Symbolic Quantifier Elimination for Robust Feedback System Design

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Copyright © IFAC Robust Control Design Milan, Italy, 2003

IFAC PUBLICATIONS www.elsevier.com/locate/ifac

SYMBOLIC QUANTIFIER ELIMINATION FOR ROBUST FEEDBACK SYSTEM DESIGN Peter Dorato and Isao Sakamaki

Department of Electrical and Computer Engineering University of New Mexico Albuquerque, NM 87131-1356, USA peter@eece. unm. edu

Abstract: In this paper the status of symbolic quantifer elimination techniques for robust feedback system design is surveyed. Many practical robust design problems can be reduced to quantified multivariate polynomial inequalities (MPIs), with logic quantifier "for all" operating on uncertain plant parameters. The elimination of the logic quantifier produces a boolean function which is free of the quantified variable, and defines the set of design parameters that satisfy the design specifications. The computational complexity of symbolic quantifier eliminations (QE) limits the complexity of robust design problems that can be solved, but some problems can be solved that otherwise would be difficult to solve. Currently available computer software for QE is also surveyed. Copyright © 2003 IFAC Keywords: Robust feedback design; linear systems; symbolic quantifier elimination, multivariate polynomial inequalities.

1. INTRODUCTION

We will focus in this presentation on linear feedback systems with plant transfer functions, G(s,p), and compensator transfer functions, C(s, q). Uncertain plant parameters are represented by the vector p, and compensator design parameters are represented by design vector q. A "practical" robust design problem is defined here as one that includes the following elements: 1. Fixed Compensator Structure. Normally the compensator will be of low order, to permit on-line tuning of design parameters, if necessary. For example, a PID controller structure. 2. Significant Plant Uncertainty. Plant plant parameters lying within, possibly large, intervals. 3. Multiple Design Specifications. Multiple performance measures which must all be met, robustly, including, of course, closed-loop stability.

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In the next section we show how practical robust design problems can be reduced to quantifierelimination problems, where the object is to eliminate the "for all" logic quantifer (V) in a boolean expression of the form V(p E P)[(vI(p,q) > 0)

f\

(V2(p,q) > 0)

f\ ...

1(1)

In expression (1), the functions Vi(P, q) represent multivariate polynomial (MP) functions (functions that are ordinary polynomials in any component of the vectors p and q when all the other components are held fixed), and where f\ represents the logic "and" operator. Symbolic quantifierelimination removes the "for all" quantifier, and produces a quantifier-free boolean expression (expression containing logic operators and mutivariate polynomial inequalities), lII(q) , which can then be used to compute the set a design-parameter vectors Q D which represents the set of all design

parameters that meet all the design specifications. If the logic quantifier "there exists", 3, operating on q, is added to expression (1), quantifier elimination return a "true of false" which then specifies if a robust design exists, or does not exist.

ratio of two MP functions, Nz(p, q) and Dz(p, q), and that the denominator function D 2 (p, q) is positive if the the transfer function T(s) is stable. See, G.C. Newton et al. (1957) for details. This means that (6) can be reduced to the following MPI,

2. REDUCTION OF PRACTICAL ROBUST SPECIFICATIONS TO MULTIVARIATE POLYNOMIAL INEQUALITIES

The transfer function T(s) can be any closed-loop transfer function of interest, e.g. tranfer function relating output sensor noise to plant output.

We focus here on three design specifications that lead to MPIs. The first is robust closed-loop stability. The closed-loop characteristic polynomial of a unity feedback system with compensator C(s, q) and plant G(s,p) is the numerator polynomial of the rational function, 1 + C(s,q)j)(s,p)

The third specification that will be considered here is an Hoo-type norm of the form, J oo

= IT(jwW < Qoo, forall wEn

(8)

(2) The function J oo can be expessed as a ratio of two MP functions, Noo(P,q;w) and Doo(p,q;w) and the specifiction given in (8)can be reduced to the MPI,

denoted,

v(p, q; w)

It is assumed to start that the coefficients ai(p, q) are MP functions. If the Lienard-Chipart (LC) stability test is used to determine if n( s) is a hurwitz polynomial (all roots with negative realparts), one obtains the following MPIs, for n odd and ao > 0,

Doo(p, q; w)Q oo

Noo(p, q; w)

> 0(9)

When dealing with frequency constraints of the form given (8), an additional "for all" quantifier must be added to (1), operating on the frequency variable w. Frequency specification of the type outlined above, are applicable to tracking and disturbance rejection problems.

an > 0, an z > 0, ... ; Hz > 0, H 4 > 0, ... (4)

where the Hi are the so-called hurwitz determinants a3 a5 ao az a4 al a3 ao az

=

3. SURVEY OF SYMBOLIC QUANTIFIER-ELIMINATION METHODS

al

Hi

=

det

°°

In Tarski (1951), it was first shown that decision problems (quantifier elimination problems with the "there exists" quantifier) could be solved in a finite number of steps. In Anderson et al. (1975), Tarski's ideas were applied to the problem of static-output feedback stabilization, a problem know to be computationally very difficult (See, for example Syrmos et al. (1997)). This was one of the first applications of symbolic quantifier elimination theory to feedback system design. In Collins and Hong (1991) an algorithm was proposed for quantifier elimination which was more efficient than the algorithm proposed by Tarksi. The Collins-Hong algorithm was referred to as a partial cylindrical algebraic decompositon algorithm. The algorithm was latter programmed into a software package referred to as QEPCAD (Quantifer Elimination via Partial Cylindrical Algebraic Decomposition). The monograph Caviness and Johnson (1998) includes a number of studies of quantifier elimination methods based on cylindrical algebraic decomposition. In Dorato et al. (1997) QEPCAD software was applied to

(5)

For details see, Gantmacher (1959). Note that the functions Hi are MP functions if the ai(p, q) are MP functions. Thus robust stability can be reduced to quantified MPIs as in expression (1), where P represents the uncertainty set for the plant parameter vector p. The second design specification we consider here is a Hz-type norm of the form, (6)

This specification can be used to deal with random disturbances, including sensor noise. It can be show that the integral in (6) can be computed as a

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V2(P, q) = 1 + P + q > 0 V3(P, q) = 7 16p 16q + 16 p 2 + 16q2 > 0 Note that the QE software requires that all MP coefficients be expressed as integers or ratio of integers. Decimal notation is not admissible. The Mathematica code to solve this problem looks like the following,

practical robust feedback design problems for linear systems. The application of QEPCAD software was extended to robust design of nonlinear systems is in Ref. Dorato et al. (1999). In Jirstrand (1998), symbolic quantifier elimination methods were explored for various control system design problems. A compilation of symbolic methods for control system design may be found in the monograph Munro (1999). Unfortunately, while symbolic QE methods solve problems in a finite number of steps, the number of steps increases very rapidly with the complexity of the problem (doubly expoentaially in most problem variables). In the survey article Dorato (2000), a hierarchy of methods is presented for practical robust feedback design, starting with symbolic quantifier elimination methods, followed by deterministic branch-and-bound methods, and finally ending with random Monte Carlo methods, with the random approach applicable to the highest level of problem complexity. Recently software for symbolic quantifier elimination has been added to the software in Mathematica. This software will be discussed in the next section, and an application of the software to a "practical" robust control design will be presented in the section that follows.

< < Experimental' vI = 9 + 48 q + 48 P + q v2 = 1 + P + q v3 = 7 16 P 16 q + 16 P 1\ 2 + 16 q 1\ 2 F=ForAll[p,O :<::: p :<::: 1 && q E Reals, vI > 0 && v2 > 0 && v3 > 0] Resolve[F] Note that multiplication requires a space between terms. Also all unquantified variable must be declared as "belonging to the reals" as seen in the above code. The output produced by Mathematica is the quantifier-free expression, W(q)

= {(

3/16 < q < 1/4)

11

(q > 3/4)}

which defines the design set Q D to be, in decimal notation, the disjoint set, QD

= {(

.1875 < q < 0.25) U (q > 0.75)}

In this case the unquantified variable q is a scalar, so that computing Q D from the boolean formula W(q) is simple. When q is a vector, the set QD is not so simply seen. For the vector case, for example for the case where q is made up of two components (ql, q2), if "Resovle[F]" is labelled, i.e. replaced by say "P=Resolve[FJ" , and the following lines are added to the code above,

4. COMPUTER SOFTWARE

In Mathematica (versions starting with 4.2) first typing < < Experimental' on the screen, initiates the symbolic quantifier elimination program. Ending with

<
Resolve[FJ a boolean formula will be produced with a nested set of conditions on the design parameters, so that the design set may be more easily identified. For two design-parameter problems, adding the lines

where F is a quantified expression such as (1), terminates the quantifier-elimination program. Then typing "shift-enter" starts the computation. In writing boolean functions in Mathematica one may use "&&" for the logic "and" operator, and "I I" for the logic "or" operator. Consider the example taken from Dorato (2000), where symbolic quantifier elimation is used to select a scalar design parameter q given plant uncertainty 0 :<::: P :<::: 1 such that the closed-loop characteristic polynomial,

n(s)

< < Graphics' InequalityG raphics' InequalityPlot[Q, {q1, 11, u1}, {q2, 12, u2}] where li and ui are lower and upper bounds on the range of design parameter values that are to be plotted, produces a plot of the design set QD.

+ (1 + P + q)s2 + (1 + P + q)s + (.5 + 3p + 3q + 2pq + p2)

5. EXAMPLE

= s3

The robust design example we consider here is taken from the position cruise control problem presented in Example 3.15 of the text Franklin et al. (2002). The position of the car is given by the transfser function,

is robustly stable. With p = 1/4, the LC stability criterion requires the satisfaction of the inequalities, VI (p, q) = 9 + 48p + 48q + 32pq > 0

587

PI ( ) s S+ P2 and the compensator is assumed to be a PI controller of the form _ qlS + q2 C( s,q ) s where uncertain parameters are are assumed to be in the intervals

G(S,p) =

\l1(q)

The two design specification considered here are, robust closed-loop stability and robust H 2 performance. The transfer function selected for H 2 performance is the tranfer function,

6. CONCLUSIONS While commercial software is now available for the application of symbolic quantifier elimination for the design of robust feedback systems, only problems of limited complexity can be sovled. Of course, super-computer systems can extend the level of complexity, but the level is likely to saturate on problems where the order of combined plant and compensator is greater than five or six. Because of the doubly-exponential nature of symbolic QE methods, a computational saturation will occur at some point, no matter how good a computer is used. Nevertheless symbolic quantifier methods may produce a design set for problems, that otherwise would be difficult to compute. Also symbolic QE determines the design set Q D exactly, while branch-and-bound and Monte Carlo methods require apriori assumptions about the design set (for example intervals for exploration of design parameters), and in a finite number of steps, produce only approximate answers. In the case of Monte Carlo answers are only in terms of probabilistic guarantees.

T(s) _

C(s)G(s) - 1 + C(s)G(s)

which represents the transfer function between sensor noise and plant output. The H 2 performance level is assumed to be
=

s3

+ P2s2 + PlqIS + PIq2

and the value of h is given by

cidod3 + c6 d2d3 2dod3(d l d 2 dod3) which is computed from the formula in table E.2l(page 372) of G.C. Newton et al. (1957) for the above T(s), given in this example by

h =

T(s) = N 2 (p,q) D 2 (p, q) where, N 2(p, q) = cidod3 + C6 d2d3 D 2(p,q) = 2dod3(d l d 2 dod3) with, d3 = 1, d 2 = P2, d l Cl = PlqI, Co = Plq2

= Plql, do = Plq2

REFERENCES B.D.O. Anderson, N.K. Bose, and E.!. Jury. Output feedback stabilization and related problems- solution via decision methods. IEEE Tran. Automat. Control, AC-20:53-66, 1975. B.F. Caviness and J.R. Johnson. Quantifier Elimination and Cylinrical Algebraic Decomposition. Springer, Wien, 1998. G.E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. J. Symbolic Computation, 12:299-328, 1991. P. Dorato. Quantified multivariate polynomial inequalities. the mathematics of practical control design problems. IEEE Control Systems Magazine, 20:48-58, 2000. P. Dorato, D. Famularo, C.T. Abdallah, and Wei Yang. Robust nonlinear feedback design via quantifier elimination. 1nl. J. of Robust and Nonlinear Control, 9:817-822, 1999. P. Dorato, Wei Yang, and C. Abdallah. Robust multi-objective feedback design by quantifier

The LC stability criterion in this case produces the MPIs,

Vl(P,q) = Plq2 > 0 V2(p,q) = P2 > 0 V3(P, q) = PIP2ql Plq2

>0

The H 2 performance specification produces the MPI,

Mathematica Quantifer elimination for the expression, V(1/1000 S; PI [VI > 0 /\ V2 > results (using quantifier-free

ql < 2/5 && 0 < q2 < 1/21O(2ql 5q~)}

From \l1(q) given above it is easy to see that the design set in the qlq2 plane (with verticle axis q2) is simply the points between the parabolic curve q2 = 1/21O(2ql 5qf), and the interval, o < qI < 2/5. For more complicated expressions, the Matematica function "InequalityPlot" can be used to see the design set.

0.001 :::; PI :::; 0.005, 0.01:::; P2 :::; 0.08

n(s)

= {O <

:::; 5/1000,1/100 S; P2 S; 8/100) 0 /\ V3 > 0 /\ V4 > 0] up to "InequalitySolve") in the formula,

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elimination. J. Symbolic Computation, 24:153159,1997. G.F. Franklin, J.D. Powell, and Abbas EmamiNaeini. Feedback Control of Dynamic Systems, 4th Edition. Upper Saddle River, NJ, 2002. Matrix Theory, Vol. II. F. R. Gantmacher. Chelsea, New York, 1959. Jr. G.C. Newton, L.A. Gould, and J.F. Kaiser. Analytic Design of Linear Feedback Controls. John Wiley & Sons, New York, 1957. M. Jirstrand. Constructive Methods for Inequality Constraints in Control. Dissertation No. 527. Link6ping University, Link6ping, 1998. N. Munro. Symbolic Methods in Control System Analysis and Design. IEE, London, 1999. V.L. Syrmos, C.T. Abdallah, P. Dorato, and K. Grigoriadis. Static output feedback: A survey. Automatica, 33:125-137, 1997. A. Tarski. A Decision Method for Elementary Algebra and Geometry, 2nd Edition. University of California Press, Berkeley, 1951.

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