The Convergence of Parameter Estimates Is Not ... - IEEE Xplore
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThA10.5
The Convergence of Parameter Estimates is Not Necessary for a General Self-Tuning Control System– Stochastic Plant Weicun Zhang Abstract— This paper is concerned with the stability and convergence of a general stochastic self-tuning control (STC) system, which consists of arbitrary control strategy and arbitrary estimation algorithm. The necessary conditions required for global stability and convergence are relaxed, i.e., the convergence of parameter estimates is removed. The key point is that with the help of Virtual Equivalent System (VES) concept, the original nonlinear dominant (nonlinear in structure) problem of stochastic STC is converted to a linear dominant (linear in structure) problem— stochastic slow switching control system. Index Terms— stochastic self-tuning control, stability, convergence, virtual equivalent system, slow switching.
I. INTRODUCTION As representative works in the proof of stability and convergence of stochastic STC, G. C. Goodwin, P. J. Ramadge and P. E. Caines proved that the self-tuning control system comprising stochastic gradient (SG) estimates and minimum-variance control is stable and convergent without requiring the convergence of parameter estimates [1]; L. Guo and H. F. Chen obtained a similar conclusion for the selftuning control system based on recursive extended leastsquare (ELS) estimates and minimum-variance control [2]. Recently, some new results on stability and convergence of STC are still emerging, see, e.g., [3][4]. Despite the fundamental progress achieved so far, a general theory of STC is still absent. We are still a long way from having a full understanding of this important class of control strategies [5]. Some similar remarks can also be found in [69]. In the bulk of literature, almost all the existing works have been specific to particular estimation algorithms and control schemes, while only a few attempts have been made toward a unified analysis of the subject from a general perspective, e.g., [10-13]. Clearly additional theoretical research is needed to expand the classes for which stability can be proven; thus, relaxation of the necessary conditions and extension to even more general classes are still to be investigated [14]. Such situation is mainly due to the lack of appropriate methodology and corresponding analysis tools. Motivated by this observation, the author proposed a Virtual Equivalent System (VES) concept and a corresponding analysis approach. The idea of VES was originated from [15] and gradually adapted later in [16-20]. VES is an artificial This work was supported by the Science Fund of State Key Laboratory of Automotive Safety and Energy (KF09071) Weicun Zhang is with the Department of Automation, University of Science and Technology Beijing, Beijing 100083 P. R. China
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
system equivalent to a self-tuning control system in the input-output sense. From another point of view, by VES we actually convert the original nonlinear dominant (nonlinear in structure) problem to a linear dominant problem (linear in structure). In return for such disposal, the analysis of stability and convergence becomes more direct and simple. Consequently, the convergence of parameter estimates is removed from the necessary conditions to guarantee the stability and convergence of a general self-tuning control system. Additionally, it has been known that the model reference adaptive control, another important ingredient of adaptive control, is actually equivalent to self-tuning control [6-8][2125]. Therefore, a general theory of adaptive control can be expected. The reminder of the paper is organized as follows. Section 2 introduces the construction of VES for stochastic STC systems. The main results are presented in Section 3. Section 4 includes some perspectives on stochastic STC systems. Finally some concluding remarks are presented in section 5. II. CONSTRUCTION OF VES For simplicity of description, we consider a simple stochastic plant with white color noise. Fortunately, as we will see later in Section III, the properties of VES for other complex cases will remain the same. A(q −1 )y(k) = q −d B(q −1 )u(k) + ω(k)
(1)
with A(q −1 )
=
1 + a1 q −1 + · · · + an q −n
B(q −1 )
= b0 + b1 q −1 + · · · + bm q −m
where y(k), u(k) and ω(k) are the output, input, and disturbance, respectively, y(k) = 0, u(k) = 0, ω(k) = 0 for k < 0, A(q −1 ) and B(q −1 ) are polynomials in backwardshift operator q −1 with unknown coefficients and with known orders or upper bounds of orders n, m. The plant equation can be rewritten as y(k) = φT (k − d)θ + ω(k) φT (k − d) = [y(k − 1), · · · , y(k − n), u(k − d), · · · , u(k − d − m)] θT = [−a1 , · · · , −an , b0 , b1 , · · · , bm ]
(2) (3) (4)
ˆ We use Pm (k) and θ(k) to denote the estimated model and estimated parameters respectively.
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ThA10.5 Let C(k) denote the adaptive controller, it may be designed by any existing principle, such as by pole-assignment. To be specific, C(k) is determined by a mapping from model parameters into controller parameters, i.e. f :P→C
∆u0 (k) yr (k)-+
⊗ - C(tk )
(5)
− 6
where P is the set of estimated models including the true model of the plant. C is the corresponding controllers set. Then we have C(k) = f (Pm (k)) (6) or ˆ θc (k) = f (θ(k))
(7)
where θc (k) is the parametrization of C(k). Then we have u(k) = φTc (k)θc (k)
(8)
where φTc (k) = [y(k), y(k − 1), · · · , y(k − s1 ), u(k − 1), · · · , u(k − s2 ), yr (k), · · · , yr (k − s3 )]
(9)
The elements of φc (k) and the limited integers s1 , s2 , s3 depend on specific control strategy, yr (k) is a given bounded reference signal of the system, yr (k) = 0 for k < 0. The self-tuning control system is shown in Fig. 1 (the estimator is omitted for simplicity), its virtual equivalent systems are shown in Fig. 2 and Fig. 3. ω(k) yr (k)-+
⊗
− 6
- C(k)
u(k)
III. MAIN RESULTS For simplicity of description, we will first give the result for a stochastic plant with white noise. And then we extend the result to a stochastic plant with colored noise. Finally we consider a general case, i.e., the result for an arbitrary stochastic plant (time-invariant or time varying, linear or nonlinear). All the limit operations in this section are in the sense of probability one. Theorem 1: If a general self-tuning control system of a stochastic plant with known structure information, i.e., n, m, and d, has the following properties: 1) A control strategy is well defined to stabilize the parameter known model and tracking the bounded reference signal yr (k) in mean square sense; ˆ ˆ − θ(k ˆ − l)k → 0 , l is a finite 2) kθ(k)k ≤ M < ∞ , kθ(k) integer. 3) The parameter estimation error satisfies
(10)
(11)
ˆ − θ(t ˆ k )] + φ (k − d)[θ(k) T
yr (k)-+
⊗
− 6
- C(k)
u(k)
(12)
ω(k)
e(k)
? - Pm (k)
+ ? + -⊗
kφ(k−d)k2 )
k=1
Remark 1: Condition 2) does not imply the convergence of ˆ θ(k). Remark 2: Condition 3) in Theorem 1 is identical to n X
Further, we obtain an equivalent system as shown in Fig. 3, where
∆u0 (k) = φTc (k)[θc (k) − θc (tk )]
n X
(13) 4) The mapping from the model parameters into the controller parameters is continuous at infinity. Then it is stable and convergent.
A stochastic self-tuning control system as shown in Fig. 1 is obviously equivalent to the system as shown in Fig. 2, where
ˆ k ) − ω(k) ei (k) = y(k) − φT (k − d)θ(t ˆ − ω(k) = y(k) − φT (k − d)θ(k)
2 ˆ ky(k)−φT (k−d)θ(k)−ω(k)k = o(1+
k=1
Figure 1 Stochastic self-tuning Control System
ˆ − ω(k) e(k) = y(k) − φT (k − d)θ(k)
? + ? + y(k) +? + -⊗ - Pm (tk ) -⊗ u(k)
Figure 3 Stochastic VES II
n X
y(k) -
? - P
ei (k)
ω(k)
kˆ ω (k) − ω(k)k2 = o(1 +
k=1
n X
kφ(k − d)k2 )
(14)
k=1
in most other literatures. Proof: First, according to Lemma 1, VES II (Fig. 3) can be decomposed into three subsystems, as shown in Fig. 4, Fig. 5, and Fig. 6, respectively, y 0 (k) = 0, u0 (k) = 0, y 00 (k) = 0, u00 (k) = 0, y 000 (k) = 0, u000 (k) = 0 for k < 0. ω(k)
y(k) -
yr (k)-+
⊗
− 6
- C(tk )
u0 (k)
? - Pm (tk )
Figure 4 Subsystem 1 of stochastic VES II
Figure 2 Stochastic VES I
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y 0 (k) -
ThA10.5 ei (k) 0
u00 (k)
+ -⊗ - C(tk ) − 6
- Pm (tk )
+ ? + y 00 (k) -⊗ -
Further, according to Condition 1) we know that for each given tk , C(tk ) and Pm (tk ) constitute a stable closedloop system. Then, based on the existing results [27-32], we know that all the three subsystems are stable. That means in subsystem 1 (Fig. 4) n
1X e kφ1 (k)k2 < ∞ n
Figure 5 Subsystem 2 of stochastic VES II
In subsystem 2 (Fig. 5), with input signal given by (11), taking account of Conditions 2) and 3), we have
∆u0 (k) 0
+? + -⊗ - Pm (tk ) u000 (k)
+ -⊗ - C(tk ) − 6
(23)
k=1
n X
000
y (k) -
kei (k)k2 = o(1 +
k=1
= o(1 +
n X
kφ(k − d)k2 )
k=1 n X
(24)
2 e kφ(k)k )
k=1
Then it follows that n n 1X e 1 X 00 ky (k)k2 = o( kφ(k)k2 ) n n
Figure 6 Subsystem 3 of stochastic VES II Then by superposition principle, we have y(k) = y 0 (k) + y 00 (k) + y 000 (k) 0
00
000
u(k) = u (k) + u (k) + u (k)
(15)
˜ φ(k) = [y(k), · · · , y(k − n), u(k − 1), · · · , u(k − d − m), yr (k), · · · , yr (k − s3 )]
φ˜1 (k) = [y 0 (k), · · · , y 0 (k − n), u0 (k − 1), · · · , u0 (k − d − m), yr (k), · · · , yr (k − s3 )] φ˜2 (k) = [y 00 (k), · · · , y 00 (k − n), u00 (k − 1), · · · , u00 (k − d − m), 0, · · · , 0] φ˜3 (k) = [y 000 (k), · · · , y 000 (k − n), u000 (k − 1), · · · , u000 (k − d − m), 0, · · · , 0]
e φ(k) = φe1 (k) + φe2 (k) + φe3 (k) e kφc (k)k = O(kφ(k)k),
(21)
e kφ(k − d)k = O(kφ(k)k) (22)
The choice of sequence tk is the same as in the deterministic situation [26], which together with Condition 2) guarantees the VES II (Fig. 3) is a ”slow switching” system with large enough dwell time and a compact index set of switching. Then subsystem 1 (Fig. 4) is a stochastic slow switching system, and the other two subsystems (Fig. 5 and Fig. 6) are deterministic slow switching systems.
k=1
k=1
n
n
k=1
k=1
e ∆u0 (k) = o(kφc (k)k) = o(kφ(k)k) Then we have n 1X n
(26)
(27)
(28)
n
ky 000 (k)k2 = o(
k=1
1X e kφ(k)k2 ) n
(29)
k=1
n
n
k=1
k=1
n
n
k=1
k=1
1X e 1 X 000 ku (k)k2 = o( kφ(k)k2 ) n n
(19)
Then we have
n
(25)
In subsystem 3 (Fig. 6), by Conditions 2) and 4), and (12), its input signal has the following property
(18)
(20)
n
1X e 1X e kφ2 (k)k2 = o( kφ(k)k2 ) n n
(17)
Similarly, define its counterparts φ˜1 (k) in subsystem 1 (Fig. 4), φ˜2 (k) in subsystem 2 (Fig. 5), and φ˜3 (k) in subsystem 3 (Fig. 6), respectively, i. e.
k=1
1X e 1 X 00 ku (k)k2 = o( kφ(k)k2 ) n n
(16)
˜ To facilitate the proof, we need to define a new vector φ(k), whose elements are the union of that of φ(k − d) and φc (k). Without loss of generality, we assume s1 < n, s2 < m, then ˜ φ(k) takes the form of
k=1
1X e 1X e kφ3 (k)k2 = o( kφ(k)k2 ) n n
(30)
(31)
By Lemma 2, regarding φe2 (k) and φe3 (k) as one variate, we come to the conclusion that the self-tuning control system is stable. Next we proceed to prove the convergence of the selftuning control system. First we examine the tracking performance of subsystem 1. Denote the tracking error of each P ”frozen” system {C(t¯k ), Pm (t¯k ), ω(k)} by et¯k (k) = yt¯k (k) − yr (k) where t¯k represents the ”frozen” tP k. Similarly, the tracking error of {C(tk ), Pm (tk ), ω(k)}, i.e., subsystem 1 (Fig. 4), is denoted by
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etk (k) = ytk (k) − yr (k) = y 0 (k) − yr (k)
ThA10.5 Then we know that for each ”frozen” system P {C(t¯k ), Pm (t¯k ), ω(k)}, there exists a stable dynamic system et¯k (k + 1) = ft¯k (et¯k (k), ω(k), k) Subsequently, by MLF methodology, we know that for P {C(tk ), Pm (tk ), ω(k)}, there exists a stable ”slow switching” dynamic system. etk (k + 1) = ftk (etk (k), ω(k), k) According to Condition 1), for each t¯k , we have n
lim sup
n→∞
1X ket¯k (k)k2 = R n k=1
where R is a constant. It is easy to see that etk (k) takes values in the set of et¯k (k), then by squeezing, it follows that n
1X lim sup ketk (k)k2 = R n→∞ n k=1
That implies the ” slow switching” subsystem 1 is tracking, i.e. n 1X 0 lim sup ky (k) − yr (k)k2 = R n→∞ n k=1
Next, by Lemma 3, regarding y 00 (k) and y 000 (k) as one variate, we see that the self-tung control system is convergent, i.e., lim sup
n→∞
1 n
n X
ky(k) − yr (k)k2
k=1 n
= lim sup n→∞
1X 0 ky (k) − yr (k) + y 00 (k) + y 000 (k)k2 n k=1
=R (32) That completes the proof of Theorem 1. We now extend the above result, i.e., Theorem 1 to a stochastic plant with colored noise. Consider A(q −1 )y(k) = q −d B(q −1 )u(k) + D(q −1 )ω(k)
(33)
where A(q −1 )
=
B(q −1 )
= b0 + b1 q −1 + · · · + bm q −m
D(q
−1
)
=
1 + a1 q −1 + · · · + an q −n 1 + d1 q −1 + · · · + ds q −s
We have the following result. (The proof is omitted.) Corollary 1: If a general self-tuning control system of stochastic plant (33) with known structure information, i.e., n, m, s, and d, has the following properties: 1) A control strategy is well defined to stabilize the parameter known model and tracking the bounded reference signal yr (k) in mean square sense; ˆ ˆ − θ(k ˆ − l)k → 0 , l is a finite 2) kθ(k)k ≤ M < ∞ , kθ(k)
integer. 3) The parameter estimation error satisfies n X
2 ˆ ky(k)−φT (k−d)θ(k)−ω(k)k = o(1+
k=1
n X
kφ(k−d)k2 )
k=1
(34) 4) The mapping from the model parameters into the controller parameters is continuous at infinity. Then it is stable and convergent. Recalling the proof procedures of Theorem 1, we actually didn’t rely on the structure information of plant P , because plant P is replaced by Pm (k) and e(k) in Fig. 2, or Pm (tk ) and ei (k) in Fig. 3, respectively. Thus we know that Theorem 1 and Corollary 1 hold for an arbitrary stochastic plant. (Details are omitted due to space limitations) IV. CONCLUDING REMARKS With the help of VES concepts and methodology, we presented some new results on the stability and convergence of a general stochastic self-tuning control system. The necessary conditions for the stability and convergence of STC have been relaxed, i.e., the convergence of parameter estimates has been removed. To some extent, these results parallel the more familiar ones on stability of slowly time-varying systems [33-35]. R EFERENCES
[1] G. C. Goodwin, P. J. Ramadge and P. E. Caines, ”Discrete time stochastic adaptive control”, SIAM J. Control and Optimization, Vol. 19, pp. 829-853, 1981 [2] L. Guo and H. F. Chen, ”The Astrom-Wittenmark self-tuning regulator revisited and ELS-based adaptive trackers”, IEEE Trans. Automatic Control, Vol. 36, pp. 802-812, 1991 [3] M. Prandini and M.C. Campi. Adaptive LQG control of input-output systems - A cost-biased approach. SIAM Journal on Control and Optimization, Vol. 39, pp. 1499-1519, 2001 [4] A. Patete1, K. Furuta and M. Tomizuka, ”Stability of self-tuning control based on Lyapunov function”, Int. J. Adapt. Control and Signal Process, Vol. 22, pp. 795-810, 2008 [5] G. C. Goodwin, A. Feuer, ”Adaptive control: where to now?”, Adaptive Systems for Signal Processing,Communications, and Control Symposium, Proc. IEEE, pp. 165-170, 2000 [6] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness, Prentice Hall: Englewood Cliffs, New Jersey, 1989 ˚ om and B. Wittenmark, Adaptive control, Addison-Wesley, [7] K. J. Astr¨ 1995 [8] P. A. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall: Englewood Cliffs, New Jersey, U. S. A, 1996 [9] Sajjad Fekri, Michael Athans and Antonio Pascoal, ”Issues, progress and new results in robust adaptive control”, Int. J. Adapt. Control and Signal Process, Vol. 20(10), pp. 519-579, 2006 [10] P. R. Kumar, ”Convergence of adaptive control schemes using least squares parameter estimates”, IEEE Trans. Automatic Control, Vol. 35, No. 4, pp. 416-424, 1990 [11] J. H. Van Schuppen, ”Tuning of gaussian stochastic control systems”, IEEE Trans. Automatic Control, Vol. 39, No. 11, pp. 2178-2190, 1994 [12] Karim Nassiri-Toussi and Wei Ren, ”A Unified Analysis of Stochastic Adaptive Control: Asymptotic Self-tuning”, Proc. of the 34th IEEE Conference on Decision and Control, pp. 2932-2937, 1995 [13] A. S. Morse, ”Towards a unified theory of parameter adaptive controlpart II: certainty equivalence and implicit tuning”, IEEE Trans. Automatic Control, Vol. 37, No. 1, pp. 15-29, 1992 [14] T. Katayama, T. McKelvey, A. Sano, C. G. Cassandras, M. C. Campi, ”Trends in systems and signals: Status report prepared by the IFAC Coordinating Committee on Systems and Signals”, Annual Reviews in Control, Vol. 30, pp. 5-17, 2006
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ThA10.5 [15] Weicun Zhang, ”Theoretical research and application of robust adaptive control”, Ph. D Thesis, Tsinghua University, 1993 [16] Weicun Zhang, Jin Young Choi, ”On the Stability and Convergence of Self-Tuning Control Systems (I)– Deterministic Plant”, International Conference on Control, Automation and Systems, Seoul, Korea, 2007 [17] Weicun Zhang, Jin Young Choi, ”On the Stability and Convergence of Self-Tuning Control Systems (II)– Stochastic Plant”, International Conference on Control, Automation and Systems, Seoul, Korea, 2007 [18] Weicun Zhang, Xiaoli Li, Jin Young Choi, ”A unified analysis of switching multiple model adaptive control - Virtual equivalent system approach”, 17th IFAC World Congress, Seoul, Korea, 2008. [19] Weicun Zhang, Xiaoli Li, Tianguang Chu, ”Virtual equivalent system method for analysis of stochastic self-tuning control”, The 6th World Congress on Intelligent Control and Automation, Dalian, China, 2006 [20] Weicun Zhang, Tianguang Chu, Long Wang, ”A new theoretical framework for self-tuning control”, International Journal of Information Technology, Vol. 11(11), pp. 123-139, 2005 [21] B. Egardt, ”Unification of some discrete time adaptive control schemes”, IEEE Trans. Automatic Control, Vol. 25, pp. 693- 697, 1980 [22] P. J. Gawthrop, ”Some interpretations of the self-tuning controller”, Proc. IEEE, Vol. 124, pp. 889-894, 1977 [23] L. Ljung, I. D. Landau, ”Model reference adaptive systems and self-tuning regulators - some connections”, Proc. 7th IFAC World Congress, Vol. 3, pp. 1973-1980, 1978 [24] K. S. Narendra, L. S. Valavani, ”Direct and indirect adaptive control”, Automatica,Vol. 15, pp. 663-664, 1979 [25] G. C. Goodwin, K. S. Sin, Adaptive filtering, prediction and control, Prentice Hall, 1984. [26] Weicun Zhang, ”The Convergence of Parameter Estimates is Not Necessary for a General Self-Tuning Control System – Deterministic Plant”, Technical Report, University of Science and Technology Beijing, 2009 [27] D. Liberzon, A. S. Morse, ”Basic Problems in Stability and Design of Switched Systems”, IEEE Control Systems Magazine, Vol. 19, pp. 59-70, 1999 [28] R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, ”Stability Criteria for Switched and Hybrid Systems”, SIAM Review, Vol. 49, pp. 545592, 2007 [29] D. Chatterjee, D. Liberzon, ”Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions”, SIAM Journal on Control and Optimization, Vol. 45, No. 1, pp. 174-206, 2006 [30] D. Chatterjee and D. Liberzon, ”On stability of stochastic switched systems”, Proceedings of 43rd Conference on Decision and Control, Vol. 4, 2004, pp. 4125-4127 [31] M. Prandini, ”Switching control of stochastic linear systems: Stability and performance results”, Proc. 6th Congress of SIMAI, Chia Laguna, Cagliari, Italy, May 2002 [32] M. Prandini and M.C. Campi, ”Logic-based switching for the stabilization of stochastic systems in presence of unmodeled dynamics”, Proc. 40th CDC, Orlando, USA, Dec. 2001 [33] C. A. Desoer, ”Slowly varying system xi+1 = Ai xi ”, Electronics Letters, Vol. 6, pp. 339-340, 1970 [34] C. A. Desoer, ”Slowly varying system x˙ = A(t)x”, IEEE Trans. Automatic Control, Vol. 14, No. 6, pp. 780-781, 1970 [35] V. Solo, ”On the stability of slowly time-varying linear systems”, Math. Control Signals Systems, Vol. 7, pp. 331-350, 1994
Second, assume (A.1) and (A.2) hold for k, k − 1, ..., 1. Considering Fig.3, Fig. 4, Fig. 5, and Fig. 6, respectively, we have ˆ k ) + ω(k + 1) + ei (k + 1) (A.3) y(k + 1) = φT (k − d + 1)θ(t ˆ k ) + ω(k + 1) y 0 (k + 1) = φT1 (k − d + 1)θ(t
(A.4)
ˆ k ) + ei (k + 1) y 00 (k + 1) = φT2 (k − d + 1)θ(t
(A.5)
000
y (k + 1) =
φT3 (k
ˆ k) − d + 1)θ(t
(A.6)
where φT1 (k − d + 1) = [y 0 (k), · · · , y 0 (k − n + 1), u(k − d + 1), · · · , u0 (k − d − m + 1)] (A.7) φT2 (k − d + 1) = [y 00 (k), · · · , y 00 (k − n + 1), u00 (k − d + 1), · · · , u00 (k − d − m + 1)] (A.8) φT3 (k − d + 1) = [y 000 (k), · · · , y 000 (k − n + 1), u000 (k − d + 1), · · · , u000 (k − d − m + 1)] (A.9) According to the assumption, it is obvious that φT1 (k−d+1)+φT2 (k−d+1)+φT3 (k−d+1) = φT (k−d+1) (A.10) Thus, we obtain y 0 (k + 1) + y 00 (k + 1) + y 000 (k + 1) ˆ k ) + ω(k + 1) + ei (k + 1) = y(k + 1) = φT (k − d + 1)θ(t (A.11) Next we observe u0 (k + 1), u00 (k + 1) and u000 (k + 1). According to the definition of u(k), i.e., equation (8), we have in Fig. 3 u(k + 1) = φTc (k + 1)θc (tk+1 ) + ∆u0 (k + 1)
(A.12)
Similarly, in Fig. 4, Fig. 5, and Fig. 6, we have
000
u0 (k + 1) = φTc1 (k + 1)θc (tk+1 )
(A.13)
u00 (k + 1) = φTc2 (k + 1)θc (tk+1 )
(A.14)
u (k + 1) =
φTc3 (k
0
+ 1)θc (tk+1 ) + ∆u (k + 1)
(A.15)
where φTc (k + 1) = [y(k + 1), y(k), · · · , y(k − s1 + 1),
A PPENDIX I L EMMAS AND P ROOF
u(k), u(k − 1), · · · , u(k − s2 + 1), (A.16)
Lemma 1: Virtual equivalent system, as shown in Fig. 3, can be decomposed into three subsystems, as shown in Fig. 4, Fig. 5, and Fig. 6, i. e., the input and output of the virtual equivalent system are identical to the sum of the inputs and outputs of the decomposed subsystems.
yr (k + 1), · · · , yr (k − s3 + 1)] φTc1 (k + 1) = [y 0 (k + 1), y 0 (k), · · · , y 0 (k − s1 + 1), u0 (k), u0 (k − 1), · · · , u0 (k − s2 + 1), yr (k + 1), · · · , yr (k − s3 + 1)] (A.17)
Proof: We give the proof by mathematical induction. First, we know that for k ≤ 0 ,
φTc2 (k + 1) = [y 00 (k + 1), y 00 (k), · · · , y 00 (k − s1 + 1),
y(k) = y 0 (k) + y 00 (k) + y 000 (k)
(A.1)
u(k) = u0 (k) + u00 (k) + u000 (k)
(A.2)
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u00 (k), u00 (k − 1), · · · , u00 (k − s2 + 1), 0, · · · , 0] (A.18)
ThA10.5 φTc3 (k + 1) = [y 000 (k + 1), y 000 (k), · · · , y 000 (k − s1 + 1), 000
000
000
u (k), u (k − 1), · · · , u (k − s2 + 1), 0, · · · , 0] (A.19) Based on (A.11), it is obvious that φTc1 (k + 1) + φTc2 (k + 1) + φTc3 (k + 1) = φTc (k + 1) (A.20) Then we get
Pn This together with (A.23) and the fact n1 k=1 kφe1 (pk )k2 < ∞, by squeezing, yields an absurd result Pn 2 e k=1 kφ(pk )k →0 Pn 2 e k=1 kφ(pk )k Therefore, we conclude our original hypothesis is false. Then we have n 1X e kφ(k)k2 < ∞ (A.25) n k=1
u0 (k + 1) + u00 (k + 1) + u000 (k + 1)
That completes the proof of Lemma 2.
= φTc (k + 1)θc (tk+1 ) + ∆u0 (k + 1)
(A.21) Lemma 3: Given
= u(k + 1)
1 n
Pn
k=1 ky
00
(k)k2 = o(1)
n
Consequently, we know that (A.1) and (A.2) hold for arbitrary k. That completes the proof of Lemma 1. e Lemma 2: Given φ(k) = φe1 (k) + φe2 (k)
lim sup
n→∞
1X 0 ky (k) − yr (k)k2 < ∞ n k=1
Then n n 1X 0 1X 0 00 2 ky (k) − yr (k) + y (k)k → ky (k) − yr (k)k2 n n k=1
n
1X e kφ1 (k)k2 < ∞ n
k=1
Proof: It is obvious that
k=1
n
n
n
k=1
k=1
1X 0 ky (k) − yr (k) + y 00 (k)k2 n
1X e 1X e kφ2 (k)k2 = o( kφ(k)k2 ) n n
k=1
n
=
Then
k=1 n X
n
1X e kφ(k)k2 < ∞ n
2 + n
k=1
n
k=1
k=1 n X
1X e 1X e kφ(k)k2 = kφ1 (k) + φe2 (k)k2 n n ≤
2 n
n
kφe1 (k)k2 +
k=1
Suppose, by contradiction, that is Punbounded, there must exist n 1 2 e k=1 kφ(pk )k → ∞, satisfying n n
2X e kφ2 (k)k2 n k=1 (A.22) 1 n
n
k=1 n
k=1 n
n
k=1
2 e subsequence k=1 kφ(k)k
1X e 1X e kφ(pk )k2 = kφ1 (pk ) + φe2 (pk )k2 n n k=1
[y 0 (k) − yr (k)]y 00 (k)
By the Cauchy inequality, we know that )2 ( n 1X 0 00 [y (k) − yr (k)]y (k) 0≤ n k=1 ( n ) ( n ) (A.27) 1X 0 1 X 00 2 2 ≤ [y (k) − yr (k)] . [y (k)] n n
Pn
a
k=1
1X 0 2X 0 → ky (k) − yr (k)k2 + [y (k) − yr (k)]y 00 (k) n n k=1 k=1 (A.26)
Proof: Making use of the triangle inequality and the fact 2ab ≤ a2 + b2 , we obtain n
n
1X 0 1 X 00 [y (k) − yr (k)]2 + [y (k)]2 n n
k=1
→0 Then by squeezing we obtain ( n )2 1X 0 [y (k) − yr (k)]y 00 (k) →0 n
(A.28)
k=1
n
2X e 2X e kφ1 (pk )k2 + kφ2 (pk )k2 ≤ n n k=1 k=1 (A.23)
That is
n
1X 0 [y (k) − yr (k)]y 00 (k) → 0 n k=1
Substituting this into (A.26), we have Besides, by the fact that the limits of sequence and its subsequences are the same, we have n
n
1X e 1X e kφ2 (pk )k2 = o( kφ(pk )k2 ) n n k=1
(A.24)
n
n
k=1
k=1
1X 0 1X 0 ky (k) − yr (k) + y 00 (k)k2 → ky (k) − yr (k)k2 n n That completes the proof of Lemma 3.
k=1
3494
ThA10.5
The Convergence of Parameter Estimates is Not Necessary for a General Self-Tuning Control System– Stochastic Plant Weicun Zhang Abstract— This paper is concerned with the stability and convergence of a general stochastic self-tuning control (STC) system, which consists of arbitrary control strategy and arbitrary estimation algorithm. The necessary conditions required for global stability and convergence are relaxed, i.e., the convergence of parameter estimates is removed. The key point is that with the help of Virtual Equivalent System (VES) concept, the original nonlinear dominant (nonlinear in structure) problem of stochastic STC is converted to a linear dominant (linear in structure) problem— stochastic slow switching control system. Index Terms— stochastic self-tuning control, stability, convergence, virtual equivalent system, slow switching.
I. INTRODUCTION As representative works in the proof of stability and convergence of stochastic STC, G. C. Goodwin, P. J. Ramadge and P. E. Caines proved that the self-tuning control system comprising stochastic gradient (SG) estimates and minimum-variance control is stable and convergent without requiring the convergence of parameter estimates [1]; L. Guo and H. F. Chen obtained a similar conclusion for the selftuning control system based on recursive extended leastsquare (ELS) estimates and minimum-variance control [2]. Recently, some new results on stability and convergence of STC are still emerging, see, e.g., [3][4]. Despite the fundamental progress achieved so far, a general theory of STC is still absent. We are still a long way from having a full understanding of this important class of control strategies [5]. Some similar remarks can also be found in [69]. In the bulk of literature, almost all the existing works have been specific to particular estimation algorithms and control schemes, while only a few attempts have been made toward a unified analysis of the subject from a general perspective, e.g., [10-13]. Clearly additional theoretical research is needed to expand the classes for which stability can be proven; thus, relaxation of the necessary conditions and extension to even more general classes are still to be investigated [14]. Such situation is mainly due to the lack of appropriate methodology and corresponding analysis tools. Motivated by this observation, the author proposed a Virtual Equivalent System (VES) concept and a corresponding analysis approach. The idea of VES was originated from [15] and gradually adapted later in [16-20]. VES is an artificial This work was supported by the Science Fund of State Key Laboratory of Automotive Safety and Energy (KF09071) Weicun Zhang is with the Department of Automation, University of Science and Technology Beijing, Beijing 100083 P. R. China
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
system equivalent to a self-tuning control system in the input-output sense. From another point of view, by VES we actually convert the original nonlinear dominant (nonlinear in structure) problem to a linear dominant problem (linear in structure). In return for such disposal, the analysis of stability and convergence becomes more direct and simple. Consequently, the convergence of parameter estimates is removed from the necessary conditions to guarantee the stability and convergence of a general self-tuning control system. Additionally, it has been known that the model reference adaptive control, another important ingredient of adaptive control, is actually equivalent to self-tuning control [6-8][2125]. Therefore, a general theory of adaptive control can be expected. The reminder of the paper is organized as follows. Section 2 introduces the construction of VES for stochastic STC systems. The main results are presented in Section 3. Section 4 includes some perspectives on stochastic STC systems. Finally some concluding remarks are presented in section 5. II. CONSTRUCTION OF VES For simplicity of description, we consider a simple stochastic plant with white color noise. Fortunately, as we will see later in Section III, the properties of VES for other complex cases will remain the same. A(q −1 )y(k) = q −d B(q −1 )u(k) + ω(k)
(1)
with A(q −1 )
=
1 + a1 q −1 + · · · + an q −n
B(q −1 )
= b0 + b1 q −1 + · · · + bm q −m
where y(k), u(k) and ω(k) are the output, input, and disturbance, respectively, y(k) = 0, u(k) = 0, ω(k) = 0 for k < 0, A(q −1 ) and B(q −1 ) are polynomials in backwardshift operator q −1 with unknown coefficients and with known orders or upper bounds of orders n, m. The plant equation can be rewritten as y(k) = φT (k − d)θ + ω(k) φT (k − d) = [y(k − 1), · · · , y(k − n), u(k − d), · · · , u(k − d − m)] θT = [−a1 , · · · , −an , b0 , b1 , · · · , bm ]
(2) (3) (4)
ˆ We use Pm (k) and θ(k) to denote the estimated model and estimated parameters respectively.
3489
ThA10.5 Let C(k) denote the adaptive controller, it may be designed by any existing principle, such as by pole-assignment. To be specific, C(k) is determined by a mapping from model parameters into controller parameters, i.e. f :P→C
∆u0 (k) yr (k)-+
⊗ - C(tk )
(5)
− 6
where P is the set of estimated models including the true model of the plant. C is the corresponding controllers set. Then we have C(k) = f (Pm (k)) (6) or ˆ θc (k) = f (θ(k))
(7)
where θc (k) is the parametrization of C(k). Then we have u(k) = φTc (k)θc (k)
(8)
where φTc (k) = [y(k), y(k − 1), · · · , y(k − s1 ), u(k − 1), · · · , u(k − s2 ), yr (k), · · · , yr (k − s3 )]
(9)
The elements of φc (k) and the limited integers s1 , s2 , s3 depend on specific control strategy, yr (k) is a given bounded reference signal of the system, yr (k) = 0 for k < 0. The self-tuning control system is shown in Fig. 1 (the estimator is omitted for simplicity), its virtual equivalent systems are shown in Fig. 2 and Fig. 3. ω(k) yr (k)-+
⊗
− 6
- C(k)
u(k)
III. MAIN RESULTS For simplicity of description, we will first give the result for a stochastic plant with white noise. And then we extend the result to a stochastic plant with colored noise. Finally we consider a general case, i.e., the result for an arbitrary stochastic plant (time-invariant or time varying, linear or nonlinear). All the limit operations in this section are in the sense of probability one. Theorem 1: If a general self-tuning control system of a stochastic plant with known structure information, i.e., n, m, and d, has the following properties: 1) A control strategy is well defined to stabilize the parameter known model and tracking the bounded reference signal yr (k) in mean square sense; ˆ ˆ − θ(k ˆ − l)k → 0 , l is a finite 2) kθ(k)k ≤ M < ∞ , kθ(k) integer. 3) The parameter estimation error satisfies
(10)
(11)
ˆ − θ(t ˆ k )] + φ (k − d)[θ(k) T
yr (k)-+
⊗
− 6
- C(k)
u(k)
(12)
ω(k)
e(k)
? - Pm (k)
+ ? + -⊗
kφ(k−d)k2 )
k=1
Remark 1: Condition 2) does not imply the convergence of ˆ θ(k). Remark 2: Condition 3) in Theorem 1 is identical to n X
Further, we obtain an equivalent system as shown in Fig. 3, where
∆u0 (k) = φTc (k)[θc (k) − θc (tk )]
n X
(13) 4) The mapping from the model parameters into the controller parameters is continuous at infinity. Then it is stable and convergent.
A stochastic self-tuning control system as shown in Fig. 1 is obviously equivalent to the system as shown in Fig. 2, where
ˆ k ) − ω(k) ei (k) = y(k) − φT (k − d)θ(t ˆ − ω(k) = y(k) − φT (k − d)θ(k)
2 ˆ ky(k)−φT (k−d)θ(k)−ω(k)k = o(1+
k=1
Figure 1 Stochastic self-tuning Control System
ˆ − ω(k) e(k) = y(k) − φT (k − d)θ(k)
? + ? + y(k) +? + -⊗ - Pm (tk ) -⊗ u(k)
Figure 3 Stochastic VES II
n X
y(k) -
? - P
ei (k)
ω(k)
kˆ ω (k) − ω(k)k2 = o(1 +
k=1
n X
kφ(k − d)k2 )
(14)
k=1
in most other literatures. Proof: First, according to Lemma 1, VES II (Fig. 3) can be decomposed into three subsystems, as shown in Fig. 4, Fig. 5, and Fig. 6, respectively, y 0 (k) = 0, u0 (k) = 0, y 00 (k) = 0, u00 (k) = 0, y 000 (k) = 0, u000 (k) = 0 for k < 0. ω(k)
y(k) -
yr (k)-+
⊗
− 6
- C(tk )
u0 (k)
? - Pm (tk )
Figure 4 Subsystem 1 of stochastic VES II
Figure 2 Stochastic VES I
3490
y 0 (k) -
ThA10.5 ei (k) 0
u00 (k)
+ -⊗ - C(tk ) − 6
- Pm (tk )
+ ? + y 00 (k) -⊗ -
Further, according to Condition 1) we know that for each given tk , C(tk ) and Pm (tk ) constitute a stable closedloop system. Then, based on the existing results [27-32], we know that all the three subsystems are stable. That means in subsystem 1 (Fig. 4) n
1X e kφ1 (k)k2 < ∞ n
Figure 5 Subsystem 2 of stochastic VES II
In subsystem 2 (Fig. 5), with input signal given by (11), taking account of Conditions 2) and 3), we have
∆u0 (k) 0
+? + -⊗ - Pm (tk ) u000 (k)
+ -⊗ - C(tk ) − 6
(23)
k=1
n X
000
y (k) -
kei (k)k2 = o(1 +
k=1
= o(1 +
n X
kφ(k − d)k2 )
k=1 n X
(24)
2 e kφ(k)k )
k=1
Then it follows that n n 1X e 1 X 00 ky (k)k2 = o( kφ(k)k2 ) n n
Figure 6 Subsystem 3 of stochastic VES II Then by superposition principle, we have y(k) = y 0 (k) + y 00 (k) + y 000 (k) 0
00
000
u(k) = u (k) + u (k) + u (k)
(15)
˜ φ(k) = [y(k), · · · , y(k − n), u(k − 1), · · · , u(k − d − m), yr (k), · · · , yr (k − s3 )]
φ˜1 (k) = [y 0 (k), · · · , y 0 (k − n), u0 (k − 1), · · · , u0 (k − d − m), yr (k), · · · , yr (k − s3 )] φ˜2 (k) = [y 00 (k), · · · , y 00 (k − n), u00 (k − 1), · · · , u00 (k − d − m), 0, · · · , 0] φ˜3 (k) = [y 000 (k), · · · , y 000 (k − n), u000 (k − 1), · · · , u000 (k − d − m), 0, · · · , 0]
e φ(k) = φe1 (k) + φe2 (k) + φe3 (k) e kφc (k)k = O(kφ(k)k),
(21)
e kφ(k − d)k = O(kφ(k)k) (22)
The choice of sequence tk is the same as in the deterministic situation [26], which together with Condition 2) guarantees the VES II (Fig. 3) is a ”slow switching” system with large enough dwell time and a compact index set of switching. Then subsystem 1 (Fig. 4) is a stochastic slow switching system, and the other two subsystems (Fig. 5 and Fig. 6) are deterministic slow switching systems.
k=1
k=1
n
n
k=1
k=1
e ∆u0 (k) = o(kφc (k)k) = o(kφ(k)k) Then we have n 1X n
(26)
(27)
(28)
n
ky 000 (k)k2 = o(
k=1
1X e kφ(k)k2 ) n
(29)
k=1
n
n
k=1
k=1
n
n
k=1
k=1
1X e 1 X 000 ku (k)k2 = o( kφ(k)k2 ) n n
(19)
Then we have
n
(25)
In subsystem 3 (Fig. 6), by Conditions 2) and 4), and (12), its input signal has the following property
(18)
(20)
n
1X e 1X e kφ2 (k)k2 = o( kφ(k)k2 ) n n
(17)
Similarly, define its counterparts φ˜1 (k) in subsystem 1 (Fig. 4), φ˜2 (k) in subsystem 2 (Fig. 5), and φ˜3 (k) in subsystem 3 (Fig. 6), respectively, i. e.
k=1
1X e 1 X 00 ku (k)k2 = o( kφ(k)k2 ) n n
(16)
˜ To facilitate the proof, we need to define a new vector φ(k), whose elements are the union of that of φ(k − d) and φc (k). Without loss of generality, we assume s1 < n, s2 < m, then ˜ φ(k) takes the form of
k=1
1X e 1X e kφ3 (k)k2 = o( kφ(k)k2 ) n n
(30)
(31)
By Lemma 2, regarding φe2 (k) and φe3 (k) as one variate, we come to the conclusion that the self-tuning control system is stable. Next we proceed to prove the convergence of the selftuning control system. First we examine the tracking performance of subsystem 1. Denote the tracking error of each P ”frozen” system {C(t¯k ), Pm (t¯k ), ω(k)} by et¯k (k) = yt¯k (k) − yr (k) where t¯k represents the ”frozen” tP k. Similarly, the tracking error of {C(tk ), Pm (tk ), ω(k)}, i.e., subsystem 1 (Fig. 4), is denoted by
3491
etk (k) = ytk (k) − yr (k) = y 0 (k) − yr (k)
ThA10.5 Then we know that for each ”frozen” system P {C(t¯k ), Pm (t¯k ), ω(k)}, there exists a stable dynamic system et¯k (k + 1) = ft¯k (et¯k (k), ω(k), k) Subsequently, by MLF methodology, we know that for P {C(tk ), Pm (tk ), ω(k)}, there exists a stable ”slow switching” dynamic system. etk (k + 1) = ftk (etk (k), ω(k), k) According to Condition 1), for each t¯k , we have n
lim sup
n→∞
1X ket¯k (k)k2 = R n k=1
where R is a constant. It is easy to see that etk (k) takes values in the set of et¯k (k), then by squeezing, it follows that n
1X lim sup ketk (k)k2 = R n→∞ n k=1
That implies the ” slow switching” subsystem 1 is tracking, i.e. n 1X 0 lim sup ky (k) − yr (k)k2 = R n→∞ n k=1
Next, by Lemma 3, regarding y 00 (k) and y 000 (k) as one variate, we see that the self-tung control system is convergent, i.e., lim sup
n→∞
1 n
n X
ky(k) − yr (k)k2
k=1 n
= lim sup n→∞
1X 0 ky (k) − yr (k) + y 00 (k) + y 000 (k)k2 n k=1
=R (32) That completes the proof of Theorem 1. We now extend the above result, i.e., Theorem 1 to a stochastic plant with colored noise. Consider A(q −1 )y(k) = q −d B(q −1 )u(k) + D(q −1 )ω(k)
(33)
where A(q −1 )
=
B(q −1 )
= b0 + b1 q −1 + · · · + bm q −m
D(q
−1
)
=
1 + a1 q −1 + · · · + an q −n 1 + d1 q −1 + · · · + ds q −s
We have the following result. (The proof is omitted.) Corollary 1: If a general self-tuning control system of stochastic plant (33) with known structure information, i.e., n, m, s, and d, has the following properties: 1) A control strategy is well defined to stabilize the parameter known model and tracking the bounded reference signal yr (k) in mean square sense; ˆ ˆ − θ(k ˆ − l)k → 0 , l is a finite 2) kθ(k)k ≤ M < ∞ , kθ(k)
integer. 3) The parameter estimation error satisfies n X
2 ˆ ky(k)−φT (k−d)θ(k)−ω(k)k = o(1+
k=1
n X
kφ(k−d)k2 )
k=1
(34) 4) The mapping from the model parameters into the controller parameters is continuous at infinity. Then it is stable and convergent. Recalling the proof procedures of Theorem 1, we actually didn’t rely on the structure information of plant P , because plant P is replaced by Pm (k) and e(k) in Fig. 2, or Pm (tk ) and ei (k) in Fig. 3, respectively. Thus we know that Theorem 1 and Corollary 1 hold for an arbitrary stochastic plant. (Details are omitted due to space limitations) IV. CONCLUDING REMARKS With the help of VES concepts and methodology, we presented some new results on the stability and convergence of a general stochastic self-tuning control system. The necessary conditions for the stability and convergence of STC have been relaxed, i.e., the convergence of parameter estimates has been removed. To some extent, these results parallel the more familiar ones on stability of slowly time-varying systems [33-35]. R EFERENCES
[1] G. C. Goodwin, P. J. Ramadge and P. E. Caines, ”Discrete time stochastic adaptive control”, SIAM J. Control and Optimization, Vol. 19, pp. 829-853, 1981 [2] L. Guo and H. F. Chen, ”The Astrom-Wittenmark self-tuning regulator revisited and ELS-based adaptive trackers”, IEEE Trans. Automatic Control, Vol. 36, pp. 802-812, 1991 [3] M. Prandini and M.C. Campi. Adaptive LQG control of input-output systems - A cost-biased approach. SIAM Journal on Control and Optimization, Vol. 39, pp. 1499-1519, 2001 [4] A. Patete1, K. Furuta and M. Tomizuka, ”Stability of self-tuning control based on Lyapunov function”, Int. J. Adapt. Control and Signal Process, Vol. 22, pp. 795-810, 2008 [5] G. C. Goodwin, A. Feuer, ”Adaptive control: where to now?”, Adaptive Systems for Signal Processing,Communications, and Control Symposium, Proc. IEEE, pp. 165-170, 2000 [6] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness, Prentice Hall: Englewood Cliffs, New Jersey, 1989 ˚ om and B. Wittenmark, Adaptive control, Addison-Wesley, [7] K. J. Astr¨ 1995 [8] P. A. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall: Englewood Cliffs, New Jersey, U. S. A, 1996 [9] Sajjad Fekri, Michael Athans and Antonio Pascoal, ”Issues, progress and new results in robust adaptive control”, Int. J. Adapt. Control and Signal Process, Vol. 20(10), pp. 519-579, 2006 [10] P. R. Kumar, ”Convergence of adaptive control schemes using least squares parameter estimates”, IEEE Trans. Automatic Control, Vol. 35, No. 4, pp. 416-424, 1990 [11] J. H. Van Schuppen, ”Tuning of gaussian stochastic control systems”, IEEE Trans. Automatic Control, Vol. 39, No. 11, pp. 2178-2190, 1994 [12] Karim Nassiri-Toussi and Wei Ren, ”A Unified Analysis of Stochastic Adaptive Control: Asymptotic Self-tuning”, Proc. of the 34th IEEE Conference on Decision and Control, pp. 2932-2937, 1995 [13] A. S. Morse, ”Towards a unified theory of parameter adaptive controlpart II: certainty equivalence and implicit tuning”, IEEE Trans. Automatic Control, Vol. 37, No. 1, pp. 15-29, 1992 [14] T. Katayama, T. McKelvey, A. Sano, C. G. Cassandras, M. C. Campi, ”Trends in systems and signals: Status report prepared by the IFAC Coordinating Committee on Systems and Signals”, Annual Reviews in Control, Vol. 30, pp. 5-17, 2006
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ThA10.5 [15] Weicun Zhang, ”Theoretical research and application of robust adaptive control”, Ph. D Thesis, Tsinghua University, 1993 [16] Weicun Zhang, Jin Young Choi, ”On the Stability and Convergence of Self-Tuning Control Systems (I)– Deterministic Plant”, International Conference on Control, Automation and Systems, Seoul, Korea, 2007 [17] Weicun Zhang, Jin Young Choi, ”On the Stability and Convergence of Self-Tuning Control Systems (II)– Stochastic Plant”, International Conference on Control, Automation and Systems, Seoul, Korea, 2007 [18] Weicun Zhang, Xiaoli Li, Jin Young Choi, ”A unified analysis of switching multiple model adaptive control - Virtual equivalent system approach”, 17th IFAC World Congress, Seoul, Korea, 2008. [19] Weicun Zhang, Xiaoli Li, Tianguang Chu, ”Virtual equivalent system method for analysis of stochastic self-tuning control”, The 6th World Congress on Intelligent Control and Automation, Dalian, China, 2006 [20] Weicun Zhang, Tianguang Chu, Long Wang, ”A new theoretical framework for self-tuning control”, International Journal of Information Technology, Vol. 11(11), pp. 123-139, 2005 [21] B. Egardt, ”Unification of some discrete time adaptive control schemes”, IEEE Trans. Automatic Control, Vol. 25, pp. 693- 697, 1980 [22] P. J. Gawthrop, ”Some interpretations of the self-tuning controller”, Proc. IEEE, Vol. 124, pp. 889-894, 1977 [23] L. Ljung, I. D. Landau, ”Model reference adaptive systems and self-tuning regulators - some connections”, Proc. 7th IFAC World Congress, Vol. 3, pp. 1973-1980, 1978 [24] K. S. Narendra, L. S. Valavani, ”Direct and indirect adaptive control”, Automatica,Vol. 15, pp. 663-664, 1979 [25] G. C. Goodwin, K. S. Sin, Adaptive filtering, prediction and control, Prentice Hall, 1984. [26] Weicun Zhang, ”The Convergence of Parameter Estimates is Not Necessary for a General Self-Tuning Control System – Deterministic Plant”, Technical Report, University of Science and Technology Beijing, 2009 [27] D. Liberzon, A. S. Morse, ”Basic Problems in Stability and Design of Switched Systems”, IEEE Control Systems Magazine, Vol. 19, pp. 59-70, 1999 [28] R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, ”Stability Criteria for Switched and Hybrid Systems”, SIAM Review, Vol. 49, pp. 545592, 2007 [29] D. Chatterjee, D. Liberzon, ”Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions”, SIAM Journal on Control and Optimization, Vol. 45, No. 1, pp. 174-206, 2006 [30] D. Chatterjee and D. Liberzon, ”On stability of stochastic switched systems”, Proceedings of 43rd Conference on Decision and Control, Vol. 4, 2004, pp. 4125-4127 [31] M. Prandini, ”Switching control of stochastic linear systems: Stability and performance results”, Proc. 6th Congress of SIMAI, Chia Laguna, Cagliari, Italy, May 2002 [32] M. Prandini and M.C. Campi, ”Logic-based switching for the stabilization of stochastic systems in presence of unmodeled dynamics”, Proc. 40th CDC, Orlando, USA, Dec. 2001 [33] C. A. Desoer, ”Slowly varying system xi+1 = Ai xi ”, Electronics Letters, Vol. 6, pp. 339-340, 1970 [34] C. A. Desoer, ”Slowly varying system x˙ = A(t)x”, IEEE Trans. Automatic Control, Vol. 14, No. 6, pp. 780-781, 1970 [35] V. Solo, ”On the stability of slowly time-varying linear systems”, Math. Control Signals Systems, Vol. 7, pp. 331-350, 1994
Second, assume (A.1) and (A.2) hold for k, k − 1, ..., 1. Considering Fig.3, Fig. 4, Fig. 5, and Fig. 6, respectively, we have ˆ k ) + ω(k + 1) + ei (k + 1) (A.3) y(k + 1) = φT (k − d + 1)θ(t ˆ k ) + ω(k + 1) y 0 (k + 1) = φT1 (k − d + 1)θ(t
(A.4)
ˆ k ) + ei (k + 1) y 00 (k + 1) = φT2 (k − d + 1)θ(t
(A.5)
000
y (k + 1) =
φT3 (k
ˆ k) − d + 1)θ(t
(A.6)
where φT1 (k − d + 1) = [y 0 (k), · · · , y 0 (k − n + 1), u(k − d + 1), · · · , u0 (k − d − m + 1)] (A.7) φT2 (k − d + 1) = [y 00 (k), · · · , y 00 (k − n + 1), u00 (k − d + 1), · · · , u00 (k − d − m + 1)] (A.8) φT3 (k − d + 1) = [y 000 (k), · · · , y 000 (k − n + 1), u000 (k − d + 1), · · · , u000 (k − d − m + 1)] (A.9) According to the assumption, it is obvious that φT1 (k−d+1)+φT2 (k−d+1)+φT3 (k−d+1) = φT (k−d+1) (A.10) Thus, we obtain y 0 (k + 1) + y 00 (k + 1) + y 000 (k + 1) ˆ k ) + ω(k + 1) + ei (k + 1) = y(k + 1) = φT (k − d + 1)θ(t (A.11) Next we observe u0 (k + 1), u00 (k + 1) and u000 (k + 1). According to the definition of u(k), i.e., equation (8), we have in Fig. 3 u(k + 1) = φTc (k + 1)θc (tk+1 ) + ∆u0 (k + 1)
(A.12)
Similarly, in Fig. 4, Fig. 5, and Fig. 6, we have
000
u0 (k + 1) = φTc1 (k + 1)θc (tk+1 )
(A.13)
u00 (k + 1) = φTc2 (k + 1)θc (tk+1 )
(A.14)
u (k + 1) =
φTc3 (k
0
+ 1)θc (tk+1 ) + ∆u (k + 1)
(A.15)
where φTc (k + 1) = [y(k + 1), y(k), · · · , y(k − s1 + 1),
A PPENDIX I L EMMAS AND P ROOF
u(k), u(k − 1), · · · , u(k − s2 + 1), (A.16)
Lemma 1: Virtual equivalent system, as shown in Fig. 3, can be decomposed into three subsystems, as shown in Fig. 4, Fig. 5, and Fig. 6, i. e., the input and output of the virtual equivalent system are identical to the sum of the inputs and outputs of the decomposed subsystems.
yr (k + 1), · · · , yr (k − s3 + 1)] φTc1 (k + 1) = [y 0 (k + 1), y 0 (k), · · · , y 0 (k − s1 + 1), u0 (k), u0 (k − 1), · · · , u0 (k − s2 + 1), yr (k + 1), · · · , yr (k − s3 + 1)] (A.17)
Proof: We give the proof by mathematical induction. First, we know that for k ≤ 0 ,
φTc2 (k + 1) = [y 00 (k + 1), y 00 (k), · · · , y 00 (k − s1 + 1),
y(k) = y 0 (k) + y 00 (k) + y 000 (k)
(A.1)
u(k) = u0 (k) + u00 (k) + u000 (k)
(A.2)
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u00 (k), u00 (k − 1), · · · , u00 (k − s2 + 1), 0, · · · , 0] (A.18)
ThA10.5 φTc3 (k + 1) = [y 000 (k + 1), y 000 (k), · · · , y 000 (k − s1 + 1), 000
000
000
u (k), u (k − 1), · · · , u (k − s2 + 1), 0, · · · , 0] (A.19) Based on (A.11), it is obvious that φTc1 (k + 1) + φTc2 (k + 1) + φTc3 (k + 1) = φTc (k + 1) (A.20) Then we get
Pn This together with (A.23) and the fact n1 k=1 kφe1 (pk )k2 < ∞, by squeezing, yields an absurd result Pn 2 e k=1 kφ(pk )k →0 Pn 2 e k=1 kφ(pk )k Therefore, we conclude our original hypothesis is false. Then we have n 1X e kφ(k)k2 < ∞ (A.25) n k=1
u0 (k + 1) + u00 (k + 1) + u000 (k + 1)
That completes the proof of Lemma 2.
= φTc (k + 1)θc (tk+1 ) + ∆u0 (k + 1)
(A.21) Lemma 3: Given
= u(k + 1)
1 n
Pn
k=1 ky
00
(k)k2 = o(1)
n
Consequently, we know that (A.1) and (A.2) hold for arbitrary k. That completes the proof of Lemma 1. e Lemma 2: Given φ(k) = φe1 (k) + φe2 (k)
lim sup
n→∞
1X 0 ky (k) − yr (k)k2 < ∞ n k=1
Then n n 1X 0 1X 0 00 2 ky (k) − yr (k) + y (k)k → ky (k) − yr (k)k2 n n k=1
n
1X e kφ1 (k)k2 < ∞ n
k=1
Proof: It is obvious that
k=1
n
n
n
k=1
k=1
1X 0 ky (k) − yr (k) + y 00 (k)k2 n
1X e 1X e kφ2 (k)k2 = o( kφ(k)k2 ) n n
k=1
n
=
Then
k=1 n X
n
1X e kφ(k)k2 < ∞ n
2 + n
k=1
n
k=1
k=1 n X
1X e 1X e kφ(k)k2 = kφ1 (k) + φe2 (k)k2 n n ≤
2 n
n
kφe1 (k)k2 +
k=1
Suppose, by contradiction, that is Punbounded, there must exist n 1 2 e k=1 kφ(pk )k → ∞, satisfying n n
2X e kφ2 (k)k2 n k=1 (A.22) 1 n
n
k=1 n
k=1 n
n
k=1
2 e subsequence k=1 kφ(k)k
1X e 1X e kφ(pk )k2 = kφ1 (pk ) + φe2 (pk )k2 n n k=1
[y 0 (k) − yr (k)]y 00 (k)
By the Cauchy inequality, we know that )2 ( n 1X 0 00 [y (k) − yr (k)]y (k) 0≤ n k=1 ( n ) ( n ) (A.27) 1X 0 1 X 00 2 2 ≤ [y (k) − yr (k)] . [y (k)] n n
Pn
a
k=1
1X 0 2X 0 → ky (k) − yr (k)k2 + [y (k) − yr (k)]y 00 (k) n n k=1 k=1 (A.26)
Proof: Making use of the triangle inequality and the fact 2ab ≤ a2 + b2 , we obtain n
n
1X 0 1 X 00 [y (k) − yr (k)]2 + [y (k)]2 n n
k=1
→0 Then by squeezing we obtain ( n )2 1X 0 [y (k) − yr (k)]y 00 (k) →0 n
(A.28)
k=1
n
2X e 2X e kφ1 (pk )k2 + kφ2 (pk )k2 ≤ n n k=1 k=1 (A.23)
That is
n
1X 0 [y (k) − yr (k)]y 00 (k) → 0 n k=1
Substituting this into (A.26), we have Besides, by the fact that the limits of sequence and its subsequences are the same, we have n
n
1X e 1X e kφ2 (pk )k2 = o( kφ(pk )k2 ) n n k=1
(A.24)
n
n
k=1
k=1
1X 0 1X 0 ky (k) − yr (k) + y 00 (k)k2 → ky (k) − yr (k)k2 n n That completes the proof of Lemma 3.
k=1
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