varying the input to the heater (PWR). Both pumps (N1 & N2) are maintained at pre-set values throughout the run.Figure 3 - Desired Temperature ProfileTo achieve the objective, a learning controller based on filter-based ILC will be designed using systematic analysis on the approximate model. This learning controller will be effortlessly augmented on top of a PI controller with minimal implementation requirements.3. Iterative Learning ControlA.The Schematic of Filter-based ILCBecause of the effectiveness for enhancing repeated tracking tasks, ILC has drawn increased attention and many schemes have been developed. In this paper, a d y : Desired output profile i e : Error at i th iterationi fb u: Feedback signal at i th iteration i ff u: Feed-forward signal at i th iterationi y: Plant output at i th iterationAs seen from Fig. 5, the learning update law and the overall control signal is )(*)()(1k u h k u k u i fb i ff i ff γ+=+ (3))()()(k u k u k u i fb i ff i +=(4)where k represents the k th time sample of the respective signal, γis the filter gain and *h is the filter operator: moving averager (* denotes convolution)Figure 5 - Block Diagram of Filter-based ILC For this paper, the (non-causal zero-phase) filter *h γ is a simple moving averager with 2 parameters M and γ, related to the filter length and filter gain respectively.∑-=++=MMj ifb i fb j kuM k u h )(12)(*γγ(5)Basically, filter-based ILC attempts to store the desired control signal in the memory bank as the feed-forward signal. With convergence, the feed-forward signal will tend to the desired control signal so that, in time, it will relieve the burden of the feedback controller. As for all ILC schemes, it is important that the plant output converges to the desired profile along with iterations. This is shown in the convergence analysis.B. Frequency Domain Convergence Analysis ofFilter-based ILCThe time-domain convergence analysis of the filter-based ILC (3),(4) is fully presented by Chen et al [9] for quite general non-linear systems. For this paper, the focus is on FOPDT processes (2). Thus, it is sufficient and convenient to provide frequency domain convergence analysis for systematic design.Compared with the time domain analysis, frequency domain analysis offers more insight into the effects of the plant, the PI controller and the learning filter on ILC performance and allows the systematic design of M and γ based on the linearized FOPDT plant model.Since ILC can only be implemented digitally, the frequency analysis should actually be done on sampled systems. Provided sampling is fast compared to the system time constant, the problem of aliasing is avoided and a zero-order hold filter will reconstruct the correct output signal to the plant. Assuming this is so, the analysis can proceed as though for continuous systems. Consider linear systems with )(ωj G b and )(ωj G o as the transfer functions of the feed b ack controller and the o pen-loop plant respectively. The c losed loop transfer function is)()(1)()()(ωωωωωj G j G j G j G j G o b o b c +=(6)Writing the learning update law (3) in the frequency domain gives)()()()(1ωγωωj U w H j U j U i fb i ff i ff +=+ (7)In the following ωj is omitted for brevity. By considering the transfer function from )(t y d and from)(t u i ff to )(t u i fb , i fb U can be written as)(1i ff d c i fb i ffc d o b bi fb U U G U U G Y G G G U -=-+=(8)where d Y and d U are the Fourier transforms of thedesired profile and the desired control respectively. Thus, (7) becomes the following difference equation in the frequency domaind c i ff c i ff U HG U HG U γγ+-=+)1(1(9)From (9) and assuming 0)(0=t u ff , it is interesting toobserve that c G acts like a filter on )(t u d while H γ is similar to an "equaliser" used to compensate the "channel" filter c G .Also, it is noted that, where H or 0=c G , the feed-forward signal maintains at its initial value. (H and c G are the noise filter and the "closed loop filter" respectively). This means that learning will not take place at these filtered frequencies. One way to get around this is to initialise the feed-forward signal with an estimate of the required control signal. Iterating in i yieldsd ic ff i c i ff U HG U HG U ])1(1[)1(01γγ--+-=+(10)Clearly, the convergence condition is1|)()(1|<-ωωγj G H c , ∀ω present in 0ff U and d U (11)The converged value is given byd ff U U =∞(12)Therefore, in linear systems, convergence requiresthat (11) is satisfied, meaning that the Nyquist plot of c HG γ-1 must fall within the unit circle for all frequencies in )(t u d . Optimally, the curve should be close to the origin to give a high rate of learning for all relevant frequencies.4. Control of the Water Heating Plant usingFilter-based ILCA. Experimental Set-up of the Water HeatingPlantFig. 6 shows the hardware block diagram of the water heating plant - PCT23 Process Plant Trainer. In the experiment, the console is interfaced to a PC via the MetraByte DAS-16 AD/DA card. The plant is controlled by a user-written C DOS program in the PC. This program is interrupt driven and serves to command the required control signal to the plant as well as collect and store plant readings.Control and reading are done at a rate of 1 Hz (which is more than adequate for the system and the ILC bandwidths – see Section 4.3).B. Design of the PI ControllerFilter-based ILC is used to augment the PI feedbackcontroller. The PI controller is tuned using relay experiments [10] according to Ziegler Nichols rule.The ultimate gain K u and period P u are 125.0o C/W and 512.2s , giving the P gain and integral time as:8.562.2==up K K & 8.4262.1==u i P TC. Design of M and γ in Filter-based ILCM is designed with 2 opposing considerations inmind – the noise rejection and the learning rate at high frequencies. High frequency noise is more effectively rejected with a small filter bandwidth, when M is large. On the other hand, as seen in the convergence analysis, H and c G , as low-pass filters, limit the learning rate at high frequencies. Thus, a large M and hence a small filter bandwidth is detrimental to high frequency learning. To reduce the impact of H on high frequency learning, H is designed so that its bandwidth is slightly larger than c G . For the plant, the bandwidth of c G is 0.0045 rad/s. M =10 and 100 give filter bandwidths of 0.14 and 0.014 rad/s respectively. Obviously, M=100 will provide better noise rejection. At the same time, M=100 also gives a bandwidth slightly larger than G c. Thus, M =100 is chosen. The noise rejection effectiveness of this filter will be verified empirically. From the bandwidths of c G and H , it is seen that the sampling frequency of 1 Hz is adequate to prevent aliasing in the signals. Thus, frequency convergence analysis presented in Section 3.B is applicable and the design of γ can be done using Nyquist plots.Using the FOPDT model (1) and the PI controller just obtained, the Nyquist plot (Fig. 7) of c HG γ-1 with100=M is obtained for γ=0.25, 0.50, 0.75 and 1.00.Note that the size of the curves, i.e. the heart-shaped lobe,increases with γ. Also, each curve starts at 1+0j for ω=-πf s and traces clockwise back to 1+0j for ω=πf s where f s =1Hz , the sampling interval.90270180Figure 7 - Nyquist Plot of 1-γHG c (M=100)Thus, all curves have portions outside the unit circle. Thus, convergence is not guaranteed for ω/2π>0.0046, 0.0040, 0.0034 & 0.0028Hz for γ=0.25, 0.5, 0.75 & 1 respectively. However, Fig. 8 shows that the frequency content of the desired control signal (Fig. 9) is negligible for ω/2π>0.003Hz . As a trade-off between stability (location in unit circle) and learning rate (proximity to origin), γ=0.5 is chosen.1010101234567891Figure 8 – Frequency Content of the Desired ControlSignalFigure 9 - Approximate Model Inverse of the DesiredTemperature ProfileD. Filter-based ILC Results for γ=0.5 and M=100Fig. 10 shows the plant output at the 1st iteration - the tracking performance of the PI controller only. From Fig. 11, after 8 ILC iterations, tracking improves vastly. This is also obvious from the RMS error trend in Fig. 12.Figure 10 - T2 with PI Control (K p =56.8 & T i =426.8)Figure 11 - T2 after 8 iterations (γ=0.5 and M=100)Figure 12 - RMS Error for γ=0.5 and M=100In Fig. 11, overshoot and undershoot occur at the turn in )(t u d (Fig. 9). The high frequency components of)(t u d are filtered out from )(t u iff by c G . This resultsin "ringing" at the turn seen in Fig. 13, thus leading to overshoot and undershoot in the output.Figure 13 -)(8t u fffor γ=0.5 and M=100From Fig. 13, it is also seen that the feed-forward signal is relatively smooth and noiseless. This implies that the ILC filter is effective in rejecting noise, making the scheme robust to random perturbations in the system.E. Profile Segmentation with Feed-forwardInitialisationFeed-forward InitialisationTo improve tracking at the turn, a variation (Fig. 14) is attempted. The desired control (Fig. 9) is piecewise smooth - a ramp and a level. Near the turn, due to the ILC filter's "window averagin g” effect, the feed -forward signal is derived from these 2 radical control efforts - one "correct", the other "wrong".Thus, the original profile is divided into 2 entirely smooth profiles. ILC proceeds separately for each - the ILC filter is not applied around the turn. At the same time, the integral part of the PI controller is reset to 0 at the start of each profile. Effectively, it is as though a brand new batch process is started at the turn.In addition, it is easy to estimate, from the first iteration, the static control effort required for the level profile. At the second iteration, the feed-forward signal for this profile is initialised to this estimate. From then on, ILC proceeds normally for all subsequent iterations, further alleviating any inaccuracy in the estimate.Compared with Fig. 12, the RMS error trend in Fig. 15 shows that the improvement from the first to the second iteration is more significant due to the feed-forward initialisation. In addition, the error settles to a smaller value.Figure 15 - RMS Error for Profile Segmentation withFeed-forward InitialisationThe smaller error is due to better tracking at the turn seen in Fig. 16. Compared to Fig. 11, there is very much less overshoot and undershoot.Figure 16 - T2 after 8 iterations (Profile Segmentationwith Feed-forward Initialisation)Through segmentation, the filter’s "window averaging" effect at the turn is eliminated. In general, the system closed-loop bandwidth still limits the frequency components successfully incorporated into the feed-forward signal. This limitation is somewhat compensated by initialising the signal with the control estimate.F. Initial Resetting ConditionOne very important property of ILC is that initial plant reset is required - the feed-forward signal is meaningful only if the plant starts from the same initial conditions in all iterations, this being known as the initial reset condition. Obviously, this is not guaranteed with profile segmentation except for the first section.In other words, initial reset is not satisfied (for the 1st few iterations) for sections except the first. Given convergence in the first section, this occurs for only the first few iterations after which the first section converges to the desired profile and reset is satisfied for (at least) the 2nd section. Note that the end point of the 1st section is exactly the initial point of the 2nd one. Likewise, this applies successively to all subsequent sections.Due to the lack of initial reset for the 1st few iterations, a deviation ∆ will be erroneously incorporated into the feed-forward signal during these iterations. When initial reset is finally satisfied at the i th iteration,∆+=d i ff U U(13)From (9),∆-+=+∆+-=++)1())(1(11c d i ff d c d c i ff HG U U U HG U HG U γγγ (14)Provided |1|c HG γ-<0 for all frequencies of interest, ∆ is reduced with further iterations till it finally becomes negligible. Thus, there is no strict requirement for initial reset in the first few iterations.Obviously, convergence in each section is independent of those sections following it. As long as the first section converges, reset will be observed for the second section after a few iterations. In the same manner, this extends to all sections that follow, provided those sections preceding them converge.5. ConclusionFilter-based ILC is used to improve trajectory tracking in FOPDT repetitive plants, in particular a water heating plant. The ILC scheme is easily implemented on top of a PI controller using simple memory components and easy numeric operations. Trajectory tracking is vastly enhanced with this effortless add-on ILC scheme. With the approximate FOPDT model, the systematic design of M and γ in this scheme can be approached from the frequency perspective. It is found that 100=M is effective in eliminating noise from the feedback signal while not affecting tracking performance. For γ, design is based on Nyquist curves.Filter-based ILC handles smooth profiles best. Thus, for even further improvement in tracking, the piecewise-smooth profile is divided into smooth sections with ILC applied separately to each. Also, to hasten convergence, the feed-forward signal is initialised to the estimated desired control signal, easily achievable if this signal is simple. These modifications give the best tracking.References[1] Z. Bien, J. X. Xu, Iterative Learning Control –Analysis, Design, Integration and Applications , Kluwer Academic Publishers, 1998.[2] T. Y. Kuc, J. S. Lee, K. Nam, An iterative learningcontrol theory for a class of non-linear dynamic systems , Automatica, vol. 28, no. 6, pp. 1215-1221, 1992.[3] H. S. Lee, Z. Bien, A note on convergence propertyof iterative learning controller with respect to sup norm , Automatica, vol. 33, pp. 1591-1593, 1997. [4] K. S. Lee, S. H. Bang, K. S. Chang, Feedback-assisted iterative learning control based on an inverse process model , Journal of Process Control, vol. 4, no. 2, pp. 77-89, 1994.[5] R. W. Longman, Designing Iterative Learning andRepetitive Controllers , Iterative Learning Control – Analysis, Design, Integration and Application, Kluwer Academic Publishers, pp. 107-145, 1998. [6] K. L. Moore, Iterative Learning Control - AnExpository Overview , Applied & Computational Controls, Signal Processing and Circuits, pp. 1-42, 1998.[7] J. Phan, J. Juang, Designs of learning controllersbased on an auto-regressive representation of a linear system , AIAA Journal of Guidance, Control and Dynamics, vol. 19, no. 2 , pp. 355-362, 1996. [8] S. Arimoto, S. Kawamura, F. Miyazaki, Betteringoperation of robots by learning , Journal of Robotic Systems, vol. 1, no. 2, pp. 123-140, 1984.[9] Y. Q. Chen, T. H. Lee, J. X. Xu, S. Yamamoto, Non-Causal Filtering Based Design of Iterative Learning Controller , In Proc. of 1st International Workshop on Iterative Learning Control, Tampa, Florida, pp. 63-70, 1998.[10] K. J. Astrom, T. Hagglund, Automatic Tuning of PIDControllers , Instrument Society of America, 1988.1. OverviewA method of enhancing trajectory tracking in repetitive processes is presented. Tracking enhancement is achieved through the i terative l earning c ontrol (ILC) scheme known as filter-based ILC. In this scheme, only 2 ILC parameters are involved, both of which are open to systematic design in the frequency domain. Also, with its minimal practical requirements, the scheme is very easily implemented on top of PID controllers.This scheme is empirically tested on a PID-controlled water heating plant. Through theoretical modelling and empirical test, it is found that this plant can be approximated with a f irst o rder p lus d ead-t ime (FOPDT) model. This model is used in the design of the ILC parameters. The empirical results show good improvements in tracking.For even better tracking performance, the trajectory is segmented into piecewise smooth sections with ILC applied separately to each. For constant level sections, the feed-forward signal is initialised to the desired control estimated from previous iterations. These modifications are known as profile segmentation and feed-forward initialisation. Empirical test on the heating plant shows that these modifications give the best tracking results.2. Filter based ILCFilter based ILC scheme uses non-causal zero-phase filtering, which simply involves 2 parameters - the filter length M and the learning gain γ. This scheme is robust to random system noise.d y: Desired output profile i e : Error at i th iterationi fb u: Feedback signal at i th iteration i ff u: Feed-forward signal at i th iteration i y: Plant output at i th iterationThe learning law and the overall control signal is )(*)()(1k u h k u k u i fb i ff i ff γ+=+ (15))()()(k u k u k u i fb i ff i +=(16)where k represents the k th time sample of the respective signal, γis the filter gain and *h is the filter operator: moving averager (* denotes convolution)3. Frequency Analysis of Filter-based ILCThe convergence analysis of this scheme is done in the frequency domain. Using this analysis and the approximate model of the plant to be controlled, the systematic design of M and γ can be approached from the frequency perspective.From the analysis, M is found to be critical in eliminating noise from the feedback signal as well as in the tracking performance at high frequencies. A compromise in the value of M must be sought between these 2 opposing considerations.The design of γ is based on Nyquist curves. From the frequency analysis, γ affects the convergence of ILC as well as its convergence rate. Likewise, a compromise must be sought between these 2 opposing factors.4. Empirical Test on a Water Heating PlantFilter based ILC is implemented on a PID-controlled water heating plant. Through theoretical modelling and actual empirical testing, this plant is found to be well approximated by a FOPDT model. This information is used in the design of M and γ.The empirical results show that filter based ILC gives good improvements in tracking.5. Profile Segmentation & Feed-forwardInitialisationFrom the empirical results, it is easy to see that filter-based ILC handles smooth profiles best. Thus, for further improvement in tracking, any piecewise-smooth profile can be divided into smooth sections with ILC applied separately to each - profile segmentation.Also, to hasten convergence, the feed-forward signal is initialised to the estimated desired control signal. This is known as feed-forward initialisation.Coupled with these modifications, filter-based ILC now gives much better tracking performance.Enhancing Trajectory Tracking for a Class of Process Control Problems usingIterative LearningJian Xin Xu, Tong Heng Lee, Yang Quan Chen and Hou TanNational University of Singapore Department of Electrical Engineering10 Kent Ridge Crescent Singapore 119260Singapore(E-mail: elexujx@.sg)。
Systems&Control Letters54(2005)429–434/locate/sysconleSuboptimal control for nonlinear systems:a successiveapproximation approachଁGong-You Tang∗College of Information Science and Engineering,Ocean University of China,Qingdao266071,ChinaReceived11June2002;received in revised form15July2004;accepted24September2004Available online10November2004AbstractThis paper presents a successive approximation approach(SAA)designing optimal controllers for a class of nonlinear systems with a quadratic performance index.By using the SAA,the nonlinear optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value(TPBV)problems.The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence.By using thefinite-step iteration of the nonlinear compensation sequence,we can obtain a suboptimal control law.Simulation examples are employed to test the validity of the SAA.©2004Elsevier B.V.All rights reserved.Keywords:Nonlinear systems;Optimal control;Suboptimal control;SAA1.IntroductionThe investigation of the optimal control is of im-portance in modern control theory.The theory andthe application of optimal control for linear time-invariant systems have been developed perfectly. However,as for nonlinear systems,synthesis prob-lems that are solved by classic control theory lead todifficult computations.People have studied optimalଁResearch supported by the National Natural Science Founda-tion of China(60074001)and the Natural Science Foundation of Shandong Province(Y2000G02).∗Tel.:+865325901980;fax:+865325901225.E-mail address:gtang@(G.-Y.Tang).0167-6911/$-see front matter©2004Elsevier B.V.All rights reserved. doi:10.1016/j.sysconle.2004.09.012control of nonlinear systems for decades.Becerra and Roberts[2]gave out the optimal parameters ap-proximation for the optimal control of the discrete nonlinear systems.Stojanovic[7]focused on the op-timal attenuation control of ellipse nonlinear systems. Sokhin[6]studied the optimal control for a nonlinear system described by afluctuate equation.Hager[3] considered an optimal control method based on the multiplicator machine.Teo and Jennings[10]applied an optimal control approach to nonlinear systems with inequality constraint.Alt[1]analyzed the stability of nonlinear optimal control system with constraint. But all the results mentioned above are still limited to the descriptive ones.These results are limited in real application.430G.-Y.Tang/Systems&Control Letters54(2005)429–434 For the convenient implementation,many subopti-mal control methods have risen which do not pursuethe optimal control performance indexes.Nagurka andYen[4]developed the method of suboptimal controlfor nonlinear system based on the Fourier series.New-man and Souccar[5]addressed the suboptimal controland the robust analysis for second-order nonlinear sys-tems.Xi and Geng[11]presented a predictable controlmethod for suboptimal control of nonlinear systems.Tang,Qu,and Gao[8]proposed a sensitivity approachfor suboptimal control of nonlinear systems.In this paper,a successive approximation approach(SAA)of suboptimal control for a nonlinear system isproposed.Wefirst treat the nonlinear term in the sys-tem as an additional disturbance.Then using the SAAof differential equation theory,we transform the non-linear two-point boundary value(TPBV)problem intoa sequence of nonhomogeneous linear TPBV prob-lems.The optimal control law obtained consists of anaccurate linear feedback term and a nonlinear com-pensation term which is the limit of an adjoint vectorsequence.By using thefinite-stepiteration of the non-linear compensation sequence,we can obtain a sub-optimal control law.Finally,simulation examples areemployed to test the validity of the SAA.2.Problem statementConsider nonlinear systems described by˙x(t)=Ax(t)+Bu(t)+f(x),t>t0,x(t0)=x0,(1)where x∈R n,u∈R r are the state vector and thecontrol vector,respectively.A and B are real constantmatrices of appropriate dimensions,x0is the initialvector,f:C1(R n)→U⊂R n,f(0)≡0whichsatisfies the Lipschitz conditions on R n.The quadraticcost functional of system(1)isJ=12x T(t f)Q f x(t f)+tft0[x T(t)Qx(t)+u T(t)Ru(t)]d t,(2)where Q f,Q∈R n×n are positive-semidefinite ma-trices,R∈R r×r is a positive-definite matrix.The op-timal control problem is tofind a control law u∗(t)which minimizes the quadratic cost functional(2)sub-ject to the dynamic equality constraint(1).According to the optimal control theory and the necessary condi-tions for the optimality,we can obtain the following nonlinear TPBV problem−˙ (t)=Qx(t)+A T (t)+f x (t),˙x(t)=Ax(t)−BR−1B T (t)+f(x),t∈t T=(t0,t f],(t f)=Q f x(t f),x(t0)=x0,(3) and the optimal control law can be written asu∗(t)=−R−1B T (t).(4) Unfortunately,the analytical solution of this nonlinear TPBV problem in(3)is difficult to be solved gener-ally.Therefore,it is necessary tofind the approximate approaches for solving the optimal problem of nonlin-ear systems.In this paper we will propose a SAA. 3.PreliminariesConsider the time-varying nonlinear system˙x(t)=A(t)x(t)+f(x),t∈t T,x(t0)=x0,(5) where x∈R n is the state vector,A∈R n×n,x0is the initial state vector,f:C1(R n)→U,f(0)≡0which satisfies the Lipschitz conditions on R n.Define vector function sequence{x(k)(t)}asx(0)(t)= (t,t0)x0,t∈t T,x(k)(t)= (t,t0)x0+tt0(t,r)f(x(k−1)(r))d r, t∈t T,x(k)(t0)=x0,k=1,2,...,(6) where is the state transfer matrix with respect to the matrix A(t).Lemma1.Sequence(6)uniformly converge to the so-lution of system(5).Proof.Consider{x(k)(t)}as a sequence of C Nt0,t f, from(6)x(1)(t)−x(0)(t)=tt0(t,r)f(x(0)(r))d r,t∈t T.(7)G.-Y.Tang /Systems &Control Letters 54(2005)429–434431Because f satisfies the Lipschitz conditions on R n ,it followsM =sup t ∈R T(t,t 0) ,= z 0 ,sup t ∈R T f (z,t) z ,z ∈U,sup t ∈R Tf (v,t)−f (w,t) v −w ,v,w ∈U,(8)where M , , ,and are some positive constants, · denotes any appropriate vector or matrix norm.Noting that (t 0,t 0) = I =1,hence M 1.From (7)and (8)we obtainx (1)(t)−x (0)(t) Mtt 0x (0)(r) d r M 2(t −t 0),t ∈t T .(9)From (6)one gets x(2)(t)−x(1)(t)=tt 0(t,r)f (x (1)(r))−f (x (0)(r)) d r,t ∈t T(10)and thenx (2)(t)−x (1)(t)M t t 0 f (x (1)(r))−f (x (0)(r)) d rM tt 0x (1)(r)−x (0)(r) d r12!M 3(t −t 0)2,t ∈t T .(11)By the mathematics induction,we obtain x(k)(t)−x(k −1)(t)k −1Mk +1(t−t 0)kk !,t ∈t T ,k =1,2, (12)According to trigonometry inequality,for any jx (k +j)(t)−x (k)(t)k +ji =k +1i −1 Mi +1(t −t 0)i i !k M k +2(t−t 0)k +1exp ( M(t −t 0)),t ∈t T ,k =0,1,2,....(13)Thus {x (k)(t)}is a Cauchy sequence in C N [t 0,t f ].This sequence is uniformly convergent [9].For j is random,the limit of this sequence is the solution of system (5).The proof is complete. 4.SAA designing processConstruct the set of TPBV problems(0)(t)=f (x (0))=0,t ∈t T ,−˙(k)(t)=Qx (k)(t)+A T (k)(t)+(f x (k −1)(x (k −1)) x (k −1)(t),˙x (k)(t)=Ax (k)(t)−BR −1B T (k)(t)+f (x (k −1)),t ∈t T ,(k)(t f )=Q f x(t f ),x (k)(t 0)=x 0,k =1,2,...(14)and the corresponding optimal control sequence u (k)(t)=−R −1B T (k)(t).(15)For the k th optimal problem,optimal state trajectoryand optimal control law are x (k)(t)and u (k)(t),respec-tively.We give out the following theorem.Theorem 1.Assume that {x (k)(t)}and {u (k)(t)}are the solution sequence of (14).Then {x (k)(t)}and {u (k)(t)}uniformly converge to the optimal state tra-jectory x ∗(t)and optimal control law u ∗(t)for system (1)with the quadratic cost functional (2),respectively .Proof.Let(k)(t)=P (t)x (k)(t)+g (k)(t),t ∈t T ,k =1,2,...,(16)where P ∈R n ×n is unknown positive-semidefinite function matrix.g (k)∈R n is the k th adjoint vector.Calculating the derivative to the both side of the (16),we get˙ (k)(t)=˙P (t)x (k)(t)+P (t)˙x (k)(t)+˙g (k)(t)= ˙P(t)+P (t)A −P (t)SP (t)x (k)(t)−P (t)Sg (k)(t)+P (t)f (x (k −1))+˙g (k)(t),(17)432G.-Y.Tang /Systems &Control Letters 54(2005)429–434where S =BR −1B T .Substituting (16)into the second equation of (14)and comparing with (17),one can ob-tain the following Riccati matrix differential equation:−˙P(t)=P (t)A +A TP (t)−P (t)SP (t)+Q,P (t f )=Q f (18)and adjoint vector differential equations˙g (k)(t)=−[(A −SP (t))]T g (k)(t)−P (t)f (x (k −1))−f x (k −1)(x (k −1)) (k −1),g(k)(t f )=0,k =1,2,3, (19)We can get the unique positive-semidefinite matrix solution P (t)from (18).Note that P (t),f (x (k −1)),f x (k −1)(x (k −1))and (k −1)(t)are known functions in (19).So (19)is a nonhomogeneous linear vector dif-ferential equation for some certain k .Then we can obtain g (k)(t)by using reversing integration:g (0)(t)=0,g (k)(t)=t ft(t,r)[P (r)f (x(k −1)(r))+f x (k −1)(x (k −1)(r)) (k −1)(r)]d r,k =1,2,3,...,(20)where is the state transfer matrix with respect to the matrix (SP −A)T .Substituting (16)into (15),we obtain the k th optimal control lawu (k)(t)=−R −1B T [P (t)x (k)(t)+g (k)(t)].(21)Substituting (16)into the third equation of (14),we can get the k th optimal closed-loop system ˙x (k)(t)=[A −SP (t)]x (k)(t)−Sg (k)(t)+f (x (k −1)),x (k)(0)=x 0.(22)According to Lemma 1,the solution sequences{g (k)(t)},{x (k)(t)}of (20)and (22)are uniformlyconvergent.The control sequence {u (k)(t)}is only related to {x (k)(t)},{g (k)(t)},so it is also uniformly convergent.Define g(t)and u ∗(t)as the limits of sequences {g (k)(t)}and {u (k)(t)},respectively.Ac-cording to Lemma 1,the limit of sequence {x (k)(t)}is the optimal state trajectory x ∗(t)of the optimalcontrol problem (1)with the quadratic cost functional(2).Then we get the optimal control law u ∗(t)=−R −1B T [P (t)x(t)+lim k →∞g (k)(t)].(23)The proof is complete.In fact,we cannot calculate the optimal control lawin (23).We may find a suboptimal control law in prac-tical applications by replacing ∞with N in (23)u N (t)=−R −1B T [P x(t)+g (N)(t)].(24)Remark 1.In (24)x(t)is an accurate solution in case of k →∞,only g (N)is an approximation by substi-tuting a finite-stepiteration of g (N)for g (∞).So the suboptimal control law u N (t)in (24)is closer to the optimal control law than u (N)(t)in (21).Algorithm 1.Suboptimal control law of system (1)Step 1:Solve the positive-semidefinite matrix P (t)from Riccati matrix differential equation (18).Let x 0(t)=g 0(t)=0,J 0=0and k =1.Step 2:Obtain the k th adjoint vector g (k)(t)from (20).Step 3:Letting N =k ,calculate u N (t)from Eq.(24).Step 4:Find J N fromJ N =12x T(t f )Q f x(t f )+ t f[x T (t)Qx(t)+u T N (t)Ru N (t)]d t .(25)Step 5:If |(J N −J N −1)/J N |< ,then stopand out-put u N (t).Step 6:Calculate x (k)(t)from (22).Step 7:Letting k =k +1,go to step2.Remark 2.If t f →∞,the quadratic cost functional (2)is rewritten asJ =12∞0[x T (t)Qx(t)+u T (t)Ru(t)]d t.(26)The algorithm presented also can be used.In this sit-uation,Riccati matrix differential equation (18)is re-duced as the following Riccati matrix equation:A T P +P A −P BR −1B T P +Q =0.(27)G.-Y.Tang/Systems&Control Letters54(2005)429–434433Fig.1.Simulation curve of the system when k=1,2,3,and4.5.A simulation exampleConsider the nonlinear system described by˙x1˙x2=01−11x1x2+x1x2x22+1u,x1(0) x2(0)=−0.8.(28)The quadratic cost functional is chosenJ=12 10(x21+x22+u2)d t.(29)Simulation results are presented in Fig.1.The cost functional values at the different iteration times are listed in Table1.From Fig.1,it can be seen that the more iterates we take,the better are the approx-imations to both the state and the control functions. This leads to a better approximation of the optimal cost shown in Table1.If we choose =0.01,then the relative error of the cost functional values satisfies |(J4−J3)/J4|< .It indicates the4th suboptimal con-trol law u4is very close to the optimal control law u∗.Table1Cost functional values at the different iteration timesIteration time k1234 Cost functional J8.5557 6.9407 6.5392 6.52576.ConclusionsA successive approximation algorithm has been used to generate the suboptimal solutions of nonlinear systems.It consists of solving a Riccati equation and a vector linear differential equation at each step.The results of the simulation show the validity of the ap-proach mentioned.The proposed method is promising and easy to implement.References[1]W.Alt,Stability of solutions to control constrained nonlinearoptimal control problems,Appl.Math.Optim.21(1)(1990) 53–68.[2]V.M.Becerra,P.D.Roberts,Dynamic integrated systemoptimization and parameter estimation for discrete time optimal control of nonlinear systems,Internat.J.Control63(2)(1996)257–281.434G.-Y.Tang/Systems&Control Letters54(2005)429–434[3]W.W.Hager,Multiplier methods for nonlinear optimal control,SIAM J.Numer.Anal.27(4)(1990)1061–1080.[4]M.L.Nagurka,V.Yen,Fourier-based optimal control ofnonlinear dynamic systems,Trans.ASME J.Dyn.Syst.Meas.Control112(1)(1990)17–26.[5]W.S.Newman,K.Souccar,Robust,near time-optimal controlof nonlinear second-order systems:theory and experiments, Trans.ASME J.Dyn.Syst.Meas.Control113(3)(1991) 363–370.[6]A.S.Sokhin,Some optimal-control problems for a nonlinearcontrol system described by a wave equation,Differential Equations17(3)(1981)346–353.[7]S.Stojanovic,Optimal damping control and nonlinear ellipticsystems,SIAM J.Control Optim.29(3)(1991)594–608.[8]G.-Y.Tang,H.-P.Qu,Y.-M.Gao,Sensitivity approach ofsuboptimal control for a class of nonlinear systems,J.Ocean Univ.Qingdao32(4)(2002)615–620(in Chinese).[9]A.E.Taylor,y,Introduction to Functional Analysis,second ed.,Wiley,New York,1980.[10]K.L.Teo,L.S.Jennings,Nonlinear optimal control problemswith continuous state inequality constraints,J.Optim.Theory Appl.63(1)(1989)1–22.[11]Y.Xi,X.Geng,The suboptimality analysis of predictivecontrol for continuous nonlinear systems,Acta Automat.Sinica25(5)(1999)673–676.。
第 63 卷第 2 期2024 年 3 月Vol.63 No.2Mar.2024中山大学学报（自然科学版）（中英文）ACTA SCIENTIARUM NATURALIUM UNIVERSITATIS SUNYATSENI基于Lyapunov-MPC方法的非合作目标近距离抵近控制*杜兴瑞，孟云鹤，陆璐中山大学人工智能学院，广东珠海 519082摘要：针对航天器近距离视线抵近同时存在轨道与姿态机动的非合作目标特定方位控制问题，提出了一种基于控制Lyapunov函数（CLF）的模型预测控制方法（LMPC）。
首先，根据视线坐标系下的轨道动力学方程，建立了满足视线指向要求的航天器相对轨道动力学模型，并推导了期望轨道的解析表达式。
其次，利用MPC方法设计控制器进行在线优化控制，并通过纳入基于李雅普洛夫方法的非线性反步控制的显式特征，构建收缩约束式，以确保闭环稳定性。
接着，对基于LMPC的控制方法的递归可行性和闭环稳定性进行了证明。
最后，仿真结果证明了所设计的LMPC轨迹跟踪方法的有效性和鲁棒性。
关键词：非合作目标；近距离视线抵近；非线性动力学系统；控制Lyapunov函数；模型预测控制中图分类号：V19 文献标志码：A 文章编号：2097 - 0137（2024）02 - 0085 - 10Non-cooperative target proximity control based onLyapunov-MPC methodDU Xingrui， MENG Yunhe， LU LuSchool of Artificial Intelligence， Sun Yat-sen University， Zhuhai 519082， ChinaAbstract：A model predictive control method based on Lyapunov function （LMPC）is proposed to control the specific orientation of non-cooperative spacecraft with orbit and attitude maneuvers. Firstly，according to the orbital dynamics equation in the line-of-sight coordinate system，the relative orbital dynamics model of the spacecraft satisfying the requirements of line-of-sight direction is established，and the analytical expression of the expected orbit is derived. Secondly， the MPC method is used to de‐sign the controller for online optimization control， and the contraction constraint formula is constructedto ensure the closed-loop stability by incorporating the display features of the nonlinear backstepping control based on the Lyapunov method. Then，the recursive feasibility and closed-loop stability of LMPC based control method are proved. Finally， the simulation results prove the effectiveness and ro‐bustness of the proposed LMPC trajectory tracking method.Key words：non-cooperative target； close line of sight； nonlinear dynamic system； control Lyapunov function；model predictive controlDOI：10.13471/ki.acta.snus.2023B047*收稿日期：2023 − 07 − 09 录用日期：2023 − 08 − 02 网络首发日期：2023 − 09 − 26基金项目：国家自然科学基金（61673390）作者简介：杜兴瑞（1997年生），男；研究方向：智能控制；E-mail：****************通信作者：孟云鹤（1978年生），男；研究方向：智能控制；E-mail：*****************第 63 卷中山大学学报（自然科学版）（中英文）航天器的自主抵近控制是实现一系列空间近距离操作任务的关键技术，在空间态势感知、空间攻防、在轨服务等领域都有着广泛应用。
