V. Blondel and J. N. Tsitsiklis, "NP-hardness of some linear control design problems," SIAM J. on Control and Optimization, vol. 35, no. 6, pp. 2118--2127, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... Many classical problems of robustness and stability in control theory aim to move the eigenvalues of a parametrized matrix into some prescribed domain of the complex plane (see [7] and [9] for instance) A particularly important example ( perhaps the most basic problem in control theory [3]) is stabilization by static output feedback: given matrices A, B, and C , is there a matrix K such that the matrix A BKC is stable (i.e. has all its eigenvalues in the left half plane) In a 1995 survey [2] experts in control systems theory described the characterization of those triples (A, ....

....(i.e. has all its eigenvalues in the left half plane) In a 1995 survey [2] experts in control systems theory described the characterization of those triples (A, B, C) allowing such a K as a major open problem. With interval bounds on the entries of K , the problem is known to be NP hard [3]. The static output feedback problem is a special case of the general problem of choosing a linearly parametrized matrix X so that its eigenvalues are as far into the left half plane as possible. This optimizes the asymptotic decay rate of the corresponding dynamical system u Xu (ignoring ....

V. Blondel and J. N. Tsitsiklis, NP-hardness of some linear control design problems, SIAM J. Control Optim., 35 (1997), 2118--2127.

.... of the symmetric part of the matrix (minimizing the initial growth rate of the associated system) Regarding the rst of these extremes, optimization of the spectral abscissa: variational analysis of this non Lipschitz function is well understood [15, 12] global optimization is known to be hard [5], and some progress has been made in local optimization methods [14] Regarding the second extreme: optimization of this convex function over a polyhedral feasible set is a semide nite programming problem, and the global minimum can be found by standard methods [4, 38] Intermediate choices of ....

.... , which is the desired contradiction. 2 We see from this result that, for example, if the set F is a polyhedron, then the limiting version of the optimization problem inf F as 1 is a computationally straightforward, convex minimization problem, whereas when = 0 the problem may be hard [5]. The idea of the H1 norm of a transfer matrix is also closely related to the complex stability radius. Consider the linear time invariant dynamical system p = Ap u; where p denotes the state vector (in this simple case coinciding with the output) and u denotes the input vector. The ....

V. Blondel and J.N. Tsitsiklis. NP-hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35:2118-2127, 1997.

....O(jx 1 j 1=2 ) though still non Lipschitz. This simple example illustrates how challenging spectral abscissa minimization is in general, even locally. More complicated spectral abscissa minimization problems often arise in control applications. A particularly important example (described in [BT97] as perhaps the most basic problem in control theory ) is stabilization by output feedback : given an n n matrix A, an n r matrix B and an s n matrix C, nd (if possible) an r s matrix K such that C is stable, i.e. has all its eigenvalues in the left half of the complex ....

....known only for very special cases of this problem, such as r = n; B = I or s = n; C = I, when the problem reduces to standard pole placement [Bar84] However, in general, this problem is hard. When interval bounds on the entries of K are speci ed, the general problem is known to be NP hard [BT97, Nem93]. The complexity of the general problem without bounds on K is not known. Since the map K 7 A C is ane in K, stabilization by output feedback can be expressed as a spectral abscissa global minimization problem. In a 1995 survey [BGL95] experts in control theory described the ....

V. Blondel and J.N. Tsitsiklis. NP-hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35:2118-2127, 1997.

....equalities, the problem in general is very hard. The reason is that the objective function, i.e. the spectral radius of a matrix, is in general not a convex function; indeed it is not even Lipschitz continuous (see, e.g. 33] Some related spectral radius minimization problems are NP hard [6, 27]. We can also formulate the FDLA problem, with per step convergence factor, as the following spectral norm minimization problem: minimize IlW 11T II subject to W,S, lrW 1 r, W1 1. In contrast to the spectral radius formulation (13) this problem is convex, and can be solved efficiently and ....

....we seek an edge weight vector with as many as zero entries as possible, subject to a prescribed maximum for the convergence factor. This is a difficult combinatorial problem, but one very effective heuristic to achieve this goal is to minimize the 1 norm of the vector of edge weights; see, e.g. [11, 6] and [17] For example, given the maximum allowed asymptotic convergence factor r TM, the 1 heuristic for the sparse graph design problem (with symmetric edge weights) can be posed as the convex problem minimize 1 Iwl (29) subject to r axI I A diag(w)n 11T n rnxI. It is also possible ....

V. Blondel and J. N. Tsitsiklis. NP-hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35:2118 2127, 1997.

....rate of at least a. The goal is to design the feedback gain matrLx 5K C R x such that the control law u 5try gives an additional damping of 5a in the closed loop system, while the controller gains satis the interval constraints for i 1, m, and j 1, n. This problem is known to be NP hard [12]. By simple Lyapunov theory (see, e.g. 13] this problem is equivalent to the existence of P G SR x such that ( 5C) 5C) which is a BMI in the variables P and The linearization method for solving the BMI (1) can be explained as follows. Since the open loop system has a decay ....

V.D. Blondel and J. N. Tsitsiklis. NP-hardness of some linear control design problems. SIAM J. on Control and Optimization, 35:2118 27, 1997.

....switching signal can still be implemented with minor modifications. Observe that the converse result of Proposition 12 is only known to hold for the case of two systems. We note that the problem of identifying stable convex combinations (of matrices with rational coefficients) is NP hard [61]. A discussion of computational issues associated with some problems related to the one addressed in this section, as well as relevant bibliography, can be found in Chapters 11 and 14 of [62] Unstable Convex Combinations The previous example suggests that even when there exists no stable convex ....

V. Blondel and J. N. Tsitsiklis, "NP-hardness of some linear control design problems," SIAM J. Control Optim., vol. 35, pp. 2118-2127, 1997.

....service types. The problem of finding a set of feasible SINR levels that are somehow optimal, in the sense that they are as high as possible considering the priorities of each service type, has been proposed in [8] This problem is closely related to the Stable Rank One Perturbed Matrix problem [1], except that it refers to discrete time stability and not continuous time stability. Henceforth, we denote the discrete time problem as Power P. One method to show that Power P is NP hard is by using the problem Stable Matrix In Unit Interval Family problem [1] for discrete time, hereafter ....

....One Perturbed Matrix problem [1] except that it refers to discrete time stability and not continuous time stability. Henceforth, we denote the discrete time problem as Power P. One method to show that Power P is NP hard is by using the problem Stable Matrix In Unit Interval Family problem [1] for discrete time, hereafter denoted SMIUIF. The principal undertaking of this paper is to show that SMIUIF is NP hard. This problem, in turn, can be used to show NP hardness for some communications problems, as discussed in Section 3, including the problem mentioned above, Power P. We shall ....

[Article contains additional citation context not shown here]

V. Blondel and J. Tsitsiklis. "NP-Hardness of Some Linear Control Design Problems ". SIAM Journal on Control Optimization, 35(6), pp. 2118-2127, November 1997

....There exists a switching sequence such that system (4.1) is quadratically stable iff there exists such that (4.4) is a stability matrix, i.e. eigenvalues of lie in the open left half complex plane. Conditions for finding are in [35] with the caveat that general convex combinations are NP hard [99], 100] A brute force approach is simply to plot the eigenvalues of for . When exists and is 1074 PROCEEDINGS OF THE IEEE, VOL. 88, NO. 7, JULY 2000 known, then for arbitrary , one can construct [79] 86] satisfying i.e. for appropriate switching , satisfies condition (4.3) above. In order ....

....of is less than one. The condition given in [20, Theorem 1] follows. For all practical purposes, this pulse width modulation scheme results in an average control that asymptotically stabilizes system (4.1) Unfortunately, finding the convex combination of (4. 9) even when it exists, is NP hard [99], 100] Moreover, there is a large class of systems for which (4.9) is never satisfied, yet there exists a stabilizing . Example 3 with the , given by (1.3a) is one such example: the system is stabilizable but no common exists. General structural conditions on the matrices that guarantee ....

[Article contains additional citation context not shown here]

V. Blondel and J. Tsitsiklis, "NP-hardness of some linear control design problems," SIAM J. Contr. Optimiz., vol. 35, no. 6, pp. 2118--2127, 1997.

....apply these results to obtain more efficient algorithms which probabilistically guarantee stability and robustness levels when designing robust controllers for uncertain systems. 1 Introduction It has recently become clear that many control problems are too difficult to admit analytic solutions [2]. New results have also emerged to show that the computational complexity of some solved control problems is prohibitive [3] Many of these linear and nonlinear control problems can be reduced to decidability problems or to optimization questions, both of which can then be reduced to the ....

....algorithms and exploiting the fact that the sample complexity is itself a random variable. This has allowed us to present Algorithm 1 as an efficient design methodology for fixed order robust control design problems [11] Recall for example that the Static Output Feedback (SOF) was shown in [2] to be NP hard when the gains of the feedback matrix were bounded, but that Algorithm 1, is well suited to address the SOF problem exactly under those conditions. It should be noted that the methodology presented in this paper can be used in many other application areas: one only needs to have an ....

V. Blondel and J. Tsitsiklis. NP-hardness of some linear control design problems. SIAM Journal of Control and Optimization, 35:2118--2127, 1997.

....have many potential applications in stability and control theory. If the domain of the semistable program is restricted to the Hermitian matrices, the problem reduces to a semide nite program. Semistable programs are, of course, not convex, and it is known that nding the global maximum is NP hard [2,21]. However, local optimality conditions may be addressed by means of the analysis developed in this paper. Here, we give a rst order necessary condition for local optimality. Other optimality conditions may also be derived but we leave these for future work. Theorem 10.1 (First order necessary ....

V. Blondel and J.N. Tsitsiklis. NP-Hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35:2118-2127, 1997.

....control, it is not surprising that many problems for such systems are known or conjectured to be computationally intractable (NP hard) or even undecidable. For example, the stability problem of a linear time varying system (which is a simple asynchronous system) cannot be solved in polynomialtime [1]. Therefore, it is very unlikely to formulate control problems for asynchronous systems exactly as computationally e#cient (polynomial time) optimization problems. It is expected, however, to develop some semi heuristic methods that are very e#ective on certain types of problems. By semi heuristic ....

V. D. Blondel and J. N. Tsitsiklis. NP-hardness of some linear control design problems. SIAM J. on Control and Optimization, pages 2118--27, 1997.

....switching signal can still be implemented with minor modifications. Observe that the converse result of Proposition 12 is only known to hold for the case of two systems. We note that the problem of identifying stable convex combinations (of matrices with rational coefficients) is NP hard [61]. A discussion of computational issues associated with some problems related to the one addressed in this section, as well as relevant bibliography, can be found in Chapters 11 and 14 of [62] Unstable Convex Combinations The previous example suggests that even when there exists no stable convex ....

V. Blondel and J. N. Tsitsiklis, "NP-hardness of some linear control design problems," SIAM J. Control Optim., vol. 35, pp. 2118--2127, 1997.

....#### ######### ########### ### ######### ######## ######## ########### ######### ########## ########## ###### ######## ###### ########### ####### ######### ############ ####### I. Introduction It has recently become clear that many control problems are too dicult to admit analytic solutions [12] [15], 17] 56] New results have also emerged to show that the computational complexity of some solved control problems is prohibitive [16] 24] 61] Many of these (linear and nonlinear) control problems can be reduced to decidability problems or to optimization questions [10] both of which ....

V. Blondel and J. Tsitsiklis. NP-hardness of some linear control design problems. SIAM Journal of Control and Optimization, 35:2118-2127, 1997.

....algorithms and exploiting the fact that the sample complexity is itself a random variable. This has allowed us to present Algorithm 3 as an ecient design methodology for xed order robust control design problems [43] Recall for example that the Static Output Feedback (SOF) was shown in [14] to be NP hard when the gains of the feedback matrix were bounded, but that Algorithm 3, is well suited to address exactly the SOF problem under those conditions. It should be noted that the methodology presented in this paper can be used in many other application areas: one only needs to have an ....

V. Blondel and J. Tsitsiklis. NP-hardness of some linear control design problems. SIAM J. of Control and Opt., 35(6):2118-2127, 1997.

....z Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A. Email: overton cs.nyu.edu. Research supported by NSF, DOE. 1 of the complex plain (see [7] and [9] for instance) A particularly important example ( perhaps the most basic problem in control theory [3]) is stabilization by static output feedback: given matrices A, B, and C, is there a matrix K such that the matrix A BKC is stable (that is, has all its eigenvalues in the left halfplane) In a 1995 survey [2] experts in control systems theory described the characterization of those triples (A; ....

....(that is, has all its eigenvalues in the left halfplane) In a 1995 survey [2] experts in control systems theory described the characterization of those triples (A; B;C) allowing such a K as a major open problem . With interval bounds on the entries of K, the problem is known to be NP hard [3]. The static output feedback problem is a special case of the general problem of choosing a linearly parametrized matrix X so that its eigenvalues are as far into the left halfplane as possible. This optimizes the asymptotic decay rate of the corresponding dynamical system u = Xu (ignoring ....

V. Blondel and J.N. Tsitsiklis. NP-hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35:2118-2127, 1997.

....complexity analysis of the problem is not quite complete yet. Anderson, Bose and Jury have shown in [1] that the problem Internet: elghaoui ensta.fr. y Work supported by DRET scholarship. z Work supported by DRET grant #92017 BC 14. 1 is decidable. A nice result of Blondel and Tsitsiklis [3] shows that the problem of finding a static output feedback controller with prespecified bounds on the controller matrix K is NP hard. Of course, this does not prove that the SOF problem is NP hard, since no a priori bounds on the controller matrix K are imposed. A number of numerical procedures ....

V. Blondel and J. N. Tsitsiklis. NP-hardness of some linear control design problems. European Jour. Control, 1, 1995.

....stability is usually associated with the left half plane. If the domain of the semistable program is restricted to the Hermitian matrices, the problem reduces to a semide nite program. Semistable programs are, of course, not convex, and it is known that nding the global minimum is NP hard [BT97, Nem93]. However, local optimality conditions may be addressed by means of the analysis developed in this paper. Here, we give a rst order necessary condition for local optimality. Other optimality conditions may also be derived but we leave these for future work. 28 J.V. BURKE AND M.L. OVERTON ....

V. Blondel and J.N. Tsitsiklis. NP-Hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35:2118-2127, 1997.

....slower, O( x 1 ) though still non Lipschitz. This simple example illustrates how challenging spectral abscissa minimization is in general, even locally. More complicated spectral abscissa minimization problems often arise in control applications. A particularly important example (described in [11] as perhaps the most basic problem in control theory ) is stabilization by output feedback: given an C, find (if possible) an r s matrix K such that C is stable, i.e. has all its eigenvalues in the left half of the complex plane. Efficient algorithms are known only for very special cases ....

....algorithms are known only for very special cases of this problem, such as r I or s C I , when the problem reduces to standard pole placement [3] However, in general, this problem is hard. When interval bounds on the entries of K are specified, the general problem is known to be NP hard [11,23]. The complexity of the general problem without bounds on K is not known. Since the map K # # C is affine in K, stabilization by output feedback can be expressed as a spectral abscissa global minimization problem. In a 1995 survey [5] experts in control theory described the characterization ....

V. Blondel, J.N. Tsitsiklis, NP-hardness of some linear control design problems, SIAM J. Control Optim. 35 (1997) 2118--2127.

....the family contain a stable matrix This problem is NP hard. It remains so even if the interval family is symmetric and all the entries of the family are fixed, except for some of the entries of a single row and a single column, which can take values between 1 and 1. The proof is given in [Blondel and Tsitsiklis, 1997a] and involves again a reduction of the partition problem. In contrast to the corresponding analysis question discussed in Section 3.2, it is not known whether stable matrix in interval family is NP hard in the strong sense. A variation of stable matrix in interval family is the problem in which ....

.... the problem with interval constraints remains NP hard for the special case of state feedback (given A and B, determine whether there exists some K within a given interval family such that A BK is stable) even though it can be solved in polynomial time in the absence of interval constraints [Blondel and Tsitsiklis, 1997a] Let us also mention some related problems that have been shown to be NP hard in the same reference. These are the problems of simultaneous stabilization by output feedback (see also [Toker and Ozbay, 1995] decentralized output feedback stabilization using a norm bounded controller, and ....

[Article contains additional citation context not shown here]

Blondel, V. D. and J. N. Tsitsiklis (1997). NPhardness of some linear control design problems, SIAM J. Control and Optimization, 35, 2118-2127.

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V. Blondel and J. N. Tsitsiklis, "NP-hardness of some linear control design problems," SIAM J. on Control and Optimization, vol. 35, no. 6, pp. 2118--2127, 1997.

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V. Blondel and J. N. Tsitsiklis. NP-hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35:2118--2127, 1997.

No context found.

V. Blondel, J. N. Tsitsiklis, NP-hardness of some linear control design problems, SIAM J. Control Optim., vol. 35, 1997, pp. 2118--2127.

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V. Blondel and J.N. Tsitsiklis. NP-Hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35:2118-2127, 1997.

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Blondel, V., Tsitsiklis, J.N. (1997): NP-Hardness of some linear control design problems. SIAM J. Control Optim. 35, 2118--2127

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V. Blondel and J.N. Tsitsiklis. NP-hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35:2118--2127, 1997.

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