Implicit self-tuning control for a class of nonlinear systems

Controlador auto-ajustable para una clase sistemas no lineales

One goal of control theory is to propose mathematical tools and algorithms to auto-regulate real process, given some desired specifications. Also one of the goals of several control theory researches is to find a control law that works for a large group of real process: linear or nonlinear, Single Input Single Output (SISO) or Multi Inputs Multi Outputs (MIMO), time invariant or time variant, and so on. Howev-er to find this type of controller is a hard work and is what keeps most of the control theory researches continuously working on it. On the other hand, the close-loop stability of controlled process is one of the most important issues to as-sure and prove when a new control algorithm is proposed.

The stability of implicit self-tuning control has been proved, for the linear discrete-time case, by the use of a Lyapunov function in (Patete y col., 2008a; 2008b), and for those systems, it suffices to use linear functions of the data to predict the system output response. The proposed algorithm was extended to the case where the linear discrete-time system is subject to white noise (Patete y col., 2008c), i.e. ARX (AutoRegresive with eXternal input) model. Several real systems have multi inputs multi outputs and, for that type of systems the results given in (Patete y col., 2008a; 2008b) were extended to the MIMO case in (Sugiki y col., 2008) and (Furuta y col., 2011).

It has been shown under relatively mild conditions that a large class of nonlinear systems can be approximated with arbitrary precision using bilinear models with finite number of coefficients. Bilinear systems are the simplest class of nonlinear systems and can also be regarded as a practical starting point for the study of other nonlinear systems. In addition, many concepts associated with linear systems can be extended to the bilinear case. A new algorithm was pro-posed, based on the results in (Patete y col., 2008a; 2008b), for the self-tuning control combining recursive parameters estimation and generalized minimum variance criterion, for a class of bilinear systems in (Patete y col., 2008d; 2011a), and also for an extended and more relaxed class of bilinear systems, where the control action could be presented only in the bilinear term in (Patete y col., 2010; 2014). However real world systems are mostly nonlinear systems and it is of interest to extend the proposed algorithm to a more complex class of nonlinear models. In general, it may be desirable, to consider the use of nonlinear functions to get good predic-tions and hence good control performance.

The paper is organized as follows: section 2 presents the problem to be solved; in section 3, the nonlinear system class to deal with is defined. Section 4 presents the generalized minimum variance criterion for the defined system class and, in section 5 the recursive implicit self-tuning algorithm based on the generalized minimum variance criterion is studied and, the main results are given by the theorem and proof which assure closed-loop system global stability. Some remarks conclude the paper.

Consider the general, Single Input Single Output (SI-SO), structure in the discrete-time case of a nonlinear sys-tem model as in (1),

(1)

where y_k is the output signal of the process, u_k is the input signal, z denotes the time shift operator: A (z,q) and are polynomial of the form: B( z, q )

q in the general case is a function of the input and output signal of the process as in (2),

(2)

Let’s consider now the first order model general case,

(3)

with q=h(y_k). If a (q) and b (q) are constant values independents from the output signal y_k, i.e ɑ(q)= ɑ₀ and bq= b₀ , then the case is the same as for linear, first order, systems considered in (Patete, 2008a). If ɑ(q) = ɑ₀ and b(q)= b₀+b₁y_k, then the case is the same as for bi-linear, first order, systems considered in (Patete y col., 2008d; 2011a; 2010; 2014).

From the above explanation, the first step to deal with this type of nonlinear systems structure is to define how to choose function q=h(y_k) (or q=h(y_k,u_k) in the gen-eral case), which is to said how to choose a(q) and b(q) for the first order case (3).

For example, for the first order system (3), if a(q) =a₀ + a₁y_k an b(q)=b₀ then (4),

(4)

Or the case when and , then (5):

(5)

and for these cases, (4) and (5), no results have been given.

Consider the general nonlinear system model structure as in (1), and q is defined as in (2), where h( y_k, u_k is any function (linear or nonlinear).

In this paper, to define the nonlinear class to deal with, the function h(y_k , u_k) is restricted to be a function depend ing only of the output process data, i.e. h (y_k, then q is as

follows

(6)

When (6) is substituted in (1), a nonlinear model with polynomial structure is obtained. For example consider a nominal model of a first order SISO time invariant nonlinear system as in (7),

(7)

where the functions a₀ (q) and b₀ (b)are polynomial and depend only of the output process signal as shown in (8) and (9):

(8)

(9)

Using (8) and (9) in (7), (10) is obtained,

(10)

where d=1and:

Note in (10) that the nonlinear terms are polynomial (depend on ) and there are bilinear terms (depend on and ).

In general, the class of nonlinear systems is defined as a SISO time invariant model (11) with the following structure:

(11)

with q as in (6).

In this section a generalized minimum variance control in (Åströmy col., 1989) (Chang y col., 1989), based on the concept of discrete-time sliding mode control (Furuta, 1990; 1993, Slotine, Li, 1991), is proposed for the defined class of nonlinear systems.

Consider de general nonlinear model (11), if:

(12)

then, substituting (12) in (11), writing the equation in the present time k and grouping terms, the following (13) is obtained

(13)

where d= n and:

Assumptions 1:

1. 1) There are no common factors in:

1. 2) The order of the system (n ,m ) in (1) is known.

2. 3) The time delay, d, is known.

3. 4) To compute the nominal control law, the polynomials

The control objective of the control law is to minimize the variance of the linear controlled sliding mode variable ^s_k+d , defined as (14):

(14)

5

where polynomials C(z^-1) and Q (z^-1)are define as in (15) and (16) respectively,

6

(15)

(16)

Remark 1: Polynomial 1 C(z ) is designed to be Schur (i.e is designed by assigning all characteristic roots inside the unit disk of the z-plane for discrete-time systems) and polynomial Q(z^-1) must be designed as in (16) for the reference tracking to be assured (Patete 2008a; 2008b).is designed to be Schur (i.e is designed by assigning all characteristic roots

Polynomials C(z^-1) and Q (z^-1) are to be designed,

so the error signal ek, defined as (17):

(17)

Where rk is the reference signal. The idea of proposing (14) defining the error signal as in (17) is based on the discrete-time sliding mode control (Furuta 1990; 1993).

To derive the nominal control law, general model (13) is multiplied by E (z-1), then:

(18)

where E(z^-1 )is a polynomial of the form:

Using the Diophantine equation (Patete 2008a; 2008b) (Chang y col., 1968):

(19)

where,

equation (18) is rewritten as:

(20)

where:

Combining (20) and (14), the variable S_k+_d is:

(21)

where G(z^-1)=E(z^-1)B(z^-1)+Q(z^-1).

Then, the generalized minimum variance control input required to vanish S_k+_d in (14) is given by:

(22)

5 Self-tuning control for the defined nonlinear system class

As it is known, parameters of real world process may not be accurately known or precise measured, or even worst parameters may change in time, and a good performance controller should be able to keep the overall system stability in such a case. System (13) is considered as a system with the same structure, however parametric uncertainties is tak-ing into consideration in this section. For the implicit self-tuning controller, the parameter of the nominal control law (22) are estimated each sampled time, under the following assumptions,

Assumptions 2:

1. 1) The order of the system (13) is known.

2. 2) The time delay, d , is known.

3. 3) Polynomial C(z^-1)is Schur.

4. 4) Polynomial Q(z^-1) is designed as in (16).

5. 5) The considered system with parametric uncertainties is in the class of systems which can be stabilized by the polynomials Q(z^-1)and C(z^-1)designed for the nominal system model (13) (Patete, 2008a; 2008b).

6. 6) The reference signal rk is bounded, i.e. for all k , where ȣ is a positive constant.

(23)

And

(24)

Where

(25)

is the vector containing measured output and control signal data,

(26)

is the vector containing the controller parameters, and

(27)

is the estimate of . ϴ

The controller uses identified parameters as follows:

(28)

Theorem 1: Given a positive definite matrix and the initial parameters vector , if the estimate of the controller (28) satisfies the recursive equations (23) and (24), under the set of Assumptions 2, then the close-loop system, combined by the self-tuning controller (28), (23) and (24) for the class of nonlinear system (13) with para-metric uncertainties is global stable.

Proof: sk +d is written as:

(29)

Where

Using the control law (28), (29) is rewritten as:

(30)

Consider the candidate Lyapunov function:

(31)

The time difference of (31) is:

(32)

(33)

(34)

(35)

From (30), S_k is:

(36)

Substituting (36) into (35), the following relation is de-rived:

(37)

The term:

in (37) can be made equal to zero as follows:

(38)

(39)

(40)

that yields (24) by the matrix inversion lemma (Åström y col., 1989).

The term:

in (37) also can be made equal to zero as described be-low:

(41)

(42)

(43)

(44)

From (21):

(45)

thus (23) is derived.

Using the recursive equations (23) and (24) in (37), for , the following relation is obtained: K=1

(46)

Initially then V1-V0<0 which gives that in V1

(47)

For , k=3,

(48)

using (47) and (48), the following is obtained:

(49)

Then, for K=N , where N is large, the following relation is derived:

(50)

(51)

For any K=N(K>2), inequality (51) holds. Equa-tion (51) implies that s_N and vanish as N ap-proaches infinity, thus ∆V_k is negative semi-definite for all K and the generalized minimum variance is minimized, which proves the overall closed-loop system stability.

As a result from the above proof , is bounded. This means that:

are bounded for all k . Furthermore as , and which means that goes to a constant value (not necessary the real value ).

The actual value y_k is shown to be bounded as fol-lows:

Multiplying (14) by B(z^-1)

(52)

(53)

and using (13):

(54)

(55)

Where T(z^-1) is defined as:

(56)

The signal S_K was proved to go to zero as

The signal rk is assumed to be bounded for all k and the signal was proved to be bounded from the boundeness of vector . From the set of Assumptions 2, number 5 means that the closed-loop characteristic polynomial, considering the described plant with parametric uncertainties, in (1), 1 T(z^-1), is Schur. Thus, k y in closed-loop is proved to be bounded. Furthermore, the error e_k= y_k-r_kis bounded.

To reinforce this proof, we use the proof proposed in (Ohata y col., 2014), as follows:

In time, k

(57)

Equation (57) is rewritten as in (54), where

(58)

Defining as:

(59)

Using the new Lyapunov function:

(60)

The time difference of (60) is:

(61)

(62)

If the terms:

And

are satisfied, then:

(63)

This means that converge to zero and converge to zero as , which implies that and are bounded. Then approaches zero because of the following relation.

(64)

(65)

(66)

(67)

(68)

Thus, the output Ykapproaches the reference r_kas because C(z^-1)is Schur. The global stability of the considered closed-loop system using the implicit selftuning controller is proved.

This algorithm represent a more general implicit (or al-so called direct) self-tuning control, based on the nominal generalized minimum variance control, using the discrete-time sliding mode control concept. The algorithm may be applied to linear SISO system model defined as in (Patete y col., 2008a; 2008b), bilinear and a more relaxed class of bi-linear systems models given by (Patete y col., 2008d; 2011a; 2010; 2014), and for the defined nonlinear system class exposed in this paper.

The algorithm presented in this work may be also ex-tended to linear ARX SISO system model presented in (Patete y col., 2008c), time-variant systems model given in (Patete y col., 2007; 2011b), and MIMO system model as in (Sugiki y col., 2008) (Furuta y col., 2011), combining the proof given in this paper and the proof given in each refer-ence, respectively.

A more general implicit self-tuning control algorithm, based on the nominal generalized minimum variance con-trol, using the discrete-time sliding mode control concept was presented. The algorithm may be applied to linear SISO system models, bilinear models, and for the defined nonlin-ear system class defined in this paper. The proposed self-tuning approach enables controller parameters to be esti-mated. The closed-loop global stability of the proposed im-plicit self-tuning control for the defined class of nonlinear systems was proved. Control stability and reference track-ing are shown to be assured. The given algorithm is based on the idea of the discrete-time sliding mode control con-cept. As a future work, the proposed algorithm is to be ap-plied to some nonlinear process model to show its perfor-mance.

This research was supported by Grants-in-Aid for Sci-entific Research (B) (Grant Number 24360166) of JSPS, Japan. The authors acknowledge the comments and sugges-tions given by Dr. Akihiko Sugiki of Nikki Denso Co., Ltd, Japan, and Akira Ohata of Toyota Motor Corporation, Japan.