Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThB02.2
Variable Structure Model Reference Adaptive Control of Unknown Switched Linear Systems with Relative Degree Greater Than One Ming-Li Chiang and Li-Chen Fu Abstract— In this paper, we consider the model reference adaptive control (MRAC) problem of switched linear systems in which the parameters and switching time instants are all unknown. We apply the output feedback variable structure (VS) based adaptive controller to switched linear systems with general relative degrees and show error convergence and signal boundedness properties by multiple Lyapunov functions. Specifically, a sufficient condition on the switching signal is obtained to ensure stability and error convergence of the system. Incidentally, the robustness and performance of VS adaptive controllers of the systems with jump parameters, which are only seen by simulations in existing literatures, are shown by thorough theoretical analysis. Simulation results are provided to validate the analysis.
and some simulations are provided to validate the proposed analysis. This paper is organized as follows. Notations and problem formulation are given in section II. VS based adaptive control of switched systems with relative degree greater than one are proposed in section III. Stability analysis and the conditions on switching signals are discussed. Signal boundedness and convergence of the tracking error are guaranteed under suitable conditions in terms of the switching signals. In section IV, numerical simulation results are presented and the conclusions are given in section V. II. N OTATIONS AND P ROBLEM S TATEMENT
I. INTRODUCTION Stability analysis and stabilization of switched systems have received a fair amount of attention and attracted many researches working on the topics in recent years. In real world, switched systems occur in many situations, for example, when operating environment suddenly varies or its parameters change promptly. Tutorial references of switched systems can be found in papers [1], [2], and the book [3]. In this paper, we discuss the model reference adaptive control (MRAC) problem for switched linear systems with unknown parameters. Given a class of switched system, we want to design an adaptive controller such that the output of the switched system can track the output of a given reference model. Note that the switching signal of switched plant is governed by the model or the corresponding environment and thus we can not control it. We applied an output feedback variable structure (VS) based adaptive controller to switched single-input singleoutput (SISO) linear systems with relative degree one in [4]. Under the derived condition on switching signals and using the proposed controller, signal boundedness is guaranteed and output error will converge to zero. In this paper, we extend the results to switched systems with relative degree greater than one. We show that if the switching signals satisfy some conditions, all signals in system are bounded and the tracking error will converge to a residue set whose size depends on the designed filter parameter. Stability properties are analyzed by multiple Lyapunov functions (MLFs) [5] This work was supported by National Science Council under the grant NSC 97-2221-E-002-178-MY3. M. L. Chiang is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan.
[email protected] L. C. Fu is with the Department of Electrical Engineering and Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan.
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
A. Notations The switched system is represented by x˙ = fσ (t) (x),
σ (t) ∈ P = {1, 2, . . . , P}
(1)
which consists of subsystems fi (x), i = 1, 2, ...P, and the piecewise constant switching signal σ : [0, ∞) → P. We denote the values of switching signals by {(T1 , σ (T1 )), . . . , (Tr , σ (Tr )), . . .}, where σ (T1 ), σ (T2 ), . . . ∈ P are indices of active subsystems and T1 , T2 , . . . are time instants at which the system is switching. Let T0 = t0 be the initial time and σ (T0 ) be the initial active subsystem. Throughout this paper, the switching signal is assumed to be right continuous. The switching signals are nonzeno, which means that the number of switchings will be finite in any finite time interval, and the switching durations ∆ τr = Tr+1 − Tr > 0 for all positive integers r. Moreover, the switching signals will not stop switching after finite switches. A switching signal has dwell-time τd if all time intervals during switchings are greater than τd . Let Nσ (t,t0 ) denote the number of switchings of the switching signal σ during the time interval (t0 ,t). We call the switching signal σ has average dwell-time τa if, given any time interval (t0 ,t), it satisfies t − t0 Nσ (t,t0 ) − N0 ≤ , (2) τa where N0 is a given positive constant. Detailed classifications of switching signals are introduced in [6]. In this paper, |x| denotes the Euclidean norm of a scalar or a vector x ∈ Rn . The L p induced norm for functions of time is denoted by k · k p , p ∈ [1, ∞], and k · k is particularly denoted as the L∞ norm. The L pe norm is denoted by kxt k p , where kxt k := sup |x(τ )| for 0 ≤ τ ≤ t.
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ThB02.2 B. Problem statement Consider the SISO switched linear system in observer canonical form x˙ p
= A pσ x p + k pσ b pσ u
yp
= hT x p = [1, 0, ..., 0]x p
error to zero. Now we extend the controller to the case of n∗ = 2. For each subsystem with index i ∈ P, we know that ([7], [8]) there exists θi∗ = [ki∗ , θ1i∗T , θ0i∗ , θ2i∗T ]T ∈ R2n such that, if u = θi∗T ω with ω = [r, ω1T , y p , ω2T ]T ∈ R2n , where
ω˙ 1 ω˙ 2
(3)
where k pσ ∈ R, u ∈ R, b pσ ∈ Rn , x p ∈ Rn ,
A pσ
−aσn−1 −aσ n−2 .. = . −aσ 1 −aσ0
1 0 0 1 .. .
··· ··· .. .
0 0 0 0
··· ···
0 0 .. .
0 .. .
1 σ = , b pσ b m−1 1 . .. 0 bσ0
,
(4) and σ : [0, ∞) → P is the switching signal that governs the switching sequence of the switched system. This switched system has the corresponding input-output representation y p = Wpσ (t) (s)[u] = k pσ (t)
Z pσ (t) (s) [u], R pσ (t) (s)
where r is the reference input and ym is the reference output. The control purpose is to design the output feedback control u such that all signals in the switched system are bounded and the output of switched plant tracks the reference output as well as possible, i. e., making the output error e1 = y p −ym as small as possible. In this MRAC problem of switched systems, we do not have the information about the time instants at which the switchings occurred nor the knowledge of the active subsystem. The following assumptions for MRAC are made [7]: (A1) For all the transfer functions Wpi , i ∈ P, R pi (s) is of order n and Z pi (s) is of order m, which means that Wpi has relative degree n∗ := n − m; (A2) The reference model has the same relative degree n∗ as the plant Wpi (s); (A3) All the plants and the reference model are completely controllable and observable; (A4) Wpi is minimum phase for all i ∈ P; (A5) The signs of k pi and km are all positive. III. VS MRAC OF S WITCHED L INEAR S YSTEMS WITH R ELATIVE D EGREE G REATER T HAN O NE
e1 =
(7)
1 Wm (s)[θ˜σT ω ]. kσ∗
(8)
Let x p = [xTp , ω1T , ω2T ]T ∈ R3n−2 , then the state space representation of the closed-loop system should be A pσ + b pσ θ0∗σ hT b pσ θ1∗T b pσ θ2∗T σ σ xp Λ + l θ1∗T l θ2∗T x˙ p = l θ0∗σ hT σ σ T lh 0 Λ b pσ b pσ + l kσ∗ r + l (θ − θσ∗ )T ω 0 0 yp
= [hT , 0, 0]x p ,
(9)
or, x˙ p yp
= Amσ x p + Bmσ r + B pσ (θ˜σT ω ) = CT x p ,
(10)
where CT (sI − Ami )−1 Bmi = Wm (s),
∀i ∈ P.
(11)
Note that for all i ∈ P, Bmi = ki∗ B pi , and ki∗ = kkmpi . From (11), the reference model can be realized by the nonminimal state space representation x˙ m
= Amσ xm + Bmσ r
ym
= CT xm ,
(12)
and if we define e = x p − xm , then the error equation of (8) can be realized as e˙ e1
A. VS MRAC of switched systems with relative degree two In [4], we have proposed a VS based adaptive controller for the switched linear systems with relative degree one and sufficient conditions on the switching signals are obtained to guarantee system stability and convergence of the output
= Λω2 + ly p ,
Λ ∈ R(n−1)×(n−1) , det(sI − Λ) = λ (s) = sn−1 + λn−2 sn−2 + .. + λ1 s + λ0 is a designed monic Hurwitz polynomial that contains Zm (s) as a factor, ω1 , ω2 ∈ Rn−1 , l = [1, 0, ...0]T ∈ Rn−1 , and then Wpi (s)u = Wm (s)r. Since parameters of the plants are unknown, θi∗ are unknown. Use θ as the estimate of θi∗ , and θ = [k, θ1T , θ0 , θ2T ]T . Then, with the following certainty equivalence principle controller u = θ T ω , provided there is no switching, we have y p = Wm (s)[r + k1∗ (θ˜iT ω )], i where θ˜i = θ − θi∗ , i ∈ P. Now consider the switched plant case with the switching signal σ (t). Let e1 = y p − ym , then use of u = θ T ω will lead to the error equation
(5)
where Z pi (s) = sm + bim−1 sm−1 + .... + bi1 s + bi0 and R pi (s) = sn + ain−1 sn−1 + ... + ai1 s + a0 , i = 1, 2, ..., P. The transfer functions Wpi (s), i ∈ P, are strictly proper and parameters of them are all unknown. Only the input and output can be measured, and we do not know when the plant switchings occur. Given a reference model Zm (s) ym = Wm (s)[r] = km [r], (6) Rm (s)
= Λω1 + lu
= Amσ e + B pσ (θ˜σT ω ) = CT e
(13)
Define L1 (s) = (s + α1 ) where α1 is a positive constant such that Wm L1 is strictly positive real (SPR). Let u = L1 (s)u f and define u f = θ T (L1−1 ω ) + L−1 uvs , where uvs will be
4241
ThB02.2 designed. Then, the error equation of the system can be represented as e1
=
1 Wm L1 (s)[θ˜σT (L1−1 ω ) + L1−1 uvs ]. kσ∗
(14)
Note that ∆(e1 ) corresponds to the filtered weighted error due to approximation of the derivative of f (e1 ). When max (|e1 (t)|, |e1 (t − ∆t)|) < ε f , the bound on ∆(e1 ) can be estimated as
From (11), a realization of (14) can be represented as e˙ e1
= Amσ e + = CT e
1 Bmσ (s + α1 )[θ˜σT (L1−1 ω ) + L1−1 uvs ] kσ∗ (15)
Let φ = L1−1 (s)ω and eˆ = e − B pσ (θ˜σT φ + L1−1 uvs ), then we have 1 eˆ˙ = Amσ eˆ + ∗ Bm′ σ (θ˜σT φ + L1−1 uvs ) kσ e1 = CT eˆ , (16) where Bm′ σ = Amσ Bmσ + α1 Bmσ . Note that u = L1 (s)u f = θ T ω + θ˙ T φ + uvs .
(17)
∆t 1 |∆(e1 )| ≤ (k(e˙1 )t k + ρd ) 2kψt k + kψ˙ t k + c1 (26) εf α1 where ρd and c1 are positive constants and c1 accounts for the initial condition effect of the integration (see [9]). If all signals in the closed-loop system are bounded, ∆(e1 ) can be bounded by δ ∆t, where δ is a positive constant. In this case, we can choose ∆t small enough such that ∆(e1 ) is kept as small as we need. Moreover, from (21) and (25), ∆(e1 ) will be exponentially decay when min(|e1 (t)|, |e1 (t − ∆t)|) > ε f . Moreover, by (23) and the fact φ˙ = −α1 φ + ω , dtd ψ (|φ |) will be implemented as
If we design L1−1 (s)uvs θ˙
= −sgn(e1 )(β1 |φ | + β2 ),
(18)
= −(e1 φ + γθ ),
(19)
where β1 , β2 are positive constants with β1 ≥ maxi∈P |θi∗ | and γ > 0 symbolizes some leakage constant, then it can be verified that the dwell time conditions in [4] are satisfied. Thus, we can obtain the results that eˆ is bounded and converges to zero exponentially if the dwell time is large enough, as we have proved in the relative degree one case in our earlier result [4]. However, uvs in (18) is not implementable since it will have unboundedness problem due to differentiation of the sign function. To avoid differentiation of the sign function and |φ | in (18), we design uvs similar to [9] in the following: = α1 f (e1 )ψ (|φ |) + f (e1 )
uvs
+
d ψ (|φ |) dt
1 ˜ f (t)ψ (|φ |) ∆t
(20)
where |e1 | f (e1 ) = −sat( )= εf
−sgn(e1 ) if |e1 | ≥ ε f − εe1f if |e1 | < ε f ,
f˜(t) = f (e1 (t)) − f (e1 (t − ∆t)),
ψ (|φ |) =
β1 |φ | + β2 2 β1 ( |2φε|φ
+
εφ 2
if |φ | ≥ εφ ) + β2
if |φ | < εφ .
Remark 1. Note that ψ (|φ|) is differentiable since when 2 ε |φ | = εφ , dtd (β1 |φ | + β2 ) = dtd β1 ( |2φε|φ + 2φ ) + β2 = βεφ1 φ T φ˙ . Moreover, ψ (|φ |) satisfies β1 |φ | + β2 ≤ ψ (|φ |) ≤ β1 |φ | + β2′ , ε where β2′ = 2φ + β2 . Now we have the following result: Theorem 1: For the switched systems (3) with relative degree n∗ = 2. If the controller is designed as (17), (19), and (20), and the switching signal has average dwell-time τa ≥ lnmM2 2 for some positive constants M2 and m2 , then all signals in the closed-loop system are bounded and the output error converges to a residue set whose size depends on the design parameter ε f . Proof. Since Wm (s)L1 (s) is SPR, from MKY lemma (see T [7]) we know that for all i ∈ P, there exists Pi = Pi > 0 and Qi > 0 such that ATmi Pi + Pi Ami = −2Qi < 0,
(22)
(23)
Here, ∆t, ε f and εφ are sufficiently small positive constants that can be specified. Then, L1−1 (s)uvs can be expressed as L1−1 (s)uvs = f (e1 )ψ (|φ |) + ∆(e1 )
f˜(t) − s[ f (e1 (t))] ψ (|φ |) . ∆t
Pi Bm′ i = C.
(28)
Choose the multiple Lyapunov functions as Vi = 12 (ˆeT Pi eˆ + 1 T k∗ θ θ ), i = 1, 2, ..., P. Then, we have i
V˙i
(24)
where ∆(e1 ) = L1−1 (s)
(27)
(21)
and (
−β1 φ T (α1 φ − ω ) if |φ | ≥ εφ d |φ | ψ (|φ |) = T −β1 φ (α1 φ − ω ) if |φ | < εφ . dt εφ
(25)
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= −ˆeT Qi eˆ + eˆ T Pi B p′ i (θ˜iT φ + L−1 uvs ) 1 − ∗ θ T (e1 φ + γθ ) ki 1 = −ˆeT Qi eˆ + ∗ e1 [(θ − θi∗ )T φ + f (e1 )ψ (|φ |) + ∆(e1 )] ki 1 T − ∗ θ (e1 φ + γθ ). (29) k
ThB02.2 When |e1 | > ε f , V˙i
1 |e1 |(β1 |φ | − |θi∗T φ | + β2 ) ki∗ 1 γ − ∗ θ T θ + ∗ e1 ∆(e1 ) ki ki 1 1 γ ≤ −ˆeT Qi eˆ − ∗ θ T θ − ∗ β2 |e1 | + ∗ e1 ∆(e1 ) ki ki ki γ 1 2 2 ≤ −qi |ˆe| − ∗ |θ | − ∗ |e1 |(β2 − ε ′ (t)) ki ki 1 (30) ≤ −m2Vi − ∗ |e1 |(β2 − ε ′ (t)), ki ≤ −ˆeT Qi eˆ −
where ε ′ > ∆(e1 ) is an exponentially decaying term. Denote by pi1 = λmin (Pi ), pi2 = λmax (Pi ), qi1 = λmin (Qi ), qi2 = λmax (Qi ), where λmax (A) and λmin (A) stand for the maximum and the minimum of eigenvalues of matrix A, respectively. Then, we can define qi1 (31) , γ }, m2 = min{min i∈P pi2 and M2 = max{
maxi∈P pi2 1 , max ∗ }. mini∈P pi1 i∈P ki
(32)
Design β2 large enough and if τa > lnmM2 , then the system 2 will be stable using similar analysis as relative degree one case in [4]. When |e1 | < ε f , 1 −|e1 | V˙i ≤ −m2Vi + ∗ |e1 | |φ |( β1 + |θ ∗ |) + ∆t δ (t) ki εf (33) ≤ −m2Vi + c2 |φ | where c2 is a positive constant which is dependent on ε f and ∆t. Note that (26) is used to obtain the second inequality. For (33), we can use the result of input-to-state stability (ISS) of switched systems in [10] to show signal boundedness of the system. Let eˆ ′ = [ˆeT , θ ]T . Thus from Theorem 3.1 in [10], we can have
When |e1 | > ε f , we know that the states are decreasing from (30) and eventually e1 will fall into the interval [−ε f , ε f ]. When |e1 | < ε f , we have the relation (36) where c3 is dependent on ε f . Hence, the output error e1 will converge to a residual set whose size depends on the parameter ε f . Remark 2. We believe that the restricted dwell-time condition can be relaxed to some extent based on the insight gained from analysis on non-switched systems with controller (17). Equivalently speaking, the proposed VS based adaptive controller can in fact accommodate more strict dwell-time condition by increase of β2 referring to (30). B. VS MRAC of switched systems with arbitrary relative degree To extend the structure of controller (17) for systems with relative degree greater than two, the expected advantage may not justify the drastic increase of controller complexity. Therefore, we adopt an alternative VS based adaptive controller with constant parameter estimate as proposed in [12] for the general case where each plant Wpi has arbitrary relative degree n∗ . To simplify the analysis, let the high frequency gain k pi = km , ∀i ∈ P, that is, kσ∗ = 1, ∀ t ≥ 0. Let u = θ T ω + uvs , where θ is a constant vector and uvs will be designed, then we know that e1 = Wm (s)(θ˜σT ω + uvs ). Define the Hurwitz polynomial N i=1
ea e0
:= Wm (s)L(s)[L−1 (s)uvs − u0 ], := e1 − ea = Wm (s)L(s)[L−1 (s)[θ˜σT ω ] + u0 ],
eai
:= Li−1 (s)[ui ] − F −1 [ui−1 ]
kφt k ≤ c5 k(e1 )t k + c6 ≤ c7 kˆet′ k + c6 ,
(35)
where c5 , c6 , c7 , are positive constants. Thus from (34), (35), and small gain theorem [11], if we choose ∆t and ε f small enough such that 1 − c3 c7 > 0, then c3 c6 + c4 kˆet′ k ≤ < ∞. (36) 1 − c3 c7
(37)
Li−1 (s)[ui + F −i (s)(Li+1 ..LN (s))−1 [θ˜σT ω ]] + εi ,
:=
LN−1 [uN ] − F −1 (s)[uN−1 ] LN−1 [F −N (s)[θ˜σT ω ] + uN ] + εN ,
i = 1, 2, ..., N − 1, eaN
(38) (39)
= (34)
where β 1 is a class K L function and c3 , c4 , are positive constants and c3 is dependent on c2 . Using similar arguments of Lemma A.2 in [9], it can be shown that
αi > 0
i=1
where N = n∗ − 1 and L(s) is chosen such that Wm (s)L(s) is SPR. The controller is designed as follows: Define the augmented error ea and auxiliary errors e0 , ea1 , ..., eN as
kˆet′ k ≤ β 1 (|ˆe′ (t0 )|,t − t0 ) + c3 kφt k ≤ c3 kφt k + c4 ,
N
L(s) := ∏ Li (s) = ∏(s + αi ),
=
(40) (41)
and the average filter F −1 (s) := (τ s + 1)−1 with τ being sufficiently small. By algebraic manipulations, signals ε1 , .., εN can be obtained as
ε1 εi
= F −1 (s)(Wm (s)L(s))−1 [−e0 ], = Li−1 F −1 (s)[εi−1 − eai−1 ], i = 2, ..., N.
(42) (43)
The signals ui , i = 0, ..., N, are designed as
Then, boundedness of eˆ and hence φ are guaranteed. From definition of eˆ and (24), we know that e and ω are bounded. Consequently, all signals in the closed loop system are bounded.
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u0 ui i
= −sgn(e0 ) f0 ,
f0 > |L−1 [θ˜σT ω ]|,
= −sgn(eai ) fi , = 1, ..., N − 1,
−i
(44) T ˜ fi > |F (s)(Li+1 ..LN ) [θσ ω ]|, (45) −1
ThB02.2 and fN > |F −N (s)[θ˜σT ω ]|.
uN = −sgn(eaN ) fN ,
(46)
Let uvs = uN and the controller is designed as T
u = θ ω + uN ,
Amσ (T ) (t−Tk ) Amσ (Tk−1 ) (Tk −Tk−1 )
e(t) = e
= Amσ xe + Bm′′ σ (L−1 [θ˜σT ω ] + u0 ) = CT xe = [1, 0, ..., 0]xe ,
(48) (49)
where xe ∈ R3n−2 . Define the multiple Lyapunov functions as 1 Vi = xTe Pi xe , i = {1, 2, ..., P}, 2 where Pi are defined by the MKY Lemma such that for some Qi > 0, ATmi Pi + Pi Ami = −2Qi , and Pi Bm′′ i = C. Then, the time derivative of Vi along solution trajectories of the ith subsystem is V˙i
= −xTe Qi xe + xTe Pi Bm′′ i (L−1 [θ˜iT ω ] + u0 ) ≤ −xTe Qi xe − |e0 |( f0 − |L−1 [θ˜iT ω ]|) qi1 ≤ −qi1 |xe |2 ≤ − Vi pi2
Let m1 = mini∈P qpi1i2 and M1 = definition of Vi , we have V˙i ≤ −m1Vi ,
maxi∈P pi2 mini∈P pi1 .
(50)
Then, from (50) and
Vi ≤ M1V j , ∀i, j ∈ P.
(51)
Thus, by stability analysis in [4] and [13], if the average dwell-time τa ≥ lnmM1 1 , then xe , and hence e0 , converge to zero asymptotically. Since F −1 (Wm L)−1 (s) is a stable proper transfer function, ε1 will converge to zero exponentially. From (40) and (45), it is clear that ea1 will converge to zero exponentially. With similar argument, we can conclude that εi and eai , i = 1, 2, ..N, all converge to zero exponentially. Let πN = LN−1 (F −N − 1)(s)[θ˜σT ω ] and eaN = eaN − εN − πN . Define e = e − Bmσ (eaN ), then e˙ = Amσ e + Bmσ eaN
(52) LN−1 (F −N − 1)(s) is and θ˜σ is piecewise
where Bmσ = Amσ Bmσ + αN Bmσ . Since a strictly proper stable transfer function constant, we have k(πN )t || ≤ cτ kωt k + EXP, where c is a positive constant and EXP is an exponentially decaying signal due to the initial condition. Hereafter, symbol c is used to denote any suitably defined positive constant and EXP the exponentially decaying signals. Since eaN and εN are exponentially decaying terms, we have k(eaN )t k ≤ cτ kωt k +
e
k
k−1 Z T j+1 Amσ (T ) (T j+1 −s) j
(47)
where θ is a constant vector that can be designed from a priori knowledge of the plants. Then, we have the following results: Theorem 2: For the switched plant (3) with relative degree n∗ and controller (47), if the switching signal has average dwell-time τa ≥ lnmM1 1 for some positive constants M1 and m1 , then all signals in the closed loop system are bounded and the output error e1 will converge to a residue set with radius proportional to τ . Proof. From (8), and (13), the state space representation of e0 in (39) can be written as x˙ e e0
EXP. Now we discuss the stability property of (52). Suppose that up to time t there are k switchings occurred, then we have
+∑ +
e
j=0 T j Z t Amσ (T ) (t−s)
e
k
Tk
Amσ (T ) (T1 −T0 )
..e
0
e(t0 )
Bmσ (T j ) eaN (s)ds
Bmσ (Tk ) eaN (s)ds
≤ β 2 (e(t0 ),t − t0 ) + ck(eaN )t k,
(53) (54)
where β 2 is a class K L function. The homogeneous part of (52) is asymptotically stable when switching signal satisfies the dwell-time condition in [4]. The zero-state part of (52) is bounded by a positive constant since Ami are exponentially stable and the dwell-time condition are satisfied. Thus, we have ket k ≤ cke(t0 )k + cτ kωt k, and hence ket k ≤ cke(t0 )k + cτ kωt k.
(55)
Now we show that kωt k ≤ cket k + c. Since the switching signal is non-zeno, there exists a non-zero time interval such that the system is not switching. Consider the behavior of the system during the first non-switching time interval [T0 , T1 ). Let σ (T0 ) = i and ni (s) be a Hurwitz polynomial such that Wpi (s)ni (s) is proper. Then, we can see that during this nonswitching interval, y p (s) = Wpi (s)ni (s)( ni1(s) u) and (Wpi (s)ni (s))−1 y p = (
1 u) := uc ni (s)
Since Wpi (s) is minimum phase and (Wpi (s)ni (s))−1 is stable, we know that k(uc )t k ≤ ck(y p )t k + c, ∀t ∈ [T0 , T1 ). Since e1 = y p − ym and ym is bounded, we have k(uc )t k ≤ ck(e1 )t k + c. By Lemma 2.8 in [8], if we can show that u grows at most exponentially fast, then kut k ≤ ck(uc )t k + c and hence k(u)t k ≤ ck(e1 )t k + c, for t ∈ [T0 , T1 ). Note that u = θ T ω + uvs , x p = [x p , ω1 , ω2 ]T , and A pi 0 0 b pi Λ 0 xp + l u (56) x˙ p = 0 0 lhT 0 Λ
Since θ is bounded and by definition of u (u is a function of ω and θ ), we know that (56) satisfies k(˙x p )t k ≤ ck(x p )t k + c.
Thus, ω , y p and u can grow at most exponentially fast. This proves that kut k ≤ ck(uc )t k + c and thus kut k ≤ ck(e1 )t k + c, for t ∈ [T0 , T1 ). From (7), we know that k(ω1 )t k ≤ ckut k + c and k(ω2 )t k ≤ ck(y p )t k + c since Λ is a Hurwitz polynomial. Then from the definition of ω = [r, ω1T , y p , ω2T ]T , we have kωt k ≤ ck(e1 )t k + c ≤ cket k + c,
∀t ∈ [T0 , T1 ).
(57)
Since the output is continuous, by time concatenation we can have similar results for time t ≤ ∞. From (55) and hence (57), ket k ≤ cτ ket k + cτ + cke(t0 )k. Thus for sufficiently small τ such that 1 − cτ > 0,
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ket k ≤
cke(t0 )k + cτ , 1 − cτ
(58)
ThB02.2 which implies that e is bounded. From (57), ω is bounded and we can conclude that all signals are bounded. Moreover, e will converge to a residue set whose size depends on the design parameter τ . Remark 3. For k pσ 6= km , we will need some modifications of the controllers and auxiliary error equations as in [12]. Nevertheless, using similar analysis approach, we can derive the same results as we made for k pσ = km .
tracking error
0.1 error of VS gen VS error of new VS
0.05 0 -0.05 -0.1
0
10
20
30 (a)
40
50
60
25 | | of VS gen VS VS VS | | of new
20 | |
15 10 5 0
IV. SIMULATION RESULTS
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In this section, we give some simulations to show the effect of plant switchings and performance of the VS based adaptive controller. Consider the relative degree two case Wp2 = s3 −4ss+3 and P = {1, 2}. Let Wp1 = s3 −ss+1 2 −3s+2 , 2 +s+4 , 1 and Wm = (s+2)2 . The reference input is r(t) = 2 sin 5.9t. Given a persistently switching signal as in Fig. 1(c). Tracking error of the switched system using conventional integral type gradient adaptive law is oscillating as shown in Fig. 1(a), and estimation of parameters is given in Fig. 1(b). Under the same switching signal, performance of the VS based adaptive controller is shown in Fig. 2. Here we choose L1 (s) = (s+4), β1 = 50, β2 = 20, γ = 10, ε f = 0.01, and ∆t = 0.001. Figure 2 shows the performance of the proposed VS controller in subsection III.A and the general VS controller in subsection III.B. We can see that the tracking error of the general VS adaptive controller has some chattering which is due to fast switching of the sign function. Note that simulations show that even the time intervals between switchings are shorter than the dwell-time we derived, the error can still converge. This shows that the dwell-time condition is very conservative and is only sufficient, not necessary.
error of switched sysetm
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Fig. 1.
Tracking error of conventional MRAC of switched systems
V. CONCLUSIONS In this paper, we extend the result of VS MRAC of switched linear systems in [4] to the case of relative degree greater than one. For relative degree two case, we derive the output feedback VS based adaptive controller from our previous work with some modifications. For general case, we use the VS adaptive controller proposed in [12] due to relatively simpler structure for higher relative degree. Theoretical stability analysis using switched system theories are given for both controllers. We show that the output
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Fig. 2. Tracking error of switched systems using VS based adaptive control
error converges to a residue set with radius depends on the designed filter parameter and all signals are bounded for the switched systems, if the switching signal satisfies the derived average dwell-time condition. For MRAC of switched linear systems, there are still many issues to be discussed. It is assumed in this paper that the sign of high frequency gains of all the subsystems are all the same. If not, we have to detect the change of the signs. Besides, the dwell time condition derived here is very conservative and can not be obtained since plant parameters are unknown. How to relax the restrictive condition on dwelltime is a practical consideration. These problems are under investigation. R EFERENCES [1] H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear systems: a survey of recent results,” IEEE Trans. Automat. Contr., vol. 54, no. 2, pp. 308-322, Feb. 2009. [2] A. N. Michel and B. Ho, “Towards a stability theory of general hybrid dynamical systems,” Automatica, vol. 35, pp. 371-384, 1999. [3] D. Liberzon, Switching in Systems and Control, Birkhauser, ¨ 2003. [4] M. L. Chiang and L. C. Fu, “Variable structure based switching adaptive control for a class of unknown switched linear systems,” in Proc. American Control Conference, pp. 3019-3024, 2009. [5] M. S. Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,” IEEE Trans. Automat. Contr., vol. 43, no. 4, pp. 475-482, April 1998. [6] J. P. Hespanha, “Uniform stability of switched linear systems: Extensions of Lasalle’s invariance principle,” IEEE Trans. Automat. Contr., vol. 49, no. 4, pp. 470-482, April 2004. [7] P. A. Iannou and J. Sun, Robust Adaptive Control, Upper Saddle River, NJ: Prentice-Hall, 1996. [8] K. S. Narendra and A. Annaswamy, Stable Adaptive Systems. Englewood Cliffs, NJ: Prentice-Hall, 1989. [9] L. C. Fu, “A new robust MRAC using variable structure design for relative degree two plants,” Automatica, vol. 28, no. 5, pp. 911-925, 1992. [10] L. Vu, D. Chatterjee, and D. Liberzon, “Input-to-state stability of switched systems and switching adaptive control,” Automatica, vol. 43, no. 4, pp. 639-646, Apr. 2007. [11] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. New York: Academic, 1975. [12] L. Hsu, F. Lizarralde, and A. D. de Ara´ujo, “New results on outputfeedback variable structure model-reference adaptive control: Design and stability analysis,” IEEE Trans. Automat. Contr., vol. 42, no. 3, pp. 386-393, March 1997. [13] J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” in Proc. 38th IEEE Conf. on Decision and Control, pp. 1655-2660, 1999.
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