Delay-dependent Robust Kalman Filtering for Interval Systems with Time delay

Proceedings of the 42nd IEEE Conference on Decision and Control

Maui, Hawaii USA, Derrmber 2003

FrE13-6

Delay-dependent Robust Kalman Filtering

for

Interval Systems with Time

delay

'Chien-Yu Lu, *Jason Sheng-Hong Tsai, Te-Jen Su, and Gwo-Jia Jong 'Department of Electrical Engineering, National Cheng-Kung University,

Tainan City, Taiwan 701, R.O.C. e-mail:[email protected] e-mail:[email protected]

Department of Electronic Engineering, National Kaohsiung University of Applied Sciences Kaohsiung, Taiwan 807, R.O.C. e-mail:[email protected]

Absrrud- This paper studies the problem of Kalman filtering for a class of linear continuous-time interval systems with delay dependent conditions. By employing a Lyapunov-Krasovskii functional approach, it is proven that the dynamics of the estimation m o r is stochastically exponentially stable in the mean square. Suffcient conditions are proposed to guarantee the existence of the desired robust Kalman filters by solving linear matrix inequality which is delaydependent. A numerical example is worked out to illustrate the validness of the theoretical results.

1. Introdnetion

Recently, the robust state estimation arises from the desire to estimate unmeasurable state variables when the plant model has uncertain parameters. A Kalman filtering with a

H ,

norm constraint [I] has been studied. A robust Kalman filtering estimation has also been introduced by [Z] for linear systems with norm-bounded parameter uncertainty in both the state and measurement matrices. The problem of digital filtering with an H , - l i e performance for a linear system has been tackled in [3].

The purpose of this paper is to consider the state-estimation problem for a class of linear continuous timedelay interval systems. Moreover, our attention is focused on the design of a linear state estimato1; the dynamics of the estimation m o r is stochastically exponentially stable in the mean square, dependeat of the time delay. Suficient conditions are proposed to guarantee the existence of desired robust exponential filtering.

2. Problem Formulation and Assumptions Kpresented by

.i(Q = A i x ( 0 + A&(f-h(W+ El w(0 9 At) = C r W + D, x(f

-

h(f)) + E2 Y f ) ,

on f > O , where A , ~ u ~ I ,

hec4zl.

c,.&,:.Cl,

D,eaz,.

Weintroduce

We consider a class of interval time-delay systems (1) (2)

A

yL+-j)

1 and

.

Clearly, all the

elements of A- are nonnegative. Moreover, A, can be writtenas A,=A + A A with + A n . ~ n l .

Similarly, we introduce &,, C , C.. AC, D ,

D, and AD. Then(1) and(2) canbe written as (3) and

(4)

x ( f ) = [ A + M l r O ) + [ A d + A A I I X ( f - ~ ( r ) ) + E I W ( ' ) . (3)

y ( t ) = [ C + AClx(f)+[D+ADlx(f- h ( f ) ) + & v " (4) where X ( ~ ) E ~ " is the state,,q)ER- is the measured

output, * ( I ) E R " and y ( r ) s p are the process and measurement noises, respectively. A, A ~ , c , D ,

and E x are known constant matrices with appropriate parts of A,, and D , , respectively. denotes the delay in the state and it is assumed that there exist positive numbers h such that o s h ( t ) r i , 1 ; ( 1 ) ~ d s I . In fact, M will take a particular deterministic ma& within but we just do not h o w which p d c u l a r one, and similarly for A A ~ , AC , AD. when we talk about the solution of (3) and (4), we mean the solution when M , A A , AC and AD take pfuiicular matrices within their matrix intervals 1141. We consider the full-order linear filter of the form

where G and K are constant matrices to be designed with apptopiate dimensions.

Let the m o r state be

Afier substituting (3)-(4) and (9) into (to), one has dimensions. AA, A A d , AC and AD are the UnCUtah

20)

= G2(t)

+

Ky(l)

,

(5)

e ( t ) = x ( f ) - l i ( f ) . (6)

e(r) = Ge(l)+[(A + M ) - K ( C + AC) -

W f )

+((Ad+ A A & W + ".dt-h(f)))I

+&%+KE><Ol (7)

We introduce the extended state vector

where n(r) is a stationary zero-mean noise signal with identity covariance matrix.

3. Main results 3. I Filler Analysis

Theorem I: Let the Kalman filter parameters

G

and

K be given. If there exist positive scalars d ,

zi

>

0 ,

Proceedings of the 42nd IEEE Conference on Deeidoo and Controt

Mad, Hawaii USA, December 2003

FrE14-1

Using Dynamic Optimization

for

Control

of

Real

Rate CPU Resource Management

Applications’

Varin Vahia*, Ashvin Goel**, Jonathan Walpole***,

Molly

H. Shor*

Abstract-In this paper, we design a proportional-period controtler for allocating CPU to real rate multimedia appli- cations on a general-purpose computer system. We model this computer system problem in state space form. We design a predictive controller to allocate the proportion of the CPU to the threads when the long-term time deriation from the current time stamp is small or positive. When it is negative and exceeds a certain threshold, w e switch to a controller designed using dynamic optimization

LQR

tracking techniques, to drive the ermr (short-term and long-term time deviations) to a small value. The challenges in the problem include the coarse granularity (quantization) of the time-stamp markings of the video frames, the unpredictable decoding completion time of the frames, the variable decoding times of the frames, and control actuation being limited to positive values.

I. INTRODUCTION

General-purpose operating systems can be designed for computers to satisfy the CPU and network needs of real-rate multimedia and sensor-based real-time applications. Multi- media and sensor-based applications generate real-rate flows, which have bounded end-to-end delay, and other jitter and dithering, requirements.

Alternate approaches that guarantee bounded end-to-end delay include dedicated hardware designed for a particular application and reservation-based schemes in real-time op-

erating systems. These approaches perform well but have their own limitations. Dedicated hardware is

an

expensive solution if the patticular application is not the primary one to be solved. Reservation based schemes often result in un- derutilization of resources when the resource requirements of

a task vary significantly over time. To avoid underutilization, Abeni, et al., designed a feedback control scheme to adapt the reservations in

a

real-time operating system based on on- line measurements of a task’s usage of the CPU resource L11.

To adapt a general-purpose operating system to address end-to-end delay requirements of tasks requires two problems to be addressed. First, the resources required in a particular time period to keep up with the end-toend delay require- ments must not exceed the total resources available in the

.

’This work was supported in part by DARPMTO under the Informa-

tion Teehnology Expeditions, Ubiquitous Computing, Quorum, and PCE programs, NSF grant3 ECS-9988435 and CCR-9988440 and EIA -0130334, and by Intel.

‘School of Elecmcal Engineering and Computer Science. Oregon State

University, Cowallis, Oregon, USA

“‘Electical and Computer Engineering k p m m e n t . University 01

Toronto, Tornnto. Canada

***OGI School of Science and Engineering, Oregoon Health and Sciences University, Beavenon. Oregon, USA

system over that period. Second, a mechanism must be in place to allocate the appropriate amount of resource to each task in the system in

a

manner that guarantees that the end- to-end delay requirements are met.

The first problem can be addressed in

a

number of ways, including restricting which new tasks to accept in the system (admissions control) or having the tasks adapt their resource requirements if insufficient resources are available in the system (selective data dropping). Admissions control may be appropriate if the type of applications present in the system cannot reasonably adapt their requirements while still functioning acceptably. However, admissions control is an all

or nothing solution, which may be a problem if the task being excluded is time-critical.

Fro

instance, if there is

a

video surveillance camera and the

task

of sending the datalvideo from the camera is excluded, a significant event may be missed by someone monitoring a situation. In multimedia $stems, the resolution and/or frame-rate can be reduced if there are not sufficient system resources to process the full video. Krasic and Wdpole have implemented selective priority-based adaptation of video quality to address this problem (21.

In this paper, we

assume

that

an

intelligent data-dropping

scheme is implemented to prevent the total resource re- quirements from exceeding those available and focus on

the second problem of smart resource allocation. We ex- tend the work of Goel, al., who designed a few different proportion-pericd controllers for network and CPU resource allocation [3,4]. Goel, et al., modified the standard Linux scheduler in a number of important ways 151 and developed and implemented proportional-period feedback controllers for CPU scheduling in Linux O B . This generated a general purpose, open source O/S with improved support for time sensitive applications. Time Sensitive Linux (TSL) exhibits improvements of 99.9% in timer latency, 98% in preemption latency and 95% in TCP output buffer latency compared to standard Lmux 01s

151.

Goel’s methods for controller design are based on classical PI controller, and on the use of

feedback to cancel directly known (or predicted) errors. In this paper, we demonstrate an alternative methodology for the controller design for this problem. The details of the problem set up and general controller structure are discussed in the next section. There are a number of challenges for the control design. The output that is measured is not continually updated but rather is updated only in response to a discrete event (the completion of a job). There is a random variation in a system parameter (the amount of resource required to