LMI-Based Reset Unknown Input Observer for State Estimation of Linear Uncertain Systems

This paper proposes a novel kind of Unknown Input Observer (UIO) called Reset Unknown Input Observer (R-UIO) for state estimation of linear systems in the presence of disturbance using Linear Matrix Inequality (LMI) techniques. In R-UIO, the states of the observer are reset to the after-reset value based on an appropriate reset law in order to decrease the $L_2$ norm and settling time of estimation error. It is shown that the application of the reset theory to the UIOs in the LTI framework can significantly improve the transient response of the observer. Moreover, the devised approach can be applied to both SISO and MIMO systems. Furthermore, the stability and convergence analysis of the devised R-UIO is addressed. Finally, the efficiency of the proposed method is demonstrated by simulation results

Analysis and synthesis of Markov Jump Linear systems with time-varying delays and partially known transition probabilities

In this note, the stability analysis and stabilization problems for a class of discrete-time Markov jump linear systems with partially known transition probabilities and time-varying delays are investigated. The time-delay is considered to be time-varying and has a lower and upper bounds. The transition probabilities of the mode jumps are considered to be partially known, which relax the traditional assumption in Markov jump systems that all of them must be completely known a priori. Following the recent study on the class of systems, a monotonicity is further observed in concern of the conservatism of obtaining the maximal delay range due to the unknown elements in the transition probability matrix. Sufficient conditions for stochastic stability of the underlying systems are derived via the linear matrix inequality (LMI) formulation, and the design of the stabilizing controller is further given. A numerical example is used to illustrate the developed theory. © 2008 IEEE.published_or_final_versio

Stability of hybrid model predictive control

In this paper we investigate the stability of hybrid systems in closed-loop with Model Predictive Controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability and exponential stability. A general theory is presented which proves that Lyapunov stability is achieved for both terminal cost and constraint set and terminal equality constraint hybrid MPC, even though the considered Lyapunov function and the system dynamics may be discontinuous. For particular choices of MPC criteria and constrained Piecewise Affine (PWA) systems as the prediction models we develop novel algorithms for computing the terminal cost and the terminal constraint set. For a quadratic MPC cost, the stabilization conditions translate into a linear matrix inequality while, for an 1-norm based MPC cost, they are obtained as 1-norm inequalities. It is shown that by using 1-norms, the terminal constraint set is automatically obtained as a polyhedron or a finite union of polyhedra by taking a sublevel set of the calculated terminal cost function. New algorithms are developed for calculating polyhedral or piecewise polyhedral positively invariant sets for PWA systems. In this manner, the on-line optimization problem leads to a mixed integer quadratic programming problem or to a mixed integer linear programming problem, which can be solved by standard optimization tools. Several examples illustrate the effectiveness of the developed methodology

Non-smooth model predictive control: stability and applications to hybrid systems

In this report we investigate the stability of hybrid systems in closed-loop with Model Predictive Controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability and exponential stability. A general theory is presented which proves that Lyapunov stability is achieved for both terminal cost and constraint set and terminal equality constraint hybrid MPC, even though the considered Lyapunov function and the system dynamics may be discontinuous. For particular choices of MPC criteria and constrained Piecewise Affine (PWA) systems as the prediction models we develop novel algorithms for computing the terminal cost and the terminal constraint set. For a quadratic MPC cost, the stabilization conditions translate into a linear matrix inequality while, for an ∞-norm based MPC cost, they are obtained as ∞-norm inequalities. It is shown that by using ∞-norms, the terminal constraint set is automatically obtained as a polyhedron or a finite union of polyhedra by taking a sublevel set of the calculated terminal cost function. New algorithms are developed for calculating polyhedral or piecewise polyhedral positively invariant sets for PWA systems. In this manner, the on-line optimization problem leads to a mixed integer quadratic programming problem or to a mixed integer linear programming problem, which can be solved by standard optimization tools. Several examples illustrate the effectiveness of the developed methodology

Observer design for a class of nonlinear systems combining dissipativity with interconnection and damping assignment

A nonlinear observer design approach is proposed that exploits and combines port-Hamiltonian systems and dissipativity theory. First, a passivity-based observer design using interconnection and damping assignment for time variant state affine systems is presented by applying output injection to the system such that the observer error dynamics takes a port-Hamiltonian structure. The stability of the observer error system is assured by exploiting its passivity properties. Second, this setup is extended to develop an observer design approach for a class of systems with a time varying state affine forward and a nonlinear feedback contribution. For a class of nonlinear systems, the theory of dissipative observers is adapted and combined with the results for the passivity-based observer design using interconnection and damping assignment. The convergence of the compound observer design is determined by a linear matrix inequality. The performance of both observer approaches is analyzed in simulation examples

Observer design for a class of nonlinear systems combining dissipativity with interconnection and damping assignment

A nonlinear observer design approach is proposed that exploits and combines port-Hamiltonian systems and dissipativity theory. First, a passivity-based observer design using interconnection and damping assignment for time variant state affine systems is presented by applying output injection to the system such that the observer error dynamics takes a port-Hamiltonian structure. The stability of the observer error system is assured by exploiting its passivity properties. Second, this setup is extended to develop an observer design approach for a class of systems with a time varying state affine forward and a nonlinear feedback contribution. For a class of nonlinear systems, the theory of dissipative observers is adapted and combined with the results for the passivity-based observer design using interconnection and damping assignment. The convergence of the compound observer design is determined by a linear matrix inequality. The performance of both observer approaches is analyzed in simulation examples