Generalized ${{\mathcal{H}}}_{2}$ Control of a Linear Continuous-Discrete System on a Finite Horizon

Abstract

This paper considers a linear continuous-discrete time-varying system described by a set of differential and difference equations on a finite horizon. For such a hybrid system, the concept of the generalized ${{\mathcal{H}}}_{2}$ norm is introduced, representing the induced norm of a linear operator generated by the system under consideration. This norm is characterized in terms of Lyapunov difference equations and also in terms of recursive linear matrix inequalities. Discrete time-varying optimal controllers, including multiobjective ones, that minimize the generalized ${{\mathcal{H}}}_{2}$ norm of the closed loop system are designed.

Appendices

Appendix

Proof of Theorem 1. For operator (3) consider its dual operator ${{\mathcal{S}}}^{* }$ defined as

$${{\mathcal{S}}}^{* }:\{{z}_{k}\}\mapsto \left({x}_{0},{\xi }_{0},v(t),\{{w}_{k}\}\right),$$

(A.1)

where

$$z\in {l}_{1}([0,N],{{\mathbb{R}}}^{n}),\quad ({x}_{0},{\xi }_{0},v,w)\in {{\mathbb{R}}}_{2}^{{n}_{x}}\times {{\mathbb{R}}}_{2}^{{n}_{\xi }}\times {L}_{2}([{t}_{0},{t}_{N}],{{\mathbb{R}}}^{{n}_{v}})\times {l}_{2}([0,N-1],{{\mathbb{R}}}^{{n}_{w}}).$$

The norm of the operator ${{\mathcal{S}}}^{* }$ is given by

$${\left\Vert {{\mathcal{S}}}^{* }\right\Vert }_{(R,2)/1}=\sup \left\{{\left\Vert \left({x}_{0},{\xi }_{0},v,w\right)\right\Vert }_{(R,2)}:{\parallel}z{\parallel }_{{l}_{1}}\le 1\right\};$$

due to duality, it has the property

$${\left\Vert {\mathcal{S}}\right\Vert }_{\infty /(R,2)}={\left\Vert {{\mathcal{S}}}^{* }\right\Vert }_{(R,2)/1}.$$

(A.2)

Thus, the norm of the operator ${\mathcal{S}}$ can be calculated through the norm of its dual operator ${{\mathcal{S}}}^{* }$, which is much easier to calculate; see below.

For obtaining an expression for the operator ${{\mathcal{S}}}^{* }$, consider an element $z\in {l}_{1}([0,N],{{\mathbb{R}}}^{n})$. Then

$$\langle z,{\mathcal{S}}y\rangle ={\langle {{\mathcal{S}}}^{* }z,y\rangle }_{(R,2)},$$

(A.3)

where y = (x₀, ξ₀, v, w) and the scalar product in the left-hand side is given by

$$\langle z,\zeta \rangle =\mathop{\sum }\limits_{k=0}^{N-1}{z}_{k}^{\top }{\zeta }_{k}.$$

The scalar product in the right-hand side has the form

$${\langle {y}_{1},{y}_{2}\rangle }_{(R,2)}={x}_{0,1}^{\top }{R}_{x}^{-1}{x}_{0,2}+{\xi }_{0,1}^{\top }{R}_{\xi }^{-1}{\xi }_{0,2}+\mathop{\sum }\limits_{k=0}^{N-1}{w}_{1,k}^{\top }{w}_{2,k}+\mathop{\int}\limits_{{t}_{0}}^{{t}_{N}}{v}_{1}^{\top }(\tau ){v}_{2}(\tau )d\tau ,$$

which matches well the definition of norm (4).

Rewrite system (1) in the semi-discrete form. This means that the continuous variable x is subject to discretization, whereas the continuous external disturbances v remains untouched:

$$\begin{array}{rcl}{\zeta }_{k+1}&=&{\widehat{A}}_{k}{\zeta }_{k}+{{\mathcal{B}}}_{k}{\omega }_{k},\\ {z}_{k}&=&{\widehat{C}}_{k}{\zeta }_{k},\end{array}$$

(A.4)

where ω_k = column(v(t), w_k) and

$$\begin{array}{rcl}{{\mathcal{B}}}_{k}&:&{L}_{2}\left([{t}_{k},{t}_{k+1}),{{\mathbb{R}}}^{{n}_{v}}\right)\times {{\mathbb{R}}}^{{n}_{w}}\to {{\mathbb{R}}}^{{n}_{x}+{n}_{\xi }}\\ &:&\left[\begin{array}{c}v(t)\\ {w}_{k}\end{array}\right]\mapsto \left[\begin{array}{c}\mathop{\int}\limits_{{t}_{k}}^{{t}_{k+1}}\Phi ({t}_{k+1},\tau ){B}_{c}(\tau )v(\tau )d\tau \\ {B}_{d,k}{w}_{k}\end{array}\right].\end{array}$$

(A.5)

Now write a closure connecting the vectors

$$\widetilde{y}={\rm{column}}\,({\zeta }_{0},{\omega }_{0},\ldots ,{\omega }_{N-1})\quad {\rm{and}}\quad \widetilde{z}={\rm{column}}\,({z}_{0},\ldots ,{z}_{N}).$$

For this purpose, take advantage of relation (A.4). As a result,

$$\widetilde{z}=\widetilde{C}\widetilde{A}\widetilde{{\mathcal{B}}}\widetilde{y},$$

(A.6)

where

$$\widetilde{C}={\rm{diag}}\left({\widehat{C}}_{0},{\widehat{C}}_{1},\ldots ,{\widehat{C}}_{N}\right),\quad \widetilde{{\mathcal{B}}}={\rm{diag}}\left(I,{{\mathcal{B}}}_{0},\ldots ,{{\mathcal{B}}}_{N-1}\right),\widetilde{A}=\left[\begin{array}{ccccc}I&0&0&\cdots \ &0\\ {\widehat{A}}_{0}&I&0&\cdots \ &0\\ {\widehat{A}}_{1}{\widehat{A}}_{0}&{\widehat{A}}_{1}&I&\cdots \ &0\\ \vdots &\vdots &\vdots &\ddots &\vdots \\ {\widehat{A}}_{N-1}\ldots {\widehat{A}}_{0}&{\widehat{A}}_{N-1}\ldots {\widehat{A}}_{1}&{\widehat{A}}_{N-1}\ldots {\widehat{A}}_{2}&\cdots \ &I\end{array}\right].$$

Thus, the operator ${\mathcal{S}}$ can be represented as ${\mathcal{S}}=\widetilde{C}\widetilde{A}\widetilde{{\mathcal{B}}}$. In view of expression (A.3), the dual operator ${{\mathcal{S}}}^{* }$ can be obviously written as

$${{\mathcal{S}}}^{* }={\widetilde{{\mathcal{B}}}}^{* }{\widetilde{A}}^{\top }{\widetilde{C}}^{\top },$$

(A.7)

where ${\widetilde{{\mathcal{B}}}}^{* }={\rm{diag}}\left({R}^{-1},{{\mathcal{B}}}_{0}^{* },\ldots ,{{\mathcal{B}}}_{N-1}^{* }\right)$ and

$${{\mathcal{B}}}_{k}^{* }:{{\mathbb{R}}}^{{n}_{x}+{n}_{\xi }}\to {L}_{2}\left([{t}_{k},{t}_{k+1}),{{\mathbb{R}}}^{{n}_{v}}\right)\times {{\mathbb{R}}}_{2}^{{n}_{w}}:\left[\begin{array}{c}x\\ \xi \end{array}\right]\mapsto \left[\begin{array}{c}{B}_{c}^{\top }(t){\Phi }^{\top }({t}_{k+1},t)x\\ {B}_{d,k}^{\top }\xi \end{array}\right].$$

Next, the following expression can be easily checked using representation (A.7):

$${\parallel}{{\mathcal{S}}}^{* }z{\parallel }_{(R,2)}^{2}={\parallel}{{\mathcal{S}}}^{* }\widetilde{z}{\parallel }_{(R,2)}^{2}={\widetilde{z}}^{\top }\widetilde{C}\,W{\widetilde{C}}^{\top }\widetilde{z},$$

where

$$W=\widetilde{A}\widetilde{Q}{\widetilde{A}}^{\top },\quad \widetilde{Q}={\rm{diag}}\left({R}^{-1},{\widehat{Q}}_{0},\ldots ,{\widehat{Q}}_{N-1}\right).$$

Since the matrix $\widetilde{C}$ is block-diagonal, consider an auxiliary block-diagonal matrix $\widetilde{Y}={\rm{diag}}\left({Y}_{0},\ldots ,{Y}_{N}\right)$ with the same blocks on the leading diagonal as the matrix W. The blocks Y_k satisfy the linear recursive equation

$${Y}_{0}={R}^{-1},\quad {Y}_{k+1}={\widehat{A}}_{k}{Y}_{k}{\widehat{A}}_{k}^{\top }+{\widehat{Q}}_{k},$$

which coincides with Eq. (9). Moreover, the following transformations are valid:

$$\begin{array}{ccc}{\left\Vert {{\mathcal{S}}}^{* }\right\Vert }_{(R,2)/1}^{2}=\sup \left\{{\left\Vert {{\mathcal{S}}}^{* }z\right\Vert }_{(R,2)}^{2}:{\parallel}z{\parallel }_{{l}_{1}}\leqslant 1\right\}\\ =\sup \left\{{\widetilde{z}}^{\top }\widetilde{C}\,W{\widetilde{C}}^{\top }\widetilde{z}:{\parallel}z{\parallel }_{{l}_{1}}\leqslant 1\right\}\\ =\sup \left\{{\widetilde{z}}^{\top }\widetilde{C}\,\widetilde{Y}{\widetilde{C}}^{\top }\widetilde{z}:{\parallel}z{\parallel }_{{l}_{1}}\leqslant 1\right\}=\mathop{\max }\limits_{k=0,\ldots ,N}{\lambda }_{\max }\left({\widehat{C}}_{k}{Y}_{k}{\widehat{C}}_{k}^{\top }\right).\end{array}$$

(A.8)

The last expression in this chain coincides with (8). The proof of Theorem 1 is complete.

Proof of Theorem 2. For deriving expression (10), assume that the value of the generalized ${{\mathcal{H}}}_{2}$ norm is equal to γ^* and is achieved at a time instant k = k^*. In this case, there exists an element

$${\widetilde{z}}^{* }={\rm{column}}\,(0,\ldots ,0,{z}_{{k}^{* }},0,\ldots ,0),\quad {\parallel}{\widetilde{z}}^{* }{\parallel }_{{l}_{1}}=1$$

such that

$${\gamma }^{* }={\left\Vert {{\mathcal{S}}}^{* }{\widetilde{z}}^{* }\right\Vert }_{(R,2)}.$$

The equality ${\parallel}{\widetilde{z}}^{* }{\parallel }_{{l}_{1}}=1$ indicates that ${z}_{{k}^{* }}={e}_{\max }\left({\widehat{C}}_{{k}^{* }}{Y}_{{k}^{* }}{\widehat{C}}_{{k}^{* }}^{\top }\right)$, where ${\widetilde{y}}^{* }={{\mathcal{S}}}^{* }{\widetilde{z}}^{* }={\rm{column}}\,({\zeta }^{* },{\omega }_{0}^{* },\ldots ,{\omega }_{N-1}^{* })$ is the vector composed of the worst-case initial conditions and external disturbances; in addition, ${\parallel}{\widetilde{y}}^{* }{\parallel }_{(R,2)}={\gamma }^{* }$. Note that for calculating ${\widetilde{y}}^{* }$ it is necessary to select the k^*th column from the matrix representation of the operator ${{\mathcal{S}}}^{* }$:

$${\widetilde{y}}^{* }=\left[\begin{array}{c}{\zeta }^{* }\\ {\omega }_{0}^{* }\\ \vdots \\ {\omega }_{{k}^{* }}^{* }\\ \vdots \\ {\omega }_{N-1}^{* }\end{array}\right]=\left[\begin{array}{c}{R}^{-1}{\widehat{A}}_{0}^{\top }\ldots {\widehat{A}}_{{k}^{* }}^{\top }{\widehat{C}}_{{k}^{* }}^{\top }{z}_{{k}^{* }}\\ {{\mathcal{B}}}_{0}{\widehat{A}}_{1}^{\top }\ldots {\widehat{A}}_{{k}^{* }}^{\top }{\widehat{C}}_{{k}^{* }}^{\top }{z}_{{k}^{* }}\\ \vdots \\ {{\mathcal{B}}}_{{k}^{* }}{\widehat{A}}_{{k}^{* }}^{\top }{\widehat{C}}_{{k}^{* }}^{\top }{z}_{{k}^{* }}\\ \vdots \\ 0\end{array}\right]=\left[\begin{array}{c}{R}^{-1}{\Psi }_{{k}^{* },0}^{\top }{\widehat{C}}_{{k}^{* }}^{\top }{z}_{{k}^{* }}\\ {{\mathcal{B}}}_{0}{\Psi }_{{k}^{* },1}^{\top }{\widehat{C}}_{{k}^{* }}^{\top }{z}_{{k}^{* }}\\ \vdots \\ {{\mathcal{B}}}_{{k}^{* }}{\Psi }_{{k}^{* },{k}^{* }}^{\top }{\widehat{C}}_{{k}^{* }}^{\top }{z}_{{k}^{* }}\\ \vdots \\ 0\end{array}\right].$$

Finally, for obtaining expression (10), just multiply the vector ${\widetilde{y}}^{* }$ by 1/γ^*, because the vector of the worst-case initial conditions and external disturbances in the definition of the generalized ${{\mathcal{H}}}_{2}$ norm (5) satisfies the condition ${\parallel}{\widetilde{y}}^{* }{\parallel }_{(R,2)}=1$. The proof of Theorem 2 is complete.

Proof of Theorem 4. Write the equation of the closed loop system (12), (13):

$$\begin{array}{rcl}{\zeta }_{k+1}&=&\left({\widehat{A}}_{k}+{\widehat{H}}_{k}{\widehat{\Theta }}_{k}\right){\zeta }_{k}+{{\mathcal{B}}}_{k}{\omega }_{k},\\ {z}_{k}&=&\left({\widehat{C}}_{k}+{D}_{k}{\widehat{\Theta }}_{k}\right){\zeta }_{k}.\end{array}$$

(A.9)

In accordance with Theorem 3, the generalized ${{\mathcal{H}}}_{2}$ norm of system (A.9) can be calculated as the solution of problem (11) in which the matrices ${\widehat{A}}_{k}$ and ${\widehat{C}}_{k}$ are replaced by ${\widehat{A}}_{k}+{\widehat{H}}_{k}{\widehat{\Theta }}_{k}$ and ${\widehat{C}}_{k}+{D}_{k}{\widehat{\Theta }}_{k}$, respectively. Introducing the new variables ${Z}_{k}={\widehat{\Theta }}_{k}{Y}_{k}$, finally arrive in inequalities (14). The proof of Theorem 4 is complete.

Proof of Theorem 6. For proving this result, take notice that equality (17) can be written as

$$G(\Theta )=\mathop{\max }\limits_{j=1,\ldots ,m}\mathop{\max }\limits_{k=0,\ldots ,N}{\alpha }_{j}^{-1}{\lambda }_{\max }^{1/2}\left({\widehat{C}}_{k}^{(j)}{Y}_{k}{\widehat{C}}_{k}^{(j)\top }\right)=\mathop{\max }\limits_{j=1,\ldots ,m}\mathop{\max }\limits_{k=0,\ldots ,N}{\lambda }_{\max }^{1/2}\left({\alpha }_{j}^{-2}{\widehat{C}}_{k}^{(j)}{Y}_{k}{\widehat{C}}_{k}^{(j)\top }\right).$$

Hence, in inequalities (14) it suffices to replace the matrices ${\widehat{C}}_{k}$ and ${\widehat{D}}_{k}$ by the matrices ${\alpha }_{j}^{-1}{\widehat{C}}_{k}^{(j)}$ and ${\alpha }_{j}^{-1}{\widehat{D}}_{k}$, which yields inequalities (18). The proof of Theorem 6 is complete.

Funding

This work was supported by the Russian Foundation for Basic Research, projects nos. 18-41-520002, 19-01-00289, project no. 0729-2020-0055, and by the Research and Education Mathematical Center “Mathematics for Future Technologies.”