EXTENDED STABILITY MARGINS ON CONTROLLER DESIGN FOR NONLINEAR INPUT DELAY SYSTEMS. Otto J. Roesch, Hubert Roth, Asif Iqbal

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1 EXTENDED STABILITY MARGINS ON CONTROLLER DESIGN FOR NONLINEAR INPUT DELAY SYSTEMS Otto J. Roech, Hubert Roth, Aif Iqbal Intitute of Automatic Control Engineering Univerity Siegen, Germany {otto.roech, Abtract: In thi paper, upper bound for time-varying delay in the real-time control of a lab helicopter in the ene of practical tability analyi are derived. Sub-optimal controller parameter for thi non-linear plant with a given maximum time-delay are calculated with the help of Lyapunov-Razumikhin method uing LMI (Linear Matrix Inequalitie). Thee parameter are then optimized for extended time-delay margin by conidering deign pecification like ettling-time and overhoot, uing a novel 3- dimenional earch algorithm in the vicinity of the previouly identified value. Trade-off between ettling-time and overhoot i achieved with the help of appropriate weighting factor in an objective function baed approach. Copyright 25 IFAC Keyword: Time delay, Lyapunov Stability, Remote Control, State Space Realization, Nonlinear Analyi INTRODUCTION The preence of time delay in indutrial procee i an important phenomenon and mut be conidered in the tability analyi and controller deign. Delay could occur becaue of data communication or by material flow in machine. They can be claified a contant or time-varying, bounded or unbounded, ditributed or not, determinitic or tochatic. Control of thee time-delay ytem i an active reearch area. Normally the delay may appear either in the tate, or in the input (or output) of the correponding ytem (Marhal, 979). Stability of uch ytem i the mot important property to be conidered and become hard to etablih keeping in mind the fact that introduction of time-delay in the tate or input caue the ytem to have infinite number of pole or zero (Marhall, 992) for cloed loop ytem. The reported tability could either be delay-independent or delay-dependent, the former being more retrictive than the later and more difficult to enure (Bouka, 22). In thi paper, ignal delay due to data-tranmiion are conidered, which could occur e.g. by large/long bu-ytem or by the ue of the internet (with TCP/IP or UDP a protocol). If the control-ytem and the plant are patially eparated, the delay occur two time in the control loop, once from the plant to the controller (forward direction), and econdly from the controller back to the plant (backward direction), ee Fig.. Since time delay in the control loop have a major influence on tability of control ytem, they mut be carefully conidered in the mathematical decription of the ytem dynamic. Zhang et al. (2) give an excellent overview of tability iue in networked control ytem. They divide the tability iue into two categorie, i) developing new network protocol for control application, and ii) to take the network a it i, and focu on control algorithm to achieve tability. For a general cla of ytem, the author ugget to mark tability region uing imulation while continuing to increae the time delay. In our work, we firt find a conervative controller uing the Razumikhin theorem, and then we go further and receive with a imulation etup optimal control parameter for a table control ytem.

2 Thi contribution i baed on the work by Niculeu et. al. (997), the delay-dependent tability with an input delay of a twin rotor laboratory helicopter i conidered. For a remote control, two kind of trategie can be conidered for uch a plant. In the firt one, a control law i deigned remotely and i ent to the plant over ome communication link, which i executed locally on the plant. In thi cae there i no input delay in the ytem and hence no bound on delay. The only condition i that the ytem mut be tabilizable. In the econd cae, the output of the controller toward the plant i delayed. Thi caue an input delay and thu introducing the concept of tability margin baed on time-delay, which will be the focu of thi work. An LMI (Linear Matrix Inequality) baed approach i ued to find the parameter of a third-order tatefeedback controller. A Lyapunov-Razumikhin function maximize the time-delay (in ub-optimal ene) of the cloed loop ytem for a table ytem like the one conidered in Fig.. Once the tability of thi ub-optimal controller i achieved, it become of practical interet to ee how much the parameter of thi tabilizing controller can be varied while remaining within the tability margin of the ytem. Furthermore, can the upper bound on the input delay be improved for uch a perturbation? Thee quetion will be dealt with in detail within the following ection. In addition, pecific conideration on the performance of the plant, uch a ettling-time, overhoot, etc., can alo be taken care of in the deign of the controller a will be hown later. 2 PROBLEM DEFINITION In contrat to ytem free of delay, a time-delay ytem conider not only the current ytem tate, but alo the pat tate which lead to a ytem decribed by delay differential equation (DDE): ( ) xt () = f t, xt (), xt ( τ ), () n with x, τ >. And appropriate initial condition derived on the interval [t -τ, t ]. Due to the tremendouly increaing uage of tele-control over the internet, uch time delay control ytem get more and more important in practical application. A general repreentation i hown in Fig., with K a the controller, G the plant, w the deired input and x the plant output. The time delay occur from the controller to the plant and again from the plant backward to the control unit. r + - e x d K u delay delay 2 Fig.. Cloed loop ytem with time-delay in the communication channel y G x In general, control deign algorithm do not conider time delay apect in the control loop, like root-locu, Nyquit plot or pole placement approache. Stability analyi of time delay ytem are eaier to handle in the time domain than in the frequency domain, epecially in the cae of time-varying uncertaintie and nonlinearitie (Gu, 23). Baed on the econd Method of Lyapunov, the theorem uch a the Kraovkii and the Razumikhin provide tability condition for time-delay ytem in the time-domain. Thee method deal excluively with tate pace decription and the tability tet amount to ufficient condition that can be poed a olution of LMI problem. Thee robut control approache are very conervative and have large tability margin, a it wa eventually experimentally teted. Baed on a controller, calculated with the Razumikhin theorem, a 3-D earch algorithm i impoed for further improving the ytem repone. 3 DEFINITION OF THE LYAPUNOV- RAZUMIKHIN FUNCTION The Razumikhin theorem i an extenion to the claical Lyapunov tability theorem. It i ued in the derivation of the following controller deign by Niculecu, et al. (997), the definition can be found in (Hale, et al., 993) or (Gu, 23). 3. Solution Approach Controller Algorithm Baed on a ytem preented in tate pace form, we can derive the equation with an input delay of the following form: The ytem x () t = Ax() t + Bu() t conider the following input ut () = Kx( t τ () t) a a control law. Thi reult in a cloed loop decription to x () t = Ax() t + BKx( t τ () t ). Applying the Razumikhin theorem for uch an input delay ytem, the following theorem found in Niculecu, et al. (997), reult in: If the following inequalitie hold imultaneouly T QA + AQ τ * + BW + + ( β + β ) Q 2 W B W B T BW Q 2 < (2) β Q+ AQA (3)

3 Q BW β 2 + W B Q (4) nxn with Q being poitive-definite and ymmetric, mxn a matrix W and the poitive calar β and β 2 hold, then the cloed loop ytem i uniformly aymptotically table. The input of the form ( τ ) ut () = WQ x t () t (5) for all the delay τ () t atifie τ ( t) τ * with K - = WQ i a ubtituting controller. For a complete proof ee the excellent paper of Niculecu, et al. (997). min λ ( W, β, β ) 2.t. W, β, β2 (3), (4), (6) (8) hold for Q> (9) Step three: Solve the LMI ytem (equation (3)(4)(6)(7)(8)) wherea W, β and β 2 are fixed. min λ ( Q) Q> t.. (3), (4), (6) (8) hold for fixed W,β,β 2 () Return to tep 2 until the convergence of λ i attained within a deired preciion. 3.2 Controller Calculation In order to olve inequalitie (2), (3) and (4), it i neceary to convert the bilinear condition into a LMI ytem, baed on the algorithm propoed by Niculecu, et al. (997). (Thi i neceary becaue the variable β and β 2 are multiplicatively aociated with Q, and therefore not olvable with available LMI olver). To olve a o called generalized eigenvalue problem (gevp), like equation (2), a calar (here: λ = τ *) get minimized by holding the et of inequalitie. The optimization variable mut be an iolated outer-factor of a LMI-matrix. LMI-Solver (e.g. LMI-Toolbox in Matlab) can not handle equation (2) directly. It mut be firt converted into the generalized eigenvalue problem (gevp) form, wherea the LMI of equation (2) yield to ( β + β ) Q BW 2 Y < W B Q 2 [ ] > defined! (6) T Y < λ( )( QA + AQ+ BW + W B ) (7) T ( QA AQ BW W B ) > (8) with λ =. τ * Now the upper bound for the time delay τ * can be maximized by minimizing λ. Algorithm from Niculecu, et al. (997), to olve the nonlinear gevp problem Step one: Solve equation (8) with repect to Q >. Step two: Solve the LMI ytem (equation (3)(4)(6)(7)(8)) wherea Q i fixed. 4 HELICOPTER SYSTEM The ytem under conideration i repreented by a helicopter model from Quaner (24), ee Fig. 2. It conit of a fixed bae, on which a rotary arm i mounted. The arm carrie the helicopter body on one end, and a counterweight on the other. The arm can make an elevation motion around the angle epilon. An encoder mounted on the axi allow meauring the elevation angle of the arm. The helicopter body i fixed at one end of the arm and i free to elevate in a certain range. Two motor with propeller mounted on the helicopter arm can generate a force proportional to the voltage applied to the motor. The force generated by the propeller caue the helicopter body to lift off the ground. The purpoe of the counterweight i to reduce the power requirement on the motor around equilibrium. The correponding nonlinear mathematical model look like the following: J ε = g y ( M + m) in( ε) + 2 k r v () ge The dimenion parameter can be een in Fig. 2, where k t =.5 N/V repreent the motor contant and v the voltage to the motor, g the gravity contant and J ge the moment of inertia around the rotating point, m i the ma of the helicopter blade incluive motor and the fixing device, M i the counterweight. Pleae note, that if yt () i not available, we can contruct a claical (aymptotic) oberver or to approximate yt () by ome delay term yt () yt ( θ ) yt (), a uggeted by Niculecu and θ Michiel (24) for a different control problem. To apply the Lyapunov-Razumikhin control algorithm preented in the previou ection, the model mut be linearized around the quiecent point ε = and tranferred into the tate pace form x = Ax + Bu (2) y = Cx t

4 ε x = ε ε (3) to 3 equentially to conider all poible combination (thi range of the gain wa previouly analyed). The Matlab Simulink imulation block diagram i given in Fig. 3. M+ F xg Rotation point ε y l r F h F xm F ym F gm Delay Motor Voltage tate vector Nonlinear Helicopter Model GainK GainK em K epilon integral Elevation Angle Deired Angle2 F vg F gg GainK Fig. 2. Schematic drawing of the 3D Helicopter model with fixed travel and pitch axi GainK2 GainK3 K Integrator A third tate wa added to repreent the integral of the elevation angle, which i neceary to reach teady tate accuracy of the elevation angle. Parameter identification enabled the analyi of the damping coefficient for an improvement of the mathematical model. Thi wa done by the actuation of a tep input to the real ytem while the decaying ocillation wa oberved, for which a damping factor got derived. The correponding numerically computed tate pace model reult finally in: A = ; B = C = (4) I 3 ; By applying the Razumikhin control algorithm from the previou ection, the following control vector i obtained K = [ ], (5) for a maximum time delay of τ * =.6 econd. By applying thi control vector to the real plant, we can ee that the controller i very conervative and how even a table behaviour for time delay up to.4 econd. The analyi of the tability margin on the controller deign wa neceary for a further improvement of the ytem, even for mall delay. 5 NONLINEAR SIMULATION DEVELOPMENTS For a tability margin extenion, a imulation of the nonlinear mathematical model of the helicopter i et up in Simulink. The above calculated controller parameter from the Razumikhin approach are ued a a tarting point. By varying thee controller parameter ucceively with additional gain factor and conidering the ettling time and the overhoot, a further improvement of the control ytem i derived. Thee additional gain factor were increaed from. GainK3 Fig. 3. Simulink imulation for the tability margin extenion with varying additional gain factor with the nonlinear helicopter model A hown in Fig. 3, the additional gain parameter (GainK, GainK2 and GainK3) are multiplied with the parameter from (5). The Elevation Angle i conidered for the analyi of the ettling time and overhoot. Delay wa fixed for each pa-through to get the bet controller parameter. 6 RESULTS The imulation reult obtained in ection 5 are hown in Table, where the delay i varied in tep from. to.7 econd, with an increment of. econd. It mut be noted that in obtaining thee parameter, the objective i to minimize the ettling-time for a given value of time delay. Settling time and overhoot againt the gain are plotted in Fig. 4. Sytem repone in term of elevation angle for τ =.4 ec i hown in Fig Settling time 3 [ec] Elevation-Angle.3.2 Fig. 4. Settling time and elevation angle v. gain, for an elevation tep-input of.25 rad K3 2 3 x 4 Index of the Gain-Vector

5 that give weight to the overhoot in addition to the ettling time. Thi objective function i then evaluated over the whole pace of admiible gain parameter. Minimum value of thi objective function will provide the optimal gain parameter while atifying the performance requirement. Radian imulation reult approx. function.5 Careful examination of the data in Table reveal a teady decreae in the gain parameter for increaing time delay. Thi can be explained in the context of paivity theory. Lowering the gain on control input i equivalent to diipating the additional energy induced into the ytem by time-delay (Anderon and Spong, 989). At the ame time, we ee an increae in the ettling time for increaing delay which i quite normal. It ha to be noted that the Razumikhin theorem provide the gain parameter for time-varying delay o the ytem will remain table with given gain value even for time-varying delay a far a the delay remain bounded by a certain given value. In order to obtain a continuou range of gain for intermediate value of delay (in the range from to.7 econd), the identified value of gain are linearized afterward, a hown by the plot in Fig. 6 where the additional gain parameter GainK i diplayed. The dot are the actual value, produced by the imulation, wherea the olid line i obtained by linear curve fitting, providing gain that are finally ued in the real-time implementation, which how good performance on actual non-linear plant. Table : Simulation reult for the additional GainKx parameter Time Delay (ec) Time (econd) Fig. 5. Elevation angle ytem repone example with minimal ettling time for a contant delay of τ =.4 econd Min. Settling Time (ec) Max. Overhoot (rad) Gain K Gain K2 Gain K Furthermore, if it i deired to improve the overhoot behavior in the repone of the ytem, it can be achieved by firt contructing an objective function Gain K delay in econd Fig. 6. Linearization of the additional gain parameter, GainK i hown here, with a delay of to.7 econd. The following expreion i ued a the objective function to be minimized f ( t, θ ) = α t + β θ, (6) OS OS where t and θ OS are ettling-time and overhoot repectively. α and β are weighting factor Objective Function Gain x 4 Fig. 7. Objective function f( t, θ ) v. Gain If the deired function give 75% weight to ettling time and 25% to overhoot then α =.75 and β =.25. Applying thi performance requirement to the cae where τ =.4, τ being the time delay, and conducting a earch over admiible control, one get new value for the ettling time and the overhoot t = ( t = 6.38), old θ =.25 ( θ =.32) OS OS, old Thee value indicate that a % increae in ettlingtime can be traded-off againt a 58.75% decreae in OS

6 overhoot. Previou and new gain obtained through the weighted objective function in (6) are given below: K = [..4.] new (7) K = old [ ] A plot of the objective function v. gain i hown in Fig CONCLUSIONS In thi work, the Lyapunov-Razumikhin approach i ued to identify ub-optimal controller parameter for a nonlinear delayed input ytem while maximizing the time-delay and enuring the tability. Thee controller parameter are then optimized uing a combination baed on 3D earch in order to achieve certain performance requirement like ettling time and overhoot. The delay bound given by Razumikhin theorem are extended uing our approach in the ene of practical tability. An approach baed on weighted objective function i further ued to achieve a performance balance between minimum ettling time and maximum overhoot allowed. The earch algorithm preented ue combination of parameter value to find the optimal value with an increment of. in the pace of admiible gain. Fig. 7 how that we are able to find the global minimum in term of ettling time. At the ame time, it can be concluded that even more accurate parameter can be obtained if the incremental tep ize i reduced, which become demanding in term of computational requirement. For future work, it i therefore uggeted to ue different heuritic approache like genetic algorithm and imulated annealing to optimize the earch for controller parameter. REFERENCES Anderon, R.J.; Spong, M.W. (989). Aymptotic tability for force refecting teleoperator with time delay. Proceeding of the IEEE International Conference on Robotic and Automation, page Bouka, E.-K.; Liu, Z.-K. (22). Determinitic and Stochatic Time Delay Sytem, Control Engineering, Birkhaeuer Gu, K.; Kharitonov, V.L.; Chen, J. (23). Stability of Time-Delay Sytem, Birkenhäuer, 23 Hale, J.K.; Renardy Y.; Wheeler, R.L. (993). Introduction to functional differential equation, In: Volume 99 of Applied Mathematical Science, Springer-Verlag. Marhall J.E.; Gorecki H.;,Korytowki A.;, Walton, K., (992). Time Delay Sytem, Stability and Performance Criteria with Application, Elli Horwood Pre. Marhall, J.E. (979). Control of time-delay ytem, Peter Peregrinu, Ltd. Niculecu, S.I.; Michiel, W. (24). Stabilizing a Chain of Integrator Uing Multiple Delay, IEEE Tranaction on Automatic Control, Vol.49, No.5. Niculecu, S.-I.; Fu, M.; Li, H. (997). Delay Dependent Cloed Loop Stability of Linear Sytem with Input Delay: An LMI Approach, Proceeding of the 36th Conference on Deciion & Control, San Diego, California, USA Quaner (24) 3D Helicopter Experiment with DAQ Board and WinCon Software, Zhang, W.; Branicky, M.S.; Phillip, S.M. (2) Stability of Networked Control Sytem, IEEE Control Sytem Magazine. For a practical uage, the above preented procedure for the derivation of a controller mut be automated in ome way. For example, by pecifying the tranfer function and the maximum time delay (which could occur in the ytem), a control engineer could directly get the controller. Of coure it i not known a priori if a table controller for the pecified maximum time delay can be found by the preented algorithm a it depend on the plant and the maximum delay of the ytem. The function for approximation of gain parameter baed on input delay (ee Fig. 6) i only a linear function. Experimental reult how that it can further be improved by uing a higher order function. The approach decribed in thi work, make the following aumption for ignal RTT (Round Trip time), a i generally done in network control ytem literature RTT = delay + delay2 (ee Fig. ).