Proc. of the IEEE International Conference on Smart Instrumentation, Measurement and Applications (ICSIMA) 26-27 November 2013, Kuala Lumpur, Malaysia

978-1-4799-0843-1/13/$31.00 ©2013 IEEE

Sampled-data Regulation of a Class of Time Varying Systems Based on a Realizable Reconstruction Filter

Muwahida Liaquat#1, Mohammad Bilal Malik*2 #* Department of Electrical Engineering, College of Electrical and Mechanical Engineering, National University of Sciences and

Technology Peshawar Road, Rawalpindi, Pakistan

1muwahida@ee.ceme.edu.pk 2mbmalik@ceme.edu.pk

Abstract—A sampled-data regulation problem for a class of

continuous time varying systems is presented. The main constraint is the availability of only the samples of the output for measurement. The particular focus is on such linear time varying systems that can be transformed into linear time invariant systems through sinusoidal transformation. This converts the problem of tracking a constant signal into a sinusoidal signal. Typical to the sampled-data control problems, the control input is generated by a discrete controller along with a discrete observer for state and disturbance estimation. A realizable reconstruction filter, which is a reduced order specialized generalized hold device, connects the discrete controller with the continuous input of the system. The resultant linear time invariant system along with the realizable reconstruction filter is thus presented as a linear impulsive system. The rest of the problem is similar to the classical regulator theory. As an example the control of gyroscope in both perturbed and unperturbed scenarios is presented.

Keywords—discretization; disturbance rejection; impulsive systems; linear systems; output regulation; sampled-data control.

I. INTRODUCTION Output regulation is the problem of asymptotic tracking /

rejection of signals generated by an exosystem (whose model is usually known but the states are not accessible). The main issue is to find an internal model based controller with limited information about the tracking error to steer the tracking error to zero. The complete solution of output regulation for continuous linear time invariant systems is already available in the conventional sense [1]-[2]. This problem has been extended to time-varying linear systems dealing with time-varying references originating from linear exogenous systems through differential regulator equations [3]-[4]. The major drawback in this technique is the difficulty to solve the regulator differential equations. Another approach is to apply if possible a coordinate transformation to convert the problem of time varying regulation into a time invariant one.

This paper focuses on linear time varying signals that can be converted into continuous time invariant linear systems by applying a sinusoidal transformation. For such systems, the output regulation is dealt by converting the tracking of a constant signal into tracking of sinusoidal signal generated by a

linear time invariant exosystem whose model is known but

only the sampled measurements are accessible. This problem is transformed into a sampled-data regulation problem due to the availability of only the sampled measurements.

Sampled-data output regulation has been the focus of extensive research with the increased use of microprocessors in the controllers [5]-[10]. Conventionally two approaches are used to solve this problem. One way is to design the controller / observer in continuous domain and then discretize the whole system [8]. While the other approach is to first discretize the complete system and then design the controller / observer in discrete domain. Sampled-data regulation is solved by using the later approach in [11] where the solution is presented by solving the regulator equations in discrete domain. The main draw back in this scheme is the failure of inter-sampling behavior of the controlled continuous system. The solution to this problem is to incorporate generalized hold device (GHD) in the controller design.

GHDs can be viewed as interpolators with advantages of state feedback controllers without using state estimation techniques. Inspired by the GHD methodology a realizable reconstruction filter (RRF) is presented in [9],[12]. RRF is modeled as an impulsive system. Since, it is assumed that the exact model of the exosystem is known, the design of RRF is based on the continuous time internal model of the exosystem. It acts like a map between the discrete input (generated by the discrete controller) to the continuous input signal to the linear system. In this way the continuous control input can be used in the design of the control scheme to overcome the inter-sampling behavior of the closed loop system. Thus the overall problem is simplified which makes it an efficient technique producing better performance as compared to conventional GHD schemes. This approach is especially useful for the tracking control of gyroscope which is presented in this paper. Using RRF as digital to analog converter, the sampled-data regulation control of the x and y axis of gyroscope is discussed with and with perturbations. In the end the working of the proposed scheme is demonstrated through the control of gyroscopes using the proposed methodology.

This work is part of the PhD research work of Muwahida Liaquatsponsored by National University of Sciences and Technology Pakistan,

II. PROBLEM FORMULATION Consider the following linear time varying (LTV) system

that can be represented by the state space model given by

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )[ ] [ ] [ ],

x t A t x t B t u t P t d t

y t C t x t

y k C k x k

δ = + +

=

= (1)

where nx R∈ is the state, py R∈ is the output and mu R∈ is the input. It is assumed that only the sampled measurements of the output [ ]y k are available for measurement. The disturbance present in the system is modeled by ( )d t .

A. State Variable Change Linear time varying system (1) is transformed into an LTI

system through state variable change [5]. Suppose a new vector is defined by

( ) ( ) ( )1 ,Tz t P t x t−= (2)

where the n n× matrix ( )TP t is invertible and continuously differentiable at each time instant t . It is given by

( ) ( )( ) ( )

cos sin( )

sin cosT

t tP t

t tω ωω ω

−= (3)

The transformation matrix ( )TP t is used to perform sinusoidal transformation on the time varying system (1) to time invariant system (4). The state equation for the resultant LTI in terms of

( )z t is given by

( ) ( )( ) ( )[ ] [ ]

( ) ( )

.

z z z

z z

z z

z t A z t B u t P w t

y t C z t

y k C z k

= + +

=

=

(4)

Such that

( ) ( ) ( ) ( ) ( )1 1z T T T TA P t A t P t P t P t− −= − , ( ) ( )1

z TB P t B t−= ,

( ) ( )1z TP P t P tδ

−= , ( ) ( )z TC P t x t= .

The goal is to design an output feedback control with access to only sampled error measurements given by

[ ] [ ] [ ],z refe k y k y k= − (5)

such that asymptotic regulation is achieved i.e.

( )lim ( ) .z reft

y t y t→∞

=

The reference signal ( )refy t is generated by a linear exosystem. The inverse transformation of (2) is then applied to

( )zy t in order to obtain ( )y t .

B. Linear Exosystem Sinusoidal transformation of the LTV system into an LTI

system leads to converting a problem of constant reference signal tracking into a time varying sinusoidal signal tracking. The well known properties of linear systems [13] suggest that the disturbance present in the linear system is of the same model as reference. However, if the exact model for the disturbance estimation is not known, choosing a suitable polynomial model can be used for the estimation purposes. Thus, the reference signal and the disturbance introduced can be modeled by an exosystem given by

( ) ( )

0

0

( )( ) ( )

( )

[ ] ( )( ) ( ),

r r

d

r r r

r r

d

S

S

w tt t

w t

y t Q w ty k y kTd t P w t

δ

δ

= =

===

w w

(6)

where rS Sδ = . The reference signal is given by ( , )r rS Q and the disturbances introduced into the system are modeled by z zP B= .

III. REALIEZABLE RECONSTRUCTION FILTER (RRF)

The control input ( )u t is to be realized through a special generalized hold device termed as a realizable reconstruction filter (RRF) [9] such that

( ) ( )( ) ( ) [ ]( ) ( ),

h

h

h

t A t t kk k B u k t k

u t C t

η η τη τ η τ τ

η−

= ≠= + =

= (7)

where the continuous states ( )tη is subject to jumps at time instances .kτ Throughout this paper the time instances are assumed to be at integer multiples of the sampling time τ . The discrete input to the impulsive system (7) is given by [ ]u k and the output is a continuous signal ( )u t . The design of RRF is based on the dynamics of the exosystem resulting in

.rhA S=

Designing hold function such a way has the advantage of state feedback instead of estimation [8]. Therefore the states of RRF

( )tη and its samples [ ]kη are available for measurement. RRF guarantees that the signals [ ]u k and ( )u t are of the same dimensions. If

rS is the minimal realization of (6) then the resulting structure of RRF is expected to be of reduced order as compared to other GHDs [6]. Also the structure of RRF ensures the controllability and observability of the overall design.

IV. SAMPLED-DATA OUTPUT REGULATION WITH RRF LTI system (4) along with RRF (7) can be treated as an

impulsive system. Let,

[ ], , , .hz

zz z

h

hhC

A BB CB C A

= = = 00

= Β0

0A P C

Then the corresponding impulsive is given by

( ) ( )( ) ( )

( )( ) ( ) ( ) ,

( )

[ ] ,

( ) ( ).

tt t w t t k

z t

kk k u k t k

z k

t t

ητ

η ττ τ τ

τ−

= = + ≠

= = + =

=

x Ax P

x x B

y Cx

(8)

The continuous-time state ( )tx experience jumps at fixed integer multiples of a τ . The continuous output required to be tracked is ( )ty and the control input is a discrete signal [ ]u k .

To further simplify the given problem, sampled-data system (8) and linear exosystem (6) can be casted as an impulsive system

( ) ( )( ) ( )

( )( ) ( ) ,

( )

[ ] ,

( ) ( ).

tt t t k

w t

kk k u k t k

w k

t t

τ

ττ τ τ

τ−

= = ≠

= = + =

=

x

x

X AX

X X B

Y CX

(9)

Where,

[ ], , .00

QS

= ==B

CA P

CA B

Then the state response of (9) for 1[ , )k kt T T +∈ can be given as

0 01

( ) ( , ) ( ) ( , ) [ ].k

jj

t t t t t T u j=

= Φ + ΦX X B (10)

The zero-input and zero state response can be represented by (10). For the sampled data impulsive system (9) the following assumptions are made

A1. ( , )d dA B is stabilizable.

A2. The pair ( ),cC e τA is detectable.

A3. { }( ) : | 1 .bSe Cτσ λ λ⊂ = ∈ ≥

A4. There exist matrices sΠ and sΓ which satisfy the discrete regulator equations

0 ,s d d s d d s

d s d

SQ

Π = Π + + Γ= Π +

A P BC

such that

, , , .d d d d d de τ= = = =AA B A B P A P C C

While the discrete exosystem is represented by dS and dQ .

A. Discrete Controller / Observer We propose the following discrete observer / controller for

sampled data output regulation for the impulsive system (9). In this context a discrete observer followed by a discrete controller is proposed such that the closed loop system is asymptotically stable. We define

[ ], and 0 .0 0

h

d

A P BCC

S=A B C= =

Then the discrete observer and controller are given as

( )

[ ]

ˆ ˆ ˆ[ 1] [ ] [ ]

ˆ[ ]ˆ

s s

k e e u k H y k

u k F Fw

τ τ

η

+ = + + −

= Γ − Π

A A B C

,

0 0 0

0

X X X

X (11)

Design of discrete controller gain F is carried out by choosing the eigen values of pair ( )d d F+A B fall within the unit circle. On the same lines, discrete observer gain H is chosen such that ( )e Hτ −A C is exponentially stable. Also, the estimated

states for the exosystem w are obtained through the standard discrete observer [13].

B. Stability Convergence The stability of the sampled data regulation using RRF is

established in [7]. The stability analysis is based on the following assumptions for continuous system (4) and (6).

B1. ( , )z zA B is stabilizable.

B2. The pair [ ]( , )0

z zz r

A PC Q

Sδ is

detectable.

To ensure the stabilizability and detectability under discretization involving RRF It is essential to ensure that Assumptions B1 and B2 guarantee A1 and A2. We assume

The sampling period τ is non-pathological

for0

z zA PSδ

. That is, for any pair of eigen values ,λ λi j

their difference is not an integer multiple of 2πτ .

C. Theorem Under assumptions B1, B2 and B3 output regulation with

internal stability is achievable via sampled error feedback if it is achievable via continuous error feedback.

Proof : See [7].

V. EXAMPLE The given technique is illustrated by designing control of a

gyroscope that can be represented by the given state-space model [14]

( ) ( ) ( ) ( )( ) ( )[ ] [ ] [ ]

'

.m

x t Ax t B t u t

y t C x t

y k C k x k

= +

=

=

(12)

The states of the system are given by 4x R∈ and 2u R∈ is the continuous control input. The given system is to be controlled using only the sampled measurements of the output my R∈ .

While the continuous output 2y R∈ needs to be regulated to follow a constant reference

Tx yr rr = .

The system matrices corresponding to (12) are

( )

[ ]0

0 1 00 0 1

0 0 0 /0 0 / 0

00

sin( )cos( )

( ) cos( ) sin( ) sin( ) cos( ) .

i

AH J

H J

B t Ktt

C t K t t t t

ωω

ωω

ω ω ω ω ω ω

−=

−

=

−

= −

(13)

The variables and their numerical values used in (13) are

( )2

2

angular momentum

moment of inertia

moment of inertia about z-axis

spin frequency of gyroscope about z-axis

scaling factor that depends on mechanism of ge

0.01Kg-m

0.017 Kg-m200 rad/sec

z

z

o

H J J

J

J

K

ω

ω π

= = −

= =

= == == nerating ( )

torque constant

0.10.01

m

i

t

K=

= =y

The gyroscope is assumed to be spinning about z axis with a constant angular speed ω . The angular positions of gyroscope around x and y axis are represented as the states 1x and 2x in the state-space model (12). The spin frequency about x and y axis corresponds to the derivatives of first and second states

respectively. The requirement is to secure the position of x

and y axis to a [ ]0.3 0.2 T=r . The initial conditions of the gyroscope are

[ ][0] 0.01 0.02 0.01 0.01 T=x .

The sampling time is chosen to be close to nyquist rate i.e., 0.04 secT = . It is apparent from (13) that the system (12) is

uncontrollable. To resolve this problem the time varying

system (12) is transformed into an LTI system using sinusoidal transformation given by (3). Using (3) and (13) the LTV system (12) is converted into LTI system given by

_

( ) ( ) ( ) ( )( ) ( ),

z z z

m z z

z t A z t B u t P w ty t C z t

= + +=

(14)

The corresponding system matrices are given by

[ ]0

0 1 00 0 1

0 0 0 /0 0 / 0

0001

0 0 1 .

z i

z

AH J

H J

B K

C K

ωω

ω

−=

−

=

−

= −

(15)

Thus, the problem of tracking a constant reference is transformed into tracking a sinusoidal signal whose model is known and is given by

( ) ( )

( ) [ ] ( )

00

1 0[ ] ( ).

r

r r

w t w t

y t w ty k y kT

ωω

−=

==

(16)

The actual system is obtained by applying the

transformation

( ) ( ) ( ).Tx t P t z t= (17)

Then [ ] [ ]1 2 x yT Tx x = axis. The control input

( )1 1u t×

is thus generated by RRF and [ ]u k is controlled

through the discrete controller (11). The controller is designed by placing eigen values for the pair ( ),z zA B

close to ω at

[ ]0.56 0.41 0 .56 0.41 0 .56 0 .41i i i− ± − ± − ± . Regulation is

carried out for both unperturbed and perturbed systems.

A. Tracking of Unperturbed System For the system without perturbations the observer (11) is

used along with the exosystem (16). The discrete observer is thus designed for estimating the sates of the system only. This is done by placing eigen values of the pair ( ),z zA C at

[ ]0.48 0.35 0.48 0.35− ± − ± . For this case, the desired tracking performance is achieved i.e., the error (5) is asymptotically reduced to zero. This is shown in the the following figures.

The tracking of the sinusoidal signal through transformed system (14) is shown in Figure 1. The asymptotic tracking of the sinusoidal signal along with the error plots are also shown. It can be seen that the proposed scheme was able to achieve the required results successfully.

Fig. 1. Tracking control of transformed system _ ( )z my t in z-coordinates.

Figure 2 shows the output of the actual system after the reverse transformation (17) has been applied to it. The result is a constant signal shown below. The output of the x and y axis can be seen by using the following equation.

( ) ' ( ),

1 0 0 0( ).

0 1 0 0

y t C x t

x t

=

= (18)

The signals corresponding to the control of x and y axis are also shown in figure 3.

B. Tracking of Perturbed System The regulation was carried out by introducing perturbations

as well. The perturbations were assumed to be sinusoidal with the following initial conditions

Fig. 2. Output ( )my t in the actual system cooridicates represented by (Equation (12)).

Fig. 3. Tracking of x and y axis with respect to reference [0 .3 0 .2 ]Tr =

Fig. 4. Tracking control of transformed system ( )zy t in z-coordinates.

[ ](0) 0.3 0.2 .Td =

This leads to using the observer (11) with the exosystem (6) for the control design. The observer is designed for estimating the sates of the system and the disturbance introduced by placing eigen values at [ ]0.48 0.35 0.48 0.35 0.48 0.35i i i− ± − ± − ± . Figure 4, shows the tracking control of the transformed system (14). The signal to be tracked is a sinusoid signal. Since, for the linear systems the perturbations introduced at the input are of the same frequency as the reference signal [13]; the sinusoidal perturbations along with their transformed values are shown in figure 6. It was required to regulate the x and y axis (18) of the time varying system (12) to the given fixed reference. This can be seen in the figure 5.The estimated disturbances in both time varying and transformed z-coordinates are shown in figure 6. It can be seen that the transformed disturbance signals are sinusoidal signals of the same frequency as the actual signal.

Fig. 5. Tracking of x and y axis with respect to reference [0.3 0.2]Tr = .

Fig. 6. Estimated sinusoidal disturbance ( )zd t and ( )xd t .

VI. CONCLUSION The sampled-data regulation problem for a class of linear

time varying systems is discussed in this paper. Sinusoidal transformation of a certain class of time varying linear systems converts them into linear time varying systems. This results in tracking of constant signals to be converted into sinusoidal tracking. Using RRF as digital to analog converter, along with the resultant linear time invariant plant are modeled as a linear impulsive system. The availability of the exact discrete time model for such linear systems along with the presented discrete controller / observer converts the overall problem in complete discrete domain. The result is a simple, efficient and implementable discrete sampled-data regulation scheme. This approach is especially useful for the tracking control of x and y axis of gyroscope for both ideal and perturbed cases. The effectiveness of the presented scheme is shown through simulations.

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