Chapter 3
Nash game based Mixed H 2 / H ∞ MPC - State Feedback Case
3.1 Introduction
Design of robust MPC is one of the most demanding and active areas of research amongst control engineers, researchers and practitioners since the past three decades.
MPC, in general, being an open-loop optimal control strategy, the problem of ad- dressing the robustness of the system, against model uncertainties and/or unknown disturbance, are of much concern in MPC design.
There are various approaches proposed so far to address this robustness issue, of which the feedback MPC design proposed in [96] is an interesting proposal, which recognises the need for feedback in the constrained optimal control problem solved online. Very recently, the first attempt has been made to bring the mixed H2/H∞
control design into receding horizon control strategy in the seminal paper [121], by extending the LMI based robust MPC control algorithm [96] solved online.
Mixed H2/H∞ control design has its own significance in the control commu- nity since its inception in [14]. The multi-objective optimal control design of mixed H2/H∞ design motivates different approaches to solve the control problem. In gen- eral, the desire to achieve conflicting objectives of the desired closed-loop perfor-
3.1. Introduction 54
mance specifications and robustness against external disturbances or uncertainty due to plant-model mismatch and/or noises, can be better handled by mixed H2/H∞
design. Nash game based approach is a well known strategy to obtain an optimal equilibrium (Nash equilibrium) in such conflicting objective decision making prob- lems. Thus the approach in [98] seems to be more appropriate for the mixedH2/H∞ design with the Nash game interpretation. Motivated by the work in [98], in the present work, the idea has been extended for the first time to the MPC paradigm.
However, this extension is not straight forward, since in [98] the design has been made for continuous time systems. Furthermore, it is known from LQ dynamic game the- ory [58, 82] that such an approach results in cross-coupled Riccati equations as that in [98]. Although, the conversion from continuous to discrete domain is possible, in general, using bilinear transformation, to the best of the author’s knowledge, there is no literature for such a conversion for cross-coupled Riccati equations resulting from linear-quadratic (LQ) Nash game into its equivalent discrete form. An LQ Nash game usually results in coupled non-symmetric Riccati equations which have to be solved to obtain the optimal solution or Nash equilibria. Nevertheless, the problem of solving non-symmetric Riccati equations is itself of much theoretical im- portance and it is still an active research area. A good survey of the methods to handle such non-symmetric Riccati equations can be found in [56, 57, 82]. The pair of discrete time non-symmetric cross-coupled algebraic Riccati equations (cAREs) obtained while solving an open-loop Nash game is dealt in [58]; where the invariant subspace method is adopted, for which necessary conditions are provided. A method of solving the cAREs is proposed in [58], where sufficient conditions are not provided and not elsewhere too to the best of the author’s knowledge. Hence, obtaining a unique stabilizing solution cannot be ensured.
On the other hand, the mixed H2/H∞ control that are usually available in the literature [98, 37] have dealt with the conventional feedback control systems and not in the setting of MPC. However, the development in the present work is for the discrete time systems in the MPC framework. The present work differs from [121] by exploring the mixedH2/H∞ MPC from a LQ game theory perspective and
3.1. Introduction 55
addressing its related issues. So once a two player LQ Nash game based mixed H2/H∞ robust controller for discrete time system has to be designed, a suitable means to solve the resulting discrete time non-symmetric coupled algebraic Riccati equations would be of much significance. One such suitable means, to obtain the optimal solution of the control problem, is dealt in the present work. In [82], a similar kind of problem is dealt as a soft constrained Nash game. However, the use of hard constraints, such as input actuator saturation constraints, which is quite common practically due to the actuator limitations, has not been dealt in [82]. Optimal control problem of linear systems subjected to hard constraints such as actuator constraints makes the control problem in hand nonlinear.
Optimal control for nonlinear systems are often designed as open-loop control framework. Achieving the required closed-loop stability and performance of non- linear systems make their control critical. Generally, in receding horizon control framework, which is an open-loop control strategy, finite horizon control input that satisfies the stability conditions is computed at every time instance. Only the first control value of the computed control law is implemented while the other values are discarded. However, on the other hand, infinite horizon closed-loop optimal control satisfies the closed-loop nominal stability. Hence, it is always preferred that an infi- nite horizon optimal control problem is solved online in order to ensure guaranteed stability. Owing to the resulting infinite-dimensionality, such infinite horizon control problem becomes computationally intractable. In contrast, the robust MPC design for saturated control input systems as in [96, 121] basically solves an infinite horizon control problem, where the feedback gain is calculated online at every time instance in the receding horizon framework. This results in a newly computed optimal state feedback gain matrix at every time instance. The location of the closed-loop state transition matrix poles in the stable region at every time instance ensures both closed-loop stability and optimal performance. Moreover, the feedback MPC, which satisfies the stability requirements implicitly, overcomes the dimensionality issues too, making it quite promising.