Chris Meissen

Performance certification of interconnected systems using decomposition techniques

We propose a unified framework for the certification of stability and input-output performance of interconnected dynamical systems. In our approach, we seek local dissipativity certificates for each subsystem such that when they are combined, the performance of the entire interconnected system is certified. We also demonstrate, by the use of numerical simulations, that the Alternating Direction Method of Multipliers (ADMM) is a promising computational approach for solving such problems.

Compositional performance certification of interconnected systems using ADMM

A compositional performance certification method is presented for interconnected systems using subsystem dissipativity properties and the interconnection structure. A large-scale optimization problem is formulated to search for the most relevant dissipativity properties. The alternating direction method of multipliers (ADMM) is employed to decompose and solve this problem, and is demonstrated on several examples.

Performance certification of interconnected nonlinear systems using ADMM

We present a compositional performance certification method for interconnected nonlinear systems, using dissipativity properties of the subsystems along with the interconnection structure. To select the most relevant dissipativity properties, we formulate a large-scale optimization problem, and solve it with the alternating direction method of multipliers (ADMM). The dissipativity properties are allowed to depend on an unknown equilibrium, enabling us to certify performance without explicit knowledge of the equilibrium for the interconnected system. The effectiveness of the algorithm is demonstrated on two examples, including a model of vehicle platoons.

Symmetry Reduction for Performance Certification of Interconnected Systems

We exploit symmetries in the interconnection topology of a networked system to provide a dimensionality reduction in the certification of performance. The certification method exploits the dissipativity properties of the subsystems; thus the conservatism introduced by the reduction is minimal when the subsystems possess similar dissipativity characteristics. We combine this reduction with distributed optimization techniques to be able to analyze large interconnections efficiently.

Comparison to Other Input/Output Approaches

Throughout the book we employed a state-space approach with the help of the dissipativity concept, generalized in Chap. 8 to dynamic supply rates. In this final chapter, we make connections to other input/output approaches that treat dynamical systems as operators mapping inputs to outputs in function spaces. We start with classical input/output techniques, and next relate the dynamic supply rates of Chap. 8 to integral quadratic constraints (IQCs) . We conclude by pointing to further results that are complementary to those presented in the book.

Brief Review of Dissipativity Theory

This chapter reviews concepts from dissipativity theory that are foundational for the subsequent chapters. It first gives the general definition and, next, discusses special types of dissipativity that describe finite \(L_2\) gain, passivity, and output strict passivity properties, followed by a graphical interpretation of each type for memoryless systems. The next topic is differential characterization of dissipativity and its specialization to linear systems. This is followed by numerical techniques for certifying dissipativity via semidefinite programming. The final section illustrates the use of dissipativity for reachability and Lyapunov stability analysis.

Stability of Interconnected Systems

This chapter introduces a class of interconnected systems and derives a stability test from the dissipativity characteristics of the subsystems. This test has the form of a linear matrix inequality (LMI) which means that it can be checked with standard convex optimization packages. The decision variables of this LMI serve as the coefficients of a Lyapunov function constructed from the storage functions associated with the dissipativity properties of the subsystems. The rest of the chapter specializes this stability test to subsystems with first finite \(L_2\) gain and, next, passivity properties. Further elaborating on passivity, the last section identifies special interconnection structures where the feasibility of the LMI has analytical characterizations that give insight into the interplay between interconnection structure and stability properties.

Searching Over Subsystem Dissipativity Properties

The stability and performance tests in earlier chapters used a fixed dissipativity property for each subsystem. In this chapter, we take a more flexible approach and employ a conical combination of several dissipativity certificates known for each subsystem. Next, we take a more exhaustive approach and combine the stability and performance tests with a simultaneous search for feasible subsystem dissipativity properties. We employ the alternating direction method of multipliers (ADMM) algorithm, a powerful distributed optimization technique, to decompose and solve this problem.

Equilibrium Independent Stability Certification

One of the difficulties in analyzing the stability of networks is that the equilibrium point depends on all subsystems. Thus, adding or removing subsystems change the equilibrium and require cumbersome iterations of the analysis. To achieve a truly compositional technique that does not require such iterations, in this chapter we introduce the notion of equilibrium independent dissipativity (EID). Instead of relying on the knowledge of the equilibrium, this notion requires dissipativity with respect to any point that has the potential to become an equilibrium when the system is interconnected with others. We present a numerical technique to certify EID and conclude with a test that exploits the EID properties of the subsystems to establish stability without exact knowledge of the equilibrium.

From Stability to Performance and Safety

In this chapter we extend the compositional approach from stability analysis to certifying performance and safety. Performance is defined as a desired dissipativity property for the interconnection, such as a prescribed gain from a disturbance input to a performance output. A storage function for the interconnection is synthesized as a weighted sum of subsystem storage functions and the weights are left as decision variables in a linear matrix inequality. Next we address the safety problem where the goal is to guarantee that trajectories starting in a prescribed set do not intersect a given unsafe set under a set of admissible disturbances. This time a weighted sum of subsystem storage functions serves as a barrier function for the interconnection.