pp robot.pdf

Seventh International Conference on Control, Automation,

Robotics And Vision (lCARCV’O2),Dec 2002, Singapore

Adaptive Pole Placement for a MIMO Quasi-Linear

Parameter Varying Mobile Robot

Antonios Tsourdos, John T. Economou, Brian A. White

Department of Aerospace, Power and Sensors

Royal Military College of Science

Cranfield University

Shrivenham,SN6 8LA

England,UK

Email: A.Tsourdos@rmcs.cranfield.ac.uk , J.T.Economou@rmcs.cradeld.ac.uk,

B.A. White@rmcs.cranfield.ac.uk

Abstract

In this paper, an indirect adaptive control scheme IS

proposed for a multiple-input multiple-output differen-

tially steered mobile robot. The resulted closed-loop

system has an effective control for both the vehicle

longitudinal velocity and the vehicle yaw rate. The

closed-loop system stabilityis examined via LMI quad-

tratic stability test. The proposed scheme shows effec-

tiveness for a variaty of realistic simulation scenarios.

become adaptive and quasi-linear, to provide perfor-

mance over a greater range, in the face of changing

operating conditions.

The tracking performance of a mobile robot is also

dependent on the location within the steering envelope

and varies with factors such as yaw rate and acceler-

ation. Several approaches, including adaptive control

[l], [2], nonlinear control [3], and gain scheduling [4]

have been used to alleviate these tracking problems.

One of the most popular methods for applying

linear time-invariant (LTI) control theory to time-

varying and/or quasi-linear systems is gain schedul-

ing [5]. This strategy involves obtaining Taylor lin-

earised models for the plant at fhitely many equilibria

(“set points”), designing an LTI control law (“point

design”) to satisfy local performance objectives for

each point, and then adjusting (“scheduling”)the con-

troller gains in real time as the operating conditions

vary. This approach has been applied successfullyfor

many years, particularly for aircraft and process con-

trol problems. Relatively recent examples (some of

which involve modern control design methods) include

active suspensions [6], high-speed drives [7].

Despite past success of gain scheduling in practice,

until recently little has been known about it theoret-

ically as a time-varying and/or quasi-linear control

technique. Also, determining the actual scheduling

routine is more of an art than a science. While ad hoc

approaches such as linear interpolation and curve fit-

ting may be sufficient for simple static-gain controllers,

doing the same for dynamic multivariable controllers

can be a rather tedious process.

An early theoretical investigation into the perfor-

mance of parameter-varying systems can be found in

[8]. During the 1980’s, Rugh and his colleagues de-

veloped an analytical framework for gain scheduling

using extended linearisation [5]. Also, Shamma and

Athans [9] introduced linear parameter-varying (LPV)

systems as a tool for quantifying such heuristic design

rules as “the resulting parameter must vary slowly”

and “the scheduling parameter must capture the non-

linearities of the plant”. Shahruz and Behtash [lo]

suggested using LPV systems for synthesising gain-

scheduled controllers.

NOMENCLATURE

mobile robot longitudinal velocity

mobile robot yaw rate

motor armature voltage

common-mode artuator voltage

differential actuator volt

mobile robot mass

mobile robot Inertia

resistive moment of inertia

effective tyre force

Motor gearbox ratio

PMDC motor torque constant

Motor armature resistance

Wheel radius

C.G. to left track

C.G. to right track

C.G to front track

C.G to rear track

1 Introduction

The performance of vehicle and robotics systems is

hghly dependent on the capabilities of the guidance,

navigation and control systems. To achieve improved

performance in such robotic system systems, it is im-

portant that more sophisticated control systems be de-

veloped and implemented. In particular, as the perfor-

mance envelope is expanded, the control schemes must

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Attention has since turned to performance and de-

sign of parameter-dependent controllers for LPV sys-

tems. Various design methods which have been pro-

posed share several common features, e.g., the cur-

rent methods are based on extended state-space ap-

proaches to B, optimal control for LTI systems [12],

[U], and LTV systems [13]. Performance is usually

measured in terms of the induced Lz-norm, and con-

trollers are designed for certain classes of parameter

variations, e.g., real or complex values, arbitrarily fast

or bounded rates of variation, shape of the parame-

ter envelope etc. The resulting parameter-dependent

controllers are scheduled automatically, so that the of-

ten arduous task of scheduling a complex multivariable

controller a posteriori is avoided.

In this paper adaptive pole-placement control de-

sign technique is applied to the motion control for the

model. The mobile robot motion is modelled to be

quasi-linear with varying parameters. Based on the

quasi-linear model, we adopt for design procedure the

adaptive pole-placement method. In this scheme, un-

known parameters are estimated and based on these

estimates, control parameters are updated. Computer

simulations show that this approach is very promising

to apply the motion control design for mobile robots,

which are highly quasi-linear in dynamics.

2 Mobile Robot model

(4)

The quasi linear parameter varying model inputs &e

transformed using the common-mode 6.2 and differ-

ential 6 - demands as shown in 6 using the transfor-

mation% 5.

Assume that the left and right vehicle side actuators

are operating at the same armature voltages respec-

tively. The derived QLPV model structure is shown

Figure 1: Vehicle wheels force diagram

The

given-in 7 where x is the state miable vector to be

determined in terms of x and the reference signal.

in 1.

U, = -K(p)Tx (7)

It should be noted that K(p) is determined recur-

sively but its structure and in particular the values

of the lognitudinal and lateral controllers Kl(p) and

Kz(p) are obtained using the pole-placement tech-

nique.

Substituting the control law in the state equation

yields:

591

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The state space now is described by:

x = A'x (8)

with the augmented matrix A' to be given by A' =

[ ] = [ -rc- -;

A(P) 0

] [ ] + [ ] [ :zi 3 + [ f ] [ r: ]

( 12)

Ab) - BK(p)*.

The characteristic equation of the augmented sys-

The compensated system therefore becomes:

tem can now be determined:

The coefficients Ai,j are parameter dependant.

4 Closed-Loop Responses

The adaptive controller effectiveness is shown by

means of three reallistic scenarios.

Equating the above mathematical expression of the

characteristic polynomial of the augmented system

with the one of the desired (obtained using the de-

4.1 Scenario 1: Rectilinear motion

els are easily obtained.

But this controller would result only to a desired

transient of all- local models by placing-the.poles of

all the local systems within a specified area. However

since our aim is good tracking for the mobile robot

we should include to the design specification except

peak overshoot and settling time, zero steady-state er-

ror. Ths can be achieved with an integral term in the

forward path as it is shown at the block diagram 2.

In this case the recilinear system mode is excited and

therefore thd = 1.5ms-1 and I'd = 0. Figure 3(a)

clearly shows the velocity convergence to the demand.

In figure 4(b) the Jacobian system coefficients are

shown which are non-zero for the longitudinal model

J11 # 0, Jzz # 0. However coeflicients 512 = 521 = 0

which clearly identify a decoupled sytem. For this

robotic vehicle the centre of gravity is positioned so

that wl = wr and hence the yaw rate is zero. For cer-

tain applications in which the opposite holds wl # wT

even at tbd f 0,rd = 0 demands the mobile robot

would tend to drift towards the direction that the cen-

tre of gravity is shifted.

, ... . .

. . . , .. . .

TUEINs

MH,

Figure 2: Adaptive MIMO Pole Placement Structure

Figure 3: Longitudinal Velocity and Jacobian Matrix

coeflicients

The new augmented model would contain for this

system two more state variables to account for this

integral term. These new state variables are defined

as:

4.2 Scenario 2: Pivot Turn motion

Another simulation scenario is for ud = O,rd/neqO

in which the yaw rate mode is excited. Figure 4(a)

shows the yaw rate convergence for a step demand of

Td = 2.5~~-'. In figure 4(b) similarly to the JXO-

bian coefficients of the rectilinear motion scenario the

system is decoupled due to the parameters properties.

xi = 1: edt = 1; ([ :: ] - [ :

I) dt

(10)

Therefore,

(11)

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sired performance characteristics) the coefficients of

the pole-placement controller for each of the local mod-

... .. .., .....

.. ...........

..............................

...

-.... 211:

.... =

................................

- -

THINS

Figure 4: Yaw rate and Jacobian Matrix Coefficients

4.3 Scenario 3: Combined motion

During this scenario both modes are instanteneously

excited and therefore the coupled system generates

non zero Jacobian elements as shown in figure 4(d).

The demanded inputs are tbd = 1.5ms-' and Td =

0.5~~-1

respectively. The resulting step responses con-

vergence are shown in figure 4(a)(b). The associated

state variable erros are shown in figure 4(c).

Figure 6: Controller Gains Space and Combined Mo-

tion Gain Adaptation

".I-,

...............................

the variations in the parameters of the system. In this

section we consider a system defined by

.................... .........

......... ............

..................... .....

.....

.......

................

. =

x = A(p)z (14)

where the state matrix Ab) is a function of a real pa-

rameter vector p = @I,. .. ,pk) E Rk. Let X = R*

be the state space of this system. One can think of

this (autonomous) system as a feedback interconnec-

tion of a plantsand a control system. We will analyze

the stability of the equilibrium point x* = 0 of this

system. More precisely, we analyze to what extent

the equilibrium point x* = 0 is asymptotically stable

when p varies in a prescribed set, say P, of uncertain

or varying parameters.

There are two particular cases of this stability prob-

lem that are of special interest.

1. the parameter vector p is a fixed but known ele-

ment of a parameter set P C Rk.

2. the parameter vector p is a time wrying p : R +

Rk which belongs to some set P of functions in

(Rk)R. The differential equation (14) is then to

be interpreted as $(t) = A(p(t))z(t).

The first case typically appears in models in which

the physical parameters are fixed but only approxi-

mately known up to some accuracy. Note that for

these parameters (14) defines a time-invariant system.

The second case involves time-varying parameters.For

this case one can in addition distinguish between the

situation where P consists of one element only (known

time varying perturbations) and the situation where P

is a higher dimensional set of time functions (arbitrary

time varying perturbations). Stability against time-

varying parameters is generally a more demanding re-

quirement for the system than stability against time-

TIUEINs

Figure 5: Longitudinal Velocity, Yaw Rate and Error

Responses and Jacobian Matrix Coefficients

4.3.1 Adaptive Gains

The adaptive gain space for the pole placement con-

troller are shown in figures 6(a)(b)(c) and (d). For the

combined motion the actual adaptive gain trajectories

are also superimposed on the gain space for the 30s

test run.

5 Stability analysis

An important issue in the design of control systems

involves the question to what extent the stability and

performance of the controlled system is robust against

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.... .,

invariant unknown parameter.This is because second

case is obviously a special case of first one.

Since A is a continuous function of parameter p E P,

and P is compact, it is clear that the condition (16)

implies that the left hand side is uniformly negative

definite. That is, there exists a scalar 6 > 0, such that

for all p E P, AT@)P + PA@) 5 41n.

The different necessary and sufficient conditions for

quadratic stability are given in following theorem.

Theorem.[l5]Given a compact set P C RS, and

an integer m > 0, then the following conditions are

equivalent:

1. function A(p) is quadratically stable over P, i.e.

there exists a matrix P E S;xn, such that for all

PEP

5.1 Quadratic Stability

A sufficient condition for x* = 0 to be an asymptoti-

cally stable equilibrium point of (14) is the efistence

of a quadratic Lyapunov function

V(s) = ZTPZ

with P = PT > 0 such that

AT(p)P + PA@) < 0

2. for any C E Co(R6,PXm),

along state trajectories x of (14) that originate in a

neighbourhood of the equilibrium x* = 0. The system

(14) is said to be quadratically stable for perturbations

P if there exists a matrix P = PT > 0 such that

A@(t))TP + PA(p(t)) < 0

for all perturbations p E P. If the system (14) is

quadratically stable for perturbations P then V(x) =

xTPx is a quadratic Lyapunov function for (14) for

all p E P. The existence of a quadratic Lyapunov

function implies that the equilibrium point x* = 0 is

asymptotically stable. Quadratic stability for pertur-

bations A is therefore equivalent to the existence of

a quadratic Lyapunov function V(x) = xTPx,P > 0

such that

there exists a matrix

X E STxn, such that for all p E P

AT(p)X + XA(p) + XC(p)CT(p)X < 0

there exists a matrix

Y E S:xn, such that for all p E P

AT(p)Y + YA@) + YB@)BT(p)Y < 0

Definition [15]: For LPV systems (15), if A@(t))

is quadratically stable over P, then (15) is a quadrat-

ically stable LPV system.

The test requires that A@(t))be modeled as a con-

vex polytope of matrices, from the possible set of d-

ues of A@(t)). To construct the polytopic model of

A@(t)), we evaluate A@@)) over the extreme values

of the parameters and their derivatives. If there are

total 1 parameters and their derivatives, where 1 2 k;

then ACp(t)) may be considered as being contained

within a convex set with 2' vertices.

for all p E P.

Note that in general quadratic stability of the sys-

tem for an uncertainty class P places an infinite num-

ber of constraints on the symmetric matrix P.

A(&)) E 4{Ai,Az,..-,Az}

The conservative assumption of the quadratic Lya-

punov stability test is that the system can change in-

hitely fast. For example, the values of the matrix

A(p(t)) could instantly change from those of the ma-

trix A1 to those of another matrix, say Al. As im-

plemented by [l6], the test amounts to searching for

the matrix solution, P, to the following linear matrix

inequality over the vertices of the convex set

5.2 Quadratic Stability forQLPV Sys-

Recently developed stability tests may be employed to

assess the stability of the parameter varying systems

such as (15) [14].

x= ACp(t))z + B@(t))u

Y = C@(t))x +aP(t)).

p(t) E P,W 2 0 (15)

The quadratic Lyapunov test is a general, although

conservative test applicable to the systems like the one

encountered in this paper.

DefinitiomGiven a compact set P C RS, and a

function A E C"(R", Rnxn

a<o (17)

If a solution exists, the Lyapunov function

V(x(t),p(t)) demonstrates stability by satisfying

the function A is quadrati-

cally stable over P if there exists a matrix P E STxn,

such that for all p E P

A~@)P+PA~)

(16)

594

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3. for any B E CO(Rs,

Rmxm),

tems

ATP+PAi < 2d

P 1 I for i=l--.

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