Output Feedback Stochastic Model Predictive Control for Linear Systems with Convex Optimization Approach

Abstract

In this paper, a stochastic model predictive controller is designed for discrete time linear time invariant systems, considering additive disturbance and stochastic constraints. As we know, in practical applications, measuring all state information of a system is not generally possible or affordable. So, in this work, an output feedback law is assumed as the control law. By utilizing the Chebyshev inequality and Schur complement, it is tried to convert a stochastic non-convex optimization problem into a deterministic convex optimization problem. Simulation results demonstrate the effectiveness of the proposed methodology.

Appendix (Proof of Recursive Feasibility)

To characterize the properties of the proposed algorithm, we establish the feasibility set of the SMPC control problem as follows:

$\Xi =\{$(${\overline{y} }_{0}, {Y}_{0}): \exists {\overline{u} }_{0,\dots ,N-1},{K}_{0,\dots ,N-1}$ such that (13a), (13b) hold for all $k=0,\dots ,N-1$ and are verified ${\overline{y} }_{t+N}\in {Y}_{f}$ and ${Y}_{{\text{t}}+{\text{N}}}\le \overline{Y }$}.

As $\overline{Y }$ is a terminal condition of $Y$, it plays the role of an invariant set for the variance. Assume that at time $t$, a feasible solution of SMPC is available as (${\overline{y} }_{t|t}, {Y}_{t|t})\in\Xi$ with optimal sequence {${\overline{u} }_{t|t}, \dots ,{\overline{u} }_{t+N-1|t}\}$ and ${\{K}_{t|t}, \dots , {K}_{kt+N-1|t}\}.$ We prove that, at time $t+1$, at least a feasible solution to SMPC exists (${\overline{y} }_{t+1|t}, {Y}_{t+1|t})$ with feasible possibly suboptimal, sequences {${\overline{u} }_{t+1|t}, \dots ,{\overline{u} }_{t+N-1|t}\}$ and ${\{K}_{t+1|t}, \dots , {K}_{kt+N-1|t}\}.$

Constraint (13a) is verified for all pairs (${\overline{y} }_{t+1+k|t}, {Y}_{t+1+k|t}) , k=0, \dots , N-2$ in view of the feasibility of SMPC at time$t$. Furthermore, in view of ${\overline{y} }_{t+N}\in {Y}_{f}$ and ${Y}_{{\text{t}}+{\text{N}}}\le \overline{Y }$, we have

$$h_{y,j} \overline{y}_{t + N|t} \le \left( {1 - 0.5\varepsilon } \right)g_{y,j} - \frac{{h_{y,j}^{2} Y_{k} }}{{2\varepsilon g_{y,j} \beta_{j} }} \le \left( {1 - 0.5\varepsilon } \right)g_{y,j} - \frac{{h_{y,j}^{2} Y_{t + N|t} }}{{2\varepsilon g_{y,j} \beta_{j} }}$$

(29)

For all $j=1,\dots ,P$ constraint (13a) is verified.

Analogously, constraint (13b) is verified for all pairs (${\overline{u} }_{t+1+k|t}, {U}_{t+1+k|t})$,$k=0, \dots , N-2,$ in view of the feasibility of SMPC at time $t$. Furthermore, in view of ${\overline{y} }_{t+N}\in {Y}_{f}$ and ${Y}_{{\text{t}}+{\text{N}}}\le \overline{Y }$ and the condition (13b), we have that

$$h_{y,j} k\overline{y}_{t + N|t} \le \left( {1 - 0.5\varepsilon } \right)g_{y,j} - \frac{{h_{y,j}^{2} U_{k} }}{{2\varepsilon g_{y,j} \beta_{j} }} \le \left( {1 - 0.5\varepsilon } \right)g_{y,j} - \frac{{h_{y,j}^{2} YU_{t + N|t} }}{{2\varepsilon g_{y,j} \beta_{j} }}$$

(30)

For all all $j=1,\dots ,P$ constraint (13b) is verified.