Energy and power based perspective of memristive controllers Ramkrishna Pasumarthy, Gourav Saha, Faruk Kazi, Navdeep Singh Abstract— The use of Casimir functions towards control of physical systems is well known, both in the context of Energy shaping as well as Power shaping techniques. In this paper we show that by use of a Memristive element in the controller design enables us to generate additional Casimir functions relating the state of the memristor to the plant state. This additional conserved quantity manifests itself in the control law in form of a state-modulated gain. We present our results with examples in the context of control by Casimir generation in the port-Hamiltonian framework, which essentially deals with shaping the energy of the system. We also present the applicability of the results towards control by power shaping of electrical circuits in the Brayton-Moser framework for modeling of electrical networks. I. INTRODUCTION The last two decades have seen considerable research towards passivity based control of physical systems. The idea is to find a feedback law such that the system becomes passive with respect to a storage function having a strict local minimum at the desired equilibrium point. Since the storage function is nonincreasing along solutions of the closed loop system, it acts as a Lyapunov function and has direct implication towards stability analysis of systems. In the early nineties the framework of port-Hamiltonian systems was introduced as a tool for modeling and control of complex physical systems, see [5], [17]. The passivity based control techniques translate into the port-Hamiltonian framework as assigning a closed-loop Hamiltonian which achieve the desired stability properties. This was done either via the generation of Casimir functions [6], [15], [17] or the IDA-PBC method [14]. Central to the port-Hamiltonian formulation is the role of energy. The port-Hamiltonian formulation also relates to the Brayton-Moser (BM) description of electrical networks, see [1], [5]. The BM formulation relies on the existence of the mixed potential function, which is essentially a power function. Thus, the BM formulation is sometimes also re- ferred to the power based formulation. The BM equations capture the passivity property by using the power function as the storage function, thus enabling control by “power shaping” of systems. Few illustrations towards power shaping stabilization can be found in [8], [13], [9]. Modeling of electrical circuits essentially was based on identifying elements relating four basic physical quantities, Ramkrishna Pasumarthy and Gourav Saha are with Department of Electrical Engineering, IIT Madras, India. ramkrishna@ee.iitm.ac.in, ee13s005@ee.iitm.ac.in Navdeep Singh and Faruk Kazi are with Department of Electrical Engineering, Veermata Jijabai Technological Institute, Mumbai, India. fskazi@vjti.org.in, nmsingh59@gmail.com namely, flux, charge, voltages and currents. While Inductor is modeled as a relation between flux linkages and current, the Capacitor defines a relation between charge and voltages. Resistor was modeled as a relation between voltages and cur- rents. However, there was a “missing element”, which would define an element relating the charge and flux linkages. Thus came the idea of introducing the memristor by Chua [2], as a missing element of circuit theory. In this paper we investigate the impact of memristors towards passivity based control of physical systems. The rest of the paper is organized as follows: In Section II we present the framework of port-Hamiltonian systems which incorporates the presence of memristive elements [11]. We further show that the Casimir functions resulting due to the presence of memristive dynamics introduces modulated damping in the system. Similar effect due to the presence of memristor elements using the IDA-PBC design method has been reported in [4]. Section III deals with the Brayton- Moser framework for modeling and stabilization of electrical networks. We show with the help of an example that by introduction of memristive elements, similar effects in terms of a state modulated gain can be observed in the control law. Finally, we conclude with some remarks on mixed potentials for systems with memristive components. II. PORT-HAMILTONIAN SYSTEMS AND ENERGY SHAPING A physical system is usually modeled as consisting of a set of energy storing elements, set of energy dissipating elements and two sets of external ports, one capturing the control action of the system, together with presence of sources and the second external port describing the interaction of the system with its environment. The interaction between these ports is captured by a geometrical object called the Dirac structure, which basically captures the property of power conservation, in such a way that the total power associated with the port variables is zero. To define the notion of Dirac structures for finite dimen- sional systems, we start with a space of power variables F×F * , for some linear space F , with power defined by P =<e | f >, (f,e) ∈F×F * where <e | f> denotes the duality product, that is, the linear functional e ∈F * acting on f ∈F . F is called the space of flows and F * the space of efforts, with the power of a signal (f,e) ∈F×F * denoted as <e | f>. There exists on F×F * a canonically defined bilinear form <<, >>,