Comments on A Time Delay Controller for Systems with Uncertain Dynamics

Transcription

1 Comments on A Time Delay Controller for Systems with Uncertain Dynamics Qing-Chang Zhong Dept. of Electrical & Electronic Engineering Imperial College of Science, Technology, and Medicine Exhiition Rd., London, SW7 2BT UK 27th January 23 Astract The introduction of time delay control (TDC) has initiated a series of researches. However, the artificially-introduced delay rings difficulties into system analysis and causes some drawacks. This paper presents an alternative method using an uncertainty and disturance estimator (UDE) and shows that the delay element is not needed at all to otain similar performances. The two inherent drawacks of TDC (oscillations in the control signal and the need of measuring the derivative of the states) have also een eliminated. The roust staility of LTI-SISO systems is analyzed and simulations are given to show the effectiveness of the UDE control and the comparison with TDC. The conclusion is that there is no need of TDC in most cases. Keywords: Time delay control (TDC), uncertainty and disturance estimator (UDE), parametric uncertainty, roust control, interval plants Introduction Aout 5 years ago, Youcef-Toumi and Ito, 2 proposed a roust control method for systems with unknown dynamics, which is called time delay control (TDC). TDC is ased on the assumption that a continuous signal remains unchanged during a small enough period and hence the past oservation of uncertainties and disturances can e used to modify the control action directly, rather than to adjust controller gains like gain scheduling or to identify system parameters like adaptive control. During the last decade, TDC has een widely studied, see e.g 3, 4, 5 and the references therein. It has een applied to DC servo motors 6, four-wheel steering systems 7, hyrid position/force control of roots 8, rushless DC motors 9, overhead cranes, root trajectory control and so on. However, TDC inherently requires that all the states and their derivatives e availale for feedack. This impose very strict limitations on the applications. Another inherent drawack of the TDC is that oscillations always exist in the control signal. As is well known in the control community, delay is not good for control. In particular, it rings difficulties into system analysis and tends to destailize the system. Although it has een shown in the aove-cited papers that TDC has very good potential to improve system performances, it is still desirale to get rid of delays, if possile, so as to simplify system analysis. The assumption used in TDC is a time-domain assumption. In this paper, we use an assumption in the frequency domain to propose an alternative control strategy to otain similar performances as TDC. The major part of the controller is called an uncertainty and disturance estimator (UDE). The two inherent drawacks of TDC disappear and the delay disappears, too. There is no need to measure the derivative of the states and there is no oscillation in the control signal. The system staility is easy to e analyzed. In addition to all these advantages, the system performance is very similar to that can e otained y TDC. Hence, there is no need of TDC in many cases. In order to keep clear comparison with the paper y Youcef-Toumi and Ito 2, a similar structure as the paper is used for this paper. In Section 2, the general UDE control law for uncertain systems is derived. More aspects of the control K. Youcef-Toumi and O. Ito, Journal of Dynamic Systems, Measurement, and Control. vol 2, pp.33-42, 99. Tel: , Fax: , zhongqc@ic.ac.uk, URL:

2 strategy are analyzed in Section 3 for LTI-SISO systems. The control strategy is then applied to two examples studied in the paper 2 in Section 4. The advantages of the proposed control strategy are very clear with comparison to TDC. Finally, conclusions are made in Section 5. 2 UDE-ased Control Law 2. System description The system to e considered is formulated as: ẋ = (A + F)x + Bu(t) + d(t), () where x = (x,..., x n ) T is the state vector, u(t) = (u (t),..., u r (t)) T is the control input, A is the known state matrix, F is the unknown state matrix, B is the control matrix having full column rank and d(t) is the unpredictale external disturances. 2.2 Reference model and structural constraint Assume that the desired specifications can e descried y the reference model ẋ m = A m x m + B m c(t), (2) where c(t) = (c (t),..., c r (t)) T is the set-point. The control ojective is to make the state error e etween the system and the reference model converges to zero. In other words, the error dynamics is stale, where K is called the error feedack gain matrix. Comine (), (2), (3), and (4), we otain Then the control action u(t) is otained as e = ( x m x x mn x n ) T = xm x, (3) ė = (A m + K)e (4) A m x + B m c(t) Ax Fx Bu(t) d(t) = Ke. (5) u(t) = B + A m x + B m c(t) Ax Fx d(t) Ke, (6) where B + = (B T B) B T is the pseudo inverse of B. Since u(t) given in (6) is the least-square solution rather than the accurate solution, (4) and (5) will only e met under the following structure constraint: I BB + A m x + B m c(t) Ax Fx d(t) Ke =. (7) Oviously, if B is invertile, the aove structural constraint is always met. If not, the choice of the reference model and the error feedack gain matrix is somewhat restricted. However, as shown in 2, the systems descried in the canonical form meet this constraint. We assume that this constraint is satisfied in the sequel. 2.3 UDE and the control law The control law in (6) can e represented in s-domain y using Laplace transform (assuming zero-initial states) as U(s) = B + A m X(s) + B m C(s) KE(s) AX(s) + B + FX(s) D(s), (8) where the Laplace transformation is denoted y the corresponding capital letter. The first part in (8) is known while the second part, denoted hereafter y U d (s), includes the uncertainties and the external disturance. According to the system dynamics (), the second part U d (s) can e rewritten as, 2

3 U d (s) = B + FX(s) D(s) = B + (A si)x(s) + BU(s). (9) In other words, the unknown dynamics and the disturances could e oserved y the system states and the control signal. However, it cannot e used in the control law directly. TDC adopts an estimation of this signal y using a small delay in time domain 2. Here, we use a different estimation strategy in the frequency domain. Assume that G f (s) is a strictly proper low-pass filter with unity steady-state gain and road enough andwidth, then U d (s) can e approximated y UDE = B + (A si)x(s) + BU(s) G f (s). () This is called the uncertainty and disturance estimator (UDE). The UDE only uses the control signal and the states to oserve the uncertainties and the unpredictale disturances. Hence, U(s) = B + A m X + B m C KE AX + UDE = B + A m X + B m C KE AX( G f ) sg f X + G f BU. The UDE control law is then derived as U(s) = (I B + BG f ) B + A m X + B m C KE AX( G f ) sg f X. () The control signal is formed y the current states X(s), the low-pass filter, the reference model and the error feedack gain. It has nothing to do with the uncertainty and the disturance. Since G f is strictly proper, there is no need of measuring the derivative of states (sg f is physically implementale). Assume that the frequency range of the system dynamics and the external disturance is limited y ω f, then the ideal low-pass filter G f (s) has a low-frequency gain (ω < ω f ) of and a high-frequency gain (ω > ω f ) of zero. Hence, the estimation error of the uncertainty and the disturance B + (A si)x(s) + BU(s) ( G f ) is zero for all the frequency range ecause G f = for the low-frequency range and (A si)x(s) + BU(s) = for the high-frequency range. A practical low-pass filter is G f (s) = T s +, (2) where T = /ω f. Although this will cause some error in the estimation, the steady-state estimation error is still zero ecause G f () = (the initial value of G f (s) can always e chosen to e zero). So far, the analysis is ased on linear systems. However, this control strategy can also e generalized to nonlinear systems with unknown dynamics, using similar expositions as in 2. In fact, all the simulations will e done for nonlinear systems. 3 Special Case: LTI-SISO systems In this Section, we consider LTI-SISO systems with uncertainties and disturances descried in the canonical form, i.e. the corresponding matrices have the following partitions: ( ) ( ) ( ) ( ) In A = ; F = ; B = ; d(t) =, (3) A F d(t) where A = ( a, a 2,..., a n ) and F = ( f, f 2,..., f n ) are n row vectors, and f i f i f i (i =,, n) are uncertain parameters. I n is the (n ) (n ) identity matrix and. The output of the system is assumed to e y = x. 3

4 Figure : The equivalent structure of UDE-controlled LTI-SISO systems 3. Control scheme The reference model and the error feedack gain matrix are partitioned as ( ) ( ) ( In A m = ; B A m = ; K = m m K ), (4) where A m = ( a m, a m2,..., a mn ) and K = ( k, k 2,..., k n ) are n row vectors. Sustitute the matrices into (), the control law is U(s) = a mi X i + m C + k i E i sg f X n + a i X i. (5) G f (s) It can e further simplified as follows: U(s) = = = where G m (s) = G f (s) G f (s) G f (s) = P m(s) + P k (s) G f (s) = P m(s) + P k (s) G f (s) m P = m(s) i= (a mi + k i )X i + m C + k i X mi s n G f (s)x + a i X i i= i= i= (a mi + k i )s i X + m C + k i s i X m s n G f (s)x + i= i= { } m P m (s) + P k (s) Y + m C + P k (s) P m (s) C + sn G f (s) Y + m P m (s) C Y + sn Y + a i s i Y i= m P m (s) C Y i= i= a i X i i= a i s i X + P a(s) Y (6) m s n +a mns n + +a m is the transfer function of the reference model, P a (s) = s n + n i= a is i is the characteristic polynomial of the known system dynamics and P k (s) = k n s n + + k is the error feedack polynomial. The equivalent structure of the UDE control scheme is shown in Figure. The UDE controller can e divided into m P m(s) four parts: the reference model to generate a desired trajectory; a local positive-feedack loop via the low-pass filter (ehaving like a PI controller); the polynomial P m (s) + P k (s) for the tracking error and the polynomial P a (s) for the output. The last two perform the derivative effect. Oviously, UDE control uses more derivative information than the common PID controller and hence UDE control has the potential to otain etter performances than the common PID controller. It is worth noting that this structure is the equivalent one of UDE-ased control, ut not the structure for implementation. The control law should e implemented according to (5) under the assumption that all the states are availale for feedack. i= 4

5 3.2 Staility analysis The transfer function of the LTI-SISO plant given y (3) is G(s) = Y (s) U(s) = s n + (a n + f n )s n + + (a + f ) = From (6), the closed-loop transfer function of the system is derived to e Y (s) C(s) = m P m (s) P (s). (7) P m + P k P m + P k + ( G f )(P P a ). (8) Since the reference model is always chosen to e stale, only the staility of the second part in the aove equation is to e discussed hereafter. If the low-pass filter is chosen as given in (2), then the characteristic polynomial of the closed-loop system is, P c (s) = (T s + )(P m + P k ) + T s(p P a ) = (P m + P k ) + T s(p m + P k + P P a ) (9) Oviously, the staility depends on the reference model, the error feedack gain, the time constant of the filter and, of course, the plant itself. The reference model is chosen to otain the desired specifications and the time constant of the low-pass filter is chosen to have road enough andwidth. Hence, the error feedack gain K can e used to gurantee the roust staility of the closed-loop system. Intuitively, if T is small enough, the system is always stale. However, this might cause the prolem of implementation and hence the value of T has to e compromised. The roust staility of the polynomial (9) can e guaranteed y the following theorem, which is otained directly using the ox theorem 2 (a generalization of the well-known Kharitonov s theorem 3) for parametric uncertain systems. Theorem (Roust staility) The closed-loop system is stale iff the 4 polynomials P m (s)+p k (s)+t sp pq (s) (p, q =, 2) are stale, in which P pq (s) = P p (s) + Q q (s) (p, q =, 2) are the Kharitonov polynomials of the uncertain polynomial P m + P k + P P a = s n + n i= (a mi + k i + f i )s i with P (s) = (β + f ) + (β 3 + f 3 )s 2 + (β 5 + f 5 )s 4 + } = (β +i + j i min {j i f +i, j i f +i ) s i, i=,even P 2 (s) = (β + f ) + (β 3 + f 3 )s 2 + (β 5 + f 5 )s 4 + } = (β +i + j i max {j i f +i, j i f +i ) s i, i=,even Q (s) = (β 2 + f 2 )s + (β 4 + f 4 )s 3 + (β 6 + f 6 )s 5 + } = (β +i + j i min {j i f +i, j i f +i ) s i, i=,odd Q 2 (s) = (β 2 + f 2 )s + (β 4 + f 4 )s 3 + (β 6 + f 6 )s 5 + } = (β +i + j i max {j i f +i, j i f +i ) s i, i=,odd where β i = a mi + k i (i =,, n), β n+ =, f n+ = f n+ = and j =. 4 Examples In order to show clear comparisons with TDC, we use two examples studied in 2. 5

6 .5 x m x.5 u (a) the output and the control signal..5.5 error () the tracking error Figure 2: The nominal response 4. A first-order nonlinear system Consider the following first-order nonlinear system, ẋ = f cos x + u + d where,.5 f 2 is an uncertain parameter with the nominal value f =. The nominal system is the same as the one in 2 and d =.5 (t.5) is a step disturance. The reference model is chosen as ẋ m = x m + c and the error feedack gain is k =. The low-pass filter is G f (s) = T s+ with T =.s. According to (), the control law is U(s) = G f (s) X(s) + C(s) sg f (s)x(s). The measurement of ẋ is not needed. The command signal is c(t) = (t) in the simulations. The nominal response is shown in Figure 2 and the response when f = 2 is shown in Figure 3. Comparing with the responses in 2, the performance is much etter. Although the uncertain parameter f is changed to f = 2, the performance is not degraded too much. Another advantage is that there is no oscillation in the control signal of UDE-controlled system ut there are oscillations in that of the TDC system. The andwidth of the low-pass filter is very important for the system performance. Three cases with different time constants are shown in Figure 4. The smaller the time constant, the roader the andwidth and the etter the performances of oth trajectory and disturance-rejection. However, the time constant might e limited y the computation capaility and the measurement noise in practice. 4.2 A servo motor positioning system Consider the servo motor positioning system studied in 2. The inherent unknown dynamics are viscous and dry frictions. Additional uncertainty is simulated y an elastic spring that is physically attached to the motor load. The dynamic equation of the system is 6

7 x m x u (a) the output and the control signal..5.5 error () the tracking error Figure 3: Response when f = 2.2 x.8.6 T=.5s T=.s T=.s Figure 4: Effect of the time constant on the performance 7

8 = k s J s J d s sgn( ) +, (2) J where is the motor rotation angle, is the motor torque, k s is the unknown spring constant, s is the unknown viscous friction coefficient, d s is the unknown dry friction coefficient, and J = 4. 4 kg m 2 is the total inertia. The reference model is chosen as the following second order system ( ) ( m = m ωn 2 2ζω ) ( ) ( m + m ωn 2 ) c, (2) where ω n = 5rad/s and ζ = are the same as those in 2 and the command c = rad. The error feedack gain is chosen to e. The UDE control law is (s) = (s) G f (s) (s) + 25C(s) sg f (s) (s), (22) where the low-pass filter G f (s) is T s+ with T =.5s. Oviously, the angular acceleration signal is not needed for feedack control, unlike the case of time delay control 2. In order to compare with TDC, simulations are also done using the following TDC control law given in 2: (t) = (t L) + J 25(t) (t) + 25c(t) (t) (t L), L where L =.5s Nominal performance When the frictions are omitted and the spring is not attached to the motor load, k s =, s =, d s =. The responses are shown in Figure 5. They are almost the same as the reference model under the UDE control or time delay control. There are oscillations in the control signal of time delay control. In order to see the oscillations clearly, the control signals of oth cases are re-drawn in Figure Roust performance In this case, we assume that k s =.7Nm/rad, s =.Nms/rad and d s =.s. The responses are shown in Figure 7. They are all degraded ecause the time delay or the time constant is not small enough. It can e seen that the two control signals are different. There are oscillations in the case of TDC while the UDE control signal is very smooth. When the time constant or time delay is chosen as.s, the responses match the reference response very well and are almost not affected y the uncertainties, as shown in Figure 8. 5 Conclusions This paper presents an uncertainty and disturance estimator (UDE) for systems with uncertainties and disturances. Similar performance can e otained without the use of a delay element. Moreover, the inherent drawacks of the TDC has een eliminated. There is no need of measuring the derivative of the states and there is no oscillation in the control signal. It is also easier to analyze the staility of a UDE control system than to analyze the staility of a TDC control system. In a word, there is no need of time delay control in most cases. However, this does not mean that the research work proposed in 2 is not important. As mentioned in the Introduction, TDC has initiated a series of studies, including the studies reported in this paper. Further work should e done to show if the delay is not needed either for the input-output linearization using time delay control, as reported in e.g. 4, 5. Acknowledgment The author would like to thank Dr. Leonid Mirkin for proposing this issue. 8

9 m (a) UDE control.2 m () Time delay control Figure 5: Nominal response (a) UDE control () Time delay control Figure 6: The control signals 9

10 m (a) UDE control m () Time delay control Figure 7: Roust performance m (a) UDE control.8 m () Time delay control Figure 8: Roust performance when L = ms or T = ms

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