Quantier Elimination and Applications in Control

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Quantier Elimination and Applications in Control

Mats Jirstrand

Department of Electrical Engineering

Linkoping University, S-581 83 Linkoping, Sweden WWW:

http://www.control.isy.liu.se/ matsj

email:

matsj@isy.liu.se

Reglermote '96, Lulea

Abstract

Quantier elimination is a method for simplifying formulas that consist of poly- nomial equations, inequalities, and quantiers. We give a brief introduction to this method and apply it to two problems in control theory.

1 Introduction

Many problems in control theory can be stated in terms of formulas involving polyno- mial equations and inequalities. If we also allow quantiers (

⁹ ⁸

) and boolean operators (

^_ ^{^} ^:

) in such formulas we have a very expressive power at our hands. A surprising fact is that any formula of the above type is equivalent to a formula without any quantiers, i.e., the quantied variables can be eliminated 6]. This simplication process is denoted quantier elimination (QE) and there are algorithms implemented in computer algebra systems that perform QE.

To demonstrate how to utilize QE in control theory we show how to formulate and solve two problems concerning stability of SISO feedback systems with static nonlinearities.

In the rst case we use the circle criterion and in the second case the describing function method to express the problems as formulas in the above framework. Both problems involve algebraic conditions on the real and imaginary part of the Nyquist curve of the linear part

^G(s)

of the feedback system. If

^G(s)

is parameterized in terms of design or control parameters it is easy to compute conditions on these parameters such that the feedback system becomes stable. We can also compute numerical values of the parameters that satisfy the derived conditions.

The aim of the paper is both to demonstrate some applications of the QE method but also the possibilities of symbolic computations in general for problems in modeling and control of dynamical systems.

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2 Quantier Elimination

By a quantier free Boolean formula we mean a formula consisting of polynomial equations (

^fⁱ^(x)⁼⁰

) and inequalities (

^f^j^(x)^<⁰

) combined using the Boolean operators

^{^}

(and),

^_

(or),

^:

(not) and

^!

(implies). A formula is an expression in the variables

^x ⁼^(x¹ ^:^:^: ^xⁿ⁾

of the following type

(Q

1 x

1 :::Q

s x

s )F(f

1

(x) ::: f

r (x))

where

^Qⁱ

is one of the quantiers

⁸

(for all) or

⁹

(there exists) and

^F(f¹^(x) ^:^:^: ^f^r^(x))

is a quantier free Boolean formula. In addition, there is always an implicit assumption that all variables are considered to be real.

Example 2.1 The following is an example of a formula:

8x h

x>0 ! x 2

+ax+b>0 i

which can be read as:

For all real numbers

^x

, if

^x^>⁰

then

^x²⁺^ax⁺^b^>⁰

.

Performing quantier elimination we obtain the following equivalent expression

h

4b-a 2

>0 _ a0 ^ b 0] i

which are the conditions on the coe cients of a second order polynomial such that it only

takes positive values for

^x^>⁰

.

²

The above problem is called a general quantier elimination problem since there are both free variables and variables bound by quantiers. If all variables are bound by quantiers we have a decision problem, i.e., to decide if the formula is true or false.

As was mentioned in the introduction it is always possible to eliminate the quantied variables and get an equivalent formula in the free variables only.

After performing quantier elimination one obtains a system of equations and inequal- ities. Solutions to such systems can, e.g., be computed by cylindrical algebraic decompo- sition (CAD) 2, 5], which in many cases is a part of the quantier elimination algorithm.

Hence, a number of solutions is often produced directly.

To perform quantier elimination in the examples of this paper we have used the software

^qepcad

3].

3 Examples

Stability of a feedback system in the presence of static nonlinearities can be investigated using the small gain theorem and its implications such as the circle and Popov criterion, see Figure 1. The circle criterion gives a su cient condition for stability on the Nyquist curve of the linear part of the feedback system 7]. The circle criterion can be stated as a formula involving quantiers and by QE we can compute conditions on, e.g., design parameters of the linear system or limits on the nonlinearity that guarantees stability.

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+ (-) (+)

G(s)

f()

2

k

1 x f(x)

Figure 1: Left: The system conguration for the circle and Popov criterion. Right: Limits on the static nonlinearity (

^k¹ ⁼⁰

in the Popov criterion.)

The circle criterion states that the closed loop system is stable if the Nyquist curve

G(i!)

does not enclose or enter the disc

D,

z2C

z+

k

2 +k

1

2k

1 k

2

k

2 -k

1

2k

1 k

2

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where

^k² ^> ^k¹ ^> ⁰

. The following example shows how QE can be used to compute bounds on controller parameters for a linear system in a feedback scheme with a static nonlinearity.

Example 3.2 Consider the feedback system in Figure 1 where

G(s)=

K

P +

K

I

s

!

1

s 2

+s+1

and the bounds on the static nonlinearity

^f()

is

^k¹ ⁼¹

and

^k² ⁼²

. Suppose that we are interested in bounds on the PI-controller parameters

^(K^P ^K^I⁾

that ensure stability of the feedback system.

The real and imaginary part of the Nyquist curve

^G(i!)

are

x

1

= -K

P

! 2

+K

P -K

I

! 4

-! 2

+1

x

2

= (K

I -K

P )!

2

-K

I

!(!

4

-! 2

+1) :

The Nyquist curve remains outside the disc

^D

, corresponding to the given

^(K^P ^K^I⁾

values, if the following inequality is satised for all

^!

x

1 +

3

4

!

2

+x 2

2

>

1

4 2

:

Clearing denominators and simplifying this expression gives the following equivalent for- mula

8!

h

!>0!

! 6

-(3K

P

+1)!

4

+(1+3K

P +2K

2

P -3K

I )!

2

+2K 2

I

>0

^ K

P

>0 ^ K

I

>0

i

(2)

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where we also have included two additional conditions on

^K^P

and

^K^I

. Performing quantier elimination in (2) gives the following equivalent formula

h

K

P

>0 ^ K

I

>0 ^

108K 4

I

-324K

P K

3

I

-216K 3

I

+315K 2

P K

2

I

+414K

P K

2

I

+127K 2

I

-36K 4

P K

I

-198K 3

P K

I

-294K 2

P K

I

-162K

P K

I

-30K

I -4K

6

P

+12K 5

P

+71K 4

P

+108K 3

P

+74K 2

P

+24K

P

+3>0 i

:

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The solution set to this system of inequalities is shown in Figure 2. Observe that we have not given any conditions that exclude the case when the Nyquist curve enclose

^D

. The upper of the two disjoint parts of the solution set corresponds to parameter values for Nyquist curves that enclose

^D

. This can be prevented in various ways, e.g., by the additional constraints

^x¹ ^>^-1 ^_ ^x² ⁶⁼⁰

. Including these in (2) before QE results in the additional constraints

^K^P^>⁰ ^{^} ^K^I ^<^K^P⁺¹

, i.e., only the lower of the disjoint parts of the original solution set corresponds to a Nyquist curve satisfying the circle criterion. To the right in Figure 2 the disc

^D

and two Nyquist curves are shown, one from each part of

the original solution set.

²

0 1 2 3 4 5 6

K

I

K

P K

I

=K

P +1

-2 -1

-3 -2 -1 Re

Im

G (i!) G

(i!)

D

Figure 2: Left: Region of

^(K^P^K^I⁾

-values that guarantees closed loop stability (gray-shaded).

Right: The disc

^D

and two Nyquist curves.

The describing function method is another approach to detect instability or limit cycles in feedback systems with static nonlinearities. Conditions on design parameters to avoid instability can be calculated using QE as in the above example. According to the describing function method the negative reciprocal of the describing function

^;^: ^Y^-1f

(C) C2

0 1]

corresponds to

^-1

in the Nyquist criterion 1]. Hence, the system is stable if the Nyquist curve do not enclose or cross this curve. The curve

^;

can be embedded in a disc, ellipsoid, or any region in the complex plane that can be described by a system of polynomial inequalities. As in the previous example conditions on design parameters that guarantees stability can be computed using QE. We demonstrate this in the following example.

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Example 3.3 Consider the same linear system

^G(s)

as in Example 3.2 and let the nonlinearity be a saturation at the input of

^G(s)

. Suppose we want to calculate conditions on

^K^P

and

^K^I

that guarantees stability of the closed loop system. The describing function for a saturation

f(e)= 8

>

<

>

:

1 e>1

e jej1

-1 e<-1

is given by

Y

f (C)=

2

arcsin

⁽^C¹⁾⁺^C¹^q¹^-^C¹² ^C^>^1:

The curve

^;

for this nonlinearity corresponds to

^x¹ ^-1 ^{^} ^x² ⁼ ⁰

, where

^x¹

and

^x²

are the real and imaginary part, respectively. Hence, to avoid crossing

^;

the real and imaginary part of the Nyquist curve have to satisfy

^x¹ ^> ^-1 ^_ ^x² ⁶⁼ ⁰

, i.e., the same condition we imposed on the Nyquist curve in the previous example. Performing QE then

gave us the conditions

^K^P ^>⁰ ^{^} ^K^I ^<^K^P⁺¹

.

²

Observe that design specications on a system in terms of gain and phase margin, resonance peak, and low frequence gain requirements all can be described by specifying

\forbidden" regions (described by systems of polynomial inequalities) in the Nyquist di- agram. Hence, QE can be used to compute the imposed conditions on various design parameters.

4 Conclusions

We have given a brief presentation of a symbolic method known as quantier elimination which can be used to simplify formulas consisting of polynomial equations, inequalities, boolean operators, and quantiers. Many problems in control theory seems to t into this framework and we refer the interested reader to 4] for more applications. To demon- strate the method we carried out some computations related to stability analysis of linear feedback systems with static nonlinearities. It was shown that su cient conditions for stability could be derived expressed as a system of polynomial inequalities in the controller parameters. Both the circle criterion and the describing function method were used.

The formulation of a control problem in the above framework implies that it, at least theoretically, can be solved in a nite number of steps. However, at present the complexity limits its usefulness for large scale applications.

Acknowledgement

This work was supported by the Swedish Research Council for Engineering Sciences (TFR), which is gratefully acknowledged. The author is also grateful to Dr. Hoon Hong at RISC, Austria for providing him with a copy of the

^qepcad

program.

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References

1] D. P. Atherton. Nonlinear Control Engineering. Van Nostrand Reinhold Company Ltd., 1982.

2] G.E. Collins. Quantier elimination for real closed elds by cylindrical algebraic de- composition. In Second GI Conf. Automata Theory and Formal Languages, Kaiser- slauten, volume 33 of Lecture Notes Comp. Sci., pages 134{183. Springer, 1975.

3] G.E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantier elimination. J. Symbolic Comput., 12:299{328, Sep 1991.

4] M. Jirstrand. Algebraic Methods for Modeling and Design in Control. Licentiate thesis, Dept. of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden, 1996.

Quantier Elimination and Applications in Control