Time delay limits in active control systems with closed-loop linear algorithm

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1 Time delay limits in active control systems with closed-loop linear algorithm A. Baratta/" O. Corbi^ Department of "Scienza delle Costruzioni", University of Naples "Federico II". Piazzale Tecchio 80, Naples, Italy it Abstract The purpose of the paper is to investigate the problem of delayed active control, with reference to a s d.o.f. system controlled by a closed-loop linear algorithm and under the action of a dynamic forcing function. The exceeding of specified limits of the time gap elapsing between the sensors' reading of the excitation and / or of the response variables and the starting of the control actuation devices can, actually, vanify the control system effectiveness with reference to structural response attenuation or, even, lead quickly to system instability. Besides, this circumstance can also be met for particular values of the occurred time delay, deeply linked to the structure mechanical characteristics and to the control parameters. As many researches have been focused on the objective of reduction of the undesirable effects produced on the structures and their response by the above described conditions, the aim of the present work is to analyse these effects induced on the response of the sample structural model considered by the introduction in the control law of assessed critical values of time delay. A comparison is then presented between the numerical results obtained by optimising the performance index of the delayed control system for the identified critical delay values and the results one can obtain by searching for the minimum value of a suitably defined norm of the delay-controlled response. 1 Introduction Particular values assumed from the time delay usually occurring in active structural control systems or its exceeding of specified limits can result in a lack of effectiveness for the control system itself.

686 Earthquake Resistant Engineering Structures When, by means of extremely sophisticated hardware, it would be admissible to think to small delay values, one could get an effective compensation by

2 686 Earthquake Resistant Engineering Structures When, by means of extremely sophisticated hardware, it would be admissible to think to small delay values, one could get an effective compensation by the introduction in the circuit of one or more elements (phase lag compensators) suitably calibrated in order to amplify the phase of the system transfer function. Theoretical and experimental have tested the effectiveness of such methods in the steady-state field, as the introduction of time delay can actually, by reducing the system phase margin, lead to instability when this has rolled off to zero. Critical time delay values can, thus, immediately be determined for a given control law, by deducing them from the limit condition of instability occurrence, and phase-shift approaches to compensate for time lag effects produced on the stationary structural response can be adopted. A different method to search for the optimal control parameters in the case of delay occurrence even in the transient phase, can be pursued by the optimisation of the controlled response with the control force constrained under a prefixed threshold (Baratta et al, 1992, 1994). The reliability of this approach for critical delay values is studied, showing that the procedure, based on the research of the optimal values of the control coefficients by the mapping of the norm solution of the controlled system response, constitutes a really effective control method, transient- steady state reliable with respect to time delay effects. Furthermore, in the following, a comparison is lead between the results obtained by adoptimg the two described procedures. 2 Delayed equation of the motion The equation of the motion for a s.d.o.f. controlled by means of a linear delayed control law can be written as follows c'm(r-r)-^w(r-r) (1) where superimposed dots denote time-derivatives, m, c and h are respectively the structural system mass, damping capacity and stiffness, u(f) is the structure's displacement variable, c'and k' are the control system parameters, T is the introduced time delay and/(t) the external excitation. In standard form «(/)+24>XO+.*«(0= -2 'a'u(t-t)-a>"u(t-t) (2) with <^,&^and,co' respectively the damping coefficient and frequency of the structure and of the control force. As the steady-state response of a delayed control system subjected to a harmonic excitation f^e""*' of amplitude f^ and frequency co ^ can be analysed by operating a Fourier transform of the equation, of motion eqn (1) with the delayed control force one can deduce the amplitude of the steady-state response of the controlled system in the case of time delay occurrence (Hou and Iwan, 1992)

3 Earthquake Resistant Engineering Structures 687 l - S~ ^ (3) where <5 = ^, /l = -, // =. (4) ft?,, k con Assume that the control parameters are chosen adopting any procedure for ideal synchronous control. The conditions for which the denominator of the amplitude X^ for given structural characteristics and control parameters, becomes zero, define the resonance frequencies ^corresponding to the instability condition for the specific control law. The critical values of time delay can then be identified as those for which the system phase reaches the instability phase condition Q)fT-(f/( f)= (2j + l)r (5), V -^M - (6) withy is an arbitrary integer and %/the system phase The two conditions correspond, in terms of Bode representation system transfer function, to the case when the phase of the synchronously controlled system reaches -180 at the frequency where the logarithmic gain of the system transfer function is equal to zero (the cross-over frequency ft>,,), that's to say when the phase-margin (P.M.} becomes zero. The phase reduction introduced by time delay in control actuation produces the following decrease of the system P.M. (Suner, Nagchaudhun and Utku, 1995) P.M. =/20 + ^ -» P.M. = /<30 + o^ -a^r (7) with a the system phase at the crossover frequency a>^ 3 Identification of the optimal control parameters 3.1 Optimal Linear Control An optimal linear control law is adopted by Iwan and Hou to find the synchronous control coefficient for an undamped s.d.o.f. with unit mass and stiffness. For a M-d.o.f system, the procedure requires the solution of the following problem

4 688 Earthquake Resistant Engineering Structures ^ 7,, = minj(u,c) Au(t)+Bc(t)+Dq(t)-u(t)=0 (8) where J is the quadratic index of performance, w(t) the state vector, q(i) and c(t) are the excitation and control vector and A, B, D suitably defined state matrices. In the following, J is assumed to have the form (9) being Ry and Rf unit weight coefficients. The individuated critical values of time delay for the resulting control law expression have then been used to search for the new parameters that minimize, for a given forcing function, the performance index rewritten, in the case of the steady-state response, as follows 3.2 Norm Optimisation Control A different approach for the localisation of the optimal control coefficients can be adopted by expressing the controlled response function by means of the norm of the response parameters rather than by the determination of the detailed system response; this way the control algorithm results to be independent on the forcing function characteristics. By expressing the norm of the displacement function u(t) by the norm of the impulsive response function h(f) and of the excitation q(t) =/(/)/w, one can find an upper bound for the maximum displacement that occurs and hence for the maximum shear force S(f) and maximum control force c(t) (10) with gr(f the excitation norm and H(CO', ') andl(a>, ) the norm operator of the control force and of the response, this one being valid for the uncontrolled case with = and co =co and the controlled one with *= and co =<% (12) As the norms valid for the controlled case, L, and H, have the form

5 Earthquake Resistant Engineering Structures 689 the problem can then be set as follows '(x)dx, H*(CD', ')= \(2 h (x) + h (x))*dx. (13) 0 ^' C<O, (14) with ^ =^,/ 4 C = c_/w Q=x^. (15) being v a prefixed percentage of the uncontrolled maximum shear force S^ For the search of the optimal control parameters for the specified critical value of time delay, an analogous procedure is followed. By suitably adimentionalising the equation of the motion in the form of eqn (2) (9 =^r, * =d (16) where a and 77 are the redefined adimensional control parameters, and (p(ff) the adimenzionalised forcing function, a discretisation is operated and a step-bystep procedure (Baratta and Zuccaro,1994) is adopted to inviduate the upper bounds of the system displacements, shear force and control force with the energy of the forcing function cp(ff) and the norms of the response operator and of the control force, c(77^^=^(2#',4.^,m (18) being A3 the time distance between the elements 9. of a discrete sequence of i instants on the duration of the external disturbance and /z/, the elements of the discretised form of the impulsive response function iteratively determined. Then the whole problem can be defined as follows mm I.W# C<Q (19)

6 690 Earthquake Resistant Engineering Structures being van arbitrary integer expressing the percentage of the norm 5(0,0 /</\) of the uncontrolled system shear force. 4. Numerical results For an undamped structural system with unit mass and stiffness, that' s to say with ^=0 and co^\ rad/sec, a comparison has been provided between the numerical results obtained by Hou and Iwan by following the procedure illustrated in section 3.1 and the results one can obtain by adopting the norm optimisation control method. The case of ideal synchronous control has been considered and, for the obtained values of optimal parameters, the families of critical delay values have been investigated to check the non-effectiveness of the control algorithm whereas such a delay is introduced in the system. Of course, the structural response is highly unstable for such critical values, as one can observe by looking at Figure 1. The phase shift approach, proposed for compensation of the steady-state effects produced by time delay, is based on the introduction in the system of elements able to provide an increase of the system phase; the derivative part of the linear control law represents actually a phase compensator whose importance appears, hence, considerable. By the phase shift approach, the original control gains are modified such that both the real system and the ideal system have the same active mechanical characteristics. From Figure 1 the norm optimisation control algorithm appears clearly to be reliable even in correspondence of the introduced critical delay value r = 0.75; (20) Uncontrolled i 1 response N., \ ' Norm & 00,, npn-synchr.control C<So/2 ~ g 6.00 non-synchrdunqus,^ ix ' \' ;i j Norm 4.00 : < ' \ \\ synchr. control 2 QQ#lay-coinp.\; y delay-norm /optimized control Figure 1: Amplitude function of the steady-state response for T= 0.75 versus the ratio &

7 Earthquake Resistant Engineering Structures delay-olc / for cof = c 'c5 O / I delay-norm Ctrl' delay-olc for cop = l 2.00 C<So/3/ _^y/ delay-olc for (0(.=0.5 x delay-norm Ctrl - / XX _\X/ delay-norm Ctrl S^ C<So Figure!: Amplitude function of the steady-state response for r= 0.75 versus (Of the steady-state response obtained with the time delay compensation performed by means of the quadratic index in the form of eqn (10) - practically strictly closed to the one obtained for the synchronous control- would seem to be better than the response deduced by searching for the optimal control parameters for the specified time delay. Actually, while the norm minimisation provides control coefficients suitable for every forcing function and thus permit with these only two values to draw the whole corresponding amplitude curve, the minimisation of the index J in the form depending on the system gain function, is linked to the knowledge of the excitation frequency and, thus, presents, varying control parameters for each point of its amplitude curve. By applying the control parameters deduced with the performance criterium under the specified delay for particular values of coj to the whole range of possible excitation frequencies, the effectiveness of the compensated OLC algorithm for 8 values not corresponding to the one for which the control system has been adjusted, has been analysed. In Figure 2 the curves obtained by using the control parameters valid for a>f=\ and Q)f = 0.5 seems to work quite well, whilst the ones valid for 6^=1.5 lead quickly the system to instability. Even for a%= 0.5 and o)f= 1 the peaks of the OLC gain function result to be higher than those relative to the case of norm optimisation.

8 692 Earthquake Resistant Engineering Structures olc, UNOC 4, uncontrolled delay-olc 2 for% = delay-olc for T = 3 \ ' , uncontrolled response t\ 1x10* _20- for % = -1x10.4 _2Q -40 : - -2x Time (sec) Time (sec) (a) (b) Figure 3: Uncontrolled DO, OLC-controlled UQLC and Norm-controlled time response to a sine wave with ^=1 for: (a) r= 0.75 sec, (b) r=3 sec. A direct comparison can be made between the OLC gain for GJ, =1 and the Norm control gain with O<S\ as the energy of the supplied control force is identical for the two cases. The time response of the delay-controlled structural system has been observed; for a harmonic excitation and low values of delay occurring in the control system, the two control procedures are almost equivalent as shown in Figure 3(a) for r= With an higher time delay the OLC algorithm seems to fail, as illustrated for r=3 in terms of time response in Figure 3(b) and of gain function in Figure uncontrolled i;esp )nse 8.00! 'I i 6.00 ; 3 i delay-olc i=3 /*«O 4.00,' / 2.00 J delay Norm Ctrl,, for T= Figure 4. Amplitude function of the steady-state response for r=3 ratio 5. versus the

9 Earthquake Resistant Engineering Structures d^lay-olc uncontrolled response J delay-olc 0.20 ")h, uncontrolled response 11 A '. I o.oo»mt\m.k 0.00 "H, on i V '.i 5 '''- ' i';.' I-, '1 ' " ' i ',' delay-norm Ctrl I delay-norm Ctrl *' for T=3 \l for 1= Time (sec) (a) Tima/sec) (o) Figure 5: Uncontrolled and controlled time response to a white noise for: (a) T= 0.75 sec, (b) r=3 sec. The simulations of structural response to a white noise show, in Figure 5 (a) and (b), respectively for T= 0.75 and r=3, how the OLC response is worth when the forcing function has not a harmonic form; attention has to be paid to the circumstance that one has considered an energy of the control force identical for the two control procedures in the case of r =0.75, whilst, for r=3, it is, for the Norm Optimisation Control method, equal to one third of that relative to the OLC method. To test the response of the structural system controlled with each of the two algorithms under more than only one particular excitation a statistical analysis has been carried out. A sample consisting of a hundred of random forcing functions has been considered to deduce some statistically significant parameters of the structural response. Table 1 illustrates, for the two different values of time delay previously introduced, the averages /^ of the uncontrolled response, /^ of the controlled system by adopting OLC and Norm Optimisation Control, and & of the control force. The obtained numerical results show that, for small time delays, a more effective control with a lower energy expense is achieved when adopting the method of constrained minimisation of the response norm operator and that this effect is enhanced when bigger values of time lag are introduced in the control system. Table 1. Statistical analysis of the OLC for r and r-3 sec. and Norm-controlled response OLC Norm Vx fy Vc r Vx fy Vc T

10 694 Earthquake Resistant Engineering Structures 5 Conclusions A comparison has been presented between two different approach of search of the optimal control parameters under the hyphotesis of time delay occurrence in the control system. The reliability of the linear control algorithm that provides a constrained minimisation of the response operator norm is tested with reference to some critical values of time delay. Simulations and statistical analysis of the controlled structural response point out that the Norm Optimisation procedure represents a really effective control method, transient-steady-state reliable with respect to effects of even higher time delay values. Furthermore the particular control algorithm results to be able to provide a better structural response attenuation in comparison to the one deriving from the adoption of a delay compensated OLC approach, whilst accomplishing the purpose of strongly containing the control force energy. 6 Acknowledgements Paper supported by grants of the Italian Ministry for University and Scientific Research (M.U.R.S.T., 40%). References 1. Baratta, A., Papa, F., & Zuccaro, G., An Optimal Design Procedure for Delayed Control of Linear Structures. Journal of Structural Control, Vol.1 n.1-2, Baratta, A., & Voiello, G., Progettazione di Sistemi di Controllo Strutturale Antisismico Mediante Soluzioni in Norma, Proc. of VI Congresso Nazionale "L 'Ingegneria sismica in Italia", Perugia, Hou, Z., & Iwan, W.D., Reliability problem of Active Control Algorithms Caused by Time Dleay. Proc. of the Iff* World Conference on Earthquake Engineering, Masri, S.F., Seismic Response Control of Structural Systems:Closure. ProcefdzMgj q/v/zc ^ ftce^ Vol.VIII, Soong, T.T., Active Structural Control: Theory and Practice. Longman Scientific and Technical, Suner, A., Nagchaudhuri, A., & Utku, S., Effect of Time Lag and Use of Compensators in the Active Control of Buildings Subjected to Earthquake Excitations. Journal of Structural Control, n.2, 1995.

(b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications:

(b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications: 1. (a) The open loop transfer function of a unity feedback control system is given by G(S) = K/S(1+0.1S)(1+S) (i) Determine the value of K so that the resonance peak M r of the system is equal to 1.4.