Complexity Issues and Decision Methods in

J. Symbolic Computation

(1995)

11, 1{12

Complexity Issues and Decision Methods in Control Systems V. Blondely , C.T. Abdallahz , and G.L. Heilemanz

y

Institutez of Mathematics, University of Liege D1 Avenue des Tilleuls 15, B-4000 Liege, Belgium. Department of EECE, University of New Mexico, Albuquerque, NM 87131, USA (Received 30 September 1995) This paper addresses the computational diculty associated with speci c control problems. Using notions from Decision and Computational Complexity theories, it emphasizes the fact that some control problems are undecidable, and that some are decidable but computationally costly.

1. Introduction

Many control theory problems can be cast as decision problems. For example, the existence of a controller to meet some closed-loop speci cations can be appropriately framed as a \yes" or \no" question. Answering this question, however, is only half the battle, since control designers are also interested in exhibiting a solution once the existence question is settled. Recently, research has been conducted to address the computational diculty associated with exhibiting a solution to many control and systems theory problems. This paper reviews the state-of-the-art in this area. We use complexity-theoretic techniques to show that the problem of determining whether linear, time-invariant (LTI) systems can be stabilized using dierent controller structures is NP -hard, implying that an unreasonable amount of time may be needed to nd the desired feedback parameters in large LTI systems. Speci cally, under the strongly held assumption that P does not equal NP , these results demonstrate that a polynomial time (in the order of the LTI system) algorithm does not exist for solving the general versions of these control problems. We will not however deal with Discrete-Event systems (Tsitsiklis 1989), and nonlinear systems (Siegelmann and Sontag 1995, Sontag 1995), both of which have recently been studied from the perspective of computational complexity. The paper is organized as follows: Section 2 presents an overview of decision methods as they apply to control problems. Section 3 reviews ideas from computational complexity. In section 4, we frame control existence problems as decision problems, while section 5 deals with the complexity of some control design problems. In section 6 we present some open control problems, while section 7 contains our conclusions.

2. Decision Methods

In the seventeenth century, Leibnitz envisioned 2 goals for Science and Philosophy: 0747{7171/90/000000 + 00 $03.00/0

c 1995 Academic Press Limited

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V. Blondel, C.T. Abdallah and G.L. Heileman

(i) To discover a precise symbolic language in which all scienti c statements can be made. (ii) To discover a method to manipulate the statements of the symbolic language in order to elucidate their meaning and validity. The latter part of the nineteenth century and the earlier part of the twentieth witnessed a urry of activities towards achieving those two goals in Mathematics. In particular, Hilbert in his famous 1900 Paris address (Hilbert 1900) advanced a list of problems, the tenth of which had to do with the solvability of polynomial equations (Diophantine) with integer coecients. Hilbert did not even question whether or not a decision procedure for determining the solvability would exist, and was in fact attempting to nd one into the 1920's. It was therefore shocking that in 1931, Godel showed that such a plan was doomed to failure (Godel 1931). Eectively, there are some statements over the integers whose truthfulness can never be determined. In 1970, it was nally shown that Hilbert tenth problem was unsolvable over the integers (Matiyasevich 1993). In contrast to the undecidability of Arithmetic, Tarski showed that the rst-order theory of the real numbers is decidable in 1930, but his complete proof did not appear until 1948 (Tarski 1951). Tarski's idea was to eliminate the quanti ers in order to arrive at a semi-algebraic expression which is equivalent to the original quanti ed expression over the reals. In this section, we review the general quanti er elimination (QE) problem. A more detailed treatment may be found in (Tarski 1951, Basu et al. 1994). Given the set of polynomials with integer coecients Pi (X; Y ); 1  i  s where X represents a k dimensional vector of quanti ed real variables and Y represents a l dimensional vector of unquanti ed real variables, let X [i] be a block of ki quanti ed variables, Qi be one of the quanti ers 9 (there exists) or 8 (for all), and let (Y ) be the quanti ed formula (Y ) = (Q1 X [1]; :::; Qw X [w])F (P1 ; :::; Ps );

(2.1)

where F (P1 ; :::; Ps ) is a quanti er free Boolean formula, that is a formula containing the Boolean operators ^ (and) and _ (or), operating on atomic predicates of the form Pi (Y; X [1]; :::; X [w])  0 or Pi (Y; X [1]; :::; X [w]) > 0 or Pi (Y; X [1]; :::; X [w]) = 0. We can now state the general QE problem

General Quanti er Elimination Problem: Find a quanti er-free Boolean formula (Y ) such that (Y ) is true if and only if (Y ) is true.

An important special problem is the QE problem with no unquanti ed variables (free variables), i.e. l = 0. This problem is referred to as the General Decision Problem.

General Decision Problem: With no unquanti ed variables, i.e. l = 0, determine if the quanti ed formula given in (2.1) is true or false.

As discussed above, algorithms for solving the general QE problem were rst given by Tarski (Tarski 1951) and Seidenberg (Seidenberg 1954), and are commonly called Seidenberg-Tarski decision procedures. Tarski showed that the QE problem is solvable, but his algorithm and later modi cations require a computational time that is exponential in the size of the problem. Researchers in Control Theory have been aware of

Complexity Issues and Decision Methods in Control Systems

3

Tarski's results and their applicability to Control problems since the 1970's, but the tedious operations made the technique very limited (Anderson et al. 1975). Later, Collins (Collins 1975) introduced a theoretically more ecient QE algorithm that uses a cylindrical algebraic decomposition (CAD) approach. However, this algorithm was not capable of eectively handling nontrivial problems. Later Collins and Hong (Collins and Hong 1991), and Hong (Hong 1992) introduced a signi cantly more ecient partial CAD QE algorithm. More recently, an algorithm which lends itself to parallelization was given in (Basu et al. 1994).

3. Computational Complexity Overview

Until recently, it was felt that decidable problems are practically solved and thus not very interesting. The introduction of Computational complexity theory has since changed this misconception. Computational complexity theory is often used to establish the tractability or intractability of computational problems, and is concerned with the determination of the intrinsic computational diculty of these problems (Garey and Johnson 1979). One important concept in this theory is that of a polynomial-time algorithm. In practice, such an algorithm can be feasibly implemented on a real computer. This is in contrast to an exponential-time algorithm, which is only feasible if the problem being solved is extremely small. The complexity class P consists of all decision problems that can be decided in polynomial-time, using a Turing machine model of computation. The simplicity of the Turing machine model appears to make it of little practical value; however, the ChurchTuring Thesis holds that the class of problems solvable on a Turing machine in polynomial time is robust across all other reasonable models of computation (including the computers we use). The complexity class NP consists of all decision problems that can be decided algorithmically in nondeterministic polynomial-time. An algorithm is nondeterministic if it is able to choose or guess a sequence of choices that will lead to a solution, without having to systematically explore all possibilities. This model of computation is not realizable, but it is of theoretical importance. Many practical problems belong to NP and it is as of yet unknown whether P = NP . In other words, these two complexity classes form an important boundary between the tractable and intractable problems. A problem is said to be NP -hard if it is as hard as any problem in NP . Thus, if P 6= NP , the NP -hard problems can only admit deterministic solutions that take an unreasonable (i.e., exponential) amount of time, and they require (unattainable) nondeterminism in order to achieve reasonable (i.e., polynomial) running times. The central idea used to demonstrate NP -hardness evolves around the NP -complete problems. A problem is said to be NP -complete if every decision problem in NP is polynomial-time reducible to it. This means that the NP -complete problems are as hard as any decision problem in NP . Given two decision problems 1 and 2 , 1 is said to be polynomial-time reducible to 2 (written as 1 p 2 ), if there exists a polynomial time algorithm R which transforms every input x for 1 into an equivalent input R(x) for 2 . By equivalent we mean that the answer produced by 2 on input R(x) is always the same as the answer 1 produces on input x. Thus, any algorithm which solves 2 in polynomial time can be used to solve 1 on input x in polynomial time by simply computing R(x), and then running 2 . In order to show that a particular (control) decision problem 2 is NP -complete, one

4

V. Blondel, C.T. Abdallah and G.L. Heileman

starts with a problem 1 in NP -complete , and attempts to show that 1 p 2 . This shows that 2 is NP -hard. To complete the proof that 2 is NP -complete, it must be demonstrated that a candidate solution can be veri ed in polynomial time. In control theory, researchers have followed this \reduction" method to study the computational diculty of some decidable problems.

4. Control Design As Decision Problems

Researchers in control theory are pursuing an analytical study of general open problems. For a control theorist, a problem is considered completely solved once testable existence conditions (preferably of the necessary and sucient variety) have been found, and an algorithm is presented to design the controller when such existence conditions are satis ed. These existence conditions could be part of a decision procedure, but the actual algorithmic procedure has to be studied from the computational complexity point of view. In the following, we let A = [aij ] be matrix with entries aij , and In be the identity matrix of size n  n. We also consider linear systems described by x_ = Ax + Bu y = Cx (4.1) where x 2 IRn , B 2 IRnm , C 2 IRpn . Note for example, that the static output feedback (SOF) problem is often recast as a problem of satisfying a system of multivariable inequalities. Speci cally, if the gain matrix K in the control law u = Ky is parameterized in terms of nitely many real coecients, then the stability conditions can be expressed as a characteristic polynomial involving the coecients of the system and the controller. Applying the Routh-Hurwitz test to the resulting characteristic polynomial leads to the generation of a set of multivariable polynomial inequalities whose satisfaction, for some choice of controller coecients, guarantees a stable closed-loop system. The exponential-time Tarski-Seidenberg elimination method (Tarski 1951) can be used to determine whether or not a solution to the multivariable polynomial inequalities exists (Anderson et al. 1975). This answers the question of the decidability of the problem, but it does not address the more practical problem of whether or not ecient (i.e., polynomial-time) methods exist for solving the problem. In the sequel, we refer to to the usual closed-loop, negative feedback used in control theory. This is shown in Figure 1. The functions C (s; q) and G(s; p) are the Laplacetransform transfer functions of the compensator and the plant, respectively, and p is a vector of plant parameters pi while q is a vector of deign parameters qi . We assume here that our systems are linear, time-invariant, and lumped, so that these two functions are rational functions in the transform variable s. Uncertainty in plant parameters are characterized by quanti ed formulas of the type 8(pi) [pi  pi  pi ] where pi and pi are rational numbers. The plant may also be characterized in state-space form, i.e. x_ = A(p)x + B (p)u; y = C (p)x (4.2) ? 1 with the transfer function given by G(s; p) = C (p)[sI ? A(p)] B (p). The general control problem stated as decision problem is then as follows:Given G(s; p), does there exist a vector q such that 8(pi) [pi  pi  pi ], the closed-loop system (4.3) T (s; p; q) = 1 + GG(s;(s;p)pC) (s; q)

Complexity Issues and Decision Methods in Control Systems r

e

+

C (s; q)

u

G(s; p)

5 y

−

Figure 1. Block Diagram Of Feedback System satis es some performance objectives? In such control problems, the unquanti ed variables are generally the compensator parameters, represented by the parameter vector Y = q, and the quanti ed variables are the plant parameters, represented by the plant parameter vector p, and the frequency variable !. The quanti er-free formula (q) then represents a characterization of the compensator design. See the paper (Dorato et al. 1995a) for a detailed use of QE to solve a robust multti-objective control problem. Note nally that in the early eighties, Sontag advocated similar decision methods in studying nite-horizon control problems for piecewise-linear systems (Sontag 1981, Sontag 1982). Note however that in some cases, a QE approach may not be best for a given control decision problem. For one thing, the problem may be undecidable to begin with (Toker 1995a). For another, decidable problems may be solved using speci c approaches. As an example, the linear quadratic regulator (LQR) problem is best solved using the Riccati equation approach, whenever a testable set of necessary and sucient conditions guarantee the existence of the controller (Dorato et al. 1995).

5. The Complexity of Some Control Problems

Recently, many decidable control problems have been shown to be NP -complete (or NP -hard) (Poljak and Rhon 1993, Nemirovskii 1993, Coxson 1993, Blondel and Tsitsiklis

1994, Toker and O zbay 1995). The following lists some related linear Algebra problems which may then be re-cast in a control setting: (i) To test whether all symmetric matrices A where a?ij  aij  a+ij , aij rationals is positive semide nite is NP -hard by reduction from the Partition problem (Nemirovskii 1993). (ii) To test whether all matrices A where a?ij  aij  a+ij , aij rationals satisfy AT A  In is NP -hard by reduction from the Partition problem (Nemirovskii 1993). (iii) To test whether all matrices A where a?ij  aij  a+ij , aij rationals are nonsingular is NP -hard by reduction from the Partition problem (Nemirovskii 1993) or from the Max-Cut problem Poljak and Rohn 1993). (iv) To test the robust stability of A where aij is either xed or a?ij  aij  a+ij is NP -hard by reduction from the Partition problem (Nemirovskii 1993). (v) To test the existence of one stable member of the family of A's where aij is either xed or a?ij  aij  a+ij is NP -hard by reduction from the Partition problem (Blondel and Tsitsiklis 1994). This problem will be referred to as the Stable matrix in unit interval family.

6

V. Blondel, C.T. Abdallah and G.L. Heileman

(vi) To check the existence of n real numbers q1 ;


; qn such that for given Ai , A0 + q1 A1 +


+ qn An is stable is NP -hard by reduction from a modi ed version of the Knapsack problem (Toker and O zbay 1995) and from the Stable matrix in unit interval family in (Goldberg et al. 1995). (vii) Given a matrix valued function F (:; :) which is bilinear (i.e. linear in each of its 2 matrix arguments). The problem of checking the existence of real matrixes X; Y such that F (X; Y ) > 0 is NP -hard by reduction from a modi ed version of the  Knapsack problem (Toker and Ozbay 1995). (viii) To recognize  for real uncertainty is NP -harda (Poljak and Rohn 1993) by reduction from Max-Cut problem. (ix) To recognize  for mixed uncertainty is NP -hard (Braatz et al. 1994) by reduction from the Indefinite quadratic programming problem. (x) To recognize  for complex uncertainty is NP -hard (Toker 1995b) by reduction from the Knapsack problem. In the next section we give a detailed presentation of some complexity results in Linear Algebra and Control problems. 5.1. Robust Stability of Matrices

The rst problem we present was studied in (Blondel and Tsistiklis 1994): Stable matrix in unit interval family

Instance: A positive integer n, a partition of I = f(i; j ) j 1  i; j  ng into disjoint sets I1 and I2 , rational numbers aij for (i; j ) 2 I1 . Question: Does the set A of n  n matrices de ned by

A = fA = aij j aij = aij for (i; j ) 2 I1 ; aij 2 [?1; 1] for (i; j ) 2 I2 g contain at least one stable matrix? Notice that some constraints are placed on all of the elements of A|each element is either a xed rational number, or a rational number in the interval [?1; 1]. As discussed before, Blondel and Tsitsiklis (Blondel and Tsistiklis 1994) proceeded to show that Stable matrix in unit interval family 2 NP -Hard. A related problem which relaxed the assumptions on aij was studied in (Goldberg et al. 1995) and is described next. Stable linear combination of matrices

Instance: A set of n  n matrices A0 ; A1 ; :::; Am Question: Do there exist real numbers q1 ; :::; qm such that the sum A0 + q1 A1 + q2 A2 +


+ qm Am has all its eigenvalues in the left-hand side of the complex plane (i.e.. the real part of each eigenvalue must be negative). theorem 5.1. The Stable linear combination of matrices problem is NP -complete. Proof. This problem is obviously in NP since a candidate solution may be veri ed in polynomial time. We prove that it is NP -hard by reduction from the Stable matrix in unit interval family problem. Given an instance of Stable matrix in unit interval

Complexity Issues and Decision Methods in Control Systems

7

family, we show how to construct (in polynomial time) an instance of Stable linear combination of matrices, such that the answer to the associated question is the same.

We are given n, I1 and I2 , together with rational values for the elements of I1 . We let

jIi j denote the size or number of elements in the set Ii . The instance of Stable matrix in unit interval family that we will construct will consist of a set of p = 1 + jI2 j matrices Ar each having a size (n + 2jI2 j)  (n + 2jI2 j). Let A0 be the 3p  3p matrix whose entries are equal to

( a

ij for (i; j ) 2 I1 ; (A0 )ij = ?1 for i = j , i > n; 0 otherwise. Let I2 = f1 ; :::; m g, i.e. let m = jI2 j. For 1  r  m, de ne a matrix Ar as follows:

81 > < (Ar )ij = > 1 : ?1

for r = (i; j ); for i = j , i = n + 2r; for i = j , i = n + 2r ? 1; 0 otherwise. This concludes the construction. It is of size polynomial in the size of the instance of Stable matrix in unit interval family, since m is at most n2 . Now we claim that the constructed instance has a solution if and only if the instance of Stable matrix in unit interval family has a solution. Let B be the instance of Stable matrix in unit interval family. The instance of Stable linear combination of matrices constructed from B is the problem of nding values for the qi which makes the following matrix stable:

0B BB ?q1 ? 1 q1 ? 1 BB ?q2 ? 1 BB q2 ? 1 BB BB @

...

?qm ? 1

qm ? 1

1 CC CC CC CC CC A

where each elements of I2 in the sub-matrix B is a unique qi . This is a \yes" instance provided that there exist real values for the qi such that the eigenvalues of this matrix are stable. The eigenvalues of this matrix are the eigenvalues of B together with the values ?qi ? 1 and qi ? 1, for 1  i  m. Then the stability requirement implies that we must have ?1 < qi < 1 for 1  i  m. Given this constraint, we see that the problem of nding an appropriate set of values for the qi is equivalent to the original instance of Stable matrix in unit interval family which is known to be NP -complete. 2  The problem of Stable matrix was also shown in (Toker and Ozbay 1995) to be NP Hard even if no bounds are placed on the variations of aij , by a reduction from a modi ed version of the Knapsack problem in (Garey and Johnson 1979). That also led to a dierent proof of the NP -Hardness of Stable linear combination of matrices.

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V. Blondel, C.T. Abdallah and G.L. Heileman

5.2. Output Feedback

The static output feedback problem is one of the most important open questions in control theory (Bernstein 1992). The question many researchers have begun asking is whether it is worth spending any more time looking for an analytical solution to the SOF problem. In fact, many of them have pointed out that algorithmic and numerical solutions may be called upon to solve the problem in many interesting cases. The hope is then that someone can come up with an algorithm that can solve most of the SOF problems encountered in practice. In this section we review recent results which suggest that such hope may not be realistic, at least for moderate and large size problems. 5.2.1. Stabilization by Static Output Feedback

As discussed before, for the static output feedback problem, the exponential-time Tarski-Seidenberg elimination method can theoretically be used to determine whether or not a solution to the multivariable polynomial inequalities (obtained from the RouthHurwitz test) exists (Anderson et al. 1975). This answers the question of the decidability of the problem, but it does not address the more practical problem of whether or not ecient (i.e., polynomial-time) methods exist for solving the problem. In the language of computational complexity theory, the SOF problem is formulated as follows: Static Output Feedback

Instance: A LTI plant of the form x_ (t) = Ax(t)+ Bu(t); y(t) = Cx(t), under the in uence of static output feedback of the form u(t) = Ky(t) + v(t). Question: Does there exist a real gain matrix K which guarantee the closed-loop stability of the LTI plant? To nd K where kij?  kij  kij+ and A + BKC stable turns out to be NP -hard (Blondel and Tsitsiklis 1994). Using the Stable matrix in unit interval family problem, Blondel and Tsitsiklis show that the SOF problem when the entries of K are constrained to lie in some intervals is NP -Hard, by reduction from the Partition problem (Garey and Johnson 1979). Therefore, to nd a xed-order dynamic compensator to simultaneously stabilize n plants is NP -hard. Blondel and Tsitsiklis also conjectured that the computational complexity of their problems remains the same even in the absence of constraints on K (Blondel and Tsitsiklis 1994). To date however, no such result is available. 5.2.2. Decentralized Static Output Feedback

We now impose some structure on the feedback gains. Consider a linear system of the form

x_ (t) = Ax(t) + yi (t) = Ci x(t);

k X i=1

Bi ui (t) i = 1; : : : ; k;

and suppose that we are interested in a static decentralized controller of the form ui (t) = Ki yi (t); i = 1; : : : ; k:

Complexity Issues and Decision Methods in Control Systems

The closed loop system is

x_ (t) = (A +

k X i=1

9

Bi Ki Ci )x(t);

which is of the same form as in stabilization by static output feedback except that several of the entries of K are forced to zero. This leads us to the problem of nding conditions on the triplet of real matrices (A; B; C ) under which there exists a matrix K with a given structure such that A + BKC is stable. The problem can be further constrained by requiring the matrix structure to be block diagonal, the blocks to have a bounded norm, or the blocks to be identical. This problem was shown to be NP -hard in (Blondel and Tsitsiklis 1994) as a by-product of the Stable matrix problem. 5.2.3. Simultaneous Stabilization by Static Output Feedback

The problem of simultaneous stabilization of k given linear systems with a possibly LTI dynamic compensator is open for k > 2 (Blondel 1994). In fact, using real complexity theory, Blondel showed that the problem is rationally undecidable. However, restricting the stabilizing compensator to be static (or dynamic but of a given order) makes the problem decidable using the Tarski-Seidenberg approach. Unfortunately, it was shown by  (Blondel and Tsitsiklis 1994 ) and by (Toker and Ozbay 1995) that given (Ai ; Bi ; Ci ); i = 1;


N the problem of nding a real matrix K such that Ai + Bi KCi ; i = 1;


N is stable is NP -Hard. The problem however may be solved using LMIs (Boyd et al. 1994) if Ci = I and Bi = B and is therefore in P . 5.2.4. Dynamic Output Feedback

The case where a dynamical output compensator of order nf  n is used may be brought back to the static output feedback case as follows: Suppose the dynamic compensator is given in state-space form as

x_ f (t) = Af xf (t) + Bf y(t) (5.1) u(t) = Cf xf (t) + Df y(t) + v(t) (5.2) Then, an augmented state-system is obtained when uf (t) = x_ f (t), and yf (t) = xf (t) by  x_ (t)   A 0  x(t)   0 B  u (t)  f x_ f (t) = 0 0 xf (t) + I 0 u(t)  y (t)   0 I  x(t)  f = (5.3) y(t) C 0 xf (t) so that the feedback law is now static and given by

 u (t)   A B  y (t)   0  f f = f f + v(t) u(t)

Cf Df

y(t)

I

or in a more compact description x~_ (t) = A~x~(t) + B~ u~(t); y~(t) = C~ x~(t); u~(t) = K~ y~(t) + v~(t)

(5.4) (5.5)

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V. Blondel, C.T. Abdallah and G.L. Heileman

where

x~ = A~ = C~ =

0 y  u   x  f f ; u~ = u ; y~ = y ; v~ = v  Axf 0  0 B 

0 0 ;  0 I ; C 0

B~ = I 0 A B ~ K = Cf Df f f



(5.6)

Therefore, the problem of stabilizing a given system using a compensator of a given order is equivalent to that of stabilizing another system using static output feedback. The comments of the previous section therefore apply. 5.2.5. Minimum-order Controller Design

In general, it is known that for a minimal system of order n (i.e. controllable and observable), and with p outputs, a stabilizing compensator of order nf = n ? p will always exist. It is however unknown whether a lower order stabilizing compensator exist. In fact, the solution of the previous problem ( nding a stabilizing compensator of a given-order) may be used to solve the minimum-order problem by simply decreasing the order of the compensator from n ? p until the compensator can no longer stabilize. The two problems are therefore equivalent and related to the SOF problem. 5.3.  Analysis

In (poljak and Rohn 1993), t was shown that the recognition problem of the structured singular value in NP -hard if the uncertainty is purely real. In other words, in order to answer the question: Is   q, for a given q is NP -hard by reduction from the MaxCut problem. In (Braatz et. al. 1994), it was shown that the recognition problem of the structured singular value in NP -hard if the uncertainty is either real or a combination of real and complex parts. In (Young 1993), it was shown that the rank one mixed  problem becomes easy (in P ) and is related to Kharitonov-like results. Similar results also appear in (Coxson 1993). Finally, Toker proved in (Toker 1995b) that the recognition problem for purely complex uncertainty remains NP -hard by reduction from the Knapsack problem.

6. Open Questions

We list here two open questions as collected by Toker on http://eewww.eng.ohiostate.edu/ tokero/comp.html 1 Is the \Static Output Feedback" problem NP -hard? 2 Is the \Simultaneous Stabilization" problem decidable?

7. Conclusions

As discussed in this overview paper, many control problems can be phrased as decision problems. Such problems can be solved in principle using the methods of quanti er

Complexity Issues and Decision Methods in Control Systems

11

elimination pioneered by Tarski and are currently available in software packages. Unfortunately, the computational complexity of such an approach is prohibitive as the size of the problem grows beyond simple academic examples. Moreover, a decision procedure may not be the best approach to study a given control existence problem, as witnessed in the LQR approach. Moreover, even when a decision procedure is used, the special features of a particular problem (rank one case for  and matrix stability results) should be exploited.

References

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