Санкт-Петербургский государственный университет, г. -ПСаентекртбург, Россия

Abstract

Данная статья посвящена задаче синтеза H∞-оптимального управления для объектов, заданных линейной стационарной системой со скалярными управляющим воздействием и внешним возмущением, и невозмущенным измерением по нескольким переменным. Решение этой задачи не единственно, что дает возможность выбирать структуру наблюдателя и обеспечивать его дополнительные свойства. Отсутствие шумов в измерениях делает невозможным решение данной задачи стандартными средствами теории H-оптимизации, основанными на решении уравнений Риккати или линейных матричных неравенств. Вместо этого применяется спектральный подход, основанный на факторизации полиномов, что снижает вычислительную сложность синтеза. Продемонстрированы теоретическое обоснование подхода и пример его практического применения. H∞-оптимизация; параметризация; закон управления; устойчивость. This paper is devoted to H∞-optimization problem for LTI systems with scalar control and external disturbance signals and with no noisy multivariate measurement signal. The solution of this problem is not unique, that provides possibility to construct the controller with desired structure and additional properties. Besides, the absence of the measurement noise makes solution of this problem with implementation of standard H-theory methods, such as various modifications of 2-Riccati or LMI technique, impossible. A special analytical spectral approach in frequency domain based on polynomial factorization can be used that increases computational efficiency. Its theoretical description and example of practical implementation are presented in this paper. H∞-optimization; parametrization; control law; stability.

Veremey E.I., Knyazkin Y.V.

H∞-OPTIMAL SYNTHESIS PROBLEM WITH NONUNIQUE SOLUTION

plants, operated in real-time regime, especially with adoptive changeover of control laws, mentioned in [4].

This work is a development of concepts, proposed in [5-7]. The considered LTI system is affected a scalar control, scalar external disturbance and measurement signal is noiseless. The similar problem of synthesis of H2 – optimal controller has already been considered in [6] and the paper [7] is devoted to solving SISO (Single InputSingle Output) H∞ – synthesis problem with implementation of interpolation technique.

The paper is organized as follows. In the next section, equations of a controlled plant are presented and the problem of H∞-optimal synthesis is posed. Section 3 is devoted to computation of the transfer functions of H∞optimal closed-loop system. In Section 4, we consider calculation of the optimal controller. In section 5, we give illustrative examples of synthesis. Finally, Section 6 concludes this paper by discussing the overall results and future perspective of research in this area.

2. Mean-square Optimization Problem

Let us introduce a linear time invariant plant

Hurwitбz id (s) is an auxiliary signal with unknown structure. Let us use where x  Rn is the state space vector, which can beu amndeads(utr)eda,re the scalar control and external disturbance respectively.

All components of the mAa,tbri,cces are known constants, the Ap,abir is controllable.

External disturbance d(t) for the system ( 1 ) is treated as an output of the following system x  Ax  bu  cd(t), d(s)  S1(s)id (s) , S1(s)  N(s) T (s) , is stable.

3. H∞-Optimal Transfer Matrices

Solving of the stated problem directly is significantly obstructed by the fact that dependency of the functional J of the transfer function W (s) is nonlinear. This difficulty can be avoided by using of the parametrization technique, proposed in [3]. Let introduce the function parameter

(s)  α(s)Fx (s)  (s)Fu (s) , where α(s)  1(s) 2 (s)  n (s) , i (s), (i  1, n) , (s) where α and  are polynomials suthchat Q(s)  A(s)(s)  α(s)B(s) is where W(s)  W1(s) / W2(s), W1 (s)  W11(s) W12 (s) ... W1n (s) , W1i , (i  1, n) , W2 are polynomials. The choice of the transfer matrix function should 0 Note that the exact value of the functional ( 4 ) cannot be calculated, because structure of the external disturbance is not completely known, but it is possible to minimize its upper bound. Now we rewrite the expression ( 4 ) in frequency domain: firstly, introduce the transfer function H (s) , such as

I  I (W)  lim

T  T provide fmuinncitmiounmal of 1 T

  x' (t)Rx(t)  k 2u 2 (t)dt .

H (s)H (s)  FxT (s)RFx (s)  k 2Fu (s)Fu (s) , where Fx (s)  (Es  A  bW)1c,

F (s)  W(Es  A  bW)1c. where the

polynomiaNlsand T are following notatioвn дальнеи

шем The controller is to be designed in the form u  W(s)x ,

Sd (s)  S1(s)S1(s) . where is the set of conWtro(sll)e,rs such ththaet characteristic polynomial of the– locloopsedsystem1)-( 3 ) ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) (8) (10)

The auxiliary transfer f

The initial problem (8) can be transformed to minimization of the functional

~ 

J2  J2 ()  H ()S1 2 , J2  mi~n , ~ where ~ is set of raltiofnraactions(s) with Hurwitz denominators. In accordance with the formulae (12)

we consider a transfer function and calculate the valZu(egsi ) . It can be seen that Z (gi )  di , i  1, n , where the complex vadluieasre determined by the form17u)l.as Th(e initial problem can be so–lvoende as NP where Z(s) is the transfer function to be des-ignnegedat.iviNtyon of the Pick mLha(tr)ix  lij () necessary Fx  Fx ()  P 1c  BQ 1~ , Fu  Fu ()  AQ 1~ , ~    P 1c,

~ ~ H (s)H (s)  (T1  T2)(T1  T2)  T3 , где T1  B*sRCs / AsG , T2  G / Q , T3  C*sRC  (C*sRBsB*sRCs ) /( As AsGG ).

~ ~ J ()  H (s,)S1(s) 2  sup H ( j)S1( j) 2  

[0,)   sup (T1  T2)S1 s j  T3( j)Sd ( j)  min .

~ 2 [0,)  ~ J a  sup T3 ( j)Sd ( j) ,

[0,) ( G ~  B*s RCs ) N (s)G(s) Q A(s)G(s) R(s) 2

 1, Z (s)  (

G ~  B*s RCs ) N (s)G(s) Q A(s)G(s) R(s) (11) (12) (13) (14) (15) (16) (17) (18) (20) (21) (22) ~ It is evident, Tt3h(ast) is independent of the param(est)earnd the value

~ Is the lower bound of the fJu(nc)ti.onSaol

~ J ()  2, 2  Ja ,

~ ~ we can consider search of the paramsuetcehr htat: instead solving of the initial problem inequality.

Let formulate the necessary and sufficient conditions of problem solvability, using the Pick's theorem.

Theorem 1. The problem (16) is solvable if and only if the value  is such that Hermitian matrix ( 7 ). -PNicekvanilnintnerapolation can be implemented to guarantee this Lh (2)  lij (2) is non-negative, where lij  1  did j  gi  g j , di  BTs (gi )RCs (gi )N (gi )  As (gi )R(gi ),i, j  1, n, Here g i (i  1, n) are the roots of polynGo(misa)l(we assume that they all for simplicity) and the polyno R(s) is result of the following factorization

2  T3 (s)Sd (s)  R(s)R(s) /G(s)G(s) .

Proof. It is necessary and sufficient to design function    such that

H ( j)H ( j)Sd ( j)  2 , [0, ) , or (T1  T2~ )(T1  T2 ~ )Sd ( j)  2  T3Sd ( j) .

Let us execute the following factorization

2 T3 (s)Sd (s)  R (s)R (s) /T (s)T (s) G(s)G(s) , (19) where R (s) is a Hurwitz polynoamndial rewrite the expression (18) as

Fu  m1(s) A(s)R (s)  m2 (s)N (s)B*sRCs ,

G(s)G(s)m2 (s)N (s) Fx   Csm2 (s)N (s)G(s)G(s)  Bsm1(s) A(s)R (s)  Bsm2 (s)N (s)B*sRCs   

G(s)G(s) A(s)m2 (s)N (s) Bsm 1(s)R (s)  m2 (s)N (s)k 2 A(s)Сs  γ(s)RB(s)

G(s)G(s)m2 (s)N (s) , where the

polynomialtrimxa(s) is result of division γ(s)  C(s)BT (s)  B(s)CT (s) A(s) , and division is done totally [6].

Proof. Let us consider, that the condition of the theorem 1 выполняется is and the transfer function Z(s)  m1 (s) / m2 (s) is solution of the Nevanlinna-Pick interpolation problem (20). As a result and sufficient condition of solvability of the problem (18) that proves the theorem. Theorem 2. The dynamics of the optimal closed-loop system ( 1 ), ( 2 ) can be described by the following transfer functions (23) (24) (25) (26) (

Q G G ~ 0 

B*sRCs ) N (s)

 As

R (s) m1(s) m2 (s)

, and ~ 0 

Q(s)m1(s) A(s)R (s)  m2 (s)N (s)B*sRCs  ,

G(s)G(s) A(s)m2 (s)N (s) One can check that the expression in the square -b(r2a4c)ketiss ienqua(l23)to zero in thse gpi:oints this follows that divisionG(tos) is

done totally. Substitution (25) to the formulae (11)-(2r4e)s.ults in (23) Remark. Note that transfer functions of the optimal control process can be improper, that is undesirable. One ~ ~ of ways to overcome this difficulty is to deform the spectral power density, using S1 (s)  N (s) T (s) , where ~  N (s)  N (s)N d (s) and polynomial Nˆ (s) is Hurwitz.

4. Transfer Matrix of the Optimal Controller

As a result, the optimal transfer functions (23), (24) of the closed loop system are calculated, but it is necessary to design controller ( 3 ), providing such dynamics of the closed – loop system. Optimum condition, proved in [6], can be used for its computation.

Theorem 3 [6].: The controller ( 3 ) provides the optimal transfer functions (37), (38) for the closed loop system ( 1 ), ( 3 ) if and only if its transfer matrix W(s)  W1(s) /W2 (s) satisfies the following main polynomial equation (MPE): where

W1(s)fx (s) W2(s) fu (s)  0 , (26) polynomial colu mfxnand polynomial fu represent the numeratofrs the optimal transfer functions (23), (24).

Now we specially note that, the polynomial equation (39) has infinitely many solutions, if there are no common roots of the items f xi (s) of the column f x (s) and the polynomial fu (s) , which implies that the optimization problem has no unique solution. Also, it is necessary to mention that the functions F~x , Fu can have the common ~ multiplier C0 (s) . It can be seen that C0 (s) is divider of the characteristic polynomial (s) of the closed-loop system and it must be a Hurwitz polynomial to provide solvability of the problem.

Let us implement one way of construction of the optimal controller ( 3 ), described in [6], accepting the following particular structure of the controller ( 3 ) transfer matrix:

W(s)  W1 (s) / W2 (s), W1 (s)  k wW0 (s) , (27) where W0 (s) is the polynomial kawndis n-dimensional real raw such that the pCol0y(nso)mial C0(s)  kwC(s) , is

Hurwitz. The

(s)  N(s)G(s)C0 (s) , polynoWm0 (ias)lsand W2 (s) can be calculated

as follows W0 (s)  fu (s), W2 (s)  k w B(s) fu (s)  N (s)m2 (s)G(s)C(s) A(s). (28) of (28) to the expressiochnaraocftertihsteic

polynomial(s) (8) results in i.e. stability of the -lcolosped systems is guaranteed. It is easy to verify that the designed controller s optimality conditions (26).

5. Examples of Synthesis

Let us use Theorems 3 – 5 to design the optimal controller with the model ( 1 ) of a control plant, having the following matrices:

Then calculate the polynomial G(s)  0.005s3  0.014s2  0.018s  0.0024 , the value Ja  0.0000063 and the polynomial R(s)  0.0054s7  0.0189s6  0.0326s5  0.024s4  0.0135s3  0.005s2  0.0015s  0.0001 . Now let us Δ(s)  s11 + 13.47 s10 + 73.91 s9 + 219.6 s8 + 390.1 s7 + 418.9 s6 + 258.7 s5 +  85.1 s4 + 15.43 s3 + 1.56 s2 + 0.083s+0.002.

It can sbeen that roots of the polyn(so)mcioalincides

with ones of the polyGno(sm) ,iaNlsd (s) , m2(s) .

Let us choose the vector k w , such that the polynomial C0(s)  kwC(s) is Hurwitz C0(s)  s  2 . As a result, we receive kw  -0.318 17.141 98.081 and the optimal controller (28), where

Let us demonstrate the frequency response: Ad ()  H ( j)S1( j) 2 on the Figure 1. It can be seen that

The dynamics of minimized variable x3(t) before and after activation of the controller at 250 s is shown on the figure 2. It can be seen that oscillation is successfully suppressed.

A novel special approach in frequency domain to H∞-optimization for LTI controlled plants is proposed and described in details. The demonstrated method is not absolutely universal and can be implemented only for systems with scalar control. Despite this flaw, the mentioned approach is of importance for a wide range of practical control applications, such as marine autopilots, tokamak plasma control, mobile robots etc.

The solution of the presented problem is not unique, because there is no measurement noise, i.e. it is irregular. Irregularity is the main property of the stated problem, making its solving by popular methods impossible and defines suitability to design an alternative special spectral approach in frequency domain that is free from this disadvantage and can be successfully implemented for this problem investigation. Model of the plant is presented in a polynomial form and transfer functions of the closed loop system are parameterized in accordance to the special method. Then we use Nevanlinna-Pick interpolation and obtain transfer functions of the H∞ – optimal system. Finally, we can compute transfer function of the optimal controller, providing dynamics.

Using approach, based on the polynomial factorization instead implementation of the well-known methods, makes an optimal synthesis procedure significantly easier, especially for the plants with small dimension. This property is very important for onboard control systems.

The working capacity and effectiveness of the proposed method is demonstrated with numerical example: H∞ optimal controller is designed for the control plant of 3-th order in one of the easiest variants of synthesis

Finally, let us note that the proposed approach has one serious disadvantage: it cannot be used for plants with multiple control signals. Overcoming of this difficulty is the problem of the future research. Also the polynomial presentation is very suitable for investigation of the additional properties of the control system, which means that robust features of the controller, disturbance with non-fractional representation of the specter, task with transport or time delays can be paid serious attention in the future.

References Об авторах: Веремей Евгений Игоревич, доктор физ и-мкаотематических наук, профессор, заведующий кафедрой компьютерных технологий и сифстаекмультета прикладной математи-пкриоцессов управления, Санк-тПетербургский государственный университeе_тv,[e-mail] Князькин Ярослав Вячеславович, аспирант кафедры компьютернеыхнхолтогий и систем факультета прикладной математик-ипроцессов управления, Сан-Пкеттербургский государственный университет, [e-mail]

Note on the authors:

Veremey Evgeny I., Doctor of Science, Full Professor, head of the department of computer applications and systems faculty of applied mathematics and Control Processes, Saint-Petersburg State University, [e-mail] Knyazkin Yaroslav V., Graduate student of department of computer applications and systems faculty of applied mathematics and Control Processes, Saint-Petersburg State University, [e-mail]

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