Parametric optimization and optimal control using algebraic geometry methods

Fotiou, Ioannis A.; Rostalski, Philipp; Parrilo, Pablo A.; Morari, Manfred

doi:10.1080/00207170600726592

Cited by 70 publications

(45 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The initial set is a rectangle such that 2.49 ≤ x 1 ≤ 2.51 and 1.49 ≤ x 2 ≤ 1.51. The results obtained using the two methods are shown in Figure 1 which are coherent with the phase portrait in [8]. We can see that the method using a change of variables achieved better precision since the reachable set it computed is include in the set computed by the other method.…”

Section: A Control Systemsupporting

confidence: 65%

“…Note that the above functions to optimize are polynomials. This problem is computationally difficult, despite recent progress in the development of methods and tools for polynomial programming (see for example [8] and references therein). An alternative solution is to find their affine bound functions, in order to replace the polynomial optimization problem by a linear programming one, which can be solved more efficiently (in polynomial time) using well-developed techniques, such as Simplex and interior point techniques [21].…”

Section: Template Polyhedramentioning

confidence: 99%

“…The first example we present is the Duffing oscillator taken from [15,8]. This is a nonlinear oscillator of second order, the continuous-time dynamics is described byÿ (t) + 2ζẏ(t) + y(t) + y(t)…”

Section: A Control Systemmentioning

confidence: 99%

“…The damping coefficient ζ = 0.3. In [8], using a forward difference approximation with a sampling period h = 0.05 time units, this system is approximated by the following discrete-time model:…”

Section: A Control Systemmentioning

confidence: 99%

“…In [8], an optimal predictive control law u(k) was computed by solving a parametric polynomial optimization problem. We model this control law by the following switching law with 3 modes:…”

Section: A Control Systemmentioning

confidence: 99%

See 4 more Smart Citations

Image Computation for Polynomial Dynamical Systems Using the Bernstein Expansion

Dang

Salinas

2009

Computer Aided Verification

View full text Add to dashboard Cite

This paper is concerned with the reachability computation problem for polynomial dynamical systems. Such computations constitute a crucial component in algorithmic verification tools for hybrid systems and embedded software with polynomial dynamics, which have found applications in many engineering domains. We describe two methods for overapproximating reachable sets of such systems; these methods are based a combination of the Bernstein expansion of polynomial functions and a representation of reachable sets by template polyhedra. Using a prototype implementation, the performance of the methods was demonstrated on a number of examples of control systems and biological systems.

show abstract

Section: A Control Systemsupporting

confidence: 65%

Section: Template Polyhedramentioning

confidence: 99%

Section: A Control Systemmentioning

confidence: 99%

Section: A Control Systemmentioning

confidence: 99%

“…In [8], an optimal predictive control law u(k) was computed by solving a parametric polynomial optimization problem. We model this control law by the following switching law with 3 modes:…”

Section: A Control Systemmentioning

confidence: 99%

See 3 more Smart Citations

Image Computation for Polynomial Dynamical Systems Using the Bernstein Expansion

Dang

Salinas

2009

Computer Aided Verification

View full text Add to dashboard Cite

show abstract

Other Developments in Multi‐parametric Optimization

2020

Multi‐Parametric Optimization and Control

View full text Add to dashboard Cite

Explicit model predictive control of hybrid systems and multiparametric mixed integer polynomial programming

Charitopoulos

Dua

2016

AIChE Journal

View full text Add to dashboard Cite

Hybrid systems are dynamical systems characterized by the simultaneous presence of discrete and continuous variables. Model-based control of such systems is computationally demanding. To this effect, explicit controllers which provide control inputs as a set of functions of the state variables have been derived, using multiparametric programming mainly for the linear systems. Hybrid polynomial systems are considered resulting in a Mixed Integer Polynomial Programming problem. Treating the initial state of the system as a set of bounded parameters, the problem is reformulated as a multiparametric Mixed Integer Polynomial optimization (mp-MIPOPT) problem. A novel algorithm for mp-MIPOPT problems is proposed and the exact explicit control law for polynomial hybrid systems is computed. The key idea is the computation of the analytical solution of the optimality conditions while the binary variables are treated as relaxed parameters. Finally, using symbolic calculations exact nonconvex critical regions are computed.

show abstract

Parametric optimization and optimal control using algebraic geometry methods

Cited by 70 publications

References 13 publications

Image Computation for Polynomial Dynamical Systems Using the Bernstein Expansion

Image Computation for Polynomial Dynamical Systems Using the Bernstein Expansion

Other Developments in Multi‐parametric Optimization

Explicit model predictive control of hybrid systems and multiparametric mixed integer polynomial programming

Contact Info

Product

Resources

About