Optimal and Robust Control, 2nd Edition

Appendix B. Algebraic Riccati Equations

Algebraic Riccati equations appear in many problems discussed in this text. In this section we summarize the main algebraic Riccati equations for continuous-time systems.

Algebraic Riccati Equations for Optimal Control and Optimal Observer

$$\begin{array}{l}\text{PA}+{\text{A}}^T\text{P}-{\text{PBR}}^{-1}{\text{B}}^T\text{P}+\text{Q}=0\hfill \end{array}$$(B.1)

$$\begin{array}{l}\text{A}\Pi +\Pi {\text{A}}^T-\Pi {\text{C}}^T{\text{M}}_v^{-1}\text{C}\Pi +{\text{M}}_d=0\hfill \end{array}$$(B.2)

CARE and FARE for Closed Loop Balancing

$$\begin{array}{l}{\text{A}}^T\text{P}+\text{PA}-{\text{PBB}}^T\text{P}+{\text{C}}^T\text{C}=0\hfill \end{array}$$(B.3)

$$\begin{array}{l}\text{A}\Pi +\Pi {\text{A}}^T-\Pi {\text{C}}^T\text{C}\Pi +{\text{BB}}^T=0\hfill \end{array}$$(B.4)

CARE and FARE for Closed Loop Balancing for Not-strictly Proper Systems

$$\begin{array}{l}{\text{A}}^T\text{P}+\text{PA}-(\text{PB}+{\text{C}}^T\text{D}){(\text{I}+{\text{D}}^T\text{D})}^{-1}({\text{B}}^T\text{P}+{\text{D}}^T\text{C})+{\text{C}}^T\text{C}=0\hfill \end{array}$$(B.5)

$$\begin{array}{l}\text{A}\Pi +\Pi {\text{A}}^T-(\Pi {\text{C}}^T+{\text{BD}}^T){(\text{I}+{\text{DD}}^T)}^{-1}(\text{C}\Pi +{\text{DB}}^T)+{\text{BB}}^T=0\hfill \end{array}$$(B.6)

Algebraic Riccati Equations for Positive-real Systems

$$\begin{array}{l}\text{PA}+{\text{A}}^T\text{P}=-(-\text{PB}+{\text{C}}^T){(\text{D}+{\text{D}}^T)}^{-1}{(-\text{PB}+{\text{C}}^T)}^T\hfill \end{array}$$(B.7)

$$\begin{array}{l}\Pi {\text{A}}^T+\text{A}\Pi =-(-\Pi {\text{C}}^T+\text{B}){(\text{D}+{\text{D}}^T)}^{-1}{(-\Pi {\text{C}}^T+\text{B})}^T\hfill \end{array}$$(B.8)

Algebraic Riccati Equations ...