31 research outputs found
Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares
In this paper, we explore the merits of various algorithms for polynomial
optimization problems, focusing on alternatives to sum of squares programming.
While we refer to advantages and disadvantages of Quantifier Elimination,
Reformulation Linear Techniques, Blossoming and Groebner basis methods, our
main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and
Handelman's theorem. We first formulate polynomial optimization problems as
verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's
algorithm, Bernstein's algorithm and Handelman's algorithm reduce the
intractable problem of feasibility of semi-algebraic sets to linear and/or
semi-definite programming. We apply these algorithms to different problems in
robust stability analysis and stability of nonlinear dynamical systems. As one
contribution of this paper, we apply Polya's algorithm to the problem of
H_infinity control of systems with parametric uncertainty. Numerical examples
are provided to compare the accuracy of these algorithms with other polynomial
optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series
Automated and Sound Synthesis of Lyapunov Functions with SMT Solvers
In this paper we employ SMT solvers to soundly synthesise Lyapunov functions
that assert the stability of a given dynamical model. The search for a Lyapunov
function is framed as the satisfiability of a second-order logical formula,
asking whether there exists a function satisfying a desired specification
(stability) for all possible initial conditions of the model. We synthesise
Lyapunov functions for linear, non-linear (polynomial), and for parametric
models. For non-linear models, the algorithm also determines a region of
validity for the Lyapunov function. We exploit an inductive framework to
synthesise Lyapunov functions, starting from parametric templates. The
inductive framework comprises two elements: a learner proposes a Lyapunov
function, and a verifier checks its validity - its lack is expressed via a
counterexample (a point over the state space), for further use by the learner.
Whilst the verifier uses the SMT solver Z3, thus ensuring the overall soundness
of the procedure, we examine two alternatives for the learner: a numerical
approach based on the optimisation tool Gurobi, and a sound approach based
again on Z3. The overall technique is evaluated over a broad set of benchmarks,
which shows that this methodology not only scales to 10-dimensional models
within reasonable computational time, but also offers a novel soundness proof
for the generated Lyapunov functions and their domains of validity
Formal Synthesis of Lyapunov Neural Networks
We propose an automatic and formally sound method for synthesising Lyapunov
functions for the asymptotic stability of autonomous non-linear systems.
Traditional methods are either analytical and require manual effort or are
numerical but lack of formal soundness. Symbolic computational methods for
Lyapunov functions, which are in between, give formal guarantees but are
typically semi-automatic because they rely on the user to provide appropriate
function templates. We propose a method that finds Lyapunov functions fully
automaticallyusing machine learningwhile also providing formal
guaranteesusing satisfiability modulo theories (SMT). We employ a
counterexample-guided approach where a numerical learner and a symbolic
verifier interact to construct provably correct Lyapunov neural networks
(LNNs). The learner trains a neural network that satisfies the Lyapunov
criteria for asymptotic stability over a samples set; the verifier proves via
SMT solving that the criteria are satisfied over the whole domain or augments
the samples set with counterexamples. Our method supports neural networks with
polynomial activation functions and multiple depth and width, which display
wide learning capabilities. We demonstrate our method over several non-trivial
benchmarks and compare it favourably against a numerical optimisation-based
approach, a symbolic template-based approach, and a cognate LNN-based approach.
Our method synthesises Lyapunov functions faster and over wider spatial domains
than the alternatives, yet providing stronger or equal guarantees
Efficient Control Approaches for Guaranteed Frequency Performance in Power Systems
Due to high penetration of renewable energy, converter-interfaced sources are increasing in power systems and degrading the grid frequency response. Synthetic inertia emulation and guaranteed primary frequency response is a challenging task. Still, there is high potential for application of highly controllable converter-interfaced devices to help performance. Renewable energy sources and demand side smart devices also need to be equipped with innovative frequency control approaches that contribute to frequency regulation operations.
First, the wind turbine generator is chosen to represent an example of a converter- interfaced source. An augmented system frequency response model is derived, including the system frequency response model and a reduced-order model of the wind turbine generator representing the supportive active power due to supplementary inputs. An output feedback observer-based control is designed to provide guaranteed frequency performance. System performance is analyzed for different short circuit ratio scenarios where a lower bound to guarantee the performance is obtained.
Second, the load side control for frequency regulation with its challenges is introduced. 5G technology and its potential application in smart grids are analyzed. The effect of communication delays and packet losses on inertia emulation are investigated to show the need of using improved communication infrastructure.
Third, a robust delay compensation for primary frequency control using fast demand response is proposed. Possible system structured uncertainties and communication delays are considered to limit frequency variations using the proposed control approach. An uncertain governor dead-band model is introduced to capture frequency response characteristics. Guaranteed inertial response is achieved and compared with a PI-based Smith predictor controller to show the effectiveness of the proposed method.
Fourth, set theoretic methods for safety verification to provide guaranteed frequency response are introduced. The Barrier certificate approach using a linear programming relaxation by Handelman’s representation is proposed with its application to power systems.
Finally, the Handelman’s based barrier certificate approach for adequate frequency performance is studied. The computational algorithm is provided for the proposed method and validated using power system benchmark case studies with a discussion on a safety supervisory control (SSC)