Lecture 4: Controllability and observability

Lecture 4: Controllability and observability Lecture 4: Controllability and observability p.1/9

Part 1: Controllability Lecture 4: Controllability and observability p.2/9

Example Two inverted pendula mounted on a chart. Length of the pendula:, respectively. force Lecture 4: Controllability and observability p.3/9

Example Two inverted pendula mounted on a chart. Length of the pendula:, respectively. force Defines a system with behavior satisfies Newton s laws Lecture 4: Controllability and observability p.3/9

By physical reasoning: if, then does not depend on the external force : if for, then also for, regardless of the external force. Lecture 4: Controllability and observability p.4/9

By physical reasoning: if, then does not depend on the external force : if for, then also for, regardless of the external force. Hence: there is no with while at the same time. No trajectory with can be steered to a future trajectory with. Lecture 4: Controllability and observability p.4/9

Assume now that the lengths of the pendula are unequal: It turns out (more difficult to prove) that in that case it is possible to connect any past trajectory with any future trajectory: Lecture 4: Controllability and observability p.5/9

Controllability is called controllable if for all and such that there exists Lecture 4: Controllability and observability p.6/9

Controllability is called controllable if for all and such that there exists concatenating trajectory time Lecture 4: Controllability and observability p.6/9

Controllability in terms of kernel representations Suppose. is represented in kernel representation by How to decide whether is controllable? Lecture 4: Controllability and observability p.7/9

Controllability in terms of kernel representations Suppose. is represented in kernel representation by How to decide whether is controllable? Theorem: Let, and let be such that is a kernel representation of. Then is controllable if and only if for all equivalently, if and only if is the same for all. Note: is the rank of as a matrix of polynomials. Lecture 4: Controllability and observability p.7/9

Examples 1. represented by, (single input/single output system). Here,,. is controllable if and only if for all equivalently, the polynomials are coprime. Lecture 4: Controllability and observability p.8/9

Examples 1. represented by, (single input/single output system). Here,,. is controllable if and only if for all equivalently, the polynomials are coprime. 2. represented by. Obviously, this is a kernel representation, with. is controllable if and only if for all (Hautus test). Lecture 4: Controllability and observability p.8/9

Controllability and image representations Let and let. If there exists such that then we call an image representation of. Lecture 4: Controllability and observability p.9/9

Controllability and image representations Let and let. If there exists such that then we call an image representation of. is then the image of the mapping Lecture 4: Controllability and observability p.9/9

Controllability and image representations Let and let. If there exists such that then we call an image representation of. is then the image of the mapping Question: Which s in have an image representation? Theorem: Let if is controllable.. has an image representation if and only Note: Relation with the notion of flat system. Lecture 4: Controllability and observability p.9/9

Part 2 Observability Lecture 4: Controllability and observability p.10/9

Example Consider a point mass with position vector, moving under influence of a force vector. This defines a system, represented by For a given, many s will satisfy the system equation: the actual will of course depend on and. Lecture 4: Controllability and observability p.11/9

Example Consider a point mass with position vector, moving under influence of a force vector. This defines a system, represented by For a given, many s will satisfy the system equation: the actual will of course depend on and. In other words: does not determine uniquely. This is expressed by saying that in, is not observable from. Lecture 4: Controllability and observability p.11/9

Observability Let variable, and be a partition of the manifest. We will say that in, the component is observable from the component if is uniquely determined by, i.e., if observed to-be-deduced Lecture 4: Controllability and observability p.12/9

Example Let,. 1. Let be represented by,. Clearly, in, is observable from : for given, is given by. Lecture 4: Controllability and observability p.13/9

Example Let,. 1. Let be represented by,. Clearly, in, is observable from : for given, is given by. 2. Let be represented by,. This time, in, is not observable from : determines only, so up to a constant. Lecture 4: Controllability and observability p.13/9

Observability in terms of kernel representations Suppose, with is represented in kernel representation by. Partition. Accordingly, partition, so that is represented by. How do we check whether, in, is observable from? Theorem: in, is observable from if and only if for all i.e., has full column rank for all. Lecture 4: Controllability and observability p.14/9

Observability in terms of kernel representations Suppose, with is represented in kernel representation by. Partition. Accordingly, partition, so that is represented by. How do we check whether, in, is observable from? Theorem: in, is observable from if and only if for all i.e., has full column rank for all. In that case, there exists a polynomial left inverse of such that ), and we have (i.e. Lecture 4: Controllability and observability p.14/9

Example Consider the system, with, represented by Under what conditions is observable from? Lecture 4: Controllability and observability p.15/9

Example Consider the system, with, represented by Under what conditions is observable from? Clearly, the equations can be re-written as Lecture 4: Controllability and observability p.15/9

Example Consider the system, with, represented by Under what conditions is observable from? Clearly, the equations can be re-written as Hence: observable from full column rank for all. (Hautus test) Lecture 4: Controllability and observability p.15/9

Part 3: Stabilizability and detectability Lecture 4: Controllability and observability p.16/9

Stabilizability is called stabilizable if for all such that there exists for,. Lecture 4: Controllability and observability p.17/9

Stabilizability is called stabilizable if for all such that there exists for,. time Lecture 4: Controllability and observability p.17/9

Stabilizability in terms of kernel representations Suppose. is represented in kernel representation by How to decide whether is stabilizable? Lecture 4: Controllability and observability p.18/9

Stabilizability in terms of kernel representations Suppose. is represented in kernel representation by How to decide whether is stabilizable? Theorem: Let, and let be such that is a kernel representation of. Then is stabilizable if and only if for all equivalently, if and only if ( ). is the same for all Lecture 4: Controllability and observability p.18/9

Detectability Let variable, and be a partition of the manifest. We will say that in, the component is detectable from the component if If is detectable from, then determines asymptotically. observed to-be-deduced Lecture 4: Controllability and observability p.19/9

Detectability in terms of kernel representation Suppose that, with is represented in kernel representation by. Partition. Accordingly, partition, so that is represented by. How do we check whether, in, is detectable from? Lecture 4: Controllability and observability p.20/9

Detectability in terms of kernel representation Suppose that, with is represented in kernel representation by. Partition. Accordingly, partition, so that is represented by. How do we check whether, in, is detectable from? Theorem: in, is detectable from if and only if for all i.e., has full column rank for all. Lecture 4: Controllability and observability p.20/9

Summarizing A system is controllable if the past and the future of any two trajectories in can be concatenated to obtain a trajectory in. Lecture 4: Controllability and observability p.21/9

Summarizing A system is controllable if the past and the future of any two trajectories in can be concatenated to obtain a trajectory in. Controllability is a property of the system. Given a kernel representation of the system, controllability can be effectively tested. Given a system and a partition, is called observable from if the condition determines uniquely. Lecture 4: Controllability and observability p.21/9

Summarizing A system is controllable if the past and the future of any two trajectories in can be concatenated to obtain a trajectory in. Controllability is a property of the system. Given a kernel representation of the system, controllability can be effectively tested. Given a system and a partition, is called observable from if the condition determines uniquely. Observability is a property of the system and a partition of its variables. Given a kernel representation of the system, observability can be effectively tested. Lecture 4: Controllability and observability p.21/9

A system is stabilizable if the past of any trajectory in can be concatenated with the future of a trajectory in that converges to zero, to obtain a trajectory in. Lecture 4: Controllability and observability p.22/9

A system is stabilizable if the past of any trajectory in can be concatenated with the future of a trajectory in that converges to zero, to obtain a trajectory in. Given a system and a partition, is called detectable from if the condition determines asymptotically as. Lecture 4: Controllability and observability p.22/9