Extended hybrid model reference adaptive control of piecewise affine systems

Extended hybrid model reference adaptive control of piecewise affine systems

Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21 Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: ww...

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Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

Contents lists available at ScienceDirect

Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Extended hybrid model reference adaptive control of piecewise affine systems Mario di Bernardo a,∗ , Umberto Montanaro b,∗ , Romeo Ortega c , Stefania Santini a a

Department of Electrical Engineering and Information Technology, University of Naples Federico II, Italy

b

Department of Industrial Engineering, University of Naples Federico II, Italy

c

Laboratoire de Signaux et Systemes (SUPELEC), Paris, France

article

info

Article history: Received 21 May 2015 Accepted 7 December 2015 Keywords: Adaptive control Hybrid and switching control Switched systems

abstract We discuss an extension to the adaptive control strategy presented in di Bernardo et al. (2013) able to counter eventual instabilities due to disturbances at the input of an otherwise L2 stable closed-loop system. These disturbances are due to the presence of affine terms in the plant and reference model. The existence of a common Lyapunov function for the linear part of the PWA reference model is used to prove global convergence of the error system, even in the presence of sliding solutions, as well as boundedness of all the adaptive gains. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction As notably highlighted in [1], adaptive control of switched systems is still an open problem. Recently, a novel model reference adaptive strategy has been presented in [2,3] that allows the control of multi-modal piecewise linear (PWL) plants. Specifically, a hybrid model reference adaptive strategy was proposed able to make a PWL plant track the states of an LTI or PWL reference model even if the plant and reference model do not switch synchronously between different configurations. While stability is guaranteed for PWL systems, for affine systems the presence of a non-square integrable disturbance term in the error equations and the possible occurrence of sliding solutions can render the proof of stability inadequate. We wish to emphasize that the problem of large state excursions and instabilities caused by constant input disturbances on the closed loop system is a common problem of adaptive control systems seldom highlighted in the literature (see for example [4,5], and [6, Sec. 4.4.4 p. 173]). Indeed, adaptive systems can be represented as the negative feedback interconnection of a passive system (defined by the estimator) and a strictly positive real (SPR) transfer function. A simple application of the passivity theorem establishes that the overall system is L2 -stable. However, this property does not ensure that the system will remain stable in the presence of external disturbances which are not L2 . The same problem is also true for recent extensions of MRAC schemes to switched systems including the hybrid extension of the Minimal Control Synthesis MRAC algorithm presented in [2]. The aim of this note is to present a modification of the control strategy presented in [2] able to guarantee asymptotic stability of the closed loop system even in the presence of sliding mode trajectories and bounded L∞ perturbations due to the affine terms in the description of the plant and/or reference model. The idea is to add an extra switching action to the controller in [2] able to compensate the presence of such a disturbance. The proof of stability is obtained by defining an appropriate common Lyapunov function and analyzing its



Corresponding authors. Tel.: +39 081 7685953; fax: +39 081 7683186. E-mail addresses: [email protected] (M. di Bernardo), [email protected] (U. Montanaro), [email protected] (R. Ortega), [email protected] (S. Santini). http://dx.doi.org/10.1016/j.nahs.2015.12.003 1751-570X/© 2015 Elsevier Ltd. All rights reserved.

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M. di Bernardo et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

properties along the closed-loop system trajectories within each of the phase space regions where the plant and reference model are characterized by different modes, and along their boundaries. We show that, even in the presence of sliding mode trajectories, the origin of the closed-loop error system is rendered asymptotically stable by the extended strategy presented in this paper. A preliminary version of the algorithm suitable to control bimodal piecewise affine system can be found in [7,8], while experimental validation results are reported in [9]. A possible extension to discrete-time piecewise-affine (PWA) plants of the approach can be found in [10]. 2. Problem statement and definitions Assume that the state space Rn is partitioned by some smooth boundaries into M non-overlapping domains, say {Ωi }i∈M M −1 with M = {0, 1, . . . , M − 1} such that i=0 Ωi = Rn and that, given any two generic indexes i1 and i2 ∈ M (with i1 ̸= i2 ), Ωi1 ∩ Ωi2 = ∅. Let the plant be described by an n-dimensional multi-modal PWA system whose dynamics are given by: x˙ = Ai x + Bu + Bi

if x ∈ Ωi , i ∈ M ,

(1)

where x ∈ R is the state vector, u ∈ R is the scalar input, and the matrices Ai , B, Bi (i = 0, 1, . . . , M − 1) are assumed to be in control canonical form, i.e. n

   Ai =   

0

1

0

0

.. .

(1)

ai

.. .

(2)

ai

··· .. . ···

0



..  .  ,  1  (n)

ai

0 0

   B=  ..  , .

b

0

  0  Bi =   ..  , .

(2)

bi

with b > 0. Note that all entries on the last row of the plant matrices Ai , B and Bi are supposed to be unknown. Also, notice that if the plant is not into canonical form, a transformation presented by the authors in [11] can be used, under certain assumptions, as part of the design process to recast the plant in the required form. In many cases, though, electromechanical systems modeled via a Lagrangian approach are structurally already into the required canonical form. The problem we wish to solve is to find an adaptive piecewise feedback law u(t ) to ensure that the state variables of the plant track asymptotically the states, say  x(t ), of a reference model independently from their initial conditions. Here, we assume that the reference model can be either an LTI system, or a multi-modal PWA system:

i , i ∈ M ,  x + Br +  Bi if  x∈Ω (3) x˙ =  Ai      domains obtained by some smooth −1 , Ω i   is a partition of Rn into M  , 0, 1, . . . , M where the state  x ∈ Rn , M i∈M boundaries and r ∈ R is the input to the reference model. Note that the reference model may possess a number of modes  ̸= M. Furthermore, we assume that the reference model defined as in (3) is chosen so different from the one of the plant, M as not to exhibit sliding solutions and that it is well-posed given the initial condition  x(0) =  x0 . In many practical cases, the

aim of the control action can be that of compensating the discontinuous nature of the plant. In these situations, the control design presented above offers a simple and viable solution for this to be achieved by simply choosing a smooth or smoother reference model. This often corresponds to the conventional choice of an asymptotically stable LTI reference model in the case of smooth systems. As for the plant, the matrices of the reference model are chosen to be in the companion form given by ( i = 0, 1, . . . ,  − 1): M



0

1

  0 0  Ai =  ..  ..  . . (1) (2)   a a i i

··· .. .

0



..  .  ,  1  (n) ···  a i

0 0   B=  ...  ,  b

0 0   Bi =   ...  i b

(4)

with  b > 0. In what follows, we use the standard notation in [12] (also adopted in [13]), for both the switching instants of the plant and reference model. More precisely, the switching sequence of the plant is given by:

  Σ = {x0 , (i0 , t0 ) , (i1 , t1 ) , (i2 , t2 ) . . . ip , tp . . . | ip ∈ M, p ∈ N},

(5)

where t0 = 0 is the initial time instant and x0 is the initial state. Note that, as in [12], when t ∈ [tp ; tp+1 ), x(t ) belongs to Ωip by definition and, thus, the ip th subsystem is active. Obviously, the switching sequence Σ may be finite or infinite. If there is a finite number of switchings, say p, then we set tp+1 = ∞.

M. di Bernardo et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

13

For any j ∈ M , using the notation first introduced in [13], we denote the sequence of switching times when the jth subsystem is switched on as:

 Σ /j = tj1 , tj2 , . . . tjs , . . .

ijs = j and s ∈ N ,



|

(6)

where ijs = j denotes that, at every instant in the sequence, the activated mode is always j. Correspondingly, the endpoints of the time intervals when the jth subsystem is active can be listed as:



tj1 +1 , tj2 +1 , . . . , tjs +1 , . . .

|

ijs = j and s ∈ N .



(7)

Analogously, we define the switching sequence of the reference model as:

          =  , p ∈ N , x0 ,  i0 , t0 ,  i1 , t1 ,  i2 , t2 . . .  ip , tp . . . |  ip ∈ M Σ

(8)

i by definition and theip th subsystem is active. with  t0 = 0. Hence, when t ∈ [ tp ; tp+1 ) then  x(t ) ∈ Ω p  the sequence of switching times when thejth subsystem of the reference model is switched on can be For any j ∈ M analogously defined as:   /j =  tjs , . . . tj2 , . . . tj1 , Σ

 | ijs = j, s ∈ N ,

(9)

with the endpoints of the intervals where the jth mode is active being:

  tj +1 , tj +1 , . . . , tj +1 , . . . 1

2

s

 | ijs = j, s ∈ N .

(10)

 as: We define the ‘‘switching signals’’ σ : R+ → M and  σ : R+ → M σ (t ) = i if x (t ) ∈ Ωi ,

i  σ (t ) = i if  x (t ) ∈ Ω

(11)

and the indicator functions σi (t ) and  σi (t ), as:

σi (t ) =  σi (t ) =



1 0



1 0

if x (t ) ∈ Ωi , elsewhere,

(12)

i , if  x (t ) ∈ Ω elsewhere,

(13)

 − 1. with i = 0, 1, . . . , M − 1 and i = 0, 1, . . . , M Also, en ∈ Rn is defined as the basis vector 

en = 0

···

0

T

1

.

(14)

3. Control strategy The control problem described in Section 2 can be solved by means of an extended switched adaptive strategy as described in the rest of this section. The proposed approach extends the work presented in [2] by exploiting an additional adaptive switching control gain to cope with the presence of the bounded piecewise constant input acting on the closed-loop system when the plant and/or reference model are PWA. Assumption 1. Assume there exists a matrix P = P T > 0 such that T  − 1. P Ai +  A P <0  i = 0, 1, 2, . . . , M i

(15)

Notice that the assumption that a CLF exists for the linear part of the PWA reference model is not too restrictive given that the adaptive control strategy will be mostly deployed to stabilize the plant or address tracking problems where the reference model (chosen by design) is realistically going to be selected as an asymptotically stable switched model possessing a CLF. Moreover we expect the reference model to be often selected as an LTI system in order to make the switched plant behave as a stable smooth system for which the matrix P can be easily found. Given the above assumption our main result can be stated as follows.

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M. di Bernardo et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

Theorem 1. Consider a PWA plant of the form (1) and a PWA reference model of the form (3). If the dynamic matrices  Ai of the reference model verify Assumption 1, then the piecewise smooth adaptive control law: u(t ) = KR (t )r (t ) + KFB (t )x(t ) + KA (t ),

(16)

where KR (t ) = α

t



ye (τ ) r (τ ) dτ + β ye (t ) r (t ) ,

(17)

0

KΣ (t ) , KFB (t ) = K0 (t ) + KΣ (t ) + 

(18)

KA (t ) = K0A (t ) + KΣ A (t ) +  KΣ A ( t )

(19)

ye , Ce xe ,

(20)

with

K0 (t ) = α

xe ,  x − x,

Ce , eTn P ,

t



ye (τ ) xT (τ ) dτ + β ye (t ) xT (t ) ,

(21)

0 M −1

KΣ ( t ) =



Kj (t ) ,

 KΣ  (t ) =

j =1

 −1 M



 Kj (t ) ,

(22)

j=1

  t ρ ye (τ ) xT (τ ) dτ , if x ∈ Ωj , Kj (t ) = tjs  0 elsewhere,   t  ρ j , ye (τ ) xT (τ ) dτ , if x ∈ Ω  Kj (t ) =  tj s   0 elsewhere,  t K0A (t ) = ρ ye (τ ) dτ ,

(23)

(24)

(25)

0 M −1

KΣ A ( t ) =

 j =1

M −1

KAj ,

 KΣ A (t ) =



 KAj

(26)

j =1

  t ρ ye (τ ) dτ , if x ∈ Ωj , KAj (t ) =  tjs 0 elsewhere,   t  ρ j , ye (τ ) dτ , if x ∈ Ω  KAj (t ) =  tj s   0 elsewhere,

(27)

(28)

and α , β and ρ being some positive scalar constants, guarantees that the state tracking error xe (t ) between the plant states x(t ) in (1) and the reference trajectory  x(t ) in (3) converges asymptotically to zero, i.e. limt →∞ xe (t ) = 0. Remarks

• The adaptation law presented above consists of three gains KR , K0 and K0A that remain switched on whatever the modes which the plant and reference model are evolving in, together with some gains Kj ,  Kj , KAj and  KAj that are switched on only

when the trajectories of the plant or reference model enter certain domains in phase space. Specifically, the switching gains Kj and KAj are associated to changes of the mode of the plant, whereas the switching gains  Kj and  KAj are associated to those of the reference model. Furthermore, the gains KR and K0 have the same structure of the gains in the Minimal Control Synthesis (MCS) approach [14], an application of Landau’s Model Reference PI Adaptive Control scheme [15]. • In order to compensate the bounded disturbance acting as an input onto the closed-loop error system, the extended strategy exploits the additional adaptive term KA (t ) when compared to the previous version of the algorithm presented in [2], which is a set of switching integral actions used to properly compensate the affine term in each region. • At the generic tjs th commutation instant, the adaptive gains Kj and KAj are initialized to the last value assumed by that gain when the trajectory of the plant x(t ) last exited from region Ωj (or zero otherwise). Analogously, the adaptive gains

M. di Bernardo et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

15

 Kj and  KAj at the generic  tjs th commutation, are initialized with the last value assumed by that gain when the trajectory j (or zero otherwise). Hence, according to the notation used for the switching instants, we have:  x(t ) left the cell Ω     (29) Kj tjs = Kj tjs−1 +1 , s ≥ 2,     KAj tjs = KAj tjs−1 +1 , s ≥ 2, (30)      (31) tjs−1 +1 , s ≥ 2, Kj  tjs =  Kj       (32) tjs−1 +1 , s ≥ 2. KAj  tjs =  KAj          tj1 = 0, Kj  Note that at the first transition the adaptive gains are set to zero, i.e. Kj tj1 = 0, Kj tj1 = 0, Kj tj1 = 0,      Kj tj1 = 0. Furthermore, the integral part of the adaptive gains KR and K0 in (17) and (21) are set to zero at time zero.

• Both control gains KR and K0 in (17) and (21) have integral and proportional terms. It is worth remarking that the use of integral plus proportional adaptation has a beneficial effect upon the convergence of the generalized state error vector in comparison to the use of integral adaptation, specially at the beginning of the adaptation process [15]. PI adaptation has also been used in [4]. • It is worth pointing out that a discussion on the robustness of the proposed approach to uncertainties and delays as well as a comparison with the classical minimal control synthesis MRAC algorithm can be found in previous work by the authors reported in [7,9,2]. Studying robustness of switched control systems to delays is an open problem (see [16] for an example). A more general investigation of how to improve the robustness of the proposed approach to delays is left for future work. 4. Proof of stability We now give the proof of Theorem 1 which is based on constructing an appropriate Common Lyapunov Function (CLF) for the closed-loop system. Note that, due to the presence of discontinuities in the closed-loop system dynamics, the error state dynamics xe (t ) is evaluated in the sense of Filippov [17] and hence, to prove convergence, the Lyapunov function is also analyzed during possible instances of sliding motion. Proof. As the reference model (3) does not admit sliding solutions by construction, we consider for the switched closed-loop error system the following time-varying domains:

 c  Ωi (t ) i∈M = {xe ∈ Rn :  x(t ) − xe ∈ Ωi }i∈M ;

(33)

with M = {0, 1, . . . , M − 1}. Furthermore we define as ∂ (t ) their boundaries. (Note that when the error trajectory xe (t ) evolves along ∂ Ωic (t ), the plant dynamics exhibit sliding motion along the surface ∂ Ωi .) From definition (33), it is now possible to express the indicator function σi (t ) in (12) as a function of the state tracking error as:

Ωic

σi (t ) =



1 0

if xe (t ) ∈ Ωic (t ), elsewhere.

(34)

From (23)–(24) and (27)–(28), it trivially follows that, at any given time instant, only one of the pairs of adaptive gains

(K1 KA1 ), (K2 KA2 ), . . . , (KM −1 KAM −1 ) and one of the pairs ( K1  KA1 ), ( K2  KA2 ), . . . , ( KM KA M  −1   −1 ) can be different from zero. Hence, Eqs. (22) and (26) can be easily rewritten as: M −1

KΣ ( t ) =



σj (t ) Kj (t ) ,

 KΣ  (t ) =

j =1

KΣ A (t ) =

 −1 M



 σj (t )  Kj (t ) ,

j=1

M −1

M −1





σj (t ) KAj ,

j =1

 KΣ A (t ) =

(35)

 σj (t )  KAj ,

j=1

σj defined as in (34) and (13), respectively. with σj and  Given expressions (35), the adaptation law of the control gains (23), (24), (27), (28) can be rewritten in terms of the  − 1): indicator functions as (i = 1, . . . , M − 1,  i = 1, . . . , M K˙ i = ρ ye (t )xT (t )σi (t ),

(36)

 K˙ i = ρ ye (t )xT (t ) σi (t ),

(37)

K˙ Ai = ρ ye (t )σi (t ),

(38)

 K˙ Ai = ρ ye (t ) σi (t ).

(39)

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M. di Bernardo et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

Now, from the plant and the reference dynamics given in (1) and (3), respectively, by using the definition of the control strategy in (16), after some algebraic manipulations the dynamics of the state tracking error can be rewritten as follows: M −1 T x˙ e =  x˙ − x˙ =  A σ (t ) xe + en ψI w − en bβ ye w w + en

 i=1

 + en ψA0 +

 −1 M



Ai  (ψ σ i (t )) +

 −1 M



i x  σ i ψ

i=1



M −1



σi ψi x + en

(ψAi σi (t )) ,

(40)

i=1

i=1

where

 T w , xT r ,   ..  I I , ψI , eT   A − A − bK . b − bK 0 0 n 0 R  T  ψi , en 1Ai − bKi , 1Ai , A0 − Ai ,   i , eTn 1 Ai ,  Ai −  A0 , ψ Ai − b Ki , 1

(41) (42) (43) (44)

ψA0 , ( b0 − b0 ) − K0A ,   ψAi , δ bi − b KAi , δ bi , ( bi −  b0 ),

(45)

ψAi , δ bi − bKAi ,

(47)

(46)

δ bi , (bi − b0 ),

 − 1. Now, by with , being the integral part of K0 in (21) and KR in (17), respectively, and i = 1, . . . , M − 1, i = 1, . . . M i , ψA0 , ψAi and ψ Ai means of the definition of the adaptive gains given in (17), (21), (25), (36)–(39), the dynamics of ψI , ψi , ψ in (42)–(47) can be written as: K0I

KRI

ψ˙ IT = −bα ye w,

(48a)

ψ˙ = −bρ ye xσi ,

(48b)

˙ T = −bρ y x  ψ i e σ i,

(48c)

T i

ψ˙ A0 = −ρ ye b, ψ˙ Ai = −ρ ye bσi ,

(48d) (48e)

˙ = −ρ y b  ψ e σ A i i.

(48f)  +M )−1 (n+1)(M

be the state vector embedding the adaptive gain dynamics (48) as well as the state Note that, letting z (t ) ∈ R tracking error (40) the evolution of the closed-loop system (40), (48) can be recast in a more compact form as the following set of differential equations with discontinuous right-hand side: z˙ (t ) = fi (z )

xe (t ) ∈ Ωic

i = 0, . . . , M − 1

(49)

where fi are the vector fields defined as the right-hand sides of the closed-loop system (40), (48).

T  and let V : R(n+1)(M +M )−1 → R be the Lipschitz, regular [18] Now, let P Ai +  A P = −Qi for some Qi = QT > 0, i∈M i i and positive definite candidate Lyapunov function given by:



V = xTe Pxe +

1

αb

ψI ψIT +

M −1 1 

ρb

i =1

ψi ψiT +

 −1 M 1 

ρb

i=1

M −1 M −1 1  2 1  2 2 i ψ T + 1 ψA0  . ψ + ψ + ψ i Ai ρb ρ b i=1 Ai ρ b 



(50)

i =1

To prove asymptotic stability of (49) in what follows we will first evaluate V˙ in the interior of each generic region Ωic and then along the generic surfaces ∂ Sl resulting from intersections of the manifolds ∂ Ωic (here l = 1, . . . , L; L being the number of manifolds where sliding is possible) [19]:

∂ Sl =

H 

∂ Ωicd ,

d=1

with H ≤ (M − 1).

(51)

M. di Bernardo et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

17

Evaluation of V˙ in Ωic In the interior of each region xe (t ) ∈ Ωic , the error system (49) is a smooth set of differential equations composed by Eqs. (40), (48) with σ and  σ taking finite constant values associated to the active modes of the plant and reference model in that region. The time derivative of V along the trajectories (40) can be computed as: V˙ = −xTe Q  σ (t ) xe +

2

αb

ψI ψ˙ IT +

M −1 2 

ρb

ψi ψ˙ iT +

i =1

 +

2xTe P

 −1 M 2 

ρb

i=1

M −1

en ψI w − en bβ ye w w + en T



σi ψi x+ en

i=1

 +

2xTe Pen

M −1

ψA0 +



(ψAi σi (t )) +

i=1

+

2

ρb

2 

ρb



 −1 M



 i x  σ i ψ

i=1

 Ai  (ψ σ i (t ))

i=1

M −1

ψA0 ψ˙ A0 +

 −1 M

T

˙ i ψ  ψ i

ψAi ψ˙ Ai +

i=1

 −1 M 2 

ρb

˙ . Ai ψ  ψ A i

(52)

i=1

Substituting (48a)–(48f) into (52) and taking into account that xTe Pen = eTn Pxe = ye , after some algebraic manipulations we have: 2 T 2 V˙ = −xTe Q  σ xe − 2bβ ye w w ≤ − min(λmin [Qi ])∥xe ∥ = −W (xe ),

(53)

i∈M 

where λmin [Qi ] is the smallest eigenvalue of Qi . Evaluation of V˙ along the manifolds ∂ Sl In this case two different situations may occur; (i) the trajectory xe (t ) crosses the generic manifold ∂ Sl (51) over a time interval of zero Lebesgue measure, or (ii) it exhibits sliding solutions. In the former case, the crossing has no effect on the stability analysis. Therefore, we focus below on the case where sliding occurs. In particular, when sliding takes place, solutions should be interpreted in the sense of Filippov [17]. Using Filippov convex method, we consider the sliding vector field, say fF , obtained by the convex combination [20]: M −1

fF :=



fi (z )γi (z ),

with γi (z ) ≥ 0;

(54)

i =0

where fi are the vector fields defined in Eq. (49) and i=0 γi (z ) = 1. Note that in the general case this is an underdetermined system of equations, hence there is no uniquely defined Filippov sliding vector. Since the following stability analysis does not depend on any particular choice of Filippov vector field, we do not consider this issue. From (54), after some algebraic manipulations it is possible to write the closed-loop dynamics (40), (48) during the sliding motion as:

M −1

T x˙ e =  A σ (t ) xe + en ψI w − en bβ ye w w + en

 −1 M

 i=1

i x + en  σ i ψ

 −1 M



i=1

M −1

Ai + en ψA0 + en  σ i ψ



γi (ψi x + ψAi )

(55a)

i=1

ψ˙ IT = −bα ye w,

(55b)

ψ˙iT = −γi bρ ye x,

(55c)

T

˙ = −bρ y x  ψ i e σ i,

(55d)

ψ˙ A0 = −ρ ye b, ψ˙ Ai = −γi bρ ye ,

(55e)

˙ = −ρ y b  ψ e σ A i i

(55g)

(55f)

where only quantities depending on discontinuities due to plant switchings are convexified as the reference model is assumed not to exhibit sliding. Evaluating the derivative of V (50) (again in the sense of Filippov) along trajectories of (55), some algebraic manipulations yield: V˙ ≤ − min(λmin [Qi ])∥xe ∥2 = −W (xe ). i∈M 

(56)

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M. di Bernardo et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

Stability of the closed-loop adaptive system From (53) and (56), it follows that for almost all t V˙ ≤ −W (xe ) < 0

i , ψA0 , ψAi , ψ Ai . a.e. t , ∀xe ∀ψI , ψi , ψ

(57)

The derivative of the Lyapunov function along Filippov closed-loop solutions is negative, hence the origin of the closed-loop system is globally stable in the Filippov sense [18]. Now following the approach in [21], from (57) for any closed-loop trajectory we have

∀t ≥ 0,

sup V ≤ C

(58)

t ∈[0+∞)

with C being a sufficient large positive constant. From (57) and (58), it follows ∞



W (xe (t )) dt ≤ C .

(59)

0

Since W (xe ) is a continuously differentiable positive-definite function, W (xe (t )) is uniformly continuous. Exploiting Barbalat’s Lemma, W (xe (t )) converges to 0 as t → ∞, hence the state tracking error xe (t ) converges to 0.  5. Numerical validation To illustrate the effectiveness of the proposed control strategy, we present its application to a system of practical meaning, a switched model of the Colpitts oscillator often used in Electronic Engineering [22]. Specifically, we consider the classical schematic for the Colpitts oscillator that contains a bipolar junction transistor T as gain element, and a resonant network, consisting of the inductor L, two capacitors C1 and C2 , and a resistor R. The circuit bias is provided by the supply voltage Vcc and the current source I0 . (For the sake of brevity we omit the circuit schematic and the derivation of the model equations that can be found in [23].) In its original form the Colpitts oscillator can be described as a PWA model of the form:

 z˙ =

M0 z + N0 + Gu, M1 z + N1 + Gu,

if z2 > 0, if z2 < 0

(60)

where Mi , Ni and G are matrices of appropriate dimension that are not in canonical form and whose expression (omitted here for the sake of brevity) can be found in [23] and u is an additional input here introduced for controlling the circuit dynamics. Using the same parameters and the transformation into canonical form presented in [11,3], the plant equations can be easily recast into the required form (1) with matrices

 A0 =

0 0 0

1 0 −1

0 1 , −0.0316



0 0 3.1626

 A1 =

0 0 , 1

1 0 −1

0 1 , −0.0316



(61)

  B0 = B1 = B =

(62)

and Ω0 := {x ∈ R3 : H T x ≤ 0}, Ω1 := {x ∈ R3 : H T x > 0} with H T = [−3.162 0 0]. In what follows, we will numerically investigate the performance of the proposed adaptive control scheme for two different choices of the reference model. Firstly we will use a smooth LTI reference model and then a PWA one. In the former case, we assume the goal is to make the switching circuit track the output of a smooth reference model. Specifically, we choose as reference model the system: 0 0 −0.75

  x˙ =

1 0 −2.75

0 0 1  x + 0 r (τ ). −3 2.5







(63)

We select the reference input r (τ ) to be the sinusoidal signal r (τ ) = 10 sin(τ ), while the initial conditions are chosen as 3.75 2.96]T . Furthermore, the scalar quantities α , β and ρ , which scale the adaptive gains, have been chosen heuristically as a trade off between convergence time and reactivity of the control action as α = 10 and β = 1, and ρ = 1. As shownin Fig. 1, under the adaptive control action proposed in this paper, excellent tracking of all state variables is achieved while all gains adaptive gains remain bounded. Fig. 2 shows a comparison with the case where a classical MCS algorithm is used to solve the same control problem. As it can be clearly seen by contrasting Fig. 1(a) with Fig. 2 the extension presented in this paper is essential to guarantee an acceptable tracking performance.

 x(0) = [2.41

M. di Bernardo et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

19

Fig. 1. LTI reference model. (a) Phase portrait of the plant trajectory (blue dashed line) converging to the steady-state periodic solution of the LTI reference (1) (2 ) model (red solid line); (b) evolution of the feedforward adaptive gain KR ; (c) feedback adaptive gains for mode zero of the plant, K0 (blue solid line), K0 (red (3)

dashed line) and K0

(1 )

(dashed dotted line), and K0A (magenta dotted line); (d) feedback adaptive gains for mode one of the plant, K1 (3 )

(red dashed line) and K1 /2 (dashed dotted line), and K1A /5 (magenta dotted line). The plant initial conditions are set to x(0) = [0.7 (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(2 )

(blue solid line), K1

−1.18

14.13]T .

To further validate the effectiveness of the strategy we next assume that the reference model is itself a PWA system of the form (3) where 0 0 −0.75

  A0 =

1 0 2.75

  B = − B0 =  B1

0 0 , 2.5

0 1 , −3



0 0 −5.25

  A1 =

1 0 −8.75

0 1 , −12



(64)



(65)

0 := { 1 := { and Ω x ∈ R3 :  H T x ≤ 0}, Ω x ∈ R3 :  H T x > 0} with  H T = [1.5 2 3]. The reference signal is r (τ ) = 5 sin(τ ), the initial conditions of the reference model are assumed to be  x(0) = [1.9 2.34 9.4]T and control weights α , β and ρ are chosen as in the previous case. The state space evolution of the trajectories of the plant and reference model are shown in Fig. 3 where the efficacy of our approach is again confirmed. (For the sake of brevity we omit the evolution of the adaptive gains that, as in the previous case, remain bounded.) Finally, note that the complexity of the control algorithm in terms of the number of control modes needed to cope with multi-modal plants and/or reference models scales linearly with the number of modes. Indeed, when moving from the single-mode LTI reference model to a bimodal PWA one, only one additional switching control gain was required to close the control loop. Further insights on the cost of the practical implementation of the strategy on low-cost micro controllers was explored in [9]. 6. Conclusions We have presented an extension of the hybrid adaptive strategy introduced in [2] aimed at compensating possible instabilities due to the presence of sliding mode trajectories and bounded perturbations acting on the closed loop error system. The external disturbances are due to the presence of affine terms in the plant and reference model dynamics. Using an appropriate common Lyapunov function, we have shown that the extended strategy guarantees asymptotic convergence of the tracking error, even in the presence of sliding solutions, as well as boundedness of all the adaptive gains. The theoretical

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M. di Bernardo et al. / Nonlinear Analysis: Hybrid Systems 21 (2016) 11–21

Fig. 2. LTI reference model. Phase portrait, plant trajectory (blue dashed line) failing to converge to the steady-state periodic solution of the LTI reference model (red solid line) in the case where the plant is controlled by a classical MCS [14] rather than the approach presented in this paper. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. PWA reference model. Phase portrait, plant trajectory (blue dashed line) converging to the steady-state periodic solution of the LTI reference model −2.9 −2.74]T . (For interpretation of the references to color in this figure legend, (red solid line). The plant initial conditions are set to x(0) = [0.1 the reader is referred to the web version of this article.)

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