The quasi-Weierstraß form for regular matrix pencils

AbstractRegular linear matrix pencils A-E∂∈Kn×n[∂], where K=Q, R or C, and the associated differential algebraic equation (DAE) Ex˙=Ax are studied. The Wong sequences of subspaces are investigate and invoked to decompose the Kn into V∗⊕W∗, where any bases of the linear spaces V∗ and W∗ transform the matrix pencil into the quasi-Weierstraß form. The quasi-Weierstraß form of the matrix pencil decouples the original DAE into the underlying ODE and the pure DAE or, in other words, decouples the set of initial values into the set of consistent initial values V∗ and “pure” inconsistent initial values W∗⧹{0}. Furthermore, V∗ and W∗ are spanned by the generalized eigenvectors at the finite and infinite eigenvalues, resp. The quasi-Weierstraß form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of A-E∂ lead to the well-known Weierstraß form. So the latter can be viewed as a generalized Jordan form. Finally, it is shown how eigenvector chains constitute a basis for the solution space of Ex˙=Ax

Funnel control of nonlinear systems

Tracking of reference signals is addressed in the context of a class of nonlinear controlled systems modelled by $r$-th order functional differential equations, encompassing inter alia systems with unknown "control direction" and dead-zone input effects. A control structure is developed which ensures that, for every member of the underlying system class and every admissible reference signal, the tracking error evolves in a prescribed funnel chosen to reflect transient and asymptotic accuracy objectives. Two fundamental properties underpin the system class: bounded-input bounded-output stable internal dynamics, and a high-gain property (an antecedent of which is the concept of sign-definite high-frequency gain in the context of linear systems)

Quasi feedback forms for differential-algebraic systems

We investigate feedback forms for linear time-invariant systems described by differential-algebraic equations. Feedback forms are representatives of certain equivalence classes. For example state space transformations, invertible transformations from the left, and proportional state feedback constitute an equivalence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a quasi proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a quasi proportional and derivative feedback form. Similar advantages hold

Funnel control of nonlinear systems

Tracking of reference signals is addressed in the context of a class of nonlinear controlled systems modelled by r-th-order functional differential equations, encompassing inter alia systems with unknown "control direction" and dead-zone input effects. A control structure is developed which ensures that, for every member of the underlying system class and every admissible reference signal, the tracking error evolves in a prescribed funnel chosen to reflect transient and asymptotic accuracy objectives. Two fundamental properties underpin the system class: bounded-input bounded-output stable internal dynamics, and a high-gain property (an antecedent of which is the concept of sign-definite high-frequency gain in the context of linear systems)

On stability of time-varying linear differential-algebraic equations

We develop a stability theory for time-varying linear differential algebraic equations (DAEs). Standard stability concepts for ODEs are formulated for DAEs and characterized. Lyapunov’s direct method is derived as well as the converse of the stability theorems. Stronger results are achieved for DAEs which are transferable into standard canonical form; in this case the existence of the generalized transition matrix is exploited

Dedicated to the memory of Christopher I. Byrnes

The Byrnes-Isidori form with respect to the relative degree is studied for time-varying linear multi-input, multi-output systems. It is clarified in which sense this form is a normal form. (A,B)-invarianttime-varying subspaces are defined and the maximal(A,B)-invariant time-varying subspace included in the kernel of C is characterized. This is exploited to characterize the zero dynamics of the system. Finally, a high-gain derivative output feedback controller is introduced for the class of systems with higher relative degree and stable zero dynamics. All results are also new for time-invariant linear systems.MSC 93C05, 93D1

Time-varying linear DAEs transferable into standard canonical form

We introduce a solution theoryfor time-varying linear differential-algebraic equations(DAEs) E(t)˙x = A(t)x which can be transformed into standard canonical form (SCF), i.e. the DAE is decoupled into an ODE ˙z1 = J(t)z1 and a pure DAE N(t)˙z1 = z1 , where N is pointwise strictly lower triangular. This class is a time-varying generalization of time-invariant DAEs where the corresponding matrix pencil is regular. It will be shown in which sense the SCF is a canonical form, that it allows for a transition matrix similar to the one for ODEs, and how this can be exploited to derive a variation of constants formula. Furthermore, we show in which sense the class of systems transferable into SCF is equivalent to DAEs which are analytically solvable, and relate SCF to the derivative array approach, differentiation index and strangeness index. Finally, an algorithm is presented which determines the transformation matrices which put a DAE into SCF

The Quasi-Weierstraß form for regular matrix pencils

Dedicated to Heinrich Voß on the occasion of his 65th birthdayRegular linear matrix pencils A - E \partial \in \mathbb{K}^{n \partial n} [\partial], where \mathbb{K} = \mathbb{Q}, \mathbb{R} or \mathbb{C}, and the associated differential algebraic equation (DAE) E\dot{x} = Ax are studied. The Wong sequences of subspaces are tackled and invoked to decompose the \mathbb{K}^n into \mathcal{V}^* \oplus \mathcal{W}^*, where any bases of the linear spaces \mathcal{V}^* and \mathcal{W}^* transform the matrix pencil into the Quasi-Weierstraß form. The Quasi-Weierstraß form of the matrix pencil decouples the original DAE into the underlying ODE and the pure DAE or, in other words, decouples the set of initial values into the set of consistent initial values \mathcal{V}^* and "pure" inconsistent initial values \mathcal{W}^* \ \{0\} . Furthermore, \mathcal{V}^* and \mathcal{W}^* are spanned by the generalized eigenvectors at the finite and infinite eigenvalues, resp. The Quasi-Weierstraß form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of A − E \partial lead to the well-known Weierstraß form. So the latter can be viewed as a generalized Jordan form. Finally, it is shown how eigenvector chains constitute a basis for the solution space of E \dot{x} = Ax

Zero dynamics and funnel control of linear differential-algebraic systems

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

Normal forms, high-gain and funnel control for linear differential-algebraic systems

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