What does Reachability mean in Control Theory, and how is it related to Controllability? How is it expressed in terms of Cayley-Hamilton Theorem In control theory, reachability refers to the ability to steer a system from an initial state to any desired state within a given time frame. It is closely related to [[Controllability]], which is the property of being able to manipulate and control the behavior of a system using appropriate inputs. Reachability is expressed in terms of the Cayley-Hamilton theorem, which states that any square matrix M satisfies its own characteristic equation. In control theory, this theorem is used to determine if a system is reachable or controllable. To express reachability using the Cayley-Hamilton theorem, consider a linear time-invariant (LTI) system described by the state-space representation: dx/dt = Ax + Bu where x represents the state vector, u represents the input vector, A is the system matrix, and B is the input matrix. The reachability of this system can be determined by examining the controllability matrix: R = [B | AB | A^2B | ... | A^(n-1)B] where n is the dimension of the state vector x. If R has full rank (i.e., rank(R) = n), then the system is said to be reachable. This means that it is possible to find appropriate inputs u such that any desired state can be reached within a finite time frame. The full rank condition ensures that all states are controllable. The Cayley-Hamilton theorem comes into play when analyzing reachability by relating powers of matrix A with linear combinations of lower powers. By using this theorem, it can be shown that if R has full rank, then A can be expressed as a linear combination of its lower powers up to degree n-1: A^n + c_(n-1)A^(n-1) + ... + c_1A + c_0I = 0, where I represents the identity matrix and c_i are coefficients. This equation is known as the Cayley-Hamilton equation. The reachability of a system can then be expressed by examining the coefficients c_i in the Cayley-Hamilton equation. If all coefficients c_i are nonzero, the system is reachable. However, if any coefficient c_i is zero, it indicates a loss of reachability and certain states cannot be reached. In summary, reachability in control theory refers to the ability to steer a system from an initial state to any desired state. It is related to controllability and can be expressed using the Cay # References ```dataview Table title as Title, authors as Authors where contains(subject, "reachability") or contains(subject, "Reachability") ```