Greville Formula Research Articles

Pole Placement for MIMO LTI Systems via extended Ackermann and Greville Formulae

The Ackermann's formula of pole placement for controllable linear time invariant (LTI) systems is extended to multi input LTI systems by employing generalized inversion of the system's controllability matrix instead of square inversion in the procedure of deriving the formula. The nullspace of the controllability matrix is affinely and redundantly parameterized in the modified Ackermann's formula by means of the Greville formula for solutions of consistent over-determined linear systems of algebraic equations. The redundant nullspace parameterizing variables are constrained such that the closed loop system matrix satisfies its characteristic equation. The infinite number of solutions for the multi input gain matrix are obtained explicitly in terms of the constrained nullspace parameterizing variables. Two examples are provided to illustrate the procedure.

Improving MRAC Transient Response Using Generalized Dynamic Inversion and Nullspace Parametrization

We demonstrate how the transient response of Model Reference Adaptive Control (MRAC) is improved by utilizing Generalized Dynamic Inversion (GDI) and the corresponding nullspace parametrization in the powerful Greville formula. The proposed approach neither alters the reference model structure, nor it modifies it through feedback. Rather, the proposed approach utilizes the control redundancy in matching the reference model for the purpose of designing an MRAC control law that yields good transient closed loop adaptive control response. The adaptive GDI-MRAC control law is composed of two parts: a particular part that enforces the reference model dynamics, and an auxiliary part that is affine in a null control vector. The particular and the auxiliary parts of the adaptive control law are confined in the range space of the transposed control matrix and its orthogonal complement nullspace, respectively. The null control vector is projected onto the nullspace of the control matrix via a nullprojection matrix. Hence, the null control vector design does not interfere with the reference model following, yet it can be designed via state feedback to improve closed loop state transient response. An improved Lyapunov control function is used to design the adaptation gains such that asymptotic vanishing of the state error dynamics is guaranteed. Extensive simulations are conducted to demonstrate the efficacy of the proposed method.

Canonical Kane’s Equations of Motion for Discrete Dynamical Systems

The generalized inversion-based canonical equations of motion (CEOMs) for ideally constrained discrete dynamical systems are introduced in the framework of Kane’s method. Upon selection of the inertia-scaled canonical generalized acceleration variables, the proposed formulation employs the acceleration form of constraints and the generalized inversion Greville formula for parameterized solutions of underdetermined algebraic equations together with the nonminimal constrained momentum balance equations. The resulting CEOMs are explicit and full order in the acceleration variables. Moreover, the geometry of constrained motion is revealed by the CEOMs intuitively by partitioning the canonical accelerations column matrix into two portions at every time instant: a portion that drives the dynamical system to abide by the constraints, and a portion that generates the momentum balance dynamics such that the system abides by the Newton–Euler laws of motion. Some insightful geometrical perspectives of the CEOMs are illustrated via vectorial visualizations, which lead to verifying Gauss’s principle of least constraints and its Udwadia–Kalaba interpretation. The procedure is illustrated by formulating a third-order CEOMs dynamic model of two particles experiencing a single degree of freedom, as well as a sixth-order CEOMs dynamic model of a three-degrees-of-freedom disk that is rolling without slipping on an inertial plane under the effect of friction.

Robust launch vehicle’s generalized dynamic inversion attitude control

PurposeTo design a robust attitude control system for the ascent flight phase of satellite launch vehicles (SLVs).Design/methodology/approachThe autopilot is based on generalized dynamic inversion (GDI). Dynamic constraints are prescribed in the form of differential equations that encapsulate the control objectives, and are generalized inverted using the Moore-Penrose Generalized Inverse (MPGI) based Greville formula to obtain the control law. The MPGI is modified via a dynamic scaling factor for assuring generalized inversion singularity-robust tracking control. An additional sliding mode control (SMC) loop is augmented to robustify the GDI closed-loop system against model uncertainties and external disturbances.FindingsThe robust GDI control law allows for two cooperating controllers that act on two orthogonally complement control spaces: one is the particular controller that realizes the dynamic constraints, and the other is the auxiliary controller that is affined in the null control vector, and is used to enforce global closed-loop stability.Practical implicationsOrthogonality of the particular and the auxiliary control subspaces ensures noninterference of the two control actions, and thus, it ensures that both actions work toward a unified goal. The robust control loop increases practicality of the GDI control design.Originality/valueThe first successful implementation of GDI to the SLV control problem.

Generalized Dynamic Inversion Scheme for Satellite Launch Vehicle Attitude Control

This paper presents a generalized dynamic inversion attitude control design for satellite launch vehicle. Dynamic constraints that encapsulate the attitude tracking objectives are prescribed in the form of asymptotically stable differential equations in the attitude variables deviations from their reference trajectories. The constraint differential equations are evaluated along the vehicle's attitude trajectories and are inverted to solve for the control variables using the Greville formula for underdetermined algebraic systems. The associated null control vector is constructed using Lyapunov stability principle, and a delaying dynamic scaling factor is augmented in the involved Moore-Penrose generalized inverse for assured singularity avoidance and stable attitude tracking. The effectiveness of the proposed control scheme is shown by its evaluation on a 6DOF model of a satellite launch vehicle in the presence of various external disturbances.

Aircraft Multi-Constrained Generalized Dynamic Inversion Control

The new and evolving paradigm of generalized dynamic inversion (GDI) is applied to aircraft maneuvering control design, through the development of a new multiple constraint formulation. The flexibility of GDI in utilizing nullspace of the controls coefficient matrix is exploited to eliminate the problems of excessive control effort and limited performance that are associated with classical dynamic inversion. Scalar deviation functions of the aircraft state error variables are differentiated along the aircraft mathematical model solution trajectories, and linear time varying stable dynamics in these deviation functions are manipulated to obtain linear algebraic constraint equations in the control vector. The constraint equations are inverted using the Greville formula, resulting in the control law. The particular part of the control law drives the states to their desired values, and the auxiliary part acts to stabilize the vehicles inner dynamics by means of the null-control vector. The null-control vector is constructed via a novel class of positive semidefinite Lyapunov functions and nullprojected Lyapunov equations. The closed loop control system is assessed by performing a fully coordinated simultaneous heading-velocity change maneuver. The new type of GDI control laws shows to require low control effort, in addition to providing excellent transient response and reasonable steady-state tracking accuracy.

Inertia-independent generalized dynamic inversion feedback control of spacecraft attitude maneuvers

The generalized dynamic inversion control methodology is applied to the spacecraft attitude trajectory tracking problem. It is shown that the structure of the skew symmetric cross product matrix alleviates the need to include the inertia matrix in the control law. Accordingly, the proposed control law depends solely on attitude and angular velocity measurements, and it neither requires knowledge of the spacecraft's inertia parameters nor it works towards estimating these parameters. A linear time-varying attitude deviation dynamics in the multiplicative error quaternion is inverted for the control variables using the generalized inversion-based Greville formula. The resulting control law is composed of auxiliary and particular parts acting on two orthogonally complement subspaces of the three dimensional Euclidean space. The particular part drives the attitude variables to their desired trajectories. The auxiliary part is affine in a free null-control vector, and is designed by utilizing a semidefinite control Lyapunov function that exploits the geometric structure of the control law to provide closed loop stability. The generalized inversion singularity avoidance is made by augmenting the generalized inverse with an asymptotically stable fast mode that is driven by angular velocity error's norm from reference angular velocity. Asymptotic tracking is achieved for detumbling maneuvers as the stable augmented mode subdues singularity. If the steady state desired quaternion trajectories are time varying, then asymptotic tracking is lost in favor of close ultimately bounded tracking because the stable augmented mode continues to be excited during steady state phase of response. A rest-to-rest slew and a trajectory tracking maneuver examples are provided to illustrate the methodology.

Inertia-Independent Generalized Dynamic Inversion Spacecraft Control

AbstractThe generalized dynamic inversion control methodology is applied to the spacecraft attitude trajectory tracking problem. It is shown that the structure of the skew symmetric cross product matrix alleviates the need to include the inertia matrix in the control law. Accordingly, the proposed control law depends solely on attitude and angular velocity measurements, and it neither requires knowledge of the spacecraft's inertia parameters nor it works towards estimating these parameters. A linear time varying attitude deviation dynamics in the multiplicative error quaternion is inverted for the control variables using the generalized inversion-based Greville formula. The resulting control law is composed of auxiliary and particular parts acting on two orthogonally complement subspaces of the three dimensional Euclidean space. The particular part drives the attitude variables to their desired trajectories. The auxiliary part is affine in a free null-control vector, and is designed by utilizing a semidefinite control Lyapunov function that exploits the geometric structure of the control law to provide closed loop stability. The generalized inversion singularity avoidance is made by augmenting the generalized inverse with an asymptotically stable fast mode that is driven by angular velocity error's norm from reference angular velocity. Asymptotic tracking is achieved for detumbling maneuvers as the stable augmented mode subdues singularity. If the steady state desired quaternion trajectories are time varying, then asymptotic tracking is lost in favor of close ultimately bounded tracking because the stable augmented mode continues to be excited during steady state phase of response. A rest-to-rest slew and a trajectory tracking maneuver examples are provided to illustrate the methodology.

Asymptotic generalised dynamic inversion attitude control

This study introduces a generalised dynamic inversion control methodology for asymptotic spacecraft attitude trajectory tracking. An asymptotically stable second-order servo-constraint attitude deviation dynamics is evaluated along spacecraft equations of motion, resulting in a linear relation in the control vector. A control law that enforces the servo-constraint is derived by generalised inversion of the relation using the Greville formula. The generalised inverse in the particular part of the control law is scaled by a decaying dynamic factor that depends on desired attitude trajectories and body angular velocity components. The scaled generalised inverse uniformly converges to the standard Moore-Penrose generalised inverse, causing the particular part to converge uniformly to its projection on the range space of the controls coefficient generalised inverse, and driving spacecraft attitude variables to nullify attitude deviation. The auxiliary part of the control law acts on the controls coefficient nullspace, and it provides the spacecraft internal stability with the aid of the null-control vector. The null-control vector construction is made by means of novel semidefinite nullprojection control Lyapunov function and state-dependent null-projected Lyapunov equation. The generalised dynamic inversion control signal is multiplied by an exponential factor during transient closed-loop response to enhance the control signal in terms of magnitude and rate of change. Illustrating examples show efficacy of the methodology.

SERVO-CONSTRAINT GENERALIZED INVERSE DYNAMICS FOR ROBOT MANIPULATOR CONTROL DESIGN

Novel concept of inverse dynamics for robot manipulator control is developed. The concept is based on generalized inversion, and therefore it avoids limitations and shortcomings of square inversion that are frequently encountered in conventional inverse dynamics. A servo-constraint on the manipulator is defined, the satisfaction of which implies that the desired robot kinematics is satisfied. A prescribed stable linear second-order dynamics in a servo-constraint deviation norm measure variable is evaluated along trajectory solutions of the manipulator, resulting in a linear algebraic relation in the control vector. Generalized inversion of the relation via the Greville formula yields a control law that is linearly parameterized in a free null-control vector. The null-control vector is projected onto nullspace of the controls coefficient in the linear algebraic relation, and therefore it does not affect servo-constraint dynamics. The design freedom offered by the null-control vector is utilized to stabilize manipulator's internal dynamics, and to provide a perturbed feedback linearizing transformation of the global closed loop system dynamics. Generalized inversion singularity avoidance is made by replacing the Moore-Penrose generalized inverse in the Greville formula by a damped generalized inverse, resulting in uniformly ultimately bounded robot manipulator trajectory tracking.

Computation of the Para-Pseudoinverse for Oversampled Filter Banks: Forward and Backward Greville Formulas

Frames and oversampled filter banks have been extensively studied over the past few years due to their increased design freedom and improved error resilience. In frame expansions, the least square signal reconstruction operator is called the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dual</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">frame,</i> which can be obtained by choosing the synthesis filter bank as the para-pseudoinverse of the analysis bank. In this paper, we study the computation of the dual frame by exploiting the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Greville</i> formula, which was originally derived in 1960 to compute the pseudoinverse of a matrix when a new row is appended. Here, we first develop the backward Greville formula to handle the case of row deletion. Based on the forward Greville formula, we then study the computation of para-pseudoinverse for extended filter banks and Laplacian pyramids. Through the backward Greville formula, we investigate the frame-based error resilient transmission over erasure channels. The necessary and sufficient condition for an oversampled filter bank to be robust to one erasure channel is derived. A postfiltering structure is also presented to implement the para-pseudoinverse when the transform coefficients in one subband are completely lost.

Variants of the Greville Formula with Applications to Exact Recursive Least Squares

In this paper, we present an order-recursive formula for the pseudoinverse of a matrix. It is a variant of the well-known Greville [SIAM Rev., 2 (1960), pp. 578--619] formula. Three forms of the proposed formula are presented for three different matrix structures. Compared with the original Greville formula, the proposed formulas have certain merits. For example, they reduce the storage requirements at each recursion by almost half; they are more convenient for deriving recursive solutions for optimization problems involving pseudoinverses. Regarding applications, using the new formulas, we derive recursive least squares (RLS) procedures which coincide exactly with the batch LS solutions to the problems of the unconstrained LS, weighted LS, and LS with linear equality constraints, respectively, including their simple and exact initializations. Compared with previous results, e.g., Albert and Sittler [J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), pp. 384--417], our derivation of the explicit recursive formulas is much easier, and the recursions take a much simpler form. New findings include that the linear equality constrained LS and the unconstrained LS can have an identical recursion---their only difference is the initial conditions. In addition, some robustness issues, in particular, during the exact initialization of the RLS are studied.

An Alternative Proof of the Greville Formula

A simple proof of the Greville formula for the recursive computation of the Moore–Penrose (MP) inverse of a matrix is presented. The proof utilizes no more than the elementary properties of the MP inverse.

On the Continuity of the Generalized Inverse

On the Continuity of the Generalized Inverse