Method for testing robust control quality for linear one-dimensional discrete control systems Текст научной статьи по специальности «Компьютерные и информационные науки»

ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА

2024 Управление, вычислительная техника и информатика № 66

Tomsk: State University Journal of Control and Computer Science

УПРАВЛЕНИЕ ДИНАМИЧЕСКИМИ СИСТЕМАМИ CONTROL OF DYNAMICAL SYSTEMS

Original article UDK 517.9

doi: 10.17223/19988605/66/1

Method for testing robust control quality for linear one-dimensional discrete control systems

Andrej N. Parshukov

Industrial University of Tyumen, Tyumen, Russian Federation, [email protected]

Abstract. The article deals with one-dimensional linear control objects in discrete time. It is assumed that the operators of the control object have structural and parametric inaccuracies in the description. In this paper, a criterion for the robust quality of control is obtained for a system consisting of a discrete control object with structural perturbations and a modal controller. Based on this criterion, a numerical method for checking the robust quality of control has been developed. The effectiveness of the method is illustrated by an example. Keywords: structural perturbations; modal controller; robust control quality.

For citation: Parshukov, A.N. (2024) Method for testing robust control quality for linear one-dimensional discrete control systems. Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaja tehnika i informatika -Tomsk State University Journal of Control and Computer Science. 66. pp. 4-12. doi: 10.17223/19988605/66/1

Научная статья

doi: 10.17223/19988605/66/1

Метод проверки робастного качества управления для линейных одномерных дискретных систем управления

Андрей Николаевич Паршуков

Тюменский индустриальный университет, Тюмень, Россия, [email protected]

Аннотация. Рассматриваются одномерные линейные объекты управления в дискретном времени. Предполагается, что операторы объекта управления имеют структурно-параметрические неточности в описании. Получен критерий робастного качества управления для системы, состоящей из дискретного объекта управления со структурными возмущениями и модального регулятора. На основе данного критерия разработан численный метод проверки робастного качества управления. Эффективность метода проиллюстрирована примером. Ключевые слова: структурные возмущения; модальный регулятор; робастное качество управления.

Для цитирования: Паршуков А.Н. Метод проверки робастного качества управления для линейных одномерных дискретных систем управления // Вестник Томского государственного университета. Управление, вычислительная техника и информатика. 2024. № 66. С. 4-12. doi: 10.17223/19988605/66/1

© A.N. Parshukov, 2024

Introduction

Technological processes with different tempo components are widespread [1, 2]. A characteristic feature of the models of such processes is the allocation in the model of operators of the so-called "basic dynamics", which describes that part of the control object that is subject to regulation, and operators of "structural disturbances" - they include those parts of the control object that already have the properties of stability and a given quality management [3. pp. 29-30]. When synthesizing a controller, structural perturbations, as a rule, are not taken into account [4-5], as a result, uncertainty arises in the transfer function (TF) of a closed system. Since the properties of stability and quality of control of the system are determined by the location of the poles of its PF, the question arises: under what operators of structural perturbations in the control object will the closed system still retain the properties of stability (robust stability) and quality of control (robust quality of control)?

In the modal control scheme [3. pp. 8-21, 6. pp. 5-20] the quality of control is given as an area S on the complex plane; the S area determines the desired location of the PF poles. Consequently, the issues of studying (testing) robust stability and robust quality of control can be considered from a unified standpoint: is it required to check whether the zeros of a given family of polynomials belong to the area S?

The problem of studying robust stability and robust quality of control is widely presented in the literature. There are three main directions in which this problem is solved: 1) the principle of "zero exclusion" [7 -12]; 2) H theory [13-14]; 3) LMI method [15-17]. However, the general formulation of the criterion of robust control quality (independent of the shape of the area S) has not yet been obtained [11. p. 227].

In this article, in development of the results obtained in [12], we develop a method for testing the robust quality of control.

The following notation is adopted:

= is equal by definition;

Rn is the n-dimensional space of real numbers;

Cn is the n-dimensional space of complex numbers;

C1 is the complex plane; T is the transposition operation;

E is the identity matrix of the corresponding dimension;

0 is a column vector consisting entirely of zeros;

j is the imaginary unit; 5 is a variable (generally complex);

5* is the number complex conjugate of 5;

t is discrete time (t = 0, ±1, ±2, ...);

z is advance operator by one cycle: zx(t) = x(t +1);

S is the area on C1;

dS is the boundary of the area S;

int S is the interior of the area S.

Let fx) be a function of the vector argument x defined on the area X c Rn; denote

|/|+=max|/(4 |/|>min|/(x)|. (1)

View operator

n

f (n, z) • z!, (2)

i = 0

is called a polynomial operator of degree n. Replacing z with 5 in (2), where 5 is a variable, we obtain an algebraic polynomial

n

f (n, s) =nft • ^ .

i=0

The set of zeros of the polynomial fn, 5) is denoted by Af):

A(/) = {*., eC1 :/(»,*.,) = (), iel^i}.

Family of operators of the form

f(n, F, z) = jf(n, f, z)=tfl - z! : f = f fn )g F |, (3)

F = {f g R"+1: f g/-Af; f0 + Af],

fn0 * 0, Af. > 0, i = On } is called an interval polynomial operator of degree n.

Replacing z with 5 in (3), we obtain an interval polynomial:

f (n, F, s) = {f(n, f, s)=ift - s1' : f g F |. (4)

Interval polynomial (4) can be represented as

f (n, F, s) = f0 (n, f0, s)+ Af (n, AF, s), (5)

where

n

r(n,f°,-sj0^, r),

i=0

ordinary polynomial, and

Af(n,AF,s) =|/(n,6f,s)= i bft -sl :5f = (5/0,...,5/„) g AfJ,

AF = {Sf g Rn+1: f g [-Af; A/..], i = 0n },

interval polynomial with symmetric coefficient uncertainty intervals.

Let A be a matrix whose coefficients are complex numbers, then we denote by ARe and Aim the matrices whose coefficients are, respectively, the real and imaginary parts of the coefficients of the matrix A, in this sense we denote

A Re = Re(A)> A Im = Im(A).

1. Synthesis of a modal controller for linear control objects with structural perturbations and statement of the problem of testing the robust quality of control

Let a linear one-dimensional discrete control object be given by a model of the form:

v(l,V,z)-a(n, A,z)y(t) = w(h,W,z)-b (m,B,z)u (t), n >m, l > h, (6)

here u is the input (control) signal, y is the output (controlled) signal, a(n, A, z), b(m, B, z), v(l, V, z), w(h, W, z) are interval operators of the form (5) such that

a0 = 1, Aan = 0, v°° = 1, Av0 = 0, w° = 1, Aw0 = 0.

Model

a0 (n,a0,z)y (t) = b0 (m,b0,z)u (t), (7)

belonging to the family of models (6), we will call "nominal"; interval operators a(n, A, z), b(m, B, z) call "basic dynamics"; operators v(l, V, z), w(h, W, z) call "structural perturbations".

The quality of control will be assigned in the form of an area S, which determines the admissible location of the PF poles on Cl. We assume that the area S satisfies the following conditions: it is located inside a circle of unit radius; simply connected; also holds s* g S for any point s g S .

In addition, based on the meaning of the problem, we require that the operators of structural perturbations v(l, V, z), w(h, W, z) satisfy the expressions:

A(v)c int s, A(w)c int s. (8)

Note that the fulfillment of expressions (8) is easy to check using the methods described in [12. pp. 7-10].

Since the requirements for the quality of control are expressed in root quality indicators, the controller will be calculated according to the modal control scheme. Following the method of synthesizing a modal

controller (described, for example, in the monograph [6. pp. 5-20]), the controller for the nominal model (7) is sought in the form of a difference equation of the n-1 order:

P(n -1,z)u(t) = a(n -1,z)y (t) + Z(n -1,z)g(t), P^ = 1, (9)

here g is a given program signal. In the modal control scheme, the coefficients of the operators P(n - 1, z) and a(n - 1, z) of the controller (9) are calculated from the condition of conversion to the identity of the equation

ast (2n -1,s) = a0 (n,a0,s)p(n -1,s)-b° (m,b0,s)a(n -1,s),

au = 1, (io)

where ast(2n - 1, 5) is the given characteristic polynomial of the standard control system (hereinafter referred to as the "standard"); the choice of standard is limited by the condition

A (ast )c int (11)

It is obvious that the choice of the polynomial x(n - 1, 5) does not affect the properties of robust stability and the robust quality of control, so the issue of calculating the polynomial x(n - 1, 5) is not considered in this paper.

The method for calculating the coefficients of polynomials P(n - 1, 5) and a(n - 1, 5) is given in [3. pp. 8-21], it is reduced to solving a system of 2n - 1 linear algebraic equations with respect to 2n - 1 unknown coefficients of polynomials P(n - 1, 5) and a(n - 1, 5). This system is uniquely solvable if the zeros of the polynomial a0(n, 5) do not coincide with the zeros of the polynomial b0(m, 5).

After closing the original object (6) by the controller synthesized according to the scheme (9)-(11), it is easy to obtain the following expression for the characteristic polynomial of the closed-loop system

a° (2n +1 -1, A,B,V,W,s) = {ac (2n +1 -1,a,b, v,w,s) :

Va g A, b g B, v eV, w eW}, (12)

here

a° (2n + 1 -1, a,b, v,w,s) = v(7, v,s) •£?(«,a,s) -p(« -l,s) --w(h, w,s)• b(m,b,s)-a(n -1,s).

The set of polynomials a°(2n + l - 1, A, B, V, W, 5) is called the "family of characteristic polynomials" of a closed-loop system. A set of

A( ac ) = {À, : 3a e A, 3b e B, 3 v eV,

3 w eW, ac ( 2n +1 - 1,a,b, v,w, À.) = 0, i = 1,2n +1 -1}

we will call the "set of zeros" of the family of characteristic polynomials (or the "set of poles" of the PF) of a closed-loop system.

We will assume that a closed system with characteristic polynomial (12) has a robust quality of control if the set A(a°) lies inside the area S, that is, the condition is satisfied (hereinafter, the condition of robust quality of control)

A(ac )c int (13)

In the presence of structural disturbances in the control object, it is impossible to guarantee in advance that the modal controller calculated by formulas (9) - (11) will ensure the fulfillment of condition (13). Thus, the problem of synthesizing a modal controller in the presence of structural disturbances in the control object consists of the following steps:

1) synthesis of a modal controller for a nominal object (7) (according to formulas (9)-(11));

2) subsequent verification of the fulfillment of condition (13) for a given family of characteristic polynomials (12) (the problem of testing the robust quality of control).

The next section presents a numerical method for checking the robust quality of control for a family of characteristic polynomials of the form (12).

2. Numerical method for checking the robust quality of control

2.1. Robust control quality criterion

Here and below (where this will not cause misunderstandings), we will omit the arguments of polynomials that are not essential for reasoning.

Theorem 1. Let the family of polynomials (12) satisfy conditions (8), (11). Then, in order for this family to satisfy the condition of robust control quality (13), it is necessary and sufficient that

0 g Ac (s), V s eOS, (14)

where set

Ac (s) = (flc(2n + /-l,i):VaGi,beB,VGF)weff|cC1)

is the "geometric image' of the family ofpolynomials (12) for the point s e OS. Proof. Polynomial

ac0(2« + /-l,a0,b0,v0,w0,S) = v(/,v0,S)-a(«,a°,S)x

xp(n -1,s)- w(h,w0,s)• b(m,b0,s)-a(n -1,s) (15)

belongs to the family (12). By conditions (8) and (11), all zeros of polynomial (15) lie inside the given area S. All polynomials of family (12) have the same degree and the coefficient at the highest degree is equal to 1, therefore, an arbitrary polynomial of family (12) can be considered as the result variations of the coefficients (a e A, b e B, v e V and w e W) with respect to the coefficients of the polynomial (15).

A change in the number of zeros of the polynomial lying inside the area S can occur if, as a result of the variation of the coefficients, at least one zero of the polynomial crosses the boundary of the area S and condition (14) is violated.

Thus, the fulfillment of conditions (8), (11), and (14) is necessary and sufficient for the robust control quality condition (13) to be satisfied. Q.E.D.

The polynomials in the family (12) contain products of variable parameters; therefore, the set ^°(s) on C1 can be non-convex (and even non-simply connected). Thus, the direct construction of the nonconvex set ^c(s) and the subsequent verification of expression (14) are difficult problems. Below we obtain sufficient conditions for robust control performance by estimating the distance from the set ^°(s) to the point (0, 0/). For a point s e OS, this distance is (taking into account notation (1)):

|ac (2n +1 -1,a,b, v, w,s)| ,

thus, expression (14) takes the form

\ac (2n +1 -1, a, b, v, w, s)| > 0, Vs eOS.

The expression on the left side of the last inequality satisfies the estimate

\ac (2n +1 -1,a,b,v,w,s)| = (v0(l,v0,s) + 5v(l,8v,s))x

x (ast (2n -1, s) + 5a (n -1,5a, s) •p (n -1, s) - 5b (m,5b, s) x

x a (n -1, s)) + (v0 (l, v0, s) + 5v(l, 5v, s) - w0 (h, w0, s) - 5w (h,5w, s)) x

x (b0 (m, b0, s) + 5b (m,5b, s)) • a (n -1, s)| > |v0 (l, v0, s) +

+5v (l, 5v, s )|_ • |ast (2n -1, s ) + 5a (n -1,5a, s ) • p (n -1, s ) -

-5b(m,5b,s)-a(n -1,s)| - v° (l, v0,s) + 5v(l,5v,s)-

-w0 (h,w0,s)-5w(h,5w,s) • |b0 (m,b0,s) + 5b(m,5b,s)| ^|a(n -1,s)|. The above reasoning can be considered as a non-rigorous proof of the following theorem.

Theorem 2. Let the family ofpolynomials (12) satisfy conditions (8), (11) and

p(s) = p1(,s)-p2(,s)>0, VsgôS,

(16)

where

Pl(s)i v°(/,v°,s) + ôv(/,ôv,s) ■\ast(2n-\,s) + +5a(n -1, Sa,s) • p(n -1,s) -56(m, 5b,s) • a(n -1,s)| , P2(S) = |V0(/,v0,S) + ÔV(/,ôv,S)-W0(A,w°,S)-ÔW(A,ôw,S) 60 ( m, b0, s ) + 56 ( m,5b,s) ^|a(n-1,s)|.

(17)

Then the family of polynomials (12) satisfies (13).

The technology for checking condition (16) is described below.

2.2. Method for checking the robust quality of control

The calculation of the functions pi(s) and p2(s) at the point s is reduced to solving 4 quadratic programming problems (hereinafter referred to as QP problems)

1

Jk ( xk )=1 xT Hk xk + qj xk

2

• min, k e 1,4,

E X - Zk ^ 0,

where

xj=5v, Axj =co/(AVj,---,AV;), =

e ! =v°(/,vV), x2=co/(ô a, ôb), Ax2 = col ( Aa0, • • •, AaB j, A60, • • •, A6m ), =c0l{^>(n-\,s),---,sn l -P(n-1,5),

-a(n -1, s )

-5 -a «-1,5

(n -1 s )),

<p2=Re(^2), v|/2 = Im(^2), 92 ¿ast(2«-l,s), x3=co/(Sv, Sw), Ax3 =co/(Avj, •••, Avt, Aw1; •••, Awh),

43=CO/(l, 5, •••, s', -1, -5, •••, V),

q>3=Re(^3), v|/3=Im(^3), e3=v0(/,v0,S)-w0(6,w°,S), x4=ôb, Ax4 = col (A60, •••, A6m), \,=col( 1, 5, •••, sm), q>4=Re(^4), \|/4=Im(^4), 04 =6°

H k = <

+2(Wk • w! + Vk • VT ), k e1,2, ), k e 3~4,

2(Wk • wl + Vk • VT

Qk =

+2 (Re (9k )• wk + Im (9k )• Vk ), k e1,2, 2(Re(9k)• Wk + Im(9k)• Vk), k e 3,4, ^ = Re2 (9k ) + Im2 (9k ),

(18)

X

+

and Ek and Zk are block matrices and vectors

Ek =(E E )T, Zk = col(Axk, Axk), k ep. For calculate Ek the dimension of the matrix E must be coordinated with the dimension of the vector xk. It is easy to verify that

HT = Hk, ^ > 0, k e 1,4. In notation (18), the functions p1(s) and p2(s) take the form:

Pl (s ) = J (x1 )• j\ (x2), P2 (s ) = 4 (x3 )• J4 (x4) • |a( n -1, s )|, (19)

x;=arg_min fk(xk), fk (xt) = + qTxi. (20)

Problem QP (20) is among the solved ones. In the case of constraints in the form of linear inequalities, this problem can be solved only by numerical methods.

Further, for convenience of presentation, we assume that the boundary dS is replaced by a finite number of points

n = {s,. eOS, i e1,N}, (N <<»), here the points si included in the set Q are chosen on the boundary dS based on the condition:

max A arg (®-s )|™ ^ <-1--— radian.

seos v /im=si- 2n +1 -1 2

Based on the above considerations, we can formulate the following algorithm for checking the robust quality of control.

2.3. The algorithm for checking the robust quality of control

Step 1. At each point s e Q(N) solve four QP problems (18), and use formulas (19)-(20) to calculate

the values of p1(s) and p2(s).

Step 2. Check the fulfillment of condition (16). If condition (16) is satisfied, then for a given family of characteristic polynomials, the robust quality of control is satisfied; if not, the modal controller should be calculated using the complete model of the control object (6).

3. Example of checking the robust quality of control

Let the control object be given by a model of the form:

(v0 ( 2, z) + Av ( 2, z )) • (a0 ( 2, z ) + Aa (1, z )) y (t) =

= ( w0 ( 2, z ) + Aw ( 2, z )) • (b0 (1, z) + Ab (1, z )) u (t),

here

a0 (2, z) = z2 -1,918z + 0,923, b0 (1, z) = 0,232z - 0,179, Aa (1, z) = 10 4 [-1; 1] z +10-4 [-2; 2], Ab (1, z) = 10-2 [-1; 1] z +10-2 [-1; 1], v0 (2,z) = 1,830z2 - 2,707z +1, w0 (2,z) = 1,848z2 - 2,717z +1, Av ( 2, z) = 104 [-5; 5] z2 +103 [-1; 1] z, Aw ( 2, z ) = 104 [-5; 5] z2 +103 [-1,5; 1,5] z. Zeros of the polynomials v°(2, z) and w0(2, z):

A( v0 ) = {0,715, 0,764} c C1,

A(w0) = {0,735 + 0,027j, 0,735 - 0,027j} c C1. Requirements for the quality of management are set by the area

S = {s: < |s| <^1> |arg(s> where n = 0,607, n = 0,961, 9 = n/4 radian. Note that the sets A(v0) and A(w0) lie inside the area S. We choose the characteristic polynomial of the standard

ast (3,z) = z3 -2,715z2 + 2,456 z -0,741. For the nominal model of control object and the standard ast(3, z), the modal controller is calculated

(z - 0,833) u (t ) = (-0,158z + 0,157) y (t ) + x0 g (t). The family of characteristic polynomials of a closed-loop system is

ac (5,s) = (v0 (2,s) + Av(2,s))x

x(ast (3,s) + Aa(1,s)^(1,s)-Ab(1,s)^a(1,s)) +

+(v0 (2, s) + Av(2, s)- w0 (2, s)-Aw(2, s))x

x(b0 (1, s) +Ab (1, s ))-a(1, s). (21)

To check the properties of the robust quality of control on the boundary of the area S, 42 points were chosen, starting from the point (0,961 + 0/) at an equal distance from one another. The smallest value of the function p(s) is 6,136^10-4, it is reached at the point (0,607 + 0/). Since the value p = 6,136^ 10-4 is greater than zero, therefore, according to Theorem 2, the family of polynomials (21) has a robust control quality.

Conclusions

In this article, a criterion for the robust quality of control is obtained for a closed-loop system consisting of a linear one-dimensional control object with structural disturbances and a modal controller. Theorem 2 establishes a criterion for the set of zeros to belong to the family of characteristic polynomials of a given area S on the complex plane.

Based on this theorem, a numerical method for checking the robust quality of control has been developed, which consists in solving four quadratic programming problems for each point s on the boundary of the area S.

The function p, defined by formulas (16)—(17), makes it possible to compare various closed systems with each other. Thus, the function p is a measure of robust properties, which can be further used to formulate the problem of improving the robust quality of control, for example, using the freedom to choose the characteristic polynomial of the standard ast(2n-1, s) within the limits of condition (11).

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Information about the author:

Parshukov Andrej N. (Candidate of Technical Sciences, Associate Professor, Industrial University of Tyumen, Tyumen, Russian Federation). E-mail: [email protected]

The author declares no conflicts of interests.

Информация об авторе:

Паршуков Андрей Николаевич - доцент, кандидат технических наук, доцент кафедры «Электроэнергетика» Тюменского индустриального университета (Тюмень, Россия). E-mail: [email protected]

Авторы заявляет об отсутствии конфликта интересов.

Received 07.10.2023; accepted for publication 05.03.2024 Поступила в редакцию 07.10.2023; принята к публикации 05.03.2024