## Implicit linear systemsThese notes are an introduction to implicit models of linear dynamical systems, with applications to modelling, control system design, and identification, intended for control-system engineers at the beginning graduate level. Because they are non-oriented, the models are particularly useful where causality is unknown or may change. They are implicit in all variables and closed under the algebraic operations, and hence are useful for computer-aided analysis and design. They possess the vector-matrix conceptual simplicity and computational feasibility of state-space equations, together with the generality of matrix-fraction descriptions, and admit of canonical forms for which the joint identification of system parameters and dynamic variables is linear. The notes simplify, generalize, and complement much recent work on "singular" or "descriptor" models, but do not duplicate it. Sections are included on realizations, canonical forms, minimal representations, algebraic design applications, quadratic optimization, identification, large-scale systems, and extensions to multi-dimensional and time-varying systems. |

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### Contents

System models | 1 |

The Kronecker form | 19 |

Analysis of singularities | 25 |

Copyright | |

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algebraic design algorithm analysis applied arbitrary assumed block equation canonical form Chapter closed-loop poles coefficient matrix column echelon form computation Consider construction contains controller corresponding defined definition design constraints design problem diag differential equations dimension discussed eigenvalues entries equivalent example external behaviour external variables external vector Figure full column rank full row rank given gives identical implicit equations implicit models implicit systems independent initial conditions inverse Jacobian matrix Kalman filter kernel Kronecker form linear systems LU factorization matrix-fraction models methods minimal system non-singular matrix observer obtained operations optimal optimal control p-vector parameters pencil permutation matrix permuted state-space form plant polynomial polynomial matrix Property realization representation result Riccati equation root-locus row echelon form satisfy scalar Section sequence singular solved stable state-space models state-space systems structure subspace subsystems system matrix system models system poles time-varying systems transfer matrix transformation zero placement