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)

The basic equations

• 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 \[ \Sigma\sim \bigl(I,A,\begin{bmatrix}B_1\pd&B_2\pd\end{bmatrix},\begin{bmatrix}C_1\pd\\C_2\pd\end{bmatrix}, \begin{bmatrix}D_{11}&0\\0&0\end{bmatrix} \bigr) \]

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

The projected³ Riccati equation read

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

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 \(D_{11}=0\),

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

Thank you

Dankeschön

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

www.janheiland.de

heiland@mpi-magdeburg.mpg.de