Ph. D. Project
Optimal output control
2019/09/30 - 2022/10/30
1. General problematic, Context

Many results exist in controllability and optimal controllability. For instance, the controllability of finite dimensional linear systems is ensured by the Kalman rank condition. Let us point out that a similar result exists for the controllability of the system output. Under this last condition, in any positive time, every initial state can be steered to a target output.
It is obvious that state controllability implies output controllability.

When the linear system is controllable, the pole placement Theorem ensures the existence of a regulator K such that the control u=Kx stabilises the system state. Moreover, if we aim to minimize a quadratic cost in state and control, then the optimal control is a feedback control, u=Kx, where K is obtained through a Riccati equation. Note that this matrix K is independent of the initial state of the system.

When we only know the system output, it is commonly used to build a state observer to estimate the state value. Then this estimate is used to compute the control. But this strategy does not ensure the optimality of the control.
In this Ph.D. Thesis, we do not aim to use state observers, but we aim to use directly the system output.
We thus aim to find an optimal control of the form u=Fy.

2. Goals

In a first time, we will focus to linear control systems. Extensions to non-linear affine control system can be considered in a second time. In addition, the questions posed above can also be treated with discrete time systems.

The goal of this Ph.D. Thesis is to explore in details questions about optimal output controllability. Some natural questions are:
-- Is a quadratic cost in state and control well-defined for every initial condition?
In fact, if the system is state controllable, then one can steer any initial condition to zero in some positive time and then take the null control for later times. But, if the system is only output controllable, this strategy is not possible. In fact one can only steers the system output to zero and this does not imply that all the system state is null. Then, the output controllability is not enough to ensure the well-posedness of the cost.
-- If the minimum of the cost is finite, is it possible to express the optimal controls as output feedbacks?
If this is the case, can the output regulator be expressed as a solution of a Riccati equation? Is it possible to give the relation between the output regulator and the system initial condition?

It has been shown that it is generically impossible to find an optimal control of the form u=Fy, with F independent of the initial condition, cf. [1]. A strategy to overcome this difficulty, is to try to find an averaged regulator, the averaged being taken with respect to the initial condition. This type of approach has been for instance considered in [2] and [3]. However, this strategy is not completely convincing with regard to the optimality of the initial cost (which is dependent of the initial state).

Another goal is to prescribe a system output. More precisely, what are the conditions ensuring that the system output stays in the neighborhood or on a target? Such a result, combined with the previous goal, would allow us to do optimal output regulation.

In this Ph.D. Thesis, we are going to consider the above questions. In particular, we will try to obtain necessary and sufficient conditions for the existence of an optimal control, the expression of this control as an output feedback...
The results obtained during this thesis could be for instance used in the context of averaged control, cf. [4].

More details on this Ph.D. offer can be found at:

[1] L. Huang and Z. Li. Solvability of quadratic optimal control via output feedback. Science in China Series A-Mathematics, Physics, Astronomy & Technological Science , 33(10) :12381245, 1990.
[2] Z.-G. Yang and X.-H. Wang. Fundamental theorem for optimal output feedback problem with quadratic performance criterion. In 2006 6th World Congress on Intelligent Control and Automation , volume 1, pages 18001804, June 2006.
[3] W. Levine and M. Athans. On the determination of the optimal constant output feedback gains for linear multivariable systems. IEEE Transactions on Automatic Control , 15(1) :4448, February 1970.
[4] E. Zuazua. Averaged control. Automatica , 50(12) :30773087, 2014.
Output controllability, Optimal control, Averaged control, Riccati Equation
Control Identification Diagnosis