Mario Sassano

Brief paper: Dynamic scaling and observer design with application to adaptive control

A constructive algorithm for designing an observer for a class of nonlinear systems is presented. We follow the invariant manifold based approach which allows one to shape the dynamics of the estimation error. However, this shaping relies on the solution of a partial differential equation, which becomes difficult for multi-output systems. In this paper we remove this restriction by adding to the reduced-order observer an output filter and a single dynamic scaling parameter. We show that this method can be applied to systems with unknown parameters, leading to a new class of adaptive controllers. As an application, we consider two examples: an induction motor with unknown load and a longitudinal controller for an aircraft with unknown aerodynamic properties.

Dynamic Approximate Solutions of the HJ Inequality and of the HJB Equation for Input-Affine Nonlinear Systems

The solution of most nonlinear control problems hinges upon the solvability of partial differential equations or inequalities. In particular, disturbance attenuation and optimal control problems for nonlinear systems are generally solved exploiting the solution of the so-called Hamilton-Jacobi (HJ) inequality and the Hamilton-Jacobi-Bellman (HJB) equation, respectively. An explicit closed-form solution of this inequality, or equation, may however be hard or impossible to find in practical situations. Herein we introduce a methodology to circumvent this issue for input-affine nonlinear systems proposing a dynamic, i.e., time-varying, approximate solution of the HJ inequality and of the HJB equation the construction of which does not require solving any partial differential equation or inequality. This is achieved considering the immersion of the underlying nonlinear system into an augmented system defined on an extended state-space in which a (locally) positive definite storage function, or value function, can be explicitly constructed. The result is a methodology to design a dynamic controller to achieve L2-disturbance attenuation or approximate optimality, with asymptotic stability.

Necessary and sufficient conditions for output regulation in a class of hybrid linear systems

In this paper, the problem of output regulation for a class of hybrid linear systems is considered. Necessary and sufficient conditions for its solution are provided, both in the full information and error feedback case, under the additional constraint that the regulator is a hybrid linear time invariant system from the same class. A stronger version of the internal model principle is also shown, requiring that the regulator must contain a copy of the zero dynamics of the plant in addition to the usual copy of the exosystem dynamics.

Constructive

In this paper, a class of infinite-horizon, nonzero-sum differential games and their Nash equilibria are studied and the notion of εα-Nash equilibrium strategies is introduced. Dynamic strategies satisfying partial differential inequalities in place of the Hamilton-Jacobi-Isaacs partial differential equations associated with the differential games are constructed. These strategies constitute (local) εα-Nash equilibrium strategies for the differential game. The proposed methods are illustrated on a differential game for which the Nash equilibrium strategies are known and on a Lotka-Volterra model, with two competing species. Simulations indicate that both dynamic strategies yield better performance than the strategies resulting from the solution of the linear-quadratic approximation of the problem.

\epsilon

The problem of output regulation for a class of hybrid linear systems characterized by periodic jumps is considered in this paper. Necessary and sufficient conditions for its solution are provided both in the full information and error feedback case. By a detailed analysis of such conditions, several interesting properties are derived, including the fact that the regulator must contain a (suitably defined) copy of the flow zero dynamics of the plant in addition to the usual copy of the exosystem dynamics, and the fact that the output regulation problem is generically solvable for fat plants (having more inputs than outputs) and generically not solvable for square plants. In addition, semi-classical solutions to the hybrid output regulation problem are introduced and discussed. Such solutions are characterized by the property that the required steady-state input can be generated as a simple linear function of the state of the exosystem (as in the non-hybrid case), although the associated steady-state response has a genuinely hybrid structure.

-Nash Equilibria for Nonzero-Sum Differential Games

The Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) problem for port-controlled Hamiltonian systems is revisited. We propose a methodology that exploits the novel notion of algebraic solution of the so-called matching equation. This notion is instrumental for the construction of an energy function, defined on an extended state-space, which does not rely upon the solution of any partial differential equation. This yields, differently from the classical solution, a dynamic state feedback that stabilizes a desired equilibrium point. In addition, conditions that allow to preserve the port-controlled Hamiltonian structure in the extended closed-loop system are provided. The theory is validated on two physical systems: the magnetic levitated ball and a third order food-chain system. A dynamic control law is constructed for both these systems by assigning a damping factor that cannot be assigned by the classical IDA-PBC.

Hybrid Output Regulation for Linear Systems With Periodic Jumps: Solvability Conditions, Structural Implications and Semi-Classical Solutions

In this paper, a special class of hybrid regulation problems is considered, which can be solved by using a class of steady-state inputs essentially coinciding with the class used in standard output regulation problems (for non hybrid systems), although the corresponding state evolution happens on a genuinely hybrid invariant manifold; hence the name “semiclassical”. The main advantage of using such solutions lies in easier implementation and the possibility of robust design.

Constructive Interconnection and Damping Assignment for Port-Controlled Hamiltonian Systems

A reduced-order globally convergent observer to estimate the depth of an object projected on the image plane of a camera is presented, assuming that the object is planar or has a planar surface and the orientation of the plane is known. A locally convergent observer can be obtained when the plane unit normal is unknown, and the latter is estimated together with the depth of the object. The observer exploits the image moments of the object as measured features. The estimation is achieved by rendering attractive and invariant a manifold in the extended state space of the system and the observer. The problem is reduced to the solution of a system of partial differential equations. The solution of the partial differential equations can become a difficult task, hence it is shown that this issue can be resolved by adding to the observer an output filter and a dynamic scaling parameter.

On semiclassical solutions of hybrid regulator equations

In this paper, the problem of linear time invariant state feedback stabilization for a class of hybrid systems is dealt with. The considered class of systems has received a considerable attention in the last years especially as a benchmark for hybrid output regulation, and in this context it turns out to be quite crucial to have stabilization approaches working under minimal hypotheses meanwhile providing linear time invariant solutions. After showing that static linear time invariant state feedback stabilizers might not exist even in the considered simple setting, a new solution is provided by formulating and solving a linear quadratic optimal control problem, which turns out to be a static time varying linear state feedback. It is then shown how such a feedback can be implemented via a stable dynamic time invariant linear state feedback, by exploiting a dynamic extension implementing the stabilized optimal costate dynamics.

Brief paper: Observer design for range and orientation identification

Lyapunov functions are a fundamental tool to investigate stability properties of equilibrium points of linear and nonlinear systems. The existence of Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. Nevertheless the actual computation (of the analytic expression) of the function may be difficult. Herein we propose an approach to avoid the computation of an explicit solution of the Lyapunov partial differential inequality, introducing the concept of Dynamic Lyapunov function. These functions allow to study stability properties of equilibrium points, similarly to standard Lyapunov functions. In the former, however, a positive definite function is combined with a dynamical system that render Dynamic Lyapunov functions easier to construct than Lyapunov functions. Moreover families of standard Lyapunov functions can be obtained from the knowledge of a Dynamic Lyapunov function by rendering invariant a desired submanifold of the extended state-space. The invariance condition is given in terms of a system of partial differential equations similar to the Lyapunov pde. Differently from the latter, however, in the former no constraint is imposed on the sign of the solution or on the sign of the term on the right-hand side of the equation. Several applications and examples conclude the paper.