- Email: [email protected]
Control of uncertain linear systems with a state delay based on an uncertainty and disturbance estimator
Proceedings of the 6th IFAC Symposium on Robust Control Design Haifa, Israel, June 16-18, 2009
Control of uncertain linear systems with a state delay based on an uncertainty and disturbance estimator Alon Kuperman ∗ and Qing-Chang Zhong ∗ ∗
Department of Electrical Engineering and Electronics University of Liverpool, UK Emails: [email protected] and [email protected]
Abstract In this paper, a robust control strategy based on the uncertainty and disturbance estimator (UDE) is extended and applied to uncertain linear continuous-time systems with state delays and disturbances. It does not require the knowledge of the bounds of uncertainties and disturbances. Both known and unknown delay cases are considered. In the case of an unknown time delay, the term with the delay is treated as an additional disturbance to the system. The proposed algorithm shows excellent tracking and disturbance rejection capabilities. The robust stability of LTI-SISO systems is analysed and simulations are given to show the effectiveness of c the UDE-based control. Copyright 2009 IFAC. Keywords: Uncertainty and disturbance estimator (UDE), time-delay systems, parametric uncertainty, robust control, 1. INTRODUCTION Dynamic behaviour of many physical processes such as traffic networks, cold rolling mills, national economic systems, chemical processes etc. contains inherent uncertainties and time delays. In the recent 20-25 years there has been a substantial increase of interest in time delay systems in the systems and control society; see for example Gu and Niculescu [2003], Zhong [2006]. Some of the time-delay systems can be treated as state-delayed plants. Stability and stabilisation issues of linear state-delay systems were discussed in [Kharitonov and Niculescu, 2003, Niamsup et al., 2008]. State feedback controllers are common in solving stabilisation problems of state-delay systems. A memoryless state feedback controller was proposed in [Lee, 1998], while robust H∞ state feedback controllers were treated in [Ge et al., 1996]. Sliding mode control of linear state delay systems via LMI was proposed in Xia and Jia [2003], Gouaisbaut et al. [2002] and optimal sliding mode approach was used in [Tang et al., 2007] to control time delay systems with sinusoidal disturbances. Guaranteed cost approach to the control of uncertain time-delay plants was reported in Wang and Zhao [2007], Mahmoud [2004], while observer-based control was treated in Bengea et al. [2004]. Robust control Oucheriah [1999] is a widely employed approach to control of time-delay systems with uncertainties, often via H∞ Suplin et al. [2006] and optimal Basin et al. [2007] approaches. Recently, the adaptive control of state-delay systems has become popular Evesque et al. [2003], Oucheriah [2001], Momeni and Aghdam [2007]. Several different approaches were combined to improve the control performance Nounou and Mahmoud [2005], Xia et al. [2008]. The Uncertainty and Disturbance Estimator (UDE) technique proposed in [Zhong and Rees, 2004] was one of them and
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then successfully used in input-output linearisation Patel et al. [2007], Talole and Phadke [2007], combining with sliding mode control Shendge and Patre [2007], Chandrasekhar and Dewan [2007], Talole and Phadke [2008]. The UDE is based on the assumption that a continuous signal can be approximated when it passes through a low-pass filter with a unity static gain and a short time constant. In this paper we present a state feedback controller for linear uncertain continuous plants with state delays and disturbances. The terms containing time delay (in case the information about the delay is absent in the controller) and the uncertainties are considered as an additional disturbance and then we employ an UDE to quickly estimate the overall disturbance and thus to robustify the state feedback controller. In Section 2, the UDE-based control law for uncertain linear plants with delay and disturbances is derived, following the procedures developed in [Zhong and Rees, 2004]. More aspects of the control strategy are analysed in Section 3 for LTI-SISO systems. Simulation examples are given in Section 4 and conclusions are made in Section 5. 2. UDE-BASED CONTROL 2.1 System description The system to be considered is formulated as: ˙ x(t) = (A1 + ∆A1 )x(t) + (A2 + ∆A2 )x(t − τ ) +(B + ∆B)u(t) + d(t), (1) T where x(t) = (x1 (t), ..., xn (t)) is the state, u(t) = (u1 (t), ..., ur (t))T is the control input, A1 and A2 are the known state matrices, ∆A1 and ∆A2 are the unknown
10.3182/20090616-3-IL-2002.0033
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
state matrices, B is the known control matrix having full column rank, ∆B is the unknown control matrix having full column rank, d(t) is the unpredictable external disturbances and τ is the time delay. 2.2 Reference model and structural constraint Assume that the desired specification can be described by the reference model x˙ m (t) = Am xm (t) + Bm c(t), (2) T where c(t) = (c1 (t), ..., cr (t)) is the uniformly bounded and piecewise continuous reference input. The control objective is to determine a state feedback control u(t) for the system (1) which forces the state error e(t) = xm (t) − x(t) between the system and the reference model e(t) = ( xm1 (t) − x1 (t) · · · xmn (t) − xn (t) )
(3)
converges to zero. In other words, the error dynamics ˙ (4) e(t) = (Am + K)e(t) is stable, where K is called the error feedback gain. Since the reference model is often chosen to be stable, K may be chosen as 0. If different error dynamics is desired or is required to guarantee robust stability (see Subsection 3.2), then common control strategies, such as pole placement, can be used to choose K. Combining (1), (2), (3), and (4), we obtain
U(s) = B+ [Am X(s) + Bm C(s) − KE(s) −A1 X(s) − A2 X(s)e−sτ − ∆A1 X(s) (8) −∆A2 X(s)e−sτ − ∆BU(s) − D(s)], where the Laplace transformation of a signal is denoted by the corresponding capital letter. We define a constant ε according to the availability of τ . In case τ is given, the first five terms in (8) are known and ε = 1; otherwise only the first four are known and ε = 0. The remaining terms, denoted hereafter by Ud (s), include the uncertainties and the external disturbance. Ud (s) can be written as Ud (s) = B+ [(ε − 1)A2 X(s)e−sτ − ∆A1 X(s) −∆A2 X(s)e−sτ − ∆BU(s) − D(s)] = B+ (A1 + εA2 e−sτ − sI)X(s) + BU(s) (9) In other words, the unknown dynamics and the disturbances Ud (s) can be observed by the system states and the control signal. However, it cannot be used in the control law directly. The UDE technique adopts an estimation of this signal in the frequency domain. Assume that Gf (s) is a strictly proper low-pass filter with a unity steady-state gain and a broad enough bandwidth, then Ud (s) can be accurately approximated by (10) UDE = Ud (s) · Gf (s). This is called the uncertainty and disturbance estimator (UDE). Hence, U(s) = B+ · [Am X(s) + Bm C(s) − KE(s)
Am x(t) + Bm c(t) − A1 x(t) − A2 x(t − τ )
−A1 X(s) − εA2 X(s)e−sτ ] + UDE
−Bu(t) − ∆A1 x(t) − ∆A2 x(t − τ )
= B+ · [Am X(s) + Bm C(s) − KE(s)
−∆Bu(t) − d(t) = Ke(t)
(5)
−sGf (s)X(s) + Gf (s)BU(s)], which gives the UDE-based control law as
Then the control action u(t) can be obtained as u(t) = B+ [Am x(t) + Bm c(t) − A1 x(t)
U(s) = −B+ (A1 + εA2 e−sτ )X(s) +
−A2 x(t − τ ) − ∆A1 x(t) − ∆A2 x(t − τ ) −∆Bu(t) − d(t) − Ke(t)],
(6)
where B+ = (BT B)−1 BT is the pseudo inverse of B. Notice that u(t) given in (6) is only an approximate solution of (5). Hence (4) and (5) will only be fully met under the following structural constraint:
I − BB+ · [Am x(t) + Bm c(t) − A1 x(t)
−A2 x(t − τ ) − ∆A1 x(t) − ∆A2 x(t − τ ) −∆Bu(t) − d(t) − Ke(t)] = 0.
−(A1 + εA2 e−sτ )X(s)(1 − Gf (s))
(7)
Obviously, if B is invertible, then this constraint is always met. If not, the choice of the reference model and the error feedback gain matrix will be restricted. However, according to [Youcef-Toumi and Ito, 1990], the systems described in canonical form meet this constraint. The following assumes that this constraint is satisfied. 2.3 UDE and the control law
1 B+ 1 − Gf (s)
×[Am X(s) + Bm C(s) − KE(s) − sGf (s)X(s)]
(11)
The control signal is formed by the state, the low-pass filter, the reference model and the error feedback gain. It has nothing to do with the uncertainty and the disturbance. Since Gf is strictly proper, sGf is implementable and there is no need of measuring the derivative of the states. Assuming that the frequency range of the system dynamics and the external disturbance is limited by ωf , a practical low-pass filter can be chosen as 1 Gf (s) = , (12) Ts + 1 where T = 1/ωf > 0. The steady-state estimation error is always zero because Gf (0) = 1 (the initial value of Gf (s) can always be chosen to be zero). It is interesting to see that 1 1 =1+ , 1 − Gf (s) Ts which means that there is an integral action. 3. SPECIAL CASE: LTI-SISO SYSTEMS
The control law in (6) can be represented in the s-domain by using the Laplace transform (assuming zero-initial states) as
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In this section, we consider LTI-SISO systems with uncertainties and disturbances described in the canonical form,
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
i.e. the corresponding matrices in (1) have the following partitions: 0 In−1 0 In−1 ; A2 = ; A1 = Ap1 Ap2 0 0 ∆A1 = ; ∆A2 = ; (13) F1 F2 0 0 0 , ; d(t) = ; ∆B = B= d(t) ∆b b where Ap1 = (−a11 , −a12 , ..., −a1n ), Ap2 = (−a21 , −a22 , ..., −a2n ), F1 = (−f11 , −f12 , ..., −f1n ) and F2 = (−f21 , −f22 , ..., −f2n ) are 1 × n row vectors, fi ≤ fji ≤ fi (i = 1, · · · , n; j = 1, 2) are uncertain parameters, In−1 is the (n − 1) × (n − 1) identity matrix, b 6= 0 and ∆b ≤ ∆b ≤ ∆b is an uncertain parameter. The output of the system is assumed to be y(t) = x1 (t). 3.1 Control scheme The reference model and the error feedback gain matrix are partitioned as 0 In−1 0 0 Am = ; Bm = ;K = , (14) bm K1 Am1 where Am1 = (−am1 , −am2 , ..., −amn ) and K1 = (−k1 , −k2 , ..., −kn ) are 1 × n row vectors. Substituting the matrices into (11), the control law is U (s) =
n X 1 ami Xi (s) + bm C(s)] [− b(1 − Gf (s)) i=1
n
1X (a1i + εa2i e−sτ )Xi (s). b i=1
The transfer function of the LTI-SISO plant given by (13) is Y (s) b + ∆b = , (17) U (s) P (s) where n X (a1i + f1i ) + (a2i + f2i )e−sτ si−1 . P (s) = sn + i=1
From (16), the closed-loop transfer function of the system is derived to be bm 1 Y (s) . = · b P (s)−Pa (s) C(s) Pm (s) 1 + (1 − G (s)) b+∆b f Pm (s)+Pk (s)
(18)
Theorem 1. Assume that (i) the reference polynomial Pm (s) = sn + amn sn−1 + · · · + am1 is chosen to be stable, (ii) the low-pass filter is chosen as (12), (iii) the error feedback polynomial Pk (s) = kn sn−1 + · · · + k1 is chosen such that Pm (s) + Pk (s) is stable. Then the closed-loop system is robustly stable if b P (jω) − Pa (jω) b+∆b ∀ω ∈ [0, ∞) (19) < 1, Pm (jω) + Pk (jω) holds on the space of uncertain parameters.
n X 1 + ki Ei (s) − sGf (s)Xn (s)] [ b(1 − Gf (s)) i=1
+
3.2 Stability analysis
(15)
Similarly as in as [Zhong and Rees, 2004], it can be further simplified as Pm (s) + Pk (s) bm U (s) = C(s) − Y (s) b [1 − Gf (s)] Pm (s) Pa (s) + Y (s), (16) b where Pm (s) = sn +amn sn−1 +· · ·+am1 is the denominator bm n of the reference model Gm (s) = Pm (s) , Pa (s) = s + Pn −sτ i−1 )s is the characteristic polynomial i=1 (a1i + εa2i e of the known system dynamics and Pk (s) = kn sn−1 + · · ·+ k1 is the error feedback polynomial. The UDE controller is divided into four parts: the referbm ence model Pm (s) to generate a desired trajectory; a local positive-feedback loop via the low-pass filter (behaving like a PI controller); the polynomial Pm (s) + Pk (s) for the tracking error and the polynomial Pa (s) for the output. The last two perform the derivative effect. Since the UDE controller uses more derivative information than the common PID controller, it has the potential to obtain better performances than the common PID controller. Note that the structure of (16) is the equivalent one of UDE-based control, but not the one used for implementation. The control law should be implemented according to (15).
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Proof. Since k1 − Gf (s)k∞ < 1 and Gm (s) is stable, the closed-loop system is robustly stable if
b P (s) − Pa (s)
b+∆b
< 1,
Pm (s) + Pk (s) ∞
according to the well-known small-gain theorem. The H∞ norm of a stable single-input-single-output transfer function is always obtained over the imaginary axis and hence the Theorem is proved. Remark 2. Since the uncertain parameters and delay terms only appear in the numerator of (19), the verification of the condition is not difficult, although it might involve heavy computation. 4. SIMULATION EXAMPLE Consider the following first-order system, x(t) ˙ = (a1 + ∆a1 )x(t) + (a2 + ∆a2 )x(t − τ ) + (b + △b)u(t) + d(t),
where a1 = 1, a2 = 1 and b = 1 are known parameters; 1 ∆a P 1 = 0.8sin(t), ∆a2 = 0.5sin(2t − 4 π) and ∆b = 0.5 [1(t − 2k) − 1(t − 1 − 2k)] are the uncertain time k
varying parameters (1(t) is the the step signal). d(t) is a negative unit-step disturbance, acting at t = 3.5sec and τ = 1sec.
The reference model is chosen as x˙ m (t) = −xm (t) + c(t) and the error feedback gain is chosen as k = 0. The lowpass time constant is chosen as T = 0.001sec. Although there are time-varying uncertainties in the system, the control law can be derived following the proposed procedure. According to (11), the control law is
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
U (s) =
1 [C(s) − X(s) − sGf (s)X(s)] 1 − Gf (s)
−(1 + εe−sτ )X(s). In the simulations, c(t) is chosen to be the step signal 1(t). 4.1 Nominal Performance The nominal response (the uncertain time varying parameters and the disturbance are zero) for the case with the delay τ unknown (ε = 0) is shown in Figure 1 and the nominal response for the case with the delay τ known (ε = 1) is shown in Figure 2. The system state x tracks the desired state xm very well in both scenarios. In the case of known τ the tracking error is practically zero at all times, while in the unknown time delay case there is a tracking error change at τ = 1 because the term containing the time delay acts like an unknown input to the system.
(a) the output signal (b) the tracking error (c) the control signal Figure 3. Example 1: The robust response when ε = 0
(a) the output signal (b) the tracking error (c) the control signal (a) the output signal (b) the tracking error (c) the control signal
Figure 4. Example 1: The robust response when ε = 1
Figure 1. Example 1: The nominal response when ε = 0
Figure 5. Example 1: The robust responses for different values of T
(a) the output signal (b) the tracking error (c) the control signal
u (there is a jump in u when the disturbance starts at t = 3.5sec) that the controller estimates the disturbance very quickly.
Figure 2. Example 1: The nominal response when ε = 1 4.2 Robust Performance The robust response for the case with unknown τ is shown in Figure 3 and the robust response for the case with known τ is shown in Figure 4. When the uncertain parameters are nonzero, the performance does not degrade significantly because the controller can well estimate the uncertainty. It can also be seen from the control signal
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The bandwidth (time constant) of the low-pass filter has an obvious influence on the system performance. The responses of three cases with different time constants are shown in Figure 5. The smaller the time constant (the broader the bandwidth), the better the tracking (i.e., the better the performance). However, in practice, the time constant might be limited by the computational capability and the measurement noise.
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
5. CONCLUSIONS This paper presents an uncertainty and disturbance estimator (UDE) for uncertain linear continuous systems with state delays and disturbances. The approach can be seen as a robust state-feedback control law. It does not require the knowledge of the bounds of uncertainties and disturbances and is continuous. The proposed algorithm shows excellent tracking and disturbance rejection capabilities almost independently on the availability in the controller of the information about the time delay. In case of the unknown time delay, the term with the delay is treated as an additional disturbance input to the system and is quickly estimated by the controller. The robust stability of LTI-SISO systems is analysed. Simulations are given to show the effectiveness of the UDE-based control. REFERENCES M. Basin, J. Rodriguez-Gonzalez, and L. Fridman. Optimal and robust control for linear state-delay systems. Journal of the Franklin Institute, 344:830–845, 2007. S. Bengea, X. Li, and R. DeCarlo. Combined controllerobserver design for uncertain time delay systems with application to engine idle speed control. Journal of Dyn. Sys. Meas. Con. - Trans. ASME, 126:772–780, 2004. T. Chandrasekhar and L. Dewan. Sliding mode control based on TDC and UDE. International Journal of Informatics and Systems Sciences, 3(1):36–53, 2007. S. Evesque, A. Annaswamy, S.-I. Niculescu, and A. Dowling. Adaptive control of a class of time-delay systems. Journal of Dyn. Sys. Meas. Con. - Trans. ASME, 125: 186–193, 2003. J.-H. Ge, P. Frank, and C.-F. Lin. Robust Hinf state feedback control for linear systems with state delay and parameter uncertainty. Automatica, 32(8):1183–1185, 1996. F. Gouaisbaut, M. Dambrine, and J. Richard. Sliding mode control of linear time delay systems: a design via LMI. IMA J. Math. Control Inform., 19:83–94, 2002. K. Gu and S.-I. Niculescu. Survey on recent results in the stability and control of time-delay systems. Journal of Dyn. Sys. Meas. Con. - Trans. ASME, 125:158–165, 2003. V. Kharitonov and S.-I. Niculescu. On the stability of linear systems with uncertain delay. IEEE Trans. Automat. Control, 48(1):127–132, 2003. C.-H. Lee. Simple stabilizability criteria and memoryless state feedback control design for time-delay systems with time-varying perturbations. IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications, 45(11):1211–1215, 1998. M. Mahmoud. Control of uncertain state-delay systems: guaranteed cost approach. IMA J. Math. Control Inform., 148:453–468, 2004. A. Momeni and A. Aghdam. An adaptive tracking problem for a family of retarded time-delay plants. International Journal of Adaptive Control and Signal Processing, 21: 885–910, 2007. P. Niamsup, K. Mukdasai, and V. Phat. Linear uncertain non-autonomous time-delay systems: stability and sta-
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bilizability via Riactti equations. Electronic Journal of Differential Equations, 2008(26):1–10, 2008. H. Nounou and M. Mahmoud. Variable structure adaptive control for a class of continuous time-delay systems. IMA J. Math. Control Inform., 150:453–463, 2005. S. Oucheriah. Robust tracking and model following of uncertain dynamic delay systems by memoryless linear controllers. IEEE Trans. Automat. Control, 44(7):1473– 1477, 1999. S. Oucheriah. Adaptive robust control of a class of dynamic delay systems with unknown uncertainty bounds. International Journal of Adaptive Control and Signal Processing, 15:53–63, 2001. A. Patel, R. Neelgund, A. Wathore, J. Kolhe, M. Kuber, and S. Talole. Robust control of flexible joint robot manipulator. In Proc. IEEE International Conference on Industrial Technology (ICIT2006), pages 649–653, 2007. P. Shendge and B. Patre. Robust model following load frequency sliding mode controller based on UDE and error improvement with higher order filter. IAENG International Journal of Applied Mathematics, 37(1), 2007. V. Suplin, E. Fridman, and U. Shaked. Hinf control of linear uncertain time-delay systems: a projection approach. IEEE Trans. Automat. Control, 51(4):680– 685, 2006. S. Talole and S. Phadke. Robust input-output linearization using UDE. In Proc. International Conference on Advances in Control and Optimization of Dynamical Systems (ACODS2007), pages 79–82, 2007. S. Talole and S. Phadke. Model following sliding mode control based on uncertainty and disturbance estimator. Journal of Dyn. Sys. Meas. Con. - Trans. ASME, 130, 2008. G.-Y. Tang, S.-S. Lu, and R. Dong. Optimal sliding mode control for linear time-delay systems with sinusoidal disturbances. Journal of Sound and Vibration, 304:263– 271, 2007. R. Wang and J. Zhao. Guaranted cost control for a class of uncertain switched delay systems: an average dwelltime approach. Cybernetics and Systems, 38(1):105–122, 2007. Y. Xia, J. Chen, G.-P. Liu, L. Wang, and D. Rees. Robust adaptive sliding mode control for uncertain time-delay systems. International Journal of Adaptive Control and Signal Processing, 2008. Y. Xia and Y. Jia. Robust sliding-mode control for uncertain time-delay systems: an LMI approach. IEEE Trans. Automat. Control, 48(6):1086–1092, 2003. K. Youcef-Toumi and O. Ito. A time delay controller for system with unknown dynamics. Journal of Dyn. Sys. Meas. Con. - Trans. ASME, 112:133–142, 1990. Q.-C. Zhong. Robust Control of Time-delay Systems. Springer, 2006. ISBN 1-84628-264-0. Q.-C. Zhong and D. Rees. Control of uncertain LTI systems based on an uncertainty and disturbance estimator. Journal of Dyn. Sys. Meas. Con. - Trans. ASME, 126:905–910, 2004.
Control of uncertain linear systems with a state delay based on an uncertainty and disturbance estimator Alon Kuperman ∗ and Qing-Chang Zhong ∗ ∗
Department of Electrical Engineering and Electronics University of Liverpool, UK Emails: [email protected] and [email protected]
Abstract In this paper, a robust control strategy based on the uncertainty and disturbance estimator (UDE) is extended and applied to uncertain linear continuous-time systems with state delays and disturbances. It does not require the knowledge of the bounds of uncertainties and disturbances. Both known and unknown delay cases are considered. In the case of an unknown time delay, the term with the delay is treated as an additional disturbance to the system. The proposed algorithm shows excellent tracking and disturbance rejection capabilities. The robust stability of LTI-SISO systems is analysed and simulations are given to show the effectiveness of c the UDE-based control. Copyright 2009 IFAC. Keywords: Uncertainty and disturbance estimator (UDE), time-delay systems, parametric uncertainty, robust control, 1. INTRODUCTION Dynamic behaviour of many physical processes such as traffic networks, cold rolling mills, national economic systems, chemical processes etc. contains inherent uncertainties and time delays. In the recent 20-25 years there has been a substantial increase of interest in time delay systems in the systems and control society; see for example Gu and Niculescu [2003], Zhong [2006]. Some of the time-delay systems can be treated as state-delayed plants. Stability and stabilisation issues of linear state-delay systems were discussed in [Kharitonov and Niculescu, 2003, Niamsup et al., 2008]. State feedback controllers are common in solving stabilisation problems of state-delay systems. A memoryless state feedback controller was proposed in [Lee, 1998], while robust H∞ state feedback controllers were treated in [Ge et al., 1996]. Sliding mode control of linear state delay systems via LMI was proposed in Xia and Jia [2003], Gouaisbaut et al. [2002] and optimal sliding mode approach was used in [Tang et al., 2007] to control time delay systems with sinusoidal disturbances. Guaranteed cost approach to the control of uncertain time-delay plants was reported in Wang and Zhao [2007], Mahmoud [2004], while observer-based control was treated in Bengea et al. [2004]. Robust control Oucheriah [1999] is a widely employed approach to control of time-delay systems with uncertainties, often via H∞ Suplin et al. [2006] and optimal Basin et al. [2007] approaches. Recently, the adaptive control of state-delay systems has become popular Evesque et al. [2003], Oucheriah [2001], Momeni and Aghdam [2007]. Several different approaches were combined to improve the control performance Nounou and Mahmoud [2005], Xia et al. [2008]. The Uncertainty and Disturbance Estimator (UDE) technique proposed in [Zhong and Rees, 2004] was one of them and
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then successfully used in input-output linearisation Patel et al. [2007], Talole and Phadke [2007], combining with sliding mode control Shendge and Patre [2007], Chandrasekhar and Dewan [2007], Talole and Phadke [2008]. The UDE is based on the assumption that a continuous signal can be approximated when it passes through a low-pass filter with a unity static gain and a short time constant. In this paper we present a state feedback controller for linear uncertain continuous plants with state delays and disturbances. The terms containing time delay (in case the information about the delay is absent in the controller) and the uncertainties are considered as an additional disturbance and then we employ an UDE to quickly estimate the overall disturbance and thus to robustify the state feedback controller. In Section 2, the UDE-based control law for uncertain linear plants with delay and disturbances is derived, following the procedures developed in [Zhong and Rees, 2004]. More aspects of the control strategy are analysed in Section 3 for LTI-SISO systems. Simulation examples are given in Section 4 and conclusions are made in Section 5. 2. UDE-BASED CONTROL 2.1 System description The system to be considered is formulated as: ˙ x(t) = (A1 + ∆A1 )x(t) + (A2 + ∆A2 )x(t − τ ) +(B + ∆B)u(t) + d(t), (1) T where x(t) = (x1 (t), ..., xn (t)) is the state, u(t) = (u1 (t), ..., ur (t))T is the control input, A1 and A2 are the known state matrices, ∆A1 and ∆A2 are the unknown
10.3182/20090616-3-IL-2002.0033
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
state matrices, B is the known control matrix having full column rank, ∆B is the unknown control matrix having full column rank, d(t) is the unpredictable external disturbances and τ is the time delay. 2.2 Reference model and structural constraint Assume that the desired specification can be described by the reference model x˙ m (t) = Am xm (t) + Bm c(t), (2) T where c(t) = (c1 (t), ..., cr (t)) is the uniformly bounded and piecewise continuous reference input. The control objective is to determine a state feedback control u(t) for the system (1) which forces the state error e(t) = xm (t) − x(t) between the system and the reference model e(t) = ( xm1 (t) − x1 (t) · · · xmn (t) − xn (t) )
(3)
converges to zero. In other words, the error dynamics ˙ (4) e(t) = (Am + K)e(t) is stable, where K is called the error feedback gain. Since the reference model is often chosen to be stable, K may be chosen as 0. If different error dynamics is desired or is required to guarantee robust stability (see Subsection 3.2), then common control strategies, such as pole placement, can be used to choose K. Combining (1), (2), (3), and (4), we obtain
U(s) = B+ [Am X(s) + Bm C(s) − KE(s) −A1 X(s) − A2 X(s)e−sτ − ∆A1 X(s) (8) −∆A2 X(s)e−sτ − ∆BU(s) − D(s)], where the Laplace transformation of a signal is denoted by the corresponding capital letter. We define a constant ε according to the availability of τ . In case τ is given, the first five terms in (8) are known and ε = 1; otherwise only the first four are known and ε = 0. The remaining terms, denoted hereafter by Ud (s), include the uncertainties and the external disturbance. Ud (s) can be written as Ud (s) = B+ [(ε − 1)A2 X(s)e−sτ − ∆A1 X(s) −∆A2 X(s)e−sτ − ∆BU(s) − D(s)] = B+ (A1 + εA2 e−sτ − sI)X(s) + BU(s) (9) In other words, the unknown dynamics and the disturbances Ud (s) can be observed by the system states and the control signal. However, it cannot be used in the control law directly. The UDE technique adopts an estimation of this signal in the frequency domain. Assume that Gf (s) is a strictly proper low-pass filter with a unity steady-state gain and a broad enough bandwidth, then Ud (s) can be accurately approximated by (10) UDE = Ud (s) · Gf (s). This is called the uncertainty and disturbance estimator (UDE). Hence, U(s) = B+ · [Am X(s) + Bm C(s) − KE(s)
Am x(t) + Bm c(t) − A1 x(t) − A2 x(t − τ )
−A1 X(s) − εA2 X(s)e−sτ ] + UDE
−Bu(t) − ∆A1 x(t) − ∆A2 x(t − τ )
= B+ · [Am X(s) + Bm C(s) − KE(s)
−∆Bu(t) − d(t) = Ke(t)
(5)
−sGf (s)X(s) + Gf (s)BU(s)], which gives the UDE-based control law as
Then the control action u(t) can be obtained as u(t) = B+ [Am x(t) + Bm c(t) − A1 x(t)
U(s) = −B+ (A1 + εA2 e−sτ )X(s) +
−A2 x(t − τ ) − ∆A1 x(t) − ∆A2 x(t − τ ) −∆Bu(t) − d(t) − Ke(t)],
(6)
where B+ = (BT B)−1 BT is the pseudo inverse of B. Notice that u(t) given in (6) is only an approximate solution of (5). Hence (4) and (5) will only be fully met under the following structural constraint:
I − BB+ · [Am x(t) + Bm c(t) − A1 x(t)
−A2 x(t − τ ) − ∆A1 x(t) − ∆A2 x(t − τ ) −∆Bu(t) − d(t) − Ke(t)] = 0.
−(A1 + εA2 e−sτ )X(s)(1 − Gf (s))
(7)
Obviously, if B is invertible, then this constraint is always met. If not, the choice of the reference model and the error feedback gain matrix will be restricted. However, according to [Youcef-Toumi and Ito, 1990], the systems described in canonical form meet this constraint. The following assumes that this constraint is satisfied. 2.3 UDE and the control law
1 B+ 1 − Gf (s)
×[Am X(s) + Bm C(s) − KE(s) − sGf (s)X(s)]
(11)
The control signal is formed by the state, the low-pass filter, the reference model and the error feedback gain. It has nothing to do with the uncertainty and the disturbance. Since Gf is strictly proper, sGf is implementable and there is no need of measuring the derivative of the states. Assuming that the frequency range of the system dynamics and the external disturbance is limited by ωf , a practical low-pass filter can be chosen as 1 Gf (s) = , (12) Ts + 1 where T = 1/ωf > 0. The steady-state estimation error is always zero because Gf (0) = 1 (the initial value of Gf (s) can always be chosen to be zero). It is interesting to see that 1 1 =1+ , 1 − Gf (s) Ts which means that there is an integral action. 3. SPECIAL CASE: LTI-SISO SYSTEMS
The control law in (6) can be represented in the s-domain by using the Laplace transform (assuming zero-initial states) as
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In this section, we consider LTI-SISO systems with uncertainties and disturbances described in the canonical form,
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
i.e. the corresponding matrices in (1) have the following partitions: 0 In−1 0 In−1 ; A2 = ; A1 = Ap1 Ap2 0 0 ∆A1 = ; ∆A2 = ; (13) F1 F2 0 0 0 , ; d(t) = ; ∆B = B= d(t) ∆b b where Ap1 = (−a11 , −a12 , ..., −a1n ), Ap2 = (−a21 , −a22 , ..., −a2n ), F1 = (−f11 , −f12 , ..., −f1n ) and F2 = (−f21 , −f22 , ..., −f2n ) are 1 × n row vectors, fi ≤ fji ≤ fi (i = 1, · · · , n; j = 1, 2) are uncertain parameters, In−1 is the (n − 1) × (n − 1) identity matrix, b 6= 0 and ∆b ≤ ∆b ≤ ∆b is an uncertain parameter. The output of the system is assumed to be y(t) = x1 (t). 3.1 Control scheme The reference model and the error feedback gain matrix are partitioned as 0 In−1 0 0 Am = ; Bm = ;K = , (14) bm K1 Am1 where Am1 = (−am1 , −am2 , ..., −amn ) and K1 = (−k1 , −k2 , ..., −kn ) are 1 × n row vectors. Substituting the matrices into (11), the control law is U (s) =
n X 1 ami Xi (s) + bm C(s)] [− b(1 − Gf (s)) i=1
n
1X (a1i + εa2i e−sτ )Xi (s). b i=1
The transfer function of the LTI-SISO plant given by (13) is Y (s) b + ∆b = , (17) U (s) P (s) where n X (a1i + f1i ) + (a2i + f2i )e−sτ si−1 . P (s) = sn + i=1
From (16), the closed-loop transfer function of the system is derived to be bm 1 Y (s) . = · b P (s)−Pa (s) C(s) Pm (s) 1 + (1 − G (s)) b+∆b f Pm (s)+Pk (s)
(18)
Theorem 1. Assume that (i) the reference polynomial Pm (s) = sn + amn sn−1 + · · · + am1 is chosen to be stable, (ii) the low-pass filter is chosen as (12), (iii) the error feedback polynomial Pk (s) = kn sn−1 + · · · + k1 is chosen such that Pm (s) + Pk (s) is stable. Then the closed-loop system is robustly stable if b P (jω) − Pa (jω) b+∆b ∀ω ∈ [0, ∞) (19) < 1, Pm (jω) + Pk (jω) holds on the space of uncertain parameters.
n X 1 + ki Ei (s) − sGf (s)Xn (s)] [ b(1 − Gf (s)) i=1
+
3.2 Stability analysis
(15)
Similarly as in as [Zhong and Rees, 2004], it can be further simplified as Pm (s) + Pk (s) bm U (s) = C(s) − Y (s) b [1 − Gf (s)] Pm (s) Pa (s) + Y (s), (16) b where Pm (s) = sn +amn sn−1 +· · ·+am1 is the denominator bm n of the reference model Gm (s) = Pm (s) , Pa (s) = s + Pn −sτ i−1 )s is the characteristic polynomial i=1 (a1i + εa2i e of the known system dynamics and Pk (s) = kn sn−1 + · · ·+ k1 is the error feedback polynomial. The UDE controller is divided into four parts: the referbm ence model Pm (s) to generate a desired trajectory; a local positive-feedback loop via the low-pass filter (behaving like a PI controller); the polynomial Pm (s) + Pk (s) for the tracking error and the polynomial Pa (s) for the output. The last two perform the derivative effect. Since the UDE controller uses more derivative information than the common PID controller, it has the potential to obtain better performances than the common PID controller. Note that the structure of (16) is the equivalent one of UDE-based control, but not the one used for implementation. The control law should be implemented according to (15).
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Proof. Since k1 − Gf (s)k∞ < 1 and Gm (s) is stable, the closed-loop system is robustly stable if
b P (s) − Pa (s)
b+∆b
< 1,
Pm (s) + Pk (s) ∞
according to the well-known small-gain theorem. The H∞ norm of a stable single-input-single-output transfer function is always obtained over the imaginary axis and hence the Theorem is proved. Remark 2. Since the uncertain parameters and delay terms only appear in the numerator of (19), the verification of the condition is not difficult, although it might involve heavy computation. 4. SIMULATION EXAMPLE Consider the following first-order system, x(t) ˙ = (a1 + ∆a1 )x(t) + (a2 + ∆a2 )x(t − τ ) + (b + △b)u(t) + d(t),
where a1 = 1, a2 = 1 and b = 1 are known parameters; 1 ∆a P 1 = 0.8sin(t), ∆a2 = 0.5sin(2t − 4 π) and ∆b = 0.5 [1(t − 2k) − 1(t − 1 − 2k)] are the uncertain time k
varying parameters (1(t) is the the step signal). d(t) is a negative unit-step disturbance, acting at t = 3.5sec and τ = 1sec.
The reference model is chosen as x˙ m (t) = −xm (t) + c(t) and the error feedback gain is chosen as k = 0. The lowpass time constant is chosen as T = 0.001sec. Although there are time-varying uncertainties in the system, the control law can be derived following the proposed procedure. According to (11), the control law is
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
U (s) =
1 [C(s) − X(s) − sGf (s)X(s)] 1 − Gf (s)
−(1 + εe−sτ )X(s). In the simulations, c(t) is chosen to be the step signal 1(t). 4.1 Nominal Performance The nominal response (the uncertain time varying parameters and the disturbance are zero) for the case with the delay τ unknown (ε = 0) is shown in Figure 1 and the nominal response for the case with the delay τ known (ε = 1) is shown in Figure 2. The system state x tracks the desired state xm very well in both scenarios. In the case of known τ the tracking error is practically zero at all times, while in the unknown time delay case there is a tracking error change at τ = 1 because the term containing the time delay acts like an unknown input to the system.
(a) the output signal (b) the tracking error (c) the control signal Figure 3. Example 1: The robust response when ε = 0
(a) the output signal (b) the tracking error (c) the control signal (a) the output signal (b) the tracking error (c) the control signal
Figure 4. Example 1: The robust response when ε = 1
Figure 1. Example 1: The nominal response when ε = 0
Figure 5. Example 1: The robust responses for different values of T
(a) the output signal (b) the tracking error (c) the control signal
u (there is a jump in u when the disturbance starts at t = 3.5sec) that the controller estimates the disturbance very quickly.
Figure 2. Example 1: The nominal response when ε = 1 4.2 Robust Performance The robust response for the case with unknown τ is shown in Figure 3 and the robust response for the case with known τ is shown in Figure 4. When the uncertain parameters are nonzero, the performance does not degrade significantly because the controller can well estimate the uncertainty. It can also be seen from the control signal
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The bandwidth (time constant) of the low-pass filter has an obvious influence on the system performance. The responses of three cases with different time constants are shown in Figure 5. The smaller the time constant (the broader the bandwidth), the better the tracking (i.e., the better the performance). However, in practice, the time constant might be limited by the computational capability and the measurement noise.
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
5. CONCLUSIONS This paper presents an uncertainty and disturbance estimator (UDE) for uncertain linear continuous systems with state delays and disturbances. The approach can be seen as a robust state-feedback control law. It does not require the knowledge of the bounds of uncertainties and disturbances and is continuous. The proposed algorithm shows excellent tracking and disturbance rejection capabilities almost independently on the availability in the controller of the information about the time delay. In case of the unknown time delay, the term with the delay is treated as an additional disturbance input to the system and is quickly estimated by the controller. The robust stability of LTI-SISO systems is analysed. Simulations are given to show the effectiveness of the UDE-based control. REFERENCES M. Basin, J. Rodriguez-Gonzalez, and L. Fridman. Optimal and robust control for linear state-delay systems. Journal of the Franklin Institute, 344:830–845, 2007. S. Bengea, X. Li, and R. DeCarlo. Combined controllerobserver design for uncertain time delay systems with application to engine idle speed control. Journal of Dyn. Sys. Meas. Con. - Trans. ASME, 126:772–780, 2004. T. Chandrasekhar and L. Dewan. Sliding mode control based on TDC and UDE. International Journal of Informatics and Systems Sciences, 3(1):36–53, 2007. S. Evesque, A. Annaswamy, S.-I. Niculescu, and A. Dowling. Adaptive control of a class of time-delay systems. Journal of Dyn. Sys. Meas. Con. - Trans. ASME, 125: 186–193, 2003. J.-H. Ge, P. Frank, and C.-F. Lin. Robust Hinf state feedback control for linear systems with state delay and parameter uncertainty. Automatica, 32(8):1183–1185, 1996. F. Gouaisbaut, M. Dambrine, and J. Richard. Sliding mode control of linear time delay systems: a design via LMI. IMA J. Math. Control Inform., 19:83–94, 2002. K. Gu and S.-I. Niculescu. Survey on recent results in the stability and control of time-delay systems. Journal of Dyn. Sys. Meas. Con. - Trans. ASME, 125:158–165, 2003. V. Kharitonov and S.-I. Niculescu. On the stability of linear systems with uncertain delay. IEEE Trans. Automat. Control, 48(1):127–132, 2003. C.-H. Lee. Simple stabilizability criteria and memoryless state feedback control design for time-delay systems with time-varying perturbations. IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications, 45(11):1211–1215, 1998. M. Mahmoud. Control of uncertain state-delay systems: guaranteed cost approach. IMA J. Math. Control Inform., 148:453–468, 2004. A. Momeni and A. Aghdam. An adaptive tracking problem for a family of retarded time-delay plants. International Journal of Adaptive Control and Signal Processing, 21: 885–910, 2007. P. Niamsup, K. Mukdasai, and V. Phat. Linear uncertain non-autonomous time-delay systems: stability and sta-
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