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Lecture

Multivariable Control: Eigenvalue Assignment and Ackermann's Formula

In course

ME-422: Multivariable control

This course covers methods for the analysis and control of systems with multiple inputs and outputs, which are ubiquitous in modern technology and industry. Special emphasis will be placed on discrete-time systems due to their relevance to digital and embe ...

Description

This lecture covers the taxonomy of control problems, eigenvalue assignment, and the Ackermann's formula. It delves into control schemes, state feedback, regulation, tracking, and the stabilization of the origin. The lecture also discusses the solution to the eigenvalue assignment problem for single-input systems, canonical controllability form, and controller design.

Instructor

Giancarlo Ferrari Trecate

Official source

Ontological neighbourhood

Information engineering

Control theory and engineering: Control theory

Related concepts (28)

Controllability

Controllability is an important property of a control system and plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. Controllability and observability are dual aspects of the same problem. Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.

Control theory

Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality. To do this, a controller with the requisite corrective behavior is required.

State-space representation

In control engineering, model based fault detection and system identification a state-space representation is a mathematical model of a physical system specified as a set of input, output and variables related by first-order (not involving second derivatives) differential equations or difference equations. Such variables, called state variables, evolve over time in a way that depends on the values they have at any given instant and on the externally imposed values of input variables.

Optimal control

Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure.

Feedback

Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to feed back into itself. The notion of cause-and-effect has to be handled carefully when applied to feedback systems: Simple causal reasoning about a feedback system is difficult because the first system influences the second and second system influences the first, leading to a circular argument.

Related lectures (32)

Multivariable Control: State-Feedback and Eigenvalue Assignment

Covers state-feedback controller design for multivariable systems and discusses simplified methods for MIMO systems.

Eigenvalue Assignment in Multivariable Control

Explores Eigenvalue Assignment in multivariable control, emphasizing the effects of discretization and the challenges in preserving system structure.

State-Space Representation: Controllability and Observability

Explores state-space representation, controllability, observability, and regulator calculation using the Ackermann method.

Controllability and Reachability

Explores reachability and controllability in multivariable control systems, discussing tests, proofs, and their implications.

Multivariable Control: Eigenvalue Assignment and Disturbance Estimation