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: [email protected]
Reglermo te '96, Lule
a
Abstract
Quantier elimination is a method for simplifying formulas that consist of polynomial 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 polynomial 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.
1
2 Quantier Elimination
By a quantier free Boolean formula we mean a formula consisting of polynomial equations
(fi(x) = 0) and inequalities (fj(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
(Q1 x1 : : : Qs xs )F (f1 (x)
fr (x))
:::
where Qi is one of the quantiers 8 (for all) or 9 (there exists) and F (f1(x) : : : fr(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
!
2
x
i
+ ax + b
> 0
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
2
-a
> 0
_
a
0
^
b
0]
i
which are the conditions on the coecients 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 inequalities. Solutions to such systems can, e.g., be computed by cylindrical algebraic decomposition (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 sucient 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.
2
k2 x
(+)
f(x)
G(s)
+
k1 x
(-)
f( )
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,
z
2C
z
+
k2
+ k1 2k1 k2
k2 - k1 2k k
1 2 (1)
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)
=
!
KP
KI
+
1
2+
s
s
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 (KP KI) that ensure stability of the
feedback system.
The real and imaginary part of the Nyquist curve G(i!) are
x1
=
-KP !2 + KP - KI
!4
x2
- !2 + 1
=
(KI - KP )!2 - KI
!(!4
- !2 + 1)
:
The Nyquist curve remains outside the disc D, corresponding to the given (KP
if the following inequality is satised for all !
x1
+
3
4
!2
2
+ x2
>
1
2
4
KI )
values,
:
Clearing denominators and simplifying this expression gives the following equivalent formula
8!
h
! > 0
!
!
6
4
2
2
2
- (3KP + 1)! + (1 + 3KP + 2KP - 3KI )! + 2KI
^
3
KP > 0
> 0
^
i
KI > 0
(2)
where we also have included two additional conditions on KP and KI. Performing quantier
elimination in (2) gives the following equivalent formula
h
KP > 0
^
KI > 0
4
108KI -
^
3
324KP KI
4
3
2
2
2
2
- 216KI + 315KP KI + 414KP KI + 127KI
3
2
6
- 36KP KI - 198KP KI - 294KP KI - 162KP KI - 30KI - 4KP
5
4
3
2
+ 12KP + 71KP + 108KP + 74KP + 24KP + 3
i
(3)
> 0 :
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 KP > 0 ^ KI < KP + 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
6
G (i!)
5
KI
KI
4
-3
= KP + 1
-2
-1
D
Im
Re
G (i!)
3
-1
2
1
-2
0
0
1
2
3
KP
4
5
6
Figure 2: Left: Region of (KP KI )-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 ; : Yf-(1C) C 2
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.
4
Example 3.3 Consider the same linear system
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 KP and KI that guarantees stability of the closed loop system. The describing function
for a saturation
f(e)
=
8>
<
>:
1
G(s)
e > 1
j j
e
e
-1
e <
1
-1
is given by
Yf(C)
2
= arcsin( C1 ) + C1
q
1
- 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 KP > 0 ^ KI < KP + 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 diagram. 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 demonstrate the method we carried out some computations related to stability analysis of linear
feedback systems with static nonlinearities. It was shown that sucient 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
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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.
5] B. Mishra. Algorithmic Algebra. Texts and Monographs in Computer Science.
Springer-Verlag, 1993.
6] A. Tarski. A Decision Method for Elementary Algebra and Geometry. University of
California Press, second edition, 1948.
7] M. Vidyasagar. Nonlinear Systems Analysis. Prentice Hall, second edition, 1993.
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