Manuel Jimenez-Lizarraga

Open-loop Nash equilibrium in polynomial differential games via state-dependent Riccati equation

This paper studies finite- as well as infinite-time horizon nonzero-sum polynomial differential games. In both cases, we explore the so-called state-dependent Riccati equations to find a set of strategies that guarantee an open loop-Nash equilibrium for this particular class of nonlinear games. We demonstrate that this solution leads the game to an ε - or quasi-equilibrium and provide an upper bound for this ε quantity. The proposed solution is given as a set of N coupled polynomial Riccati-like state-dependent differential equations, where each equation includes a p-linear form tensor representation for its polynomial part.We provide an algorithm for finding the solution of the state-dependent algebraic equation in the infinite-time case based on a Hamiltonian approach. A numerical procedure is detailed to find the solution for this set of strategies. Numerical examples are presented to illustrate the effectiveness of the approach.

Quasi-equilibrium in LQ differential games with bounded uncertain disturbances: robust and adaptive strategies with pre-identification via sliding mode technique

A finite time multi-persons linear-quadratic differential game with bounded disturbances and uncertainties is considered. When players cannot measure these disturbances, it is demonstrated that the standard feedback Nash strategies bear to a quasi Nash-equilibrium depending on an uncertainty upper bound that confirms the robustness property of such standard strategies. In the case of periodic disturbances, another concept, namely adaptive concept, containing three different versions is suggested. They are sliding modes, second order sliding modes and window integration. All of them realize the identification of unknown periodic disturbances during an “identification period” when all participants apply the standard feedback Nash strategies with the, so-called, “shifting signal” generated only by a known external exciting signal. After that period the complete standard strategies with “pre-identification” including the recalculated shifting signal are activated. A numerical example dealing with a two participants game shows that the cost functional for each player achieves better values when the adaptive approach is applied.

Leader-follower strategies for a multi-plant differential game

In this paper the formulation of a concept for a type of robust leader-follower equilibrium for a multi-plant or multiple scenarios differential game is developed. The game dynamic is given by a family of N different possible differential equations (multi-model representation) with no information about the trajectory which is realized. The robust leader-follower strategy for each player must confront with all possible scenarios simultaneously. The problem of each player is the designing of min-max strategies for each player which guarantee an equilibrium for the worst case scenario. Based on the robust maximum principle, the conditions for a game to be in robust leader-follower equilibrium are presented. As in the Nash equilibrium case the initial min-max differential game may be converted into a standard static game given in a multidimensional simplex. A numerical procedure for resolving the case of linear quadratic differential game is presented.

HOSM observer for robust output regulator in uncertain nonlinear polynomial systems

This paper studies the problem of a robust optimal regulation for a nonlinear polynomial system based only on output information. The parameters describing the dynamics of the nonlinear polynomial plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and the application of a certain control input is associated with the worst or least favorable value of the unknown parameter. A high order sliding mode observer is designed for the nonlinear plant in such a way that the worst case control can be applied for a system with incomplete information. Additionally a matched type uncertainty is also compensated by means of the same output signal. A numerical example is given to illustrate the effectiveness of the approach.

Robust output Nash strategies based on sliding mode observation in a two-player differential game

This paper tackles the problem of a two-player differential game affected by matched uncertainties with only the output measurement available for each player. We suggest a state estimation based on the so-called algebraic hierarchical observer for each player in order to design the Nash equilibrium strategies based on such estimation. At the same time, the use of an output integral sliding mode term (also based on the estimation processes) for the Nash strategies robustification for both players ensures the compensation of the matched uncertainties. A simulation example shows the feasibility of this approach in a magnetic levitator problem.

Robust Control of a Nonlinear Electrical Oscillator Modeled by Duffing Equation

Abstract This paper studies the control of a nonlinear electrical circuit exemplified by a Duffing equation, which contains parameter uncertainties in the right hand side of this nonlinear differential equation. Even though this type of circuit has been object of a variety of control strategies in the past, very few papers have been devoted to the design of an optimal control law with a quadratic performance index and subject to that type of uncertainties. We develop simulation examples that show that not only an optimal control strategy can be applied to such a system, in case of an optimal control strategy without uncertainties, but an optimal control of the mini-max type can also be implemented in case of the uncertainty in the model.

Equilibrium in Linear Quadratic Stochastic Games with Unknown Parameters

In this paper we present a linear quadratic stochastic game in which some parameters as well as some game states cannot be determined by any of the players. Such a problem is not solvable using standard approach. Therefore, we develop strategies based on a technique that allows each player to simultaneously estimate both unknown states and parameters. The separation principle, which holds for this problem, is then applied to synthesize strategies that ensure the feedback Nash equilibrium. Stochastic differential games, estimations, Nash equilibirum.

Formation flight of fixed-wing UAVs based on linear quadratic affine game

In this paper we propose a model of zero-sum differential games (ZSG) to perform the formation flight of a group of three fixed-wing Unmanned Aerial Vehicles (UAVs). The dynamics of these UAVs are described in the linearized model around nominal values. For the formation strategy, the game scheme for one of the UAVs acts as the leader vehicle following a freely designed trajectory meanwhile the two vehicles just follow the leader. A reduced dimension Riccati equation is solved in order to achieve the desired formation. A numerical example is provided to prove the strategy of the formation fight.

Higher order sliding modes manifold design via singular LQ control

Abstract The paper shows how the solution of Singular Optimal Stabilization Problem (SOSP) for nominal system can be used for the sliding manifold design for higher order sliding mode control (HOSMC) for uncertain system. This paper consists of two main parts. In the first part, the natural connection between the order of singularity of SOSP and the order of sliding mode is established, and as a conclusion it is shown that the Singular Optimal Manifold (SOM) can be considered as the sliding surface for HOSMC of the corresponding order ensuring the sliding modes for uncertain system. In the second part we have designed a feasible curve to arrive to the SOM in prescribed time taking into account actuators limitations. It is shown that integral HOSMC ensures the exact tracking to this curve starting from the initial time moment in spite of uncertainties.

Robust Mini-Max regulator for uncertain nonlinear polynomial systems

In this paper, we present a solution to the problem of the quadratic Mini-Max regulator for polynomial uncertain systems. The main characteristic of this type of problems is that the parameters describing the dynamic of the nonlinear plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and the solution is given in terms of the worst case scenario. That is to say, the result of the application of a certain control input (in terms of the cost function value) is associated with the worst or least favorable value of the unknown parameter. Based on the general necessary conditions for mini-max optimality, a closed form for the control is provided, which makes use of a p-linear form tensor representation of the polynomial system. In its turn, this allows one to present the solution in a way similar to the so called Riccati technique. The final control is shown to be a convex combination (with some weights) of the optimal controls for each fixed parameter of the polynomial system. A numerical example for the robust regulation of the well-known Duffing equation which represents a typical (and challenging to control) nonlinear polynomial system is presented, to show the effectiveness of our approach.