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The long standing problem is discussed of how to deflate the part associated with the eigenvalue infinity in a structured matrix pencil using structure preserving unitary transformations. We derive such a deflation procedure and apply this new technique to symmetric, Hermitian or alternating pencils and in a modified form to (anti)-palindromic pencils. We present a detailed error and perturbation analysis of this and other deflation procedures and demonstrate the properties of the new algorithm with several numerical examples.
Enforcing solvability of a nonlinear matrix equation and estimation of multivariate ARMA time series
(2013)
The matrix equation $X+AX^{-1}A^T=B$, arising in parameter estimation of certain time series models,
is solvable only for certain values of the matrices $A,B$.
We present a numerical method to modify $A,B$ in order to make the matrix equation solvable.
Since solvability depends on the location of the eigenvalues of the palindromic matrix polynomial $\lambda^2 A+\lambda B+A^T$,
our method works by moving those eigenvalues to specified locations using first order spectral perturbation theory.
The method is heuristic but works in practice, as is supported by several compelling numerical examples.
These examples arise from parameter estimation of a common time series model, the multivariate ARMA(1,1).
Dissipativity is an important property of individual systems
that guarantees a stable interconnected system.
However, due to errors in the modeling process weakly non-dissipative
models may be constructed.
In this paper we introduce a method to perturb a non-dissipative LTI system in order to enforce dissipativity using spectral perturbation results for
para-Hermitian pencils.
Compared to earlier algorithms the new method
is applicable to a wider class of problems,
it utilizes a simpler framework, and
employs a larger class of allowable perturbations
resulting in smaller perturbations. Moreover, system stability can be enforced as well.
Numerical examples are provided to show the effectiveness of the new approach.
The well-known Kalman-Yakubovich-Popov Lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality. In this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of a so-called Lur'e equation, which mainly is a linear matrix inequality with a rank minimizing condition. Finally, we apply the result to standard systems to obtain the well-known result about the solvability of the algebraic Riccati equation.
We consider linear differential-algebraic m-input m-output systems with positive
strict relative degree or proper inverse transfer function; in the single-input single-output case these
two disjoint classes make the whole of all linear DAEs without feedthrough term. Structural properties
- such as normal forms (i.e. the counterpart to the Byrnes-Isidori form for ODE systems), zero
dynamics, and high-gain stabilizability - are analyzed for two purposes: first, to gain insight into the
system classes and secondly, to solve the output regulation problem by funnel control. The funnel
controller achieves tracking of a class of reference signals within a pre-specified funnel; this means in
particular, the transient behaviour of the output error can be specified and the funnel controller does
neither incorporate any internal model for the reference signals nor any identification mechanism, it
is simple in its design. The results are illuminated by position and velocity control of a mechanical
system encompassing springs, masses, and dampers.
We introduce a~numerical method for the numerical solution of the Lur'e matrix equations that arise, for instance, in linear-quadratic infinite time horizon optimal control. The method is based on the characterization of the solutions in terms of deflating subspaces of a suitable even matrix pencil. Via a Cayley transformation, the problem is transformed to the discrete-time case. This leaves us with a symplectic problem with several Jordan blocks of eigenvalue 1 and even size, which arise from the remaining eigenvalues at infinity of the original problem. For the solution of this modified problem, we use the {\em structure-preserving doubling algorithm} (SDA), an iterative scheme for the solution of dense continuous- and discrete-time algebraic Riccati equations. Unlike other iterative schemes, this algorithm converges also when the pencil has eigenvalues on the unit circle, as is the case in our problem. Implementation issues such as the choice of the parameter $\gamma$ in the Cayley transform are discussed. The numerical examples presented confirm the effectiveness of this method.
We study the class of linear differential-algebraic m-input m-output systems
which have a transfer function with proper inverse.
A sufficient condition for the transfer function to have proper inverse
it that the system has 'strict and non-positive relative degree'.
We present two main results:
First, a so called 'zero dynamics form' is derived: this form is - within the class of system equivalence - a simple "almost normal" form of the DAE; it is a counterpart to the well-known Byrnes-Isidori form for
ODE systems with strictly proper transfer function.
The 'zero dynamics form' is exploited to characterize structural properties such as
asymptotically stable zero dynamics,
minimum phase, and high-gain stabilizability.
The zero dynamics are characterized by (A,E,B)-invariant subspaces.
Secondly, it is shown that the 'funnel controller' (that is a static nonlinear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.
Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.
Model Reduction for a Class of Nonlinear Electrical Circuits by Reduction of Linear Subcircuits
(2010)
We analyze a model reduction approach for a class of electrical circuits containing nonlinear resistance.
In our approach
the linear subcircuits are extracted and replaced with linear passive reduced-order models. The resulting nonlinear
reduced-order model preserves admissibility and passivity. Moreover, we derive a-priori bounds for the error between
the input-output map of the original circuit equations and that of our reduced-order model. These bounds
are valid for all inputs. Since only linear subcircuits are reduced, our approach is effective when the number of nonlinear resistances is relatively small. The performance of our approach is illustrated numerically.