adaptive stabilization of discrete linear systems via

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93C55 Discrete time systems, 93C40 adaptive control, 34D23 global stability. ... In [3] and [4], Narendra et al. use a continuous-time multiestimation scheme to.
ADAPTIVE STABILIZATION OF DISCRETE LINEAR SYSTEMS VIA A MULTIESTIMATION SCHEME A. Ibeas, M. de la Sen, S. Alonso-Quesada Instituto de Investigación y Desarrollo de Procesos FACULTAD DE CIENCIAS UNIVERSIDAD DEL PAIS VASCO Campus de Leioa ( Bizkaia ) Apdo. 644 Bilbao SPAIN Abstract. A pole-placement based adaptive controller synthesized from a multiestimation scheme is designed for linear time-invariant plants. A higher level switching structure between the various estimation schemes is used to supervise the reparametrization of the adaptive controller in real time. The basic usefulness of the proposed scheme is to improve the transient behavior so that the closed loop stability is guaranteed. The scheme becomes specifically attractive when extended to linear plants whose parameters are piecewise constant while changing abruptly to new constant parametrizations or when they are time-varying. A complete description of the controller architecture with its associated multiestimation scheme is given. Also, the proofs of boundedness of all the relevant closed-loop signals are given so that the closed-loop system is stable. Keywords. Discrete adaptive control, multiestimation, stability, switching techniques, supervisory control AMS Classification Numbers. 93C55 Discrete time systems, 93C40 adaptive control, 34D23 global stability.

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I.

INTRODUCTION

The challenge of control theory nowadays is to develop control system schemes able to achieve a good performance in terms of speed, accuracy and stability for increasingly complex systems, including the presence of large uncertainties in the system to be controlled. This paper deals with the problem of improving the transient response of adaptive systems in the case when the plant to be controlled has large uncertainties in its parameter values or these can change abruptly, using a multiestimation based adaptive controller. Multiestimation strategies are useful to improve the adaptation transients by accomodating the estimation/controller parametrization pairs through online switchings between the estimators to determine the most covenient parameter estimation. The switches are organized after minimum residence time periods in such a way that the adaptive controller is parametrized by the estimator which generates the estimated output which is closer to the true one [1] [2]. In [3] and [4], Narendra et al. use a continuous-time multiestimation scheme to show how adaptive multiestimation control can achieve a good performance in the transient of adaptive control systems. They consider N e adaptive models ( or estimation schemes ) and an adaptive controller associated with each model. Each pair identification scheme-adaptive controller can be indexed by an integer i ∈ {1,2,..., N e } . There is a supervisory index J s( i ) associated with each individual identification scheme. The identification scheme that parametrizes the adaptive controller is chosen from a switching rule obtained by evaluating the minimum of the J s( i ) , i = 1,2,..., N e . At a finite or infinite sequence of decision times, the supervisory index of each identifier is compared and the controller corresponding to the best index is selected. The sequence of controllers is not required to converge and, in general, will not do so ( specially if the plant is time-varying ). In [5], a switching supervisory control scheme is used for adaptive stabilizing an unknown plant. The mathematical proofs of closed-loop stability are given and it is shown that a good control can be achieved asymptotically even if the switching mechanism does not stop in finite time. It is shown in [6] that there exists convergent decision rules, for discrete multiestimation systems, under which the supervised control system converges to a "good" identifier. These approaches use, in general, a higher-order level switching rule, which acts as a supervisor of the basic parametrized controller, whose feature is to compare on-line the performance indexes of all the identifiers in order to choose the best controller parametrized by such identifier [7]. Other approaches have been reported in the literature as those based in a predefined switching route over the set of controllers, [1], [8]. Several proposals have been given about the choice of the supervisory index, like in [3] and [4], where an accumulated identification error with forgetting factor for continuous time systems is used. In [9], a discrete time version of this index is used to switch between set point problem controllers. In [7], the "performance signal" is the norm-squared value of the identification error and in [10] the time average of the squared identification error is used. Although the specific form of the supervisory index may be different from one work to another, it is usual to define it based on the identification error, which is a natural choice since that index can reflect how far a specific identifier is from the real plant behaviour. The main objective of this work is to extend the results of [3], [4] and [5], concerning the continuous time case, to the discrete time case while showing how a significant transient response improvement can also be achieved. Mathematical proofs of closed-loop signals boundedness, which have not been reported before in the literature for the discrete case, will also be given. Simulations that corroborate the efficiency of the use of the multiestimation scheme in the proposed way are included. Also, the influence of the free-design parameters of the estimation scheme in the closedloop performance will be discussed. The paper is organized as follows. Section II deals with the system to be controlled and the multiestimation and adaptive controller architecture together with the basic assumptions needed for 2

stability and convergence purposes. In Section III, the main properties about identification algorithms, control law and closed-loop stability are given. In Section IV, some computer simulations and their corresponding discussion are presented and finally, conclusions end the paper. There are two appendixes containing the mathematical proofs of the main theorems about the stability of the closed-loop. Notation. u k is the plant input sequence, y k is the plant output sequence, •

N e is the number of pairs of identification models ( identifiers )-adaptive controllers



running in parallel, c : Z 0+ → Κ = {1,2,..., N e } is the switching map that defines the integer i associated with the identification model-adaptive controller pair connected in operation, (i ) (i) (i) ˆy k is the i-th identifier's estimated output, ek = y k − ˆy k is the identification error between the i-th identifier's output and the real plant output, θ is the real plant parameter vector, θˆ k( i ) is the i-th estimated parameter vector, θ~k( i ) = θ k − θˆ k( i ) is the i-th error parameter vector.



The input/output measures vector ( or regressor ) ϕ k is defined by



ϕ kT = [− y k −1

− y k −2

− y k −n

u k + m−n

u k − n ],

u k + m−n −1



C n×n is the vector space of square matrices over the complex field, J p is the closed-loop



performance index used for comparing the basic and the multiestimation based adaptive control schemes, J s( i ) is the supervisory performance index of the i-th identification algorithm used for defining the switching rule, τ D is the dwell time, N D denotes the dwell samples, τ is the



sampling period, p k( i ) is the i-th controller parameter vector, Κ = { 1,2 ,..., N e } the overall set of identification models-adaptive controller pairs, k p ( z 2 + b1 z + b0 ) H( z ) = 3 is the plant transfer function used in examples whose z + a 2 z 2 + a1 z + a0

[

associated parameter vector is θ T = a 2 •

a1

a0

kp

k p b1

]

k p b0 ,

‘Big O ’ and ‘Small o’ notation: let be f , g : ℜ0+ → ℜ0+ , it is said that f = O (g ) if there exist real finite constants k1 , k 2 ≥ 0 such that f ≤ k1 g + k 2 . It is said that f = o(g ) if f  f = O (g ) and lim g →0   = 0 . g II.

PROBLEM STATEMENT

A) Discrete plant Consider the linear and time invariant discrete plant A( q −1 ) y k = B( q −1 )u k

(1)

where u k and y k are the input and the output sequences respectively, q −1 is the one-step delay operator and the degrees of polynomials A* (q) = q n A(q −1 ) ( monic ) and B* (q ) = q n B(q −1 ) are n and m ( n > m) respectively, where q is the one-step forward operator. 3

Assumption A.1. It is assumed that n ≥ n , m ≥ m and d = n − m are known. All unstable plant zeros ( if any ) are known and are also zeros of the reference model. The first part in Assumption A.1 means that an upper bound of the polynomial degrees n, m and the plant delay ( or relative order ) d are known and the second part implies that the plant zeros polynomial B( z ) can be expressed as B = B + B − , where B − contains all zeros of B that satisfy z ≥ 1 − δ for some δ ∈ [0 ,1) ( in particular, B − includes all unstable roots of B ) and B + is monic.

We have fixed δ = 0 ( i.e. B − only contains unstable and critically stable roots of B ) and B − is assumed to be known. Assumption A.2. The reference model H m ( z ) = Bm ( z ) / Am ( z ) is exponentially stable ( i.e. all roots of Am ( z ) satisfy z ≤ 1 − σ for some σ ∈ (0,1] ).

Assumption A.1 implies that Bm ( z ) can be written as Bm = Bm' B − , where Bm' is a free design polynomial. The plant polynomials A and B + may be unknown and then estimated ( while B − is known ) and the control objective is to synthesize a model-following adaptive controller which achieves an acceptable transient response for practical applications. The way used in this paper for improving transient response of adaptive systems is based on the implementation of a multiestimation scheme with appropriate switchings between the various estimators so as to reparametrize on-line the adaptive controller. B) Parallel multiestimation scheme In the case when the plant (1) is not perfectly known, parameter estimation has to be used. It is typical in control problems to know a compact subset of the parameter space D ⊆ ℜ n +m where the real plant parameter vector belongs to. This knowledge allows the use of projections of the estimates within such a domain. Assumption A.3. There exists a known convex and compact subset D ⊆ ℜ n + m of the parameter space where the real plant parameter vector belongs to so that for all plant parametrization in D, the polynomials A and B are coprime. If the estimation algorithm starts running with and estimated vector far away from the real plant parameter vector, then the transient will have large deviations from the desired output resulting in a bad performance. In this work we have chosen a parallel multiestimation scheme to improve the transient response of the adaptive system. The architecture of the multiestimation scheme is represented in Figure 1:

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Figure 1. Multiestimation scheme There exist N e estimation algorithms running in parallel ( i.e. at each sampling time t k every algorithm gives the estimated parameter vector θˆ k( i ) and the estimated plant output ˆy k( i ) , i ∈ Κ based on past plant input and output measurements ). Each algorithm is different from each other in what is concerned with the estimated parameter vector initialization and/or the kind of the estimation algorithm and integrate the so-called multiestimation scheme. There also exist N e adaptive controllers ( only one being in operation at each time ) such that the i-th adaptive controller is parametrized at every instant t k by the i-th estimation algorithm. Thus, every pair identification algorithm-adaptive controller is indexed with only one integer i ∈ Κ . Denote by c k the integer that defines de controller ( parametrized by its respective identification algorithm ) which is active ( i.e. connected to the plant for control purposes ) at time t k . Note that a mapping c, of image c k , may be defined which is called the switching map. A switching rule based on the identification errors ek( i ) = y k − ˆy k( i ) = ϕ kT θ~k( i ) of the N e estimation algorithms chooses at each sampling time t k the individual estimation scheme ( identifier ) which parametrizes the controller at time kτ which is in fact connected in feedback to the plant. Remarks. 1. At each time t k , only one adaptive controller is connected to the plant generating the control input. 2. All the j-th estimation algorithms are always ( i.e. at every time t k ) running to calculate all the estimated plant outputs. Also, each respective adaptive controller is updated for all time although only the i-th controller is generating the plant input.

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C) Adaptive controller All the controllers are based on pole-placement ( see for instance, [11] ), whose basic scheme is displayed in Figure 2:

Figure 2. Basic adaptive controller Then, we will consider for each controller i ∈ Κ the polynomials Rk( i ) , S k( i ) ,T ( T only depends on the reference model zeros polynomial which is of constant coefficients ) where T = Bm' and Rk(i ) (monic ), S k(i ) are the unique solutions with degrees fulfilling deg( Rk( i ) ) = n , deg( S k( i ) ) = n − 1 and deg( Am ) = 2 n

(2)

of the polynomial diophantine equation ˆ k( i ) Rk( i ) + Bˆk( i ) S k( i ) = Am A0 A

(3)

with deg( Aˆ k( i ) ) = n and deg( Bˆ k( i ) ) = n − 1 , for all i = 1,2,..., N e at every time t k and for all i ∈ Κ with A0 = 1 . Assumption A.3 is extended to the multiestimation scheme by using projection when necessary as follows. ˆ k( i ) , Bˆk( i ) are coprime Assumption A.4. It is assumed that θˆk( i ) ∈ Di for all k ≥ 0 and i ∈ Κ , then A over Di for all k ≥ 0 . It is also feasible to use different known compact and convex subsets Di for coprimeness of each pair ( Aˆ k( i ) , Bˆ k( i ) ) , i = 1,2,..., N e . This would be useful provided that each pair estimator-adaptive controller is associated with a different plant operation point what is often the case, for instance, in some chemical engineering processes. From Assumption A.4, equation (3) has a unique solution for all k ≥ 0 under the degree constraints (2). The control law is given by Rk u k = Tu ck − S k y k 6

where ( Rk , S k ) ∈ {( Rk( i ) , S k( i ) ); i ∈ Κ} , i.e. at each time, the pair ( Rk , S k ) is defined by ( Rk( i ) , S k( i ) ) for some i ∈ Κ and then, the control input is generated by the corresponding i-th controller uk = uk(i ) . We summarize in the following controller parameter vector all the parameters of each controller i ∈ Κ : p k( i )T = t m( i ) t 0( i ) s n( i−)1,k s0( i,k) rn(−i 1) ,k r0(,ik) . Now, it is necessary to elucidate how to choose the current adaptive controller ( or, in other words, which is the active controller at t k ) from the family of parallel controllers such that the adaptation transients are practically acceptable while the closed-loop scheme is maintained globally stable. Thus, the basic pole-placement adaptive controller ( see Figure 2 ) is reparametrized by one of the estimators of the multiestimation scheme during appropriate time intervals. A higher-order level switching law ( supervisor ) calculates the switching times between the various estimators what is used as a mechanism to on-line reparametrize the basic adaptive controller in operation to generate the control input. The operation mode of such a supervisor is discussed in the sequel.

[

]

C.1 Supervision Supervisor's duty is to value the performance of the possible controllers connected to the plant with the aim of choosing the current controller from the set of parallel controllers. The supervision scheme is a high level mechanism for calculating the switching times between the various estimators to on-line reparametrize the basic adaptive controllers. The proposed specific performance index has the form: J s( i ) ( k ) = α

k

∑ λk − ( y

=k − M

− ˆy ( i ) ) 2 + β ( ˆy k( c+k1−1 ) − ˆy k( i+)1 ) 2

(4)

where

[

yˆ k( c+k1−1 ) = − y k and

[

yˆ k( i+)1 = − y k

− y k −n +1 uk( ck −1 )

− y k −1

− y k −n +1 uk( i )

− y k −1

uk −1

uk −1

]

uk −n +1 ⋅ θˆk( ck −1 )

]

uk −n +1 ⋅ θˆk( i )

and ck ∈ Κ denoting the identifier-controller pair in operation at sample k. M is an integer number large enough to give sense to the performance evaluation. Note that (4) has two parts. The first one is a measure of the long-term accuracy of each identification algorithm, where the forgetting factor λ ( which, in general, can be sample-dependent ) establishes the effective memory of the index in rapidly changing environments. The second one weights the output jump associated with the controller switching. If the switching between two controllers causes an abrupt variation of the plant output, then this second term acts either to avoid switching or to make the system to switch to a controller closer to the current one in operation in order to reduce the plant’s output variation. Weights α , β determine the contribution of each term to the global index and are such that α , β ≥ 0 , α + β = 1 . Now, the switching rule for the basic adaptive controller reparametrization is obtained from the performance index (4) as follows. If the switching sampling times sequence is denoted by {t (1) , t ( 2 ) ,..., t (π ) } ( where π , which may be finite or infinite countable, is the number of switchings ), then the identifier c Ni that parametrizes the basic adaptive controller at time t ( i ) = N iτ , i = 1,2,..., π is determined by c Ni ∈ { j ∈ Κ

J s( j ) ( N i ) = min{J s( r ) ( N i ) with r ∈ Κ}} 7

where the switching sampling sequence is such that t ( i +1) − t ( i ) ≥ τ D where τ D is a dwell ( or residence ) time which appears because of stability considerations ( see Theorem 3 in the next Section ). The c-mapping is constant between two switching sampling times ( i.e. for all integer ∈ {N i ,..., N i +1 − 1} , the image of c is constant for all t ∈ t ( i ) , t ( i +1) ) c = c Ni , and then the same

[

identifier parametrizes the adaptive controller ). Then, we can resume the control law as: Rk( ck )u k = Tu ck − S k( ck ) y k

(5)

where c k is defined above. III.

PROPERTIES OF THE MULTIESTIMATION SCHEME AND CLOSED-LOOP STABILITY

A) Boundedness and convergence results of the multiestimation scheme In this work, all the recursive identification algorithms associated with the multiestimation scheme will be of standard least-squares type. The difference between the various ones consists of the different initialization of the estimates vector for each algorithm. Theorem 1. The standard least-squares single estimation algorithm has the following properties: (i) (ii)

(iii)

θˆk is bounded for all integer k ≥ 0 and has finite limit, provided that θˆ0 is bounded. ek2 ~ = 0 where ek = y k − yˆ k = ϕ kT θ~k −1 , with θ k = θ − θˆk being the parameter T 1 + ϕ k Pk −1ϕ k error vector.  +N  If an asymptotic persistence excitation condition  ∑ ϕ k ϕ kT  > µI > 0 is satisfied for some  k=    real constant µ > 0 , N arbitrary and large enough, over the plant input, then θˆk → θ for k → ∞. lim k →∞

Proof. See for instance [11]. For the multiestimation scheme, the next two results follow: Theorem 2. The combined estimated parameter vector from the multiestimation scheme θˆ k = θˆ k( ck ) has the following properties: (i) (ii)

θˆk is bounded for all integer k ≥ 0 provided that θˆ0 is bounded. If there is a real constant k 0 ≥ 0 such that ∀k ≥ k 0 ck is constant or if all estimators have the same limit: θˆk( i ) → θˆ for k → ∞ and ∀i ∈ Κ , then the combined estimated vector has finite limit.

(iii)

ek2 ~ lim k →∞ = 0 where ek = y k − yˆ k = ϕ kT θ k −1 , with θ~k = θ − θˆk being the combined T 1 + ϕ k Pk −1ϕ k parameter error vector.

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(iv)

If the persistence excitation condition defined in Theorem 1 is fulfilled, then the combined estimator has a finite limit and converges asymptotically to the real plant parameter vector: θˆ k → θ for k → ∞ .

Proof. The proof is trivial from Theorem 1. By Assumption A4 and Theorem 1, every controller parameter vector p k( i ) is bounded ∀k ≥ 0 and has a finite limit as k → ∞ , ∀i ∈ Κ . B) Closed-loop stability Now, let Aˆ k = Aˆ k( ck ) , Bˆ k = Bˆ k( ck ) , Rk = Rk( ck ) , S k = S k( ck ) the plant estimation and controller polynomials, ck ∈ Κ being the current estimator and controller in operation. In order to prove the closed-loop stability, the following auxiliary linear time-varying extended system will be used for subsequent analysis: x k = Gk x k −1 + ϑ1 ek + ϑ 2Tu ck where x kT = y k

[

y k −1 … y k − n +1

(6)

u k( ck ) … u k( c−kn)+ 2

u k( c−kn)+1

]

(7.a)

is the real 2n state vector,

Gk = Gk( ck )

− aˆ n −1,k   1  0   = 0   − sn −1,k  0    0 

− aˆ n −2,k





bˆn −1,k 0

bˆn −2,k 0



0

0

0

− s0,k

− rn −1,k

− rn −2,k

0

1

0

… … …

0

0

0



− aˆ 0,k



0

0

1

0

0



− s n − 2 ,k

− sn −3,k

0

0

1 … … − s1,k 0 …

0

0



0

bˆ0,k   0     0  …  − r1,k − r0,k  0 0    1 0  (7.b) …

is the dynamical matrix associated with the current estimator-controller pair in operation, and

ϑ1T = [1 0

 ϑ 2T = 0 

0] ( ck )

with ek = y k − yˆ k and yˆ k = yˆ k

0

1 n +1

0

 0 

(7.c)

where ck ∈ Κ for each sampling time t k = kτ .

This system is built as follows. First, the discrete plant is given by equation (1). This equation may be equivalently rewritten as ˆ k y k = Bˆk u k + ek A

(8)

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by using the identification error. Equation (8) together with the control law (5), can be represented in state variables in the form (6) with the state vector defined in (7). Now, introduce the next new technical assumption. Assumption A.5. max k ≥k0 Gk( +j 1) − Gk( j ) ≤ ∆ for all j ∈ Κ , with ∆ sufficiently small and k 0 large enough. From the structure of the matrices Gk( j ) , Assumption A.5 means that the estimates parameter vector does not change, asymptotically, much from one sample to another, condition which is satisfied trivially by the least-squares identification algorithm because it converges to a finite limit. Consider that there are finite real constants, C ≥ 1, 0 < ρ < 1 such that an activated controller ln C remains active a minimum number of samples given by η min > . This condition states that a ln ρ minimum dwell ( or residence ) time given by τ D = η minτ is required ( τ being the sample period ) to guarantee the stability of the system (6)-(7), i.e. when a controller becomes active at time t k , that controller must be kept in operation at least until time t k + τ D . Then, if the performance index associated with another possible controller is less than the active one, the system can switch to it. However, the basic adaptive controller must be kept within the time period [t k ,t k + τ D ) from the same parametrizing estimation algorithm of the multiestimation scheme. The existence of such a dwell time allows us to guarantee the stability of the closed-loop system. It is proved in Appendix A that the above auxiliary system (6)-(7) is globally stable under Assumptions A1-A5 together with the dwell time requirement. This is an intermediate result in the proof of the stability of the closed-loop system and its perfect asymptotic tracking properties. From the results proved in Appendix A, one may calculate the value of the constants ρ and C such that the dwell time parameter can be tuned before the control scheme starts running while all signals remain bounded. Although an analytical characterization of the dwell time may be obtained before starting the estimation process based on ‘a priori’ knowledge, the technical way in which the dwell time requirement is implemented in practice, is through an on-line norm calculation-based algorithm. This algorithm works as follows:

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where k is the current sample, j is an algorithmic variable, is updated to the current active controller and SWITCH is a variable that allows ( or not ) switching. At each sampling time the transition matrix ( which is composed by the left-hand products of the dynamical matrices Gk( ck ) ) norm is evaluated; if the value of that norm is larger than or equal to one, the activated identifier must be kept active. If that norm has value less than one, then the system is allowed to switch to another identifier with an associated reparametrization of the adaptive controller. This algorithm is directly inspired on the demonstration technique developed in Appendix A. The above discussion may be summarized in the subsequent main result of this section, concerned to the stability and tracking properties: Theorem 3. Assume that Assumptions A1-A5 hold and that η min >

ln C ( i.e. there exist a ln ρ

minimum dwell time given by τ D = η minτ ). Then, the following two propositions hold: (i)

( Global stability ) The linear time-varying extended system (6)-(7) is globally stable. Thus, the closed-loop system is globally stable as well.

( ii )

( Perfect tracking asymptotic property ) The closed-loop system’s output asymptotically tracks perfectly the reference model output.

The proof of Theorem 3( i ) is given in Appendix A, and the proof of Theorem 3( ii ) is given in Appendix B.

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IV.

SIMULATED EXAMPLES

A) Examples Some simulation examples, in which we can appreciate the advantages of the control scheme are now presented. It will be shown how the multiestimation scheme may be used to improve the performance of conventional discrete adaptive control systems. From a mathematical perspective, there is little agreement on how to determine the system’s performance, and several quality indexes have been proposed. The following index has been chosen as a measure of the closed-loop performance: k

J p ( k ) = ∑ ( y − y m )2

(9)

=1

which measures the discrepancies between the real output of the plant y k and the desired one y mk . The simulated examples start with an example referred to an unknown invariant plant and continue with plant whose parameters change abruptly. When the plant parameter vector changes, once the change is detected, related to a prescribed threshold for the supervisory performance index (4), it is convenient to reset the estimated parameter vectors θˆk( i ) to their original starting values θˆ0( i ) to identify the new plant parameter vector as fast as possible. Thus, a reset mechanism has been included in Examples 2-3, where the plant parameters change abruptly in order to detect when the plant parameters have changed and reset the estimated parameter vectors to their original locations. Finally we include a time-varying plant based example, case for which the multiestimation scheme reveals promising. In all simulations, λ from equation (4) is constant and fixed to λ = 0.95 . The unique estimator is initialized in θˆ0(1) in all cases. The objective of the simulations is to corroborate that the multiestimation based control scheme is able to improve the transient response of adaptive systems, i.e. it gets a better performance compared with a conventional one-identifier based adaptive control. Example 1. Consider the third order plant with two complex conjugate poles given by: H( z ) =

( z − 0.3 )( z − 0.6 ) ( z − 0.5 )( z 2 − 2 z + 5 )

and the reference model’s discrete transfer function Hm( z ) =

( z − 0.25 )( z − 0.45 ) ( z − 0.1 )( z − 0.2 )( z − 0.3 )( z − 0.4 )

We assume that plant parameters belong to the known compact set D = { k p , a 2 , a1 , a 0 , b1 , b0

0.8 ≤ k p ≤ 1.1, −3.5 ≤ a 2 ≤ −0.5, 3 ≤ a1 ≤ 7, −3.5 ≤ a 0 ≤ −0.5,

− 1 ≤ b1 ≤ −0.75, 0.12 ≤ b0 ≤ 0.2 } The multiestimation scheme has four identifiers running in parallel uniformly initialized over the parameter space D. We include simulations for different weights and dwell times with N D = τ D / τ being an integer number. The model reference input is a unity step.

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Initializations:

θˆ0(1)T = [− 0.8 3.5 − 0.8 0.84 − 0.75 0.12] θˆ 0( 2 )T = [− 1.7 4.75 − 1.7 0.9 − 0.82 0.15] θˆ0(3)T = [− 2.7 5.75 − 2.7 0.96 − 0.9 0.17] θˆ0( 4 )T = [− 3.4 7.5 − 3.4 1.06 − 0.98 0.19]

Results ( Figures 3.x ) 8

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3.10

3.11 Single identifier: output

15

30

40

samples

3.12 Single identifier: performance index

400 350

10

300 5

250 Ind ex 200

yk 0

150

-5

100 -10

-15 0

50 10

20

30

40

0

50

0

10

20

samples

3.13 30

40

50

Four identifiers: control law with ND = 5 and alpha = 0.95

25

30

20

20

15

10 co ntr 0 ol

co 10 ntr ol 5

-10

0

-20 -30

-5

-40

-10

-50 0

40

3.14

Single identifier: control law

50

30 samples

10

20

30

40

-15 0

50

10

20

30

samples

samples

3.15

3.16

40

50

Example 2. Consider the following piecewise time invariant third order discrete plant with two complex conjugate poles H ( z) = H ( z) =

( z − 0.7)( z − 0.85) ( z − 1.5)( z 2 − z + 0.5)

0 ≤ k < 100

( z 2 − 1.4 z + 0.5) ( z − 1.1371)( z 2 − 1.163z + 0.6166) ( z 2 − 1.35z + 0.47) H ( z) = ( z − 1)( z 2 − 1.15z + 0.55)

H ( z) =

100 ≤ k < 200 200 ≤ k < 300

( z 2 − 1.3z + 0.45) ( z − 1.05)( z 2 − 1.0708 z + 0.4148)

whose parameters belong to the known compact set: 14

300 ≤ k ≤ 400

50

D = { k p , a 2 , a1 , a 0 , b1 , b0

0.9 ≤ k p ≤ 1.05, −2.8 ≤ a 2 ≤ 2, 1.5 ≤ a1 ≤ 2.2, −0.8 ≤ a0 ≤ −0.45,

− 1.6 ≤ b1 ≤ −1.4, 0.45 ≤ b0 ≤ 0.6 } The reference model and its input are the same as in Example 1. The plant changes its parameters abruptly among values in D. There are four identifiers running in parallel initialized as:

θˆ0(1)T = [− 2.1 1.5 − 0.45 0.9 − 1.4 0.45] θˆ0( 2 )T = [− 2.3 1.75 − 0.57 0.94 − 1.47 0.5] θˆ0(3)T = [− 2.5 2 − 0.7 0.98 − 1.54 0.55] θˆ0( 4 )T = [− 2.7 2.2 − 0.8 1.03 − 1.6 0.6] Results ( Figures 4.x ) Four identifiers: output with ND = 2 and alpha = 0.95

Four identifiers: switching map with ND = 2 and alpha = 0.95

3

Four identifiers: performance index with ND = 2 and alpha = 0.95

3

30

2.5 25

2 1.5

20 ck

1 yk 0.5

Ind ex 15

2

0 10

-0.5 -1

5

-1.5 -2

0

50

100

150

200 samples

250

300

350

1

400

0

5

10

15

20

4.1 12

x 10

15

25 30 samples

35

40

45

0

50

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50

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150

4.2

Four identifiers: output with ND = 2 and alpha = 1

200 samples

250

300

350

400

350

400

350

400

4.3

Four identifiers: switching map with ND = 2 and alpha = 1

Four identifiers: output with ND = 5 and alpha = 0.95

3

3

10

2.5

8

2

6

1.5 ck

4

1

yk

yk 2

2

0.5

0

0

-2

-0.5

-4

-1

-6

-1.5

-8

0

50

100

150

200 samples

250

300

350

400

1

0

50

100

150

4.4

200 samples

250

300

350

-2

400

0

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150

4.5

Four identifiers: switching map with ND = 5 and alpha = 0.95

250

300

4.6

Four identifiers: performance index with ND = 5 and alpha = 0.95

3

200 samples

30

2.5

x 10

16

Four identifiers: output rwith ND = 5 and alpha = 1

2 25 1.5 20 ck

1 Ind ex 15

2

yk 0.5 0

10 -0.5 5 -1 1

0

10

20

30 samples

4.7

40

50

0

0

50

100

150

200 samples

4.8

15

250

300

350

400

-1.5

0

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200 samples

4.9

250

300

Four identifiers: switching map with ND = 5 and alpha = 1

Single identifier: output

3

Single identifier: performance index

7

250

6 200

5 4 ck

150 yk 3

Ind ex

2 2

100 1 0

50

-1 1

0

50

100

150

200 samples

250

300

350

-2

400

0

50

100

150

4.10

200 samples

250

300

350

0

400

0

50

100

150

4.11 Single identifier: control law

6

250

300

4.12 4

4

200 samples

Single identifier: control law with ND = 2, alpha = 0.95

2

2 0

0

Co -2 ntr -4 ol

Co -2 ntr ol -4

-6 -8

-6

-10 -8

-12 -14 0

50

100

150

200 samples

250

300

350

-10 0

400

4.13

50

100

150

200 samples

250

300

350

400

4.14

Example 3. We consider again a third order linear piecewise time invariant plant but particularly severe initial conditions. The plant changes abruptly its parameters, and the ( two zeroes; three poles ) evolution is:

(0.3

0.6 ; 0.5 1 ± 2 j )

(0.23 (0.2

0.52; 0.47 0.77 ± 1.92 j )

0.5; 0.46 0.67 ± 1.85 j )

(0.165 (0.16

0 ≤ k < 100

0.485; 0.42 0.54 ± 1.8 j ) 0.44 ; 0.35 0.43 ± 1.8 j )

100 ≤ k < 200 200 ≤ k < 300 300 ≤ k < 400 400 ≤ k ≤ 500

There are thirty identifiers running in parallel. The first one is initialized in θˆ0(1)T = [1 1 1 0.2 1 1], which is an arbitrary vector with the aim of testing the multiestimation scheme in a highly severe condition. The other ones are initialized in the form:

θˆ0( m )T = θˆ0(1)T + m ⋅ [− 0.12 0.18 − 0.12 0.28 − 0.063 − 0.027] The reference model and its input are identical to those in Example 1. Results ( Figures 5.x )

16

350

400

8

Single identifier: output

4000

Single identifier: performance index

x 10

2 1.8

3000

200

1.6 2000

1.4

yk 1000

1.2 Ind ex 1

0

100 0 yk -100 -200

0.8

-300

0.6

-1000 -2000 -3000 0

100

200

300

400

0.4

-400

0.2

-500

0

500

0

100

200

samples

300

400

-600 0

500

100

200

samples

5.1

400

500

5.3

Thirty identifiers: switching map detail. ND = 2 and alpha = 0.95

30

300 samples

5.2

Thirty identifiers: switching map with ND = 2 and alpha = 0.95

30

Thirty identifiers: output with ND = 2 and alpha = 0.95

300

Thirty identifiers: switching map detail . ND = 2 and alpha = 0.95 20

25

19

25

18 ck

20

ck

20

15

15

10

10

5

5

ck 17 16 15 14 13

0

0

100

200

300

400

0

500

12 11 0

5

10

15

samples

5.4 5

x 10

7

20 samples

25

30

35

305

5.5

Thirty identifiers: performance index. ND = 2 , alpha = 0.95

20

3.5

x 10

315

320 325 samples

330

335

5.6

Thirty identifiers: output with ND = 2, alpha = 1

Thirty identifiers: switching map with ND = 2 and alpha = 1

30

3

6

25

2.5 5

2

ind4 ex

ck

1.5

yk

1

3

20

15

0.5 10

0

2

-0.5 1 0

310

5

-1 0

100

200

300

400

500

-1.5 0

100

200

samples

5.7 500

300

400

500

0

0

100

200

samples

5.8

Thirty identifiers: output with ND = 5 and alpha = 0.95

25

300

400

500

samples

5.9 6

Thirty identifiers: switching map with ND = 5 and alpha = 0.95

8

x 10

Thirty identifiers: performance index. ND = 5 and alpha = 0.95

7

0

20 6

-500

ck

yk

5 Ind de 4 x

15

-1000 10

3

-1500

-2500 0

2

5

-2000

1 100

200

300

400

500

0

0

100

200

300

400

500

0

0

100

200

300

samples

samples

samples

5.10

5.11

5.12

17

400

500

4

2

Thirty identifiers: control law with ND = 2 and alpha = 0.95

Single identifier: control law

x 10

1000 500

1.5

0

1

co ntr ol

-500

0.5

Co -1000 ntr ol -1500

0 -0.5

-2000

-1

-2500

-1.5 -2 0

-3000 100

200

300

400

-3500 0

500

100

200

300

samples

samples

5.13

5.14

400

500

Example 4. Consider now the time-varying plant whose parameters change with time in the following way:

θ RT ( k = 0 ) = [− 2.5 6 − 2.5 1 − 0.9 0.18 ] θ RT ( k ) = θ RT ( k = 0) + [− 0.05 0.12 − 0.05 0.02 − 0.018 0.0036]⋅ sin(kτ ) There are thirty identifiers running in parallel initialized in the way:

θˆ0(1)T = [− 2 4.5 − 2 0.8 − 0.7 0.15]

θˆ0( m )T = θˆ0(1)T + m ⋅ [− 0.033 0.0833 − 0.033 0.01 − 0.001 0.0017] The reference model is the same as those of the above examples and its input is a square wave. Results ( Figures 6.x ) 2

Thirty identifiers: output with ND = 2, alpha = 0.95

15

Thirty identifiers: switching map with ND = 2, alpha = 0.95

45

Thirty identifiers: index with ND = 2, alpha = 0.95

40

1.5

35

1 ck

0.5

10

30

yk

Ind25 ex

0

20 -0.5

5

15

-1

10

-1.5 -2 0

5 100

200

300

400

500

0

0

10

20

samples

30 40 samples

6.1

0

60

0

100

200

-1.8

90 2

400

500

6.3

Single identifier: index

100

300 samples

6.2

Single identifier: output

3

50

Real and estimated coefficient with ND = 2, alpha = 0.95

-1.9

80 -2 70

1

co -2.1 effi cie-2.2 nt

60 Ind ex 50

yk 0

40 -1

-2.3

30

-2.4

20

-2

-2.5

10 -3 0

100

200

300

400

500

0

0

100

200

300

400

500

-2.6 0

100

200

300

samples

samples

samples

6.4

6.5

6.6

18

400

500

B) Discussion Basically, there are two parameters which can be tuned to accurate the transient response of the adaptive system: the dwell time τ D ( or equivalently the dwell number of samples N D ) and the weight α ( β is determined from the relationship α + β = 1 ). If α is fixed to a constant value and change τ D , it can be seen that if τ D grows, then the performance index grows as well, giving therefore a worse performance. That is because the dwell time forces the system to keep the same controller active for that time interval avoiding potential switching to another controller which could achieve a better performance. Thus, it is recommended to choose the least dwell time available ( according with the stability considerations of Theorem 4 ) in order to achieve the best performance. This optimum dwell time is determined by the algorithm whose flux diagram is exposed in Section III.B. Now, if τ D ( or equivalently N D ) is fixed and α is modified, it is observed that there is a large difference between the values α = 1 ( no weight over the output jump in (4) since β = 0 ) and α < 1 ( there is a weight over the output jump ). The difference is more significant in Examples 2 and 3 where the plant parameters change abruptly. It is shown how the weight α acts either avoiding switching in the estimation scheme or making the system to switch between a sequence of controllers until it arrives to the best one in order to reduce the abrupt variation of the output. Thus, it is recommended to choose a value for α such that 0 < α < 1 and by computer simulations to get a fine tuning of α according to the specific requirements of each problem. Note that there is a positive lower bound of α as well. With the switching logic proposed, more relevance is given to the identification errors than to the output jumps, just to see which controller is the best one in order to control the system. Thus, it is recommended to choose α ∈ (0.5,1) , the finest tuning being elucidated, if desired, from the closed-loop behaviour. It is observed that an important transient response improvement can be achieved. First, the performance index proposed is lower in the multiestimation scheme than in the conventional adaptive control scheme and second, the plant output deviations from the desired output during the transient are lower in the multiestimation case as well. It can be concluded that the performance improvement is achieved in two ways: first in the particular case of smaller overshoot peaks in the transient output and second in the accumulated sense defined by the performance index. In Examples 2 and 3, the plant parameters change abruptly. Taking into account that an identification algorithm designed for time invariant plants is being used, the changes must be far enough one from each other. It is observed that the performance index of the multiestimation scheme is lower than the conventional adaptive control in spite of no switching occurs. This happens because the proposed control scheme has a supervision which resets the adaptation gain matrices when a change in the parameters, which causes at least one per cent of change of the supervisory indexes (4), is detected. Thus, after a change the multiestimation scheme with adaptive control is more capable to be accommodated to the new circumstances than the conventional adaptive control scheme, improving the transient response. It can be seen that the output tracks more quickly and accurately the desired output with the multiestimation scheme. In Example 4, the plant parameter vector changes sinusoidally around a constant vector. Such a change must be slow enough for a potential validity of the proposed method. In other words, the sinusoidal frequency is small enough because of the same reason presented above. It can be observed that a lower performance index is obtained by the use of the multiestimation control scheme compared to the use of a single estimation scheme. This improvement is achieved fundamentally during the transient, where switching occurs regularly. The above comments in the first paragraph about the dwell time and the weight α apply to the Examples 2-4 as well. 19

V

CONCLUSIONS

In this work, a multiestimation scheme for discrete adaptive control has been presented. The scheme has been proved to guarantee the closed-loop stability if the switchings between the various estimators are subject to a minimum dwell time, which can be estimated either from ‘a priori’ knowledge or through an ‘ad hoc’ on-line computation algorithm. It has been proved that a judicious choice of the switching rule allows the designer to obtain relevant improvements in the identification and closed-loop control performances. It can be concluded that the multiestimation scheme is an effective option for the development of high performance transient response adaptive controllers, specially for plants that are highly uncertain ( i.e. the compact set D where the real plant parameter vector belongs to is very large ). For such plants, an important transient response improvement can be achieved in two ways: the output overshoot peaks and the accumulated deviations from the desired output are smaller in this case than in the conventional adaptive control based scheme. The scheme also achieves an acceptable performance when applying to plants whose parameters change abruptly with time. In those cases, the multiestimation scheme gives a better performance than the conventional adaptive configuration not only because of the switching but also due to the supervision, which acts resetting the identification algorithms and the adaptation gain matrices. The proposed scheme reveals to be promising for the adaptive control of time-varying discrete systems as well, where a transient response improvement is also achieved. Simulations that corroborate the efficiency of the use of the multiestimation scheme in the cases of time-invariant and slowly time-varying plants are also included. A closed-loop performance quality index is introduced in order to compare the performances of the proposed scheme and the conventional adaptive control scheme. ACKNOWLEDGEMENTS The authors are very grateful to MCYT by its partial support of this work through grant DPI20000244. REFERENCES [1]

M. Fu and B. R. Barmish, Adaptive stabilization of linear systems via switching control, IEEE Transactions on Automatic Control, 31 (1986) no. 12, 1097-1103.

[2]

M. H. Chang and E. J. Davison. Adaptive switching control of LTI MIMO systems using a family of controllers approach, Automatica, 35 (1999) 453-465.

[3]

K. S. Narendra and J. Balakrishnan. Improving transient response of adaptive control systems using multiple models and switching, IEEE Transactions on Automatic Control, 39 (1994) no. 9, 1861-1866.

[4]

K. S. Narendra and J. Balakrishnan. Adaptive control using multiple models, IEEE Transactions on Automatic Control, 42 (1997) no. 2, 171-187.

[5]

J. Hocherman-Frommer, S. R. Kulkarni and P. J. Ramadge. Controller switching based on output prediction errors, IEEE Transactions on Automatic Control, 43 (1998) no. 5, 596607.

20

[6]

S. R. Kulkarni and P. J. Ramadge. Model and controller selection policies based on output prediction errors, IEEE Transactions on Automatic Control, 41 (1996) no. 11, 1594-1604.

[7]

A. S. Morse. Supervisory control of families of linear set-point controllers-Part 1: Exact matching, IEEE Transactions on Automatic Control, 41 (1996) no. 10, 1413-1431.

[8]

F. M. Pait and A. S. Morse. A cyclic switching strategy for parameter-adaptive control, IEEE Transactions on Automatic Control, 39 (1994) no. 6, 1172-1183.

[9]

D. Borrelli, A. S. Morse and E. Mosca. Discrete time supervisory control of families of two degrees of freedom linear set-point controllers, IEEE Transactions on Automatic Control, 44 (1999) no. 1, 178-182.

[10]

E. Mosca and T. Agnoloni. Inference of candidate loop perfomance and data filtering for switching supervisory control, Automatica, 37 (2001), 527-534.

[11]

K. J. Aström and B. Wittenmark. Computer-controlled Systems, Prentice-Hall (1990).

[12]

R. A. Horn and C. R. Johnson. Matrix Analysis, Cambridge University Press (1996).

[13]

K. Ogata. Discrete Time Control Systems, Prentice Hall (1996).

[14]

G. C. Goodwin and K. S. Sin. Adaptive Filtering and Control, Prentice-Hall Inc., Englewood Cliffs (1984).

21

APPENDIX A Stability of the auxiliary extended system Before proving Theorem 4, three technical lemmas needed forward for the proofs of the main result are established. Lemma A.1. Let A ∈ C n×n and ε > 0 be given. Thus, there is an induced matrix norm ⋅ such that

ρ ( A) ≤ A ≤ ρ ( A) + ε , where ρ ( A ) is the spectral radius of A. ( For the proof of this Lemma, see for example [12] ). Lemma A.2. ( Dwell time existence and upper bound calculation ) (i)

Let A ∈ C n×n be such that ρ ( A ) < 1 , i.e. A is a convergent matrix ( namely, stable in a discrete context ). Then, for every matrix norm ⋅ : C n×n → ℜ there are real constants a > 0 ( finite ) and 0 < b < 1 such that A k ≤ ab k ∀k > 0 .

(ii)

Let J ∈ C n×n

J1  be a Jordan block diagonal matrix J =    0

0   ∈ C n×n with   Jm

J2

1 0 λ   1 λ   ∈ C σ p ×σ p ∀p = 1,2,..., m and ∀λ ∈ σ (J ) with Jp =  1   λ   0 σ ( J ) = {λi }ir=1 being the spectrum of J and each eigenvalue is such that λi < 1 and has multiplicity ν i ∀i = 1,2,..., r . Then, for any matrix norm ⋅ : C n×n → ℜ there exist real finite constants a > 0 , 0 < b < 1 and a finite integer k 0 such that J k ≤ ab k ,

∀k ≥ k 0

and hence J k → 0 as k → ∞ . Proof. (i) Consider a real constant b such that 0 < ρ ( A ) < b < 1 ( we can do so because A is convergent ). Now we fix ε = b − ρ ( A) > 0 . By Lemma A.1, there is an induced matrix norm ⋅ α such that A α ≤ ρ ( A) + ε = b < 1 . Using matrix norm properties, A k

k

α

≤ A α ≤ bk

∀k > 0 .

Since C n×n is a finite dimensional vector space, all norms are equivalent and for any other matrix norm ⋅ β defined over C n×n there exists a finite constant a > 0 such that A β ≤ a A α ∀A ∈ C n×n and then A k

β

≤ a Ak

α

≤ ab k

∀k > 0

22

 J 1k  k We have that J =     0

(ii)

J

k 2

0   . Let evaluate J k for any i i   J rk 

0 0 ν i −1 k  k k     J ik = ∑  λki − Bi = ∑  λki − Bi ∀k ≥ 2ν i , where Bi =   =0   =0   0 0 following upper bound for the equation above is obtained:

1 0 0 1 0 0 0 0

0 0  , since Biν i = 0 . The  1 0

k  k  J ik ≤ ν i max  λki − Bi ≤ ν i M max  λki − ∀k ≥ 2ν i where M = max max Bi . 0≤ ≤ν i −1   0≤ ≤ν i −1   0≤ ≤ν i −1 0 ≤ i ≤ r k  k −ν +1 1−ν Since J is a convergent matrix, then J ik ≤ ν i M λi i max   . If it is set M = M λi i and 0≤ ≤ν i −1   k  k note that max   ≤ k ν i ∀k ≥ 2ν i one has, J ik ≤ ν i M λi k ν i ∀k ≥ 2ν i . If it is defined 0≤ ≤ν i −1   now ν = max{ν i , i = 1,2,..., r} and 1 > ρ ≥ max{ λi , i = 1,2,..., r} , one finally gets J ik ≤ νMρ k k ν

∀k ≥ 2ν y ∀i = 1,2,..., r

(A.1)

Since ρ k k ν → 0 exponentially as k → ∞ , there exist real finite constants a > 1 and 1 > b > ρ such that J k ≤ ab k ∀k ≥ k 0 from (A.1). Remark A.1. Lemma A.2( i ) is an easy-proof result that guarantees the existence of the dwell time and part (ii) allows the practical calculation of an upper-bound of it. Lemma A.3. Let be {Bi }iN=1 a family of square matrices such that the following assumptions hold: (i)

All matrices have the same eigenvalues { λi }in=1 with

(ii)

max 1≤ j ≤ N −1 B j +1 − B j ≤ δ with δ sufficiently small

λi < 1 ∀i = 1,2 ,..., n

Then, there exist real finite constants C > 0 , 0 < ρ < 1 and δ sufficiently small such that,

∏B

i

≤ Cρ N + o(δ )

1≤i ≤ N

Proof. If we consider for each matrix Bi its Jordan form J i we can write the finite matrix product

∏B

i

as:

1≤i ≤ N

23

∏B

i

1≤i ≤ N

≡ BN BN −1

B1 =

∏ [Q J Q ] i

i

−1 i

1≤i ≤ N

If Assumption (ii) holds, then Jordan forms of two consecutive matrices may differ slightly and non-singular transformation matrices would be very similar and so Qi−1Qi −1 ≈ I , i.e. we will be able to write that Qi−1Qi −1 = I + Pi where Pi is a perturbation matrix such that Pi ≤ ε i with 0 < ε i 0

≤ M e( i ) ∀i , k

and for each controller according to properties stated in Theorem 2 that ∃ M (pi ) > 0

p k(i )

≤ M p( i ) ∀i, k

Now, choose M e = max {M e( i ) } and M p = max {M p( i ) } so that θˆk( i ) 1≤i ≤ N e

1≤i ≤ N e

≤ Me

and

p k(i )

≤Mp

∀i, k . In this point, the solution of the matrix difference equation (6) corresponding to the linear auxiliary extended system is expanded. For this purpose, let us introduce the state transition matrix between the time samples k and k + N in the following way:

Ψ (k + N , k ) = p

k+



∑η 0 arbitrarily small there exists an integer k ≥ 0 large enough such that the differences aˆ (p ,jk) + − aˆ (p j,k) , bˆ p( ,jk)+ − bˆ p( ,jk) , s (p j,k) + − s (p ,jk) and rp( ,jk)+ − rp( ,jk) are less than or equal to µ ,

for

≥ 0 finite, in

particular when = 1 , i.e. when considering two consecutive terms of the sequence and so asymptotically Assumption ( ii ) from Lemma A.3 holds and the lemma can be applied here, where δ is related to µ .

26

If each controller parametrization is active during a minimum number of samples η min , such that ln C , all the products in the right-hand side of (A.6) involve Cρ ηmin < 1 , or equivalently η min > ln ρ k → 0 ( or for k finite and N → ∞ ), all the N above term tends to zero. The dwell time is selected as τ D = η minτ . Taking norms in (A.3) leads to, numbers less than unity and then as k, N → ∞ with

k + N −1

xk + ∑

x k + N ≤ Ψ (k + N , k )

j =k

(

( cj )

Ψ( k + N , j + 1) ϑ1 e j

)

+ ϑ 2Tu cj + o( δ )

(A.7)

and introducing bounds from equations (A.4) (A.5) and (A.6) in (A.7), one obtains for k finite,

xk + N ≤ C ρ

ηp

    η η ( j)  ∏ Cρ ν  C ρ 1 x k +  M T + sup e   1≤ν < p  k ≤ ≤k + N 1≤ j ≤ N e 

   M Ψ + o(δ )  

(A.8)

k + N −1   where M Ψ ( k , N ) = ∑  ∏ C ( cλ ) ρ ην  is a bounded constant which depends on k and N, with j = k  1≤ν < p j +1, k + N  p j +1,k + N being the number of switches between samples j + 1, and k + N. Now, with N → ∞ , applying Lemma A.4 to equation (A.8), from Theorems 1 and 2 and the dwell time constraint, it can be obtained that { xk } is a bounded sequence and lim k →∞ ek( j ) = 0 for all j = 1,2,..., N e , then the

state vector is bounded and the output of the plant and the control law as well. Then, the auxiliary system (6)-(7) is globally stable. Since the sequences {uk }k∞=1 ,{ y k }∞k =1 are bounded, then the closedloop system is globally stable and Theorem 3( i ) has been fully proved. APPENDIX B Perfect tracking asymptotic property Proof of Theorem 3( ii ). We can always write that y k = yˆ k + ek , i.e. y k = y mk +

Rk e Am k

(B.1)

by using (3) and the control law (5). It has to be proved that the second term of the right-hand side of equation (B.1), associated to the identification error tends to zero as k → ∞ . First, rewrite this 1 term as y ek = v k where v k = Rk ek = Rk ϕ kT θ~k −1 . Note that ν k does not depend explicitly on y k Am ~ since ϕ T θ ( ck ) depends on y , y ,... from ϕ 's definition with deg( R ) = n and deg( A ) = 2n . k

k −1

k −1

k −2

k

k

m

By choosing as state vector = [y e ,k y e ,k −2 n +1 ] , it follows that the above system (B.1) of output yek can be rewritten in state variables as: x kT

k

x k = Hx x −1 + βv k = H k x0 + ∑ H k − j βv j j =1

27

all k ≥ 1

(B.2)

where − a 2mn −1   1 H = 0    0

− a 0m  1   0   0 0 ; β =          0  0 

− a 2mn −2 0 1 0

2n

by using recursion with the model polynomial given by Am ( z ) = z 2 n + ∑ a 2mn −i z 2 n −i . Since H is a i =1

constant matrix whose eigenvalues are the roots of the polynomial Am ( that are stables by Assumption A.2 ) then, there are finite constants C ≥ 1 and ρ < 1 such that H ≤ Cρ and for H k powers , according to Lemma A.2, we will have that H k ≤ Cρ k subject to the given constraint for the dwell ( or residence ) time. Taking norms in equation (B.2), one gets: k

x k ≤ Cρ x0 + ∑ Cρ k

k− j

j =1

ν j ≤ Cρ x0 + C sup ν k

1≤ ≤ k

k

∑ρ

k− j

1− ρk 1− ρ

≤ Cρ x 0 + C sup ν k

1≤ ≤ k

j =1

(B.3) k

since

∑ ρ k− j = j =1

k

1− ρ . 1− ρ

Now, since v = R e

,

sup ν

≤ ( n + 1 )M p sup e (ji ) so that for any sample k 0 ≥ 0 : 1≤i ≤ N e 0≤ j ≤ k

1≤ ≤ k

x k ≤ Cρ

k − k0

x k0

1 − ρ k − k0 +C ( n + 1 )M p 1− ρ

sup

e (j i )

1≤i ≤ N e k0 − n ≤ j ≤ k

k0 → 0 , it follows from Theorems 1 and 3( i ) that there exists a k bounded convergent sequence {mk }∞k =0 mk ≥ 0 with mk → 0 for k → ∞ such Then, as k , k 0 → ∞

that

sup

with

e (j i ) ≤ mk . Then, there exists a bounded convergent sequence { d k }k∞=0

d k ≥ 0 with

1≤i ≤ N e k0 − n ≤ j ≤ k

d k → 0 for k → ∞ which bounds the state vector norm, namely, x k ≤ d k → 0 as k → ∞ . Thus, ek = ϕ kT θ~k → 0 as k → ∞ since ϕ k is bounded for all k ≥ 0 from Theorem 3( i ), so that Rk e → 0 as k → ∞ for any bounded initial conditions since Am is stable and Rk is bounded Am k from Theorem 2. Thus, Theorem 3( ii ) has been proved since y k → y mk as k → ∞ from (B.1).

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