Tutorial on “Positive Systems and Large Scale Control” - CDC 2018
Organizer: M. Elena Valcher, Univ. Padova, Italy
Participants: P. Colaneri, Politecnico di Milano, Italy
Y. Ebihara, Kyoto Univ., Japan
A. Rantzer, Lund University, Sweden
M.E. Valcher, Univ. Padova, Italy
General description of the Session Topic
Positive systems are an important class of systems that arise in diverse application areas, such as chemical process industry, electronic circuit design, communication networks, energy networks, transportation and traffic modeling, pharmacokinetics and biology, to mention a few. The distinctive trait of these systems is to be characterized by describing variables that take only nonnegative values, and this feature makes them the appropriate modeling tool for physical systems whose describing variables are concentrations, pressures, population levels, queue levels, etc.
The peculiar nature of these systems has called for ad hoc tools in order to explore fundamental control problems like stability, positive realization, controllability, state estimation, optimal control. The theory of positive systems (and more in general the theory of monotone systems), in fact, relies on fundamental building blocks as the Perron-Frobenius theorem, graph theory, linear copositive functions, cone theory and linear programming. In particular, positive system theory has gained renewed interest from the view point of convex optimization, and fruitful results have been obtained for the analysis and synthesis of positive systems in terms of L1- and L-infinity-induced norms. Even though the computation of these induced norms is hard for general (non-positive) systems, the internal positivity allows us to compute the norms using linear programming. Moreover, the matrices defining the linear program inherit sparsity properties of the original system.
The properties described above are of tremendous importance in the study of large-scale interconnected systems, since they enable stability analysis and control synthesis for systems of almost arbitrary size. A reason for this is that the power iterations needed to compute a Perron-Frobenius eigenvector can be carried out by parallel processing, with one processor for each matrix row. The matrix size does not matter as long as the number of non-zero elements on each row is bounded. Another way to explain the scalability is that stability analysis of positive systems can be carried out with linear Lyapunov functions, while non-positive systems in general require quadratic Lyapunov functions.
Positive switched systems naturally arise when providing mathematical descriptions of physical systems that exhibit two distinguished features: the describing variables are subject to some nonnegativity constraint and the physical laws that govern the system dynamics change according to the operating conditions.
Examples of physical systems whose behaviors are well-captured by positive switched systems can be found in TCP congestion control, in HIV therapy, in biochemical networks, in traffic modeling, etc... The theory of positive switched systems is still in a relative infancy, but a few fundamental problems have already been the object of extensive research, in particular stability and stabilization. Other problems, like reachability and optimal control, are still at an early stage, also due to their intrinsic complexity.
The aim of this tutorial session is to provide a general introduction to positive systems and positive switched systems, and to focus on some fundamental topics that are of great interest for their applications. Specifically, we will start with a general introduction to positive systems and positive switched systems, to allow the audience to become familiar with the main results and the mathematical tools, in particular with those used in the following talks. We will then focus on H-infinity optimal distributed control and L1-induced norm for large scale and interconnected positive systems with application to transport and energy networks. Finally, the state of the art regarding stabilizability and optimal control of positive switched system will be presented, together with the application of these results to HIV mitigating therapy, traffic congestion control and distributed allocation of power in a public battery charging station.
Tutorial Session Organization
First Talk: Elena Valcher (30 min)
Positive systems and positive switched systems: basic theoretical results and main applications. An overview.
In the talk we will start by providing some real-world motivating examples that stimulated the research in this field. The talk will provide then an overview of the main theoretical tools adopted in the investigation of linear state-space models under the positivity constraint, e.g., Perron-Frobenius theory, linear copositive Lyapunov functions, diagonal Lyapunov functions, L1-induced norm. Stability and stabilizability of both positive systems and positive switched systems will be discussed, by pointing out known results and open problems.
Second Talk: Anders Rantzer (30 min)
Control synthesis for large scale positive systems, with application to infrastructure networks
Positive systems appear naturally in modeling of many large network problems. In this presentation, we will show how to exploit the positivity property to simplify analysis and synthesis of controllers for such networks. In particular, we will explain how synthesis of H-infinity optimal distributed controllers can achieve the same performance as centralized Riccati equation-based controllers, at much lower computational cost. For controllers with integral action, internal positivity of the closed loop system will be lost, but the crucial benefits can still be retained by exploiting external positivity. Motivated by transportation networks, we will also a take brief look at the effect of capacity constraints, where the notion of positivity needs to be replaced by its non-linear counterpart, monotonicity.
Third Talk: Yoshio Ebihara (30 min)
L1-induced norm analysis of positive systems and its application to stabilization of large-scale interconnected positive systems.
In this talk we start from the basics about linear copositive Lyapunov functions for positive systems, followed by the characterization of the L1-induced norm of positive systems by means of linear inequalities (linear programming problems). We show that, a slightly generalized version, weighted L1-induced norm, is useful for the stability analysis of large-scale interconnected systems constructed from positive subsystems. More precisely, we show that the interconnected system is stable if and only if there exists a set of weighting vectors that renders the weighted L1-induced norm of each positive subsystem smaller than one. We can readily apply this result to decentralized stabilizing state-feedback controller synthesis for large-scale interconnected systems, where we can design each local controller optimally and purely locally without knowing global information about the whole interconnected system.
Fourth Talk: Patrizio Colaneri (30 min)
Optimal scheduling of positive switched systems: application examples.
In the talk we briefly illustrate three application-oriented problems: 1) Theraphy scheduling for HIV load mitigation, 2) Traffic light scheduling in road junctions, 3) AIMD based distributed car battery charging in public parking. The three problems are closely connected with the theory of optimal control of positive switched systems. It is shown that under some assumptions one can exploit the convexity of the cost function, thus paving the way to easy numerical solutions. Another fundamental tool is played by the so-called “arg-min” switching theory inherited by co-positive Lyapunov-Metzler inequalities. Such inequalities generate a switching rule capable to stabilize the system and provide an upper bound on the optimal cost. Finally the classical AIMD (additive increasing multiplicative decreasing) algorithm is presented and its effectiveness in distributed scheduling problems is illustrated via a simple resource allocation problem.
