# Riccati-based $$\mathcal H _ \infty$$-control for DAEs

Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

ECC 2020

# Introduction

## Descriptor systems

$\def\pd{^{\mathsf{d}}} \def\pa{^{\mathsf{a}}} \def\Hinf{\mathcal{H} _ \infty} \def\indoe{\begin{bmatrix}I&0\\0&0\end{bmatrix}} \def\indoa{\begin{bmatrix}A&0\\0&I\end{bmatrix}} \def\genbobt{\begin{bmatrix}B_1 & B_2\end{bmatrix}} \def\gencoct{\begin{bmatrix}C_1 \\ C_2\end{bmatrix}} \def\tgenbobt{\begin{bmatrix}\tilde B_1 & \tilde B_2\end{bmatrix}} \def\tgencoct{\begin{bmatrix}\tilde C_1 \\ \tilde C_2\end{bmatrix}} \def\mybobt{\begin{bmatrix}B_1\pd & B_2\pd \\ B_1\pa & 0\end{bmatrix}} \def\mycoct{\begin{bmatrix}C_1\pd & C_1\pa \\ C_2\pd & 0\end{bmatrix}} \def\mybo{\begin{bmatrix}B_1\pd \\ B_1\pa \end{bmatrix}} \def\mybt{\begin{bmatrix}B_2\pd \\ 0 \end{bmatrix}} \def\myco{\begin{bmatrix}C_1\pd & C_1\pa \end{bmatrix}} \def\myct{\begin{bmatrix}C_2\pd & 0 \end{bmatrix}} \def\gpmt{\gamma^{-2}} \newcommand\tnsqrd[1]{ \| #1 \| _ 2^2} \def\sqrod{\tilde D^{\frac{1}{2}}} \def\msqrod{\tilde D^{-\frac{1}{2}}} \def\Xinf{\mathcal X _ \infty} \def\xinf{X _ \infty}$

Descriptor systems have an ODE part and an algebraic part

$\begin{split} \begin{bmatrix} I & 0 \\ 0 & N \end{bmatrix} \dot x & = \begin{bmatrix} A & 0 \\ 0 & I \end{bmatrix} x + \begin{bmatrix} B^{\mathsf{d}} \\ B^{\mathsf{a}} \end{bmatrix}u \\ y & = \begin{bmatrix} C^{\mathsf{d}} & C^{\mathsf{a}} \end{bmatrix}x \end{split}$

(Note that $$N$$ is nilpotent and not invertible)

## Transfer functions

$\begin{split} G(s) &= \begin{bmatrix} C^{\mathsf{d}} & C^{\mathsf{a}} \end{bmatrix} \begin{bmatrix} sI-A & 0 \\ 0 & sN-I \end{bmatrix}^{-1} \begin{bmatrix} B\pd \\ B\pa \end{bmatrix}\\ & \quad = C\pd(sI-A)^{-1}B\pd + C\pa\sum _ {i=0}^{\nu}(sN)^iB\pa \\ & \quad = G\pd(s) + G\pa(s) \end{split}$

• ODE part $$G\pd$$ -- the strictly proper part

• algebraic part $$G\pa$$ -- polynomial part

## Control with DAE-Riccati

• Typically done: control the ODE part, keep the polynomial part

• for control -- OK, as long as the polynomial part is zero

• in fact, in Benner, Heiland, Weichelt: $$B\pa=0$$
• if not, the Riccatis need to consider the algebraic part
• see, e.g., Möckel, Reis, and Stykel (2011)

# $$\Hinf$$-control

## The basic equations

\begin{align*} \mathcal E\dot{x} &= \mathcal Ax\phantom{_1} + B_1w\phantom{_1} + B_2 u, \\ z &= C_1 x + D_{11} w + D_{12} u, \\ y &= C_2 x + D_{21} w + D_{22} u. \end{align*}
• generally written as $$\Sigma \sim (\mathcal E,\mathcal A,[B_1,B_2],[C_1,C_2],D)$$

## The $$\mathcal H _ \infty$$ control problem

• For now, we assume that we can do state-feedback.

• Then the suboptimal $$\Hinf$$ control problem reads

1. find $$\gamma$$ and $$K$$ such that $$(\mathcal E,\mathcal A-B_2K)$$ is admissible1

2. and such that, with $$u=-B_2Kx$$, the map of the perturbance to the performance output is bounded, i.e. $\|z\|_2 < \gamma \|w\|_2$

• Assumption: $$(\mathcal E, \mathcal A)$$ is of index 1

• wlog: Weierstraß Canonical Form $\Sigma\sim \bigl(\indoe,\indoa,\genbobt,\gencoct,D\bigr)$

• wlog: $$D=0$$: $\Sigma\sim \bigl(\indoe,\indoa,\tgenbobt,\tgencoct,0 \bigr)$

• Assumption: $$B_2\pa=0$$, $$C_2\pa=0$$ $\Sigma\sim \bigl(\indoe,\indoa,\mybobt,\mycoct,0 \bigr)$

• Equivalence: to a standard LTI system with feedthrough $\Sigma\sim \bigl(I,A,\begin{bmatrix}B_1\pd&B_2\pd\end{bmatrix},\begin{bmatrix}C_1\pd\\C_2\pd\end{bmatrix}, \begin{bmatrix}D_{11}&0\\0&0\end{bmatrix} \bigr)$

# Equivalence of DAE and ODE Riccati Feedback

## Main results

In this index-1 case,

1. Non-symmetric Riccati Feedback is the standard $$\Hinf$$-Riccati-Feedback for the equivalent ODE system with feedthrough $$D_{11}$$.

2. The (projected) symmetric DAE Riccati simply neglects the feedthrough.

## LTI with feedthrough

Consider

$\dot x = Ax + B_1w + B_2u, \quad x(0)=0$

and $z = \begin{bmatrix} C_1 x + D _ {11}w \\ u \end{bmatrix}.$

## The modified Hamiltonian

With $$X _ \infty$$ being a stabilizing solution to the Riccati equation associated with the Hamiltonian pencil2 $\small \begin{split} &\begin{bmatrix} -sI+A & 0 \\ {C_1\pd}^*{C_1\pd} & -sI - A^* \end{bmatrix} \\ &-\begin{bmatrix} - B_{1}\pd (- \gamma^2 + D_{11}^* D_{11})^{-1} D_{11}^* C_{1}\pd & B_{2}\pd {B_{2}\pd}^* + B_{1}\pd (- \gamma^2 + D_{11}^* D_{11})^{-1} {B_{1}\pd}^*\\ - {C_{1}\pd}^* D_{11} (- \gamma^2 + D_{11}^* D_{11})^{-1} D_{11}^* C_{1}\pd & {C_{1}\pd}^* D_{11} (- \gamma^2 + D_{11}^* D_{11})^{-1} {B _ {1}\pd}^* \end{bmatrix} \end{split}$ the feedback $$u=-B_2X _ \infty x$$ solves the robust regulator problem.

## As a descriptor system

$\indoe \dot x = \indoa x + \mybo w + \mybt u$ with $z = \begin{bmatrix} \myco x \\ u \end{bmatrix}.$ (note that there is no $$D$$!)

## The DAE Riccati

Define the feedback as $$u = -\mybt^ * \Xinf x$$, where $$\Xinf$$ is an admissible solution to the nonsymmetric generalized Riccati equation $\begin{split} \mathcal A^* X + X^* \mathcal A - X^*(B_2B_2^* - \gpmt B_1B_1^*)X + C_1^*C_1 = 0, \\ \quad \mathcal E^*X = X^*\mathcal E. \end{split}$

For this Riccati equation, with $$\mathcal E=\indoe$$ and $$\mathcal A=\indoa$$:

### Lemma

If $$\sigma_{\max{}}(C_1\pa B_1\pa) = \sigma_{\max{}}(D_{11}) < \gamma^2$$ and there exists a $$\gamma$$-stabilizing controller, then

1. $$\Xinf$$ exists, is admissable, and looks like $$\begin{bmatrix} X_\infty & 0 \\ X_{21} & X_{22} \end{bmatrix}$$, where

2. $$\xinf$$ is the stabilizing solution associated with the Hamiltonian pencil of the LTI with $$D_{11}=-C_1\pa B_1\pa$$.

Thus,

• the feedback $$u = -\mybt \Xinf x = -B_2\pd X_\infty x\pd$$ is the same as in the standard case,

• the $$\Hinf$$-performance bound, $\tnsqrd{z} = \tnsqrd{C_1\pd x\pd-C_1\pa B_1\pa w}+\tnsqrd{u} \leq \gamma^2\tnsqrd{w}$ follows

• directly from examining $$\frac{d}{dt}x^*\mathcal E^*\Xinf x$$
• or from the equivalence to the LTI system and the feedback.

## Symmetric Riccati Equation

$\mathcal A^*X\mathcal E + \mathcal E^*X\mathcal A - X(B_2B_2^* - \gpmt B_1B_1^*)X + PC_1^*C_1P^* = 0.$

With $$(\mathcal E, \mathcal A)$$ in the Weierstraß Canonical Form, we infer that

• $$P=P^*=\indoe$$ and $$\mathcal X = \begin{bmatrix}X_{11} & 0 \\ 0 & 0 \end{bmatrix}$$
• and that the $$X_{11}$$ of a stabilizing solution $$\mathcal X$$
• coincides with $$X_\infty$$ with $$D_{11}=0$$,
• so that the feedback $$u=-B_2\mathcal Xx$$ ensures $$\gamma$$-stability
• only for the case of no feed through.

# Conclusion

## Conclusion

• For control, a Riccati equation has to respect the algebraic components

• If $$(\mathcal E,\mathcal A)$$ is index-1, then the Descriptor system is equivalent to a standard system LTI with feedthrough

• For state-feedback the suboptimal $$\Hinf$$-controller can be defined and estimated explicitly

• The non-symmetric Riccati approach coincides with the standard results

## Спасибо

### Dankeschön

www.mpi-magdeburg.mpg.de/823023/cacsd

www.janheiland.de

heiland@mpi-magdeburg.mpg.de

## References

Benner, P., and T. Stykel. 2014. “Numerical Solution of Projected Algebraic Riccati Equations.” SIAM J. Numer. Anal. 52 (2): 581–600. doi:10.1137/130923993.

Möckel, J., T. Reis, and T. Stykel. 2011. “Linear-Quadratic Gaussian Balancing for Model Reduction of Differential-Algebraic Systems.” Internat. J. Control 84 (10): 1627–43. doi:10.1080/00207179.2011.622791.

Zhou, K., J. C. Doyle, and K. Glover. 1996. Robust and Optimal Control. Upper Saddle River, NJ: Prentice-Hall.

1. i.e. index-1 and stable in the ODE part

2. see, e.g., Zhou, Doyle, and Glover (1996), Ch. 17.4

3. see, e.g., Benner and Stykel (2014)