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Revista Técnica de la Facultad de Ingeniería Universidad del Zulia

versión impresa ISSN 0254-0770

Rev. Téc. Ing. Univ. Zulia vol.34 no.2 Maracaibo abr. 2011

 

Stability of implicit self-tuning controllers for time-varying systems based on Lyapunov function 

Estabilidad de controladores auto-ajustables para sistemas variantes en el tiempo basada en funciones Lyapunov 

Anna Patete1, Katsuhisa Furuta2 

1Escuela de Ingeniería de Sistemas, Facultad de Ingeniería, Universidad de Los Andes, Núcleo La Hechicera. Av. Alberto Carnevalli, Mérida 5101, Venezuela.
Telf: 58-274-2402986, Fax: 58-274-2402847. apatete@ula.ve
2Department of Computer and Systems Engineering, Tokyo Denki University.
Saitama, Hatoyama, Hikigun, Saitama, 350-0394, Japan.
furuta@k.dendai.ac.jp 

Abstract 

The important issue on self-tuning control includes the stability, performance and convergence of involved recursive algorithms. Based on a Lyapunov function, this paper proves the stability of implicit self-tuning controllers, combining recursive parameters estimation with a forgetting factor and generalized minimum variance criterion, for time-varying systems. The system parameters are considered to be changing continuously but slowly or changing abruptly but infrequently. The analysis is extended to the case where the system model is subject to system and measurement noises. The main results are the theorems which assure the overall stability of the closed-loop system, which are proved in a straight way compared with previous stability analysis results. 

Key words:  Self-tuning control, generalized minimum variance control, sliding-mode control, discrete-time systems, time-varying systems, Lyapunov function. 

Resumen 

Los problemas más importantes de los controladores auto-ajustables, se refieren a la estabilidad, desempeño y convergencia de los algoritmos recursivos involucrados. Basándose en funciones Lyapunov, este trabajo prueba la estabilidad de los controladores auto-ajustables implícitos, combinando la estimación recursiva de los parámetros del controlador incluyendo el factor de olvido con el criterio de variancia mínima generalizada, aplicados a los sistemas variantes en el tiempo. Se considera que los parámetros del sistema cambian continua pero lentamente o cambian abrupta pero infrecuentemente. El análisis también se extiende al caso donde el modelo del sistema incluye ruido. Los principales resultados presentados en este trabajo son los teoremas que aseguran la estabilidad global del sistema en lazo cerrado, los cuales son probados en una manera mucho más directa comparando con resultados previos en el área. 

Palabras clave:  controladores auto-ajustables, control de varianza mínima generalizada, control por régimen deslizante, sistemas discretos, sistemas variantes en el tiempo, función de Lyapunov. 

Recibido el 8 de Diciembre de 2009 

En forma revisada el 10 de Enero de 2011 

1. Introduction 

Åström [1] showed, in a stochastic way, that a self-tuning minimum variance control is optimal for a general CARMA (Controlled Auto-Regressive Moving Average) model, even though the noise parameters are not explicitly estimated; however the proof begins with the assumptions that parameters convergence is assured (not necessarily to the optimal values) and the system model is minimum phase. 

By extending the minimum variance criterion (MVC) of [1], Clarke [2] propose the generalized minimum variance control (GMVC) for non-minimum phase systems by the use of a cost function which incorporates system input and set-point variation, and a control law was derived for a system model with known parameters. The parameters of the control law for the real systems with unknown parameters are estimated using a recursive least-squares (RLS) algorithm. Furuta [3] proposed a discrete-time variable structure system (VSS) approach to the case where system parameters are unknown; the VSS is designed based on MVC or GMVC, and a recursive estimator of controller parameters is applied. 

Based on key technical lemmas, the global convergence of implicit self-tuning controllers was studied for discrete-time minimum phase linear systems in a seminal paper by Goodwin [4] and for explicit self-tuning controllers in the case of non-minimum phase systems by Goodwin [5]. From the viewpoint of sliding mode control (SMC), Patete [6, 7] gave a complete proof for the stability of implicit self-tuning controllers based on GMVC for minimum or non-minimum phase systems by the use of a Lyapunov function. However, all these researches have been done for time-invariant systems (TIS). Clark [8] studied the stability of self-tuning controllers for time-invariant systems subject to noise, based on the idea of describing the system in a feedback form and using the notion of dissipative-real systems. However no rigorous stability proof was given. 

The purpose of this paper is to analyze the stability of the implicit self-tuning controller for discrete time-varying systems (TVS) and discrete time-varying systems subject to system and measurement noises. The criterion considered is the minimization of an auxiliary controlled variable based on the concept of sliding mode control to yield the system stability. 

The paper is organized as follows: in section 2, the GMVC based on the sliding mode control concept [3] is reviewed. Section 3 studies the recursive estimation of controller parameters based on GMVC to deal with time-varying systems. Simulation examples are given in section 4. Some remarks conclude the paper. 

2. Generalized minimum variance control 

The controller design with the GMVC based on the sliding mode control concept, in the case of time-invariant systems [3, 6, 7], is reviewed in this section. The discrete-time single-input single-output (SISO) time-invariant system is considered. The representation of the nominal system with input _VP_EQN_0.GIF and output _VP_EQN_1.GIF is given by: 

_VP_EQN_2.GIF,     (1) 

where _VP_EQN_3.GIF and _VP_EQN_4.GIF have no common factors and z denotes the time-shift operator _VP_EQN_5.GIF. In the Laplace transformation, the time-shift operator is described as _VP_EQN_6.GIF where _VP_EQN_7.GIF is the sampling period (for simplicity, and without loss of generality, _VP_EQN_8.GIF is assumed). In order to derive the nominal control law the polynomials _VP_EQN_9.GIF and _VP_EQN_10.GIF are assumed to have constant and known parameters, represented by: _VP_EQN_11.GIF and _VP_EQN_12.GIF, where _VP_EQN_13.GIF. The delay step, d, is also assumed to be known. The control objective is to minimize the variance of the controlled variables _VP_EQN_14.GIF, defined in the deterministic case as: 

_VP_EQN_15.GIF,     (2) 

where the polynomials _VP_EQN_16.GIF  _VP_EQN_17.GIF and _VP_EQN_18.GIF are to be designed, so that the specifications written below should be satisfied. The error signal _VP_EQN_19.GIF is defined as _VP_EQN_20.GIF, where _VP_EQN_21.GIF is the reference signal. The idea is similar to the discrete-time sliding mode control [3]. 

The polynomial _VP_EQN_22.GIF is chosen Schur and should be designed by assigning all characteristic roots inside the unit disk in the z-plane. Equation (2) is rewritten as: 

_VP_EQN_23.GIF,     (3) 

where the polynomial _VP_EQN_24.GIF is defined as _VP_EQN_25.GIF, i.e. _VP_EQN_26.GIF  _VP_EQN_27.GIF and the polynomials _VP_EQN_28.GIF and _VP_EQN_29.GIF  _VP_EQN_30.GIF satisfy the equality: 

_VP_EQN_31.GIF.     (4) 

Then the GMVC input required to vanish _VP_EQN_32.GIF in Eq. (2) is given by: 

_VP_EQN_33.GIF,     (5) 

where the polynomials _VP_EQN_34.GIF and _VP_EQN_35.GIF are chosen to make the control system satisfy the following Lemma. 

Lemma [9]: The necessary and sufficient condition for the control input to make _VP_EQN_36.GIF stable is that all the roots of the polynomial _VP_EQN_37.GIF

_VP_EQN_38.GIF,     (6) 

belong to the open unit disk, and the polynomials _VP_EQN_39.GIF, _VP_EQN_40.GIF, _VP_EQN_41.GIF have no common zeros outside of the unit disk. 

The uncertainty in system characteristics leads to a certain family of models rather than to a single system model to be considered. In the case where the uncertainties come from parametric perturbations, we have a family of closed-loop characteristic polynomials _VP_EQN_42.GIF instead of a single nominal characteristic polynomial _VP_EQN_43.GIF. Defining _VP_EQN_44.GIF, which maps the stable zone inside the unit circle into the outside in the z-plane, then _VP_EQN_45.GIF is defined as: 

_VP_EQN_46.GIF,         (7) 

where g is a positive constant representing the margin of perturbation and _VP_EQN_47.GIF gives a set of admissible perturbations defined as _VP_EQN_48.GIF. For robust stability analysis, we may use the method by Tsypkin [10] for closed-loop discrete-time systems, which involves the modified characteristic locus criterion. 

Criterion [10]: For robust stability of closed-loop discrete-time parametric systems, it is sufficient that 

_VP_EQN_49.GIF,     (8) 

does not enclose and does not intersect the critical circle o of radio g and centered at the origin, i.e. _VP_EQN_50.GIF, when w moves from 0 to _VP_EQN_51.GIF. _VP_EQN_52.GIF and _VP_EQN_53.GIF are the range of parametric perturbations, defined as _VP_EQN_54.GIF, _VP_EQN_55.GIF. Where _VP_EQN_56.GIF, _VP_EQN_57.GIF and _VP_EQN_58.GIF, _VP_EQN_59.GIF are the upper and lower bounds of _VP_EQN_60.GIF and _VP_EQN_61.GIF, respectively. 

3. Self-tuning control of time-varying systems based on GMVC 

It has been proved [6, 7] that the following self-tuning algorithm assures the overall stability for SISO time invariant systems, when the system constant parameters are not accurately known, by the recursive estimation of the controller parameters _VP_EQN_62.GIF and _VP_EQN_63.GIF, under the following assumptions. 

Assumptions 1: a) The order of the system in Eq. (1) is known. b) The delay step d is known. c) Polynomial _VP_EQN_64.GIF is Schur. d) Lemma 1 is satisfied. e) Criterion 1 is satisfied. f) The given reference signal r is bounded. 

The self-tuning control based on GMVC algorithm is given by the following recursive estimation equations: 

_VP_EQN_65.GIF ×
       _VP_EQN_66.GIF,     (9), 

_VP_EQN_67.GIF    (10) 

where _VP_EQN_68.GIF is the vector containing measured output and control signal data. _VP_EQN_69.GIF is the vector containing the parameters of _VP_EQN_70.GIF and _VP_EQN_71.GIF, and _VP_EQN_72.GIF is the estimate of q. 

Then the controller includes identified parameters as follows: 

_VP_EQN_73.GIF, (11) 

where _VP_EQN_74.GIF and _VP_EQN_75.GIF are estimates of _VP_EQN_76.GIF and _VP_EQN_77.GIF, respectively. 

In several adaptive problems it is of interest to consider the situation in which the parameters are time-varying. From now on, system parameters are assumed to change abruptly but infrequently or changing continuously but slowly. Then, the family of system models is represented by: 

_VP_EQN_78.GIF,    (12) 

where (1) is the nominal system model of (12). The self-tuning control based on GMVC algorithm presented is extended to the case where a forgetting factor is introduced into the recursive estimate equations (9) and (10) to deal with this type of time-varying systems. 

Assumptions 2: a) The unknown time- varying system parameters are assumed all uniformly bounded away from infinity. b) The time-varying controller parameters vector _VP_EQN_79.GIF has the following model, _VP_EQN_80.GIF, where _VP_EQN_81.GIF is a zero mean time-varying signal. Thus, _VP_EQN_82.GIF _VP_EQN_83.GIF with _VP_EQN_84.GIF denoting the expectation with respect to _VP_EQN_85.GIF

Theorem 1 

Recursive estimates of controller parameters based on generalized minimum variance criterion with a forgetting factor: Given a positive definite matrix _VP_EQN_86.GIF, a parameter _VP_EQN_87.GIF and an initial parameters vector _VP_EQN_88.GIF, if the estimate _VP_EQN_89.GIF of the controller in Eq. (11) is given by the recursive equations: 

_VP_EQN_90.GIF ×
       _VP_EQN_91.GIF,     (13) 

_VP_EQN_92.GIF,     (14) 

under Assumptions 1 and 2, then the overall closed-loop time-varying system combining equations (11), (13), (14) and equation (12) gives the overall stability in the sense of the expectation with respect to _VP_EQN_93.GIF

Proof: Using the control law in Eq. (11), _VP_EQN_94.GIF may be rewritten as: 

_VP_EQN_95.GIF,         (15) 

where, 

_VP_EQN_96.GIF.         (16) 

The candidate Lyapunov function is given by: 

_VP_EQN_97.GIF.     (17) 

The time difference of Eq. (17) is considered as follows: 

_VP_EQN_98.GIF,     (18) 

where µ is the forgetting factor, _VP_EQN_99.GIF. Then, for _VP_EQN_100.GIF, the following is derived: 

_VP_EQN_101.GIF
         _VP_EQN_102.GIF        (19) 

_VP_EQN_103.GIF
             _VP_EQN_104.GIF
             _VP_EQN_105.GIF              (20) 

_VP_EQN_106.GIF
             _VP_EQN_107.GIF
             _VP_EQN_108.GIF .           (21) 

From Eq. (15), _VP_EQN_109.GIF is: 

_VP_EQN_110.GIF.         (22) 

Substituting Eq. (22) into Eq. (21), the following relation can be obtained: 

_VP_EQN_111.GIF
             _VP_EQN_112.GIF
             _VP_EQN_113.GIF
            (23) 

The third and fourth terms in the right-hand side of Eq. (23) may be equal to zero as follows: 

From the third term on the right-hand side of Eq. (23), 

_VP_EQN_114.GIF,     (24) 

_VP_EQN_115.GIF.     (25) 

Equation (25) yields Eq. (14) by the matrix inversion lemma. From the fourth term of Eq. (23): 

_VP_EQN_116.GIF.     (26) 

Using Eq. (3) and Eq. (16), and substituting them into Eq. (26) the following is obtained: 

_VP_EQN_117.GIF
       _VP_EQN_118.GIF,         (27) 

_VP_EQN_119.GIF
                                  _VP_EQN_120.GIF
                                 _VP_EQN_121.GIF.
            (28) 

Finally, by taking the expectation with respect to _VP_EQN_122.GIF in (28), Eq. (13) is derived. 

Then, by using of the recursive equations (13) and (14) for a positive bounded _VP_EQN_123.GIF, _VP_EQN_124.GIF is proved negative semi-definite, i.e. _VP_EQN_125.GIF, as follows: 

Using the recursive equations (13) and (14) into (23), for _VP_EQN_126.GIF the following relation is obtained 

_VP_EQN_127.GIF
_VP_EQN_128.GIF.     (29) 

Initially _VP_EQN_129.GIF, then _VP_EQN_130.GIF, which gives _VP_EQN_131.GIF. For _VP_EQN_132.GIF

_VP_EQN_133.GIF
_VP_EQN_134.GIF.         (30) 

Then, for a large N the following relation is derived: 

_VP_EQN_135.GIF  × _VP_EQN_136.GIF.     (31) 

Equation (31) implies that _VP_EQN_137.GIF approaches to zero as N goes to infinity; then the left-hand side of Eq. (31) will also vanish. Thus, _VP_EQN_138.GIF and _VP_EQN_139.GIF vanish as N approaches to infinity. 

Since the polynomial _VP_EQN_140.GIF is designed to satisfy Eq. (6) and Eq. (7) for a bounded reference _VP_EQN_141.GIF, both input _VP_EQN_142.GIF and output _VP_EQN_143.GIF are shown to be bounded, i.e. multiplying Eq. (2) by _VP_EQN_144.GIF and using Eq. (12), the following expression for the control signal _VP_EQN_145.GIF is derived: 

_VP_EQN_146.GIF
       _VP_EQN_147.GIF.     (32) 

_VP_EQN_148.GIF vanishes in the sense of expectation with respect to _VP_EQN_149.GIF as k goes to infinity, as shown in Eq. (31). Since Eq. (7) is designed robust stable then, for a bounded reference _VP_EQN_150.GIF, _VP_EQN_151.GIF is proved to be bounded for all k; thus from Eq. (2) _VP_EQN_152.GIF is bounded, furthermore _VP_EQN_153.GIF are shown bounded for all k, and the overall stability of the closed- loop time-varying system is assured in the sense of expectation with respect to _VP_EQN_154.GIF. Especially for a constant reference _VP_EQN_155.GIF, when _VP_EQN_156.GIF approaches to zero in the sense of expectation with respect to _VP_EQN_157.GIF as k approaches to infinity, from Eq. (32) the control signal _VP_EQN_158.GIF approaches to a constant, i.e. _VP_EQN_159.GIF. This implies that, _VP_EQN_160.GIF. Then, from Eq. (2), the output signal _VP_EQN_161.GIF approaches to _VP_EQN_162.GIF; furthermore the error signal _VP_EQN_163.GIF approaches to zero, and the output signal _VP_EQN_164.GIF convergence to the constant reference signal _VP_EQN_165.GIF is assured in the sense of expectation with respect to _VP_EQN_166.GIF.n 

Remarks: The usage of a parameter µ in the difference of the Lyapunov function Eq. (18) is similar to the introduction of a forgetting factor in the least-squares error function [9], which implies that a time-varying weighting of the data is introduced. The most recent data is given unit weight, but data that is t time units old is weighted by _VP_EQN_167.GIF

We do not prove, or claim, that _VP_EQN_168.GIF converges to its true values q. Instead, each element of _VP_EQN_169.GIF approaches to constant values in the sense of expectation with respect to _VP_EQN_170.GIF

In general, real systems are also subject to noise, and it is of interest to ensure the overall closed-loop stability in presence of system and measurement noises. The proposed algorithm is extended to the case where system and measurement noises are considered. The white noise signal _VP_EQN_171.GIF is defined as a bounded independent random sequence, which has the following properties: 

_VP_EQN_172.GIF  with  _VP_EQN_173.GIF

where _VP_EQN_174.GIF is a zero mean uncorrelated random signal with standard deviation s and _VP_EQN_175.GIF is the expectation with respect to noise _VP_EQN_176.GIF

The nominal system model and the family of system models to be considered in this section are represented as: 

_VP_EQN_177.GIF,     (33) 

_VP_EQN_178.GIF,     (34) 

respectively, where _VP_EQN_179.GIF represents the system and measurement noise (white noise) signal. This model is so-called AR (Auto Regressive) model. 

Using the definition of _VP_EQN_180.GIF given in Eq. (2), and including equations (4) and (33), _VP_EQN_181.GIF is rewritten as 

_VP_EQN_182.GIF
          _VP_EQN_183.GIF,         (35) 

If the control law in Eq. (11) is used for the exactly known system, then: 

_VP_EQN_184.GIF.         (36) 

The degree of polynomial _VP_EQN_185.GIF is _VP_EQN_186.GIF, which implies that _VP_EQN_187.GIF depends only on future states of x. Therefore, _VP_EQN_188.GIF gives the minimum variance control for _VP_EQN_189.GIF

Theorem 2 

Recursive estimates of controller parameters based on generalized minimum variance criterion for auto regressive system models with a forgetting factor: Given a positive definite matrix _VP_EQN_190.GIF, a parameter µ _VP_EQN_191.GIF and the initial parameters vector _VP_EQN_192.GIF, the estimate _VP_EQN_193.GIF of the controller in Eq. (11) satisfies the recursive equations (13) and (14) for a white zero mean noise under Assumptions 1 and 2; thus the overall closed-loop time-varying system combining equations (11), (13), (14) and equation (34) gives the overall stability in the sense of the expectation with respect to system and measurement noise _VP_EQN_194.GIF and with respect to _VP_EQN_195.GIF

Proof: The proof follows as the given proof in Theorem 1, using equations (33)-(36), combined with the proof given in Patete [11]-Theorem 2 for auto regressive time-invariant systems.n 

4. Simulation results 

In the following, the proposed algorithm in Eq. (13) and Eq. (14) is denoted by STC-TVS- GMVC. The initial condition _VP_EQN_196.GIF (where I is the identity matrix) and the parameter _VP_EQN_197.GIF (forgetting factor) are chosen. The reference signal is set to the unit step. 

As an academic example, consider the following non-minimum phase system model with d = 2: 

_VP_EQN_198.GIF,     (37) 

where the parameter intervals are given as _VP_EQN_199.GIF _VP_EQN_200.GIF _VP_EQN_201.GIF, _VP_EQN_202.GIF, _VP_EQN_203.GIF. For _VP_EQN_204.GIF 

_VP_EQN_205.GIF, _VP_EQN_206.GIF, _VP_EQN_207.GIF, _VP_EQN_208.GIF 

thus, the nominal system model is represented by: 

_VP_EQN_209.GIF,     (38) 

and _VP_EQN_210.GIF, _VP_EQN_211.GIF, _VP_EQN_212.GIF

For the nominal controller design using the GMVC presented in section 2, the following polynomials are chosen: 

_VP_EQN_213.GIF,     (39) 

_VP_EQN_214.GIF,         (40) 

which lead to the following polynomials for the controller law: _VP_EQN_215.GIF and _VP_EQN_216.GIF _VP_EQN_217.GIF, these are used as initial estimates of the controller parameters. 

In the first, for the simulation example, the real system model is assumed to be represented by: 

_VP_EQN_218.GIF.     (41) 

The robust stability analysis of the closed-loop system in presence of parametric interval uncertainties is shown in Figure 1. As shown, _VP_EQN_219.GIF does not intersect with the critical circle _VP_EQN_220.GIF, which implies that the sufficient condition for robust stability is satisfied. Figure 2 shows the output responses when, after 100 samples, the system model in Eq. (41) abruptly changes to the following system model: 

_VP_EQN_221.GIF.     (42) 

Figure 1. Robust stability analysis for system (37):~ T(e j,), o(6,0) and 6 1.

5. Conclusions 

The overall stability of a self-tuning control algorithm, based on recursive controller parameter estimation including a forgetting factor and generalized minimum variance criterion, for a class of time-varying systems, has been proved based on the discrete-time sliding mode control theory. The results have been extended to the case where system and measurement noises are considered into the system model. The validity of the proposed algorithm was also demonstrated through simulation results. The principal contribution of the obtained stability results is to assure the overall stability if the presented control algorithm is implemented on a real system with time-varying parameters, even in the presence of system and measurement noises. 

 Frame3.JPG

Figure 2. yk vs. rk: GMVC (dashed-line) and STC-TVS-GMVC (solid-line) algorithms applied to system
in Eq. (37), when after 100 samples the system changes from Eq. (41) to Eq. (42), I and 08. .

Acknowledgments 

This research was supported by the COE Grant in Aid of MEXT, Japan. The authors acknowledge the comments and suggestions given by Dr. Akihiko Sugiki of Tokyo Denki University, Japan. 

References 

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3.  Furuta, K.: “VSS Type Self-Tuning Control”, IEEE Trans. Industrial Electronics, Vol. 40, No. 4 (1993) 37-44. 

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8.  Clarke, D., Phil D., and Gawhrop P.: “Self-Tuning Control”, IEE Proceedings, Vol. 126 No. 6 (1979) 633-640. 

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10.  Tsypkin, Y. and Furuta, K.: “Frequency-Domain Criteria of Robust Stability: Discrete-Time and Continuous-Time Systems, A unified Approach”, International Journal of Robust and Nonlinear Control, Vol. 5 (1995) 207-222. 

11.  Patete, A., Furuta, K. and Tomizuka, M.: “Self-tuning control based on Generalized Minimum Variance Criterion for Auto Regressive Models”, Automatica, Vol. 44, No. 8 (2007) 1970-1975.