# Delta models in adaptive control of nonlinear process.

1. INTRODUCTION

Control of a nonlinear or t-variant process using fixed set PID controllers causes many negative effects which usually impair the quality of regulation and reduce the lifetime of a controlled facility. We describe one way how to reduce these problems in this paper. The main aim is the approximation of a time-continuous model of controlled process by a discrete [delta]-model and continually identifying of its parameters.

2. DELTA MODELS

Delta models represented a link between t-continuous [delta]-models and discrete Z-models. The parameters of this discrete models are convergented to the parameters of t-continuous models for a short sample time period (with regard to a dynamism of process).

When we are identifying the parameters we will be look for the [delta]-model whose structure will be same as the structure of T-continuous model. We suppose that the period of identification process is much shorter then dynamism of process. The parameters of the 8-model are much closed to the parameters of t-continuous model of controlled process. EXTERNAL LINEAR 8-MODEL OF CONTROLLED PROCESS

T-continuous external linear model (ELM) is choosen on the basis of fundamental knowledge of dynamics of nonlinear control process. This model is described by a difference equation in time domain.

a([sigma])y(t) = b([sigma])u(t) (1)

and in a complex domain by a transfer function:

G(s) = b(s)/a(s) (2)

with conditions of tightness deg b [less than or equal to] deg a.

DELTA MODEL OF PROCESS

We establish the [delta]-operator which is defined like:

[delta] = q - 1/[T.sub.0] (3)

where: q presents a shift operator and To a sample time.

When the sample time period is shorted, then the [delta]-operator approximates the derivative operator a so that it applies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

and [delta]-model

a'([delta])y(t') = b'([delta])u(t') (5)

approximates the t-continuous model. In (5) operator t' is a discrete time and a', b' are [delta]-polynomials.

3. SPEED SERVOMECHANISM MODEL

T-continuous model of second order has been choosen as:

[??](t) + [a.sub.1][??](t) + [a.sub.0] y(t) = [b'sub.0] u(t) (6)

Delta model which is corresponding with (6) has a form:

[[delta].sup.2] y(t') + [a'.sub.1] [delta]y(t') + [a'.sub.0]y(t') = [b'.sub.0] u(t') (7)

Regression vector has a form:

[[PHI].sup.T.sub.[delta]] (k - 1) = [-([[phi].sub.y](k - 2 ) - [[delta].sub.y](k - 1) [[delta].sub.u] (k - 2)] (8) where:

[[delta].sub.y](k - 2) = y(k - 2) (9)

[[delta].sub.y](k - 1) = y(k - 1) - y(k - 2)/[T.sub.0] (10)

[[delta].sub.u](k - 2) = u(k - 2) (11)

Vector of delta model parameters

[[PHI].sup.T.sub.[delta]](k) = [[a'.sub.0] [a'.sub.1] [b'.sub.0]] (12)

is recursively estimated from the equation:

[[delta].sub.y](k) = [[PHI].sup.T.sub.[delta]](k) = [[THETA].sup.T.sub.[delta]] (k) [[PHI].sub.[delta] (k - 1) + [epsilon] (k) (13)

where:

[[delta].sub.y](k) = y(k) - 2y(k - 1) + y(k - 2)/[T.sup.2.sub.0] (14)

For the identification of parameters has been used recursive identification algorithm with directional forgetting. More information can be found for example in (Bobal, 2008).

4. TWO FEEDBACK CONTROLLERS (TFC)

[FIGURE 1 OMITTED]

Transfer in a control loop

G--linear input-output model of control process Q, R--feedback controllers

Signals in a control loop

w--reference signal e--control error [u.sub.vl], [u.sub.v2]--control action

v--system intput disturbance u--process-input-control action y--output signal

Transfer functions of both feedback controllers are considered in the form:

Q(s) = [??](s)/[??](s) R(s) = r(s)/[??](s) (15)

where [??](s), [??](s) a r(s) are relative prime polynomials which are expressed in terms of complex variable.

In this case, both input signals w a v (reference signal and disturbance) are considered only as a step function with the images:

W(s) = [W.sub.0]/s V(s) = [v.sub.0]/s (16)

For the output signal and for the disturbance can be derived (the argument is omitted for clarity):

Y(s) = b/d [rW(s) + [??]V(s)] (17)

E(s) = 1/d [(a[??] + b[??])W(s) - b[??]V(s)] (18)

where:

d(s) = a(s)[??](s) + b(s) x (r(s) + [??](s)) (19)

where d(s) is a characteristic polynomial with roots that are represented the poles of the closed control loop.

Now we choose the polynomial t(s) which meets a condition:

t(s) = t(s) + [??](s) (20)

Polynomial t(s) we substitute into (19). The stability condition will be satisfied if the polynomials [??](s) and t(s) are given by a solution of a Diophantine equation:

a(s)[??](s) + b(s) x t(s) = d(s) (21)

with a stable polynomial d(s) on the right side.

The asymptotic tracking of a reference signal and the elimination of a disturbance will be guaranteed for a step input signals if the conditions a[??] + b[??] and p contain s.

These conditions will be fulfilled if the polynomials p and q are in the form:

[??](s) = sp(s) (22)

[??](s) = sq(s) (23)

The transfer function of controllers we can consider in the form:

Q(s) = q(s)/p(s) R(s) = r(s)/s x p(s) (24)

The stability of controllers is garanted by a stable polynomial t(s) in a denominator of (24).

The degree of polynomials q(s) and r(s) must satisfy the following inequality to meet the conditions of inner purity control system:

deg q [less than or equal to] deg p (25)

deg q [less than or equal to] deg p + 1 (26)

The polynomial t(s) we rewrite to the form:

t(s) = r(s) + s x q(s) (27)

When we considering the solvability of (22) and conditions (26) and (27), the degree of polynomials may be identified as:

deg t = deg r = deg a (28)

deg q [greater than or equal to] deg a - 1 (29)

deg p [greater than or equal to] deg a - 1 (30)

deg d [greater than or equal to] 2deg a (31)

If then the polynomials t, r and q are in the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

where the coefficients [r.sub.i], [q.sub.i] and [t.sub.i] are considered the conditions:

[r.sub.0] = [t.sub.0]

[r.sub.i] + [q.sub.i] = [t.sub.i] pro i = 1, ...,n (34)

Unknown coefficients [r.sub.i] and [q.sub.i] can be calculated by using a optional factors [[beta].sub.i] [member of] <0,l>:

[r.sub.i] = [[beta].sub.i] [t.sub.i] (35)

[q.sub.i] = (1 - [[beta].sub.i]) x [t.sub.i] pro i = 1,... , n (36)

The coefficients distribute a weight between numerators of transfer functions Q and R. It is true, if the value of is increased, the response to a step change is accelerating. Notice: If [[beta].sub.i] for all i, the TFC engagement corresponding with 1DOF system control configuration and if ([beta] = 0 for all i and setpoint and disturbance are a step functions, it corresponding with 2DOF system control configuration.

5. REAL MEASUREMENTS

The transfer functions of controllers have a form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

Characteristic polynomial: d(s) = n(s)[(s + 0.9).sup.2] where the polynomial n(s) is a product of spectral factorization of the polynomial a(s). Controller parameters: [beta] = 0 Sample time period: [T.sub.0] = 0.025s Duration of the control: t = 300s Identification process parameters: [C.sub.ii] = 1000, [delta](0) = 1,

[lambda](0) = 0.001, v(0) = [10.sup.-6], [??] = 0.99

[FIGURE 2 OMITTED]

6. ACKNOWLEDGEMENTS

This work was partly supported by the Ministry of Education of the Czech Republic under the grant MSM 7088352101 and by TBU in Zlin under the grant IGA/55/FAI/10/D.

7. REFERENCES

Babik, Z. (2009). Pole placement method in control of linear continuous-time SISO systems, Diploma work, Thomas Bata University, Faculty of Aplied Informatic, Zlin

Dostal, P., Bobal V., Gazdos F. (2005). Adaptive control of a nonlinear process by two feedback controllers, In: 13th Mediterranean Conference on Control and Automation, Limassol, 946-951, ISBN 0-7803-8937-9, Cyprus

Dostal, P. (2006). State and algebraic control theory, textbooks, Thomas Bata University, Faculty of Aplied Informatic, Zlin

Bobal, V. (2008). Adaptive and predictive control, 1. release, Thomas Bata University, Faculty of Aplied Informatic, ISBN 80-7318-662-3, Zlin

Vojtesek, J. (2003). Simulation and control of a nonlinear system--continuous stirred tank reactor (CSTR). Research study, Thomas Bata University, Faculty of Technology, Zlin
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