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NONLINEAR CONTROL OF A CONTINUOUSLY VARIABLE

TRANSMISSION USING HYPERSTABILITY THEORY

Martin S. Sackmann, Volker G. Krebs

Institut fr Regelungs- und Steuerungssysteme, Universitt Karlsruhe (TH),

Kaiserstr. 12, 76131 Karlsruhe, Germany

fax: +49 (0)721 / 608-2707 and e-mail: {sackmann, krebs}@irs.etec.uni-karlsruhe.de

Keywords: Nonlinear control, hyperstability theory, subopti-

mal control, Hamilton matrix, continuously variable transmis-

sion.

Abstract

Nonlinear control design methods using Popovs hyperstabil-

ity theory which have been developed so far are applicable for

systems with linear feedforward components only. In this

contribution the existing criterion of hyperstability is extended

to systems with nonlinear feedforward blocks. A new designstrategy is proposed taking advantage of particular properties

of the Hamilton matrix used in LQ control. The efficiency of

the method is demonstrated by the controller design of a con-

tinuously variable transmission of a car.

1 Introduction

Nonlinear control has become more and more a focus in re-

search since linear system models mostly are not appropriate

to describe real world processes, and thus nonlinear models

have to be used. To cope with nonlinear dynamics, numerousdifferent nonlinear controller design approaches are proposed

in the literature. There are e.g. nonlinear design methods

based on differential geometric tools such as input-output

linearization [Isid95], input-state linearization [NiSc90] or the

'Flat Systems' approach [Flie95]. However, the applicability

of these methods is limited due to restrictive conditions. When

using input-output linearization to guarantee asymptotic sta-

bility, it is necessary that either the relative degree of the plant

is equal to its order or the zero dynamics is asymptotically

stable. For input-state linearization a diffeomorphism, i.e. a

continuously differentiable map with a continuously differen-

tiable inverse, is used to transform the nonlinear system into

nonlinear controllability standard form. However, a seriousdisadvantage of this approach results from the fact, that in

general it is quite difficult to calculate the transformation law

for higher order systems [Allg96]. The difficulty associated

with the 'Flat Systems' method is the necessary construction of

a flat output. Further nonlinear approaches are based on Ly-

apunovs method [Foel93a], [GaGl96], sliding mode control

[Khal96], or backstepping [GaGl96] to implicitly guarantee

asymptotic stability and robustness under system uncertain-

ties. In this paper we present a new approach for controlling

nonlinear plants using the concept of hyperstability theory to

obtain globally asymptotically stable systems.

The paper is organized as follows. In the second section themain aspects of Popovs hyperstability theory are reviewed. In

section 3 the existing stability criterion is extended to systems

with nonlinear feedforward blocks and a new condition for

asymptotic hyperstability for nonlinear systems is derived.

Due to the analogy with the performance index used in LQ

control a new method is proposed to satisfy this condition. In

section 4 it is shown that the eigenvalues of the so called

Hamilton matrix are the positive and negative eigenvalues of

the controlled linear time-invariant system using an infinite

optimization interval. Therefore, optimal control can be

achieved by pole assignment, where the eigenvalues of the

controlled linear SISO-system are shifted to the negative ei-

genvalues of the Hamilton matrix. This result is extended tothe nonlinear case in order to obtain a suboptimal control

strategy. Based on this general result, in section 5 a particular

nonlinear controller is designed for a continuously variable

transmission to achieve asymptotic hyperstability under con-

sideration of state space restrictions. In the final section the

results are summarized.

2 Hyperstability Theory

In 1973 V. M. Popov introduced the notion of hyperstability

as a generalization of the absolute stability for nonlinear sys-tems [Popo73]. Roughly speaking, hyperstability comprises

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two stability concepts: input-output stability and Lyapunov

stability [Foel93b]. In particular, if a system is asymptotically

hyperstable it is globally asymptotically stable in the sense of

Lyapunov, too [Foel93b]. Thus, on the one hand hyperstabil-

ity theory is a useful tool for the analysis of feedback systems

and on the other hand it profitably can be applied to controllerdesign. Necessary and sufficient conditions for asymptotic

hyperstability were defined in [Foel93b] for the standard

multivariable feedback system (Fig. 1) with the nonlinear

controller r(y) as follows:

Fig. 1. Standard feedback system

x n

u m

y n

:

:

:

1

1

1

Definition 1: The nonlinear feedback system (Fig. 1) with the

equilibrium point xR=0 is asymptotically hyperstable iff:

For the linear, time-invariant, completely controllable and

completely observable system in the feedforward block, there

exists a positive definite symmetric matrix P, a regular matrix

L and an arbitrary matrix V satisfying the Kalman-

Yakubovich-Equations (KYEs):

A P PA LL

C PB LVD D V V

T T

T

T T

+ =

=+ =

,

, (1)

and the Popov integral inequality

( ) ( )v y dTt

0

0

2 (2)

holds for all t0 and a positive 0 .

Obviously, the relationship to Lyapunov stability is given by

the first Kalman-Yakubovich equation, which shows the first

time derivative of a Lyapunov function with a positive defi-

nite matrix P for an autonomous linear system.

Until now this criterion of asymptotic hyperstability exclu-

sively applies to systems with a linear feedforward block. In

the following section we will extend this stability definition

and the conditions for hyperstability to systems with nonlinear

feedforward blocks.

3 Extension of the asymptotic hyperstability

criterion

In this section we will consider stabilizable nonlinear systemswith a feedforward block represented by the equations

( ) ( )&

, , .

x f x G x u

y x

x u yn m n

= +

=

and

(3)

The equilibrium point is xR=0 and the function f(x) satisfies

f(0) = 0. The nonlinear feedback system including the nonlin-

ear controller r(x) is given in Fig. 2.

The system considered now is converted to the standard feed-

back system of Fig. 1 by means of a suitable structure trans-

formation.

Fig. 2. Nonlinear feedback system

The feedback system is extended by a constant matrix AE with

negative eigenvalues and AE, as illustrated in Fig. 3.

Fig. 3. Extended nonlinear feedback system

Combining the matrices AE, G(x)r(x), and f(x) , accordingto the dashed line, a new feedback block is created. Introduc-

ing the input , in the feedforward block we obtain the as-

ymptotically stable linear system

&$ $

$ $ $ ,

x A x Bu

y C x B C I I

E= +

= = =

; ; identity matrix n n(4)

and in the feedback block the static nonlinearity

( ) ( ) ( )$v x x f x= A + G x r .E (5)To achieve asymptotic hyperstability of the transformed ex-

tended nonlinear system, definition 1 given in section 2 mustbe satisfied.

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First we prove that the KYEs

A P PA LL

C PB LV

D D V V

E

T

E

T

T

T T

+ =

=

+ =

,

$ $ ,

are satisfied with positive definite P and regular L. Using the

notation of (4) and choosing D=0 the KYEs are reduced to

A P PA LL

I PI

E

T

E

T+ = =

,

.0

Obviously, P has to be the identity matrix I nn and there-fore, P is positive definite. The remaining equation

A A LL QET

E

T+ = = (6)

is satisfied by choosing a suitable matrix AE with negative ei-

genvalues to get a positive definite, symmetric matrix Q=LLT.

A regular L can be determined by means of Cholesky factori-

zation.

Second, the Popov integral inequality

( ) ( ) ( ) ( )J v y d v x dT

t

T

t

= = $ $0 0

0

2

has to hold for all t 0 and a positive 0 .

We substitute $v by (5) using the fact that

( )x A x x A x x A xT ET

E

TT

E

T= = ,

and the system (3) with ( )u r x= which yields

( ) ( ) ( )&x f x G x r x= .From (6) we obtain

( )x A x x A A x x Q xT ET

E E

T T= + = 1

2

1

2

and thus, the integral inequality is given by

J v x d x x x Q xd

x x x Q x d

T

t

T

t

T

T t T

t

= = +

=

$ &

0 0

0

0

0

2

1

2

1

2

1

2

.

To satisfy this inequality it is necessary to prove the bounded-

ness of the integral for t, because in this case the otherterm on the left side of the inequality is bounded, too. For that

reason, the design of a controller which guarantees that the

integral inequality

J x Q xdT

t

= < 0

0

2

holds for all t 0 is necessary and sufficient to obtain an as-

ymptotically hyperstable closed loop system. Therefore, wehave the following theorem.

Theorem 2: The nonlinear system

( ) ( )&

, ,

x f x G x u

y x

x u yn m n

= +

=

and

(7)

with the equilibrium point xR = 0, and f(0) = 0 is asymptoti-

cally hyperstable, iff there exists a continuous control

u = r(x) satisfying

J x Q x dT

t

= 0

0

2 (8)

with a positive definite, symmetric matrix Q, a positive

0 < , an arbitrary initial condition x(t=0), and all t0.

Proof: Using the Lemma of Barbalat [Popo73] it is trivial to

prove that

( )limt

x t = 0

since Q is positive definite and J is limited.

The derivation of a criterion for asymptotic hyperstability for

time discrete nonlinear feedforward blocks is carried out in

the same way as it was demonstrated for the case of continu-

ous systems.

One possibility to achieve the boundedness of the integral is

to use results from linear quadratic optimal control. Compar-

ing this integral with the quadratic index of performance

( )J x Q x u S u dT T

t

= +0 (9)used in optimal control theory, where Q and S are positive

definite matrices, the similarity is evident. Since the integrand

of (9) is equal or greater than the integrand of (8), it is suffi-

cient to prove the boundedness of (9) for all t 0.

In the next section we apply a relation between the eigenval-

ues of the Hamilton matrix and the optimal closed loop sys-

tem to calculate an optimal controller without solving the Ric-

cati equation, explicitly.

4 Optimal control of linear systems

Linear quadratic regulation (LQR) is one of the most efficient

linear control synthesis methods. By means of LQR a desired

dynamical behavior for a linear time-invariant, controllable

system

&x A x Bu= +is achieved by minimizing the quadratic index of performance

(9). We consider the case of infinite time horizon and the ma-

trices Q and S are assumed to be positive definite. Thus, the

cost function reads

J x Qx u S u dT T

= +

0 minimum. (10)

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Usually, the performance index is minimized by solving the

Hamilton-Jacobi equations [Foel94]. This leads to the Riccati

equation

A P PA PBS B P QT T+ + =1 0 , (11)

which has to be solved to obtain a positive definite matrix P

for the construction of the optimal control law

u R x S B P xT= = 1 . (12)

Obviously, (11) is a nonlinear matrix equation, and therefore,

in general it can only be solved numerically. To overcome this

restriction we will focus on a remarkable property of the

Hamilton matrix [Foel94]

HA BS B

Q AA

T

T=

1

. (13)

The eigenvalues of the optimal controlled system

( ) ( )&x A BR x A BS B P x A xT R= = =1

are the negative eigenvalues of the Hamilton matrix HA, and,

moreover, the eigenvalues of HA are symmetric to the imagi-

nary axis.

Proof: From linear matrix theory it is known that similar ma-

trices have identical eigenvalues. The matrices E and F are

called similar [ZuFa84] if there exists such a regular matrix T,

that the matrix equation E T = T F holds. Setting up

E H FA BS B

AT

I

P IA

R

T

R

T= =

=

; ;

1

0

0

with the (nn) unity matrix I and the matrix F, possessing thepositive and negative eigenvalues of AR, the similarity can be

showed, if the Riccati equation (11) is satisfied.

Due to this symmetry of the eigenvalues the characteristic

polynomial of the Hamilton matrix has to be an even function.

Consequently, the optimal control problem can be solved

analytically for second order linear SISO-systems, because a

reduction of the fourth order polynomial to a quadratic func-

tion is possible by an appropriate substitution. Therefore, wecan explicitly calculate the eigenvalues of the Hamilton ma-

trix.

For this reason, the optimal controller will be determined by

pole assignment using the negative eigenvalues of the Hamil-

ton matrix. Moreover, this method can be used for nonlinear

SISO-systems in the state-dependent coefficient (SDC) form

( ) ( )&x A x x g x u= + , too. However, in general only suboptimal

control is obtained [ClDS96], and only local asymptotic sta-

bility is guaranteed. Nevertheless, we will see in section V

that it is possible to determine a suitable Lyapunov function to

prove asymptotic stability in a certain region.

5 Controller design for a continuously variable

transmission

On account of more and more strict regulations the reduction

of fuel consumption and exhaust emissions of automobiles are

one of the major challenges in car industry, nowadays. Thereare several approaches to tackle this problem [GuSc95]. Here

we will focus on the application of continuously variable

transmissions (CVTs). A CVT allows to continuously vary the

gear-ratio in contrast to a conventional transmission, which

offers 4 or 5 gears only. This property enables efficient and

smooth power transmission from a vehicles engine to its

wheels by providing optimal transmission ratios to maximize

fuel efficiency. However, a CVT is an extremely complex,

nonlinear system, and therefore, classical control methods

achieve the required performance by a huge expenditure in

controller design only [GuSc95]. This fact is the motivation

for the design of a nonlinear controller for CVT of a 1.3 liter

four cylinder spark-ignition engine with compression ratio

10:1.

The derivation of the plant model (Fig. 4) and further details

are given in [GuSc95].

Fig. 4. Plant model

The nonlinear dynamics of the second order plant is repre-

sented by

( ) ( ) ( )( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

& , ,

,

&

w w w w w

w w

t f t t g t t u t

h t t d t

t u t

= +

+

=

(14)

where the functions fw(), gw() and hw() are given by

( )

( ) ( )

( )

( )

f

a c

g

h

w w

w w

w e

w w

w e

w e

w w

w e

, ,

, ,

, ,

=

+ +

+

= +

= +

3 2 2

2

2

2

1

with the wheel velocity w, the continuously variable gear-ratio, the external disturbance d and the control input u. Theremainder of the parameters and their constant values are

given in (Guzzella, Schmid, 1995). All functions are smooth

since the denominator never becomes zero for positive , w,

e and real. Moreover, the system (14) is confined to a cer-tain region X 2, which is given by the inequalities

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w w w

e w e

,min ,max

min max

,min ,max

,

,

.

(15)

The set of all equilibrium points in X for d = 0 is described by( ) ( ){ } w wX, , = where

( )( )

( ) ( )

( )

=

+

+ + + +

2

3

2 2 2 4

3

2

4 4 4

2

a

a c a c

a.

This set is represented by the curve in Fig. 5.

According to [GuSc95] in the region X the controllability

matrix of the system is regular. Our aim is to design a subop-timal controller, which takes into account the given state

space restrictions (15), and achieves asymptotic stability with

regard to the equilibrium point xR = [w,R, R]T

for all x0 =

[w,0 ;0]T X.

In the first step we introduce the distance between the state

vector and the equilibrium point as new coordinates

z = [w,,]T [w,R,R]

T. To transform the nonlinear system

into the SDC form, it is necessary to determine the elements

of the matrix A. The elements a11 and a12 are defined by

( )

( ) ( )( )

( )

af z z

z

z

af z z a z

z

z

w w R R

R

w w R R

110

1 2

1

3

1

12

1 2 11 1

2

2=

+

=

lim, , ,

, , ,.

,

,

=

=a + 2 a

+

R

2

R

3

w,R

w R

2

e ,

The resulting SDC form reads

( ) ( )&z A z z b z ua a

zb

u= + =

+

11 12 1

0 0 1,

where( )( )

( )( )b =

z +1

2 R

w

+

+ +

z

z

w R e

R e

1

2

2

,

.

The coefficients a11, a12 and b1 are continuous functions.

In the next step the eigenvalues of the Hamilton matrix HA

with S = s > 0 and a positive definite matrix Qq q

q q=

11 12

12 22

are calculated. The negative eigenvalues 1 and 2 of thecharacteristic polynomial

( )( )( )( )p c t c t c c c cHA = + + = + + 4

1

2

2 1 1 2 2

with

( )

( )

t q + q b +q b +s a s

t = a q a +q a +q a s

1 22 12 1 11 1

2

11

2

2 11 1 2 12 22 11

2

11 12

2

2

2

=

/ ,

/ ,

are determined to

1 21 1

2

22 2

, =

+ t t

t .

By pole assignment with u r z r z= 1 1 2 2 the poles of thematrix AR(z) are shifted to the values 1 and 2 where r1 and r2are given by

r =a a + a

a + a b

r =a a b + a a

a + a b

11 2 11 1 11 2 11

2

12 11 1

21 12 2 12 1 1 2 11 12

12 11 1

.

The limiting values of r1 and r2 exist and are finite if a12 tends

to a11b1. Therefore, the control input u is continuous. Tokeep the states inside the prescribed region according to (15)

we need a suitable positive definite matrix Q and a scalar s,

which can be chosen as

( )q q q

s

w

w R

w

w R

11 12 22

90

150

= = =

=

, ,

; ; ;R

1 21+ +

where

( )( )( )

( )( )( )

1

w

w

2

2

w

min

2

=2

- 0.97 / + + 0.1

=2

- 1.03 + 0.05

e R,max

,

are suitable positive factors to force the state space restric-

tions starting from the curve . In Fig. 5 the resulting systembehavior is shown in the -w-plane for a bundle of trajecto-ries in the region X.

Fig. 5. Trajectories in the-w-plane

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A positive definite matrix P is determined by the Riccati

equation (11) for the equilibrium point z = 0. Using the quad-

ratic Lyapunov function V=zT

P z one can prove asymptotic

hyperstability of the equilibrium point in the region X. Fig. 6

illustrates a simulation of the closed loop behavior starting at

x0 = [w,nom ,nom]T

.

0 2 4 6 8 10 12 14125

130

135

140

0 2 4 6 8 10 12 142

3

4

5

0 2 4 6 8 10 12 14-2

0

2

4

u

w

time (sec)

Fig. 6. Simulation of the closed loop behavior

Moreover, numerous simulations verify the robustness of the

proposed design method using hyperstability theory against

parameter uncertainties.

6 Conclusions

In this contribution a new definition of and applicable condi-

tions for asymptotic hyperstability for systems with nonlinear

feedforward blocks are presented. This approach enables the

design of nonlinear controllers for specified closed loop prop-

erties, e.g. input-output-stability and Lyapunov stability. Fur-

thermore, this new criterion of asymptotic hyperstability can

be derived for nonlinear time discrete feedforward blocks in a

similar way. In conjunction with particular properties of the

Hamilton matrix suboptimal control can be achieved for non-

linear second order SISO-systems without solving the Riccati

equation. Moreover, it is possible to obtain robustness againstparameter uncertainties and to consider state space restric-

tions. The efficiency and the advantages of this method are

demonstrated by the design of a nonlinear controller for a

continuously variable transmission of a spark-ignition engine.

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