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Characterization of feedback Nash equilibria for multi-channel systems via a set of non-fragile stabilizing state-feedback solutions and dissipativity inequalities

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Abstract

We consider the problem of state-feedback stabilization for a multi-channel system in a game-theoretic framework, where the class of admissible strategies for the players is induced from a solution set of the individual objective functions that are associated with certain dissipativity properties of the system. In such a framework, we characterize the feedback Nash equilibria via a set of non-fragile stabilizing state-feedback solutions corresponding to a family of perturbed multi-channel systems. Moreover, we show that the existence of a weak-optimal solution to a set of constrained dissipativity problems is a sufficient condition for the existence of a feedback Nash equilibrium, whereas the set of non-fragile stabilizing state-feedbacks solutions is described in terms of a set of dilated linear matrix inequalities.

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Notes

  1. Recently, a similar dilated LMIs condition has been investigated by Fujisaki and Befekadu [13] in the context of reliable decentralized stabilization for multi-channel systems (see [10, 29] and references therein for a review of dilated LMI technique).

  2. Note that the existence of a solution for state trajectories is well defined and it is always upper semicontinuous in \(x_0\) (e.g., see [9] and [21]).

  3. Note that the upper bound \(\hat{\rho }\) continuously depends (in the weak sense) on \(x_0\) and \(K_j\), \(j=1, 2, \ldots , N\).

  4. In this paper, the game is essentially defined in the framework of an incomplete information, since the \(j\)th player’s objective function involves different uncertainty information, i.e., \(u_{\rho _j}\), about the system. However, we remark that the \(j\)th player decides his own strategy by solving the optimization problem with the opponents’ strategies \((K_{1}^*,\ldots ,K_{j-1}^*, K_{j+1}^*,\ldots ,K_{N}^*)\) fixed.

  5. In general, simultaneously solving a set of optimization problems, i.e., solving (28) together with (27), is not easy since it is a non-convex optimization problem which involves finding a solution satisfying at the intersection of a set of constrained quadratic functionals [42] (c.f. Remark 3, Sect. 2 above).

  6. Here we remark that a strong version of fixed-point theorem is required to establish the existence of feedback Nash equilibria for the game \(\Gamma \), which is defined on compact topological spaces with continuous objective functions (e.g., see [14]). To this end, if we introduce the following continuous map  \(\Phi _{[x_0, u_{\hat{\rho }}]} :\mathcal K _N \times \mathcal K _N \rightarrow \mathbb R \) defined by

    $$\begin{aligned} \Phi _{[x_0, u_{\hat{\rho }}]} (K, \bar{K}) = \sum _{j=1}^N \left(J_j(x_0, u_{\hat{\rho }_j}, K) - J_j(x_0, u_{\hat{\rho }_j}, K_{\lnot j}) \right), \end{aligned}$$

    where \(K = ( K_1, K_2, \ldots , K_N ) \in \mathcal K _N \), \(\bar{K} = (\bar{K}_1, \bar{K}_2, \ldots , \bar{K}_N ) \in \mathcal K _N\), \(K_{\lnot j} = (K_1, \ldots , \bar{K}_j \ldots , K_N) \in \mathcal K _N\), \(j=1, 2, \ldots , N\) and \(u_{\hat{\rho }} \triangleq (u_{\hat{\rho }_1}, u_{\hat{\rho }_2}, \ldots , u_{\hat{\rho }_N}) \in \prod _{j=1}^N [-\hat{\rho }_j, \hat{\rho }_j]\). Note that for such a map whose fixed-point is an equilibrium is called a Nash map for the game \(\Gamma \), i.e., if the \(N\)-tuple \( (K_1^*, K_2^*, \ldots , K_N^*) \) is a feedback Nash equilibrium, then \(J_j(x_0, u_{\hat{\rho }_j}, K_{\lnot j}^*) \le J_j(x_0, u_{\hat{\rho }_j}, K^*)\) with \(K_j \in \mathcal K _j\) for all \(j \in \{1, 2, \ldots , N\}\). This further shows that the map \(\Phi \) satisfies \(\Phi _{[x_0, u_{\hat{\rho }}]} (K^*, K) \ge 0\) for any arbitrary \(K=(K_1, K_2, \ldots , K_N)\in \mathcal K _N\). Therefore, the feedback Nash equilibrium \(K^*\) is an equilibrium point, i.e., a fixed point, for the map \(\Phi _{[x_0, u_{\hat{\rho }}]}(.,.)\).

  7. Note that the stability behavior is considered here over an infinite-time horizon.

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Acknowledgments

This work was supported in part by the National Science Foundation under Grant No. CNS-1035655; G. K. Befekadu acknowledges support from the Moreau Fellowship of the University of Notre Dame.

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  1. Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Getachew K. Befekadu, Vijay Gupta & Panos J. Antsaklis

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  1. Getachew K. Befekadu
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Befekadu, G.K., Gupta, V. & Antsaklis, P.J. Characterization of feedback Nash equilibria for multi-channel systems via a set of non-fragile stabilizing state-feedback solutions and dissipativity inequalities. Math. Control Signals Syst. 25, 311–326 (2013). https://doi.org/10.1007/s00498-012-0105-z

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