## Control applications of nonlinear convex programming (1998)

Venue: | the 1997 IFAC Conference on Advanced Process Control |

Citations: | 7 - 3 self |

### BibTeX

@INPROCEEDINGS{Boyd98controlapplications,

author = {Stephen Boyd and Cesar Crusius and Anders Hansson},

title = {Control applications of nonlinear convex programming},

booktitle = {the 1997 IFAC Conference on Advanced Process Control},

year = {1998},

pages = {5--6}

}

### OpenURL

### Abstract

Since 1984 there has been a concentrated e ort to develop e cient interior-point methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interior-point methods (beyond their e ciency for LP): they extend gracefully to nonlinear convex optimization problems. New interior-point algorithms for problem classes such as semide nite programming (SDP) or second-order cone programming (SOCP) are now approaching the extreme e ciency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to e ciently solve nonlinear convex optimization problems opens up new applications. In the rst example we show how SOCP can be used to solve robust open-loop optimal control problems. In the second example, we show how SOCP can be used to simultaneously design the set-point and feedback gains for a controller, and compare this method with the more standard approach. Our nal application concerns analysis and synthesis via linear matrix inequalities and SDP. Submitted to a special issue of Journal of Process Control, edited by Y. Arkun & S. Shah, for papers presented at the 1997 IFAC Conference onAdvanced Process Control, June 1997, Ban. This and related papers available via anonymous FTP at

### Citations

904 |
Linear Programming and Extensions
- Dantzig
- 1963
(Show Context)
Citation Context ...e vector x is the optimization variable and ai, bi, c, F , and g are problem parameters. Linear programming has been used in a wide variety of elds since Dantzig introduced the simplex method in 1948 =-=[Dan63]-=-. In control, for example, Zadeh and Whalen observed in 1962 that certain minimum-time and minimum-fuel optimal control problems could be (numerically) solved by linear programming [ZW62]. In the late... |

687 | A new polynomial-time algorithm for linear programming
- Karmarkar
- 1984
(Show Context)
Citation Context ...L97]. In 1984, Karmarkar introduced a new interior-point algorithm for linear programming which was more e cient than the simplex method in terms of worstcase complexity analysis and also in practice =-=[Kar84]-=-. This event spurred intense research ininterior-point methods, which continues even now. In the last year or so several books have been written on interior-point methods for linear programming, e.g.,... |

505 | Primal-Dual Interior-Point Methods
- Wright
- 1997
(Show Context)
Citation Context ...nt spurred intense research ininterior-point methods, which continues even now. In the last year or so several books have been written on interior-point methods for linear programming, e.g., byWright =-=[Wri97]-=- and Vanderbei [Van97]. Moreover, several high quality, e cient implementations of interior-point LP solvers have become available (see, e.g., [Van92,Zha94,CMW97,GA94,MR97]). The most obvious advantag... |

499 | Interior point methods in semidefinite programming with applications to combinatorial optimization - Alizadeh - 1995 |

295 | Robust convex optimization
- Ben-Tal, Nemirovski
- 1998
(Show Context)
Citation Context ...e system. The observation that some robust convex optimization problems can be solved using new interior-point methods has been made recently by El Ghaoui and Lebret [EL96] and Ben Tal and Nemirovski =-=[BTN96]-=-. In this section we have considered a speci c example, using a peak tracking error criterion, but the method is quite general. It can, for example, be used to nd the worst case quadratic cost (see Lo... |

263 |
Interior point polynomial methods in Convex Programming: theory and applications
- Nesterov, Nemirovski
- 1994
(Show Context)
Citation Context ...nonlinear convex optimization problems, whereas the simplex method, which is based very much on the linear structure of the problem, does not. This observation was made rst by Nesterov and Nemirovski =-=[NN94]-=-, who developed a general framework for interior-point methods for a very wide variety of nonlinear convex optimization problems. Interior-point methods for nonlinear convex optimization problems have... |

249 | Control System Synthesis: A Factorization Approach - Vidyasagar - 1985 |

183 | Determinant maximization with linear matrix inequality constraints - Vandenberghe, Boyd, et al. - 1998 |

156 | Linear programming: Foundations and extensions
- Vanderbei
- 1997
(Show Context)
Citation Context ...earch ininterior-point methods, which continues even now. In the last year or so several books have been written on interior-point methods for linear programming, e.g., byWright [Wri97] and Vanderbei =-=[Van97]-=-. Moreover, several high quality, e cient implementations of interior-point LP solvers have become available (see, e.g., [Van92,Zha94,CMW97,GA94,MR97]). The most obvious advantages of interior-pointov... |

152 | Linear ‘Controller Design Limits of Performance
- Boyd, Barratt
- 1991
(Show Context)
Citation Context ...the exposition we consider the static case, i.e., the case in which all signals are vectors that do not change in time. The method discussed here does, however, extend to the dynamic case; see, e.g., =-=[BB91]-=-. For some other references on static feedback control, see, e.g., [SP96,KKB94]. The system we consider is shown in gure 6. The signal w 2 R nw is the exogenous input, which might include command sign... |

144 | System analysis via integral quadratic constraints - Megretski, Rantzer - 1997 |

135 | Model predictive control: Past, present and future
- Morari, Lee
- 1999
(Show Context)
Citation Context ...ing horizon control), in which linear or quadratic programs are used to solve an optimal control problem at each time step. Model predictive control is now widely used in the process control industry =-=[ML97]-=-. In 1984, Karmarkar introduced a new interior-point algorithm for linear programming which was more e cient than the simplex method in terms of worstcase complexity analysis and also in practice [Kar... |

92 | A primal-dual potential reduction method for problems involving matrix inequalities - Vandenberghe, Boyd - 1995 |

64 | Robust performance of linear parametrically varying systems using parametrically–dependent linear feedback - Becker, Packard - 1994 |

61 | A linear matrix inequality approach to H1 control - Gahinet, Apkarian - 1994 |

56 |
Industrial applications of model based predictive control
- Richalet
- 1993
(Show Context)
Citation Context ...or example, Zadeh and Whalen observed in 1962 that certain minimum-time and minimum-fuel optimal control problems could be (numerically) solved by linear programming [ZW62]. In the late 70s, Richalet =-=[Ric93]-=- developed model predictive control (also known as dynamic matrix control or receding horizon control), in which linear or quadratic programs are used to solve an optimal control problem at each time ... |

35 | Semide nite programming - Vandenerghe, Boyd - 1996 |

34 | Robust model predictive control - Campo, Morari - 1987 |

34 | Interior point methods for optimal control of discrete time systems - Wright - 1993 |

31 | Applying new optimization algorithms to model predictive control - Wright - 1996 |

30 | PCx user guide - Czyzyk, Mehrotra, et al. - 1996 |

23 | LOQO user’s manual - Vanderbei - 2000 |

21 | Ecient convex optimization for engineering design - Boyd, Vandenberghe, et al. - 1994 |

19 | Explicit controller formulas for lmi-based H1 synthesis - Gahinet - 1996 |

17 | Robust Stability of Constrained Model Predictive Control - Zheng, Morari - 1993 |

14 | Interior point methods in semide nite programming with applications to combinatorial optimization - Alizadeh - 1995 |

14 | sdpsol: A Parser/Solver for Semide nite Programming and Determinant Maximization Problems with Matrix Structure. User's Guide, Version Beta - Wu, Boyd - 1996 |

13 | On linear programming and robust modelpredictive control using impulse response, Systems and control - Allwright, Papavasiliou - 1992 |

12 | LMI Lab: A Package for Manipulating and Solving LMIs. INRIA - Gahinet, Nemirovskii - 1993 |

10 | SOCP – Software for secondorder cone programming. User’s guide - Lobo, Vandenberghe, et al. - 1997 |

10 | A.Packard, “Optimal control of perturbed linear static systems - Smith - 1996 |

9 | sp: Software for Semide nite Programming. User's Guide, Beta Version - Vandenberghe, Boyd - 1994 |

8 |
Second-order cone programming: interior-point methods and engineering applications. Linear Algebra and Appl
- Lobo, Vandenberghe, et al.
- 1997
(Show Context)
Citation Context ...es the Euclidean norm, i.e., kzk = p z T z. SOCPs include linear and quadratic programming as special cases, but can also be used to solve a variety of nonlinear, nondi erentiable problems; see, e.g.,=-=[LVBL97]-=-. Moreover, e cient interior-point software for SOCP is now available [LVB97,AHN + 97]. In this paper we will examine three areas of control where our ability to numerically solve nonlinear, nondi ere... |

7 | sdppack User’s Guide, Version 0.9 Beta. NYU - Alizadeh, Haeberly, et al. - 1997 |

7 |
On optimal control and linear programming
- Zadeh
- 1962
(Show Context)
Citation Context ...hod in 1948 [Dan63]. In control, for example, Zadeh and Whalen observed in 1962 that certain minimum-time and minimum-fuel optimal control problems could be (numerically) solved by linear programming =-=[ZW62]-=-. In the late 70s, Richalet [Ric93] developed model predictive control (also known as dynamic matrix control or receding horizon control), in which linear or quadratic programs are used to solve an op... |

5 | SDPA (Semide nite Programming Algorithm) { User's Manual - Fujisawa, Kojima, et al. - 1999 |

3 | sdppack User's Guide, Version 0.8 Beta. NYU - Alizadeh, Haeberly, et al. - 1997 |

3 |
Complexity of elementary hybrid systems
- Blondel, Tsitsiklis
- 1997
(Show Context)
Citation Context ... T Qx(t) dt where the maximum (supremum) is taken over all trajectories of (21). Even the question whether J wc < 1, i.e., whether the system is stable, is interesting (and challenging; it is NP-hard =-=[BT97]-=-). Before proceeding let us mention one reason why we might be interested in the linear time-varying system described above. Suppose we have a nonlinear, 19stime-varying system where f(0;t)=0and for a... |

2 | A Convex Characteristization of GainScheduled H1 Controllers - APKARIAN, GAHINET - 1995 |

2 | Robust Least Squares and Applications
- Ghaoui, Lebret
- 1996
(Show Context)
Citation Context ... or `drives' the uncertainty in the system. The observation that some robust convex optimization problems can be solved using new interior-point methods has been made recently by El Ghaoui and Lebret =-=[EL96]-=- and Ben Tal and Nemirovski [BTN96]. In this section we have considered a speci c example, using a peak tracking error criterion, but the method is quite general. It can, for example, be used to nd th... |

2 | Improving static performance robustness of thermal processes - Kabuli, Kosut, et al. - 1994 |

2 | User's guide to LIPSOL: a matlab toolkit for linear programming interior-point solvers - Zhang - 1994 |

1 | csdp, a C library for semide nite programming - Borchers - 1997 |