How do you compare and contrast different state space methods such as modal, canonical, or diagonal forms?

1 Modal form

Modal form is a state space method that uses the eigenvalues and eigenvectors of the system matrix to define the state variables. In modal form, the system matrix is diagonal, and each state variable corresponds to a mode of the system. The modes are the natural frequencies and damping ratios of the system, which determine how it responds to disturbances and inputs. Modal form is useful for studying the stability and transient behavior of the system, as well as for designing modal controllers that assign desired poles to the system.

2 Canonical form

Canonical form is a state space method that uses a standard structure for the system matrix and the input and output matrices. There are different types of canonical forms, such as controllable, observable, or Jordan forms, but they all have the same characteristic polynomial as the original system. Canonical form is useful for simplifying the calculations and manipulations of the state space equations, as well as for testing the controllability and observability of the system. Controllability and observability are important properties that determine whether the system can be controlled or measured by the inputs and outputs.

3 Diagonal form

Diagonal form is a state space method that uses a transformation matrix to diagonalize the system matrix. In diagonal form, the system matrix has only nonzero elements on the main diagonal, and the input and output matrices are transformed accordingly. Diagonal form is useful for decoupling the state variables and reducing the complexity of the system. It also makes it easier to design state feedback controllers that place the poles of the system at desired locations.

4 Comparison and contrast

The different state space methods have advantages and disadvantages depending on the application and the goals of the control system design. Modal form reveals the intrinsic dynamics of the system and allows for modal control, but it may not be possible to find a modal form for every system. Canonical form simplifies the state space equations and allows for controllability and observability tests, but it may not be unique or intuitive for every system. Diagonal form decouples the state variables and allows for state feedback control, but it may not be feasible or optimal for every system. Therefore, it is important to understand the characteristics and limitations of each method and choose the most suitable one for your problem.