Numerical Construction of LISS Lyapunov Functions under a Small Gain Condition

In the stability analysis of large-scale interconnected systems it is frequently desirable to be able to determine a decay point of the gain operator, i.e., a point whose image under the monotone operator is strictly smaller than the point itself. The set of such decay points plays a crucial role in checking, in a semi-global fashion, the local input-to-state stability of an interconnected system and in the numerical construction of a LISS Lyapunov function. We provide a homotopy algorithm that computes a decay point of a monotone op- erator. For this purpose we use a fixed point algorithm and provide a function whose fixed points correspond to decay points of the monotone operator. The advantage to an earlier algorithm is demonstrated. Furthermore an example is given which shows how to analyze a given perturbed interconnected system.Comment: 30 pages, 7 figures, 4 table

Stability, observer design and control of networks using Lyapunov methods

We investigate different aspects of the analysis and control of interconnected systems. Different tools, based on Lyapunov methods, are provided to analyze such systems in view of stability, to design observers and to control systems subject to stabilization. All the different tools presented in this work can be used for many applications and extend the analysis toolbox of networks. Considering systems with inputs, the stability property input-to-state dynamical stability (ISDS) has some advantages over input-to-state stability (ISS). We introduce the ISDS property for interconnected systems and provide an ISDS small-gain theorem with a construction of an ISDS-Lyapunov function and the rate and the gains of the ISDS estimation for the whole system. This result is applied to observer design for single and interconnected systems. Observers are used in many applications where the measurement of the state is not possible or disturbed due to physical reasons or the measurement is uneconomical. By the help of error Lyapunov functions we design observers, which have a so-called quasi ISS or quasi-ISDS property to guarantee that the dynamics of the estimation error of the systems state has the ISS or ISDS property, respectively. This is applied to quantized feedback stabilization. In many applications, there occur time-delays and/or instantaneous jumps of the systems state. At first, we provide tools to check whether a network of time-delay systems has the ISS property using ISS-Lyapunov-Razumikhin functions and ISS-Lyapunov-Krasovskii functionals. Then, these approaches are also used for interconnected impulsive systems with time-delays using exponential Lyapunov-Razumikhin functions and exponential Lyapunov-Krasovskii functionals. We derive conditions to assure ISS of an impulsive network with time-delays. Controlling a system in a desired and optimal way under given constraints is a challenging task. One approach to handle such problems is model predictive control (MPC). In this thesis, we introduce the ISDS property for MPC of single and interconnected systems. We provide conditions to assure the ISDS property of systems using MPC, where the previous result of this thesis, the ISDS small-gain theorem, is applied. Furthermore, we investigate the ISS property for MPC of time-delay systems using the Lyapunov-Krasovskii approach. We prove theorems, which guarantee ISS for single and interconnected systems using MPC

Relaxed ISS Small-Gain Theorems for Discrete-Time Systems

In this paper ISS small-gain theorems for discrete-time systems are stated, which do not require input-to-state stability (ISS) of each subsystem. This approach weakens conservatism in ISS small-gain theory, and for the class of exponentially ISS systems we are able to prove that the proposed relaxed small-gain theorems are non-conservative in a sense to be made precise. The proofs of the small-gain theorems rely on the construction of a dissipative finite-step ISS Lyapunov function which is introduced in this work. Furthermore, dissipative finite-step ISS Lyapunov functions, as relaxations of ISS Lyapunov functions, are shown to be sufficient and necessary to conclude ISS of the overall system.Comment: input-to-state stability, Lyapunov methods, small-gain conditions, discrete-time non-linear systems, large-scale interconnection

Strong iISS: combination of iISS and ISS with respect to small inputs

International audienceThis paper studies the notion of Strong iISS, which imposes both integral input-to-state stability (iISS) and input-to-state stability (ISS) with respect to small inputs. This combination characterizes the robustness property, exhibited by many practical systems, that the state remains bounded as long as the magnitude of exogenous inputs is reasonably small but may diverge for stronger disturbances. We provide three Lyapunov-type sufficient conditions for Strong iISS. One is based on iISS Lyapunov functions admitting a radially non- vanishing (class K) dissipation rate. However we show that it is not a necessary condition for Strong iISS. Two less conservative conditions are then provided, which are used to demonstrate that asymptotically stable bilinear systems are Strongly iISS. Finally, we discuss cascade and feedback interconnections of Strong iISS systems

Input-to-State Stabilization in H1H^1-Norm for Boundary Controlled Linear Hyperbolic PDEs with Application to Quantized Control

International audienceWe consider a system of linear hyperbolic PDEs where the state at one of the boundary points is controlled using the measurements of another boundary point. For this system class, the problem of designing dynamic controllers for input-to-state stabilization in $H^1$-norm with respect to measurement errors is considered. The analysis is based on constructing a Lyapunov function for the closed-loop system which leads to controller synthesis and the conditions on system dynamics required for stability. As an application of this stability notion, the problem of quantized control for hyperbolic PDEs is considered where the measurements sent to the controller are communicated using a quantizer of finite length. The presence of quantizer yields practical stability only, and the ultimate bounds on the norm of the state trajectory are also derived

Variational and Time-Distributed Methods for Real-time Model Predictive Control

This dissertation concerns the theoretical, algorithmic, and practical aspects of solving optimal control problems (OCPs) in real-time. The topic is motivated by Model Predictive Control (MPC), a powerful control technique for constrained, nonlinear systems that computes control actions by solving a parameterized OCP at each sampling instant. To successfully implement MPC, these parameterized OCPs need to be solved in real-time. This is a significant challenge for systems with fast dynamics and/or limited onboard computing power and is often the largest barrier to the deployment of MPC controllers. The contributions of this dissertation are as follows. First, I present a system theoretic analysis of Time-distributed Optimization (TDO) in Model Predictive Control. When implemented using TDO, an MPC controller distributed optimization iterates over time by maintaining a running solution estimate for the optimal control problem and updating it at each sampling instant. The resulting controller can be viewed as a dynamic compensator which is placed in closed-loop with the plant. The resulting coupled plant-optimizer system is analyzed using input-to-state stability concepts and sufficient conditions for stability and constraint satisfaction are derived. When applied to time distributed sequential quadratic programming, the framework significantly extends the existing theoretical analysis for the real-time iteration scheme. Numerical simulations are presented that demonstrate the effectiveness of the scheme. Second, I present the Proximally Stabilized Fischer-Burmeister (FBstab) algorithm for convex quadratic programming. FBstab is a novel algorithm that synergistically combines the proximal point algorithm with a primal-dual semismooth Newton-type method. FBstab is numerically robust, easy to warmstart, handles degenerate primal-dual solutions, detects infeasibility/unboundedness and requires only that the Hessian matrix be positive semidefinite. The chapter outlines the algorithm, provides convergence and convergence rate proofs, and reports some numerical results from model predictive control benchmarks and from the Maros-Meszaros test set. Overall, FBstab shown to be is competitive with state of the art methods and to be especially promising for model predictive control and other parameterized problems. Finally, I present an experimental application of some of the approaches from the first two chapters: Emissions oriented supervisory model predictive control (SMPC) of a diesel engine. The control objective is to reduce engine-out cumulative NOx and total hydrocarbon (THC) emissions. This is accomplished using an MPC controller which minimizes deviation from optimal setpoints, subject to combustion quality constraints, by coordinating the fuel input and the EGR rate target provided to an inner-loop airpath controller. The SMPC controller is implemented using TDO and a variant of FBstab which allows us to achieve sub-millisecond controller execution times. We experimentally demonstrate 10-15% cumulative emissions reductions over the Worldwide Harmonized Light Vehicles Test Cycle (WLTC) drivecycle.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155167/1/dliaomcp_1.pd