resulting in the reference model transfer function being. ( ). ( ). ( ). ( ). ( )z. Az. A ...... [28] T.M. Apostol, Mathematical Analysis, 2nd Edition, Addison- Wesley Co., ...
MODEL MATCHING VIA MULTIRATE SAMPLING WITH FAST SAMPLED INPUT GUARANTEEING THE STABILITY OF THE PLANT ZEROS. EXTENSIONS TO ADAPTIVE CONTROL
M. De la Sen and S. Alonso-Quesada Department of Electricity and Electronics. Faculty of Science and Technology University of Basque Country. Campus of Leioa. Aptdo. 644- Bilbao 48080- Bilbao. SPAIN
Abstract. This paper investigates the model matching problem in discrete systems with free- design zeros of the reference model for any fractional order-hold and a wide set of admissible values of the running sampling period used in the discretization process of a linear and time-invariant single-input single-output continuous-time plant. For such a purpose, a fast input sampling rate at sufficiently small sampling period is used with an appropriate set of scalar gains to generate them within the slow running sampling period. This makes possible to stabilize the discrete plant zeros what allows their cancellation in the model matching problem synthesis irrespective of either the absence of continuous-time zeros or the presence of continuous-time critically stable or unstable ones. The design philosophy is extended to model matching based adaptive control schemes by using appropriate partitions of the parameter space so that a finite number of sets of scalar gains is precalculated to be stored and then used to online accommodate the numerator of the estimated transfer functions so that it might be a stable time-varying polynomial with no common factors with the estimated plant poles.
Keywords: Adaptive control, fractional-order holds, model matching, pole-placement, multirate sampling 1. Introduction It is well- known that the presence of unstable plant zeros is an important drawback in the synthesis of model- matching based controllers in both the non-adaptive and adaptive cases since such zeros have to be cancelled by the controller if they are not present as zeros of the reference model, [1-15]. The requirement of the stability of the plant zeros prior to their cancellation or the need to transmit them to the reference model may be overcome in model- matching based schemes by using alternative techniques like, for instance, either estimation modification or sliding-mode based controllers (see, for instance, [6] and [8-13]) or with the use of multirate control, [16-19]. Alternatively, the reference model has to be constrained to possess the unstable plant zeros, if any. In the discrete case, the use of fractional or generalized sampling and hold devices might improve the stability of such zeros related to the use of the standard zero-order hold, [19-23]. It is well-known that multirate sampling allows the accommodation of the various sampling rates to the real needs for each
1
sampled variable in discrete systems while it may be used for noise filtering purposes if suited, [1619]. The filtering design issues are performed by taking advantage of the fact that usually the noise spectrum and the unmodeled dynamics contributions are out of the range of the dominant frequencies of interest in the closed-loop behaviour. The ratio of the various sampling rates combined with a selection of an “ad-hoc” set of scalar gains has been proposed recently in [24] as a useful tool for the achievement of stable discretized zeros under rather weak structural stabilizability conditions. Such a philosophy is very relevant of the synthesis of model matching based controllers in both nonadaptive and adaptive designs since all the zeros of the reference model may be fixed arbitrarily because all the discrete plant zeros, since stable, may be cancelled if suited. It should be pointed out that, except in particular cases, critically stable discretization zeros appear and also unstable or critically stable intrinsic discrete zeros may appear for ranges of sampling rates usual in practical designs. This problem can be partly solved by using either Fractional- Order Holds of appropriate correcting gains β ∈ [ −1, 1 ] ( β -FROH) or Generalized Sampled- Data Hold Functions, [19-24]. Note, in particular, that the set of β -FROH ´s include the popular Zero-Order Hold ( β = 0) used in most designs and the First-Order Hold ( β =1) as particular cases. The main objectives of this paper are: 1)
To obtain an appropriate input-output model for multirate sampling with two sampling periods with fast sampled input of single-input single-output continuous linear time-invariant plants, for any β -FROH, and any running sampling period (i.e. the slow sampling rate operating the output) used for discretization.
2)
To select the set of scalar gains used to generate the set of fast sampled inputs in-between the running sampling period from the basic value generated by the controller at each (running) sampling instant so that:
a) All the discrete plant zeros are stable while being zero/pole cancellation free. b) The closed-loop model matching problem may be solved in a suitable way for any arbitrary prefixed stable reference model (including full arbitrariness in the selection of its zeros).
3)
If the plant is unknown with only slight “a priori” knowledge on its parametrization structure then the above Points 1-2 may be extended by incorporating an adaptive scheme for any β -FROH used for discretization.
The paper is organized as follows. Section 2 formulates a discrete state-space description under fast input sampling to then obtain an input-output discrete transfer function for the running slow sampling rate, namely, that acting on the state/ output signals. The selection of the scalar gains that generate the fast sampled input so that the discrete plant zeros are stable is focused on depending on the continuous- time plant parametrization in Section 3. Section 4 discusses the synthesis of a model-
2
matching based controller with a possible potential free design of all the zeros of the reference model. Section 5 extends the proposed design to not fully known plants via use of adaptive control. Simulated examples which highlight the proposed design philosophy are provided in Section 6 and, finally, conclusions end the paper.
2.Discrete state-space description and ARMA model Assume the continuous-time plant:
S c : x& ( t ) = A x ( t ) + b u ( t ) ; y ( t ) = c T x ( t )
(1)
where x(t) is the n- state vector with initial condition x (0) = x 0 , u (t) and y(t) are the scalar input and output, and A∈R n × n , b∈ R and c ∈ R parametrize the system. The transfer function of S c is
G c ( s )= c T( s I − A ) − 1 b =
c T Adj ( s I − A ) b Det ( s I − A )
of order n (i.e. the number of poles), m ≤ n −1 zeros and relative order (i.e. pole-zero excess) n r = n – m, where Adj (.) and Det (.) stand for the Adjoint Matrix and Determinant of the square matrix (.). Assume that T is the running sampling period acting on the output which parametrizes the discrete system while T ´= T / N is the input sampling period for some integer N ≥ 1 . If N ≥ 2 then the discrete system is a multirate sampling system with fast input sampling period T ´. In the following, any finite subset of the set of positive integers {1, 2 , ... , r } is denoted by r . If N ≥ 2 and the input to the discrete plant is generated with sampling rate T ´= T / N via a β -FROH involving a signal reconstruction over the fast sampling intervals then such an input becomes:
u(t )= u
j
(kT )+
( ( k T ) − u ( k T ) ) ⎛⎜⎜ t − ⎛⎜ k + jN− 1 ⎞⎟ T ⎞⎟⎟
βN u T
j −1
j
⎡⎛ j −1 ⎞ ⎛ j ⎞ ∀ t∈⎢⎜ k+ ⎟T ,⎜ k + ⎟T N ⎠ ⎝ N⎠ ⎣⎝ u
j
( k T ) : = u ⎛⎜ k T + ( j −1)T ´ ⎝
⎝
⎝
⎠
(2)
⎠
⎞ ⎟⎟ , j∈ N , where the plant input at the fast sampled instants are : ⎠
⎞ ⎟=u ⎠
[( k N + j −1 )T ´ ] = u ⎡⎢ ⎛⎜⎝ k + jN−1 ⎞⎟⎠ T⎤⎥ = α ⎣
⎦
j
u k ( j∈ N ) (3)
for some set α i ∈ R ( i ∈ N ) subject to the constraint:
(
)
⎡⎛ N −1 ⎞ ⎤ u 0 ( k T ) : = u k T − T ´ = u N [ ( k −1 ) T ] = u ⎢ ⎜ k −1 + ⎟ T = α N u k −1 N ⎠ ⎥⎦ ⎣⎝ The auxiliary real sequence
{u }
Note that u 1( kT ) = α 1 u ( kT ) ≠ u
k
k
∞ 0
will be then generated by the synthesized discrete controller.
unless α 1 = 1 .
3
Remarks. 1. The notation u
j
( k T ) = u [ k T + ( j −1)T ´ ] = α j u k ,
proposed in [24], facilitates the
problem description for the fast sampling rate T ´ = T / N , within the slow running sampling rate T, to
{ }
quantify the input value at the fast sampling instants from the real sequence u k
∞ 0
via the set of
gains α i ∈ R ( i ∈ N ).
(
2. Note that u 0 ( k T ) = u k T − T
´
) = u [ ( k − 1 ) T ]= u N
⎡⎛ N−1 ⎞ ⎤ ⎢ ⎜ k − 1 + N ⎟ T ⎥ = α N u k − 1 is useful ⎠ ⎦ ⎣⎝
[
to describe the input reconstruction within the fast sampling intervals k T , k T + T ´
) when a β -
FROH is used with β ≠ 0 .
3. If a Zero-Order Hold (ZOH) is used (i.e. β = 0) then u ( t ) = u
[
j
( k T ) = u ( k T + ( j −1 )T ´ )= α
ju k
;
)
∀ t ∈ k T + ( j −1 ) T ´ , k T + j T ´ , ∀ j ∈ N with u ( k T ) = u 1 ( k T ) = α 1 u k .
4. The auxiliary input sequence will be used to store the sampled values generated by the controller to then calculate the plant input u ( t ) via (2)-(3) within each running (slow) sampling interval
[ k T , ( k +1) T )
from the set u
j
( k T ) = u ( k T + ( j −1) T ´ ) = α j u k
; α j ∈ R , ∀ j ∈ N and the given
β -FROH.
As pointed out before, u ( k T ) ≠ u k if α 1 ≠ 1 but, since any polynomial may be made monic via appropriate normalization by its leading coefficient while maintaining identical all its zeros, α 1 may be fixed to unity if suited without loosing the essential features of the design methodology described in the sequel.
2.1 Discrete plant One gets from (1) the following state- trajectory:
[
]
⎛ j −1 ⎞ x k T + ( j −1 )T ´ + τ = Φ ⎜ T+ τ ⎟ x (kT )+ N ⎝ ⎠
⎛ j −1 ⎞ T + τ⎟ x ( k T ) + =Φ ⎜ ⎝ N ⎠ l =1
T N
j
∑∫
[
0
∫
j −1 ⎞ ⎛ ⎜ k+ ⎟T + τ N ⎠ ⎝ kT
⎡⎛ ⎤ j −1 ⎞ Φ ⎢⎜ k + ⎟ T + τ − s⎥ b u ( s )d s N ⎠ ⎣⎝ ⎦
⎤ l −1 ⎞ ⎞ ⎡⎛ ⎛ j− l ⎞ ⎛ j− l T + τ−s ⎟ U ⎜ T + τ − s ⎟b u ⎢⎜ k + Φ⎜ ⎟ T + s⎥ d s N ⎠ ⎠ ⎣⎝ ⎝ N ⎠ ⎝ N ⎦ (4)
)
∀ τ ∈ 0 , T ´ , all integer k ≥ 0 , ∀ j∈ N , where U (t) = 1 (t) is the (unity step) Heaviside function. The substitution of (2)-(3) into (4) yields:
[
]
⎧ ⎛ j −1 ⎞ x k T + ( j −1 ) T ´ + τ = Φ ( τ ) ⎨ Φ ⎜ T ⎟ x (kT )+ ⎠ ⎩ ⎝ N
4
j −1
∑ l =1
⎡ αl ⎢ ⎢⎣
∫
T N 0
⎛ j− l ⎞ ⎤ T − s ⎟ ds ⎥ b u k Φ⎜ ⎝ N ⎠ ⎥⎦
+
j −1 β N ⎡ ⎛⎜ ⎢ T ⎢ l = 1 ⎜⎝ ⎣
+
βN ⎛ ⎜ T ⎝
∑ ∫
∫
T N 0
⎞ ⎛ j− l ⎞ Φ⎜ T − s⎟ s ds⎟ b v ⎟ ⎝ N ⎠ ⎠
⎡⎛ l ⎞ ⎤ ⎤ ⎪⎫ ⎛ ⎢ ⎜k + N ⎟T ⎥ ⎥ ⎬ + α j ⎜ ⎝ ⎪ ⎝ ⎠ ⎣ ⎦⎦⎭
⎞
τ
∫ Φ ( τ − s ) d s ⎟⎠ b u 0
⎡⎛ j −1 ⎞ ⎤ Φ ( τ − s ) s d s ⎞⎟ b v ⎢ ⎜ k + ⎟T N ⎠ ⎥⎦ ⎠ ⎣⎝
τ 0
k
(5)
where Φ ( t ) = e A t for all t
(
)
α l − α l − 1 u k if N ≥ l ≥ 2 ⎧ ⎡⎛ l ⎞ ⎤ ⎪ v⎢⎜ k+ ⎡⎛ N −1 ⎞ ⎤ ⎟T = ⎨ ⎟ T = α 1 u k − α N u k − 1 if l =1 N ⎠ ⎥⎦ ⎪ α 1 u k − u ⎢ ⎜ k −1 + ⎣⎝ N ⎠ ⎥⎦ ⎣⎝ ⎩
so that if x k ( y k ) is used as an abbreviated notation for x ( k T ) ( y ( k T ) ) , one gets from (5) and (1): ⎧⎪ ⎡⎛ ⎤ j −1 ⎞ x ⎢⎜ k + ⎟ T + τ ⎥ = Φ ( τ ) ⎨Ψ N ⎠ ⎪⎩ ⎣⎝ ⎦ +
j −1
⎡ j−1 βN ⎛⎜ α − α Φ( τ ) ⎢ l ⎝ T ⎣⎢ l = 2
∑ (
j −1
x k+
∑α
l
l =1
l −1
)Ψ
j − l −1
j − l −1
Ψ
Γ ´ u k +Ψ
Γu
j− 2
k
⎫⎪ ⎬ + α j Γ( τ ) u k ⎪⎭
(
Γ ´ α 1 u k− α
)
⎤⎞ N u k −1 ⎥ ⎟ ⎦⎠
⎫⎪ ⎬ ⎪⎭
(6.a) ⎡⎛ ⎤ ⎡⎛ ⎤ j −1 ⎞ j −1 ⎞ T y ⎢⎜ k + ⎟T + τ⎥ ⎟ T + τ ⎥ = c x ⎢⎜ k + N ⎠ N ⎠ ⎣⎝ ⎦ ⎣⎝ ⎦
(6.b)
where Φ( τ )=e
Aτ
; Γ (τ ) = ∫ e A ( τ − s ) b d s ; Γ ´ ( τ ) = ⎛⎜ ∫ e A ( τ − s ) s d s ⎞⎟ b ; τ
τ
⎝
0
⎠
0
AT j N
⎛ j ⎞ T⎟=e Ψ j =Φ ⎜ ⎝ N ⎠ ⎡ T/N ⎛ T ⎞ ⎤ Γ≡ Γ T ´ = ⎢ Φ⎜ − s ⎟ d s⎥ b 0 N ⎝ ⎠ ⎦ ⎣
( ) ∫
[
;
Γ ´≡ Γ
´
( T )= ⎡⎢ ∫ ⎣ ´
T/N 0
⎤ ⎛ T ⎞ Φ ⎜ − s ⎟ s d s⎥ b N ⎝ ⎠ ⎦
]
for all τ ∈ 0 , T ´ , ( j ∈ N ) since T ´= T / N. 2.2 ARMA model for the ZOH ( β =0) One gets from (6) by taking β =0, j = N and τ = T ´ : x
k +1 = Ψ
N
xk+
N
∑ l =1
y
k
=cTΨ
N
α lΨ
N−l
Γu k
(7)
x k + y fk
(8)
5
y fk = c T x k
]
x
0
=0
⎛ =⎜ ⎜ ⎝
N
∑
α
l =1
l
cTΨ
N−l
⎞ ⎡ N Γ⎟ u k = ⎢ ⎟ ⎠ ⎣⎢ l = 1
∑
(
α l c T qI − Ψ
N
)
−1
Ψ
N−l
⎤ Γ⎥ u ⎦⎥
k
(9) where y fk = y k ] x 0 = 0 is the forced response to zero initial conditions with q being the (slow) running sampling rate advance operator run by the nonnegative integer k ; i.e. σ k + 1 = q σ k for any sampled signal σ k = σ ( k T ) . Note that (9) is a deterministic input-output ARMA model describing the discretized system at the running sampling instants which can be rewritten equivalently through the parametrization:
(
S 0 d : A 0 ( q ) y k = B ´0
T
)
(
(q) g u k = B
´ T 0
)
(q) g u k = B 0 ( q ) u k
(10)
provided that x 0 = 0 with associate discrete transfer function G β ( z ) : = G 0 ( z ) = n β = n 0 = m 0 + 1 = n poles and m
B0(z ) of A 0(z )
= m 0 = n −1 zeros (since β = 0 and the continuous transfer
β
function is strictly proper), where
(
A 0 ( q ) = Det q I − Ψ
) =q +∑ a n
N
n
i
q n −i
i =1
B 0 ( q ) = Β ´0
B ´0
T
T
( q ) g = B ´0 ( q ) g T
( q ) = [ B 0 1 ( q ), B 0 2 ( q ), ... , B 0 N ( q ) ] ;
n { ⎛ ⎞ ⎜ ⎟ T T T g = ⎜ g , ..... , g ⎟ ; g ⎜ ⎟ ⎝ ⎠
(
T
T
( q ) = [ B 0T, n −1 , q B 0T, n − 2 , ... , q n −1 B 0T0 ]
= ( α 1 , ....., α N )
)
B 0 l ( q ) = c T Adj q I − Ψ N Ψ N − l Γ =
[
B ´0
n −1
∑b( i=0
0 li
)
q n −1 − i ; l ∈ N
]
B 0 Tl = b 1( l0 ) , .... , b (N0l) ; l ∈ m 0 ∪ { 0 }
(11)
The discrete transfer function is of unity relative order and it might possess zero-pole cancellations if the continuous transfer function is not cancellation-free (i.e. if the continuoustime state-space realization S c is not minimal) and /or if the sampling periods are chosen so that the discrete model losses its controllability. The discrete transfer function possesses m 0 = n-1 zeros of which m are intrinsic zeros associated with the discretization of the continuous ones while the remaining m 0 − m = n − m −1 (if any) are discretization zeros being associated with the discretization process of the continuous plant, [19-24].
6
2.3 ARMA model for the β-FROH ( β ≠ 0) Now, Eq. 9 becomes replaced with: ⎡ N y fk = ⎢ ⎢⎣ l =1
∑
+
(
α l c T qI − Ψ
(
βN T c qI − Ψ T
N
)
N
)
−1
⎤ Γ⎥ u ⎥⎦
N−l
Ψ
⎧ N αl−α ⎨ ⎪⎩ l = 2
−1 ⎪
∑(
l −1
k
)Ψ
N−l
⎛ 1⎞ + ⎜⎜ α 1 − α N ⎟⎟ Ψ q⎠ ⎝
N −1
⎫⎪ ´ ⎬Γ u ⎪⎭
k
(12) which leads to the subsequent ARMA model for x 0 = 0 : ⎡ S β d : A 0 ( q ) y k = ⎢ B ´0 ⎣
T
(q) g +
βN ´ T βN ⎤ B β (q )g ´⎥ u k − α T T ⎦
N
B
´ β1
( q ) u k −1
(13)
or, equivalently, S βd : A β ( q )y
k
= B β ( q )u
(14)
k
whose associate discrete transfer function is G β ( z ) =
Bβ(z )
A β(z )
=
Bβ(z )
z A 0(z )
, where B ´0
T
( q ) is
defined as in (11), B ´β ( q ) g ´ = B ´β ( q ) g ´ with T
T
⎡ A β ( q ) = q A 0 (q ) ; B β ( q ) = q ⎢ B ´0 ⎣
B ´β
T
( q ) = [ B β´ 1 , B β´2 , ... , B β´, N ] ;
g
´T
(
(q) g +
B ´β
n { ⎛ ⎞ ⎜ ⎟ ´ T ´ T ´ T = ⎜ g , ..... , g ⎟ an g ⎜ ⎟ ⎝ ⎠
with
T
´T β
( q ) g ´ ⎤⎥ − ⎦
βN α T
N
B
´ β1
(q )
( q ) = [ B β´ T, n − 1 , q B ´βT, n − 2 , ... , q n −1 B ´βT0 ]
extended
vector
with
repeated
components
)
= α 1 , α 2 − α 1 , ..., , α N − α N −1 , and
(
)
B ´β l ( q ) = c T Adj q I − Ψ N Ψ N − l Γ ´ = B
T
βN B T
´T βl=
[b
´( β 1l
)
n −1
∑b i=0
´( β li
)
q n − 1− i ; l ∈ N
]
, .... , b ´N( lβ ) ; l ∈ m 0 ∪ { 0 }
(15)
Note that the transfer of S β d for β ≠ 0 possesses n β = m β +1= n +1= m 0 + 2 discrete poles (of which at least one being located at the origin) and m
β
= n discrete zeros (since β ≠ 0) of which
m are intrinsic zeros while n-m ≥ 1 (since the continuous transfer function is strictly proper) are
7
of
discretization ones, [20-24]. Note that the model (14)-(15) generalizes the well-known fact that if N=1 (i.e. single rate sampling is used) and a β- FROH is used with β ≠ 0 then a single discretization pole at the origin appears irrespective of the sampling period used. Note that the total number of discrete zeros m β = n-1 if β = 0 and m β = n otherwise, irrespective of the number m of zeros of the continuous plant. The following result follows trivially from the ARMA models (10)-(11) and (14)-(15):
(
)
Theorem 1. Assume that N ≥ N * β : = Max 2 , m β . Then, the polynomial of zeros of S β d is
ρ-Schur-stable (i.e. all its zeros are located in the interior of the open circle of radius 0 < ρ ≤ 1 in the complex plane) if and only if B β ( z ) ≠ 0 for all complex z satisfying z ≥ ρ for the given
β- FROH correcting gain β∈ [ −1, 1 ] , running sampling period T and sampling ratio N. An equivalent condition is that B β ( z
ρ
) ≠0
for all complex z ρ : = z / ρ with z
ρ
≥ 1 .The discrete
plant zeros are stabilizable via multirate sampling with fast input sampling for the given β and T if and only if there is a N-real vector g = ( α 1 , α 2 , ... , α
N
) T of gains generating the fast input
sequence within the slow running sampling period such that B β ( z ) = 0 if and only if z < ρ , or equivalently, B β ( z
ρ
) = 0 if and only if
z ρ m β = n β − 1 . Finally, note that for strictly proper first-order continuous-time systems, no specific benefit can be obtained from the use of multirate sampling with fast input sampling and ZOH since no discrete zero exists, [29].
8
However, the use of β-FROH of appropriate correcting nonzero gains might stabilize the appearing single discrete zero which is not present in β = 0. The next Section discusses the existence and selection of sets α
i
( i∈ N )
of gains which guarantee the stabilization of the
discrete zeros under rather weak structural conditions on the continuous plant parametrization, the running sampling period T and the sampling ratio N. 3. Existence and selection of valid vectors of α i -gains for achievement of stable plant discrete zeros
3.1 Existence of α i -gains stabilizing the discrete zeros The direct identity below in the indeterminate q holds for any polynomial vectors µ and ν of real coefficients of dimension being identical to the order δ of the real square matrix ∆, [27]:
⎡ z I − ∆ ν ( q )⎤ µ T Adj ( z I − ∆ ) ν = Det ⎢ T 0 ⎥⎦ ⎣− µ (q )
(
) zero/ pole cancellations if δ
Such an expression includes δ − δ ´
´
= deg [ Det ( z I − ∆ ) ] < δ .
Using this relation in (10) for β = 0, and (13)-(14) for β ≠ 0, one gets directly :
⎡ M 0 ( q ) M ´β ( q ) ⎤ B β ( q ) = Det ⎢ ⎥ T 0 ⎣ −c ⎦
(16)
where M
0
( q )= ( q I − Ψ N )
; f
β
( q ) = ⎧⎨
⎩1
βN ⎡ M ´β( q ) = f β (q ) ⎢ C Γ g + C T ⎣
⎧ = f β (q) ⎨ Ψ ⎩ − ⎧⎪ = f β (q ) ⎨ ⎪⎩
N −1
∑
l =1
, Γ+
⎤ β N N −1 ´ Ψ Γ αN g´ ⎥ − T ⎦
∑
N−1
N− l
Γ ´α
N −1
N− l
⎤ ⎫ ⎡⎛ βN ´ ⎞ βN ´ ⎢ ⎜ Γ + T Γ ⎟ α l − T Γ α l −1 ⎥ ⎬ ⎠ ⎦ ⎭ ⎣⎝
N
βN ´⎞ βN ⎛ Ψ Γ ⎟− ⎜Γ + T T ⎝ ⎠ Γ´ α
βN ´⎞ Γ ⎟,Ψ ⎜Γ + T ⎝ ⎠
N −1 ⎛
´
N
βN Ψ T
− ⎡ C e β = ⎢Ψ ⎣
⎡ ⎢Ψ ⎣
Γ
βN ´⎞ ⎛ Γ ⎟ α1 + Ψ ⎜ Γ+ T ⎝ ⎠ l=2
βN Ψ T
N −1
if β ≠ 0 if β = 0
q
N
N −2
= f β( q ) C
eβ
N − l −1
⎤ βN ⎛ ⎞ Γ ´⎥ α l +⎜Γ + Γ´⎟ αN T ⎝ ⎠ ⎦
g - Z ´β α N
βN ´⎞ βN ⎛ Γ ⎟− Ψ ⎜Γ + T T ⎝ ⎠
βN ´ ⎤ βN Ψ Γ ⎥ ; Z ´β = T T ⎦
N−1
⎫⎪ ⎬ ⎪⎭
Γ´
9
N −3
βN ´⎞ βN ´ ⎛ Γ ´, ... , Ψ ⎜ Γ + Γ ⎟− Γ T ⎝ ⎠ T
(
C Γ = Ψ N −1 , Ψ
N−2
)
, .... , I Γ ; C
Γ´
(
= Ψ N −1 , Ψ
N−2
)
, .... , I Γ ´ (17)
Note that the above relations (16)-(17) imply that, f β( q ) C e β g − Z ´β α N ⎤ ´ ⎥ , Z 0 = 0; C Γ = Z 0 ⎦
⎡ qI − Ψ N B β ( q ) = Det ⎢ T ⎣ −c
T 0
, and
M ´0 = C Γ g = Z T0 g . The following result follows directly from (16) subject to (17):
Theorem 2. The subsequent items hold: (i) A necessary condition for Schur- stabilizability of the discrete zeros of S β d for any β-FROH is that the continuous transfer function G (s) has no unstable or critically stable zero-pole cancellations, i.e. the state-space realization S c eq. 1 of the continuous-time system is stabilizable and detectable. A sufficient condition is that (A, b, c) be a controllable and observable triple so that the state-space realization of S c is minimal so that its associate transfer function G (s) is zero/pole cancellation-free.
(ii) A necessary condition for the existence of g = ( α 1 , α 2 , ... , α N
) T ∈ R N ( ∀ N ≥ N * 0 ) such
(
that S 0 d possesses a Schur-stable polynomial B 0 ( z ) is that the discrete pairs Ψ
(Ψ
N
T
,c
) be both stabilizable, for the given (T, N). If ( Ψ
necessary condition for B 0 ( z )
) to be Schur-stable is that the pair ( c N
N
)
, Γ and
, Γ is a controllable pair then a T
,Ψ
N
) be detectable, for
the given (T, N). A sufficient condition for B 0 ( z ) to be Schur-stable for some g ∈ R pairs
( Ψ , Γ ) and
(Ψ
N
, cT
is that the discrete
) be controllable and detectable, respectively, for the given (T,
N). A necessary and sufficient condition for B 0 ( z )
⎡ zI −Ψ N rank ⎢ T ⎣ −c
N *0
to be Schur stable is that
CΓg⎤ N ⎥ = n + 1 for some g ∈ R , all complex z with z ≥ 1 , and all 0 ⎦
integer N ≥ N * 0 . (iii) A sufficient condition for B 0 ( z ) to be Schur-stable for some g ∈ R N ( ∀ N ≥ N * 0 ) is that (A, b, c) be a controllable and observable triple and the
10
pair (T, N) satisfy
Im ( λ i − λ j )≠
2k π N ´ for both N ´ =1 and N ´ = N ≥ N * 0 , any integer k, and all complex T
conjugate pairs of eigenvalues of A satisfying Re λ i = Re λ j .
Proof. (i) The necessary condition follows since any cancellation in the continuous transfer function is transmitted to the discrete one for any β-FROH. Since the discrete pole associated to an unstable (critically stable) continuous one is unstable (critically stable), i.e. it lies outside the open unit circle, the discrete cancellation is still unstable (critically stable). The sufficiency-type condition guaranteeing the above necessary one is obvious.
(ii) To simplify the proof, first assume that N ≥ n . The necessity is proved proceeding by contradiction by using appropriate versions of Popov- Belevitch-Hautus test for controllability (stabilizability)/observability (detectability) as follows [27]. Since
⎡ zI − Ψ N B 0 ( z ) = Det ⎢ T ⎣ −c
CΓg⎤ ⎥ from (16)-(17), then B 0 ( z ) = 0 for some complex z with 0 ⎦
[
z ≥ 1 , so that it is not Schur-stable, if rank ζ I − Ψ
(Ψ
of Ψ N satisfying ζ ≥ 1 (i.e. if
(Ψ
N
,cT
[
)
rank z I − Ψ
N
T
,c
T
]
, c ≤ n − 1 for some eigenvalue ζ
) is not stabilizable which is equivalent to
not being detectable) or , if N
]
[
]
⎛ ⎡I , C Γ g = rank ⎜⎜ q I − Ψ N , C Γ ⎢ n ⎣0 ⎝
⎛ ≤ Min ⎜⎜ rank z I − Ψ ⎝
[
[
rank ξ I − Ψ
N
N
,C
], rank ⎡⎢ I0
0⎤ g ⎥⎦
⎣
]
⎞ ⎟ ≤ n −1 ⎟ ⎠
detectability c T , Ψ
N
(Ψ
⎡ zI −Ψ N ⎣
g
if
Γ
)
rank ( C Γ ) < n ; i.e. if and only if the pair
(
, Γ ) controllable ⇒ Ψ N , C
−c
T
Γ
)
stabilizable and then
CΓg⎤ ⎥ has full rank equalizing (n+1) for all complex 0 ⎦
[
number z with z ≥ 1 for infinitely many values of g if rank z I − Ψ
(
eigenvalue z of Ψ N with z ≥ 1 (i.e. c T , Ψ
,Γ
any
) is a necessary condition for Schur- stabilizability of zeros . To prove the
sufficiency part, note that ⎢
(Ψ
for
, C Γ ≤ n − 1 some eigenvalue ξ of Ψ N satisfying ξ ≥ 1 , i.e. if ( Ψ N , C
, Γ ) is not controllable, thus,
(
⎞ ⎟ ⎟ ⎠
0⎤ g ⎥⎦ n
Γ
is not stabilizable but this can hold if and only if
(Ψ
N
N
) is
N
]
, c = n for all complex
detectable) provided that rank [ C Γ ] = n (i.e.
) is controllable). The proof has been performed for N ≥ n . If N =
11
T
N * 0 = n-1, then the proof
remains valid since deg
( B 0 (z ) ) = m 0 = n − 1 implies that the polynomial
B 0 ( z ) possesses n real
coefficients of which only (n-1) are sufficient to prefix its zeros so that one of the components of g might be fixed arbitrarily. Also, it remains valid for all integer N ≥ N * 0 provided the rank condition holds for N = N * 0 from the structure of the involved matrix. The final necessary and sufficient
⎡ zI −Ψ N condition is obvious since rank ⎢ T ⎣ −c
CΓg⎤ N ⎥ = n + 1 for some g ∈ R with N = N * 0 0 ⎦
and all complex number z with z ≥ 1 ⇔ B 0 (z ) ≠ 0 for all z ≥ 1 and such a g ∈ R N which is the stabilizing vector of fast sampled input gains. If a g ∈ R
N
exists then a g ∈ R N
*0
always trivially exists for each integer N ≥ N * 0 .
(
)
(iii) (A, B, c) controllable and observable and Im λ i − λ j ≠ for all any integer k,
if Re λ i = Re λ j imply that
2k π N ´ for N ´ =1 and N ´ = N , T
(c
( Ψ , Γ ) and
T
,Ψ
) are controllable and
N
observable pairs, respectively, and then stabilizable and detectable as well since no hidden oscillatory modes can occur for the used sampling periods. The proof follows for N ≥ n while its validity for
N = m 0 = n − 1 follows using the same reasoning as in (ii). From Theorem 2 (ii), the following result follows directly.
Corollary 1. Assume that N ≥ N * 0 . A necessary and sufficient condition for B 0 ( z ) to be ρ-Schur
⎡ zI −Ψ N
stable for a given real constant ρ ∈ (0 , 1] is that rank ⎢
⎣
−c
CΓg⎤ ⎥ = n + 1 for some 0 ⎦
T
g ∈ R N and all complex z with z ≥ ρ . The condition holds for N ≥ N * 0 if and only if it holds for N = N * 0 .
For β ≠ 0 , Theorem 2 (ii) –(iii) and Corollary 1 are extended directly as follows for any β:
Theorem 3. Assume that N ≥ N * β . A necessary and sufficient condition for Bβ ( z ) to be ρ- Schur-
⎡ zI − Ψ N stable for a given real constant ρ ∈ (0 , 1] is that rank ⎢ T ⎣ −c
g r = (α 1 , α 2 , ...., α N −1 ) T ∈ R
N * β −1
defined in such a way that
C
r eβ
g
0
r
⎤ ⎥ = n +1 for some ⎦
and all complex z with z ≥ ρ where the matrix C er β is
C er β g r = C e β g − Z β r .
If
[
]
rank C er β = n
then
a necessary and sufficient condition for the ρ- Schur- stabilizability of B β ( z ) is the existence of
12
g ∈R *
N *β −1
⎡ zI −Ψ N such that rank ⎢ T ⎣ −c
g*⎤ ⎥ = n + 1 for all complex z with z ≥ ρ . The 0 ⎦
condition holds for N ≥ N * β if and only if it holds for N = N * β .
Outline of Prof. The extension of the proof for N ≥ N * β is trivial if it holds for N = N * β . First assume, for exposition simplicity purposes, that N ≥ n + 1 .The analysis of the discrete zeros may be performed by fixing one of the α i ( i∈ N ) gains equal to unity with no loss in generality. Note that since N ≥ 2 ,
α N =1
⎡ zI − Ψ N Det ⎢ T ⎣ −c
f
β
β≠0 ,
and
( q ) ( C er β g r + Z β r ) − Z ´β 0
contribution of α N = 1 through C exists
g * ∈R
g r ∈R N *β −1
N
*β
the
−1
such
that
eβ
⎤ ⎥ ⎦
zeros where
of
B β (z)
g r = ( α 1 , ... , α
N −1
are
)
and
those Z βr
is the
. The above determinant is nonzero for all z ≥ ρ if and only it
⎡ zI − Ψ N rank ⎢ T ⎣ −c
C
r eβ
g
r
0
⎤ ⎥ = n +1 . ⎦
Then,
it
fulfilling g * = C er β g r if rank ( C er β ) = n, which is unique for each given g
⎡ zI −Ψ N that rank ⎢ T ⎣ −c
of
*
exists , such
g*⎤ ⎥ = n + 1 for all complex z with z ≥ ρ . The validity of the proof for 0 ⎦
N = n follows directly from taking into account the degree of B β ( z ) under a similar reasoning as that used in Theorem 2.
3.2 Modification of the inter-sample plant input reconstruction A slight modification of the reconstruction of the inter-sample plant input may be performed for with (2) where the plant input at the fast sampled instants u
j
( k T ) are
defined as in (3) for all j∈ N but
with u 0 ( k T ) =0 for all integer k ≥ 0 .That means that the β-FROH operates with the fast sampling period
only
within
⎡ 1 ⎞ ⎛ t ∈ ⎢k T , ⎜ k + ⎟ T N⎠ ⎝ ⎣
the
running
sampling
intervals
but
βN ⎞ ⎛ u( t )= ⎜ 1+ ⎟u 1( kT ) T ⎠ ⎝
⎞ ⎟⎟ . This modification becomes in an alternative input reconstruction on the ⎠
first fast interval of the running sampling period for β ≠ 0 while it remains unaltered related to (2)-(3) for β = 0. In this case, the discretization procedure does not add a zero at the origin, f β( q ) = 1 and Z β r = 0 in (17) so that C
eβ
= C er β and g = g r . As a result, Theorem 3 becomes simplified in the
sense that the necessary and sufficient condition for the stabilizability of the discrete zeros
13
⎡ zI − Ψ N is rank ⎢ T ⎣ −c
C
g⎤ ⎥ = n +1 0 ⎦
eβ
g = (α 1 , α 2 , .... , α N −1 ) T ∈ R
[
rank C
eβ
] = n holds
N * β −1
for
.
all
Also,
if
with z ≥ ρ
z
complex the
controllability-like
and
some
condition
then the discrete zeros are ρ- Schur- stabilizable for some vector of fast
(
sampled input gains if and only if c T , Ψ
) is a stabilizable pair.
N
3.3 Compact ARMA description under discretization of a β-FROH for any β ∈ [−1, 1 ] The ARMA description of the discrete plant S β d is now rewritten in a simpler and compact way for any value of the gain β in order to describe the discrete system in a linear parametrized way which is useful for the subsequent analysis and then for estimation purposes. Note directly from (10), subject to eqns. 11, for β = 0 and from (14), subject to eqns. 15, for β ≠ 0 that the following ARMA description may be used irrespective of the value of β : y k =−
n
∑
a i y k −i +
i =1
n
∑θ
´(β ) i + 1 u u k −1 −i
= θ ´ ( β ) T ( g ) ϕ ´k( β−1) = θ ( β ) T ϕ (kβ−)1 ( g )
(18)
i=0
where the vector g of fast sampled input gains may be transferred either to the parameter vector θ
´ (β )
while removing it from the regressor ϕ ´k(−β1 ) or to the regressor ϕ (kβ−1) ( g ) by removing it
from the parameter vector θ ( β ) , where T ´ ´ ´ ´ θ ( β ) ( g ) = ⎛⎜ θ Ty , θ u( β ) ( g ) ⎞⎟ T = ⎛⎜ θ Ty , θ ´1(uβ ) , ... , θ n(uβ ) , θ n(β+1), u ⎞⎟ T ⎝ ⎠ ⎝ ⎠
(
θ (β ) = θ Ty , θ (uβ )
T
)
T
= ⎛⎜ θ Ty , θ (1βu) , ... , θ (nβ u) , θ ⎝ T
(
ϕ ´k( −β1) = ϕ Ty , k −1 , ϕ ´u(,βk)−T1 ϕ
y , k −1 =
(y
)
T
(
ϕ ´u( β, k) −1 = u k − 1 , u k − 2 , ... , u θ y = ( − a 1 , − a 2 , ... , − a
θ (βu )
T
( g ) = ⎛⎜ θ (1βu) ⎝
⎡ θ 1´ (uβ ) ( g ) = ⎢ b ⎣
(
( 0) T 0
T
n
(
)
k−n
)T
, u k − n −1
)
T
; ϕ
( β) u , k −1
(β ) T ⎤ 0 ⎥g
)
)
T
( g ) = ( u Tk −1 , u Tk − 2 , ..., u Tk − n , α N u (kβ− )n −1 )T
(
T
b (lβ ) = b 1´l(β ) , b ´2(lβ ) , ... , b ´N(βl )
T
; g = α 1 , α 2 , ... , α N
T
βN b T
⎞T ⎟ ⎠
T
, θ (2βu) , ... , θ (nβ u) , θ
+
(β ) n +1 , u
ϕ (kβ−)1 ( g ) = ϕ Ty , k −1 , ϕ (uβ,)k − 1 ( g )
;
k − 1 , y k − 2 , ... , y k − n
T
⎦
(β ) n +1 , u
)
T
⎞ T ; θ ´ (β ) = α ⎟ n +1 , u ⎠
⎡ ; θ ´l(uβ ) ( g ) = ⎢ b ⎣
T
14
( 0) T l −1
+
N
θ (nβ+)1, u ; θ
βN b T
(β ) T ⎤ l −1 ⎥ g
⎦
(β ) n +1 , u = −
−α N
β N ´( β ) b 1, n −1 T
β N ´(β ) b 1, l − 2 ( 2 ≤ l ≤ n ) T
β N ´(β ) ⎛ ´ ( 0 ) β N ´(β ) ´ (0 ) ´ ( 0 ) β N ´(β ) ⎞ θ (1β u) = ⎜ b 1 0 + b 1 0 , ... , b N − 1, 0 + b N − 1 , 0, b N 0 + b N0 ⎟ T T T ⎝ ⎠
T
)
(
β N ´ (β ) β N ´(β ) ⎛ ´ ( 0 ) β N ´(β ) ⎞ ´ (0 ) ´ (0 ) θ (lβ )u = ⎜ b 1, l − 1 + b 1, l − 1, ... , b N − 1, l − 1 + b N − 1, l − 1 , b N , l −1 + b N , l − 1 − b ´1,(lβ−)2 ⎟ T T T ⎝ ⎠
T
( 2≤l≤ n ) u
k −l
( g ) = ( α 1 u k − l , α 2 u k − l , ... , α N u k − l ) T ( l ∈ n ) (19)
where u (k0− )n − 1 = 0 and u (kβ− )n − 1 = u k − n − 1 for β ≠ 0 . These identities reduce the dimensionalities of the relevant regressor and parameter vectors when β = 0, compared to β ≠ 0 , from (19) while keeping invariant the structure of the ARMA model. 3.4 On the selection of a vector g of gains on the parameter space for Schur stabilization of B β ( z ) Theorems 2 and 3 dictate that the existence of g ∈ R
n
such that B β ( z ) is Schur stable is a generic
property in the sense that it only requires simple “ a priori” conditions on the continuous system and the running sampling period as well as on ratio of sampling periods which translate into controllability/observability and stabilizability/ detectability properties of the discrete system. By the continuity properties of the zeros of a discrete system, it turns out that the existence of a stabilizing g implies the existence of infinitely many stabilizing ones located in appropriate neighborhoods of the first one. Roughly speaking, if a stabilizing g does not exist for arbitrarily chosen β-FROH then either unstable or critically stable zero-pole cancellations are present in the continuous system or, otherwise, the pair (T, N) is not set appropriately so as to avoid hidden oscillatory modes. It is now discussed how a finite set of stabilizing vectors of gains might be chosen to guarantee the stabilization of the discrete zeros for a whole compact parameter space to which the parametrization associated with the discrete zeros belongs to. This feature will be then especially relevant in the extension of the formulation to adaptive control in the case when the plant is not fully known. In this case, a finite set of stabilizing vectors of gains may be stored a priori according to each location of the parameter estimates through time to be then used to stabilize the estimated discrete zeros according to the online calculated values of the estimates. The number of parameters of the discrete systems S 0 d eqns. 10-11 and S β d ( β ≠ 0 ), eqns. 13-15 is calculated as follows: .
For β = 0, there are n poles since deg A
0
( z )=
(
)
(
)
n and deg B 0 i (z ) = n-1 i∈ N . Each
polynomial has a number of coefficients equating a unity excess on its degree so that the total number
(
of parameters is p 0 = (N+1) n ≥ p * 0 = n N
15
*0
)
+ 1 excluding the unity leading coefficient of
A
0
( z ) if monic. Then, there are
p
0B
≥ p
*
0B
= n N * 0 parameters associated with B
the remaining n ones are associated with the monic polynomial A
.
For β ≠ 0, deg ( A
β
0
0
(z )
while
( z ).
( z ) ) = n+1 with one of the poles being always located at z=0, and (n+1)
(
)
(
)
parameters for each polynomial B ´β i ( z ) with deg B ´β i ( z ) = n i∈ N . Since each polynomial has a number of coefficients equating a unity excess on its degree so that the total number of parameters is p
β
= (n+1)N+n ≥ p * β = ( n + 1) N
monic. From them, p
βB
≥ p
remaining n with the monic A
*
β
βB
*β
+ n excluding the unity leading coefficient of A β ( z ) if
= ( n + 1) N
*β
parameters are associated with B
β
(z )
while the
( z ).
3.5 Condition of stabilizability of the discrete zeros Note that the parametrization of the state-space description S β d contains, in general, a larger number of parameters in real vectors θ s( β ) ∈ H s( β ) ⊂ R
p βS
than that associated with the input-output ARMA
model parametrized by real vectors θ ( β ) ∈ H ( β ) = H (Aβ ) × H ( Bβ ) ⊂ R
pβ A
×R
pβB
≡ R
pβ
whose
dimension p β = p β A + p β B ≤ p S β equates the total number of coefficients of the polynomials
A β ( z ) and B β ( z ) , less one since A β ( z ) is monic, defined , respectively, by real p β A -vectors p p θ (Aβ ) ∈ H (Aβ ) ⊂ R β A and real p β B -vectors θ (Bβ ) ∈ H ( Bβ ) ⊂ R β B . Note also that the mapping
f : H s( β )→ H ( Bβ ) is surjective but, in general, non-injective. The subsequent result addresses the
conditions for testing the ρ- Schur stabilizability of B β ( z ) via multirate sampling with fast sampled input for a given real ρ ∈ ( 0 , 1 ] and sampling periods T (running slow) and T ´ = T / N all θ s( β ) ∈ H s( β ) ⊂ R
pβS
β
(fast) and
from the state-space parametrization associated with S β d .
Theorem 4. Assume that θ s( β ) ∈ H s( β ) ⊂ R
pβS
, where H s( β ) is a bounded parameter space, for the
given real β ∈[ −1, 1 ] . Then, the following three items hold:
(i) The polynomial B β ( z ) is ρ-Schur stabilizable for all θ s( β ) ∈ H s( β ) if and only if it exists a real N - vector g θ β of fast input gains for each θ s( β ) , such that: ⎡ z I − Ψ N *β rank ⎢ −c T ⎣⎢
(
)
C er β N * β g θ β ⎤ ⎥ = n +1 0 ⎦⎥
(20)
16
for all complex z with z ≥ ρ where C
r eβ =
C
r eβ
( N ) depends on N with C e 0
= CΓ.
(ii) B β ( z ) is ρ- Schur stabilizable for any parametrization of S β d in H s( β ) for some real ρ ∈( 0, 1 ] if and only if it suffices a finite set G
H
θ (Sβ ) in H s( β ) for (at least) a g θ β ∈ G
(iii) Assume that C
r eβ
(β) B
H
of fast input gains to guarantee that (20) holds for each and all complex z with z ≥ ρ .
(β) B
( N ) is full row rank. *β
Then, B β ( z ) is ρ- Schur stabilizable in H s( β ) for
some real ρ ∈( 0, 1 ] if and only if it suffices a finite set G
θβ
of fast input gains to guarantee that
(20) holds for each θ (Sβ ) in H s( β ) for (at least) a g θ β ∈ G Ψ
N*β
H
(β) B
and all eigenvalue z of
satisfying z ≥ ρ .
Proof. (i) It is obvious since Eq. 20 holds for some g θ β ∈ R
stable for each θ s( β ) ∈ H s( β ) ⊂ R
pβS
N *β
if and only if B β ( z ) is ρ- Schur
.
(ii) If (20) holds for some g θ β for each θ (Sβ ) ∈ H s( β ) ⊂ R
open finite neighborhood Ω ( g θ β ) of R
p
*β
p
βS
from item (i) then it also holds in some
centred at g θ β since the matrix rank function is a
piecewise constant function of the matrix entries (since the matrix eigenvalues are continuous functions of the matrix entries). Since H s( β ) is bounded and H s( β ) ⊂ H s( β ) (the closure of H s( β ) ) which is bounded and closed (i.e. a compact set) so that H s( β ) is included in an open cover C H ( β ) in
R
N*β
, it exists as well a finite sub-cover
∪
i ∈ Ca rd ( G
θβ
)
Ω i ( g θ β i ) ⊇ C H ( β ) ⊇ H s( β ) ⊇ H s( β ) of
H s( β ) , g θ β i ∈ G H (β ) ( i ∈ Ca rd (G H (β ) ) , the cardinal of G H (β ) ) from Heine-Borel theorem, [28]. This implies that Ca rd ( G H (β ) ) is finite. (iii) If
C
r e β is
⎡ z I − Ψ N *β rank ⎢ −c T ⎢⎣ ⎡ z I − Ψ N *β rank ⎢ −c T ⎣⎢
full row rank and
C
r eβ
( N )g *β
0
θβ
[ (
rank z I − Ψ
N *β
)
T
]
, c =n
for all
z ≥ρ
then
⎤ N *β and all z ≥ ρ if and only if ⎥ = n + 1 for some g θ β in R ⎥⎦
N *β h θβ ⎤ and all z ≥ ρ since the mapping by the ⎥ = n + 1 for some h θ β in R 0 ⎦⎥
represented by the matrix C
r eβ
is injective. As a result, it always exist g θ β fixing a real vector h θ β
17
in
such
a
⎡ z I − Ψ N *β ⇒ rank ⎢ −c T ⎣⎢ where ( z I − Ψ
N *β
way
[
rank z I − Ψ
that
N *β
]
⎡ z I − Ψ N *β ⎤ h θ β = rank ⎢ ⎥= n − c T ⎥⎦ ⎢⎣
h θβ ⎤ ⎥ = n + 1 for all z ≥ ρ provided that the rank conditions hold only 0 ⎦⎥
) is singular with z ≥ ρ ; i.e. at the eigenvalues of Ψ
N *β
fulfilling z ≥ ρ .
It turns out that any item of Theorem 4 holds for all integer N ≥ N * β if and only it holds for N = N * β . Specific conditions on the state-space realizations have been given in Theorems 2-3 and Corollary 1 imply and/or be implied by (20) for ρ=1. The subsequent result addresses the conditions for
testing
the
ρ-
Schur
stabilizability
of
Bβ (z )
with
parameter
vector
p θ (Bβ ) ∈ H (Bβ ) ⊂ R β B associated with B β ( z ) from the input-output ARMA description without
requiring a test on the parameter space H s( β ) of the state-space description. It is concerned with the existence of a finite number of gains α
i
(i ∈ N )
for each N ≥ N * β which stabilize the zeros of
B β ( z ) for all discrete plant parameter vector θ (Bβ ) within some compact set H (Bβ ) .
Corollary 2. Assume that θ
(β) B
∈ H (Bβ ) ⊂ R
p
*β
, where H (Bβ ) is a bounded parameter space, for the
given real β ∈[ −1, 1 ] . Then, the following two items are equivalent: (i) B β ( z ) is ρ- Schur stabilizable in H (Bβ ) for any given real constant real ρ ∈( 0, 1 ] . (ii) There is a sufficiently large integer µ
0
≥ 1 such that it exist finite numbers µ ∈ [ µ
0
, ∞ ) of
finite sets G H (β ) of real N-vectors (and then infinitely many uncountable finite sets as a result ) in
R N ( N ≥ N *β )
which
guarantee that (20) holds for each θ
(β) B
in H (Bβ ) for (at least) a
g θ β ∈G H (β ) and all complex z fulfilling z ≥ ρ . Proof. From Theorem 4 [(i)-(ii)], B β ( z ) is ρ- Schur stabilizable in a bounded set H s( β ) of
admissible values of its coefficients for a given correcting gain β ∈[ −1, 1 ] and any ρ ∈ ( 0 , 1 ] if and only if it exists a finite (extendable to infinity) number of gain vectors of finite dimension N ≥ N *β
(
in sets G H (β ) of cardinal subject to ∞ ≥ Card G
H (β )
)≥ µ ≥1 0
. It turns out that µ 0 is a
sufficiently large finite integer number, which guarantee that (20) holds for each θ S(β ) in H s( β ) for (at
18
least) a g θ β ∈G H (β ) and all complex z fulfilling z ≥ ρ . Since the mapping H s( β ) → H (Bβ ) is
surjective then H s( β ) bounded (compact) ⇒ H (Bβ ) bounded (compact). As a result, Items (i)-(ii) are equivalent for existing fast sampled input vector gains of dimension N ≥ N *β belonging to stabilization sets of finite cardinal lower-bounded by µ
0
for all parameter vector θ (Bβ ) ∈ H (Bβ ) .
Note that the proof of Corollary 2 still holds trivially if the real intervals C i H l are semi–open intervals of the form C
ij
[
=h
− ij
,h
+ ij
) for
j ∈ p β B − 1 and C
iH
p βB
[
= h
− ip
βB
,h
+ ip
βB
] (i∈ µ
0)
which are the disjoint and components of C i . A useful practical implementation of the test of Corollary 2 is as follows.
3.6 Practical testing algorithm using Corollary 2 Assume that B β ( z ) is ρ- Schur stabilizable in a bounded H (Bβ ) for some given real constant
(
ρ ∈( 0, 1 ] . The problem is to find a vector g 0 θ
(β ) B
) of sampled input gains which stabilize all the
zeros of B β ( z ) in a circle of radius ρ ∈ ( 0 , 1 ] for each given θ (Bβ ) ∈ H (Bβ ) . By continuity of the zeros of a polynomial with respect to continuous variations of their coefficients, it follows that
g 0 (θ
(β ) B
)
also ρ- Schur stabilize infinitely many polynomials B β ( z ) associated with parameter
vectors θ ´ (Bβ ) ∈ H (Bβ ) in some open neighborhood of θ (Bβ ) . It also follows that infinitely many parameter vectors of gains g in a neighborhood of g 0 ρ- Schur stabilize infinitely many polynomials B β ( z ) associated with parameter vectors θ ´ (Bβ ) ∈ H (Bβ ) in some open neighborhood of θ (Bβ ) . A finite set of vector gains may be designed constructively so that B β ( z ) is ρ- Schur stabilizable for at least one of them (and then for infinitely many ones neighbors of it) for each θ (Bβ ) in H (Bβ ) . An ¨ad hoc¨ algorithm is now given for the case when H (Bβ ) is connected and compact (or bounded by using its closure).
3.7 Algorithm 1 (Calculation of a finite set of stabilizing vector gains)
[
]
Step 1. Take real intervals H i = h i− , h i+ such that H = H 1 × H 2 × ... × H p β ⊇ H (Bβ ) ; i.e. each
i- th component of θ β B is in H i for i ∈ p
βB .
Fix test integer numbers µ i ≥ 1 , i ∈ m β + 1 starting
19
with test values and built real intervals
(
)
Step 2. Define µ =
[
− ij
]
, h i+j , which depend on µ i ,
(H )=
h i−, j + 1 = h i−, j j ∈ µ i −1 so that H i = U
j∈ µ
[h ij
i
[
i
]
βB
and
]
i
∏ ( µ ) and build all the bounded sets
with H i i j = h i− i j , h i+ i j ⊂ H i ; i j ∈ µ i ; j ∈ p
i
U h i−j , h i+j ; i ∈ p β B .
j∈ µ
p βB i =1
( j ∈µ )
H i =H
; i ∈ µ and H =
[
ii 1
× H i i 2 × ... × H i i m β defined
U (H
1
i∈µ
)
×H 2 ×...× H p β ⊇ H (Bβ )
]
(i.e. each set H i is formed by the Cartesian product of h i−i j , h i+i j , for some i j∈ µ i for i = 1, 2, … , p β B ). Step 3. If for each H i there is a real N- vector g i ( i ∈ µ ) which stabilizes B β (z) for all θ (Bβ ) in
H i then the set g i ( i ∈ µ ) is a stabilizing set for the zeros of S β d of cardinal µ. Else increase µ i ← µ i + 1 to make a refinement by reducing the length of the intervals
[h
− ij
]
, h i+j for at least one i ∈ p
βB
and go to Step 1 until a sufficiently tight interval partition for all
the components of the parameter vector has been found such that a stabilizing set of gains exists. Note that the assumption that the parameter set of interest is connected is not very strong since a connected set including the set of parameters might always be defined. Note also that for a sufficiently tight partition, the above algorithm always stop since the parameter set is bounded and any stabilizing gain of a test parameter vector always stabilizes zeros in a neighbourhood in the parameter space of such a test parameter vector. The algorithm is not difficult to use for sufficiently tight intervals whose central point define a nominal parameter vector which is stabilized by a vector of gains. By continuity arguments of the zeros of a polynomial, such a vector of gains also stabilizes parametrizations in intervals around those central points. This is a starting point to define the test intervals in the parameter space as well as the related integers µ i . The main interest in the use of the above Algorithm relies in the case when the system has changing parameters and, specifically, in the adaptive case addressed in Section 5. 3.8 Conditions for arbitrarily prefixing the discrete zeros The conditions of stabilizability/ detectability of the involved discrete pairs used in Theorems 2-4 and Corollary 2 are weak but they may lead to expensive computational time and memory storage requirements in the adaptive case. The reason is that the vectors of gains stabilizing zeros have to be calculated and then pre-stored in strips of the parameter space to be then used. The following result is concerned with arbitrary zero-placement under stronger controllability/observability– type conditions than the stabilizability / detectability invoked in the previous results. Under those conditions, the zeros may be located at arbitrary fixed conditions in both the non-adaptive and adaptive cases,
20
Theorem 5. Assume that θ s( β ) ∈ H s( β ) ⊂ R
pβS
, where H s( β ) is a bounded parameter space, for the
given real β ∈[ −1, 1 ] . Then, the following three items hold: (i) The zeros of the polynomial B β ( z ) might be assigned to a set of arbitrary complex numbers
Λ z ⊂ C for all θ s( β ) ∈ H s( β ) if and only if it exists a real N * β - vector g θ β of fast input gains for each θ s( β ) , such that: ⎡ z I − Ψ N *β rank ⎢ −c T ⎣⎢
C
r eβ
( N )g *β
0
for all complex z in C / Λ
⎡ z I − Ψ N *β rank ⎢ −c T ⎢⎣
θβ
C
r eβ
⎤ ⎥ = n +1 ⎦⎥
(21)
and
z
( N )g *β
θβ
0
⎤ ⎥ ≤ n for all z ∈ Λ ⎥⎦
z
.
(ii) The zeros of the polynomial B β ( z ) might be assigned to a set of arbitrary complex numbers
[
Λ z ⊂ C for a given θ s( β ) ∈ H s( β ) if and only if rank C
(
observable and Ψ
N
*β
[ (Ψ
that
N
*β
N*β
)
*β
(c
T
,Ψ
N
*β
)
is
)
z
]
is the set of zeros. Item (ii) follows directly since
N ⎡ g = rank ⎢ z I − Ψ * β ⎣
, g and
( N )] = n ,
, g is controllable for some a real N * β - vector g θ β .
Proof. Item (i) is obvious since Λ rank z I − Ψ
r eβ
(c
T
,Ψ
N
*β
⎤ c ⎥ = n for all complex z implies and is implied by ⎦
T
) being
controllable and observable pairs (from the Popov-
Belevich- Hautus controllability and observability tests, [27]) and g = C
r eβ
( N )g *β
θβ
may be
generated to an arbitrary value from a vector of gains g θ β since the associated mapping is injective since C
[
r eβ
( N ) is full row rank. As a result ,
rank z I − Ψ
*β
N *β
]
N g = rank ⎡⎢ z I − Ψ * β ⎣
T
c ⎤⎥ = n ⇒ ⎦
⎡ z I − Ψ N *β rank ⎢ −c T ⎣
21
C eβ ( N *β ) g θ β ⎤ ⎥ = n + 1 , ∀z∈ C 0 ⎦
for some g
[
θβ
rank z I − Ψ
∈R
N *β
N *β
if C
r eβ
( N )g *β
with C
θβ
]
N g = rank ⎡⎢ z I − Ψ * β ⎣
T
r eβ
(N ) *β
being full row rank
while
c ⎤⎥ = n , ∀z∈ C is a necessary condition for (21) ⎦
to hold by construction.
4. Model – matching
The transfer function of continuous-time plant (1) G c ( s ) = c T ( s I − A ) −1 b in the Laplace transform variable s yields to a discrete one G A
β
β
(z )=
Bβ(z )
Aβ (z )
in the Z-transform variable z with
( z ) = f β ( z ) A 0 (z ) for β ≠ 0 from (13)-(15).
4.1 Control objective The goal is to match a discrete reference model whose zero-pole cancellation free transfer function is G β m ( z )=
B β m (z )
A βm ( z )
, which may depend on β if suited, under the subsequent assumptions (see, for
instance, [1-6] and references therein). Assumptions. (1). The discrete plant ARMA model is parametrized in a known compact set H ( β ) . (2) A β m ( z ) is monic and ρ m –Schur stable for some prescribed design real constant ρ
(
)
(
m
∈ ( 0 ,1]
)
and B β m ( z ) is prefixed arbitrarily with deg B β m ( z ) ≤ deg A β m (z ) − 1 . (3)
For
each
θ∈H (β ) ,
there
exists
a
g θ ∈ ( α 1 , α 2 , ... , α N ) T ∈ R N
with
N ≥ N * β = Max ( 2 , m β ) such that B β m ( z ) is ρ- Schur stable for some real constant ρ ∈ ( 0 , 1 ] .
Assumption 1 facilitates the exposition. According to the results in the previous section, the parameter set might be, in general, any bounded (rather than compact) set so that all the subsequent results apply mutatis-mutandis over some compact set which contains it. The stability of the reference model in Assumption 2 is a basic structural requirement for closed-loop stability since the uncancelled closed-loop poles are the reference model poles. The degree constraint reflects the fact that the pole-zero excess of the reference model transfer function has to be non less than the pole-zero excess in the plant transfer function which is unity for all β since the continuous plant is strictly proper. This constraint ensures the realizability of pole-placement based and model-matching based synthesized controllers. Finally, Assumption 3 is guaranteed if the polynomial of zeros of the discretized plant is Schur- stabilizable via multirate sampling for any parameter vector in the parameter space. Explicit ¨ad hoc¨ conditions have been given in Theorems 2-3 and Corollary 2. An important consequence of Assumption 3 is that the reference model zeros might be fixed arbitrarily
22
via cancellations by the controller of unsuitable plant zeros. In other words, the maintenance as closed-loop zeros of the critically stable and unstable plant zeros is not required in model-matching designs.
4.2 Control law and controller synthesis for known plant Now assume that the particular plant parameter vector θ ∈ H ( β ) is known. Then, the proposed control law is generated by the difference equation: R β ( q)u k = S β ( q) y k − T β ( q )r k ;
u ik ≡ u
i
( k T) =
αiuk
(22)
with u k ≡ u ( k T ) and r k ≡ r ( k T ) are , respectively, the plant input and reference signal at the running sampling period while y
mk
= G β m ( q ) r k is the reference model output under zero initial
conditions for any uniformly bounded reference input sequence
{ r k } ∞0 .
The discrete controller
consists of a feedback compensator and a precompensator of respective transfer functions:
F f (z )=
Sβ (z ) R β( z )
F p (z )=
;
T β(z )
(23)
R β( z )
Assume that the plant zeros are uniquely factorized as B β ( z ) = B β c ( z ) B β t ( z ) where B β c ( z ) is the monic polynomial of cancelled plant zeros (which equates unity if no zero is cancelled) and B β t ( z ) is the polynomial of transmitted zeros (which equates unity if all the plant zeros are cancelled).
Now,
the
numerator
of
the
reference
model
transfer
function
is
B β m ( z ) = B ´β m ( z ) B β t ( z ) where B ´β m ( z ) contains the free-design reference model zeros. If B β t ( z ) is unity then the reference model zeros are fixed arbitrarily. The numerator and denominator polynomials of (23) are synthesized as follows: R β ( z ) = B β c ( z ) R ´β ( z ) = B β c ( z )( z − 1 ) l R ´β 1
The pair
(R
´ β1
( z ), S β ( z ) )
(z ) ;
T β ( z ) = B ´β m ( z ) A β 0 ( z )
(24)
is the unique solution of the Diophantine equation of polynomials
( z − 1 ) l A β ( z ) R ´β1 ( z ) + Bβ t ( z ) S β ( z ) = A β 0 ( z ) A β m (z )
(25)
where A β 0 ( z ) is a ρ m -Schur stable monic polynomial of zero-pole closed-loop cancellations. If A β 0 ( z ) is unity then there are no closed-loop zero-pole cancellations. If it is of degree at least unity then the reference model transfer function includes additional stable zero-pole cancellations
23
resulting in the reference model transfer function being G β m ( z ) =
B β m (z ) A β 0 ( z ) A β m ( z )A β 0 ( z )
. The integer
number l ≥ 0 takes account of integrators supplied by the controller. If l ≥ 1 , it defines
the
multiplicity of a discrete integrator in the controller which ensure good tracking properties of polynomial signals of degree (l − 1 ) as well as an acceptable rejection to load disturbances. Under the current formulation, all the plant zeros may be stabilized and then cancelled by the controller under weak conditions [see Theorems 2-3 and Corollary 2]. For coherency of the synthesis problem with the realizability of the controller as well to guarantee the existence and uniqueness of the solution of the Diophantine equation (25) , the subsequent degree constraint are required, [4], [15]:
(
)
(
deg R ´β1 ( z ) = deg A
(
deg ( S β ( z )) = deg A
(
)
( z )) +
deg A
( z )) + l
−1 ;
β0
β
(
deg A β 0 ( z ) ≥ 2 deg A
β
(
( z )) −
(
βm
deg A
( z )) − deg ( A β ( z )) − l
deg ( T βm
β
( z )) = deg ( B ´β m ( z ) ) + deg ( A β 0 ( z ) )
( z )) − deg ( B β t ( z )) + l −1 (26)
Note that the last degree constraint holds without using closed-loop zero/pole cancellations (i.e. with
A β 0 ( z ) equal to unity) if the reference model possesses a sufficiently large order satisfying
(
deg A
βm
( z )) ≥ 2 deg (A β ( z ))
(
)
− deg B β t ( z ) + l −1 .
5. Adaptive control
If the plant parameter vector θ ∈ H ( β ) is unknown but Assumptions1-3 still hold, then all the above design in Section 4 remains valid if the plant parameter vector is estimated via an estimation algorithm while the controller parametrization and control law are subject to (22)-(26) by replacing the known plant parameter vector by its estimated one, [2-6], [17-18]. Since the compact set H ( β ) is known the ´´a priori´´ estimated parameter vector is projected onto the boundary H ( β ) when necessary. The input gains which stabilize the discrete zeros of the estimated model are collected ˆ ( β ) ( i ∈ µ ) of H ( β ) . Such a from the set of gains that stabilizes the estimated zeros for each strip H i set is calculated by using Corollary 2 and Algorithm 1, or alternatively, a sample-dependent vector
g
k
is updated such that the estimated plant zeros are located at a prescribed fixed Schur- stable
polynomial if Theorem 5 holds.
5.1 Algorithm 2 (Estimation algorithm and adaptive controller synthesis) Step 1 (´´A priori ´´estimation). For each running sampling period integer index k, compute the ´´a
priori´´ plant parameter vector of estimates from a recursive algorithm (for instance, a least-squares
24
algorithm) by using its value θˆ 0k − 1 at the preceding running sampling instant and covariance matrix
P k −1 : θˆ 0k = θˆ
0 k −1
+
P k = P k −1 −
P k −1 ϕ 1+ ϕ
T k −1
P
k −1
ek
k −1
ϕ
(27.a) k −1
P k − 1 ϕ k − 1 ϕ Tk − 1 P 1+ ϕ
T k −1
P
k −1
k −1
ϕ k −1
, P 0 = P T0 > 0
(27.b)
by using the regressor ϕ k −1 built with previous input and output measurements with the previous vector
of
fast
sampled
e k = y k − yˆ k = y k − θˆ
input
gains
g k −1
where
the
estimation
error
is
0 T k − 1 ϕ k −1 .
Step 2 ( ´´A posteriori´´ modification of the estimates ). Calculate the ´´a posteriori´´ estimated vector
by using projection on H ( β ) ≡ H 1( β ) × ...× H (pββ ) from the parameter estimated vector obtained in Step
( )
1, i.e. θˆ k = Pr oj H ( β ) θˆ 0k component-wise defined by: θˆ k i = θˆ 0ki if θˆ θˆ
ki
0 ki
∈ H i( β )
(
)
= θ k i ∈ Boundary H i( β ) ∩ R : θˆ 0k i − θ
ki
≤
θˆ 0k i − θ ´i
(
)
∀ θ ´i ∈ H i( β ) ∩ R
Step 3 (Calculation of the vector of fast sampled input gains).
- If there is some zero-pole cancellation in the ´´a posteriori´´ estimation model then modify locally such a vector gain so that: (1) the stability of the estimated zeros is maintained (2) no zero-pole cancellation takes place in the ´´a posteriori ´´ estimated model.
- Locate θˆ k in one of the µ strips Hˆ i( β ) ( i ∈ µ ) of the parameter set H ( β ) for which a set of vector gains g i( β ) ( i ∈ µ
) stabilizing the cancelled zeros of the ´´a posteriori´´ plant estimation model
exists.
- Make g k ← g
( β) i
. [If Theorem 5 holds, one can alternatively calculate g
k
such that the ¨a
posteriori¨ estimated plant numerator is prefixed, Schur- stable and constant].
Step 4 (Adaptive controller parametrization). Parametrize the adaptive controller for each current running sampling instant by using (23)-(25) , subject to the degree constraints (26), by first replacing the known plant parameter vector by its ´´a posteriori´´ estimated model from Step 2 and using g
k
from Step 3.
25
Step 5 (Adaptive control law). Implement the adaptive control law by using (22) modified via the
time-varying controller parametrization of Step 4. Step 6. Make k ← k + 1 and go to Step 1.
The following result might be proved without difficulty by using the boundedness properties of the recursive least-squares estimation algorithm since the parameter set is compact and the estimated plant zeros are bounded and Schur stable while the estimation algorithm avoids ´´a posteriori´´ zero – pole cancellations in the discrete plant estimation model and the reference model is stable.
Theorem 6 (Estimation convergence and closed-loop stability). Under Assumptions 1- 3, the
following items hold: (i) The ´´a priori´´ and ´´a posteriori´´ estimated plant vectors are bounded for all running
sampling instants and converge asymptotically to finite limits as k → ∞ . The time-varying parameter vector which parameterizes the adaptive controller is uniformly bounded for all running sampling instants. (ii) The closed loop system is stable and all the signals in the loop are bounded.
Outline of proof. The boundedness and convergence to a finite limit of the ´´a priori´´ estimates is
proved under standard calculations from the discrete Lyapunov´s-like sequence candidate ~ T ~ ~ V k = θ 0k P k− 1 θ 0k of the ´´a priori ´´ parametrical error θ 0k = θ k − θˆ 0k , where θ k is the discrete parameter vector which depends on constant parameters and the preceding vector of gains g
k −1 ,
by using (27.a) combined with the matrix inversion lemma in (27.b). This makes V
k
to be
uniformly bounded for bounded initial conditions and to asymptotically converge to a finite limit. As a result, the identification error converges asymptotically to zero, and the ´´a priori´´ estimated vector is bounded and converges asymptotically to a finite limit. The ´´a posteriori´´ estimated vector is bounded and converges to a finite limit since it is obtained from a projection of the ´´a priori´´ one (which converges asymptotically to a finite limit) on a compact set for all samples. Then, the adaptive controller parameters are also bounded and converge to finite limits from (23) to (25), modified for the ´´ a posteriori´´ estimates and subject to (26), since the resulting adaptive control law (22) is nonsingular because there are no zero/ pole cancellations in the ´´a posteriori´´ estimated plant model transfer function from Step 3 of Algorithm 2. The sequence of vector gains
{g } k
∞ 0
converges
asymptotically from Step 3 of Algorithm 2 if the set of µ stored vector gains is sufficiently large which also makes θˆ k to asymptotically converge to a finite limit. The estimation error is uniformly bounded and it tends asymptotically to zero while the plant input and output are bounded since the
26
closed-loop system is stable and reference model matching is ensured. The tracking error between the plant and reference model outputs, i.e. ε k = y k − y M k , is also a uniformly bounded sequence which converges asymptotically to zero.
6. Simulation examples Some simulation results which illustrate the effectiveness of the proposed method are shown in
the current section. A continuous-time unstable plant with an unstable zero of transfer function G( s )=
s − 0 .5 is considered. ( s − 2 )( s + 1 )
In a first group of simulations, a suitable multirate
scheme with fast input sampling through a ZOH device is used to place the zero of the discretized plant into the stability region and a discrete-time controller is synthesized so that the discrete-time closed-loop system matches a reference model. The results for the case of known plant parameters are presented in a first example and then three examples with different adaptive control strategies are considered. The difference among such adaptive control strategies lies in the way of updating the multirate parameters for ensuring the stability of the estimated discretized plant zero. In a second group of simulations the same problem is addressed with a multirate input and a discretization based on a FOH device. In such a case, the discretized plant has two zeros since the holder device adds one pole/zero pair. Then, the multirate input has to ensure the stability of both zeros.
6.1. Multirate input and ZOH device A. Known plant parameters
The discretization of the continuous-time plant with a multirate, N=2, and a ZOH device for a sampling time T=0.2 s. is performed leading to the discrete transfer function G ( z )=
B (z ) ( 0.1107 α1 + 0.1029 α 2 ) z − 0.1196 α1 − 0.1163 α 2 = . The zero of such a A( z ) z 2 − 2.3106 z + 1.2214
discretized plant can be fixed in the stability region via a suitable choice of the multirate parameters. In this example the multirate parameters are α1=2989.6 and α2=-3116.8, then B(z)=10z+5 and the zero is placed in z1=-0.5. The control objective is to cancel the discretized plant zero and add the zero of the reference model to the discrete-time closed-loop system to match the reference model transfer function G
m
z + 0.728 (z )= B m (z ) = 3 . A m ( z ) z + 0. 6 z 2 + 0.12 z + 0.008
The values of the control parameters to meet the objective proposed in the example are r1=2.9106, s0=5.6236 and s1=-3.5470. Figure 6.1 displays the time evolution of the closed-loop system output, its values at the sampling instants and the sequence of the discrete-time reference model output. Figure 6.2 shows the plant input signal. Note that perfect model matching is
27
achieved without any constraints in the choice of the zeros of the reference model, in spite of the continuous-time plant possess an unstable zero.
Fig. 6.1: Plant and reference model output signals
Fig 6.2: Plant input signal.
B. Unknown plant parameters
Now, the plant is unknown so that an adaptive version of the discrete-time controller designed in the previous example is considered with the initial estimated parameter vector
(3.4658, − 1.8321, 0.166, 0.1544 , − 0.1794 , − 0.1745) T and
the 6-th order covariance matrix is
initialized to Diag ( 10 5 ). Three different methods are considered to update the multirate parameters. The first one consists of updating the multirate parameters at all the slow sampling times so that the discretized zero is maintained constant into the stability region. The second one consists of changing the value of the multirate parameters only when the discrete zero, which is time-varying, is going out of the stability region. Otherwise, the values for the multirate parameters are maintained equal to those of the previous slow sampling time. Finally, the third method selects a set of possible values for the multirate parameters pair (α1, α2) as stated in the theoretical framework of the paper. Each pair ensures the stability of the discrete zero when the estimated vector is located into certain region in the parametrical space. This method requires “a priori” knowledge about the domain into which the real parameter vector belongs to.
B1. Method 1:Discretized plant zero maintained constant Figure 6.3 displays the time evolution of the closed-loop system output, its values at the sampling instants and the sequence of the discrete-time reference model output. Figure 6.4 shows the plant output signal. Figure 6.5 shows the plant input signal generated from the multirate with the ZOH applied to the control sequence u(k). Finally, figures 6.6 and 6.7 display, respectively, the time evolution of the multirate and the adaptive controller parameters. The discrete-time model matching is reached after a transient time interval and the continuoustime plant output signal is bounded for all time as it can be observed from figures 6.3 and 6.4.
28
The multirate and the adaptive control parameters are time-varying until they converge to constant values.
Fig. 6.3: Plant and reference model output signals.
Fig 6.4: Plant output signal.
Fig. 6.5: Plant input signal.
Fig 6.6: Multirate parameters.
Fig. 6.7: Adaptive control parameters.
B2. Method 2:Discretized plant zero is time-varying The multirate parameters are maintained constant to their values at the previous (slow) sampling instant until the discrete plant zero is going out of the stability region. When this happens the multirate parameters are calculated to place the discrete zero in z d=0.5 or z d=-0.5, according to the region towards the discrete zero was going out of the stability region. The
29
discrete-time model matching is reached after a transient time interval and the continuous-time plant output signal is bounded for all time as it can be observed from figures 6.8 and 6.9. The maximum value reached by the continuous-time signal is smaller than that obtained with the previous method (B1) for updating the multirate parameters. Figure 6.10 shows the plant input signal generated from the multirate with the ZOH applied to the control sequence u(k). Figures 6.11 and 6.12 display, respectively, the evolution of the multirate and controller parameters. The adaptive control parameters are time-varying until they converge to constant values while the multirate parameters are maintained constant for all time since the time-varying discretized plant zero is not going out of the stability region. The evolution of such a zero and the parameters of the time-varying polynomial of plant zeros are shown in Figure 6.13.
Fig. 6.8: Plant and reference model output signals.
Fig. 6.10: Plant input signal.
Fig. 6.12: Adaptive control parameters.
Fig 6.9: Plant output signal.
Fig 6.11: Multirate parameters.
Fig. 6.13: Numerator parameters of the discretized plant and evolution of its zero.
B3. Method 3: Discretized plant zero being time-varying
30
The following set of the multirate parameter pairs is selected since each set stabilizes the plant zeros forcertain regions of the parameter space defined together with each of those sets: (α1 (k ), α 2 (k ) ) = (1993, −2077.9 )
if
(α1 (k ), α 2 (k ) ) = ( 4013.4, −4206.3)
if
{bˆ (k ) ≥ 0.1, bˆ ≥ 0.1, bˆ (k ) ≥ −0.181 and bˆ (k ) < −0.174} {bˆ (k ) ≥ 0.1, bˆ (k ) ≥ 0.1, bˆ (k ) < −0.181 and bˆ (k ) ≥ −0.174} {bˆ (k ) ≥ 0.1, bˆ (k ) ≥ 0.1, bˆ (k ) ≥ −0.181 and bˆ (k ) ≥ −0.174} (0) 11
(0) 12
(0) 11
(0) 12
(0) 21
(0) 22
(0) 21
(0) 22
(0) (0) (0) (α1 (k ), α 2 (k ) ) = ( 3734.3, −3909.4 ) if 11(0) 12 21 22 It is assumed that the whole covered parameter space is known “a priori”. The discrete-time
model matching is reached after a transient time interval and the continuous-time plant output signal is bounded for all time as it can be observed from figures 6.14 and 6.15. The maximum value reached by the continuous-time signal is smaller than those obtained with the previous methods (B1 and B2) for updating the multirate parameters. Figure 6.16 shows the plant input signal generated from the multirate with the ZOH applied to the control sequence u(k). Figures 6.17 and 6.18 display, respectively, the time evolution of the multirate and the adaptive controller parameters. The three predefined pairs of values for the multirate parameters are used in the simulation of the system. The evolution of the discretized plant zero and the parameters of the time-varying polynomial are shown in Figure 6.19.
Fig. 6.14: Plant and reference model output signals
Fig. 6.16: Plant input signal.
Fig 6.15: Plant output signal.
Fig 6.17: Multirate parameters.
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Fig. 6.18: Adaptive control parameters. Fig. 6.19: Numerator parameters of the discretized plant and evolution of its zero. 6.2. Multirate input and FOH device A. Known plant parameters
The discretization of the continuous-time plant with a multirate, N=3, and a FOH device for a sampling time T=0.2 s. yields: G( z) =
B ( z ) (0.0057α1 + 0.0074α 2 + 0.2374α 3 ) z 2 − (0.0027α1 + 0.0026α 2 + 0.3075α 3 ) z + 0.0399α 3 = A( z ) z ( z 2 − 2.3106 z + 1.2214)
The zeros of such a discretized plant can be fixed in the stability region via a suitable choice of the multirate parameters. In this example the multirate parameters are α1=-76176, α2=57232 and α3=125, then B(z)=20z2+20z+5 and the zeros are placed in z1=z2=-0.5. The values of the control parameters to meet the objective proposed in the example are: r1=2.7106, s0=5.1015, s1=-3.3067 and s2=0.0001. Figure 6.20 displays the time evolution of the closed-loop system output, its values at the sampling instants and the sequence of the discrete-time reference model output. Figure 6.21 shows globally the time evolution of the closed-loop system output. Figure 6.22 displays the plant input signal generated from the multirate with the FOH applied to the control sequence u(k). These results illustrate the perfect matching between the discrete-time closedloop system and the reference model, without restriction in the choice of such a reference model, in spite of the continuous-time plant possess an unstable zero.
Fig. 6.20: Plant and reference model output signals.
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Fig 6.21: Plant output signal.
Fig. 6.22: Plant input signal.
B. Unknown plant parameters The parameters of the plant to be controlled are unknown. The estimated discretized plant
obtained from the multirate, with N=3, and the FOH device is used. The 10-th order covariance matrix is initialized to Diag ( 10 5 ) and the initial vector of estimates is: θˆ(0) = [3.4658 −1.8321 0.0086 0.0112 0.3562 −0.0041 −0.0039 −0.4057 −0.0555 0.059]
T
The multirate parameters are also time-varying since they are calculated from the estimated parameters vector. Two different methods are considered to update the multirate parameters. The first one consists of updating the multirate parameters at all sampling times so that the discretized zeros are maintained constant into the stability region. The second one consists of changing the value of the multirate parameters only when at least one of the discretized zeros, which are time-varying, is going out of the stability region. Otherwise the values for the multirate parameters are maintained as those of the previous sampling time. The results corresponding to each alternative are presented below.
B1. Method 1:Discretized plant zeros constant Figure 6.23 displays the time evolution of the closed-loop system output, its values at the sampling instants and the sequence of the discrete-time reference model output. Figure 6.24 shows globally the plant output signal. Figure 6.25 shows the plant input signal generated from the multirate with the FOH applied to the control sequence u(k). Finally, figures 6.26 and 6.27 display, respectively, the time evolution of the multirate and the adaptive controller parameters. The discrete-time model matching is reached after a transient time interval and the continuoustime plant output signal is bounded for all time as it can be observed from figures 6.23 and 6.24. The multirate and the adaptive control parameters are time-varying until they converge to constant values.
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Fig. 6.23: Plant and reference model output signals.
Fig 6.24: Plant output signal.
Fig. 6.25: Plant input signal.
Fig. 6.26: Multirate parameters.
Fig. 6.27: Adaptive control parameters.
B2. Method 2: Discretized plant zero time-varying The multirate parameters are maintained constant to their values in the previous sampling instant until at least one of the discretized plant zero is going out of the stability region. When this happens the multirate parameters are calculated to place the discretized zeros in z1=z2=-0.5. The discrete-time model matching is reached after a transient time interval and the continuoustime plant output signal is bounded for all time as it can be observed from figures 6.28 and 6.29. The maximum value reached by the continuous-time signal is smaller than that obtained with the previous method (B1) for updating the multirate parameters. Figure 6.30 shows the plant input signal generated from the multirate with the FOH applied to the control sequence
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u(k). Figures 6.31and 6.32 display, respectively, the time evolution of the multirate and the adaptive controller parameters. The adaptive control parameters are time-varying until they converge to constant values while the multirate parameters are piecewise constant and they only vary when it is necessary to maintain both discretized zeros into the stability region. The evolution of such zeros and the parameters of the time-varying polynomial of estimated zeros are, respectively, shown in Figures 6.33 and 6.34.
Fig. 6.28: Plant and reference model output signals.
Fig 6.29: Plant output signal.
Fig. 6.30: Plant input signal.
Fig. 6.31: Multirate parameters.
Fig. 6.32: Adaptive control parameters.
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Fig. 6.33: Real and imaginary components of the the discretized plant zeros.
Fig. 6.34: Numerator parameters of the discretized plant.
7.Conclusions
This paper has investigated the model matching problem in discrete systems under fully free- design zeros of the reference model for any fractional order-hold and any running sampling period used in the discretization process of a linear and time-invariant single-input single-output continuous-time plant. The mechanism utilized for a free design of the reference model zeros is the use of a fast input sampling rate at sufficiently small sampling period with an appropriate associate set of scalar gains calculated so that all the discrete plant zeros are stable under rather weak structural conditions. This makes possible the cancellation of a part of all of them, if suited, irrespective of the locations of the continuous-time zeros, if any. The design philosophy is extended to model matching based adaptive control schemes by using appropriate partitions of the parameter space so that a finite number of sets of scalar gains is pre-calculated and then stored for on-line accommodation of the numerator of the estimated plant transfer functions. Asymptotic model matching is guaranteed. ACKNOWLEDGMENTS The authors are very grateful to the Spanish Ministry of Education and to University of Basque Country by their partial support of this work through Research Grants DPI 2003-0164 and 9/ UPV/ EHU 00I06.I06-15263/2003, respectively. They are also grateful to Mr. A. Ibeas by his useful comments and suggestions. REFERENCES [1] M. de la Sen, ´´On pole-placement controller for linear time-delay systems with commensurate point delays´´, Mathematical Problems in Engineering, Vol. 2005, No. 1, Jan. 2005, pp. 123-140, 2005. [2] M. de la Sen, ´´Adaptive stabilization of continuous- time systems through a controllable modified estimation algorithm ´´, Mathematical Problems in Engineering, Vol. 2004, No. 2, pp. 133-144, June 2004. [3] A. Ibeas, M. de la Sen and S. Alonso- Quesada, ´´ Stable multiestimation model for single-input singleoutput discrete adaptive control systems´´, Int. J. of Systems Science , Vol. 35, No.8, pp. 479-501, July 10, 2004.
36
[4] M. de la Sen, ´´ Robust stable pole-placement control of continuous linear systems with two parametrical estimation schemes ´´, Journal of the Franklin Institute- Engineering and Applied Mathematics, Vol. 341, No. 3, pp. 231-254, May 2004. [5] M. de la Sen, J. J. Minambres, A.J. Garrido, A. Almansa and J.C. Soto, ´´Basic theoretical results for expert systems . Application to the supervision of adaptation transients in planar robots´´, Artificial Intelligence, Vol. 152, No. 2, pp. 173-211, Feb. 2004. [6] S. Alonso-Quesada and M. de la Sen, ´´ Robust adaptive control of discrete nominally stabilizable plants ´´, Applied Mathematics and Computation, Vol. 150, No. 2, pp. 555- 583, Jan. 15, 2004. [7] N. M. Filatov, U. Keuchel and H. Unbehauen, ´´ Dual control for an unstable mechanical plant ´´, IEEE Control Systems, Vol. 16, No.4, pp. 31-37, 1996. [8] F. Zheng and P. M. Frank, ´´ Finite dimensional variable structure control design for distributed delay systems´´, Int. J. of Control, Vol. 74, No. 4, pp. 398-408, March 2001. [9] Y. Niu, J. Lam and X. Wang ´´ Sliding-mode control for uncertain neutral delay systems´´, Int . J. of Systems Science, Vol. 35, No. 2, pp. 139-150, Feb. 15, 2004. [10] Y. Niu, J. Lam, X. Wang and Ho Dwc, ´´ Observer- based sliding mode controller for nonlinear statedelayed systems ´´, IEE Proceedings- Control theory and Applications , Vol. 151, No. 1, pp. 38-44, Jan. 2004. [11] P. Richard, F. Gouaisbaut and W. Perruquetti, ´´Sliding mode control in the presence of delay´´, Kybernetica, Vol. 37, No. 3, pp. 277-294, 2001. [12] C.H. Chou and C.C. Cheng, ´´ Design of adaptive variable structure controllers for perturbed time-varying state delayed systems ´´, Journal of the Franklin Institute , Vol. 338, No.1, pp. 35-46, 2001. [13] S. Oucheriah, ´´ Exponential stabilization of linear delayed systems using sliding-mode controllers´´, IEEE Transactions on Circuits and Systems I. Fundamental Theory and Applications, Vol. 50, No. 6, pp. 826-830, June 2003. [14] S. P. Bhattacharaya, H. Chapellat and C. H. Keel, Robust Control. The Parmeter Approach, Prentice- Hall, Upper Saddle River, N.J., 1995. [15] A.D.S. Lordelo, E.A. Juzzo and P.A.V. Ferreira, ´´ On the design of robust controllers using the interval Diophantine equation ´´, Proceedings of the 2004 Conference on Computer Aided Design- Control Systems Design, Taipei, Taiwan, pp. 173-178, September 2-4, 2004. [16] K.K. Moore, S.P. Bhattacharaya and M. Dahleh, ´´Capabilities and limitations of multirate control systems¨, Automatica, Vol. 29, No. 4, pp. 941-951, 1993. [17] R.L. Morris and C.P. Neuman, ´´Model reference adaptive control with multiple samples between parameter adjustments ´´, IEEE Transactions on Automatic Control, Vol. 26, No. 2, pp. 534-537, 1981. [18] M. de la Sen, ´´Multirate hybrid adaptive control ´´, IEEE Transactions on Automatic Control, Vol. 31, No. 6, pp. 582-586, 1986. [19] P.T. Kabamba, ´´Control of linear systems using generalized sampled-data hold functions´´, IEEE Transactions on Automatic Control, Vol. 32, No. 7, pp. 772-783, 1987. [20] R. Barcena, M. de la Sen and I. Sagastabeitia, ´´Improving two stability properties of the zeros of sampled systems with fractional order hold ´´, IEE Proceedings- Control Theory and Applications, Vol. 147, No. 4, pp. 456- 464, Jul. 2000.
37
[21] S. Liang and M. Ishitobi, ´´ Two stability properties of the zeros of sampled models for time-delay systems in fractional order hold case´´, Dynamics of Continuous, Discrete and Impulsive Systems- Series BApplications and Algorithms, Vol. 19, No. 3, pp. 299-312, Jun. 2004. [22] S. Liang, M. Ishitobi and Q. Zhu, ´´ Improvement of stability of zeros in discrete–time multivariable systems using fractional-order hold ´´, Int. J. of Control, Vol. 76, No. 1, pp. 1699-1711, Nov. 20, 2003. [23] J.T. Chan , ´´On the stabilization of discrete system zeros´´, Int. J. of Control, Vol. 69, No. 6, pp. 789-796, April 1998. [24] S. Liang and M. Ishitobi, ´´ Properties of zeros of discretised systems using multirate input and hold ´´, IEE Proceedings Control Theory and Applications, Vol. 15, No. 2, pp. 180-184, March 2004. [25] C. Van Loan, ´´ How near is a stable matrix to an unstable matrix´´, Contemporary Mathematics, Vol. 47, pp. 465-477, 1985. [26] D. Hinrichsen and A.J. Pritchard, ´´ Stability radii of linear systems´´, Systems and Control Letters, Vol. 7, pp. 1-10, 1986. [27] T. Kailath, Linear Systems, Prentice- Hall Inc., Englewood Cliffs, N. J. 1980. [28] T.M. Apostol, Mathematical Analysis, 2nd Edition, Addison- Wesley Co., Reading, MA, 1974. [29] M. De la Sen, R. Barcena and A.J. Garrido, ´´ On the intrinsic limiting zeros as the sampling period tends to zero´´, IEEE Transactions on Circuits and Systems I : Fundamental Theory and Applications, Vol. 48, No. 7, pp. 898-900, 2001.
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