Alexander Lanzon

Stability Robustness of a Feedback Interconnection of Systems With Negative Imaginary Frequency Response

A necessary and sufficient condition, expressed simply as the dc loop gain (i.e., the loop gain at zero frequency) being less than unity, is given in this note to guarantee the internal stability of a feedback interconnection of linear time-invariant (LTI) multiple-input multiple-output systems with negative imaginary frequency response. Systems with negative imaginary frequency response arise, for example, when considering transfer functions from force actuators to colocated position sensors, and are commonly important in, for example, lightly damped structures. The key result presented here has similar application to the small-gain theorem, which refers to the stability of feedback interconnections of contractive gain systems, and the passivity theorem, which refers to the stability of feedback interconnections of positive real (or passive) systems. A complete state-space characterization of systems with negative imaginary frequency response is also given in this note and also an example that demonstrates the application of the key result is provided.

Feedback Control of Negative-Imaginary Systems

This paper investigates the robustness of positive-position feedback control of flexible structures with colocated force actuators and position sensors. In particular, the theory of negative-imaginary systems is used to reveal the robustness properties of multi-input, multi-output (MIMO) positive-position feedback controllers and related types of controllers for flexible structures. The negative-imaginary property of linear systems can be extended to nonlinear systems through the notion of counterclockwise input-output dynamics.

A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems

The note is concerned with linear negative imaginary systems. First, a previously established Negative Imaginary Lemma is shown to remain true even if the system transfer function matrix has poles on the imaginary axis. This result is achieved by suitably extending the definition of negative imaginary transfer function matrices. Secondly, a necessary and sufficient condition is established for the internal stability of the positive feedback interconnections of negative imaginary systems. Meanwhile, some properties of linear negative imaginary systems are developed. Finally, an undamped flexible structure example is presented to illustrate the theory.

A Feedback Linearization Approach to Fault Tolerance in Quadrotor Vehicles

Abstract In this paper the control problem of a quadrotor vehicle experiencing a rotor failure is investigated. First we derive a nonlinear mathematical model for the quadrotor including both translational and rotational drag terms. Then we use a feedback linearization approach to design a controller whose task is to make the vehicle enter a constant angular velocity spin around its vertical axis, while retaining zero angular velocities around the other axis. These conditions can be exploited to design a second control loop, which is used to perform trajectory following. The proposed double control loop architecture allows the vehicle to perform both trajectory and roll/pitch control when a rotor failure is present.

Stability Analysis of Interconnected Systems With “Mixed” Negative-Imaginary and Small-Gain Properties

In this letter, an analytical framework is proposed to examine stability of two stable, linear time invariant systems interconnected in positive feedback where the systems have “mixed” properties of negative-imaginary and small-gain. Using the notion of dissipativity, the interconnection of systems is guaranteed to be finite-gain stable under the condition that the dc loop gain is contractive. This work builds on Griggs, and exploits a new set of frequency dependent triplets that was introduced in above reference to “mix” two unconditional stability statements, i.e., small-gain and passivity. Unlike the above reference the present work explores the important question of how a conditional stability statement as needed when two negative-imaginary systems are connected in a feedback loop can be “mixed” with an unconditional stability statement as needed when two contractive systems are connected in a feedback loop. The usefulness of the proposed analytical framework is demonstrated via a numerical example.

Computing the Positive Stabilizing Solution to Algebraic Riccati Equations With an Indefinite Quadratic Term via a Recursive Method

An iterative algorithm to solve algebraic riccati equations with an indefinite quadratic term is proposed. The global convergence and local quadratic rate of convergence of the algorithm are guaranteed and a proof is given. Numerical examples are also provided to demonstrate the superior effectiveness of the proposed algorithm when compared with methods based on finding stable invariant subspaces of Hamiltonian matrices. A game theoretic interpretation of the algorithm is also provided.

Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures With Free Body Motion

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented.

A “mixed” small gain and passivity theorem in the frequency domain ☆

Abstract We show that the negative feedback interconnection of two causal, stable, linear time-invariant systems, with a “mixed” small gain and passivity property, is guaranteed to be finite-gain stable. This “mixed” small gain and passivity property refers to the characteristic that, at a particular frequency, systems in the feedback interconnection are either both “input and output strictly passive”; or both have “gain less than one”; or are both “input and output strictly passive” and simultaneously both have “gain less than one”. The “mixed” small gain and passivity property is described mathematically using the notion of dissipativity of systems, and finite-gain stability of the interconnection is proven via a stability result for dissipative interconnected systems.

Flight Control of a Quadrotor Vehicle Subsequent to a Rotor Failure

In this paper, the problem of designing a control law in case of rotor failure in quadrotor vehicles is addressed. First, a nonlinear mathematical model for a quadrotor vehicle is derived, which includes translational and rotational dynamics. Then a robust feedback linearization controller is developed, which sacrifices the controllability of the yaw state due to rotor failure to linearize the closed-loop system around a working point, where roll and pitch angles are zero and the angular speed around the vertical axis is a nonzero constant. An H∞ loop shaping technique is adopted to achieve regulation of these variables around the chosen working point. Finally, an outer loop is proposed for achieving control of the linear displacement under the assumption of small angles approximation for the pitch and roll angles. The proposed control strategy allows the vehicle to use the remaining three functional rotors to enter a constant angular speed around its vertical axis, granting stability and representing an ...

Validating Controllers for Internal Stability Utilizing Closed-Loop Data

We introduce novel tests utilizing a limited amount of experimental and possibly noisy data obtained with an existing known stabilizing controller connected to an unknown plant for verifying that the introduction of a proposed new controller will stabilize the plant. The tests depend on the assumption that the unknown plant is stabilized by a known controller and that some knowledge of the closed-loop system, such as noisy frequency response data, is available and on the basis of that knowledge, the use of a new controller appears attractive. The desirability of doing this arises in iterative identification and control algorithms, multiple-model adaptive control, and multi-controller adaptive switching. The proposed tests can be used for SISO and/or MIMO linear time-invariant systems.