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## Control and verification of high-dimensional systems via DSOS and SDSOS optimization (2014)

Venue: | In Proceedings of the 53rd IEEE Conference on Decision and Control |

Citations: | 1 - 1 self |

### Citations

1408 |
Integer and combinatorial optimization
- Nemhauser, Wolsey
- 1988
(Show Context)
Citation Context ...ndustry-motivated field of integer programming, the cutting-plane approaches used on real-life problems are almost exclusively based on linear programming (LP) or second order cone programming (SOCP) =-=[14]-=-, [15]. Even though semidefinite cuts are known to be stronger, they are too expensive to be used even at the root node of branchand-bound techniques for integer programming. In the field of SOS optim... |

384 | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization
- Parrilo
- 2000
(Show Context)
Citation Context ...sublevel set of V (x) is an inner approximation of the ROA. We choose our Lyapunov function to be the cost-to-go function V (x) = xTSx of the LQR controller and attempt to maximize ρ. As described in =-=[2]-=-, under the assumption that the Hessian of V̇ (x) is positive definite at the origin, the following is a sufficient condition for (2): (xTx)(V (x)− ρ) + L(x)V̇ (x) ≥ 0. (3) Here, L(x) is a “multiplier... |

250 |
Practical Methods for Optimal Control Using Nonlinear Programming
- Betts
- 2001
(Show Context)
Citation Context ... shown in Figure 4. An open-loop controller that swings up the system from the downright configuration to the upright one was designed using the direction collocation trajectory optimization approach =-=[27]-=-. A time-varying LQR controller was then designed to correct for deviations from this nominal trajectory. At the end of the trajectory, the robot switches to our balancing controller. We performed 30 ... |

244 | Second-order cone programming.
- Alizadeh, Goldfarb
- 2003
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Citation Context ... in Theorem 3.1 is equivalent to the diagonal elements Mii,Mjj , along with the determinant MiiMjj −M2ij , being nonnegative. This is a rotated quadratic cone constraint and can be imposed using SOCP =-=[20]-=-. IV. DSOS AND SDSOS POLYNOMIALS We now introduce naturally motivated cones that are inner approximations of POSn,d and that lend themselves to LP and SOCP. Definition 3 ( [1]): • A polynomial p is di... |

154 |
Some NP-complete problems in quadratic and nonlinear programming
- Murty, Kabadi
- 1987
(Show Context)
Citation Context ...that many questions in control theory can be posed as checks on nonnegativity of functions. However, checking nonnegativity is NP-hard even when the functions are restricted to the set of polynomials =-=[9]-=-. The SOS relaxation relies on the ability to efficiently check if a polynomial can be expressed as a sum of squares of other polynomials. This search for a sum of squares decomposition can be cast as... |

154 | Some concrete aspects of Hilbert’s 17th problem, in: Real Algebraic Geometry and Ordered Structures, in
- Reznick
- 2000
(Show Context)
Citation Context ...rform the semidefinite programs resulting from SOS. Example 4.1: Consider the polynomial, p(x) = x41x 2 2 + x42x 2 3 + x 4 3x 2 1 − 3x21x22x23. This polynomial is nonnegative but not a sum of squares =-=[21]-=-. However, there is an LPbased nonnegativity certificate since one can show that p ∈ 1-DSOS. Hence, 1-DSOS * SOS. The following two theorems provide asymptotic guarantees on r-dsos (and hence r-sdsos)... |

135 |
The swingup control problem for the acrobot
- Spong
- 1995
(Show Context)
Citation Context ...ign high degree polynomial feedback controllers using high degree Lyapunov functions for smaller systems with benefits in terms of running time as compared to SOS programming. We consider the Acrobot =-=[24]-=-, which is a benchmark for control of underactuated systems. The system is a special case of the N -link pendulum examined in Section V-A (with N = 2) and is actuated only at the joint between the two... |

89 | Safety verification of hybrid systems using barrier certificates - Prajna, Jadbabaie - 2004 |

76 |
System Identification Toolbox, for use with Matlab. The Mathworks Inc., 2006. FrB03.1 Avdelning, Institution Division, Department Division of Automatic Control Department of Electrical Engineering Datum Date 2010-03-03 Språk Language Svenska/Swedish Eng
- Ljung
(Show Context)
Citation Context ...ned with SDSOS programming (as described below). System identification for the hardware platform was performed using the prediction error minimization method in MATLAB’s System Identification Toolbox =-=[25]-=- in order to identify parameters of the model presented in [24]. The dynamics were then Taylor expanded around the equilibrium to degree 3 in order to obtain a polynomial vector field. We use the meth... |

35 | Local stability analysis using simulations and sum-of-squares programming
- Topcu, Packard, et al.
(Show Context)
Citation Context ...ty of problems including feedback control synthesis, safety verification and computation of regions of attraction, invariant sets, and reachable sets for a broad class of nonlinear and hybrid systems =-=[3]-=-–[8]. The key observation behind the approach is that many questions in control theory can be posed as checks on nonnegativity of functions. However, checking nonnegativity is NP-hard even when the fu... |

30 | Some controls applications of sum of squares programming
- Jarvis-Wloszek, Feeley, et al.
- 2003
(Show Context)
Citation Context ... problem (7) is not convex in general since it involves conditions that are bilinear in the decision variables. However, problems of this nature are common in the SOS programming literature (see e.g. =-=[4]-=-) and are typically solved by iteratively optimizing groups of decision variables. Each step in the iteration is then a DSOS program (or a SDSOS/SOS program if the constraints in (7) are replaced by D... |

24 | Convex computation of the region of attraction of polynomial control systems
- Henrion, Korda
(Show Context)
Citation Context ...f problems including feedback control synthesis, safety verification and computation of regions of attraction, invariant sets, and reachable sets for a broad class of nonlinear and hybrid systems [3]–=-=[8]-=-. The key observation behind the approach is that many questions in control theory can be posed as checks on nonnegativity of functions. However, checking nonnegativity is NP-hard even when the functi... |

18 |
Exploiting algebraic structure in sum of squares programs, Positive polynomials
- Parrilo
- 2005
(Show Context)
Citation Context ...tems of higher dimension have been addressed using SOS programming in certain cases, they involve exploiting special structure (e.g., symmetry, sparsity) of the particular problem under consideration =-=[10]-=-–[13]. For many real-world control applications, we would like to be able to handle problems of high dimensionality even when such structure is limited or not available. Further, even for smaller prob... |

13 | Control design along trajectories with sums of squares programming
- Majumdar, Ahmadi, et al.
- 2013
(Show Context)
Citation Context ...ntify parameters of the model presented in [24]. The dynamics were then Taylor expanded around the equilibrium to degree 3 in order to obtain a polynomial vector field. We use the method presented in =-=[26]-=- to design a balancing controller for the system. In particular, we search for a degree 8 Lyapunov function V (x) and a degree 3 feedback controller u(x) in order to maximize the size of the region of... |

9 | Algebraic relaxations and hardness results in polynomial optimization and Lyapunov analysis - Ahmadi - 2011 |

7 | Lower bounds for polynomials using geometric programming
- Ghasemi, Marshall
- 2012
(Show Context)
Citation Context ...mial nonnegativity that are perhaps stronger than a SOS decompostion, but still provide useful solutions for various control applications. A previous approach that has a similar spirit is the work in =-=[18]-=-, which tackles the specific task of finding a lower bound on the minimum of a polynomial using geometric programming (GP). However, these GP-based conditions seem to be too strong and we show in [1] ... |

6 |
A network decomposition approach for efficient sum-of-squares programming based analysis
- Anderson, Papachristodoulou
- 2010
(Show Context)
Citation Context ...ructure, such as sparsity or symmetry of the underlying polynomials, to reduce the size of the SDPs. Examples include applications to network problems where connectivity information is known a priori =-=[11]-=-, dynamical systems that can be decomposed and analyzed using smaller subsystems [12], and analysis of delayed linear systems with a low-rank delay coefficient matrix [13]. Another approach which also... |

6 |
50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-art
- Jünger, Liebling, et al.
- 2010
(Show Context)
Citation Context ...y-motivated field of integer programming, the cutting-plane approaches used on real-life problems are almost exclusively based on linear programming (LP) or second order cone programming (SOCP) [14], =-=[15]-=-. Even though semidefinite cuts are known to be stronger, they are too expensive to be used even at the root node of branchand-bound techniques for integer programming. In the field of SOS optimizatio... |

6 |
Systems polynomial optimization tools (SPOT), available online: http://web.mit.edu/ameg/www
- Megretski
- 2010
(Show Context)
Citation Context ...mality of our approach with the SOS approach in cases where this is possible (i.e., smaller instances of problems). A software package written using the Systems Polynomial Optimization Toolbox (SPOT) =-=[22]-=- includes a complete implementation of the presented methods and is available online1. The toolbox features very efficient polynomial algebra and allows us to setup the large-scale LPs and SOCPs arisi... |

6 | Complexity of ten decision problems in continuous time dynamical systems,
- Ahmadi, Majumdar, et al.
- 2013
(Show Context)
Citation Context ...rocessors with a clock speed of 3.4 GHz and 16 GB RAM. A. Regions of Attraction In our first example, we consider the computation of regions of attraction (ROA), which is known to be a NPhard problem =-=[23]-=-. The system we examine is the N -link pendulum depicted in Figure 2. This system has 2N states x = [θ1, . . . , θN , θ̇1, . . . , θ̇N ] composed of the joint angles and their derivatives. There are N... |

5 | Interior-point algorithms for sum-of-squares optimization of multidimensional trigonometric polynomials
- Roh, Dumitrescu, et al.
- 2007
(Show Context)
Citation Context ... this direction include the work in [16], which proposes a method for solving large scale robust stability problems in a parallel computing environment with a customized interior point solver, and in =-=[17]-=-, which offers a customized interior point algorithm for optimizing over the set of nonnegative trigonometric polynomials. The approach we take in this paper for enhancing scalability is orthogonal to... |

5 |
On factor width and symmetric H-matrices, Linear Algebra and its
- Boman, Chen, et al.
- 2005
(Show Context)
Citation Context ...atrix D such that AD (or equivalently, DAD) is dd. The set of n×n sdd matrices will be denoted by SDDn. We note that sdd matrices are sometimes referred to as generalized diagonally dominant matrices =-=[19]-=-. Remark 3.2: The fact that diagonal dominance is a sufficient condition for positive semidefiniteness follows directly from Gershgorin’s circle theorem. This also implies that sdd matrices are psd si... |

4 | Dynamical system decomposition for efficient, sparse analysis - Anderson, Papachristodoulou - 2010 |

3 |
Solving large-scale robust stability problems by exploiting the parallel structure of Polya’s theorem
- Kamyar, Peet, et al.
- 2013
(Show Context)
Citation Context ... which also holds promise has been to design customized solvers for special classes of SDPs and avoid resorting to off-the-shelf interior point solvers. Examples in this direction include the work in =-=[16]-=-, which proposes a method for solving large scale robust stability problems in a parallel computing environment with a customized interior point solver, and in [17], which offers a customized interior... |

2 |
DSOS and SDSOS optimization: More tractable alternatives to SOS optimization
- Ahmadi, Majumdar
- 2014
(Show Context)
Citation Context ...le SDPs have many appealing features, current SDP solvers do not approach the scalability or numerical maturity of LP and SOCP solvers. Our approach is based on the recent work of Ahmadi and Majumdar =-=[1]-=-, which replaces the positive semidefiniteness constraint inherent in the SOS approach with stronger conditions based on diagonal dominance and scaled diagonal dominance. This leads to the DSOS and SD... |

2 | editorial: Special issue on positive polynomials in control. Automatic Control - Chesi, Henrion, et al. |

1 |
Reducing the complexity of the sum-of-squares test for stability of delayed linear systems
- Zhang, Peet, et al.
- 2011
(Show Context)
Citation Context ...of higher dimension have been addressed using SOS programming in certain cases, they involve exploiting special structure (e.g., symmetry, sparsity) of the particular problem under consideration [10]–=-=[13]-=-. For many real-world control applications, we would like to be able to handle problems of high dimensionality even when such structure is limited or not available. Further, even for smaller problems,... |