## Constrained model predictive control, state estimation and coordination

- Author(s): Yan, Jun
- et al.

## Abstract

In this dissertation, we study the interaction between the control performance and the quality of the state estimation in a constrained Model Predictive Control (MPC) framework for systems with stochastic disturbances. This consists of three parts: (i) the development of a constrained MPC formulation that adapts to the quality of the state estimation via constraints; (ii) the application of such a control law in a multi-vehicle formation coordinated control problem in which each vehicle operates subject to a no-collision constraint posed by others' imperfect prediction computed from finite bit-rate, communicated data; (iii) the design of the predictors and the communication resource assignment problem that satisfy the performance requirement from Part (ii). Model Predictive Control (MPC) is of interest because it is one of the few control design methods which preserves standard design variables and yet handles constraints. MPC is normally posed as a full-state feedback control and is implemented in a certainty-equivalence fashion with best estimates of the states being used in place of the exact state. However, if the state constraints were handled in the same certainty-equivalence fashion, the resulting control law could drive the real state to violate the constraints frequently. Part (i) focuses on exploring the inclusion of state estimates into the constraints. It does this by applying constrained MPC to a system with stochastic disturbances. The stochastic nature of the problem requires re-posing the constraints in a probabilistic form. Using a gaussian assumption, the original problem is approximated by a standard deterministic constrained MPC problem or the conditional mean process of the state (the prediction). The state estimates' conditional covariances appear in tightening the constraints as measuring the necessary standoff from the bound on the real state. ̀Closed-loop covariance' is introduced to reduce the infeasibility and the conservativeness caused by using long-horizon, open-loop prediction covariances. The resulting control law is applied to a telecommunications network traffic control problem as an example. The idea of posing and transforming a probabilistic MPC problem works well, but not limited to, linear systems. In Part (ii), we consider applying constrained MPC as a local control law in a coordinated control problem of a group of distributed autonomous systems. Interactions between the systems are captured via constraints. First, we inspect the application of constrained MPC to a completely deterministic case. Formation stability theorems are derived for the subsystems and conditions on the local constraint set are derived in order to guarantee local stability or convergence to a target state. If these conditions are met for all subsystems, then this stability is inherited by the overall system. For the case when each subsystem suffers from disturbances in the dynamics, own self- measurement noises, and quantization errors on neighbors' information due to the finite-bit-rate channels, the constrained MPC strategy developed in Part (i) is appropriate to apply. Disturbance attenuation, or ̀s̀tring stability", is studied in this framework and it is shown that inactivity of the MPC constraints implies stability. This then provides a connection between control objective, communications resource assignment and performance. A one-dimensional vehicle example is computed to crystallize ideas. The application of this part is not restricted to linear systems. In Part (iii), we discuss the local predictor design and bandwidth assignment problem in a coordinated vehicle formation context. The MPC controller used in Part (ii) relates the formation control performance and the information quality in the way that large standoff implies conservative performance. If the communication channels used to exchange local information are noiseless, but have only finite bit-rate, the bits assigned to each variable in the information package will change the prediction error covariance, and hence the control performance, via the quantization errors which can be regarded as measurement noises. In this part, we aim at deriving the minimal communication resource and the corresponding bit-rate assignment strategy the corresponding stable state predictors that is used to formulate the MPC constraints. We first develop an LMI (Linear Matrix Inequality) formulation for cross-estimator design in a simple two-vehicle scenario with non-standard information: one vehicle does not have access to the other's exact control value applied at each sampling time, but to its known, pre-computed, coupling linear feedback control law. Then a similar LMI problem is formulated for the bandwidth assignment problem that minimizes the total number of bits by adjusting the prediction gain matrices and the number of bits assigned to each variable. This LMI formulation takes care of the constraint on steady state prediction error covariance imposed by the formation performance requirement, the constraint on the limited total bandwidth, and the constraint on the predictors being stable. Some linear approximation is used to include the bandwidth assignment variables in the LMI formulation. The solution of the resulting LMIs guarantees the feasibility of the bandwidth assignment scheme and stable predictors, but not optimality. An example of a three- vehicle formation is also provided. The LMI formulation here is restricted to linear systems