State feedback control design for multivariable discrete time systems with required accuracy via mean-square criterion

Proceedings of the 9th IFAC Symposium Advances in Control Education The International Federation of Automatic Control Nizhny Novgorod, Russia, June 19-21, 2012

State feedback control design for multivariable discrete time systems with required accuracy via mean-square criterion Vladimir N. Chestnov*, Zhanna V. Zatsepilova** 

*V.A.Trapeznikov Institute of Control Science, Russian Academy of Sciences Russia (e-mail: [email protected]) **National University of Science and Technology «MISIS» Russia (e-mail: [email protected]). Abstract: We formulate the discrete-time controller design problem where the desired accuracy of controlled variables in the mean-square sense is guaranteed in the presence of bounded polyharmonic disturbances with a priori unknown number of harmonics, amplitudes, and frequencies. The amplitudes of the harmonics must satisfy a condition that results in the boundedness of the power of each polyharmonic component. The concept of mean-square radius of the steady state of the closed-loop system is introduced which accounts for the bounds on the mean-square values of both the controlled variables and control inputs (for each controlled and control variable). The attainment of the desired accuracy of the control system is formulated as the problem of ensuring the required mean-square radius of the steady state. The solution is based on the discrete-time H∞-optimization procedure by properly choosing the corresponding weighting matrices of the minimax cost criterion. Keywords: linear multivariable systems, bounded disturbances, discrete-time systems, state feedback, Hinfinity optimization, Riccati equation. 

it is required to design a controller of a prescribed accuracy, which uses the minimum amount of information about the disturbances.

1. INTRODUCTION In standard control theory courses the H∞-optimization is usually considered only in terms of optimal damping of l2norm bounded external disturbances (finite energy signals) which deflect a perturbed motion from the unperturbed one, in conformity with the concept going back to A.M. Lyapunov. The present work shows that the possibilities of this approach to a synthesis of controllers are much wider and allows a future engineer on control systems to use these procedures in case of actions of the unknown and powerbounded external disturbances which is of undoubted practical interest.

In this work, by this information we mean the knowledge of the mean-square values for each component of the disturbance. However unlike the standard LQG-problem, a deterministic function of time, namely, polyharmonic functions with arbitrary number of harmonics with unknown amplitudes and frequencies, will be considered rather than any stochastic characteristics. The accuracy is estimated via use of the mean-square values of the controlled variables and the control inputs. This work continues the line of research conducted in Aleksandrov A.G., Chestnov V.N. (1998, b) and Chestnov V.N. (1998) for the continuous-time case and Chestnov V.N. (2005) for the discrete-time case.

The offered approach to the synthesis of controllers is successfully applied in the educational process in preparation of automation engineers. In practice, automatic systems are usually affected by unknown exogenous bounded disturbances which cause the errors in the controlled variables. One of the main problems of automatic control theory is to ensure the required tolerances for these control errors.

To solve the problem, we use H∞-optimization procedures which were developed for the case of complete measurement of the state vector in the works of Iglesias P.A., Glover K. (1991, а.), Yaesh I., Shaked U. (1991), Grimble M. J. (2006). The solution of the problem of prescribed accuracy relies on the choice of the weighting matrices in the minimax quadratic functional of H∞ optimization.

For random disturbances, the accuracy of the control system is evaluated via the mean-square value of the controlled variables and, given the spectral characteristics of the noise, the well-developed apparatus of the stochastic control theory (LQG (H2)-optimization) can be exploited for the solution (Kwakernaak H., Sivan R., 1972)). However, in practice it is often the case that the assumption of the standard stochastic control theory are not fulfilled. This leads to problems where 978-3-902823-01-4/12/$20.00 © 2012 IFAC

2. STATEMENT OF THE PROBLEM Consider a linear discrete-time system described by the equations

96

10.3182/20120619-3-RU-2024.00011

9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

x([ k  1]T )  Ax(kT )  B1w( kT )  B2 u (kT ), z ( kT )  C1 x(kT ).

see (Chestnov V.N., 1998). It is obvious that for

(1)

conditions (5) and (6) are valid.

u (kT )  Kx(kT ), k  0,1, 2...,

(2)

We thus arrive to the following problem. Problem. For the given number   0 , find a stabilizing controller (2) such that the inequality

where x( kT )  R n is the state of plant (1), u ( kT )  R m is the control input, z (kT )  R

m1

rst2  1 ,

is the controlled output,

rst2   2

w(kT )  R  is the nonmeasurable exogenous disturbance, T is the sampling time, A, B1, B2 , C1, C2 are known matrices, and K is an unknown matrix of controller (2). The pair ( A, B2 ) is assumed controllable and the pair (C1 , A) is

(8)

is fulfilled for all disturbances in the class (3), (4). If such a controller does not exist, it is required to find a controller that gives the minimum possible value for    * for which the requirement (8) is fulfilled.

observable. The components of the vector of exogenous disturbances are bounded polyharmonic functions:

3. A LEMMA ON MEAN SQUARES

p

 wis sin[ s kT  is ], i  1,  ,

wi (kT ) 

1  p  . (3)

s1

In the exposition to follow, the lemma on mean squares is a key tool for obtaining the results in this paper. Below, this lemma is formulated for the discrete-time case.

Here, the amplitudes wis , phases  is ( i  1,  , s  1, p ), and distinct frequencies  s ( s  1, p ) are not known as well as the number p of frequencies, 1  p   . Finally, it is assumed that the mean square value (power) of each component of the disturbance satisfies the conditions N

p





1 1 wi2 (kT )   wis2  wi*2 , N  N  1 2 s 1 k 0

wi2  lim

Consider the following asymptotically stable discrete-time system given by

~ ~ ~ x ([ k  1]T )  A~ x (kT )  B w ( kT ), ~ z ( kT )  C~ x ( kT ), k  0,1, 2 ...

(9)

(4) where w is the vector of exogenous disturbances in the class (3), (4) and z is the vector of controlled outputs. Denote by

where wi* ( i  1,  ) are given positive numbers (regarding Eq. (4), see, for example, Tsypkin Ya. Z. (1964)).

T zw (e jT ) the disturbance-output frequency transfer matrix having dimension ( l1  l 2 ). Assume that this matrix satisfies

The problem of ensuring the prescribed accuracy via the mean-square criterion consists in finding a stabilizing controller (2) such that the mean-square values of the controlled and control variables of system (1), (2) subjected to disturbances in the class (3), (4) satisfy the requirements

the following frequency matrix inequality:

1 N 2 zi (kT )  zi*2 , i  1, m1 , N  N  1 k 0

zi2  lim



1 ui2 (kT )  u i*2 , i  1, m , N  N  1 k 0

where

~



( i  1, m1 , ) and

u i*

m1





z i2

 

* 2 i 1 z i

m





ui2

 

* 2 i 1 u i

,

~

are some positive-definite diagonal matrices of appropriate sizes.

(5)

We introduce the vector w *  [ w1* , w2* , ... , wl*2 ]T composed of the numbers that appear in the right-hand sides of inequalities (4) for the signal w and define its Euclidean

(6)

norm by w * 

( w * )T w * . The following lemma holds.

( i  1, m ) are given numbers. Lemma (on mean squares). Let the frequency inequality (10) be valid. Then, for any input signal from the class (3), (4), the mean squares of the steady-state values of the output variables of the stable system (9) belong to the set specified by the inequality

To solve the problem, we introduce the notion of the meansquare radius of the steady state of the closed-loop system (1), (2). The value of this radius is given by

rst2

(10)

where Q  diag[q1 , q2 , ... , ql1 ] and R  diag[r1 , r2 , ... , rl2 ]

N

u i2  lim zi*

~ ~ TzTw (e  jT )  Q  Tzw (e jT )  R ,   [0,  / T ],

(7)

l1

 i 1

97

q~i zi2 

l2

 ~ri  wi*2 , i 1

(11)

9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

where zi ( i  1, l1 ) are the components of z .

Similarly,

Proof. As k   , the forced oscillations of the output of system (9) are given by

we

z (kT )  Tzw (e jwsT )w( s )e js kT , ~ ~ ~ Tzw (e  jwsT )  C (e  jwsT I  A) 1 B .

p

zi ( kT ) 

 ai ( s ) sin s (kT )  i ( s ),

w ( kT )  w( s )e  js kT

for

obtain

where

By superposition, for the s -th harmonic of the input signal w with components satisfying (3), we can write

i  1, l1 , (12)

s 1

z  z Tzw (e jsT ) w( s )e js kT  Tzw (e jsT ) w( s)e js kT  . 2j 2j

ai ( s )  0 and  i ( s ) ( i  1, l1 , s  1, p, 1  p   ) are, respectively, the amplitudes and phases of the output oscillations generated by the s -th harmonic of the l 2 -dimensional input signal w of the form (3). The amplitudes of oscillations of each coordinate of the vector z where

It is obviously that ai2 (s )  z i z i , where zi and are the i -th components of the vectors z respectively.

z i and z ,

in (12) are absolute values of the appropriate components of  jT

Tzw (e

)w(s)

where

(s)

and w  [w1s e

 j 1s

w(s)

[w1se

Then, by taking into account a diagonal structure of the

Tzw (e jT ) w( s ) and

the complex conjugate vectors j1s

w2se

j2s

w2s e  j 2 s ... wl2 s e

... wl2se

~

matrix Q , we obtain

jl2s T

]

l1

 j l2 s T

~

~

q~iai2 (s )  zTQz  w(s)TTzTw (e j T )QTzw (e j T )w(s) , (15)

] .

s

s

i 1

It is immediate to check that the s -th harmonic of the input signal w with components (3) can be represented as

w( s ) e js kT

w( s ) e  js kT

 2j

where s  1, p. Summing up equalities (15) over all frequencies and taking into account inequalities (10), the diagonal structure of the

. We then find the particular

~

matrix R and the inequalities (4), for w we thus obtain

solutions of the difference equation (9) and the associated values

of

w ( kT ) 

the

output

w( s ) e js kT

and

substitute w ( kT )  w( s ) e obtain

z

vectors

w ( kT ) 

js kT

z

and

w( s ) e  js kT

find

its

particular

p l1

We

q~i ai2 (s ) 

p

~ w(s)T Rw(s)T 

l2





~riwi*2 .

s 1 i 1

s 1

i 1

(16)

into the first formula (9) to

~ ~ ~ x ([k  1]T )  A~ x (kT )  Bw( s )e js kT , and

.

for

Changing the order of summation in (16) and passing on to the mean squares and accounting for the prescribed power bounds (4) on every component of the vector w (3), we arrive at the desired inequality (11).

(13)

solution

in

the

form

We stress that this lemma is of the general form, and estimate (11) is attainable.

~ x (kT )  x s e js kT , were xs  С т is a vector of complex numbers.

4. THE STATE FEEDBACK PROBLEM We then have

Let ~ z be the extended vector of the controlled output of the

~ x ([k  1]T )  x s e js ( k 1)T  x s e jsT e jskT . Substituting

(14)

into

~ ~ xse jsT e js kT  A xs e js kT  Bw( s)e js kT , ~ ~ x s  (e jsT I  A) 1 Bw( s ) .

1/ 2

(14)

 R

(13): we

quadratic cost functional (Basar T., Bernhard P., 1991) 

J  min max

ul 2 wl 2

 [ z T ( kT )Qz ( kT ) u T ( kT ) Ru ( kT )  k 0

2 T

  w (kT ) w( kT )] , which are chosen by the designer.

~

Since z (kT)  C~ x(kT) we arrive at z(kT)  Tzw(e jwsT )w(kT) ,

~

u 

C1x  T  , where Q  Q  0 and R u   1/ 2

R  R T  0 are weighting matrices of the minimax

obtain

~ ~ Hence, ~ x (kT )  (e jsT I  A) 1 Bw( s )e js kT . ~

1/ 2

Q z  Q form ~ z   1/ 2   

~

Assume that the control law (2) guarantees the satisfaction of the cost inequality

where Tzw (e jws T )  C (e jwsT I  A)1 B .

T~z w

98





(17)

9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

m1

where T~z w is the discrete-time transfer function matrix of the closed-loop system from the exogenous signal w to output ~z , and  is a given positive number.

 qi i 1

u (kT )  ( R 

 PB ( I 

 C1T QC1

Clearly, by choosing the entries of the weighting matrices Q and R as

w*

(19)

qi 

and B   1B1 .

2

( zi* ) 2

w* , i  1, m1 ;

ri 

2

(ui* ) 2

, i  1, m ,

from (23) we obtain the inequality

The conditions (18) and (19) are necessary and sufficient for (17) to hold (Yaesh I., Shaked U., 1991; Grimble M.J., 2006).

m1



 

m





ui2

 

* 2 i 1 ui

  2 , which is equivalent to the cost

condition (8) in the formulated problem with state controller (18). 5. A NUMERICAL EXAMPLE Let us consider a numerical example of finding the state feedback controller. To illustrate, we use the interconnected model of the shaping and grooving mills. A structural diagram of the corresponding electric drives is shown in Fig.1.

1/ 2

~z  Q z   T~ w  Q Tzw  w.  1/ 2   1/ 2  z w  R u   R Tuw 

zi2

* 2 i 1 zi

In this section, we are aimed at choosing the weighting matrices Q and R from the condition that guarantees the solution of the formulated problem with state-space controller (18). To this end, we transform the cost condition (17) with account for the form of the ~ z -vector and the structure of the disturbance-output transfer function matrix: 1/ 2

(23)

i 1

Proof: Using the lemma on mean squares, inequality (23) can be obtained from (22).

where P  P T  0 is the solution of the discrete-time Riccati equation

BT PB ) 1 BT P

2

ui2   2 w* ,

components in the right-hand side of (4).

(18)

AT PA  P  AT PB2 ( R  B2T PB2 )1 B2T PA 

m

 ri

where w * - is the Euclidean norm of the vector w* with

It is well-known that the control low has the form (Yaesh I., Shaked U., 1991; Chestnov V.N., 2005; Grimble M.J., 2006)

B2T PB2 )1 B2T PAx (kT ) ,

zi2 

(20)

Here, Tzw is the discrete-time transfer matrix of the closeloop system from the external signal w to the controlled output z ; Tuw is the discrete-time transfer function matrix from the vector w to the control input u . In terms of frequency transfer matrices, the equivalent of inequality (17) has the form

T~zTw (e  jT )  T~z w (e jT )   2 I ,   [0,  / T ].

(21)

Accounting for (20), from (21) we arrive at the inequality T Tzw (e  jT )  Q  Tzw (e jT )  T  Tuw (e  jT )  R  Tuw (e jT )   2 I ,   [0, / T ].

Fig. 1. The structural diagram of the interconnected electric drives in the shaping and grooving mills.

(22)

Here, x1 , x 2 are the deviations of the output voltages of the thyristor converters, which are fed to the motors’ armature circuit; x3 , x 4 are the deviations of currents of the drive

Let the weighting matrices Q and R have diagonal structure

Q  diag[q1, q2 , ... , qm1 ] ,

qi  0 ,

i  1, m1 ,

motors; x5 is the angle speed deviation of the motor shafts

R  diag[r1 , r2 , ... , rm ] , r j  0 , j  1, m .

rotation; u1 ,u 2 are the deviations of the control voltages fed to the thyristor converters from the drive control system; M 1 , M 2 are the deviations of the electromagnetic torque of the motor from the nominal values; M is the deviation of the loading torque (drag); Tc1 , Tc 2 are the time constants of the

Theorem 1. The steady-state values of the controlled variables and control inputs in the closed-loop system (1), (18) satisfy the following frequency inequality:

99

9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

thyristor converters; k c1 , k c 2 are the transfer coefficients of

the desired steady state error in the speed of the shaft

the thyristor converters; R1 , R2 are he ohmic resistances of

( z1  1 ), and the limits on the load-torque deviation from the nominal value (20% of the nominal motor torque

*

the motors’ armature circuit; Te1 , Te 2 are the electromagnetic constants of the motors’ armature circuit; J is the total equivalent moment of inertia of the system; C e1 , C e2 , C M 1 , C M 2 are constructive constants of the

w *  600 ). In other words, the matrix Q is chosen according to (30) and R=I, i.e.

motors. Finally, z  [ z1 , z 2 , z 3 ]T  [ x3 , x 4 , x5 ]T is the controlled output. In that case, the matrices of the continuoustime plant are taken in the following form:

600 0 0  1 0 . Q   0 1,6 0  , R   0 1    0 0 1,6

0 0 0 0   100    0  83 , 3333 0 0 0  , A  137,8105 0  11,2866 0  1123,155   0 132, 4591 0  11,0653  1101,133    0 0 0,24867 0, 25364 0

The sampling time is T  0.01 s. The associated matrix of the discrete-time controller (23) is given by

- 0,0050 K  0,0012

0   0  16120     13702  0   0  , B2   0 B1   0 0 ,     0   0   0   0,0307   0 0 

0,0004

- 0,0023

- 0,0124

0,0021

0,0006

0,0066 . - 0,0130 0,1232

The next figure illustrates that the worst exogenous disturbances is a step.

Tz1w ( jv)

0 0 0 0 1  C1  C 2  0 0 1 0 0 . 0 0 0 1 0 Given matrices of the discrete-time plant are: 0 0 0 0  0,3679   0  0,4346 0 0 0  , A   0,8114  0,0049 0,8804  0,0132  10,5223    0,0047 0,8395  0,0127 0,8823  10,3272  0,0012 0,0012 0,0023 0,0024 0,9732  0    0   , B1   0,0017     0,0016    0,0003

v Fig.3 The peak amplitude-pseudo frequency characteristic close-loop system from w to z1 (scaled by amplitude 600 of exogenous disturbances w). Figure 3 below depicts the step response of the currents and angular speed deviation of the motors under the step change of the load torque w  600 .

0  101,8978   0 92 , 9656  , B2   78,3032  0,1754     0,2003 67,1784   0,0707 0,0611 

0 0 0 0 1  C1  C 2  0 0 1 0 0 . 0 0 0 1 0

The diagonal weighting matrix Q is chosen so as to satisfy the conditions on the admissible deviations of the drive *

*

motors currents from the nominal values ( z 2  z3  375 ), 100

9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

Use of procedures of H∞-optimization. Automation and Remote Control, V. 59, N. 8, 1153-1164. Basar T., Bernhard P. (1991) H∞-Optimal Control and related minimax design problems: A dynamic game approach, Boston, MA Birkhauser. Chestnov V.N. (1998) Synthesis of multidimensional systems of prescribed accuracy by the mean-square criterion. Automation and Remote Control, V. 59, N. 12, 17861792. Chestnov V.N. (2005) Design of Digital State H∞-Controllers of the MIMO Systems with Prescribed Precision. Automation and Remote Control. V. 66, N. 8, pp. 1233 – 1238. Grimble M.J. (2006) Robust Industrial Control Systems Optimal Design Approach for Polynomial Systems. John Wiley & Sons, Ltd. Iglesias P.A., Glover K. (1991). State-space discrete-time H∞ control theory. Eur. Control Conf. Grenoble. France, V. 2, pp. 1730-1735. Iglesias P.A., Glover K. (1991). State-space approach to discrete-time H∞ control. Int. J. Control, V. 54. N. 5, pp. 1031-1073. Kwakernaak H., Sivan R. (1972) Linear Optimal Control Systems, New York: Wiley-Intersciene. Tsypkin Ya.Z. (1964) Sampling Systems Theory and Its Applications. V. I, II. Oxford: Pergamon Press. Yaesh I., Shaked U. (1991). A transfer function approach to the problem of discrete-time systems: H∞-optimal linear control and filtering. IEEE Trans. Automat. Control, V. 36, N. 11, pp. 1264-1271.

0,09 0,1 Time (sec)

Fig.3 Step response of currents and speed of the motor. From the figures, it is seen that the currents of the motors are close to each other. This means that the values of the armature currents are also close to each other and that the steady state deviations of the currents and shaft speed from the nominal values do not exceed the prespecified limits. 6. CONCLUTION This work is devoted to the design of discrete-time controllers that ensure the required accuracy of the controlled variables in the mean-square sense in the presence of bounded polyharmonic disturbances with arbitrary number of harmonics with unknown amplitudes and frequencies. The amplitudes of the harmonics must satisfy a condition that results in the boundedness of the power of each polyharmonic component. The concept of mean-square radius of the steady state of the closed-loop system is introduced which accounts for the bounds on the mean-square values of both the controlled variables and control inputs (for each controlled and control variable). The attainment of the desired accuracy of the control system is formulated as the problem of ensuring the required mean-square radius of the steady state. The solution is based on the discrete-time H∞ optimization procedure by properly choosing the corresponding weighting matrices of the minimax cost criterion. The results of this work can be easily generalized to the output feedback case with H∞ controllers. Practical efficiency of the offered method of synthesis of controller is illustrated by controlling the interconnected model of the shaping and grooving mills. REFERENCES Aleksandrov A.G., Chestnov V.N. (1998). Synthesis of multivariable systems of prescribed accuracy. Part I. Use of procedures of LQ-optimization. Automation and Remote Control, V.59, N. 7, 1998, 973-983. Aleksandrov A.G., Chestnov V.N. (1998) Synthesis of multivariable systems of prescribed accuracy. Part II.

101