Input-to-State Stability Mapping for Nonlinear Control Systems Using Quantifier Elimination

This project contains the QEPCAD source files for the test of nonlinear systems regard to input-to-state stability (ISS). The project allows the verification of the results presented in the follwing paper:

Voßwinkel, R.; Röbenack, K.; Bajcinca, N.:
Input-to-State Stability Mapping for Nonlinear Control Systems Using Quantifier Elimination.
European Control Conference (ECC), Limassol, Cyprus, June 12-15, 2018, pp. 906-911.

Prerequisites

You need to install the program QEPCAD B, which is a version of QEPCAD (Quantifier Elimination by Partial Cylindrical Algebraic Decomposition). The package is contained in the repositories of typical Linux distributions. Alternatively, QEPCAD B can be installed from the source code:

https://www.usna.edu/CS/qepcadweb/B/QEPCAD.html

Example 1

We consider the one-dimensional system

with the state x and the input w. To test the system regard to ISS we use the Lyapunov candidate function V(x)=qx² with q>0.

Detailed ISS Test without parameters

We use additional comparison functions with the coefficients p,r,c,d>0 to ensure that V is globally positive definite.

QEPCAD script iss_ex1a.q:

[ Example 1 without parameters ]
(q,p,r,c,d,x,w)
0
(E q)(E p)(E r)(E c)(E d)(A x)(A w)
[q>0 /\ p>0 /\ r>0 /\ c>0 /\ d>0 /\ 
 p x^2 <= q x^2 /\
 q x^2 <= r x^2 /\
 2 q x (-x-(1/10)x^2-x^3+(1/10)w)<=-d x^2+c w^2].
finish

The script yields the result TRUE.

Simplified ISS Test without parameters

A quadratic form with a positive constant or a positive definite matrix, respectively, is always globally positive definite. This allows a simplification of the ISS test:

QEPCAD script iss_ex1b.q:

[ Example 1 without parameters, simplified ]
(q,c,d,x,w)
0
(E q)(E c)(E d)(A x)(A w)
[q>0 /\ c>0 /\ d>0 /\ 
 2 q x (-x-(1/10)x^2-x^3+(1/10)w)<=-d x^2+c w^2].
finish

The script yields the result TRUE.

Simplified ISS Test with parameter k

Now we consider the nonlinear systen

with the additional parameter k acting as a free variable. We investigate the question, for which parameters k the system is ISS.

QEPCAD script iss_ex1c.q:

[ Example 1 with parameter k (free variable) ]
(k,q,c,d,x,w)
1
(E q)(E c)(E d)(A x)(A w)
[q>0 /\ c>0 /\ d>0 /\ 
 2 q x (-x-k x^2-x^3+(1/10)w)<=-d x^2+c w^2].
finish

The script yields the result k - 2 < 0 /\ k + 2 > 0. This is an equivalent quntifier-free formula of the prenex formula above. Hence, for -2<k<2 the system is ISS.

The src directory contains the source QEPCAD files listed in the next table.

File	Description
`iss_ex1a.q`	Detailed ISS test without parameters
`iss_ex1b.q`	Simplified ISS test without parameters
`iss_ex1c.q`	Simplified ISS test with the parameter k as free variable

Licence

This project is licensed under the GNU General Public License v3.0 - see the LICENSE file for details.

TUD-RST / qe-iss