Closed-loop Lyapunov Function Research Articles

Geometric-dissipative control of exothermic continuous reactors

The problem of robustly stabilizing through output-feedback control an open-loop unstable exothermic continuous reactor with temperature measurement is addressed. The combination of advanced geometric control and classical mechanics methods yields: (i) solvability of the detailed model-based nonlinear output-feedback problem in terms of passivity, observability and dissipativity, (ii) open-loop energy and closed-loop Lyapunov functions in analytic form, and (iii) the tradeoff between response speed, robustness, and control effort. On the basis of simplified model tailored according to the passivity-observability-dissipativity structure, the advanced geometric observer-based nonlinear controller is realized with an industrial-like PID controller. The methodology is illustrated with numerical simulations.

Local linear dynamics assignment in IDA-PBC

In this paper, the technique of local linear dynamics assignment is presented, which complements the powerful Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) methodology. In IDA-PBC, nonlinear state feedback controllers are designed by matching the system’s dynamics with a desired Port-Hamiltonian (PH) state representation. The latter consists of an energy function, which serves as the closed-loop Lyapunov function, as well as matrices, describing the virtual internal exchange and dissipation of the energy. A major difficulty in IDA-PBC is how to determine reasonable values for the large number of free design parameters. Local linear dynamics assignment offers a solution to this problem with a number of advantages. (i) Invoking the closed-loop Jacobian linearization to fix the parameter values provides transparency with respect to the resulting local dynamic behavior. (ii) An appropriate state transformation isolates the coordinates available for energy shaping. (iii) A related local linear state transformation makes the resulting system of design equations linear. (iv) Assigning a Hurwitz closed-loop Jacobian and ensuring positive semi-definiteness of the closed-loop dissipation matrix, the tedious definiteness check of the energy is omitted. The design steps are illustrated with the Ball on Wheel example.

Control Design for Electromagnetic Actuators Based on Backstepping and Landing Reference Governor

Electromechanical actuation systems are widely used in a variety of industrial applications. Several control challenges in these applications stem from the need to achieve the soft contact of the moving parts of the actuators, also known as soft landing where the main goal is to reduce the system noise and wear, and prevent the relevant parts from damage and breaking. In this paper we propose a control design which can be employed to guarantee the enforcement of the required soft landing constraints. Our design approach is based on an application of the landing reference governor which gradually adjusts a set-point to a nominal controller employing nonlinear inversion and backstepping techniques. The closed-loop Lyapunov function of the nominal controller is used to estimate and avoid high landing velocities. We consider a single-mass dynamic model of the actuator and report the corresponding simulation results.

EXTENSION OF KALMAN FILTER THEORY TO NONLINEAR SYSTEMS WITH APPLICATION TO WING ROCK MOTION

ABSTRACTA formulation for nonlinear Kalman filter theory is developed in this paper. It is shown that the solution of the nonlinear Kalman filter problem is governed by a Hamilton‐Jacobi‐Bellman inequality (HJBI). Choosing the closed loop Lyapunov function in a symmetric matrix form of the state vector results in the reduction of the HJBI to an algebraic Riccati inequality along with several other algebraic inequalities. These inequalities are formulated into a series of closed loop Lyapunov inequalities which are turned into equalities by adding a positive state vector function. Closed loop Lyapunov functions are then obtained successively by solving these eqauations based on the powers of the state vector. Control of a nonlinear wing rock motion is employed as an example to illustrate the theory.

Optimal induced l1-norm state feedback control

State feedback synthesis is considered for linear discrete-time systems to minimize an upper bound of the closed-loop induced l 1-norm. A sufficient condition, called Generalized Bounded Real Inequality (GBRI), is presented to establish such a bound. An algorithm similar to the Invariant Kernel Algorithm (Shamma, IEEE Trans. Automat. Control 41 (1996) 533–544) and the contractive set algorithm (Blanchini, IEEE Trans. Automat. Control 39 (1994) 428–433) reduces the analysis and synthesis problems to linear programming. If the problem is feasible, our algorithm gives a polyhedral set that induces a closed-loop Lyapunov function and leads to feasible control laws. An example is given for which the optimal induced l 1-norm achieved by a linear controller is achieved by our synthesis approach.

On the Control of Linear Systems Using Nonlinear Optimal Feedback

Optimal nonlinear feedback control of a linear system is presented in this paper. The non-negative cost function is designed to Find the optimal nonlinear controller. The Hamilton-Jacobi-Bellman (HJB) inequality is employed to achieve this objective. It is shown that the closed loop Lyapunov function (CLLF) is required to have an even order such that the CLLF is always positive with non-positive time derivative. Therefore, the second method of Lyapunov theory is satisfied. This will allow the controlled system to be stable in the large and the problem of saturation in the control input is reduced. An undetectable marginally stable case is solved for illustration.

Optimal feedback control of a nonlinear system - Wing rock example

A procedure is presented for optimizing a state feedback control law for a nonlinear system with respect to a positive performance index. The Hamilton-Jacobi-Bellman equation is employed to derive the optimality equations wherein this performance index is minimized. The closed-loop Lyapunov function is assumed to have the same matrix form of state variables as the performance index. The constant interpolated terms of these matrix forms are easily determined so as to guarantee their positive definitenesses. The optimal nonlinear system is asymptotically stable in the large, as both the closed-loop Lyapunov function and performance index are positive definite. An unstable wing rock equation of motion is employed to illustrate this method. It is shown that the wing rock model using nonlinear state feedback is asymptotically stable in the large. Both optimal linear and nonlinear state feedback cases are evaluated.

Robust input-output feedback linearization

To make input-output feedback linearization a practical and systematic design methodology for single-input nonlinear systems, two problems need to be addressed. One is to handle systematically the difficulties associated with the internal dynamics or zero-dynamics when the relative degree is less than the system order. The other is to account for the effect of model uncertainties in the successive differentiations of the output of interest. While the first problem has recently received considerable attention, the second has been largely unexplored. This paper represents a preliminary study of a systematic methodology to account robustly for parametric uncertainties in the original system model. The approach is based on combining sliding control ideas with the recursive construction of a closed-loop Lyapunov function, and is illustrated with a simple example.

Gramian assignment based on the Lyapunov equation

A theorem on the assignability of the controllability Gramian using state feedback is presented. The approach is based on the Lyapunov equation. Specifically, it shows that the set of Gramian matrices which are assignable using state feedback lies in a vector space and is precisely the intersection of that vector space with the set of positive definite matrices. The results show that it may be possible to reshape the structure of the system indirectly by specifying a desired closed-loop Lyapunov function, rather than by directly assigning eigenvalues or eigenvectors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>