Compositional stability criteria based on cyclically neutral supply conditions

In this paper we consider stability of large scale interconnected nonlinear systems that satisfy a strict dissipativity property in terms of local storage and supply functions. Existing compositional stability criteria certify global stability by constructing a global Lyapunov function as the (weighted) sum of local storage functions. We generalize these results by unifying spatial composition, i.e., (weighted) sum of local supply functions is neutral, with temporal composition, i.e., (weighted) sum of supply functions over a time cycle is neutral. Two benchmark examples illustrate the benefits of the developed compositional stability criteria in terms of reducing conservatism and constrained distributed stabilization.</p

A general dissipativity constraint for feedback system design, with emphasis on MPC

A ‘General Dissipativity Constraint’ (GDC) is introduced to facilitate the design of stable feedback systems. A primary application is to MPC controllers when it is preferred to avoid the use of ‘stabilising ingredients’ such as terminal constraint sets or long prediction horizons. Some very general convergence results are proved under mild conditions. The use of quadratic functions, replacing GDC by ‘Quadratic Dissipation Constraint’ (QDC), is introduced to allow implementation using linear matrix inequalities. The use of QDC is illustrated for several scenarios: state feedback for a linear time-invariant system, MPC of a linear system, MPC of an input-affine system, and MPC with persistent disturbances. The stability that is guaranteed by GDC is weaker than Lyapunov stability, being ‘Lagrange stability plus convergence’. Input-to-state stability is obtained if the control law is continuous in the state. An example involving an open-loop unstable helicopter illustrates the efficacy of the approach in practice.National Research Foundation Singapor

Mini-Workshop: Applied Koopmanism

Koopman and Perron–Frobenius operators are linear operators that encapsulate dynamics of nonlinear dynamical systems without loss of information. This is accomplished by embedding the dynamics into a larger infinite-dimensional space where the focus of study is shifted from trajectory curves to measurement functions evaluated along trajectories and densities of trajectories evolving in time. Operator-theoretic approach to dynamics shares many features with an optimization technique: the Lasserre moment–sums-of-squares (SOS) hierarchies, which was developed for numerically solving non-convex optimization problems with semialgebraic data. This technique embeds the optimization problem into a larger primal semidefinite programming (SDP) problem consisting of measure optimization over the set of globally optimal solutions, where measures are manipulated through their truncated moment sequences. The dual SDP problem uses SOS representations to certify bounds on the global optimum. This workshop highlighted the common threads between the operator-theoretic dynamical systems and moment–SOS hierarchies in optimization and explored the future directions where the synergy of the two techniques could yield results in fluid dynamics, control theory, optimization, and spectral theory

Heisenberg Picture Approach to the Stability of Quantum Markov Systems

Quantum Markovian systems, modeled as unitary dilations in the quantum stochastic calculus of Hudson and Parthasarathy, have become standard in current quantum technological applications. This paper investigates the stability theory of such systems. Lyapunov-type conditions in the Heisenberg picture are derived in order to stabilize the evolution of system operators as well as the underlying dynamics of the quantum states. In particular, using the quantum Markov semigroup associated with this quantum stochastic differential equation, we derive sufficient conditions for the existence and stability of a unique and faithful invariant quantum state. Furthermore, this paper proves the quantum invariance principle, which extends the LaSalle invariance principle to quantum systems in the Heisenberg picture. These results are formulated in terms of algebraic constraints suitable for engineering quantum systems that are used in coherent feedback networks

Heisenberg Picture Approach to the Stability of Quantum Markov Systems

Quantum Markovian systems, modeled as unitary dilations in the quantum stochastic calculus of Hudson and Parthasarathy, have become standard in current quantum technological applications. This paper investigates the stability theory of such systems. Lyapunov-type conditions in the Heisenberg picture are derived in order to stabilize the evolution of system operators as well as the underlying dynamics of the quantum states. In particular, using the quantum Markov semigroup associated with this quantum stochastic differential equation, we derive sufficient conditions for the existence and stability of a unique and faithful invariant quantum state. Furthermore, this paper proves the quantum invariance principle, which extends the LaSalle invariance principle to quantum systems in the Heisenberg picture. These results are formulated in terms of algebraic constraints suitable for engineering quantum systems that are used in coherent feedback networks

Contributions to fuzzy polynomial techniques for stability analysis and control

The present thesis employs fuzzy-polynomial control techniques in order to improve the stability analysis and control of nonlinear systems. Initially, it reviews the more extended techniques in the field of Takagi-Sugeno fuzzy systems, such as the more relevant results about polynomial and fuzzy polynomial systems. The basic framework uses fuzzy polynomial models by Taylor series and sum-of-squares techniques (semidefinite programming) in order to obtain stability guarantees. The contributions of the thesis are: ¿ Improved domain of attraction estimation of nonlinear systems for both continuous-time and discrete-time cases. An iterative methodology based on invariant-set results is presented for obtaining polynomial boundaries of such domain of attraction. ¿ Extension of the above problem to the case with bounded persistent disturbances acting. Different characterizations of inescapable sets with polynomial boundaries are determined. ¿ State estimation: extension of the previous results in literature to the case of fuzzy observers with polynomial gains, guaranteeing stability of the estimation error and inescapability in a subset of the zone where the model is valid. ¿ Proposal of a polynomial Lyapunov function with discrete delay in order to improve some polynomial control designs from literature. Preliminary extension to the fuzzy polynomial case. Last chapters present a preliminary experimental work in order to check and validate the theoretical results on real platforms in the future.Pitarch Pérez, JL. (2013). Contributions to fuzzy polynomial techniques for stability analysis and control [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/34773TESI

On Observer-Based Control of Nonlinear Systems

Filtering and reconstruction of signals play a fundamental role in modern signal processing, telecommunications, and control theory and are used in numerous applications. The feedback principle is an important concept in control theory. Many different control strategies are based on the assumption that all internal states of the control object are available for feedback. In most cases, however, only a few of the states or some functions of the states can be measured. This circumstance raises the need for techniques, which makes it possible not only to estimate states, but also to derive control laws that guarantee stability when using the estimated states instead of the true ones. For linear systems, the separation principle assures stability for the use of converging state estimates in a stabilizing state feedback control law. In general, however, the combination of separately designed state observers and state feedback controllers does not preserve performance, robustness, or even stability of each of the separate designs. In this thesis, the problems of observer design and observer-based control for nonlinear systems are addressed. The deterministic continuous-time systems have been in focus. Stability analysis related to the Positive Real Lemma with relevance for output feedback control is presented. Separation results for a class of nonholonomic nonlinear systems, where the combination of independently designed observers and state-feedback controllers assures stability in the output tracking problem are shown. In addition, a generalization to the observer-backstepping method where the controller is designed with respect to estimated states, taking into account the effects of the estimation errors, is presented. Velocity observers with application to ship dynamics and mechanical manipulators are also presented

Distributed control of deregulated electrical power networks

A prerequisite for reliable operation of electrical power networks is that supply and demand are balanced at all time, as efficient ways for storing large amounts of electrical energy are scarce. Balancing is challenging, however, due to the power system's dimensions and complexity, the low controllability and predictability of demand, and due to strict physical and security limitations, such as finitely fast generator dynamics and finite transmission-line capacities. The need for efficient and secure balancing arrangements is growing stronger with the increasing integration of distributed generation (DG), the ongoing deregulation of production and consumption of electrical energy, and thus, also the provision of many of the ancillary services that are essential for network stability. DG is mostly based on renewable, intermittent sources such as wind and sun, and consequently, it is associated with a much larger uncertainty in supply than conventional, centralized generation. Moreover, with the emergence of deregulated energy markets as core operational mechanism, the prime goal of power system operation is shifted from centralized minimization of costs to the maximization of individual profit by a large number of competing, autonomous market agents. The main objective of this thesis is to investigate the control-technical possibilities for ensuring efficient, reliable and stable operation of deregulated and badly predictable electrical power networks. Its contributions cover aspects of power system operation on a time scale ranging from day-ahead trading of electrical energy to second-based load-frequency control. As a first contribution, we identify the maximization of security of supply and market efficiency as the two main, yet conflicting objectives of power system operation. Special attention is paid to congestion management, which is an aspect of power system operation where the tension between reliability and efficiency is particularly apparent. More specifically, the differences between locational pricing and cost-based congestion redispatch are analyzed, followed by an assessment of their effects on grid operation. Next, we demonstrate that the current synchronous, energy-based market and incentive system does not necessarily motivate producers to exchange power profiles with the electricity grid that contribute to network stability and security of supply. The thesis provides an alternative production scheduling concept as a means to overcome this issue, which relies on standard market arrangements, but settles energy transactions in an asynchronous way. Theoretical analysis and simulation results illustrate that by adopting this method, scheduling efficiency is improved and the strain on balancing reserves can be reduced considerably. A major part of this thesis is dedicated to real-time, i.e., closed-loop, balancing or load-frequency control. With the increasing share of badly predictable DG, there is a growing need for efficient balancing mechanisms that can account for generator and transmission constraints during the operational day. A promising candidate solution is model predictive control (MPC). Because the large dimensions and complexity of electrical power networks hamper a standard, centralized implementation of MPC, we evaluate a number of scalable alternatives, in which the overall control action is computed by a set of local predictive control laws, instead. The extent of inter-controller communication is shown to be positively correlated with prediction accuracy and, thus, attainable closed-loop performance. Iterative, system-wide communication/coordination is usually not feasible for large networks, however, and consequently, Pareto-optimal performance and coupled-constraint handling are currently out of reach. This also hampers the application of standard cost-based stabilization schemes, in which closed-loop stability is attained via monotonic convergence of a single, optimal system-wide performance cost. Motivated by the observations regarding non-centralized MPC, the focus is then shifted to distributed control methods for networks of interconnected dynamical systems, with power systems as particular field of application, that can ensure stability based on local model and state information only. First, we propose a non-centralized, constraint-based stabilization scheme, in which the set of stabilizing control actions is specified via separable convergence conditions for a collection of a-priori synthesized structured max-control Lyapunov functions (max-CLFs). The method is shown to be non-conservative, in the sense that non-monotonic convergence of the structured functions along closed-loop trajectories is allowed, whereas their construction establishes the existence of a control Lyapunov function, and thus, stability, for the full, interconnected dynamics. Then, an alternative method is provided in which also the demand for a monotonically converging full-system CLF is relaxed while retaining the stability certificate. The conditions are embedded in an almost-decentralized Lyapunov-based MPC scheme, in which the local control laws rely on neighbor-to-neighbor communication only. Secondly, a generalized theorem and example system are provided to show that stabilization methods that rely on the off-line synthesis of fixed quadratic storage functions (SFs) fail for even the simplest of linear, time-invariant networks, if they contain one or more subsystems that are not stable under decoupled operation. This may also impede the application of max-CLF control. As key contribution of this thesis, to solve this issue, we endow the storage functions with a finite set of state-dependent parameters. Max-type convergence conditions are employed to construct a Lyapunov function for the full network, whereas monotonic convergence of the individual SFs is not required. The merit of the provided approach is that the storage functions can be constructed during operation, i.e., along a closed-loop trajectory, thus removing the impediment of centralized, off-line LF synthesis associated with fixed-parameter SFs. It is shown that parameterized-SF synthesis conditions can be efficiently exploited to obtain a scalable, trajectory-dependent control scheme that relies on non-iterative neighbor-to-neighbor communication only. For input-affine network dynamics and quadratic storage functions, the procedure can be implemented by solving a single semi-definite program per node and sampling instant, in a receding horizon fashion. Moreover, by interpolating a collection of so-obtained input trajectories, a low-complexity explicit control law for linear, time-invariant systems is obtained that extends the trajectory-specific convergence property to a much stronger guarantee of closed-loop asymptotic stability for a particular set of initial conditions. Finally, we consider the application of max-CLF and parameterized SFs for real-time balancing in multimachine electrical power networks. Given that generators are operated by competitive, profit-driven market agents, the stabilization scheme is extended with the competitive optimization of a set of arbitrarily chosen, local performance cost functions over a finite, receding prediction horizon. The suitability of the distributed Lyapunov-based predictive control and parameterized storage function algorithms is evaluated by simulating them in closed-loop with the 7-machine CIGRÉ benchmark system. The thesis concludes by summarizing the main contributions, followed by ideas for future research