Adaptive control of Hydraulic Drive

1 MASTER’S THESIS

ADAPTIVE CONTROL SYSTEM OF HYDRAULIC DRIVE

Examiner Professor Erkki Lähderanta Supervisor Professor Heikki Handroos

Author Frumkin Konstantin Lappeenranta

2 Abstract

Lappeenranta University of Technology Faculty of Technology

Department of Mathematics and Physics Frumkin Konstantin

ADAPTIVE CONTROL SYSTEM OF HYDRAULIC DRIVE Master’s thesis

2014

85 pages, 30 pictures, 0 tables.

Examiners: Professor Erkki Lähderanta Supervisors: Professor Heikki Handroos

Keywords: adaptive control, Lyapunov, hydraulic drive, control system.

Adaptive control systems are one of the most significant research directions of modern control theory. It is well known that every mechanical appliance’s behavior noticeably depends on environmental changes, functioning- mode parameter changes and changes in technical characteristics of internal functional devices. An adaptive controller involved in control process allows reducing an influence of such changes. In spite of this such type of control methods is applied seldom due to specifics of a controller designing.

The work presented in this paper shows the design process of the adaptive controller built by Lyapunov’s function method for the Hydraulic Drive. The calculation needed and the modeling were conducting with MATLAB^® software including Simulink^® and Symbolic Math Toolbox™ etc. In the work there was applied the Jacobi matrix linearization of the object’s mathematical model and derivation of the suitable reference models based on Newton’s characteristic polynomial. The intelligent adaptive to nonlinearities algorithm for solving Lyapunov’s equation was developed.

Developed algorithm works properly but considered plant is not met requirement of functioning with. The results showed confirmation that adaptive systems application significantly increases possibilities in use devices and might be used for correction a system’s behavior dynamics.

3 Acknowledgements

I would like to thank my supervisor, consultant and also Hamid Roozbahani for helping in process of solving of plenty issues met during the work. A lot of thanks to everybody who does not disturb me and to all Lappeenranta University staff provided motivating and comfortable environment for performing the work.

Particular thanks to my family, folks and the whole universe for giving opportunities and emotions which many do not have.

4 Table of Contents

Table of Contents ... 4

ABBREVIATIONS AND SYMBOLS ... 6

INTRODUCTION ... 7

1. THEORETICAL OVERVIEW ... 8

1.1 Adaptive control system structures with reference models ... 8

1.2. Direct adaptive control systems with reference model and parametric algorithms settings for linear stationary objects ... 11

1.3. Identifiability and stability to the additive perturbations of parametric tuning algorithms ... 16

1.4. Roughening methods, robustness and dissipativity of algorithms of parametric settings. Estimation of limit sets ... 19

2. SYSTEM DESCRIPTION ... 26

2.1 Mathematical model ... 26

2.1 External leakage model ... 32

2.3 State space representation ... 33

3. THE PROBLEM OF FINDING THE MODEL PARAMETERS WITH MCMC METHOD ... 35

4. THE PID TUNING ... 36

4.1 Characteristics of the system with PID-controller involved ... 38

5. ADAPTIVE CONTROLLER DESIGN ... 41

5.1 Check of controller operability ... 41

5.1.1 Common structure of a controller ... 41

5.1.2 Check of algorithm operability ... 42

5.2 Formulation of the aim of adaptive control for the hydraulic drive ... 48

5 5.3 Development of the intelligent adaptive controller ... 48 5.4 Intelligent adaptive controller structure ... 49 5.5 Check of intelligent adaptive controller operability ... 50 5.6 Structure of the system of hydraulic drive with intellectual adaptive

controller involved ... 54 5.7 The modeling of the system with intelligent adaptive controller ... 55 5.8 The design of adaptive controller with the reference model based on a

Newton polynomial as the characteristic polynomial ... 58 CONCLUSION ... 62 APPENDIX 1: BLOCK DIAGRAM OF THE HYDRAULIC DRIVE WITH PID- CONTROLLER ... 63 APPENDIX 2: SCRIPT FOR CALCULATING COEFFICIENTS OF MEMBERS OF NEWTON’S POLYNOMIAL ... 64 APPENDIX 3: SCRIPT FOR CALCULATING JAKOBI MATRICES AND RESULT OF EXECUTION ... 65 APPENDIX 4: FUNCTION FOR CALCULATING LYAPUNOV’S EQUATION ... 75

6 ABBREVIATIONS AND SYMBOLS

ODE – Ordinary differential equation PID – Proportional-Integral-Derivative MCMC – Markov Chain Monte Carlo PSO - Particle Swarm Optimization

7 INTRODUCTION

Nowadays one of the most appreciated direction of development of automation control theory and practice of technical plants controlling is the adaptive approach. A large body of data concerning different adaptive control methods has been reported. Adaptive control designed with Lyapunov-function method is well studied approach.

In this work the adaptive controller has been designed and applied to the Hydraulic Drive. Most of Hydraulic drives should have high performance and accuracy, however because of wearout, these indexes could be decreasing with time flowing. Applying of an adaptive control system to a hydraulic drive could keep behavior of a plant in satisfying conditions and even improve dynamics.

For reaching results in this work were designed Adaptive controller on the Lyapunov’s function method based on the reference model of replicating real model with desired parameters and the on reference model being a system with transfer function consist of the Newton polynomial as characteristic polynomial.

The Modal Control method was considered to be applied in addition to adaptive control.

The system of Hydraulic Drive is described by the 7th order nonlinear ODE.

This mathematical model accurately describes real system with taking in account many physical processes.

This paper firstly presents a theoretical review and description of applied methods. Then the modeling, control design and dynamic simulations of the hydraulic system. The experiments are done in the Matlab®.

8 1. THEORETICAL OVERVIEW

1.1 Adaptive control system structures with reference models

Mathematical models of nonlinear nonstationary plants and setting of purpose of adaptive control

The most common model of continuous determined nonlinear nonstationary plant is described as vector differential equation in Cauchy form



^{, ,t}





x f x u . (1.1)

Where

T

1 n

x x

x[ ] – state vector, xRⁿ – n-order real space with the Euclidean metric,

T

1 m

u u



u [ ] – input signal vector, ^uRm, mn_{. Real} functions

T

( ) [ ( )f1  f_n( )]

f – defined in scope



0 0



Г_t  x,t: x η; η const( 0) или η    ;t t t, R,u( ) U , (1.2) is continuous by ,x u, ^t and have continuous partial derivatives by x_i, u_j

, i = 1, 2, …, n, j = 1, 2, …, m, is uniform over ^t limited in every compact subset of this area. That means, that for any β η exists ^{c( )β} such, that

( , , ) ( , , )

t t ( )

  c

 

 ^T  ^T

f x u f x u

x x β

when x ^; u ^{; t}^t0, where ^{T T}

 , 

 

f f

x u – Jacobi matrixes ; U – set uniform over t limited piecewise-continuous allowed inputs ^{uu x( , )t} , ^ – vector or matrix Euclidean norm; ^t – time; T – transpose sign. All said above, provides in the scope (1.2) existence, uniqueness, nonlocal duration and

continuous dependence on the initial data ^{x( )t0} and time ^t0 solutions of the equation (1.1) for every allowed [19] .

A wide class of objects allows the permitted explicitly about the vector control ^u, namely: ^{u x( , )t} xf x^{( , )}t B x( , ) ( , )t u x t , (1.3)

9 where ^{f( )} , ^{u( )} are the same in (1), but ^{f( )} does not depend on ^{u( )} , and ( , )t

B x – the m-order matrix of scalar functions ^{bij( , )x t} continuous and

everywhere (global) limited in the area ^Гt, i.e. ^{B x( , )t b b, const(0)} for any ,t

x from the area (1.2).

Obviously, that due to the previously listed properties of the vector function ( , )t

f x equation (1.3) can also be written in an equivalent representation with the state vector of the linearly extracted ^{x( )t}

( , )t ( , ) ( , )t t

 

x A x x B x u x , (1.4)

where the representation f x( , )t A x( , )t x always (and controversial) is admissible, and the matrix function A x( , )t with elements a_ij( , )x t , Unlike the matrix function B x( , )t , As a rule, is not globally restricted but locally restricted, for example, with constants , c, as previously uniformly over t. Equation

( , )t  ( , )t

f x A x x can be represented [20] as

1

( , )t 



0 ( , )t d

A x J xΘ Θ, (1.5)

where ( , )_T

( , ) t

t  

 J x f x

x – Jacobi matrix of the vector function f x( , )t ,  – auxiliary variable.

Application of linearization (Taylor) in a neighborhood of a solution of the equation (1.4), generated by some input u , leads to a non-stationary linear * equation in deviations   x x x , *   u u u*, где x*(t) – solution (1.4) for a given u (leave the old symbols of variables *  x x;  u u):

( )t ( ) ( , )t t

 

x A x B u x , (1.6)

and even * constu  matrix function of time A( )t и B( )t retain the properties of non stationarity of the original object (1.4).

The stationary approximation equation will be used (1.6)

0 0

 

,t

 

x A x B u x , (1.7)

10 where the constant matrices A₀, B₀ are identical in structure to the matrices ( , )t

A x , B x( , )t in the sense of same location of their nonzero and zero elements and meet some of the time-averaged parameters of the original object (1.4) or derived from the matrixes (1.6) at a fixed time t t_.

The considered classes of nonlinear nonstationary (1.1), (1.3), (1.4), linear time-dependent (1.6) and linear time-invariant (1.7) objects controllable at least in the sense of fulfillment of certain conditions of complete controllability (Kalman [17] , [18] ) for the set of linear stationary approximation (1.7) of nonlinear objects (1.1), (1.3) or (1.4) uniformly by x,t across the predetermined area (1.2). There are constant matricesA₀, B₀, are the result of linearization at any pointt_,

* const

x , where t_ – any fixed time. This condition is not sufficient for complete controllability of a non-linear process for any of the definitions discussed in [3] , but, at least, is required for any of them.

Statement of the problem of adaptive control:

a) defining the degree of uncertainty of the object, and in most cases it is believed that the object is known up to a vector of unknown parameters ξ(t)R^d, constant or time-varying, where d – dimension of the parameter vector, and many

 R, containing the vectorξ( )t , defines a class of uncertainty of the object;

b) Setting the control objective in the form of a target inequality

c) determining the adaptive control law u x_а( , , )t θ (Law on the main loop [4] ), which depends on some parameters of the vector θ( )t , tunable during the work of the adaptive system;

d)

defining the rules, called settings algorithms and being as specified above, or differential algebraic equations

θ( )t F θ x u( , , , )t ,

( )t  ( , , , )t

θ Ф θ x u . (1.8)

11 1.2. Direct adaptive control systems with reference model and parametric algorithms settings for linear stationary objects

Let the class adaptable objects specified with stationary approximation (1.7), where the matrices A B₀, ₀ unknown (vector composed of elements of the matrix and has a dimension d n² nm), and the vector x( )t fully accessible for

measuring, and let the desirable performance of adaptive systems are characterized by the behavior of the reference model in the form

0 м  м м  м ( )t

x A x B u . (1.9)

Where u( )t u_а( )t u⁰( )t , u⁰( )t – program control signal, and u_а( )t – adaptive control signal, being defined; A_м, B_м – n n - и nm-order constant matrices, A_м – Hurwitz matrix; u⁰( )t U⁰ – set of allowed program control signals – globally bounded functions of time, i.e. u⁰( )t  h const  , and such that excited by them and the initial data t₀, x_м( )t₀ program (for reference [3] ), the trajectories of the form xм( )t xм



t; xм( ),t0 t0, u⁰( )t



Г_t (all trajectories lie in the allowable region (1.2)). Setting as a goal the convergence of the plant’s

solutions x( )t and the model x_м( )t with time in specified class of adaptive systems (all controllable plants as (1.7)) and under any initial condition from the scope (1.2), allows to consider the goal function as positive definite quadratic form

( ) 0.5 T( ) ( );

Q t  e t Pe t e( )t x( )t x_м( )t , (1.10) where PP^T 0 – constant positive definite symmetric matrix. Structure of the searchless adaptive law for this case is known, ( [22] , [4] , [10] –[11] , [42]) and is a linear function of x( )t и u( )t :

( )t  ( ) ( )t t  ( )t 0( )t

А A B

u K x K u , (1.11)

where K K_A, _B – matrixes of configurable parameters of appropriate dimensions. The implementation of the goal function taken as limit relation

lim ( ) 0

t Q t

  , (1.12)

12 leads to asymptotical stability of the trivial solution ^e

 

^{t 0} of adaptive system (1.7), (1.9), (1.11) in General (i. e. when  , when the object’s domain area (1.3) is the whole space Rⁿ), by variables e( )t , uniform by the initial

coordinate e( )t₀ and «parametric» perturbations. Structure of the law (1.11) is related to the so-called invariance conditions to the reference model (1.9) [12] , obtained by equating the right sides of the object (1.7) and the substituted in his law (in [4] , [21] is named as generalized configurable plant):



A0 B K0 _A( )t

 

x B0B K0 _B( )t



u⁰( )t A x м B uм ⁰( )t . (1.13) From (1.13) is obvious, that conditions of attainability (asymptotic

convergence object’s and model’s trajectories) is equivalent to the existence of such constant values of the custom matrices K^*_A, K^*_B, that

* *

М  0  0 _A, М  0  0 _B

A A B K B B B K , (1.14) is called [13] also concurrency conditions. Condition (1.14) ) is equivalent to equations

0 м 0 0 м 0 0

rank { B B B,  }  rank {B A A,  } rank B , (1.15) called in [4] , [50], [51] adaptability conditions, or following equations



B B0 0^  E_n

 ^

A A 0  м

^

0;



B B0 0^  E B_n



м  0, (1.16) known as Erzberger’s condition [33], where m n -matrix B B B₀^  ( ^T_{0 0})^1B^T₀ is pseudoinverse of rectangular m n -matrix B B B E₀, ₀^ ₀  _m, E_n,E_m – identity matrices with orders n and m, rankA – rank of matrix А.

The requirement of asymptotic stability suggests way to determine algorithms for parameters setting of the control law (1.11) by the method of Lyapunov functions

(second, or direct,

Lyapunov’s method

).

Application of Lyapunov's second method for the synthesis of adaptive

systems become

very popular

[10] .

The next theorem could be formulated and proved within Lyapunov’s function method.

13 Theorem 1.1.

Given a linear stationary object (1.7) with unknown constant matrices

A₀, B₀,

reference model (1.9) and searchless adaptive control law (1.11).

Цель Adaptive control goal (1.12) is performed in the adaptive system class,

determined by the set of matrices

{A B₀, ₀}, satisfying adaptability condition (1.16),

and bounded input signals

u⁰( )t U⁰,

if algorithms of parametric tuning of adaptive control law (1.11) is chosen in the form of matrix differential equations of the form

 

^T

T T T 0

0 0

( )t   , ( )t  

A A B B

K Γ B Pex K Γ B Pe u , (1.17) where Γ Γ_A, _B – randomly selected mm-order symmetrical (particularly diagonal) positive definite matrix coefficients gain settings, and the matrix P satisfies the well-known equation of Lyapunov

T

м  м  

A P PA G, (1.18)

having a unique solution PP^T 0 due to the matrixA_м is Hurwitz matrix [20] for any symmetric positive definite matrix GG^T 0. Note that this

statement is equivalent to saying that the adaptive system of the form (1.7), (1.9), (1.11), (1.17) has the trivial solution^e

 

^{t 0}, asymptotically stable in the region (1.2), or in a whole (in the variables^e

 

^t ) under the condition (1.16).

Proof. The proof is based on the choice of positive definite Lyapunov functions as

   

T T 1 T 1

( ) 0.5 tr tr

V t  e Pe δ Γ δ^{A A A}  δ Γ δ^{B B B}  , (1.19) where «tr» means trace of square matrix (the sum of the elements of its main diagonal). Introduced m n - и m m -order matrixes parametric mismatch

   

   

* *

0 0 м 0 0 м

* *

0 0 м 0 0 м

( ) ( ) ( ), ,

( ) ( ) , ,

t t t

t t

 

 

        

        

A A A A A

B B B B B

δ B A A K K K K B A A

δ B B B K K K K B B B (1.20)

14 taking place under the fulfilling conditions (1.14)–(1.16). Error equation is written by (1.10), subtracting equation (1.9) from (1.7) with taking in account (1.11), ase A e  _м (A₀ A_мB K_{0 A})x B( ₀B_м B K u_{0 B}) ⁰. Then the equation is modified with pseudoinverse matrixB₀^ as

 

   ^{ } 

⁰

м  0 0^ 0 м   0 0^ 0 м  ,

    ^A     ^B 

е А е B B А А K x B B B B K u (1.21)

or, taking in account (1.20), the error equation (1.21) eventually is written as:



⁰



м 0

  _A  _B

e A e B δ x δ u . (1.22)

Following the computing of derivative V t( ) by equation (1.22) and by (1.18):

   

T T 1 T 1

( ) tr tr

V t e Pe   δ Γ δ _A ^_{A A}  δ Γ δ _B ^_{B B} 

 

T T T T T 1

м  0 tr ^ 

 e A Pe x δ B Pe ^T  ^A  δ Γ δ^{A A A} 

 

^{0 T T T0}



^{tr T 1}



  

 

 u δ B Pe^B δ Г δ^{B B B}  . (1.23) Given that there are the following identities (

 

^{u0 T u0T):}

   

T T T T T T 0 T T T T T 0T

0 0 0 0

δ tr δ , δ tr δ ,

δ , δ ,

 

 

A A B B

A A B B

x B Pe B Pex u B Pe B Peu

K K

From (1.18) entails, that^{e A PeT Tм 0.5eT}



^{A PTм PAм}



^{e 0.5e GeT , and}

choosing functions δ_A, δ_B so, that the brackets in the expression is eliminated (1.23), stated equation settings is obtained (1.17). Performing (1.17) make the expression on ^{V t}

 

be eventually transformed to:

T 2

( ) 0.5 , или ( ) 0.5λ ( )

V t   e Ge V t   G e . (1.24)

The last inequality follows from the estimating inequalities for positive definite quadratic form [26]

 

^{2  T }

 

^{2, 2  T ,  T 0}

λ G e e Ge Λ G e e e e G G ,

where ^

 

^{G 0 and} 

 

G 0 – the smallest and largest eigenvalues

15 of the matrix G (having all the eigenvalues are real positive because of its positive definiteness). Thus, it is proved that the adaptive control system for the class of linear stationary objects (1.7) with reference model (1.9), adaptive law (1.11) and parametric tuning algorithms (1.17) when the structural conditions of adaptability

(1.14), (1.16) performed, unexcited motion e( )t 0 asymptotically (and even exponentially) stable by the variable ^e

 

^t uniformly ont₀, e( )t₀ in the scope (1.2) or in a whole (whenη ). This result follows from the corresponding theorems of Lyapunov method with Lyapunov's functions, « satisfying the

inequalities specific for quadratic forms » [7] . At the same time, the unperturbed motion e( )t 0, δA

 

^{t 0}, δB

 

^{t 0} of the adaptive system has Lyapunov’s stability (by variables δA

 

^t , δB

 

^t ) in the same area, as for the derivative of the function (1.19), calculated according to the system (1.22), there is strict inequality

0

V  only on the argument e( )t 0 and the simple inequality V 0 on parametric mismatch δ_A( )t , δ_B( )t . Theorem is proved.

Remark. The algorithms (1.17) contain the unknown matrixB₀, and due to they are not realizable. In practice, the unknown matrixB₀ replace the well-known matrixB_м, and thus for a wide range of applications, they remain operational, although such substitution make the requirement of asymptotic stability be sacrificed, since Theorem 1.1 can not be proved. Theoretical justification for the substitution B_м instead B₀ in the algorithms (1.17), making them realizable, is an unsolved problem of theory.

16 1.3. Identifiability and stability to the additive perturbations of parametric tuning algorithms

Previously there have been specified the essential conditions of strict constancy of the unknown matrices A₀,B₀ and the absence of external perturbations. Otherwise, the effect of non-robustness property of simple

Lyapunov’s stability [7] the control goal (1.12) in the built adaptive system can be achieved only theoretically when the coefficients of the matrices K_A( ),t K_B( )t are infinitely increasing and, thus, stability is violated (in common) on the variables

( )t

δA , δ_B( )t . Therefore, because of the always existing disturbances such a system should recognize fundamentally unworkable.

Remark. Out of this situation is the providing the uniform asymptotic stability of the system (1.7), (1.9), (1.11), (1.17) over the entire ensemble of variablese( )t ,δ_A( )t ,δ_B( )t . Then, in view of the known results of the stability theory outlined in [7] , [27] , adaptive system will remain the property of

asymptotic stability of the trivial solution ^e

 

^{t 0,} δA

 

^{t 0,} δB

 

^{t 0} (is rough on the definition in [7] ) under the affecting to the object (1.7) additive, and

parametric coordinate perturbation expressed as the n-order function σ x( , )t

0 0 ( , ) t ( , )t

  

x A x B u x σ x , (1.25) if these perturbations satisfy the following conditions [7] : there exists the scalar continuous function ( )x , that in the area (1.2) for any t the inequality of the following form is performed

( , )t  ( ), ( )0, 0, (0)0

σ x ηx ηx x η . (1.26)

Conditions for asymptotic stability of the adaptive system (1.7), (1.9), (1.11), (1.17) on variablesδ_A( )t , δ_B( )t are considered in many papers on adaptive control [10] , [14] , [15] , [24] , but in the most general case of non-linear process of the form (1.3), linear by the inputs considered in [4] , [6] , [28] and consist in the implementation of the so-called identifiability conditions of adaptive control algorithms. While this is happening there is performing the stronger goal than the

17 control goal (1.12) of convergence of solutions of the object and models, namely, the goal of convergence in a sense of parameters is set, generating their equations [9] . In our case this means the requirement that in the area (1.2) with (1.12) the limit ratio is performed

lim ( ) , lim ( )

t t t t

K_A K  ^*_A K_B K ^*_B. (1.27) Theorem 1.2. In the adaptive system (1.9), (1.11), (1.17) with the perturbed object (1.25) to achieve the goal (1.27) (identifying properties tuning algorithms), it is sufficient that the vector-valued function Φ x( )t   ^T_м( ),t u^0T( )t ^T is integrally nondegenerate that means by [4] , there exist positive integers L, , that for any

t t 0

   

^{T α}

t L

n t

  d





^{Φ Φ τ} ^{E . (1.28)}

In [4] is formulated several identifiability criterias equivalent to (1.28). In particular, for systems with the reference model condition (1.28) is performed if the couple A_м, B completely controllable, and "function spectrum _м u⁰( )t contains at least n frequencies ». This theorem is proved in [4] .

Moreover, from the asymptotic stability the stronger properties can be obtained under additional assumptions. For practical problems is not enough to establish the total decay property of the perturbed motions semi-infinite interval [ , t0  ), and decisive is the fact of decaying to some -space (distance), circling unperturbed (reference or program) movement, where the quantity  determined by the required accuracy of maintaining the program mode, and the subsequent behavior of trajectories, immersed in -space, may not be regulated and, in particular, the requirement of asymptotic properties of solutions - is superfluous.

Then the workability of the adaptive system can be determined by well-known property so-called stability under constantly acting perturbations, including bounded in an average interval [7] , [27] , [29] .

18 Definition. Let the plant with taking in account perturbations is described as (1.25), and the vector function ^{σ x( , )t} is not necessarily continuous, does not satisfy (1.26) and never becomes zero at the origin, In addition, the reference model (1.9) are not the solutions of the equation (1.25), at least under certain simplifying conditions (that is the behavior of the plant (1.25) is not like the linear one). Then the trivial solution of the adaptive system (1.9) (1.11), (1.17) with the object (1.25) will be called stable under constantly acting perturbations, if for any

0

ε there are two numbers, ξε( )0 and  ( )  0, the choice of which depends on the choice of the number  and such that for any t₀, initial values z( )t₀ ξ, where z e  ^T,δ δ^TA, ^TB^T, and under all perturbations ^

 

^{x,t } in the area (1.2) the inequality z( ; , ( ), )t t₀ z t₀ σ  is performed.

Further extension of a perturbations class allows the function σ x( , )t to have on small enough intervals big enough spikes, that is making it satisfying the inequality (τ) τ η, 0

t T

t

d T





^Φ   ^{, (1.29)}

where ( )t – any positive continuous function, satisfying the condition ( , )t ( )t

 x  ,  – formal integration variable, and then the adaptive system as in [29] , can be called stable under constantly acting perturbations bounded on

average.. Where T can be any number, but the choice of  , depends on  and T .

In conclusion, there can be stated the following theorem.

Theorem 1.3. Trivial solution ^e

 

^{t 0}_, δA

 

^{t 0}

, δB

 

^{t 0}

of the adaptive system (1.9), (1.11), (1.17) with the plant (1.25) is (exponentially) stable under constantly acting perturbations (including limited in an average interval) with respect to the earlier definitions, if it is identifiable, and Lyapunov's function of the

19 system satisfies the so-called ([7] ) inequalities, characteristic of the quadratic forms. ■

This result follows immediately from the corresponding general theorems of Lyapunov methods [7] , [29] , [30] , and first formulated in [15] .

Remark. Terms of identifiability is not fully determined by structural properties of the adaptive system, and therefore are not always available for a designer. Therefore operability of adaptive systems in terms of the parameters drift and influence of other uncontrollable pertrubations (untouching the impact of non- linearities) shall be provided with more powerful tools - a modification of

adaptation algorithms, giving them property, greater than roughness, – dissipativity.

1.4. Roughening methods, robustness and dissipativity of algorithms of parametric settings. Estimation of limit sets

Suppose that in (1.2) together with the main nonlinear plant (1.3) there is considered to have an additive effect with the pertrubations ^{σ x}

 

^,t

( ,t ( , ) ( , )t t ( , )t

  

x f x ) B x u x σ x , (1.30) in the particular case, there is form (1.25).

Definition. As in [8] and [27] , the plant (system) (1.30) is considered as dissipative, if the unperturbed plant (system) when σ x( , ) t  0 allows the trivial solution x0, σ x( , )t   _σ const and in the area (1.2) exists the scope

const ( )

D D 

  

x , that for every t₀ and 0 x( )t₀  , ^{ 0}



^{ D}



^every

solution ^x



^{t t; ,0 x}

 

^t0



is satisfied the limit ration (for the upper limit lim) as



0

 

0



lim ; ,

t t t t D

 x x  . In other words, there is independent of the choice t₀, x( )t₀ the number T 0 such that after the time t^*   t₀ T t₀ the inequality

0 0

( ; , ( ),t t t )  D

x x Is performed for any t  *t . D- area is called the limit set of perturbed motions of the plant (1.30), and -area – dissipative area.

20 Similarly, there are introduced definitions of dissipativity of adaptive systems. If the dissipativity domain is the whole area (1.2) or space Rⁿ and dissipativity is generated by the Lyapunov's function, satisfying the system (1.30), is considered without perturbances ( = 0), inequalities characteristic for quadratic forms [7] , then the name of such dissipativity is exponential dissipativity in a large or whole, and while the synthesis of dissipative adaptive systems, and it is always kept in mind further.

Remark. Solutions of a dissipative plant (system) is sometimes called the finally limited [32] , and the unique bounded solution^x



^{t t; ,0 x}

 

^t0



, for which the following limit relations is performed.



⁰

 

⁰

 ^{ }

lim ; ,

t t t t ^ t D

 x x x  ,

is in some sense the limit mode of the plant (system). Obviously, that in the specified definition of dissipativity the limit mode is the trivial (zero) solution of the plant (system) (1.30). Further, the plant having solution, which is dissipative in a whole, is called dissipative system.

Remark. It is important to note that if the adaptive control algorithms of a dissipative system provide the ability to control the size of the limit set D and time T, after which all trajectories come into the D-tube, bounding program mode, then the property of exponential dissipatively meets engineering ideas of a well-

designed system with given accuracy performance (D-area) and speed performance (time T), and estimations D and T can be written explicitly as inequalities (in a scope of synthesis method on Lyapunov’s functions).

Definition. In [13] the term "robust adaptive system" is defined as an adaptive control system, which has the linearized error equation is asymptotically stable, whereupon it keeps operability in the vicinity of the steady (nominal) mode in the so-called "unstructured" [13] perturbation, substantially corresponding to either the conditions (1.26), when such a system in the consideration of classical theory of stability [7] is named as rough (see the remark at the beginning of 1.3),

21 or stronger conditions of type (1.29), when, according to the same classical

terminology, the system is stable under constantly acting perturbations, including limited on average interval (see Theorem 1.3). In [4] it is noted that in the modern scientific and technical literature there is a tendency to use the term "robustness" to the systems in which unstructured (uncontrolled and not amenable to mathematical description) perturbations lead to such deviations from the program modes for which can be written various majorize evaluation. There is the consideration of roughening methods (regularization), giving the properties of robustness to the parameter adaptation algorithms in a class of linear and not stationary plants (1.6), in the general case , affected by unstructured additive perturbations (x, t), only limited by the norm in (1.2) (it is less restrictive requirement than even (1.29)), namely, the equation of the plant

( )t ( )(t ( )t 0( ))t ( , )t

  _A  

x A x B u u σ x . (1.31)

It's also assumed that the matrix functions of time in (1.31) are bounded and have continuous and time-limited derivatives in semi-infinite interval [t ,  ), namely

1 1

( ) t a; ( ) t b; ( ) t a; ( ) t b

A B A B ; (1.32) for all

1 1

( , ) t  _; , , , b a b  0 (const)

σ x .

Obviously, for such object does not exist steady state values of the tunable coefficients of the matricesK^_A, K^_B  const, and algorithms (1.17) do not provide operability (stability in the above sense) of adaptive control system throughout the region (1.2).

Many papers consider methods of regularization of algorithms for parametric tuning in the form (1.17), providing dissipation of unsteady plant (1.6), when imposed constraints (1.32) (see [10] , [14] , [15] , [25]

, [55]) but they have been studied in most detail in [5] , [6] , [23] , [31] .

22

Widespread

method of roughening the additive feedback, and, following [4] , the algorithm (1.17) in the following form can be modified:

T T

γ 0 ξ( )

 

    

K Γ B Pex K , (1.33) where – m m - matrix of tunable parameters,  0 (const);

T 0

 

Γ Γ ; ξ( )K – m m - an additive feedback matrix, satisfying the condition that there are symmetrical, in particular, the diagonal positive definite matrix

0, ' 0

    such that



^{T *}



²

tr ξ K( ) [K K ] Λ K K ^* Λ', (1.34) where, matrix norm - Euclidean for some matrix К is equal

tr ( T )



K K K . Condition (1.34) expresses negativity of additive feedback in (1.33), which prevents excessive increase in values KK^{ 2}, here K^ – some a priori estimation of the final value of the matrix K( )t for operating mode,

specified by the designer with the specific considerations, such as K^  0. For the condition (1.34) the following form is satisfied, for example, linear negative

feedback in the following form.

( )  (  ^)

ξ K Λ K K or ξ K( )  Λ K, (1.35) widely used in the methods of roughening, and then modified algorithms (1.17) take the form

T T

  0 

A A A A

K Γ B Pex Λ K ;

T 0T

  0 

B B B B

K Γ B Peu Λ K , (1.36)

where, Λ_A, Λ_B> 0 – is constant symmetric (in particular, the diagonal) positive definite matrices.

In [4] is specified the disadvantage of the approach of roughening by (1.35), making the distortion in control at low values of the norm e(t)xТ(t), and relay method of regulation is proposed, in which the feedback turned off inside the

K

23 sphere {K(t): ||K (t) –K^|| < d > 0 (const)}, and indefinitely increase the penalty for a leaving of this sphere:

 

sign col when ,

ξ(col )

0 when ,

d d

 



   

 

  



K K K K K

K K

(1.37)

where colK – parameter vector "stretched" in the column of the matrix

;   0

K (const), and sign (relay) function of a vector quantity is understood component-wise, i.e., if f 



f1, , f_n



^T, then signf 



sign ,f1 , sign f_n



^T. In a system with relay roughening (1.37) can occur so-called sliding (real) modes.

There, in [4] , provides another way of roughening - method of introducing deadband by goal condition (1.12), when the control process is turned off within the tube e  ,   0 (const):

T T T 0T

0 , Δ, 0 , Δ,

0, Δ, 0, Δ,

    

 

 

 

 

 

A B

A B

Γ B Pex e Γ B Peu e

K K

e e (1.38)

However, the implementation of (1.38) can also make real sliding modes.

In [71], the method of regularization of algorithms with nonlinear feedback:

T T

ξ( ( )) K t  Λ B Pe0 ( )t K( ),t Λ Λ 0. (1.39) Remark. The purpose of wider varying of limit set size (D – area) can be reached by introducing constant parameters matrix K^, in the considered above algorithms of tuning,

for example, the algorithms (1.36) modified to an

T T

0 T 0

( ( ) );

( ( ) ).

t

 t

   

   

*

A A A A A

0 *

B B B B B

K Γ B Pex Λ K K

K Γ B Peu Λ K K (1.40) Obviously, that taking in account the algorithms (1.40), the expressions of estimations of limit sets become other, and parameters of algorithms might enter into them as more effective from the viewpoint of their influence on the size of D- area, although, of course, designed for each specific system it should be clarified by modeling. However, in this case, additional arbitrary coefficients K^ are

24 introduced in an adaptive system, which are selected by the dissipativity condition, ie, their choice depends essentially on operability of the system, therefore

introducing them into the tuning algorithms (1.40) complicates the implementation.

■

Proof of dissipativity properties of the plant (1.6) with parameter adaptation algorithms (1.17) with the roughening as additive feedback (1.33) obeying the negativity condition (1.34) for all their modifications options discussed here, except (1.39). in general, it follows from the general results in [4] , [6] , [30] for the speed gradient algorithm. However, there are general theoretical results proving the sufficiency of the properties of dissipativity trivial solutions of the nonlinear unperturbed plant in sufficiently general form in the presence of

unstructured additive perturbations. These results were obtained using the methods of Lyapunov's functions and regulates by the criteria of T. Ioshidzavy and others [8] , based on negative determination calculated according to the derivative of the Lyapunov's function, endowed with the necessary properties out some open sphere.

For example, there is the following general theorem.

Theorem 1.4. Suppose that there is given the non-linear non-stationary object (1.30) with acting upon it the additive perturbations^{σ x}

 

^,t . For dissipativity of object (1.30) (in the sense given in the beginning of definition 1.4 ) the existence of a Lyapunov's function is sufficient V( , )x t , positively defined in the area (1.2) when    , the derivative is calculated by equation (1.30) and subject to the inequality in the same area V( , )x t  V( , )x t V^ x( , )t , (1.41)

where   0,   0,   [0, 1) ([8] , [30]). And there is exponential dissipativity of the trivial solution x0 (in a whole), uniform by t t₀, ( )x ₀ in the area (1.2). ■

Proof. Supposing that in the inequality (1.41)   0. Let V t( ) V t( ) ,

  0,   0, Then, after integration with ^{t 0}



^t₀ ^0



^{there is}

25

0

( ) (0 )

t

t t

V t e^ V  e^



e^sds, and since

t

0

e^s ds  



^{when t  ,}

then there is limit relation for the upper limit of the form

lim ( ) lim 0 lim

t s

t

t t

t t t

e ds V t e

e e





 

  

 

   



  

.

Thus, when 0 from (1.41) it follows that for the upper limit of the function V( , )x t

lim ( )

t V t 



  .

If V  1/2x Px P^T ,  P^T, P  0, т. е. V –

is positive definite quadratic form, it is easy to proceed with the evaluation of the limit set for the trajectories

x(t),

namely

:

lim ( ) 2 ,

( ) ( )

t t  D 

   

 x  

P P , (1.42)

where ( ) P  0 – minimum eigenvalue of the matrixP.

In the general case, inequality (1.41) when  0 (0  1) there is ([30], [30] ) estimation of the limit set of the form

1 1

lim ( )

t V  ^







    

t , (1.43)

and again in the case of quadratic forms with constant coefficients for V the radius of the limit set is defined as

1

2 1 1

lim ( )

( )

t t  ^

 





    

x P ;

1 2(1 )

1

D ( )  ^

 

  

 P    . □

26 2. SYSTEM DESCRIPTION

2.1 Mathematical model

The system description is taken from [3] . The studied system comprises a directly operated proportional servo solenoid valve with position control, cylinder, power unit, four pressure sensors, and a single displacement sensor (Fig. 1). Since the type of control is of influence on the outcome of the output, e.g., the position of the mass, the system is in closed loop positional control, i.e., the position of the mass as feedback was summed with the signal from the pulse generator to form the input signal to the system. The mathematical model of the system involves a large number of parameters, which may be completely unknown or only known within certain ranges.

Fig. 1. Schematic diagram of the servo hydraulic system.

Voltage u (V) is the valve input. When the input is applied to the valve, spool is shifted and openings are produced. The shift of the spool, namely position

displacement xs (mm), is in both directions. This displacement is small (of the scale 10^-1 mm) and not measureable; the full displacement is also not available. But actuator and the spool are connected to the linear variable differential transducer

27 (LVDT). The range of the LVDT signals us (V) is 10 V for an input of 10 V and u_s is testable. In this study, voltage u_s is measured and directly used for providing information of the spool displacement. Using the normalised spool displacement, i.e., us/10, would be another option.

The main spool of the valve is a mass held in position by a spring system. The main spool is the key component of the flow divider and is highly responsible for the outcome of the transfer function.

The relation between the simulated valve spool position xs (m) and the input voltage u can be of first order as: Gv(s) = xs(s)/u(s) = τ1/(s+τ2), or x_s    t u₁ t₂ x_s, where term τ₁ has no unit and term τ₂ has unit (1/sec or s^-1). But term τ₁ should have unit as (ms^-1V^-1) since τ₁ multiply by u is measured in (ms^-1). Similarly we can also represent the transfer function of the valve dynamics, between us and u, using the first order system, type 1, as the following:

1 2

s s

u    t u t u (2.1)

Where term t1 is the gain (s^-1) and t2 the time constant (s^-1).

The relationship between u_s and u can also be given as:

( ) /

s s

u  K u u T (2.2)

where K is the gain (no physical unit) and T the time constant (s).

A first order model can only be applied in case of limited frequency range, well below the natural frequency of the valve; the second order model responds the servovalve dynamics through a wider frequency range. A linearized model for an electrohydraulic servo system with a two-stage flow control servovalve and a double ended actuator has revealed that the higher order model fits closer to the experimental data because of the reduced unmodelled dynamics.

When a second order transfer function is used to represent the valve model, type 2, the valve’s dynamics could be as the following:

2 2

s n 2 n s n s

u  k     u    u  u , (2.3)

28 where k is the gain (no physical unit),  the damping ratio (no physical unit), and _n the natural angular frequency (radian/s).

The valve flow gain depends upon the rated flow and input current. The rate of change of input signal is also limited, in such control boards in order to provide a well behaving response of the valve. In addition, the servo solenoid valve under study has an on-board electronics (OBE), providing position feedback of the spool of the valve. Disturbances as friction or flow forces on the spool are rejected.

Using the Newton’s second law, the equation of motion for the servo hydraulic system becomes:

1 1 2 2 .

p f

m x  p A  p A F (2.4)

Where, m denotes the mass weight (kg), xp the displacement of piston (m), A1 and A2 the piston areas (m²), p1 and p2 the pressures (Pa), and Ff the friction force (N).

The friction force is defined as:

0 1 ,

f v p

F z dz k x

  dt

      (2.5)

( ) ,

p p

p

dz x

x z

dt   g x (2.6)

2

0

( ) 1 ( ) ,

p s

x v

p c s c

g x F F F e



 

 

 

 

 

   

 

 

(2.7)

where 0 is the flexibility coefficient (N/m), 1 the damping coefficient (Ns/m), k_v the viscous friction coefficient (Ns/m), z the internal state, F_C the

Coulomb friction level (N), FS the static friction force level (N), and vs the Stribeck velocity (m/s).

The pressures at valve ports were described as: