In this paper we have introduced a new digital adaptive control design algorithms based on the theory of matrix polynomials. the main contribution of this procedure is based on the so called block pole placement gathered with a MIMO RLS estimator, the advantages of this control are the non-interacted behavior, simplicity in control design and allowing to relocate not only the eigenvalues but both eigenstructure are adaptively assigned altering both stability and the rate convergence.
tive block pole placement (or adaptive matrix polynomial
placement) via MIMO digital compensator design. In the
Design of controllers for multivariable systems requires fourth section we have illustrated an application example
an assessment of structural properties of transfer matri- which is the discrete adaptive control of winding process
ces and matrix polynomials see [
The paper is organized as follow, the first section be an introductory to the present work. Then the second section will include some exist MIMO identification algorithms.
It is followed with the section which deals with
adapIn the present work a new MIMO adaptive compensator
design procedure is proposed which ofers the designer a
larger degree of freedom[
A MIMO ARMAX (autoregressive moving average with exogenous excitation) mode (− 1)[] = (− 1)[] + (− 1)[]
(1) [] = (− 1)− 1(− 1)[] + (− 1)− 1(− 1)[]
(2) Where [] ∈ and [] ∈ are input and output vectors of the system respectively, while [] ∈ is a white-noise signal and the polynomial matrices (− 1), (− 1) and (− 1) have the following structure (− 1) = + 1− 1 + · · ·
+ − (− 1) = 1− 1 + · · · + − (− 1) and (− 1) assuming in this case (− 1) = . the following matrices.
Taking the transpose of equation (1) and expanding (− 1) and (− 1) yield: (− 1) = [] =
=
...
[, :] − [, :] ...
⎤ ⎥ ⎥ ⎥ ⎦ ⎥ ⎥ , Φ = ⎥ ⎦ · · · · · · [︃Φ ... Φ
]︃ ⎡ − [ − + 1, :] ⎤ ⎤ ⎢⎢⎢ ... ⎣
⎡ [︂ 0.5015 0.3047 ]︂ ⎤ = ⎢⎢⎢⎣⎡ 121 ⎥⎥⎥⎦⎤ = ⎢⎢⎢⎢⎣⎢⎢⎢ [︂[︂ − −−− 0000.4...020099909946247 − 0000...462.5090925557733 ]︂]︂ ⎥⎥⎥⎦⎥⎥⎥⎥
− 0.8998 0.5021 ⎥⎥⎥⎦ ⎥⎥⎥⎦ W6× he1r,ed:imT(he)d=im6e n×si1o,ndoimf (the) m=a6tr× ice6s, dairme(di)m=( 6)× =2 − [ − 1, :] · · · − [ − , :] and dim() = 6 × 6.
⎡ ⎡ [, :] ⎤ · · · ⎡ [ − + 1, :] ⎤ ⎤ Comments: Φ = ⎢⎢⎢⎣ ⎢⎢⎢⎣ ... ⎥⎥⎥⎦ · · · ⎢⎢⎢⎣ ... ⎥⎥⎥⎦ ⎥⎥⎥⎦ ■puTthaetioMnI MloOadRaescsuorcsiiavteedlewasitthsqMuaIMreOsrleedaustcessquthaerecsobmy [ − 1, :] · · · [ − , :] casting it in recursive form which is useful for On-line with = and is the number of / data. To system identification. avoid a large space memory and the large dimension ma- ■ This basic RLS can be improved by introducing a fortrix inversion taken by the simple least secure a MIMO getting factor [31] in order to give more weights to the recursive least square algorithm can be handled and elab- most recent data. orated to be used in digital software preserving the memory space. 2.2. MIMO Recursive Least Squares A recursive implementation of the MIMO least squares can be written as an algorithm: Algorithm (MIMO RLS Algorithm) 1- Initialize to zero 2- Let = × Where c is a constant 3- For = : − 1, = [︁− [, :] · · · − [ − + 1, :], [, :] · · · [ − + 1, :]]︁ = ( )(1 + )− 1 = + ( ( + 1, :) − ) = ( − ) end Example1: consider the next dynamical system with 2.3. MIMO Maximum Likelihood For the previous given A MIMO ARMAX model the equation (1) can be developed to yield (− 1)[] = ([] + 1[ − 1] + · · ·
+ [ − ]) − (1[ − 1] + · · ·
+ [ − ]) Using the Kronecker operator we can be rewrite it as: [] = ⊗ []() − [ , , ]
(4) ⊗ [ − 1] + · · ·
+ ⊗ [ − ] = − ⊗ [ − 1] − · · · − = − ⊗ [ − 1] − · · · − ⊗ [ − ] ⊗ [ − ] = 1000 is used to excite the system. A simulation experiment has been performed for signal to noise ratio equal to 20 for both outputs. = [ , , ]
= [︁(1 ) · · · ( ) ]︁ = [︁(1 ) · · · ( ) ]︁ = [︁(1 ) · · · ( ) ]︁ as:
The best estimate of the parameter vector ˆ can be obtained using a numerical minimization algorithm such ∙ Steepest descent method: +1 = − ∇ ∙ Gauss Newton method: +1 = − (∇ ∇)− 1∇ ∇ = ⎢⎢⎢ ⎢ ⎢ ⎣ ⎡ [ + 1] ⎤ . .
. [ ] ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
For = + 1 to ■ Compute the prediction error ⎡ [ + 1] ⎤
and finally we can form = ⎢⎢ ⎢ ⎢ ⎣ . .
. [ ] ⎥ ⎥ ⎥ ⎥ ⎦ [] ( ) [] ( ) [] ( ) = = = = [] [] [] , , [] [] [] , Where it elements can be computed through MIMO IIR (Infinite Impulse Response) digital filtering using the updated matrix coeficients estimates polynomial: 1,ˆ2 of the matrix ˆ ˆ(− 1) = 2 + ˆ1− 1 + ˆ2− 2 Then using the Gauss Newton method to update the parameter vector gives the results shown below: data.
in
LMFD (− 1) = (− 1) = (− 1) = ˆ[ − ] ˆ[] = ˆ(− 1)[] − ˆ(− 1)[] − 1ˆ[ − 1] · · · − ■ Compute the partial derivatives of []: []
[] can be computed through MIMO IIR (Infinite Impulse Response ) digital filtering using the polynomial ˆ(− 1
) updated matrix coeficients estimates ˆ of the matrix Step2: Estimate the parameter vector using +1 = − ∇ or +1 = − (∇ ∇)− 1∇ with = , 0 < <
1 and is the number of / Step3: If no convergence, go to step1.
process by
its
︂[
The aim is to estimate the matrix polynomials , and (− 1 ) (− 1 ) and (− 1
) from I/O data contami- is based on the fact that the Transfer Function Matrices nated by white noise. A PRBS data sequence of length 3.2. Nonadaptive Compensator Design
Consider now the unity feedback system in the next (5) figure. The plant is described by a × proper rational matrix (RMFD)
(− 1) = (− 1)(− 1)− 1 The compensator to be designed is required to have a × proper rational matrix (LMFD).
(− 1) = (− 1)− 1(− 1) Hence the closed-loop transfer matrix is: (9) (10) (− 1) and (− 1) of a MIMO system described by the vector diference equation
[] = (− 1)[] + (− 1)[] can be represented as ratio of two polynomial matrices.
However, because matrices do not commute in general,
we note that there are two representations for the
transfer function matrix (− 1) (or (− 1)) as a ratio
of two polynomial matrices [
(− 1) = (− 1)(− 1)− 1 ∙ Left Matrix Fraction Description (LMFD)
(− 1) = (− 1)− 1(− 1) Where the matrix polynomials (− 1), (− 1), (− 1) and (− 1) have the following structures (6) (7) (− 1) = + 1− 1 + ... + − (− 1) =0 + 1− 1 + ... + − (− 1) =0 + 1− 1 + ... + − (− 1) = + 1− 1 + ... + − ∈ R× The matrix coeficients have the following dimensions:
R× , ∈ R× , ∈ R× and ∈ Remark1: it is possible to obtain either LMFD or RMFD from the other only by solving the following matrix equation (− 1)(− 1) = (− 1)(− 1) (8) This last matrix equality can be expanded and rewritten in more compact form after rearrangement into
=
Where:
is the Silvester matrix and
= [︁1 , 2 , · · · , , − 1 , · · · , −
]︁
= [︁1 , 2 , · · · , , × , × , · · · , × ]︁
and the solution vector is
= +
Hence the design problem becomes: Given (− 1) and
(− 1) and an arbitrary (− 1) , find (− 1) and
(− 1) to satisfy this compensator equation. We note
that the roots of (− 1) are the poles of the closed-loop
transfer matrix (− 1) , and the solvents of (− 1)
are block-poles of (− 1). The compensator design,
to achieve arbitrary block pole placement for the
feedback configurations described previously, requires the
solution of the compensator equation (14). Various
numerical algorithms, for solving the Diophantine equation,
have been developed and diferent approaches have been
attempted [33] and [34]. The method proposed in this
section is developed from the results obtained by Chen
[
the construction of a Sylvester matrix (or a generalized -Enter the values of: , , , = . resultant matrix of {(− 1), (− 1)}). The solution is -Enter the nominal values of the ∈ × and obtained by applying searching algorithms for linearly ∈ × dependent rows of the obtained matrix. -Initiate ˆ(− 1) and ˆ(− 1) by the values of ∈ × and ∈ × Given a set of -dimensional rows 1, 2, ..., , an × matrix () is determined recursively for = 1, 2, ..., 1. initialize (0) = ( × identity matrix) 2. for = 1, 2, ... do if ( − 1) ̸= 0 , then
︀[ ( − 1) ]︀ [︀ ( − 1) ]︀ () = ( − 1) −
( − 1)
and is linearly independent of the previous rows
else () = ( − 1)
and is linearly dependent.
3.3. Adaptation Mechanism Devolvement
Classical controllers cannot solve the problem of
uncertainties in dynamic systems, because the change in
the process parameters cause a change in that operating
conditions which leads to technical matters in the system
(Instability and undesired performance) see [
Enter the desired Block poles ∈ × to be placed and construct the corresponding matrix polynomial (− 1) Then compose the Diophantine equation as: (− 1) = ︁( ˆ(− 1)ˆ(− 1) + ˆ (− 1)ˆ(− 1))︁ Now solve the Diophantine equation using recursive search algorithm we obtain ˆ(− 1) and ˆ (− 1)
Give the desired trajectory sequence [].
Compute the closed loop output and the control law by: [] = ˆ(− 1)(︁ ˆ(− 1)ˆ(− 1) + ˆ (− 1)ˆ(− 1))︁ − 1ˆ (− 1)[] [] = ˆ(− 1)− 1ˆ (− 1)[] [] = [] − [] Identify the plant parametrs using MIMO-RLS: [ − + 1, :], [, :] · · · [ − + 1, :]]︁ Controller
D/A
P lant H(s)
A/D Nois ΔH On_Line P arameter
Estimation
Calculation Of T he
Controller P arameters 4. Application to Winding process Winding systems are encountered in a wide variety of industrial plants such as rolling mills in the steel industry, plants involving web conveyance including coating, paper-making and polymer film extrusion processes. The main role of a winding process is to control the web conveyance in order to avoid the efects of friction and sliding, as well as the problems of material distortion and can also damage the quality of the final product [ 35]. The illustrative example used here is modeled by identification of RMFD using constrained PEM see [36].
Remark2: In order to simplify the control procedure let we chose a fixed structure compensator of 1 order With constant gain pre-compensator ⎢⎢⎢ ˆ1 ⎥⎥ ⎢ ⎥⎥⎥ = ⎢⎢⎢ 1 ⎢⎢⎣ ˆ 0 ⎢⎣ 2
⎥ ⎦
3 ⎡ 3 3 3 1 2 3 1 2 3 3 ⎤− 1 ⎡ 3
⎤ 3 ⎥
⎥ 1 ⎥⎥
⎦ 2 ⎢⎢ 1 ⎥ ⎢ ⎥ ⎢ 2 ⎥⎥ ⎣ ⎦ 3 The nominal values of the compensator coeficients are obtained according to this last equation.
Remark3: assuming that the system uncertainties are of 7% of the nominal one, means that (− 1) = 0(− 1) + ∆ (− 1) Now starting the adaptive block pole placement algorithm we obtain the next results as shown in figure It can be observed from the above simulation results that the algorithm developed in this paper able to assign block poles guaranteing the system stability even if some sudden uncertainties occurs and with smaller tracking errors. The influence of the parameter change don’t afect the designed digital compensator due to the goodness of its adaptation mechanism, also simulation results show that relatively small interactions for the closed-loop system 2 g n i k1 a r T y r o0 t c e j a −1 r T e h T −2 −30 10
Time 2(0sec) 40 when the setpoint of one of the variables is changed. Which means that the closed loop system is perfectly and complectly decoupled, This is because the control action produced by the MFD in both variables acts simultaneously on both manipulated variables as soon as a change in the reference of any of them is detected.
In this paper, a new adaptive Block pole-placement control for MIMO discrete-time systems has been considered. This control scheme includes the MIMO RLS estimation algorithm and Block pole-placement control, which is used not only to identify the unknown plant parameters but to achieve some specified performances. The proposed control scheme indeed improves both regulation and tracking error. 6. Declaration on Generative AI During the preparation of this work, the authors used ChatGPT, Grammarly in order to: Grammar and spelling check, Paraphrase and reword. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the publication’s content.